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Abstract

Recent precise measurements of Jupiter’s and Saturn’s gravity fields help constraining the properties of the zonal flows in the outer envelopes of these planets. The link is provided by a simplified dynamic equation, which connects zonal flows to related buoyancy perturbations. These can result from density perturbations but also from the gravity perturbations. Whether the latter effect, which we call dynamic self-gravity (DSG), must be included or is negligible has been a matter of intense debate. We show that the second-order differential equation for the gravity perturbations becomes an inhomogeneous Helmholtz equation when assuming a polytrope of index unity for density and pressure. This equation can be solved semi-analytically when using modified spherical Bessel functions for describing the radial dependence. The respective solutions allow us to quantify the impact of the DSG on each gravity harmonic, practically independent of the zonal flow or the details of the planetary interior model. We find that the impact decreases with growing spherical harmonic degree ℓ. For degrees ℓ = 2 to about ℓ = 4, the DSG is a first-order effect and should be taken into account in any attempt of inverting gravity measurements for zonal flow properties. For degrees of about ℓ = 5 to roughly ℓ = 10, the relative impact of DSG is about 10 per cent and thus seems worthwhile to include, in particular since this comes at little extra cost with the method presented here. For yet higher degrees, it seems questionable whether gravity measurements or interior models will ever reach the precision required for disentangling the small DSG effects, which amount to only a few per cent at best.