ARTICLE IN PRESS

Planetary and Space Science 57 (2009) 1706–1713

Contents lists available at ScienceDirect

Planetary and Space Science

0032-06

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/pss

Model of Saturn’s internal planetary magnetic field basedon Cassini observations

M.E. Burton a,�, M.K. Dougherty b, C.T. Russell c

a Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USAb Imperial College of Science and Technology, London, UKc University of California, Los Angeles, CA, USA

a r t i c l e i n f o

Article history:

Received 18 November 2008

Received in revised form

9 April 2009

Accepted 14 April 2009Available online 3 May 2009

Keywords:

Saturn magnetic field

Magnetosphere

Planetary magnetic fields

33/$ - see front matter & 2009 Elsevier Ltd. A

016/j.pss.2009.04.008

esponding author. Tel.: +1818 3547375.

ail address: marcia.burton@jpl.nasa.gov (M.E.

a b s t r a c t

We have derived a model of Saturn’s internal planetary magnetic field from data obtained during the

first three years of the Cassini Mission. This model is based on the most complete set of observations yet

obtained in Saturn’s magnetosphere and includes data from forty-five periapsis passes at a wide variety

of geometries. Due to uncertainties in the rotation rate of the planet the model is constrained to be

axisymmetric. To derive the model, the external currents are modeled explicitly as an equatorial ring

current centered on Saturn’s equator and the internal planetary magnetic field is derived using standard

inversion techniques. The field is adequately described by a model of degree 3 and the spherical

harmonic coefficients are g10 ¼ 21;162, g20 ¼ 1514, g30 ¼ 2283. Units are nanoTeslas (nT) and are based

on a planetary radius of 60,268 km. The model is consistent with a northward offset of the magnetic

equator from the rotational equator of 0.036 Saturn radii. Reanalysis and comparison with data obtained

by Pioneer-11 and Voyager-1 and -2 shows little evidence for secular variation in the field in the almost

thirty years since those data were obtained.

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Based on data obtained during the three brief flybys of Pioneer-11 and Voyager-1 and -2 almost thirty years ago, Saturn’s internalmagnetic field has been characterized as highly axisymmetric. Theinitial orbits of the Cassini spacecraft confirmed this result(Dougherty et al., 2005; Giampieri et al., 2006b). In such amagnetosphere, periodic signatures in the magnetic field, plasmadata and radio emissions are not expected. However, periodicemission of Saturn kilometric radiation (SKR) was detected byVoyager-1 (Desch and Kaiser, 1981). The period (10 h 39 min 24 s�7 s) was presumed to represent the rotation period of the deepinterior and was used to define the official InternationalAstronomical Union (IAU) rotation rate. Subsequent observationsof SKR by the Ulysses spacecraft revealed that the period waslonger by approximately 1% (Galopeau and Lecacheux, 2000).Further observations from Cassini confirmed this result andindicate the SKR period varies over much shorter time scales aswell (Kurth et al., 2007).

In a reanalysis of data obtained by Pioneer and Voyager, amodulation of the magnetic field close to that of the planetaryrotation period was detected (Espinosa and Dougherty, 2000). The

ll rights reserved.

Burton).

proposed mechanism was that a magnetic anomaly close to theplanet creates a compressional wave that propagates radiallyoutward across the background magnetic field (the so-calledcamshaft model) (Espinosa et al., 2003). Periodicities such asthese now appear to be a ubiquitous feature of Saturn’s magneticfield. In an initial study based on spectral analysis of magneticfield data from the first fifteen orbits of the Cassini mission,Giampieri et al. (2006a) reported the detection of a timestationary magnetic signal with a period of 10 h 47 min 6 s�40 s. The signal was found to be stable in period, amplitude andphase over fourteen months of observations, suggesting a closeconnection with the conductive region inside the planet. Thisperiod was shown later to correspond to the variable SKR periodover the same time interval (Kurth et al., 2007). Cowley et al.(2006) confirmed that the magnetic oscillations are consistentwith a model in which phase fronts rotate with the planet andradiate outward at a speed comparable to the equatorial Alfvenspeed, consistent with the ‘camshaft model’. Southwood andKivelson (2007) have suggested that the periodic signature maybe due to a rotating non-axisymmetric system of field alignedcurrents. Noting that the plasma and magnetic field in the innermagnetosphere rotate synchronously with the variable SKR,Gurnett et al. (2007) emphasize an external source. They proposethat the observed electron density modulation in the inner regionof the plasma disk acts as the ‘cam’ that drives other rotationallymodulated phenomenon and suggest the origin is linked to the

ARTICLE IN PRESS

M.E. Burton et al. / Planetary and Space Science 57 (2009) 1706–1713 1707

plasma disk that originates when Enceladus variable neutral cloudis ionized. Subsequently, Andrews et al. (2008) examined 23 of theCassini equatorial orbits and found that the phase of the magneticoscillations are well-organized by a period close to that of thevariable SKR.

