## 1. Introduction

[2] 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.

[3] Without accurate knowledge of Saturn's rotation rate, it is not possible to derive an internal magnetic field model that includes non-axial terms. Planetary magnetic field models based on Pioneer and Voyager data [*Davis and Smith*, 1990; *Connerney et al.*, 1984; *Giampieri and Dougherty*, 2004a] as well as initial models based on Cassini data [*Dougherty et al.*, 2005] were necessarily axisymmetric since they were based on a rotation period now thought to be incorrect by several minutes [*Galopeau and Lecacheux*, 2000]. In those models any non-axial field would have been ‘smeared’ over wide range of longitudes reducing its contribution. Subsequent models were constrained to be strictly axisymmetric because of this lack of knowledge [*Burton et al.*, 2009] yet the periodic character of the magnetic field in Saturn's inner magetosphere is evident [*Giampieri et al.*, 2006; *Southwood and Kivelson*, 2007; *Andrews et al.*, 2008].

[4] 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.

[5] Given the high degree of symmetry, less direct methods have been used to estimate Saturn's rotation rate. *Anderson and Schubert* [2007] 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.* [2009] 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.

[6] 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.

[7] 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.