A model of Saturn's internal planetary magnetic field based on data from the Cassini prime mission has been derived. In the absence of a determination of the rotation rate, the model is constrained to be axisymmetric. Non-axisymmetric models for a range of plausible planetary rotation periods have also been derived and we evaluate upper limits on the asymmetry of the internal magnetic field based on those models. We investigate whether a maximum in the non-axisymmetric magnetic field can be identified at a particular rotation rate thus providing insight into the rotation rate of the planet's interior. No such peak can unambiguously be identified. An axisymmetric octupole model is adequate to fit the data and addition of higher order terms does not improve the goodness of fit. The largest value of the dipole tilt obtained from non-axisymmetric models (<0.1°) confirm the high degree of symmetry of Saturn's magnetic field.
 Since the beginning of the Cassini mission in July 2004, the spacecraft has completed more than 130 orbits in a wide variety of geometries in Saturn's magnetosphere. The periapsis distance of thirty-seven of those orbits during the prime mission was closer than Enceladus's orbit at 3.95 Rs. As Cassini continues to orbit in Saturn's magnetosphere and more data are obtained close to the planet in a variety of geometries, our knowledge of Saturn's magnetosphere improves and it is necessary to update earlier models of Saturn's internal magnetic field.
 For Jupiter, given the substantial contribution by non-axial field, a direct method of determining the rate of rotation is possible by examining the periodic variation in the tilt of the magnetic dipole axis [Russell et al., 2001]. Saturn's magnetic field has long been known to have negligible dipole tilt, making this direct determination difficult.
 Given the high degree of symmetry, less direct methods have been used to estimate Saturn's rotation rate. Anderson and Schubert  used a procedure that minimized the wind-induced dynamic heights in the atmosphere to determine a period of 10 hours, 32 minutes, and 35 seconds ± 13 seconds. Read et al.  proposed that Saturn's rotation rate can be determined by considering the dynamical stability of the planet's jet streams. They used observational estimates of winds and temperatures to derive a rotation period of 10 hours, 34 minutes, and 13 seconds ± 20 seconds based on a minimization of potential vorticity.
 However, even given the uncertainties in rotation period associated with these models (13–20 seconds), knowledge of the planetary longitude is highly uncertain over a short period of time. For an uncertainty of 13 seconds, the planetary longitude is uncertain by 90 degrees over a years time and by almost a full rotation over the four year time span of the Cassini of the prime mission.
 In this analysis, using Cassini magnetometer data obtained on all close Saturn flybys, we update the spherical harmonic coefficients for an axisymmetric model of Saturn's internal planetary magnetic field. We also assess the range of possible values of the non-axial coefficients and extent of non-axisymmetry of the magnetic field for plausible planetary rotation rates.
 One-minute averages of Cassini fluxgate magnetometer data were used in to derive the model of Saturn's internal magnetic field. These data are described in detail by Dougherty et al. . The measured field is likely to include that due to external current systems, such as the ring current, tail currents, magnetopause currents as well as the internal planetary field. In this analysis we model both the internal and external magnetic field using the standard formulation in which the components of the field are given by:
where a is Saturn's equatorial radius, r is the radial distance to the center of the planet, θ is the colatitude, ϕ is the longitude and Pnm (cos θ) are the Schmidt-normalized associated Legendre function of degree n and order m. The magnetic field data were analyzed using standard singular value decomposition of the inversion matrix to determine the model parameters [Connerney et al., 1981]. The basic modeling procedure is outlined by Burton et al.  but differs slightly since here we model the external field as terms in the spherical harmonic series as opposed to modeling the external ring current explicitly in the earlier analysis.
