Dynamical tides in Jupiter as revealed by Juno
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Dynamical tides in Jupiter as revealed by Juno
Benjamin Idini and David J. Stevenson Division of Geological and Planetary Sciences, California Institute of Technology1200 E California Blvd, MC 150-21Pasadena, CA 91125, USA (Received Dec. 4, 2020; Revised Jan. 11, 2021; Accepted Feb. 15, 2021)
Submitted to PSJABSTRACTThe Juno orbiter continues to collect data on Jupiter’s gravity field with unprecedented precisionsince 2016, recently reporting a non-hydrostatic component in the tidal response of the planet. At themid-mission perijove 17, Juno registered a Love number k = 0 . ± .
006 that is − ±
1% (1 σ ) fromthe theoretical hydrostatic k ( hs )2 = 0 . k to the well-known hydrostatic k . Exploiting the analytical simplicity of a toy uniform-densitymodel, we show how the Coriolis acceleration motivates the negative sign in the ∆ k observed byJuno. By simplifying Jupiter’s interior into a core-less, fully-convective, and chemically-homogeneousbody, we calculate ∆ k in a model following an n = 1 polytrope equation of state. Our numericalresults for the n = 1 polytrope qualitatively follow the behaviour of the uniform-density model, mostlybecause the main component of the tidal flow is similar in each case. Our results indicate that thegravitational effect of the Io-induced dynamical tide leads to ∆ k = − ± Keywords: dynamical tides — Jupiter’s interior — Juno — gravitational fields INTRODUCTIONThe interior structure of a planet or star closely corresponds with its origin and evolution story. Seismology providesthe tightest constraints on the interior structure of Earth (Dahlen & Tromp (1998) and ref. therein), Saturn (Marley &Porco 1993; Fuller 2014), the Sun (Christensen-Dalsgaard et al. 1985) and other distant stars (Aerts et al. (2010) andref. therein). In particular, Saturn’s ring seismology facilitates estimates of the planet’s rotation rate (Mankovich et al.2019) and possible dilute core (Mankovich et al., 2020). Unlike Saturn, Jupiter lacks extensive optically-thick ringswith embedded waves that are excited by resonance of ring particle motions with internal normal modes. Alternativelyto ring seismology, Doppler imaging reveals a suggested seismic behavior in Jupiter, limited to radial overtones of p-modes. At best, the current Doppler imaging data resolves the spacing in frequency space of low-order p-modes,providing a loose constraint compatible with simple interior models (Gaulme et al. 2011). Future efforts based onsimilar techniques promise revealing additional information on Jupiter’s seismic behavior.In the current absence of detailed seismological constraints, the Juno orbiter (Bolton et al. 2017) emerges as thealternative directed to reveal Jupiter’s interior by employing gravity field measurements of global-scale motions. Based
Corresponding author: Benjamin [email protected] a r X i v : . [ a s t r o - ph . E P ] F e b Idini & Stevenson on radiometric observations, Juno produces two kinds of gravity field measurements sensitive to Jupiter’s interiorstructure: the zonal J (cid:96) and tesseral C (cid:96),m gravity coefficients. The odd J (cid:96) +1 coefficients reflect contributions fromzonal flows, including atmospheric zonal winds and zonal flow in the dynamo region. The even J (cid:96) coefficients containJupiter’s response to the centrifugal effect responsible for Jupiter’s oblateness, with minor contributions from zonalwinds in the atmosphere (Iess et al. 2018) and the dynamo region (Kulowski et al. 2020). A time-dependent subset of C (cid:96),m coefficients contains Jupiter’s tidal response to the gravitational pull from its system of satellites. Closely relatedto the time-dependent C (cid:96),m , the Love number k represent the non-dimensional gravitational field of tides evaluatedat the outer boundary of the planet (Munk & MacDonald 1960). One common interpretation relates k to the degreeof central concentration of the planetary mass (e.g., compressibility of the planetary material or the presence of acore). In a quadrupolar gravitational pull, the leading term in the tidal response relates to the Love number k , whichcorresponds to the (cid:96) = m = 2 spherical harmonic.Following linear perturbation theory, the Love number k breaks into two contributions: one hydrostatic and theother dynamic. The hydrostatic k ignores the time–dependence of tides. The dynamical tide represents the tidal flowand perturbed tidal bulge solving the traditional equation of motion F T = M ¨ u , rather than F T = 0, where F T is thesatellite-induced tidal force, diminished by the opposing self-gravity of the perturbed planet.Early studies of the Love numbers in the gas giant planets relied on purely hydrostatic theory, assisted by simplethermodynamic principles alongside loosely constrained interior models (Gavrilov & Zharkov 1977). With the assis-tance of historical astrometric data, the Cassini mission provided the first occasion to test the accuracy of those firsthydrostatic models, finding a ∼
10% discrepancy between the theoretical and observed k of Saturn (Lainey et al.2017). Hydrostatic theory adjusted to Cassini’s observation after incorporating the oblateness produced by the cen-trifugal effect into the tidal model, not requiring to invoke the gravitational effects of Saturn dynamical tides. Theoblateness produced by the centrifugal effect is large due to Saturn’s fast rotation, leading to major higher-order crossterms neglected in the earlier theory (Wahl et al. 2017a). This effect remains hydrostatic provided that rotation occurson cylinders, which allows the centrifugal force to be represented as the gradient of a potential.Unlike Cassini, the Juno orbiter recently detected a 3 σ deviation in Jupiter’s observed k from the revised hydrostatictheory that accounts for the interaction of tides with oblateness (Notaro et al. 2019; Durante et al. 2020). Importantly,the difference between the observed k and hydrostatic theory cannot be attributed to a failure to correctly constrainthe hydrostatic number. Hydrostatic tides are well-constrained (i.e., the k error in Wahl et al. (2020) is ± .
02% ofthe central value) because the effect of oblateness of the planet on the zonal gravity coefficients J (cid:96) is known to a highprecision by Juno. Juno’s non-hydrostatic detection motivates a more careful consideration of neglected effects thatcontribute to k , particularly the gravitational field related to dynamical tides.Here we evaluate dynamical tides as a potential explanation to Juno’s non-hydrostatic detection. We concentrateon the contribution of dynamical tides to the overall gravity field while ignoring their contribution to dissipation. Inother words, we implicitly assume that the imaginary part of k is too small to affect our results for the real part, inagreement with observations of the orbital evolution of Io ( Q = −| k | / Im( k ) ∼ ; Lainey et al. (2009)).We are motivated in our efforts by the prospect of finding an additional contribution to k coming from Jupiter’s core.A traditional model of a Jupiter-like planet consists of a discreet highly-concentrated central region of heavy elements(i.e., a core made of rock and ice) surrounded by a chemically homogeneous and adiabatic envelope of hydrogen-richfluid material (Stevenson 1982). Jupiter’s observed radius indicates a super-solar abudance of heavy elements thataccounts for a total of ∼ M E . However, whether heavy elements reside in a traditional core or are distributedthroughout the envelope is less clear. The observed J and J require some tendency towards central concentrationbut do not require a traditional core. In a subsequent investigation, we exploit our results of the dynamical tidepresented here to answer questions about the origin and evolution of Jupiter by including in our model an enrichmentof heavy elements that increases with depth (i.e., a dilute core; Wahl et al. (2017b)). Using the information containedin k , we plan to provide answers to the following questions about Jupiter: Whether solid or fluid, does Jupiter havea traditional core? Alternatively, do the heavy elements in Jupiter spread out from the center forming a dilute core?The remainder of this manuscript is organized as follows. In Section 2, we describe the Juno non-hydrostaticdetection and develop the mathematical formalism used in the calculation of the fractional dynamical correction to k . In Section 3, we calculate the fractional dynamical correction to k using simple tidal models, which leads toan unambiguous explanation to the non-hydrostatic Juno detection. In Section 4, we deliver a discussion on thelimitations of our analysis, other physical processes potentially altering the Love number k , and future directions ofinvestigation. In Section 5, we outline the conclusions and implications of our study. upiter dynamical tides JUPITER’S LOVE NUMBER2.1.
