Saturn's Very Axisymmetric Magnetic Field

Earth and Planetary Science Letters 304 (2011) 22–28

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Earth and Planetary Science Letters

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Saturn's very axisymmetric magnetic ﬁeld: No detectable secular variation or tilt

Hao Cao a,⁎, Christopher T. Russell a, Ulrich R. Christensen b, Michele K. Dougherty c, Marcia E. Burton d a UCLA Institute of Geophysics and Planetary Physics, Los Angeles, CA 90095, USA b Max Planck Institute for Solar System Research, Max-Planck-Strasse 2, 37191 Katlenburg-Lindau, Germany c Blackett Laboratory, Imperial College London, SW72AZ, UK d Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA article info abstract

Article history: Saturn is the only planet in the solar system whose observed magnetic field is highly axisymmetric. At least a Accepted 20 February 2011 small deviation from perfect symmetry is required for a dynamo-generated magnetic field. Analyzing more Available online 5 March 2011 than six years of magnetometer data obtained by Cassini close to the planet, we show that Saturn's observed field is much more axisymmetric than previously thought. We invert the magnetometer observations that Editor by: T. Spohn were obtained in the “current-free” inner magnetosphere for an internal model, varying the assumed unknown rotation rate of Saturn's deep interior. No unambiguous non-axially symmetric magnetic moment is Keywords: Saturn detected, with a new upper bound on the dipole tilt of 0.06°. An axisymmetric internal model with Schmidt- magnetic dynamo normalized spherical harmonic coefficients g10=21,191±24 nT, g20=1586±7 nT. g30=2374 ±47 nT is interior structure derived from these measurements, the upper bounds on the axial degree 4 and 5 terms are 720 nT and 3200 nT respectively. The secular variation for the last 30 years is within the probable error of each term from degree 1 to 3, and the upper bounds are an order of magnitude smaller than in similar terrestrial terms for degrees 1 and 2. Differentially rotating conducting stable layers above Saturn's dynamo region have been proposed to symmetrize the magnetic field (Stevenson, 1982). The new upper bound on the dipole tilt implies that this stable layer must have a thickness LN=4000 km, and this thickness is consistent with our weak secular variation observations. © 2011 Elsevier B.V. All rights reserved.

1. Introduction gravitational and atmospheric data (Anderson and Schubert, 2007; Read et al., 2009). Secular variation of the terrestrial magnetic field The tilt of the magnetic dipole with respect to the rotation axis for has been widely studied (Bloxham and Gubbins, 1985; Jackson et al., solar system planets other than Saturn varies from values of five 2000) as a useful tool to investigate the fluid motion in the Earth's degrees or less at Mercury (Anderson et al., 2010), to roughly ten outer core. One of the most well defined features is the dipole moment degrees at Earth and Jupiter, and to up to 60° at Uranus (Russell and currently changing about 5% per 100 years (Barton, 1989). At the Dougherty, 2010). A tilted dipole magnetic field enables a very outer planets, constraints have been placed on the Jovian magnetic sensitive measurement of the rotation rate of the dynamo region deep field (Connerney and Acuna, 1982; Yu et al., 2010). For Saturn, direct in the interior of a planet. At Earth, the variation in the rotation rate of measurements of the magnetic field were made by Pioneer 11 in 1979, the dipole at a period of 60 years has detected the torsional oscillation Voyager 1 and 2 in 1980 and 1981 respectively. Several axisymmetric of the core and its correlation with the length of the day (Roberts et al., magnetic field models were proposed from these observations 2007). At Jupiter it has been used to make measurements to (Connerney et al., 1982; Davis and Smith, 1990). In this paper we millisecond accuracy of the System III rotation rate (Yu and Russell, show that there is no unambiguous non-axisymmetric intrinsic 2009). In contrast, at Saturn the signal that was originally thought to magnetic moment which would allow us to determine the rotation be due to a rotating internal field was found to be changing (Galopeau rate of Saturn's interior. The dipole tilt must be less than 0.06°. No and Lecacheux, 2000), and hence due to exterior, not interior sources. secular variation in the quarter century from the Pioneer 11 to the This so-called SKR period is hence analogous to the Jupiter System IV Cassini observations is detected exceeding the probable errors in the period (Sandel and Dessler, 1988) and the rotation period of the inversions. interior of Saturn remains unknown, except for estimates based on 2. Cassini observation and data selection