 Because of significant effect of Enceladus on Saturn's magnetosphere [Kivelson, 2006] we use only data obtained on the thirty-seven orbits inside the L-shell of Enceladus. Figure 1 shows the geometry of the close orbits, latitude versus longitude, color-coded according to radial distance (1 Rs = 60,268 km). The longitude is based on the official IAU planetary rotation period of 10 hours 39 minutes 22.4 seconds. There were nine unique short-period (∼7 day) orbits at the end of the Cassini prime mission in June and July, 2008 (orbits numbered 70–78 in Figure 1). They were highly inclined with maximum latitudes reaching more than 74 degrees in both the southern and northern hemispheres and periapsis distances of 2.7 Rs near the equatorial plane. These orbits covered a wide range of latitudes close to the planet, ideal for sampling the planetary field, especially the higher order moments. (There was a data outage on orbit 71 and no data was obtained.)
 The coefficients for the axisymmetric model derived in this analysis as well as the root mean square error are shown in Table 1. Only terms to degree 3 (dipole, quadrupole and octupole) are required to model the internal field. Including terms of degree 4 and above do not improve the fit to the data. By the use of partial solutions [Connerney et al., 1991], successively increasing the number of parameters included in the solution and examining the improvement in residual field, it is clear that only terms to degree 3 are necessary to model Saturn's field. Additional parameters increases the parameter uncertainty without improving the solution. The external field is modeled using only terms of degree 1 (G10, H11, H11). Model coefficients from earlier analyses are also shown for comparison.
Table 1. Spherical Harmonic Coefficients for the Axisymmetric Model Derived in This Studya
 The degree of nonaxisymmetry of the magnetic field can be estimated by deriving a number of magnetic field models that include non-axial terms for a range of plausible rotation rates. Because the true planetary longitude is unknown due to the uncertainty in rotation period, we vary the rotation rate over a range of likely values, calculate a new ‘pseudo-longitude’ based on that rate and invert the data to obtain model parameters, gnm and hnm for each rotation rate. Distributions of the non-axial (m = 1) coefficients for a set of models based on rotation periods from 10 hours 30 minutes and 10 hours 40 minutes are shown in Figure 2.
 Next we assess whether there is a single rotation rate that can be associated with a peak in the non-axial magnetic field. The premise underlying the analysis is that the contribution by any non-axial magnetic field would be a maximum for a model based on the correct planetary rotation period. To evaluate the relative contribution by the non-axial field we evaluate the mean square field at the planet's surface [Lowes, 1974].
 The m = 0 terms represent the axisymmetric field (axial dipole, quadrupole, octupole) and the m ≠ 0 are the non-axisymmetric terms. To detect a signal in a non-axial field we evaluate the ratio of the non-axial to total ‘power’ given by:
(where N represents the maximum degree of the inversion, in this case).
 We perform the inversion to obtain a magnetic field model based on a range of probable rotation rates. The rotation period is incremented in one-second steps and a new model derived. Using data obtained on all thirty-seven close orbits we detect no signal. Using data obtained on the unique orbits 70–78 in 2008 we detect a peak in the non-axial power. Figure 3a shows the ratio of non-axial to total power for the range of planetary rotation periods investigated. The peak occurs at a rotation period of 10 hours 34 minutes 26 seconds. The ratio of non-axial to total power in the octupole (degree 3) field is shown in the second panel and that peak is almost coincident with peak in total non-axial power. With further investigation however, we find that for this particular rotation period, orbits 70–78 which are well separated in longitude based on the IAU period, converge to cover similar swaths of planetary latitude and longitude. Thus they do not provide the spatial separation implied by Figure 1. Accordingly the condition number is high and the fit to the data to the data is poor. Thus although the signal is intriguing, we cannot conclude that this particular period represents the rotation rate of the interior where the magnetic field is produced. Absence of a detection at a particular rotation rate is consistent with results obtained by others [Sterenborg and Bloxham, 2010], however our analysis is based on a more complete set of orbits.
 Various characteristics of Saturn's internal field based on these models can be investigated. An upper limit can be placed on the tilt of the magnetic dipole axis given by tan−1 [(g11)2 + (h11)2)1/2]/g10. The distribution based on models from 10 hours and 30 minutes to 10 hours and 40 minutes derived in this study (Figure 2) show the almost complete absence of a dipole tilt. The mean and median values of the tilt are both .03 degrees.