A correction to the hydrostatic k The main objective of this manuscript is to evaluate the hypothesis that Juno captured a systematic deviation of k from the hydrostatic number and that most of the deviation can be explained by the neglected gravitational effectof dynamical tides. The mean k Juno estimate at the time of perijove 17 is 0 .
565 (Durante et al. 2020), establishinga −
4% deviation from the theoretical hydrostatic number k ( hs )2 = 0 .
590 (Wahl et al. 2020) . The correction to thehydrostatic k due to the rotational bulge is included in k ( hs )2 and is of order q = Ω R J / G M J ∼ .
1, the ratio ofcentrifugal effects to gravity at the equator (Wahl et al. 2020); R J is the equatorial radius and M J Jupiter’s mass.The satellite-independent 3 σ uncertainty (confidence level ≈ . k ( hs )2 at perijove 17(Durante et al. 2020), close to the mean of observed deviation. At the end of the prime mission, the satellite-independent3 σ uncertainty in Juno’s observation is projected to decrease to 1% (William Folkner, personal communication, April8, 2020). Following the optimistic assumption that the mean deviation remains the same at the end of the primemission, k ( hs )2 will require a non-hydrostatic correction from − −
3% to be reconciled with 3 σ observations.In our tidal models, we evaluate a dynamical correction to the hydrostatic k as: k k ( hs )2 = 1 + ∆ k + O (1%), (1)where ∆ k is the fractional dynamical correction calculated for a spherical planet using perturbation theory and comesfrom the inertia terms in the equation of motion. The fractional dynamical correction is of order ∆ k ∼ ω / π G ¯ ρ ∼ . ρ is the mean density of the planet and ω the forcing frequency related to the tide. The theoretical hydrostaticnumber k ( hs )2 includes the effect of the oblateness of the planet, a realistic equation of state, and a density profileconsistent with the zonal gravitational moments J and J . From assuming that the number k ( hs )2 is perfectly known,we aim to evaluate how the gravitational contribution of dynamical tides perturbs k ( hs )2 .Instead of adding dynamical effects into the already complicated numerical model used to calculate k ( hs )2 , we useperturbation theory to isolate the dynamical effects in a much simpler interior model defined by an n = 1 polytropicequation of state. The n = 1 polytrope p = Kρ closely follows the equation of state of a H-He mixture (Stevenson2020) and is chosen for computational simplicity but is not crucial to the ∆ k calculation. The density distributionin a non-rotating n = 1 polytrope is ρ = ρ c j ( kr ). The central density ρ c is set to fit Jupiter’s total mass; j isthe zero-order spherical Bessel function of second kind; k = 2 π G /K is a normalizing constant for the radius, where K = 2 . · (cgs) for a H/He cosmic ratio; G is the gravitational constant; r is the radial coordinate. Exploitingthe compact equation of state and density profile of the n = 1 polytrope, we calculate the dynamical Love numberby accounting for the dynamical terms in the equation of motion (Section 3.2.2). The fractional dynamical correction∆ k comes from comparing the hydrostatic and dynamical Love number in the polytrope; however, as the correctionis expressed in fractional terms, ∆ k calculated this way introduces dynamical effects into any hydrostatic model, toleading order approximation. The hydrostatic Love number in a spherical planet following an n = 1 polytrope isobtained analytically (Fig. 1; Appendix A). For example, the degree-2 hydrostatic Love number of a spherical planetfollowing an n = 1 polytrope is k = 15 /π − ≈ . k ( hs )2 = 0 .
590 produce little effect on the dynamical tide and can be ignored fornow.The uncertainty O (1%) in equation (1) is an order-of-magnitude estimate of the neglected cross term that accountsfor the effect of the centrifugal effect on dynamical tides. Individually, the centrifugal effect and the dynamical effectare both of order ∼ ∼ ω and 2Ω that matters, with or without oblateness.The remainder of this manuscript deals with the calculation of the dynamical Love number.2.2. Equations of tides in an adiabatic gas giant planet
To calculate the Love number k , we require to compute the tidal gravitational potential φ (cid:48) on Jupiter. The Lovenumber k represents the ratio of φ (cid:48) over the gravitational pull of the satellite φ T evaluated at the outer boundary of Idini & Stevenson
Figure 1.
Angular patterns and radial functions describing the 3-D structure of the gravitational field of tides. The radialfunctions are the normalized hydrostatic gravitational potential | φ | in an n = 1 polytrope (A9). The sign of the hydrostatic (cid:96) − tide is ( − (cid:96)/ . the planet: k (cid:96),m = (cid:32) φ (cid:48) (cid:96),m φ T(cid:96),m (cid:33) r = R p . (2)At a distance r from the center of a planet of radius R p , the potential φ T from a satellite orbiting in a circular orbitaligned with the equatorial plane of the planet is: φ T = (cid:88) (cid:96) =2 (cid:96) (cid:88) m = − (cid:96) U (cid:96),m (cid:18) rR p (cid:19) (cid:96) Y m(cid:96) ( θ, ϕ ) e iωt , (3)where U (cid:96),m are a numerical constants and Y m(cid:96) normalized spherical harmonics (Appendix A). The tidal frequency ω = | m (Ω − ω s ) | represents the frequency of a standing wave as observed from the perspective of an observer rotatingwith the planet at spin rate Ω, where ω s is the orbital frequency of the satellite and m the order of the tide. We follow upiter dynamical tides ω is always positive and retrograde tides are represented by a negative order m that flips thecoordinate frame. The simplifications applied to the orbit are consistent to first order with the observed eccentricities e < .
01 and inclinations i < . ◦ of the Galilean satellites.We calculate the tidal response of a rigidly-rotating planet from a problem defined by the linearly-perturbed mo-mentum, continuity, and Poisson’s equations, respectively: − iω v + 2 Ω × v = − ∇ p (cid:48) ρ + ρ (cid:48) ρ ∇ p + ∇ ˜ φ (cid:48) , (4) ∇ · ( ρ v ) = iωρ (cid:48) , (5) ∇ ˜ φ (cid:48) = − π G ρ (cid:48) . (6)The tidal response of the planet produces adiabatic perturbations to the gravitational potential φ (cid:48) , the density profile ρ (cid:48) , and pressure p (cid:48) . The potential of the gravitational pull φ T and the tidal gravitational potential φ (cid:48) are combinedinto ˜ φ (cid:48) = φ T + φ (cid:48) for analytical simplicity. Adiabatic perturbations in an adiabatic planet follow the thermodynamicstatement (e.g., Wu (2005)): p (cid:48) p = Γ ρ (cid:48) ρ = c s ρ (cid:48) p , (7)where Γ is the first adiabatic index (Aerts et al. 2010). Unprimed pressure and density represent the unperturbedstate of the planet in hydrostatic equilibrium. We rewrite the momentum equation of an adiabatic planet as: − iω v + 2 Ω × v = ∇ (cid:18) ˜ φ (cid:48) − p (cid:48) ρ (cid:19) = ∇ (cid:18) ˜ φ (cid:48) − c s ρ (cid:48) ρ (cid:19) = ∇ ψ . (8)In equation (8), hydrostatic tides follow ψ = 0 (Appendix A).The tidal flow becomes a function of the potential ψ after operating the divergence and curl on the momentumequation (Wu 2005; Goodman & Lackner 2009): v = − iω − ω (cid:18) ∇ ψ + 2 iω Ω × ∇ ψ − ω Ω ( Ω · ∇ ψ ) (cid:19) . (9)After replacing the flow into the continuity equation, the governing equations of tides in an adiabatic planet reduceto: ∇ · (cid:18) ρ (cid:18) ∇ ψ + 2 iω Ω × ∇ ψ − ω Ω ( Ω · ∇ ψ ) (cid:19)(cid:19) = (cid:18) − ω π G (cid:19) ∇ ˜ φ (cid:48) , (10) ψ = c s π G ρ ∇ ˜ φ (cid:48) + ˜ φ (cid:48) . (11)Perturbation theory allows us to decouple the weakly coupled potentials ψ and ˜ φ (cid:48) in equations (10) and (11).According to perturbation theory, the tidal gravitational potential splits into a static and dynamic part: φ (cid:48) = φ + φ dyn , (12)where φ corresponds to the gravitational potential of the hydrostatic tide after solving equation (11) with ψ = 0(Fig. 1; Appendix A). By definition, the sound speed in an n = 1 polytrope follows c s = 2 Kρ , which reduces thehydrostatic version of equation (11) to: ∇ φ k + φ + φ T = 0. (13)As a good approximation, we ignore the contribution from dynamical tides to the potential ψ by setting ∇ ˜ φ (cid:48) ≈ ∇ φ in the right-hand side in equation (10). According to equation (10), the potentials satisfy ψ ∼ φ (cid:48) ω / π G ¯ ρ , which leadsto φ (cid:48) (cid:29) ψ given a tidal frequency ω (cid:28) π G ¯ ρ . By continuity, the approximation means that the tidal flow mostlyadvects the mass in the hydrostatic tidal bulge (i.e., ρ (cid:48) ≈ ρ and equation (6)). The decoupled tidal equations simplyto: ∇ · (cid:18) j ( kr ) (cid:18) ∇ ψ + 2 iω Ω × ∇ ψ − ω Ω ( Ω · ∇ ψ ) (cid:19)(cid:19) = (cid:18) − ω π G ρ c (cid:19) ∇ φ , (14) Idini & Stevenson ψ = ∇ φ dyn k + φ dyn . (15)We obtain the dynamical gravitational potential φ dyn first solving ψ from equation (14) and then using the result tocalculate φ dyn from equation (15).The boundary condition at the center of the planet imposes a finite solution for both potentials, allowing us to discardthe divergent term characteristic of problems that include the Laplace operator. As required for a free planetaryboundary, the condition at the outer boundary sets the Lagrangian perturbation of pressure equal to zero (e.g.,Goodman & Lackner (2009)): v r = v · ˆ n = − iω p (cid:48) ∂ r p = − iωρ (cid:48) c s ρg , (16)or: ˆ n · ∇ ψ + 2 iω ˆ n · ( Ω × ∇ ψ ) − ω (ˆ n · Ω )( Ω · ∇ ψ ) = − (cid:18) − ω π G (cid:19) c s ρg ∇ ˜ φ (cid:48) = − (cid:18) − ω g (cid:19) ( ψ − ˜ φ (cid:48) ), (17)where v r is the radial component of the tidal flow, ˆ n is a unitary vector normal to the outer boundary, and g thegravitational acceleration at the outer boundary. The sound speed and density nearly vanish near the outer boundaryof a compressible body, resulting in a finite radial flow (e.g., v r = − Kωρ (cid:48) /g in an n = 1 polytrope). The outerboundary condition (17) indicates ψ/R J ∼ ( ˜ φ (cid:48) − ψ ) ω /g . The Jupiter-Io system prescribes 1 /R J (cid:29) ω /g , which leadsto ˜ φ (cid:48) (cid:29) ψ and simplifies equation (17) into:ˆ n · ∇ ψ + 2 iω ˆ n · ( Ω × ∇ ψ ) − ω (ˆ n · Ω )( Ω · ∇ ψ ) = (cid:18) − ω g (cid:19) ( φ + φ T ). (18)Also at the outer boundary, the gravitational potential should be continuous in amplitude and gradient with a gravi-tational potential external to the planet that decays as r − ( l +1) .In the following section, we solve the tidal equations for a uniform-density model and an n = 1 polytrope model,with and without the Coriolis effect. DYNAMICAL TIDES IN A GAS GIANT PLANETFollowing our simplified model of dynamical tides, we calculate ∆ k in a coreless, chemically homogeneous, andadiabatic Jupiter-like model. The thermal state becomes almost adiabatic in a convecting fluid planet with homoge-neous composition. From the point of view of tidal calculations, the deviation from adiabaticity is negligible in theinterior because the superadiabaticity required to sustain convection is a tiny fraction of the adiabatic temperaturegradient, despite the possible inhibitions arising from rotation and convection. A fluid parcel in an adiabatic interiorthat is adiabatically displaced by a tidal perturbation will find itself at a new state that is essentially unchanged indensity and temperature from the unperturbed state at that pressure. This definition of neutral stability begins tobreak down near the photosphere, where the density is low and the radiative time constant is no longer huge for blobswith spatial dimension of order the scale height. However, that region represents only a tiny fraction of the planet anddoes not produce enough gravity to significantly alter the real part of the Love number k . We discuss hypotheticalcontributions to k from a core and depth-varying chemical composition in Section 4.As a matter of simplifying the arguments presented in this section, we mostly concentrate on the Love number at (cid:96) = m = 2, commonly known as k . Correspondingly, k is forced by the degree-2 component in the gravitational pull: φ T = 316 G m s a r sin θe − i ( ωt +2 ϕ ) . (19)Dynamical effects scale with the satellite-dependent ω . We concentrate in the dynamical effects caused by Io, theGalilean satellite with the dominant gravitational pull on Jupiter.3.1. A non-rotating gas giant
To an excellent approximation, dynamical tides in a non-rotating planet represent the forced response of the planetin the fundamental normal mode of oscillation (f-mode) (Vorontsov et al. 1984). Despite Cassini suggesting that upiter dynamical tides ∝ r (cid:96) ). 3.1.1. The harmonic oscillator analogy
In the following, we use the forced harmonic oscillator as an analog model to tidally-forced f-modes. In this model,the fractional dynamical correction to k acquires a simple analytical form. The equation of motion of a mass M connected in harmonic motion to a spring of stiffness K and negligible dissipation is: − M ω u + K u = F T , (20)where ω is the forcing frequency and F T the tidal forcing. F-modes oscillate at frequencies ω that are much higher thanthe forcing tidal frequency, meaning that tidal resonances with f-modes are highly unlikely. Assuming that dynamicaleffects are small so that the tidal forcing is mostly balanced by static effects (i.e., F T ≈ K u s ), the displacement of themass is: u = u s (cid:18) ω ω − ω (cid:19) . (21)The mass assumes the static equilibrium position u s as the forcing frequency tends to zero. The displacement u isanalogous to the Love number k , thus the fractional dynamical correction becomes:∆ k = u − u s u s = ω ω − ω . (22)3.1.2. The Coriolis-free n = 1 polytrope To verify the analogy of the forced harmonic oscillator to tidally-forced f-modes, we calculate the tidal response ofa non-rotating n = 1 polytrope directly from the governing equations of tides. When Ω = 0, the governing equation(14) reduces to: j ( kr ) ∇ ψ − j ( kr ) ∂ r ( ψ ) = − (cid:18) ω π G ρ c (cid:19) ∇ φ . (23)For the potential ψ at (cid:96) = m = 2, the boundary condition at the outer boundary r = R p (17) is: gω ∂ r ψ − ψ = − j ( kR p ). (24)For the same degree and order, the continuity of the gravitational potential and its gradient at the outer boundaryrequires: ∂ r φ dyn = − (cid:32) φ dyn + 5 R p (cid:33) . (25)At the center of the planet r = r →
0, we find the following scaling: ∇ φ ∼ j ( kr ) ∼
0, and j ( kr ) ∼ constant.A finite potential ψ satisfying equation (23) is ψ ∼ r near r , or: ∂ r ψ − r ψ = 0. (26)Similarly, a finite gravitational potential of dynamical tides is φ dyn ∼ r at the center of the planet, satisfying: ∂ r φ dyn − r φ dyn = 0. (27)We compute the fractional dynamical correction to k first projecting the tidal equations into spherical harmonics(Appendix B) and later solving for the relevant potentials using a Chebyshev pseudo-spectral numerical method Idini & Stevenson
Table 1.