Since the Saturn orbit insertion of Cassini on 30 June 2004 ⁎ Corresponding author at: 595 Charles E. Young Dr. East 6862 Slichter Hall Los Angeles, CA 90095, USA. Tel.: +1 310 825 4321; fax: +1 310 206 8042. (Dougherty et al., 2005), the magnetometer onboard made continu- E-mail address: [email protected] (H. Cao). ous measurements of Saturn's magnetic ﬁeld over a wide range of

1000 0 [nT] r -1000 B -2000

1000 [nT]

0 10 [nT] B

0 φ B -10 0 2000 4000 6000 8000 Pseudo Time

Fig. 1. Cassini's magnetometer measurements of three magnetic ﬁeld components inside L=3.8 Rs from Rev 003 to Rev 126. The azimuthal component is clearly almost two orders of magnitude smaller than the radial and meridional components. “Pseudo Time” here is the sequential count of the measurements used in the analysis, a small fraction of the total data obtained by the magnetometer.

latitude and planetocentric distance. In particular, revolutions 6, 28, 3. Inversion technique and an axisymmetric model 46, 53, 68–78, 116, 125, 126 achieved both high inclinations and low periapsis altitudes. These passes together with the lower inclination To deal with outliers in the data, a robust fitting procedure using passes provide an excellent data set in order to examine both the tilt iteratively reweighted least squares linear regression (Holland and of the dipole (or any non-axially symmetric magnetic moment) and Welsch, 1977) is applied to all inversions in this analysis. The weights the rotation rate of the planet. Fig. 1 shows the three magnetic field at each iteration are calculated by applying a Cauchy weighting components from Cassini's magnetometer measurements inside fi dipole L-shell=3.8 Rs (dipole L-shell here is the magnetic eld lines Table 1 of a spin-axisymmetric dipole field which intersect the equatorial Trajectory information of the 37 Cassini orbits adopted in this analysis, the first column plane at distance L, 1 Rs=60,268 km as the IAU standard definition of shows the revolution number of each orbit, the second and third columns show the the Saturn equatorial radius) from Rev 3 (February 2005) to Rev 126 start time and end time of each orbit, the fourth column shows the periapsis distance. (February 2010) in KRTP, spherical polar Saturn centered coordinates Rev Start time End time Periapsis distance with K signifying Kronian, and R, T and P signifying the radial, (UT) (UT) (Rs) meridional and azimuthal directions. The measurements adopted in 003 2005-Feb-01 04:00:00 2005-Feb-27 06:54:59 3.50 this study are regarded to be current-free since they avoid the current 004 2005-Feb-27 06:55:00 2005-Mar-19 16:54:59 3.50 generated by the Enceladus interaction with the rotating magneto- 005 2005-Mar-19 16:55:00 2005-Apr-06 23:29:59 3.51 006 2005-Apr-06 23:30:00 2005-Apr-23 23:59:59 2.60 spheric plasma near the dipole L-shell=3.95 Rs (Jia et al., 2010). The 007 2005-Apr-24 00:00:00 2005-May-12 05:23:59 3.59 azimuthal component is almost two orders of magnitude smaller than 008 2005-May-12 05:24:00 2005-May-30 08:15:59 3.60 the radial and meridional components, as one would expect if the field 009 2005-May-30 08:16:00 2005-Jun-17 13:09:59 3.59 were nearly axisymmetric. Table 