Using gravitational data, along with Pioneer and Voyagerradio occultation and wind data, Anderson and Schubert(2007) proposed that a rotation period of 10 h 32 min and 35�13 s represents the rotation rate of the deep interior. Theyfound that a mean geoid that matches the gravitationaldata and minimizes the wind-induced dynamic heights ofthe 100 mbar isosurface could be derived based on this rotationrate.

Based on the most optimistic uncertainty cited for any of theseproposed rotation rates (�6 s) the longitude could be in error by180� over the course of the four year mission. This uncertainty inthe rotation rate and the nature and origin of the observedperiodicities has impeded the development of a planetarymagnetic field model and at the same time has called intoquestion the previously accepted notion of a strictly axisymmetricmagnetic field. Any magnetic field model that includes non-axialterms is untenable if the assumed rotation rate is incorrect.

Table 1Times, distance and latitudes of the periapsis passes used in this study.

Orbit Day of year Date

1 O 183 2004-07-01

2 A 302 2004-10-28

3 B 350 2004-12-15

4 C 16 2005-01-16

5 3 48 2005-02-17

6 4 68 2005-03-09

7 5 88 2005-03-29

8 6 104 2005-04-14

9 7 123 2005-05-03

10 8 141 2005-05-21

11 9 159 2005-06-08

12 10 177 2005-06-26

13 11 196 2005-07-14

14 12 214 2005-08-02

15 13 232 2005-08-20

16 14 248 2005-09-05

17 15 266 2005-09-23

18 16 285 2005-10-12

19 17 302 2005-10-29

20 18 331 2005-11-27

21 19 358 2005-12-24

22 20 17 2006-01-17

23 21 56 2006-02-25

24 22 79 2006-03-20

25 23 118 2006-04-28

26 24 142 2006-05-22

27 25 181 2006-06-30

28 26 204 2006-07-23

29 27 228 2006-08-16

30 28 252 2006-09-09

31 29 268 2006-09-25

32 30 284 2006-10-11

33 31 301 2006-10-28

34 32 313 2006-11-09

35 33 325 2006-11-20

36 34 336 2006-12-02

37 35 349 2006-12-15

38 36 365 2006-12-31

39 41 82 2007-03-23

40 42 98 2007-04-08

41 43 114 2007-04-24

42 44 130 2007-05-10

43 45 147 2007-05-27

44 46 163 2007-06-12

45 47 179 2007-06-28

Despite this uncertainty, it is useful and instructive to derive amagnetic field model based only on axisymmetric terms, sincesuch a model is independent of the assumed rotation rate and canbe compared with earlier models based on Pioneer and Voyagerwhich were constrained in just such a way (Connerney et al.,1982). Using similar analysis and inversion techniques to thoseused to obtain the earlier Pioneer/Voyager models, we havederived an updated axisymmetric model of Saturn’s magneticfield based on data obtained during the first three years of theCassini orbital tour.

2. Analysis

2.1. Data used in this study

Our analysis is based on data obtained during the first threeyears of the Cassini mission (July 1, 2004–July 1, 2007). Closestapproach occurred at a variety of radial distances, latitudes andlongitudes and are given in Table 1. All orbits with periapsisdistances less than 10Rs, 45 in all, were included in the analysis.Saturn orbit insertion (SOI) (Orbit 0), on July 1, 2004 was the

Time UTC Periapsis distance (Rs) Latitude (deg)

