AMD-Stability and the Classification of Planetary Systems

Astronomy & Astrophysics manuscript no. 30022 hk rev c ESO 2017 March 23, 2017

AMD-stability and the classiﬁcation of planetary systems.

J. Laskar and A. Petit

ASD/IMCCE, CNRS-UMR8028, Observatoire de Paris, PSL, UPMC, 77 Avenue Denfert-Rochereau, 75014 Paris, France e-mail: [email protected] Accepted XXX. Received YYY; in original form ZZZ

ABSTRACT

We present here in full detail the evolution of the angular momentum deficit (AMD) during collisions as it was described in (Laskar 2000). Since then, the AMD has been revealed to be a key parameter for the understanding of the outcome of planetary formation models. We define here the AMD-stability criterion that can be easily verified on a newly discovered planetary system. We show how AMD- stability can be used to establish a classification of the multiplanet systems in order to exhibit the planetary systems that are long-term stable because they are AMD-stable, and those that are AMD- unstable which then require some additional dynamical studies to conclude on their stability. The AMD-stability classification is applied to the 131 multiplanet systems from The Extrasolar Planet Encyclopaedia database (exoplanet.eu) for which the orbital elements are sufficiently well known. Key words. Celestial mechanics -Planets and satellites: general - Planets and satellites: formation

1. Introduction cal works usually use empirical criteria based on the Hill radius proposed by Gladman (1993) and The increasing number of planetary systems has refined by Chambers et al. (1996); Smith & Lis- made it necessary to search for a possible classifica- sauer (2009), and Pu & Wu (2015). These crite- tion of these planetary systems. Ideally, such a clas- ria of stability usually multiply the Hill radius by sification should not require heavy numerical analy- a numerical factor ∆sep empirically evaluated to a sis as it needs to be applied to large sets of systems. value around 10. They are extensions of the an- Some possible approaches can rely on the stability alytical results on Hill spheres for the three-body analysis of these systems, as this stability analysis is problem (Marchal & Bozis 1982). Works on chaotic also part of the process used to consolidate the dis- motion caused by the overlap of mean motion res- covery of planetary systems. The stability analysis onances (MMR, Wisdom 1980; Deck et al. 2013; can also be considered a key part to understanding Ramos et al. 2015) could justify the Hill-type cri- the wider question of the architecture of planetary teria, but the results on the overlap of the MMR systems. In fact, the distances between planets and island are valid only for close orbits and for short- other orbital characteristic distributions is one of term stability. the oldest questions in celestial mechanics, the most Another approach to stability analysis is to con- famous attempts to set laws for this distribution of sider the secular approximation of a planetary sys- planetary orbits being the Titius-Bode power laws tem. In this framework, the conservation of the (see Nieto 1972; Graner & Dubrulle 1994 for a re- semi-major axis leads to the conservation of another view). quantity, the angular momentum deficit (AMD; Recent research has focused on statistical anal- Laskar 1997, 2000). An architecture model can be ysis of observed architecture (Fabrycky et al. 2014; developed from this consideration (Laskar 2000). Lissauer et al. 2011; Mayor et al. 2011), eccentricity The AMD can be interpreted as a measure of the distribution (Moorhead et al. 2011; Shabram et al. orbits’ excitation (Laskar 1997) that limits the plan- arXiv:1703.07125v2 [astro-ph.EP] 22 Mar 2017 2016; Xie et al. 2016), or inclination distribution ets close encounters and ensures long-term stability. (Fang & Margot 2012; Figueira et al. 2012); see Therefore a stability criterion can be derived from Winn & Fabrycky (2015) for a review. The analysis the semi-major axis, the masses and the AMD of of these observations has been compared with mod- a system. In addition, it can be demonstrated that els of system architecture (Fang & Margot 2013; the AMD decreases during inelastic collisions (see Pu & Wu 2015; Tremaine 2015). These theoreti- section 2.3), accounting for the gain of stability of a

