observing hot stars in all four stokes parameters

The William‐Wehlau Spectropolarimeter: Observing Hot Stars in All Four Stokes ParametersAuthor(s): Thomas Eversberg, Anthony F. J. Moffat, Michael Debruyne, John B. Rice,Nikolai Piskunov, Pierre Bastien, William H. Wehlau, and Olivier ChesneauSource: Publications of the Astronomical Society of the Pacific, Vol. 110, No. 753 (November1998), pp. 1356-1369Published by: The University of Chicago Press on behalf of the Astronomical Society of the PacificStable URL: http://www.jstor.org/stable/10.1086/316253 .

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1356

Publications of the Astronomical Society of the Pacific, 110:1356–1369, 1998 Novemberq 1998. The Astronomical Society of the Pacific. All rights reserved. Printed in U.S.A.

The William-Wehlau Spectropolarimeter: Observing Hot Starsin All Four Stokes Parameters

Thomas Eversberg,1,2 Anthony F. J. Moffat,1,2 Michael Debruyne,2,3 John B. Rice,4 Nikolai Piskunov,3,5

Pierre Bastien,1 William H. Wehlau,3,6 and Olivier Chesneau7

Received 1998 February 13; accepted 1998 July 22

ABSTRACT. We introduce a new polarimeter unit, which, mounted at the Cassegrain focus of any telescopeand fiber-connected to a fixed CCD spectrograph, is able to measure all Stokes parameters I, Q, U, and V acrossspectral lines of bright stellar targets and other point sources in a quasi-simultaneous manner. Applying standardreduction techniques for linearly and circularly polarized light, we are able to obtain photon noise–limited linepolarization. We briefly outline the technical design of the polarimeter unit and the linear algebraic Muellercalculus for obtaining polarization parameters of any point source. In addition, practical limitations of the opticalelements are outlined. We present first results obtained with our spectropolarimeter for four bright, hot-star targets.We confirm previous results for Ha in the bright Be star g Cas and find linear depolarization features across theemission-line complex C iii/C iv (5696/5808 A) of the WR 1 O binary g2 Vel. We also find circular linepolarization in the strongly magnetic Ap star 53 Cam across its Ha absorption line. No obvious line polarizationfeatures are seen across Ha in the variable O star v1 Ori C above the % instrumental level.j ∼ 0.2

1. INTRODUCTION

Obtaining polarization spectra of stars is a relatively youngtopic. Only a few instruments exist for mid- and large-sizetelescopes—needed to detect the generally low polarizationlevels. Most popular is the use of a rotatable half-wave retarderplate along with a crystal polarizer to measure linearly polarizedlight. Examples of where this is done are the Anglo-Australian(Bailey 1989) and Herschel (Tinbergen & Rutten 1992) tele-scopes. At Keck the so-called “dual-waveplate” method is used(see, e.g., Cohen et al. 1996; Goodrich, Cohen, & Putney 1995)consisting of a quarter-wave and a half-wave plate. Using aquarter-wave plate (QWP), e.g., at ESO/CASPEC, allows oneto measure circularly polarized light (Mathys & Stenflo 1976).However, there are only a few instruments able to conveniently

1 Departement de Physique, Universite de Montreal, C.P. 6128, Succ. Centre-Ville, Montreal, QC H3C 3J7, Canada, [email protected]; and Ob-servatoire du Mont Megantic.

2 Visiting Astronomer, University of Toronto Southern Observatory, LasCampanas, Chile.

3 Department of Physics and Astronomy, University of Western Ontario,London, ON N6A 3K7, Canada.

4 Brandon University, 270-18th Street, Brandon, MB R7A 6A9, Canada.5 Uppsala University, Box 5151, S-751 20 Uppsala, Sweden.6 Deceased.7 Observatoire de la Cote d’Azur, 2130 route de l’Observatoire, Caussols,

06460 St. Vallier de Thiey, France.

measure both linearly and circularly polarized light together.A very recent example is the new spectropolarimeter at theBernard Lyot Telescope at Pic Du Midi (Donati et al. 1997).

Some stars are well known to show intrinsic linear (see, e.g.,St-Louis et al. 1987) as well as circular (see, e.g., Borra &Landstreet 1980) continuum polarization. The main sources ofintrinsic continuum polarization are electron (Thomson) and(less important) Rayleigh scattering in stellar atmospheres andenvironments, whereas nonrelativistic gyrating electrons inmagnetic fields yield broadband linear and circular polarization(see, e.g., Schmidt 1988). Interstellar grains normally lead tosmoothly wavelength-dependent linear polarization (Ser-kowski, Mathewson, & Ford 1975), with low-amplitude cir-cular polarization crossing over at linear peak (Martin & Camp-bell 1976). Intrinsic line polarization can occur due to theZeeman effect (circular and, to a lesser extent, linear, dependingon the magnetic field strength and orientation) and asymmetriesbetween line and continuum sources (linear). For example,emission lines formed further out in a stellar wind are generallyless polarized, leading to linear depolarization across the lineif asymmetries such as wind flattening are present (Schulte-Ladbeck, Meade, & Hillier 1992).

Spectropolarimetry is relatively rarely used in stellar astron-omy. In this paper we focus on spectropolarimetry of hot starsonly, for which the main difficulties are the following:

1. The degree of hot-star (from K for an A9T 5 7500eff



WILLIAM-WEHLAU SPECTROPOLARIMETER 1357

1998 PASP, 110:1356–1369

Fig. 1.—Simple sketch of the William-Wehlau Spectropolarimeter

star to 50,000 K for an O3 star) polarization is often much lessthan 1% in the continuum, 0.1% in lines (excluding Ap stars)which requires long exposure times and/or large telescopes.

2. If in a stellar atmosphere the source of electron (or Ray-leigh) scattering shows radial symmetry, most of the polari-zation cancels out. As a consequence, deviations from sphericalsymmetry or a structured wind are the fundamental conditionsfor detection of intrinsic, nonmagnetic linear polarization.

3. A global stellar magnetic dipole is the field configurationwith the highest probability of detection in circular polarization.Detection of higher order or localized dipole field geometriesis more difficult, requiring high spectral resolution aided bystellar rotation or wind expansion.