02:38 1.33 17.1

10:19 6.2 12.0

05:51 4.8 4.1

06:36 4.9 6.4

00:59 3.5 0.11

11:39 3.5 0.11

23:42 3.5 0.12

23:20 2.6 3.25

01:51 3.6 12.1

06:09 3.6 12.2

10:42 3.6 12.3

15:46 3.6 12.4

22:12 3.6 12.5

05:21 3.6 12.6

11:09 3.6 12.7

11:29 2.8 8.2

20:36 3.1 0.26

01:41 3.1 0.34

22:56 4.6 0.11

11:24 4.6 0.16

21:14 4.6 0.16

07:06 5.6 0.03

10:47 5.6 0.03

20:14 5.5 �0.15

23:59 5.5 �0.15

09:01 5.5 �0.34

13:05 5.4 �0.35

21:48 4.2 �10.1

20:54 4.2 �10.1

17:38 2.9 �12.4

19:32 4.0 �21.1

22:51 5.5 �29.4

00:17 4.7 �25.9

00:03 4.7 �25.9

23:06 4.7 �25.9

21:44 4.7 �25.9

00:18 7.7 �38.0

05:15 9.8 �44.6

12:02 9.7 �66.1

16:49 7.5 �40.3

20:13 5.7 �31.4

22:55 4.2 �23.3

00:37 3.2 �15.2

00:51 2.7 �9.2

00:53 2.5 �1.0

ARTICLE IN PRESS

0 60 120 180 240 300 360−60

−40

−20

0

20

40

60

1.5 2 2.5 3 4 5 6SOI - Orbit 46 (July 1, 2004 - June 12, 2007)

Range (Rs)

Latit

ude

(deg

rees

)

Longitude (degrees)

3.5 4.5 5.5

Fig. 1. Latitude vs. longitude of data used in this study. Data is color-coded

according to radial distance range in units of Saturnian radii, Rs

(1Rs ¼ 60;268 km). Only data obtained within 6 RS is shown. Longitude is based

on the IAU rotation period.

M.E. Burton et al. / Planetary and Space Science 57 (2009) 1706–17131708

closest at only 1.33Rs. Five orbits came within three Saturnianradii and another 13 were closer than 4Rs, providing excellentspatial coverage close to the planet. Fig. 1 depicts latitude versuslongitude of the trajectory within 6Rs, color-coded according todistance from the planet. The longitude is based on the IAUrotation period.

One-minute averages of magnetic field data acquired by theCassini fluxgate magnetometer (Dougherty et al., 2004) wereused to derive the model. Data obtained within 20Rs wereused to model the field due to external sources and data within10Rs were used to model the internal field. The data weresurveyed for evidence of magnetopause or bowshock crossingsand those data were excluded from the study. Measurementsnear close flybys of the icy satellites were excluded as well. Themodel is derived from more than 78,000 1-min averages of themagnetic field.

2.2. Modeling the ring current field

A large-scale eastward flowing ring current is known to existin Saturn’s magnetosphere and has been studied extensivelyusing Cassini observations (Bunce et al., 2007). Since itscontribution close to the planet can be significant, it must beaccounted for when modeling the planetary field. We use theanalytical expression derived by Giampieri and Dougherty (2004)based on the simple axisymmetric equatorial current sheetcentered on the planet’s equator, first described by Connerneyet al. (1981) for the Jovian magnetosphere. In a cylindricalcoordinate system ðz;r;fÞ with the z-axis parallel to Saturn’smagnetic dipole axis, the disk is confined to the region jzjoD, anda prob. The azimuthal current density is distributed asJf ¼ I0=r. The cylindrical components, z and r of the magneticfield are given by

Bzðz;r; aÞ ¼m0I0

2logðzþ Dþ xrÞðz� Dþ ZrÞ

" #þX1k¼1

ð�1Þkð2k� 1Þ!a2k

22kðk!Þ2

(

�P0

2k�1½ðz� DÞ=ðZrÞ�Z2kr

�P0

2k�1½ðzþ DÞ=ðxrÞ�

x2kr

24

359=; (1)

Brðz;r; aÞ ¼m0I0

2r 2 signðzÞðminjzj;DÞ þ Zr

8<:

� xrX1k¼1

ð�1Þkð2k� 2Þ!a2kr22kðk!Þ2

�P1

2k�1½ðz� DÞ=ðZrÞ�Z2kr

�P1

2k�1½ðzþ DÞ=ðxrÞ�

x2kr

24

359=; (2)

where

xr ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðzþ DÞ2 þ r2

q; Zr ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðz� DÞ2 þ r2

q

Eqs. (1) and (2) describe the ring current field for an outerboundary at infinity. For a finite disk of inner radius a and outerradius b, the disk field is calculated with the outer radius, b

replacing a in Eqs. (1) and (2) and Bz ¼ BzðaÞ � BzðbÞ and a similarexpression for Br. I0 has units of current per unit length. Thus thetotal current is given by I ¼ 2I0D logðb=aÞ. Pm

n are the associatedLegendre functions.