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lower multiplicity system. Here we extend the previous analysis of (Laskar 2000), and derive more H = H0 + H1 (4) precisely the AMD-stability criterion that can be with used to establish a classification of the multiplane- n 2 n tary systems. 1 X kr˜ik X m0mi H0 = − G (5) In the original letter (Laskar 2000), the de- 2 mi ri tailed computations were referred to as a preprint i=1 i=1 to be published. Although this preprint has been and in nearly final form for more than a decade, 2 1 ku˜ 0k X mim j and has even been provided to some researchers H = − G (6) 1 2 m ∆ (Hernández-Mena & Benet 2011), it is still unpub- 0 1≤i< j i j lished. In Sects. 2 and 3 we provide the fundamen- The integrable part, H , is the Hamiltonian of tal concepts of AMD, the full description, and all 0 a sum of disjoint Kepler problems of a single planet proofs for the model that was described in (Laskar of mass m around a fixed Sun of mass m . A set of 2000). This material is close to the unpublished i 0 adapted variables for H will thus be given by the preprint. Section 4 recalls briefly the model of plan- 0 elliptical elements, (a , e , i , λ , $ , Ω ), where a is etary accretion based on AMD stochastic trans- k k k k k k k the semi-major axis, e the eccentricity, i the in- fers that was first presented in (Laskar 2000). This k k clination, λ the mean longitude, $ the longitude model provides analytical expressions for the av- k k of the perihelion, and Ω the longitude of the node. eraged systems architecture and orbital parameter k They are defined as the elliptical elements associ- distribution, depending on the initial mass distri- ated to the Hamiltonian bution (Table 2). 2 In Sect. 5, we show how the AMD-stability cri- 1 kr˜kk mk H0k = − µ (7) terion presented in section 3 can be used to develop 2 mk rk a classification of planetary systems. This AMD- stability classification is then applied to a selec- with µ = Gm0. tion of 131 multiplanet systems from The Extra- solar Planet Encyclopaedia database (exoplanet.eu) 2.2. Angular momentum deficit (AMD) with known eccentricities. Finally, in Sect. 6 we dis- cuss our results and provide some possible extension The total linear momentum of the system is null in for this work. the barycentric reference frame Xn Xn 2. Angular momentum deficit L = miu˙ i = u˜ i = r˜0 = 0 . (8) i=0 i=0 2.1. Planetary Hamiltonian Let G be the total angular momentum. Its ex-

Let P0, P1,..., Pn be n + 1 bodies of masses pression is not changed by the linear symplectic (u, u˜) −→ (r, r˜) m0, m1,..., mn in gravitational interaction, and let change of variable . We have thus O be their centre of mass. For every body Pi, we Xn Xn −−→ G = u ∧ u˜ = r ∧ r˜ (9) denote ui = OPi. In the barycentric reference frame i i i i with origin O, Newton’s equations of motion form i=0 i=1 a differential system of order 6(n + 1) and can be When expressed in heliocentric variables, the an- written in Hamiltonian form using the canonical co- gular momentum is thus the sum of the angular ordinates (ui, u˜ i = miu˙ i)i=0,n with Hamiltonian momentum of the Keplerian problems of the un- perturbed Hamiltonian H0. In particular, if the an- n 2 gular momentum direction is assumed to be the axis 1 X ku˜ ik X mim j H = − G , (1) z, the norm of the angular momentum is 2 mi ∆i j i=0 0≤i< j Xn q 2 G = Λk 1 − ek cos ik, (10) where ∆i j = kui − u jk, and G is the constant of k=1 gravitation. The reduction of the centre of mass is √ achieved by using the canonical heliocentric vari- where Λk = mk µ ak. For such a system, the AMD ables of Poincaré (ri, r˜i) (Poincaré1905; Laskar & is defined as the difference between the norm of the Robutel 1995), defined as angular momentum of a coplanar and circular system with the same semi-major axis values and the r0 = u0; ri = ui − u0 for i , 0 (2) norm of the angular momentum (G), i.e. (Laskar, 1997, 2000) r˜0 = u˜ 0 + u˜ 1 + ··· + u˜ n; r˜i = u˜ i for i , 0 (3) Xn q This Hamiltonian can then be split into an in- 2 C = Λk 1 − 1 − ek cos ik (11) tegrable part, H0, and a perturbation, H1, k=1 Article number, page 2 of 18 Laskar & Petit: AMD-stability