On the other hand, the step from measuring broadband tomeasuring narrowband (i.e., spectral) polarization allows oneto better disentangle the global structure of hot star winds and/or to provide more detailed information about magnetic fieldgeometries. Furthermore, it is of great interest to measure bothlinear and circular polarization at the same (or nearly so) time,since, e.g., in hot stars, the magnetic configuration producingcircular line polarization is expected to be associated with windasymmetries yielding linear polarization. For a more detailedreview about the use of spectropolarimetry in hot stars we referto Eversberg (1996).

2. THE POLARIMETER UNIT AND THE MUELLERCALCULUS

The newly built William-Wehlau Spectropolarimeter is acombination of a retarder consisting of two rotatable QWPsand a Wollaston prism as beam-splitter and double-beam po-

larizer, leading light from stellar point sources via a doublefiber-feed into a fixed CCD slit spectrograph. The instrumentwas developed and built at the University of Western Ontarioin collaboration with Brandon University (Manitoba) and Univ-ersite de Montreal (Quebec). Figure 1 shows the basic designof the polarimeter unit.

The WW Spectropolarimeter performs polarimetric analysisof starlight. It is designed to be mounted at the ≈f/8 Cassegrainfocus of medium- or large-size telescopes. Incoming starlightpasses through an input aperture (pinhole, normally with adiameter mm) and is collimated by an f/8 system ofD 5 200lenses. The collimated light then passes through the analyzingsection of the instrument consisting of two rotatable achromaticAR coated QWPs (controlled by a personal computer) and afixed Wollaston prism as polarizer. The independent rotationof the QWPs relative to the Wollaston prism determines thepolarization components that are being measured. The Wol-laston prism separates the components into two linearly po-larized beams whose polarization orientations are parallel (or-dinary ray, Ik) and perpendicular (extraordinary ray, I⊥) to thefast axis of the prism.

The two beams exit the Wollaston prism separated by anangle of 07.67, with each beam deviating the same amount fromthe optical axis. A zoom lens system then focuses the aperturefor each beam to mm at f/4 onto twin mmD 5 100 D 5 150optical fibers (silica, step-indexed, high OH, polymide buffer),which are separated by 1 mm. Therefore, light that is always≈100% polarized parallel to the fast axis is focused onto onefiber, while light that is always ≈100% polarized perpendicularto the fast axis is focused onto the other fiber. Note that adeviation between the orientation axes of the Wollaston prism



1358 EVERSBERG ET AL.

1998 PASP, 110:1356–1369

and the zero positions of the QWPs is necessary so that thiscan occur (see below).

The fibers then feed the light for each beam to the slit of aspectrograph, where the ≈f/4 emerging light from the fibers isreimaged to f/8 at the slit. The spectra of the light from eachfiber are imaged so that they are parallel and adjacent, withsufficient space between them on the CCD detector of thespectrograph. The two components can be combined over anobserving sequence to obtain all four Stokes parameters, whichfully define the polarization of the light. When the light isdispersed by the spectrograph, the polarization of the starlightmay be calculated as the wavelength-dependent quantities I(l),Q(l), U(l), and V(l) via the Mueller calculus (see below).

For calibration purposes and locating the zero position ofthe QWPs,8 a removable Glan-Taylor prism, which producesnearly 100% linearly polarized light at all optical wavelengths,can be inserted in front of the retarders. The fast axes of theQWPs must first be aligned with the symmetry axis of theWollaston prism to obtain simple orientation relations for thepolarimetric output. This can be entirely done automaticallyusing a computer algorithm or step by step by rotating eachQWP to locate intensity extrema of the 100% polarized beamexiting the Glan-Taylor prism. The polarization axis of theGlan-Taylor prism is normally aligned with the axis of theWollaston prism. This is necessary to get a simple output for100% polarized light (maximum in one beam, minimum in theother). The fiber positions are aligned with the Wollaston axisby shining light back through the fibers, whose orientation aswell as that of the Wollaston axis is adjusted so that one seestwo colinear pairs of beams arriving at the aperture plane(viewed by reflection from behind). The two inner beams ofeach pair must superimpose onto the aperture for correctalignment.

Following the Mueller calculus and the rules for matrix al-gebra, we can calculate the four Stokes parameters for thisarrangement with retardance t (ideally 907) for both QWPs.With A the input Stokes vector (I, Q, U, V) and A9 the outputStokes vector (I9, Q9, U9, V9) after passing the retarder plateswith matrices R1 and R2, and the polarizer with matrix P, wehave (writing A and A9 in vertical form)

′A 5 P 3 R 3 R 3 A. (1)2 1

Note that we can only actually measure intensities I9 of theoutput beams. The general forms of R1,2 and P (assumed to be

8 According to the manufacturer’s specifications, the currently installed ach-romatic QWPs have ideal retardances ( ) at two wavelengths only,Ct 5 90

and 6000 A. Between, and out to several 100 A beyond these values,l 5 4300the retardance varies in a simple but nonlinear fashion between 857 and 957.

well aligned in the beam) are given by Serkowski (1962):

R 5 [X X X X ], (2)1 2 3 4

where

10

X 5 ,1 0( )0

02cos 2w 1 sin 2w cos

X 5 ,2 (1 2 cos t) cos2w sin 2w( )sin 2w sin t

0(1 2 cos t) cos 2w sin 2w

X 5 ,3 2 2sin 2w 1 cos 2w cos t( )2 cos 2w sin t

02 sin 2w sin t

X 5 ,4 cos 2w sint( )cos t

and

1 cos 2J sin 2J 021 cos 2J cos 2J cos 2J sin 2J 0

P 5 , (3)2sin 2J cos 2J sin 2J sin 2J 02 ( )0 0 0 0

where w is the rotation angle of the retardation plate axis withrespect to the polarizer axis measured counterclockwise as seenlooking toward the sky, and J is the transmission angle of thepolarizer ( for k, for ⊥).C CJ 5 0 J 5 90

Following these expressions for different angular positionsof the QWPs and adopting for both plates, we get aCt { 90number of resulting equations for the final, observed Stokesparameters, of which the intensity I9 (after allowance for var-ious gain factors) contains all the information we need to deriveI, Q, U, and V of the original beam A. If we indicate the intensityin either of the detected output beams , with re-′ ′I , Iw ,w ,k w ,w ,⊥1 2 1 2

spect to the angular values of the two retarders, and orientationof the beams from the polarizer (k or ⊥), in this order, F is atime-dependent varying function, which is the same for eachbeam, here (e.g., atmospheric seeing, transparency), andFw ,w1 2