To model the ring current contribution to the magnetic fieldwe used 1-min averages obtained outside of 4Rs to minimizethe contribution by the planetary field and inside of 20Rs inorder to minimize the contribution due to currents flowingon the magnetopause or tail currents. All passes with a closestapproach distance less than 10Rs during the first three yearsof the tour were used in the analysis, a total of 45 orbits.Current sheet structure and characteristics are known to varywith local time (Arridge et al., 2008) and temporal variations inthe solar wind and magnetosphere are likely to occur overtime scales corresponding to that of a periapsis pass (severaldays). To minimize these effects, the inbound and out-bound legs of each orbit were analyzed separately result-ing in 86 cases (four passes had insufficient data, due to datagaps).

First, to eliminate the estimated contribution due to internalsources we subtracted the axisymmetric SPV model magneticfield (Davis and Smith, 1990) from the observations. The residualfield (observed� internal) was transformed into cylindricalcoordinates, Br and Bz. Eqs. (5) and (6) were used to fit the dataand obtain the geometrical parameters (a,b,D) and the currentparameter, m0I0 using a large-scale non-linear least-squares fitalgorithm based on the reflective Newton method (Giampieri andDougherty, 2004). We have made no attempt to model othercurrents systems that exist in Saturn’s magnetosphere, such as tailor magnetopause currents.

2.3. Modeling the planetary field

After the best-fit ring current parameters were obtainedfor each orbit segment as described, the estimated contribu-tion was calculated (Eqs. (1) and (2)) and subtracted fromthe measured field. The residual field was assumed to originatein Saturn’s interior and was used to determine the modelparameters.

The internal planetary magnetic field in spherical polarcoordinate system (r, y;f) is given by the standard equations

Br ¼ �@V

@r

¼X1n¼1

Xn

m¼0

ðnþ 1ÞRp

r

� �ðnþ2Þ

½gmn cosðmfÞ þ hm

n sinðmfÞ�

" #

�Pmn ðcos yÞ (3)

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M.E. Burton et al. / Planetary and Space Science 57 (2009) 1706–1713 1709

By ¼ �1

r

@V

@y

¼ �X1n¼1

Xn

m¼0

Rp

r

� �ðnþ2Þ

½gmn cosðmfÞ þ hm

n sinðmfÞ�

" #

�dPm

n ðcos yÞdy

(4)

Bf ¼ �1

r sin y@V

@f¼

1

siny

�X1n¼1

Xn

m¼0

Rp

r

� �ðnþ2Þ

½gmn sinðmfÞ � hm

n cosðmfÞ�

" #

�Pmn ðcos yÞ (5)

where Rp is the planetary radius (60,268 km), Pmn ðcos yÞ are the

Schmidt quasi-normalized associated Legendre function of degreen and order m traditionally used in magnetic field modeling andgm

n and hmn are internal spherical harmonic coefficients.

To model the planetary magnetic field we have used thegeneralized inverse method of Lanczos (1971) that utilizessingular value decomposition (SVD). Details of the use of thistechnique to solve linear least-squares problems are givensuccinctly in Press et al. (1982) and an example of its applicationto planetary magnetic field modeling can be found in Connerney(1981) thus only a brief sketch of the method is provided here. Theequations describing the planetary magnetic field (5) can bewritten in matrix form as

Br1

By1

Bf1

..

.

Bri

Byi

Bfi

..

.

Brn

Byn

Bfn

0BBBBBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCCCCA

¼

Xr11 Xr12 � � � Xr1m

Xy11 Xy12 � � � Xy1m

Xf11 Xf12 � � � Xf1m

..

. ... . .

. ...

Xri1 Xri2 � � � Xrim

Xyi1 Xyi2 � � � Xyim

Xfi1 Xfi2 � � � Xfim

..

. ... . .

. ...

Xrn1 Xrn2 � � � Xrnm

Xyn1 Xyn2 � � � Xynm

Xfn1 Xfn2 � � � Xfnm

0BBBBBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCCCCA

g1

g2

gn

h1

..