2.3. AMD and collision of orbits (18) The instabilities of a planetary system often result Thus, the Keplerian energy of the system decreases in a modification of its architecture. A planet can during collision. Part of the kinetic energy is dis- be ejected from the system or can fall into the star; pelled during the collision. As expected, there is in both cases this results in a loss of AMD for the no loss of energy when m2r˜1 = m1r˜2, that is, as system. The outcome of the AMD after a planetary m1m2 , 0, when u˙ 1 = u˙ 2. As an immediate con- collisions is less trivial and needs to be computed. sequence of the decrease of energy during the colli- Assume that among our n + 1 bodies, the (totally sion, we have m inelastic) collision of two bodies of masses 1 and 1 m 1 − m m2, and orbits O1, O2 occurs, forming a new body ≥ + (19) (m3, O3). During this collision we assume that the a3 a1 a2 other bodies are not affected. The mass is conserved with m3 = m1 + m2, (12) m1 m1 and the linear momentum in the barycentric refer- m = = (20) m1 + m2 m3 ence frame is conserved so u˜ 3 = u˜ 1 + u˜ 2, and also r˜ = r˜ + r˜ ; (13) 3 1 2 2.3.2. AMD evolution during collision On the other hand, r3 = r1 = r2 at the time of √ the collision, so the angular momentum is also con- Let f (x) = 1/ x. As f 0(x) < 0 and f 00(x) > 0, we served have, as f is decreasing and convex, r3 ∧ r˜3 = r1 ∧ r˜1 + r2 ∧ r˜2 (14) ! ! ! ! It should be noted that the transformation of 1 m 1 − m 1 1 f ≤ f + ≤ m f + (1 − m) f the orbits (m1, O1) + (m2, O2) −→ (m3, O3) during the a3 a1 a2 a1 a2 collision is perfectly defined by Eqs. (12, 13). The (21) problem which remains is to compute the evolution of the elliptical elements during the collision. thus √ √ √ 2.3.1. Energy evolution during collision m3 a3 ≤ m1 a1 + m2 a2 (22) Just before the collision, we assume that the or- During the collision, the angular momentum is con- bits (m1, O1) and (m2, O2) are elliptical heliocentric served (14), and so is the conservation of its normal orbits. At the time of the collision, only these two component, that is bodies are involved and the other bodies are not √ q affected. The evolution of the orbits are thus given 2 m3 µa3 1 − e cos i3 = by the conservation laws (12, 13). The Keplerian 3 energy of each particle is √ q √ q − 2 − 2 2 m1 µa1 1 e1 cos i1 + m2 µa2 1 e2 cos i2 . 1 kr˜ik mi mi hi = − µ = −µ (15) (23) 2 mi ri 2ai We deduce that in all circumstances we have a de- At collision, we have r = r = r = r; we have thus 1 2 3 crease of the angular momentum deficit during the the conservation of the potential energy collision, that is m3 m1 + m2 m1 m2 −µ = −µ = −µ − µ . (16) C3 ≤ C1 + C2 (24) r3 r r1 r2 with The change of Keplerian energy is thus given by the √ q change of kinetic energy 2 Ck = mk µak(1 − 1 − ek cos ik)(k = 1, 3) (25) 2 2 2 1 kr˜3k 1 kr˜1k 1 kr˜2k δh = h3 − h1 − h2 = − − ; (17) The equality can hold in (24) only if m1 = 0, 2 m3 2 m1 2 m2 m2 = 0, or a1 = a2 and u˙ 1 = u˙ 2, that is when one of that is, with Eqs. (12, 13), the bodies is massless, or when the two bodies are on the same orbit and at the same position (at the 2m1m2m3δh time of the collision, we also have r1 = r2). = m m (r˜ + r˜ )2 − m (m + m )r˜2 − m (m + m )r˜2 The diminution of AMD during collisions acts 1 2 1 2 2 1 2 1 1 1 2 2 as a stabilisation of the system. A parallel can be 2 2 2 2 = −m2r˜1 − m1r˜2 + 2m1m2r˜1 · r˜2 made with thermodynamics, the AMD behaving for 2 the orbits like the kinetic energy for the molecules = −(m2r˜1 − m1r˜2) of a perfect gas. The loss of AMD during collisions ≤ 0 can thus be interpreted as a cooling of the system. Article number, page 3 of 18 A&A proofs: manuscript no. 30022 hk rev

3. AMD-stability We say that a planetary system is AMD-stable if the angular momentum deﬁcit (AMD) amount in the system is not suﬃcient to allow for planetary collisions. As this quantity is conserved in the secular system at all orders (see Appendix B), we con- collisions jecture that in absence of short period resonances, the AMD-stability ensures the practical1 long-term stability of the system. Thus for an AMD-stable system, short-term stability will imply long-term stability. The condition of AMD-stability is obtained when the orbits of two planets of semi-major axis a, a0 cannot intersect under the assumption that the total AMD C has been absorbed by these two planets. It can be seen easily that the limit condition of collision is obtained in the planar case and can thus be written as

a(1 + e) = a0(1 − e0) (26) √ √ p √ m µa(1 − 1 − e2) + m0 µa0(1 − 1 − e02) = C, (27)

where (m, a, e) are the parameters of the inner orbit and (m0, a0, e0) those of the outer orbit (a ≤ a0). collisions 3.1. Collisional condition We assume that a, a0, m, m0 are non-zero. Denoting α = a/a0, γ = m/m0, the system in Eq. (27) becomes

D(e, e0) = αe + e0 − 1 + α = 0 (28)

