Dynamical Tides in Jupiter As Revealed by Juno

Dynamical Tides in Jupiter as Revealed by Juno

Benjamin Idini and David J. Stevenson Division of Geological and Planetary Sciences, California Institute of Technology, 1200 E. California Boulevard, MC 150-21, Pasadena, CA 91125, USA [email protected] Received 2020 December 4; revised 2021 February 2; accepted 2021 February 15; published 2021 April 6

Abstract The Juno orbiter has continued to collect data on Jupiterʼs gravity field with unprecedented precision since 2016, recently reporting a nonhydrostatic component in the tidal response of the planet. At the mid-mission perijove 17, Juno registered a Love number k2 = 0.565 ± 0.006 that is −4% ± 1% (1σ) from the theoretical hydrostatic ()hs k2 = 0.590. Here we assess whether the aforementioned departure of tides from hydrostatic equilibrium represents the neglected gravitational contribution of dynamical tides. We employ perturbation theory and simple tidal models to calculate a fractional dynamical correction Δk2 to the well-known hydrostatic k2. Exploiting the analytical simplicity of a toy uniform-density model, we show how the Coriolis acceleration motivates the negative sign in the Δk2 observed by Juno. By simplifying Jupiter’s interior into a coreless, fully convective, and chemically homogeneous body, we calculate Δk2 in a model following an n = 1 polytrope equation of state. Our numerical results for the n = 1 polytrope qualitatively follow the behavior of the uniform-density model, mostly because the main component of the tidal flow is similar in each case. Our results indicate that the gravitational effect of the Io- induced dynamical tide leads to Δk2 = − 4% ± 1%, in agreement with the nonhydrostatic component reported by Juno. Consequently, our results suggest that Juno obtained the first unambiguous detection of the gravitational effect of dynamical tides in a gas giant planet. These results facilitate a future interpretation of Juno tidal gravity data with the purpose of elucidating the existence of a dilute core in Jupiter. Unified Astronomy Thesaurus concepts: Tides (1702); Jupiter (873); Gravitational fields (667); Galilean satellites (627); Planetary interior (1248)

1. Introduction coefficients contain Jupiter’s response to the centrifugal effect responsible for its oblateness, with minor contributions from zonal The interior structure of a planet or star closely corresponds winds in the atmosphere (Iess et al. 2018) and the dynamo region with its origin and evolution history. Seismology provides the (Kulowski et al. 2020). A time-dependent subset of Cℓ tightest constraints on the interior structure of Earth (Dahlen & ,m coefficients contains Jupiter’s tidal response to the gravitational Tromp 1998 and references therein),Saturn(Marley & Porco pull from its system of satellites. Closely related to the time- 1993; Fuller 2014),theSun(Christensen-Dalsgaard et al. 1985), dependent Cℓ , the Love number kℓ represent the nondimen- and other distant stars (Aerts et al. 2010 and references therein).In ,m ,m sional gravitational field of tides evaluated at the outer boundary particular, Saturnʼs ring seismology facilitates estimates of the of the planet (Munk & MacDonald 1960). One common planetʼs rotation rate (Mankovich et al. 2019) and possible dilute k core (Mankovich 2020). Unlike Saturn, Jupiter lacks extensive interpretation relates to the degree of central concentration of the planetary mass (e.g., compressibility of the planetary material optically thick rings with embedded waves that are excited by ) resonance of ring particle motions with internal normal modes. As or the presence of a core . In a quadrupolar gravitational pull, the leading term in the tidal response relates to the Love number k2, an alternative to ring seismology, Doppler imaging reveals a ℓ = = suggested seismic behavior in Jupiter, limited to radial overtones which corresponds to the m 2 spherical harmonic. of p-modes. At best, the current Doppler imaging data resolve the Following linear perturbation theory, the Love number k2 spacing in frequency space of low-order p-modes, providing a breaks down into two contributions, one hydrostatic and the ( other dynamic. The hydrostatic k2 ignores the time dependence loose constraint compatible with simple interior models Gaulme fl et al. 2011). Future efforts based on similar techniques promise to of tides. The dynamical tide represents the tidal ow and ’ perturbed tidal bulge solving the traditional equation of motion reveal additional information on Jupiter s seismic behavior. T T = T In the current absence of detailed seismological constraints, the FMu= ,̈ rather than F 0, where F is the satellite-induced Juno orbiter (Bolton et al. 2017) emergesasthealternative tidal force, diminished by the opposing self-gravity of the directed to reveal Jupiter’s interior by employing gravity field perturbed planet. measurements of global-scale motions. Based on radiometric Early studies of the Love numbers in the gas giant planets observations, Juno produces two kinds of gravity field measure- relied on purely hydrostatic theory, assisted by simple thermodynamic principles alongside loosely constrained interior models ments sensitive to Jupiter’s interior structure: the zonal Jℓ and fi fi fl (Gavrilov & Zharkov 1977). With the assistance of historical tesseral Cℓ,m gravity coef cients. The odd J2ℓ+1 coef cients re ect fi contributions from zonal flows, including atmospheric zonal astrometric data, the Cassini mission provided the rst occasion to test the accuracy of those first hydrostatic models, finding an winds and zonal flow in the dynamo region. The even J2ℓ ∼10% discrepancy between the theoretical and observed k2 of Saturn (Lainey et al. 2017). The discrepancy was solved after Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further incorporating the oblateness produced by the centrifugal effect distribution of this work must maintain attribution to the author(s) and the title into the tidal model, not requiring to invoke the gravitational of the work, journal citation and DOI. effects of Saturn dynamical tides. The oblateness produced by the

1 The Planetary Science Journal, 2:69 (15pp), 2021 April Idini & Stevenson centrifugal effect is large due to Saturn’s fast rotation, leading to 2. Jupiter’s Love Number major higher-order cross-terms neglected in the earlier theory 2.1. A Correction to the Hydrostatic k (Wahl et al. 2017a). This effect remains hydrostatic provided that 2 rotation occurs on cylinders, which allows the centrifugal force to The main objective of this paper is to evaluate the hypothesis be represented as the gradient of a potential. that Juno captured a systematic deviation of k2 from the Unlike Cassini, the Juno orbiter recently detected a 3σ hydrostatic number and that most of the deviation can be deviation in Jupiter’sobservedk from the revised hydrostatic explained by the neglected gravitational effect of dynamical tides. 2 ( ) theory that accounts for the interaction of tides with The mean k2 Juno estimate at the time of perijove 17 PJ17 is ( ) 0.565 (Durante et al. 2020), establishing a −4% deviation from oblateness Notaro et al. 2019; Durante et al. 2020 . ()hs ( the theoretical hydrostatic number k2 = 0.590 Wahl et al. Importantly, the difference between the observed k2 and 2020). The correction to the hydrostatic k2 due to the rotational hydrostatic theory cannot be attributed to a failure to correctly ()hs 2 3 bulge is included in k2 and is of order qRM=WJ  J ~0.1, constrain the hydrostatic number. Hydrostatic tides are well ( ( ± the ratio of centrifugal effects to gravity at the equator Wahl et al. constrained i.e., the k2 error in Wahl et al. 2020 is 0.02% of ) ’ ) 2020 ; RJ is the equatorial radius, and MJ is Jupiter s mass. The the central value because the effect of the oblateness of the satellite-independent 3σ uncertainty (confidence level ≈99.7%) in fi planet on the zonal gravity coef cients J2ℓ is known to a high Juno’s observation is 3% of k ()hs at PJ17 (Durante et al. 2020), ’ 2 precision by Juno. Juno s nonhydrostatic detection motivates close to the mean of the observed deviation. At the end of the a more careful consideration of neglected effects that prime mission, the satellite-independent 3σ uncertainty in Juno’s fi contribute to k2, particularly the gravitational eld related to observation is projected to decrease to 1% (W. Folkner 2021, dynamical tides. personal communication, 2020 April 8). Following the optimistic Here we evaluate dynamical tides as a potential explanation for assumption that the mean deviation remains the same at the end of ʼ ()hs Juno s nonhydrostatic detection. We concentrate on the contrib- the prime mission, k2 will require a nonhydrostatic correction ution of dynamical tides to the overall gravity field while ignoring from −5% to −3% to be reconciled with 3σ observations. their contribution to dissipation. In other words, we implicitly In our tidal models, we evaluate a dynamical correction to assume that the imaginary part of k2 is too small to affect our the hydrostatic k2 as results for the real part, in agreement with observations of the k orbital evolution of Io (Q =-∣∣kkIm () ~ 105 ; Lainey et al. 2 =+D+11%,1k () () 222 ()hs 2 2009). k2 We are motivated in our efforts by the prospect of finding where Δk2 is the fractional dynamical correction calculated for an additional contribution to k2 coming from Jupiter’score.A traditional model of a Jupiter-like planet consists of a discreet, a spherical planet using perturbation theory and comes from highly concentrated central region of heavy elements (i.e., a the inertia terms in the equation of motion. The fractional 2 core made of rock and ice) surrounded by a chemically dynamical correction is of order Dk2 ~~wpr40.1¯ , where homogeneous and adiabatic envelope of hydrogen-rich fluid r¯ is the mean density of the planet and ω is the forcing material (Stevenson 1982).Jupiter’s observed radius indicates frequency related to the tide. The theoretical hydrostatic ()hs a supersolar abundance of heavy elements that accounts for a number k2 includes the effect of the oblateness of the planet, fi total of ∼20 ME. However, whether heavy elements reside in a realistic equation of state, and a density pro le consistent a traditional core or are distributed throughout the envelope is with the zonal gravitational moments J2 and J4. From assuming ()hs less clear. The observed J2 and J4 require some tendency that the number k2 is perfectly known, we aim to evaluate toward central concentration but do not require a traditional how the gravitational contribution of dynamical tides per- ()hs core. In a subsequent investigation, we exploit our results of turbs k2 . the dynamical tide presented here to answer questions about Instead of adding dynamical effects into the already ()hs the origin and evolution of Jupiter by including in our complicated numerical model used to calculate k2 , we use model an enrichment of heavy elements that increases with perturbation theory to isolate the dynamical effects in a much depth (i.e., a dilute core; Wahl et al. 2017b).Usingthe simpler interior model defined by an n = 1 polytropic equation 2 information contained in k ,weplantoprovideanswerstothe of state. The n = 1 polytrope p = Kρ closely follows the 2 – ( ) following questions about Jupiter. Whether solid or fluid, does equation of state of an H He mixture Stevenson 2020 and is chosen for computational simplicity but is not crucial to the Jupiter have a traditional core? Alternatively, do the heavy Δ = elements in Jupiter spread out from the center, forming a k2 calculation. The density distribution in a nonrotating n 1 polytrope is ρ = ρ j (kr). The central density ρ is set to fit dilute core? c 0 c Jupiter’s total mass; j is the zero-order spherical Bessel The remainder of this paper is organized as follows. In 0 function of the second kind; k2 = 2p K is a normalizing Section 2, we describe the Juno nonhydrostatic detection and constant for the radius, where K = 2.1 × 1012 (cgs) for an develop the mathematical formalism used in the calculation of H/He cosmic ratio;  is the gravitational constant; and r is the fractional dynamical correction to k2. In Section 3,we the radial coordinate. Exploiting the compact equation of state calculate the fractional dynamical correction to k2 using simple and density profile of the n = 1 polytrope, we calculate the tidal models, which leads to an unambiguous explanation of the dynamical Love number by accounting for the dynamical terms nonhydrostatic Juno detection. In Section 4, we deliver a in the equation of motion (Section 3.2.2). The fractional discussion on the limitations of our analysis, other physical dynamical correction Δk2 comes from comparing the hydro- processes potentially altering the Love number k, and future static and dynamical Love number in the polytrope; however, directions of investigation. In Section 5, we outline the as the correction is expressed in fractional terms, Δk2 conclusions and implications of our study. calculated this way introduces dynamical effects into any

