Jupiter's Dynamical Love Number

Recent observations by the {\it Juno} spacecraft have revealed that the tidal Love number $k_2$ of Jupiter is $4\%$ lower than the hydrostatic value. We present a simple calculation of the dynamical Love number of Jupiter that explains the observed"anomaly". The Love number is usually dominated by the response of the (rotation-modified) f-modes of the planet. Our method also allows for efficient computation of high-order dynamical Love numbers. While the inertial-mode contributions to the Love numbers are negligible, a sufficiently strong stratification in a large region of the planet's interior would induce significant g-mode responses and influence the measured Love numbers.


INTRODUCTION
The Juno spacecraft recently found an "anomaly" in Jupiter's tidal Love number: the measured k 2 = 0.565±0.006 (Durante et al. 2020) appears to be smaller than the theoretical hydrostatic value k (hs) 2 = 0.590 (Wahl et al. 2020) by 4%. This discrepancy may be explained in terms of dynamical tides, i.e., Jupiter's response to the finite-frequency tidal forcings from the Galilean moons (Idini & Stevenson 2021). Here we present a simple calculation that explains this Love number "anomaly" quantitatively. Naive expectation would suggest a 1/(ω 2 α − ω 2 ) enhancement (where ω α is the f-mode frequency of the planet) of the tidal response due to the finite tidal frequency (ω) as compared to the hydrostatic (ω = 0) response. The key to obtain the correct answer is to treat the rotational (Coriolis) effect on the modes of a rotating planet and their tidal responses in a self-consistent way. Our general method also allows for efficient computation of high-order dynamical Love numbers k lm , as well as the inclusion of the contributions to k lm from the inertial modes (due to planetary rotation) and g-modes (due to stable stratification in the planetary interior).

DYNAMICAL LOVE NUMBER AND NORMAL MODES
Consider a planet (mass M , radius R and spin angular frequency Ω s ) orbited by a satellite (mass M ) in a circular orbit with semi-major axis a and orbital frequency Ω orb . We assume the spin axis is aligned with the orbital axis. In the frame corotating with the planet, the (lm)-component of the tidal potential produced by M on the planet is where A lm = (GM /a l+1 )W lm (with W lm a dimensionless constant; W lm = 0 when l + m =even), r = (r, θ, φ) specifies the position vector (in spherical coordinates) measured from the center of the planet, and is the tidal forcing frequency. It suffices to consider only m > 0. The relevant non-zero tidal components are (lm) = (2, 2), (3, 1), (3, 3), (4, 2), (4, 4) etc. The linear response of the planet to the tidal forcing is specified by the Lagrangian displacement, ξ(r, t), of a fluid element from its unperturbed position. In the rotating frame of the planet, the equation of motion takes the form where C is a self-adjoint operator (a function of the pressure and gravity) acting on ξ (see, e.g., Friedman & Schutz 1978). A free mode of frequency ω α (in the ro- where {α} denotes the mode index, which includes the azimuthal number m. We carry out phase-space mode expansion (Schenk et al. 2002) Using the orthogonality relation ξ α , 2iΩ s × ξ α + (ω α + ω α ) ξ α , ξ α = 0 (for α = α ), where A, B ≡ d 3 x ρ (A * · B), we find (Lai & Wu 2005) where and we have used the normalization ξ α , ξ α = 1. In Eq. (7), δρ α is the Eulerian density perturbation associated with the eigenfunction ξ α . Equation (6) has stationary solution The gravitational perturbation associated with the density perturbation δρ(r, t) = α c α (t)δρ α (r), evaluated at the planet's surface (r = R), is Thus the tidal Love number is In the above equation, the tidal overlap coefficientQ α,lm and the mode frequenciesω α andε α are in units where G = M = R = 1, i.e.,ω α = ω α /(GM/R 3 ) 1/2 , etc. Note that for a given m > 0, the sum in Eq. (11) includes modes with positive ω α and negative ω α , corresponding to prograde (with respect to the planet's rotation) and retrograde modes.

