1.3 planets in scattered light

Research Collection

Doctoral Thesis

Characterization of planetary systems in scattered light withdifferential techniques

Author(s): Buenzli, Esther

Publication Date: 2011

Permanent Link: https://doi.org/10.3929/ethz-a-006838186

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Diss ETH No. 19917

Characterization of Planetary Systems inScattered Light with Differential Techniques

A dissertation submitted to

ETH Zurich

for the degree of

DOCTOR OF SCIENCES

presented by

ESTHER BUENZLI

Dipl. Phys. ETH

born May 22, 1983citizen of Fehraltorf (ZH), Switzerland

accepted on the recommendation of

Prof. Dr. M. R. MeyerPD Dr. H. M. Schmid

Dr. F. Menard

Zurich, 2011

—

To Adrian

—

Abstract

This thesis is devoted to the study of scattered optical and near-infrared lightreceived from planetary system objects, in particular solar and extrasolar giantplanets and debris disks.

The light is originally emitted by the parent star and subsequently redis-tributed by the smaller bodies in a manner characteristic for their scattering parti-cles. The measured scattered light therefore provides information on the size andcomposition of the individual particles, as well as on geometry and arrangementof an ensemble of particles. In this thesis, these are in particular gas and haze par-ticles in planetary atmospheres, and dust particles in circumstellar debris rings.

A challenge in measuring scattered light from exoplanetary systems is thebrightness of the central star, whose halo outshines the much fainter, smallerobjects at optical and near-IR wavelengths by many orders of magnitude. Thestellar light must be removed to reveal the planets or dust, and a powerfulmethod to achieve this is to use differential techniques. Differential polarimetrymakes use of the fact that the emitted starlight is generally unpolarized, whilescattering processes usually induce some amount of polarization. Subtraction oforthogonal polarization states therefore effectively removes unpolarized stellarlight while preserving scattered polarized light. A second such technique isangular differential imaging, where the observed field rotates around the starduring the observations. The pupil plane, and therefore the stellar point spreadfunction with its structure distortions introduced by the atmosphere and optics,remains fixed. Subtraction of optimally chosen rotated frames removes the stellarpoint spread function while preserving the fainter objects’ signals.

PART ONE investigates in detail the diagnostic potential of polarimetry for thecharacterization of giant planet atmospheres. Models are calculated for the po-larization depending on atmospheric parameters and constituents for extrasolarand solar system gas giants with a Monte Carlo multiple scattering code.

First, a parameter study is performed with a large grid of simple modelsto determine the influence of scattering layer thickness, absorption and planetphase on intensity and polarization. Rayleigh scattering, isotropic scattering, andHenyey-Greenstein phase functions are considered. The disk-integrated polar-ization for phase angles typical for extrasolar planet observations, as well as thelimb polarization effect observable for solar system objects near opposition, arediscussed. The polarization as a function of wavelength is compared for a planetat quadrature and opposition, and predictions are made for broadband polari-metric observations.

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Abstract

In a second step, a detailed model of the atmosphere of Uranus is constructedto interpret spectropolarimetric observations of the limb polarization of Uranusfor the wavelength range 530 to 930 nm. For the first time, polarization propertiesof atmospheric constituents of Uranus are derived. The limb polarization is dom-inated by Rayleigh scattering on molecules. It is influenced by the polarization ofa vertically extended tropospheric haze with wavelength dependent polarizationproperties, as well as a thin, highly polarizing stratospheric haze layer. From thelimb polarization model, the polarization phase curve of Uranus and the spec-tropolarimetric signal at large phase angles is calculated in order to predict thepolarization and detectability of an Uranus-like extrasolar planet.

Finally, a model of Jupiter’s polar haze is made for spatially resolved spec-tropolarimetry, focusing on the polarimetric signal at 600 nm in a slit spanningfrom the North to the South pole. The strong radial polarization at the poles, witha seeing corrected maximum of more than 10%, is well explained by stronglypolarizing and forward scattering fractal aggregate haze particles.

PART TWO describes observations of scattered light from the debris diskaround the star HD 61005. Ground-based high-contrast imaging data in H-bandare reduced with optimized angular differential imaging. The observations areof higher resolution than previous observations by the Hubble Space Telescope,and the disk is newly revealed to be a narrow, highly inclined ring. The ringcenter is found to be offset from the star by approximately 3 AU, which couldbe a result of a planetary companion that perturbs the remnant planetesimalbelt. An upper mass limit for companions that excludes any object above thedeuterium-burning limit for angular separations down to 0.35′′ is found. From apreviously imaged swept-back outer feature, the likely result of interaction withthe interstellar medium, we see two distinct streamers originating at the ansaeof the ring. The ring shows a strong brightness asymmetry along both the majorand minor axis. The brightness difference between the ring ansae can only partlybe explained by the ring center offset, possibly suggesting density fluctuations inthe ring.

This thesis shows that scattered light observations with differential techniquesare promising methods to detect and characterize planet atmospheres and de-bris disks as demonstrated on specific examples. These observations are verycomplementary to thermal light observations, and the sophisticated differentialtechniques make them feasible from large ground-based telescopes. In the out-look sections, ongoing observing programs are described that were initiated as aresult of this thesis, in particular polarimetric observations of a hot Jupiter andfollow-up observations of the HD 61005 debris disk for a characterization of thegrain size distribution and a deeper planet search. Future prospects are discussedwith a main emphasis on the upcoming 2nd generation instrument SPHERE forthe Very Large Telescope.

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Zusammenfassung

Diese Dissertation befasst sich mit dem Studium des optischen und nah-infraroten Streulichts von Objekten in Planetensystemen, insbesondere von so-laren und extrasolaren Riesenplaneten und Trummerscheiben.

Das Licht wird ursprunglich vom Zentralstern ausgesendet und danach vonden kleineren Objekten je nach Streueigenschaften ihrer Bestandteile weiterver-teilt. Das gemessene Streulicht enthalt daher Informationen uber die Grosse undZusammensetzung der einzelnen Teilchen und uber die Geometrie und die An-ordnung der Ansammlung. In dieser Arbeit sind dies Gas- und Aerosolteilchenin Planetenatmospharen und Staubteilchen in zirkumstellaren Trummerringen.

Eine Schwierigkeit beim Messen von Streulicht extrasolarer Planetensystemeist die Helligkeit des Zentralsterns, dessen Halo das Streulicht der schwacherenObjekte um ein Vielfaches uberstrahlt. Differentielle Techniken sind wirksameMethoden, um das Sternenlicht zu entfernen und Planeten oder Staub sichtbarzu machen. Die differentielle Polarimetrie nutzt aus, dass Sternenlicht im Allge-meinen unpolarisiert ist, wahrend Streuprozesse meist einen Teil des Lichts po-larisieren. Subtrahiert man Messungen in orthogonalen Polarisationsrichtungenvoneinander, verschwindet das unpolarisierte Sternlicht, wahrend polarisiertesStreulicht erhalten bleibt. Eine zweite Technik ist “Angular differential imaging”.Dabei rotiert das beobachtete Bildfeld bei den Aufnahmen. Wahrenddessen bleibtdie Pupillenebene stabilisiert, und somit auch die stellare Abbildungsfunktion,welche durch die Atmosphare und die Optik deformiert wird. Subtraktion vonoptimal ausgewahlten, gegeneinander rotierten Bildern entfernt die Abbildungdes Sterns, wahrend die schwachen Objekte ubrig bleiben.

DER ERSTE TEIL ist eine detaillierte Untersuchung des diagnostischen Potenti-als der Polarimetrie zur Charakterisierung der Atmospharen von Riesenplaneten.Polarisationsmodelle in Abhangigkeit von Atmospharenparametern werden fursolare und extrasolare Gasriesen mit einem Monte Carlo Streucode berechnet.

Zuerst wird eine Parameterstudie mit einem grossen einfachen Modellgitterdurchgefuhrt, um den Einfluss von Streuschichtdicke, Absorption und Planeten-phase auf Intensitat und Polarisation zu bestimmen. Rayleighstreuung, isotro-pe Streuung und Henyey-Greenstein-Phasenfunktionen werden betrachtet. Diescheibenintegrierte Polarisation fur Phasenwinkel typisch fur extrasolare Plane-ten sowie der Randpolarisationseffekt fur raumlich aufgeloste Sonnensystempla-neten in Opposition werden diskutiert. Die Polarisation als Funktion der Wel-lenlange wird fur Planeten in Halbphase und Vollphase verglichen und Vorher-sagen fur die Breitbandpolarisation werden gemacht.

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Zusammenfassung

Im zweiten Schritt wird ein detailliertes Modell der Atmosphare von Uranuserstellt, um spektropolarimetrische Beobachtungen der Randpolarisation vonUranus im Wellenlangenbereich 530 bis 930 nm zu interpretieren. Zum ersten Malwerden polarimetrische Eigenschaften der Atmospharenbestandteile von Uranusbestimmt. Die Randpolarisation wird durch Rayleighstreuung an Molekulen do-miniert. Ausserdem wird sie durch eine vertikal ausgedehnte Aerosolschicht mitwellenlangenabhangigen Polarisationseigenschaften und eine dunne, hochpola-risierende stratospharische Aerosolschicht beeinflusst. Aus den Randpolarisati-onsmodellen wird die Polarisationsphasenkurve und das spektropolarimetrischeSignal von Uranus bei grossen Phasenwinkeln berechnet, um die Polarisationund Beobachtbarkeit eines Uranus-ahnlichen Exoplaneten vorherzusagen.

Schliesslich wird ein Modell fur die polaren Aerosole von Jupiter erstellt furraumlich aufgeloste Spektropolarimetrie, mit Fokus auf die Wellenlange 600 nmin einem Spalt vom Nord- zum Sudpol. Die hohe radiale Polarisation an denPolen, nach Seeingkorrektur hoher als 10%, wird gut durch hochpolarisierende,vorwartsstreuende, fraktal zusammengesetzte Aerosolteilchen erklart.

DER ZWEITE TEIL beschreibt Beobachtungen des Streulichts derTrummerscheibe um den Stern HD 61005. Bodengestutzte Hochkontrastbilderim H-Band werden mit optimiertem “Angular differential imaging” reduziert.Die Beobachtungen sind hoher aufgelost als fruhere Bilder des Hubble Welt-raumteleskops, und die Scheibe wird neu als schmaler, stark inklinierter Ringgesehen. Das Ringzentrum ist vom Stern um ca. 3 AU versetzt, was auf einenplanetaren Begleiter hinweisen konnte, welcher den Ring von Planetesimalenstort. Die obere Massengrenze fur Begleiter mit einem Winkelabstand von mehrals 0.35′′ schliesst Objekte uber der Deuteriumbrennlimite aus. Von einem schonfruher abgebildeten, ausgedehnten, verformten Teil, welcher wohl durch Inter-aktion mit dem interstellaren Medium geformt wurde, sieht man zwei Bander,welche von den Randern des projizierten Rings ausgehen. Der Ring besitzt starkeHelligkeitsasymmetrien entlang beider Halbachsen. Die Unterschiede zwischenden beiden Ringseiten konnen nur teilweise durch das verschobene Ringzentrumerklart werden, was zusatzlich Dichteschwankungen im Ring vermuten lasst.

Diese Dissertation zeigt, dass Streulichtbeobachtungen mit differenti-ellen Techniken vielversprechend sind, um planetare Atmospharen undTrummerscheiben zu entdecken und zu charakterisieren, wie anhand mehrererBeispiele demonstriert wird. Diese Beobachtungen sind komplementar zu Mes-sungen der thermischen Strahlung. Die differentiellen Techniken ermoglichendie Messung mit grossen erdgebundenen Teleskopen. Im Ausblick werden lau-fende Beobachtungsprogramme beschrieben, welche aufgrund der Resultatedieser Arbeit initiiert wurden. Dies sind polarimetrische Beobachtungen vonheissen Jupitern und Nachfolge-Beobachtungen der HD 61005 Trummerscheibefur die Bestimmung der Teilchengrossenverteilung sowie eine tiefergehendePlanetensuche. Zukunftsaussichten werden mit Hauptbezug auf das kommendeZweitgenerationeninstrument SPHERE fur das Very Large Telescope diskutiert.

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Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Zusammenfassung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The discovery of planetary systems . . . . . . . . . . . . . . . . . . . 2

1.1.1 Exoplanet detection techniques . . . . . . . . . . . . . . . . . 31.2 Scattering of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.1 Physics of scattering . . . . . . . . . . . . . . . . . . . . . . . 71.2.2 Scattering on different particles types . . . . . . . . . . . . . 11

1.3 Planets in scattered light . . . . . . . . . . . . . . . . . . . . . . . . . 161.3.1 The structure of planetary atmospheres . . . . . . . . . . . . 161.3.2 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.4 Debris disks in scattered light . . . . . . . . . . . . . . . . . . . . . . 231.4.1 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.4.2 Current status of debris disks observations in scattered light 24

1.5 Differential techniques for high-contrast imaging of planetary sys-tems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.5.1 Instruments for high contrast imaging . . . . . . . . . . . . . 251.5.2 Angular Differential Imaging . . . . . . . . . . . . . . . . . . 261.5.3 Polarimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.6 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Part I Polarimetry of gaseous planets 35

2. A grid of polarization models for Rayleigh scattering planetary atmo-spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.2 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.2.1 Intensity and polarization parameters . . . . . . . . . . . . . 422.2.2 Atmosphere parameters . . . . . . . . . . . . . . . . . . . . . 432.2.3 Geometric parameters . . . . . . . . . . . . . . . . . . . . . . 432.2.4 Monte Carlo simulations . . . . . . . . . . . . . . . . . . . . . 45

2.3 Model results for a homogeneous Rayleigh-scattering atmosphere . 46

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Contents

2.3.1 Phase curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.3.2 Radial dependence for resolved planetary disks at opposition 502.3.3 Parameter study for quadrature phase and opposition . . . 53

2.4 Models beyond a Rayleigh scattering layer with a Lambert surface 592.4.1 Atmospheres with Rayleigh and isotropic scattering . . . . . 592.4.2 Forward-scattering phase functions . . . . . . . . . . . . . . 602.4.3 Models with two polarizing layers . . . . . . . . . . . . . . . 63

2.5 Wavelength dependence . . . . . . . . . . . . . . . . . . . . . . . . . 642.6 Special cases and diagnostic diagrams . . . . . . . . . . . . . . . . . 67

2.6.1 Fractional polarization versus intensity . . . . . . . . . . . . 682.6.2 Polarization near quadrature versus limb polarization . . . 702.6.3 Broadband polarized intensity . . . . . . . . . . . . . . . . . 70

2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722.A Model grid tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3. Polarization of Uranus: Constraints on haze properties and predictionsfor analog extrasolar planets . . . . . . . . . . . . . . . . . . . . . . . . . 793.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803.2 Spectropolarimetric data . . . . . . . . . . . . . . . . . . . . . . . . . 813.3 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.3.1 Atmospheric structure and haze properties . . . . . . . . . . 833.3.2 Radiative transfer code . . . . . . . . . . . . . . . . . . . . . . 86

3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883.4.1 Rayleigh scattering and methane absorption . . . . . . . . . 883.4.2 Tropospheric haze . . . . . . . . . . . . . . . . . . . . . . . . 893.4.3 Stratospheric haze . . . . . . . . . . . . . . . . . . . . . . . . 92

3.5 Predictions for the polarimetric signal of Uranus at large phase an-gles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.6 Detectability of an Uranus analog around a nearby M dwarf . . . . 963.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

3.7.1 Polarimetric properties of Uranus . . . . . . . . . . . . . . . 983.7.2 Limb polarization measurements for Uranus . . . . . . . . . 993.7.3 Prospects for exoplanet polarimetry . . . . . . . . . . . . . . 99Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4. A polarimetric model for Jupiter’s polar haze . . . . . . . . . . . . . . . 1034.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.2 Polarimetric data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.3 Polarization model for the poles of Jupiter . . . . . . . . . . . . . . . 1074.4 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . 110

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

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Contents

5. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.1 Prospects with SPHERE/ZIMPOL and beyond . . . . . . . . . . . . 1155.2 Hot Jupiter polarimetry . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.2.1 Polarimetric search for WASP-18 b . . . . . . . . . . . . . . . 119Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Part II Angular differential imaging of a debris disk 123

6. Dissecting the ’Moth’: Discovery of an off-centered ring in the HD 61005debris disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1256.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1266.2 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.3 Data reduction and PSF subtraction . . . . . . . . . . . . . . . . . . 1286.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6.4.1 Surface brightness of ring and streamers . . . . . . . . . . . 1316.4.2 Ring geometry and center offset . . . . . . . . . . . . . . . . 1326.4.3 Background objects and limits on companions to HD 61005 135

6.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

7. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1437.1 Further observations of the Moth . . . . . . . . . . . . . . . . . . . . 143

7.1.1 A search for companions inside the ring . . . . . . . . . . . . 1437.1.2 Resolved disk observations at different wavelengths . . . . 144

7.2 A detailed model for the Moth . . . . . . . . . . . . . . . . . . . . . 1467.3 Future prospects for debris disks imaging . . . . . . . . . . . . . . . 147

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

List of Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

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Chapter 1

Introduction

The question whether life is unique to Earth or ubiquitous in the universe has oc-cupied humankind for millennia. The astronomers’ task in answering this ques-tion is to determine whether planets around other stars are actually a commonphenomenon, whether for such planets conditions for habitability are met, andhow signs of life can be detected by remote observations. Taking a spectrum of apotentially habitable Earth-like planet is probably one of the greatest challengesof modern astronomy. Before we are capable of this endeavor, many subsequentsmaller steps can be taken towards this ultimate goal. Giant planets are mucheasier to detect and characterize, and it is reasonable to develop the know-howfor planet detection and characterization by studying large planets. At the sametime, these studies can determine how diverse planetary systems are, provideclues to their formation and evolution, and help us establish the origin of ourown solar system.

During the past two decades it has been well established that planets aroundother stars exist with a wide range of masses and semi-major axes. Indeed, evi-dence points towards the fact that in particular smaller planets are quite commonin the galaxy. Most planets have so far been found by indirect observing tech-niques, where the effect of the planet on the star is measured, rather than the lightfrom the planet itself. This has the disadvantage that the planet’s atmosphere can-not be characterized. More recently, direct measurements have become feasible.There, the difficulty lies in separating the bright starlight from the much fainterplanetary light. This is currently achieved with differential techniques, where thesystem is imaged in two ways, in which the star remains largely identical, buta planet property changes such that a subtraction removes the star and revealsthe planet. The two techniques applied in this thesis are polarimetric differentialimaging (PDI) and angular differential imaging (ADI).

The planetary light has two components: a scattered light component that isreflected starlight, and a thermal light component, which is the radiation emittedby the planet itself. Young planets are quite warm and are therefore intrinsicallybright, but the luminosity drops quickly with increasing age as the planet coolsoff. The amount of scattered light is independent of the planet age, rather it isinversely proportional to the squared distance between the star and the planet.Scattered light observations would constrain the planet albedo, the ratio betweenincoming and outgoing light, which is important for the energy balance of the at-mosphere. Also, they would constrain particle properties, on which the scattering

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Chapter 1. Introduction

strongly depends. A scattered light search would target mainly nearby stars, be-cause while the planet should be close to the star, it should also be well separatedin angular distance. Also, many photons are required to have very low noise,because a high precision must be achieved to obtain the small star-planet con-trast. The nearest brightest stars are therefore the ideal targets. Finding a planetin these systems would be extremely interesting for follow-up observations withfuture instruments.

Not only planets can be imaged in scattered light. Debris rings, brighterKuiper belt or asteroid belt analogs, are actually much easier targets. These ob-servations directly reveal the dust geometry and can also show indirect evidenceof planetary companions that sculpt the dust. The same differential techniquesas for planet searches are applicable.

In this thesis it is discussed how scattered light observations from planets anddebris rings made with differential techniques can be interpreted. The introduc-tion covers the fundamental background to this topic, starting from how plane-tary systems are discovered. The physics of light scattering and observables forplanet atmospheres and debris disks are discussed, and the differential imagingtechniques are introduced. Finally, a detailed overview over the content of thethesis is given.

1.1 The discovery of planetary systems

Before the development of optical instruments, five planets were known in addi-tion to the Earth. Mercury, Venus, Mars, Jupiter and Saturn were easily visible bynaked eye and recognizable as planets by their unusual motion with respect tothe “fixed” stars. After the development of the telescope, two additional planetswere discovered: Uranus in 1781 and Neptune in 1846. Many other objects werefound to be a part of the Solar System that do not meet the currently accepted def-inition of a planet1. These include moons, asteroids, Kuiper belt objects, Comets,and small dust particles. While not planets, these objects still provide many cluestowards the formation and evolution of our planetary system.

The first evidence of circumstellar matter around a main sequence star with-out significant mass-loss was found in 1983, when the Infrared AstronomicalSatellite (IRAS) discovered unexpectedly strong infrared radiation beyond 20 µm(Aumann et al. 1984) from Vega. The radiation was stronger than could be ex-pected from the star alone, and this excess was quickly attributed to origin fromsmall cool dust particles. The first scattered light image of such a dust disk wasobtained when Smith & Terrile (1984) used a stellar coronagraph to block the lightof the star β Pictoris, revealing an extended edge-on disk.

1 From the International Astronomical Union (IAU) resolution B5: A planet [in the solar Sys-tem] is a celestial body [excluding satellites] that is in orbit around the Sun, has sufficient mass forits self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearlyround) shape, and has cleared the neighborhood around its orbit.

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1.1. The discovery of planetary systems

The discovery of an actual extrasolar planet happened in 1992, when Wol-szczan & Frail (1992) discovered three low-mass worlds orbiting the pulsar PSRB1257+12 by measuring the variation in the pulsar period induced by the plan-ets. It is however likely that these planets accreted from ejected material after thesupernova explosion that turned the star into a neutron star. The first extraso-lar planet orbiting a sun-like star was found around 51 Peg by Mayor & Queloz(1995). A big surprise was the fact that 51 Peg b is a Jupiter-mass planet orbitingits star with a period of only 4.2 days, much shorter than the orbital period ofMercury. This discovery led to a new era of planet detection using several differ-ent techniques, and today more than 500 extrasolar planets and more than 1000additional planet candidates have been found. These discoveries show a hugediversity in planetary systems, with planet masses, separations and eccentricitiesoften unlike those of the planets in the Solar System.

1.1.1 Exoplanet detection techniques

The currently successful exoplanet detection techniques do not detect light scat-tered by the planet. Rather, most often they measure the effect the planet hason its host star. This effect can be the gravitational pull of the planet on thestar (radial velocity and astrometry method), or the photometric dimming of thehost star (transit method) or brightening (microlensing method) of a backgroundsource. Direct measurements have been made of the thermal emission of hotJupiters (secondary eclipse measurements) and young self-luminous planets (di-rect imaging). Recent reviews of all techniques can be found e.g. in Seager (2011).

Radial Velocity

In the radial velocity (RV) or Doppler spectroscopy method, a spectrum of the host staris measured at high resolution over the time of a planet’s orbit. The radial com-ponent of the motion of the star around the common center of gravity of star andplanet can be detected through a periodic shift of the stellar absorption lines be-cause of the Doppler effect. Today, radial velocity differences of less than 1 m/scan be measured for example with the HARPS instrument (Mayor et al. 2003),which has proven to be the most successful planet hunting instrument. The RVmeasurements do not deliver the true mass m of the planet, but only the projectedmass m sin i, because only the line-of-sight component of the stellar motion pro-duces a Doppler shift. Objects in a near face-on orbit will therefore have a truemass that is much higher than the measured minimum mass. However, this con-figuration is statistically rare, assuming all planet orbits are randomly oriented inspace. Therefore, most of the ∼ 500 planetary objects detected by the RV methodwill be true planets. Only for ∼ 100 transiting planets the inclination is known,such that the true mass can be found. The RV method additionally delivers thesemi-major axis and eccentricity of the planet’s orbit. It is most suited for the de-tection of planets at small separations around quiet F to M-type stars with manyabsorption lines. Active and variable stars produce significant noise in the radial

3


velocity signal that makes the detection of low-mass planets difficult. Planets atlarge separations, with periods of several years, need a long observing sequenceuntil the planetary signal can be extracted. A long-term goal is to be able to mea-sure radial velocities at precisions below 10 cm/s, which is required to find anEarth-mass planet in the habitable zone of a solar type star.

Transits

The transit method takes advantage of the fact that planets with an orbit that iswell aligned with the line-of-sight will pass in front of the host star periodically,blocking a significant amount of stellar light, and thus dimming the star. Thefirst discovered transiting planet, HD 209458, was found by Charbonneau et al.(2000). The transit method delivers the radius, and in conjunction with RV mea-surements the true mass and therefore density of the planet. A first characteriza-tion of the planet structure can thus be achieved, and has revealed a wide rangeof densities, from rocky planets (e.g. Kepler-10b, Batalha et al. 2011) to very puffygas giants (e.g. Wasp-17b Anderson et al. 2010). During the transit, some stel-lar light passes through the atmosphere at the limb of the planet. Dependingon the location of absorbers present in the atmosphere, the transit radius willvary with wavelength. Transit spectroscopy can therefore measure constituents ofthe planet’s atmosphere. Up to now, reliable detections were made for atomicsodium (Charbonneau et al. 2002), potassium (Sing et al. 2011) and hydrogen(Ly-α, Vidal-Madjar et al. 2003).

When the planet passes behind the star in the secondary eclipse, the thermallight of the planet disappears and only the stellar light remains. Subtracting thislight from a measurement made before the secondary eclipse leaves only the plan-etary light. Such a measurement can therefore be considered a direct detection ofthe planetary atmosphere (Charbonneau et al. 2005). If the planet is followed forthe whole orbit, a thermal phase curve can be constructed (Fig. 1.1), and a basiclongitudinal map of the planet temperature derived (Knutson et al. 2007).

While a lot of information can be gleaned from a transiting planet, strict geo-metric requirements must be fulfilled for the planet to transit. The transit proba-bility drops quickly for increasing planet separation from & 10% for hot jupitersto only ∼ 2% for a jupiter-sized planet in a 50-day orbit around a solar-type star(Kane & von Braun 2009). Additionally, transits occur much less frequently for along-period planet. A very large number of stars must therefore be continuouslymonitored. This is among others systematically done with the Kepler satellite(Borucki et al. 2003), which has in the first third of its mission already detectedmore than 1000 planet candidates (Borucki et al. 2011).

Microlensing

The microlensing technique relies on the fact that an object can act as a lens thatmagnifies the light of a well-aligned background star, resulting in a well-definedlight curve. If a planet orbits the lens star, a secondary peak or lensing light curve

4

1.1. The discovery of planetary systems

Figure 1.1: Left: Schematic orbit of an eclipsing planet. Primary transit with transmit-

ted light through the atmosphere, thermal phase variations and secondary eclipse. The

planet size is not to scale. Right: Thermal phase curve with primary and secondary

eclipse for HD 189733 b measured with Spitzer at 8 µm Knutson et al. (2007).

distortions can be present in the light curve, giving the mass ratio between starand planet and the planet separation. These events are rare and unique, andcannot be followed-up. The method is sensitive to low-mass planets and planetsat large separations. A large survey, monitoring millions of stars in the galacticbulge, can therefore provide unique statistical information about the occurrenceof such planets. At this time, among others, a planet with a mass as low as 5Mo plus (Beaulieu et al. 2006), a Jupiter-Saturn analog system (Gaudi et al. 2008)and a large number of giant planets at separations > 10 AU, some possibly evenfree-floating, (Sumi et al. 2011) have been detected.

Astrometry

The astrometry method accurately measures the position of the star in the planeof the sky with respect to background stars, providing the projected orbit of thestar around the center of mass of the star-planet system. The method is mainlysensitive to massive planets at large separations, for which long-term observa-tions are necessary. At this time, no planet has been detected by astrometry, butseveral have been confirmed (e.g. GJ876 b, Benedict et al. 2002). The astromet-ric measurements then deliver the true mass and orbit of the planet. The spacemission Gaia (Perryman et al. 2001) is expected to find thousands of giant planetsat separations between 1 and 4 AU by measuring high-precision astrometry for∼300,000 stars within 200 pc (Casertano et al. 2008).

Direct imaging

The most direct way of probing a planet’s atmosphere is through direct imaging,where an actual image of the planet is taken. This method is very difficult becausethe star is orders of magnitudes brighter than the planet and the planets are typ-ically located at angular separations < 1′′. Even with the augmented resolutionprovided by an adaptive optics (AO) system and blocking the stellar light with

5


a coronagraph, the stellar halo from the star’s PSF drowns the planetary signal.Sophisticated data reduction techniques are required to remove the stellar light.Such techniques are described in more detail in Sect. 1.5.2 and applied in Sect.6. Favorable for direct detection are young planets that are still self-luminousbecause of gravitational contraction. In the near-infrared the star-planet contrastfor young massive planets is typically of order 10−5 − 10−6 and therefore alreadyin the range of today’s instruments at 8 m class telescopes for separations & 20AU. Older planets like in the Solar s System, that are barely self-luminous andmainly shine from reflected starlight, show a contrast of only ∼ 10−8 and aremuch harder to discover. One method feasible for the detection of such planetsin the near future is polarimetry of scattered light. This technique is explained inSect. 1.5.3 and its planet characterization potential discussed in Sect. 2 to 4.

The first successful direct detection of a planetary system was achieved byMarois et al. (2008), who discovered three planetary mass objects (between 5 and12 MJ) in wide orbits (24 to 68 AU) around the A-type star HR 8799 (Fig. 1.2).Further reasonably firm direct detections of planets are β Pic b (∼9 MJ) at ∼8 AULagrange et al. 2010) and 1RXS J160929.1-210524 b, a planetary mass object at anunusually large orbit of 330 AU (Lafreniere et al. 2010).

Direct measurements in multiple filters allow a determination of the planet’sspectral type and thus a temperature, which together with the luminosity thenprovides a radius. Spectrophotometry or spectroscopy give a first indication ofthe composition of the planet’s atmosphere (e.g. Bowler et al. 2010; Currie et al.2011, for HR 8799 b, indicating a cloudy atmosphere). The mass of the planetscannot be determined directly, but if the stellar (and thus planetary) age is known,evolutionary models (e.g. Baraffe et al. 2003) can give an estimate. Since the agesof stars are often not well known and the evolutionary models not yet well cal-ibrated in the planetary mass range, the derived masses often have a relativelylarge uncertainty. Often enough, a potential planetary mass object is found to bea brown dwarf once the stellar age is determined more precisely (e.g. GJ 758 b,Thalmann et al. 2009; Currie et al. 2010).

1.2 Scattering of light

When an electromagnetic wave encounters a particle, be it a single electron, atom,molecule or a larger liquid or solid particle, the electric field of the wave sets elec-tric charges in the particle into oscillatory motion. The electric charges are accel-erated and therefore radiate electromagnetic energy (Fig. 1.3). This secondaryradiation is commonly called “scattered light”. Even reflection and refractionon surfaces and interfaces and diffraction on slits, gratings or edges are actu-ally scattering events on many coupled molecules. This thesis only treats elasticscattering, where the frequency of the scattered light remains constant. Inelasticscattering effects, e.g. Raman scattering, are not discussed.

The excited charges may also convert part of the incident energy into otherforms, for example thermal energy. This process is called “absorption”. A good

6

1.2. Scattering of light

Figure 1.2: The HR 8799 system with three directly imaged planets. The colorful pattern

in the center are the speckle residuals after subtraction of the star’s PSF. Figure by Marois

et al. (2008)

review on the physics of scattering and absorption by particles is given in Bohren& Huffman (1983).

1.2.1 Physics of scattering

In a single particle, the oscillating field induces a dipole moment in each regionof the particle by perturbing the electron clouds of the atoms or molecules. Eachdipole oscillates with the same applied frequency and emits light into all direc-tions. At a distant point in some direction, the total scattered light is producedby a superposition of the single wavelets. Because dipole radiation is coherent,the phases have to be taken into account. The phase relations generally vary withscattering direction, depending on the size and shape of the particle. Both theamplitude and phase also depend on the particle material. Therefore, the finalscattered intensity will strongly vary with direction, depending on the particleproperties.

For an incident monochromatic plane wave with an electric field

Ei = E0ei(k·x−ωt), (1.1)

the scattered field in the far-field region kr ≫ 1 is approximately transverse andcan be written with respect to the incident field as

(

E‖s

E⊥s

)

=eik(r−z)

−ikr

(

S2 S3

S4 S1.

)(

E‖i

E⊥i

)

(1.2)

E‖ and E⊥ are the components parallel and perpendicular to the scattering planedefined by the scattering direction er and the direction of the incident beam ez. S

7


Figure 1.3: Scattering of a light wave on a small particle, inducing dipole oscillations that

reradiate energy at the same frequency. Figure by D. W. Hahn, Light Scattering Theory

http://plaza.ufl.edu/dwhahn/RayleighandMieLightScattering.pdf.

is the amplitude scattering matrix whose elements depend on the particle prop-erties and scattering direction.

A scattering process does not only influence the intensity of the light in a par-ticular direction, but also its polarization. The polarization is the direction inwhich the electric field vector Ei(x, t) or Es(x, t) oscillates. Each wave has a cer-tain polarization direction of its own, but the polarization of light, i.e. an ensem-ble of many waves, describes the degree to which the electric field vectors of thesewaves are aligned in a particular direction. If the orientation of the E-field vectorsis randomly distributed, we speak of unpolarized light. There exist three typesof polarization: linear, i.e. oscillation in a plane, circular, i.e. a rotating electricfield vector at fixed z position, and elliptical polarization, which is a combina-tion of the two. In this work, we focus solely on linear polarization, which is thedominant polarization for scattering from planetary atmospheres or dust grainsin disks.

Stokes formalism

Polarized light can be described with the four Stokes parameters, typically rep-resented in the form of the Stokes vector I, where I is the intensity, Q and Uthe linear polarization in horizontal/vertical and diagonal direction, and V thecircular polarization (Fig. 1.4).

I =

IQUV

=

E20x+ E2

0y

E20x− E2

0y

2E0x E0y cos (δx − δy)2E0x E0y sin (δx − δy)

=

I0 + I90

I0 − I90

I45 − I135

IR − IL

. (1.3)

x and y denote the components of the amplitude E0 and phase δ of the electriccomponent of the wave in these directions, with the propagation direction beingz. IX are the intensities measured through a polarization filter with throughput

8

http://plaza.ufl.edu/dwhahn/Rayleigh and Mie Light Scattering.pdf


Figure 1.4: Left: Electric field vector evolution of elliptically polarized light, decomposed

into circular polarization Ecirc (blue) and linear polarization components Ex (red) and Ey

(green). Right: Stokes components for linear polarization Q and U. The direction of +Q is

defined arbitrarily, but commonly in sky north direction. +U is defined as the direction

rotated by 45 counter-clockwise. After rotation of 180 the polarization is back in the

same state.

for polarized light in direction X. R and L stand for right- and lefthand circularpolarization.

The Stokes vector is not actually a vector, but rather a one-column matrix. Thefollowing relation always holds:

I2 ≥ Q2 + U2 + V2 ≥ 0, (1.4)

where the first equal sign holds for fully polarized light and the second for un-polarized light. Intermediate cases are called partially polarized. The degree ofpolarization is

p =

√

Q2 + U2 + V2

I, (1.5)

and the polarization angle θ for linear polarization is given by

tan 2θ =U

Q, (1.6)

counted in counter-clockwise direction with θ = 0 in positive Q direction.A scattering process modifies the full Stokes vector of incoming light. Equa-

tion 1.2 can be reformulated as

Is =

Is

Qs

Us

Vs

=1

k2r2

F11 F12 F13 F14

F21 F22 F23 F24

F31 F32 F33 F34

F41 F42 F43 F44

Ii

Qi

Ui

Vi

. (1.7)

with F the scattering matrix composed of the real components Fij that can becalculated from Sk in Eq. 1.2. For most scattering cases some degree of symmetryis present, and not all 16 matrix elements are independent.

9


For scattering on a sphere or on a group of randomly oriented particles whichhave a plane of symmetry, only six independent matrix elements remain:

F =

F11 F12 0 0F12 F22 0 00 0 F33 −F34

0 0 F34 F44.

(1.8)

Neglecting circular polarization, i.e. setting any Fi4 component equal to 0, theremaining elements are given by:

F11 =1

2(|S2|2 + |S3|2 + |S4|2 + |S1|2), (1.9)

F12 =1

2(|S2|2 − |S3|2 + |S4|2 − |S1|2), (1.10)

F22 =1

2(|S2|2 − |S3|2 − |S4|2 + |S1|2), (1.11)

F33 = ℜ(S2S∗1 + S3S∗

4). (1.12)

These scattering matrix coefficients are discussed for different particle types inSect. 1.2.2. F11 is usually referred to as the (single) scattering phase function,while −F12/F11 corresponds to the single scattering fractional polarization.

