Astronomy & Astrophysics manuscript no. charon ©ESO 2020 December 7, 2020 Tidal evolution of the Pluto–Charon binary Alexandre C. M. Correia1; 2 1 CFisUC, Department of Physics, University of Coimbra, 3004-516 Coimbra, Portugal 2 ASD, IMCCE, Observatoire de Paris, PSL Université, 77 Av. Denfert-Rochereau, 75014 Paris, France December 7, 2020; Received; accepted To be inserted later ABSTRACT A giant collision is believed to be at the origin of the Pluto–Charon system. As a result, the initial orbit and spins after impact may have substantially differed from those observed today. More precisely, the distance at periapse may have been shorter, subsequently expanding to its current separation by tides raised simultaneously on the two bodies. Here we provide a general 3D model to study the tidal evolution of a binary composed of two triaxial bodies orbiting a central star. We apply this model to the Pluto–Charon binary, and notice some interesting constraints on the initial system. We observe that when the eccentricity evolves to high values, the presence of the Sun prevents Charon from escaping because of Lidov-Kozai cycles. However, for a high initial obliquity for Pluto or a spin- orbit capture of Charon’s rotation, the binary eccentricity is damped very efficiently. As a result, the system can maintain a moderate eccentricity throughout its evolution, even for strong tidal dissipation on Pluto. Key words. planets and satellites: dynamical evolution and stability — minor planets, asteroids: individual (Pluto, Charon) 1. Introduction nation of the orbital plane of Charon, that is, 122˝. Indeed, maps of the surface have been created using HST images and “mutual In 1978, a regular series of astrometric observations of Pluto events” (e.g., Drish et al. 1995; Stern et al. 1997; Young et al. revealed that the images were consistently elongated, denounc- 1999; Buie et al. 2010), and although the authors assumed the ing the presence of Pluto’s moon, Charon (Christy & Harring- above obliquity in the creation of these maps, the solution would ton 1978). The orbital parameters determined for this system not have held together if the obliquity was completely incorrect. show that the two bodies evolve in an almost circular orbit with Assuming equal densities, the normalized angular momen- a 6.387-day period, and that the system also shows an impor- ˝ tum density of the Pluto–Charon pair is 0.45 (McKinnon 1989), tant inclination of about 122 with respect to the orbital plane exceeding the critical value 0.39, above which no stable rotating of Pluto around the Sun (e.g., Stern et al. 2018). Charon has an single object exists (e.g., Lin 1981; Durisen & Tohline 1985). important fraction of the mass of the system (about 12%), and The proto-planetary disk is not expected to produce such sys- therefore can be considered more as a binary planet rather than a tems, and so alternative theories have been proposed for their satellite. Indeed, the barycenter of the Pluto–Charon system lies origin. Harrington & van Flandern(1979) first suggested that outside the surface of Pluto. Later, it was found that four addi- Pluto and Charon could be escaped satellites of Neptune after tional tiny satellites move around the barycenter of the system, an encounter with another planet, an unlikely scenario because also in nearly circular and coplanar orbits (Weaver et al. 2006; Triton is on a retrograde orbit (McKinnon 1984). More reliable Brozovic´ et al. 2015). hypotheses were proposed that take into account the excess of The brightness of Pluto varies by some tens of percent with angular momentum in the system, such as binary fission of a a period of 6.387 days (e.g., Walker & Hardie 1955; Tholen & rapidly rotating body (e.g., Lin 1981; Nesvorný et al. 2010) or Tedesco 1994; Buie et al. 2010). Although this period coincides the accumulation process of planetesimals in heliocentric orbits with the orbital period of Charon, it has been identified as the (e.g., Tancredi & Fernández 1991; Schlichting & Sari 2008). rotation of Pluto, since Charon itself is too dim to account for the The above-mentioned theories have some limitations and it amplitude of the variation. Therefore, at present, the rotation of is more commonly accepted that Charon resulted from the giant Pluto is synchronous with the orbit of Charon, keeping the same collision of two proto-planets in the early inner Kuiper belt (e.g., arXiv:2012.02576v1 [astro-ph.EP] 4 Dec 2020 face toward its satellite. The present configuration likely resulted McKinnon 1989; Canup 2005; Rozner et al. 2020). This scenario from the action of tidal torques raised on