.

hm

0BBBBBBBBB@

1CCCCCCCCCA

(6)

where Xl;n;m are the elements of the n�m design matrix made upof basis functions, (n, number of data points, m, number of modelparameters), y ¼ Bi is the data vector of magnetic fieldmeasurements, and m is the model parameter vector (the gn;m

and hn;m model coefficients) to be determined. The index l refers tothe r, y, or f component of the field. As a simple example, theelements of the design matrix associated with the r, y, and fcomponents of the g10 term are

X1;1;1 ¼ 2Rp

r

� �3

P10 ¼ 2Rp

r

� �3

cos y

X2;1;1 ¼Rp

r

� �3 @P10

@y¼

Rp

r

� �3

sin y

X3;1;1 ¼ 0 (7)

Table 2Spherical harmonic coefficients for magnetic field models in units of nT based on a pla

Multipole term SPV Z3 Cassini (SOI)

g10 21,225 21,248 21,084

g20 1566 1613 1544

g30 2332 2683 2150

CN

Condition number, CN (dimensionless) for the models derived in this study is also sho

Singular value decomposition relies on the fact that any n�m

matrix A can be written as

A ¼ U �w � VT (8)

where U is an n�m column-orthogonal matrix, V is an m�m

orthogonal matrix and w is an m�m diagonal matrix of positiveor zero elements, the singular values. The SVD solution can beshown to be the least-squares solution (Aster et al., 2005) and canbe written as

m ¼ Vð1=wÞUT� y (9)

or alternately

m ¼Xm

i¼1

Ui � y

wiVi (10)

where Ui denotes the columns of U each one a vector of length N

and Vi denotes the columns of V, each one a vector of length M.This equation states that the fitted model parameters are linearcombination of the columns of V with coefficients obtained byforming the dot products of the columns of U and the data vectory weighted by the singular values. The variance in the estimate ofthe model parameter is given by

s2ðmjÞ ¼Xmi¼1

Vi

wi

� �2

(11)

3. Results

We have implemented the analysis techniques described inSections 2.2 and 2.3 to derive an axial model of degree 3 forSaturn’s internal magnetic field based on all data obtained duringthe first three years of the Cassini orbital tour. The modelcoefficients are shown in Table 2 (column 5). The conditionnumber, the ratio of the largest to smallest singular value and ameasure of the sensitivity of the solution to inaccuracies in thedata, is also given. It reflects the spatial coverage of the data andthus will improve as the mission progresses. The conditionnumber for the design matrix associated with our model is 4.36,thus errors associated with the g30 term are approximately fourtimes larger than those associated with the g10 term.

For comparison, coefficients from earlier, axial models basedon a more limited subset of Cassini data are also shown in Table 2.A model based on Cassini Saturn orbit insertion data only(Dougherty et al., 2005) is shown in column 1 and one based ondata obtained during the first year of the Cassini mission(Giampieri et al., 2006b) is shown in column 2. These modelparameters can be directly compared with those from our analysissince they were derived using similar methods for assessing theexternal sources and similar inversion techniques. All three modelparameters derived in our analysis using three years of data arewithin a percent of those based on data obtained during the firstyear only. The dipole term is only slightly smaller and thequadrupole and octupole terms slightly larger. The table can bethought of as a representation of the evolution of the planetary

netary radius of 60,268 km.

Cassini (SOI - 8/05) Cassini (SOI - 7/07) Cassini, P11,V1,V2

21,169 21,162 21,145

1504 1514 1546

2264 2283 2241

5.08 4.36 3.81

wn.

ARTICLE IN PRESS

M.E. Burton et al. / Planetary and Space Science 57 (2009) 1706–17131710

field model over the course of the Cassini mission as more data atvaried geometries has been acquired.

To assess how well the model fits the observations, we havecalculated the root-mean-square misfit or residual on a pass-by-pass basis. The calculation of the misfit does not include the Bfcomponent which is dominated by the ‘cam’ signal in the innermagnetosphere. The mean of that value for all passes is 1.67 andthe median is 1.41 indicating that the model fits the data quitewell. The misfit based on data obtained close to the planet ðo6RsÞshows a modest improvement over that based on the entireanalysis interval of 10Rs, with a mean of 1.5 and the median of1.27 nT. On a pass-by-pass basis the misfit of the model based onCassini data alone (Cassini, SOI �7=07) in Table 2 is depicted bythe open blue circles in the bottom panel of Fig. 2. The top panelshows the closest approach distance of each orbit. The range oflatitudes covered in the analysis interval is depicted by the extentof the vertical line in the second panel and the symbol representsthe value at closest approach. There is a similar representation forthe coverage in local time.

Fig. 3 shows how well the model fits the data for four of theclosest periapsis passes, Rev 6 with a closest approach distance of2.6Rs, Rev 28 (2.9Rs), Rev 46 (2.7Rs) and Rev 47 (2.5Rs). Themodel is shown in blue and the data in black and on the scale ofthese figures, the two are indistinguishable. The periodicsignature in the azimuthal component of the field, Bf, and thereis evidence of the ‘cam’ signal in all four examples. The radialdistance and latitude of each pass are also shown.

