ARTICLE IN PRESS
Planetary and Space Science 57 (2009) 1706–1713
Contents lists available at ScienceDirect
Planetary and Space Science
journal homepage: www.elsevier.com/locate/pss
Model of Saturn’s internal planetary magnetic field based
on Cassini observations
M.E. Burton a,, M.K. Dougherty b, C.T. Russell c
a
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA
Imperial College of Science and Technology, London, UK
c
University of California, Los Angeles, CA, USA
b
a r t i c l e in f o
a b s t r a c t
Article history:
Received 18 November 2008
Received in revised form
9 April 2009
Accepted 14 April 2009
Available online 3 May 2009
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 g 10 ¼ 21; 162, g 20 ¼ 1514, g 30 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.
Keywords:
Saturn magnetic field
Magnetosphere
Planetary magnetic fields
1. Introduction
Based on data obtained during the three brief flybys of Pioneer11 and Voyager-1 and -2 almost thirty years ago, Saturn’s internal
magnetic field has been characterized as highly axisymmetric. The
initial orbits of the Cassini spacecraft confirmed this result
(Dougherty et al., 2005; Giampieri et al., 2006b). In such a
magnetosphere, periodic signatures in the magnetic field, plasma
data and radio emissions are not expected. However, periodic
emission of Saturn kilometric radiation (SKR) was detected by
Voyager-1 (Desch and Kaiser, 1981). The period (10 h 39 min 24 s
7 s) was presumed to represent the rotation period of the deep
interior and was used to define the official International
Astronomical Union (IAU) rotation rate. Subsequent observations
of SKR by the Ulysses spacecraft revealed that the period was
longer by approximately 1% (Galopeau and Lecacheux, 2000).
Further observations from Cassini confirmed this result and
indicate the SKR period varies over much shorter time scales as
well (Kurth et al., 2007).
In a reanalysis of data obtained by Pioneer and Voyager, a
modulation of the magnetic field close to that of the planetary
rotation period was detected (Espinosa and Dougherty, 2000). The
Corresponding author. Tel.: +1818 354 7375.
E-mail address: [email protected] (M.E. Burton).
0032-0633/$ - see front matter & 2009 Elsevier Ltd. All rights reserved.
doi:10.1016/j.pss.2009.04.008
proposed mechanism was that a magnetic anomaly close to the
planet creates a compressional wave that propagates radially
outward across the background magnetic field (the so-called
camshaft model) (Espinosa et al., 2003). Periodicities such as
these now appear to be a ubiquitous feature of Saturn’s magnetic
field. In an initial study based on spectral analysis of magnetic
field data from the first fifteen orbits of the Cassini mission,
Giampieri et al. (2006a) reported the detection of a time
stationary magnetic signal with a period of 10 h 47 min 6 s
40 s. The signal was found to be stable in period, amplitude and
phase over fourteen months of observations, suggesting a close
connection with the conductive region inside the planet. This
period was shown later to correspond to the variable SKR period
over the same time interval (Kurth et al., 2007). Cowley et al.
(2006) confirmed that the magnetic oscillations are consistent
with a model in which phase fronts rotate with the planet and
radiate outward at a speed comparable to the equatorial Alfven
speed, consistent with the ‘camshaft model’. Southwood and
Kivelson (2007) have suggested that the periodic signature may
be due to a rotating non-axisymmetric system of field aligned
currents. Noting that the plasma and magnetic field in the inner
magnetosphere rotate synchronously with the variable SKR,
Gurnett et al. (2007) emphasize an external source. They propose
that the observed electron density modulation in the inner region
of the plasma disk acts as the ‘cam’ that drives other rotationally
modulated phenomenon and suggest the origin is linked to the
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M.E. Burton et al. / Planetary and Space Science 57 (2009) 1706–1713
plasma disk that originates when Enceladus variable neutral cloud
is ionized. Subsequently, Andrews et al. (2008) examined 23 of the
Cassini equatorial orbits and found that the phase of the magnetic
oscillations are well-organized by a period close to that of the
variable SKR.
Using gravitational data, along with Pioneer and Voyager
radio 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. They
found that a mean geoid that matches the gravitational
data and minimizes the wind-induced dynamic heights of
the 100 mbar isosurface could be derived based on this rotation
rate.
