# The lost meaning of Jupiter’s high–degree Love numbers

Benjamin Idini Division of Geological and Planetary Sciences, California
Institute of Technology
1200 E California Blvd, MC 150-21
Pasadena, CA 91125, USA  David J. Stevenson Division of Geological and
Planetary Sciences, California Institute of Technology
1200 E California Blvd, MC 150-21
Pasadena, CA 91125, USA

###### Abstract

NASA’s Juno mission recently reported Jupiter’s high–degree ( $=4,2$) Love
number $k_{42}=1.289\pm 0.063$ the hydrostatic $k_{42}=1.743\pm 0.002$ is
$7\sigma$ 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 ($\ell>m$, $m\geq 2$) 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

Solar system gas giant planets (1191) — Galilean satellites (627) — Planetary
interior (1248) — Tides (1702)

††software:  Mathematica (Wolfram, 1999)

##  1 Introduction

NASA’s Juno mission recently registered Jupiter’s tidal response in a set of
Love numbers, observing

The Love number $k_{42}$ corresponds to the tidal gravitational potential of
Jupiter normalized by the tidal forcing produced by the satellite, both in the
$\ell,m=4,2$ term of the spherical harmonics projection. The oblate figure of
a rotating planet distorts the traditional meaning attributed to Love numbers
by introducing spherical harmonic coupling; the tidal forcing at given $\ell$
produces a tidal response in multiple spherical harmonics. In particular,
Jupiter’s Love number $k_{42}$ contains a small contribution from the tidal
response to the $\ell,m=4,2$ tidal forcing, plus a dominant contribution from
the coupled tidal response to the $\ell=m=2$ tidal forcing. For the sake of
brevity, we partially omit further references to the order $m$, which should
always be considered $m=2$ throughout this paper.

The oblate figure of Jupiter enhances Jupiter’s hydrostatic Love number
$k_{42}$ by an order of magnitude when compared to a hypothetical nonrotating
Jupiter (Wahl et al., 2020). To order of magnitude, we can estimate
$k_{42}\sim qk_{2}(a/R)^{2}$, where $a$ is the semimajor axis of the
satellite, $R$ is the planetary radius, $k_{2}$ is the $l=m=2$ Love number,
and $q$ is the adimentional rotational parameter,

$q=\frac{\Omega^{2}s^{3}}{\mathcal{G}M}\textnormal{,}$ (1)

where $M$ is the mass of the planet, $\Omega$ is the planet’s rotational
frequency, $s$ is the average planetary radius, and $\mathcal{G}$ is the
gravitational constant. In the case of Jupiter-Io, we obtain $q\approx 0.09$
and $k_{42}\sim 1.9$. The coupled tidal response to the $\ell=2$ tidal forcing
that contributes to $k_{42}$ is of order $\sim qk_{2}$. The factor $(a/R)^{2}$
describes how much smaller the tidal forcing is at $\ell=4$ when compared to
$\ell=2$. the resulting hydrostatic Love number $k_{42}$ varies greatly among
the Galilean satellites according to the semimajor axis of each orbit, a
result that contradicts the traditional concept of the hydrostatic Love number
defined as a property of the planet.

Here, we use first–order perturbation theory to analytically explain the
correction to the hydrostatic Love number introduced by rotation, a result
only known so far via implementation of numerical strategies (Wahl et al.,
2017). As a response to rotation, the oblate figure of the planet promotes
mixing in the tidal response at different zonal degree $\ell$, causing a
$\sim+10\%$ correction to $k_{2}$ observed both in Saturn (Lainey et al.,
2017; Wahl et al., 2017) and Jupiter (Durante et al., 2020; Wahl et al., 2020;
Idini & Stevenson, 2021), and an order of magnitude increment in $k_{42}$ that
is key to correctly interpreting Juno’s $k_{42}$ observation.

The remainder of this paper is organized as follows. In Section 2, we derive
the general solution for the hydrostatic tidal response in the interior of a
gas giant planet mostly made of H-He. In Section 3, we use first–order
perturbation theory to obtain the hydrostatic Love numbers while including the
oblate figure of the planet introduced by rotation. In Section 4, we use our
theoretical results from Section 3 to calculate Jupiter’s hydrostatic
$k_{42}$. In Section 5, we discuss the implications of our results. In Section
6, we summarize our conclusions.

