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GAS GIANT PLANETS

Saturn’s magnetic field revealedby the Cassini Grand FinaleMichele K. Dougherty*, Hao Cao, Krishan K. Khurana, Gregory J. Hunt,Gabrielle Provan, Stephen Kellock, Marcia E. Burton, Thomas A. Burk,Emma J. Bunce, Stanley W. H. Cowley, Margaret G. Kivelson,Christopher T. Russell, David J. Southwood

INTRODUCTION: Starting on 26 April 2017,the Grand Finale phase of the Cassini missiontook the spacecraft through the gap betweenSaturn’s atmosphere and the inner edge of itsinnermost ring (the D-ring) 22 times, endingwith a final plunge into the atmosphere on15 September 2017. This phase offered an op-portunity to investigate Saturn’s internal mag-netic field and the electromagnetic environmentbetween the planet and its rings. The internalmagnetic field is a diagnostic of interior struc-ture, dynamics, and evolution ofthe host planet. Rotating convec-tive motion in the highly electrical-ly conducting layer of the planetis thought to maintain the mag-netic field through the magneto-hydrodynamic (MHD) dynamoprocess. Saturn’s internal magnet-ic field is puzzling because of itshigh symmetry relative to the spinaxis, known since the Pioneer 11flyby. This symmetry prevents anaccurate determination of the rota-tion rate of Saturn’s deep interiorand challenges our understandingof the MHD dynamo process be-cause Cowling’s theorem precludesa perfectly axisymmetric magneticfield being maintained through anactive dynamo.

RATIONALE: The Cassini fluxgatemagnetometer was capable ofmea-suring themagnetic fieldwith a timeresolution of 32 vectors per s and upto 44,000 nT, which is about twicethepeak field strength encounteredduring theGrandFinale orbits. Thecombination of star cameras and gy-roscopes onboardCassini providedthe attitudedetermination requiredto infer thevector components of themagnetic field. External fields fromcurrents in themagnetosphereweremodeled explicitly, orbit by orbit.

RESULTS: Saturn’s magnetic equator, wherethe magnetic field becomes parallel to the spinaxis, is shifted northward from the planetaryequator by 2808.5 ± 12 km, confirming thenorth-south asymmetric nature of Saturn’smagnetic field. After removing the systematicvariation with distance from the spin axis, thepeak-to-peak “longitudinal” variation in Saturn’smagnetic equator position is <18 km, indicatingthat the magnetic axis is aligned with the spinaxis to within 0.01°. Although structureless in

the longitudinal direction, Saturn’s internalmagnetic field features variations in the lat-itudinal direction across many different char-acteristic length-scales. When expressed inspherical harmonic space, internal axisymmetricmagnetic moments of at least degree 9 areneeded to describe the latitudinal structures.Because there was incomplete latitudinal cov-erage during the Grand Finale orbits, whichcan lead to nonuniqueness in the solution,

regularized inversion tech-niques were used to con-struct an internal Saturnmagnetic field model upto spherical harmonicdegree 11. This modelmatches Cassini measure-

ments and retains minimal internal magneticenergy. An azimuthal field component twoorders of magnitude smaller than the radialand meridional components is measured onall periapses (closest approaches to Saturn).The steep slope in this component and mag-netic mapping to the inner edge of the D-ringsuggests an external origin of this component.

CONCLUSION: Cassini Grand Finale ob-servations confirm an extreme level of axi-

symmetry of Saturn’s internalmagnetic field. This implies thepresence of strong zonal flows (dif-ferential rotation) and stable strat-ification surrounding Saturn’s deepdynamo. The rapid latitudinal var-iations in the field suggest a sec-ond shallow dynamo maintainedby the background field from thedeep dynamo, small-scale helicalmotion, and deep zonal flows inthe semiconducting region closerto the surface. Some of the high-degree magnetic moments couldresult from strong high-latitudeconcentrations of magnetic fluxwithin the planet’s deep dynamo.The periapse azimuthal field orig-inates from a strong interhemi-spherical electric current systemflowing along magnetic field linesbetween Saturn and the inner edgeof the D-ring, with strength com-parable to that of the high-latitudefield-aligned currents (FACs) asso-ciated with Saturn’s aurorae.▪

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A meridional view of the results of the Cassini magnetometerobservations during the Grand Finale orbits. Overlain on thespacecraft trajectory is the measured azimuthal field from the firstGrand Finale orbit, revealing high-latitude auroral FACs and a low-latitudeinterhemispherical FAC system. Consistent small-scale axisymmetricinternal magnetic field structures originating in the shallow interior areshown as field lines within the planet. A tentative deep stable layer and adeeper dynamo layer, overlying a central core, are shown as dashedsemicircles. The A-, B-, C-, and D-rings are labeled, and the magneticfield lines are shown as solid lines. RS is Saturn’s radius, Z is the distancefrom the planetary equator, r is the perpendicular distance from thespin axis, and Bf is the azimuthal component of the magnetic field.

The list of author affiliations is available in the fullarticle online.*Corresponding author. Email: m.dougherty@imperial.ac.ukCite this article as M. K. Dougherty et al.,Science 362, eaat5434 (2018). DOI: 10.1126/science.aat5434

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RESEARCH ARTICLE◥

GAS GIANT PLANETS

Saturn’s magnetic field revealedby the Cassini Grand FinaleMichele K. Dougherty1*, Hao Cao2,3, Krishan K. Khurana4, Gregory J. Hunt1,Gabrielle Provan5, Stephen Kellock1, Marcia E. Burton6, Thomas A. Burk6,Emma J. Bunce5, Stanley W. H. Cowley5, Margaret G. Kivelson4,7,Christopher T. Russell4, David J. Southwood1

During 2017, the Cassini fluxgate magnetometer made in situ measurements of Saturn’smagnetic field at distances ~2550 ± 1290 kilometers above the 1-bar surface during22 highly inclined Grand Finale orbits. These observations refine the extreme axisymmetryof Saturn’s internal magnetic field and show displacement of the magnetic equatornorthward from the planet’s physical equator. Persistent small-scale magnetic structures,corresponding to high-degree (>3) axisymmetric magnetic moments, were observed.This suggests secondary shallow dynamo action in the semiconducting region of Saturn’sinterior. Some high-degree magnetic moments could arise from strong high-latitudeconcentrations of magnetic flux within the planet’s deep dynamo. A strong field-alignedcurrent (FAC) system is located between Saturn and the inner edge of its D-ring, withstrength comparable to the high-latitude auroral FACs.