G is the time-independent gain factor for each beam (e.g., fiber




1998 PASP, 110:1356–1369

transmission, pixel sensitivity), we obtain, for example:9

1′I 5 (I 1 Q)F G , (4a)0,0,k 0,0 k2

1′I 5 (I 2 Q)F G , (4b)0,0,⊥ 0,0 ⊥2

1′I 5 (I 2 Q)F G , (4c)45,45,k 45,45 k2

1′I 5 (I 1 Q)F G , (4d)45,45,⊥ 45,45 ⊥2

1′I 5 (I 1 U)F G , (4e)0,45,k 0,45 k2

1′I 5 (I 2 U)F G , (4f)0,45,⊥ 0,45 ⊥2

1′I 5 (I 2 U)F G , (4g)0,245,k 0,245 k2

1′I 5 (I 1 U)F G , (4h)0,245,⊥ 0,245 ⊥2

1′I 5 (I 1 V )F G , (4i)245,0,k 245,0 k2

1′I 5 (I 2 V )F G , (4j)245,0,⊥ 245,0 ⊥2

1′I 5 (I 2 V )F G , (4k)45,0,k 45,0 k2

1′I 5 (I 1 V )F G . (4l)45,0,⊥ 45,0 ⊥2

With these equations we easily obtain the intensity-normalizedStokes parameters q, u, and :v

Q R 2 1Qq { 5 , (5a)I R 1 1Q

U R 2 1Uu { 5 , (5b)I R 1 1U

V R 2 1Vv { 5 , (5c)I R 1 1V

with

′ ′I I0,0,k 45,45,⊥R 5 , (6a)ÎQ ′ ′I I0,0,⊥ 45,45,k

′ ′I I0,45,k 0,245,⊥R 5 , (6b)ÎU ′ ′I I0,45,⊥ 0,245,k

′ ′I I245,0,k 45,0,⊥R 5 . (6c)ÎV ′ ′I I245,0,⊥ 45,0,k

9 Intermediate QWP positions are also usable, in principle, though requiringmore complicated calculations.

Note that these double ratios10 are impervious to both time-dependent variations and time-independent gain factors (no flat-fielding necessary), as long as the two beams are measuredsimultaneously on the same part of the detector each time. Thisis normally the case. They should therefore be purely photon-noise limited. The intensity I, within a constant factor (whichwill vary smoothly with wavelength, as for any spectrograph),is easy to get from a simple addition of both beams on a givenimage, after appropriate determination of the gain factors byflat-fielding. Since q, u, and are independent of these gainvfactors, it is clear that one can obtain much higher precisionfor them compared to I. Also note that any of the angles w ofthe l/4 plates can be replaced by with identical re-Cw 5 180sults, providing the surfaces of the plates are not inclined tothe optical axis (Serkowski 1974). This is shown in two-di-mensional mode in Table 1 for the two QWP position anglesw1 and w2 and the corresponding output .′ ′I /Ik ⊥

If t deviates from 907, the matrix R introduces cross-talk inthe output Stokes parameters. In our case, measured values forthe output using the Glan-Taylor prism deviate slightly′ ′I /Ik ⊥from expected values. This is shown in Table 2, where eachmatrix element consists of the theoretically expected valueabove the measured output value in parentheses.′ ′I /Ik ⊥

While modern achromatic polarizing elements (e.g., Glan-Taylor prisms, Wollaston prisms) can be manufactured to veryhigh tolerances, the same cannot be claimed for retarders, evenif “superachromatic” (V. Vats 1997, private communication).In an attempt to calculate the real retardance for each QWP ata fixed wavelength, we start with

′A 5 P 3 R 3 A. (7)

Using the Mueller matrices P and R, we have for the ordinarybeam

F Gw k′ 2 2I 5 [I 1 Q(cos 2w 1 sin 2w cos t)k 2

1U(1 2 cos t) cos 2w sin 2w 2 V sin 2w sin t], (8a)

and for the extraordinary beam

F Gw ⊥′ 2 2I 5 [I 2 Q(cos 2w 1 sin 2w cos t)⊥ 2

2U(1 2 cos t) cos 2w sin 2w 1 V sin 2w sin t]. (8b)

With 100% polarized light in Q (i.e., with the aligned Glan-

10 This is referred to as self-calibration by Miller et al. (1988).




1998 PASP, 110:1356–1369

TABLE 1Output Matrix for the Ratio and Different QWP Positions′ ′I /Ik ⊥

w2

w1

2907 2457 07 457 907 1357 1807 2257

21807 . . . . . .I1QI2Q

I1VI2V

I1QI2Q

I2VI1V

I1QI2Q

I1VI2V

I1QI2Q

I2VI1V

21357 . . . . . .I2UI1U

I1QI2Q

I1UI2U

I2QI1Q

I2UI1U

I1QI2Q

I1UI2U

I2QI1Q

2907 . . . . . . .I1QI2Q

I1VI2V

I1QI2Q

I2VI1V

I1QI2Q

I1VI2V

I1QI2Q

I2VI1V

2457 . . . . . . .I1UI2U

I2QI1Q

I2UI1U

I1QI2Q

I1UI2U

I2QI1Q

I2UI1U

I1QI2Q

07 . . . . . . . . . . .I1QI2Q

I1VI2V

I1QI2Q

I2VI1V

I1QI2Q

I1VI2V

I1QI2Q

I2VI1V

457 . . . . . . . . .I2UI1U

I1QI2Q

I1UI2U

I2QI1Q

I2UI1U

I1QI2Q

I1UI2U

I2QI1Q

907 . . . . . . . . .I1QI2Q

I1VI2V

I1QI2Q

I2VI1V

I1QI2Q

I1VI2V

I1QI2Q

I2VI1V

1357 . . . . . . . .I1UI2U

I2QI1Q

I2UI1U

I1QI2Q

I1UI2U

I2QI1Q

I2UI1U

I1QI2Q

Notes.—Output matrix for the ratio and different QWP positions according to eqs. (4a)–(4l), generalized to include all′ ′I /Ik ⊥useful angles that measure one Stokes polarization parameter at a time, but neglecting Gk/G⊥. The QWP angles refer to idealvalues w, not wobs; see text. Values of w do not run from 07 to 3607 for each plate, due to limitations in rotation of the QWPsin the WW spectropolarimeter (cf. Table 2).