Based on the most optimistic uncertainty cited for any of these
proposed rotation rates (6 s) the longitude could be in error by
180 over the course of the four year mission. This uncertainty in
the rotation rate and the nature and origin of the observed
periodicities has impeded the development of a planetary
magnetic field model and at the same time has called into
question the previously accepted notion of a strictly axisymmetric
magnetic field. Any magnetic field model that includes non-axial
terms is untenable if the assumed rotation rate is incorrect.
1707
Despite this uncertainty, it is useful and instructive to derive a
magnetic field model based only on axisymmetric terms, since
such a model is independent of the assumed rotation rate and can
be compared with earlier models based on Pioneer and Voyager
which were constrained in just such a way (Connerney et al.,
1982). Using similar analysis and inversion techniques to those
used to obtain the earlier Pioneer/Voyager models, we have
derived an updated axisymmetric model of Saturn’s magnetic
field based on data obtained during the first three years of the
Cassini orbital tour.
2. Analysis
2.1. Data used in this study
Our analysis is based on data obtained during the first three
years of the Cassini mission (July 1, 2004–July 1, 2007). Closest
approach occurred at a variety of radial distances, latitudes and
longitudes and are given in Table 1. All orbits with periapsis
distances less than 10Rs, 45 in all, were included in the analysis.
Saturn orbit insertion (SOI) (Orbit 0), on July 1, 2004 was the
Table 1
Times, distance and latitudes of the periapsis passes used in this study.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
Orbit
Day of year
Date
Time UTC
Periapsis distance (Rs)
Latitude (deg)
O
A
B
C
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
41
42
43
44
45
46
47
183
302
350
16
48
68
88
104
123
141
159
177
196
214
232
248
266
285
302
331
358
17
56
79
118
142
181
204
228
252
268
284
301
313
325
336
349
365
82
98
114
130
147
163
179
2004-07-01
2004-10-28
2004-12-15
2005-01-16
2005-02-17
2005-03-09
2005-03-29
2005-04-14
2005-05-03
2005-05-21
2005-06-08
2005-06-26
2005-07-14
2005-08-02
2005-08-20
2005-09-05
2005-09-23
2005-10-12
2005-10-29
2005-11-27
2005-12-24
2006-01-17
2006-02-25
2006-03-20
2006-04-28
2006-05-22
2006-06-30
2006-07-23
2006-08-16
2006-09-09
2006-09-25
2006-10-11
2006-10-28
2006-11-09
2006-11-20
2006-12-02
2006-12-15
2006-12-31
2007-03-23
2007-04-08
2007-04-24
2007-05-10
2007-05-27
2007-06-12
2007-06-28
02:38
10:19
05:51
06:36
00:59
11:39
23:42
23:20
01:51
06:09
10:42
15:46
22:12
05:21
11:09
11:29
20:36
01:41
22:56
11:24
21:14
07:06
10:47
20:14
23:59
09:01
13:05
21:48
20:54
17:38
19:32
22:51
00:17
00:03
23:06
21:44
00:18
05:15
12:02
16:49
20:13
22:55
00:37
00:51
00:53
1.33
6.2
4.8
4.9
3.5
3.5
3.5
2.6
3.6
3.6
3.6
3.6
3.6
3.6
3.6
2.8
3.1
3.1
4.6
4.6
4.6
5.6
5.6
5.5
5.5
5.5
5.4
4.2
4.2
2.9
4.0
5.5
4.7
4.7
4.7
4.7
7.7
9.8
9.7
7.5
5.7
4.2
3.2
2.7
2.5
17.1
12.0
4.1
6.4
0.11
0.11
0.12
3.25
12.1
12.2
12.3
12.4
12.5
12.6
12.7
8.2
0.26
0.34
0.11
0.16
0.16
0.03
0.03
0.15
0.15
0.34
0.35
10.1
10.1
12.4
21.1
29.4
25.9
25.9
25.9
25.9
38.0
44.6
66.1
40.3
31.4
23.3
15.2
9.2
1.0
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M.E. Burton et al. / Planetary and Space Science 57 (2009) 1706–1713
8
SOI - Orbit 46 (July 1, 2004 - June 12, 2007)
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
m I0 <
Br ðz; r; aÞ ¼ 0
2 signðzÞðminjzj; DÞ þ Zr
2r :
xr
Range (Rs)
60
2
4
Latitude (degrees)
40
20
1
X
ð1Þk ð2k 2Þ!a2k r
22k ðk!Þ2
k¼1
P 12k1 ½ðz DÞ=ðZr Þ
Z2k
r
39
P 12k1 ½ðz þ DÞ=ðxr Þ =
5
;
x2k
(2)
r
where
0
xr ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðz þ DÞ2 þ r2 ;
Zr ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðz DÞ2 þ r2
−20
−40
−60
0
60
120
180
240
300
360
Longitude (degrees)
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.
