SEARCH

SEARCH BY CITATION

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Generalized Inversion of Magnetic Field Data
  5. 3. Method and Results
  6. 4. Discussion
  7. Acknowledgments
  8. References
  9. Supporting Information

[1] Analysis of magnetic field data from the Pioneer 11, Voyager 1 and Voyager 2 spacecraft produced the first models of Saturn's internal field. One striking feature of these models is the lack of non axisymmetric terms, which poses strong constraints on the dynamo mechanism. Resolution of the internal field using magnetic field data from different epochs is complicated by the uncertainty in our knowledge of the planetary rotation rate. By reanalyzing the flyby data, using modern inversion techniques, we derive the first tentative direct measurement of the rotation rate of the magnetic field. The measured rotation period agrees within 0.6 s with the value obtained from remote radio emission measurements, and its uncertainty is reduced to ±2.4 s. From an inversion of all available magnetic field data, we conclude that it is premature to exclude the presence of non-axisymmetric terms when describing the internal planetary field, and in particular we find a significant dipole tilt of 0.17°.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Generalized Inversion of Magnetic Field Data
  5. 3. Method and Results
  6. 4. Discussion
  7. Acknowledgments
  8. References
  9. Supporting Information

[2] The discovery and analysis of the internal planetary field at Saturn was left to the flybys of the planet by Pioneer 11 and Voyagers 1 and 2 [Smith et al., 1980a, 1980b; Acuña and Ness, 1980; Acuña et al., 1981; Ness et al., 1981; Connerney et al., 1982], due to the fact that Saturn's radio source is not easily detected from Earth. These analyses resulted in two very similar magnetic field models [Acuña et al., 1983; Davis and Smith, 1990], known as the Z3 and the SPV model, respectively. A remarkable and surprising aspect of these two models is the close alignment of the dipole and rotation axes (<1°), as well as the high degree of axisymmetry of the field. Such an axisymmetry is remarkable in that it is the only such planetary dynamo known, and is in apparent contradiction with Cowling's theorem on hydromagnetic dynamos [Cowling, 1934]. Although this contradiction can easily be explained, the field's near axisymmetry clearly has implications for the interior of the planet, and a number of ideas have been proposed to explain the absence of non-axisymmetric terms. For example, Stevenson [1980] suggested that a non-axisymmetric component can be suppressed by the sheared motion of a highly conducting shell outside of the dynamo region, although Love [2000] showed that this mechanism not always works. Moreover, the axisymmetry is also mysterious since there are a number of phenomena observed within the magnetosphere of Saturn which require a longitudinal asymmetry of the magnetic field, such as the rotational modulation of Saturn kilometric radio emission [Kaiser et al., 1980] and polar auroral brightening [Bhardwaj and Randall, 2000], as well as periodic variation of the optical spoke activity on the B ring [Grün et al., 1984].

[3] The issue of the axial symmetry of the internal planetary field model is closely related to the uncertainty in the planetary rotation rate. Saturn is differentially rotating with the rotation period depending on both latitude and radius, however the source region of the magnetic field is predicted to be at around 0.5 Rs [Hubbard and Stevenson, 1984], and it is in fact the rotation period of this source which we consider here. A perfectly symmetric field does not allow a determination of the rotation rate to be carried out from magnetic field observations alone. On the other hand, when the rotation period is poorly known, a substantial longitude smearing is occurring if any comparison between data sets from different epochs is carried out. For the case of Saturn, the prior conclusion that the internal magnetic field is axisymmetric has been obtained from observations taken at different times, but this requires a better knowledge of the rotation rate of the planet than currently available. In preparation then for the upcoming Cassini flybys of Saturn [Dougherty et al., 2004], we have reanalysed the previous flyby data sets from Saturn, with the double intent of addressing the rotation rate uncertainty, and testing the axial symmetry of the internal planetary field.

[4] In section 2 we illustrate the available data sets, and the problems which arise when combining them. Then, in section 3 we describe the procedure used in our analysis, derive our best estimate of the rotation period, and obtain the internal magnetic field model. Finally, in the last section we discuss the implications concerning the internal field modelling and future experiments.