The internal magnetic field of a planet pro-vides a window into its interior (1). Saturn’smagnetic field was first measured by thePioneer 11 magnetometers (2, 3) and shortlyafterward by Voyagers 1 and 2 (4, 5). Those

observations (2–7) revealed that Saturn’s magneticaxis is tilted with respect to the planetary spinaxis by less than 1°. The axisymmetric part ofSaturn’s internal field is more complex than adipole (commonly denoted with the Gauss co-efficient g1

0), with the axisymmetric quadrupoleand octupole magnetic moment contributions(coefficients g2

0 and g30, respectively) amount-

ing to ~10% of the dipole moment contributionwhen evaluated at the 1-bar surface of the planet.Three decades later, Cassini magnetometer (MAG)measurements (8–12) made before the GrandFinale phase refined Saturn’s dipole tilt to <0.06°(11). Secular variation is at least an order of mag-nitude slower than at Earth (11), and magneticflux is expelled from the equatorial region towardmid-to-high latitude (12). Magnetic moments be-yond spherical harmonic degree 3, with Gausscoefficients on the order of 100 nT, were sug-

gested by the MAG measurements during CassiniSaturn Orbit Insertion (SOI) in 2004 (12) butwere not well resolved because of the limitedspatial coverage of SOI, with periapsis (closestapproach) at 1.33 RS (where 1 RS = 60,268 kmis the 1-bar equatorial radius of Saturn) andlatitudinal coverage within ±20° (12).Saturn’s main magnetic field is believed to

be generated by rotating convective motion inthe metallic hydrogen (13, 14) layer of Saturnthrough the magnetohydrodynamic (MHD) dy-namo process (1, 15–17). The highly symmetricnature of Saturn’s internal magnetic field withrespect to the spin axis defies an accurate de-termination of the rotation rate of Saturn’s deepinterior and challenges our understanding of theMHD dynamo process because Cowling’s theorem(18) precludes the maintenance of a perfectlyaxisymmetric magnetic field through an activeMHD dynamo. The possibility that we are ob-serving Saturn’s magnetic field at a time withno active dynamo action can be ruled out giventhe sizable quadrupole and octupole magneticmoments because these would decay with timeat a much faster rate than that of the dipole [(19),chapter 2]. In addition to the main dynamoaction in the metallic hydrogen layer, deepzonal flow (differential rotation) and small-scaleconvective motion in the semiconducting regionof Saturn could lead to a secondary dynamoaction (20).We analyzed magnetometer measurements

made during the last phase of the Cassini mis-sion. The Cassini Grand Finale phase began on26 April 2017, with the spacecraft diving throughthe gap between Saturn’s atmosphere and theinner edge of its D-ring 22 times, making a finaldive into the atmosphere on 15 September 2017.

Periapses were ~2550 ± 1290 km above the at-mosphere’s 1-bar level, equivalent to ~1.04 ±0.02 RS from the center of Saturn (fig. S1). Thelow periapses, high inclination (~62°), and prox-imity to the noon-midnight meridian in localtime provided an opportunity to investigateSaturn’s internal magnetic field and the mag-netic environment between Saturn and its rings(figs. S1 and S2). The fluxgate magnetometer (8)made continuous (32 measurements per second)three-component magnetic field measurementsuntil the final second of spacecraft transmission.For analysis, we averaged the data to 1-s reso-lution. During ~±50 min around periapsis, theinstrument used its highest dynamic range mode,which can measure magnetic field componentsup to 44,000 nT, with digitization noise of 5.4 nT(8). This mode had not been used since an Earthflyby in August 1999 (21) and therefore requiredrecalibration during the Grand Finale orbits.Four spacecraft roll campaigns and numerousspacecraft turns enabled this recalibration. Thespacecraft attitude needed to be known to ahigh accuracy of 0.25 millirads, but intermit-tent suspension of the spacecraft star identi-fication system during phases of instrumenttargeting resulted in gaps in the required in-formation. These gaps were filled by reconstruct-ing spacecraft attitude using information fromthe onboard gyroscopes. Attitude reconstructionis currently available for nine of the first 10 orbits,so we restricted our analysis to those orbits.Individual orbits of Cassini are commonly re-ferred to with a rev number; these nine orbitsare rev 271 to 280, excluding rev 277.

External magnetic field from sources inthe magnetosphere and ionosphereof Saturn

In deriving the internal planetary magnetic fieldfrom in situ MAG measurements, it is necessaryto separate the internal planetary field from ex-terior sources of field (currents in the magneto-sphere and ionosphere). In the saturnian system,the simplest external source to correct is the mag-netodisk current, which flows in the equatorialregion far out in the magnetosphere (22–26). Likethe terrestrial ring current, the magnetodisk cur-rent produces a fairly uniform depression in thefield strength close to the planet and is normallytreated as axisymmetric (fig. S3). At Saturn, themagnetodisk current contributes ~15 nT of mag-netic field parallel to the spin axis of Saturn (BZ)in the inner magnetosphere (22–26). Magneto-pause and magnetotail currents, which deter-mine the properties of the outer magnetosphere,contribute ~2 nT BZ field in the inner magneto-sphere of Saturn (22, 26). More complex are theubiquitous planetary period oscillations (PPOs),global 10.6- to 10.8-hour oscillations, at close tothe expected planetary rotation period that arepresent in all Saturn magnetospheric parameters[reviewed in (27, 28)]. The PPOs arise from arotating electrical current systems with sourcesand sinks in Saturn’s ionosphere and magneto-sphere and closure currents that flow along mag-netic field lines (commonly referred to as FACs)

