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 k₄₂ = 1.289 ± 0.063 (1σ), an order of magnitude above the hydrostatic k₄₂ obtained in a nonrotating Jupiter model. After numerically modeling rotation, the hydrostatic k₄₂ = 1.743 ± 0.002 is still 7σ 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 k₄₂ result analytically. We use a simple Jupiter equation of state (n = 1 polytrope) to obtain the fractional change in k₄₂ when comparing a rotating model with a nonrotating model. Our analytical result shows that the hydrostatic k₄₂ 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 k₄₂ introduces an orbital factor (a/s)² into k₄₂, where a is the satellite semimajor axis and s is Jupiter's average radius. As a result, different Galilean satellites produce a different k42. 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 k₄₂. After including the coupling from the well-understood ℓ = 2 dynamical tides (Δk₂ ≈ −4%), Jupiter's hydrostatic k₄₂ requires an unknown dynamical effect to produce a fractional correction Δk₄₂ ≈ −11% in order to fit Juno's observation within 3σ. Future work is required to explain the required Δk₄₂.