TABLE 2Example Output Matrix

w2

w1

2907 2457 07 457 907 1357 1807 2257

21807 . . . . . .`

(141)1

(0.87)`

(66)1

(0.98)`

(33)1

(1.09)`

(435)1

(0.81)

21357 . . . . . .1

(1.06)`

(132)1

(1.52)0

(0.01)1

(0.79)`

(73)1

(1.16)0

(0.002)

2907 . . . . . . .`

(179)1

(0.92)`

(109)1

(0.88)`

(34)1

(1.14)`

(455)1

(0.75)

2457 . . . . . . .1

(1.26)0

(0.009)1

(0.85)`

(222)1

(1.63)0

(0.027)1

(1.16)`

(135)

07 . . . . . . . . . . .`

(476)1

(0.68)`

(62)1

(1.15)`

(90)1

(0.84)`

(370)1

(1.00)

457 . . . . . . . . .1

(0.94)`

(204)1

(1.32)0

(0.022)1

(0.69)`

(233)1

(1.04)0

(0.008)

907 . . . . . . . . .`

(93)1

(1.12)`

(556)1

(0.72)`

(29)1

(1.36)`

(112)1

(0.62)

1357 . . . . . . . .1

(1.20)0

(0.021)1

(0.81)`

(667)1

(1.52)0

(0.046)1

(1.15)`

(101)

Notes.—Example output matrix with the Glan-Taylor prism inserted ( , ) for the ratio and different QWP′ ′Q 5 I U 5 V 5 0 I /Ik ⊥positions in a narrow band centered near Ha. Each matrix element consists of the theoretical output (with ) and theG /G 5 1k ⊥measured value in parentheses. Angles have been corrected to coincide with appropriate extrema (w, not wobs, see Fig. 2).

Taylor prism) we have and , and we getU 5 V 5 0 Q 5 I

F G Iw k′ 2 2I 5 (1 1 cos 2w 1 sin 2w cos t) (9a)k 2

and

F G Iw ⊥′ 2 2I 5 (1 2 cos 2w 2 sin 2w cos t). (9b)⊥ 2

For , we have the ideal case:Ct 5 90

F G Iw k′ 2I 5 (1 1 cos 2w) (10a)k 2

and

F G Iw ⊥′ 2I 5 sin 2w. (10b)⊥ 2

However, and deviate slightly from the expected ideal′ ′I Ik ⊥output functions at certain angular positions. In detail withinthe constants Gk and G⊥, is conserved, whereas maxima′ ′I 1 Ik ⊥of and minima of appear to follow a simple sine wave′ ′I I⊥ k

with rotation angle. This behavior must obviously be causedby small deviations of t from 907 for both QWPs. By using

cos t 5 c 1 c sin w 1 c cos w, (11)1 2 3

we fitted equations (9a) and (9b) via x2 minimization, leadingto , ,(F G I/2, F G I/2) 5 (5303, 4559) c 5 0.050 c 5 2w k w ⊥ 1 2

, and for QWP 1 and0.049 c 5 20.0013

( , , ,F G I/2, F G I/2) 5 (5330, 4700) c 5 0.160 c 5 0.017w k w ⊥ 1 2

and for QWP 2. The fit also yielded a slight shiftc 5 0.0023

of the extrema from their theoretically expected positions, lead-ing to corrections 7.12 for QWP1 andw 5 0.994w 2 10 w 5obs

7.87 for QWP2. The deviation from unity of the0.999w 2 13obs




1998 PASP, 110:1356–1369

Fig. 2.—Output of the two beams with the Glan-Taylor prism, QWP 1 (left) and QWP 2 (right) and the Wollaston prism, in a narrow band centered near Ha.(Top panel) Open and filled circles, respectively, represent the measured average intensity in the ordinary beam ( ) and extraordinary beam ( ). Curves represent′ ′I Ik ⊥the fits using eqs. (9a), (9b), and (11) after allowing for a shift in angle to match best our data points. (Bottom panel) Top: residuals from a x2 minimum fit for

, as indicated for both QWPs. bottom: residuals from a x2 minimum fit using eq. (11). Values obtained for c1, c2, and c3 are indicated.t 5 constant

slopes is likely a consequence of imperfections in the steppingmotors.

To illustrate this for the WW polarimeter, we show in Figure2 for each QWP separately, combined with the aligned Glan-Taylor and Wollaston prisms, how the outputs and vary′ ′I Ik ⊥compared with predictions.

We discovered the reason for this behavior by sending alaser beam through each QWP independently. In transmission,the outcoming beams were geometrically highly stable whilerotating the plates, whereas in reflection both plates showedone or two beams that rotated in elliptical patterns around ageometrically stable central beam.

The variation of the effective retardation of the QWPs withrotation angle arises because of the varying angle of inclinationthat the collimated beam has with the QWP (or one of itsindividual layers) as the QWP is rotated. If the QWP weremounted in its cell so that the surfaces were not normal to theaxis of the cell and the cell axis were properly aligned withthe optical axis of the collimated beam, then that one misa-lignment by itself would produce no change in inclination asthe cell is rotated. If the cell were rotated about an axis thatis misaligned with the optical axis but the QWP were properlymounted normal to the axis of the cell then again there wouldbe no variation in the inclination angle with rotation as a resultof that misalignment alone. However, if both misalignmentsare present, i.e., the orientation of the QWP in its cell is notnormal to the axis of rotation and the axis of rotation is not

aligned with the optical axis, then we have an oblique rotator.As the cell is rotated, the angle of inclination will nod withrespect to the optical axis and the effective retardation willvary. If the misalignments are only a few minutes of arc, thenthe variation will be small, of course. In our case we foundthat the front surface of each multielement QWP was normalto the rotation axis of the cell but that the air gap betweensome of the additional components was a wedge so that someparts of the QWP participated in the oblique rotator behavior.This varying inclination introduces small angle-dependent de-viations from the mean retardance (see eqs. [8.3.2] and [8.3.3]in Serkowski 1974). A sine-wave behavior for t (or forcos t

t moderately close to 907) is thus quite reasonable, as foundabove.