closest at only 1.33Rs. Five orbits came within three Saturnian
radii and another 13 were closer than 4Rs, providing excellent
spatial coverage close to the planet. Fig. 1 depicts latitude versus
longitude of the trajectory within 6Rs, color-coded according to
distance from the planet. The longitude is based on the IAU
rotation period.
One-minute averages of magnetic field data acquired by the
Cassini fluxgate magnetometer (Dougherty et al., 2004) were
used to derive the model. Data obtained within 20Rs were
used to model the field due to external sources and data within
10Rs were used to model the internal field. The data were
surveyed for evidence of magnetopause or bowshock crossings
and those data were excluded from the study. Measurements
near close flybys of the icy satellites were excluded as well. The
model is derived from more than 78,000 1-min averages of the
magnetic field.
2.2. Modeling the ring current field
A large-scale eastward flowing ring current is known to exist
in Saturn’s magnetosphere and has been studied extensively
using Cassini observations (Bunce et al., 2007). Since its
contribution close to the planet can be significant, it must be
accounted for when modeling the planetary field. We use the
analytical expression derived by Giampieri and Dougherty (2004)
based on the simple axisymmetric equatorial current sheet
centered on the planet’s equator, first described by Connerney
et al. (1981) for the Jovian magnetosphere. In a cylindrical
coordinate system ðz; r; fÞ with the z-axis parallel to Saturn’s
magnetic dipole axis, the disk is confined to the region jzjoD, and
a prob. The azimuthal current density is distributed as
J f ¼ I0 =r. The cylindrical components, z and r of the magnetic
field are given by
( "
#
1
X
ðz þ D þ xr Þ
m I0
ð1Þk ð2k 1Þ!a2k
log
þ
Bz ðz; r; aÞ ¼ 0
ðz D þ Zr Þ
2
22k ðk!Þ2
k¼1
2
39
P 02k1 ½ðz DÞ=ðZr Þ P02k1 ½ðz þ DÞ=ðxr Þ =
4
5
(1)
;
Z2k
x2k
r
r
Eqs. (1) and (2) describe the ring current field for an outer
boundary at infinity. For a finite disk of inner radius a and outer
radius b, the disk field is calculated with the outer radius, b
replacing a in Eqs. (1) and (2) and Bz ¼ Bz ðaÞ Bz ðbÞ and a similar
expression for Br. I0 has units of current per unit length. Thus the
total current is given by I ¼ 2I0 D logðb=aÞ. Pm
n are the associated
Legendre functions.
To model the ring current contribution to the magnetic field
we used 1-min averages obtained outside of 4Rs to minimize
the contribution by the planetary field and inside of 20Rs in
order to minimize the contribution due to currents flowing
on the magnetopause or tail currents. All passes with a closest
approach distance less than 10Rs during the first three years
of the tour were used in the analysis, a total of 45 orbits.
Current sheet structure and characteristics are known to vary
with local time (Arridge et al., 2008) and temporal variations in
the solar wind and magnetosphere are likely to occur over
time scales corresponding to that of a periapsis pass (several
days). To minimize these effects, the inbound and outbound legs of each orbit were analyzed separately resulting in 86 cases (four passes had insufficient data, due to data
gaps).