Pluto by Charon. Tidal provides the system with its large angular momentum and can torques raised on Charon by Pluto are even stronger, and so the also explain the additional small moons in the system (Canup satellite is also assumed to be synchronous with Pluto, which 2011). The outward migration of Neptune may have instigated corresponds to a final equilibrium situation (e.g., Farinella et al. huge perturbations in previously stable zones of the Kuiper belt, 1979; Cheng et al. 2014). and oblique low-velocity collisions between similarly sized ob- From photometric observations, Andersson & Fix(1973) jects should have been frequent at the time (e.g., Malhotra 1993). found the angle between the spin of Pluto and its orbit around the Such a collision probably produced an intact Charon, although Sun to be approximately 90˝ ˘40˝. The uncertainty on this value it is also possible that a disk of debris orbited Pluto from which is significant, but it clearly suggests a high obliquity. Because of Charon later accumulated. The resulting system is a close binary the complete tidal evolution that is evident in the system, the in an eccentric orbit, with the separation at periapse not exceed- obliquity has usually been assumed to be the same as the incli- ing many Pluto radii (Canup 2005, 2011). Article number, page 1 of 14 A&A proofs: manuscript no. charon angular velocity and the rotational angular momentum vectors m1 Ω of the body are given by Ω “ pΩi; Ω j; Ωkq and L “ pLi; L j; Lkq, 1 respectively, which are related through the inertia tensor I as 1 r L “ I ¨ Ω ô Ω “ I´ ¨ L ; (1) Ω0 r 12 where I11 I12 I13 rS I “ » I12 I22 I23 fi ; (2) I I I m r m — 13 23 33 ffi 0 02 2 – fl Fig. 1. Jacobi coordinates, where r is the position of m1 relative to m0 2 I22I33 ´ I23 I13I23 ´ I12I33 I12I23 ´ I22I13 (inner orbit), and rs is the position of m2 relative to the center of mass 1 of m and m (outer orbit). The bodies with masses m and m are con- ´1 2 0 1 0 1 I “ » I13I23 ´ I12I33 I11I33 ´ I13 I12I13 ´ I11I23 fi ; sidered oblate ellipsoids with angular velocity Ω, and body m2 is con- ∆I 2 sidered a point-mass. — I12I23 ´ I22I13 I12I13 ´ I11I23 I11I22 ´ I ffi — 12 ffi – (3)fl Most previous studies on the past orbital evolution of the and Pluto–Charon system (Farinella et al. 1979; Lin 1981; Mignard 2 2 2 1981; Dobrovolskis et al. 1997; Cheng et al. 2014) assume that ∆I “ I11I22I33 ` 2I12I13I23 ´ I11I23 ´ I22I13 ´ I33I12 : (4) both spin axes are normal to the binary orbital plane (2D model), and therefore limit the evolution to the rotations. Although this The gravitational potential of the ellipsoidal body at a is the expected outcome of tidal evolution, after a large colli- generic position r relative to its center of mass is given by (e.g., sion the obliquity of Pluto can take any value (e.g., Dones & Goldstein 1950) Tremaine 1993; Kokubo & Ida 2007; Canup 2011). Previous Gm 3G 1 studies also assume that the Pluto–Charon binary is alone. How- Vpm; I; rq “ ´ ` rˆ ¨ I ¨ rˆ ´ trpIq ; (5) r 2r3 3 ever, the Sun exerts a torque on the system that causes both the „ obliquities and the orbital plane of the binary to precess (Do- where G is the gravitational constant, rˆ “ r{r “ pxˆ; yˆ; zˆq is brovolskis & Harris 1983). For the Earth–Moon system, Touma the unit vector, and trpIq “ I11 ` I22 ` I33. We neglect terms & Wisdom(1994, 1998) showed that the presence of the Sun is in pR{rq3, where R is the mean radius of the body (quadrupo- critical to understand its early evolution. Some preliminary work lar approximation). Adopting the Legendre polynomial P2pxq “ on the Pluto–Charon system (Carvalho 2016) suggests that the p3x2 ´ 1q{2, we can rewrite the previous potential as obliquity and the Sun may also play a role. Finally, Cheng et al. Gm G (2014) showed that the inclusion of the gravitational harmonic V m; I; r I I P y I I P z p q “ ´ ` 3 22 ´ 11 2pˆq ` 33 ´ 11 2pˆq coefficient C22 in the analysis allows smooth, self-consistent r r ” evolution to the synchronous state. It is therefore important to ` 3 I xˆyˆ˘` I xˆzˆ ``I yˆzˆ :˘ (6) simultaneously take into account the effect of the obliquity, the 12 13 23 Sun, and the residual C22, in order to obtain a more realistic de- ` ˘ı scription of the past history of the Pluto–Charon system. 2.2. Point-mass problem In Sect.2, we first derive a full 3D model (for the orbits and spins) that is suitable to describe the tidal evolution of a We now consider that the ellipsoidal body orbits a point-mass M hierarchical three-body system, where the inner two bodies are located at r. The force between the two bodies is easily obtained assumed to be triaxial ellipsoids.