1 shows the start time, end time, and 012 2005-Jul-24 07:24:00 2005-Aug-11 08:51:59 3.61 periapsis distance for each orbit used in this analysis ordered by 013 2005-Aug-11 08:52:00 2005-Aug-28 11:53:59 3.59 014 2005-Aug-28 11:54:00 2005-Sep-14 16:45:59 2.88 revolution number. Fig. 2 shows the spatial coverage of these orbits 015 2005-Sep-14 16:46:00 2005-Oct-02 23:59:59 3.00 inside L-shell=3.8 Rs. The upper panel shows the latitude versus 016 2005-Oct-03 00:00:00 2005-Oct-21 00:00:00 3.00 radial distance, which is unaffected by any uncertainty in our 028 2006-Aug-28 19:14:01 2006-Sep-17 17:19:00 2.96 knowledge of the rotation rate of the planet. The lower panel shows 045 2007-May-19 05:19:01 2007-Jun-04 06:04:00 3.24 046 2007-Jun-04 06:04:01 2007-Jun-20 06:20:00 2.75 the latitude versus longitude. The longitude is calculated by assuming 047 2007-Jun-20 06:20:01 2007-Jul-09 08:59:00 2.46 the rotation period is 10h33m00s, which lies between two values 053 2007-Nov-25 11:25:01 2007-Dec-11 12:11:00 2.54 suggested by Anderson and Schubert (2007) and Read et al. (2009). 054 2007-Dec-11 12:11:01 2007-Dec-27 01:28:00 3.03 The longitude coverage would change as the rotation rate changes. 057 2008-Jan-24 07:08:01 2008-Feb-02 20:08:00 3.26 The plot here shows good longitude coverage near the most likely 058 2008-Feb-02 20:08:01 2008-Feb-14 19:18:00 3.28 059 2008-Feb-14 19:18:01 2008-Feb-25 22:02:00 3.27 rotation period. 068 2008-May-14 05:14:01 2008-May-21 23:31:00 3.21 Only measurements inside the L-shell=3.8 Rs, rather than inside a 069 2008-May-21 23:31:01 2008-May-29 09:15:00 3.21 sphere with R=3.8 Rs, are selected to perform this analysis. Field- 070 2008-May-29 09:15:01 2008-Jun-05 12:35:00 2.80 aligned currents associated with the Saturnian magnetosphere- 072 2008-Jun-12 15:44:01 2008-Jun-19 18:54:00 2.70 ionosphere coupling has been detected at several highly inclined 073 2008-Jun-19 18:54:01 2008-Jun-26 20:45:00 2.69 074 2008-Jun-26 20:45:01 2008-Jul-03 21:34:00 2.69 orbits of Cassini (Talboys et al., 2009). The azimuthal magnetic 075 2008-Jul-03 21:34:01 2008-Jul-10 22:22:00 2.69 perturbation (Bφ) due to these field-aligned currents reaches ~30 nT 076 2008-Jul-10 22:22:01 2008-Jul-17 23:11:00 2.69 compared with a background (Bφ) ~5 nT (as shown by Fig. 2(a) in 077 2008-Jul-17 23:11:01 2008-Jul-25 07:25:00 2.69 Talboys et al. 2009). A spherical region with radius 3.8 Rs would 078 2008-Jul-25 07:25:01 2008-Jul-31 19:52:00 2.69 fi 116 2009-Aug-03 15:20:02 2009-Aug-19 11:05:01 2.94 contain these eld-aligned currents at high latitudes, violating the 122 2009-Nov-11 16:53:02 2009-Nov-30 17:33:01 3.20 current-free assumption. These magnetic perturbations would bias 123 2009-Nov-30 17:33:02 2009-Dec-19 00:00:01 3.20 the inversion of the internal field. The region inside L-shell=3.8 Rs 124 2009-Dec-19 00:00:02 2010-Jan-03 03:52:01 2.59 excludes all these field-aligned currents and thus is a more 125 2010-Jan-03 03:52:02 2010-Jan-19 04:16:01 3.00 appropriate region in which to invert the internal field of the planet. 126 2010-Jan-19 04:16:02 2010-Feb-04 22:25:01 2.57 24 H. Cao et al. / Earth and Planetary Science Letters 304 (2011) 22–28