Radiative transfer with scattering

When electromagnetic radiation propagates through a medium, the Stokes vectorI varies because of absorption, emission and scattering in the medium. Mathe-matically, this is expressed by the equation of radiative transfer:

dI

dS= −αI + ǫ, (1.13)

where α = µa + µs is the extinction coefficient, the sum of absorption and scat-tering coefficient, and ǫ the emission coefficient that includes intrinsic emissionand scattered radiation. Here the focus is only on absorption and scattering ata specific frequency ν, the intrinsic emission is neglected, which is reasonable inthe short-wavelength limit. Then,

dIν

dS= −(µa,ν − σs,ν)Iν +

µs,ν

4π

∫

ΩF(θ)IνdΩ, (1.14)

where F is the scattering phase matrix and θ the scattering angle.There exist several methods for solving this equation to calculate the flux and

polarization scattered from an atmosphere into a particular direction. A fewspecial cases can be calculated analytically, to be found for example in Chan-drasekhar (1950). For most cases however, numerical methods must be ap-plied. A popular method to calculate the intensity and polarization of a multiply-scattering atmosphere is the “adding-doubling method” (Hovenier 1971). In this

10


technique, the reflection and transmission properties of a plane-parallel slab arecalculated by starting from a thin homogeneous layer with known properties,doubling it to the desired thickness, or adding thin dissimilar slabs. Its disadvan-tage is that it is only suited for horizontally homogeneous planet atmospheres.A more flexible and straight-forward method is to use a Monte Carlo code. Pho-tons are followed on their path through the atmosphere, where they scatter orabsorbe with some probability depending on the cross sections and abundances.Scattering directions are deduced from probability density functions constructedfrom the phase matrix. A detailed introduction to the specifics of the method ap-plied to planet atmospheres in this thesis can be found in Schmid (1992), and asummary in Chapter 2.

1.2.2 Scattering on different particles types

Rayleigh scattering

When the scattering particle is very small compared to the wavelength of thelight, the Rayleigh scattering approximation holds. This type of scattering is veryimportant in atmospheres, because all gas molecules fall in this regime. It is alsovalid for very small haze particles. The mathematical conditions for Rayleighscattering are,

x =2πa

λ≪ 1, |m|α ≪ 1 (1.15)

where x is the size parameter, a the particle radius, λ the incident wavelength inthe surrounding medium (λ = λ0/m0 with λ0 the wavelength in vacuum andm0 the refractive index of the medium), and m = n − iκ the complex refractiveindex of the particle. The real part n of the refractive index indicates refraction oflight, while the imaginary part corresponds to absorption. For a dielectric particle(κ = 0) there is only scattering, and no absorption.

A Rayleigh scattering particle scatters light like an oscillating dipole. If thepolarizability of the particle is isotropic, e.g. for a small sphere, the scatteringmatrix components (Eq. 1.9 to 1.12) are given by:

1

k2r2F11 =

x6

k2r2

∣

∣

∣

∣

m2 − 1

m2 + 2

∣

∣

∣

∣

2

· 0.5(1 + cos2 θ), (1.16)

1

k2r2F12 =

x6

k2r2

∣

∣

∣

∣

m2 − 1

m2 + 2

∣

∣

∣

∣

2

· 0.5(cos2 θ − 1), (1.17)

1

k2r2F33 =

x6

k2r2

∣

∣

∣

∣

m2 − 1

m2 + 2

∣

∣

∣

∣

2

· cos θ, (1.18)

The angular distribution of scattered light depends on the incident polariza-tion (Fig. 1.5). For incident unpolarized light, forward and backward scatter-ing are enhanced with respect to right angle scattering. For scattering at an an-gle θ = 90 the polarization fraction p = −F12/F11 = 100% and the direction

11


Figure 1.5: Angular distribution (normalized) of the light scattered by a sphere small

compared to the wavelength for incident unpolarized light (solid), polarized parallel

(dashed) and polarized perpendicular (dash-dot) to the scattering plane. Figure reprinted

from Bohren & Huffman (1983).

perpendicular to the scattering plane. For molecules the polarizability is gen-erally not isotropic, and a small depolarisation is introduced, depending on themolecule shape and number of electrons. For the major ingredients of planet at-mospheres, this effect is of order 1 − 2%. This small depolarization is neglectedin the Rayleigh scattering calculations in this thesis.

Very typical for Rayleigh scattering is the wavelength dependence of the scat-tering cross section: if m is independent of λ, which nearly holds for molecules,

then σ(λ) ∝ x6

k2 ∝ λ−4. Rayleigh scattering is therefore much stronger at shorterwavelengths than at longer wavelengths, which is the main reason why theEarth’s sky is blue. Rayleigh scattering in planet atmospheres is the main focusof Chapter 2.

Mie scattering on spherical particles

Scattering on a spherical particle can be solved exactly using Mie theory (Mie1908). The basis of the solution is the expansion of the plane wave in sphericalharmonics. The lengthy calculation of incident and scattered field is given inBohren & Huffman (1983).

The matrix components S1 to S4 (Eq. 1.2) are

S1(θ) =∞

∑n=1

2n + 1

n(n + 1)(anπn(cos θ) + bnτn(cos θ)) (1.19)

S2(θ) =∞

∑n=1

2n + 1

n(n + 1)(bnπn(cos θ) + anτn(cos θ)) (1.20)

S3(θ) = S4(θ) = 0 (1.21)

12


Figure 1.6: Single scattering phase function and polarization of example Mie scattering

particles as a function of scattering angle. F11 is normalized to the value at 0 scattering

angle. Lines indicate: water droplets with effective size parameter xe f f = 23 and “stan-

dard” (Hansen & Travis 1974) size distribution (blue, Karalidi et al. 2011), haze particles

(n = 1.66) with xe f f = 4.5 and standard size distribution (green, Stam et al. 2004), spheres

with xe f f = 15 with refractive index m = 1.53 + 0.008i and power-law size distribution

(red, Mishchenko & Travis 2003), and the Rayleigh scattering limit (black).

πn(cos θ) and τn(cos θ) are functions of Legendre polynomials. an and bn are ex-pressed through Riccati-Bessel functions and also depend on the size parameterx (Eq. 1.15). From the coefficients follow also the scattering and extinction crosssections,

Csca =2π

k2

∞

∑n=1

(2n + 1)(|an|2 + |bn|2), (1.22)

Cext =2π

k2

∞

∑n=1

(2n + 1)ℜ(an + bn). (1.23)

For scattering on multiple particles a sum over all particles per unit volumemust be performed to get the total scattering and extinction coefficients. Thephase functions usually show a strong forward scattering peak, which increaseswith particle size. Phase functions of a collection of spherical particles of size sim-ilar to the wavelength often show strong ripples because of interference of multi-ply refracted rays. These disappear for larger particles, or are smoothed out if awider size distribution of particles is present. The polarization can also be partlynegative (parallel to scattering plane). Some spheres, such as water droplets, haveone or more narrow peaks at large scattering angles, corresponding to a rainbowfeature. The wavelength dependence of the scattering cross section is generallyweak, which explains why clouds and fog are white.

Numerical calculations are straightforward and many codes are available forthe calculation of these coefficients from Mie theory, e.g. bhmie2 given in the Ap-pendix of Bohren & Huffman (1983). A sample of phase functions is shown inFig. (1.6).

2 Code available in several programming languages at http://code.google.com/p/scatterlib/

13

http://code.google.com/p/scatterlib/


Figure 1.7: An aggregate particle constructed by diffusion-limited aggregation seen from

two different sides. The particle is made of 170 monomers with each monomer consisting

of 22 dipoles. Each dipole is depicted as a small sphere. Figure from West (1991).

Scattering on fractal aggregate particles

While Mie theory is only exact for spherical particles, e.g. water droplets, it isoften used as a first approximation for scattering on non-spherical particles, suchas dust or aerosol particles. The phase function is often quite well reproduced,with a strong forward scattering peak and reduced back scattering compared toRayleigh scattering. However, with more polarization measurements of planetatmospheres or dust disks available, it was soon found that the full phase matrixis often poorly reproduced by Mie theory. This was clearly demonstrated for thehaze particles at the poles of Jupiter with Pioneer observations (West 1991). Thepolarization phase function resembles strongly the Rayleigh polarization phasefunction, with a peak of almost 100% near 90 phase angle, while the phase func-tion is strongly forward scattering (Fig. 1.8). This combination cannot be repro-duced by Mie theory. It was theorized that these particles are fractal aggregateparticles (see Fig. 1.7 and Sect. 1.3.1 and 4), for which the forward scattering partis reproduced by the average projected area, while the high polarization arisesfrom the smaller monomer components. The phase functions of such particlescan only be calculated with approximations and the calculations are computa-tionally intensive. Two commonly used methods are the Discrete Dipole Approx-imation (DDA, e.g. Yurkin et al. 2007) and the T-matrix method (e.g. Mishchenkoet al. 1996), for which a simpler approximate parametrization has been used byTomasko et al. (2008).

Absorption

The absorption along a beam of light penetrating an absorbing material generallyfollows the Beer-Lambert law, sometimes simply called Beer’s law,

I(λ, z) = I0e−µa(λ)z. (1.24)

The validity of the law breaks down at large abundances, in particular if the mate-rial is also highly scattering. In some instances, non-linear effects can also inducea deviation. In this thesis, the Beer-Lambert law is considered valid.

14


Figure 1.8: Single scattering phase function and polarization of a sample fractal aggregate

dust particle composed of silicate and graphite as a function of scattering angle for dif-

ferent wavelengths, calculated with the discrete dipole approximation. F11 is normalized

to the value at 0 scattering angle. For the longest wavelength, the Rayleigh scattering

limit holds. Figure adapted from Shen et al. (2009).

The absorption coefficient of a particle is derived from the complex part κ ofthe index of refraction m = n− iκ. Replacing the wave number in the propagationdirection kz in the electromagnetic wave (Eq. 1.1) with the relation kz = ωm/c,

E(z, t) = E0e−ωκz

c ei(kzz−wt) (1.25)

This is therefore equivalent with a wave without a complex index of refractionand a damping factor. The absorption coefficient is then,

µa =ωk

c=

4πκ

λ. (1.26)

If only the transmission of light into a particular direction is regarded, thenlight losses by scattering must also be taken into account. In that case,

I(λ, z) = I0e−(µa(λ)z+µs(λ)z) = I0e−αz, (1.27)

where α is the extinction cross section.While dust, haze and cloud particles can be absorbing to some degree, the

main absorbers in the solar system gas giant atmospheres are molecules withtransitions in the optical or near-infrared. Absorption through dipole transitionswhich occur in the optical and infrared only work for polar molecules with apermanent dipole moment. For the cool gas giants, the dominant absorber at op-tical wavelengths is methane. No laboratory measurements exist for the methaneabsorption coefficients at the temperature and pressure conditions of the gas gi-ant planets. The best available absorption coefficients were measured empirically

15


from the Solar System gas giant’s albedo spectra by Karkoschka (1998) and im-proved by Karkoschka & Tomasko (2010) with in-situ measurements of Huygenson Titan. For hot atmospheres like hot Jupiters, pressure-broadened atomic fea-tures, such as Na or K can be very strong (e.g. Burrows et al. 2008).

Another form of absorption that is significant especially in the cloud-free at-mospheres like Uranus and Neptune is collision-induced absorption (CIA). Thistype of absorption occurs when the electron cloud of a molecule is displacedby collision with another molecule, thus forming a temporary dipole moment.The light wave can then modulate the dipole moment to shift the molecule be-tween different rotovibrational (in the visible and near-infrared regions) or roto-translational (far-infrared and microwave regions) states. Then, part of the lightis absorbed in some frequency bands. This mechanism works for non-polarmolecules, and in gaseous atmospheres it is strongest for collisions between H2

molecules. The collision-induced absorption coefficients are proportional to theproduct of densities of the collision partners. Absorption profiles can be calcu-lated by numerical integration of Schroedinger’s equation with the appropriatedipole and potential models. The calculated profiles can also be represented insimpler, analytical expressions that allow a faster evaluation for desired wave-lengths, temperatures and gas densities and are reasonably accurate. Such cal-culations of absorption coefficients for all significant collision partners and fre-quency bands in planetary and stellar atmospheres were performed by A. Bo-rysow3, e.g. in Borysow (2002).

1.3 Planets in scattered light

1.3.1 The structure of planetary atmospheres

The fundamental parameters of a planetary atmosphere are its pressure, tem-perature and composition. These vary strongly with height and often also withlocation on the planet. For simplicity we here assume only the 1D case where asingle vertical pressure, temperature and abundance profile describes each pointon the planet sphere.

The atmospheric structure has been investigated in detail for the Solar Systemplanets, both remotely and in-situ, while first observations of exoplanet atmo-spheres already provide insight into vastly different atmospheres. A review onSolar System gas giant atmospheres can be found in Irwin (2003), and on exo-planet atmospheres in Seager (2010), on both of which this chapter is based.

Pressure and Temperature

If the vertical wind velocities are much smaller than the horizontal velocities, theassumption of hydrostatic equilibrium is very accurate. This is generally the case

3 Tables and codes for all calculations are publicly available at http://www.astro.ku.dk/ abo-rysow/programs/index.html

16

1.3. Planets in scattered light

in planetary atmospheres. Then the pressure difference dp through a slab of airwith density ρ and thickness dz subject to gravitational acceleration g is

dp = −ρgdz. (1.28)

From the ideal gas law, which is adequate for planetary atmospheres,

P = nkT =ρkT

µmH(1.29)

with n the number density, k the Boltzmann constant, µ the mean molecularweight and mH the mass of a hydrogen atom. Inserting Eq. 1.29 into Eq. 1.28 andintegrating under the assumption that the temperature is constant with height, itfollows that

p = p0e−z/H (1.30)

where p0 is the pressure at z = 0 and

H = kT/µmHg (1.31)

is the scale height. Because many giant planets rotate very rapidly and are there-fore oblate, the gravitational acceleration g and thus the scale height H vary sig-nificantly between equator and polar regions. Additionally, the mean molecularweight may vary with height if some species condense, and the temperature isgenerally not constant with height.

In planetary atmospheres the competing modes of energy transport aremainly radiation and convection. If the optical depth is high, radiation cannoteasily escape and the atmosphere is convective. This is generally the case in thelower troposphere. The temperature gradient is then adiabatic,

dT

dz= − g

cp= −Γd (1.32)

with cp the specific heat capacity at constant pressure and Γd the dry adiabaticlapse rate. If clouds condense the temperature drop with height is a bit slowerwith the saturated adiabatic lapse rate Γs. At higher altitudes, the optical depthtowards space becomes smaller and radiation can more easily penetrate and es-cape. The atmosphere becomes radiative and the temperature is determined byradiative equilibrium. In the upper troposphere the temperature decreases downto a certain level at the radiative-convective boundary (tropopause). In the strato-sphere, the atmosphere is highly stable against convection, the air forms stratifiedlayers and the temperature raises with height because of the absorption of sun-light (see e.g. Fig. 3.3 for the atmospheric structure of Uranus). For the giantplanets, the temperature structure of the stratosphere and tropopause dependsstrongly on the photochemical processes.

17


Atmospheric composition

The composition of atmospheres is mainly dominated by three components: gas,clouds and haze. The distinction between clouds and haze is not always clearin the literature. A simple distinction could be that clouds are formed throughcondensation of gaseous species, while haze is formed in the upper atmospherelayers through photochemical processes and may then gradually move lowerthrough eddy transport.

Gas abundances are usually given as the (volume) mixing ration fi = ni/ntot,with ni the number density of the ith gas and ntot the total number density ofgases. For giant planets, the dominant gases are H2 and Helium, with the absorp-tion features in the spectra of the Solar System gas giants dominated by methane.

A gas condenses when its partial pressure equals the saturated vapor pressureand cloud condensation nuclei are present. The approximate condensation levelof a cloud can be estimated from thermodynamics with the Clausius-Clapeyronequation,

dp

dT=

∆S

∆V≈ Lp

RT2(1.33)

with L the latent heat of vaporization per mole and R the molar gas constant. Thisequation can be integrated depending on the temperature dependence of L.

Photochemistry is induced by UV radiation from the host star, which disso-ciates molecules in the upper layers of the atmosphere. The fragments can thenrecombine into different molecules. For example, dissociated methane may formethane or other hydrocarbons, such as chains of polyacetylene that then assembleto aggregate particles. Another important dissociation mechanism are chargedparticles. For example at the poles of Jupiter, auroral activity is believed to playan important part in the formation of fractal aggregate particles. Figure 1.9 de-picts a plausible scheme for the formation, coagulation and settling of particles inthe atmosphere at the poles of Jupiter (Friedson et al. 2002).

1.3.2 Observables

When observing a planet inside or outside of the Solar System, we measure theflux received from the planet. Analyzing the flux variations with wavelength orpolarization state and with respect to the incoming radiation from the star, manyphysical properties of the planet can be derived. The influence of different pa-rameters on these observables is discussed extensively in Chapter 2 and appliedto two specific cases in Chapters 3 and 4.

Reflectivity

A key concept for scattered light observations from planets is the albedo, which isa measure of the reflectivity of the planet. It is given by the ratio of light scatteredby the planet to that received. A high albedo indicates strong scattering, e.g. ongas or clouds, while for a low albedo the planet mainly absorbs light. The albedo

18


Figure 1.9: Schematic view of production, coagulation and settling of aggregate haze at

the poles of Jupiter through microphysical processes. Figure reprinted from Friedson

et al. (2002).

controls the energy balance of the planet and its effective temperature, becauseabsorbed light contributes to the thermal pool of photons and is reemitted asheat at thermal wavelengths. There are different ways of specifying the planetaryalbedo, and it is important to clarify which albedo is meant in a specific situation.

The spherical albedo AS is the reflectivity in all directions at a single wavelength.For AS = 1 all incoming radiation at that wavelength is scattered back into space.The bond albedo AB is the spherical albedo integrated over all wavelengths, andthus the total radiation scattered back into space over all directions and wave-lengths. AB is weighted by the incoming radiation and therefore depends on thestellar spectrum.

The Bond and Spherical albedos are not ideal quantities to describe extrasolarplanets, since the radiation cannot generally be measured over all directions. It isfavorable to express the reflectivity as a function of the phase angle α. The phaseangle is the angle between the star, the planet, and the observer. At α = 0 thefully lit hemisphere of the planet is turned towards to the observer. Extrasolarplanets are hidden behind their star at that point, while the outer Solar Systemplanets are fully visible at opposition. At α = 180 only the dark side is visibleand the planet is eclipsing the star. An extrasolar planet in a face-on system will

19


Figure 1.10: Schematic phase and polarization of a planet on an orbit with a moderate

inclination. The solid lines indicate the orientation and strength of the polarization for a

Rayleigh scattering atmosphere. The dotted circle approximately shows where the stellar

halo will prevent planet detection.

always be seen at a phase angle α = 90. More generally,

cos α = (sin i · sin θ) (1.34)

where i is the inclination of the system and θ the angle of the planet with respectto the time t0 when the radial velocity is maximal. For circular orbits θ = 2π(t −t0)/T with T the orbital period.

The geometric albedo Ag(λ) is defined as the reflectivity at α = 0 at a particularwavelength λ, normalized to the reflectivity of a flat Lambert surface that sub-tends the same solid angle. A Lambert surface scatters all incoming radiationisotropically, and shows equal brightness when viewed from any direction. ALambert sphere has an Ag = 2/3. In principle, an object can be more stronglyreflecting into a particular direction than a Lambert surface, and therefore thegeometric albedo can be larger than 1. This is for example the case for Saturn’smoon Enceladus for which Ag = 1.38 (Verbiscer et al. 2007).

The phase function f (α) expresses the reflectivity at any phase angle. The rela-tion between this reflectivity and the spherical albedo is then,

As(λ) = 2∫ π

0f (α) sin αdα (1.35)

Polarization

Not only the reflectivity changes with phase angle, but also the polarization frac-tion. If single scattering dominates, then the polarization phase function willbasically follow the single scattering polarization function (−F12/F11 in Eq. 1.7).If multiple scattering occurs, the polarization directions of the scattered photonswill be more randomly oriented, and the resulting polarization is lower. Bumpsin the single scattering polarization function (e.g. rainbows) are smoothed out.For Rayleigh scattering, the maximum polarization still occurs near 90, but it isonly about 30% (see Chapter 2). Since the polarization is generally perpendicularto the scattering plane (and therefore positive), the orientation of the polariza-tion also depends on phase angle (Fig. 1.10). If the system is seen near face-on,

20


Figure 1.11: Illustration of the limb polarization effect for planets near opposition.

Backscattered light from near the limb is either unpolarized (immediate backscattering)

or polarized perpendicular to the limb after two or more scatterings. See text for expla-

nation. Figure from Joos & Schmid (2007).

the phase angle stays 90 and the polarization fraction p(α) will remain approxi-mately constant. However, the polarization direction will vary and the polariza-tion can still be measured differentially between different phases of the orbit inStokes Q/I and U/I. If the orbit is near edge-on, the polarization direction willstay the same along the orbit, but the polarization fraction will vary between 0 at0 phase angle and maximum near 90 (for Rayleigh scattering).

At 0 phase angle, the polarization integrated over a planetary disk is zero ifthe planetary disk is uniform and/or symmetric. For extrasolar planet observa-tions, this is not relevant. A planet is behind the star, and therefore not observ-able, in that situation. The outer Solar System gas giants are however always seenat very small phase angles from Earth and only a very small polarization is ex-pected. However, a second order scattering effect, the limb polarization effect (vande Hulst 1980), produces a linear polarization near the limb of the planet that ismeasurable if the planet can be spatially resolved. Figure 1.11 illustrates why thiseffect occurs. An incoming unpolarized photon that is singly scattered backwards(at point 1) remains unpolarized. Photons that are scattered in a plane approxi-mately tangential to the visible limb (at point 1) and, with one or more scatterings,back towards the observer (point 2) will have scattered predominantly perpendic-ular to the scattering plane and are therefore polarized perpendicular to the limb.Photons that are scattered perpendicular to the visible limb (at point 1) will havea higher chance of being absorbed if scattered downwards, or of escaping if scat-tered upwards because of the different densities. Therefore, a larger number ofpolarized photons scattered from the limb returns to the observer and a net radialpolarization perpendicular to the limb is visible. Low spatial resolution degradesthe polarization fraction, because overlapping areas of negative (horizontal) andpositive (vertical) polarization cancel out.

21


Contrast

When observing an extrasolar planet in scattered light, we cannot measure thealbedo and polarization fraction directly. Rather, we measure the contrast be-tween the star and the planet. This number depends also on the number of pho-tons from the star incident on the planet, which directly depends on the distanceof the planet to the star and the size of the planet. The intensity contrast,

CI(α) =Fp

Fs=

πR2

4πa2AS

4 f (α)

AS=

R2

a2f (α) (1.36)

where R is the planet radius, a its separation from the star, AS the sphericalalbedo, and f (α) the phase function with f (0) the geometric albedo. The factorof 4 f (α)/AS gives the solid angle into which the observed radiation is scatteredat the planet surface with respect to 4π.

From the intensity contrast, the polarization contrast is simply calculated bymultiplying with the polarization fraction,

Cp(α) = CI(α)p(α) =R2

a2f (α)p(α) (1.37)

with p(α) the phase dependent polarization fraction. While the polarizationcontrast is lower than the intenstiy contrast because p ≤ 1, the liming contrastreached with polarimetry is significantly better than only in intensity thanks tothe differential method. Effectively, a much fainter signal can be recovered.

Thermal emission

All planets emit some amount of thermal radiation, mostly in the infrared, in ad-dition to the reflected starlight. The thermal radiation is generally unpolarized.If the thermal contribution is significant with respect to the reflected light at aparticular wavelength, the polarization degree of the planet at that wavelengthwill be lower than without thermal radiation. The wavelength range where sig-nificant thermal radiation is emitted depends strongly on temperature. For thecool Solar System gas giants, the thermal radiation is negligible with respect tothe reflected light below 2 µm. For very young, hot self-luminous planets, orfor strongly irradiated hot Jupiters, the thermal radiation can be significant evenin the optical wavelength range. Reflected light searches therefore do not targetvery young planets. For hot Jupiters, the planet cannot be separated from thestar, and additional contrast enhancing techniques like using a coronagraph orAngular Differential Imaging (cf. Sect. 1.5 are useless. The final contrast limit,that determines whether a planet of given contrast is measurabe, must thereforebe established entirely from the polarimetrc accuracy of the instrument.

The additional unpolarized thermal contribution of the planet is insignificantfor polarization measurements, because only polarized reflected light is counted.For intensity searches of scattered light, e.g. phase curve measurements withCoRot (Snellen et al. 2009), it is however not possible to separate the scatteredfrom the thermal light, making an albedo measurement more difficult.

22

1.4. Debris disks in scattered light

1.4 Debris disks in scattered light

Circumstellar dust in planetary systems holds a much smaller mass than planets,but because of the much larger surface area it produces strong observational sig-nals in scattered light or from thermal reemission of photons. The easiest way todetect a dusty disk is by measuring the spectral energy distribution (SED) of thestar in the thermal infrared. The thermal dust radiation provides an excess on topof the stellar spectrum. The IRAS satellite measured such excess for over a hun-dred nearby main-sequence stars (e.g. Backman & Gillett 1987) and the Spitzersatellite expanded the known sample (e.g. Hillenbrand et al. (2008)). The diskdetections are sensitivity (or calibration) limited and only relatively bright debrisdisks can be detected with current instruments. An extrasolar analog of the So-lar System Kuiper belt or asteroid belt would remain yet undetected, therefore itcan be reasonably assumed that circumstellar dust and planetesimals occur veryfrequently.

From the shape of the SED excess some constraints on the dust temperature(s)and inner disk radius can be made. The geometry and grain properties of thedust are however quite degenerate. For example, depending on the dust prop-erties one or multiple narrow rings can have a very similar infrared signature asmore extended dust region. It is therefore crucial to additionally obtain resolvedimages of the disk which clearly constrain the geometry and give additional cluesabout the dust properties to break the degeneracies, either from scattered light ob-servations or thermal infrared observations as performed by the Herschel satelliteMatthews et al. (e.g. 2010). A detailed review on debris disks focusing on infraredobservations and evolution is Wyatt (2008). More focus on direct imaging of disksin scattered light is given in Kalas (2010). The link of debris disks to planetesimalsand planets is described in detail in Krivov (2010).

1.4.1 Observables

The infrared brightness of a disk is usually expressed in terms of the fractionalluminosity f = Lir/L∗, the ratio between disk and star infrared luminosity. Fordebris disks f < 10−2. Scattered light observations give the surface brightnessof the disk as a function of separation from the star projected into 2 dimensions,usually expressed in Jy/arcsec2 or mag/arcsec2. With observations at multiplewavelengths and additional polarization measurements, grain properties such assize distribution or porosity can be much better constrained (e.g. Fitzgerald et al.2007; Graham et al. 2007).

Additionally, the geometry of the scattering disk is strongly constrained. Theinclination with respect to the line of sight is immediately evident, which is ofparticular interest if an additional companion discovered with the radial velocitymethod is present that might have coplanar orbit with the disk. The system ar-chitecture in radial, vertical and/or azimuthal can also be assessed. Most oftenthe debris disks were revealed to be quite narrow rings with large inner holes.The truncation slope of the inner or outer boundary or an offset of the disk center

23


Figure 1.12: The Fomalhaut debris ring (NASA press release for Kalas et al. 2005).

Figure 1.13: The edge-on AU Mic debris disk with polarization vectors on top of the

Stokes I image (Graham et al. 2007).

from the star may hint toward an unseen companion. This is discussed in detailfor the Fomalhaut system in Kalas et al. (2005, see Fig. 1.12), and for the HD 61005in Chapter 6 of this thesis. The scale height can be used to determine the verticalvelocity dispersion. Azimuthal asymmetries are quite common, usually in theform of spiral arms or clumpy structures (e.g. Reche et al. 2008).

1.4.2 Current status of debris disks observations in scattered

light

More than 10 debris disks have currently been resolved in scattered light4 be-tween 0.6 and 2 µm. Another ∼ 15 have been resolved with lower spatial resolu-tion at longer, thermal wavelengths. Most scattered light detections were madeby HST coronagraphy with NICMOS in the near-infrared or ACS in the opti-cal because the much more stable PSF in space was easier to remove than forground-based observations. Only now, sophisticated reduction techniques forground-based high-contrast imaging from large telescopes can deliver similar oreven superior results as HST for special cases (see Sect. 1.5 and 6). For edge-ondisks like AU Mic (Fig. 1.13) the geometry is less clear. In that case, polarimetricobservations also made with HST could resolve degeneracies between disk ge-ometry and scattering phase function, showing that the disk is also a ring withan inner hole (Graham et al. 2007).

4 Resolved circumstellar disk library: http://www.circumstellardisks.org

24

http://www.circumstellardisks.org

1.5. Differential techniques for high-contrast imaging of planetary systems

1.5 Differential techniques for high-contrast

imaging of planetary systems

1.5.1 Instruments for high contrast imaging

Current high-contrast imagers in operation that have successfully detected plane-tary companions and debris disks, or are capable of this, include VLT/NaCo, Sub-aru/HiCIAO, Keck/NIRC2, Gemini-N/NIRI, Gemini-S/NICI and MMT/CLIO.These systems all share a number of common features that are necessary to imagean extrasolar planet. They are using a large telescope (6.5 − 10 m diameter) for ahigh resolution and a large photon collecting area. The instruments are coupledto an adaptive optics (AO) system that corrects distortions by atmospheric tur-bulence in order to achieve a near diffraction-limited performance rather than aseeing limited image. A wavefront sensor measures the distortion of the incom-ing wavefront of a guide star (usually the target itself) by atmospheric turbulence,and a deformable mirror e.g. with piezo-elastic actuators is reshaped to removethe wavefront errors. Measurement and correction happen in a cycle with a du-ration of order 1 ms, faster than the typical characteristic speckle variation timescale (atmosphere coherence time scale τ0 of several ms).

The cameras operate in the near infrared, typically from 1 − 5 µm (J to Mband). Different types of coronagraphs, e.g. Lyot coronagraphs, apodized Lyotcoronagraphs, four quadrant phase masks (4QPM) or apodized phase plates(APP), are available to block the starlight as much as possible, or saturated imag-ing is possible by taking many images with very short exposure times at highefficiency. Some instruments also provide a low resolution spectroscopic mode tofollow-up planetary candidates and provide a crude spectrum for a first charac-terization of the atmosphere.

Two upcoming instruments with a primary goal of detecting extrasolar plan-ets through direct imaging are SPHERE (Spectro-Polarimetric High-Contrast Ex-oplanet REsearch) for the VLT (Beuzit et al. 2006) and GPI (Gemini Planet Imager)for Gemini-S (Macintosh et al. 2008). They are extreme adaptive optics systemswith more than 1000 actuators, providing a Strehl ratio of ≈ 90% in H-band. GPIconsists of only one instrument, an Integral Field Spectrograph (IFS) that takes alow resolution spectrum between 0.95 and 2.5 µm at any image point in the fieldof view (FoV) of 2.4”. It also features a polarimetric mode. The result is a 3Dcube of images, with an image at each wavelength or polarization state. The PSFof the star can then be removed with differential methods, e.g. spectral differen-tial imaging, angular differential imaging or polarimetry (see next section for lasttwo). The system is due for first light in late 2011.

SPHERE is a 2nd generation instrument for the VLT due for first light in thesummer of 2012. It consists of three different parts: the near-infrared dual-beamspectral and polarimetric differential imager and low resolution spectrographIRDIS (0.95-2.3 µm), an integral field spectrograph (IFS, 0.95-1.7 µm) and a differ-ential polarimeter (ZIMPOL, 0.55-0.9 µm). IRDIS is well suited for the discoveryof planets and imaging of disks with a large FoV of 11”. IRS can be used simulta-

25


neously for atmospheric characterization of planets in the innermost region witha FoV of 1.77”. ZIMPOL is a fast-modulating polarimeter (see Sect. 1.5.3) witha FoV of about 3” optimized for the detection of polarized reflected light fromplanets around the most nearby stars.

1.5.2 Angular Differential Imaging

For high-contrast observations, the PSF noise is a quasistatic speckle noise patterncaused by imperfect optics and slowly evolving optical alignments. To removethis pattern, a reference PSF must be subtracted. Conventionally, a reference starwith similar properties and position angle was observed close in time to the tar-get, and its PSF was used for the subtraction. While this procedure can attenuatethe stelar PSF to some degree, the remaining residuals are still quasistatic. A bet-ter method is to use the target star’s own PSF as reference, by avoiding that theplanetary companion is subtracted together with the PSF. This is achieved with atechnique called Angular Differential Imaging (ADI, Marois et al. 2006). The obser-vations are performed in pupil tracking mode, i.e. the field derotator of the alt-aztelescope is switched off (for Cassegrain instruments) or adjusted (for Nasmythinstruments). The instrument and telescope optics remain aligned, but the fieldof view rotates on the detector. The field rotation rate in deg/min is given by

ψ = 0.2596cos A cos φ

sin z(1.38)

with A the telescope azimuth, φ the telescope latitude and z the zenith angle ofthe object. It is most favorable to observe an object near meridian transit, wherethe field rotation is maximal.

While the stellar PSF remains fixed, the companion slowly rotates around thestar. For each short-exposure image, a reference PSF can be constructed by select-ing images where the companion has rotated sufficiently. Removing this PSF willstrongly reduce the quasistatic speckle noise and reveal faint companions afterthe residual images are derotated and combined (Fig. 1.14. The speckle noise at-tenuation is stronger for longer observing times, its main limitation is the amountof field rotation captured.

There are a number of approaches on how to best select the images for theconstruction of the reference PSF. The simplest is to median-combine all imagesof the sequence. Advantages are that the same reference PSF is subtracted fromall images and self-subtraction effects and pixel-to-pixel noise are minimized. Butsince the PSF evolves with time, this “simple” ADI method is usually not suffi-cient. The result is improved if for each image, a further, more optimized refer-ence frame for subtraction is constructed. For each image a combination of onlya few frames (typically 2-4) is used that were taken closely in time, but wherethe planet has rotated sufficiently (typically 1-2 FWHM) to avoid significant self-subtraction. Because the amount of field rotation needed to rotate away the signaldiffers with distance from the rotation center, the image is divided into annuli,

26

1.5. Differential techniques for high-contrast imaging of planetary systems

Figure 1.14: Schematic representation of the frame combination in the simple angular dif-

ferential imaging method. The first row is a sequence of images taken in pupil-stabilized

mode where the stellar PSF and speckles (gray) remain fixed while the companion (red)

rotates around the center. The median of the images is subtracted, the images are dero-

tated and combined to reveal the companion. Figure provided by C. Thalmann.

and for each annuli different frames are chosen such that the separation criterionis met at that radial distance.

An even more optimized PSF subtraction is provided by the LOCI (LOcalizedCombination of Images, Lafreniere et al. 2007) algorithm. Each image is dividedinto segments, and for each segment, frames that have sufficiently rotated arechosen and linearly combined in an optimized way to minimize the subtractionresiduals in that segment, i.e.

σ2 = ∑i

(

OTi − ∑

k

ckOki

)2

, (1.39)

with OTi the optimization segment of the ith image, and Ok

i the same segment inthe kth image which has rotated sufficiently with respect to the ith image. Boththe separation criterion and the segment size and geometry are adjustable param-eters that can be chosen for a specific case, with a trade-off between subtractionresidual noise and object self-subtraction.

1.5.3 Polarimetry

The intrinsic polarization of stars is very small, at least integrated over theirwhole surface. For the sun it is . 10−7 (Kemp et al. 1987). The measured polariza-tion of stars is usually produced by interstellar dust in the line-of-sight. Starlightreflected by a planetary atmosphere however will generally be polarized to somedegree, in particular if Rayleigh scattering or haze scattering dominate. This of-

27


fers the possibility to discern the faint reflected and polarized light of the planetfrom the bright unpolarized halo of the star.

Polarization cannot be measured directly. A polarimeter measures the inten-sity of the light in orthogonal polarization states, from which the polarization canthen be calculated. The stellar PSF will look the same when measured in differentpolarization states, while the small planet dot will be stronger in one state. Sub-tracting an image of opposite polarization state will therefore suppress the PSFwhile bringing out the planet if the polarization is sufficient.

The most common setup in conventional instruments with a polarimetricmode (e.g. VLT/NaCo) is the combination of a half- or quarterwave plate witha Wollaston prism in dual-beam polarimeter. The halfwave plate selects the lin-ear polarization state (Q, U) to be measured, the quarterwave plate selects thecircular polarization V. The Wollaston prism then splits the incoming inten-sity into two beams of orthogonal polarization state, for example IQ,‖ and IQ,⊥,which are measured on separate parts of a detector. Then, I = IQ,‖ + IQ,⊥ andQ = IQ,‖ − IQ,⊥. To lessen the effects of irregularities on the detector, an ad-ditional image is taken with the halfwave plate rotated by 45. For a halfwaveplate rotation angle γ the polarization direction is rotated by 2γ, therefore a 45

switches the polarization direction of the two beams such that I′Q,‖ = IQ,⊥. The

data are combined to obtain a polarization fraction Q/I with either the differencemethod (see Fig. 1.15),

Q

I=

1

2

(

I‖ − I⊥I‖ + I⊥

+I′⊥ − I′‖I′⊥ + I′‖

)

, (1.40)

or with the ratio method,

Q

I=

R − 1

R + 1, with R2 =

I‖/I⊥I′‖/I′⊥

(1.41)

For high-precision polarimetry, the dual-beam method is not sufficiently pre-cise. It is much more favorable to measure the two states on the same detectorpixels. In that case, the orthogonal polarization states have to be measured se-quentially in time, and must therefore be taken more quickly than the typicaltime scales on which the atmospheric speckles change (≈ 10 ms). This poses achallenge because detector read-out times are generally slower. ZIMPOL (ZurichImaging POLarimeter, e.g. Povel 1995) uses a fast polarization modulation princi-ple that circumvents this problem and has been successfully used to measure po-larization to a precision better than 10−5 for solar applications (e.g. Stenflo 1996).The polarization of the incoming beam is modulated with a frequency in the kHzrange by ferro-electric liquid crystal modulators. A polarizer selects a linear po-larization state, which is measured on a chip that demodulates in real-time byshifting the measured charges to other, covered detector rows while the otherstate is measured, and then shifts them back. After many modulation periods,the signal is finally read out and the intensity and polarization can be constructedfrom the sum and difference of the two measured polarization states.