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1Physics Department, The Blackett Laboratory, ImperialCollege London, London, SW7 2AZ, UK. 2Department of Earthand Planetary Sciences, Harvard University, 20 OxfordStreet, Cambridge, MA 02138, USA. 3Division of Geologicaland Planetary Sciences, California Institute of Technology,1200 E California Boulevard, Pasadena, CA 91125, USA.4Department of Earth, Planetary, and Space Sciences,University of California, Los Angeles, Los Angeles, CA 90025,USA. 5Department of Physics and Astronomy, University ofLeicester, Leicester LE1 7RH, UK. 6Jet Propulsion Laboratory,California Institute of Technology, 4800 Oak Grove Drive,Pasadena, CA 91109, USA. 7Department of Climate andSpace Sciences and Engineering, University of Michigan,Ann Arbor, MI 40109, USA.*Corresponding author. Email: m.dougherty@imperial.ac.uk

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Fig. 1. Vector magnetic field measurements from nine Cassini GrandFinale orbits. (A to C) The IAU System III Saturn-centered spherical polarcoordinates are adopted here. The peak measured magnetic fieldstrength is ~18,000 nT, whereas the azimuthal component of the field(plotted on a different scale) is ~1/1000 of the total field outside the

high-latitude auroral FACs region, which are labeled. Vertical dashedlines indicate the MAG range 3 (jBj > 10;000 nT) time period. (C)The vertical dotted lines mark the inner edge of the D-ring mappedmagnetically, and the solid vertical solid lines mark the outer edgeof the D-ring mapped magnetically, as described in the text.

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and map to the auroral regions. Normally, twoperiods are present at any one time, with onedominant in the northern hemisphere and theother in the southern (29–33), which during theinterval studied here were 10.79 and 10.68 hours,

respectively (33). The rotation periods, ampli-tudes, and relative amplitudes of the two PPOsystems vary slowly over Saturn’s seasons (34–36).These seasonal changes indicate that the sourceof the periods is most likely to be of atmospheric

origin rather than deep within the planet (37, 38).Moreover, the PPO periods are a few percentlonger compared with any likely rotation periodof the deep interior of the planet (39–41). Thereis also a nonoscillatory system of strong FACs inthe auroral region with current closure throughthe planetary ionosphere, and there are dis-tributed FACs at other latitudes (42–44). Thisnonoscillatory current system is associated withmagnetospheric plasma that rotates more slowlythan the planet (42). During the Grand Finalephase, the Cassini spacecraft was close enoughto the planet during periapse to directly detectcurrents in the ionosphere. We describe resultson these external current systems first becausethey must be considered during analysis of theinternal field.Shown in Fig. 1 is the vector magnetic field

measurements in International AstronomicalUnion (IAU) Saturn System III right-hand spheri-cal polar coordinates, with r, q, and ϕ denotingradial, meridional, and azimuthal directions, re-spectively (45) from the nine Cassini Grand Fi-nale orbits for which accurate spacecraft attitudeinformation is available. This coordinate systemhas its origin at the center of mass of Saturn, andthe spin axis of Saturn is the polar axis (zenithreference). The peak magnetic field strengthencountered during these Grand Finale orbitsis ~18,000 nT, which is well within the rangeof the fluxgate magnetometer (8). The peakfield was not encountered during periapsis butat mid-latitude in the southern hemisphere.The dominant radial and meridional magneticfield components, Br and Bq, which exhibitsimilar behavior on each orbit, are shown inFig. 1, A and B. This repeatability, combined

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Fig. 2. Cassini’sGrand Finale trajec-tory shown in themeridional plane.The trajectory ofCassini GrandFinale orbit rev 275(black/blue tracewith arrows) in ther-Z cylindricalcoordinate system—

where r is the per-pendicular distancefrom the spin axis,and Z is distance fromSaturn’s planetaryequator—is overlainon the averagepositions of Saturn’sauroral FACs region(gray region) and thenewly discoveredlow-latitude FACs (redregion). Thin graytraces with arrows aremagnetic field lines.The interval along thetrajectory during which the measured magnetic field was >10,000 nT is highlighted in blue. Smallcircles are at 3-hour intervals, and the beginning of day 142 of year 2017 is shown.

Fig. 3. Saturn’s magnetic equator position as directly measured alongthe Cassini Grand Finale orbits. (A) The measured cylindrical radialcomponent of the magnetic field versus distance from Saturn’s planetaryequator from nine Cassini Grand Finale orbits. (B) The distribution of themagnetic equator northward displacements (triangles) versus cylindricalradial distance compared with predictions from existing degree-3 internal

field models (lines) (4, 7, 10, 11). The Z3 model is from (4), the Cassini 3model is from (11), the SPV model is from (7), the Cassini (SOI, June 2007)model, and the Cassini, P11, V1, V2 model are both from (10). The Cassini(SOI, June 2007) model were derived from Cassini SOI data to July 2007data, and the Cassini, P11, V1, V2 model is derived from Cassini SOI toJuly 2007 data combined with Pioneer 11 and Voyager 1 and 2 data (10).

