On the equilibrium figure of close-in planets and satellites
aa r X i v : . [ a s t r o - ph . E P ] A p r On the equilibrium figure of close-in planets and satellites
Alexandre C.M. CorreiaDepartamento de F´ısica, I3N, Universidade de Aveiro, Campus de Santiago, 3810-193Aveiro, Portugal;ASD, IMCCE-CNRS UMR8028, Observatoire de Paris, UPMC, 77 Av. Denfert-Rochereau,75014 Paris, FranceandAdri´an Rodr´ıguezInsituto de Astronomia, Geof´ısica e Ciˆencias Atmosf´ericas, IAG-USP, Rua do Mat˜ao 1226,05508-090 S˜ao Paulo, BrazilReceived ; accepted 2 –
ABSTRACT
Many exoplanets have been observed close to their parent stars with orbitalperiods of a few days. As for the major satellites of the Jovian planets, thefigure of these planets is expected to be strongly shaped by tidal forces. How-ever, contrarily to Solar System satellites, exoplanets may present high values forthe obliquity and eccentricity due to planetary perturbations, and may also becaptured in spin-orbit resonances different from the synchronous one. Here wegive a general formulation of the equilibrium figure of those bodies, that makesno particular assumption on the spin and/or orbital configurations. The gravityfield coefficients computed here are well suited for describing the figure evolutionof a body whose spin and orbit undergo substantial variations in time.
Subject headings: celestial mechanics — planetary systems — planet-star interactions— planets and satellites: general — planets and satellites: physical evolution
1. Introduction
All the main satellites of the giant planets in the Solar System are in synchronousrotation, and locked in a nearly zero-obliquity Cassini state; their orbits are nearly circular,with orbital periods less than 16 days, and lie in the equatorial plane of the central planet .These configurations are the result of the long-term evolution of the spin and orbits dueto tidal forces raised by the central planet (e.g. Correia 2009). The influence of the tidaldeformation on the shape of the satellites is also appreciable, because the satellites’ figuresare supposed to be in hydrostatic equilibrium with the tidal and centrifugal potentials (e.g.Schubert et al. 2004). This assumption results from the fact that their interiors are eitherhot (Io and Europa) or plastic (icy satellites, e.g., Ganymede, Callisto, and Titan).About 40% of the presently known exoplanets are found in orbits with periods below16 days, and at least more than 25% of them are not alone in their systems . As for thesatellites above, we can assume that these planets acquire an equilibrium shape dictated byits internal gravity and a perturbing potential. However, due to planetary perturbationsfrom additional companions, close-in exoplanets may present non-zero obliquity (e.g.Laskar & Robutel 1993; Levrard et al. 2007) or eccentric orbits (e.g. Correia & Laskar 2004;Mardling 2007). In addition, they may also be captured in spin-orbit resonances differentfrom the synchronous one (e.g. Goldreich & Peale 1966; Rodr´ıguez et al. 2012).Previous works have shown that non-synchronous satellites present a different figure(Zharkov & Leont’ev 1989; Giampieri 2004). In addition, if the orbit has some eccentricity,the low order gravity coefficients can be separated into static and periodic components(involving different time-scales), and only the static part contributes to the permanent http://ssd.jpl.nasa.gov/ http://exoplanet.eu/ 4 –deformation (Rappaport et al. 1997; Giampieri 2004). Therefore, since exoplanets mayevolve in very eccentric orbits, their figures are most likely different from what we observein Solar System satellites.In this Letter we generalize previous studies to any eccentricity value. We also includethe effect of an arbitrary obliquity, that may change completely the gravity field coefficients.We adopt a vectorial formalism, which is different from the traditional expansion of thegravitational potential in spherical harmonics.
2. The gravitational potential
The gravitational potential of a body of mass m at a generic point P is given by: V ( r ) = − G Z dm | r − r ′ | (1)where G is the gravitational constant, r is the position of P , and r ′ the position of a masselement dm with respect to center of mass of the body. Assuming r ′ ≪ r , we can developthe potential limited to the second order, which gives (e.g. Tisserand 1891; Smart 1953): V ( r ) = − Gmr + 3 G r (cid:18) ˆ r · I · ˆ r − tr ( I ) (cid:19) , (2)where ˆ r = r /r , I = I I I I I I I I I (3)is the itertia tensor, and tr ( I ) = I + I + I its trace. We chose as reference thenon-inertial frame ( i , j , k ) fixed to the planet, where k is the direction of the spin axis(Fig. 1). 5 – Line of Nodes
Ω ψθ n Fig. 1.— Reference frames for the definition of the reference angles. n is the normal vectorto the orbital plane of the perturbing body at r ⋆ . The reference frame ( i , j , k ) is fixed withrespect to the planet’s figure, k being the rotation axis, that is tilted by an angle ε (theobliquity) with respect to n . The rotation angle, θ , and the position of the perturber, ψ ,are measured from the line of nodes between the equatorial and orbital planes, along theirrespective planes. Ω is the rotation rate. 6 –The above potential is also often expressed alternatively using the gravity fieldcoefficients J , C n , S n , as V ( r ) = − Gmr − GmR r h C P ( k · ˆ r )+ C i · ˆ r )( k · ˆ r ) + S j · ˆ r )( k · ˆ r )+ C i · ˆ r ) − ( j · ˆ r ) ) + S i · ˆ r )( j · ˆ r ) i , (4)where P ( x ) = (3 x − /
2, and J = − C = I − ( I + I ) mR , (5) C = − I mR , S = − I mR , (6) C = I − I mR , S = − I mR . (7)
3. The perturbing potential
The mass distribution in the body, characterized by the inertia tensor (Eq. 3), is aresult of the self gravity, but also of the body’s response to any perturbing potential.For a planet with rotation rate Ω = Ω k , a mass element dm is also subject to thecentrifugal potential (e.g. Goldstein 1950) V c ( r ′ ) = −
12 ( Ω × r ′ ) = −
12 Ω r ′ (cid:0) − ( k · ˆ r ′ ) (cid:1) . (8)In addition, since the planet moves around a central star of mass m ⋆ , we also need toconsider the tidal potential (e.g. Lambeck 1980) V t ( r ′ , r ⋆ ) = − Gm ⋆ r ⋆ (cid:18) r ′ r ⋆ (cid:19) P (ˆ r ′ · ˆ r ⋆ ) , (9) 7 –where r ⋆ is the position of the star with respect to the planet center of mass. In the frameof the planet, we can express ˆ r ⋆ as (e.g. Correia 2006): r ⋆ r ⋆ = cos ε sin ψ sin θ + cos ψ cos θ cos ε sin ψ cos θ − cos ψ sin θ − sin ε sin ψ , (10)where ε is the obliquity of the planet (the angle between the spin axis and the normal tothe orbit), θ is the rotation angle, ψ = ω + ν is the angle between the line of nodes and thedirection of the star, ω is the argument of the periapse, and ν is the true anomaly (Fig. 1).The resulting perturbing potential is then given by V p ( r ′ , r ⋆ ) = V c ( r ′ ) + V t ( r ′ , r ⋆ ) , (11)that can also be rearranged as V p ( r ′ , r ⋆ ) = (cid:18) Gm ⋆ r ⋆ − Ω (cid:19) r ′ G r ′ (ˆ r ′ · I p · ˆ r ′ ) , (12)where I p = I c + I t , (13)with I c mr ′ = Ω r ′ Gm , (14)and I t mr ′ = − m ⋆ m (cid:18) r ′ r ⋆ (cid:19) (cid:20) r ⋆ r ⋆ (cid:21) T (cid:20) r ⋆ r ⋆ (cid:21) . (15)The last term in the above equation is directly obtained from expression (10), where T denotes the transpose. 8 –
4. Equilibrium figure