28

1.6. Thesis overview

Figure 1.15: Illustration of the data reduction for polarimetric imaging taken with a dual-

beam polarimeter using the difference method. Figure adapted from Hinkley et al. (2009).

Polarimetric Differential Imaging (PDI) can deliver a contrast enhancement of∼ 10−4 to 10−5. The planet polarization of typical SPHERE targets is only ∼ 10−8.The rest of the contrast is gained in intensity by the use of a coronagraph and ADI,which can be combined with PDI at least in a simplified form with natural fieldrotation. Pupil tracking is not foreseen for SPHERE/ZIMPOL.

1.6 Thesis overview

This thesis investigates which parameters can be derived from scattered lightobservations of planetary systems, i.e. planets and circumstellar material, withdifferential techniques, in particular polarimetry and angular differential imag-ing. These studies were executed in the broad context of the science case ofSPHERE/ZIMPOL and SPHERE/IRDIS for scattered light observations of plane-tary systems. They make predictions and suggest interpretations for signals thatmay be detectable with SPHERE, and take advantage of similar observations al-ready available or feasible with currently existing instruments that can alreadyprovide interesting scientific results in their own right.

The first part of the thesis is dedicated to the study of planetary atmospheresin polarized light.

In Chapter 2, a parameter study of polarimetric models for planet atmospheresis presented, that discusses the effects of the most important atmosphere pa-rameters on the intensity and polarization phase curve and the limb polariza-tion. The study is focused on Rayleigh scattering, but also considers scatter-ing on haze particles. The connection between the limb polarization as mea-surable for the Solar System gas giants and the polarization at large phase an-gles as will be measured for extrasolar planets is discussed. Diagnostic diagramsare presented which could be useful for a first coarse analysis of broadband po-larimetic differential imaging observations of extrasolar planets in scattered lightwith SPHERE/ZIMPOL. Also, a large grid of models is provided for a homoge-

29


neous Rayleigh scattering layer above a diffuse cloud layer that allows an easycalculation of polarization spectra. This work was presented in the paper “A gridof polarization models for Rayleigh scattering planetary atmospheres” by E. Buenzli &H. M. Schmid, published in 2009 in Astronomy & Astrophysics 504, 209.

In Chapter 3 the polarization models are applied to the specific case of the at-mosphere of Uranus. Spectropolarimetric observations of Uranus are modeled toderive polarimetric haze properties and atmospheric structure. The best-fit modelis then used to predict the polarization signal of Uranus from an extrasolar view-point and the detection possibility for an ice giant around a nearby M-Dwarf isdiscussed. This work is presented in the paper “Polarization of Uranus: Constraintson haze properties and predictions for analog extrasolar planets” by E. Buenzli & H.M.Schmid, submitted for publication in Icarus, 2011.

Chapter 4 is a brief investigation of the polar haze of Jupiter. A haze model isderived for observations at a single wavelength. This work is a modified verisonof a part of the paper “Long slit spectropolarimetry of Jupiter and Saturn” by H.M.Schmid, F. Joos, E. Buenzli & D. Gisler, published in 2010 in Icarus 212, 701. Thechapter focuses on my contribution to the paper, the haze modeling, after a shortpresentation of the data obtained and described by F. Joos and H. M. Schmid.

The first part concludes with an outlook on the polarimetric search for re-flected light from planets around the nearest stars with SPHERE/ZIMPOL, andin the more distant future with E-ELT/EPOL. Additionally, the application of po-larimetry for hot Jupiter observations is discussed.

The second part of the thesis investigates scattered light from circumstellarmaterial by focusing on observations and analysis of a specific debris disk.

In Chapter 6, ground-based high-contrast observation of the debris diskHD 61005, less formally known as ‘the Moth’, are presented at unprecedentedangular resolution achieved through angular differential imaging. The disk isresolved as a debris ring with an outer faint component that appears to be in-teracting with the ambient interstellar medium. An analysis of the ring geometryand surface brightness is shown, which indicates that the ring center is offset fromthe star, and additional strong brightness asymmetries are present. Detection lim-its are given for potential planetary candidates that could be responsible for thering perturbation, and detected point sources were confirmed to be backgroundobjects. This study was carried out in a collaboration within the framework ofa large observing program searching for thermal emission from planetary masscompanions around nearby young stars. Reduction and modeling codes wereprovided by several people. Detection limits and confirmation of background ob-jects were done by collaborators. I led the study, performed the reduction, mod-eling and analysis of the disk and wrote the paper. Comments and suggestionswere provided by co-authors. This work was presented in the letter “Dissectingthe Moth: Discovery of an off-centered ring in the HD 61005 debris disk” by E. Buenzliet al., published in 2010 in Astronomy & Astrophysics 524, L1.

The final chapter discusses ongoing and potential future follow-up observa-tions of the HD 61005 debris disk as well as an idea for future modeling efforts,in particular for the faint outer component composed of particles blown-out from

30

BIBLIOGRAPHY

the parent body ring. Additionally, some perspectives on disk observations inscattered light with other techniques and future instruments are given.

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33

Part I

Polarimetry of gaseous planets

35

Chapter 2

A grid of polarization models for Rayleighscattering planetary atmospheres∗

E. Buenzli1 and H. M. Schmid1

Abstract

Context. Reflected light from giant planets is polarized by scattering, offering thepossibility of investigating atmospheric properties with polarimetry. Polarimet-ric measurements are available for the atmospheres of solar system planets, andinstruments are being developed to detect and study the polarimetric propertiesof extrasolar planets.Aims. We investigate the intensity and polarization of reflected light from plan-ets in a systematic way with a grid of model calculations. Comparison of theresults with existing and future observations can be used to constrain parametersof planetary atmospheres..Methods. We present Monte Carlo simulations for planets with Rayleigh scat-tering atmospheres. We discuss the disk-integrated polarization for phase an-gles typical of extrasolar planet observations and for the limb polarization effectobservable for solar system objects near opposition. The main parameters in-vestigated are single scattering albedo, optical depth of the scattering layer, andalbedo of an underlying Lambert surface for a homogeneous Rayleigh scatteringatmosphere. We also investigate atmospheres with isotropic scattering and for-ward scattering aerosol particles, as well as models with two scattering layers.Results. The reflected intensity and polarization depend strongly on the phaseangle, as well as on atmospheric properties, such as the presence of absorbersor aerosol particles, column density of Rayleigh scattering particles and cloudalbedo. Most likely to be detected are planets that produce a strong polarizationflux signal because of an optically thick Rayleigh scattering layer. Limb polariza-tion depends on absorption in a different way than the polarization at large phaseangles. It is especially sensitive to a vertical stratification of absorbers. From limbpolarization measurements, one can set constraints on the polarization at largephase angles.

∗ This chapter has been published by Astronomy & Astrophysics (2009) 504, 2091 Institute of Astronomy, ETH Zurich, CH-8093 Zurich, Switzerland

37

Chapter 2. Polarization models for planetary atmospheres

Conclusions. The model grid provides a tool for extracting quantitative resultsfrom polarimetric measurements of planetary atmospheres, in particular on thescattering properties and stratification of particles in the highest atmosphere lay-ers. Spectropolarimetry of solar system planets offers complementary informa-tion to spectroscopy and polarization flux colors can be used for a first character-ization of exoplanet atmospheres.

2.1 Introduction

Light reflected from planetary atmospheres is generally polarized. The reflectionis the result of different types of scattering particles with characteristic polariza-tion properties. Polarimetric observations therefore provide information on theatmospheric structure and on the nature of scattering particles that complementsother observations. Systematic model calculations are required to interpret theavailable polarimetry from solar system planets and prepare for future polari-metric measurements of extrasolar planets.

Scattering processes. Rayleigh scattering occurs on particles much smaller thanthe wavelength of the scattered light. This process produces 100% polarizationfor a single right angle scattering. Rayleigh scattering is much stronger for shortwavelengths because the cross section behaves like σ ∝ 1/λ4, and it favors for-ward and backward scattering, which are both equally strong. The blue sky inEarth’s atmosphere is a well known example of Rayleigh scattering by molecules.

Aerosol haze particles with a size roughly comparable to the wavelength canproduce strongly forward directed scatterings. Depending on the structure of theparticle, a high (p > 90%) or low (p ≈ 20%) fractional polarization results fora scattering angle of 90. For example, the maximum polarization for scatteringby optically thin zodiacal or cometary dust is not more than ≈ 30% (e.g. Leinertet al. 1981; Levasseur-Regourd et al. 1996), while a polarization close to 100% isinferred for single scattering of haze particles in Saturn’s moon Titan (Tomaskoet al. 2008)).

Liquid droplets in clouds produce a polarization because of refraction andreflection, which can be particularly high (> 50%) for scattering angles of about140 for spherical water droplets, corresponding to the primary rainbow (see e.g.Bailey 2007). Clouds made of ice crystals reflect and refract light in many differentways, and no distinct polarization features like rainbows are expected, exceptlocally, where ice crystals may have very similar structures.

Multiple scatterings in planetary atmospheres randomize the polarization di-rection of the single scatterings and lower the observable polarization signifi-cantly. Therefore the net polarization of the reflected light depends not only onthe scattering angle and the properties of the scattering particles, but also on theatmospheric structure. For this reason it is not surprising that a large diversity ofpolarization properties exists for the solar system planets.

38

2.1. Introduction

Observations. Venus shows a low (< 5%) negative polarization, which is a po-larization parallel to the scattering plane, for most phase angles . In the blue andUV, a rainbow feature with a positive polarization of several percent is present(e.g. Coffeen & Gehrels 1969; Dollfus & Coffeen 1970), indicating that the reflec-tion occurs mainly from droplets in optically thick clouds (Hansen & Hovenier1974).

For the giant planets, only observations near opposition are possible withearth-bound observations. Near opposition the disk-integrated polarization islow because single back-scattering is unpolarized and multiple scattering polar-ization cancels for a symmetric planet.

With disk-resolved observations of Jupiter, Lyot (1929) first detected that theJovian poles show a strong limb polarization of order 5-10%. To understand thiseffect one has to consider a back-scattering situation at the limb of a sphere, wherelocally we have a configuration of grazing incidence and grazing emergence (fora plane parallel atmosphere) for the incoming and the back-scattered photons,respectively. Photons scattered upwards will mostly escape without a secondscattering, and photons scattered down have a low probability of being reflectedtowards us after the second scattering, but a high probability of being absorbedor undergoing multiple scatterings. Thus photons that are reflected towards usby two scatterings travel predominantly parallel to the surface. Because the po-larization angle induced in a single dipole-type scattering process, like Rayleighscattering, is perpendicular to the propagation direction of the incoming photon,a polarization perpendicular to the limb is produced.

Measurements at large phase angles (≈ 90) for Jupiter with spacecrafts de-tected a polarization of ≈ 50% for the poles while the polarization is much lower(< 10%) for the equatorial region (Smith & Tomasko 1984). The high polarizationat the poles can be explained by reflection from a scattering aerosol haze layer,while the polarization at the equator is low because of reflection from clouds. To-wards short wavelengths (blue) the polarization at the equator increases strongly,indicating that also Rayleigh scattering contributes to the resulting polarization.

For Saturn the polarization is qualitatively similar to Jupiter with an enhancedpolarization at the poles at short wavelengths (blue). In the red the polarizationlevel of the poles is lower than for Jupiter (Tomasko & Doose 1984).

Uranus and Neptune display a strong limb polarization along the entirelimb (Schmid et al. 2006b; Joos & Schmid 2007). Albedo spectra (e.g. Baines &Bergstralh 1986) and the polarization indicate that Rayleigh scattering is predom-inant in these atmospheres.

An interesting case is Saturn’s moon Titan, which has a thick scattering layerof photochemical haze that produces a very high disk-integrated polarization of∼ 50 % in the B and R band (Tomasko & Smith 1982). More recently the Huygensprobe measured the scattering and polarization properties of the aerosol parti-cles in great detail during its descent through Titan’s atmosphere (Tomasko et al.2008).

The observations show that Rayleigh scattering is an important polarigenericprocess in atmospheres of solar system objects, in particular for Uranus and Nep-

39


tune, and for the equatorial regions of Jupiter and Saturn. Besides Rayleigh scat-tering one has to consider the reflection from haze particles (aerosols). Scatteringby small aerosol particles (d < λ) may be approximated by Rayleigh scattering.For large particles, d & λ, the strong forward scattering effect and the reduced po-larization for right angle scattering cause significant differences when comparedto Rayleigh scattering.

Clouds dominate in the atmosphere of Venus, and at longer wavelengths (red)also in Saturn and Jupiter. The reflection from clouds produces only a low posi-tive or even negative polarization signal in Venus, Saturn or Jupiter, typically ata level p < 5 %. In a first approximation one may therefore treat clouds like adiffusely scattering layer producing no polarization.

Polarimetric measurements of stellar systems with known extrasolar planetswere attempted, but up to now no convincing detection of the polarized reflectedlight from an extrasolar planet has been made (Lucas et al. 2009; Wiktorowicz2009). The deduced upper limits on the polarization flux from the close-in planetindicate that these objects are not covered with a well reflecting Rayleigh scatter-ing layer.

Model calculations. The classical theory for the analytic solution of the multi-ple scattering problem is treated in the seminal work of Chandrasekhar (1950),from which the polarization of conservative (non-absorbing) Rayleigh scatteringplanets can be derived. van de Hulst (1980)) gives a comprehensive overview ontheoretical work up to that time including many numerical model results.

Schmid et al. (2006b) put together available model results useful for parameterstudies of the polarization from Rayleigh scattering atmospheres. This includesthe following model results:

• Phase curves for the disk-integrated intensity and polarization for finite,conservative (no absorption) Rayleigh scattering atmospheres for differentoptical thicknesses and ground albedos from Kattawar & Adams (1971),

• the limb polarization at opposition for semi-infinite Rayleigh scattering at-mospheres with different single scattering albedos derived from formu-las and tabulated functions given in Abhyankar & Fymat (1970; 1971) andChandrasekhar (1950),

• the limb polarization at opposition for finite, conservative (no absorp-tion) Rayleigh scattering atmospheres for different optical thicknesses andground albedos from tabulations given in Coulson et al. (1960).

For Venus detailed models for the reflection from clouds were developed,which demonstrate nicely the diagnostic potential of polarimetric measurements(e.g. Hansen & Hovenier 1974)). More recent modeling of the polarization fromplanets was performed mainly to analyze and reproduce polarimetric observa-tions of Jupiter and Titan from spacecrafts (e.g. Smith & Tomasko 1984; Braaket al. 2002; Tomasko et al. 2008)

40

2.2. Model description

Another line of investigation now concentrates on the expected polarizationof extrasolar planets. The Rayleigh and Mie scattering polarization of close-inplanets was investigated by Seager et al. (2000). These calculations consider plan-ets which are unresolved from their central star and the polarization signal isstrongly diluted by the unpolarized stellar light.

Stam et al. (2004) modeled the polarization of a Jupiter-like extrasolar planetwith methane absorption bands for three special cases and presented polariza-tion spectra and wavelength integrated phase curves. Also monochromatic phasecurves for a non-absorbing clear and a hazy atmosphere are available (Stam et al.2006). Other studies determined the expected polarization from clouds of terres-trial planets (e.g. Bailey 2007) or the polarization of extrasolar analogs to Earth(Stam 2008).

Despite all these models systematic model calculations are sparse in the liter-ature. For finite Rayleigh scattering atmospheres, polarization phase curves havebeen calculated only for few selected cases. No results are available for the limbpolarization of atmospheres with finite thickness and absorption.

It is the goal of this paper to present a grid of model results for Rayleigh scat-tering models with absorption and to explore the model parameter space in asystematic way. The results should allow a comparison with observations andprovide a tool for their interpretation. Additionally effects of selected deviationsfrom simple Rayleigh scattering models will be discussed.

In the next section the paper describes our scattering model and the MonteCarlo simulations. Section 2.3 presents the results from a comprehensiveRayleigh scattering model grid covering the three atmosphere parameters: sin-gle scattering albedo ω, optical thickness of the Rayleigh scattering layer τsc, andalbedo of the underlying reflecting surface AS. In Sect. 2.4 we explore the ef-fects of a mixture of isotropic and Rayleigh scattering, of particles with a forwardscattering phase function, and of two polarizing layers. In Sect. 2.5 we discussspectral dependences. Section 2.6 highlights some special cases and diagnosticdiagrams which may be of particular interest for the interpretation of observa-tional data. A discussion and conclusions are given in the final section. Appendix2.7 describes the tables with the numerical results of our calculations of intensityand polarization phase curves for a grid of 333 model parameter combinations.These are available in electronic form at the CDS.

2.2 Model description

Our planet model consists of a spherical body of radius R, illuminated by a par-allel beam. This geometry is appropriate for not rapidly rotating planets witha large separation, d ≫ R, from the parent star. Each surface element is ap-proximated by a plane parallel atmosphere. This simplification is reasonable forplanets without an extended, tenuous atmosphere.

41


2.2.1 Intensity and polarization parameters

The intensity and polarization of the reflected light is described by the Stokesvector I = (I, Q, U, V). The linear polarized intensity or polarization flux is de-fined by the parameters Q = I0 − I90 and U = I45 − I135, where the indices standfor the polarization direction with respect to a specified direction in the selectedcoordinate system. In this paper only processes producing linear polarizationare studied and therefore the Stokes parameter V for the circular polarization isomitted. We express the fractional polarization by the symbols

q =Q

I, u =

U

I, p =

√

Q2 + U2

I, (2.1)

and the polarized intensity

p · I =√

Q2 + U2 . (2.2)

For the study of the limb polarization in resolved solar system planets at op-position we introduce the radial Stokes parameter Qr, which is positive for anorientation of the polarization parallel to the radius vector r (perpendicular tothe limb) and negative for an orientation perpendicular to r. The Stokes Ur pa-rameter is the polarization direction ±45 to the radius vector (see e.g. Schmidet al. 2006b, for an illustrative description of the radial polarization). The polar-ization fraction is represented by qr and ur.

The radial polarization curves qr(r) and Qr(r) can only be observed if theplanetary disk is well resolved. The measured radial profile depends strongly onthe achieved spatial resolution. Because of the limited spatial resolution of mostobservations it is very hard to exactly measure the polarization near the limb. It ismuch less difficult to evaluate a disk-integrated polarization or polarization fluxand to estimate and correct the degradation of the observed value with respectto the intrinsic value with a simulation of the observational resolution or pointspread function. This approach is described in detail in Schmid et al. (2006b) forseeing limited polarimetry of Uranus and Neptune.

Therefore we mainly discuss the intensity weighted polarization 〈qr〉 =〈Qr〉/I, which is the equivalent to the disk-integrated radial polarization∫

Qr(r)2π r dr normalized to the geometric albedo. The geometric albedo Ag

is the disk-integrated reflected intensity of a given model at opposition normal-ized to the reflection of a white Lambertian disk. It corresponds to I(0) in ourcalculations.

The radial polarization curves are qualitatively similar for most models. Theshape of the intensity curve varies significantly from limb darkening to limbbrightening for different model parameters and cannot solely be described bythe geometric albedo. Additionally we choose the Minnaert law exponent k asfit parameter for the shape of the center-to-limb intensity curve. The Minnaertlaw for opposition is I(r) = Ir=0µ2k−1, where µ(r) = (1 − r2)1/2. This yields thefollowing one-parameter fit curve I(r) = Ir=0(1 − r2)k−1/2.

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2.2.2 Atmosphere parameters

The plane parallel atmosphere is assumed to consist of a homogeneous scatter-ing layer that is either semi-infinite or finite with a reflecting (cloud or ground)Lambertian surface layer with a surface albedo. The basic model atmospheres aredescribed by three parameters:

• the single scattering albedo ω,

• the (vertical) optical thickness for scattering, τsc, of the scattering layer,

• the albedo AS of the surface below the scattering layer.

The single scattering albedo ω is defined by the ratio between the scatteringcross section σ and the sum of absorption cross section κ and scattering crosssection σ, with the cross sections multiplied by the fractions of scattering or ab-sorbing particles ( fsc or fabs)

ω =fscσ

fabsκ + fscσ. (2.3)

The value ω = 1 indicates pure scattering (no absorption) while ω = 0 is theother extreme of no scattering and just absorption (e.g. black dust). Similarly, asurface albedo of AS = 0 corresponds to a black surface, while a perfectly whiteLambertian surface is defined by AS = 1.

The optical depth for scattering τsc follows from the column density Z of thescattering layer: τsc = Z · σ, where σ is the scattering cross section per particle.The semi-infinite case corresponds to τsc = ∞. We treat absorption like an addi-tion of absorption optical depth to a layer with a given scattering optical depthτsc, which is equivalent to reducing the single scattering albedo. This approach issuited for discussing the reflected intensity and polarization inside and outsideof absorption features like CH4 or H2O-bands, where κ differs dramatically whileσ is essentially equal. Then the total optical thickness τ of the layer including ab-sorption κ is given by

τ = ( fscσ + fabsκ) · Z =τsc

ω. (2.4)

The basic model grid (Sect. 3) considers only Rayleigh scattering (σ = σRay,τsc = τRay) as scattering process and Lambert surfaces with an albedo AS belowthe scattering layer. Extensions, such as including non-polarizing isotropic scat-tering where σ = σRay + σiso, haze layers or more than one scattering layer arediscussed in Sect. 4.

2.2.3 Geometric parameters

The geometric parameters describe the location of the considered surface pointP and the escape direction of the photons (Fig. 2.1). A global coordinate system

43


E

S

P

+Q

+Q

−Q

0

z

δ

ϑ

ϕ

α

δ

S’

ϑ

−Q

Figure 2.1: Model geometry. The dashed line represents the trajectory of a reflected pho-

ton.

describes the orientation of the planet with respect to the star. Its polar axis isthe surface normal at the sub-stellar point S′, and the location of each point Pis described by polar angle δ and azimuthal angle θ (not drawn). δ is also thephoton’s angle of incidence at point P. The escape direction, i.e. the location ofthe observer, is given by a polar angle α and azimuthal angle χ (not drawn). α isequivalent to the phase angle defined by the three (central) points: star or sun S,planet 0, and observer E (Earth).

For the description of the scattering processes, a local coordinate system is setup at point P for the plane parallel atmosphere with surface normal z perpendic-ular to the planet surface in P, polar angle ϑ and azimuthal angle ϕ.

In general, each point P can have individual atmospheric properties. Then themodel outputs, the Stokes vector components I, Q and U, depend each on sevenparameters:

I(δ, θ, α, χ, τsc(δ, θ), ω(δ, θ), AS(δ, θ)) .

This description allows calculation of the reflected intensity and polarizationof each surface point on the illuminated hemisphere viewed from any direction.Obviously this large parameter space needs to be simplified for a first parameteranalysis. If we adopt the same atmospheric structure everywhere on the planet,τsc, ω and AS are no longer functions of δ and θ and we obtain a rotationally

44


symmetric model geometry with respect to the line S − 0, which is independentof the azimuthal angle χ.

For extrasolar planets it will not be possible to resolve the disk in the near fu-ture. For disk-integrated results we can eliminate the dependence of the reflectedintensity and polarization on the surface point parameters δ and ϑ. Because ofthe rotational symmetry of the geometric model, the intensity and the polariza-tion then only depend on the polar viewing angle or phase angle α. Moreoverthe orientation of the polarization signal is either parallel or perpendicular to thescattering plane (the plane S-0-E), which we call the Q polarization direction. Qis defined positive for a polarization perpendicular to the plane S-0-E and neg-ative for a polarization parallel to this plane. The U-polarization is zero in thiscoordinate system for symmetry reasons.

For the full disk the integrated intensity and polarization signals from a planetdepend on the following parameters:

I(α, τsc, ω, AS) , Q(α, τsc, ω, AS) .

For solar system planets at opposition we obtain a rotationally symmetric scat-tering geometry (viewing direction is identical to the axis of symmetry of the ge-ometric model). We then have a scattering model which depends only on δ or thenormalized projected radius r = sin δ, and which is independent of θ. The result-ing polarization will be in the radial direction either parallel or perpendicular tothe radius vector r and therefore our model output is the radial Stokes parameterQr (cf. Sect. 2.2.1). The Stokes Ur parameter again has to be zero for a sphericallysymmetric planet.

For exact opposition the dependences of the scattering model results can bedescribed by the following parameters:

I(r, τsc, ω, AS) , Qr(r, τsc, ω, AS) .

These are the center to limb intensity curve and the center to limb radial polar-ization curves which both depend only on the atmospheric parameters.

2.2.4 Monte Carlo simulations

For our simulations we used the Monte Carlo code described in (Schmid 1992),which was slightly adapted for the case of light reflection from a planet. Basi-cally the code calculates the random walk histories of many photons in the planetmodel atmosphere until the photons have escaped or are destroyed by an absorp-tion process. After a sufficiently large number have escaped, the scattering inten-sity and polarization of the reflected light can be established for different lines ofsight. In our calculations we assume that despite multiple scatterings the escap-ing photons emerge at the same point where they penetrated into the planet. Ineach scattering process the photon undergoes a direction and polarization changecalculated from the appropriate phase matrix. The linear polarization of the pho-tons in the simulations is defined by the orientation γ of the electric vector for the

45


photon’s electromagnetic wave. In a given coordinate system we can then eval-uate the contribution to the Stokes intensity for each photon in Q ∝ cos 2γ andU ∝ sin 2γ direction.

The escaping photons have to be collected in discrete direction bins (in ourmodels phase angles α = 2.5, 7.5, . . . with a finite bin width ∆α = 5) to evalu-ate I(α) and Q(α). These are then a mean photon intensity and polarization forthat bin. ∆α should be small to resolve any structure in I(α) and Q(α), but alsosufficiently large to collect enough photons for results with small statistical errors.The aim of our simulations is to reach at least the expected precision of observa-tional data. The rotational symmetry imposed on our models helps to increasethe bin size for the phase curve interval αk, which behaves like αk ∝ sin α. Thismeans that we have to divide the photon count per bin by the factor 2π sin α∆α.The intensity is obtained by normalizing with the reflectivity of a white Lamber-tian disk. For a given simulation the relative statistical errors (photon shot noise)are particularly good for α ≈ 90, much less favorable for α = 2.5 and very badfor α = 177.5 where only a few photons will be collected, because the irradiatedhemisphere of the planet is almost invisible for this phase angle. For the center-to-limb curves we bin uniformly in δ = arcsin(r) with a bin size of ∆δ = 5,which requires an additional normalization by 2π sin δ cos δ∆δ.

The number of photons per model was chosen such that the number of re-flected photons in phase angle bins relevant for observations (α ≈ 30 − 120) areabout N ≈ 2 · 106 when integrated over the whole disk. This corresponds to anerror in polarization ∆p =

√2/N = 0.1%. For the radial curves the total number

of photons was increased such that the same precision was reached in most radialbins. No photons emerge at the exact phase angle α = 0. Therefore for the limbpolarization calculations we count all photons that are in the bin 0 < α < 5,even though the calculation for the radial polarization includes the assumptionthat α = 0. The error induced by this measure is smaller than the statistical error.

A general guideline for the Monte Carlo technique for random walk problemsis given in Cashwell & Everett (1959) and many Monte Carlo simulations for theinvestigation of light scattering are described in the astronomical literature (seee.g. Witt 1977; Code & Whitney 1995; Wolf et al. 1999). In (Schmid 1992) a detaileddescription on many aspects of the employed Monte Carlo code are given; e.g. thegeneral scheme of the code, the required transformations between the involvedcoordinate systems (star - planet, planet - plane parallel atmosphere, atmosphere- photon), the determination of the free path length, the treatment of isotropicscattering and Rayleigh scattering according to the Rayleigh phase matrix, anassessment of statistical errors, and a comparison with analytical calculations.

2.3 Model results for a homogeneous

Rayleigh-scattering atmosphere

This section discusses the model grid results for simple homogeneous Rayleighscattering atmospheres described by parameters ω, τsc and AS (cf. sec. 2.2.2) We

46

2.3. Model results for a homogeneous Rayleigh-scattering atmosphere

Figure 2.2: Left: Phase dependence of the intensity I, fractional polarization q and polar-

ized intensity Q for Rayleigh scattering atmospheres. Right: Radial dependence of the

intensity I, radial polarization qr and radial polarized intensity Qr at opposition. Line

styles denote: Semi-infinite case τsc = ∞ (solid) for single scattering albedos ω = 1

(thick), 0.1 (thin) and finite atmosphere (τsc = 0.3) with ω =1 (dashed) and 0.6 (dash-dot)

for surface albedos AS = 1 (thick) and 0 (thin). Also shown is the intensity curve for

conservative semi-infinite isotropic scattering (dotted).

discuss phase curves (Sect. 2.3.1) and radial profiles (Sect. 2.3.2) for selected casesand explore the full parameter space for disk-integrated results at α = 90 andα = 0 (Sect. 2.3.3).

Many of the general dependences of these model results on atmospheric pa-rameters were already discussed in previous studies mentioned in the introduc-tion (Sect. 2.1). Compared to these our calculations are much more comprehen-sive and the extensive model grid results are provided in electronic form (seeAppendix 2.7). An overview of the dependence of observable quantities, suchas intensity, fractional polarization and polarized intensity, on atmosphere pa-rameters is presented in diagrams which may be useful for the interpretation ofobservational data.

The results presented in this section are in very good agreement with the pre-vious calculations in Kattawar & Adams (1971); Stam et al. (2006) and Schmidet al. (2006b).

2.3.1 Phase curves

For the investigation of extrasolar planets, the phase dependence of the disk-integrated polarization is of interest. We discuss the phase curve for selected

47


model cases (Fig. 2.2, left): a semi-infinite and a finite scattering layer with differ-ent absorption properties of the scattering and surface layers.

The semi-infinite, conservatively scattering layer is a good reference case foran illuminated sphere and is often used for scattering atmospheres. All irradi-ated light is reflected after one or several scatterings and the spherical albedo isequal to 1. An intensity phase curve for isotropic scattering is given in van deHulst (1980), and Bhatia & Abhyankar (1982) published a polarization curve forRayleigh scattering in graphical form, but no tabulated values could be found inthe literature.

In our Monte Carlo simulation we treat the semi-infinite atmosphere as τsc =30 and AS = 1, which yields essentially the same results as an infinite layer butavoids infinite scattering of some photons. Our results of this case are tabulatedin Table 2.1.

Intensity: The intensity phase curves I(α) have their maximum at α = 0, whenthe whole illuminated hemisphere is visible, and they decrease steadily to zeroat α = 180, where only the dark side of the planet is seen. The intensity I(0)is equivalent to the geometric albedo. It is 0.7975 for the semi-infinite Rayleighscattering atmosphere (Prather 1974), higher than for the semi-infinite isotropicscattering model, (Iiso(0

) = 0.690 van de Hulst 1980) or a white Lambertiansphere (ILam(0) = 2/3), because the Rayleigh scattering phase matrix favorsforward and backward scattering. On the other hand the Rayleigh scatteringintensity curve is lower for the range α ≈ 52 − 120.

Of course, the reflected intensity decreases with absorption (with lower singlescattering albedo ω) in the atmosphere and with the albedo AS of the underlyingsurface layer. The effect of absorption in the scattering atmosphere is importantfor thick layers, while the albedo of the underlying surface is important if theoptical depths of the scattering region above is small. A quantitative descriptionof these dependences is given in Dlugach & Yanovitskij (1974) and Sromovsky(2005b)).

Polarization fraction. The disk-integrated polarization fraction q(α) is alwayszero for phase angles α = 0 and α = 180 because of the imposed rotational sym-metry. The polarization maximum is near the right-angle scattering configurationα ≈ 90.

The polarization for the semi-infinite, conservative Rayleigh scattering layerreaches a maximum of q = 32.6% for α = 95. For reduced scattering albedo,e.g. due to absorption in a molecular band, q(α) increases (see e.g. van de Hulst1980). This happens because absorption strongly reduces the fraction of multi-ply scattered photons in the reflected light which have randomized polarizationdirections. If the absorption is very strong then the reflected light consists essen-tially only of photons that made one single Rayleigh scattering. The polarizationphase curve then approaches the Rayleigh scattering polarization phase functionp(α) = (1 − cos2 α)/(1 + cos2 α) with a polarization close to 100 % at α = 90.

48


Table 2.1: Reflectivity I(α), polarization fraction q(α) and polarized intensity Q(α) phase

curves for a very deep (τ = 30) conservative (ω = 1) Rayleigh scattering atmosphere

above a perfectly reflecting Lambert surface (surface albedo AS = 1). This model ap-

proximates well a conservative, semi-infinite Rayleigh scattering atmosphere. Addition-

ally the fit parameter a(α) for the parametrization of the polarized intensity Q(α) (Eq.

2.5) is given for relevant phase angles.

α [] I(α) q(α) [%] Q(α) a(α) α [] I(α) q(α) [%] Q(α) a(α)2.5 0.795 0.0 0.0000 92.5 0.174 32.5 0.0566 2.627.5 0.785 0.4 0.0031 97.5 0.150 32.5 0.0488 2.77

12.5 0.766 1.1 0.0084 102.5 0.130 31.8 0.0413 2.9517.5 0.740 2.1 0.0155 107.5 0.111 30.5 0.0339 3.1622.5 0.708 3.4 0.0241 1.85 112.5 0.094 28.6 0.0269 3.4227.5 0.671 5.1 0.0342 1.86 117.5 0.079 26.2 0.0207 3.7632.5 0.630 6.9 0.0435 1.87 122.5 0.066 23.4 0.015437.5 0.587 9.1 0.0534 1.89 127.5 0.054 20.3 0.011042.5 0.542 11.4 0.0618 1.91 132.5 0.043 17.0 0.007347.5 0.497 13.9 0.0691 1.94 137.5 0.033 13.7 0.004552.5 0.453 16.6 0.0752 1.98 142.5 0.025 10.4 0.002657.5 0.410 19.3 0.0791 2.03 147.5 0.018 7.3 0.001362.5 0.368 22.0 0.0810 2.08 152.5 0.013 4.4 0.000667.5 0.329 24.6 0.0809 2.14 157.5 0.008 2.0 0.000272.5 0.292 27.0 0.0788 2.21 162.5 0.005 0.0 0.000077.5 0.259 29.1 0.0754 2.29 167.5 0.002 -1.4 0.000082.5 0.228 30.7 0.0700 2.39 172.5 0.001 -1.9 0.000087.5 0.199 31.9 0.0635 2.49 177.5 0.000

The statistical error of the Monte Carlo calculation for I(α) is smaller than0.001 for all α. The uncertainty of the polarization fraction is less than 0.1% for phase angles between 5 and 165 degrees. Extrapolating the intensityI towards α = 0 with a quadratic least-squares fit to the first four points(α = 2.5, . . . , 17.5) yields a value I(0) = 0.7970. This agrees with theexact solution I(0) = 0.7975 from Prather (1974) to the third digit.

49


For finite scattering atmospheres the polarization fraction q also depends onthe albedo of the surface layer. In the models discussed in this section the po-larization is only produced in the Rayleigh scattering layer, while reflection fromthe surface layer is unpolarized. Therefore the resulting polarization is low fora high surface albedo and high for a low surface albedo (see e.g. Kattawar &Adams 1971; Stam 2008). This reflects the relative contribution of the polarizedlight from the scattering layer with respect to the unpolarized light reflected fromthe surface underneath.

The peak of the polarization curve is shifted towards large phase angles(α ≈ 110) for models with thin scattering layers and high surface albedos, as pre-viously described by Kattawar & Adams (1971). At large phase angles (α > 90),when only a planet crescent is visible, the fraction of scattered photons hittingthe planet initially under grazing incidence is relatively high. For a thin scat-tering layer grazing incidence helps to enhance the probability for a polarizingRayleigh scattering. For this reason the polarized light from the Rayleigh scatter-ing atmosphere is less diluted by unpolarized light reflected from the surface atlarge phase angles and the fractional polarization is higher.

Polarized intensity. The polarized intensity Q(α), which is the product of po-larization q and intensity I, is zero at α = 0 and 180, while the maximum of thephase curve Q(α) is near α ≈ 65, depending slightly on the model parameters.The maximum value for the polarized intensity, considering the entire parame-ters space, is Qmax = 0.0812 for the semi-infinite, conservative Rayleigh scatteringatmosphere at α = 65. It seems unlikely that another type of scattering processand model atmosphere can produce a higher polarized intensity.

The polarized intensity decreases with increasing absorption, because thedrop in intensity is stronger than the increase in fractional polarization. The po-larization flux is a rough measure for the number of reflected photons undergoingone single Rayleigh scattering. Second and higher order scatterings also add tothe polarized intensity, but only at a much lower level. Adding absorption canonly reduce the number of such scatterings and therefore diminishes the polar-ized intensity.