30 20 10 0 −10 −20

Latitude [deg] −30 −40 2.5 3 3.5 4 Radius [Rs] 30 20 10 0 −10 −20 Latitude [deg] −30 −40 −150 −100 −50 0 50 100 150 Longitude [deg]

Fig. 2. Trajectories of the Cassini measurements used in this analysis. Upper panel shows the latitude and radial distance coverage, which is not affected by any uncertainty in our knowledge of the rotation rate of the planet; bottom panel shows the latitude and longitude coverage, assuming a rotation period 10h33m00s which is close to two estimations from gravitational and atmospheric data. function to the residuals from the previous iteration. This treatment between our model and the observations. Searching for the axial grants lower weight to data points that do not fit well. Thus it is less hexadecapole and higher degree terms using the same method and sensitive to outliers in the data as compared with ordinary least dataset results in larger probable error than the absolute value of each squares regression which weights every data point equally. The coefficient (also listed in Table 2), and no improvement appears in the Cauchy weighting function takes the form root mean square (RMS) of the residual field. Upper bounds on axial degree 4 and 5 terms can be made from the probable errors; the qffiffiffiffiffiffiffiffiffiffiffiffi −2 amplitude of axial degree 4 and 5 terms must be smaller than 720 nT CW = 1 + resid = 2:385×s× 1−h ð1Þ i i i and 3200 nT respectively. The covariance matrix and correlation coefficient matrix for an axisymmetric model with internal degree 3 where ‘residi’ are the usual least-square residuals from the previous and external degree 1 are listed in Table 3. The square roots of the iteration, 2.385 is the tuning constant, hi are the leverage values from diagonal terms of the covariance matrix, which are usually regarded a least squares fit, and s is an estimate of the standard deviation of the as estimations of the uncertainty of each coefficient to be estimated, are error term s=MAD/0.6745. The quantity MAD is the median absolute ~5 times smaller than the probable error we give above. Meanwhile, deviation of the residuals from their median, the constant 0.6745 the correlation coefficients between g10, g30, and G10 are fairly large. makes the estimate unbiased for the normal distribution. The robust These large correlation coefficients indicate that the uncertainties in fitting procedure provides us a more trustworthy inversion result each term could be larger than the value given by the covariance compared with the ordinary least squares method. matrix. We believe therefore that our error estimation based on the We have derived an axisymmetric intrinsic field model including, analysis of three independent data groups is the more robust estimate. dipole, quadrupole and octupole terms as well as an external dipole We notice that other axisymmetric models (Burton et al., 2009; term, from the measurements. For the intrinsic field of Saturn, terms Dougherty et al., 2005) derived from the Cassini magnetometer to the 3rd degree are necessary, while for the external field mainly measurements are not fully consistent with the model we derived, caused by an equatorial azimuthal ring current, a uniform field model especially the quadrupole and octupole terms. We attribute the should be the best approximation throughout such a long time period. differences to the improper selection of the “current-free” region and

To resolve the probable error, Δxprob which specifies the range an inappropriate external field model in those analyses. Burton et al. P xFΔxprob which contains 50% of the values to be estimated assuming (2009) and Dougherty et al. (2005) used all observations inside 8 Rs a normal distribution, of each coefficient, we divide the measurements into three independent groups by orbit number retaining Table 2 similar radial and latitudinal coverage in each group. Independent Coefficients of axisymmetric models for Saturn based on Cassini observations inside inversion is then applied to each data group, and the average value L=3.8 Rs from Rev 3 to Rev 126. The SPV model (Davis and Smith, 1990) based on Pioneer fi 11, Voyager 1 and 2 measurements and the Z3 model (Connerney and Acuna, 1982) based and probable error for each coef cient is calculated from these three on Voyager 1 and 2 measurements are also presented here for comparison. All values independent inversions. Table 2 lists the coefficients derived for this are in units of nT (nanotesla). One Saturn radius is 60,268 km in all three models. axisymmetric model with probable errors, the SPV model (Davis and Coefficients Cassini SPV Z3 Smith, 1990) based on Pioneer 11, Voyager 1 and 2 measurements and (Rev 3–126) the Z3 model (Connerney and Acuna, 1982) based on Voyager 1 and 2 0 g1 21,191±24 21,225 21,248 measurements. In all three models presented here, a Saturn radius Rs 0 g2 1586±7 1566 1613 is 60,268 km to make the coefficients of different models comparable. 0 g3 2374±47 2332 2683 0 Fig. 3 shows the difference between the Cassini measurements (as (g4)* (−70±243) 0 − shown in Fig. 1) and the axisymmetric model we derived (as shown in (g5)* ( 148±1070) 0 − Table 2) in each magnetic component. It is clear that the differences G1 13±1 RMS 2.2 are well confined in a small range exhibiting the good agreement H. Cao et al. / Earth and Planetary Science Letters 304 (2011) 22–28 25