To compare a planetary field model based on data obtained inthe Pioneer/Voyager era with one based on Cassini data, it isnecessary to apply the same analysis methods to all data sets.A direct comparison of model parameters from earlier studieswould not be valid because of the use of different modelingmethods and data selection techniques. Although the Pioneer-11and Voyager-1 and -2 data were analyzed extensively in the years

0

5

10

r (R

s)

−500

50

Latit

ude

(deg

rees

)

0

10

20

Loca

l Tim

e

0

0 5 10 15 20

Orb

Mis

fit (n

T)

0

5

0 5 10 15 20

0 5 10 15 20

0 5 10 15 20

15

10

Fig. 2. Root-mean-square misfit for individual orbits (bottom panel), as well geometric

units of Saturnian radii, Rs (top panel) latitude in degrees and local time. The misfit in u

misfit based on a model of Cassini, Pioneer and Voyager is shown by the smaller solid

following those flybys (see, for example, Davis and Smith, 1990;Acuna et al., 1983) none of those analyzis used precisely the samemethods that we have applied here. To derive the Z3 magneticfield model, Connerney et al. (1982) combined data from Pioneer-11 and Voyager-1 and -2 and derived source coefficients (G10, G11,H11) to model the external field. Davis and Smith (1990) used datafrom all three spacecraft flybys to derive their SPV model which isbased on a least-squares approach. They used a data weightingscheme to emphasize measurements close to the planets and alsomodeled the external field using the Gauss coefficients, Table 3.(For reference, the spherical harmonic coefficients normalized fora planetary radius of 60,268 km for the SPV (Davis and Smith,1990) model are g10 ¼ 21;225, g20 ¼ 1566, and g30 ¼ 2332, andfor Z3 (Connerney et al., 1982) g10 ¼ 21;248, g20 ¼ 1613 andg30 ¼ 2683. All coefficients are in units of nT.)

We have applied the same methods as those used to model theCassini planetary field, described in Sections 2.2 and 2.3, to thedata obtained by Pioneer-11 and Voyager-1 and -2. For consis-tency, we have re-derived the ring current parameters, eventhough these too have been analyzed extensively in (Giampieriand Dougherty, 2004; Bunce and Cowley, 2003). Not surprisingly,the ring current parameters and root-mean-square residuals(shown in Table 4) compare favorably with those of Giampieriand Dougherty (2004) since they are based on precisely the sameanalysis techniques. The slight differences are likely due tosubtleties of the non-linear fitting procedure but either set ofparameters is likely a good representation of the external field.Performing the inversion on a data set comprised only of Pioneer-11 and Voyager-1 and -2 measurements leads to the modelparameters g10 ¼ 21;130 nT, g20 ¼ 1583 nT and g30 ¼ 2211 nT.We confirm, as suspected, that the model coefficients obtainedare not identical to either the SPV or Z3 models.

To investigate whether there is evidence that Saturn’s fieldhas changed over the thirty year time span between the

25 30 35 50

it Number

4540

25 30 35 504540

25 30 35 504540

25 30 35 504540

parameters for the periapsis passes including radial distance of closest approach in

nits of nT based on the Cassini model are depicted by the open blue circles and the

black dots.

ARTICLE IN PRESS

0200400600800

1000

-5

0

5

-500

50100150200

Rev 6

0200400600800

1000

02468

r(R

s)

-10

-5

0

5

104:12:00 105:00:00 105:12:00

Latit

ude

(deg

rees

)

Day of year, 2005

0

200

400

600

-4-20246

-500

-300

-100

100

Rev 28

0200400600800

1000

2345678

r(R

s)

-30-20-10

01020

252:00:00 252:12:00 253:00:00 253:12:00

Day of year, 2006

0200400600800

Bθ

(nT)

-5

0

5

Bφ

(nT)

-400-300-200-100

0100

Br (

nT)

Rev 46

0200400600800

1000

|B (

nT)

|

2345678

r(R

s)

-20-15-10-505

10

162:12:00 163:00:00 163:12:00

Day of year, 2007

0200400600800

1000

-5

0

5

-120

-80

-40

0

Rev 47

0

500

1000

02468

10

r(R

s)

-4-202

178:12:00 179:00:00 179.:12:00

Day of year, 2007

Latit

ude

(deg

rees

)

Latit

ude

(deg

rees

)

Latit

ude

(deg

rees

)B

r (nT

)B

θ (n

T)B

φ (n

T)|B

(nT

)|

Br (

nT)

Bθ

(nT)

Bφ

(nT)

|B (

nT)

|

Br (

nT)

Bθ

(nT)

Bφ

(nT)

|B (

nT)

|

Fig. 3. Examples of the data and model for a few close periapsis passes. Model is shown in blue and data is shown in black. The components of the magnetic field in a

spherical coordinate system as well as the magnitude in units of nT are shown, as well as the radial distance range at closest approach in Rs and latitude in degrees.