2 The Planetary Science Journal, 2:69 (15pp), 2021 April Idini & Stevenson with the equatorial plane of the planet is

ℓ ⎛ ⎞ℓ T ⎜ r ⎟ m itw fqj= ååUℓm, Yeℓ (),, () 3 ℓmℓ==-2 ⎝ Rp ⎠ m where Uℓ,m are numerical constants and Yℓ are normalized spherical harmonics (Appendix A). The tidal frequency ω = |m(Ω − ωs)| represents the frequency of a standing wave as observed from the perspective of an observer rotating with the planet at a spin rate Ω, where ωs is the orbital frequency of the satellite and m is the order of the tide. We follow the convention where ω is always positive and retrograde tides are represented by a negative order m that flips the coordinate frame. The simplifications applied to the orbit are consistent to first order with the observed eccentricities e < 0.01 and inclinations i < 0°.5 of the Galilean satellites. We calculate the tidal response of a rigidly rotating planet from a problem defined by the linearly perturbed momentum, continuity, and Poisson equations, respectively, ¢p r¢ -+iwvv2,4W ´=- + +¢p f˜ () r r2 Figure 1. Angular patterns and radial functions describing the 3D structure of the gravitational field of tides. The radial functions are the normalized =¢·(rwrv )i ,5 ( ) |f0| = ( ( )) hydrostatic gravitational potential in an n 1 polytrope Equation A9 . 2 ˜ The sign of the hydrostatic ℓ-tide is (−1)ℓ/2+1. ¢=-fpr4. ¢ () 6 The tidal response of the planet produces adiabatic perturbations hydrostatic model, to leading-order approximation. The hydro- to the gravitational potential f¢, density profile r¢, and pressure p¢. static Love number in a spherical planet following an n = 1 The potential of the gravitational pull fT and the tidal gravitational polytrope is obtained analytically (Figure 1; Appendix A). For potential f¢ are combined into f˜ ¢=ffT + ¢ for analytical example, the degree-2 hydrostatic Love number of a spherical simplicity. Adiabatic perturbations in an adiabatic planet follow the 2 ( ) planet following an n = 1 polytrope is k2 = 15/π − 1 ≈ 0.520. thermodynamic statement e.g., Wu 2005 Stated explicitly, our approach assumes that the realistic ()hs p¢ r¢ 2 r¢ elements included in the calculation of k2 = 0.590 produce =G1 = cs ,7() little effect on the dynamical tide and can be ignored for now. p r p ( ) The uncertainty ()1% in Equation 1 is an order-of- where Γ is the first adiabatic index (Aerts et al. 2010). magnitude estimate of the neglected cross-term that accounts 1 Unprimed pressure and density represent the unperturbed state for the centrifugal effect on dynamical tides. Individually, the of the planet in hydrostatic equilibrium. We rewrite the centrifugal and dynamical effects are both of order ∼10%. In linear perturbation theory, the cross-term is roughly the momentum equation of an adiabatic planet as ∼ multiplication of the individual terms, resulting in 1%. The ⎛ p¢⎞ uncertainty may be smaller as the dynamical correction -+iwfvv2W ´=¢-⎜ ˜ ⎟ ⎝ r ⎠ involves terms that tend to cancel each other. This cancellation ()8 is expected to remain when an account is made of the ⎛ ⎞ ˜ 2 r¢ oblateness of the planet, since (as our analysis shows below) it =⎜f ¢-cs ⎟ =y. ⎝ r ⎠ is the small difference between ω and 2Ω that matters, with or without oblateness. In Equation (8), hydrostatic tides follow ψ = 0 (Appendix A). The remainder of this paper deals with the calculation of the The tidal flow becomes a function of the potential ψ after dynamical Love number. operating the divergence and curl on the momentum equation (Wu 2005; Goodman & Lackner 2009): 2.2. Equations of Tides in an Adiabatic Gas Giant Planet iw ⎛ 24⎞ To calculate the Love number k, we must compute the tidal v =- ⎜⎟+y WWW ´-y (· y ). 22⎝ 2 ⎠ gravitational potential f¢ on Jupiter. The Love number k 4W-w iw w represents the ratio of f¢ over the gravitational pull of the ()9 f T satellite evaluated at the outer boundary of the planet: After replacing the flow into the continuity equation, the ⎛ ⎞ governing equations of tides in an adiabatic planet reduce to ⎜ f¢ℓm, ⎟ kℓm, = ⎜ ⎟ .2() ⎛ ⎛ 24⎞⎞ fT ⎜ ⎜⎟⎟ ⎝ ℓm, ⎠ +´-·(·)ry⎝ WWWy y ⎠ rR= p ⎝ iw w2 ⎠ ()10 ⎛ 4W-22w ⎞ At a distance r from the center of a planet of radius Rp, the =⎜ ⎟¢2f˜ , potential f T from a satellite orbiting in a circular orbit aligned ⎝ 4p ⎠

3 The Planetary Science Journal, 2:69 (15pp), 2021 April Idini & Stevenson 2 polytrope). The outer boundary condition (Equation (17)) cs 2 y =¢+¢ff˜˜.11() ˜ 2 – 4pr indicates y RgJ ~¢-(fyw) . The Jupiter Io system pre- scribes 1/R ? ω2/g, which leads to f˜ ¢  y and simplifies Perturbation theory allows us to decouple the weakly J Equation (17) into coupled potentials ψ and f˜ ¢ in Equations (10) and (11). According to perturbation theory, the tidal gravitational 24 nˆ · +y nnˆ ·(WWW ´-y ) (ˆ ·)(· y ) potential splits into a static and dynamic part, iw w2 ()18 fff¢=0 + dyn,12() ⎛ 4W-22w ⎞ =⎜ ⎟()ff0 + T . where f0 corresponds to the gravitational potential of the ⎝ g ⎠ hydrostatic tide after solving Equation (11) with ψ = 0 Also at the outer boundary, the gravitational potential should be (Figure 1; Appendix A).Bydefinition, the sound speed in an continuous in amplitude and gradient with a gravitational = 2 n 1 polytrope follows cs = 2Kr, which reduces the hydro- −( l+1) ( ) potential external to the planet that decays as r . static version of Equation 11 to In the following section, we solve the tidal equations for a 20f uniform-density model and an n = 1 polytrope model with and ++ff0 T =0.() 13 k2 without the Coriolis effect. As a good approximation, we ignore the contribution from 3. Dynamical Tides in a Gas Giant Planet dynamical tides to the potential ψ by setting ¢»220ff˜ on Following our simplified model of dynamical tides, we the right-hand side in Equation (10). According to calculate Δk in a coreless, chemically homogeneous, and ( ) 2 2 Equation 10 , the potentials satisfy y ~¢fw4 pr¯, which adiabatic Jupiter-like model. The thermal state becomes almost 2 leads to f¢  y given a tidal frequency w  4pr¯.By adiabatic in a convecting fluid planet with homogeneous continuity, the approximation means that the tidal flow mostly composition. From the point of view of tidal calculations, the advects the mass in the hydrostatic tidal bulge (i.e., r¢»r0 and deviation from adiabaticity is negligible in the interior because Equation (6)). The decoupled tidal equations simplify to the superadiabaticity required to sustain convection is a tiny fraction of the adiabatic temperature gradient, despite the ⎛ ⎛ 24⎞⎞ ⎜ ⎜⎟⎟ possible inhibitions arising from rotation and convection. A +´-·()jkr0 y WWWy (·)y ⎝ ⎝ iw w2 ⎠⎠ fluid parcel in an adiabatic interior that is adiabatically ⎛ ⎞ ()14 fi 4W-22w displaced by a tidal perturbation will nd itself in a new state =⎜ ⎟20f , that is essentially unchanged in density and temperature from ⎝ ⎠ 4pr c the unperturbed state at that pressure. This definition of neutral 2f dyn stability begins to break down near the photosphere, where the y = + fdyn.15() 2 density is low and the radiative time constant is no longer huge k for blobs with spatial dimension of order the scale height. We obtain the dynamical gravitational potential fdyn by first However, that region represents only a tiny fraction of the solving ψ from Equation (14) and then using the result to planet and does not produce enough gravity to significantly calculate fdyn from Equation (15). alter the real part of the Love number k. We discuss The boundary condition at the center of the planet imposes a hypothetical contributions to k from a core and depth-varying finite solution for both potentials, allowing us to discard the chemical composition in Section 4. divergent term characteristic of problems that include the As a matter of simplifying the arguments presented in this Laplace operator. As required for a free planetary boundary, the section, we mostly concentrate on the Love number at ℓ = = condition at the outer boundary sets the Lagrangian perturba- m 2, commonly known as k2. Correspondingly, k2 is tion of pressure equal to zero (e.g., Goodman & Lackner 2009), forced by the degree-2 component in the gravitational pull:

2 T 3 ms 22-+it()wj 2 p¢ cs fq= resin .() 19 vnir ==-v · ˆ wwr=-i ¢ ,16() 2 16 a3 ¶rp rg Dynamical effects scale with the satellite-dependent ω.We or concentrate on the dynamical effects caused by Io, the Galilean 24 nˆ · +y nnˆ ·(WWW ´-y ) (ˆ ·)(· y ) satellite with the dominant gravitational pull on Jupiter. iw w2 ⎛ 22⎞ 2 3.1. A Nonrotating Gas Giant 4W-w cs 2 ˜ =-⎜ ⎟ ¢f ()17 ⎝ 4pr ⎠ g To an excellent approximation, dynamical tides in a nonrotating planet represent the forced response of the planet ⎛ 4W-22w ⎞ =-⎜ ⎟(yf-¢˜ ), in the fundamental normal mode of oscillation ( f-mode; ⎝ g ⎠ Vorontsov et al. 1984). Despite Cassini suggesting that ( ) fl higher-order normal-mode overtones p-modes dominate the where vr is the radial component of the tidal ow, nˆ is a unitary gravitational field of Saturn’s free-oscillating normal modes vector normal to the outer boundary, and g is the gravitational (Markham et al. 2020), the forced response of normal modes acceleration at the outer boundary. The sound speed and density depends on the coupling of the gravitational pull and the mode nearly vanish near the outer boundary of a compressible body, radial eigenfunction. The forced response of the p-modes resulting in a finite radial flow (e.g., vKr =-2 wr ¢ g in an n = 1 contributes negligibly to the gravitational field of dynamical

4 The Planetary Science Journal, 2:69 (15pp), 2021 April Idini & Stevenson tides (Vorontsov et al. 1984) because of the bad coupling Table 1 between the zero-node radial component of the gravitational Io-induced Fractional Dynamical Correction Δk in a Coriolis-free Jupiter ’ pull and the p-modes eigenfunctions, the latter having one or Harmonic Oscillator n = 1 Polytrope more radial nodes. Conversely, the gravitational pull more Type (%)(%) efficiently excites f-modes, whose radial eigenfunctions (1)(2)(3) roughly follow the radial scaling of the gravitational pull ℓ Δk +15 +13 (∝ r ). 2 Δk42 +5 +5 Δk31 +2 +2 3.1.1. The Harmonic Oscillator Analogy Δk33 +19 +15 Δ + + In the following, we use the forced harmonic oscillator as k44 25 19 f an analog model to tidally forced -modes. In this model, ( ) ( ) the fractional dynamical correction to k acquires a simple Note. 2 See Equation 22 . The mode frequency without rotation comes from 2 Vorontsov et al. (1976). analytical form. The equation of motion of a mass M connected in harmonic motion to a spring of stiffness  and negligible 2 dissipation is potential ψ satisfying Equation (23) is ψ2 ∼ r near r0,or -+=Muw2  u FT ,20() 2 ¶-ryy2 2 =0.() 26 r0 where ω is the forcing frequency and F T is the tidal forcing. fi The f-modes oscillate at frequencies ω0 that are much higher Similarly, a nite gravitational potential of the dynamical tides dyn 2 than the forcing tidal frequency, meaning that tidal resonances is f2 ~ r at the center of the planet, satisfying with f-modes are highly unlikely. Assuming that the dynamical dyn 2 dyn effects are small so that the tidal forcing is mostly balanced by ¶-rff2 2 =0.() 27 T r0 static effects (i.e., Fu»  s), the displacement of the mass is We compute the fractional dynamical correction to k2 by first ⎛ 2 ⎞ w0 projecting the tidal equations into spherical harmonics uu= s⎜ ⎟.21() ( ) ⎝ ww2 - 2 ⎠ Appendix B and later solving for the relevant potentials 0 using a Chebyshev pseudospectral numerical method (Appendix C). After projecting Equations (15) and (23) into The mass assumes the static equilibrium position us as the forcing frequency tends to zero. The displacement u is spherical harmonics, we obtain two decoupled equations for the radial parts of the potentials ψ and f (Appendix B.1). After analogous to the Love number k; thus, the fractional dynamical numerically solving the radial equations in Appendix B.1 using correction becomes Io’s gravitational pull (ωs ≈ 42 μHz) the fractional dynamical Δ ≈ uu- w2 correction corresponds to k2 1.2%, in close agreement with D=k s = .22() 2 2 the forced harmonic oscillator analogy applied to the oscillation us ww0 - frequency of the degree-2 f-mode ω0 ≈ 740 μHz (Vorontsov et al. 1976). We observe a similar agreement between the = 3.1.2. The Coriolis-free n = 1 Polytrope harmonic oscillator and the Coriolis-free n 1 polytrope at higher-degree spherical harmonics (Table 1). Our results agree To verify the analogy of the forced harmonic oscillator to with a previously reported fractional correction to the tidally forced f-modes, we calculate the tidal response of gravitational coefficient C2,2 ∝ k2 due to dynamical tides in a a nonrotating n = 1 polytrope directly from the governing nonrotating Jupiter (Vorontsov et al. 1984). equations of tides. When Ω = 0, the governing Equation (14) When Vorontsov et al. (1984) excluded Jupiterʼs spin, they reduces to were doing something that was mathematically sensible but ⎛ ⎞ physically peculiar. Tides occur much more frequently in w2 ʼ fl jkr()-2yy jkr ()() ¶ =-⎜ ⎟ 20f .23 () Jupiter s rotating frame of reference, and the tidal ow is 0 1 r ⎝ ⎠ 4pr c accordingly much larger than if one had Jupiter at rest, which implies a much larger dynamical effect. Consequently, Δk2 For the potential ψ at ℓ = m = 2, the boundary condition at increases by an order of magnitude after partially including = ( ( )) the outer boundary r Rp Equation 17 is Jupiter’s rotation in the tidally forced response of f-modes. g Without the Coriolis effect but including Jupiter’sspinrate ¶-=-rpyy22 5.jkR2 () () 24(Ω ≈ 176 μHz) in the calculation of Io’s tidal frequency w2 (ω ≈ 270 μHz), the fractional dynamical correction in an n = 1 For the same degree and order, the continuity of the polytrope corresponds to Δk2 ≈ 13%, close to the Δk2 ≈ 15% gravitational potential and its gradient at the outer boundary from the forced harmonic oscillator analogy (Equation (22)).In requires general, for a nonrotating planet, the dynamical correction increases as the tidal frequency approaches the characteristic ⎛ ⎞ 35fdyn + ’ f (  3 ~ μ ) dyn⎜ 2 ⎟ frequency of Jupiter s -modes MRJ J 600 Hz . ¶=-rf ⎜ ⎟.25() 2 ⎝ R ⎠ p 3.2. The Coriolis Effect in a Rotating Gas Giant