Io-induced fractional dynamical correc-tion ∆ k in a Coriolis-free Jupiter.Harmonic oscillator n = 1 polytropeType (%) (%)(1) (2) (3)∆ k +15 +13∆ k +5 +5∆ k +2 +2∆ k +19 +15∆ k +25 +19 Note —(2) See equation (22). The mode fre-quency without rotation comes from Vorontsovet al. (1976). (Appendix C). After projecting equations (23) and (15) into spherical harmonics, we obtain two decoupled equationsfor the radial parts of the potentials ψ and φ (Appendix B.1). After numerically solving the radial equations inAppendix B.1 using Io’s gravitational pull ω s ≈ µ Hz), the fractional dynamical correction corresponds to ∆ k ≈ . ω ≈ µ Hz (Vorontsov et al. 1976). We observe a similar agreement between the harmonic oscillator andthe Coriolis-free n = 1 polytrope at higher-degree spherical harmonics (Table 1). Our results agree with a previouslyreported fractional correction to the gravitational coefficient C , ∝ k due to dynamical tides in a non-rotating Jupiter.(Vorontsov et al. 1984).When Vorontsov et al. (1984) excluded Jupiter’s spin, they were doing something that was mathematically sensiblebut physically peculiar: tides occur much more frequently in Jupiter’s rotating frame of reference and the tidal flow isaccordingly much larger than if you had Jupiter at rest, which implies a much larger dynamical effect. Consequently,∆ k increases by an order of magnitude after partially including Jupiter’s rotation in the tidally-forced response off-modes. Without the Coriolis effect but including Jupiter’s spin rate (Ω ≈ µ Hz) in the calculation of Io’s tidalfrequency ( ω ≈ µ Hz), the fractional dynamical correction in an n = 1 polytrope corresponds to ∆ k ≈ k ≈
15% from the forced harmonic oscillator analogy (22). In general for a non-rotating planet, thedynamical correction increases as the tidal frequency approaches the characteristic frequency of Jupiter’s f-modes( (cid:112) G M J /R J ∼ µ Hz). 3.2.
The Coriolis effect in a rotating gas giant
The Galilean satellites produce dynamical tides for which the Coriolis effect plays an important role. Followingrelatively slow orbits ( ω s (cid:28) Ω), the Galilean satellites produce tides on Jupiter with a tidal frequency ω ∼ k to be less than the predicted number for a purely hydrostatic tide(Section 2.1) and yet our analysis above produces a positive ∆ k when dynamical effects are included and the Corioliseffect neglected (see equation (22)). We must accordingly motivate the change in sign when Coriolis is included.In the following, we first calculate the gravitational effect of dynamical tides in a uniform-density sphere to revealthe fundamental behaviour of the tidal equations avoiding most of the technical difficulties related to using an n = 1polytrope. We later found that the more complicated case of an n = 1 polytrope introduces a minor quantitativedifference, but lead to the same general behaviour.3.2.1. A uniform-density sphere
First, we explain why ∆ k changes sign because of the Coriolis effect in a specially simple model with uniformdensity. We calculate the fractional dynamical correction to k in two steps: (1) we calculate the potential of the flow ψ in a uniform-density sphere, and (2) we use the ψ calculated this way to calculate the gravity potential φ (cid:48) . upiter dynamical tides c s is infinite and (10) reduces to the well-known Poincare problem(Greenspan et al. 1968): ∇ ψ − ω ( Ω · ∇ ) ψ = 0, (28)where the boundary condition at the outer boundary requires to satisfy equation (18).Following the incompressibility of a uniform-density sphere, ψ retains the symmetry and degree-2 angular structurefrom the gravitational pull in equation (19), thus acquiring exact solutions in the form: ψ ∝ ( x − iy ) . (29)The numerical factor in ψ is set by the outer boundary condition (17), corresponding to (Goodman & Lackner 2009): ψ = 3 ω (2Ω − ω )8 π G ¯ ρ ˜ φ (cid:48) = R p ω (2Ω − ω )2 g ˜ φ (cid:48) . (30)In a constant-density sphere, tides act displacing the sphere’s boundary within an infinitesimally thin shell. Accordingto the momentum equation, the tidal gravitational potential relates to the potential ψ following:˜ φ (cid:48) − ψ − p (cid:48) ¯ ρ = 0. (31)The radial tidal displacement projected into spherical harmonics is: ξ = (cid:88) (cid:96),m ξ (cid:96),m ( r ) Y m(cid:96) ( θ, ϕ ), (32)and the pressure perturbation follows: p (cid:48) = − ξ ∂p∂r = ξ ¯ ρg . (33)The gravitational potential of a thin spherical density perturbation follows directly from the definition of the gravita-tional potential and integration throughout the volume: φ (cid:48) = (cid:88) (cid:96),m π G ¯ ρ (2 (cid:96) + 1) R (cid:96) +2 p r (cid:96) +1 ξ (cid:96),m Y m(cid:96) . (34)The degree-2 tidal gravitational potential corresponds to: φ (cid:48) = 4 π G ¯ ρ R p r ξ = 35 (cid:18) R p r (cid:19) gξ . (35)We once again use perturbation theory to split the hydrostatic and dynamic contributions to the tidal displacement (i.e., ξ = ξ + ξ dyn ). We first solve the well-known problem of the hydrostatic k (i.e., ψ = 0) in a uniform-density sphere(Love 1909). At the sphere’s boundary (i.e., r = R p ), the hydrostatic gravitational potential follows φ = 3 gξ / r = R p , the potential of the gravitational pull becomes φ T = 2 gξ /
5. Following thelast two results, the Love number is k = 3 /
2, as expected.The dynamical contribution to the tidal displacement ξ dyn produces the gravitational potential φ dyn = 3 gξ dyn / ψ becomes ψ = − gξ dyn /
5. Combined with equation (30), the last result for ψ allows us to reach an expression for the fractionaldynamical correction in a uniform-density sphere:∆ k = ξ dyn ξ ≈ − (cid:18) R p g (cid:19) ω (2Ω − ω ). (36)Two effects contribute to the fractional dynamical correction: a negative contribution from the Coriolis effect ∝ ω/π G ¯ ρ and a positive contribution from the dynamical amplification of f-modes ∝ ω /π G ¯ ρ . The two contributionscancel each other at 2Ω = ω , where the tide achieves hydrostatic equilibrium. Tides become hydrostatic not only when0 Idini & Stevenson
Io EuropaGanymedeCallisto
Figure 2.
Fractional dynamical correction ∆ k in a rotating uniform-density sphere including the Coriolis effect as a functionof tidal frequency (see equation (36)). the planetary spin is phase-locked with the orbit of the satellite (Ω = ω s ), but also in planet-satellite systems wherethe central body is rotating at a rate orders of magnitude much faster than the orbit of the satellite (i.e., Ω (cid:29) ω s ). Asthe frequency of the degree-2 f-mode approximately follows ω ∼ g/R p , ∆ k in equation (36) approximately becomesthe positive fractional correction determined in Section 3.1 after setting Ω = 0.At the degree-2 Io-induced tidal frequency, the fractional dynamical correction corresponds to ∆ k ≈ − . k because their tidal frequency falls closer to hydrostatic equilibrium (Fig.2). A negative ∆ k works in the direction required by the non-hydrostatic component identified by Juno in Jupiter’sgravity field (Section 2.1).The direction of the flow provides an explanation for the negative sign of the fractional dynamical correction viathe Coriolis acceleration. By definition, a uniform-density sphere has no density perturbations in its interior, thusproduces an interior tidal gravitational potential that satisfies φ (cid:48) ∝ r (cid:96) Y m(cid:96) ∝ φ T . We adopt equation (30) as thedegree-2 potential ψ and obtain analytical solutions for the cartesian components of the resulting degree-2 tidal flow upiter dynamical tides v = − AωR p g (ˆ x ( ix + y ) + ˆ y ( x − iy )) , (37)where A is a constant depending on ξ (Appendix D). The degree-2 tidal flow purely exist in equatorial planes, showingno vertical component of motion (Fig. 3b). c.a. Non-rotating f-mode acceleration Tidal flow Coriolis acceleration b. Figure 3.