##  2 The hydrostatic tidal response

In this section, we derive the equation and general solution for the
hydrostatic tidal response of a gas giant planet mostly made of H-He fluid.
Tides in hydrostatic equilibrium follow Poisson’s equation and a simple
equation of motion

$\nabla^{2}\phi=-4\pi\mathcal{G}\rho\textnormal{,}$ (2) $\nabla
p=\rho\nabla\phi\textnormal{.}$ (3)

The potential $\phi$ represents the relevant gravitational forces, $p$ is
pressure, and $\rho$ is density. In a gas giant planet mostly made of H-He,
the equation of state of the fluid can be conveniently approximated by an
$n=1$ polytrope , which follows

$p=K\rho^{2}\textnormal{,}$ (4)

where $K$ is a constant describing the material properties. The simple result
$\nabla p=2K\rho\nabla\rho$ combined with Equations (2) and (3) allows us to
obtain

$\frac{\nabla^{2}\phi}{k^{2}}+\phi=0\textnormal{,}$ (5)

where $k^{2}=2\pi\mathcal{G}/K$.

The first approximation to the hydrostatic tidal response comes from
considering tides as a perturbation $\phi^{\prime}$ over a spherical planet
with a spherically symmetric gravitational potential $\phi_{0}$. In such
scenario, the potential $\phi$ can be written as

$\phi=\phi_{0}+\phi^{\prime}+\phi_{T}\textnormal{,}$ (6)

where the tidal forcing potential takes the form

$\phi_{T}=\sum_{\ell=2}^{\infty}\sum_{m=-\ell}^{\ell}U_{\ell}^{m}\left(\frac{r}{s}\right)^{\ell}Y_{\ell}^{m}\textnormal{,}$
(7)

$U_{\ell,m}$ is a numerical factor defined by

$U_{\ell,m}=\left(\frac{s}{a}\right)^{\ell}\left(\frac{\mathcal{G}m_{s}}{a}\right)\left(\frac{4\pi(\ell-m)!}{(2\ell+1)(\ell+m)!}\right)^{1/2}\mathcal{P}_{l}^{m}(0)\textnormal{,}$
(8)

$\mathcal{P}_{\ell}^{m}$ are the associated Legendre polynomials of degree
$\ell$ and azimuthal order $m$, $Y_{\ell}^{m}$ are spherical harmonics, and
$m_{s}$ the mass of the satellite. The tidal response that solves Equation (5)
follows

$\phi^{\prime}=\sum_{\ell=2}^{\infty}\sum_{m=-\ell}^{\ell}\left(A_{\ell}j_{\ell}(kr)-\left(\frac{r}{s}\right)^{\ell}\right)U_{\ell}^{m}Y_{\ell}^{m}\textnormal{,}$
(9)

where $j_{\ell}$ is the spherical Bessel function of the first kind.

The boundary condition for $\phi^{\prime}$ at the outer boundary of the planet
defines the coefficients $A_{\ell}$. At the outer boundary of the planet, the
tidal response $\phi^{\prime}$ should match an external potential that decays
with distance $r$ away from the planet, according to the factor
$(s/r)^{\ell+1}$. Both potentials should also match in their directional
derivative normal to the outer boundary of the planet. In a spherical planet,
the directional derivative is simply $\partial_{r}$ due to the convenient
decomposition of the tidal response into an axially symmetric factor and a
spherical harmonic. In an oblate planet, however, the directional derivative
involves additional terms that depend on the oblate figure of the planet.
Additionally, the boundary of the planet is no longer the average radius $s$,
but instead an oblate figure that varies with colatitute $\theta$ (i.e.,
roughly following a $\mathcal{P}_{2}(\cos\theta)$ figure). In the following
section, we consider those effects to calculate $A_{\ell}^{m}$ and obtain
first order corrections to the hydrostatic Love number due to the oblate
figure of a rotating planet.

