MNRAS 000,1{10 (2016) Preprint 13 August 2018 Compiled using MNRAS LATEX style file v3.0 Toward an initial mass function for giant planets Daniel Carrera,? Melvyn B. Davies, and Anders Johansen Lund Observatory, Department of Astronomy and Theoretical Physics, Lund University, Box 43, SE-221 00 Lund, Sweden Accepted XXX. Received YYY; in original form ZZZ ABSTRACT The distribution of exoplanet masses is not primordial. After the initial stage of planet formation is complete, the gravitational interactions between planets can lead to the physical collision of two planets, or the ejection of one or more planets from the system. When this occurs, the remaining planets are typically left in more eccentric orbits. Here we use present-day eccentricities of the observed exoplanet population to reconstruct the initial mass function of exoplanets before the onset of dynamical instability. We developed a Bayesian framework that combines data from N-body simulations with present-day observations to compute a probability distribution for the planets that were ejected or collided in the past. Integrating across the exoplanet population, we obtained an estimate of the initial mass function of exoplanets. We find that the ejected planets are primarily sub-Saturn type planets. While the present- day distribution appears to be bimodal, with peaks around ∼ 1MJ and ∼ 20M⊕, this bimodality does not seem to be primordial. Instead, planets around ∼ 60M⊕ appear to be preferentially removed by dynamical instabilities. Attempts to reproduce exoplanet populations using population synthesis codes should be mindful of the fact that the present population has been depleted of intermediate-mass planets. Future work should explore how the system architecture and multiplicity might alter our results. Key words: planets and satellites: dynamical evolution and stability { planets and satellites: gaseous planets { planets and satellites: formation 1 INTRODUCTION et al.(2009b); these authors also showed how the eccentric- ities can be subsequently dampened by a planetesimal belt The past twenty years of exoplanet observations have re- for less massive planets (m + m < M ). vealed a large diversity of planetary systems (e.g. Mayor 1 2 J Several authors have noticed a mass-eccentricity corre- et al. 2011; Borucki et al. 2011; Batalha et al. 2013), in- lation among exoplanets: low-mass planets tend to be on cluding a large number of planets with orbital eccentricities less eccentric orbits than high-mass planets (e.g. Ribas & much higher than those found in the solar system. These Miralda-Escud´e 2007; Ford & Rasio 2008; Raymond et al. eccentricities challenged our understanding of planet forma- 2010). Raymond et al.(2010) proposed that planet masses tion occurring in a disk, where eccentricities are dampened in the same system are correlated. Specifically, systems that by planet-disk interactions. Several authors have suggested form high-mass planets also tend to form equal-mass plan- that these high eccentricities might have a dynamical origin ets, while low-mass planet systems produce a wide diversity arXiv:1703.08647v1 [astro-ph.EP] 25 Mar 2017 (e.g. Ford et al. 2001; Juri´c& Tremaine 2008). In this view, of mass ratios. The present work is an extension of this idea. the eccentricities arise from planet-planet interactions which We use the present-day eccentricities of observed exoplanets can cause a planetary system to become dynamically unsta- to estimate the masses of the planets that were ejected or ble. During a dynamical instability, close encounters between collided. The result of this process is the exoplanet initial planets lead to either collisions or planet-planet scatterings. mass function (IMF). This dynamically active period ends only with a collision, or the ejection of one of the planets from the system. Juri´c& This paper is organized as follows. In Section2 we sum- Tremaine(2008) showed that this type of process can repro- marize the theory behind dynamical stability and discuss duce the observed distribution of exoplanet eccentricities (at previous work. In Section3 we present an introduction to least those above e = 0:2) for a wide range of possible ini- Approximate Bayesian Computation, and describe how the tial conditions. A similar result was obtained by Raymond algorithm can be adapted to the problem we want to solve. We describe our core simulations in Section4, including our data selection method. The results of these simulations ? E-mail: [email protected] are presented in Section5. In Section6 we discuss possi- © 2016 The Authors 2 Carrera et al. ble caveats and suggest opportunities for future research. and where M is the mass of the star, mk is the mass of Finally, we summarize and conclude in Section7. a planet, and a and e are the planet semimajor axis and eccentricity in barycentric coordinates. Therefore, for given planet masses and eccentricities, there is a critical value of δ that guarantees Hill stability (see also the discussion in 2 DYNAMICAL INSTABILITY Barnes & Greenberg 2006). In this section we cover some important background and key Raymond et al.