The perturbing potential (Eq. 11) acting on the planet, deforms it, and modifies theexternal gravitational potential (Eq. 1). A most convenient way of defining this deformationis through the Love number approach (e.g. Love 1927; Peltier 1974), in which the body’sresponse is evaluated in the frequency domain. As long as the distortions are small, wecan simplify the problem by ignoring the small interaction terms between the centrifugaland tidal potentials (Zharkov & Trubitsyn 1978). The additional gravitational potential,∆ V ( r ), due to the deformation of the planet in response to the perturbing potential, is thengiven at the planet’s surface ( r = R ) by (e.g. Lambeck 1980):∆ V ( R ) = k V p ( R , r ⋆ ) , (16)where k is the second Love number for potential. In general, k is a complex number,which depends on the frequency σ of the perturbation. | k | gives the amplitude of the tide,while the imaginary part gives the phase lag between the perturbation and the maximaldeformation.If we assume that the planet behaves like a Maxwell body with homogeneous density ρ we have (Henning et al. 2009): k ( σ ) = k f στ a στ b , (17)where k f is the fluid second Love number, τ a = υ/µ is the relaxation time, and τ b τ a = (cid:18) µR Gmρ (cid:19) = k f k ( ∞ ) . (18) k f is constant for a given body and corresponds to its maximal deformation (for a staticperturbation σ = 0, and hence k = k f ). For an homogeneous sphere k f = 3 /
2, but more A Maxwell body behaves like an elastic body over short time scales, but flows like a fluidover long periods of time. It is characterized by a homogenous rigidity µ and viscosity υ . 9 –generally k f can can be obtained from the Darwin-Radau equation (e.g. Jeffreys 1976): I mR = 23 − s − k f k f ! . (19)In Figure 2 we plot the dependence of | k | with the tidal frequency for the Earth. Sincestatic and elastic stresses involve very different time scales, there is no conflict between thetwo types of responses (Fig. 2). We can thus assume a static response for the centrifugepotential (Eq. 8) and an elastic one for the tidal potential (Eq. 9), which depends on thevarying position of the star with respect to a point at the planet’s surface.Combining expressions (2, 12, and 16) we thus have I = I + k I p ( R ) , (20)where I ∝ I corresponds to the moment of inertia with spherical symmetry, I beingthe identity matrix. It becomes then straightforward to compute the gravity coefficients(Eqs. 5 − I is irrelevant.For the centrifuge contribution, all terms in I c are zero except I c (Eq. 14), thus theonly non-zero gravity coefficient is J : J c = k (0) I c ( R ) mR = k f Ω R Gm . (21)For the tidal contribution, the inertia matrix is permanently modified (Eq. 15), meaningthat all gravity coefficients are non-zero. The global contribution to the gravity coefficientsis given by the sum of the centrifuge and tidal contribution (Eq. 13), i.e., J = J c + J t , (22)with J t = k m ⋆ m (cid:18) Rr ⋆ (cid:19) (cid:20) −
32 sin ε (1 − cos 2 ψ ) (cid:21) , (23) 10 – -10 -8 -6 -4 -2 | k | σ (yr -1 ) Fig. 2.— Dependence of the amplitude | k | with the tidal frequency σ for the Earth. Weuse a visco-elastic model (Eq. 17) with υ ∼ kg m − s − ( ˇC´ıˇzkov´a et al. 2012), k = 0 . k f = 0 .
933 (Yoder 1995). 11 –while for the remaining coefficients, the tidal contribution is the total one: C = − k m ⋆ m (cid:18) Rr ⋆ (cid:19) sin ε h cos ε sin θ − cos ε θ − ψ ) + sin ε θ + 2 ψ ) i , (24) S = − k m ⋆ m (cid:18) Rr ⋆ (cid:19) sin ε h cos ε cos θ − cos ε θ − ψ ) + sin ε θ + 2 ψ ) i , (25) C = k m ⋆ m (cid:18) Rr ⋆ (cid:19) h sin ε cos 2 θ + 2 cos ε θ − ψ ) + 2 sin ε θ + ψ ) i , (26) S = − k m ⋆ m (cid:18) Rr ⋆ (cid:19) h sin ε sin 2 θ + 2 cos ε θ − ψ ) + 2 sin ε θ + ψ ) i . (27)Note that, unless the planet evolves in a circular orbit, we cannot replace k by k f inexpression (23) for the terms independent of ψ , because the distance to the star also varieswith ψ as (e.g. Murray & Dermott 1999): r ⋆ = a (1 − e ) / (1 + e cos ν ), where a and e arethe semi-major axis and the eccentricity of the orbit, respectively.
