The lost meaning of Jupiter's high-degree Love numbers

Abstract: NASA's Juno mission recently reported Jupiter's high-degree (degree $\ell$, azimuthal order $m$ $=4,2$) Love number $k_{42}=1.289\pm0.063$ ($1\sigma$), an order of magnitude above the hydrostatic $k_{42}$ obtained in a nonrotating Jupiter model. After numerically modeling rotation, the hydrostatic $k_{42}=1.743\pm0.002$ is still $7\sigma$ away from the observation, raising doubts about our understanding of Jupiter's tidal response. Here, we use first-order perturbation theory to explain the hydrostatic $k_{42}$ result analytically. We use a simple Jupiter equation of state ($n=1$ polytrope) to obtain the fractional change in $k_{42}$ when comparing a rotating model with a nonrotating model. Our analytical result shows that the hydrostatic $k_{42}$ is dominated by the tidal response at $\ell=m=2$ coupled into the spherical harmonic $\ell,m=4,2$ by the planet's oblate figure. The $\ell=4$ normalization in $k_{42}$ introduces an orbital factor $(a/s)^2$ into $k_{42}$, where $a$ is the satellite semimajor axis and $s$ is Jupiter's average radius. As a result, different Galilean satellites produce a different $k_{42}$. We conclude that high-degree tesseral Love numbers ($\ell> m$, $m\geq2$) are dominated by lower-degree Love numbers and thus provide little additional information about interior structure, at least when they are primarily hydrostatic. Our results entail important implications for a future interpretation of the currently observed Juno $k_{42}$. After including the coupling from the well-understood $\ell=2$ dynamical tides ($\Delta k_2 \approx -4\%$), Jupiter's hydrostatic $k_{42}$ requires an unknown dynamical effect to produce a fractional correction $\Delta k_{42}\approx-11\%$ in order to fit Juno's observation within $3\sigma$. Future work is required to explain the required $\Delta k_{42}$.