First, to eliminate the estimated contribution due to internal
sources we subtracted the axisymmetric SPV model magnetic
field (Davis and Smith, 1990) from the observations. The residual
field (observed internal) was transformed into cylindrical
coordinates, Br and Bz . Eqs. (5) and (6) were used to fit the data
and obtain the geometrical parameters (a,b,D) and the current
parameter, m0 I0 using a large-scale non-linear least-squares fit
algorithm based on the reflective Newton method (Giampieri and
Dougherty, 2004). We have made no attempt to model other
currents systems that exist in Saturn’s magnetosphere, such as tail
or magnetopause currents.
2.3. Modeling the planetary field
After the best-fit ring current parameters were obtained
for each orbit segment as described, the estimated contribution was calculated (Eqs. (1) and (2)) and subtracted from
the measured field. The residual field was assumed to originate
in Saturn’s interior and was used to determine the model
parameters.
The internal planetary magnetic field in spherical polar
coordinate system (r, y; f) is given by the standard equations
@V
@r "
1 X
n
X
Br ¼ ¼
ðn þ 1Þ
n¼1 m¼0
P m
n ðcos yÞ
#
ðnþ2Þ
Rp
m
½g m
cosðm
f
Þ
þ
h
sinðm
f
Þ
n
n
r
(3)
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M.E. Burton et al. / Planetary and Space Science 57 (2009) 1706–1713
1 @V
r @y "
ðnþ2Þ
1 X
n
X
Rp
By ¼ ¼ n¼1 m¼0
m
dP n ðcos yÞ
r
Singular value decomposition relies on the fact that any n m
matrix A can be written as
#
m
½g m
n cosðmfÞ þ hn sinðmfÞ
A ¼ U w VT
(4)
dy
1 @V
1
¼
r sin y @f sin y
" 1 X
n
X
Rp ðnþ2Þ
Bf ¼ n¼1 m¼0
Pm
n ðcos yÞ
r
1709
#
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 positive
or zero elements, the singular values. The SVD solution can be
shown to be the least-squares solution (Aster et al., 2005) and can
be written as
m ¼ Vð1=wÞUT y
m
½g m
n sinðmfÞ hn cosðmfÞ
(5)
where Rp is the planetary radius (60,268 km), Pm
n ðcos yÞ are the
Schmidt quasi-normalized associated Legendre function of degree
n and order m traditionally used in magnetic field modeling and
m
gm
n and hn are internal spherical harmonic coefficients.
To model the planetary magnetic field we have used the
generalized inverse method of Lanczos (1971) that utilizes
singular value decomposition (SVD). Details of the use of this
technique to solve linear least-squares problems are given
succinctly in Press et al. (1982) and an example of its application
to planetary magnetic field modeling can be found in Connerney
(1981) thus only a brief sketch of the method is provided here. The
equations describing the planetary magnetic field (5) can be
written in matrix form as
1 0
1
0
X r11 X r12 X r1m
Br1
C
BB C BX
B y1 C B y11 X y12 X y1m C
C B
C
B
B Bf1 C B X f11 X f12 X f1m C
C B
C0
B
1
B . C B .
.. C
..
..
C g1
B . C B .
. CB
.
.
B . C B .
C B
CB g 2 C
B
C
B B C BX
C
X ri2 X rim C
CB
B ri C B ri1
gn C
C B
CB
B
C
B
B Byi C ¼ B X yi1 X yi2 X yim CB
(6)
C B
CB h1 C
B
C
C
BB C BX
C
B fi C B fi1 X fi2 X fim CB
.
C B
CB . C
B
B .. C B ..
.. C@ . A
..
..
C
B . C B .
.
.
.