2. Generalized Inversion of Magnetic Field Data

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Generalized Inversion of Magnetic Field Data
  5. 3. Method and Results
  6. 4. Discussion
  7. Acknowledgments
  8. References
  9. Supporting Information

[5] During the years 1979–1981, the three spacecraft Pioneer 11 (P11), Voyager 1 (V1) and Voyager 2 (V2) flew by Saturn and provided the first in situ measurements of the planet's magnetic field. Our study makes use of all available encounter data, obtained via the NASA Planetary Data System (http://pds.jpl.nasa.gov). The V1 and V2 data sets contain 48 s averages of the magnetic field, and are expressed in a Kronographic spherical system based on the Saturn Longitude System [Desch and Kaiser, 1981]. The data have been averaged from the 60 ms instrument sample rate data. The spacecraft trajectory in the IAU_SATURN reference frame was reconstructed from new ephemeris data made available as SPICE kernels by the JPL Navigation Ancillary Information Facility (http://pds-naif.jpl.nasa.gov/naif.html). The original P11 data consist of 1-minute averages, in the Pioneer Ecliptic frame, which were converted into IAU_SATURN with the use of SPICE routines. We have preprocessed all data by eliminating outliers (at 5σ), without applying any smoothing or weighting.

[6] The magnetic field potential is usually expressed as a Legendre series, whose coefficients are the Schmidt multipoles gnm and hnm, and can be obtained from the observations through the inversion of a linear relationship. Using Singular Value Decomposition (SVD) techniques, we can determine the maximum order n in the Legendre expansion of the magnetic potential, that can realistically be measured.

[7] For the three spacecraft trajectories, taken individually, we find a condition number between 20 and 30 for n = 2, and between 300 and 500 for n = 3. This implies that a single flyby can only meaningfully measure the internal field up to quadrupole (n = 2) order. Note that the authors of the Z3 and SPV models were able to determine the field up to order 3, by combining the data from multiple flybys, but at the same time by assuming the field to be axisymmetric (this assumption reduces the number of free parameters from 15 to 3, not counting a uniform external field). In principle, the full octupole (n = 3, 15 coefficients) order can still be reached, if we combine the data from the three encounters into a single data set, since in this case the condition number is reduced to ∼100. However, this procedure, based on the combination of magnetic field data taken at different epochs, requires the adoption of a common planetocentric reference frame, in which all the magnetometer data and spacecraft trajectories can be expressed. In particular, one needs to know the rotation period of the planet well enough in order for the longitude angle to be well defined. Unfortunately, past measurements of the rotation rate of the interior of the planet, based on the modulation of radio emissions detected by V1 [Desch and Kaiser, 1981], could only provide the Saturnian period with an uncertainty of ±7 s. This uncertainty translates into an error in longitude of about 65° between the epochs of P11 and V1 flybys (Δt ∼ 14 months), and 107° between P11 and V2 flybys (Δt ∼ 24 months). More recent Ulysses observations were found to disagree with the Voyager results by as much as 1% [Galopeau and Lecacheux, 2000]. Given that this discrepancy was found to vary with time, we adopted the value by Desch and Kaiser [1981], which refers to the same epoch as the magnetic field observations. Obviously, inverting the combined magnetic field data without taking the rotation's uncertainty into account is bound to produce large errors in the non-axial components of the field. On the other hand, we need non-zonal terms to directly measure the rotation period, as was done in Jupiter's case [Russell et al., 2001]. Thus, it is clear that both problems need to be solved simultaneously and consistently.

3. Method and Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Generalized Inversion of Magnetic Field Data
  5. 3. Method and Results
  6. 4. Discussion
  7. Acknowledgments
  8. References
  9. Supporting Information

[8] Here, we propose a method to obtain a direct (i.e., not based on radio observations) measurement of the rotation period of the magnetic field by minimizing the rms residuals of all flyby data sets. Before combining the flyby data, we remove known external contributions (see section 3.1), and add a time-dependent longitude correction Δϕ(t) = −(2π/Prot2) (tt0Prot, where Prot is the rotation period of the planetary magnetic field, ΔProt is its (unknown) correction, and t0 is an arbitrary reference epoch. For a given value of ΔProt, we solve for the internal field parameters using the SVD technique, and compute the rms residuals, combining the components of the field from the three flybys. The value of ΔProt which produces a minimum in the rms residuals, if it exists, represents our best estimate for the correction to the rotation period, at the same time indicating the presence of significant non-axisymmetric terms.

