tokens: task of solving equation ( 69 ). The problem then assumes the form of an inhomogeneous Helmholtz equation and the solution becomes particularly simple when expanding the radial dependence in modified spherical Bessel functions that fulfil the boundary conditions. Like in the classical gravity problem, Legendre polynomials remain the representation of choice for the latitudinal dependence. These basis functions allow a very efficient semi-analytical solution to the problem. Each of the calculations presented here required only a few seconds of run time on a standard 4-core notebook. There has been a discussion whether the DSG term could be neglected when inverting high precision gravity observations of Jupiter and Saturn for zonal flow properties. Our new formulation allows us to quantify the relative impact of the DSG for each gravity harmonic, practically independent of the considered zonal flow or background state. A special case arises for degree l = 1. For the background density with polytropic index unity, the l = 1 solution comprises the case where the two buoyancy contributions in the TGWE cancel. This corresponds to the homogeneous solution of the Helmholtz equation. Zonal flow and gravity perturbation then decouple, and it becomes impossible to draw any conclusion on the zonal flows from the respective gravity contribution. Kong et al. ( 2017 ) seem to have noticed the related problems without realizing their origin. However, this is of little practical interest since the origin is generally chosen to coincide with the centre of gravity so that l = 1 contributions vanish. Table 5 compares the relative DSG impact with the precision of most recent gravity measurement for Jupiter and Saturn. The even harmonics J 2 – J 6 are not listed since they are dominated by the rotational deformation of the planet. For Jupiter’s J 3 , J 5 , and J 7 coefficients, the relative impact of DSG is comparable to the error and should thus be taken into account when inverting gravity harmonics for zonal flow properties. This agrees with the results and conclusion by Kong et al. ( 2017 ). The error in the higher order observed moments may decrease as the Juno mission progresses. For Saturn, J 3 , J 5 , and J 10 seem precise enough to warrant including DSG effects. The estimates of Kong et al. ( 2017 ) and Galanti et al. ( 2019 ) for the relative impact of the DSG are compatible with our results. Including the DSG term generally increases the amplitude of the gravity coefficients. Table 5. Relative error of gravity harmonics for Jupiter (Iess et al. 2018 ) (second column) and Saturn (Iess et al. 2019 ) (third column). The fourth column shows S l , the relative impact of the DSG for radial profile f 3 also listed in Table 4. In addition to the TWE, Cao Stevenson ( 2017 ) also solve the Euler equation for a polytrope of index unity and a geostrophic equatorial zonal jet. The respective results not only provide the perturbed density structure but also the
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