C B
C hm
B
C B
C
B
B Brn C B X rn1 X rn2 X rnm C
C
C
B
B
B Byn C B X yn1 X yn2 X ynm C
A @
A
@
Bfn
X fn1 X fn2 X fnm
where X l;n;m are the elements of the n m design matrix made up
of basis functions, (n, number of data points, m, number of model
parameters), y ¼ Bi is the data vector of magnetic field
measurements, and m is the model parameter vector (the g n;m
and hn;m model coefficients) to be determined. The index l refers to
the r, y, or f component of the field. As a simple example, the
elements of the design matrix associated with the r, y, and f
components of the g 10 term are
3
3
Rp
Rp
X 1;1;1 ¼ 2
P10 ¼ 2
cos y
r
r
3
3
Rp @P 10
Rp
X 2;1;1 ¼
¼
sin y
r
@y
r
X 3;1;1 ¼ 0
(8)
(7)
(9)
or alternately
m¼
m
X
Ui y
i¼1
wi
Vi
(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 linear
combination of the columns of V with coefficients obtained by
forming the dot products of the columns of U and the data vector
y weighted by the singular values. The variance in the estimate of
the model parameter is given by
s2 ðmj Þ ¼
m X
Vi 2
i¼1
(11)
wi
3. Results
We have implemented the analysis techniques described in
Sections 2.2 and 2.3 to derive an axial model of degree 3 for
Saturn’s internal magnetic field based on all data obtained during
the first three years of the Cassini orbital tour. The model
coefficients are shown in Table 2 (column 5). The condition
number, the ratio of the largest to smallest singular value and a
measure of the sensitivity of the solution to inaccuracies in the
data, is also given. It reflects the spatial coverage of the data and
thus will improve as the mission progresses. The condition
number for the design matrix associated with our model is 4.36,
thus errors associated with the g 30 term are approximately four
times larger than those associated with the g 10 term.
For comparison, coefficients from earlier, axial models based
on 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 on
data obtained during the first year of the Cassini mission
(Giampieri et al., 2006b) is shown in column 2. These model
parameters can be directly compared with those from our analysis
since they were derived using similar methods for assessing the
external sources and similar inversion techniques. All three model
parameters derived in our analysis using three years of data are
within a percent of those based on data obtained during the first
year only. The dipole term is only slightly smaller and the
quadrupole and octupole terms slightly larger. The table can be
thought of as a representation of the evolution of the planetary
Table 2
Spherical harmonic coefficients for magnetic field models in units of nT based on a planetary radius of 60,268 km.
Multipole term
SPV
Z3
Cassini (SOI)
Cassini (SOI - 8/05)
Cassini (SOI - 7/07)
Cassini, P11,V1,V2
g 10
g 20
g 30
CN
21,225
1566
2332
21,248
1613
2683
21,084
1544
2150
21,169
1504
2264
5.08
21,162
1514
2283
4.36
21,145
1546
2241
3.81
Condition number, CN (dimensionless) for the models derived in this study is also shown.
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M.E. Burton et al. / Planetary and Space Science 57 (2009) 1706–1713
field model over the course of the Cassini mission as more data at
varied geometries has been acquired.
To assess how well the model fits the observations, we have
calculated the root-mean-square misfit or residual on a pass-bypass basis. The calculation of the misfit does not include the Bf
component which is dominated by the ‘cam’ signal in the inner
magnetosphere. The mean of that value for all passes is 1.67 and
the median is 1.41 indicating that the model fits the data quite
well. The misfit based on data obtained close to the planet ðo6RsÞ
shows a modest improvement over that based on the entire
analysis interval of 10Rs, with a mean of 1.5 and the median of
1.27 nT. On a pass-by-pass basis the misfit of the model based on
Cassini data alone (Cassini, SOI 7=07) in Table 2 is depicted by
the open blue circles in the bottom panel of Fig. 2. The top panel
shows the closest approach distance of each orbit. The range of
latitudes covered in the analysis interval is depicted by the extent
of the vertical line in the second panel and the symbol represents
the value at closest approach. There is a similar representation for
the coverage in local time.
Fig. 3 shows how well the model fits the data for four of the
closest periapsis passes, Rev 6 with a closest approach distance of
2.6Rs, Rev 28 (2.9Rs), Rev 46 (2.7Rs) and Rev 47 (2.5Rs). The
model is shown in blue and the data in black and on the scale of
these figures, the two are indistinguishable. The periodic
signature in the azimuthal component of the field, Bf , and there
is evidence of the ‘cam’ signal in all four examples. The radial
distance and latitude of each pass are also shown.
To compare a planetary field model based on data obtained in
the Pioneer/Voyager era with one based on Cassini data, it is
necessary to apply the same analysis methods to all data sets.