At the center of the planet r = r0 → 0, we find the following The Galilean satellites produce dynamical tides for which the ∇2f0 ∼ ( ) ∼ fi scaling: 0, j1 kr0 0, and jkr0 ()0 ~ constant.A nite Coriolis effect plays an important role. Following relatively 5 The Planetary Science Journal, 2:69 (15pp), 2021 April Idini & Stevenson slow orbits (ωs = Ω), the Galilean satellites produce tides on The gravitational potential of a thin spherical density perturba- Jupiter with a tidal frequency ω ∼ 2Ω. Consequently, the two tion follows directly from the definition of the gravitational inertial terms responsible for dynamical tides on the left-hand potential and integration throughout the volume: side of the equation of motion (Equation (4)) have similar amplitudes. Moreover, Juno observes k2 to be less than the R ℓ+2 ( ) 4pr¯ p m predicted number for a purely hydrostatic tide Section 2.1 , f¢=å xℓm, Yℓ .34() ()21ℓ + r ℓ+1 and yet our analysis above produces a positive Δk2 when ℓm, dynamical effects are included and the Coriolis effect is neglected (see Equation (22)). We must accordingly motivate The degree-2 tidal gravitational potential corresponds to the change in sign when Coriolis is included. fi 4 ⎛ ⎞3 In the following, we rst calculate the gravitational effect of 4pr¯ Rpp3 R f ¢=xx = ⎜ ⎟ g .35() dynamical tides in a uniform-density sphere to reveal the 2 3 2 ⎝ ⎠ 2 fundamental behavior of the tidal equations, avoiding most of 5 r 5 r the technical difficulties related to using an n = 1 polytrope. We later found that the more complicated case of an n = 1 We once again use perturbation theory to split the hydrostatic ( polytrope introduces a minor quantitative difference but leads and dynamic contributions to the tidal displacement i.e., ξ = ξ0 + ξdyn) fi to the same general behavior. 2 .We rst solve the well-known problem of the hydrostatic k2 (i.e., ψ = 0) in a uniform-density sphere 3.2.1. A Uniform-density Sphere (Love 1909). At the sphere’s boundary (i.e., r = Rp), the hydrostatic gravitational potential follows f0 = 3gξ0/5. From First, we explain why Δk2 changes sign because of the Equation (31) evaluated at r = Rp, the potential of the Coriolis effect in a specially simple model with uniform T 0 density. We calculate the fractional dynamical correction to k gravitational pull becomes f = 2gξ /5. Following the last 2 = / in two steps: (1) we calculate the potential of the flow ψ in a two results, the Love number is k2 3 2, as expected. dyn uniform-density sphere, and (2) we use the ψ calculated this The dynamical contribution to the tidal displacement ξ dyn dyn way to calculate the gravity potential f¢. produces the gravitational potential f = 3gξ /5. After In a uniform-density sphere, the sound speed c is infinite, applying perturbation theory and canceling the hydrostatic s ( ) ψ ψ = and Equation (10) reduces to the well-known Poincaré problem terms in Equation 31 , the potential 2 becomes 2 (Greenspan et al. 1968), −2gξdyn/5. Combined with Equation (30), the last result for ψ allows us to reach an expression for the fractional dynamical 4 2 -2y (·)W 2y =0, () 28 correction in a uniform-density sphere: w2 dyn ⎛ ⎞ where the boundary condition at the outer boundary is required x 5Rp D=k2 »-⎜ ⎟ww()2. W- () 36 to satisfy Equation (18). x0 ⎝ 4g ⎠ Following the incompressibility of a uniform-density sphere, ψ2 retains the symmetry and degree-2 angular structure from Two effects contribute to the fractional dynamical correc- the gravitational pull in Equation (19), thus acquiring exact tion: a negative contribution from the Coriolis effect solutions in the form µW2 wpr¯ and a positive contribution from the dynamical fi µwp2 r y µ-()xiy2.29 ()ampli cation of f-modes ¯. The two contributions 2 cancel each other at 2Ω = ω, where the tide achieves The numerical factor in ψ2 is set by the outer boundary hydrostatic equilibrium. Tides become hydrostatic not only condition (Equation (17)), corresponding to (Goodman & when the planetary spin is phase-locked with the orbit of the (Ω = ω ) – Lackner 2009) satellite s but also in planet satellite systems where the central body is rotating at a rate many orders of magnitude 32ww()W- Rpww()2W- faster than the orbit of the satellite (i.e., Ω ? ωs). As the y2 = f˜ ¢= f˜ ¢.30() 8pr¯ 222g frequency of the degree-2 f-mode approximately follows w2 ~ gR, Δk in Equation (36) approximately becomes the In a constant-density sphere, tides act by displacing the 0 p 2 positive fractional correction determined in Section 3.1 after sphere’s boundary within an infinitesimally thin shell. Accord- setting Ω = 0. ing to the momentum equation, the tidal gravitational potential At the degree-2 Io-induced tidal frequency, the fractional relates to the potential ψ following dynamical correction corresponds to Δk2 ≈−7.8%. The other p¢ Galilean satellites lead to a smaller Δk2 because their tidal fy˜ ¢- - = 0.() 31 ( ) r¯ frequency falls closer to hydrostatic equilibrium Figure 2 .A negative Δk2 works in the direction required by the nonhydro- The radial tidal displacement projected into spherical harmo- static component identified by Juno in Jupiter’s gravity field nics is (Section 2.1). fl m The direction of the ow provides an explanation for the xx= å ℓm, ()rYℓ ( qj,, ) ( 32 ) negative sign of the fractional dynamical correction via the ℓm, Coriolis acceleration. By definition, a uniform-density sphere has and the pressure perturbation follows no density perturbations in its interior and thus produces an fi ℓ m T interior tidal gravitational potential that satis es f¢µrYℓ µf . ¶p We adopt Equation (30) as the degree-2 potential ψ and obtain p¢=-xxr= ¯g.33() ¶r analytical solutions for the Cartesian components of the resulting 6 The Planetary Science Journal, 2:69 (15pp), 2021 April Idini & Stevenson Despite the aforementioned difference between models, the tidal flow remains similar so that dynamical tides motivate a negative correction to k2 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 vr ∝ j2(kr)/j1(kr) when the flow has small divergence, as it does. Remarkably, the dominant contribution to the Taylor series expansion of vr is linear in r, even out to a large fraction of the planetary radius. 2 In a uniform-density sphere, the potential ψ is ψ2 ∝ r ,which leads to a tidal flow that follows v2 ∝∇ψ2; therefore, vr is also linear in r in this model. As shown, the dominant contribution to vr scales with radius as ∝r,bothinann = 1 polytrope andinauniform-densitysphere.Inann = 1 polytrope, the dominant contribution to vr is curl- and divergence-free 2 and provides the ψ2 ∝ r part of the solution to the potential ψ (Figure 4(a)).Sincethen = 1 polytrope also contains terms where ψ is of higher order in r, it produces a flow with nonzero curl and divergence, causing ψ2 to depart 2 from ψ2 ∝ r . Because high-order terms in r are smaller than the dominant term, dynamical effects on k2 in a uniform- density sphere are qualitatively similar to those in an n = 1 polytrope. Δ Figure 2. Fractional dynamical correction k2 in a rotating uniform-density We compute the fractional dynamical correction to k2 in a sphere including the Coriolis effect as a function of tidal frequency (see rotating polytrope following the same strategy used in Equation (36)). Section 3.1.2. In opposition to the Coriolis-free polytrope, solving Equation (14) is technically challenging due to the ℓ- degree-2 tidal flow using Equation (9), coupling of the potential ψℓ,m (e.g., mode mixing) promoted by ARw p the Coriolis effect. Mode mixing is also found in hydrostatic v2 =-(xixˆ() + y + yxˆ())() - iy ,37 g tides over a planet distorted by the effect of the centrifugal force (Wahl et al. 2017a). The result of projecting where A is a constant depending on ξ (Appendix D). The Equation (14) into spherical harmonics is an infinite ℓ-coupled ψ ( ) degree-2 tidal flow purely exist in equatorial planes, showing set of ordinary differential equations for ℓ,m Appendix B.2 , no vertical component of motion (Figure 3(b)). similarly observed in the problem of dissipative dynamical The Coriolis acceleration plays a major role in setting the sign tides (Ogilvie & Lin 2004). The Coriolis-promoted ℓ-coupling of the fractional dynamical correction for the Galilean satellites. comes from the sine and cosine in the spin rate of the planet Without Coriolis, the acceleration of nonrotating f-modes (W W=rˆ cosqqq -ˆ sin ), which changes the degree of the sustains a positive dynamical tidal displacement that follows spherical harmonics related to ψ. As a consequence, a given dyn 2 0 dyn 0 ξ ≈ 5Rpω ξ /4g.Aξ > 0 increases the tidal gravitational spherical harmonic from f on the right-hand side of field, which leads to a positive Δk2. Conversely, as shown in Equation (14) forces multiple spherical harmonics of the Equation (36), the fractional dynamical correction flips sign potential ψ with different ℓ. dyn when Coriolis promotes ξ < 0. A Coriolis term enters the Projected into spherical coordinates, the boundary condition momentum equation, introducing an acceleration that competes (Equation (17)) at r = Rp corresponds to with the acceleration of nonrotating f-modes, ultimately impact- ing ξdyn. According to the right-hand rule, the Coriolis 2 ⎛ ⎞ acceleration (i.e., Ω × v;Figure3(c)) opposes the direction of 24W W ¶qy 2 ¶-y ¶-y ⎜cos2 qy¶- sin q cos q ⎟ the acceleration of nonrotating f-modes (i.e., −ω ξ; Figure 3(a)). r j 2 r iRw p w ⎝ Rp ⎠ The resulting gravitational field is smaller than the hydrostatic ⎛ ⎞ field if ω < 2Ω, where the Coriolis acceleration beats the 4W-22w =⎜ ⎟()ff0 + T . acceleration of nonrotating f-modes. ⎝ g ⎠