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with the fixed local time of the orbits and theslight differences in orbital period, demonstratethat the planetary magnetic field is close toaxisymmetric.Shown in Fig. 1C is the azimuthal component

of the field, Bϕ, which is almost three orders ofmagnitude smaller than the radial andmeridionalcomponents outside the auroral FACs region. Ifit originates from the interior of the planet, Bϕ

would be expected to decay with radial distanceand also be part of a nonaxisymmetric field pat-tern (Eq. 5). However, much of the observed Bϕ

signal seems to be of external rather than inter-nal origin. The magnetic perturbations from thenorthern and southern auroral currents throughwhich the spacecraft flew are marked in Figs. 1and 2. In Fig. 2, the cylindrical coordinate systemis adopted, where r is the perpendicular distanceto the spin axis of Saturn and Z is the distancefrom the planetary equator of Saturn, whichis defined by its center of mass. These currentsheets have been crossed many times duringthe Cassini mission, but previous crossings oc-curred much further from the planet (43, 44, 46).The sharp field gradients observed are due tolocal FACs, and the short scale of the variationin the field indicates local current sources ratherthan relatively distant currents in the planet’sinterior. The scatter in the auroral FACs fromorbit to orbit is partly due to the variability of theaurora over time and partly due to the largecontribution from the rotating PPO signals,which are encountered at different phases oneach orbit.The narrow central peak in Bϕ ranging be-

tween 5 and 30 nT observed along these GrandFinale orbits was unexpected. It can be seenfrom Fig. 1C that the large change in field slopetakes place where the spacecraft crosses themagnetic field lines that map to the inner edgeof the D-ring.We adopted 1.11 RS from the centerof Saturn in the equatorial plane as the D-ringinner edge. If the azimuthal field gradient is dueto a local FAC, the current within the D-ringis interhemispheric and always flows from thenorthern to the southern hemisphere along thesenine Grand Finale orbits. There must be returncurrents either at lower altitude than the space-craft penetrates in the ionosphere or at otherlocal times. Assuming approximate axisymme-try (for azimuthal spatial scales much largerthan latitudinal scales), Ampère’s law combinedwith current continuity leads to a total interhem-ispheric current flow in the range of 0.25millionto 1.5 million ampere per radian of azimuth.Thus, this low-latitude FAC is comparable instrength with the currents observed on fieldlines connected to the high-latitude auroral re-gion (43, 44, 46). It thus may be part of a globalmagnetospheric–ionospheric coupling system.Alternatively, the systemmight be purely of ion-ospheric origin, in which case it could be due toskewed flow between northern and southernends of the field line (47). All the orbit periapsesare near local noon, so the repeatability fromperiapsis to periapsis does not rule out a depen-dency on local time.

There is a clear difference between the sig-natures of northern and southern auroral FACs(Fig. 1C), and again, magnetic mapping revealsthe source. The locations of the auroral currentsheets are well known magnetically from pre-vious Cassini measurements (Fig. 2) (43, 44, 46).During Cassini’s inbound trajectory in the north,the spacecraft is located on the field lines justpoleward of the main FAC region, as shown bythe decreasing Bϕ, which is indicative of the dis-tributed downward current on polar field lines,as previously shown (43, 44, 46). The less struc-tured Bϕ during the outbound southern tra-jectories (Fig. 1C) indicates that the southernhemisphere auroral FACs are not being fullycrossed by the spacecraft, and instead we ob-served the center of FAC activity moving dy-namically back and forth over the spacecraft.

Saturn’s internal magnetic field

Next we analyzed the radial and meridionalcomponents of the measured magnetic field,which are dominated by the axisymmetric in-ternal magnetic field. The highly inclined natureof the Grand Finale orbits allows direct determi-nation close to the planet of Saturn’s magneticequator, which is defined as where the cylindri-cal radial component of the magnetic field (Br)vanishes. The measured Br as a function of thespacecraft’s distance from the planetary equatorof Saturn is shown in Fig. 3A. The measuredvalue along these orbits shows that Saturn’smagnetic equator is 0.0466 ± 0.0002 (1 SD) RS

northward of the planetary equator, measuredat distance ~1.05 RS from the spin axis of Saturn.

This indicates a small yet non-negligible north-south asymmetry in Saturn’s magnetic field.Shown in Fig. 3B are the distribution of thedirectly measured magnetic equator position ofSaturn in the r-Z plane and predictions fromvarious preexisting degree-3 models (4, 7, 10, 11).Several degree-3 internal field models have beenderived in (10) from different combinations ofdatasets; the two shown in Fig. 3 were (i) derivedfrom data between Cassini SOI to July 2007 and(ii) data from Cassini SOI to July 2007 combinedwith Pioneer 11 and Voyager 1 and 2 data. Anominal magnetodisk field (table S2) was addedto the internal field when predicting the mag-netic equator positions from existing models. Asystematic decrease of the magnetic equatorposition with distance from the spin axis is evi-dent. This is a consequence of the axisymmetricpart of the magnetic field, as illustrated by thepredictions from existing axisymmetric models.In addition to this systematic trend, variationsof measured magnetic equator positions at sim-ilar r were also observed. These additional var-iations of the magnetic equator positions serveas a direct bound on possible nonaxisymmetryof Saturn’s internal magnetic field because anyinternal nonaxisymmetry would cause longitu-dinal variations of the magnetic equator posi-tions. The measured peak-to-peak variation atsimilar cylindrical radial distances, ~0.00030 RS

(18 km), translates to a dipole tilt of 0.0095°, aguide value for possible internal nonaxisymme-try. This value is about one order of magnitudesmaller than the upper limit on dipole tiltplaced before the Grand Finale (10, 11, 48). The

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Fig. 4. Saturn’s magnetic field beyond spherical harmonic degree 3 as directly measuredduring the Grand Finale orbits. (A and B) Residuals of the radial and meridional magnetic fields fromthe unregularized degree-3 internal field model (parameters are provided in table S1). It can be seenthat the residuals are on the order of 100 nT, which is 10 times larger than the magnetodisk field.