M.E. Burton et al. / Planetary and Space Science 57 (2009) 1706–1713 1711

Pioneer/Voyager and Cassini eras, we have solved for modelparameters using data from all four spacecraft (column 4 ofTable 2). This model fits the Pioneer and Voyager observations

quite well as Table 4 shows. The first column is the misfit based onthe Pioneer/Voyager model and the second column is the misfitbased on the model using data from all four spacecraft. There is

ARTICLE IN PRESS

Table 3Pioneer-11, Voyager-1 and -2 disk parameters.

a b D m0I0 rms

Pioneer-11 inbound 6.7 12.4 2.1 48.6 2.4

Pioneer-11 outbound 5.7 26.2 3.6 27.0 2.1

Voyager-1 inbound 7.3 15.5 3.4 35.7 1.1

Voyager-1 outbound 7.6 21.4 2.2 59.7 1.2

Voyager-2 inbound 6.3 12.1 2.2 36.4 1.7

Voyager-2 outbound 6.2 29.1 4.4 20.0 1.2

Inner radius a, outer radius b and half-thickness D are given in units of Rs and m0I0

in units of nT.

Table 4Root-mean-square misfit in units of nT for Pioneer-11 and Voyager-1 and -2.

rms misfit P-11,V1,V2 rms misfit P-11,V1,V2,Cassini

Pioneer-11 inbound 2.3 2.1

Pioneer-11 outbound 1.0 .95

Voyager-1 inbound 1.5 1.4

Voyager-1 outbound 2.8 2.6

Voyager-2 inbound 3.0 3.1

Voyager-2 outbound 0.95 1.0

Misfit is shown separately for inbound and outbound segments for each flyby. The

first column shows the misfit calculated using the model based on Pioneer-11,

Voyager-1 and -2 only. The second number is the misfit based on all available data

including Cassini.

10

M.E. Burton et al. / Planetary and Space Science 57 (2009) 1706–17131712

only a small difference between the two values. With theexception of both legs of the Voyager-2 flyby, the misfit is lowerfor the model based on all four spacecraft data. Although notshown here we have examined the misfit close to the planet, usingdata within 3, 4, 5 or 6Rs and in each case they are similar to thosebased on all data within 10Rs.

This model based on data from all four spacecraft also fits theCassini data well. The mean and median values of the misfit are1.99 and 1.43 nT. The misfit for individual orbit segments,indicated by the black dots in the bottom panel of Fig. 2 doesnot differ appreciably from the misfit based on the Cassini model.It should be noted that the model based on all four spacecraft doesnot fit the data obtained on either the inbound or outboundsegments of the Saturn orbit insertion pass particularly well,which likely contributes to the larger mean value.

0.001

0.01

0.1

1

1 1.5 2 2.5 3 3.5

Rel

ativ

e m

agnt

iude

Harmonic Degree

Fig. 4. Relative magnetic spectrum for the Cassini model at Saturn’s surface (open

circles) and at 0.39Rs (closed circles).

4. Discussion

We have derived a new axial model of Saturn’s magnetic fieldwhich is based on all available data from three years of the Cassinimission and represents the most complete description yet ofSaturn’s planetary field. The greater spatial coverage provided byan orbiting spacecraft mitigates the issue of model non-unique-ness that compromised earlier models based on data of limitedspatial extent obtained during brief planetary flybys. Our modelsubstantiates some of the basic characteristics of Saturn’splanetary field which were based on data obtained by Pioneerand Voyager. There is a relatively large northward offset of themagnetic equator from the rotational equator given by

g20

2g10

¼ :036Rs (12)

which is somewhat smaller than that calculated using either theSPV (.037Rs) or Z3 (.038Rs) models. Assuming a polar radius of54,364 km, this model leads to surface field magnitudes of80,000 nT at the north pole, 18,000 nT at the equator and66,000 nT at the south pole.