[9] Figure 1 shows the rms residuals vs. ΔProt, at the last iteration. The local minimum occurs for ΔProt ≃ 0.6 s, where rms ≃ 2.69 nT. Figure 1 indicates that i) non-axisymmetric terms are clearly present (otherwise the rms value would be insensitive to ΔProt), ii) the non-axisymmetric part of the field is fairly small (as the y-scale reveals, the rms is only reduced by less than 5% when Prot is allowed to vary), and iii) the measured rotation period is consistent with the IAU value (10 h 39 m 22.4 s, corresponding to ΔProt = 0), which is therefore confirmed to be indirectly related to the rotation of the magnetic field's source region. By fitting a (inverted) gaussian to the curve shown in Figure 1, we can conclude that ΔProt ≃ 0.6 ± 2.4 s. Thus, with this method we are able to reduce the uncertainty in Prot, and at the same time provide a first direct measurement of the rotation rate of the magnetic field. Note, however, that other minima of the rms curve exist well outside the ±7 s range. In particular, a minimum of comparable value occurs at ΔProt ≃ 20 s. Because of the fact that we are only sensitive to long-wavelength components, we cannot fully resolve this ambiguity: in fact, a 20 s correction in the rotation period translates into a longitude shift of ∼180° between P11 and V1 epochs, and ∼120° between V1 and V2 epochs. Thus, P11 and V1 would see the same quadrupole field, and V1 and V2 the same octupole field. Only extending the temporal baseline and/or increasing the truncation order n would allow removal of this partial degeneracy. In the absence of further data, it is logical to interpret the minimum which is closer to the IAU value as the most likely one.

image

Figure 1. Correction to Saturn's rotation period. The curve represents the rms of the field residuals (in nT) versus period offset (in s).

Download figure to PowerPoint

3.1. External Sources

[10] An important aspect of our analysis is the presence of external contributions to the magnetic field. In particular, close to the planet, the most important external source is the azimuthal ring current [Smith et al., 1980b; Ness et al., 1981]. In the present analysis, in order to eliminate or reduce this contribution, we have considered only data within 8Rs, and in this region we have removed an estimate of the field due to the disk by using available axisymmetric disk models [Connerney et al., 1983; Giampieri and Dougherty, 2004]. However, we should keep in mind that the numerical values of the disk's parameters quoted in Giampieri and Dougherty [2004] (namely, inner and outer radii, half thickness, and current density), depend on the assumption that the internal field is described by the SPV model. Since we are now looking for a new internal field model, we have dropped that assumption, and applied an iterative procedure: once the new internal field was found with the procedure described above, we removed its contribution from the raw data, and re-fitted the disk model. The new disk's parameters were then used to obtain a more accurate internal field model, etc. Convergence is reached after very few iterations. The new parameters for the disk, which should replace those quoted in Giampieri and Dougherty [2004], are shown in Table 1. Note that, compared to Giampieri and Dougherty [2004], we are able to marginally reduce the residuals rms, especially in the Pioneer 11 case.

Table 1. New Disk Parametersa
 μ0I0abDrms
P1145.6 (50.8)6.0 (6.4)12.4 (13.9)2.3 (1.8)2.6 (3.9)
V154.1 (54.4)7.9 (7.9)16.5 (16.4)2.6 (2.6)2.4 (2.5)
V258.2 (56.8)7.1 (7.2)13.9 (13.7)2.0 (2.1)2.9 (2.9)

[11] The axisymmetric disk model, by construction, does not contribute to the Bϕ component of the field. In order to take into account possible azimuthal contributions from magnetospheric sources, we have repeated the analysis by including also an external non-axisymmetric potential, with a different set of external Schmidt coefficients Gnm, Hnm for each flyby. While the curve shown in Figure 1, and the resulting value of Prot in particular, do not change significantly from the case without external potential, we noticed that the inclusion of these additional parameters produces a large condition number (>500 with next = 1), and larger residuals compared to the axisymmetric disk model. We conclude that, while the presence of external non-axisymmetric components cannot be excluded given the available data, they are unlikely to cause the effect shown in Figure 1.

3.2. Internal Magnetic Field Model

[12] The corrected rotation period defines a new kronographic reference frame, which, in general, can be consistently used to combine data taken at different epochs, in order to obtain the internal magnetic field multipoles up to the highest possible order. For the data considered here, this analysis is already part of the procedure used to find ΔProt, and the final iteration produces the multipoles given in Table 2. Also shown in Table 2 are the formal errors associated with the multipoles. However, because of the remaining uncertainty in the rotation period, and hence in the definition of the planetocentric reference frame, systematic errors may still be present. In order to estimate a realistic uncertainty in the multipole coefficients, we have varied ΔProt within a 1-sigma interval, and looked for the maximum variation in the estimated parameters. For each multipole coefficient, the maximum variation is interpreted as the parameter's uncertainty, which turns out to be always larger than the corresponding formal error. Note that systematic errors affect also the zonal terms, contrary to what one may expect. In addition, we have repeated the analysis with two different inversion methods: a least square inversion based on the singular value decomposition, and a robust regression. The robust regression uses an iteratively reweighted least squares algorithm, with the weights at each iteration calculated by applying the Cauchy function to the residuals from the previous iteration. This algorithm gives lower weight to points that do not fit well, and therefore the results are less sensitive to outliers in the data as compared with ordinary least squares regression. The two methods give very consistent results; in fact, the differences in the estimated parameters from the two methods are comparable to the formal errors quoted in Table 2.