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corresponding field components at spacecraftaltitude are only ~3 nT (in a background fieldof 18,000 nT), which is too small to be dis-cerned from external fields in the measured Bϕ.Such an extreme axisymmetric planetary mag-netic field is difficult to explain with a dynamomodel (18, 49–52).Although the internal magnetic field of

Saturn is highly axisymmetric (structureless inthe longitudinal direction), it exhibits latitudinalstructures across many different length scales.This is revealed by our retrieval of the axi-symmetric Gauss coefficients of Saturn’s inter-nal magnetic field from these nine Grand Finaleorbits. In this analysis, the magnetodisk fieldwas explicitly included, adopting an analyticalformula (25) with initial parameters taken from(22). The axisymmetric internal Gauss coeffi-cients and the magnetodisk current field werethen retrieved via an iterative process. To de-termine the internal Gauss coefficients, onlymeasurements from range 3 of the fluxgatemagnetometer were adopted (jBj > 10; 000 nT,r < 1.58 RS), and measurements from all nineorbits were treated as a single dataset. Thesemeasurements were outside the auroral FACsregion, as shown in Figs. 1 and 2. For retrievalof the magnetodisk current field, measurementsfrom both range 3 and range 2 of the fluxgatemagnetometer were used ( jBj > 400 nT, r <4.50 RS), and each orbit was treated separately(table S2). Along the trajectory of the close-inportion of the Cassini Grand Finale orbits, thecontributions from the magnetopause and mag-netotail currents cannot be practically separated

from the quasi-uniform magnetodisk currentcontribution and thus are subsumed into ourmagnetodisk current model. The time-varyingexternal (magnetodisk + magnetopause + magne-totail) BZ field could create a time-varyingcomponent of the internal dipole g1

0 throughelectromagnetic induction. The maximum mag-netodisk field variation we have observed alongthe nine orbits is ~10 nT, which for an in-duction depth of 0.87 RS (Methods and fig. S4)would induce a time variation in the internaldipole coefficient of ~3.3 nT. This is less thanthe digitization level of the highest range of thefluxgate magnetometer onboard Cassini (5.4 nT);thus, the detection of which is not straightforward.We first investigated the minimum param-

eter set (in terms of internal Gauss coefficients)needed to adequately describe the observations,while applying no regularization to the param-eters (Methods) (53, 54). This experiment revealedthat axisymmetric internal Gauss coefficientsup to degree 9 are required to bring the root-mean-square (RMS) residual in range 3 below10 nT (fig. S5). The residual (Br, Bq) after re-moval of the unregularized degree 3 model (co-efficients provided in table S1) are on the orderof 100 nT but have larger amplitude and largerspatial scales in the northern hemisphere ascompared with those in the southern hemi-sphere (Fig. 4). Shown in Fig. 5 is the residual(Br, Bq) as a function of latitude after removalof the unregularized degree 6 model and thebest-fitting magnetodisk model for each orbit(coefficients provided in tables S1 and S2), whichexhibits consistent small-scale features with am-

plitude ~25 nT and typical spatial scale ~25° inlatitude. The highly consistent nature (from orbitto orbit) of these small-scale magnetic perturba-tions in (Br, Bq), in contrast to the orbit-to-orbitvarying Bϕ signature, suggests a source below thehighly time-variable currents in the ionosphereof Saturn.A new internal magnetic field model for

Saturn, which we refer to as the Cassini 11 mod-el, is presented in Table 1 and Fig. 6, with 11denoting the highest spherical harmonic degreeGauss coefficient above the uncertainties (Meth-ods). Because there was incomplete latitudinalcoverage during the Grand Finale orbits, whichcan lead to nonuniqueness in the solution, aregularized inversion technique (53, 54) has beenused to construct the Cassini 11 model. With theregularized inversion technique (53, 54), we ex-plicitly sought internal field solutions that notonly match the observations but also containminimum magnetic power beyond degree 3,when evaluated at a reference radius (Methods).Presented in Table 1 are the axisymmetric Gausscoefficients, five times the formal uncertainty(5s) (definition is provided in Methods), and theRMS residual of the Cassini 11 model. The re-sulting Mauersberger-Lowes (M-L) magneticpower spectrum (55–57) of the Cassini 11 modelevaluated at the surface of Saturn is shown inFig. 6. In the Cassini 11 model, the axisymmetricGauss coefficients between degree 4 and degree 11are on the order of 10 to 100 nT, whereas thoseabove degree 11 are below the 5s uncertainty.We have also investigated whether it is pos-

sible to obtain a simpler description of Saturn’s

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Fig. 5. Saturn’s magnetic field beyond spherical harmonic degree 6 as directly measuredduring the Grand Finale. (A and B) Residuals of the radial and meridional magnetic fields from theunregularized degree-6 internal field model (parameters are provided in table S1). The fieldcorresponding to the magnetodisk current has also been removed (parameters of the magnetodiskcurrent model are provided in table S2). Latitudinally banded magnetic structures on the orderof 25 nT are evident. Orbit-to-orbit deviations in Bq between –60° and –40° latitudes are mostly dueto the influences of southern hemisphere high-latitude FACs.

Table 1. Gauss coefficients of a newmodel for Saturn’s internal magneticfield, which we refer to as the Cassini 11model, constructed from nine orbits ofCassini Grand Finale MAG data with reg-ularized inversion. The reported uncer-

tainty is five times the formal uncertaintiesassociated with the chosen regularization

(Methods).

Gauss coefficient Value

(nT)

Uncertainty

(nT)

g10 21140.2 1.0

. .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ...

g20 1581.1 1.2

. .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ...

g30 2260.1 3.2

. .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ...

g40 91.1 4.2

. .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ...

g50 12.6 7.1

. .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ...

g60 17.2 8.2

. .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ...

g70 −59.6 8.1

. .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ...

g80 –10.5 8.7

. .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ...

g90 –12.9 6.3

. .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ...

g100 15.0 7.0

. .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ...

g110 18.2 7.1

. .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ...

g120 0.3 7.7

. .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ...

RMS residual 6.2. .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... .. ... ... .. ... ... .. ... ... .. ...