F-MODE CONTRIBUTION
In most situations, the sum in Eq. (11) is dominated by f-modes since they have the largest tidal overlap Q α,lm . For planetary rotation rate Ω s much less than the breakup rate (GM/R 3 ) 1/2 , (e.g.,Ω s = 0.288 for Jupiter), the effect of rotation on the modes can be treated perturbatively (e.g. Unno et al. 1989). Let ω 0 (> 0) be the mode frequency of a non-rotating planet, then for a given m > 0, the sum in Eq. (11) includes with Table 1. Oscillation modes of non-rotating polytropic (n = 1) planet model Note-ω0 and Q l are the mode frequency and tidal overlap coefficient (Eq. 7), both in units such that G = M = R = 1, and C is defined in Eq. (14). The planet's density profile is that of n = 1 polytrope (with the equation of state P ∝ ρ 2 ). The first model has Γ1 (the adiabatic index) equal to Γ = 1 + 1/n, and we list the properties for the f-mode and the first radialorder p-mode. The second model has Γ1 = 2.4 throughout the planet, and the third model has Γ1 = 2.4 only in two regions (r/R ∈ [0.5, 0.7], [0.85, 0.93]) and Γ1 = Γ otherwise (the transition width is 0.025R; see Eq. 26), and we list the properties for the f-mode and the first three radial-order gmodes. Note that when |Q l | 1, the quoted Q l values are only accurate in 2-3 significant figures.
where ξ α,0 = ξ r (r)r + ξ ⊥ (r)r∇ ⊥ Y lm is the mode eigenvector of a non-rotating planet. To a good approximation, we can also set Q α,lm to be the non-rotating value, i.e, lm to Jupiter's tidal Love number as a funtion of the orbital frequency Ω orb of the perturbing satellite (in units of the spin frequency Ωs). All results (solid curves) are computed using the n = 1 (isentropic) polytrope model, except that the dot-dashed curve is for k2 = k22 computed using the n = 0.9 polytrope model. The vertical dashed lines specify the orbital frequencies of Io, Europa, Ganymede and Callisto (from right to left). The −4% "anomaly" of k2 observed by Juno can be explained by the planetary model with n 1.
Thus Eq. (11) reduces to For an incompressible planet model (n = 0 polytrope), the l = 2 mode (Kelvin mode) has Thus withω = 2(Ω orb −Ω s ). Giant planets are approximately described by a n = 1 polytrope (corresponding to P ∝ ρ 2 ). Table 1 list the numerical values of ω 0 , Q l and C for several nonrotating poltropic models (with different levels of stratification; see Section 5). For l = m = 2 tidal response (and n = 1), 2C 1, we have Applying to the Jupiter- This explains the 4% discrepancy between k 2 and k (hs) 2 . Note that our static k Our results depicted in Fig. 1 can be compared to those of Idini & Stevenson (2021) obtained using more complicated calculations (see their Table 2). Our δ 22 , δ 33 and δ 44 values (evaluated for the orbital frequencies of Io, Europa, Ganymede and Callisto) agree reasonably well with theirs, but our δ 31 , δ 42 values are a factor of a few smaller.
Finally, using Table 1, we can easily check that the contributions from p-modes to k lm are negligible.

INERTIAL-MODE CONTRIBUTION
In addition to f-modes and p-modes, a rotating planet possesses a spectrum of inertial modes supported by Coriolis force. For n = 1 polytrope, the m = 2 inertial modes have been computed by Xu & Lai (2017) using a spectral code. The mode properties are for the prograde mode, and for the retrograde mode, where Q ± is the tidal coupling coefficient Q α,22 . Since ω < 0, we can write the inertial mode contribution of k 2 as Defineω ≡ ω/Ω s (and similarlyω ± andε ± ) andQ ± ≡ Q ± /Ω 2 s , we have (24) For n = 1 polytrope, this gives 8.04 0.556 + |ω| + 1.82 1.10 − |ω| .