A very important property of the polarization flux Q is that it does not dependon the albedo of the surface layer AS (assumed to produce no polarization) belowthe scattering region.

2.3.2 Radial dependence for resolved planetary disks at

opposition

For the interpretation of the limb polarization of solar system objects close toopposition, we discuss the radial or center-to-limb dependence of the intensityI(r), the radial polarization qr(r) and the radial polarized intensity Qr(r) (Fig. 2.2,right) for the same model parameters as for the phase curves in Sect. 2.3.1.

50


Intensity: The radial intensity curve I(r) shows a pronounced limb darkeningin the semi-infinite conservative case. For a strongly absorbing atmosphere, e.g.within an absorption band, the I(r)-curve becomes essentially flat. Thus for anabsorbing (and homogeneous) semi-infinite atmosphere limb brightening cannotbe produced. For comparison the center-to-limb intensity curve I(r) for isotropicscattering and for a Lambert sphere (I(r) = 1/π(1 − r2)1/2) are also shown.

For finite scattering atmospheres with an optically thin layer the center-to-limb intensity curve can show a limb brightening effect. Limb brightening occursfor a highly reflective scattering layer (high ω) located above a dark surface (lowAS), e.g. a thin aerosol layer or a methane-poor layer above the methane-richabsorbing layer (e.g. Price 1978). Limb brightening is observed in solar systemplanets in deep absorption band (e.g. Karkoschka 2001; Sromovsky & Fry 2007).Limb brightening is investigated in more detail in Sect. 2.3.2.

Radial polarization fraction: The radial polarization fraction qr(r) is alwayszero in the disk center because of the symmetry of the scattering situation. Forall cases the polarization increases steadily towards the limb and reaches a max-imum value close to the limb between r = 0.95 and 1.0. The polarization qr(r) isalways positive, which means a radial polarization direction or limb polarizationperpendicular to the limb.

It is important to note that the limb polarization decreases with decreasingsingle scattering albedo ω (more absorption) in contrast to the situation at largephase angles. This indicates that the photons producing the limb polarization aremore strongly reduced by absorption than the reflected “unpolarized” photons.

The explanation is that singly scattered (i.e. backscattered) photons do notcontribute to the limb polarization, while reflected photons scattered twice or afew times are responsible for the largest part of the limb polarization. Absorp-tion implies that a larger fraction of escaping photons are singly-scattered andtherefore unpolarized at opposition. Note however that for the semi-infinite at-mosphere, the maximum radial polarization is not reached in the conservativecase. A slightly lower scattering albedo (ω ≈ 0.95) mostly reduces the amountof highest order scatterings and thus the polarization fraction is somewhat en-hanced when compared to the conservative case (see Schmid et al. 2006b), andFig. 2.5 in Sect. 2.3.3).

The fractional limb polarization qr(r) for finite scattering layers dependsstrongly on the albedo of the underlying surface AS: qr(r) is high for low AS andlow for high AS like for large phase angles. A low surface albedo decreases thephotons with multiple scatterings in the plane perpendicular to the limb, whichare polarized parallel to the limb, thus enhancing the polarization in perpendic-ular direction. Therefore the limb polarization of a bright layer over a dark onecan be even higher than for a semi-infinite atmosphere. This is discussed in moredetail in Sect. 2.3.3.

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Figure 2.3: Radial intensity curves for different model parameters normalized to the cen-

tral disk intensity, showing examples of limb darkening and limb brightening. The solid

line is the calculated model and the dotted line the best fit with the Minnaert law. (a),

(b) and (c) are for conservatively (ω = 1) scattering layers above black surfaces (AS = 0)

with scattering layer thickness τsc = 0.1 (a), 1 (b) and ∞ (c). (d) is a thin (τsc = 0.2), highly

absorbing ω = 0.2 scattering layer above a white surface (AS = 0).

Radial polarized intensity: The radial polarized intensity Qr(r) = qr(r) · I(r)increases with r from zero in the disk center to a maximum at r > 0.9 and thendrops at the very limb. For semi-infinite atmospheres, Q just decreases at all radiiwith decreasing single scattering albedo ω.

For finite atmospheres, the limb polarization flux Qr(r) depends only slightlyon the surface albedo. Decreasing AS from 1 to 0 can increase Qr(r) at most∼ 0.002 for some models, while for most models Qr(r) is virtually constant. Thisis similar to the case for large phase angles.

Limb darkening and limb brightening vs. limb polarization

For a surface with a low albedo AS below a thin scattering layer the limb can bebrighter than the disk center, an effect that is generally called limb brightening.In principle this effect should be called “a central disk darkening”, because thelow surface albedo AS does not brighten the limb. It only absorbs more light inthe center of the disk, where a higher fraction of photons reach the absorbingsurface because of their perpendicular incidence when compared to the situationof grazing incidence at the limb. Despite this fact we will retain the term “limbbrightening” and consider the limb brightness on a relative scale compared to thebrightness of the disk center.

The limb darkening and limb brightening effect can be parametrized to a firstorder using the Minnaert law I(r) = Ir=0(1 − r2)k−1/2. The Minnaert parameterk determines the shape of the curve, k = 1 corresponds to Lambert’s law, k = 0.5

52


to a flat intensity distribution I(r) = I0 and k < 0.5 to limb brightening. The kparameter was determined by fitting Minnaert’s law to all modeled intensity pro-files, fixing the intensity at the center and excluding the outermost point wherethe formula diverges for k < 0.5. With the exception of some cases mentionedbelow, most profiles can be fitted adequately.

In Fig. 2.3 different examples of limb darkening and limb brightening areshown along with the best fit of the Minnaert law. Intensities are normalizedto the central disk intensity. There are two types of limb brightening curves. Forvery thin atmospheres the maximal brightness is measured at the very edge ofthe planet, while for a moderate optical depth the intensity raises slightly up to acertain radius (e.g. 0.9Rp) and then drops very close to the limb. This second casecannot be fitted with the one-parameter Minnaert law and is approximated hereby a relatively flat curve k ≈ 0.5.

The limiting case of a conservative semi-infinite atmosphere yields a Minnaertparameter of k ≈ 0.9. For ω going towards 0, k tends to a flat intensity distributionk = 0.5. For finite atmospheres there is a strong dependence on the surface albedoAS. For a strongly absorbing atmosphere over a bright surface (ω low, AS high)absorption is more likely towards the limb (k > 1), for a bright atmosphere overa dark surface the opposite is true (k < 0.5). In the latter case the central diskintensity is very low.

Similar to limb brightening, the limb polarization is also enhanced for a brightscattering layer over a dark surface. However, there are fundamental differencesbetween these two effects. Limb polarization arises only for a polarizing processlike Rayleigh scattering, while limb brightening occurs also for non-polarizingscattering processes like isotropic scattering. Additionally limb brightening isthe stronger the thinner the upper bright layer, while limb polarization requiresa sufficiently thick scattering layer above the dark surface. Finally, limb polariza-tion can also occur for cases of limb darkening, e.g. the semi-infinite, conservativeatmosphere. Therefore, the two effects provide complementary diagnostics of thevertical structure of the atmosphere.

2.3.3 Parameter study for quadrature phase α = 90 and

opposition α = 0

This section explores the full parameter space for simple Rayleigh scattering at-mospheres. We explore the parameter space by varying one of the three parame-ters ω, τsc and AS (cf. sec. 2.2.2) while fixing the other two. We study the result-ing intensity I(90), polarization fraction q(90), and polarized intensity Q(90)(Figs. 2.4 to 2.8).

The shapes of the model phase curves for the intensity I(α), fractional polar-ization q(α), and polarized intensity Q(α) look very similar for different modelparameters (see Fig. 2.2). Therefore it is reasonable for a model parameter studyto select the results for the phase angle α = 90, considering them as represen-tative (qualitatively) for all phase angles. A phase angle α = 90 is ideal for

53


extrasolar planets because all planets will pass through this configuration twiceduring an orbit, regardless of inclination.

The same type of parameter study is presented for the limb polarization ofplanets at opposition (α = 0). For this we determine disk-integrated (averaged)quantities for the intensity and radial polarization from the model results (Figs.2.5 to 2.9). The integrated intensity I(0) is equivalent to the geometric albedo,〈qr〉 is the intensity weighted average of the fractional polarization, and 〈Qr〉 theintegrated polarized intensity on the same scale as the integrated intensity. Thesequantities are determined as described in Sect. 2.2.1.

Figures 2.4 and 2.5 show the dependence on the single scattering albedo ω.For a given scattering optical depth τsc a reduction in ω is equivalent to an en-hancement of the absorption κ in the scattering layer. Strong differences in κ(λ)occur in planetary atmospheres for molecular absorptions (e.g. due to CH4 orH2O) inside and outside the band while σ is essentially equal.

In Figs. 2.6 and 2.7 the Rayleigh scattering optical depths from τsc = 10.0 to0.01 are plotted. This illustrates quite well the possible spectral dependence fromshort to long wavelengths (left to right) for a Rayleigh scattering atmosphere.Since the Rayleigh scattering cross sections is proportional to 1/λ4, it is possiblethat a planet has τsc = 4 at 400 nm and τsc = 1/4 at 800 nm.

The effect of the albedo AS of the surface below the Rayleigh scattering layeris shown in Figs. 2.8 and 2.9.

General results from the Figures 2.4 to 2.9 are:

• lowering the Rayleigh scattering albedo ω always results in a lower inten-sity I, and lower polarized intensity Q or Qr,

• lowering the Rayleigh scattering albedo ω results in a higher polarization qat large phase angles. Contrary to this the fractional limb polarization qr isreduced for lower ω,

• lowering the Rayleigh scattering optical depth τsc produces a strong reduc-tion in the polarized intensity Q or Qr in the optically thin case τ . 2 andcauses essentially no change in Q or Qr in the optical thick case τ & 2,

• lowering the surface albedo AS lowers the intensity I and enhances the frac-tional polarization q or qr,

• changing the surface albedo AS does not change the polarized flux Q andhardly Qr.

The most important difference between the limb polarization 〈qr〉 and thedisk-averaged polarization q(90) is their opposite dependence on the Rayleighscattering albedo ω (see e.g. the middle panels of Figs. 2.8 and 2.9). This occursbecause the limb polarization at opposition is mainly caused by photons under-going two to about six scatterings rather than just one.

Another difference is the influence of τsc on the fractional polarization: q dropswith τsc for bright non-polarizing surfaces and increases for dark surfaces. It

54


Figure 2.4: Intensity, polarization and polarized intensity at quadrature as function of

single scattering albedo ω for optical depths τsc = ∞ (solid), 0.6 (dashed), 0.1 (dash-dot)

and surface albedos AS = 1 (left), 0.3 (middle), 0 (right).

Figure 2.5: Geometric albedo, disk-integrated radial polarization and polarized intensity

at opposition as function of single scattering albedo ω for optical depths τsc = ∞ (solid),

0.6 (dashed), 0.1 (dash-dot) and surface albedos AS = 1 (left), 0.3 (middle), 0 (right).

55



optical depth τsc for single scattering albedos ω = 1 (solid), 0.8 (dashed), 0.4 (dash-dot)

and surface albedos AS = 1 (left), 0.3 (middle), 0 (right).


at opposition as function of optical depth τsc for single scattering albedos ω = 1 (solid),

0.8 (dashed), 0.4 (dash-dot) and surface albedos AS = 1 (left), 0.3 (middle), 0 (right).

56



surface albedo AS for optical depths τsc = ∞ (solid), 0.6 (dashed), 0.1 (dash-dot) and

single scattering albedos ω = ωRay = 1 (left), 0.8 (middle), 0.4 (right).


at opposition as function of surface albedo AS for optical depths τsc = ∞ (solid), 1 (dot-

ted), 0.6 (dashed), 0.1 (dash-dot) and single scattering albedos ω = ωRay = 1 (left), 0.8

(middle), 0.4 (right).

57


Table 2.2: Best fit parameter b(α, τsc) for the parametrization of the polarized intensity Q

(Eq. 2.5).

τsc b(30, τsc) b(60, τsc) b(90, τsc) b(120, τsc)0.1 0.38 0.39 0.39 0.440.2 0.63 0.65 0.66 0.750.3 0.77 0.80 0.82 0.900.5 0.99 1.02 1.04 1.100.8 1.23 1.22 1.26 1.231.0 1.32 1.33 1.32 1.252.0 1.52 1.48 1.42 1.27

10.0 1.59 1.59 1.45 1.28

is more complicated at opposition: the limb polarization is highest if the darkground eliminates photons that would otherwise scatter twice perpendicular tothe limb, but the atmosphere is still thick enough to produce many photons thatescape having scattered twice parallel to the limb. The maximum possible limbpolarization 〈qr〉 = 5.25% is reached for τr = 0.8, AS = 0 and ω = 1.

From the variation of τsc shown in Fig. 2.6 it can be seen that the polarizedintensity Q(90) saturates above τ = τsc · ω & 2. Therefore Q(90) cannot probedeep atmospheric layers. For the intensity and fractional polarization, an absorb-ing ground under a conservatively scattering layer can be noticed even at τ & 10.

The polarized intensity Q(α) consists mostly of photons undergoing just onesingle Rayleigh scattering. Therefore, Q is not changed by processes which hap-pen deep in the atmosphere or by diffuse scattering on the surface. Q is onlyreduced if the number of single Rayleigh scatterings are reduced, e.g. becausethere is only a thin Rayleigh scattering layer, or because photons are efficientlyabsorbed high in the atmosphere.

We can approximate the polarized intensity Q by the following parametriza-tion:

Q(α, τsc, ω) = Q(α, ∞, 1) · (1 − e−a(α)τsc) · ωb(α,τsc), (2.5)

where a(α) and b(α, τsc) are fit parameters. Table 2.2 shows the best fit parameterb(α, τsc), while Q(α, ∞, 1) and a(α) are listed in Table 2.1.

For optically thick Rayleigh scattering atmospheres the polarized intensity Qdepends only on the single scattering albedo ω, and the parametrization reducesto:

Q(α, ω, τsc & 2) ≈ Q(α, ∞, 1) · ωb(α,τsc&2) . (2.6)

At quadrature this is

Q(90, ω, τsc & 2) ≈ 0.060 · ω1.45 (±0.002) . (2.7)

For the limb polarization flux 〈Qr(0)〉 the dependence on ω is much steeperbecause both I and qr drop with decreasing ω, as can be seen from the bottom

58

2.4. Models beyond a Rayleigh scattering layer with a Lambert surface

panels of Fig. 2.5. For thick Rayleigh scattering layers, τ & 2, the ω-dependenceof the limb polarization flux is

Qr(τsc & 2) ≈ 0.022 · ω4.23 (±0.001) . (2.8)

2.4 Models beyond a Rayleigh scattering layer with

a Lambert surface

The three parameter model grid discussed in Sect. 2.3.3 provides an overview onbasic dependences of simple Rayleigh scattering models. In this section we de-scribe a few results for particle scattering properties different from pure Rayleighscattering, or models with more than one polarizing scattering layer.

2.4.1 Atmospheres with Rayleigh and isotropic scattering

Pure Rayleigh scattering is a simplification for planetary atmospheres. Alreadyfor Rayleigh scattering by molecular hydrogen one needs to account for a weakdepolarization effect, because the diatomic molecule is non-spherical. Anotherdepolarization effect for scattered radiation occurs in dense gas because colli-sions with other particles take place frequently during the scattering process. Inaddition aerosols and dust particles can also be efficient scatterers in planetaryatmospheres and the net scattering phase matrix differs from Rayleigh scatteringand should be evaluated, e.g. by using the more general Mie theory.

A simple way for taking such effects into account in a first approximation is touse a linear combination of the Rayleigh scattering and isotropic scattering phasematrices

S = w · R + (1 − w) · I , (2.9)

where w = σRay/σ and 1 − w = σiso/σ are the relative contributions of theRayleigh scattering and isotropic scattering to the total scattering cross sectionσ = σRay + σiso. Note that the single scattering albedo ω and the scattering op-tical depth τsc now include both the Rayleigh and the isotropic scattering crosssection (cf. Sect. 2.2.2).

Isotropic scattering is non-polarizing. If the scattering in the atmosphere iscomposed of both isotropic and Rayleigh scattering, then the fractional polar-ization and the polarized intensity are reduced by isotropic scattering, while theintensity is comparable (cf. Fig. 2.2).

Figure 2.10 shows the fractional polarization p(90) at quadrature as a func-tion of σiso/σ for a few representative cases. In the single scattering limit thedecrease is linear, the strongest deviation from a linear law is found for the semi-infinite atmosphere because of the large amount of multiple scatterings. A similarbehavior is found for other phase angles, as well as for the radial limb polariza-tion at opposition.

59


Figure 2.10: Polarization of atmospheres with Rayleigh and isotropic scattering at 90 as

a function of isotropic single scattering albedo ωi, normalized to the case of pure Rayleigh

scattering ωr = 1 or ωi = 0. Plotted models are: semi-infinite atmosphere (thick solid),

τsc = 0.3, AS = 1 (dashed), τsc = 0.3, AS = 0 (dash-dotted), and τsc = 0.05 (thin solid).

All models are without absorption, i.e. ωi + ωr = 1.

2.4.2 Forward-scattering phase functions

The high polarization of Jupiter’s poles and the disk-integrated polarization ofTitan (e.g. Tomasko & Smith 1982; Smith & Tomasko 1984) has been explainedby the presence of a thick layer of polarizing haze particles. The derived singlescattering properties indicate strong forward scattering and Rayleigh-like linearpolarization with maximal polarization close to 100% at about 90 scattering an-gle. Particles that satisfy this behavior are thought to be aggregates that are non-spherical and with a projected area smaller than optical wavelengths (e.g. West1991). We investigate the polarization properties of a planet with such a hazelayer. The particle scattering properties are implemented as described in Braaket al. (2002) using a simple parametrized scattering matrix of the form

F(#) =

F11(ϑ) F12(ϑ) 0 0F12(ϑ) F11(ϑ) 0 0

0 0 F33(ϑ) 00 0 0 F44(ϑ)

, (2.10)

where ϑ is the scattering angle and

F11(ϑ) = PHG(g, ϑ) =1 − g2

(1 + g2 − 2g cos ϑ)(3/2), (2.11)

F12(ϑ)

F11(ϑ)= pm

cos2 ϑ − 1

cos2 ϑ + 1, (2.12)

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Figure 2.11: Probability density function ρ(ϑ) for Rayleigh scattering (solid), Henyey-

Greenstein function with asymmetry parameter g = 0.6 (dashed) and g = 0.9 (dotted).

F33(ϑ)

F11(ϑ)=

2 cos ϑ

cos2 ϑ + 1, (2.13)

F44 = 0 . (2.14)

F11(ϑ) or PHG(ϑ) is the Henyey-Greenstein phase function with the asymmetryparameter g (see e.g. van de Hulst 1980). g = 0 corresponds to isotropic scatter-ing, g = 1 to pure forward scattering, g < 0 to enhanced backscattering. Sincehaze particles have been shown to be strongly forward scattering, we limit ourdiscussion to the two cases g = 0.6 and g = 0.9.

Figure 2.11 shows the probability density function ρ(ϑ) for PHG(ϑ) in compar-ison with Rayleigh scattering. The probability density function for the scatteringangle ϑ is the phase function F11(ϑ) weighted by sin(ϑ) and normalized such thatthe integral over ρ(ϑ) equals 1. From this function the probability of the scat-tering angle within a certain interval is calculated by integrating ρ(ϑ) over thisinterval. One can see that for the haze models small scattering angles (forwardscattering) are greatly enhanced in comparison to Rayleigh scattering, while theprobability for backscattering is much lower.

F12(ϑ)/F11(ϑ) describes the fractional polarization of the scattered radiationas a function of the scattering angle. For scattering on haze particles it can besimilar to Rayleigh scattering scaled by a factor pm, the maximal single scatteringpolarization at 90 scattering angle. For a first qualitative analysis we set pm = 1which is an upper limit that may slightly overestimate the resulting polarization.The other matrix elements are identical to Rayleigh scattering.

Figures 2.12 and 2.13 show the phase and radial dependences for the hazemodels similar to the Rayleigh scattering case in Sect. 2.3.1 and 2.3.2.

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Figure 2.12: Phase dependence of the intensity I, fractional polarization q and polarized

intensity Q for a haze layer. Left: Semi-infinite case τsc = ∞ for single scattering albedos

ω = 1 (thick), 0.6 (thin) and g = 0.6 (solid), 0.9 (dashed). The dotted line is the Rayleigh

scattering case for comparison. Right: Finite atmosphere τsc = 0.3 with ω = 1 for surface

albedos AS = 1 (thick), 0 (thin) and line styles as for the left plot.

Figure 2.13: Radial dependence of the intensity I, radial polarization qr and radial polar-

ized intensity Qr at opposition for a haze layer. Left: Semi-infinite case τsc = ∞ for single

scattering albedos ω = 1 (thick), 0.6 (thin) and g = 0.6 (solid), 0.9 (dashed). The dotted

line is the Rayleigh scattering case for comparison. Right: Finite atmosphere τsc = 0.3

with ω = 1 for surface albedos AS = 1 (thick), 0 (thin) and line styles as for the left plot.

62


Intensity: The phase curves of the haze models differ from the Rayleigh scatter-ing models mainly at small phase angles. The geometric albedo is lower for thehaze models because backscattering is strongly suppressed compared to Rayleighscattering. This is already discussed by Dlugach & Yanovitskij (1974), who cal-culated albedos for semi-infinite hazy atmospheres. Our calculations result inslightly higher albedos because of the inclusion of polarization. At phase anglesaround 90 the intensities are very similar for all models for non-absorbing at-mospheres. An absorber greatly reduces the albedo of a planet with enhancedforward scattering, because many photons penetrate deeply into the atmosphereafter the first scattering and then have a high probability of being absorbed. Theradial intensity curves mainly reflect the lower geometric albedo, while the shapeof the curve is similar for all models.

Polarization fraction: The angle of maximal polarization is generally larger forthe haze models than for Rayleigh scattering. In the semi-infinite conservativecase it is ≈ 110 for haze as opposed to ≈ 95 for Rayleigh scattering. The shiftto larger angles is particularly enhanced for a finite haze layer over a bright Lam-bert surface. However the maximal polarization decreases with increasing g. Forstrong absorption, both in or below the scattering layer, the polarization phasecurve tends toward the single scattering function like in the Rayleigh case. Thefractional limb polarization of haze layers can be much higher than for Rayleighscattering layers, with disk-integrated values reaching 〈qr〉 ≈ 11% and peak max-ima qr(r) ≈ 20%. This is understandable because the singly scattered (backscat-tered) photons which are unpolarized are strongly reduced for foward scatteringparticles.

Polarized intensity: The polarized intensity Q(α) is significantly lower for for-ward scattering phase functions than for Rayleigh scattering in the phase anglerange α = 30 − 90 and for the limb polarization effect at opposition. It dropsstrongly with increasing g or increasing absorption. Like for Rayleigh scatteringthe polarized intensity is independent of the surface albedo AS. The phase curvesQ(α) show a shift of the maximum towards larger phase angles when comparedto Rayleigh scattering, in particular for models with thin scattering layers.

2.4.3 Models with two polarizing layers

Up to now we have treated the region below the scattering layer simply as aLambert surface with an albedo AS, which produces no polarization. In this sec-tion we explore model results for two polarizing layers with different absorptionproperties, where the lower layer can be a semi-infinite Rayleigh scattering atmo-sphere as described in Sect. 2.3.1.

We focus on the question at what depth of the upper scattering layer τsc thepolarization properties of the underlying layer are washed out by multiple scat-tering and are no longer recognizable in the reflected radiation.

63


Figure 2.14: Fractional polarization q as function of τsc of the upper layer at quadrature

for models with Rayleigh scattering (solid) or isotropic (dotted) upper layer, and semi-

infinite Rayleigh scattering lower layer with ω = 0.6 (thick) or Lambertian lower layer

with A = 0.2 (thin), which provides the same reflectivity.

Figure 2.14 compares the fractional polarization q(90) for three cases as afunction of scattering optical depth of the upper layer, τsc,u: a non-absorbingRayleigh scattering layer above a semi-infinite, low albedo Rayleigh scattering at-mosphere, the same scattering layer above a low albedo Lambertian surface, andan isotropic, non-polarizing scattering layer above the semi-infinite, low albedoRayleigh scattering atmosphere.

The reflected polarization shows no dependence on the polarization proper-ties of the underlying surface for deep scattering layers with τsc > 2. There aretoo many multiple scatterings to preserve this type of information from deeperlayers in the escaping photons. An imprint from the polarization of the lowerlayer becomes visible for thin scattering layers with τsc . 2. Particularly wellvisible is the polarization dependence on τsc for the case where a polarizing layeris located below an isotropically scattering layer. The polarizing lower layer onlybecomes apparent for τiso < 1.

The same is true for the polarized intensity, because the reflected intensityonly shows a very weak dependence on the phase function of the scattering layer.The effects are also very similar for the limb polarization at opposition.

2.5 Wavelength dependence

The wavelength dependence of the reflected intensity and polarization of a modelplanet can be calculated using wavelength dependent parameters τsc(λ), ω(λ),and AS(λ) or ωl(λ) for single or double layer models respectively. These param-

64

2.5. Wavelength dependence

eters must be derived from a model with a given column density of scatteringparticles and mixing ratios for Rayleigh-scattering and absorbing particles.

As an example we selected parameters which approximate very roughly anUranus-like atmosphere Trafton (e.g. Trafton 1976) considering only Rayleighscattering by H2 and He and absorption by CH4. In our first example we look at ahomogeneous scattering layer with methane absorption above a reflecting cloudlayer with a wavelength independent surface albedo AS = 1. This is a strong sim-plification for Uranus because the methane mixing ratio is of order 100 lower inthe stratosphere than in the troposphere (Sromovsky & Fry 2007). Neverthelessit is a useful example for discussing basic effects of the wavelength dependence.

In a second example we make a first approximation for a methane mixing ratiothat is varying with height, by having an upper layer of finite thickness withoutmethane and a lower semi-infinite layer that includes methane.

The Rayleigh scattering cross section of molecular hydrogen is given by Dal-garno & Williams (1962) as

σRay,H2(λ) =

8.14 · 10−13

λ4+

1.28 · 10−6

λ6+

1.61

λ8, (2.15)

where λ is in A and σRay,H2in cm2/molecule.

The total Rayleigh scattering optical depth is

τRay = σray,H2 ∑i

Zi(ni − 1)2

(nH2− 1)2

, (2.16)

where Zi is the column density and ni the index of refraction of the i-th con-stituent2. We use the same wavelength dependence as for the H2 cross sectionfor all constituents. Our upper scattering layer has a column density Z = ∑i Zi

= 500 km-am.3 For the atmospheric composition we adopt particle fractions of0.5% CH4 in the single layer case, and a methane free upper layer with 1 % CH4

in the lower layer in the two layer case. In all layers the He fraction is 15% andthe rest is H2.

Because of the strong wavelength dependence of the Rayleigh scattering crosssection, τsc(λ) changes significantly from the UV to the near-IR (Fig. 2.15, toppanel). Keeping ω and AS fixed (no absorber) yields the intensity and polariza-tion results given in Figs. 2.6 and 2.7 as function of τsc, which are in this caseequivalent to results as function of λ.

The wavelength dependent single scattering albedo ω(λ) follows from theCH4 absorption optical depth τCH4

= ZCH4κCH4

(λ) and the Rayleigh scatteringoptical depth according to

ω(λ) =τsc(λ)

τsc(λ) + τCH4(λ)

. (2.17)

2 nH2= 1.0001384, nHe = 1.000035, nCH4

= 1.0004413 1 km-am = 2.687 · 1024 molecules cm−2

65


Figure 2.15: Wavelength dependences of the model parameters total optical depth

τ(λ) = τsc + τCH4of the upper layer, single scattering albedo ω(λ) or ωu(λ) of the up-

per layer, surface albedo AS or single scattering albedo ωl(λ) of the lower layer. Two

cases are considered: A layer of Rayleigh scattering with CH4 absorption above a white

(AS = 1) Lambert surface (solid), Rayleigh scattering layer without absorption ωu = 1

above a deep clear atmosphere with methane absorption (dotted).

The absorption cross sections κCH4(λ) were taken from Karkoschka (1994) and

the resulting ω(λ) is shown in Fig. 2.15.The intensity I(λ), fractional polarization q(λ), and polarized intensity Q(λ)

is determined from the wavelength dependent model parameters for our twocases at quadrature and opposition (Fig. 2.16).

At quadrature both examples show similar results. In both cases the polariza-tion is enhanced and the polarized intensity is reduced within methane absorp-tion bands, only the changes are less pronounced for a non-absorbing upper layer.The polarized intensity Q(λ) also drops with wavelength, but it is overall higherin the second case because the polarizing Rayleigh scattering extends to deeperlayers. The biggest qualitative difference is seen in the continuum polarizationq(λ). In the case of an underlying reflecting cloud, q(λ) drops towards longerwavelengths because of the smaller scattering optical depth above the diffuselyscattering cloud. In the second case there is only polarizing Rayleigh scatteringand no depolarization effect, so that the increasing absorption with wavelengthin the lower layer results in a higher polarization.

Similar spectropolarimetric models but with a Jupiter-like homogeneous at-mosphere (higher column density, less methane than in our example) above both

66

2.6. Special cases and diagnostic diagrams

Figure 2.16: Model spectra for the intensity, polarization, and polarized intensity at

quadrature (left) and intensity, radial polarization, and radial polarized intensity at op-

position (right). Lines as in Figure 2.15.

a dark surface AS = 0 and a reflecting extended cloud were discussed by Stamet al. (2004) for α = 90. The qualitative behavior of intensity and polarizationwith wavelength is quite similar to our example. However, for the same columndensity and methane fraction we find a significantly lower intensity and higherpolarization within methane bands. The origin of this discrepancy is unclear. Afurther comparison with intensity calculations for a Neptune-like atmosphere bySromovsky (2005a) shows a very good agreement at all wavelengths. Based onthis we conclude that our model spectra should be correct.

Intensity and polarized intensity at opposition behave qualitatively similarto the large phase angle case. However the fractional polarization qr(λ) is com-pletely different. Absorbing particles in the upper layer tend to reduce the frac-tional limb polarization, while absorption in the lower layer enhances it. Obser-vations of the limb polarization of Uranus and Neptune (Joos & Schmid 2007)show that the fractional polarization is indeed enhanced within methane bands.Clearly for modeling limb polarization of these planets in absorption bands it isimportant to take into account the proper vertical stratification of the absorbingcomponent. A detailed model accounting for methane saturation and freeze-outto fit the observations is beyond the scope of this paper.

2.6 Special cases and diagnostic diagrams

We explore some special and extreme model cases in diagnostic diagrams of ob-servational parameters for phase angle α = 90 and opposition.

67


2.6.1 Fractional polarization versus intensity

Figure 2.17 displays the diagnostic diagram for the reflectivity I(90) and therelative polarization q(90) at phase angle α = 90. Also indicated are the iso-contours for the polarization flux Q(90).

The diagram shows points for special model cases and curves for the depen-dence on specific model parameters. The shaded area defines the area of observa-tional parameters covered by our 3-parameter model grid for Rayleigh scattering(Sect. 2.3). Including isotropic scattering (Sect. 2.4.1) or having a vertically inho-mogeneous atmosphere (Sect. 2.4.3) does not expand the covered area.

Figure 2.17 emphasizes that it is not possible to have a Rayleigh scatteringplanet with both very high albedo and polarization. A high albedo implies eithera lot of multiple or isotropic scattering, which both reduce the fractional polar-ization. On the other hand a high polarization implies mainly single scatteringand therefore strong absorption and a low reflectivity. The maximal polarizationat a fixed intensity is given by the model with a conservative (ω = 1) scatteringlayer over a dark (AS = 0) surface and appropriate τsc. The semi-infinite atmo-sphere with varying ω gives only slightly lower results than the former models.The maximum of the product Q = q · I = 0.060 is reached for the conservativesemi-infinite atmosphere (τsc = ∞). Since the polarized intensity Q is indepen-dent of the surface albedo AS, a change in AS is equivalent to a shift along theQ iso-contours in the diagram.

The diagram also indicates the location of the haze models discussed inSect. 2.4.2. Most of the haze models lie within the same area as Rayleigh scat-tering. Only for very thick haze layers with high single-scattering albedo is ittheoretically possible to get somewhat higher fractional polarization for a givenintensity.

Figure 2.18 is the same diagram at opposition for the geometric albedo I(0),the disk-integrated limb polarization 〈qr〉 and iso-contours for the radial polar-ized intensity 〈Qr〉. Like for large phase angles, the limb polarization for fixedintensity is highest for the conservative Rayleigh scattering layers over a darksurface. However, for τsc → 0 the polarization drops to 0%, while at large phaseangles with AS = 0 it raises towards 100% when the few reflected photons aremainly singly scattered. The semi-infinite models provide a distinctly lower frac-tional polarization signal than a finite conservatively scattering atmosphere overa dark surface. The fractional limb polarization for very low albedos can be sig-nificantly higher for atmospheres with haze than for Rayleigh scattering, becausethe unpolarized backscattering is greatly reduced.

For models with two polarizing layers with different absorption properties,the results are located in the same area as for one layer above a surface. Thelimiting cases are models with a completely dark lower layer (equivalent to a darksurface), and two identical layers (equivalent to a single semi-infinite layer). Foratmospheres that contain also isotropically scattering particles the polarization isalways lower and the intensity either slightly enhanced or reduced depending onα because of the different scattering phase functions.

68


Figure 2.17: Intensity vs. polarization at quadrature for the grid models. The shaded

area indicates the possible range for Rayleigh and isotropic models. The symbols and

lines indicate: semi-infinite conservative Rayleigh scattering (square), Lambert sphere

(round), and black planet (diamond). The dash-dotted line shows semi-infinite models,

the dashed and full line finite models without absorption (ω = 1) with surface albedo

AS = 1 and 0 respectively. The haze models shown in Fig. 2.12 are indicated by crosses

(high albedo) and plusses (low albedo).

Figure 2.18: Geometric albedo vs. disk-integrated radial polarization at opposition for

the grid models. The models are the same as in Fig. 2.17.

69


2.6.2 Polarization near quadrature versus limb polarization

For the prediction or the future interpretation of the polarization of extrasolarplanets it is of interest to compare the polarization at phase angles near quadra-ture with the limb polarization at opposition.

Figure 2.19 shows a diagram for the fractional polarization q(90) and thefractional limb polarization 〈qr(0)〉. Again the special models are indicated withblack symbols and lines as in Figs. 2.17 and 2.18.

A lower limit for the polarization fraction at phase angles α ≈ 90 can beset from the limb polarization at opposition 〈qr〉 for Rayleigh scattering or partlyisotropic scattering atmospheres. For example a limb polarization of 〈qr〉 ≈ 2%implies a minimal polarization of q(90) ≈ 20%. The upper limit for the polariza-tion fraction q(90) is not well constrained by 〈qr〉. The lower limit for Rayleigh orisotropic scattering may overestimate the polarization at large phase angles onlyfor very thick and bright haze layers.

A tighter correlation is obtained for the polarization flux Q(90) and the limbpolarization flux 〈Qr〉, which is shown in Fig. 2.20. All Rayleigh scattering mod-els are located in a narrow area along a line from the origin (Lambert sphere /black planet) to the semi-infinite, conservative Rayleigh scattering model. Thus,for Rayleigh scattering atmospheres, one can predict the large phase angle polar-ization flux Q(α) from the limb polarization flux 〈Qr〉 and vice versa. The areais slightly broadened if isotropic scattering is included in the models, but the re-lation still holds quite well. Only very thick and high albedo haze layers show asignificantly lower Q(90) for a given 〈Qr(0)〉.

2.6.3 Broadband polarized intensity

Color indices of observational parameters are often relatively easy to measureand they are helpful for the characterization of atmospheres. From the atmo-sphere models they are obtained by averaging spectral results (Sect. 2.5) over thefilter bandwidths.

Here we discuss the colors for a Rayleigh scattering atmosphere with methaneas a main absorber. It is investigated how the polarized intensity color changesas a function of methane mixing ratio and column density above a cloud or sur-face. Color indices are calculated by integrating Q(λ) over the wavelength rangeforeseen for filters in the SPHERE/ZIMPOL instrument (Beuzit et al. 2006). Thefilters are assumed to have flat transmission curves with cut offs at 555 and 700nm (R-band) and 715 and 865 nm (I-band). We concentrate on the color index ofthe polarized intensity QI/QR (Fig. 2.21).

The polarized intensity is higher at shorter wavelengths, and QI/QR < 1 forall models because of the decrease of the Rayleigh scattering cross section withwavelength and the general increase of the absorption cross section of methanewith wavelength. QI/QR is near 1 only for very thick atmospheres with very littlemethane or very thin atmospheres above a surface with wavelength independentscattering properties. In the former case QR and QI are very high, in the latter

70


Figure 2.19: Disk-integrated radial polarization at opposition vs. polarization at quadra-

ture for the grid. The models are the same as in Fig. 2.17. All Rayleigh scattering models

lie in the dark shaded area, (partly) isotropic models also in the light shaded area.

Figure 2.20: Disk-integrated radial polarized flux at opposition vs. polarized flux at

quadrature for the grid models. The indicated models are the same as in Fig. 2.17. All

Rayleigh scattering models lie in the dark shaded area, (partly) isotropic models also in

the light shaded area.