The solution to this problem is simple. Either be more carefulthat there is no misalignment of any of the components of theQWPs in their cells or be careful that the rotation axis is quiteprecisely aligned with the optical axis. Either will suffice butobviously if both are accomplished the effect we have seenwill be removed. It is important to note that we chose to useQWPs with an air gap so that even with variation in the angleof the surfaces of the components to the incident angle of thetransmitted beam in a collimated beam situation, there wouldbe absolutely no movement of the image formed by the cameraoptics. A QWP that is inclined to the collimated beam wouldproduce a displacement of the collimated beam but no deviationin direction. That is a consequence of the fact that a QWP must




1998 PASP, 110:1356–1369

Fig. 3.—Stokes q, u, and for Sirius (a CMa) observed with the Glan-vTaylor prism at UTSO. Average values in the wavelength range 5200–6200A are indicated. Note that should be zero for a perfect instrument.v

have, by its very nature, surfaces that are parallel to one anotherto a very high degree of precision.

It is not obvious that superachromatic retarders (althoughgiving better tolerances on t close to 907, compared to theachromats here) would behave much better (Donati et al. 1997),since they are even more complex, requiring more dielectriclayers (V. Vats 1997, private communication).

We also obtained a number of exposures of bright stars withthe Glan-Taylor prism to measure the overall efficiency andpossible cross-talk between different Stokes parameters at dif-ferent wavelengths. As seen in Figure 3, cross-talk betweendifferent polarization parameters shows wavy structures as afunction of wavelength, with an amplitude that never exceeds4%. Some of the nonzero offset in might be due to a slightumisalignment in the rotation of the Glan-Taylor prism. We can,however, exclude this as the reason for the imperfect behaviorof retardance t with angle and wavelength. We find similarbehavior also in our UTSO data in the blue.

The bottom line here is that we can probably eliminate thesedeviations by applying first-order corrections and/or using bet-ter QWPs. For our described observations with imperfectQWPs we minimize the impact on deviations in the final Stokesvalues by choosing ranges of rotation for the QWPs, where thetwo required ratio peaks are more nearly the same and overalldeviations from are smallest.Ct 5 90

3. OBSERVATIONS

In order to test the WW polarimeter on the sky, we haveobserved a number of different hot star prototypes to covervarious spectropolarimetric outputs. These include the brightestBe star visible in the sky g Cas (B0 IVe), the young variableOrion Trapezium star v1 Ori C (O7 V) and the strongly mag-netized Ap star 53 Cam (A2p). These were observed duringtwo runs at the 1.6 m telescope of the Observatoire du montMegantic (OMM) in 1997 September and December using theWilliam-Wehlau Spectropolarimeter at the Cassegrain focus.With the 600 l/mm grating of the fiber-fed spectrograph weobtained a net (FWHM) resolution of 1.9 A in the wavelengthrange 5800–7100 A, to cover mainly Ha, He i l5876, and Hei l6678.

We have also observed the 78 day WR 1 O binary g2 Vel(WC8 1 O9 I) during a five week run in 1997 February/Marchat the University of Toronto Southern Observatory (UTSO) atLas Campanas, Chile. We obtained spectra during 21 nightswithin of periastron passage. Using the GarrisonDf ∼ 50.2spectrograph the 3 pixel spectral resolution was about 6 A inthe effective wavelength region 5200–6000 A covering mainlythe He ii l5411, C iii l5696, C iv ll5802/12, and He i l5876emission lines.

The very first step for each night was the calibration of theQWP positions. For this we sent an artificial light sourcethrough the system including the Glan-Taylor prism. With100% input polarized light, we located where the intensities

and reached extreme values. Actually, our computer pro-′ ′I Ik ⊥gram is able to find these extrema by itself. However, we founda small but significant inconsistency in the computer outputpositions, probably introduced by the chromatic- and angle-dependent nature of the quarter-wave plates.

For each program star we obtained a number of exposuresequences, each containing two, four, or six spectral images atdifferent QWP positions to measure (1) , (2) q(l) and u(l),v(l)or (3) q(l), u(l), , respectively, as indicated above (cf. eqs.v(l)[4a]–[6c]). One sequence was obtained with the Glan-Taylorprism when necessary, to correct for possible cross-talk be-tween different Stokes parameters and to evaluate the overallefficiency of detecting polarization. Complete sequences fornonpolarized standard stars (e.g., b Cas at OMM and a CMaat UTSO) were also observed. Polarized standards are generallytoo faint to be observed for our purpose, compounded with aproblem of reliably measuring continuum polarization (seebelow).

A number of flat-field images, wavelength comparison spec-tra, and bias images were obtained at the beginning and endof each night. The procedures for flat-fielding and wavelengthcalibration were different at each telescope:

At UTSO we used an internal Tungsten lamp in the opticalfeeding box between the polarimeter and the telescope for flat-fielding. A Fe-Ne lamp in the same feeding box was used forwavelength calibration.

At OMM we used the auto-guider of the telescope. For thisreason we could not use the feeding optics and obtained domeflats in the usual manner. For wavelength calibration we useda Cu-Ar lamp in the spectrograph.

As one can see, light from the comparison lamp at UTSO




1998 PASP, 110:1356–1369

passes through the whole instrument, whereas at OMM thecomparison light originated in the spectrograph. Despite thesedifferent configurations, we found no significant differencesbetween these techniques.

4. DATA REDUCTION

As one can see from the final equations, a full observingsequence for all four Stokes parameters consists of six expo-sures at different QWP positions. Each exposure results in atwo-dimensional CCD image with two spectra, one for theordinary beam and one for the extraordinary beam.

After bias subtraction, the possibility of blending betweenthe two beams (see Harries 1995 in the case of an echellespectrograph, where free space between adjacent orders is lim-ited) was considered. Because of ample separation between thetwo fibers, both where they receive and deliver the light beams,blending is negligible in our case. In addition we checked forany shift of the spectral pair from one image to another on theCCD chip. This is necessary to avoid division by differentpixels, and hence different sensitivities on the chip, within thesame observing sequence. For this reason we cross-correlatedthe first image in a given fixed wavelength range with all otherimages in a direction perpendicular to the dispersion. The shiftwas always found to be negligibly small.

The next step was to check for deviations from ideal geo-metrical orientation of the spectra on the CCD chip, in orderto avoid problems in tracing the apertures when collapsing thespectra (see Donati et al. 1997). The angle between the dis-persion direction and the CCD axis was small: 17.0 at UTSOand 07.14 at OMM. With such small angles, we do not have toworry about significant spectral smearing during the extraction.