A direct comparison of model parameters from earlier studies
would not be valid because of the use of different modeling
methods and data selection techniques. Although the Pioneer-11
and Voyager-1 and -2 data were analyzed extensively in the years
following those flybys (see, for example, Davis and Smith, 1990;
Acuna et al., 1983) none of those analyzis used precisely the same
methods that we have applied here. To derive the Z3 magnetic
field model, Connerney et al. (1982) combined data from Pioneer11 and Voyager-1 and -2 and derived source coefficients (G10 , G11 ,
H11 ) to model the external field. Davis and Smith (1990) used data
from all three spacecraft flybys to derive their SPV model which is
based on a least-squares approach. They used a data weighting
scheme to emphasize measurements close to the planets and also
modeled the external field using the Gauss coefficients, Table 3.
(For reference, the spherical harmonic coefficients normalized for
a planetary radius of 60,268 km for the SPV (Davis and Smith,
1990) model are g 10 ¼ 21; 225, g 20 ¼ 1566, and g 30 ¼ 2332, and
for Z3 (Connerney et al., 1982) g 10 ¼ 21; 248, g 20 ¼ 1613 and
g 30 ¼ 2683. All coefficients are in units of nT.)
We have applied the same methods as those used to model the
Cassini planetary field, described in Sections 2.2 and 2.3, to the
data obtained by Pioneer-11 and Voyager-1 and -2. For consistency, we have re-derived the ring current parameters, even
though these too have been analyzed extensively in (Giampieri
and 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 Giampieri
and Dougherty (2004) since they are based on precisely the same
analysis techniques. The slight differences are likely due to
subtleties of the non-linear fitting procedure but either set of
parameters is likely a good representation of the external field.
Performing the inversion on a data set comprised only of Pioneer11 and Voyager-1 and -2 measurements leads to the model
parameters g 10 ¼ 21; 130 nT, g 20 ¼ 1583 nT and g 30 ¼ 2211 nT.
We confirm, as suspected, that the model coefficients obtained
are not identical to either the SPV or Z3 models.
To investigate whether there is evidence that Saturn’s field
has changed over the thirty year time span between the
r (Rs)
10
5
Misfit (nT)
Local Time
Latitude
(degrees)
0
0
5
10
15
20
25
30
35
40
45
50
0
5
10
15
20
25
30
35
40
45
50
0
5
10
15
20
25
30
35
40
45
50
0
5
10
15
20
25
30
35
40
45
50
50
0
−50
0
20
10
0
15
10
5
0
Orbit Number
Fig. 2. Root-mean-square misfit for individual orbits (bottom panel), as well geometric parameters for the periapsis passes including radial distance of closest approach in
units of Saturnian radii, Rs (top panel) latitude in degrees and local time. The misfit in units of nT based on the Cassini model are depicted by the open blue circles and the
misfit based on a model of Cassini, Pioneer and Voyager is shown by the smaller solid black dots.
ARTICLE IN PRESS
M.E. Burton et al. / Planetary and Space Science 57 (2009) 1706–1713
Br (nT)
Br (nT)
100
-100
-300
-500
600
Bθ (nT)
1000
800
600
400
200
0
400
200
0
6
4
2
0
-2
-4
Bφ (nT)
5
Bφ (nT)
Bθ (nT)
Rev 28
Rev 6
200
150
100
50
0
-50
0
1000
800
600
400
200
0
Latitude
(degrees)
r(Rs)
|B | (nT)
1000
800
600
400
200
0
8
6
4
2
0
Latitude
(degrees)
r(Rs)
|B | (nT)
-5
5
0
-5
-10
104:12:00
105:00:00
105:12:00
8
7
6
5
4
3
2
20
10
0
-10
-20
-30
252:00:00
252:12:00
Br (nT)
-40
-80
Bθ (nT)
0
1000
800
600
400
200
0
Bφ (nT)
Br (nT)
Bθ (nT)
Bφ (nT)
5
0
-5
1000
|B | (nT)
1000
800
600
400
200
0
10
0
8
6
4
2
0
r(Rs)
8
7
6
5
4
3
2
500
Latitude
(degrees)
|B | (nT)
r(Rs)
0
-120
-5
Latitude
(degrees)
253:12:00
Rev 47
Rev 46
100
0
-100
-200
-300
-400
5
253:00:00
Day of year, 2006
Day of year, 2005
800
600
400
200
0
1711
10
5
0
-5
-10
-15
-20
162:12:00
163:00:00
163:12:00
Day of year, 2007
2
0
-2
-4
178:12:00
179:00:00
179.:12:00
Day of year, 2007
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.