71


Figure 2.21: Polarized intensity QR as a function of color QI/QR where QR is the broad-

band R signal (555 to 700 nm) and QI the broad-band I signal (715 to 865 nm) for Rayleigh

scattering planets with methane at 90 phase angle. Indicated are models with constant

methane mixing ratios f (dashed) or constant atmosphere column density Z in km-am

(dotted) above a Lambert surface.

very low. For intermediately thick atmospheres the color index QI/QR mainlydepends on the methane mixing ratio, while QR mainly depends on the columndensity.

From this diagram we may predict that a color index of QI/QR ≈ 0.25 − 0.5could be typical for Rayleigh scattering atmospheres with methane absorption.Aerosol particles and absorbers other than methane are expected to have a differ-ent spectral dependence of the scattering and absorption cross sections.

2.7 Conclusions

This paper presents a grid of model results for the intensity and polarization ofRayleigh scattering planetary atmospheres, covering the model parameter spacein a systematic way. The model parameters considered are the single scatteringalbedo ω, which describes absorption, the scattering optical depth of the layer τsc,and the albedo of a Lambert surface AS. The results of these model calculationsare available in electronic form at CDS (see Appendix 2.7). In addition we exploremodels which combine Rayleigh and isotropic scattering, as well as particles withstrong forward scattering and atmospheres with vertical stratification.

Simple Rayleigh scattering models are a good first approximation to the po-larization of light reflected from planetary atmospheres because some amountof Rayleigh-like scattering by molecules or very small aerosol particles can be

72

2.7. Conclusions

expected in any atmosphere. From the model grid, which basically providesmonochromatic results, the spectropolarimetric signal can be calculated. Thisis done by considering the wavelength dependence of Rayleigh scattering andabsorption in an atmosphere with given column density and particle abundance(see Sect. 2.5).

The phase curves for the reflected intensity and polarization show a strong de-pendence on phase but they always have similar shapes (see Fig. 2.2). However,the absolute level of the phase curve is a strong function of atmospheric parame-ters such as the abundance of absorbers or aerosol particles, the optical thicknessof the Rayleigh scattering layer, or the albedo of the surface layer underneath(Sect. 2.3 and 2.4, see also Kattawar & Adams (1971)).

The model calculations demonstrate that polarimetric observations wouldprovide strong constraints on the atmospheric properties of the planetary atmo-spheres. An example is the polarization flux Q(α) of the reflected light which foroptically thick atmospheres is a simple function of the single scattering albedoroughly according to Q(α) ∝ ωb (b ≈ 1.5). If both polarization and intensity canbe measured, then one can distinguish between highly reflective and absorbingplanets with or without substantial layers of Rayleigh-like scattering particles.

According to the models a similar diagnostic is possible with observations ofthe geometric albedo, center-to-limb polarization profile, and limb polarizationfor solar system planets near opposition. The limb polarization is in additionparticularly sensitive to the vertical stratification of scattering or absorbing parti-cles located high in the atmosphere.

The diagnostic potential is further enhanced if data for different spectral fea-tures, e.g. inside and outside of absorption bands, or from different spectralwavelength regions can be combined (Sect. 2.5, see also Stam et al. (2004)).

The calculations presented in this work are based on simple atmosphere mod-els and they are therefore mainly useful for a first interpretation of data. For spec-tropolarimetric data of high quality, which are already available for solar systemplanets, one should use more sophisticated atmospheric models including a moredetailed geometric structure, accurate abundances, and better scattering modelsfor aerosol particles. With such models it might be possible for polarimetric stud-ies to make a contribution to our knowledge on the rather well known atmo-spheres of solar system objects.

Nonetheless the simple limb polarization models are of interest because theylink the model results for large phase angles, suitable for extrasolar planet re-search, to models which can be easily compared with observations of solar systemobjects. Thus it may be possible to associate polarimetric observations of extraso-lar planets to solar system objects. On the other hand it is possible to predict theexpected polarization for quadrature phase of Uranus- and Neptune-like extra-solar planets with this simple model grid based on the existing limb polarizationmeasurements of Uranus and Neptune (Fig. 2.20).

Polarimetric measurements for extrasolar planets are expected in the near fu-ture from high precision polarimeters. The measurements will first provide thepolarimetric contrast, which is the ratio of the polarization flux from the planet

73


Q(α) to the flux of the central star according to

C(α) =R2

D2· q(α) · I(α) =

R2

D2· Q(α) , (2.18)

where R is the radius of the planet and D the distance from its central star.Very sensitive polarimetric measurements of stars with known close-in plan-

ets already exist, taken e.g. with the PLANETPOL instrument (Hough et al. 2006;Lucas et al. 2009). This instrument measures the polarized intensity Q(α) of theplanet diluted by the unpolarized flux of the central star. It is then difficult toseparate the fractional polarization q and the reflected intensity I. D is knownfrom the radial velocity curve, but already the radius of the planet R may be hardto derive if the system shows no transits. For photometrically very stable starsthe reflected intensity I(α) · R2/D2 may be measurable with photometry of thephase curve with high precision instruments like MOST (e.g. Rowe et al. 2008).An uncertainty in the planet radius will affect the precision of the estimation ofthe normalized reflected intensity I(α) (or reflectivity) of the planet.

SPHERE, the future “VLT planet finder”, which includes the high precisionimaging polarimeter ZIMPOL, could provide successful polarimetric detections(Beuzit et al. 2006; Schmid et al. 2006a). This instrument will be able to spatiallyresolve nearby (d < 10 pc) star-planet systems and allow a polarimetric searchfor faint companions to stars. In a first step only the differential polarizationsignal, i.e. the polarization flux Q(α), can be measured in the residual halo ofthe central star. The measurement of the intensity signal I(α) might be possibleif further progress in techniques like angular differential imaging is achieved.Even if a determination of I(α) · R2/D2 is possible, an uncertainty remains in thetranslation to normalized intensity I(α) if the radius of the planet is not known.

Thus it may initially be quite difficult to measure intensity and radius. For thisreason it is important to investigate the diagnostic potential of the wavelengthdependence in the polarization flux in more detail. For example the R-band andI-band yield a polarization color index QI/QR from which it should be possibleto infer constraints on the Rayleigh scattering optical depth or the strength of ab-sorption bands (see Sect. 2.6.3). Another route of investigation for atmospheres ofextrasolar planets are measurements of the phase dependence of the polarizationflux. For example the location of the maximum of Q along the phase curve is sen-sitive to the presence of aerosol particles, as discussed in Sect. 2.4.2. For planetsin eccentric orbits the dependence of the polarization flux on the separation fromthe host star can be determined.

One can hope that the current rapid progress in extrasolar planet observa-tions continues, so that intensity measurements and accurate radius estimatesfor extrasolar planets become available soon after the first polarization flux de-tections, using the next generation of ground based telescopes and space instru-ments. Such instruments, if equipped with a polarimetric observing mode, wouldallow a broad range of observational programs on the reflected intensity and po-larization from planets.

74

2.7. Conclusions

Acknowledgements. This work is supported by the Swiss National Science Foun-

dation (SNF). We thank Harry Nussbaumer and Franco Joos for carefully reading the

manuscript.

2.A Model grid tables

Our extensive model grid of intensity and polarization phase curves for homoge-neous Rayleigh scattering atmospheres (Sect. 2.3) is available in electronic format CDS. Table 2.3 shows a sample of the first few lines and columns. The table isstructured as follows: Model parameters: Column 1: scattering optical thicknessτsc, Column 2: single scattering albedo ω, Column 3: surface albedo AS, Modelresults: Column 4: spherical albedo Asp, Column 5: geometric albedo I(0), Col-umn 6: limb polarization flux 〈Qr(0)〉, Column 7: I(7.5), Column 8: I(12.5),. . . ,Column 40: I(172.5) Column 41: Q(7.5),. . . , Column 74: Q(172.5). Columns 7 to74 are I(α) and Q(α) spaced in 5 degree intervals. I(0) is equivalent to I(2.5)in our calculations. Q(2.5), I(177.5) and Q(177.5) are very close to zero for allmodels and are not listed.

All results are disk-integrated. Binning, normalization and errors are de-scribed in Sect. 2.2.4. For all calculations the number of photons was cho-sen such that ∆(Q/I) < 0.1% for phase angles α = 0 − 130, and therefore(∆I)/I < 0.07% .

The model grid spans the following parameters: τsc = 99, 10, 5, 2, 1, 0.8, 0.6,0.4, 0.3, 0.2, 0.1, 0.05, 0.01, ω = 1, 0.99, 0.95, 0.9, 0.8, 0.6, 0.4, 0.2, 0.1, AS = 1, 0.3,0. Models for only three values of AS are given because the polarized intensityis independent of AS and the intensity drops nearly linearly with increasing AS.Models with τsc = 99 were calculated only for AS = 1 since the results are inde-pendent of AS. Instead of a model with ω = 1 and τsc = 99, we calculated themodel with τsc = 30 to reduce computation time, but the results are equivalent.

The spherical albedo Asp in column 4 is the ratio of reflected photons in anydirection to total incoming photons, while the geometric albedo I(0) in column5 is the disk-integrated reflected intensity at opposition normalized to the reflec-tion of a white Lambertian disk. For our sample of Rayleigh scattering models,typically I(0) = (0.80 ± 0.06)Asp.

75

Ch

ap

ter

2.

Po

lariz

atio

nm

od

els

for

pla

neta

ryatm

osp

here

s

Table 2.3: Extract of model grid results.

τsc ω AS Asp I(0) Qr(0) I(7.5) I(12.5) . . . I(172.5) Q(7.5) . . . Q(172.5)30.00 1.00 1.0 1.0000 0.7947 0.02161 0.7846 0.7661 . . . 0.0008 0.00334 . . . -0.0000199.00 0.99 1.0 0.7947 0.6378 0.02108 0.6290 0.6130 . . . 0.0007 0.00331 . . . -0.0000199.00 0.95 1.0 0.5975 0.4884 0.01686 0.4813 0.4681 . . . 0.0007 0.00304 . . . -0.0000199.00 0.90 1.0 0.4794 0.3980 0.01316 0.3918 0.3807 . . . 0.0006 0.00278 . . . -0.0000199.00 0.80 1.0 0.3438 0.2912 0.00837 0.2866 0.2779 . . . 0.0006 0.00229 . . . -0.0000199.00 0.60 1.0 0.1966 0.1707 0.00341 0.1676 0.1623 . . . 0.0004 0.00146 . . . -0.0000099.00 0.40 1.0 0.1087 0.0958 0.00118 0.0940 0.0908 . . . 0.0003 0.00084 . . . 0.0000099.00 0.20 1.0 0.0470 0.0418 0.00024 0.0410 0.0396 . . . 0.0001 0.00039 . . . 0.0000099.00 0.10 1.0 0.0221 0.0197 0.00006 0.0193 0.0186 . . . 0.0001 0.00018 . . . 0.0000010.00 1.00 1.0 1.0000 0.7949 0.02141 0.7848 0.7662 . . . 0.0008 0.00338 . . . -0.0000110.00 1.00 0.3 0.8889 0.7085 0.02227 0.6992 0.6820 . . . 0.0008 0.00338 . . . -0.0000210.00 1.00 0.0 0.8833 0.7042 0.02236 0.6950 0.6779 . . . 0.0008 0.00336 . . . -0.0000210.00 0.99 1.0 0.8057 0.6453 0.02111 0.6378 0.6214 . . . 0.0008 0.00310 . . . -0.0000110.00 0.99 0.3 0.7875 0.6326 0.02112 0.6238 0.6076 . . . 0.0007 0.00342 . . . -0.0000310.00 0.99 0.0 0.7858 0.6312 0.02107 0.6225 0.6064 . . . 0.0007 0.00344 . . . -0.00002. . .

76

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78

Chapter 3

Polarization of Uranus: Constraints on hazeproperties and predictions for analogextrasolar planets∗

E. Buenzli1 and H. M. Schmid1

Abstract

We model spectropolarimetric observations of the limb polarization of Uranusfor the wavelength range 530 to 930 nm using a Monte Carlo scattering code,and derive for the first time polarization properties of atmospheric constituentsof Uranus. We find that the limb polarization is dominated by Rayleigh scatter-ing on molecules. The observed enhancement of the fractional polarization in themethane bands can be naturally explained by a methane depletion in the strato-sphere. We obtain a very good fit to the full spectropolarimetric observations byincluding tropospheric aerosols with a wavelength dependent single scatteringpolarization, dropping from pm ≈ 0.25 at 630 nm to −0.25 at 840 nm at 90 scat-tering angle, and constant at smaller and larger wavelengths. Additionally a thin(τ ≈ 0.01 − 0.03), positively polarizing stratospheric haze layer is required toreproduce the limb polarization in the strong methane bands at 790 and 890 nm.The distribution and optical depth of the haze particles agrees well with the ex-tended haze layer model proposed by Karkoschka & Tomasko (2009), with mostof the haze continuously intermixed with gas below 1.2 bar.

From the limb polarization model we derive the polarization phase curve ofUranus and the spectropolarimetric signal at large phase angles in order to pre-dict the polarization and detectability of an Uranus-like extrasolar planet. At 90

phase angle, the polarization fraction is predicted to be ≈ 30% averaged over thewavelength range 555 to 865 nm. Within deep absorption bands the polariza-tion fraction can reach as high as 75%. We show that the direct detection of anUranus-analog around a nearby M-dwarf could be feasible with a polarimetricinstrument as proposed for the E-ELT 42 m telescope.

∗ This chapter has been submitted for publication in Icarus1 Institute of Astronomy, ETH Zurich, CH-8093 Zurich, Switzerland

79

Chapter 3. Polarization of Uranus

3.1 Introduction

Strong limb polarization signals have been detected for Uranus and Neptunewith ground based observations (Schmid et al. 2006b; Joos & Schmid 2007). Thesedata for the first time provide an observational assessment of the scattering po-larization for the atmospheres of these two planets. In this work we present scat-tering atmosphere models for the polarization of Uranus, compare the results tothe observations and predict the polarization of Uranus for large phase angles.

The structure and scattering properties of the troposphere and stratosphereof Uranus have been studied in detail through imaging and spectroscopic ob-servations (e.g. Rages et al. 1991; Baines et al. 1995). Most recently, Karkoschka &Tomasko (2009) (hereafter K09) presented comprehensive HST-STIS spectroscopycovering the wavelength range 300 to 1000 nm at high spatial resolution. One oftheir most important findings was that the methane mixing ratio appears to varywith latitude. In addition they present evidence for vertically extended haze withan altitude dependent specific optical depth, as opposed to clouds at or belowthe methane condensation level (e.g. as inferred from Keck infrared imaging andspectroscopy by Sromovsky & Fry 2007; 2008).

A complementary approach to studying properties of the upper atmosphereis given by polarimetry. Light scattered by gas, aerosols or clouds is generallypolarized to some degree, depending on size, shape and distribution of the scat-tering particles (see e.g. Buenzli & Schmid 2009, and references therein).

Polarimetric observations of planetary atmospheres are especially powerfulif a large range of phase angles can be covered. From Earth this is possible forVenus, where Hansen & Hovenier (1974) derived detailed properties of clouddroplets from polarimetriy. For the giant planets, the phase angles seen fromEarth are always small, for Uranus less than ≈ 3 degrees. Because of symmetryreasons the disk integrated polarization is essentially zero. A larger phase an-gle coverage can only be obtained from space crafts. This has been achived forJupiter, Saturn and Titan from the Pioneer 10 and 11 missions (see e.g. Smith &Tomasko 1984; Tomasko & Doose 1984; Tomasko & Smith 1982), Voyager (e.g.West et al. 1983b;a), and more recently by Galileo for Jupiter (Braak et al. 2002)and Cassini for Saturn (West et al. 2009, for preliminary results) and Titan (notyet published). In situ measurements of the haze polarization were performedby Cassini-Huygens for Titan (Tomasko et al. 2008).

The only spacecraft to visit Uranus and obtain a large phase angle coveragewas Voyager 2 in 1986. But while Voyager 2 carried a photopolarimeter, to ourknowledge no polarimetric observations of Uranus for large phase angles werepublished.

The only polarimetric information obtainable for Uranus from the ground isthrough measurement of the limb polarization effect with disk resolved polarime-try. Limb polarization is a higher order scattering effect that produces a radialpolarization signal of up to several percent near the planetary limb because ofsymmetry breaking (see e.g. Schmid et al. 2006b; van de Hulst 1980, for moredetailed descriptions of this effect).

80

3.2. Spectropolarimetric data

An advantage of the ground based measurements is that they allow for moreversatile observing modes than usually possible from a spacecraft. Spectropolari-metric observations provide the detailed dependence of polarization with wave-length, and differences inside and outside of strong absorption bands. Addi-tionally, ground based measurements are easily repeatable and could be used tomonitor temporal variability of the haze properties.

The limb polarization of Uranus was measured for the first time by Schmidet al. (2006b) from imaging polarimetry, and by Joos & Schmid (2007) from spec-tropolarimetry, in the wavelength range of 530 to 930 nm. The observationsshowed a polarization of ∼ 1 − 3% along the entire limb, dropping with wave-length, but enhanced within methane absorption bands. The observations werediscussed qualitatively and compared to simple analytic models, but no detailedmodeling considering the atmospheric structure was done. A parameter studyfor the limb polarization effect for scattering atmospheres was made in Buenzli& Schmid (2009). Up to now, no detailed modeling of the limb polarization effectfor any solar system planet has been published.

It is the goal of this paper to model the spectropolarimetric observations ofUranus with a radiative transfer scattering code, considering the atmosphericproperties derived by K09. The spatial resolution of the available polarimetricobservations is seeing limited and cannot compete with the spatial resolution andwavelength coverage of the HST-STIS observations. Therefore it is not possibleto investigate latitudinal or longitudinal structures. But because of the additionalpolarimetric information they provide an independent test of the extended hazelayer model, while at the same time constraining the polarimetric properties ofthe aerosols. Additionally, from this model the full polarization phase curve canbe predicted, which is of interest for future polarimetric observations of extrasolarplanets, e.g. with the VLT/SPHERE instrument (Beuzit et al. 2006; Schmid et al.2006a), and in particular for the follow-up instruments planned for the E-ELT orspace-based coronagraphic missions.

In Sect. 3.2 we summarize the spectopolarimetric limb polarization obser-vations that we model. Section 3.3 outlines our modeling approach, describingthe adapted atmospheric structure and particle scattering properties, and the ra-diative transfer code. In Sect. 3.4 we discuss the results of the model fit to theobservations. In Sect. 3.5 we predict the polarization signal of Uranus at largephase angles, and in Sect. 3.6 we assess the detectability of an Uranus-like planetaround a nearby M-dwarf. We conclude in Sect. 3.7.

3.2 Spectropolarimetric data

The spectropolarimetric observations of Uranus were obtained on November 29,2003 with the EFOSC2 instrument at the ESO 3.6 m telescope in La Silla, Chile.The data are described in detail in Joos & Schmid (2007). In this section we sum-marize the most important points.

The diameter of Uranus at the time of observations was 3.51”, the sub-earth

81


Figure 3.1: Orientation of the slit shown on an image of Uranus in radial Qr limb polar-

ization flux. Data from Schmid et al. (2006b).

latitude −19, the position angle of the south pole was 78, and the phase angle2.8. Long slit spectropolarimetry was taken with an 0.5” wide slit oriented inN-S and E-W direction in the celestial coordinate system. The slit in N-S directioncovers roughly the equatorial region (see Fig. 3.1), while the slit in E-W directionspans from near the South pole towards northern latitudes. The data cover thewavelength range 520-935 nm, with a wavelength scale of 0.206 nm per pixel, anda spatial scale of 0.157” per pixel. The seeing was about 1” during the observa-tions. The data were spectrally averaged into 3 nm bins, in order to improve theS/N ratio and to remove a fringe pattern at λ > 700 nm. The instrumental po-larization was found to be less than 0.2 %, and the polarization angle calibrationwas accurate to θ ≈ 2.

In this work we focus on the equatorial data, for which the atmospheric prop-erties are not expected to vary a lot over the slit, whereas the northern latitudesdiffer considerably from the southern parts of the planet because of strong sea-sonal effects (K09).

No absolute intensity calibration using standard stars was performed. In-stead, an average intensity spectrum was calculated from the two slit inte-grated spectra, which was then normalized to the full disk albedo spectrum ofKarkoschka (1998). The resulting albedo spectrum is therefore essentially identi-cal to Karkoschka (1998).

The fractional polarization shows a strong dependence along the slit, with ahigh radial polarization at the limbs and essentially no polarization in the center(see Fig. 3.1). The spectropolarimetric signal was determined for the limb regions,the central region, and the flux weighted average over the total slit. It is impor-tant to consider that the polarization is degraded due to the seeing and the finiteslit width, while the model assumes basically infinite resolution and an infinites-

82

3.3. Modeling

imally thin slit. The degradation of the polarization by the seeing can only becorrected accurately if the signal from the entire slit is considered. Therefore weconcentrate our modeling efforts on the flux weighted, slit averaged spectropo-larimetry for the equatorial slit orientation indicated in Fig. 3.1. From the seeingof about 1” and a slit width of 0.5”, a “seeing”-correction factor of 1.37± 0.15 wasderived by Joos & Schmid (2007) for the integrated polarization signal. The rela-tively large uncertainty is due to the typical but not well known seeing variationsduring the observations.

The slit averaged fractional polarization is defined as

〈Qr/I〉slit =

∫

Qr(r)dr∫

I(r)dr, (3.1)

where Qr is the radial polarization, i.e. the polarization measured in radial direc-tion from the planet center, corresponding here simply to the direction of the slit.A polarization parallel to the slit (perpendicular to the planetary limb) is definedas +Qr, while perpendicular to the slit (parallel to the planetary limb) is −Qr. TheUr component is zero within the errors of the observations, which is expected be-cause of symmetry reasons. The radial polarization is described in more detail inSchmid et al. (2006b).

Figure 3.2 shows the albedo spectrum and the seeing corrected slit averagedspectropolarimetry that we will model in the following sections. Consideringonly slit averaged data is reasonable because there is no obvious longitudinalasymmetry present, neither in the used spectropolarimetry nor in contemporaryHST imaging (Karkoschka 2001). In any case the dependence of the polarimetriccenter-to-limb profile on atmospheric parameters is rather subtle and can not beinvestigated with the seeing limited spatial resolution of our data. The uncer-tainty in the polarization data is indicated in the panel and is mainly due to thenot accurately known seeing correction factor. In addition strong photon noiseis present in the deep (low flux) absorption bands around 790 nm and between850 nm and 920 nm. The spectral features in the fractional polarization spectrumare anti-correlated to the features in the intensity spectrum. However, the polar-ization flux Q ∝ 〈Q/I〉slit · Ageo still shows the methane absorptions, but muchless pronounced than the intensity spectrum.

3.3 Modeling

3.3.1 Atmospheric structure and haze properties

As a base for our models we adopt the atmospheric structure given by model Dof Lindal et al. (1987), their most plausible model, which was derived from radiooccultation measurements of Voyager 2 in the equatorial region of Uranus. Themodel provides temperature, pressure, total number density of gas molecules,and methane fraction as a function of height for 111 layers between 2.5 · 10−4

and 2.3 bar. The helium abundance is assumed to be 15% by number density,

83


Figure 3.2: Geometric albedo, slit averaged limb polarization corrected for polarization

dilution due to seeing and slit width and radial polarization flux. Dotted lines in the

polarization fraction indicate the uncertainty of the seeing correction.

constant with altitude. The methane fraction reaches a maximum of 2.26% atthe 1.3 bar level and is assumed constant deeper in the atmosphere. Above thislevel the relative humidity of methane is constant at 30% and above 660 mbar theatmosphere is completely free of methane. The remaining gas is H2.

We extend the atmosphere downwards to 30 bar to ensure a large enoughoptical depth at all wavelengths of the model. The temperature is extrapolatedlinearly in log p. The full temperature-pressure diagram is shown in Fig. 3.3. Itis in good agreement with the model by Marley & McKay (1999) who providea T-p profile down to 10 bars. The total molecular number density is calculatedfrom the ideal gas law. The atmosphere then consists of 150 layers and is boundbelow by a Lambertian surface with an albedo of 0.8, accounting for the smallchance of absorption for photons that could penetrate below this deep level. To

84

3.3. Modeling

Figure 3.3: Temperature-pressure diagram for Uranus used in our model calculations.

Dotted lines separate the regions of constant haze optical depth per bar.

speed up the calculations we reduce the number of layers by combining adjacentlayers with very similar single scattering albedos by adding up the optical depthcontribution of the different components, resulting in model atmospheres with15 to 40 layers depending on wavelength.

Following K09 we distribute the haze into four atmospheric zones with dif-ferent haze optical depths τh per bar. These are the stratosphere (p < 0.1 bar),the upper troposphere (0.1 < p < 1.2 bar), the mid-troposphere (1.2 < p < 2bar) and the low troposphere (p > 2 bar). For the tropospheric haze particles, weadopt K09’s scattering properties that have provided a good fit to their spectro-scopic observations. These are a double Henyey-Greenstein phase function forF11(ϑ) (cf. Sect. 3.3.2, eq. 3.6) with asymmetry parameters g1 = 0.7, g2 = −0.3and wavelength dependent weighting factor f , and single scattering albedo ωh,t:

f = 0.94 − 0.47 sin4

(

1000 − λ

445

)

, (3.2)

ωh,t = 1 − 1

2 + eλ−290

37

, (3.3)

with λ in nm. In our wavelength range ωh,t > 0.998. Where ωh,t = 1 we set thevalue to 0.9999 to avoid infinite scatterings.

Since K09 did not model the polarization, we set the other arguments Fxy ofthe phase matrix (eq. 3.5) as described in Sect. 3.3.2, with the maximum singlescattering polarization pm in eq. 3.7 as a free paramter.

For the stratospheric particles, K09 adopt Mie theory, but they state that theirobservations cannot distinguish between spherical and aggregate particles. Ahigh single scattering polarization together with strong forward scattering would

85


be indicative of aggregate particles, like on Jupiter or Titan (e.g. West 1991).Therefore, for the stratospheric haze, we explore different particle types usinga parametrized phase matrix (Eq. 3.5) with a one-term Henyey-Greenstein phasefunction (Eq. 3.6 with f = 1), where the asymmetry parameter g and the maxi-mum single scattering polarization pm are free parameters. The single scatteringalbedo is set to

ωh,s = 1 − 0.25e365−λ

150 , (3.4)

which was obtained from a fit to the values calculated by K09 from Mie theory,and accounts for the darker haze at shorter wavelengths. It is ωh,s = 0.92 at 534.4nm and ωh,s = 0.994 at 924.4 nm.

3.3.2 Radiative transfer code

We use an extended version of the Monte Carlo scattering code described inBuenzli & Schmid (2009). The code calculates the random walk histories of manyphotons in the atmosphere from incidence to escape or absorption, and followsthe direction and polarization change because of scattering processes. The inten-sity and polarization are then established as a function of radial distance fromthe disk center for the backscattering situation (phase angle α ≈ 0) as needed tocompare to the Uranus data. For all other phase angles, the disk integrated sig-nals are calculated assuming a homogeneous planet (no latitudinal structures) toprovide the full intensity and polarization phase curve. The code fully includesmultiple scattering, which is essential because the photons mainly contributingto limb polarization have been scattered twice or more.

The incoming radiation is assumed to be a parallel beam of unpolarized pho-tons incident on the spherical planet. The model atmosphere consists of multiple,locally plane parallel layers, and photons emerge at the same point where theyentered into the atmosphere, despite multiple scatterings. These simplificationsare reasonable for Uranus because of its optically thick atmosphere and smallscale height with respect to the radius.

The direction and polarization changes are calculated from probability den-sity functions derived from the appropriate phase matrices of the scattering parti-cles. This approach is described in detail for Rayleigh scattering in Schmid (1992).For scattering on haze and cloud particles, our code allows for scattering matricesof the form

F(#) =

F11(ϑ) F12(ϑ) 0 0F12(ϑ) F11(ϑ) 0 0

0 0 F33(ϑ) 00 0 0 F44(ϑ)

, (3.5)

where ϑ is the scattering angle. For simplicity we use the approach ofparametrized functions, as introduced by Braak et al. (2002), because the exactshape of the phase functions is not very significant for limb polarization calcula-tions. We choose the two-term Henyey Greenstein function as the intensity phase

86

3.3. Modeling

function

F11(ϑ) = PHG(g1, g2, f , ϑ)

= f · 1 − g21

(1 + g21 − 2g1 cos ϑ)3/2

(3.6)

+(1 − f ) · 1 − g22

(1 + g22 − 2g2 cos ϑ)3/2

,

and scaled Rayleigh-like single scattering polarization dependence with positiveor negative maximum pm as variable parameter:

F12(ϑ)

F11(ϑ)= pm

cos2 ϑ − 1

cos2 ϑ + 1. (3.7)

−F12/F11 is the single scattering polarization fraction. A positive (negative) singlescattering polarization means a polarization orthogonal to the scattering plane(parallel). F33 is the same as for Rayleigh scattering,

F33(ϑ)

F11(ϑ)=

2 cos ϑ

cos2 ϑ + 1, (3.8)

and we neglect circular polarization by setting F44 = 0.We also neglect Raman scattering, which beyond 500 nm has an effect smaller

than 4% on the reflectivities (Sromovsky 2005).The Rayleigh scattering optical depth is computed from the gas column den-

sity for each layer as described by Buenzli & Schmid (2009). We include ab-sorption by methane and collision-induced absorption for H2-H2 and H2-He, al-though H2-He was found to be negligible.

Methane absorption coefficients are taken from Karkoschka (1998). The coeffi-cients provided by Karkoschka are not temperature and pressure dependent butare an average derived directly from observations of the Jovian planets and Titan.K09 found that for Uranus some minor modification for the methane windows al-low for simpler models. We adopt their corrections and find much better fits tothe geometric albedo spectrum. Recently, Karkoschka & Tomasko (2010) pub-lished new methane absorption coefficients derived from additional newer dataincluding temperature and pressure dependences. For wavelengths shorter than1000 nm the changes are small and therefore we did not recalculate our modelswith the new coefficients.

Coefficients for collision induced absorption were calculated from Fortranprograms which are available online from A. Borysow2 and are documented inBorysow (1991; 1993) for the H2-H2 fundamental band, Zheng & Borysow (1995)for the first overtone band, Borysow et al. (2000) for the second overtone band,and for H2-He in Borysow & Frommhold (1989) and Borysow et al. (1989). Wemade the calculations at temperature steps of 10 K, and linearly interpolated fortemperatures in between.

2 http://www.astro.ku.dk/ aborysow/programs/index.html

87


For each layer and wavelength we set or calculate the optical depth and thesingle scattering albedo for the gas and haze particles, which are the input toour Monte Carlo code. Models are run with 109 − 5 · 1010 photons depending onwavelength, such that at each wavelength point the statistical error of the frac-tional polarization ∆〈Q/I〉slit ≤ | ± 0.07%|.

3.4 Results

In this section we discuss our model fit and the parameter space explored withour calculations. Our best model compared to the observations is shown in Fig.3.4. The derived values for the parameters of this model are listed in Tab. 3.1.

The modeled polarization agrees with the observations within ∆〈Q/I〉slit ≈±0.2% for almost all wavelengths, and the agreement is generally better than0.1%. This lies within the uncertainty of the observations and the statistical errorsof the Monte Carlo simulation. The residuals are largest within some methanebands, and arise most likely from observational errors or possibly the uncertain-ties in the methane coefficients and in the methane abundance distribution.

The derivation and discussion of the best-fit parameters are addressed in moredetail in the following subsections. Because the full spectropolarimetric calcula-tions take up a significant amount of computation time, we could not perform afull exploration of the large parameter space. The uncertainties and limitations ofour model fit are therefore also discussed.

3.4.1 Rayleigh scattering and methane absorption

In our model we do not fit the CH4 gas abundances. Therefore we discuss their in-fluence on the polarization in this section. Rayleigh scattering on H2 and He is themain contributor to the polarization on Uranus. A semi-infinite, non-absorbingatmosphere would produce a slit integrated limb polarization 〈Q/I〉slit = 1.64%(Schmid et al. 2006b), more than is seen for Uranus at continuum wavelengths.

In the case of absorption, this value can rise up to a maximum of 3.8% if thepredominant absorption occurs below the main scattering layer, and it is smallerthan the contimuum value if the absorber is well mixed with the scattering gas(Buenzli & Schmid 2009). For Uranus, the inhomogeneous vertical methane mix-ing ratio, with a much larger methane fraction at higher pressures, naturally leadsto the enhancement of the limb polarization seen in methane bands.

A purely gaseous atmosphere with the gas mixing ratios of Uranus wouldoverestimate the observed limb polarization by a factor of 2 or more in the con-tinuum at all observed wavelengths, and in absorption bands for wavelengthsshorter than 750 nm. Additionally, the absorptions in the albedo spectrum wouldbe overestimated. Therefore the polarization of Uranus can be explained asmainly arising from Rayleigh scattering on gas, but reduced in strength by scat-tering from intermixed weakly- or non-polarizing tropospheric cloud or haze par-ticles. In the strongest methane bands, at λ = 790 and 890 nm, the observed limb

88

3.4. Results

Table 3.1: Best-fit model parameters of the haze particlesa

Param. pres. [bar] Valueτh < 0.1 0.15 / barpm < 0.1 1g < 0.1 0.6ωh,s < 0.1 (see eq. 3.4)

τh 0.1 − 1.2 (0.03 / bar)τh 1.2 − 2 1 / barτh > 2 0.3 / barpm > 0.1 see Fig. 3.5g1 > 0.1 (0.7)g2 > 0.1 (−0.3)f > 0.1 (see eq. 3.2)ωh,t > 0.1 (see eq. 3.3)

a Parameter values of the haze are given for different pressure regions. Thetop four parameters are for the stratospheric haze, the others for the tro-pospheric haze. Listed are the optical depth per bar τh, single scatteringpolarization pm (cf. eq. 3.7), parameters of the single or double Henyey-Greenstein function (g, or g1, g2 and f, cf. eq. 3.6), and single scatteringalbedo ωh,s or ωh,t. Values in brackets were not determined but set follow-ing K09.

polarization is underestimated and requires an additional thin, highly polarizingstratospheric haze layer to augment the polarization.

3.4.2 Tropospheric haze

The tropospheric haze was investigated by K09, who found a strong increase inhaze optical depth at a pressure p = 1.2 bar, corresponding to the condensationlevel of methane. Their best-fit model to the HST-STIS spectroscopy for latitudesnear the equator had optical depths τh ≈ 1.0 − 1.1 bar−1 at 1.2 < p < 2 bar, andτh ≈ 0.1 − 0.6 bar−1 at p > 2 bar, with τh constant with wavelength. We use theirfindings as a starting point for our models and vary the optical depth togetherwith the polarization properties of the haze.

In a first step we assume a wavelength independent polarization to keep thenumbers of free parameters as low as possible. Calculating the spectropolari-metric signal for a grid of haze optical depths and polarization, we cannot finda model that correctly reflects the slope of the polarization fraction with wave-length. For any fixed −1 < pm < 1, and τh(1.2 < p < 2 bar) and τh(p > 2 bar), either the polarization at wavelengths λ < 700 nm is underestimated, or over-estimated at λ > 700 nm. Keeping pm fixed but introducing a decreasing τh withwavelength similarly does not provide a good solution, because a lower τh results

89


Figure 3.4: Best model fit (red) to the observed (black dotted) geometric albedo and radial

slit averaged polarization.

in a higher polarization in methane bands. An increase in τh with wavelength onthe other hand seems implausible because for hazes and dust the optical depth istypically larger for short wavelengths. Also a change in optical depth would in-fluence the intensity. We therefore keep τh constant with wavelength, consistentwith K09, but vary pm with wavelength.

Indeed, we can now find a good fit for τh = 1 bar−1 at 1.2 < p < 2 bar, and τh

= 0.3 bar−1 at p > 2 bar, consistent with K09, with the polarization as discussedbelow. Our observations therefore support the extended haze layer model.

There are some degeneracies in the optical depth parameters. The results arealmost as good for a more equal haze distribution in the two tropospheric pres-sure regions, e.g. for τh = 0.65 bar−1 in both regions. We therefore cannot clearlydistinguish whether the haze is equally distributed in pressure, or if there is aclear drop in haze opacity below some pressure level. However, without hazein one of the two regions, the fits are clearly worse, and the models also indicatethat the haze optical depth in the upper region cannot be much lower than in thelower region.

The optical depth of the thin haze layer in the upper troposphere (0.1 < p <

90

3.4. Results

Figure 3.5: Wavelength dependence of the single scattering polarization parameter pm

(Eq. 3.7) for our best model.

1.2 bar) has very little effect on our disk- or slit-integrated model results, and istherefore not constrained by our observations. K09 derived the optical depth ofthis haze layer from UV observations and limb-darkening. We simply adopt avalue of τh = 0.03 bar−1, which is an average of the haze optical depth derived byK09 for latitudes near the equator.