The reduction procedure was carried out with IRAF and itsstandard tools. One problem was the strong asymmetric shapeand its slight variation with wavelength of one of the spectrumpairs perpendicular to the dispersion in the UTSO data. TheIRAF/APSUM task traces the line peak and averages flux per-pendicular to the dispersion direction, so that one might expectglitches in tracing the aperture as the peak shifts along thespectrum. After trying different techniques for the two 10 pixelwide spectra separated center-to-center by about 30 pixels, wedecided to choose broad, 30 pixel windows and use the firstnightly spectrum as a template for all other spectra in the samenight in the same wavelength range, in order to guarantee di-vision always by the same pixels according to equations(6a)–(6c). Note that we have ratioed the two spectra to obtainq(l), u(l), and after collapsing to one dimension, ratherv(l)than in the two-dimensional domain. This is quite acceptable(and simplifies the extraction technique), as pointed out belowin the context of obtaining I(l). The above procedure was alsoused for our OMM data, with a window width for collapsingof 30 pixels. We note that tracing the apertures was not aproblem here because of symmetric shapes of our two spectraperpendicular to the dispersion.

Cosmic-ray (CR) rejection was done iteratively. At first CRswere identified by eye on the original image and rejected byreplacing their intensities by the mean of neighboring unaf-fected pixels. After collapsing the spectra, any residual spikesin the one-dimensional spectra were deleted by using again themean of neighboring unaffected pixels after reidentifying themin the two-dimensional input images as cosmic rays. Ratioingthe spectra as necessary to obtain q, u, and has the advantagevthat cosmic rays, found usually only in one aperture at a time,appear as even stronger, easy to identify spikes relative to thenormally uncomplicated q, u, spectra, after double-ratioing.v

We also subtracted the image background, consisting mostlyof dark current, which was extremely small compared to thestellar source. For this, we used a median grid of 5 pixels inwindows of 10 pixel width centered at 25 pixels on either sideof each beam and a background fit in the dispersion directionof third order. The background level increases linearly withexposure time. This also applies to the OMM data, but with abackground window of 30 pixel width.

As a final test we extracted the spectra by using simpleaverages of image windows independent of IRAF/APSUM. Wecalculated a 20 pixel wide average on the spectra and subtracted10 pixel wide background averages on both sides of each beam.There is no significant difference between this and the IRAFresult. For this reason we took the easy route by extracting allthe spectra in one step within IRAF.

Because I(l) cannot be obtained from double ratios as canq, u, and , but must be obtained from simple additions of twovspectra in one image, we are obliged to reduce pixel-to-pixelvariations I9(l) by flat-fielding. This reduction step was doneafter collapsing the flats to one-dimensional spectra using thesame nightly template as for starlight. This is possible becausethe two beams always have the same relative pixel-to-pixelillumination, even always the same ≈100% linear polarizationinput into each of the ordinary and extraordinary beam fibers.The only thing that varies at a given wavelength is the relativeintensity of each beam, regardless of the spectral and polari-metric details of the stellar source.

The wavelength calibration was carried out for each beamseparately using the same nightly stellar extraction template.This was done before double-ratioing the spectra to avoid anyshift in wavelength space (e.g., due to the dispersion directionbeing inclined to the CCD axis) between the two spectra. Forbest results we determined individual residuals of each arc withrespect to its rest wavelength and eliminated strong deviators.A fifth-order Legendre polynomial fit related pixel positions towavelength. The wavelength drift between comparison spectrataken at the beginning and the end of the night was found tobe negligibly small. This is not surprising, in view of the stableconfiguration with the fiber-fed spectrograph fixed in the dome.

After computing all Stokes parameters for g2 Vel we foundthat the continuum polarization varies from one polarizationsequence to the next. Further examination revealed that thisoccurs for all observed stars and thus must be instrumental in




1998 PASP, 110:1356–1369

Fig. 4.—(Left) Six consecutive observing sequences of Stokes q, u, and for g2 Vel during one night at UTSO (hand-guided). (Right) Seven consecutivevsequences of g Cas during one night at OMM (auto-guided).

Fig. 5.—Three consecutive observing sequences of Stokes q, u, and forvdome flats at OMM. Average values for q, u, and in the wavelength rangev6050–6850 A are indicated. The nonzero means in q and u are likely due toa small level of linear polarization in the dome flat light.

origin. The continuum polarization varies in such a way thatstrong stochastic deviations from the mean arise, combinedwith varying gradients dq/dl, du/dl and d /dl. Two examplesvfor consecutive sequences of Stokes q, u, and obtained atvUTSO and OMM are given in Figure 4.

A typical stochastic rms scatter of ∼1% broadband contin-

uum polarization is found in q, u, and at both telescopes. Thevmost likely explanations for this are due to (1) small variationsin overall illumination of the two fibers, whose spatial surfacesensitivities are never perfectly uniform, and (2) possible me-tallic aperture edge effects, both caused ultimately by seeing/guiding fluctuations. The first effect dominates in our data,since it appears equally in q, u, and , whereas the second effectvis only expected in q and u. These two effects introduce timedependences in the gain factors, G (see eqs. [4a]–[4l] above)that are different for the two beams. These effects also appearto be worse at blue wavelengths. We believe this is due towavelength-dependent sensitivity at the fiber-face, which de-grades gradually toward shorter wavelengths, as is well knownfor fiber transmission. Note that at OMM we used the auto-guider, whereas at UTSO guiding was done by eye, leading tosomewhat better results at OMM. We find support for the as-sumed instrumental origin of this scatter by obtaining all Stokesparameters for dome flats, which illuminate the fibers moreuniformly and invariably with time. As one can see in Figure5, the broadband output is stable within ∼0.04%, i.e., the Pois-son noise level.

Donati et al. (1997) have had the same experience with theirtwin fiber-fed spectropolarimeter. This appears to be a generallimitation of such double-fiber configurations. In our polar-imeter the pinhole-aperture is imaged onto the fibers. A possible(but difficult) way to avoid variations in broadband polarizationcould be via imaging of a pupil instead of the star onto thefibers, using Fabry lenses, and via use of a nonmetallic aperture.