STABLE STRATIFICATION AND G-MODE CONTRIBUTION
In Sections 3-4 we considered fully isentropic models for Jupiter, i.e., the adiabatic index Γ 1 ≡ (∂ ln P/∂ ln ρ) s equals the polytropic index Γ ≡ d ln P/d ln ρ = 1 + 1/n. In reality, some regions of the planet may be stably stratified, with Γ 1 > Γ. Indeed, the gravity measurement by Juno and structural modeling suggest that Jupiter have a diluted core and a total heavy-element mass of 10-24 Earth masses, with the heavy elements distributed within an extended region covering nearly half of Jupiter's radius (Wahl et al. 2017;Debras & Chaberier 2019;Stevenson 2020). The composition gradient outside the diluted core would provide stable stratification, and the planet would then possess g-modes.
To explore of how g-modes influence the tidal love numbers, we consider three simple planetary models, all having a n = 1 density profile (Γ = 2), but with different adiabatic index profiles: (i) Γ 1 = 2.4 throughout the Figure 2. Same as Fig. 1, but for the n = 1, Γ1 = 2.4 planetary model, which possesses g-modes. The heavy solid curves include the contributions of f-modes and first three radial-order g-modes to k lm , the light solid curves include only f-modes (for the n = 1 isentropic model, as in Fig. 1). planet (see Table 1); (ii) Γ 1 = 2.4 only in the stable regionr = r/R ∈ [0.5, 0.7] (with a transition width of  Table 1). For each model, we compute the f-modes and g-modes of a nonrotating planet (see Table 1), and use Eqs. (12)-(13) to account for the effect of rotation on the modes. We include only the first three radial-order g-modes in our calculation of k lm . The perturbative approach of the rotational effect is approximately valid for these modes since mCΩ s is less than the mode frequency |ω 0 |. Figures 2-4 show the results for the dynamical Love numbers based on the three models. It is obvious that significant dynamical correction to the hydrostatic k (hs) lm occurs around the resonance, where ω α = ω. The "strength" of each resonance is measured by the tidal overlap coefficient, and a large |Q l | value implies that the "width" of the resonant feature is larger (see Eq. 16). For the Γ 1 = 2.4 model, the stratification is strong, the broad/strong resonance with the g 1 mode can affect k lm associated with the Galilean moons (Fig. 2). For the model with the stable stratified region restricted to r/R ∈ [0.5, 0.7] (Fig. 3), the resonance feature is much weaker/narrower, but still ∆k 2 /k (hs) 2 becomes −5% for Io. When the model further includes the stratified region at r/R ∈ [0.85, 0.93] (Fig. 4), the resonance features shift and broaden, and k 31 becomes affected for the Galilean moons. Obviously, these results are for illustrative purpose, but they indicate that resonance features due to stable stratification in the planet's interior may influence the the measured dynamical Love numbers.
Note that Figures 2-4 do not include contributions from high-order g-modes. These modes (with mode frequencies comparable to Ω s ) become mixed with inertial modes (so-called "inertial-gravity" modes; see Xu & Lai 2017) and cannot be treated using Eqs. (12)-(13). However, because of their small tidal overlap coefficients, they are unlikely to be important contributors to k lm except for the coincidence of an extremely close resonance.

CONCLUSION
We have derived a general equation (Eq. 11) for computing the dynamical Love number k lm of a rotating giant planet in response to the tidal forcings from its satellites. In most situations, the Love number is dominated by the tidal response of f-modes, and the general expression reduces to Eq. (16), which can be easily evaluated using the mode properties of nonrotating planet models (see Table 1). We show that the 4% discrepancy between the measured k 2 of Jupiter and the theoretical hydrostatic value can be naturally explained by the dynamical response of Jupiter's f-modes to the tidal forcing from Io -the key is to include the rotational (Coriolis) effect in the tidal response in a self-consistent way. We also show that the contributions of the inertial modes to the Love number k 2 are negligible.
We have also explored the effect of stable stratification in Jupiter's interior on the Love numbers. If sufficiently strong stratification exists in a large region of the planet's interior, g-mode resonances may influence the dynamical Love numbers associated with the tidal forcing from the Galilean moons. Thus, precise measurements of various k lm could provide constraints on the planet's interior stratification.