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internal magnetic field (in terms of Gauss co-efficients) in a shifted coordinate system. Weshifted the coordinate system northward fromSaturn’s center of mass along the direction ofthe spin axis, in the range of 0.03 to 0.05 RS

denoted by zS, and solved for the internal Gausscoefficients. We found that no simpler descrip-tion of the field in terms of Gauss coefficientscan be obtained in the Z-shifted coordinates be-cause of the complex nature of Saturn’s internalmagnetic field in the latitudinal direction. Onlythe quadrupole moment g2

0 can be reduced tozero in the Z-shifted coordinates with zS ~ 0.0337RS; all other even-degree moments (such as g4

0,g6

0, and g80) remain nonzero (fig. S6). In ad-

dition, g40 becomes more than an order of

magnitude larger in these shifted coordinatescompared with that in the Saturn-centered co-ordinates. The RMS residual remains unchangedas we shifted the coordinate system. We there-fore did not consider models in shifted co-ordinates any further.

Implications for Saturn’s dynamoand interior

Mathematically, downward continuation of themagnetic field measured outside of a planettoward its interior is strictly valid throughregions with no electrical currents J. Fromthe perspective of dynamo action, the downwardcontinuation of the magnetic field is possiblethrough regions with no substantial dynamoaction. For Saturn, becasue of the smooth in-crease of electrical conductivity as a functionof depth (13, 14, 58), the vigor of dynamo actionis expected to rise smoothly with depth (20) andcan be quantified by the magnetic Reynolds num-ber (Rm), which measures the ratio of magneticfield production/modification by flow in anelectrically conducting fluid against ohmic dif-fusion (Methods) (59). Assuming 1 cm s−1 flowand an electrical conductivity model from (58, 60),the local Rm reaches 10 at 0.77 RS, 50 at 0.725 RS,and 100 at 0.70 RS (fig. S9). For dynamo actionin the shallow semiconducting layer, Rm = 20 islikely to be sufficient given the existence of themain magnetic field originating from the deepdynamo. Thus, the top of the shallow dynamois likely to fall between 0.77 and 0.725 RS. Wechose 0.75 RS as an approximation for the sur-face of the shallow dynamo; the properties ofthe downward continued magnetic field arebroadly similar in this range of depths.The small-scale Br beyond degree 3 within

±60° latitude at 0.75 RS based on the Cassini 11model is shown in Fig. 7. The small-scale mag-netic structures measured along the spacecrafttrajectory map to latitudinally banded magneticstructures with amplitude ~5000 nT and typicalspatial scale ~15° in latitude at 0.75 RS, whichamount to ~5 to 10% of the local backgroundfield (fig. S7). The inferred small-scale magnet-ic field pattern is broadly similar for either aspherical surface with radius 0.75 RS or a dy-namically flattened elliptical surface with equa-torial radius 0.75 RS and polar radius 0.6998 RS

(fig. S8). The magnetic power in the low-degree

(degrees 1 to 3) part of the M-L spectrum stillgreatly exceeds that in the higher degree partwhen evaluated at 0.75 RS; the dipole power isone order of magnitude larger than the octupolepower, whereas the octupole power is almosttwo orders of magnitude larger than those ofdegrees 4 to 11. The orders-of-magnitude differ-ence between the low-degree power and high-degree power at 0.75 RS suggests that thelatitudinally banded magnetic perturbationscould originate from a secondary dynamo actionin the semiconducting region of Saturn (20).This secondary dynamo action likely only pro-duces perturbations on top of the primary deepdynamo, as suggested by a kinematic mean-fielddynamo model (20). Fully three-dimensional(3D) self-consistent giant planet dynamo modelswith deep differential rotation (zonal flows) alsoproduced latitudinally banded magnetic fields(61, 62).The dynamo a-effect, which describes the

generation of a poloidal magnetic field from atoroidal magnetic field [poloidal and toroidalmagnetic field are defined in (63)] by rotatingconvective motion, is likely to be of the tradi-tional mean-field type in the shallow dynamo(16, 17, 20). The validity of the traditional a-effect

has been extensively discussed in (20). In thisscenario, no substantial low-degree nonaxisym-metric magnetic field is expected from theshallow dynamo.From an empirical perspective, some but not

all of the high-degree magnetic moments couldstill have a deep interior origin. One extremeinterpretation of the Cassini 11 model is to regardthe magnetic moments up to degree 9 as en-tirely being of deep interior origin because themagnetic power of degrees 7 and 9 is within afactor of three of that of degree 3 when eval-uated at 0.50 RS. When discussing internalmagnetic fields well below 0.75 RS, we have(i) treated the shallow dynamo as a small per-turbation and (ii) implicitly assumed stablestratification between the surface of the deepdynamo and the shallow dynamo. Under thisassumption, there exists no local dynamo a-effectbetween the surface of the deep dynamo and theshallow dynamo. Although the w-effect, whichdescribes the generation of the toroidal magneticfield from differential rotation acting on thepoloidal magnetic field, likely operates in thestably stratified layer, the w-effect will not af-fect the poloidal component of the magneticfield. Downward continuation of the magnetic

Dougherty et al., Science 362, eaat5434 (2018) 5 October 2018 6 of 8

Fig. 6. Magnetic powerspectrum of the Cassini 11model.This new internalfield model for Saturn isconstructed from nineCassini Grand Finaleorbits with regularizedinversion. Central valuesand five times the formaluncertainties derived fromthe regularized inversionare shown (Methods). Itcan be seen that the high-degree moments betweendegree 4 and degree 11are on the order of10 to 100 nT, whereasthose above degree 11 arebelow the deriveduncertainty.

Fig. 7. Small-scale axi-symmetric magneticfield of Saturn at0.75 RS. The values ofDBr were computedby using the centralvalues of the degree 4to degree 11 Gausscoefficients (Table 1).These small-scalemagnetic perturbationsat 0.75 RS are on the order of 5000 nT, which amount to ~5 to 10% of the local background field (fig. S7).The plotted latitudinal range corresponds to the region of Cassini MAG range 3 measurements along theGrand Finale orbits.

