The Lost Meaning of Jupiter's High-degree Love Numbers

NASA's Juno mission recently reported Jupiter's high-degree (degree l, azimuthal order m = 4, 2) Love number k (42) = 1.289 +/- 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 +/- 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 l = m = 2 coupled into the spherical harmonic l, m = 4, 2 by the planet's oblate figure. The l = 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 (l > m, m >= 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 l = 2 dynamical tides (Delta k (2) approximate to -4%), Jupiter's hydrostatic k (42) requires an unknown dynamical effect to produce a fractional correction Delta k (42) approximate to -11% in order to fit Juno's observation within 3 sigma. Future work is required to explain the required Delta k (42).