 Some characteristics of Saturn's magnetic field such as the strength of the dipole moment and the northward offset of the magnetic equator are very similar to those based on earlier models derived from Pioneer and Voyager data. The average value of the strength of Saturn's dipole moment, M = a3 [(g10)2 + (g11)2 + (h11)2]1/2 is 4.66 × 1018 Tm3. Saturn's magnetic equator, g20/2g10 is offset to the north by .036 Rs resulting in a somewhat larger magnetic field at high northern latitudes in comparison with the south.
 The harmonic spectrum of Saturn's magnetic field is characterized by a ‘quadrupole-deficit’. Compared to Earth's magnetic spectrum in which the mean square surface field (equation (4)) decreases with increasing degree, the ratio of the dipole:quadrupole:octupole terms for Saturn is 1:.008:.02.
 Based on an improved understanding of processes dominating Saturn's magnetosphere, our modeling approach differs in two ways from previous analyses [Burton et al., 2009; Dougherty et al., 2005]. A simple symmetric ring current centered on the equator is no longer thought to accurately represent the current sheet at Saturn. Instead the current sheet is found to be displaced above Saturn's rotational equator and to assume the shape of a bowl or basin, referred to as a magnetodisc [Arridge et al., 2008]. Previous internal field models relied on the oversimplified ring current and the non-linear analytical expression of Giampieri and Dougherty [2004b] to evaluate the external field. The field was evaluated using that formulation on an orbit-by-orbit basis since the ring current is known to vary with solar wind conditions and local time. Saturn's magnetodisc dominates the external magnetic field and has an important role in the global configuration of the magnetosphere yet empirical models describing the location and characteristics of the current sheet are not yet readily available. Since this is the case, we estimate the external field as well as the internal field using (equations (1)–(3)) which are derived from the expression for the magnetic potential. A single set of spherical harmonic coefficients (G10, G11, H11) is obtained that describes the external field in a least squares sense.
 Measurements obtained by all Cassini fields and particles instruments demonstrate that the structure and dynamics of Saturn's inner magnetosphere are governed by plasma created at Enceladus which orbits at 3.95 Rs [Kivelson, 2006]. Earlier magnetic field models were based on data obtained at radial distances as far as 10–15 Rs, well outside Enceladus orbit. It is likely that the observed field at those radial distances is modified by processes in the inner magnetosphere and does not necessarily reflect the magnetic field generated in Saturn's interior. We restrict the data included in the analysis to measurements obtained inside the L-shell of Enceladus, to an L-value of 3.8. Accordingly, the axisymmetric model dipole, quadrupole and octupole coefficients differ somewhat from the previous model based on data obtained during the first three years of the Cassini mission which included all data within 10 Rs. The earlier approach would not eliminate the influence of currents at high northern and southern latitudes [Bunce et al., 2008].
 Models of the internal field that include non-axisymmetric terms based on a range of rotation rates have been evaluated. Non-axial models coefficients are all normally distributed about zero. The octupole coefficients are as well, but are characterized by a somewhat larger standard deviation (Figure 2).
 Some characteristics of Saturn's magnetic field remain substantially unchanged from those based on data obtained in the Pioneer-Voyager era. Models based on data obtained by the Cassini magnetometer continue to confirm the high degree of axisymmetry of the internal magnetic field. Various mechanisms have been proposed to account for this including the existence of a stably stratified layer at the top of the metallic conducting region that would attenuate any non-axisymmetric field produced in the interior [Stevenson, 1982]. Stanley  has studied the axisymmetrizing effects of stably stratified layers surrounding the dynamo using simulations based on the numerical dynamo model of Kuang and Bloxham . For a range of interior parameters (Rayleigh number, thermal boundary conditions, etc.) they find that the smallest average dipole tilt is 0.8 degrees, far greater than models based on observations indicate. The mean dipole tilt for models based on rotation rates between 10 hours 30 minutes and 10 hours 40 minutes is 0.03° (Figure 2). All values are less than 0.1°.
 The research described in this publication was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.