B 0 Δ -5

5 θ 0 Δ Β -5

B 0 Δ -5 0 2,000 4,000 6,000 8,000 Difference between Measurements and Axial Model [nT] Pseudo Time

Fig. 3. Difference between measurements used in this analysis and the axisymmetric model we derived (as shown in Table 2). Difference in all three components are well confined in a small value range. and 6 Rs respectively, running into regions where various current of magnitude. This two-step fitting procedure will result in more systems flow. To deal with these current systems, they implemented accurate non-axisymmetric terms as long as they exist because the the ring current model first derived for Jupiter, which has been proven resulting coefficients are similar in magnitude. Fig. 4 shows the to be not appropriate for Saturn. Arridge et al. (2008) has shown that amplitude of the non-axisymmetric power for the intrinsic dipole and Saturn's current sheet is displaced above Saturn's equator. Sergis et al. quadrupole as well as the RMS of the difference between the (2010) has shown that the ring current profile widely used for Saturn measurements and the non-axisymmetric model as the rotation rate is not consistent with the measurements made by Cassini MIMI and changes. Peaks and deep minima can be found in the non- CAPS. axisymmetric powers and RMS respectively. The maximum of the non-axially symmetric dipole power corresponds to a 0.06° tilt. 4. Non-axially symmetric moments and the rotation rate Close examination does not establish that this is an intrinsic signal. of the planet First, neither the maximum of the non-axisymmetric dipole power nor the maximum of the non-axisymmetric quadrupole power Before we proceed to examine the secular variation of the field we corresponds to the absolute minimum or relative minimum of the address two questions: What is the rotation rate of Saturn's interior RMS misfit. The peak of the non-axisymmetric power and a deep and does Saturn have any significant non-axisymmetric field? This minimum of the RMS should occur at the same rotation rate, if the examination is carried out by solving for the rotation rate and the misfit were due to the existence of non-axisymmetric moments. non-axially symmetric moments within the same procedure, includ- Second, the maximum of non-axisymmetric dipole power does not ing also corrections to the axisymmetric moments. If there exist non- correspond to any maximum of non-axisymmetric quadrupole power. axially symmetric moments, they would be maximum at the correct If there were internally generated non-axisymmetric field compo- rotation rate while the difference between the measurements and the nents, all degrees would show a maximum at the right rotation rate, model would be a minimum as well. We set the range of the rotation because the wrong rotation period smears the non-axially symmetric period between 10h30m00s and 10h50m00s with 0.1 s as the step fields. Third, the same tests on independent subsets of data with size, which covers all Saturn “rotation periods” reported. For each relatively good spatial coverage do not reproduce the same maxima in “rotation period”, we calculate the coefficients for fitting the residual the non-axisymmetric powers or corresponding minima in the RMS. field from the axisymmetric model that we derived to internal degree This is inconsistent with the idea that these signals are due to non- 2 and external degree 1 with all orders for each degree. The weakness axially symmetric intrinsic magnetic moments since the internal of the observed Bphi component indicates that the non-axisymmetric source would create the same signal for any subsets of measurements terms must be orders of magnitude smaller than the axisymmetric at the true rotation period. Besides these inconsistencies, the biggest terms. After removing the axisymmetric terms, the remaining terms improvement of the RMS deviation is only ~0.2 nT. This value is close for which we are solving, the non-axisymmetric terms and the to the noise level of the Cassini magnetometer in the high fields in corrections to the axisymmetric terms, are basically of the same order which most of the measurements used in this study were obtained. By

