[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 *g*_{n}^{m} and *h*_{n}^{m}, 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 Z_{3} 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 (Δ*t* ∼ 14 months), and 107° between P11 and V2 flybys (Δ*t* ∼ 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.