The mean-square field at the planet’s surface introduced byLowes (1974) is given by

RnðaÞ ¼ ðnþ 1ÞXn

m¼0

ðg2n;m þ h2

n;mÞ (13)

where a is the planetary radius. At a distance r, not at the planet’ssurface, the mean-square field is

RnðrÞ ¼ ðnþ 1Þða=rÞð2nþ4ÞXn

m¼0

ðg2n;m þ h2

n;mÞ (14)

Eq. (14) can be used to estimate the apparent depth to the dynamoregion where the spectrum is assumed to be white. Elphic andRussell (1978) used Eq. (14) to estimate this depth for Mercuryand Jupiter. For Saturn, there is an unexplained deficit associatedwith the quadrupole term in the model based on Cassini data aswell the previous SPV and Z3 model (Connerney, 1993), thuswe use the dipole and octupole terms to calculate the depth atwhich the mean-square field of these two terms are equal. Thisyields a value of 0.39 for the radius of the dynamo producingregion. Fig. 4 shows the magnetic spectrum at the surface as wellat 0.39Rs.

Our reanalysis of Pioneer and Voyager data shows an absenceof evidence for secular variation in the field in the thirty yeartime span over which those data were obtained. We haveapplied the same analysis methods to all data sets, thus removingthe dependence of model parameters on modeling methodsand data selection techniques. A model based on data fromall four spacecraft fits the Pioneer and Voyager observationsequally well as a model based on Pioneer and Voyager dataalone. The same is true for Cassini and from this we conclude thatthere is no evidence for secular variation in the planetarymagnetic field in the interval over which those data wereobtained.

We have investigated whether a model of degree 3 is adequateto describe Saturn’s field. A model of degree 4 or 5 does notappreciably improve the misfit, nor does it appreciably alter themodel parameters obtained. For a model of degree 4 or 5 thecoefficients, g40 and g50 are �100 and the mean misfit improvesonly slightly (1.66 in both cases, compared with 1.99 above).

ARTICLE IN PRESS

0

2 104

4 104

6 104

8 104

1 105

0

1

2

3

4

5

6

1 2 3 4 5 6 7

sing

ular

val

ues

UT y

Degree

Fig. 5. UTy and the singular values determined by an axial model that includes

terms to degree 6.

M.E. Burton et al. / Planetary and Space Science 57 (2009) 1706–1713 1713

A condition that insures a stable solution arises fromconsideration of Eq. (10). If the dot products of the columns of Uand the data vector decay to zero more quickly than the singularvalues, the solution should not be unstable due to the presence ofsmall singular values. This is referred to as the discrete Picardcondition (Aster et al., 2005) and inspection of UTd and thesingular values (Fig. 5) suggests that higher degree parametersmay contribute to the solution, although UTd has clearly decayedto zero by degree 4. Further investigation of the contribution byhigher degree terms will be a topic of a future study in which datafrom additional orbits of unique geometry will be included.

We note that the simple axisymmetric ring current model isnot an ideal representation of the external currents since it clearlydoes not reflect our current understanding based on Cassiniobservations. From images of energetic neutral atoms (Krimigis etal., 2007), the ring current is not uniform and symmetric, butinstead varies substantially with local time and is a highlydynamic structure and is more likely a bowl-shaped structurethan a simple axisymmetric disk (Arridge et al., 2008). We haveattempted to account for this by modeling the external fieldseparately for the inbound or outbound orbit segments. Morecomplicated modeling is however beyond the scope of this paperand we assert that, given the acceptable values for the misfit onmost of the passes (Fig. 2), the axisymmetric model does anadequate job of estimating the field due to the ring current closeto the planet.

An axisymmetric model does not incorporate the periodic‘cam’ signature observed in the data, however, as shown in Fig. 1from (Giampieri et al., 2006a), this signature rarely exceeds 5 nT inany component thus represents a small fraction of the field whichis dominated by the axial terms. In a separate study we haveinvestigated planetary field models that include non-axial termsthat are based on the proposed fixed rotation rates in thepublished literature (Giampieri et al., 2006a; Anderson andSchubert, 2007) and variable rotation rates based on the SKRperiod (Kurth et al., 2007) or the phase of the magnetic variation(Cowley et al., 2006). These models have small non-axisymmetricterms and do not represent a greatly improved fit over the axialmodel described here. Given the uncertainty in the rotation rateand periodic signature described in Section 1, the validity of any ofthese models is suspect, but will continue to be a topic of futurestudy.

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