Table 3 Covariance matrix and correlation coefficients of the inversion applied to the measurements with axisymmetric internal degree 3 and external degree 1 model. The correlation coefficients between g10, g30 and G10 are fairly large, it is not surprising since the spatial coverage of the measurement does not spread out uniformly over the entire sphere yet the field configuration of these three terms are similar to each other within the region of coverage of these measurements.

Covariance matrix of coefﬁcient estimates Correlation of coefﬁcient estimates

g10 g20 g30 G10 g10 g20 g30 G10

g10 11.4945 0.5693 42.5483 −0.1772 1.0000 0.1207 0.9409 −0.8647 g20 0.5693 1.9029 2.6968 −0.0047 0.1207 1.0000 0.1466 −0.0558 g30 42.5483 2.6968 177.9154 −0.5374 0.9409 0.1466 1.0000 −0.7114 G10 −0.1772 −0.0047 −0.5374 0.0037 −0.8647 −0.0558 −0.7114 1.0000 26 H. Cao et al. / Earth and Planetary Science Letters 304 (2011) 22–28

20 Rev 3-126

Dipole [nT] 0

100

0 Quadrupole [nT]

2.2 Non-Axially Symmetric Amplitude

RMS Fit 2

Residual [nT] 1030 1035 1040 1045 1050 Rotation Period [HRMM]

Fig. 4. The amplitude of the non-axisymmetric power of intrinsic dipole and quadrupole terms and the root mean square (RMS) residuals as the rotation rate changes from 10h30m to 10h50m. The maximum dipole power occurs at 10h32m13s while the maximum of quadrupole power occurs at 10h35m11s, and the minimum of RMS occurs at 10h33m38s. The maximum of the dipole does not correspond to a minimum of the RMS. These maxima of dipole and quadrupole powers may result from the re-arrangement of the spatial coverage as the rotation rate changes changing the condition number of the inversion. And the biggest improvement of RMS is only ~0.2 nT, close to the noise level of the magnetometer onboard Cassini in high field. comparing the non-axisymmetric model which corresponds to the position follows a Fisherian distribution centered at the rotation pole maximum dipole power and the measurements, as shown in Fig. 5,we with angular standard deviation σ, the probability to observe a tilt can find no systematic match between the model and the measure- much smaller than σ goes quadratically to zero with decreasing tilt ments. This reveals that the residual field from the axisymmetric angle. Even if we assume σ for Saturn were only one degree, much model we derived is mainly due to external sources rather than smaller than the typical tilt angle of other planetary magnetic fields, internal non-axisymmetric moments. We conclude therefore that the the probability to observe by chance an angle of below 0.06° would be maximum dipole tilt achieved in this analysis, 0.06°, can only be less than 0.4%. Most probably the very small dipole tilt of Saturn's field regarded as an upper limit and the data are equally consistent with requires a special explanation. zero tilt. There is no further evidence in the measurements that the Differential rotation in a stably stratified electrically conducting non-axially symmetric moments in this analysis arise from internal layer surrounding the dynamo region can axisymmetrize the sources. magnetic field (Stevenson, 1982). Such a layer with gradually varying helium depletion possibly exists in Saturn because of the limited 5. Interior of the planet miscibility of helium with metalized hydrogen, but its thickness is very uncertain (Morales et al., 2009). Horizontal motions in this layer The extremely high degree of axisymmetry of Saturn's magnetic can occur for example as a thermal wind circulation. The equator-to- field is challenging for dynamo theory, because a perfectly axisym- pole temperature difference at Saturn's surface can drive differential metric field cannot be generated by a dynamo (Cowling's theorem). rotation in the stable layer (Stevenson, 1980). Latitudinal differences The dipole tilt of a planetary magnetic field varies with time and in heat flow associated with the dynamics of convection in the occasionally it can be significantly smaller than its mean value. One dynamo region have the same effect (Christensen and Wicht, 2008). may speculate that by chance we observe Saturn's field at a special A simple analytical model revealed that non-axisymmetric time of very small tilt. However, assuming that the magnetic pole magnetic field components of harmonic order mN0 in the dynamo