##  3 The hydrostatic tidal response in a rotating planet

In this section, we use first–order perturbation theory to illuminate the
effect that the oblate figure of a rotating planet on the hydrostatic Love
number. To $q$ order of approximation, the oblate figure of a gas giant mostly
made of H-He follows (Appendix A)

$R(\theta)\approx
s\left(1-\frac{5}{\pi^{2}}q\mathcal{P}_{2}(\cos\theta)\right)\textnormal{.}$
(10)

At the outer boundary of the oblate planet (i.e., $r=R(\theta)$), the
gravitational tidal response requires to satisfy the boundary condition

$\nabla\phi^{\prime}(R)\cdot\hat{n}=\nabla\Theta^{\prime}(R)\cdot\hat{n}\textnormal{,}$
(11)

where $\Theta$ is an external gravitational potential that matches
$\phi^{\prime}$ at $r=R$,

$\Theta_{\ell}^{m}{{}^{\prime}}=\left(\frac{R}{r}\right)^{\ell+1}\phi_{\ell}^{m}{{}^{\prime}}(R)\textnormal{,}$
(12)
$\partial_{r}\Theta_{\ell}^{m}{{}^{\prime}}=-\frac{(\ell+1)}{r}\left(\frac{R}{r}\right)^{\ell+1}\phi_{\ell}^{m}{{}^{\prime}}(R)\textnormal{,}$
(13)
$\partial_{\theta}\Theta_{\ell}^{m}{{}^{\prime}}=\frac{(\ell+1)}{R}\left(\frac{R}{r}\right)^{\ell+1}\phi_{\ell}^{m}{{}^{\prime}}(R)\partial_{\theta}R+\left(\frac{R}{r}\right)^{\ell+1}\partial_{\theta}\phi_{\ell}^{m}{{}^{\prime}}(R)\textnormal{,}$
(14)

and $\hat{n}$ is the vector normal to the oblate surface of the planet
$R(\theta)$,

$\hat{n}=\left(1-\frac{5}{\pi^{2}}q\mathcal{P}_{2}(\cos\theta)\right)\hat{r}-\frac{15}{\pi^{2}}q\cos\theta\sin\theta\hat{\theta}\textnormal{.}$
(15)

After applying the differential operator in spherical coordinates and keeping
only terms of order $q$, the boundary condition reduces to

$\sum_{\ell}\left(1-\frac{5}{\pi^{2}}q\mathcal{P}_{2}(\cos\theta)\right)\partial_{r}\phi_{\ell}^{m}{{}^{\prime}}(R)=\sum_{\ell}-\frac{(\ell+1)}{s}\phi_{\ell}^{m}{{}^{\prime}}(R)$
(16)

Assuming that rotation only causes a small deviation from a sphere, we can
write $R=s(1-\epsilon\mathcal{P}_{2})$, where $\epsilon=5q/\pi^{2}$ is a small
parameter. We evaluate the hydrostatic tidal response at the oblate outer
boundary of the rotating planet by Taylor expansion of $\phi^{\prime}$ over
$\epsilon$,

$\phi_{\ell}^{m}{{}^{\prime}}(R)\approx\phi_{\ell}^{m}{{}^{\prime}}(s)-\epsilon
s\partial_{r}\phi_{\ell}^{m}{{}^{\prime}}(s)\mathcal{P}_{2}\textnormal{,}$
(17)
$\partial_{r}\phi_{\ell}^{m}{{}^{\prime}}(R)\approx\partial_{r}\phi_{\ell}^{m}{{}^{\prime}}(s)-\epsilon
s\partial_{r,r}\phi_{\ell}^{m}{{}^{\prime}}(s)\mathcal{P}_{2}\textnormal{.}$
(18)

From Equation (9), the hydrostatic tidal response of a sphere evaluated at
$r=s$ follows