5. Permanent deformation
In the previous section we assumed for the tidal perturbation an elastic response, butthis is not completely true for all harmonics. Indeed, although the position of the star withrespect to the planet surface may not be constant, when we average its motion over oneorbital period, some perturbations do not average to zero, and the planet can assume a 12 –different permanent figure. In order to identify the static harmonics, we need to perform anexpansion of the true anomaly ν in series of the eccentricity e and mean anomaly M :e ik θ r ⋆ = 1 a ∞ X q = −∞ G ( q, e )e i(k θ − , (28)and e i(k θ − ν ) r ⋆ = 1 a ∞ X q = −∞ H ( q, e )e i(k θ − , (29)where the functions G ( q, e ) and H ( q, e ) are power series in e (Tab. 1), and k and 2 q areintegers . Static perturbations thus correspond to frequencies for which σ = k ˙ θ − q ˙ M = 0.They can occur whenever k = q = 0, or if there is a commensurability between the rotationrate and the orbital motion (spin-orbit resonances). For an arbitrary rotation rate (such as the rotation of the Earth), only terms with k = q = 0 will contribute to a permanent deformation of the planet. By replacing equations(28) and (29) in the expressions of the gravity coefficients (Eqs. 23 − θ and the mean anomaly M , we get that all gravity coefficients becomezero, except J t : < J t > = k f m ⋆ m (cid:18) Ra (cid:19) G (0 , e ) (cid:20) −
32 sin ε (cid:21) , (30)where G (0 , e ) = (1 − e ) − / . Although for eccentric orbits the distance to the starconstantly varies with ψ , there is a permanent perturbation along the direction of the twobodies whose average is not zero. We have retained the use of semi-integers for q for a better comparison with previousworks. 13 –Table 1: Hansen coefficients G ( q, e ) and H ( q, e ) to e . q G ( q, e ) H ( q, e ) − / e + e e − / e + e e / e + e / e + e − e + e / e + e − e + e / e e − e / e e − e / e / e The exact expression of these coefficients is given by G ( q, e ) = π R π (cid:0) ar (cid:1) exp(i 2qM) dM and H ( q, e ) = π R π (cid:0) ar (cid:1) exp(i 2 ν ) exp(i 2qM) dM. 14 – When the spin is tidally evolved, it may be captured in a spin-orbit resonance, forwhich Ω = pn (e.g. Goldreich & Peale 1966; Correia & Laskar 2009). The most commonoutcome is the synchronous resonance ( p = 1), observed in the Solar System for themajority of the main satellites. However, non-synchronous configurations are also possible,as it is the case of Mercury ( p = 3 /
2) (Colombo 1965).For synchronous rotation, the planet acquires a permanent deformation alongthe direction that always point to the perturber (e.g. Ferraz-Mello et al. 2008). Fornon-synchronous resonances, in average, one direction always points to the perturber atthe periapse (Goldreich & Peale 1966). Thus, bodies with some rigidity can also acquire apermanent deformation along this direction.In order to obtain the contribution to the gravity coefficients (Eqs. 23 − θ and M . However, since the rotation rate is now in resonance, we have that φ = θ − pM is constant, and therefore we must retain the terms with argument ( q = p k/ J t is still given by expression (30),but the remaining gravity coefficients become: < C > = − k f m ⋆ m (cid:18) Ra (cid:19) sin ε h G ( p/ , e ) cos ε sin φ − H ( p/ , e ) cos ε φ − ω )+ H ( − p/
2) sin ε φ + 2 ω ) i , (31) < S > = − k f m ⋆ m (cid:18) Ra (cid:19) sin ε h G ( p/ , e ) cos ε cos φ − H ( p/ , e ) cos ε φ − ω )+ H ( − p/ , e ) sin ε φ + 2 ω ) i , (32) 15 – < C > = k f m ⋆ m (cid:18) Ra (cid:19) h G ( p, e ) sin ε cos 2 φ + 2 H ( p, e ) cos ε φ − ω )+ 2 H ( − p, e ) sin ε φ + ω ) i , (33) < S > = − k f m ⋆ m (cid:18) Ra (cid:19) h G ( p, e ) sin ε sin 2 φ + 2 H ( p, e ) cos ε φ − ω )+ 2 H ( − p, e ) sin ε φ + ω ) i . (34)Notice that the coefficients < C > and < S > can only be different from zero for“integer” spin-orbit resonances (1 /
