Radio tracking data selection
We analysed the two-way and three-way Doppler data at 60 s count time acquired by Cassini during the fly-bys devoted to gravity, including T011, T022, T033, T045, T068, T074, T089, T099, T110 and T122, using JPL’s Mission Analysis, Operations, and Navigation Toolkit Environment (MONTE50). The range-rate observations used to measure the gravity field are extracted from the Doppler shift affecting a radio-frequency signal sent from Earth in X-band (7.2 GHz) and retransmitted back in X-band (8.4 GHz) and Ka-band (32.5 GHz). As in the 2019 analysis3, we selected X/Ka-band data over X/X-band data when available and two-way over three-way data. This approach differs from the data selection used more recently16, which focused on X/X-band data. However, we also processed X/X-band data for comparison with previous works and found that this choice does not affect the results of the analysis, because the Sun–Cassini–Earth geometry did not imply propagation through the solar corona and the effects of scintillation on different frequencies is negligible. The Doppler data were averaged at 60 s as in the early analyses2,3. We also processed data at 10 s integration time, as in the latest analysis16. We observed no significant difference in the results. We included the Advanced Water Vapor Radiometer (AWVR) calibrations when available to compensate for the effects of the Earth’s troposphere on the radiometric observables. When the AWVR data were not available, we used the Tracking System Analytical Calibration (TSAC) to compensate for tropospheric and ionospheric effects. The data were weighted by the root mean square of the observation residuals individually for each tracking pass.
This work has several improvements in data analysis over previous work3,16 involving the application of signal processing techniques developed for Juno51 and InSight23,24. First, for the closest approach passes, we process the recorded open-loop data from the Radio Science Receiver (RSR) using a phase-locked loop that is optimized for thermal noise51. This produces unique frequency measurements at 1 s count-time intervals. Second, we apply a phase-averaging technique24 instead of frequency-averaging, as is typically done, to increase the compression time to 60 s. Phase compression is superior over frequency compression for low count times when thermal noise is the dominating source. Phase compression enabled a reduction of data noise of approximately 25% on average, with a peak of 30% for T122, which is consistent with its previous application on InSight radio science data23,24 (Extended Data Fig. 1).
Radio tracking data analysis
We modelled the spacecraft dynamics accounting for the gravity accelerations, in a relativistic framework, imparted by the Sun, the planets in the Solar System and all of Saturn’s large moons. For Titan and Saturn, we also accounted for the spherical harmonic expansion of their gravity fields. The input gravity field of Titan, expanded to degree and order 5, was initialized to zero and accounted for time-varying tidal effects on degree 2 parameterized by the tidal Love number k 2 . Saturn’s gravity field was assumed to be consistent with the ephemerides models52. The body-fixed reference frames for Titan and Saturn are also consistent with the ephemerides52. We accounted for solar radiation pressure, atmospheric drag using the recently revised Titan Global Reference Atmosphere Model (Titan-GRAM)53 and the acceleration given by radioactive decay in the radioisotope thermoelectric generator with navigation models used by previous studies3,16. We also accounted for the time-varying effects of Titan’s atmosphere on its gravity field by expanding the pressure fields obtained from global climate model simulations54 into the corresponding surface mass variations55,56. The technique that we used allowed us to take into account both the gravity time variations induced by the direct gravitational attraction of the atmospheric mass and those induced by the response of the moon to the surface loading. The atmospheric mass redistribution induced by the tides that Saturn exerts on Titan’s atmosphere is large, leading to gravity variations on the degree 2 gravity on the order of 10−9, larger than Venus and Mars atmospheric gravity effects by an order of magnitude. By modelling these contributions, we removed their effects from the data and obtained an estimate of k 2 that is only given by the solid moon. These effects are still too small to be detected with fly-bys and they do not markedly affect our solution (Supplementary Information).
The data were segmented into observation arcs centred around the closest approach. We fit the data arcs with a set of parameters describing the dynamics of the spacecraft using a weighted least-squares estimation filter. We integrated the trajectory and computed the Doppler observations. The differences from the raw observations (residuals) were minimized by iterating the filter. A priori uncertainties were used to regularize the least-squares inversion. For each data arc, we estimated the initial position and velocity of Cassini (with a priori uncertainty 100 km and 20 m s−1 for each direction), the atmospheric drag scale factor C D (a priori equal to 10% of its value) and a scale factor for the solar radiation pressure (10%). As in previous studies2,3, we estimated the accelerations given by the radioisotope thermoelectric generator with a priori uncertainties that are 103–104 times larger than their nominal value. Doppler radio science measurements during fly-by T110 were performed using the low-gain antenna. We estimated the location of its phase centre with respect to Cassini’s centre of mass to calibrate the effects of the spacecraft rotation on the Doppler observables. All the information from the arcs is combined to estimate the geophysical parameters, including Titan’s gravity harmonics up to degree 5, the tidal Love number k 2 and the associated time lag Δt (from which the imaginary part Im(k 2 ) is derived). We also included Titan’s gravitational mass when we estimated its orbit. All of these quantities were unconstrained in the inversion, including the higher-degree gravity field that was not constrained with a Kaula’s rule.
To be consistent with previous studies, the uncertainties are provided with solutions obtained by numerically integrating and estimating the reference position and velocity of Titan with respect to the Saturn system barycentre at the beginning of each data arc. The a priori uncertainties are given by the covariance of the SAT441 (ref. 52) solution multiplied by 10.
We note that, as acknowledged in the most recent analysis16, most of the spherical harmonic coefficients at degree 5 are compatible with zero, indicating that the Cassini radio tracking data are not sensitive to full 5 × 5 expansion. We retained the estimation of the degree 5 coefficients to be consistent with previous analyses, but their inclusion in the data fit is not necessary and it does not yield any further information about Titan’s gravity field and interior structure. When we removed these coefficients from the estimation, we were able to further reduce the uncertainties in the geophysical parameters, which supports our finding that both Re(k 2 ) and Im(k 2 ) are observable through Cassini radio tracking (Supplementary Information).
Uncertainties on estimated geophysical parameters
The reported uncertainties in the Doppler-derived geophysical parameters were obtained from the covariance matrix computed by the weighted least-squares filter. The uncertainties have several sources, the most important being Doppler data noise, the number of observations and the number of estimated parameters. Estimating a larger set of parameters increases the correlations in the solution and the resulting uncertainties. Improvements in the root mean square of the data noise translate almost linearly into improvements in the uncertainties of the retrieved geophysical parameters. When we estimated the Titan ephemerides without using phase compression in our preprocessing, we obtained uncertainties compatible with previous studies: \({\sigma }_{{\rm{Re}}({k}_{2})}=0.060\), \({\sigma }_{{\rm{Im}}({k}_{2})}=0.044\) (refs. 3,16,37). Applying phase compression reduced the data noise root mean square between 20% and 30% for all tracking passes around closest approach, which in turn improved the uncertainties on the derived parameters by about 15–20%: \({\sigma }_{{\rm{Re}}({k}_{2})}=0.048\), \({\sigma }_{{\rm{Im}}({k}_{2})}=0.035\). The solutions obtained are statistically consistent and the variations are less than 1σ.
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