Degree-2 ( (cid:96) = m = 2) tidal perturbations on a uniform-density sphere forced by the gravitational pull of a companionsatellite: (a) the non-rotating f-mode acceleration − ω ξ , (b) tidal flow as shown in equation (37), and (c) Coriolis acceleration Ω × v according to the right-hand rule. The Coriolis acceleration plays a major role in setting the sign of the fractional dynamical correction for the Galileansatellites. Without Coriolis, the acceleration of non-rotating f-modes sustains a positive dynamical tidal displacementthat follows ξ dyn ≈ R p ω ξ / g . A ξ dyn > k .Conversely, as shown in equation (36), the fractional dynamical correction flips sign when Coriolis promotes ξ dyn < ξ dyn . According to the right-hand rule, the Coriolis acceleration (i.e., Ω × v , Fig. 3c) opposes the direction of the acceleration of non-rotating f-modes (i.e., − ω ξ , Fig. 3a). The resultinggravitational field is smaller than the hydrostatic field if ω < The n = 1 polytrope In the following, we consider the more relevant case of a compressible planet that follows an n = 1 polytropicequation of state (14). In contrast to the localized tidal perturbation of a uniform-density sphere, a compressible bodyyields a tidally-induced density anomaly that arises from advection of the isodensity surfaces within the body. Theresulting tidal gravitational potential is different in each case owing to differences in the tidally-perturbed densitydistribution obtained in a uniform-density sphere and a compressible body.Despite the aforementioned difference between models, the tidal flow remains similar so that dynamical tides motivatea negative correction to k in each case. In an n = 1 polytrope, the continuity equation (5) tells us that the degree-2 radial component of the flow takes the form v r ∝ j ( kr ) /j ( kr ) when the flow has small divergence, as it does.Remarkably, the dominant contribution to the Taylor series expansion of v r is linear in r , even out to a large fractionof the planetary radius. In a uniform-density sphere, the potential ψ is ψ ∝ r , which leads to a tidal flow thatfollows v ∝ ∇ ψ ; therefore, v r is also linear in r in this model. As shown, the dominant contribution to v r scaleswith radius as ∝ r , both in an n = 1 polytrope and in a uniform-density sphere. In an n = 1 polytrope, the dominantcontribution to v r is curl-free and divergence-free and provides the ψ ∝ r part of the solution to the potential ψ (Fig. 4a). Since the n = 1 polytrope also contains terms where ψ is higher order in r , it produces a flow with non-zerocurl and non-zero divergence, causing ψ to depart from ψ ∝ r . Because high-order terms in r are smaller thanthe dominant term, dynamical effects on k in a uniform-density sphere are qualitatively similar to those in an n = 1polytrope.We compute the fractional dynamical correction to k in a rotating polytrope following the same strategy used inSection 3.1.2. In opposition to the Coriolis-free polytrope, solving equation (14) is technically challenging due to the (cid:96) − coupling of the potential ψ (cid:96),m (e.g., mode mixing) promoted by the Coriolis effect. Mode mixing is also foundin hydrostatic tides over a planet distorted by the effect of the centrifugal force (Wahl et al. 2017a). The result of2 Idini & Stevenson projecting equation (14) into spherical harmonics is an infinite (cid:96) − coupled set of ordinary differential equations for ψ (cid:96),m (Appendix B.2), similarly observed in the problem of dissipative dynamical tides (Ogilvie & Lin 2004). TheCoriolis-promoted (cid:96) − coupling comes from the sine and cosine in the spin rate of the planet ( Ω / Ω = ˆ r cos θ − ˆ θ sin θ ),which changes the degree of the spherical harmonics related to ψ . As a consequence, a given spherical harmonic from φ in the right-hand side in equation (14) forces multiple spherical harmonics of the potential ψ with different (cid:96) .Projected into spherical coordinates, the boundary condition (17) at r = R p corresponds to: ∂ r ψ − iωR p ∂ ϕ ψ − ω (cid:18) cos θ∂ r ψ − sin θ cos θ ∂ θ ψR p (cid:19) = (cid:18) − ω g (cid:19) ( φ + φ T ). (38)The outer boundary condition is also (cid:96) − coupled after projected into spherical harmonics (Appendix B.2). Figure 4.
Radial functions in an n = 1 polytrope (thick blue and orange curves) of the (a) potential ψ and the (b) dynamicalgravitational potential φ dyn . The thinner black curves in (a) represent the radial scaling of the potential ψ in a uniform-densitysphere. At the center of the planet r = r →
0, we find the following scaling: ∇ φ ∼ j ( kr ) ∼ constant. As a result,the tidal equation (14) becomes the previously solved problem of the potential ψ in a uniform-density sphere (28) nearthe center. Requiring to be finite near the center and to satisfy equation (28), the radial part of the potential ψ follows ψ (cid:96),m ∼ r (cid:96) . The boundary condition for ψ (cid:96),m near the center corresponds to: ∂ r ψ (cid:96),m − (cid:96)r ψ (cid:96),m = 0. (39)The equation for the gravitational potential of dynamical tides φ dyn remains unchanged compared to the Coriolis-free polytrope (Appendix B.1). The outer and inner boundary conditions for the gravitational potential generalize indegree as: ∂ r φ dyn(cid:96),m = − (cid:32) ( (cid:96) + 1) φ dyn(cid:96),m + (2 (cid:96) + 1) U (cid:96),m R p (cid:33) , (40) ∂ r φ dyn(cid:96),m − (cid:96)r φ dyn(cid:96),m = 0. (41)By projecting ψ (cid:96),m and φ (cid:96),m into a series of N Chebyshev polynomials oriented in the radial component (Ap-pendix C), we numerically solve (B24) and (B14) truncating the infinite series of (cid:96) − coupled equations at an arbitrary (cid:96) = L max . We choose a truncation limit L max = 50 and the number of Chebyshev polynomials N max = 100 based onnumerical evidence of convergence for k and k . upiter dynamical tides Table 2.
Jupiter Love numbers.Hydrostatic Juno PJ17-3 σ σ fractional difference ∆ k (rotating n = 1 polytrope)Type Number Number (%) (%)Io Europa Ganymede Callisto(1) (2) (3) (4) (5) (6) (7) (8) k ± -4 -2 -1 -1 k ± +7 +8 +10 +12 k ± +1 +3 +4 +5 k ± +2 +5 +7 +8 k ± +7 +11 +13 +15 Note —(2) The hydrostatic number is from Wahl et al. (2020). (3) The Juno PJ17-1 σ number is thesatellite-independent number from (Durante et al. 2020). (4) The 3 σ fractional difference representsthe minimal/maximal 3 σ non-hydrostratic fractional correction required to explain Juno observations.The fractional dynamical correction in (5-8) is valid for an n = 1 polytrope forced by the gravitationalpull of the Galilean satellites. We obtain ∆ k = − .
0% at the degree-2 Io-induced tidal frequency (Table 2), which is of slightly lower amplitudethan the estimate in a uniform-density sphere and in agreement with the k non-hydrostatic component observedby Juno at PJ17. However different models, both the uniform-density sphere and the polytrope produce fractionaldynamical corrections that fall within the order-of-magnitude estimate ∆ k ∼ ω / π G ρ ∼ .
1. As argued before, thedominant contribution to the potential ψ follows the radial scaling ψ (cid:96) ∝ r (cid:96) (Fig. 4a). Ignoring the sign, the radialscaling of the dynamical gravitational potential φ dyn (Fig. 4b) closely follows the shape of the hydrostatic gravitationalpotential (Fig. 1).Due to the essentially circular and equatorial geometry of the Galilean orbits, the spherical harmonic (cid:96) = m = 2dominates Jupiter’s tidal gravitational field. Consequently, we concentrate in comparing k Juno observation to ourmodel prediction. Of significantly higher uncertainty, the mid-mission Juno report of Love numbers at perijove 17include other spherical harmonics in addition to k (Table 2). Our polytropic model predicts an Io-induced tidalgravitational field in a 3 σ -agreement with most Love numbers observed at PJ17, save for k and k .3.2.3. Detection of dynamical tides in systems other than Jupiter-Io.
A detection of dynamical tides via direct measurement of the gravitational field will be challenging in bodies otherthan Jupiter (Fig. 5). The 1 σ uncertainty in the gravitational field of degree-2 Io tides is projected to be σ J ∼ · − m /s at the end of the proposed Juno extended mission (William Folkner, personal communication, April 8, 2020).The uncertainty in the measured tidal gravity field depends on the number and design of spacecraft orbits, theuncertainty in ephemerides, and instrumental capabilities. Assuming the uncertainty σ J , we roughly estimate thegravitational pull required to produce a detectable dynamical component in the gravity field using: φ T ( r = R ) (cid:38) σ J k | ∆ k | . (42)Our calculation indicates that detecting dynamical tides in Saturn will require a mission with a more precise deter-mination of the gravity field than that obtained by Juno (Fig. 5). A 1 σ detection of Europa-induced dynamical tidesseems plausible at the end of Juno’s extended mission, assuming that the factors determining the uncertainty in thegravity field remain similar to those of Io. We calculate a model prediction for the satellite-dependent Jupiter Lovenumber for all the Galilean satellites (Table 2). We obtain k = 0 .
