# Have gravitation multipoles of Jupiter and Saturn beyond J2 been measured or at least estimated? At least the zonal harmonics?

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 ×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.

• 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. – Steve Linton Oct 8 at 11:29
• @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. – uhoh Oct 8 at 11:38
• @SteveLinton I've added "zonal harmonics" back into the question, thanks for bringing this up. – uhoh Oct 8 at 12:33

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:
*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.