5 Measurement Non-Axial Model r

B 0 Δ -5

5 θ 0 Δ Β -5

5 Differences with Axial Model [nT] φ

B 0 Δ -5 0 2,000 4,000 6,000 8,000 Pseudo Time

Fig. 5. Comparison between measurements and the non-axisymmetric model derived corresponding to the maximum intrinsic dipole power. The modeled axisymmetric ﬁeld is removed to enable better comparison. H. Cao et al. / Earth and Planetary Science Letters 304 (2011) 22–28 27

field are damped at the top of the stable layer by a factor that depends assuming each has the same value as in our model, which may be on the parameter combination αRM (Stevenson, 1982). Here, an optimistic assumption since Pioneer 11 had less coverage in UL αRM = λ is a magnetic Reynolds number based on the thickness L latitude than Cassini but Pioneer 11 went significantly closer to the of the stable layer and the characteristic velocity U of differential planet, we would conclude that there is no detectable secular rotation (λ is magnetic diffusivity) and α≈mL/R, with R the (mean) variation at any degree for the first three intrinsic magnetic moments radius of the stable layer. The damping factor Δ depends exponen- of Saturn. This is in contrast with the rapid change of the terrestrial 1 tially on ðÞαRM 2. Assuming that the dipole tilt in the dynamo region is magnetic dipole moment at the ‘present’ stage: no rapid change is of order 10°, Δ must be less than 0.006, which requires that αRM observed for Saturn, the upper bound for the dipole change in the past exceeds approximately one hundred. While the original study 30 years, calculated from the probable errors, would be 2.8 nT/year. assumed that Saturn's dipole was tilted by one degree and inferred Based on dynamo scaling laws, Christensen (2010) proposed that that αRM must be approximately 30 (Stevenson, 1982), the larger the flow velocity in Saturn's dynamo is an order of magnitude larger value required by the new upper limit for the tilt is perfectly than in the geodynamo. Together with the approximately ten times compatible with our present (scanty) knowledge of Saturn's interior larger radius of the dynamo region, this should result in a similar time structure and dynamics. Assuming L=3500 km, R=30,000 km, λ=4 scale of secular variation for both planets in the absence of a stable m2 s− 1 and setting m=1, the velocity of differential rotation U must conducting layer above the dynamo. exceed 1 mm s− 1. The velocity amplitude due to thermal forcing from Such a conducting layer with thickness L (not necessarily the planet's surface is difficult to estimate. For internal forcing Aubert differentially rotating) damps fluctuations of the field with a − pffiffiffiffiffi (2005) found in dynamo models that the typical amplitude of characteristic time scale τ by a factor exp L= τλ . In order to pass 1 L2 F 2 τ N = differential rotation scales as U≈ð ΩÞ where F is the buoyancy without strong damping, the time scale must be λ. With the constraint on the stable layer thickness derived from the extremely flux and Ω the rotation rate. Although this applies to differential small dipole tilt, L≥ 4000 km, field variations on time scales rotation in the dynamo region itself, the flow in the stable layer significantly shorter than 105 years are eliminated. For the axial probably has a comparable magnitude. A reasonable estimate for the dipole this corresponds to a secular variation of dg0 = dt b 0:2nT=year. buoyancy flux at the top of Saturn's dynamo is F≈10-10m2 s-3, which 1 − This is consistent with our result, although an order of magnitude less results in a little less than 1 mm s 1 for U. We conclude that the stable than the upper bound. Our finding of weak secular variation provides layer must have a thickness of L≥4000 km, slightly more than an independent line of support for the hypothesis of a stable originally envisioned by Stevenson (1982). A rather thicker stable conducting layer. layer, which puts the top of the dynamo region at roughly 40% of the planetary radius, also ensures better agreement of Saturn's observed 7. Discussion and conclusions dipole moment with the prediction from scaling theory (Christensen 2010). Other recent work was also unable to identify a non-axially Numerical simulations of dynamos operating below a stably symmetric field (Sterenborg