Table 2. Saturn's Magnetic Multipoles Up to Octupole Ordera
TermValueσσfSPVZ3
  • a

    The values, in nT, result from the SVD analysis of all data. The third column (σ) represents our estimate of the uncertainty, always larger than the formal error (σf, fourth column). The last two columns give the multipoles from previous models, for comparison.

g10212326232116021535
g1123232--
h1160352--
g2015632341015601642
g21−132906--
g225476--
h215116111--
h22−1121135--
g3028219863123202743
g31−20921112--
g3228259522--
g33−156817--
h31136590735--
h32−8028913--
h331921179--

4. Discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Generalized Inversion of Magnetic Field Data
  5. 3. Method and Results
  6. 4. Discussion
  7. Acknowledgments
  8. References
  9. Supporting Information

[13] We have obtained the first direct measurement of the rotation rate of the internal magnetic field (independent of radio emission observations). We have worked under the assumption that the correct value of the rotation rate lies close to the IAU value, and that the rotation period does not change over the time span of the three flybys. Our analysis reduces the uncertainty in our knowledge of its value from 7 s per rotation period to about 2.4 s. Although the main source of uncertainty in the calculation of the internal field multipoles still resides in our rotation rate knowledge, we produce much improved results over previous work. In particular, the more precise rotation rate which we obtain can be used to reduce the error in our knowledge of the planetary longitude between the P11 and V1 and V2 flybys, allowing an inversion of the data from the three flybys to be carried out. The new internal planetary field model shows good agreement (within errors) with the SPV and Z3 models for the axial terms, with values for the dipole and octupole terms intermediate between the two previous models. Most notably, in this new model the non-axial terms are present and in particular some m = 1 sectorial coefficients appear to be significant. Also due to the clear and unambigious result in Figure 1 showing the correction to the rotation period value, we suggest that the non-axial terms cannot be excluded from the magnetic field model. However, our estimates for the non-axisymmetric terms are affected by possibly large systematic errors, entirely due to the remaining uncertainty in our knowledge of the rotation period. If the rotation period had been known exactly, the systematic errors in Table 2 would reduce to the much smaller formal errors.

[14] We note that the new octupole term g30 differs more substantially from the previous models than do the dipole and quadrupole terms. Our analysis indicates that the reason for this discrepancy is likely to reside in our treatment of external sources. In fact, in our procedure we have subtracted the disk model field from the data before the inversion, in order to try and ensure that it is purely internal effects being taken account of, whereas the SPV and Z3 models considered a uniform external field acting in the vicinity. By repeating our calculation ignoring the effect of the current sheet disk model, smaller values for the dipole and octupole terms are obtained.

[15] From Table 2, we find a dipole tilt angle of 0.17 ± 0.09 degrees, in agreement with the previously estimated upper bound of 1 degree [Smith et al., 1980a, 1980b]. Given the large uncertainty in the harmonic coefficients, we interpret this angle as an improved upper bound. The dipole's vertical displacement of around 0.4 Rs, does not depend on non-axial terms of course, thus is very similar to those values obtained by the SPV and Z3 models. However, we stress that, given the relative importance of higher order terms, it would not be appropriate to describe the internal planetary field as that of a simple displaced dipole field. In particular, we find that some of the higher order multipoles are particularly significant such as the g30 and h31 coefficients, implying that orders higher than 3 could possibly be important. If this turns out to be the case, then all models based on the inversion of a truncated Legendre series (including our own, as well as the previous SPV and Z3 models) will be inadequate, and regularization techniques will have to be applied instead [Parker, 1994]. This would obviously affect the conclusions of the present paper, especially if the higher order components do not rotate at the same rate as the lower ones. This issue could be resolved by the Cassini spacecraft which, by utilising its combined fluxgate and vector helium/scalar magnetometers, and by taking repeated measurements over 4 years, will resolve the internal field to fourth and possibly even fifth order.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Generalized Inversion of Magnetic Field Data
  5. 3. Method and Results
  6. 4. Discussion
  7. Acknowledgments
  8. References
  9. Supporting Information

[16] We are grateful to J. Wolf and S. Espinosa for their help in recovering the data, and to C. Russell, E. Smith, and A. Yu for helpful discussions.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Generalized Inversion of Magnetic Field Data
  5. 3. Method and Results
  6. 4. Discussion
  7. Acknowledgments
  8. References
  9. Supporting Information

Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Generalized Inversion of Magnetic Field Data
  5. 3. Method and Results
  6. 4. Discussion
  7. Acknowledgments
  8. References
  9. Supporting Information

Please note: Wiley Blackwell is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing content) should be directed to the corresponding author for the article.