578 in the case of Europa, a prediction testable bythe recently approved Juno extended mission. DISCUSSION4
Idini & Stevenson
TitanRheaDioneTethysEnceladusMimas
IoCallisto
GanymedeEuropa
Figure 5.
Conditions for the detection of dynamical tides evaluated for the Galilean satellites (black) and inner Saturnsatellites (white). Satellites to the right of the dashed line have favorable conditions for a detection of dynamical tides assumingan uncertainty roughly similar to that of Io’s k on Jupiter at the end of Juno’s extended mission. The fractional dynamicalcorrection ∆ k is for an n = 1 polytrope. Future updates to Juno Love number observations.
The discrepancy between our predicted k , k , and the Juno-PJ17 observations may allude to several reasons: (1)a suggestion to revise the hydrostatic (cid:96) − coupled k (cid:96),m in Wahl et al. (2020), (2) a failure of perturbation theory in ourmodel when accounting for the (cid:96) − coupled k (cid:96),m , (3) other physical reasons; for example, tidal resonance with normalmodes or the neglected correction from a dilute core. We strongly suggest a thorough analysis of these possibilities infuture investigations. Ultimately, the perijove passes required to complete the scheduled Juno mission may change thestill highly-uncertain numbers reported in Table 2. A recent revision to Juno observations at PJ29 (Daniele Durante,personal communication, November 18, 2020) suggest an agreement of our k prediction with the revised satellite-independent k = 0 . ± .
016 (1 σ ). The PJ29-revised k = 1 . ± .
095 (1 σ ) remains in disagreement with our k prediction but the difference is much narrower than that attained at PJ17.A disagreement between our predicted k (cid:96),m and high-degree Juno observations does not impair the much morerelevant agreement observed for k . Compared to the amplitude of the tidal gravitational potential related to k , upiter dynamical tides k represents an order-of-magnitude smaller contribution to the tidalgravitational field due to the factor R p /a ∼ / φ T(cid:96),m . In addition, whereas the predicted k simply depends onthe contribution from the Coriolis effect and the dynamical response of f-modes, the more complicated predicted k additionally depends on the numerical solution of the (cid:96) − coupled system of equations described in Section B.2.Ignoring for the moment other possibilities related to the k discrepancy, resonant tides have been previously invokedas a potential candidate to explain the current structure of the Laplace resonance in Saturn (Fuller et al. 2016; Laineyet al. 2020). As planets in the Solar System rotate far from break-up, there is no overlap between the frequenciesof tides and f-modes in adiabatic, non-rotating planets. However, compositional gradients (g–modes) and rotation(inertial modes) introduce additional normal modes whose frequencies can become close to the tidal frequency, eitherby chance or planetary evolution. These hypothetically resonant tides could produce high dissipation rates; thus, adetectable imaginary part in the Love number that would consequently induce a significant change in the real part ofthe Love number. Equivalently, the high dissipation rate from an hypothetically resonant tide would cause a phasebetween the gravitational pull and the degree-2 tidal bulge. However, degree-2 tidal dissipation in Jupiter due to Iotides is modest (Lainey et al. 2009). This argument does not necessarily apply for higher-degree tides ( (cid:96) >
2) thathave much smaller amplitudes and therefore whose phase shifts would be much harder to detect.We compare our Io-induced fractional dynamical corrections to the Juno-PJ17 satellite-independent observations.We justify the use of the satellite-independent k uncertainty because our results indicate small variations in the Lovenumber due to dynamical effects (Table 2), assuming the absence of degree-2 tidal resonances. In the hypotheticalof an (cid:96) = 4, m = 2 tidal resonance, the Love number k would vary significantly among satellites. In such case,we would require to use a satellite-dependent uncertainty (Durante et al. 2020), in which no a priori information isused at the time of inferring the Love number. So far, we neither confirm or deny resonant tides that may be havingan impact on k or k . An improved version of the satellite-independent k uncertainty could be obtained a prioriassuming that the Love number increases ∼
4% outward when comparing the inner to the outer satellites. A strongerconclusion on the possibility of tidal resonances observed in Juno data requires additional progress in the mission toreduce the uncertainty on k and k , plus a thorough analysis of resonances with Jupiter interior models that includea compositional gradient.A tighter constraint on the satellite-dependent k from satellites other than Io will test the prediction of our model ofsatellite-dependent dynamical tides. Despite the relatively large fractional dynamical correction obtained for the innerSaturn satellites (e.g. Mimas or Enceladus), their small mass leads to an overall small tidal disturbance that is difficultto detect in the gravity field. From all Jupiter and Saturn satellites, only Europa elevates a short-term prospect ofobtaining a new detection of dynamical tides via Juno’s extended mission (Fig. 5). A detection of dynamical tides dueto other satellites will require an uncertainty on k significantly lower than that produced by Juno.4.2. Other potential contributions to k Free-oscillating normal modes
Free oscillations of normal modes cannot explain the bulk of the non-hydrostatic Juno detection discussed here. Thesmall gravitational field of tides becomes resolvable by Juno in part because the phase of the signal is well known. Theunknown phase of non-resonant free oscillations departs from the phase of the satellite used in determining k . Evenif freely-oscillating normal modes were detected in Jupiter as they were in Saturn (Iess et al, 2019), the anticipatedhigh frequency of their gravity field would render them irrelevant to the tidal problem. In order that free oscillationsplay a role in the observed gravity, they require to avoid a rapid decay after becoming excited (i.e., a very high Q ). Itis not known whether free oscillations persist over multiple Juno perijove passes.4.2.2. Jupiter’s rheology
A central viscoelastic region in Jupiter’s interior could potentially reduce k below the hydrostatic number; however,evidence suggest that such possibility is unlikely. Viscoelastic deformation of a body produces a k between the purelyelastic and the hydrostatic number; a model that helps to explain Titan’s observed k (Iess et al. 2012). Assuming thata traditional core in Jupiter exists, the core radius should remain small (i.e., ∼ . R J ) to satisfy the constraint onthe total abundance of heavy elements and the super-solar enrichment of the envelope (Wahl et al. 2020). At this coreradius, the tidal deformation of the core does not contribute to Re( k ) (Storch & Lai 2014). Whether rigid, elastic,or viscoelastic, a small traditional core produces a small effect on Re( k ) due to the added heavy elements, alreadyincluded in the hydrostatic number (Wahl et al. 2020). Beyond the possibility of a viscoelastic traditional core, the6 Idini & Stevenson hydrogen-rich envelope most likely behaves as an inviscid fluid. The kinematic viscosity of the fluid external to thecore needs to reach ν ∼ m /s in order for its viscosity to become relevant at tidal timescales (i.e, ω ∼ ν/R J ).Such kinematic viscosity exceeds by ∼
17 orders of magnitude realistic estimates of the hydrogen-dominated fluidviscosity (Stevenson & Salpeter 1977). A similar argument applies to a dilute core, which most likely consist of amixture dominated by hydrogen, either by atomic number and even probably by mass.4.2.3.