and Bloxham, 2010). The RMS of the stratified layer reached time-average dipole tilt angles not smaller residual field in their study was 7 nT compared to our 2 nT RMS. A than 1.5° (Christensen and Wicht, 2008) and 0.8° (Stanley, 2010). possible reason for the difference is that we avoided all strong current However, in these models the values of αR were very moderate. In M systems by restricting the analysis inside the dipole L-shell=3.8 Rs Stanley's model αR ≈4, corresponding to a damping factor Δ≈0.5, M and visually inspecting all residuals. Moreover, our study used which suggests that the stable layer in this model is not the primary measurements provided directly by the Imperial College magnetom- cause for the small dipole tilt. In the case of Christensen and Wicht, eter team and not as reprocessed by the Planetary data system. As a who assumed a thicker stable layer, αRM ≈13. The associated − result, we have obtained upper limits on the intrinsic dipole tilt and damping factor Δ 1 ≈5–7 is significant and agrees with the reduction secular variation: there is no detectable secular variation from the of the tilt from 8° in a corresponding model without stable layer, but it − Pioneer 11 to the Cassini epoch; for all practical applications, the is still far from the required value Δ 1 ≥200. Dynamo models at magnetic moments of Saturn are completely aligned with the spin axis. higher values of αR are needed to determine possible limits to the M Based on the differentially rotating layer shielding scenario, we do degree of axisymmetrization, which may result from a non-zonal not expect the non-axially symmetric magnetic moments to be totally large-scale flow in the stable layer that could be driven by Lorentz shielded. One question would be, what is the lower bound on the forces or non-latitudinal thermal wind forcing. “observable” dipole tilt after the shielding. If the differentially rotating conducting layer surrounding the dynamo region has a thickness of 6. Secular variation 10,000 km, which is large and comparable to the dynamo region thickness, then the damping factor would be 0.0001. Beginning with a Since there is no significant non-axisymmetric field, we cannot 10° dipole tilt in the dynamo region, the “observable” dipole tilt would determine the spin rate. By comparing axisymmetric models derived be 0.001°. This corresponds to a maximum surface (~1 Rs) non- from data taken at different epochs, we can deduce the secular axisymmetric field around 0.4 nT, Cassini MAG probably will not be variation for Saturn. From values listed in Table 2, it can be seen that able to resolve such a small dipole tilt. However, as long as the the SPV model is very close to our model while the differences damping is not so extreme, or if non-zonal flow components in the between the Z3 model and our model are relatively large (N13% for stable layer generate non-axisymmetric field components, there is the octupole term). Considering the fact that the SPV model is able to still hope that we will deduce the non-axially symmetric moments fit the observations from Pioneer 11, Voyager 1 and 2 equally well, we and the rotation rate of the planet from the direct magnetic field investigate the secular variation based on this model. The secular measurements much closer to the planet that will be obtained near variation, inferred from the difference of our model and the SPV the end of the mission. model, would be −1.2±1.6 nT/year, 0.7±0.5 nT/year and 1.5± 3.2 nT/year for axial dipole, quadrupole and octupole coefficients for Disclosure statement Saturn. This can be compared with 19.6 nT/year, 16.7 nT/year, and 4.2 nT/year as RMS-values for corresponding coefficients of Earth's No authors have any actual or potential conflict of interest magnetic field calculating from the 11th International Geomagnetic including any financial, personal or other relationships with other Reference Field (IGRF) model for the period 1900–2010. If we people or organizations within three years of beginning the work consider the probable errors for coefficients in the SPV model submitted that could inappropriately influence (bias) this work. 28 H. Cao et al. / Earth and Planetary Science Letters 304 (2011) 22–28

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