This excellent answer has me wondering if there are any measurements or estimates of gravitational multipole moments of Jupiter or Saturn beyond J2.

Here is what I found for J2 (in unitless form) in the NSSDC factsheets for Jupiter and Saturn:

Body: J2 x 1E+06:
Earth 1,082.63
Jupiter 14,736
Saturn 16,298

Question: Have gravitation multipoles of Jupiter and Saturn beyond J2 been measured or at least estimated? At least the zonal harmonics perhaps? For estimates, please distinguish observational estimates based on natural or artificial satellite orbits from those based on planetary oscillation observations or planetary dynamical predictions.

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    $\begingroup$ I don't have the data, but hasn't gravity mapping been a major part of the Juno science mission. I'd also be amazed if Cassini hadn't produced some data for Saturn. $\endgroup$ Oct 8, 2019 at 11:29
  • $\begingroup$ @SteveLinton I'm pretty sure there is some raw data that could at least be used to estimate the zonal (but not tesseral) harmonics (the axisymmetric ones, e.g. J3, J4, J5, J6...) but as eluded to in the linked answer, the data may not be complete, or processed yet. But true mapping can only happen after an extended, systematic series of low orbits or passes. If the coverage is spotty or incomplete, whatever you extract has to be done with a set of assumptions. $\endgroup$
    – uhoh
    Oct 8, 2019 at 11:38
  • $\begingroup$ @SteveLinton I've added "zonal harmonics" back into the question, thanks for bringing this up. $\endgroup$
    – uhoh
    Oct 8, 2019 at 12:33
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    $\begingroup$ What is the used of mutipole expanssion. What avantages it as over the system. For exampe How is F = GmM/r**2 diferent from the expanded form. $\endgroup$
    – lunthang
    May 19, 2022 at 15:15
  • $\begingroup$ @lunthang that's an excellent new question post, please consider asking it so that everyone has a chance to answer. $\endgroup$
    – uhoh
    May 19, 2022 at 21:30

2 Answers 2


This article provides estimates of zonal harmonics down to $J_8$ derived from Doppler data in the Juno mission.* Tesseral harmonics are "statistically zero as expected for a fluid planet in equilibrium". The article also compares the zonal harmonics estimates to past estimates from the Pioneer and Voyager missions. The values are reported in Table 2, reported here:

Table of zonal harmonics values derived from the Juno mission

*Actually, they say that the zonal harmonics have been estimated up to the 12th order, but I haven't seen the $J_{10}, J_{12}$ reported in the article. I haven't read it thoroughly, though.

  • $\begingroup$ Grea! And thank you for the edit! $\endgroup$
    – uhoh
    Oct 9, 2019 at 16:07
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    $\begingroup$ Sure, thanks for pointing that out! $\endgroup$
    – LeWavite
    Oct 9, 2019 at 16:12
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    $\begingroup$ Trying a second time to comment. The article has supporting information that includes all the estimated values, up to $J_{12}$, for Jupiter. But only even zonal components up to $J_8$ and no tesseral/sectoral components at all are greater in magnitude than the reported uncertainties. $\endgroup$ May 12, 2020 at 23:50

For Saturn:

Data from Cassini-Huygens are given in this table from phase 76 of Reference 1

$J_2×10^6=16324.19\pm0.11$ (observed)

$J_4×10^6=-939.32\pm0.98$ (observed), $-971$ (theory)

$J_6×10^6=91\pm 5$ (observed)

$J_8×10^6=-10$ (assumed)


1. Saturn from Cassini-Huygens edited by Michele Dougherty, Larry Esposito, Stamatios Krimigis. (Springer Science & Business Media, Sep 30, 2009).

  • 1
    $\begingroup$ super, thank you! $\endgroup$
    – uhoh
    May 13, 2020 at 4:39

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