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field well below 0.75 RS can only be justified ifthere are no effective electrical currents gen-erating the poloidal magnetic field in the layerbetween the surface of the deep dynamo andthe shallow dynamo.The radial magnetic field is shown in fig. S10

as a function of latitude at 0.50 RS computedfrom the Gauss coefficients of the Cassini 11model up to degree 9. It can be seen that theradial component of Saturn’s magnetic fieldat 0.50 RS is strongly concentrated towardhigh latitude, leaving minimum magneticflux at latitudes below ±45°, a weaker version ofwhich is evident from the magnetic momentsup to degree 3 (fig. S10). The possibility of weakmagnetic field at the poles of Saturn near 0.50 RS

is suggested in fig. S10. In this alternative inter-pretation, magnetic moments beyond degree 9would be of shallower origin. This highlightsthe ambiguity in the separation of magneticmoments of shallow origin from those of deeporigin, which cannot be resolved with MAGdata alone.The main dynamo of Saturn does appear to be

deeply buried, originating from a depth deeperthan the expected full metallization of hydrogenat ~0.64 RS (13). The surface strength of Saturn’sinternal magnetic field is more than one order ofmagnitude weaker than that of Jupiter (64–67).From dynamo scaling analysis, Saturn’s magneticfield appears to be weaker than expected fromboth force-balance considerations and energy-balance considerations if the outer boundaryof the deep dynamo is much shallower than0.40 RS (60). The most widely accepted the-oretical explanation for Saturn’s highly axi-symmetric magnetic field (49, 50) suggests thatdifferential rotation in a stably stratified andelectrically conducting layer above the deepdynamo region electromagnetically filters outnonaxisymmetric magnetic moments. Although3D numerical dynamo simulations (51, 52) thatimplement stable stratification and differentialrotation have only produced an average dipoletilt of 0.6°, about two orders of magnitude largerthan the newly derived guide value 0.0095°, theRm in these 3D numerical dynamo simulations(51, 52) are only in the range of 100 to 500,which is orders of magnitude smaller than pos-sible values (~30,000) inside the metallic hydro-gen layer of Saturn (68). At present, we regardthe extreme axisymmetry of Saturn’s magneticfield as a continuing enigma of the inner work-ings of Saturn.

MethodsUn-regularized and regularized inversionof Saturn’s internal magnetic field

The internal magnetic field of Saturn outside thedynamo region is described using Gauss co-efficients (gn

m , hnm) where n and m are the

spherical harmonic degree and order, respectively:

V Int ¼X

n¼1

Xl

m¼0Rp

Rp

r

� �nþ1

gmn cos mϕþ hmn sin mϕ� �

Pmn ðcos qÞ ð1Þ

ðBIntr ;BInt

q ;BIntϕ Þ ¼ �∇V Int ð2Þ

BIntr ¼

Xn¼1

Xn

m¼0ðnþ 1Þ Rp

r

� �nþ2

gmn cos mϕþ hmn sin mϕ� �

Pmn ðcos qÞ ð3Þ

BIntq ¼ �

Xn¼1

Xn

m¼0

Rp

r

� �nþ2

gmn cos mϕþ hmn sin mϕ� � dPm

n ðcos qÞdq

ð4Þ

BIntϕ ¼

Xn¼1

Xn

m¼0

Rp

r

� �nþ2 m

sin q

gmn sin mϕ� hmn cos mϕ� �

Pmn ðcos qÞ ð5Þ

It can be seen from Eq. 5 that the azimuthalmagnetic field of internal origin, BInt

ϕ , must bepart of a non-axisymmetric (m ≠ 0) field pattern.The forward model can be represented as

data = G model (6)

For this particular problem, data representsthe magnetic field measurements (Br, Bq, Bϕ),model represents the Gauss coefficients (gn

m,hn

m), and G represents the matrix expressionof Eqs. 3 to 5.In un-regularized inversion, one seeks to

minimize the difference between the data andthe model field only

|data – G model|2 (7)

whereas in regularized inversion, additional con-straints are placed on the model parameters.Instead, one seeks to minimize

|data – G model |2 + g 2|L model|2 (8)

in which L represents a particular form of reg-ularization and g is a tunable damping param-eter adjusting the relative importance of modelconstraints.We seek to minimize the power in the sur-

face integral of the radial component of themagnetic field beyond spherical harmonicdegree 3

∫r¼rrefB2rðn > 3ÞdW ¼

Xn¼4

ðnþ 1Þ22nþ 1

Rp

rref

� �2nþ4

½ðgmn Þ2 þ ðhmn Þ2� ð9Þat a reference radius rref. Thus,

L ¼ nþ 1ffiffiffiffiffiffiffiffiffiffiffiffiffi2nþ 1

p Rp

rref

� �nþ2

ð10Þ

for n > 3 and L = 0 for n ≤ 3.After experimenting with different rref and

g, we choose rref = 1.0 and g = 0.6 which yieldsmooth magnetic field perturbations whenviewed between 1.0 RS and 0.75 RS.

In this framework, the model solution canbe computed via

model = (GTG + g2 LTL)−1GT data = GN data(11)

in which superscript T represents matrix transpose.The model covariance matrix can be com-

puted from

modcov = GN datacov (GN)T (12)

where datacov is the data covariance matrix,taken to be sdata

2I, in which sdata is the RMSof the data model misfit and I is the identitymatrix (54). The formal uncertainty of the mod-el solution is the square root of the diagonalterm in the model covariance matrix.