$\phi_{\ell}^{m}{{}^{\prime}}(s)=(A_{\ell}j_{\ell}(ks)-1)U_{\ell}^{m}Y_{\ell}^{m}\textnormal{,}$
(19)
$\partial_{r}\phi_{\ell}^{m}{{}^{\prime}}(s)=\left(A_{\ell}\partial_{r}j_{\ell}(ks)-\frac{\ell}{s}\right)U_{\ell}^{m}Y_{\ell}^{m}\textnormal{,}$
(20)
$\partial_{r,r}\phi_{\ell}^{m}{{}^{\prime}}(s)=\left(A_{\ell}\partial_{r,r}j_{\ell}(ks)-\frac{\ell(\ell-1)}{s^{2}}\right)U_{\ell}^{m}Y_{\ell}^{m}\textnormal{.}$
(21)

We replace Equations (17) and (18) into Equation (16) to obtain the final
equation for the coupled hydrostatic tidal response of an oblate rotating
planet,

$\displaystyle\sum_{\ell}\left(A_{\ell}\left(j_{\ell}(ks)\left(\frac{\ell+1}{s}\right)+\partial_{r}j_{\ell}(ks)\right)-\frac{2\ell+1}{s}\right)U_{\ell}^{m}Y_{\ell}^{m}$
$\displaystyle=\frac{15}{\pi^{2}}q\sum_{\ell}\left(A_{\ell}\left(s\partial_{r,r}j_{\ell}(ks)+(\ell+2)\partial_{r}j_{\ell}(ks)\right)-\frac{\ell(2\ell+1)}{s}\right)U_{\ell}^{m}Y_{\ell}^{m}\mathcal{P}_{2}\textnormal{.}$
(22)

We use a recursive relation based on Clebsch-Gordan coefficients to calculate
the coupling in spherical harmonics introduced by the term
$Y_{\ell}^{m}\mathcal{P}_{2}$ (Idini & Stevenson, 2021),

$Y_{\ell}^{m}\cos^{2}\theta=p_{\ell-1}p_{\ell}Y_{\ell-2}^{m}+(p_{\ell}^{2}+p_{\ell+1}^{2})Y_{\ell}^{m}+p_{\ell+1}p_{\ell+2}Y_{\ell+2}^{m}\textnormal{,}$
(23)
$p_{\ell}=\left(\frac{\ell^{2}-m^{2}}{4\ell^{2}-1}\right)^{1/2}\textnormal{.}$
(24)

Using the recursive relation above, we can write the term that couples the
spherical harmonics of the hydrostatic tidal response as

$Y_{\ell}^{m}\mathcal{P}_{2}=\frac{3}{2}\cos^{2}\theta
Y_{\ell}^{m}-\frac{Y_{\ell}^{m}}{2}=\frac{3}{2}p_{\ell-1}p_{\ell}Y_{\ell-2}^{m}+\left(\frac{3}{2}(p_{\ell}^{2}+p_{\ell+1}^{2})-\frac{1}{2}\right)Y_{\ell}^{m}+\frac{3}{2}p_{\ell+1}p_{\ell+2}Y_{\ell+2}^{m}\textnormal{.}$
(25)

In the following section, we evaluate Equations (8), (22), and (25) to obtain
$A_{\ell}$ in the case of Jupiter when tidally perturbed by the gravitational
pull of Io.

##  4 Jupiter’s hydrostatic Love numbers

For the sake of simplicity, we analyze the case of coupling between Jupiter’s
rotational and hydrostatic tidal responses in $\ell=2$ and $\ell=4$, ignoring
terms of higher degree. . From Equation (22), we can write a linear system of
equations in the form

$c_{1}A_{2}+c_{2}A_{4}=c_{3}\textnormal{,}$ (26)
$c_{4}A_{2}+c_{5}A_{4}=c_{6}\textnormal{,}$ (27)

where we obtain the $c_{n}$ coefficients from evaluating the sum in Equation
(22) truncated at $\ell=4$ and set for $m=2$.