The wavelength dependence of the polarization parameter pm is mainly deter-mined by the observed level and slope of the polarization outside of the methanebands. For a constant, non-polarizing haze, the polarization is too low at short,and too high at long wavelengths, both in the continuum and methane absorp-tion bands. This suggests a higher single scattering polarization at shorter and alower one at longer wavelengths, with some decrease in between. We thereforedescribe pm(λ) as a single analytic function which levels off at some wavelengthat both the high and low end. We find a good match for pm with a value of +0.25below 630 nm, a value of −0.25 above 840 nm, and a constant gradient in betweenas shown in Fig. 3.5.

The derived function for pm(λ) can be considered an approximation for amore realistic, smoother function. The function simply illustrates the generalbehavior of the polarization: a positive single scattering polarization near 90

scattering angle at λ < 700 nm and a negative at λ > 700 nm. Such a depen-dence might be explained by droplets or ice crystals with sizes of a few µm (seee.g. Karalidi et al. 2011, for spherical water droplets), but detailed calculations forparticles to be expected at the temperatures of Uranus are lacking. Also, for suchparticles the used Rayleigh-like scattering angle dependence is an oversimplifi-cation. The exact scattering properties of the tropospheric haze particles is notwell constrained by our measurements, because the limb polarization arises frommultiply scattered photons. However, scattering particles with generally positive(orthogonal to scattering plane) polarization pm . 0.25 at wavelengths between530 and 730 nm and negative (parallel) polarization pm & −0.25 between 730and 930 nm seems robust because we cannot find a haze in good agreement with

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K09 with different polarization properties that can fit both the intensity and thelimb polarization spectra. Another option might be a change in haze scatteringproperties with altitude, which we did not consider here.

We tested a number of models with localized, geometrically thin clouds atfixed pressure levels. A coarse grid spanning different optical depths, pressurelevels and single scattering polarization of particles for the cloud indicates thatthe slope of the continuum polarization and/or ratios of polarization in methanebands to continuum fit systematically worse than for the extendend haze layermodel. Our data therefore indicate that the extended haze layer model proposedby K09, which also fits better for the HST/STIS data, is more likely than a lo-calized cloud model. Because the parameter space is large, we cannot excludecombinations of multiple cloud levels.

3.4.3 Stratospheric haze

All models without a stratospheric haze that fit the wavelengths shorter than 750nm underestimate the polarization in the strong absorption bands at wavelengthslonger than 750 nm, for our best model by 0.2-0.3%. This discrepancy does not oc-cur outside of the deep bands. Only a polarizing haze layer situated in the strato-sphere, such that the scattering occurs above the absorbing methane, can increasethe polarization in these bands sufficiently. This thin layer does not significantlyaffect the model fit a shorter wavelengths. Evidence for a thin stratospheric hazelayer was also found by K09 in intensity measurements.

K09 derived an optical depth of about 0.05 bar−1 at the equator for the strato-spheric haze. If we adopt the same optical depth, the resulting methane bandpolarization is not increased sufficiently, even if the haze particles are set to bevery highly polarizing (pm = 1) and reduced in backscttering (g = 0.6).

A better match is achieved for τh ≈ 0.15 bar−1 (and thus a total optical depthof 0.015 for the stratospheric haze above a pressure level of 0.1 bar), for aforementioned scattering properties, even though the polarization is still ∼ 0.1% toolow in the absorption band around 790 nm. The fit is degenerate in optical depthτh and maximum single scattering polarization pm. A higher optical depth com-bined with a lower single scattering polarization (e.g. τh = 0.3 bar−1, pm = 0.5)also provide good fits for the polarization. Much higher optical depths for thestratospheric haze are unlikely because the intensity signal would be too stronglyaffected.

The uncertainties of the polarization measurements is larger in the deepmethane bands because of the low photon counts. However, the polarizationin the absorption bands between 750 and 900 nm does give strong evidence fora thin, positively polarizing stratospheric haze layer. Our observations unfortu-nately cannot constrain the particle type of the stratospheric haze. If the particlesare highly polarizing, they may either be forward scattering aggregate aerosolparticles like at the poles of Jupiter, or they are very small particles which can bedescribed by a Rayleigh-like phase function. For the case of moderate polariza-

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3.5. Predictions for the polarimetric signal of Uranus at large phase angles

tion, pm ≤ 0.5, larger spherical particles as described by Mie-theory are also apossibility.

3.5 Predictions for the polarimetric signal of Uranus

at large phase angles

Our best fit model to the limb polarization data, with the parameters summa-rized in Tab. 3.1, now allows for predictions for the intensity and polarizationspectra of Uranus at different phase angles. This is mainly of interest for futureprograms aimed at the detection and characterization of reflected light from ex-trasolar planets, in particular using polarimetry. While in the solar system theouter gas giants are always seen at small phase angles from Earth, an extraso-lar planet circling another star will cover the phase angle interval [90 − i, 90 + i],where i is the inclination of its orbit.

We calculate the disk integrated spectropolarimetric signal over the full phasecurve, assuming for the whole planet the same atmospheric structure as for theequatorial region of Uranus. This is a simplification, because methane abundanceand haze optical depths probably vary with latitude (K09). However, the maindeviations in methane abundance occur at high southern latitudes and for thehaze at pressures > 2 bar, where the influence on the disk integrated value isrelatively small. Also, discrete cloud features are small and of low optical depth.

Figure 3.6 shows the intensity and polarization spectra of our best fit modelat quadrature phase angle (α = 90) and at the angle of maximum polarizationflux (α ≈ 70). The intensity spectrum is very similar to the geometric albedospectrum, but reduced in flux because only a part of the visible planetary disk isilluminated.

The polarization fraction varies strongly with wavelength. The polarization isgreatly enhanced within methane bands, and reaches maximum values of up to≈ 75% at quadrature. Because of the strong absorption most photons are singlyscattered and therefore have a high polarization if scattered at 90 angle. If thethin stratospheric haze is not highly polarizing, the maximum value can be 10-20% lower. Outside methane bands multiple scatterings lower the polarizationfraction because the scatterings occur in differently oriented planes. The polar-ization fraction drops with wavelength outside methane bands, from ≈ 25% at550 nm (20% at 70), to ≈ 10% at 830 nm. This happens because the opticaldepth of the highly polarizing Rayleigh scattering gas drops as ∼ λ−4, meaningthat the tropospheric haze has a stronger influence on the polarization signal atlonger wavelengths. Even if the single scattering polarization of the haze did notdecrease with wavelength, the polarization would drop.

The polarized intensity, which is the product of intensity and polarizationfraction, drops strongly with wavelength, with additional dips in methane bands.

If differential polarimetry is used as a contrast enhancing tool for the detection

93


Figure 3.6: Predicted disk integrated flux I, polarization fraction Q/I and polarized in-

tensity Q as a function of wavelength at 90 phase angle (red) and at 70 phase angle

(blue) for our best fit model.

of extrasolar planets, the measured signal is the contrast

C(α) =R2

D2· p(α) · F(α) (3.9)

where R is the planet’s radius and D its distance to the central star. It is propor-tional to the polarized intensity. Therefore it is favorable to perfom these obser-vations at short wavelengths and outside of strong absorption bands.

First polarimetric detections of extrasolar planets will not deliver the spec-tropolarimetric signal, but an average polarization for broadband filters. There-fore, we have also calculated the full phase curve averaged over three filters fore-seen in the SPHERE/ZIMPOL instrument. The filters are assumed as having flattransmission curves between 555 and 700 nm (R-band), 715 and 865 nm (I-band),and 555 and 865 nm (RI-band). Figure 3.7 displays the intensity, polarization frac-tion and polarized intensity as a function of phase angle. Before the integration

94

3.5. Predictions for the polarimetric signal of Uranus at large phase angles

Figure 3.7: Predicted disk integrated flux I, polarization fraction Q/I and polarized in-

tensity Q as a function of phase angle for our best fit model, integrated over a broadband

RI filter (555 to 865 nm, green), R filter (555 to 700 nm, blue), and I filter (715 to 865 nm,

red) and weighted with the photon spectrum of a G2V star. Addtionally, the broadband

RI curve weighted with the photon spectrum of a M4V star is shown (black).

over the spectral region, the albedo and polarization spectra at each phase anglewere weighted with a template photon spectrum of a G2V star (Pickles 1998).

The intensity phase curve obviously drops with phase angle, because smallerfractions of the visible hemisphere are illuminated at larger phase angles. Fromthe spectrum it is also clear that the integrated intensity is larger at shorter wave-lengths because methane absorption is weaker.

The polarization fraction however shows only little color dependence inte-grated over the selected filter bands. The maximum is 28% at ≈ 90 − 95 forboth the shorter and longer wavelength filter. This occurs despite the stronglyincreased polarization in the absorption bands in the longer wavelength filter,because the flux in these bands is very low and therefore they do not contributestrongly to the integrated polarization.

The maximal polarized intensity is found at ≈ 70, and the FWHM is about ≈80. Therefore polarimetric observations should be taken at phase angles between30 and 110.

95


As obvious from the spectrum the polarized intensity is much larger in theshorter wavelength filter.

3.6 Detectability of an Uranus analog around a

nearby M dwarf

An upcoming instrument to search for polarized reflected light from extrasolarplanets resolved from its central star will be SPHERE at the VLT (Beuzit et al.2006). The main targets of SPHERE are planets closer than 1 AU to the centralstar, and these are generally much warmer than Uranus. Nevertheless, for a suc-cessful interpretation of a future polarization signal of an exoplanet, it is helpfulto understand the signature of a planet such as Uranus. This case would be char-acteristic for a planet with a relatively clear troposphere mixed with haze parti-cles, a vertically inhomogeneous distribution of methane, an overabundance of Cwith respect to the solar values, and strong CH4 absorption bands.

The detection of an Uranus-analog around a solar-type star is very challeng-ing and beyond the capacities of the currently planned EPICS/E-ELT instrument(Kasper et al. 2010), because the large separation of 20 AU will result in a verylow contrast. However, around cooler stars, a planet of a similar temperature canbe found much closer in. Here we estimate if a planet of the size and tempera-ture of Uranus could be detectable with the proposed polarimeter in the EPICSinstrument for the E-ELT (Kasper et al. 2010) around a nearby M-dwarf. We dis-regard for our estimation that the lower energetic stellar radiation could result ina significantly different atmospheric structure.

One of the prime targets could be Barnard’s star, a very low luminosity Mdwarf at a distance of only 1.8 pc. With a bolometric luminosity of L = 0.0035L⊙,a planet would receive the same amount of radiation as Uranus around the sun ata separation of 1.18 AU, or 0.64”, corresponding to an orbital period of about 3.2years. At this separation, the current upper mass limit for a planetary companionfrom radial velocity measurements is M sin i ≈ 30 M⊕ (Zechmeister et al. 2009),more than twice as massive as Uranus.

The predicted phase curve weighted with the photon spectrum of an M4Vdwarf compared to a G2V dwarf is also depicted in Fig. 3.7. For an M4V star,the photon flux rises strongly towards redder wavelengths, unlike for a G2V star,where the photon flux decreases slightly. Therefore, the spectral regions wherethe reflected flux is smaller are weighted more strongly, making the intensity andpolarization flux phase curve significantly lower for the broadband RI filter. Theeffect is much smaller for the R and I filter because of the narrower filter pass-bands. The polarization fraction is not affected, because it is nearly constant withwavelength outside of absorption bands.

In Fig. 3.8 we show the polarization constrast of an Uranus-analog derivedfrom our best-fit model for one period of a circular orbit around Barnard’s starat 1.18 AU for four inclinations, using the RI phase curve shown in Fig. 3.7. Themaximum contrast reached is about 6 · 10−10, which lies approximately at the de-

96

3.7. Conclusions

Figure 3.8: Predicted polarization contrast of an Uranus analog around Barnard’s star for

our best fit model, integrated over a broadband RI filter (555 to 865 nm), weighted with

the photon spectrum of an M4V star, for four inclinations: i = 90 (blue), i = 60 (green),

i = 30 (red), i = 0 (black)

tection limit that is currently planned to be achieved with an imaging polarimeterat the E-ELT (Kasper et al. 2010).

In the face-on case (i = 0), the phase angle is always equal to 90 and there-fore the polarization contrast remains constant at a relatively high level. In theedge-on case (i = 90) the contrast varies strongly because of the large phase an-gle variation, with two relatively short duration peaks, and diminishing to zeroduring transit and secondary eclipse. For an inclination of 30 the contrast re-mains very high for half the orbital period, during which the phase angles are inthe favorable range of 60 - 90 degrees.

Even though only few low luminosity M-dwarfs will be suitable for observa-tions with EPICS, this estimation shows that with differential polarimetry a directobservation of a cool, Uranus-like extrasolar ice-giant could be feasible within thenext 10-15 years. Somewhat closer-in or larger planets with a similar atmosphericstructure as Uranus could be observed and characterized quite easily with the E-ELT around many nearby stars. A big advantage would be if the orbital phaseof the planet is already known, for example from radial velocity measurements,allowing to perform the polarization observations when the planet’s polarizationsignal is expected to be strong.

3.7 Conclusions

We have modeled the polarization signal of the reflected light from Uranus. Wecompared our model with spectropolarimetric limb polarization observations ofUranus between 530 and 930 nm. It is the first full spectropolarimetric model fit

97


to observations of a solar system planet. We simulated in detail the polarimetricstructure of spectral features and derived for the first time polarimetric parame-ters of scattering particles on Uranus, in particular the scattering haze. We wereable to obtain a very good fit to the slit-integrated limb polarization in the wholewavelength range. For the vertical distribution of the haze, our model also sup-ports the extended haze layer model derived by K09.

3.7.1 Polarimetric properties of Uranus

Our model is the first to determine polarimetric properties of scattering particlesin the atmosphere of Uranus. Constraints on the haze distribution and scatteringproperties of the haze particles could be derived. The limb polarization spectrumis especially sensitive to the altitude level of absorption with respect to that ofscattering. The increase of fractional polarization within methane bands stronglysupports a decrease in CH4 abundance towards high altitudes as determined e.g.from radio occultation observations (Lindal et al. 1987). The absolute level of po-larization, which decreases quickly with wavelength, and the relative differencebetween continuum and CH4 bands require besides Rayleigh scattering on H2 atropospheric haze that is extended and intermixed with the gas and whose po-larization properties change significantly with wavelength, from positive to neg-ative single scattering polarization like for scattering by certain types of droplets.The optical depth of this haze is of order 1 bar−1 below 1.2 bar, and most likelylower below 2 bar, in good agreement with values derived by K09. The extendedhaze layer model performs better at fitting the limb polarization than more local-ized clouds.

The derived single scattering polarization function of the tropospheric hazemust be regarded as a first order approximation because its scattering angle de-pendence is not well defined for limb polarization measurements. Regardless,we can say that for a haze in accordance with K09 the haze polarizes positively atwavelengths below 730 nm, and negatively beyond, if uniform scattering proper-ties are assumed throughout the troposphere. A next step would be to calculateeffective scattering matrices for different size distributions and compositions asa function of wavelength to determine which physical haze properties match thedetermined scattering properties.

The measured limb polarization in the deep CH4 bands requires a thin (τh ≈0.015 − 0.03), positively polarizing (pm ≈ 0.5 − 1 depending on τh) stratospherichaze at pressures p < 0.1 bar. The nature of the particles, whether they are smallmolecules that essentially act as additional Rayleigh scatterers, larger sphericalparticles, or whether forward scattering fractal aggregates as found on Jupiter’spoles and Titan are required, cannot be determined with certainty from our ob-servations. Their properties could be better constrained with additional high-precision limb polarization measurements in strong near-infrared methane bands(e.g at 1.7 µm), where practically no limb polarization is expected from Rayleighscattering.

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3.7. Conclusions

3.7.2 Limb polarization measurements for Uranus

Limb polarization measurements provide the opportunity to investigate obser-vationally the polarimetric properties of the atmosphere of Uranus with Earth-based polarimetry, despite the small phase angle. This work demonstrates thatan excellent model fit to the spectropolarimetric data can be achieved.

Since this is the first detailed spectropolarimetric modeling of the limb polar-ization of a planet, the full diagnostic potential of such observations still needsto be investigated. The interpretation of limb polarization measurements alone isambiguous and therefore we have based our model analysis strongly on previousmodeling, in particular on the modeling of the HST/STIS spectrometry. In thisway the limb polarization spectrum confirms previous results and provides newconstraints on the vertical distribution and the polarimetric properties of the scat-tering particles in the atmosphere as summarized in the previous section. For aninterpretation of these results with respect to particle properties such as size andstructure, a significant amount of calculations of wavelength dependent scatter-ing matrices of haze particles must be made. Some effort into this direction wastaken by Tomasko et al. (2008) to determine the haze properties on Titan.

Limb polarization measurements should be considered as one useful tool forfuture studies of Uranus. High precision spectropolarimeters are readily avail-able for the community at several observatories, providing access to wavelengthregions from the near-UV to the near-IR, far beyond the range explored in thiswork. High spatial resolution polarimetry is possible with ground-based AO sys-tems (see Perrin et al. 2008) and HST which allow detailed studies on the latitudedependence of the limb polarization, or of differences between the morning andthe evening limb. Variations of the limb polarizaton on a large range of timescales can also be investigated. Considering all these possibilities it seems verylikely that limb polarization data can play a crucial role in future investigationsof Uranus.

It will be important to obtain well-calibrated intensity measurements togetherwith high-precision limb polarization measurements. While the polarization frac-tion is a differential measurement independent of the intensity calibration, only aprecise measurement of both parameters together over a wide spectral range willallow some degeneracies in optical depth and polarization properties in the mod-els to be resolved. In addition, a code that can calculate both the polarization andintensity combined with an algorithm for an efficient parameter space search willbe required to determine meaningful uncertainties in the large parameter space.

3.7.3 Prospects for exoplanet polarimetry

Differential polarimetry is a very promising contrast enhancing technique for de-tecting and characterizing extrasolar planets through their reflected light. Planetswith a similar atmospheric structure as Uranus are particularly well suited for po-larimetric detection because of the relatively thick, strongly polarizing gas layer,and strong absorption occuring only below ∼1 bar. In our own solar system,

99


equally well suited is only Neptune (with a quite similar atmosphere as Uranus).Additionally, planets with a thick photochemical haze layer would be promisingtargets. For example, Titan is the most strongly polarizing solar system objectbecause of a very thick, highly polarizing haze layer (Tomasko & Smith 1982;Tomasko et al. 2008). Jupiter shows a high polarization in the polar regions be-cause of a similar thick photochemical haze layer, while the polarization of theequatorial region is much lower (Smith & Tomasko 1984; Schmid et al. 2011).

The near-future polarimetric searches for extra-solar planets will only be ableto target planets at much smaller separations than the giant planets in our solarsystem. Only around M-dwarfs planets at similar temperatures as our solar sys-tem giants may be found, but because of the faintness of these stars only a verylimited sample will be accessible even with a 40 m class telecope. For planetsaround earlier-type stars, the targeted planets will be warmer. Because no ob-servations and only very few and rather basic models without polarization (Su-darsky et al. 2000; 2003; 2005; Cahoy et al. 2010) exist for such atmospheres, thepolarimetric signal to be expected from these planets is still very unclear. Fur-ther modeling efforts in this direction should be undertaken, with an emphasison highly polarizing photochemical haze particles.

For ground based observations, the lower limit of observable wavelengthsis currently set by the performance of the adaptive optics systems. With thecurrent generation of extreme AO-systems, seeing correction for wavelengthsshorter than 600 nm is difficult. Observations at shorter wavelengths wouldbe very attractive because of the strong increase of the Rayleigh scattering crosssection towards shorter wavelengths, thus an increase of the polarization fluxcan be expected for most planets. This is seen e.g. in the equatorial regions ofJupiter (Smith & Tomasko 1984), and the same is of course true for Earth (theblue sky is a direct result of Rayleigh scattering). Therefore, an advancementin AO-technology towards shorter wavelengths, or a high precision polarimeteron a space mission, could significantly improve detection prospects of planetsthrough polarimetry and provide a unique window into atmospheres of planetsreflecting similarly as the planets in the solar system.

Acknowledgements. We are grateful to Franco Joos for providing the reduced po-

larimetric observations of Uranus in electronic form. We thank Michael R. Meyer and

Francois Menard for reading the manuscript and providing comments.

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BIBLIOGRAPHY

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Chapter 4

A polarimetric model for Jupiter’s polarhaze∗

E. Buenzli1, H. M. Schmid1,F. Joos1 & D. Gisler1

Abstract

We present models of ground-based limb polarization observations of Jupiter forspatially resolved (long slit) spectropolarimetry, focusing on the polarimetric sig-nal at λ = 6000 A in a slit spanning from the North to the South pole. For thepolar region we find a very strong radial (perpendicular to the limb) fractionalpolarization with a seeing corrected maximum of about +11.5 % in the Southand +10.0 % in the North. This indicates that the polarizing haze layer is thickerat the South pole. The polar haze layers extend down to 58 in latitude. Thederived polarization values are much higher than reported in previous studiesbecause of the better spatial resolution of our data and an appropriate considera-tion of the atmospheric seeing. The model calculations demonstrate that the highlimb polarization can be explained by strongly polarizing (p ≈ 1.0), high albedo(ω ≈ 0.98) haze particles with a scattering asymmetry parameter of g ≈ 0.6 as ex-pected for aggregate particles of the type described by West (1991). The deducedparticle parameters are distinctively different when compared to lower latituderegions.

4.1 Introduction

Solar system planets have frequently been observed polarimetrically until 1990with instruments using single channel (aperture) detectors (e.g. Leroy 2000).However, almost no data were taken with “modern”, ground-based imaging po-larimeters and spectropolarimeters using array detectors. Therefore the polari-metric properties of solar system planets are still not well characterized. As part

∗ This chapter is a modified excerpt of the paper ‘Long slit spectropolarimetry of Jupiter andSaturn’ by H.M. Schmid, F. Joos, E. Buenzli & D. Gisler, published in Icarus 212, 701 (2011), focus-ing on my contribution to this work.

1 Institute for Astronomy, ETH Zurich, CH-8093 Zurich, Switzerland

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Chapter 4. Jupiter’s polar haze

of a program of “modern” ground-based polarimetric observations of solar sys-tem planets, Schmid et al. (2006) and Joos & Schmid (2007) described polarimetricdata for Uranus and Neptune, for which strong limb polarization was detected.Modeling for the data of Uranus was presented in Chapter 3 of this thesis. Inthis chapter we present models for polarimetry of Jupiter taken with the EFOSC2instrument attached to the ESO 3.6 m telescope.

Polarimetric data of Jupiter was first published in the pioneering paper of Lyot(1929), who detected a strong positive polarization of p ≈ 5 − 8 % at the poleswith an orientation perpendicular to the limb. In the disk center he measureda phase angle dependent polarization, which is essentially zero near oppositionand slightly negative (parallel to the scattering plane), p ≈ −0.4 %, for phaseangles around α = 10. These measurements were confirmed and improvedin many later observations using single aperture polarimeters (e.g. Dollfus 1957;Gehrels et al. 1969; Morozhenko 1973; Hall & Riley 1976) and some imaging po-larimetry by Carlson & Lutz (1989).

Important results on the polarization of Jupiter were achieved with the Pio-neer 10 and 11 spacecrafts, which obtained polarization maps for phase angleslarger than α = 12. The data show that the polarization in the B- and R-band forα ≈ 90 reaches a level of about p ≈ 50 % at the poles, while the polarization israther low (< 10 %) in the equatorial region (e.g. Smith & Tomasko 1984).

Here we present models for spatially resolved long slit spectropolarimetry ofJupiter, focusing on the wavelength 6000 A in a North-South oriented slit. In thenext section a description of the observational data, including the observationsand data reduction, is given. In section 4.3 the observations are compared withhaze model simulations. The results are discussed in the final section.

4.2 Polarimetric data

Spectropolarimetric observations of Jupiter were taken during the nights ofNovember 29 and 30, 2003 with EFOSC2 at the ESO 3.6m telescope at La Silla.These data originate from the same run and instrument setup as the spectropo-larimetry of Uranus and Neptune from Joos & Schmid (2007), where descriptionsof the measuring strategy and the data reduction are given. Here we provide onlya brief outline and highlight some special points.

Observational parameters for Jupiter were taken or derived from data givenin U. S. Naval Observatory & Royal Greenwich Observatory (2001) and are sum-marized in Table 4.1. The position angle (PA) of the scattering plane θ is givenwith respect to the central meridian (North-South direction) of the planet. Thebright limb is on the East for θ close to 90. If the scattering plane is tilted withrespect to the East-West direction, a perpendicular or parallel polarization withrespect to the scattering plane will not only produce a Q-polarization (in N-Sorientation), but also a significant U-polarization component. For a tilt angle of∆θ = −2 as in the case of our Jupiter observations this factor is U = −0.07 Q.The apparent diameters dN−S and dE−W are used to convert locations x from the

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4.2. Polarimetric data

Table 4.1: Parameters for the Jupiter observations. θ is the orientation of the scattering

plane, and rCM are distances from the sub-earth point on the central meridian.

parameter Jupiterdate (2003) Nov. 29Phase angle α 10.4

θ 88

polar axis incl. −1.5

lat. sub-earth point −1.6

diameter (E-W) 35.91′′

rCM limbs ±16.79′′

rCM south pole −16.78′′

rCM equator −0.5′′

disk center (= sub-earth point) along the central meridian (CM) to radial distancesrCM = x · dN−S/2 which can be converted to planetographic latitudes consideringthe ellipsoidal shape and the inclination of the planet (Table 4.1).

The EFOSC2 instrument provides long slit spectropolarimetry. A special slitmask is placed in the focal plane. It consists of a series of 19.7′′ long slitlets with aperiod of 42.2′′ to avoid overlapping of the two beams produced by the Wollastonprism. The width of the slitlets used was 0.5′′. The spatial scale was 0.157′′ andthe spectral scale 2.06 A per pixel. The spatial resolution (given by the effectiveseeing) of our data is about 1′′, as derived from the width of the spectra of thestandard stars. The spectral resolution was 6.4 A for a 0.5′′ wide slit. We focushere on the observations in N-S direction since we are mostly interested in thepolarization properties of the polar haze.

The instrumental polarization was found to be less than 0.2 % in the centralregion of the field. The polarization angle calibration should be accurate to about∆θ ≈ ±2. Solar and telluric spectral features in the intensity spectra were cor-rected with the help of Mars observations taken with the same instrument con-figuration. The overall slope of the intensity spectrum was adjusted to the albedospectrum from Karkoschka (1998). We define the Q parameter of the Stokes vec-tor as positive for a polarization parallel to the slit (N-S direction), and thus per-pendicular to the limb. In this case Q is equivalent to the radial polarization Qr.The orientation of +Ur is rotated by 45 in counter-clock wise direction (Northover East).

The long-slit spectropolarimetric measurements provide polarimetrically wellcalibrated profiles for the central meridian for all wavelengths between 5300 and9300 A. Figure 4.2 shows profiles for the continuum at 6000 A, spectrally averagedfrom 5900 to 6100 A, (solid line). Spectral averaging was done to enhance thesignal to noise ratio.

The intensity profile shows belt and zones in good agreement with previousstudies (e.g. Moreno et al. 1991; Chanover et al. 1996). The Qr/I and Qr polariza-tion profiles in Fig. 4.2 show Qr/I ≈ 10 % at the south pole. At the north pole,Qr/I reaches a maximum value just above 8 %. The peak polarization at the limb

105


Figure 4.1: Slit positions for the spectropolarimetric observations with EFOSC2 on a Q

(left) polarization flux image of Jupiter from Gisler (2005). North is up and East is left.

The gray scale is normalized to the central intensity and spans the range from −1.0 %

(black) to +1.0 % (white).

depends significantly on the spatial resolution (or the effective seeing). It is in-teresting to note that the South pole shows not only a stronger polarization thanthe North pole, but also a stronger limb brightening in the methane absorptionλ8870. The polarization is negative in the center of the disk. The sign change oc-curs at about ±12.5 arcsec, corresponding to a Jovian latitude of about ±59. Thepolarization in the disk center depends on phase and it varies from Q/I ≈ 0.0 %for α = 0 to about Q/I ≈ −0.5 % for α = 12 as described in detail in Mo-rozhenko (1973). The measurements are qualitatively in good agreement withearlier studies for the visual-red spectral region, e.g. from Lyot (1929), Dollfus(1957), or Hall & Riley (1968).

The fractional intensity I/Islit can be converted into reflectivity f . If the aver-age reflectivity along the central meridian 〈 fslit〉 is known, then the reflectivity ina bin is

f = 〈 fslit〉I/Islit

x/xslit.

From the full disk image the ratio Λ = 〈 fdisk〉/〈 fslit〉 between the average re-flectivity for the full planetary disk ( = the geometric albedo Ag) and fslit can bederived. With the geometric albedo from the literature

〈 fslit〉 =Ag

Λ.

We derive Λ = 0.92 from a ZIMPOL image (Gisler 2005) for the “continuum”filter centered at 6010 A (width 180 A). This value does not differ much fromΛLam = 0.85 for a perfectly white Lambert sphere. Karkoschka (1998) gives ageometric albedo of Ag = 0.59 for data taken in 1995 at this wavelength. Thisvalue can be used for our calibration because the global reflectivity variations ofJupiter are small (. 5 %) and the phase dependence of the reflectivity (≈ 1.5 %for α ≈ 0 − 10) can be neglected.

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4.3. Polarization model for the poles of Jupiter

The flux weighted fractional polarization Qr/I for the wavelength 6000 Adoes not depend on uncertainties in the absolute flux calibration. The uncertain-ties are mainly due to systematic errors like instrument calibration or inaccuraciesin the slit positioning. The derived Qr/I profile (Fig. 4.2) should be accurate to∆(Qr/I) ≈ ±0.1 % − 0.2 %.

4.3 Polarization model for the poles of Jupiter

Our polarimetry of Jupiter has revealed a surprisingly high limb polarization.We explore whether these observational results are compatible with simple hazescattering models. For a detailed characterization of the scattering particles inJupiter extensive modeling would be required. Unfortunately up to now thereare only few limb polarization models available – the model grid presented inChapter 2 and polarization model of Uranus in Chapter 3, as well as a few pre-vious, mostly analytic results as summarized in Schmid et al. (2006). It is not yetwell explored how the limb polarization depends on the scattering phase matrixof the haze particles, the stratification of the atmosphere, and the optical depthof absorbers. Therefore, our model fitting remains ambiguous without extensivemodel simulations which are beyond the scope of this work.

The goal of these model calculations is to explain the observed peak limb po-larization of more than 9.5 % in the V-band at the South pole. Considering thatthe seeing degrades this polarization, the maximum limb polarization must bewell above 10 %. Rayleigh scattering models yield up to 10 % limb polarization(e.g. Schmid et al. 2006), but only for highly absorbing atmosphere models, whichare not appropriate for the reflected intensity seen on the poles of Jupiter.

Detailed scattering models for the polarization of Jupiter were presented bySmith & Tomasko (1984) and Braak et al. (2002), but only for mid and low lati-tudes. No detailed scattering models exist for the polarization at the poles. Smith& Tomasko (1984) made a simple fit to the polarization measured in the red withthe Pioneer spacecraft near quadrature phase for a Rayleigh scattering layer withsingle-scattering albedo of ω = 0.983, optical thickness τ = 0.5, and a surfacealbedo of AS = 0.67. However, this Rayleigh scattering model yields a maximumlimb polarization of only 7.3 %, or ≈ 6.5 % if the degradation by the seeing is con-sidered, whereas our measurements show a much higher fractional polarization.

We calculate the polarization along the central meridian of Jupiter, consider-ing three zones: the N and S polar zones where the polarization is positive (S+and N+), and a central zone where the polarization is negative. For the S+ andN+ zones we use the Monte Carlo multiple scattering code described in Chapter2. The chosen atmosphere structure is very similar to the haze model presentedby Smith & Tomasko (1984) for low latitudes. They determined haze and gasproperties for the South Tropical Zone and South Equatorial Band with an atmo-sphere consisting of a top gas layer G1, a scattering haze layer H, a lower gaslayer G2, and an optically thick surface layer S at the bottom. We fit the polar re-gions S+ and N+ with this model. For the wavelength 6000 A the optical depths

107


for the two Rayleigh scattering gas layers are τG1 = 0.011 and τG2 = 0.018, andthe single scattering albedo is ωG = 0.976, calculated for a methane abundance of0.18 %. Thus, the haze layer is geometrically thin and located at a pressure levelof 290 mbar, while the opaque surface layer is at 760 mbar. The continuum polar-ization at 6000 A does not depend much on the exact height of the haze and cloudsurface layers. Having the haze layer at ≈ 10 mbar, as suggested by near-IR ob-servations by Banfield et al. (1998), and the cloud layer at ≈ 1.3 bar, as measuredby the Galileo probe at a different location (Ragent et al. 1998), would not notablychange the intensity and polarization. The situation is different for the polariza-tion in the CH4 bands which depends in various ways on the pressure level of thedifferent layers. The full spectropolarimetry should therefore strongly constrainthe vertical distribution of the haze and cloud particles.

The free parameters of our model are: the cloud layer albedo AS assumed to bea gray Lambert surface, the optical thickness of the haze layer τh, and the haze pa-rameters, which are single scattering albedo ωh, single term Henyey-Greensteinasymmetry parameter gh, and maximum polarization for right angle scatteringph (see Braak et al. 2002). The polar model is independent of latitude, but theincidence and viewing angles produce a latitude dependence in the reflected po-larization and intensity.

For the reflected intensity at lower latitudes the same model is used, but forthe fractional polarization an “artificial” constant value of Qr/I = −0.7 % isadopted. The negative polarization is introduced to fit the transition between thepositively polarized polar zones and the negative central zone. The borders rS

and rN are free parameters which are determined in the data fitting process.In order to describe the smearing of the signal due to atmospheric seeing and

instrumental light scattering, the “discrete” three zone model is convolved witha Moffat (Moffat 1969) point spread function (PSF). In the formalism of Trujilloet al. (2001) this PSF includes a β-parameter which describes the scattering wing.A small β implies strong wings, a Gaussian is obtained for β → ∞, while at-mospheric turbulence theory predicts β = 4.76. For our observations we deriveβ = 1 from the residual light outside the nominal limb, indicating significantscattering in the instrument.

In Fig. 4.2 the observed intensity and polarization profiles for the centralmeridian for 6000 A are compared with the model fit. At the poles for |r| > 0.8the match is satisfactory except for the fractional polarization outside the limb|r| > 1, where the statistical errors are large because I → 0. At low latitudes thereare some discrepancies because we did not try to fit the band structure for theintensity, and instead adopted a constant value for the fractional polarization.

A good fit for the limb polarization at λ = 6000 A for both poles is obtainedfor strongly polarizing ph = 1 haze with low absorption ωh = 0.99 and an asym-metry parameter gh = 0.6. Such scattering parameters are typical for (randomlyoriented) aggregate particles as proposed to be present in Jupiter and Titan byWest (1991). Small Mie scatterers with diameters much smaller than the wave-length of the scattered light have similar scattering parameters. But the scatteringcross section for small spheres is much higher in the blue and one would expect a

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4.3. Polarization model for the poles of Jupiter

Figure 4.2: Observations of Jupiter (dotted line) from November 2003 compared to model

calculations (dashed). Intensity I, fractional polarization Qr/I, and polarization flux Qr

profiles through the central meridian (N-S) are given for the continuum at 6000 A.

fractional limb polarization which decreases rapidly with wavelength. The scat-tering cross section and therefore the scattering optical depths of the haze layerare expected to decrease only slowly with wavelength for aggregate particles andthey can therefore explain the rather gentle decrease in the Q/I and Q-spectra to-wards longer wavelengths observed for the poles of Jupiter (see Fig. 4.3), whichare not discussed here in detail.

The measured limb polarization at the poles for r = ±0.96 is about 9.8 % inthe South and 8.4 % in the North. The North-South differences can be explainedby different optical depths τh(N+) = 0.72, τh(S+) = 1.1 for the haze layers.

Our model fit includes the smearing due to seeing and the modeling indicatesthat the intrinsic limb polarization reaches maxima of about 11.5 % in the Southand 10.0 % in the North. Such a high limb polarization is probably only possiblewith particles having a scattering phase function with reduced backscattering,with an asymmetry parameter comparable to g ≈ 0.6 as used in our model. Ourmodel fit is not unique. However, many parameters are already well constrained.From the parameter space explored by us it seems very likely that the asymmetry

109


parameter lies in the range 0.5 < gh < 0.8 and the single scattering polarizationis ph > 0.8.

The locations of the transitions rS and rN between the highly polarized polesand the negatively polarized central zone are at r = ±0.82 ± 0.01, identical forboth hemispheres. This corresponds to a planetographic latitude of ±58 ± 3. Inthe South the transition between the S+ and the central region is compatible withan unresolved discontinuity. In the North the transition is more gradual, with aweak wing of positive polarization towards lower latitude. This is in qualitativeagreement with Fig. 14 in Smith & Tomasko (1984), who measured the latitudinalpolarization dependence with Pioneer for large phase angles (82 and 98). Theirred filter data show a steep increase in the polarization in the South from about10 % at a latitude of −55 to about 33 % at −65 and a more gradual increase from13 % at +55 to about 25 % at +65 in the North.

The surface albedo AS = 0.75 is also quite well constrained, since a high valueAS > 0.9 would significantly reduce the resulting fractional polarization and alow value AS < 0.6 would underpredict the reflectivity at the poles. It also seemsquite safe to explain the North-South asymmetry in the polar polarization with adifference in the optical thickness of the polarizing haze layer.