3.2.2. The n = 1 Polytrope ()38 In the following, we consider the more relevant case of a ℓ compressible planet that follows an n = 1 polytropic equation The outer boundary condition is also -coupled after being of state (Equation (14)). In contrast to the localized tidal projected into spherical harmonics (Appendix B.2). = → ﬁ perturbation of a uniform-density sphere, a compressible body At the center of the planet r r0 0, we nd the following ∇2f0 ∼ yields a tidally induced density anomaly that arises from scaling: 0 and jkr0 ()0 ~ constant. As a result, the tidal advection of the isodensity surfaces within the body. The Equation (14) becomes the previously solved problem of the resulting tidal gravitational potential is different in each case potential ψ in a uniform-density sphere (Equation (28)) near the owing to differences in the tidally perturbed density distribu- center. Required to be ﬁnite near the center and satisfy tion obtained in a uniform-density sphere and a compressi- Equation (28), the radial part of the potential ψ follows ℓ ble body. ψℓ,m ∼ r . The boundary condition for ψℓ,m near the center

7 The Planetary Science Journal, 2:69 (15pp), 2021 April Idini & Stevenson

Figure 3. Degree-2 (ℓ = m = 2) tidal perturbations on a uniform-density sphere forced by the gravitational pull of a companion satellite: (a) the nonrotating f-mode acceleration −ω2ξ, (b) tidal ﬂow as shown in Equation (37), and (c) Coriolis acceleration Ω × v according to the right-hand rule.

Figure 4. Radial functions in an n = 1 polytrope (thick blue and orange curves) of the (a) potential ψ and (b) dynamical gravitational potential fdyn. The thinner black curves in panel (a) represent the radial scaling of the potential ψ in a uniform-density sphere. corresponds to the estimate in a uniform-density sphere and in agreement with the k nonhydrostatic component observed by Juno at PJ17. ℓ 2 ¶-rℓmyy, ℓm, =0.() 39 Both the uniform-density sphere and the polytrope models r0 produce fractional dynamical corrections that fall within 2 The equation for the gravitational potential of dynamical the order-of-magnitude estimate Dk2 ~~wpr40.1 .As fdyn argued before, the dominant contribution to the potential ψ tides remains unchanged compared to the Coriolis-free ℓ polytrope (Appendix B.1). The outer and inner boundary follows the radial scaling ψℓ ∝ r (Figure 4(a)). Ignoring the conditions for the gravitational potential generalize in degree as sign, the radial scaling of the dynamical gravitational potential fdyn ( ( )) ⎛ ⎞ Figure 4 b closely follows the shape of the hydrostatic ()ℓℓU+++121fdyn ( ) ( ) dyn ⎜ ℓm, ℓm, ⎟ gravitational potential Figure 1 . Due to the essentially circular ¶=-rfℓm, ⎜ ⎟,40() ⎝ Rp ⎠ and equatorial geometry of the Galilean orbits, the spherical harmonic ℓ = m = 2 dominates Jupiter’s tidal gravitational ℓ field. Consequently, we concentrate on comparing the k Juno ¶-ffdyn dyn =0.() 41 2 r ℓm, ℓm, observation to our model prediction. Of significantly higher r0 uncertainty, the mid-mission Juno report of Love numbers By projecting ψℓ,m and fℓ,m into a series of N Chebyshev at PJ17 includes other spherical harmonics in addition to k2 polynomials oriented in the radial component (Appendix C), ( ) ( ) ( ) Table 2 . Our polytropic model predicts an Io-induced tidal we numerically solve Equations B4 and B14 by truncating gravitational field in a 3σ agreement with most Love numbers the infinite series of ℓ-coupled equations at an arbitrary observed at PJ17, save for k42 and k31. ℓL= max. We choose a truncation limit Lmax = 50 and the number of Chebyshev polynomials N = 100 based on max 3.2.3. Detection of Dynamical Tides in Systems Other than Jupiter–Io numerical evidence of convergence for k2 and k42. We obtain Δk2 = −4.0% at the degree-2 Io-induced tidal A detection of dynamical tides via direct measurement of the frequency (Table 2), which is of slightly lower amplitude than gravitational field will be challenging in bodies other than

8 The Planetary Science Journal, 2:69 (15pp), 2021 April Idini & Stevenson

Table 2 Jupiter Love Numbers Δ ( = ) Hydrostatic Juno PJ17 3σ 3σ Fractional Difference k Rotating n 1 Polytrope ( ) Type Number Number (%) % Io Europa Ganymede Callisto (1)(2)(3)(4)(5)(6)(7)(8) k2 0.590 0.565 ± 0.018 −7/−1 −4 −2 −1 −1 k42 1.743 1.289 ± 0.189 −37/−15 +7 +8 +10 +12 k31 0.190 0.248 ± 0.046 +6/+55 +1 +3 +4 +5 k33 0.239 0.340 ± 0.116 −6/+91 +2 +5 +7 +8 k44 0.135 0.546 ± 0.406 +4/+605 +7 +11 +13 +15

Note. (2) The hydrostatic number is from Wahl et al. (2020). (3) The Juno PJ17 3σ number is the satellite-independent number from Durante et al. (2020). (4) The 3σ fractional difference represents the minimal/maximal 3σ nonhydrostatic fractional correction required to explain the Juno observations. The fractional dynamical correction in columns (5)–(8) is valid for an n = 1 polytrope forced by the gravitational pull of the Galilean satellites. The bold values represent the fractional dynamical correction at the tidal frequency of Io, the satellite with the dominant gravitational pull on Jupiter.

detection of Europa-induced dynamical tides seems plausible at the end of Juno’s extended mission, assuming that the factors determining the uncertainty in the gravity ﬁeld remain similar to those of Io. We calculate a model prediction for the satellite- dependent Jupiter Love number for all of the Galilean satellites (Table 2). We obtain k2 = 0.578 in the case of Europa, a prediction testable by the recently approved Juno extended mission.

4. Discussion 4.1. Future Updates to Juno Love Number Observations

The discrepancy between our predicted k42, k31, and the Juno PJ17 observations may allude to several reasons: (1) a suggestion to revise the hydrostatic ℓ-coupled kℓ,m in Wahl et al. (2020), (2) a failure of perturbation theory in our model when accounting for the ℓ-coupled kℓ,m,and(3) other physical reasons, for example, tidal resonance with normal modes or the neglected correction from a dilute core. We strongly suggest a thorough analysis of these possibilities in future investigations. Ultimately, the perijove passes required to complete the Figure 5. Conditions for the detection of dynamical tides evaluated for the scheduled Juno mission may change the still highly uncertain Galilean satellites (black) and inner Saturn satellites (white). Satellites to the numbers reported in Table 2. A recent revision to Juno right of the dashed line have favorable conditions for a detection of dynamical ’ observations at PJ29 (D. Durante 2021, personal communica- tides assuming an uncertainty roughly similar to that of Io s k2 on Jupiter at the ) end of Juno’s extended mission. The fractional dynamical correction Δk2 is for tion, 2020 November 18 suggests an agreement of our k31 an n = 1 polytrope. prediction with the revised satellite-independent k31 = 0.234 ± 0.016 (1σ).ThePJ29-revisedk42 = 1.5 ± 0.095 (1σ) remains in Jupiter (Figure 5). The 1σ uncertainty in the gravitational field disagreement with our k42 prediction, but the difference is much −2 2 −2 of degree-2 Io tides is projected to be σJ ∼ 6 × 10 m s at narrower than that attained at PJ17. A disagreement between our the end of the proposed Juno extended mission (W. Folkner predicted kℓ,m and high-degree Juno observations does not impair 2021, personal communication, 2020 April 8). The uncertainty the much more relevant agreement observed for k2. Compared to in the measured tidal gravity field depends on the number and the amplitude of the tidal gravitational potential related to k2,the design of spacecraft orbits, the uncertainty in ephemerides, and tidal gravitational potential related to k42 represents an order-of- fi instrumental capabilities. Assuming the uncertainty σJ,we magnitude smaller contribution to the tidal gravitational eld / ∼ / T roughly estimate the gravitational pull required to produce a due to the factor Rp a 1 6infℓm, . In addition, whereas the detectable dynamical component in the gravity field using predicted k2 simply depends on the contribution from the Coriolis effect and the dynamical response of f-modes, the more sJ fT ()rR=  .42() complicated predicted k42 additionally depends on the numerical kk22∣∣D solution of the ℓ-coupled system of equations described in Appendix B.2. Our calculation indicates that detecting dynamical tides in Ignoring for the moment other possibilities related to the k42 Saturn will require a mission with a more precise determination discrepancy, resonant tides have been previously invoked as of the gravity field than that obtained by Juno (Figure 5).A1σ a potential candidate to explain the current structure of the