Table 1: Rotational correction to Jupiter’s hydrostatic Love number under the
tidal perturbation of Io, Europa, and Ganymede.

| polytrope | polytrope | polytrope | CMS | CMS | CMS
---|---|---|---|---|---|---
| Io | Europa | Ganymede | Io | Europa | Ganymede
$\delta k_{2}$ | 1.11 | 1.11 | 1.11 | 1.10 | 1.10 | 1.10
$\delta k_{42}$ | 14.4 | 35.0 | 87.6 | 13.6 | 32.8 | 83.7

Note. — The rotational correction $\delta k_{\ell m}$ is the ratio between the
Love number in an oblate rotating Jupiter model, over the Love number in a
spherical nonrotating Jupiter model. Jupiter’s rotation rate follows
$q=0.0892$. (2-4) We obtain the analytical results in an $n=1$ polytrope from
the ratio between Equation (30) and Equation (29). (5-7) We calculate from
Love numbers reported in Wahl et al. (2020), the Concentric Maclaurin Spheroid
(CMS) method.

Our analytical polytropic Jupiter model approximates the rotational correction
to the hydrostatic Love number reported in Wahl et al. (2020) to $q$–order
accuracy (Table 1). In the case of tides raised by Io, we obtain a $\sim 10\%$
increment in $k_{2}$ and an order of magnitude increment in $k_{42}$, both
results previously reported in numerical calculations using the Concentric
Maclaurin Spheroids (CMS) method (Wahl et al., 2016, 2020). The order of
magnitude enhancement in $k_{42}$ comes from the the tidal response to the
$\ell=2$ tidal forcing rotationally–coupled into the $\ell=4$ gravitational
field. We calculate the rotational correction $\delta k_{\ell m}$ in Table 1
as the ratio between the Love number in an oblate rotating polytrope over the
Love number in a spherical nonrotating polytrope,

$\delta k_{\ell m}=\frac{k_{\ell}}{k_{\ell}^{(s)}}\textnormal{.}$ (28)

The Love number in a spherical nonrotating polytrope follows (Idini &
Stevenson, 2021)

$k_{\ell}^{(s)}=\left(\frac{2\ell+1}{\pi}\right)\frac{j_{\ell}(\pi)}{j_{\ell-1}(\pi)}-1\textnormal{,}$
(29)

while the Love number in an oblate rotating polytrope follows

$k_{\ell}=A_{\ell}j_{\ell}(\pi)-1\textnormal{,}$ (30)

where the $A_{\ell}$ coefficients come from solving Equations (26) and (27).
In the case of Jupiter’s rotation ($q=0.0892$) and Io’s semimajor axis, we
obtain $A_{2}=5.19$ and $A_{4}=42.2$. For the sake of comparison, the
nonrotating $n=1$ polytrope in hydrostatic equilibrium produces $A_{2}=5$ and
$A_{4}=17.3$.

We can apply our rotational corrections calculated from Equation (28) to the
nonrotating Love numbers of a Jupiter model with a more realistic equation of
state and density profile (i.e., Wahl et al. (2020)). Our results agree with
the CMS results within a margin of $2\%$ and $3\%$ for $k_{2}$ and $k_{42}$,
respectively (Table 2). The difference between both results comes from second
order effects not included in our analysis.

Table 2: Jupiter’s hydrostatic Love numbers under the tidal perturbation of
Io, Europa, and Ganymede.

| CMS |  $n=1$ polytrope | CMS | CMS | CMS
---|---|---|---|---|---
| Nonrotating | Io | Io | Europa | Ganymede
$k_{2}$ | 0.5364 | 0.60 | 0.5898 | 0.5894 | 0.5893
$k_{42}$ | 0.1279 | 1.8 | 1.7432 | 4.1975 | 10.7058

Note. — (2) Numerical results obtained with the Concentric Maclaurin Spheroid
(CMS) method applied to a nonrotating Jupiter model that follows an equation
of state derived from ab initio simulations (Wahl et al., 2020). (3) We obtain
the analytical results in an $n=1$ polytrope from applying the fractional
correction in Equation (28) to the nonrotating result in (2). (4-6) CMS
Numerical results for the Love number of a rotating Jupiter model (Wahl et
al., 2020).