Finally, it is interesting to note that the haze model for low latitudes withωh = 0.95, ph = 0.9 and gh = 0.75 from Smith & Tomasko (1984), which was alsoadopted by Braak et al. (2002), cannot fit the high limb polarization at the poles.This points to distinct differences between the haze particles at the poles and atlower latitudes.

4.4 Discussion and conclusions

We present the first polar haze model for Jupiter based on polarimetric data, fo-cusing on the wavelength region around 6000 A, where the limb polarization isexceptionally strong. The new data are of unprecedented quality and well cal-ibrated for seeing effects to allow for detailed comparisons with model calcula-tions of the scattering layers in Jupiter.

Stratospheric haze is responsible for the strong polar polarization, and thesharp transition near 60 is pointing to a well defined border in the strato-spheric circulation. We measure a resolution-corrected peak polarization of about+11.5 %. All previous studies reported lower values (p ≈ 6 − 8 %) for the max-imum polarization at the poles of Jupiter. This difference is easily explained bythe better spatial resolution (seeing ≈ 1′′) of our data and the appropriate seeingcorrection.

Comparison of the polarization flux of the entire positively polarized polarhoods with previous and future observations for long term studies would be in-teresting for investigations of the haze production, destruction and transport inthe polar stratosphere of Jupiter. Such long term polarization changes were re-ported e.g. by Starodubtseva et al. (2002) and Starodubtseva (2009).

Since we found no limb polarization models with a limb polarization higher

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4.4. Discussion and conclusions

than 10 % in the literature, we carried out exploratory model calculations. Theseindicate that a polar haze layer consisting of forward scattering and highly po-larizing aggregate particles as proposed by West (1991) is compatible with ourobservations. Similar haze particles, but with much lower optical depth, wereproposed to exist above the negatively polarizing clouds in Jupiter’s equatorialregion (Smith & Tomasko 1984; Braak et al. 2002)

This work shows that modern polarimetric measurements from the groundcan provide accurate quantitative results which strongly constrain the scatteringproperties of the atmospheres. More observations, e.g. for other wavelengths,and a lot of limb polarization modeling still needs to be done. Further modelsshould investigate the dependence of the limb polarization signal on the scatter-ing phase matrix of different populations of haze particles. In addition one needsto take into account the spectropolarimetric structure in the methane bands in or-der to derive the vertical stratification of the scattering layers. Figure 4.3 showsthe measured spectropolarimetric signal of Jupiter that we did not discuss in de-tail in this chapter. Like on Uranus (cf. Chapter 3), the polarization is enhancedin the methane bands. The reason for this enhancement however is different. OnUranus, the enhancement is primarily because of the methane depletion at lowpressures, resulting in stronger absorption below the main scattering layer andthus an increase in limb polarization. The very thin, highly polarizing strato-spheric haze layer only has a small effect in the deepest methane bands. OnJupiter, the methane mixing ratio is essentially constant with pressure. The en-hanced polarization in methane bands then results from the thick, highly polar-izing stratospheric haze layer which is not well mixed with the gas. Again, themain polarizing scattering layer is then located above the main absorber.

Scattering layers, reflecting the solar light, are an intriguing aspect of solarsystem planets. They affect the radiative transfer in these objects. For the in-vestigation of the reflected light from extra-solar planets a comprehensive under-standing of the physics of the high altitude haze layers is very important. For thisreason it is essential to carry out detailed investigations of the reflecting layersin solar system planets. Investigations based on modern polarimetric observa-tions, as presented here for Jupiter, are therefore very valuable for progress inthis direction.

Acknowledgments. We are indebted to the ESO La Silla support team at the 3.6m

telescope who were most helpful with our very special EFOSC2 instrument setup. We

are particularly grateful to Oliver Hainaut. We thank Harry Nussbaumer for carefully

reading the Icarus manuscript. We also acknowledge the many useful comments from

the referees of the Icarus paper. This work was supported by a grant from the Swiss

National Science Foundation.

111


Figure 4.3: Spectropolarimetry of Jupiter for southern (solid) and northern (dotted) polar

(positively polarized, S+ and N+) and equatorial (negatively polarized, S− and N−)

regions. The intensity I(λ) for the polar regions S+ and N+ are multiplied by a factor of

3 with respect to S− and N− for visibility reasons. The middle panel gives Qr/I(λ) and

the bottom panel the polarization flux Qr.

112

BIBLIOGRAPHY

Bibliography

Banfield, D., Conrath, B. J., Gierasch, P. J., Nicholson, P. D., & Matthews, K. 1998, Icarus,134, 11


Carlson, B. E. & Lutz, B. L. 1989, NASA Special Publication, 494, 289

Chanover, N. J., Kuehn, D. M., Banfield, D., et al. 1996, Icarus, 121, 351

Dollfus, A. 1957, Supplements aux Annales d’Astrophysique, 4, 3

Gehrels, T., Herman, B. M., & Owen, T. 1969, AJ, 74, 190

Gisler, D. 2005, PhD thesis, Eidgenoessische Technische Hochschule Zurich, Switzerland

Hall, J. S. & Riley, L. A. 1968, Lowell Observatory Bulletin, 7, 83

Hall, J. S. & Riley, L. A. 1976, Icarus, 29, 231



Leroy, J.-L. 2000, Polarization of light and astronomical observation, ed. Leroy, J.-L.

Lyot, B. 1929, Ann. Observ. Meudon, 8

Moffat, A. F. J. 1969, A&A, 3, 455

Moreno, F., Molina, A., & Lara, L. M. 1991, J. Geophys. Res., 96, 14119

Morozhenko, A. V. 1973, Soviet Ast., 17, 105

Ragent, B., Colburn, D. S., Rages, K. A., et al. 1998, J. Geophys. Res., 103, 22891

Schmid, H. M., Joos, F., & Tschan, D. 2006, A&A, 452, 657


Starodubtseva, O. M. 2009, Solar System Research, 43, 277

Starodubtseva, O. M., Akimov, L. A., & Korokhin, V. V. 2002, Icarus, 157, 419

Trujillo, I., Aguerri, J. A. L., Cepa, J., & Gutierrez, C. M. 2001, MNRAS, 321, 269

U. S. Naval Observatory & Royal Greenwich Observatory. 2001, The Astronomical Almanacfor the year 2003, ed. U. S. Naval Observatory & Royal Greenwich Observatory

West, R. A. 1991, Appl. Opt., 30, 5316

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Chapter 5

Outlook

5.1 Prospects with SPHERE/ZIMPOL and beyond

Polarimetric detection of extrasolar planets will only really begin onceSPHERE/ZIMPOL has gone on sky, currently expected in 2012. Even then, adetection will be very difficult. The largest limitation is the small target sample.Only for the very nearest, brightest stars the necessary contrast for a detectioncan be achieved. In the photon-noise limit, the noise scales with the inversesquare root of the photon flux. The photon flux goes as d−2, where d is the targetdistance in pc. The noise is therefore proportional to the stellar distance. Theplanet polarization signal is in principle independent from stellar distance forfixed physical separation from the star. However, the angular separation of theplanet scales inversely proportional with the stellar distance. At the same angularseparation, even if the same noise limit is reached, the signal of the planet is by afactor d−2 smaller. The S/N ratio of a planet at fixed angular separation thereforescales either with d−2 in the speckle-noise limit, or even d−3 in the photon-noiselimit (see also Thalmann 2008).

This strong distance dependence leaves only a handful of feasible targets.α Centauri A and B, with their distance of only 1.3 pc, are by far the best tar-gets. They are followed by Sirius, Altair, Procyon, ε Eridani, τ Ceti and ε Indi. Forα Cen A and B, there are detection limits down to M sin i ≈ 1 MJ for a < 2 AUfrom radial velocity measurements (Endl et al. 2004), and much deeper limitswith more recent measurements (yet unpublished, M. Endl, X. Dumusque, pri-vate communication). These exclude objects down to a few Earth masses in theseparation range best probed by SPHERE/ZIMPOL. A Jupiter around τ Ceti andε Indi should likely also have been detected already. Sirius, Altair and Procyonare earlier-type stars with broad lines, and ε Eri a younger, more active star, there-fore high-precision RV measurements are more difficult for these targets. With theoccurrence rate of planets with masses between 0.3 and 13 MJ at separations of0.1− 1 AU estimated from radial velocity data (Cumming et al. 2008; Heinze et al.2010), to ≈ 5%, the target sample is however not large enough to expect a detec-tion statistically. It should be noted that the statistics is determined mainly fromFGK-stars, rather than earlier-type stars. A further difficulty is the fact that mostof the target stars are actually binaries, where planet occurence, while possisble,is not statistically determined but expected to be even lower than for single stars.

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Chapter 5. Outlook

Assuming that a planet of appropriate size is present at a suitable separationaround a bright nearby star, it is still not necessarily detectable. Two other param-eters are crucial for a detection: the phase angle and the atmospheric properties(see Eq. 1.37). Only for phase angles between ≈ 30 − 110 (ideal: 50 − 90) thepolarization signal is high enough for typical polarizing atmospheres, and theplanet is separated enough from the star (see Sect. 1.3.2, 2.3.1, 3.5). Because inany orbit with inclination i & 20 detection will be impossible for parts of theorbit, a better chance of detection is obtained by making multiple observationsspaced by fractions of an orbital interval. The observing strategy must be set foreach target individually, depending on the typical orbital period of a planet in theSPHERE/ZIMPOL detection range.

The atmospheric properties of a planet are even harder to estimate than theplanet phase angle. SPHERE/ZIMPOL will pioneer direct imaging of a com-pletely new type of planet: Jupiters at separations of tenths of an AU. No com-parable planets exist in the solar system, and for extra solar planets only atmo-spheres of hot Jupiters and young planets at far separations have been measured.A few theoretical models that calculate albedo spectra or phase curves of Jupiterswith temperatures of a few 100 K have been made (Sudarsky et al. 2000; 2003;2005; Cahoy et al. 2010). These suggest that such planets are either cloudless plan-ets that are dark because of strong absorption by sodium and potassium, or thatthey are bright because of condensed water clouds, which then are not expectedto show a strong polarization. It must however be noted that these atmospheremodels are quite simple and neglect any photochemistry, which may be responsi-ble for production of bright, polarizing haze as seen on the poles of Jupiter and onTitan. Additionally, cloud condensation is generally more complicated than as-sumed. On Uranus no significant clouds have been found where expected at thecondensation level of methane. This could mean that atmospheres with a thickRayleigh scattering layer similar to that of Uranus and Neptune cannot be ex-cluded. If SPHERE/ZIMPOL should detect a polarization signature of a planet,these models must be refined to include more realistic physical and chemical pro-cesses.

The only star in the SPHERE/ZIMPOL target list for which a planet has beenpublished is ε Eridani. Radial velocity and astrometry measurements suggesteda highly eccentric, Jupiter-mass planet with a periastron of ≈ 0.3′′(Hatzes et al.2000; Benedict et al. 2006). At that point it would lie just at the edge of the AOcontrol radius and could be detectable if the atmospheric properties are favor-able. From the orbit proposed by astrometry, we have constructed a curve for theexpected planet contrast for a 1.2 RJ planet with an optimistic Rayleigh scatteringatmosphere as a function of orbital separation (Fig. 5.1). Because of the high ec-centricity, the planet will only spend a very short time of its 7-year period at closeseparations where it is detectable. Good timing of the observations is thereforecrucial. The 5σ-detection limit for a planet around ε Eri at 0.3′′separation was es-timated by Thalmann (2008) to ≈ 3 · 10−8 for a 4 hour observation. The photonnoise limit is reached with angular differential imaging for this source. There-fore, a longer integration may result in a sensitivity low enough for the detection

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5.1. Prospects with SPHERE/ZIMPOL and beyond

Figure 5.1: Polarization contrast versus angular separation of the proposed planet

around ε Eridani from an orbit provided by astrometry, assuming a radius of 1.2 RJ .

of the planet if its polarization properties are favorable. ε Eri is therefore a promis-ing candidate for the first polarimetric detection of an atmosphere. However, itshould be noted that recently, doubts about the reality of the planet have risenfrom new radial velocity data (A. Hatzes, private communication). Because theastrometric measurements require the RV measurements to constrain parameters,they cannot be viewed as an independent confirmation. This example shows theimportance of confirmation of planets through completely independent methods.Thermal high-contrast imaging excludes planets at a mass M > 3 MJ (Janson et al.2008), with the proposed planet having M = 1.55 MJ .

SPHERE/ZIMPOL can largely be seen as a test-case for the potential of planetdetection through polarimetry. If all goes well, and deep contrast limits arereached, even if there are no detections, it can be considered a success that opensthe door for a more ambitious instrument. A follow-up instrument of SPHEREis already planned for the European Extremely Large Telescope (E-ELT) with aplanned mirror diameter of 42 m. The high contrast imager EPICS (Exo-PlanetsImaging Camera and Spectrograph, Kasper et al. 2010) could also incorporatea polarimetric arm called EPOL (Keller et al. 2010). With the much larger pho-ton collecting area and the smaller inner working angle than SPHERE, a muchlarger number of targets will be available. Several already known radial-velocityplanets will be within reach, simplifying a detection because the optimal phase isalready known. It can be expected that several Jupiter and Neptune-sized plan-ets could be detected. For the closest, brightest stars, even Super-Earths could befound and characterized, providing their polarization is high enough. Unfortu-nately, Earth-like planets would still be out of reach even for α Cen A and B. The

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E-ELT could go online in ≈ 2020, but EPICS would most likely not be a first-lightinstrument. Therefore, the first large survey for polarization from planets willhave to wait for another few years. A future upgrade to SPHERE/ZIMPOL withimproved AO and new coronagraphic methods (e.g. an APP, see Sect. 7.1.1) couldimprove the chances for a detection already before the EPICS is ready. Space-based high-contrast imagers with polarimeters are not currently foreseen.

Differential polarimetry is a unique but very challenging method for directimaging of extrasolar planets. Clearly, observations in the near-infrared will bemore successful at imaging planets. These observations are however limited torelatively young planets outside of the vicinity of the sun. A direct image of aplanet in the nearest solar neighborhood would be an important step in our questtowards the ultimate goal of characterizing an Earth analog. Therefore the poten-tial of differential polarimetry should be fully explored, and SPHERE/ZIMPOLcan be seen as the first step in this direction.

5.2 Hot Jupiter polarimetry

Hot Jupiters are the planets with the smallest separations from the star, orbitingwith a period of less than a few days at a distance of a few percent of an AU. Be-cause of the inverse square dependence of the polarization signal of the planet onseparation from the star, these planets can in principle show a high polarizationsignal, about 2 orders of magnitude higher than the SPHERE targets. However,they cannot be observationally separated from the star, and methods like coron-agraphy and angular differential imaging cannot be applied to improve the con-trast. Polarimetry can disentangle the scattered (partially polarized) light of theplanet from the (unpolarized) direct light of the star as well as the (unpolarized)thermal light of the planet. Other methods that target the scattered light signalare high resolution spectroscopy (Charbonneau et al. 1999) and transit measure-ments at short wavelengths (Rowe et al. 2008, e.g.). The former method has amuch lower efficiency than polarimetry, the latter works only for transiting plan-ets and requires the stability of a space mission. Also, it cannot distinguish thethermal light from very hot planets from scattered light.

For these observations, a polarimetric sensitivity of ≈ 10−6 to 10−5 must beachieved. This requires very high stability and calibration of the instrument andis generally only possible with a dedicated, optimized instrument. Attempts todetect polarization from hot Jupiters have already been made. The first was Lu-cas et al. (2009) who searched for the known planets around τ Boo and 55 Cncwith the PLANETPOL instrument (Hough et al. 2006) at wavelengths between≈ 500 − 900 µm. They could show that τ Boo b cannot be covered by a bright,Rayleigh scattering atmosphere. The idea that hot Jupiters are dark was sup-ported by an albedo measurement by the MOST satellite for HD 209458 b, whichprovided a 3σ upper limit of Ag < 0.17 for λ between ≈ 400− 700 nm (Rowe et al.2008). Model calculations for HD 209458b are in agreement with the low upperlimits for the albedo. Burrows et al. (2008) and Fortney et al. (2008) obtain a geo-

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5.2. Hot Jupiter polarimetry

Figure 5.2: Theoretical flux spectrum of HD 209458 compared to Uranus, Jupiter, a

Rayleigh sphere and a Lambert sphere (Fortney 2009).

metric albedo Ag < 0.1 beyond 500 nm. However, they predict a strong increasetowards the blue and near-UV (see Fig. 5.2) because of Rayleigh scattering by H2,H and He. Berdyugina et al. (2008) and Berdyugina et al. (2011) reported a polari-metric detection of HD 189733 b at the 2 · 10−4 level in the UV and blue. This un-physically high polarization (higher than for a semi-infinite Rayleigh scatteringatmosphere) was contradicted by higher-precision measurements by Wiktorow-icz (2009). Their POLISH instrument (Wiktorowicz & Matthews 2008) is the onlyinstrument with demonstrated capabilities to achieve a polarimetric contrast of. 10−6 at blue wavelengths (Wiktorowicz & Graham 2011) after recent improve-ments. HD 189733 b remains an interesting target for scattered light searchesbecause HST transmission spectroscopy showed evidence of a Rayleigh scatter-ing haze layer (Sing et al. 2011). Once a detection is made, it would be inter-esting to compare with polarization models for hot Jupiter atmospheres. A fewgeneric examples with Mie scattering condensates were calculated by Seager et al.(2000) before atmospheric properties were actually measured. Polarization couldbe added to current hot Jupiter intensity models. However, these models do notyet properly treat photochemical haze layers. A first interpretation could alreadybe made with simple Rayleigh scattering models as presented in Chapter 2.

5.2.1 Polarimetric search for WASP-18 b

We have developed our own observing program to polarimetrically search forpolarized light from a hot Jupiter, which was granted time by ESO and observa-tions were successfully carried out in October 2010. Our goal was to achieve apolarimetric contrast of . 10−5 on the VLT/FORS2 instrument to detect the po-

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Chapter 5. Outlook

larization of the transiting hot Jupiter WASP-18 b, or set a meaningful upper limiton it’s albedo.

FORS2 is an imaging and spectropolarimeter at the Cassegrain focus with anabsolute instrumental polarization of ≈ 0.1% and a detector that is particularlysensitive in the UV/blue. The key issue is the polarimetric calibration of the datato a level of ≈ 0.001%. With FORS2 this can only be achieved with a “calibrationtrick”. Models and observations indicate that we can assume no (significant) po-larization signal for hot Jupiters at long wavelengths (say & 500 nm). Thus we usethe long wavelengths as zero polarization reference and search for a polarizationsignal in the blue/near- UV relative to this reference. In this way the instrumentand telescope polarization can be monitored in the red and accurately subtractedfor all wavelengths. This method requires simultaneous measurement of the fullwavelength range between ≈ 350− 600 nm. We therefore use spectropolarimetry,which has the additional advantage of smaller read-out overheads than imagingpolarimetry. The observations are strongly limited by the read-out time for thebright (V≈ 10) sources, but spreading the photons over a large detector area in-creases the efficiency. In addition to the wavelength differential, we also take ob-servations during several phases, where the expected planet signal is strong andweak and differentiate the two. This gets rid of potential interstellar polarization.

This calibration technique is new, and to check its feasibility we analyzed re-peated standard star spectropolarimetry from a previous FORS1 program. Theresults show that the (instrumental) polarization for a zero polarization starchanges continuously during the night (with parallactic angle) by up to ∼0.08%.With the 600 nm region as zero polarization reference, the polarization in theUV/blue is constant within the photon noise limit of ∼ 0.02 %. A higher precisioncan only be reached by accumulating more photons. For hot Jupiter polarimetrythe photon noise should be improved by at least another factor of 10. It cannot betested in advance whether calibration to the photon-noise limit is still possible atthat level.

The transiting planet WASP-18b (Hellier et al. 2009) is large, Rp = 1.17 RJ andin a very tight orbit with a = 0.02 AU, having a period of only 0.94 days. There-fore, its ratio R2/a2 is one of the highest of the currently known hot jupiters. Themaximum possible polarization contrast (for a semi-infinite Rayleigh scatteringatmosphere) is 6 · 10−5. Also, the system is bright enough (V= 9.3) to reach a pho-ton noise limit of < 10−5 thanks to the photon collecting power of the VLT and thehigh efficiency of the spectropolarimetric mode of FORS2, using all photons be-tween 350 and 620 nm and applying a wide spectral binning. Targeting brightersystems would not reduce the required telescope time with FORS because of de-tector read-out overheads. With twenty hours of observing time, covering highpolarization phases and zero-point phases (secondary eclipse), as well as someintermediate phases, at least twice each, the necessary amount of photons can bereached for a photon noise limit of 6 · 10−6. Comparison between equal phasestaken at different times can be used as a calibration check.

The observations were carried out in good conditions in October 2010. Apreliminary analysis shows that a polarimetric precision of 10−4 can easily be

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5.2. Hot Jupiter polarimetry

reached with simple data reduction, as suggested by the previous calibration test.To gain an additional order of magnitude of precision, a sophisticated reduction,e.g. with more precise spectral extraction and accounting for irregularities in in-dividual frames, is required and currently ongoing.

Successful polarization observations of hot Jupiters would have great poten-tial for a very complementary atmospheric characterization than provided by thetransit technique. The polarization yields strong constraints on the albedo, whilea comparison of different spectral bins allows an estimate on the abundance ofRayleigh scatterers, gray scatterers (e.g. haze or dust particles), and of absorbingparticles in the high altitude layers of the atmosphere. Most importantly, even theatmospheres of non-transiting planets like τ Boo could be investigated, and incli-nation and orbital parameters be determined. Systems seen from polar directionscould be compared with those seen in the orbital plane. The very high accuracyof polarimeters required to make such detailed measurements of hot Jupiters isstill a challenge to overcome, in particular if the planets are predominantly darkin scattered light.

121

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Part II

Angular differential imaging of a debris disk

123

Chapter 6

Dissecting the ’Moth’:Discovery of an off-centered ring in theHD 61005 debris disk∗

E. Buenzli1, C. Thalmann2, A. Vigan3, A. Boccaletti4, G. Chauvin5, J.-C. Augereau5, M.R. Meyer1, F. Menard5, S. Desidera6, S. Messina7, T. Hen-ning2, J. Carson8,,2, G. Montagnier 9, J.-L. Beuzit5, M. Bonavita10, A. Eggen-berger5, A.M. Lagrange5, D. Mesa6, D. Mouillet5, and S.P. Quanz1,

Abstract

The debris disk known as “The Moth” is named after its unusually asymmet-ric surface brightness distribution. It is located around the ∼90 Myr old G8Vstar HD 61005 at 35 pc and has previously been imaged by the HST at 1.1 and0.6 µm. Polarimetric observations suggested that the circumstellar material con-sists of two distinct components, a nearly edge-on disk or ring, and a swept-back feature, the result of interaction with the interstellar medium. We resolveboth components at unprecedented resolution with VLT/NACO H-band imag-ing. Using optimized angular differential imaging techniques to remove the lightof the star, we reveal the disk component as a distinct narrow ring at inclinationi = 84.3 ± 1.0. We determine a semi-major axis of a = 61.25 ± 0.85 AU and aneccentricity of e = 0.045± 0.015, assuming that periastron is located along the ap-parent disk major axis. Therefore, the ring center is offset from the star by at least

∗ A shorter version of this chapter is published in Astronomy & Astrophysics 524, L1 (2010)1 Institute of Astronomy, ETH Zurich, CH-8093 Zurich, Switzerland2 Max Planck Institute for Astronomy, Heidelberg, Germany3 Laboratoire d’Astrophysique de Marseille, UMR 6110, CNRS, Universite de Provence, 13388

Marseille, France4 LESIA, Observatoire de Paris-Meudon, 92195 Meudon, France5 Laboratoire d’Astrophysique de Grenoble, UMR 5571, CNRS, Universite Joseph Fourier,

38041 Grenoble, France6 INAF - Osservatorio Astronomico di Padova, Padova, Italy7 INAF - Osservatorio Astrofisico di Catania, Italy8 College of Charleston, Department of Physics & Astronomy, Charleston, South Carolina, USA9 European Southern Observatory: Casilla 19001, Santiago 19, Chile

10 University of Toronto, Toronto, Canada

125

Chapter 6. An off-centered ring in the HD 61005 debris disk

2.75 ± 0.85 AU. The offset, together with a relatively steep inner rim, could indi-cate a planetary companion that perturbs the remnant planetesimal belt. Fromour imaging data we set upper mass limits for companions that exclude any ob-ject above the deuterium-burning limit for separations down to 0.35”. The ringshows a strong brightness asymmetry along both the major and minor axis. Abrighter front side could indicate forward-scattering grains, while the brightnessdifference between the NE and SW components can be only partly explained bythe ring center offset, suggesting additional density enhancements on one side ofthe ring. The swept-back component appears as two streamers originating nearthe NE and SW edges of the debris ring.

6.1 Introduction

Dust in planetary systems is most likely produced by collisions of planetesimalsthat are frequently arranged in a ring-like structure (Wyatt 2008). These debrisdisks are therefore thought to be brighter analogs of our solar system’s Kuiperbelt or asteroid belt. Previous studies found no correlation between the presenceof known massive planets and infrared dust emission from debris disks (Moro-Martın et al. 2007; Bryden et al. 2009; Kospal et al. 2009), but several systemsare known to host both (e.g. HR 8799, HD 69830). Scattered light imaging ofdebris disks has revealed numerous structures thought to be shaped by planets.Warps in the β Pic debris disks (Mouillet et al. 1997; Augereau et al. 2001) arecaused by a directly confirmed 9± 3 MJ planet (Lagrange et al. 2009; 2010; Quanzet al. 2010). Fomalhaut hosts a debris ring with a sharp inner edge and an offsetbetween ring center and star (Kalas et al. 2005), for which dynamical models sug-gest the presence of a planet (Quillen 2006). A planetary candidate was indeedimaged (Kalas et al. 2008). Other larger scale asymmetric structures, e.g. aroundHD 32297 (Kalas 2005), are thought to result from interaction with the ambientinterstellar medium (ISM), such as movement through a dense interstellar cloud(Debes et al. 2009). Alternatively, they might be perturbed by a nearby star (e.g.HD 15115, Kalas et al. 2007).

The source HD 61005 was first discovered to host a debris disk by theSpitzer/FEPS program (formation and evolution of planetary systems, Meyeret al. 2006). The star properties are summarized in Tab. 6.1. It has the largest24 µm infrared excess with regard to the photosphere (∼110%, Meyer et al. 2008)of any star observed in FEPS . The star’s age was estimated to be 90 ± 40 Myr(Hines et al. 2007), although a more recent analysis suggests that HD 61005 couldbe a member of the 40 Myr old Argus association (Desidera et al. 2011). This sig-nificantly younger age would better match the very strong infrared excess. Thedisk was resolved with HST/NICMOS coronagraphic imaging at 1.1 µm (Hineset al. 2007, Fig. 6.1 a) that revealed asymmetric circumstellar material with twowing-shaped edges, that is the reason for in the nickname “the Moth”. The diskwas also resolved at 0.6 µm with HST/ACS imaging and polarimetry (Manesset al. 2009, Fig. 6.1 b). The polarization suggests two distinct components, a

126

6.2. Observations

Figure 6.1: (a): HD 61005 imaged at 1.1 µm with HST/NICMOS (Hines et al. 2007).

(b): HD 61005 imaged at 0.6 µm with HST/ACS with polarization vectors overlaid

(Maness et al. 2009).

nearly edge-on disk or ring and a swept-back component that interacts with theISM. Both papers focused on the properties and origin of the swept-back com-ponent. In this work, we present high-contrast ground-based imaging with un-precedented angular resolution. We discover and characterize a distinct asym-metric ring in the inner disk component and discuss the probability of a planetarycompanion.

6.2 Observations

We observed HD 61005 on February 17, 2010 with the NACO instrument (Rous-set et al. 2003; Lenzen et al. 2003) at the VLT. The observations were obtained inthe framework of the NaCo Large Program Collaboration for Giant Planet Imag-ing (ESO program 184.C0567). The images were taken in the H-band (1.65 µm)in pupil-tracking mode (Kasper et al. 2009) to allow for angular differential imag-ing (ADI, Marois et al. 2006). The field of view was 14′′ × 14′′ and the plate scale13.25 mas/pixel. We performed the disk observations without a coronagraph,and used the cube-mode of NACO to take 12 data cubes. Each cube consisted of117 saturated exposures of 1.7927 s, yielding a total integration time of 41.95 min.The saturation radius was ∼0.15”. A total of 112 of field rotation was capturedwhile the pupil remained fixed. Additionally, the star drifted by 2.4′′on the de-tector during the observations, but the drift during single exposures is negligibleand can be compensated by proper re-centering of each frame. Before and afterthe saturated observations we took unsaturated images with a neutral densityfilter to measure the photometry for the central star. The adaptive optics sys-tem provided a point spread function (PSF) with a full-width at half-maximum(FWHM) of 60 mas with ∼0.8” natural seeing in H-band (22% Strehl ratio). Asingle exposure raw frame is shown in Fig. 6.2.

127


Table 6.1: Stellar properties of HD 61005, from SIMBAD or Desidera et al. (2011).

Parameter Value Error

RA 07 h 35 m 47.46 sDEC -32 12’ 13.044”Distance 35.3 pc 1.1Spectral type G8VV mag 8.22 0.01H mag 6.578L/L⊙ 0.583 0.046R/R⊙ 0.84 0.038Te f f 5500 K 50log g 4.2 0.2Fe/H 0.01 0.v sin i 8.2 km/s 0.5Rotation period 5.04 d 0.04istar 77 +13,−15age 40 Myr −10,+80

6.3 Data reduction and PSF subtraction

The data were flat-field corrected with twilight flats and bad-pixel corrected witha 10σ filter, replacing the bad pixels by the median of surrounding pixels. Allframes were centered on the star by manually determining the center for theframe in the middle of the full time-series from the diffraction spikes and align-ing the others through cross-correlation. We removed 3 bad-quality frames wherethe AO loop was open and averaged the remaining images in groups of three fora total of 467 frames. This binning significantly speeds up the subsequent PSFsubtraction, but does not significantly affect its quality because very little move-ment has taken place during this time because of the very short exposure timesof single frames. A more limiting factor to the reduction quality is the accuracyof the centering, which we estimate to ≈ 0.5 pixels.

We then used LOCI (locally optimized combination of images, Lafreniere et al.2007) and customized ADI to subtract the stellar PSF to search for point sourcesand extended non-circular structures.

In our customized ADI reduction, each image is divided into annuli of 2FWHM width. For each frame and each annulus, an average of the two frameswhere the field object is rotated by 2 FWHM in either direction is subtracted to re-move the stellar halo. Frames too close to the beginning and end of the time-seriesto have two corresponding subtraction frames are excluded from the process. Fi-nally, all images are derotated and median combined.

In LOCI, each annulus is further divided into segments, and for each seg-

128

6.4. Results

Figure 6.2: Raw NaCo H-band observation of HD 61005. The star center is saturated and

the diffraction spikes are visible. The images is displayed in log scale. The image size is

14x14′′.

ment an optimized PSF is constructed from a linear combination of sufficientlyrotated frames. A minimum rotation of 0.75 FWHM is optimal for point sourcedetection and has led to several detections around other targets (Marois et al.2008; Thalmann et al. 2009; Lafreniere et al. 2010). To reveal the extended neb-ulosity around LkCa 15, Thalmann et al. (2010) used a much larger minimumseparation of 3 FWHM. For the nearly edge-on and therefore very narrow debrisdisk around HD 61005, we obtain an optimal result for a minimum separation of1 FWHM, but using large optimization areas of 10000 PSF footprints to lessen theself-subtraction of the disk. We also reduced the data with LOCI with a separationcriterion of 0.75 FWHM and small optimization segments of 300 PSF footprints toset hard detection limits on companions.

To check that the revealed structures are not artifacts of the ADI methods,a classical PSF subtraction was applied to HD 61005 using the unresolved starTYC-7188-571-1, observed 3 hours after HD 61005 in the same observing mode.A total of 400 frames matching the same parallactic angle variation for HD 61005and the reference star were considered. After shift-and-add, a scaling factor wasderived from the ratio of the azimuthal average of the HD 61005 median imageto the azimuthal average of the recentered median image of the PSF. The medianof the PSF sequence was then subtracted from all individual frames of HD 61005.The resulting cube was derotated and collapsed to obtain the PSF-subtracted im-age. Additional azimuthal and low-pass filtering was applied to improve the diskdetection.

129


Figure 6.3: High-contrast NACO H-band images of HD 61005, (a) reduced with LOCI,

(b) reduced with ADI. In (b) The slits used for photometry are overlaid. The curved

slit traces the maximum surface brightness of the lower ring arc, while the rectangular

boxes enclose the streamers. In all images the scaling is linear, and 1′′corresponds to a

projected separation of 35.3 AU. The arc-like structures in the background are artifacts of

the observation and reduction techniques, and are asymmetric because the field rotation

center was offset from the star. The region with insufficient field rotation is masked out.

The yellow plus marks the position of the star, the green cross the ring center.

130

6.4. Results

6.4 Results

The NACO H-band images obtained by reduction with LOCI, ADI and referencePSF subtraction are shown in Fig. 6.3 and Fig. 6.4. The circumstellar materialis resolved to an off-centered, nearly edge-on debris ring with a clear inner gapand two narrow streamers originating at the NE and SW edges of the ring. Astrong brightness asymmetry is seen between the NE and SW side and betweenthe lower and upper arc of the ring. The inner gap has not been previously re-solved by HST, where only the direction of the polarization vectors hinted at adisk-like component separate from the extended material that interacts with theISM.

LOCI provides the cleanest view of the ring geometry with respect to the back-ground because it effectively removes the stellar PSF while bringing out sharpbrightness gradients. The negative areas near the ring result from oversubtrac-tion of the rotated disk signal embedded in the subtracted PSF constructed byLOCI. In particular, the ring’s inner hole is enhanced. However, tests with ar-tificial flat disks showed that while self-subtraction can depress the central re-gions, the resulting spurious gradients are shallow and different from the steepgradients obtained from the edge of a ring. Because of significant variable fluxloss, photometry is unreliable in the LOCI image. In the ADI reduction, the self-subtraction is deterministic and can be accounted for, while the stellar PSF issubtracted adequately enough to allow photometric measurements.

In the image produced by reference PSF subtraction (Fig. 6.4) the stellar PSF isnot effectively removed. The image is unsuitable for a quantitative analysis, butit confirms the streamers and the strong brightness asymmetry, and also suggeststhe presence of a gap on the SW side.

6.4.1 Surface brightness of ring and streamers

The surface brightness of the ring and streamers (Fig. 6.5) is obtained from theADI image. We measure the mean intensity of the bright ring arc as a function ofangular separation from the star in a curved slit of 5 pixels width (≈ 65 mas, seeFig. 6.3b) following the maximum brightness determined in Sect. 6.4.2. For thestreamers the slit is rectangular and of the same width. We calculate the meanintensity in the intersection of the slit with annuli of 9 pixels width. To estimatethe self-subtraction because of ADI, we apply our ADI method to a model ring(Sect. 6.4.2). The measured flux loss in our slit is 24± 5%, where the error includesvariations with radius and between the two sides. For the true ring this valuemight differ by a few percent. For lack of a good model for the swept material,we apply the same correction factor for the streamers, though the systematic erroris likely larger. The dominant source of error are subtraction residuals, whichwe measure as the dispersion in wider slits rotated by ±45. To obtain absolutephotometry we use the observations of HD 61005 taken in the neutral densityfilter as reference. The NE arc is about 1 mag/arcsec2 or a factor of 2–3 brighterthan the SW arc, consistent with the factor of 2 brightness asymmetry seen by

131


Figure 6.4: NaCo image of HD 61005 after reference PSF subtraction using the star TYC-

7188-571-1, observed 3 hours after HD 61005 in the same observing mode.

HST at shorter wavelengths. The surface brightness of the inner 1.1” of the SWarc is of the same order as the residuals, making a power-law fit unreliable.

The two streamers are tilted by an angle of ∼ 23 with respect to the ring’ssemi-major axis. We detect material out to a projected distance of ∼140 AU (4”)from the star. Beyond 2.8′′ the S/N ratio is too low to perform a meaningfulpower-law fit. Closer, the power-law slopes agree with those by Maness et al.(2009) within errors. We do not detect the fainter, more homogeneously dis-tributed swept material seen by HST because such structures are subtracted byADI. However, the streamers are also seen in the reference PSF subtracted im-age and thus are the most visible component of the swept material. They mayrepresent the limb-brightened edges of the total scattering material.

