On the interpretation of the equatorially antisymmetric Jovian gravitational field

Since the odd zonal gravitational coe cients of Jupiter are nearly unaffected by the planet's rotational distortion, an e ective way of estimating the internal structure of the equatorially antisymmetric Jovian winds is to measure the odd coe cients induced by their equatorially antisymmetric component and then apply a mathematical theory to invert them. The thermal-gravitational wind equation (TGWE) provides this theoretical basis for interpretation. Here we show that the kernel term of the TGWE requires that its solutions satisfy a solvability condition. The thermal wind equation is a diagnostic relation that generates a \solution" for any zonal wind pro le, but that solution does not necessarily satisfy the solvability condition required for the TGWE. We develop a new approach to solving the TGWE that respects the solvability condition. We then calculate the odd zonal gravitational coe cients of Jupiter using a pro le of zonal winds that satis es the solvability condition and is equatorially antisymmetric and consistent with the observed cloud-level winds of Jupiter. We also explain the subtle but profound di erence between the TWE and the TGWE via an analogous inhomogeneous ordinary di erential equation. The developed method can be readily extended for inversion of the data soon to be acquired by the Juno spacecraft.