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 $k_{42}=1.289\pm 0.063$ ($1\sigma$), an order of magnitude above the hydrostatic $k_{42}$ obtained in a nonrotating Jupiter model. After numerically modeling rotation, the hydrostatic $k_{42}=1.743\pm 0.002$ 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 ($\ell >m$, $m\ge 2$) 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 $\Delta k_{42}$.

8 pages, 2 tables, accepted to PSJ

References in corpus (1)

• Equilibrium Tidal Response of Jupiter: Detectability by Juno

Cited by in corpus (4)

• Jupiter's interior from Juno: Equation-of-state uncertainties and dilute core extent
• Dynamical tides in Jupiter and other rotationally flattened planets and stars with stable stratification
• Rotating Love: The dynamical tides of spinning Newtonian stars
• Quantification of Tides in Giant Planets from Observations