How Departures of the "Generalized Potential" from Exact Potentials for Oblate Spheroids Affect Calculations of Giant Planet Mass, Zonal Harmonics, and Satellite Properties and Orbits

Forces external to oblate bodies are demonstrably non-central, which has important ramifications for the ice giants and their satellites. In PSS 2018 (v.152, p.68), we simplified historic formulae of Gauss and derived new analytical solutions for the gravitational potential and force outside a constant density oblate. Numerical calculations for billion element meshes confirmed our equations. Approximations addressing internal density variations were provided. We showed that the force on a general point is not directed towards the oblate's center, nor are forces simply proportional to the inverse square of that distance, despite forces in the equatorial and axial directions pointing towards the center. Only for orbits outside 6 body radii is the central forces approximation accurate; for low ellipticity, centrality is useable for 2-6 body radii. The series expansion of the generalized potential (GP) can be reconciled with the exact potential only along the special directions, but its coefficients (J_i, zonal harmonics) must take on specific values. For a constant density oblate, J_i depend only on ellipticity and a numerical constant specific to each i. Any internally densified oblate can be constructed from summing a point mass with a series of homogeneous oblates with the same ellipticity but different constant densities. Thus, internal densification proportionately reduces all J's from their theoretical values for a homogeneous oblate. For general directions, the GP series does NOT correspond with Gauss's exact potential. Hence, use of inclined orbits of satellites in GP fits has caused the masses of Uranus and Neptune to be underestimated by 0.2%. Extractions of zonal harmonics via fitting yield mutually inconsistent values (i.e., J₂ is half that expected for oblate bodies with constant density but J₄ is similar to expectations). Hence, calculated moments of inertia of the ice giants contain errors, creating flaws in gravitational models of their interiors. Our new formulae and series should improve interpretation of flyby data, and improve the accuracy of orbital and physical parameters for ice giants, their rings, and satellites.

*This article was prepared independent of Everett Criss's employment and without use of information, resources, or other support from Panasonic Avionics Corp.