Electromagnetic induction inside Saturn

Due to the rapidly radial varying nature ofelectrical conductivity inside Saturn, the threefactors controlling the electromagnetic (EM)induction at Saturn are: the frequency of theinducing field wind, the electrical conductiv-ity s(r) and the electrical conductivity scaleheightHs ¼ s=jds=drj. The EM induction insideSaturn would occur at a depth where the frequen-cy dependent local EM skin depth dðr;windÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 windm0sðrÞ=p

becomes similar to or less thanthe local electrical conductivity scaleheightHs(r),where m0 is the magnetic permeability.Figure S4 shows the frequency dependent EM

skin depth compared to the electrical conduc-tivity scaleheight inside Saturn, adopting theelectrical conductivity profile (58). Two frequen-cies have been chosen, the rotational frequency ofSaturn wr ¼ 2p�10:5 hours ∼1.7 × 10–4 rad s–1

and the orbital period of Cassini Grand Finaleorbits wo ¼ 2p�7 Earth days ∼1.0 × 10–5 rad s–1.It can be seen from fig. S4 that EM inductioninside Saturn would occur at depth 0.87 RS

for wind equal to the rotational frequency ofSaturn and at depth 0.86 RS for wind equal tothe orbital frequency of Cassini Grand Finaleorbits. Thus, the induction depth inside Saturnwould likely be 0.87 RS or less.A 10 nT inducting BZ field would create a

time-varying internal dipole field with equatorialstrength of 5 nT and polar strength of 10 nTwhen evaluated at this depth, which correspondsto a time varying dipole coefficient g1

0 of 3.3 nTwhen defined with respect to the equatorial radi-us of Saturn’s 1 bar surface.

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68. Assuming 1 cm s–1 flow and an electrical conductivity of2 × 105 S/m, Rm associated with a layer with a thickness of0.2 RS would be ~30,000.

ACKNOWLEDGMENTS

All authors acknowledge support from the Cassini Project. H.C.acknowledges Royal Society Grant RP\EA\180014 to enable anacademic visit to Imperial College London, during which some ofthe work has been carried out. Funding: Work at Imperial CollegeLondon was funded by Science and Technology Facilities Council(STFC) consolidated grant ST/N000692/1. M.K.D. is funded byRoyal Society Research Professorship RP140004. H.C. is funded byNASA’s CDAPS program NNX15AL11G and NASA Jet PropulsionLaboratory (JPL) contract 1579625. Work at the University ofLeicester was supported by STFC grant ST/N000749/1. E.J.B. issupported by a Royal Society Wolfson Research Merit Award. Workat the University of California, Los Angeles is funded by NASA JPLcontract 1409809. K.K.K. is funded by NASA JPL contract1409806:033. M.G.K. is funded by JPL under contract 1416974at the University of Michigan. M.E.B. and T.A.B. are supportedby the Cassini Project. Author contributions: M.K.D. led theinstrument team and supervised the data analysis. H.C., K.K.K.,and S.K. carried out the magnetometer calibration analysis. T.A.B.carried out the spacecraft attitude reconstruction. H.C., G.J.H.,and G.P. carried out the magnetic field data analysis. M.E.B., E.J.B.,S.W.H.C., M.G.K., C.T.R., and D.J.S. provided theoretical supportand advised on the data analysis. All authors contributed to thewriting of the manuscript. Competing interests: The authorsdeclare no competing interests. Data and materials availability:The derived model parameters are given in Table 1 and tables S1and S2. Fully calibrated Cassini magnetometer data are releasedon a schedule agreed with NASA via the Planetary Data System athttps://pds-ppi.igpp.ucla.edu/mission/Cassini-Huygens/CO/MAG;we used data from rev 271 through to rev 280, excluding rev 277.

SUPPLEMENTARY MATERIALS

www.sciencemag.org/content/362/6410/eaat5434/suppl/DC1Figs. S1 to S10Tables S1 and S2

9 March 2018; accepted 5 September 201810.1126/science.aat5434

Dougherty et al., Science 362, eaat5434 (2018) 5 October 2018 8 of 8

RESEARCH | RESEARCH ARTICLE | DIVING WITHIN SATURN ’S RINGS

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Saturn's magnetic field revealed by the Cassini Grand Finale

SouthwoodThomas A. Burk, Emma J. Bunce, Stanley W. H. Cowley, Margaret G. Kivelson, Christopher T. Russell and David J. Michele K. Dougherty, Hao Cao, Krishan K. Khurana, Gregory J. Hunt, Gabrielle Provan, Stephen Kellock, Marcia E. Burton,

DOI: 10.1126/science.aat5434 (6410), eaat5434.362Science 

, this issue p. eaat5434, p. eaat1962, p. eaat2027, p. eaat3185, p. eaat2236, p. eaat2382Scienceand directly measured the composition of Saturn's atmosphere.

identified molecules in the infalling materialet al.how they interact with the planet's upper atmosphere. Finally, Waite investigated the smaller dust nanograins and showet al.particles falling from the rings into the planet, whereas Mitchell

determined the composition of large, solid dustet al.kilometric radiation, connected to the planet's aurorae. Hsu present plasma measurements taken as Cassini flew through regions emittinget al.decay of free neutrons. Lamy

detected an additional radiation belt trapped within the rings, sustained by the radioactiveet al.the planet. Roussos measured the magnetic field close to Saturn, which implies a complex multilayer dynamo process insideet al.Dougherty

planet's upper atmosphere. Six papers in this issue report results from these final phases of the Cassini mission. spacecraft through the gap between Saturn and its rings before the spacecraft was destroyed when it entered theregions it had not yet visited. A series of orbits close to the rings was followed by a Grand Finale orbit, which took the

The Cassini spacecraft spent 13 years orbiting Saturn; as it ran low on fuel, the trajectory was changed to sampleCassini's final phase of exploration

ARTICLE TOOLS http://science.sciencemag.org/content/362/6410/eaat5434

MATERIALSSUPPLEMENTARY http://science.sciencemag.org/content/suppl/2018/10/03/362.6410.eaat5434.DC1

CONTENTRELATED

http://science.sciencemag.org/content/sci/362/6410/eaat2382.fullhttp://science.sciencemag.org/content/sci/362/6410/eaat2236.fullhttp://science.sciencemag.org/content/sci/362/6410/eaat3185.fullhttp://science.sciencemag.org/content/sci/362/6410/eaat2027.fullhttp://science.sciencemag.org/content/sci/362/6410/eaat1962.fullhttp://science.sciencemag.org/content/sci/362/6410/44.full

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