9 The Planetary Science Journal, 2:69 (15pp), 2021 April Idini & Stevenson Laplace resonance in Saturn (Fuller et al. 2016; Lainey et al. gravity field would render them irrelevant to the tidal problem. 2020). As planets in the solar system rotate far from breakup, In order for free oscillations to play a role in the observed there is no overlap between the frequencies of tides and f- gravity, they require avoiding a rapid decay after becoming modes in adiabatic, nonrotating planets. However, composi- excited (i.e., a very high Q). It is not known whether free tional gradients (g-modes) and rotation (inertial modes) oscillations persist over multiple Juno perijove passes. introduce additional normal modes whose frequencies can become close to the tidal frequency, either by chance or by 4.2.2. Jupiter’s Rheology planetary evolution. These hypothetically resonant tides could ’ produce high dissipation rates and thus a detectable imaginary A central viscoelastic region in Jupiter s interior could part in the Love number that would consequently induce a potentially reduce k2 below the hydrostatic number; however, significant change in the real part of the Love number. evidence suggests that such a possibility is unlikely. Viscoe- Equivalently, the high dissipation rate from an hypothetically lastic deformation of a body produces a k2 between the purely resonant tide would cause a phase between the gravitational elastic and hydrostatic numbers, a model that helps to explain ’ ( ) pull and the degree-2 tidal bulge. However, the degree-2 tidal Titan s observed k2 Iess et al. 2012 . Assuming that a dissipation in Jupiter due to Io tides is modest (Lainey et al. traditional core in Jupiter exists, the core radius should remain ( ∼ ) 2009). This argument does not necessarily apply for higher- small i.e., 0.15RJ to satisfy the constraint on the total (ℓ > ) abundance of heavy elements and the supersolar enrichment of degree tides 2 that have much smaller amplitudes and ( ) therefore whose phase shifts would be much harder to detect. the envelope Wahl et al. 2020 . At this core radius, the tidal ( )( We compare our Io-induced fractional dynamical corrections to deformation of the core does not contribute to Re k2 Storch & ) the Juno PJ17 satellite-independent observations. We justify Lai 2014 . Whether rigid, elastic, or viscoelastic, a small traditional core produces a small effect on Re(k2) due to the the use of the satellite-independent k2 uncertainty because our results indicate small variations in the Love number due to added heavy elements, already included in the hydrostatic ( ) dynamical effects (Table 2), assuming the absence of degree-2 number Wahl et al. 2020 . Beyond the possibility of a tidal resonances. In the hypothetical of an ℓ = 4, m = 2 tidal viscoelastic traditional core, the hydrogen-rich envelope most fl resonance, the Love number k would vary significantly likely behaves as an inviscid uid. The kinematic viscosity of 42 fl ν ∼ 11 2 −1 among satellites. In such a case, we would be required to use a the uid external to the core needs to reach 10 m s in satellite-dependent uncertainty (Durante et al. 2020), in which order for its viscosity to become relevant at tidal timescales ( 2) ∼ no a priori information is used at the time of inferring the Love i.e., w ~ n RJ . Such kinematic viscosity exceeds by 17 number. So far, we neither confirm nor deny resonant tides that orders of magnitude the realistic estimates of the hydrogen- fl ( ) may be having an impact on k or k . An improved version of dominated uid viscosity Stevenson & Salpeter 1977 .A 42 31 similar argument applies to a dilute core, which most likely the satellite-independent k2 uncertainty could be obtained a priori assuming that the Love number increases ∼4% outward consists of a mixture dominated by hydrogen. when comparing the inner to the outer satellites. A stronger conclusion on the possibility of tidal resonances observed in 4.2.3. The Dynamical Contribution of a Traditional Core Juno data requires additional progress in the mission to reduce A traditional core blocks the tidal flow from extending to the the uncertainty on k42 and k31, plus a thorough analysis center of the planet by forcing a zero-flow boundary condition of resonances with Jupiter interior models that include a at the core radius. As shown earlier, the radial tidal flow sets the compositional gradient. amplitude of the tidal gravitational potential and roughly scales A tighter constraint on the satellite-dependent k2 from with distance from the center following vr ∝ r in an n = 1 satellites other than Io will test the prediction of our model of polytrope. Consequently, the tidal flow is nearly zero in the satellite-dependent dynamical tides. Despite the relatively large area where a traditional core would exist, minimizing a fractional dynamical correction obtained for the inner Saturn potential effect of the traditional core on the fractional ( ) satellites e.g., Mimas or Enceladus , their small mass leads to dynamical correction. A thorough quantification of Δk in a fi 2 an overall small tidal disturbance that is dif cult to detect in the model with a traditional core that blocks the flow requires fi gravity eld. From all Jupiter and Saturn satellites, only Europa further investigation. We expect an effect going from negligible elevates a short-term prospect of obtaining a new detection of to small (i.e., less than +1% applied to the current estimate in ’ ( ) dynamical tides via Juno s extended mission Figure 5 .A Table 2) given the limits to traditional core size imposed by the detection of dynamical tides due to other satellites will require constrained total abundance of heavy elements. an uncertainty on k2 that is significantly lower than that produced by Juno. 4.2.4. A Dilute Core A dilute core may promote an additional departure of the 4.2. Other Potential Contributions to k2 tidal response from the hydrostatic tide to that caused by dynamical tides. The hydrostatic tide in k provides the same 4.2.1. Free-oscillating Normal Modes 2 information about the planet as J2 (Hubbard 1984). In the Free oscillations of normal modes cannot explain the bulk of presence of a dilute core, the gravity produced by tides the nonhydrostatic Juno detection discussed here. The small fundamentally differs from the J2ℓ coefficients due to the gravitational field of tides becomes resolvable by Juno in part different timescales associated with tidal perturbations and the because the phase of the signal is well known. The unknown evolution of the rotation rate. Tidal timescales are short phase of nonresonant free oscillations departs from the phase of compared to the timescale required for the tidal perturbation to the satellite used in determining k2. Even if freely oscillating equilibrate with the environment by either heat transport or normal modes were detected in Jupiter as they were in Saturn compositional evolution. By contrast, the timescale at which (Markham et al. 2020), the anticipated high frequency of their the rotation rate evolves is so long that the planet adjusts to any

10 The Planetary Science Journal, 2:69 (15pp), 2021 April Idini & Stevenson perturbation caused by the centrifugal effect. Tidal displace- Appendix A ments remain roughly adiabatic, whereas displacements Hydrostatic Tides in an Index-one Polytrope induced by changes in the rotation rate reach thermodynamic ω ≈ fl In hydrostatic tides, the tidal frequency becomes 0, and equilibrium. A uid parcel in the proximity of the dilute core the tidal flow is slow enough to set v ≈ 0. After projecting f0 responds differently depending on the timescale of the into spherical harmonics by setting a solution in the form perturbation; only an adiabatic perturbation leads to changes 0 0 ℓ -itw in the buoyancy of the fluid parcel, causing a wavelike ff= å ℓm, ()xYm ( qj,, ) e ( A1 ) oscillation known as static stability. Consequently, the dilute ℓm, core produces a signature in the tidal response of the planet not the radial part of f0 at a given harmonic that satisfies registered by J2. We address the tidal effects of a dilute core in Equation (13) follows a subsequent investigation. 2 ⎛ ℓℓ()+ 1 ⎞ ⎛ x ⎞ℓ 00⎜⎟0 ⎜⎟ ¶+¶+-xx, ffx 1 f=- ,A2() ℓ x ℓℓ⎝ x2 ⎠ ⎝ p ⎠ 5. Conclusions m fi where Yℓ are the normalized spherical harmonics de ned by Our tidal models suggest that the gravity field observed by Juno captured the dynamical tidal response of Jupiter to the m ()()!21ℓℓm+- m imj Yℓ ()qj, =  ℓ ()cosq e , () A3 gravitational pull of the Galilean satellites. We show that two 4p ()!ℓm+ fi effects contribute to the dynamical gravity eld of tides in and  m are the associated Legendre polynomials corresponding to Jupiter: the dynamical response of f-modes and the Coriolis ℓ acceleration. When the Coriolis effect is ignored, tides closely m ℓm+ m ()-1 22m d 2 ℓ follow the dynamical response of f-modes modeled as a forced  ℓ ()mm= ()11.A4-- ()()m 2ℓl! dmℓm+ harmonic oscillator. In ignoring the Coriolis effect, dynamical fi ampli cation in a harmonic oscillator accounts for the The normalized radial coordinate follows x = kr, which leads to ’ dynamical response of f-modes in the planet s interior, forced a planet with radius π. Note that Equation (A2) is nondimen- by the gravitational pull of the companion satellites. As the sional and should be scaled by the factor tidal frequency is lower than the f-mode oscillation frequency, ⎛ ⎞ℓ ⎛ ⎞12 the dynamical response of f-modes amplifies the gravity field of ⎛m ⎞ Rp 4p()!ℓm- ⎜⎟s ⎜ ⎟ m ’ Uℓm, = ⎜ ⎟  l ()0. ( A5 ) the hydrostatic tide. Motivated by Jupiter s fast rotation, we ⎝ a ⎠⎝ a ⎠ ⎝ ()()!21ℓℓm++⎠ show that the Coriolis effect leads to a significant additional contribution to the dynamical tide. When the Coriolis effect is The order m does not appear in Equation (A2), indicating a ℓ m included in our tidal models, we show that the Coriolis degeneracy on m of the hydrostatic tide. As xYℓ is a solution ’ ( 2 ℓ m ) acceleration produces a competing effect of opposite sign to Laplace s equation i.e., =()xYℓ 0 , a complete solution compared to the dynamical response of f-modes. When both to Equation (A2) is dynamical effects are considered together, they reduce the Love ⎛ x ⎞ℓ number k below the hydrostatic number if ω < 2Ω and amplify 0 ⎜⎟ 2 fℓ =+Ajℓ () x Bnℓ () x -⎝ ⎠ .A6 ( ) it otherwise. Following our theoretical prediction, dynamical p effects lead to a negative correction to Jupiter’s hydrostatic We require f0 to be finite at the center of the planet and thus set Love number k2 in the case of the Galilean satellites, for which B = 0. According to the outer boundary condition, we set a external ω < Ω the degree-2 tidal frequency is 2 . The fractional gravitational potential F0(x) that extends outward from the planet – Δ = ℓ dynamical correction for the Jupiter Io system is k2 and matches the internal tidal potential at the planetary radius as −4%. Our analysis provides an explanation for the recently observed nonhydrostatic component in the gravity field of ⎛ ⎞ℓ++1 ⎛ ⎞ℓ 1 0 ⎜⎟p 0 ⎜⎟p Jupiter tides obtained by the Juno mission. F=ℓ ()x ⎝ ⎠ fpℓ () =⎝ ⎠ (Ajℓ ℓ ()p -1. ) ( A7 ) In conclusion, our analysis proposes that the Juno nonhy- xx fi drostatic detection is the rst unambiguous measurement of the The continuity of the gradients of the internal and external gravitational effects of dynamical tides in a gas giant. Our potentials at the surface of the planet sets the constant Aℓ to conclusion depends on the assumption that the degree-2 tidal 21ℓ + frequency of the Galilean satellites is far from resonance with A = .A8() ’ ℓ Jupiter s normal modes. In a subsequent investigation, we will ppjℓ-1() utilize the results reported here to infer the extension and static stability of Jupiter’s dilute core from k . The uncertainty Consequently, the gravitational potential of hydrostatic tides at 2 degree ℓ is expected in the observed k2 at the end of the mission exceeds the uncertainty achieved by our model, suggesting that a more ⎛ 21ℓ + ⎞ jx() ⎛ x ⎞ℓ detailed tidal model will be required in the future to fully f0 = ⎜⎟ℓ - ⎜⎟ ℓ ⎝ ⎠ ⎝ ⎠ ,A9() exploit the information contained in the data provided by Juno. pppjℓ-1()