The dynamical contribution of a traditional core
A traditional core blocks the tidal flow from extending to the center of the planet by forcing a zero-flow boundarycondition at the core radius. As shown earlier, the radial tidal flow sets the amplitude of the tidal gravitationalpotential and roughly scales with distance from the center following v r ∝ r in an n = 1 polytrope. Consequently, thetidal flow is nearly zero in the area where a traditional core would exist, minimizing a potential effect of the traditionalcore on the fractional dynamical correction. A thorough quantification of ∆ k in a model with a traditional core thatblocks the flow requires further investigation. We expect an effect going from negligible to small (i.e., less than +1%applied to the current estimate in Table 2) given the limits to traditional core size imposed by the constrained totalabundance of heavy elements. 4.2.4. A dilute core
A dilute core may promote an additional departure of the tidal response from the hydrostatic tide to that causedby dynamical tides. The hydrostatic tide in k provides the same information about the planet than J (Hubbard1984). In the presence of a dilute core, the gravity produced by tides fundamentally differ from the J (cid:96) coefficientsdue to the different timescales associated to tidal perturbations and evolution of the rotation rate. Tidal timescalesare short compared to the timescale required for the tidal perturbation to equilibrate with the environment by eitherheat transport or compositional evolution. By contrast, the timescale at which the rotation rate evolves is so longthat the planet adjust any perturbation caused by the centrifugal effect. Tidal displacements remain roughly adiabaticwhereas displacements induced by changes in the rotation rate reach thermodynamic equilibrium. A fluid parcel inthe proximity of the dilute core responds differently depending on the timescale of the perturbation; only an adiabaticperturbation leads to changes in the buoyancy of the fluid parcel, causing a wave-like oscillation known as staticstability. Consequently, the dilute core produces a signature in the tidal response of the planet not registered by J .We address the tidal effects of a dilute core in a subsequent investigation. CONCLUSIONSOur tidal models suggest that the gravity field observed by Juno captured the dynamical tidal response of Jupiterto the gravitational pull of the Galilean satellites. We show that two effects contribute to the dynamical gravityfield of tides in Jupiter: the dynamical response of f-modes and the Coriolis acceleration. When the Coriolis effect isignored, tides closely follow the dynamical response of f-modes modeled as a forced harmonic oscillator. In ignoringthe Coriolis effect, dynamical amplification in a harmonic oscillator accounts for the dynamical response of f-modes inthe planet’s interior, forced by the gravitational pull of the companion satellites. As the tidal frequency is lower thanthe f-mode oscillation frequency, the dynamical response of f-modes amplifies the gravity field of the hydrostatic tide.Motivated by Jupiter’s fast rotation, we show that the Coriolis effect leads to a significant additional contribution tothe dynamical tide. When the Coriolis effect is included in our tidal models, we show that the Coriolis accelerationproduces a competing effect of opposite sign compared to the dynamical response of f-modes. When both dynamicaleffects are considered together, dynamical effects reduce the Love number k below the hydrostatic number if ω < k in the case of the Galilean satellites, which degree-2 tidal frequency is ω < k = − k . The uncertainty expected in the observed k at the end of the mission exceeds the uncertainty achieved byour model, suggesting that a more detailed tidal model will be required in the future to fully exploit the informationcontained in the data provided by Juno. upiter dynamical tides Software:
Matplotlib (Hunter 2007) APPENDIX A. HYDROSTATIC TIDES IN AN INDEX-ONE POLYTROPEIn hydrostatic tides, the tidal frequency becomes ω ≈ v ≈
0. Afterprojecting φ into spherical harmonics by setting a solution in the form: φ = (cid:88) (cid:96),m φ (cid:96),m ( x ) Y (cid:96)m ( θ, ϕ ) e − iωt , (A1)the radial part of φ at a given harmonic that satisfies equation (13) follows: ∂ x,x φ (cid:96) + 2 x ∂ x φ (cid:96) + (cid:18) − (cid:96) ( (cid:96) + 1) x (cid:19) φ (cid:96) = − (cid:16) xπ (cid:17) (cid:96) , (A2)where Y m(cid:96) are normalized spherical harmonics defined by: Y m(cid:96) ( θ, ϕ ) = (cid:115) (2 (cid:96) + 1)4 π ( (cid:96) − m )!( (cid:96) + m )! P m(cid:96) (cos θ ) e imϕ , (A3)and P m(cid:96) Associated Legendre polynomials corresponding to: P m(cid:96) ( µ ) = ( − m (cid:96) l ! (1 − µ ) m/ d (cid:96) + m dµ (cid:96) + m ( µ − (cid:96) . (A4)The normalized radial coordinate follows x = kr , which leads to a planet with radius π . Note that equation (A2) isnon-dimensional and should be scaled by the factor: U (cid:96),m = (cid:18) G m s a (cid:19) (cid:18) R p a (cid:19) (cid:96) (cid:18) π ( (cid:96) − m )!(2 (cid:96) + 1)( (cid:96) + m )! (cid:19) / P ml (0). (A5)The order m does not appear in equation (A2), indicating a degeneracy on m of the hydrostatic tide. As x (cid:96) Y m(cid:96) is asolution to Laplace’s equation (i.e., ∇ ( x (cid:96) Y m(cid:96) ) = 0), a complete solution to equation (A2) is: φ (cid:96) = Aj (cid:96) ( x ) + Bn (cid:96) ( x ) − (cid:16) xπ (cid:17) (cid:96) . (A6)We require φ to be finite at the center of the planet and thus set B = 0. According to the outer boundary condition,we set a external gravitational potential Φ (cid:96) ( x ) that extends outward from the planet and matches the internal tidalpotential at the planetary radius as:Φ (cid:96) ( x ) = (cid:16) πx (cid:17) (cid:96) +1 φ (cid:96) ( π ) = (cid:16) πx (cid:17) (cid:96) +1 ( A (cid:96) j (cid:96) ( π ) − A (cid:96) to: A (cid:96) = 2 (cid:96) + 1 πj (cid:96) − ( π ) . (A8)8 Idini & Stevenson
Consequently, the gravitational potential of hydrostatic tides at degree (cid:96) is: φ (cid:96) = (cid:18) (cid:96) + 1 π (cid:19) j (cid:96) ( x ) j (cid:96) − ( π ) − (cid:16) xπ (cid:17) (cid:96) , (A9)and the hydrostatic Love number follows: k (cid:96) = (cid:18) (cid:96) + 1 π (cid:19) j (cid:96) ( π ) j (cid:96) − ( π ) −