##  5 Discussion

###  5.1 The lost meaning of Love numbers in rotating gas giant planets

High–degree tesseral Love numbers ($\ell>m$, $m\geq 2$) lose their original
meaning in a rotating planet with an oblate figure. As first proposed by
A.E.H. Love in 1909, Love numbers represent the tidal response of a planet
normalized by the tidal forcing, both at the same $\ell,m$ spherical harmonic.
Accordingly, the hydrostatic tidal gravitational field of a spherical planet
is a sum over terms $k_{\ell,m}\phi_{\ell,m}^{T}$. In this original meaning,
the Love number represents an interior property of the planet. In the context
of gas giant exoplanets, the Love number may describe the degree of central
concentration of mass, with a lower Love number indicating a more centrally
concentrated planet (Batygin et al., 2009). However, the coupling introduced
by rotation complicates this convenient picture.

As shown in Equation (17), the hydrostatic tidal response to the forcing at a
given $\ell$ contains terms from multiple spherical harmonics. Rotation
introduces a significant term with $\ell=4$ spherical harmonic corresponding
to part of the tidal response to the $\ell=2$ tidal forcing. In fact, this
term dominates Jupiter’s $\ell=4$ tidal gravitational field, with $7\%$ of the
amplitude arising from the tidal response to the $\ell=4$ tidal forcing and
$93\%$ from the tidal response to the $\ell=2$ tidal forcing coupled by the
oblate figure of the planet (Table 1). According to this new term, the
$\ell=4$ tidal gravitational field is (Equation (17)),

$\phi_{4}^{2}{{}^{\prime}}\sim\frac{5}{\pi^{2}}q\phi_{2}^{2}{{}^{\prime}}\mathcal{P}_{2}\sim\frac{5}{\pi^{2}}q\left(\frac{r}{s}\right)^{2}k_{2}U_{2,2}Y_{4}^{2}\textnormal{.}$
(31)

When computing $k_{42}$ after normalization of Equation (31) by the tidal
forcing $\phi_{4}^{2}{{}_{T}}$, the term introduced by rotation promotes a
dependency of $k_{42}$ on the semimajor axis of the satellite,

$k_{42}\sim\frac{5}{\pi^{2}}q\left(\frac{a}{s}\right)^{2}k_{2}\textnormal{.}$
(32)

Explicitly revealed here using perturbation theory, the dependency of $k_{42}$
on semimajor axis was previously observed in numerical results obtained with
CMS (Wahl et al., 2017, 2020). The difference in $k_{42}$ among the Galilean
satellites can be explained by the orbital factor $(a/s)^{2}$, where $a/s$ is
roughly 6 for Io, 10 for Europa, and 15 for Ganymede (Table 1).

###  5.2 The $7\sigma$ discrepancy observed by Juno

Our results confirming a $7\sigma$ discrepancy between Juno’s $k_{42}$ and
hydrostatic tides . The CMS hydrostatic $k_{42}$ requires a $\Delta
k_{42}\approx-15\%$ fractional correction to fit the Juno observation within
$3\sigma$. Due to rotational coupling (Equation (32)), one part of the
required correction comes from dynamical effects on $k_{2}$ that include the
Coriolis effect. The $\Delta k_{2}\approx-4\%$ effect introduced by dynamical
tides (Idini & Stevenson, 2021; Lai, 2021) reduces the fractional correction
required by Juno to $\Delta k_{42}\approx-11\%$. This residual effect must
come from additional dynamical effects related to the $\ell=4$ tidal response,
which is only a small fraction ($7\%$) of the hydrostatic $k_{42}$. At
perijove 17, Juno $3\sigma$ uncertainty on $k_{2}$ is only $3\%$ (Idini &
Stevenson, 2021), much smaller than the required $\Delta k_{42}\approx-11\%$.
Consequently, additional hypothetical dynamical effects applied to $k_{2}$ are
constrained by Juno to be small and insufficient. We require future studies to
understand the origin of the $\Delta k_{42}$ fractional correction required to
fit Juno observations.