We acknowledge the support of NASA’s Juno mission. B.I. and the hydrostatic Love number follows thanks Erin Burkett for her comments. We acknowledge the ⎛ 21ℓ + ⎞ j ()p constructive comments from two anonymous referees. ⎜⎟ℓ kℓ = ⎝ ⎠ - 1.() A10 Software: Matplotlib (Hunter 2007). p jℓ-1()p

11 The Planetary Science Journal, 2:69 (15pp), 2021 April Idini & Stevenson Appendix B Projection of the Dynamical Tide Equations into Spherical Harmonics Here we project into spherical harmonics the equation for the potential ψ in a nonrotating (Equation (23)) and rotating (Equation (14)) n = 1 polytrope. The equation for the gravitational potential of dynamical tides is Equation (15), forced by a different potential ψ depending on rotation. We evaluate solutions in the form

m -itw yy= å ℓm, ()xYℓ ( qj,, ) e ( B1 ) ℓm,

dyn dyn m -itw ff= å ℓm, ()xYℓ ( qj,. ) e ( B2 ) ℓm, In the following, we conveniently drop the time-dependent part e i ω t out of our derivation. Notice that we normalize the radial coordinate following x = kr, leading to a body of normalized radius π = kRp.

B.1. The Coriolis-free n = 1 Polytrope We project into spherical harmonics the potential ψ (Equation (23)) and the dynamical gravitational potential (Equation (15)) of the nonrotating polytrope:

⎛ ⎛ ⎞ ⎞ ⎛ ⎞ 2 jx() ⎛ ℓℓ()+ 121⎞ ℓ + w2jx() jx()⎜¶+-¶+- (yyy ) ⎜ 1 ⎟ ()⎜⎟1 ⎟ = ⎜ ⎟ ℓ ,B3() 0 xx,, ℓm⎝ ⎠ x ℓm, ⎝ 2 ⎠ ℓm, ⎝ ⎠ ⎝ x jx0 () x ⎠ ppjℓ-1() 4pr c ⎛ ⎞ dyn 2 dyn ⎜⎟ℓℓ()+ 1 dyn ¶+¶+-xx, ()ffx () 1 fy= ℓm, .B4() ℓm, x ℓm, ⎝ x2 ⎠ ℓm,

B.2. The n = 1 Polytrope Relative to the left-hand side of Equation (14), the projections of the ﬁrst, second, and third terms, respectively, follow

⎛ ⎛ ⎞ ⎞ 21j ℓℓ()+ ·(jjyy ) =¶+-¶-⎜ ⎜ 1⎟ ⎟ Y m,B5() 0 ℓm, 0 xx, ⎝ ⎠ x2 ℓm, ℓ ⎝ x j0 x ⎠

2 2mjW 1 m ´=·(j W ylm )y ℓ, mY ,B6 ( ) iw 0 wx ℓ 2 ⎛⎛ ⎛ j ⎞ ⎞ 44W ⎜⎟0 m 2 -·(jjjWW ( · yℓm, )) =- ⎜⎜ ¶- xx, + ¶ x⎟yq ℓm, Y cos w2 0 w2 ⎝⎝ 0 ⎝ 1 x ⎠ ⎠ ℓ ()B7 ⎛ ⎞ ⎞ j0 m 22j0 j10j m j0 2 m +¶xℓmyyqqyq, Y +⎜ + - ¶xℓm⎟ , cos sin ¶qqqY +ℓm, sin ¶, Y ⎟ . x ℓ ⎝ x2 x x ⎠ ℓ x2 ℓ ⎠ The multiplication of spherical harmonics with trigonometric functions expresses a physical statement about the coupling effect that Coriolis produces in the tidal gravitational response of a rotating body. The partial derivatives in the spherical harmonics indicate changes in quantum numbers described in the following differential relations (Lockitch & Friedman 1999):

m m m sinq¶=qYℓQYℓℓ ℓ+1 ℓ+-11 -+() 1 QYℓ ℓ , () B8

m m m cosqYQYQYℓ =+ℓ+1 ℓ+-11ℓ ℓ ,() B9 where

⎛ ℓm22- ⎞12 Qℓ = ⎜ ⎟ .B10() ⎝ 41ℓ2 - ⎠ Combining the previous differential relations, we arrive at expressions for each of the angular terms in Equation (B7):

m 2 m 2 2 m m YQQYQQYQQYℓ cosq =+++ℓℓ-1 ℓ-+2 ()ℓℓℓ1 ℓℓ++12ℓ+2 , () B11

m m 2 2 m m cosqq sin¶=-+qYℓQQYℓQℓQYℓQQYℓ () 1ℓℓ-1 ℓ-+2 -+ (() 1ℓℓℓ -1 ) +ℓℓ++12ℓ+2 , () B12

12 The Planetary Science Journal, 2:69 (15pp), 2021 April Idini & Stevenson

2 m 2 m 2 2 2 m 2 m sinq¶=+qq, Yℓℓ () 1 QQYℓℓ-1 ℓ-+2 ++--+ (() 2 ℓℓQℓℓQYℓQQYℓℓℓ ()) 31 +ℓℓ++12ℓ+2 . () B13 After grouping terms with the same spherical harmonic, Equation (14) becomes an inﬁnite set of ℓ-coupled radial equations following the structure of a Sturn–Liouville problem:

()1 ()1 ()1 ()2 ()2 ()2 ((PQRℓm, ¶+xx, ℓm, ¶+x ℓm, )yyℓm, + ( PQRℓm, ¶+xx, ℓm, ¶+x ℓm, ) lm+2, ()0 ()0 ()0 +¶+¶+()PQRxx, x ylm-2, ℓm, ℓm, ℓm, ()B14 ⎛ ⎞ w22-W4 ⎛ 21ℓ + ⎞ jx() =U ⎜ ⎟⎜⎟ℓ . ℓm, ⎝ ⎠⎝ ⎠ 4pr c pjℓ-1() p The nine radial coefﬁcients correspond to

2 ()0 4W PjQQ=- ℓℓ-1 ,B15() ℓm, w2 0 ⎛ 4W2 ⎞ Pj()1 =-⎜1 () QQ2 + 2 ⎟,B16 () ℓm, 0 ⎝ w2 ℓℓ+1 ⎠

2 ()2 4W PjQQ=- ℓℓ++12,B17() ℓm, w2 0 2 ⎛ j ⎞ ()0 4W ⎜⎟0 Qℓ= ()23-+jQℓℓ-1 Q , () B18 ℓm, w2 ⎝ x 1⎠ 2j 4W2 ⎛ j ⎞ Q ()10=--j ⎜⎟02((121++ℓQQjQQ )( -2 ))( -2 +2 ) , () B19 ℓm, x 1 w2 ⎝ x ℓℓ++1 1 ℓℓ1 ⎠ 2 ⎛ j ⎞ ()2 4W ⎜⎟0 Qℓ=- ()23--jQℓℓ++12 Q , () B20 ℓm, w2 ⎝ x 1⎠ 2 ⎛ j j ⎞ ()0 4W ⎜⎟0 1 Rℓℓ=- ()-+2,QQℓℓ-1 () B21 ℓm, w2 ⎝ x2 x ⎠ ll()+ 12 j mjW 4W2 ⎛ j j ⎞ R ()10=- + 1 + ⎜⎟12((ℓQℓQ+-11,B22 )20 ) +ℓℓ ( + )( Q2 + Q2 ) ( ) ℓm, x2 wwx 2 ⎝ x ℓℓ++1 x2 ℓℓ1 ⎠ 2 ⎛ j j ⎞ ()2 4W ⎜⎟0 1 Rℓℓ=- ((++411 ) ) -- (ℓ 1 )QQℓℓ++12 . ( B23 ) ℓm, w2 ⎝ x2 x ⎠ The projection into the spherical harmonics of the boundary condition (Equation (38)) leads to

