DDA 2015 – Gravity and Tide Parameters Determined from Satellite and Spacecraft Orbits

This is one of a series of notes taken during the 2015 meeting of the AAS Division on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.

Session: Dynamics of Small Solar System Bodies II

Robert A. Jacobson (JPL)


As part of our work on the development of the Jovian and Saturnian satellite ephemerides to support the Juno and Cassini missions, we determined a number of planetary system gravity parameters. This work did not take into account tidal forces. In fact, we saw no obvious observational evidence of tidal effects on the satellite or spacecraft orbits. However, Lainey et al. (2009 Nature 459, 957) and Lainey et. al (2012 Astrophys. J. 752, 14) have published investigations of tidal effects in the Jovian and Saturnian systems, respectively. Consequently, we have begun a re-examination of our ephemeris work that includes a model for tides raised on the planet by the satellites as well as tides raised on the satellites by the planet. In this paper we briefly review the observations used in our ephemeris production; they include astrometry from the late 1800s to 2014, mutual events, eclipses, occultatons, and data acquired by the Pioneer, Voyager, Ulysses, Cassini, Galileo, and New Horizons spacecraft. We summarize the gravity parameter values found from our original analyses. Next we discuss our tidal acceleration model and its impact on the gravity parameter determination. We conclude with preliminary results found when the reprocessing of the observations includes tidal forces acting on the satellites and spacecraft.


  • Jupiter and Saturn gravity fields program at JPL
    • started with Pioneer
    • probably end with Juno (or proposed Europa) mission
    • also Earth-based
      • 1874-2014
      • Saturnrigh stellar occultations
        • pole orientation
      • Saturn ring plane crossing times
        • pole orientation
    • spacecraft:
      • radiometric tracking
      • imaging
      • VLBI
      • Saturn ring occultations
    • But no tidal forces used in any analysis so far.
  • But tidal effects are not zero
    • Lainey et al. 2009, 2012
    • Efroimsky & Lainey 2007 (JGR 112)
    • $U_{jk} = k_2^k \left(\dfrac{\mu_j}{R_k}\right)^3 \left(\dfrac{R_k}{r}\right)^3 \left(\dfrac{R_k}{r^*_{jk}}\right)^3 P_2\left(\hat{r} \cdot \hat{r}^*_{jk}\right)$
    • $r^*_{jk} = r_{jk} – \Delta t_j \left[\dot{r}_{jk} + \dot{W}_k\left(\hat{r}_{jk}\times\hat{h}_k\right)\right]$
    • Tidal lag effects
  • Put tides in fitting model
    • $\rightarrow k_2$
    • $\rightarrow$ gravity harmonic coefficients
    • tidal lags: indeterminate from existing data
    • tidal dissipation function $Q = \dfrac{2 \pi E}{\Delta E} = f(\Delta t)$
      • $E$ = max energy stored in one tidal cycle
      • $\Delta E$ = energy dissipated during that cycle
      • $f(\Delta t) = \dfrac{1}{\omega^{\alpha} \Delta t}$
  • comparison to Lainey for Jupiter:
    • indeterminate
  • comparison to Lainey for Saturn (common $Q$):
    • $\Delta t$ and $\dfrac{k_2}{Q}$ successfully detected for Mimas, Enceladus, Tethys, Dione, and Rhea, $k_2 = 0.379 \pm 0.011$
    • Lainey: $\dfrac{k_2}{Q} = 2.3\pm0.7 \times 10^{-4}$, $k_2 = 0.341$
    • JPL:$\dfrac{k_2}{Q} = 1.0\pm0.2 \times 10^{-4}$, $k_2 = 0.381 \pm 0.011$

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