(2009a) refer to the entire left-hand- results regarding the stability of planetary systems. In prin- side of Equation (3) as β, and the right-hand-side as βcrit. ciple, the eccentricity of the surviving planet is determined They showed that planet-planet scatterings naturally lead to by the need to conserve energy and angular momentum. For planetary systems at the edge of dynamical stability, with example, in the simple case of two planets, on coplanar or- β/βcrit just above 1, and that this result is in agreement with bits the energy E and angular momentum Λ of each planet observation. This is further evidence that dynamical insta- is, bilities driven by planet-planet scatterings occur frequently in planetary systems. For a system of two low-mass planets with near-circular GMm E = − (1) coplanar orbits, the Hill stability criterion (Equation (3)) is 2a p s approximately ∆ > 2 3 (Chambers et al. 1996), where ∆ is Ga¹1 − e2º the semimajor axis separation measured in mutual Hill radii, Λ = Mm ; (2) M + m a − a where M is the stellar mass, and m, a, and e are the mass, ∆ = 2 1 ; (8) R semimajor axis, and eccentricity of the planet (e.g. Davies H 1/3 et al. 2014). In practice, this has limited utility because the m1 + m2 a1 + a2 RH = ; (9) problem is degenerate, even in the simplest case of a two- 3M 2 planet system. After a dynamical instability, the escaping where m1, m2, a1, and a2 are the masses and semimajor planet carries an uncertain amount of angular momentum axes of the two planets. For systems with more than two (Ford et al. 2001). Even if the angular momentum of the planets there is no analytic solution, but Chambers et al. ejected planet was known, one would still have only two (1996) showed that these systems are probably unstable for equations for five unknowns (the mass of the ejected planet, separations up to ∆ = 10, at least for equal-mass planets. The and the initial mass and semimajor axis of the two planets). time before planets experience their first close encounter (tce) In sections3 and4 we describe the statistical approach that grows exponentially with ∆. For a given value of ∆, tce seems we use to tackle this degeneracy. to depend weakly on the number of planets (for systems with There are two common definitions of dynamical stabil- more than two planets) and the planet masses (Chambers ity. In Hill stability, the ordering of the planets, in terms of et al. 1996; Faber & Quillen 2007). For a planetary system their proximity to the central star, is fixed. In other words, with ten equal-mass planets, Faber & Quillen(2007) found the planet orbits never cross, but the outermost planet is al- lowed to escape the system. The other, more stringent defini- 1 tion, is Lagrange stability. In it, planets retain their ordering tce µ µ 12 log10 ∼ −5 − log10 + 1:44∆ ; (10) (i.e. are Hill stable), remain bound to the star, and variations yr 10−3 10−3 in semimajor axis and eccentricity remain bounded. While where µ = m/M is the planet-star mass ratio. This is an Lagrange stability is the more useful definition, it has proven extremely steep dependence on ∆, meaning that two systems more difficult to solve mathematically than Hill stability. with similar ∆ values can have very different lifetimes. There is no analytic solution for the evolution of three gravitating bodies, but in some cases it is possible to derive analytic constraints on the motion of the planets (e.g. Zare 1977). These constraints can be understood as limitations on 3 APPROXIMATE BAYESIAN angular momentum exchange between planets. For example, COMPUTATION Gladman(1993) showed that, to first order, a pair of planets We saw in section2 that it is not possible to exactly deter- on coplanar orbits are guaranteed to be Hill stable whenever mine the mass of an ejected exoplanet given the present-day observables. In this section we explain how one can use mod- −3 µ2 2 4/3 µ1 µ2 ern statistical methods together with N-body simulations to α µ1 + ¹µ1γ1 + µ2γ2δº > 1 + 3 (3) δ2 α4/3 obtain a probability distribution for the ejected planet mass. Our tool of choice is Approximate Bayesian Compu- where tation, or ABC (see the introduction by Marin et al. 2011). Like all Bayesian algorithms, ABC is a way to solve the r a problem δ = 2 (4) a1 q − 2 P¹θjDº / P¹DjθºP¹θº; (11) γk = 1 ek (5) m where D is some data set, θ is a set of model parameters, µ = k (6) k M and P¹θº is the Bayesian prior. In its simplest form, the ABC α = µ1 + µ2 (7) algorithm looks like this MNRAS 000,1{10 (2016) Giant planet IMF 3 0 for i = 1 : N do if Ci then 0 repeat m1;i m/(1 + qi º θ0 P¹θº else 0 0 D model¹θ º m1;i m until ρ¹D; D0º < end if 0 θi θ m2;i qi ∗ m1;i end for end for The function ρ is a distance measure that compares The set of values fm1;i; m2;i g approximately follow the 0 the synthetic dataset D and the observed data D, and is posterior distribution P¹m1; m2jm; eº.