As a starting point to circumvent the problem of instrumentalvariations in continuum polarization, we applied a weighted




1998 PASP, 110:1356–1369

Fig. 6.—Simulation of polarization features with artificially broadened spectra, Sl (see text). (Left) Bottom: mean spectrum of h Cen. Top: Stokes (solid line)vand Sl (dashed line). (Right) Bottom: mean spectrum of g2 Vel. Top: Stokes q, u, and (solid line) and Sl (dashed line).v

linear fit to all UTSO q(l), u(l), (l) spectra, yielding a slopevand zero point, neglecting interstellar polarization. (For theOMM data we used a parabolic fit.) Along with other obser-vations, the data in Figure 4 give the impression that the curvestend to converge to the red. In fact, we found a fairly clearcorrelation between the slopes and their corresponding zeropoints. However, when applied to the data, the noise of thiscorrelation would introduce continuum variations of up to 2%,which is not acceptable. Thus, we cannot estimate the exactcontinuum polarization. For this reason we simply subtracteda fit to the original individual Stokes q, u, and spectra inveach sequence and thus neglected broadband polarization, whencombining sequences to get mean q(l), u(l), and spectra.v(l)This procedure allows us to obtain reliable, empirical estimatesof the scatter in polarization as a function of wavelength. Onthe other hand, small-scale relative variations of polarizationwith wavelength (i.e., line polarization), the main goal of thisinstrument, are impervious to this broadband effect.

After combining to average q, u, and spectra for variousvstars of the whole UTSO run, we found Mexican-hat featuresat atomic line positions in all observed absorption lines in allStokes spectra. The amplitude was typically ∼0.1% polarizationfor narrow absorption lines of depth ∼0.9 continuum. This issmall but quite significant and difficult to see even in nightlymean spectra. Apparently, the beam that contains enhancedintensity ( , , )—even for q 5 u 5 v 5I 1 Q I 1 U I 1 V0—broadens the stellar spectrum, regardless of whether in theordinary beam or in the extraordinary beam. This induces ar-

tificial features in Stokes q, u, and at wavelengths where thevintensity spectra have strong gradients dI/dl. The origin of thisbehavior is completely unknown. It is not intrinsic to the stars,since it occurs in all lines the same way in q, u, . If it werevdue to the spectrograph, it should have cancelled out in thedouble ratio in equations (6a)–(6c), which is not the case. Wefirst presumed that one or both of the QWPs introduce thisphenomenon. However, these features do not appear in ourOMM data and it is more likely that our setup at UTSO in-troduced this behavior in a way that lacks an obviousexplanation.

In an attempt to eliminate this effect in the UTSO data, weconvolved the mean intensity spectrum Il with a Gaussian Gl

containing j as a free parameter. From this we obtained amodified spectrum Sl:

IlS 5 . (12)l I ^ Gl l

This function gives the form of a Mexican hat for narrow linesand resembles the observed spurious line polarization verywell. We obtained a best fit with pixel (∼0.68 A) forj 5 0.37all stars. Examples of this procedure are given in Figure 6.

After subtraction of Sl from individual nightly mean Stokesspectra at UTSO, we obtained the final wavelength-dependentStokes parameters q(l), u(l), and . To increase the pre-v(l)cision without significant loss in resolution we binned all Stokes




1998 PASP, 110:1356–1369

Fig. 7.—Mean rectified intensity spectrum of g2 Vel and mean normalizedStokes parameters q, u, and for the whole run combined, after removal ofvthe spectral broadening effect. The polarimetric data have been binned by a3 pixel boxcar; 2 j error bars are indicated.

Fig. 9.—Mean rectified intensity spectrum of 53 Cam and mean normalizedStokes for December 21. The polarimetric data have been binned to 5 pixels;v2 j error bars are indicated.

Fig. 10.—Mean rectified intensity spectrum including Ha, and mean Stokesq, u, and for v1 Ori C; 2 j error bars are indicated for bins of 5 pixels.v

Fig. 8.—Global mean rectified intensity spectrum including Ha, and meanStokes q, u, and for g Cas; 2 j error bars are indicated for bins of 5 pixels.v




1998 PASP, 110:1356–1369

spectra to an effective resolution of 2 A for the OMM dataand 6 A for the UTSO data.

5. RESULTS

5.1. g2 Velorum

As noted in § 3, spectropolarimetric data of g2 Vel wereobtained during an extended run at UTSO in 1997 February/March. In this nearby binary system with an orbital period of78.5 days (Schmutz et al. 1997, who also give a completedescription of the orbit), the two wind components create ashock cone around the weaker wind O star (St-Louis, Willis,& Stevens 1993). The variable excess emission in the lines ofC iii l5696 and C iv l5808 are presumed to be created byradiative cooling downstream along the cone. The opening an-gle of the cone depends on the relative wind momentum fluxesof the two stars. This shock cone introduces a deviation fromspherical symmetry in the wind so that, in principle, it couldlead to phase-dependent intrinsic continuum polarization. Evenmore important to create phase-dependent continuum polari-zation, however, is the scatter of O-star light from free electronsin the asymmetrically located WR wind (see St-Louis et al.1988 for g2 Vel and other WR 1 O binaries). On the otherhand, WR emission-line flux in a binary will be essentiallyunpolarized, so that phase-dependent variations of depolari-zation will occur at the positions of the emission lines (seeMoffat & Piirola 1994). We note that constant linear line de-polarization can also occur for a flattened WR wind (Schulte-Ladbeck et al. 1992).

Our observations were carried out over orbital phases cen-tered on periastron passage, i.e., . Figure 7 showsf 5 0.0 5 0.2all four obtained Stokes parameters I, q, u, and , averagedvover the whole run.

Orbital phase effects have probably diluted the net line po-larization, which is expected to be largest at orbital phases whenthe stars are seen at quadrature. Unfortunately, individualnightly mean spectra are not of sufficient S/N to extract thisinformation. We report significantly different values in bothStokes q and u across strong emission lines compared to thecontinuum. Since g2 Vel is known to show phase-dependentbroadband polarization, especially near periastron (St-Louis etal. 1988), it is reasonable to assume that we are seeing linedepolarization effects here. We draw attention to additionallinear depolarization in the electron scattering (redward) wingof C iii l5696.

We find no significant circular polarization across any linesalong the whole mean spectrum, above the instrumental levelof ∼0.03%.