Pioneer/Voyager and Cassini eras, we have solved for model
parameters using data from all four spacecraft (column 4 of
Table 2). This model fits the Pioneer and Voyager observations
quite well as Table 4 shows. The first column is the misfit based on
the Pioneer/Voyager model and the second column is the misfit
based on the model using data from all four spacecraft. There is
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M.E. Burton et al. / Planetary and Space Science 57 (2009) 1706–1713
The mean-square field at the planet’s surface introduced by
Lowes (1974) is given by
Table 3
Pioneer-11, Voyager-1 and -2 disk parameters.
Pioneer-11 inbound
Pioneer-11 outbound
Voyager-1 inbound
Voyager-1 outbound
Voyager-2 inbound
Voyager-2 outbound
a
b
D
m0 I0
rms
6.7
5.7
7.3
7.6
6.3
6.2
12.4
26.2
15.5
21.4
12.1
29.1
2.1
3.6
3.4
2.2
2.2
4.4
48.6
27.0
35.7
59.7
36.4
20.0
2.4
2.1
1.1
1.2
1.7
1.2
Rn ðaÞ ¼ ðn þ 1Þ
n
X
2
ðg 2n;m þ hn;m Þ
(13)
m¼0
where a is the planetary radius. At a distance r, not at the planet’s
surface, the mean-square field is
Rn ðrÞ ¼ ðn þ 1Þða=rÞð2nþ4Þ
n
X
2
ðg 2n;m þ hn;m Þ
(14)
m¼0
Inner radius a, outer radius b and half-thickness D are given in units of Rs and m0 I0
in units of nT.
Table 4
Root-mean-square misfit in units of nT for Pioneer-11 and Voyager-1 and -2.
Pioneer-11 inbound
Pioneer-11 outbound
Voyager-1 inbound
Voyager-1 outbound
Voyager-2 inbound
Voyager-2 outbound
rms misfit P-11,V1,V2
rms misfit P-11,V1,V2,Cassini
2.3
1.0
1.5
2.8
3.0
0.95
2.1
.95
1.4
2.6
3.1
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.
only a small difference between the two values. With the
exception of both legs of the Voyager-2 flyby, the misfit is lower
for the model based on all four spacecraft data. Although not
shown here we have examined the misfit close to the planet, using
data within 3, 4, 5 or 6Rs and in each case they are similar to those
based on all data within 10Rs.
This model based on data from all four spacecraft also fits the
Cassini data well. The mean and median values of the misfit are
1.99 and 1.43 nT. The misfit for individual orbit segments,
indicated by the black dots in the bottom panel of Fig. 2 does
not differ appreciably from the misfit based on the Cassini model.
It should be noted that the model based on all four spacecraft does
not fit the data obtained on either the inbound or outbound
segments of the Saturn orbit insertion pass particularly well,
which likely contributes to the larger mean value.
Eq. (14) can be used to estimate the apparent depth to the dynamo
region where the spectrum is assumed to be white. Elphic and
Russell (1978) used Eq. (14) to estimate this depth for Mercury
and Jupiter. For Saturn, there is an unexplained deficit associated
with the quadrupole term in the model based on Cassini data as
well the previous SPV and Z3 model (Connerney, 1993), thus
we use the dipole and octupole terms to calculate the depth at
which the mean-square field of these two terms are equal. This
yields a value of 0.39 for the radius of the dynamo producing
region. Fig. 4 shows the magnetic spectrum at the surface as well
at 0.39Rs.
Our reanalysis of Pioneer and Voyager data shows an absence
of evidence for secular variation in the field in the thirty year
time span over which those data were obtained. We have
applied the same analysis methods to all data sets, thus removing
the dependence of model parameters on modeling methods
and data selection techniques. A model based on data from
all four spacecraft fits the Pioneer and Voyager observations
equally well as a model based on Pioneer and Voyager data
alone. The same is true for Cassini and from this we conclude that
there is no evidence for secular variation in the planetary
magnetic field in the interval over which those data were
obtained.