6.4.2 Ring geometry and center offset

We convolve the LOCI image with the PSF and measure the ring’s inclination andposition angle by ellipse fitting through points of maximum intensity in selectedregions. Assuming that the ring is intrinsically circular, the fit yields an inclina-tion of 84.3 and position angle 70.3, with systematic errors of ∼ 1. The positionangle agrees well with the position angle of the disk-component determined byHST, while we find the inclination to be ∼ 4 closer to edge-on. To determine theseparation of the ring ansae, we create inclined ring annuli and find for each sidethe ring with maximum intensity at and close to the ansa. The error of the semi-major axis and offset is assessed by measuring the 1σ background of an equallylarge region away from the ring at equal angular separation. All values for a ando for which the corresponding ansa has an intensity within 1σ from the maximum

132

6.4. Results

Figure 6.5: H-band surface brightness of the brighter ring arc and the streamers mea-

sured from the ADI image and corrected for self-subtraction. Dotted lines indicate the

error. The dotted dark line is the 1 σ sky background. Solid dark lines are power-law fits

with the obtained slope indicated. The transition regions between ring and streamers are

not fitted.

are considered solutions within a 1σ error. This fit yields a radius of 61.25 ± 0.85AU, and a ring center offset from the star by 2.75± 0.85 AU toward SW along theapparent disk major axis. The radial extension of the ring agrees with the locationof the power-law break seen by HST at 0.6 µm.

To assess the effect of our PSF subtraction method and determine additionalring parameters, we create synthetic scattered light images of inclined rings withthe GRaTer code for optically thin disks (Augereau et al. 1999). The models areinserted into the data of our PSF reference star (cf. 6.3) and the LOCI algorithm isapplied to allow direct comparison with the observations.

We let the (unprojected) ring be intrinsically elliptical with the periastron lo-cated along the apparent disk major axis, because our data cannot constrain anoffset along the minor axis. The ring’s surface density as a function of radialdistance r to the star (corrected for eccentricity) is described as

σ(r) = σ0 × R(r)( r

a

)

, (6.1)

with the radial structure R(r) a smooth combination of two power laws, such thatthe profile rises to a peak position and then fades with distance from the star:

R(r) =

( r

a

)−2αin+( r

a

)−2αout− 1

2

, (6.2)

where a is the semi-major axis, αin > 0 and αout < 0.Directionally preferential scattering is represented by a Henyey-Greenstein

(HG) phase function with an asymmetry parameter g.

133


Figure 6.6: Comparison of a radial cut averaged over 5 pixels through the midplane for

the observation reduced with LOCI and two model disk implanted into the reference star

and reduced in the same way. a Best fit model, b Model without an offset. The intensity

of the SW-side of the model is scaled down by a factor 1.3.

Figure 6.7: As fig. 6.6 but for a model without an offset.

We compare the model constructed from the parameters derived above, aswell as a similar model without an offset, with the observation. An inclination ofi = 84.3 ± 1.0 is also a good match for an intrinsically elliptical ring. While thecenter offset leads to a small increase (factor of ∼1.2) in surface brightness on theNE side with respect to the SW side, models that are identical on the two sidesexcept for the offset still underestimate the extent of the enhancement by a factorof ∼1.3. That the brightness asymmetry is visible after all reduction methods andin the HST 0.6 µm image suggests a physical asymmetry in the density or grainproperties. Because asymmetric dust models that include the ISM interaction gobeyond the scope of this study, we focus on validating the ring geometry.

We compare a radial cut along the midplane averaged over 5 pixels (Fig. 6.6),artificially lowering the model intensity on the faint side by a factor of 1.3. Indeed,models with an offset o = a · e of 2.75 ± 0.85 AU, where a is the semi-major axisfor the peak density (61.25 ± 0.85 AU) and e the eccentricity (0.045 ± 0.015), stillprovide a decent match after reduction with LOCI. Models without offset are

134

6.4. Results

Table 6.2: Properties of the debris ring derived from the NaCo observations reduced with

LOCI

Inclination 84.3 ± 1

Position angle 70.3 ± 1

Semi-major axis 61.25 ± 0.85 AUEccentricity 0.045 ± 0.015Star center offset 2.75 ± 0.85 AU

worse particularly out to ∼63 AU for each ring side because the shift in peakintensity is missing (see Fig. 6.7). Therefore the offset does not appear to be anartifact of the data reduction.

Model comparison suggests an inner surface density power-law slope of ∼7,but a fit is difficult because reduction artifacts differ for models and observations.The outer slope (fixed to −4) is uncertain because we do not model the ISM inter-action. In any case, the inner rim appears to be significantly steeper than the outerrim. From the brightness asymmetry between the upper and lower arc we esti-mate the HG asymmetry parameter to |g| ∼0.3. This value is uncertain becausethe weak arc is strongly contaminated by reduction residuals. Additionally, theHG phase function is a simplistic model for scattering in debris disks. A posi-tive g-value, assuming that the brighter side is the front, would indicate forwardscattering grains. This may not always be the case (see e.g. Min et al. 2010).

6.4.3 Background objects and limits on companions to

HD 61005

In our full VLT/NaCo field image (Fig. 6.8) we detect six faint sources at r > 3′′,marked as fs-1 to fs-6. In addition to our VLT data, we used the HST/NICMOSobservations of Hines et al. (2007) (program 10527) obtained in November 20,2005 and June 18, 2006, to determine whether these are bound or backgroundobjects. The relative positions recorded at different epochs can be compared tothe expected evolution of the position measured at the first epoch under the as-sumption that the sources are either stationary background objects or comovingcompanions. For the range of explored semi-major axes, any orbital motion canbe considered to be of lower order compared with the primary proper and paral-lactic motions. Considering a proper motion of (µα, µδ) = (−56.09± 0.70, 74.53±0.65) mas/yr and a parallax of π = 28.95 ± 0.92 mas for HD 61005 as well as therelative positions of all faint sources at each epoch (see Table 6.3), a χ2 probabil-ity test of 2 × Nepochs degrees of freedom (corresponding to the measurements:separations in the ∆α and ∆δ directions for the number Nepochs of epochs) wasapplied. None of the six sources are comoving with HD 61005 with a probabilityhigher than 99.99%. They are found to be background stationary objects with a

135


Figure 6.8: Full field of view of our NACO H−band data reduced by derotating, adding

and spatially filtering. Six background sources are identified.

Table 6.3: Relative positions of the faint sources 1 to 6 (Fig. 6.8). A conservative astromet-

ric error of 1 pixel has been considered for the relative position measurements obtained

with HST/NICMOS and VLT/NaCo observations (i.e 75.8 mas and 13.25 mas).

Name UT Date ∆RA ∆DEC UT Date ∆RA ∆DECNICMOS (mas) (mas) VLT (mas) (mas)

fs-1 2006-06-18 -1929 2918 2010-02-17 -2189 3080fs-2 2005-11-20 -4672 4669 2010-02-17 -4449 4441fs-3 2005-11-20 -2326 9321 2010-02-17 -2068 9034fs-4 2005-11-20 -8487 -15 2010-02-17 -8222 -370fs-5 2006-06-18 -1586 -6848 2010-02-17 -1358 -7123fs-6 2006-06-18 -1148 -7671 2010-02-17 -1336 -8024

136

6.5. Discussion

probability higher than 60%. We can therefore fully exclude the possibility thatthese sources are physically bound companions of HD 61005. Maness et al. (2009)had already determined that four sources visible in their image were backgroundsources. Two of these correspond to fs-3 and fs-4, and we here confirm their re-sult. Their other two sources are outside of our field of view. We therefore providea new result for the four sources fs-1, fs-2, fs-5 and fs-6.

In the LOCI image reduced with the smaller minimum rotation (Fig. 6.9) wesearch for closer companions. After convolving the resulting image with an aper-ture of 5 pixels diameter, we calculate the noise level at a given separation asthe standard deviation in a concentric annulus. To determine the flux loss frompartial self-subtraction we implant artificial sources in the raw data and measuretheir brightness after the reduction process. The measured contrast curve is cor-rected for this flux loss to yield the final 5σ detectable contrast curve (Fig. 6.10).We translate the contrast to a mass limit using the COND evolutionary modelsby Baraffe et al. (2003) for ages 40 Myr, 90 Myr and 140 Myr. We do not detectany companion candidates, but are able to set limits well below the deuteriumburning limit.

6.5 Discussion

The high-resolution image enables us to distinguish the actual debris ring fromthe material that appears to be streaming away from the system. The results re-veal a ring center offset of ∼3 AU and an additional brightness asymmetry sug-gesting density variations. The eccentricity of the debris ring could be shaped bygravitational interaction with a companion on an eccentric orbit.

A similar system is Fomalhaut with a belt eccentricity of 0.11 (Kalas et al.2005). Mass constraints were discussed for Fomalhaut b by Chiang et al. (2009)with disk stability arguments. A planet will disrupt the disk more strongly themore massive and the closer to the ring it is. For the planet to secularly forcethe ring planetesimals on eccentric orbits without completely disrupting the de-bris ring, it is thought that the inner boundary of the ring should lie at the edgeof the chaotic zone of the planet, in which overlapping first-order mean-motionresonances are responsible for sending particles on chaotic, short-lived orbits. Auniversal empiric relation between planet mass and it’s distance from the innerring edge was found by both Quillen (2006) and Chiang et al. (2009):

∆a =|achaotic − apl |

apl= b · (

Mpl

M⋆

)27 , (6.3)

where b is a scaling factor determined to b = 1.5 by Quillen (2006) purely frommodels, and to b = 2 by Chiang et al. (2009) for an actual fit to the HST data ofthe Fomalhaut ring.

This equation is translatable to other planetary systems, even though the exactscaling factor may depend on the shape of the ring, in particular on the sharpness

137


Figure 6.9: NaCo high contrast imaging data reduced with LOCI with a minimum sep-

aration of 0.75 FWHM to search for close point sources. The debris ring is still faintly

visible. The image size is 3′′x 3′′

Figure 6.10: Contrast for companions around HD 61005 detectable at the 5σ level, and

corresponding mass limits determined from the COND models by Baraffe et al. (2003)

for ages 90 Myr (solid), 40 Myr (dotted), 140 Myr (dashed). One arcsec corresponds to a

projected separation of 35.3 AU.

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6.5. Discussion

Figure 6.11: Mass of a planet as function of separation from the star that would secularly

force the debris disk’s planetesimals into elliptical orbits. We assume the semi-major

axis of the inner ring edge to be at 52 AU. The mass is compared to the mass limits

derived from high-contrast imaging for various ages, assuming a potential planet to be

at maximum angular separation. The mass contrast curves as function of separation are

thus only a lower limit. Red: scaling factor by Quillen (2006), blue: scaling factor by

Chiang et al. (2009). Solid: Detection limit age of 90 Myr, dotted: 40 Myr, dashed: 140

Myr.

of the inner edge. It is still a reasonable first approximation to determine the al-lowed planet masses depending on the location of the planet. Figure 6.11 showsthis relation for HD 61005, assuming an inner ring edge of 52 AU and a star massof 1 M⊙. The curve is compared to the detection limits from the high-contrastimaging data, assuming the planet to be a maximum angular separation alongit’s orbit. A planet well below our detection limit of ∼2-5 MJ located beyond∼37 AU at maximum angular separation could easily perturb the ring. Because ofthe high inclination a planet of higher mass and lower semi-major axis could alsohide within the residuals at smaller projected separation from the star. Multipleepoch data taken several years apart would be required to cover the full orbitalparameter space for massive planets. The dynamical arguments therefore cur-rently do not provide a planet mass limit. Additionally, it must be rememberedthat these models only assume perturbations by a single planet, while multiplesmaller planets may induce the same effect.

Models of the spectral energy distribution by Hillenbrand et al. (2008) sug-gested the debris with multiple temperature components provided a better fitthan models with a single-temperature blackbody. This could be understood aseither an extended debris model (Rinner < 10 AU and Router >40 AU), or an innerwarm ring and an outer cool ring. The outer cool ring could coincide with thedust detected in the scattered light images. Additionally, a preliminary resolved

139


detection of an inner warm component at 10µm was made by Fitzgerald et al.(2010). Interestingly, the suggested inclination of this ring does not match theinclination of the outer, cooler ring. The detection is however still very tentative.

To explain the structure of the interacting material, Maness et al. (2009) ex-plored several scenarios, and proposed that a low-density cloud is perturbinggrain orbits because of ram pressure. The streamers would be barely bound, sub-micron sized grains on highly eccentric orbits, consistent with the observed bluecolor and the brightness profile. Their model currently cannot reproduce thesharpness of the streamers, but the observed geometry of the parent body ringmight help improve the models to validate this theory. These might then answerwhether the ring offset could also be caused by the ISM interaction rather thana planet. Obtaining colors of the ring through high-resolution imaging at otherwavelengths could indicate if a grain size difference exists between parent bodyring and swept material.

As a solar-type star, and with a debris ring at a radius not much larger thanthat of the Kuiper belt, the HD 61005 system provides an interesting comparisonto models of the young solar system. Booth et al. (2009) calculate the infraredexcess of the solar system as a function of time based on strong assumptions con-sistent with the Nice model (Gomes et al. 2005). This model of the early solar sys-tem’s history proposes that a dynamical rearrangement of the outer giant planetsthrough migration resulted in a planetesimal clearing event. This event then ledto the so called Late Heavy Bombardment (LHB) at an age of the solar systemof 880 Myr, a period of time where a strongly increased amount of impacts wasregistered in the inner solar system, visible e.g. on the surface of the moon. Thisevent also dramatically changed the luminosity evolution of the solar system’sdebris.

At an age between ∼30-100 Myr, HD 61005 has an excess ratio at 24 µm and70 µm that is about 4 times higher (Hillenbrand et al. 2008) than the ratios cal-culated for the solar system at comparable age for the assumption of a single-phase size distribution. For more realistic grain size distributions, the solar sys-tem might have been similarly bright. In any case, HD 61005 is one of the mostluminous debris disks known. Its other obvious unusual feature is the morphol-ogy of the swept component. Perhaps further modeling will provide a causalconnection between its high observed dust generation rate and this apparent in-teraction with the ISM. It is also an interesting target for future deeper searchesfor planetary mass companions as well as remnant gas that could be associatedwith the debris.

Acknowledgments. We thank David Lafreniere for providing the source codefor his LOCI algorithm. We thank J. Alcala, E. Brugaletta, E. Covino and A. Lan-zafame for discussions on the stellar properties and H.M. Schmid for comments.EB acknowledges funding from the Swiss National Science Foundation (SNSF).AE is supported by a fellowship for advanced researchers from the SNSF (grantPA00P2 126150/1)

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Augereau, J. C., Nelson, R. P., Lagrange, A. M., Papaloizou, J. C. B., & Mouillet, D. 2001,A&A, 370, 447

Baraffe, I., Chabrier, G., Barman, T. S., Allard, F., & Hauschildt, P. H. 2003, A&A, 402, 701

Booth, M., Wyatt, M. C., Morbidelli, A., Moro-Martın, A., & Levison, H. F. 2009, MNRAS,399, 385

Bryden, G., Beichman, C. A., Carpenter, J. M., et al. 2009, ApJ, 705, 1226

Chiang, E., Kite, E., Kalas, P., Graham, J. R., & Clampin, M. 2009, ApJ, 693, 734

Debes, J. H., Weinberger, A. J., & Kuchner, M. J. 2009, ApJ, 702, 318

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Fitzgerald, M. P., Kalas, P., Graham, J. R., & Maness, H. M. 2010, in Proceedings of theconference In the Spirit of Lyot 2010: Direct Detection of Exoplanets and Circumstellar Disks.October 25 - 29, 2010. University of Paris Diderot, Paris, France. Edited by Anthony Boc-caletti.

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142

Chapter 7

Outlook

The HD 61005 system is a highly interesting system that encompasses severalfeatures and lies at a distance from Earth that allows for well-resolved images al-ready with current telescopes. The debris disk itself shows a very strong infraredexcess and an unusual asymmetric structure in scattered light. Its parent plan-etesimal ring at ≈ 60 AU appears to be slightly off-centered, which may be dueto gravitational forcing by one or multiple companions inside the belt. The outerthin disk component is swept-up by interaction with the ISM and therefore wellvisible. There also exists preliminary evidence for a second, inner belt of warmdust. All these features make HD 61005 a highly interesting target for follow-up studies at multiple wavelengths. The following section describes some of theongoing and planned follow-up observations of this debris disk and its potentialplanetary mass companions. With additional data, it will also be possible to morestrongly constrain models on grain properties, dynamics of the disk/ISM interac-tion and disk/planet interaction. Finally, this section will discuss future prospectsfor debris disk imaging with upcoming extreme adaptive optics instruments.

7.1 Further observations of the Moth

7.1.1 A search for companions inside the ring

Our H-band observations did not yield any planet candidates at the 5σ-level in-side the debris ring. The derived mass limits within the ring are of order 2− 5 MJ

at projected separations beyond 0.7”, and rising up to a limit of ≈ 10 MJ at theinner working angle of ≈ 0.3′′. However, we see three candidate signals at a2 − 4σ-level inside the ring, with hypothetical masses of 4 − 10 MJ at projectedseparation of ≈ 10 − 20 AU. While chances are high that these sources are arti-facts, if any of these were confirmed as real, it would be a very valuable object ofstudy because of the strong similarities of this system with our own solar systemand its disk-planet interaction.

The NaCo Large Program collaboration has therefore initiated new obser-vations of the HD 61005 system in the L’-band at 4 µm (led by C. Thalmann),where the Strehl ratio and PSF stability are better than at shorter wavelengthsand should render the PSF removal with ADI particularly effective. Also, thebrightness contrast between planet and star, as well as planet and disk, is ex-

143

Chapter 7. Outlook

pected to be more favorable than in the H-band. The recent update of NaCo withan apodizing phase plate (APP) additionally provides the opportunity to reachsignificantly lower masses at small separations from the star. The APP modifiesthe wavefront phase in the pupil plane, suppressing the airy diffraction rings onone half of the field of view, while producing a much more distorted PSF on theother half. Making two observations with the APP rotated by 180 in between, afull image at or near the background limit can be obtained towards small (≈ 0.2′′)separations from the central star (Kenworthy et al. 2010). Quanz et al. (2010) suc-cessfully detected the companion around βPic using the APP.

The observations were performed in early 2011. Preliminary analysis suggeststhat, unfortunately, the predicted background limited performance could not bereached (C. Thalmann, private communication). In fact, the obtained planet masslimit surprisingly appears to be very similar to that in H-band, except for reach-ing a slightly smaller inner working angle. No point sources are detected, thusmaking the low-sigma H-band signals unlikely planetary candidates. It is pos-sible that further refinement of the reduction procedure may yet result in betterdetection limits or detection of point sources. No traces of the debris disk areseen.

7.1.2 Resolved disk observations at different wavelengths

Prior to our NaCo observations of HD 61005, the only resolved scattered lightimages of this debris disk had been obtained by HST. Images at 0.6 µm and 1.1µallowed a first color determination of the disk, although limited to the spatialresolution of the 1.1µ NICMOS data. This data set had resolved only the outer,swept component. While for the shorter wavelength ACS observations additionalpolarimetric data hinted towards a highly inclined disk or ring component, thering was not directly resolved. A color determination could therefore only bemade for the outer, swept-back component. It was found that this componentappears predominantly blue at these wavelengths, indicating small grains withamin < 0.3µm and a steep size distribution (Maness et al. 2009).

With the resolution of the dust within the parent-body planetesimal belt, thequestion opens whether the grain size distribution differs in the belt with respectto the outer component. The standard scenario for dust replenishment in debrisdisks involves the grinding down of planetesimals to smaller and smaller sizesin a collisional cascade. Smaller grains are forced onto orbits of much higher ec-centricity than that of the parent planetesimal belt by radiation pressure, and thesmallest grains are entirely removed from the cascade by blow-out onto hyper-bolic orbits. Observational evidence of a cascade around an intermediate massstar where dust production and removal are in steady-state is seen for the Vegadebris disk (Muller et al. 2010) from resolved images at mid-infrared to mm-wavelengths. The collisional evolution is thought to be similar for bright debrisdisks around solar-type stars (e.g. Kains et al. 2011). However, this remains ob-servationally unconfirmed.

The HD 61005 disk is an ideal target to directly confirm the blow-out scenario

144

7.1. Further observations of the Moth

for a solar-type star. The outer, usually very faint component, is well visible forthis disk in scattered light because it is swept-up by the ISM, while the parent-body ring is resolvable with large telescopes from the ground because its nearlyedge-on inclination make it well suitable for the effective ADI method for PSFremoval. We have therefore submitted a proposal (for which I am the PI) to ESOwhich has been accepted to obtain observations of HD 61005 in J- and Ks-bandwith the same observing setup as used for our H-band observations. The surfacebrightness of the ring and streamers will be measured with conservative ADIanalogous to Sect. 6.4.1. We will obtain color information for both the ring andthe faint outer component. While the uncertainty of absolute photometry is rela-tively high for these observations, we are mainly interested in relative differencesthat can be obtained confidently. Also, the J-band observations of the outer com-ponent can be calibrated with the available HST data. A radial color dependencewill be good evidence for a change in grain size distribution, with the bluer com-ponent being depleted in large grains (cf. next section), thus directly confirmingthe blow-out scenario.

Additionally, the new data will determine whether the previously measured3σ-offset between the stellar position and ring center is indeed real and revealwhether the disk asymmetries show a dependence on wavelength in geometryor brightness.

Further resolved scattered-light observations at various wavelengths havebeen obtained or are scheduled by several different groups. Fitzgerald et al.(2010) reported H- and K-band observations of HD 61005 obtained withKeck/NIRC2 at the “In the Spirit of Lyot” conference in the fall 2010. TheirH-band result, obtained through removal of a reference PSF, appears to be verysimilar to our image reduced with conservative ADI, at least in terms of ge-ometry. It will be interesting to compare the surface brightness distribution todetermine the true uncertainty of the ADI observations. Their K-band data is ap-parently of poor quality. Interestingly, they also obtained thermal imaging 10 µmobservations using Gemini-S/T-ReCS, a mid-infrared imager and spectrographat an 8 m class telescope. After a preliminary analysis, the data show a small blobof ≈0.2” diameter which is consistent with a ring of warm dust that is stronglyinclined with respect to the outer cool ring. It is unclear whether this featureis indeed real, but its presence would be in accordance with SED models thatindicate an inner warm dust component. The strong inclination with respect tothe second belt could point towards planetary companions strongly disturbingthe dust.

Other ongoing projects include HST/STIS observations (PI: G. Schneider)which are obtaining high-resolution scattered light images at optical wavelengthstowards a small inner working angle. Finally, Herschel observations (PI: V. Geers)are scheduled to search for remnant gas associated with the debris disk by look-ing for O[I] line emission.

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Chapter 7. Outlook

7.2 A detailed model for the Moth

Combining the multi-wavelength HST and above described additional ground-based observations with the SED available from Spitzer/FEPS and long wave-length observations (Roccatagliata et al. 2009), we will construct a detailed diskmodel using the radiative transfer code GraTer by Augereau et al. (1999). Thecode calculates scattered light images and SED for various debris disk geometries,grain compositions and grain size distributions. A color difference in scatteredlight images will indicate if a difference in grain size distribution exists betweenparent body ring and swept material. This is expected when collisional equilib-rium is reached in the birth ring, leading to a grain size distribution dn(a) ∝ aκ dawith a the grain radius. The theoretical power-law exponent in the birth ring isκ = −3.5. Small grains are forced outward onto large, eccentric orbits, down toa critical size amin = ablow below which they are blown out of the system on hy-perbolic orbits. Larger grains stay bound in the birth ring. Thus, in our scenariothe grain size distribution gets steeper (p smaller), meaning the color bluer, withincreasing distance to the star due to pressure forces (Augereau et al. 2001). Theflux scattered for a given grain distribution and chemical composition and stellarirradiation field is proportional to the mean scattering cross section σsca.

Figure 7.1 shows the theoretical ratio of σsca in J- and Ks- bands, which isdirectly proportional to the color index J-Ks, as a function of κ = κ(d). We con-sidered several chemical compositions, and for each we fixed amin according toa best-fit value for the mid- and far-IR SED or to the theoretical blowout size.It shows that color differences between the inner and outer disk will providedstrong constraints on the chemical composition of the grains and their size dis-tribution. We would therefore obtain a direct confirmation of the blow-out sce-nario for a solar-type star and thus the main mechanism of dust removal for mostknown systems.

Dynamical modeling will most likely present a challenge, both to explain theshape of the material that is interacting with the ISM and the eccentric shape andbrightness asymmetry of the birth ring. The currently favored model for the ISMinteraction (Maness et al. 2009) suggests that a warm, low-density cloud is per-turbing orbits of the small grains. These models are only a crude approximationto the data and may be missing fundamental processes. However, at least the pre-viously proposed cold dense cloud has been ruled out by observations of the Na Icolumn density towards the object (Maness et al. 2009), which can be translatedinto an total hydrogen (H I+H2) column density. A new program of spectroscopicobservations with HST/STIS is searching for evidence of the low-density cloud,but so far the results have been inconclusive (J. Graham, private communication)except for confirming the ruling out of a dense cloud.

Dynamical models for the potential planetary perturber are also tricky in par-ticular because the inner boundary of the ring is not well defined by the LOCIobservations. In the case of Fomalhaut, the ring shows a very sharp edge, whichdoes not appear to be the case for HD 61005. While the inner rim seems to besteeper than the outer rim, it is unclear how much LOCI smoothes or steepens

146

7.3. Future prospects for debris disks imaging

−20 −15 −10 −5κ = κ(d)

0

2

4

6

8

<σ s

ca(J

)> /

<σ s

ca(K

)>

Si (amin = 2.03) Si+ice (amin = 0.202)Si+C+ice+P (amin = 1.23)

−20 −15 −10 −5κ = κ(d)

0

2

4

6

8

<σ s

ca(J

)> /

<σ s

ca(K

)>

Si (amin = 0.203) Si+ice (amin = 0.224)Si+C+ice+P (amin = 1.11)

Figure 7.1: Theoretical ratios of scattering cross sections in J- to Ks-bands as function of

the size distribution power-law (proxy for distance from the star, larger distances having

smaller κ values, from right to left) for different grain compositions, including silicates

(Si), ice, amorphous carbon (C) and porosity (P). Cross sections are normalized such that

σsca(κ = −3.5) = 1 (i.e. in the birth ring). A larger ratio means a bluer color. Left: min.

grain size amin derived from SED model-fitting. Right: amin = ablowout. amin is given in

µm. Calculations made by J. Lebreton.

the slope and more studies on the effect of LOCI on extended structures would berequired. The HST/STIS imaging observations may provide a better constraintfor this material. Also, it is at this time unclear if the ISM interaction also hasan effect on the birth ring. The strong brightness asymmetry, which cannot beexplained simply by the eccentric shape of the ring, hints towards a more com-plicated phenomenon. A model encompassing both the ISM interaction and aplanetary perturber may be required to fit all the geometric properties of this sys-tem.

7.3 Future prospects for debris disks imaging

Possibilities for space-based observations of debris disks in scattered light havetaken a serious set-back with the failure of both NICMOS and the High Reso-lution Channel (HRC) of the ACS camera on HST. The new WFC3 instrumentdoes not include a coronagraph. The only remaining instrument for debris diskimaging on HST is STIS, which has imaged the disk around HR 4796A (Schneideret al. 2009), and has an ongoing program to image several known debris disks tosmall inner working angles (PI: G. Schneider). The disadvantages of STIS are itsnon-optimal coronagraphic mode (occulting bar and wedges) and the fact thatcoronagraphy cannot be combined with filters. The images are taken simply inthe range of the detector sensitivity between ≈ 200 − 1030 nm. With no furtherservice missions to HST planned, it is therefore unlikely that HST can continue tomake a significant contribution to debris disk imaging. Unless a small space mis-sion dedicated to direct imaging, such as EXCEDE (PI: G. Schneider), is launchedin the near future, no space-based imaging will be possible until the launch of

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Chapter 7. Outlook

the James Webb Space Telescope (JWST), currently expected in approximately2018. The future of debris disk imaging must therefore lie with ground-basedtelescopes.

Already today, ground-based imaging of debris-disks is feasible with angu-lar differential imaging techniques, as evidenced in Chapter 6 of this thesis forHD 61005. The ansae of HR 4796A have been imaged with polarimetric differen-tial imaging (Hinkley et al. 2009) on a 3.6 m telescope, and high quality imageshave been taken with SDI and ADI on Gemini-S/NICI (Liu et al., Lyot Confer-ence). The NaCo/PDI mode has been successfully used on the disk of the Her-big Ae/Be star HD 100456 (Quanz et al. 2011) and should be sufficient to alsocharacterize some of the brightest known debris disks in polarization. The ADItechnique should be applicable to all reasonably bright disks that are sufficientlyinclined towards the line of sight and therefore do not show strong angular sym-metry.

In the next 1-2 years, two ground-based instruments with strong imaging ca-pabilities will go online: SPHERE and GPI (cf. Sect. 1.5.1). For both instruments,a science case for debris disk imaging has been established and dedicated pro-grams are expected to be conducted by the instrument teams. GPI is optimizedto measure both intensity and polarization. Debris disks with infrared fractionalluminosities of a few 10−5 should be within reach (Perrin et al. 2010). SPHEREwill mainly be able to measure the polarized intensity, but over a larger wave-length range and with more filters.

Studies for both instruments could focus on the detailed characterization ofdisk geometry and sub-structures and particle properties for previously imageddisks towards small inner working angles of ≈ 0.1′′, also revealing warm dustcloser to the star than currently accessible. This could for example be done for theMoth, to confirm the inner ring suggested by Fitzgerald et al. (2010). A survey ofbright Spitzer-detected sources could resolve additional disks in scattered lightto expand the sample and achieve better statistics on debris disk grain propertiesand geometries. Additionally, more planets within the inner gaps of debris ringscould be found, showing direct evidence of disk/planet interactions and therebyconstraining planet formation theory.

With the next generation of extremely large telescopes, it could be expectedto reach the innermost part of solar system analogs where Earth-like planets mayform.

148

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149

Acknowledgments

This thesis would not have been possible without the support and help of manypeople to whom I am most grateful.

First and foremost, I would like to thank Hans Martin Schmid, my primary ad-visor, for offering me the PhD position in the exciting field of exoplanet research,even before the funding was fully secured. During the course of the thesis, he al-ways had an open door and useful advice on many problems I encountered. Hemanaged to keep me enthusiastic about the ‘diagnostic potential’ of polarimet-ric observations, but also supported my venturing off into a related field of myown choosing. He always encouraged me to attend schools and conferences, towrite papers, submit my own observing proposals, and develop my own ideas.Thank you for the always very positive and pleasant atmosphere within our smallgroup.

Secondly, I am very grateful to Michael Meyer, who also served as my advisorand main referee of this thesis. With his arrival at ETH he brought a lot of freshideas and a vibrant new group with related research interests that resulted inmany interesting discussions. Thank you for giving me a home in your groupand for being an inspiration on how to do exciting and careful scientific research.Without your support I might not have dared to take on ‘the Moth’. Thank you foryour guidance towards a scientific career without forgetting to listen to concernsand problems and always trying to find constructive solutions.

I would like to thank Francois Menard for agreeing to be the external examinerof this thesis, providing useful comments and traveling to Zurich to attend thedefense. I also thank Simon Lilly and Jan Stenflo for providing financial supportfor my position and travel during the first half of my PhD. This PhD was alsosupported by a grant from the Swiss National Science Foundation (SNSF).

I was fortunate to have two wonderful friends at the institute, Lucia Kleintand Susanne Wampfler, who shared my passion for astronomy throughout ourPhysics studies and PhD theses. It was always fun to exchange the most recentgossip, solve problems together related to IDL or general science, do public out-reach together or spend nights at the observatory hunting every barely visibledeep sky object just for fun. Lucia, we were a great team on Diavolezza duringthe Astrowoche in 2009. Thanks also for providing the template for this disserta-tion. Susanne, you were a terrific office mate. I am very grateful for the help withmany little day-to-day problems an astronomer encounters, and even more so foryour support and open ear when things did not go smoothly.

Many other people at the institute were responsible for a productive andfriendly atmosphere. Sascha Quanz often had useful tips, be it for proposals,

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papers, data reduction or job applications. Franco Joos took me along to an ob-serving run in La Silla, Chile, only 3 months into my thesis, where I learned alot about polarimetric observations. Peter Steiner always provided swift helpwith any computer related problems, while Barbara Codoni, Rajni Malhotra andMarianne Chiesi took care of administrative matters very efficiently. I had in-teresting conversations over lunch, coffee or cake about life, the universe and ev-erything, about science, teaching duties or Dungeons and Dragons, with AndreasBazzon, Daniel Gisler, Christian Monstein, Rene Holzreuter, Christian Thalmann,Marina Battaglia, Simon Bruderer, Richard Wenzel, Dominique Fluri, NadineAfram, Kevin Heng, Maddalena Reggiani, Carolin Dedes, Vincent Geers, RichardParker, Andrea Banzatti, Henning Avenhaus, Michiel Cottar, Udo Wehmeier,Alex Feller, Svetlana Berdyugina, Jan Stenflo, Harry Nussbaumer, Julien Carron,Adam Amara, Thomas Bschorr, Katharina Kovac and Monique Aller.

I am also very grateful to my external collaborators, in particular the membersof the SPHERE science team. I was readily welcomed into the team and could of-ten take part in meetings all over Europe. Most importantly, they allowed me toparticipate in the related NaCo Large program which lead to an important resultof this thesis. I am indebted to Christian Thalmann for suggesting my applica-tion for NaCo observations one fine evening in the midst of a D&D session, onceagain showing that important steps in scientific careers usually happen at unex-pected times. I thank Jean-Luc Beuzit and Gael Chauvin for selecting me as anobserver to the VLT, and Gael for teaching me about high-contrast imaging ob-servations in Paranal. The whole team gave me their confidence for taking on theexciting data of ‘the Moth’ and I am thankful to all my co-authors for the con-tributions and helpful suggestions, in particular Christian, Gael, Arthur Vigan,Anthony Boccaletti, Jean-Charles Augereau, and David Mouillet. I also thankJeremy Lebreton for his modeling efforts and input for the future observationsfor the Moth.

I was fortunate to be able to travel to many beautiful places for observing runs,conferences and work-shops, where I met a lot of interesting people, both for sci-entific conversations but also to have fun with like-minded people. In particularI had a great time at the ESA summer school in Alpbach designing an Astrobi-ology space mission, and hanging out and climbing Arecibo with a fun group ofgrad students and post-docs at the DPS meeting in Puerto Rico. I am happy to bea part of the (exo)planetary community, which consists of mostly very pleasantpeople.

I also want to thank Daniel Apai for providing me the opportunity to continuepursuing a career in astronomy for the next couple of years beyond my PhD, sothat maybe I will find my own exoplanet at some point, after all. I am lookingforward to working with you in Arizona.

While a PhD thesis is a lot of work and a rewarding way to spend four years,there fortunately also exists a life outside of astronomy. I always enjoyed theweekly lunch with the good friends and former physics student colleagues, Ros-marie, Lukas, Dominik and Patrick. A welcome diversion were also my weeklychoir rehearsals in the ACZ, which also gave me the opportunity to perform beau-

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tiful and difficult choral works in two of the greatest concert halls in Switzerland.I particularly thank Hiu, Tamara, Stefan, Christoph and Catherine for fun timesduring and after singing. I also spent many evenings in fun D&D sessions withChristian, Rene, Marina, Carolin, Leonidas and Tobias that were always excitingand challenging, in particular thanks to the expert dungeon mastering by Ath-man.

A big thank you goes to my family and friends. My parents always supportedand encouraged me. Without them, I would not have got to where I am now.My sister, my brother and my close friends were also a great support during aparticularly difficult phase.

Last, but definitely not least, I want to thank Adrian, my love. You have beenwith me from the start to the end of this thesis and went through all the highsand lows with me, constantly providing love and encouragement. Thank youfor always believing in me more than I did. And thank you for supporting mydreams. I am happy that we will face our next and probably yet biggest adventuretogether.

Esther Buenzli, Institute for Astronomy, ETH Zurich August 2011

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List of Publications

Publications in refereed journals

• Buenzli, E. & Schmid, H.M.A grid of polarization models for Rayleigh scattering planetary atmospheres2009, A&A 504, 259

• Buenzli, E., Thalmann, C., Vigan, A., Boccaletti, A., Chauvin, G., Augereau,J.C., Meyer, M.R., Menard, F., Desidera, S., Messina, S., Henning, T., Carson,J., Montagnier, G., Beuzit, J.L., Bonavita, M., Eggenberger, A., Lagrange,A.M., Mesa, D., Mouillet, D., & Quanz, S.P.Dissecting the Moth: Discovery of an off-centered ring in the HD 61005 debris disk2010, A&A 524, L1

• Buenzli, E. & Schmid, H.M.Polarization of Uranus: Constraints on haze properties and predictions for analogextrasolar planets2011, submitted to Icarus

• Schmid, H.M., Joos, F., Buenzli, E., & Gisler, D.Long slit spectropolarimetry of Jupiter and Saturn2011, Icarus 212, 701

• Thalmann, C., Janson, M., Buenzli, E., Brandt, T.D., Wisniewski, J.P., Moro-Martin, A., Usuda, T., Schneider, G., Carson, J., McElwain, M.W., Grady,C.A., Goto, M., and the SEEDS collaboration,Images of the extended out regions of the debris ring around HR 4796 A2011, ApJL in press

Publications in conference proceedings

• Buenzli, E., Schmid, H.M & Joos, F.Polarization models for Rayleigh scattering planetary atmospheres2009, Earth, Moon, and Planets, 105, 153Conference proceedings of ’Future ground based solar system research’

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List of Publications

• Joos, F.,Buenzli, E., Schmid, H.M & Thalmann, C.Reduction of polarimetric data using Mueller calculus applied to Nasmyth instru-ments2008, Proc. of SPIE, Vol. 7016, 2008

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