1. (A10) B. PROJECTION OF THE DYNAMICAL TIDE EQUATIONS INTO SPHERICAL HARMONICSHere we project into spherical harmonics the equation for the potential ψ in a non-rotating (23) and rotating (14) n = 1 polytrope. The equation for the gravitational potential of dynamical tides is equation (15), forced by a differentpotential ψ depending on rotation. We evaluate solutions in the form: ψ = (cid:88) (cid:96),m ψ (cid:96),m ( x ) Y m(cid:96) ( θ, ϕ ) e − iωt , (B11) φ dyn = (cid:88) (cid:96),m φ dyn(cid:96),m ( x ) Y m(cid:96) ( θ, ϕ ) e − iωt . (B12)In the following, we conveniently drop the time dependent part e iωt out of our derivation. Notice that we normalizethe radial coordinate following x = kr , leading to a body of normalized radius π = kR p .B.1. The Coriolis-free n = 1 polytrope We project into spherical harmonics the potential ψ (23) and the dynamical gravitational potential (15) of thenon-rotating polytrope: j ( x ) (cid:18) ∂ x,x ( ψ (cid:96),m ) + (cid:18) x − j ( x ) j ( x ) (cid:19) ∂ x ( ψ (cid:96),m ) + (cid:18) − (cid:96) ( (cid:96) + 1) x (cid:19) ψ (cid:96),m (cid:19) = (cid:18) (cid:96) + 1 πj (cid:96) − ( π ) (cid:19) ω j (cid:96) ( x )4 π G ρ c , (B13) ∂ x,x (cid:16) φ dyn(cid:96),m (cid:17) + 2 x ∂ x (cid:16) φ dyn(cid:96),m (cid:17) + (cid:18) − (cid:96) ( (cid:96) + 1) x (cid:19) φ dyn(cid:96),m = ψ (cid:96),m . (B14)B.2. The n = 1 polytrope Relative to the left-hand side in equation (14), the projection of the first, second, and third terms, respectivelyfollow: ∇ · ( j ∇ ψ (cid:96),m ) = j (cid:18) ∂ x,x + (cid:18) x − j j (cid:19) ∂ x − (cid:96) ( (cid:96) + 1) x (cid:19) ψ (cid:96),m Y m(cid:96) , (B15)2 iω ∇ · ( j Ω × ∇ ψ lm ) = 2 m Ω j ωx ψ (cid:96),m Y m(cid:96) , (B16) − ω ∇ · ( j Ω ( Ω · ∇ ψ (cid:96),m )) = − ω (cid:18)(cid:18) j ∂ x,x − (cid:18) j + j x (cid:19) ∂ x (cid:19) ψ (cid:96),m Y m(cid:96) cos θ + j x ∂ x ψ (cid:96),m Y m(cid:96) + (cid:18) j x + j x − j x ∂ x (cid:19) ψ (cid:96),m cos θ sin θ∂ θ Y m(cid:96) + j x ψ (cid:96),m sin θ∂ θ,θ Y m(cid:96) (cid:19) . (B17)The multiplication of spherical harmonics with trigonometric functions expresses a physical statement about thecoupling effect that Coriolis produces in the tidal gravitational response of a rotating body. The partial derivatives inthe spherical harmonics indicate changes in quantum numbers described in the following differential relations (Lockitch& Friedman 1999): sin θ∂ θ Y m(cid:96) = (cid:96)Q (cid:96) +1 Y m(cid:96) +1 − ( (cid:96) + 1) Q (cid:96) Y m(cid:96) − , (B18)cos θY m(cid:96) = Q (cid:96) +1 Y m(cid:96) +1 + Q (cid:96) Y m(cid:96) − , (B19) upiter dynamical tides Q (cid:96) = (cid:18) (cid:96) − m (cid:96) − (cid:19) / . (B20)Combining the previous differential relations, we arrive to expressions for each of the angular terms in equation (B17): Y m(cid:96) cos θ = Q (cid:96) − Q (cid:96) Y m(cid:96) − + ( Q (cid:96) + Q (cid:96) +1 ) Y m(cid:96) + Q (cid:96) +1 Q (cid:96) +2 Y m(cid:96) +2 , (B21)cos θ sin θ∂ θ Y m(cid:96) = − ( (cid:96) + 1) Q (cid:96) − Q (cid:96) Y m(cid:96) − − (( (cid:96) + 1) Q (cid:96) − (cid:96)Q (cid:96) +1 ) Y m(cid:96) + (cid:96)Q (cid:96) +1 Q (cid:96) +2 Y m(cid:96) +2 , (B22)sin θ∂ θ,θ Y m(cid:96) = ( (cid:96) + 1) Q (cid:96) − Q (cid:96) Y m(cid:96) − + ((2 + (cid:96) − (cid:96) ) Q (cid:96) − (cid:96) ( (cid:96) + 3) Q (cid:96) +1 ) Y m(cid:96) + (cid:96) Q (cid:96) +1 Q (cid:96) +2 Y m(cid:96) +2 . (B23)After grouping terms with the same spherical harmonic, equation (14) becomes an infinite set of (cid:96) -coupled radialequations following the structure of a Sturn-Liouville problem: (cid:16) ( P (1) (cid:96),m ∂ x,x + Q (1) (cid:96),m ∂ x + R (1) (cid:96),m (cid:17) ψ (cid:96),m + (cid:16) P (2) (cid:96),m ∂ x,x + Q (2) (cid:96),m ∂ x + R (2) (cid:96),m (cid:17) ψ l +2 ,m + (cid:16) P (0) (cid:96),m ∂ x,x + Q (0) (cid:96),m ∂ x + R (0) (cid:96),m (cid:17) ψ l − ,m = U (cid:96),m (cid:18) ω − π G ρ c (cid:19) (cid:18) (cid:96) + 1 π (cid:19) j (cid:96) ( x ) j (cid:96) − ( π ) . (B24)The nine radial coefficients correspond to: P (0) (cid:96),m = − ω j Q (cid:96) − Q (cid:96) , (B25) P (1) (cid:96),m = j (cid:18) − ω ( Q (cid:96) + Q (cid:96) +1 ) (cid:19) , (B26) P (2) (cid:96),m = − ω j Q (cid:96) +1 Q (cid:96) +2 , (B27) Q (0) (cid:96),m = 4Ω ω (cid:18) (2 (cid:96) − j x + j (cid:19) Q (cid:96) − Q (cid:96) , (B28) Q (1) (cid:96),m = 2 j x − j − ω (cid:18) j x (cid:0) (cid:96) + 1)( Q (cid:96) − Q (cid:96) +1 ) (cid:1) − j ( Q (cid:96) + Q (cid:96) +1 ) (cid:19) , (B29) Q (2) (cid:96),m = − ω (cid:18) (2 (cid:96) − j x − j (cid:19) Q (cid:96) +1 Q (cid:96) +2 , (B30) R (0) (cid:96),m = − ω ( (cid:96) − (cid:18) (cid:96) j x + j x (cid:19) Q (cid:96) − Q (cid:96) , (B31) R (1) (cid:96),m = − l ( l + 1) j x + 2 m Ω j ωx + 4Ω ω (cid:18) j x (( (cid:96) + 1) Q (cid:96) − (cid:96)Q (cid:96) +1 ) + j x (cid:96) ( (cid:96) + 1)( Q (cid:96) + Q (cid:96) +1 ) (cid:19) , (B32) R (2) (cid:96),m = − ω (cid:18) ( (cid:96) ( (cid:96) + 4) + 11) j x − ( (cid:96) − j x (cid:19) Q (cid:96) +1 Q (cid:96) +2 . (B33)The projection into spherical harmonics of the boundary condition (38) leads to: Y m(cid:96) (cid:18) ∂ x ψ (cid:96),m − m Ω ωπ ψ (cid:96),m (cid:19) − ω (cid:18) Y m(cid:96) cos θ∂ x ψ (cid:96),m − ψ (cid:96),m π sin θ cos θ∂ θ Y m(cid:96) (cid:19) = U (cid:96),m (cid:18) − ω g (cid:19) (cid:18) (cid:96) + 1 π (cid:19) j (cid:96) ( x ) j (cid:96) − ( π ) Y m(cid:96) . (B34)The previous differential relations still apply to deal with coupled spherical harmonics in the boundary condition.Grouping terms for each spherical harmonic Y m(cid:96) , we reach an (cid:96) − coupled boundary condition with the structure: (cid:88) j =0 (cid:16) ˆ Q ( j ) (cid:96),m ∂ x + ˆ R ( j ) (cid:96),m (cid:17) ψ m(cid:96) +2 j − = U (cid:96),m (cid:18) − ω g (cid:19) (cid:18) (cid:96) + 1 π (cid:19) j (cid:96) ( π ) j (cid:96) − ( π ) . (B35)0 Idini & Stevenson
The six radial coefficients correspond to: ˆ Q (0) (cid:96),m = − ω Q (cid:96) − Q (cid:96) , (B36)ˆ Q (1) (cid:96),m = 1 − ω (cid:0) Q (cid:96) + Q (cid:96) +1 (cid:1) , (B37)ˆ Q (2) (cid:96),m = − ω Q (cid:96) +1 Q (cid:96) +2 , (B38)ˆ R (0) (cid:96),m = 4Ω πω ( (cid:96) − Q (cid:96) − Q (cid:96) , (B39)ˆ R (1) (cid:96),m = − m Ω πω − πω (cid:0) ( (cid:96) + 1) Q (cid:96) − (cid:96)Q (cid:96) +1 (cid:1) , (B40)ˆ R (2) (cid:96),m = − πω ( (cid:96) − Q (cid:96) +1 Q (cid:96) +2 . (B41) C. CHEBYSHEV PSEUDO-SPECTRAL METHODWe solve the Sturn-Liouville differential problem (Boyd 2001) defined by: p ( r ) u (cid:48)(cid:48) ( r ) + q ( r ) u (cid:48) ( r ) + r ( r ) u ( r ) = f ( r ), (C42)and constrained to the boundary conditions: α u (cid:48) ( a ) + α u ( a ) = α , (C43) β u (cid:48) ( b ) + β u ( b ) = β , (C44)where a and b are the two ends of a boundary value problem. We shift the domain of equation (C42) from r ∈ [ a, b ]to the domain of Chebyshev polynomials µ ∈ [ − ,
1] and seek for a solution that is a truncated sum of an infiniteChebyshev series: u ( µ ) ≈ N max (cid:88) n =0 a n T n ( µ ), (C45)with the Chebyshev polynomials defined by: T n ( µ ) = cos( nt ), (C46)and t = arccos( µ ). Our objective is to obtain the coefficients a n by solving a linear inverse problem: La = f . (C47)The square matrix L and the vector f come from the evaluation of equation (C42) into Gauss-Lobatto collocationpoints defined by: µ i = cos (cid:18) πiN − (cid:19) , i = 1 , , . . . , N −