We have investigated whether a model of degree 3 is adequate
to describe Saturn’s field. A model of degree 4 or 5 does not
appreciably improve the misfit, nor does it appreciably alter the
model parameters obtained. For a model of degree 4 or 5 the
coefficients, g 40 and g 50 are 100 and the mean misfit improves
only slightly (1.66 in both cases, compared with 1.99 above).
10
4. Discussion
g 20
¼ :036Rs
2g 10
1
Relative magntiude
We have derived a new axial model of Saturn’s magnetic field
which is based on all available data from three years of the Cassini
mission and represents the most complete description yet of
Saturn’s planetary field. The greater spatial coverage provided by
an orbiting spacecraft mitigates the issue of model non-uniqueness that compromised earlier models based on data of limited
spatial extent obtained during brief planetary flybys. Our model
substantiates some of the basic characteristics of Saturn’s
planetary field which were based on data obtained by Pioneer
and Voyager. There is a relatively large northward offset of the
magnetic equator from the rotational equator given by
0.1
0.01
(12)
which is somewhat smaller than that calculated using either the
SPV (.037Rs) or Z3 (.038Rs) models. Assuming a polar radius of
54,364 km, this model leads to surface field magnitudes of
80,000 nT at the north pole, 18,000 nT at the equator and
66,000 nT at the south pole.
0.001
1
1.5
2
2.5
Harmonic Degree
3
3.5
Fig. 4. Relative magnetic spectrum for the Cassini model at Saturn’s surface (open
circles) and at 0.39Rs (closed circles).
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M.E. Burton et al. / Planetary and Space Science 57 (2009) 1706–1713
8 10
6 10
6
4
5
4
4
UTy
3
4 10
4
2
2 10
References
5
4
singular values
1 10
1
0
0
1
2
3
4
5
6
1713
7
Degree
Fig. 5. UT y and the singular values determined by an axial model that includes
terms to degree 6.
A condition that insures a stable solution arises from
consideration of Eq. (10). If the dot products of the columns of U
and the data vector decay to zero more quickly than the singular
values, the solution should not be unstable due to the presence of
small singular values. This is referred to as the discrete Picard
condition (Aster et al., 2005) and inspection of UT d and the
singular values (Fig. 5) suggests that higher degree parameters
may contribute to the solution, although UT d has clearly decayed
to zero by degree 4. Further investigation of the contribution by
higher degree terms will be a topic of a future study in which data
from additional orbits of unique geometry will be included.
We note that the simple axisymmetric ring current model is
not an ideal representation of the external currents since it clearly
does not reflect our current understanding based on Cassini
observations. From images of energetic neutral atoms (Krimigis et
al., 2007), the ring current is not uniform and symmetric, but
instead varies substantially with local time and is a highly
dynamic structure and is more likely a bowl-shaped structure
than a simple axisymmetric disk (Arridge et al., 2008). We have
attempted to account for this by modeling the external field
separately for the inbound or outbound orbit segments. More
complicated modeling is however beyond the scope of this paper
and we assert that, given the acceptable values for the misfit on
most of the passes (Fig. 2), the axisymmetric model does an
adequate job of estimating the field due to the ring current close
to the planet.
An axisymmetric model does not incorporate the periodic
‘cam’ signature observed in the data, however, as shown in Fig. 1
from (Giampieri et al., 2006a), this signature rarely exceeds 5 nT in
any component thus represents a small fraction of the field which
is dominated by the axial terms. In a separate study we have
investigated planetary field models that include non-axial terms
that are based on the proposed fixed rotation rates in the
published literature (Giampieri et al., 2006a; Anderson and
Schubert, 2007) and variable rotation rates based on the SKR
period (Kurth et al., 2007) or the phase of the magnetic variation
(Cowley et al., 2006). These models have small non-axisymmetric
terms and do not represent a greatly improved fit over the axial
model described here. Given the uncertainty in the rotation rate
and periodic signature described in Section 1, the validity of any of
these models is suspect, but will continue to be a topic of future
study.
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