5.2. g Cassiopeiae

We have observed g Cas during three nights in 1997 Sep-tember at the mont Megantic Observatory. g Cas is one of thebest stellar candidates to test for (relatively strong) linear linepolarization and our initial goal was to make a comparison with

previous observations. This star is well known to show lineardepolarization over its Ha line (Poeckert & Marlborough1977). This is explained by an extended equatorial disk (Han-uschik 1996) that scatters and linearly polarizes photosphericlight, combined with the emission of nonpolarized light fromrecombination lines (Schulte-Ladbeck et al. 1992). g Cas isalso a strong candidate for localized hot flares and magneticfields (Smith 1998).

Because we are not able to reliably estimate continuum po-larization, we can only give relative polarization values acrossspectral lines. We obtain values for Dq and Du across Ha,equivalent to the values observed some 20 years ago by Poeck-ert & Marlborough (1977). In Figure 8 we show the three-night average of all four Stokes parameters, with a typical rmsscatter of 0.04% for each 5 pixel bin.

5.3. 53 Cam

The Ap star 53 Cam is well known to show an extraordinarilystrong effective magnetic field, the strength of which is rota-tionally modulated between 14 kG and 25 kG, with a cor-responding dipole field of 228 kG (Borra & Landstreet 1977).Thus, this star—despite its relative faintness—was our firstchoice to test our instrument for sensitivity to rotation mod-ulated Stokes across its photospheric absorption lines. Forvthis reason we observed 53 Cam in the Ha region in 1997December at OMM. Only one night was clear enough to ob-serve this star over a sufficiently long enough interval (6 hr)to obtain high accuracy in Stokes . This night was Decemberv21/22 at phase (Landstreet 1988) after crossover. Thef 5 0.41average intensity and Stokes spectra are shown in Figure 9.v

Though 1 j error bars are about 0.1% in Stokes , we reportvsignificant rise and fall of ∼0.15% on the blue and red sides,respectively, of the center of Ha, as expected at this phase fromprevious work (Borra & Landstreet 1980).

5.4. v1 Ori C

The brightest star in the Orion trapezium is the O7 V starv1 Ori C. This star, one of the closest O stars in the sky, is theprinciple ionization source of M42 and spectroscopically var-iable in the optical (Conti 1972; Walborn 1981) as well as inthe UV (Walborn & Panek 1984). Stahl et al. (1993) and Wal-born & Nichols (1994) report an asymmetry in its wind and astable 15.4 day modulation over at least 15 years. They con-clude that this behavior is reminiscent of a magnetic obliquerotator with a surface field of about 1800 G. Reported X-raymodulation in the 15 day cycle (Gagne et al. 1997; Babel &Montmerle 1997) is compatible with a surface field of several100 G. This proposed magnetic field and the wind asymmetrymakes v1 Ori C a strong candidate for showing phase-dependentcircular as well as linear polarization effects across its lines.

We observed this star during two nights (December 21/22and 22/23) at OMM. The average intensity and Stokes q andu (second night only) and (1st night only) spectra are shownv




1998 PASP, 110:1356–1369

Fig. 11.—William H. Wehlau (1926–1995)

in Figure 10. Note that the nebular Ha emission in I(l) hasnot been removed.

No obvious line polarization features are seen above the% instrumental level.j ∼ 0.2

6. SUMMARY

We have introduced a new type of polarimeter unit, which,mounted at the Cassegrain focus of any telescope and combinedwith a standard CCD spectrograph, is able to measure all fourStokes parameters as a function of wavelength in a quasi-si-multaneous manner. The combination of two independentlyrotatable quarter-wave plates with a Wollaston polarizer asbeam splitter creates a double beam that contains all polari-metric information about the stellar target. An extraction tech-nique is devised that leads to spectra in q, u, and , whosevprecision is limited only by stochastic photon noise (Poissonerrors), whereas intensity spectra are limited by flat-field noise.We also describe different deviations from ideal for the opticalelements. We discuss the behavior of “achromatic” QWPs withrespect to chromatism and rotation position, as well as limi-tations in the use of optical fibers.

During three extended runs at two different telescopes weobtained polarimetric spectra for all four Stokes parameters ofseveral hot star prototypes with the new William-Wehlau Spec-tropolarimeter. Despite some instrumental problems, we seeboth line linear depolarization due to Thomson scattering andcircular polarization due to Zeeman splitting.

1. For the WR 1 O binary g2 Vel we report polarizationfeatures across its strong emission lines. Schulte-Ladbeck etal. (1992) showed that this is due either to a flattened WR windand/or phase-dependent asymmetries between the O-star andthe scatterers mainly in the WR wind.

2. We confirm previously measured polarization variationsacross the Ha emission line of the brightest Be star visible inthe sky g Cas, but with higher precision.

3. We detect a circular polarization feature in Ha in thestrong magnetic Ap star 53 Cam.

4. For the bright Trapezium star v1 Ori C we fail to findany polarization feature in Ha at the 0.2% level. Future at-tempts will have to improve on this precision and secure betterphase coverage.

These results establish the William-Wehlau Spectropolari-meter as a viable new instrument for measuring all wavelength-dependent Stokes parameters of stars. Some improvements willbe attempted, e.g., better QWPs, dielectric aperture, and morestable illumination of the fibers. Future preference will be givento large telescopes, because of the small amount of polarizationin stellar objects.

The authors would like to thank Bob Garrison for the gen-erous allotment of observing time at UTSO, Steve Steeles forhis continuous support during observations at UTSO, as wellas Bernard Malenfant, Ghislain Turcotte, and Johannes Vorbergat OMM, and Henry Leparskas at Elginfield Observatory. Wethank John Landstreet and Rene Racine for helpful discussionsduring the development of our instrument, and Jean-FrancoisBertrand for help in Fortran programming. T. E. is grateful forfull financial aid from the Evangelisches Studienwerk/Ger-many, which is supported by the German Government. A. F.J. M., P. B., and J. B. R. thank NSERC (Canada) for financialassistance. A. F. J. M. and P. B. also acknowledge monetaryaid from FCAR (Quebec).

T. E., A. F. J. M., M. D., J. B. R., N. P., and P. B. wish tonote that the original principal investigator for this instrument,William H. Wehlau, unfortunately passed away in 1995 Feb-ruary while attending a scientific meeting in Cape Town, SouthAfrica. We wish to express our deepest respect for his highlevel of competence and leadership during the planning andearly construction stages of the spectropolarimeter that nowcarries his name (Fig. 11).




1998 PASP, 110:1356–1369

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observing hot stars in all four stokes parameters

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