⎛ W ⎞ W2 ⎛ y ⎞ m ⎜⎟24m ⎜⎟m 2 ℓm, m Y ¶-xℓmy ,,y ℓm- YYcosqy¶-xℓm, sinqq cos ¶q ℓ ⎝ wp ⎠ w2 ⎝ ℓ p ℓ ⎠ ()B24 ⎛ 421W-22w ⎞⎛ ℓ + ⎞ jx() =U ⎜ ⎟⎜⎟ℓ Y m. ℓm, ⎝ ⎠⎝ ⎠ ℓ g ppjℓ-1() The previous differential relations still apply to deal with coupled spherical harmonics in the boundary condition. Grouping terms for m ℓ each spherical harmonic Yℓ , we reach an -coupled boundary condition with the structure

2 ⎛ ⎞ ()j ()j 421W-22w ⎛ ℓ + ⎞ j ()p (QRˆ ¶+ˆ )ym = U⎜ ⎟⎜⎟ℓ .B25() å ℓm,,22x ℓm ℓj+- ℓm, ⎝ ⎠⎝ ⎠ j=0 g p jℓ-1()p The six radial coefﬁcients correspond to

()0 4W2 QQQˆ =- ℓℓ-1 ,B26() ℓm, w2

()1 4W2 QQQˆ =-1 ()2 + 2 ,B27 () ℓm, w2 ℓ ℓ+1

13 The Planetary Science Journal, 2:69 (15pp), 2021 April Idini & Stevenson

()2 4W2 QQQˆ =- ℓℓ++12,B28() ℓm, w2

()0 4W2 RℓQQˆ = ()- 2,ℓℓ-1 () B29 ℓm, pw2

()1 24mW W2 Rˆ =- - ((ℓQℓQ+-1, )2 2 ) ( B30 ) ℓm, pw pw2 ℓ ℓ+1

()2 4W2 RℓQQˆ =- ()- 1.ℓℓ++12 () B31 ℓm, pw2

Appendix C Chebyshev Pseudospectral Method We solve the Sturn–Liouville differential problem (Boyd 2001),deﬁned by

pru()+ () r qru () ¢+ () r rrur () () = fr (),C1 ( ) and constrained to the boundary conditions

aaa012ua¢+() ua () =,C2 ( )

bbb012ub¢+() ub () =,C3 ( ) where a and b are the two ends of a boundary value problem. We shift the domain of Equation (C1) from r ä [a, b] to the domain of Chebyshev polynomials μ ä [−1, 1] and seek a solution that is a truncated sum of an inﬁnite Chebyshev series,

Nmax uaT()mm» å nn (),C4 ( ) n=0 with the Chebyshev polynomials deﬁned by

Tntn ()m = cos ( ) ( C5 ) and t = arccos()m . Our objective is to obtain the coefﬁcients an by solving a linear inverse problem:

La= f .C6() The square matrix L and vector f come from the evaluation of Equation (C1) into Gauss–Lobatto collocation points deﬁned by

⎛ pi ⎞ m = cos ⎜⎟,1,2,,1,iN=¼- () C7 i ⎝ N - 1⎠ plus the constraints from boundary conditions. The partial derivatives in Equation (C1) assume the analytical form

¶T ()m sin (nt ) n = n ,C8() ¶m sin()t

¶2T ()m cos (nt ) ⎛ ntcos()⎞ n =-n2 +⎜ ⎟ sin()nt . ( C9 ) ¶m2 sin23()t ⎝ sin ()t ⎠

Appendix D Tidal Flow in a Uniform-density Sphere We calculate the tidal ﬂow from projecting Equation (9) into Cartesian coordinates:

⎛ ⎞ ⎛ w ⎞ ⎛ W ⎞ ⎛ W ⎞ ⎛ W2 ⎞ ⎜⎟i ⎜⎟⎜⎟224i i v =- ⎜xˆˆ¶+xyyx¶+y ¶- ¶+zˆ()⎜ -¶1.⎟ z⎟y D1 ⎝ 4W-22ww⎠⎝ ⎝ ⎠ ⎝ ww⎠ ⎝ 2 ⎠ ⎠ The potential ψ depends on the potential of the gravitational pull (Equation (19)) and the tidal potential internal to the thin shell disturbed by tides. Analogous to what we did for the external potential, we obtain the tidal gravitational potential (i.e., r < Rp) from

14 The Planetary Science Journal, 2:69 (15pp), 2021 April Idini & Stevenson integration throughout the volume,

⎛ ⎞2 3 ⎜ r ⎟ fx2 ¢= g 2,D2() 5 ⎝ Rp ⎠ which leads to

Rppww()2W- AR ww()2W- 2 y2 = f˜ ¢= ()xiy- .D3 () 2g 2 2g The constant A comes from the numerical factor of the relevant potentials, corresponding to

3 m 3 15 gx A =+s 2 .D4() 3 2 16 a 20 2p Rp

As shown in Equation (D3), ψ2 is independent of z, leading to a 2D tidal ﬂow in the equatorial planes. Placing Equation (D3) into Equation (D1), the degree-2 tidal ﬂow becomes

ARw p v2 =-(xixˆ() + y + yxˆ()) - iy .D5 () g

ORCID iDs Iess, L., Jacobson, R. A., Ducci, M., et al. 2012, Sci, 337, 457 Kulowski, L., Cao, H., & Bloxham, J. 2020, JGRE, 125, e06165 Benjamin Idini https://orcid.org/0000-0002-2697-3893 Lainey, V., Arlot, J.-E., Karatekin, Ö., & Van Hoolst, T. 2009, Natur, David J. Stevenson https://orcid.org/0000-0001-9432-7159 459, 957 Lainey, V., Casajus, L. G., Fuller, J., et al. 2020, NatAs, 4, 1053 Lainey, V., Jacobson, R. A., Tajeddine, R., et al. 2017, Icar, 281, 286 References Lockitch, K. H., & Friedman, J. L. 1999, ApJ, 521, 764 Love, A. E. H. 1909, RSPSA, 82, 73 Aerts, C., Christensen-Dalsgaard, J., & Kurtz, D. W. 2010, Asteroseismology Mankovich, C. R. 2020, AGUA, 1, e00142 (Amsterdam: Springer) Mankovich, C., Marley, M. S., Fortney, J. J., & Movshovitz, N. 2019, ApJ, Bolton, S. J., Adriani, A., Adumitroaie, V., et al. 2017, Sci, 356, 821 871, 1 Boyd, J. P. 2001, Chebyshev and Fourier Spectral Methods (New York: Dover) Markham, S., Durante, D., Iess, L., & Stevenson, D. 2020, PSJ, 1, 27 Christensen-Dalsgaard, J., Gough, D., & Toomre, J. 1985, Sci, 229, 923 Marley, M. S., & Porco, C. C. 1993, Icar, 106, 508 Dahlen, F., & Tromp, J. 1998, Theoretical Global Seismology (Princeton, NJ: Munk, W. H., & MacDonald, G. J. 1960, The Rotation of the Earth; A Princeton Univ. Press) Geophysical Discussion (Cambridge: Cambridge Univ. Press) Durante, D., Parisi, M., Serra, D., et al. 2020, GeoRL, 47, e2019GL086572 Notaro, V., Durante, D., & Iess, L. 2019, P&SS, 175, 34 Fuller, J. 2014, Icar, 242, 283 Ogilvie, G. I., & Lin, D. 2004, ApJ, 610, 477 Fuller, J., Luan, J., & Quataert, E. 2016, MNRAS, 458, 3867 Stevenson, D., & Salpeter, E. 1977, ApJS, 35, 239 Gaulme, P., Schmider, F.-X., Gay, J., Guillot, T., & Jacob, C. 2011, A&A, Stevenson, D. J. 1982, AREPS, 10, 257 531, A104 Stevenson, D. J. 2020, AREPS, 48, 465 Gavrilov, S., & Zharkov, V. 1977, Icar, 32, 443 Storch, N. I., & Lai, D. 2014, MNRAS, 438, 1526 Goodman, J., & Lackner, C. 2009, ApJ, 696, 2054 Vorontsov, S., Gavrilov, S., Zharkov, V., & Leontev, V. 1984, AVest, 18, 8 Greenspan, H. P., et al. 1968, The Theory of Rotating Fluids (New York: Vorontsov, S., Zharkov, V., & Lubimov, V. 1976, Icar, 27, 109 Cambridge Univ. Press) Wahl, S. M., Hubbard, W. B., & Militzer, B. 2017a, Icar, 282, 183 Hubbard, W. B. 1984, Planetary Interiors (Princeton, NJ: Van Nostrand- Wahl, S. M., Hubbard, W. B., Militzer, B., et al. 2017b, GeoRL, 44, 4649 Reinhold) Wahl, S. M., Parisi, M., Folkner, W. M., Hubbard, W. B., & Militzer, B. 2020, Hunter, J. D. 2007, CSE, 9, 90 ApJ, 891, 42 Iess, L., Folkner, W., Durante, D., et al. 2018, Natur, 555, 220 Wu, Y. 2005, ApJ, 635, 674