1, 2 /
1, 3 /
1, etc.), since the Hansen functions G ( p/ , e )and H ( p/ , e ) are not defined when p is an half-integer (Tab. 1).When the argument of the periapse, ω , is also a fast varying angle ( ˙ ω ≫ τ − a ), whichis often the case for solid close-in planets and satellites, the resonant angle becomes γ k = φ − kω , with k = 0, ±
1, or ± ω , only retaining terms in γ k for a given k value. For moderateobliquity, the dominating term is γ = φ − ω . Thus, we get < C > = < S > = 0 , (35) < C > = k f m ⋆ m (cid:18) Ra (cid:19) H ( p, e ) cos ε γ , (36) < S > = − k f m ⋆ m (cid:18) Ra (cid:19) H ( p, e ) cos ε γ . (37)For damped librations γ is constant. Addopting γ = 0, i.e., the projection of the i -axis inthe orbital plane points to the star at periapse, we further get cos 2 γ = 1 and < S > = 0. 16 –For a planet with the spin-axis normal to the orbit ( ε = 0), and truncating the series H ( p, e ) to e , we retrieve the same results as in Giampieri (2004). In Table 2 we computethe gravity coefficients for the terrestrial planets and main satellites in the Solar Systemand compare with the observed values. We obtain a good agreement in all situations exceptin the cases of Mercury, Venus and the Moon. The observed values in these three situationscorrespond to fossilized values acquired when their rotations were much faster than today(Touma & Wisdom 1994; Correia & Laskar 2001, 2012).
6. Conclusion
Using a vectorial formalism, we derived the gravity field coefficients of a planetin hydrostatic equilibrium with the tidal and centrifugal potentials. We have made noparticular assumption on the inertia tensor, so our results are valid for any rotation rate,obliquity and orbital eccentricity. In particular, they allow us to compute the shape of theplanet for an arbitrary spin-orbit resonance.Combining expressions (21), (30) and (36) we get C J = 3 H ( p, e ) cos ( ε/ p ) + (6 − ε )(1 − e ) − / , (38)which may considerably differ from the ratio 3 /
10 observed for the satellites in the SolarSystem ( p = 1, ε = 0, e = 0) (Table 2). Therefore, if we are able to measure the ratio C /J for exoplanets in eccentric orbits, for instance, by detecting differences in the light curveat each transit (e.g. Barnes & Fortney 2003; Ragozzine & Wolf 2009), we can determinein which spin-orbit the planet is locked, assuming zero obliquity. Inversely, if we assumesynchronous rotation, we can infer the obliquity of the planet.Our work is also important for future studies on the long-term evolution of planetsand satellites. Indeed, as the shape of the planet changes from one spin-orbit resonance to 17 –Table 2: Low order gravity field coefficients ( × − ).Body k f < J > J < C > C Mercury a ± . ± . b ± .
026 - 0.539 ± . b b ± . ± . b ± . ± . c ± . ± . c ± . ± . c ± . ± . c,d ± . ± . e ± . ± . f ± . ± . < C > is computed only whenthe rotation is trapped in a spin-orbit resonance. References: a Smith et al. (2012); b Yoder(1995); c Schubert et al. (2004); d Anderson et al. (2001); e Iess et al. (2007); f Iess et al.(2010). 18 –another, the capture probabilities are considerably modified. As an example, if one supposethat the Moon acquired its present figure in the past when it was closer to the Earth, weconclude that the Moon was not synchronous at the time, since C /J = 0 .
11 (Tab. 2). Ifthe rotation was in a 3/2 spin-orbit resonance (Garrick-Bethell et al. 2006), using equation(38) we can determine that e ≃ .
17, and from expression (36) that a ≃ R ⊕ .We acknowledge support from FCT-Portugal (PTDC/CTE-AST/098528/2008,SFRH/BSAB/1148/2011, PEst-C/CTM/LA0025/2011), and FAPESP (2009/16900-5,2012/13731-0). 19 – REFERENCES