##  6 Conclusions

We used first–order perturbation theory to calculate the rotational correction
to Jupiter’s hydrostatic Love number $k_{42}$. We showed that the oblate
figure of the rotating planet forces the $\ell=m=2$ tidal response to couple
into the $\ell,m=4,2$ tidal gravitational field, increasing the hydrostatic
$k_{42}$ beyond an order of magnitude for Io and roughly by two orders of
magnitude for Ganymede. As a result, we conclude that low–degree hydrostatic
Love numbers dominate high–degree hydrostatic tesseral Love numbers ($\ell>m$,
$m\geq 2$), and thus the latter provide little additional information about
interior structure. The exception is the case where dynamical effects
particular to a given high–degree Love number acquire relevant amplitude due
to, for example, tidal resonances.

Our analysis leads to important implications for the correct interpretation of
a $7\sigma$ anomaly in Jupiter’s $k_{42}$ as observed by NASA’s Juno mission.
The Juno $k_{42}$ anomaly is slightly attenuated by the coupled $\ell=2$
dynamical tides ($\Delta k_{2}\approx-4\%$). At Juno’s mid–mision perijove 17,
Jupiter’s hydrostatic Love number $k_{42}$ requires an additional fractional
correction $\Delta k_{42}\approx-11\%$ from unknown dynamical effects
associated to its tidal response to the $\ell=4$ tidal forcing. We require
further analysis to unravel the origin of the required fractional correction.

We acknowledge the support of NASA’s Juno mission. We benefited from
constructive discussions with James Fuller, Janosz Dewberry, and Christopher
Mankovich.

##  Appendix A The oblate figure of a gas giant planet

In this appendix, we revise the classical result of calculating the
first–order figure of a gas giant planet mostly made of H-He (i.e., an $n=1$
polytrope) after perturbed by the centrifugal effect . The external potential
of a body perturbed by the centrifugal effect follows

$\phi\approx\frac{\mathcal{G}M}{r}\left(1-\left(\frac{s}{r}\right)^{2}J_{2}\mathcal{P}_{2}(\cos\theta)\right)\textnormal{,}$
(A1)

and the rotational forcing potential follows

$\phi_{R}=\frac{\Omega^{2}r^{2}}{3}\left(1-\mathcal{P}_{2}(\cos\theta)\right)\textnormal{,}$
(A2)

The outer boundary of the planet represents an equipotential surface where
pressure is constant. If $R(\theta)$ represents the outer boundary of the
planet, we require to satisfy

$\phi(R)+\phi_{R}(R)=\textnormal{constant}=\phi(\theta=0)\textnormal{,}$ (A3)

or

$\frac{\mathcal{G}M}{R}\left(1-\left(\frac{s}{R}\right)^{2}J_{2}\mathcal{P}_{2}(\cos\theta)\right)+\frac{\Omega^{2}R^{2}}{3}\left(1-\mathcal{P}_{2}(\cos\theta)\right)=\frac{\mathcal{G}M}{b}\left(1-\left(\frac{s}{b}\right)^{2}J_{2}\right)\textnormal{,}$
(A4)

where $b$ is the polar radius. We consider the first–order expansion on the
oblate figure of the planet as

$R(\theta)=\sum_{\ell=0}^{\infty}\delta
r_{2\ell}\mathcal{P}_{2\ell}(\cos\theta)\approx s+\delta
r_{2}\mathcal{P}_{2}(\cos\theta)\textnormal{,}$ (A5)

After replacing the first–order expansion of $R(\theta)$ into Equation (A4),
we obtain the classical result

$\frac{\delta r_{2}}{s}\approx-\left(J_{2}+\frac{q}{3}\right)\textnormal{,}$
(A6)

which is accurate to first order in $q$,

To obtain $J_{2}$ in an $n=1$ polytrope, we the $\ell=2$ rotational
gravitational response of the polytrope to the centrifugal effect, both
evaluated at $r=s$,

$J_{2}=\left(5j_{2}(\pi)-1\right)\frac{q}{3}\textnormal{,}$ (A7)

Finally, we obtain the first–order oblate figure of a rotating gas giant
planet perturbed by the centrifugal effect,

$R(\theta)\approx
s\left(1-\frac{5}{\pi^{2}}q\mathcal{P}_{2}(\cos\theta)\right)\textnormal{.}$
(A8)

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