√ √ √ C (e, e0) = γ α(1 − 1 − e2) + (1 − 1 − e02) (29) = C/Λ0 √ with Λ0 = m0 µa0, and where (e, e0) = C/Λ0 is C Fig. 1. Collision conditions for e = 1/α − 1. Case (a): called the relative AMD. As e and e0 are planetary 0 α < 1/2 ⇐⇒ e0 > 1. Case (b): α > 1/2 ⇐⇒ e0 < 1. eccentricities, we also have Collisions can only occur in the shaded region. 0 ≤ e ≤ 1 ; 0 ≤ e0 ≤ 1 . (30) have thus the diﬀerent cases (Fig. 1) The set of collisional conditions (Eqs. 28, 29, 30) can be solved using Lagrange multipliers. We are 1 (a) α < ⇐⇒ e0 > 1 and e ≤ 1 (31) looking for the minimum value of the relative AMD 2 0 C (e, e ) (29) for which the collision condition (28) 1 (b) α > ⇐⇒ e < 1 and e < e (32) is satisﬁed. These conditions are visualised in the 2 0 0 (e, e0) plane in Fig. 1. The collision condition (28) AB corresponds to the segment of Fig. 1. The do- and the limit case α = 1/2, for which e0 = 1. In all main of collisions is limited by the conditions (30). 0 cases, the Lagrange multipliers condition is written For e = 0, we have e0 = 1 − α < 1, and the inter- 0 section of the collision line with the axis e = 0 is λ∇D(e, e0) = ∇C (e, e0), (33) obtained for e0 = 1/α − 1. This value can be greater or smaller than 1 depending on the value of α. We which gives 1 Here practical stability means stability over a very √ √ √ 1 − e02 α 1 − e2 long time compared to the expected life of the central = (34) star. e0 γ e Article number, page 4 of 18 Laskar & Petit: AMD-stability

This relation allows e0 to be eliminated in the 3.3. Behaviour of the critical AMD collision condition (28), which becomes an equation We now analyse the general properties of the crit- in the single variable e, and parameters (α, γ): ∂F ical AMD function Cc(α, γ). As ∂e (e, α, γ) > 0, we γe can apply the implicit function theorem to the do- F(e, α, γ) = αe + q − 1 + α = 0. (35) main De,α,γ, which then ensures that the solution α(1 − e2) + γ2e2 of the collision equation (35), ec(α, γ), is a continuous function of γ (and even analytic for γ ∈]0, +∞[). D Here F(e, a, γ) is properly defined for (e, a, γ) in the Moreover, on e,α,γ, 2 domain De,α,γ defined by e ∈ [0, 1], α ∈]0, 1], γ ∈ ∂F αe 1 − e ]0, +∞[, as in this domain, 1 − e2 + γ2e2/α > 0. We (e, α, γ) = ≥ 0 . (40) ∂γ 2 2 23/2 also have α 1 − e + γ e We also have ∂F αγ (e, α, γ) = α + . (36) e 1 − e2 ∂e 2 2 23/2 ∂ec c c α(1 − e ) + γ e (α, γ) = − ≤ 0, (41) ∂γ 2 2 23/2 α 1 − ec + γ ec + γ ∂F Thus, (e, α, γ) > 0 on the domain De,α,γ and and ec(α, γ) is a decreasing function of γ. For any ∂e given values of the semi-major axes ratio α, and F(e, α, γ) is strictly increasing with respect to e for masses γ, we can thus find the critical value Cc(α, γ) e ∈ [0, 1]. Moreover, as 0 < α < 1, which allows for a collision (39). For the critical value Cc(α, γ), a single solution corresponds to the F(0, α, γ) = −1 + α < 0 ; F(1, α, γ) = 2α > 0 ; tangency condition (Fig. 1), and this solution is ob- (37) tained at the critical value ec(α, γ) for the eccentricity of the orbit O. The values of the critical relative and AMD Cc(α, γ) are plotted in Figure 2 versus α, for different values of γ. Deriving Eq. 29 with respect γe0 to γ, one obtains F(e , α, γ) = > 0 . (38) 0 q √ − 2 2 2 ∂C √ α 1 e0 + γ e0 = α 1 − 1 − e2 ∂γ (42) The equation of collision (35) thus always has a √ e ∂e e0 ∂e0 single solution ec in the interval ]0, min(1, e0)[. This +γ α √ + √ . ensures that this critical value of e will fulfil the con- 1 − e2 ∂γ 1 − e02 ∂γ dition (30). The corresponding value of the relative Using the two relations (28, 34), this reduces to 0 AMD Cc(α, γ) = C (ec, ec) is then obtained through ∂Cc(α, γ) √ q (29). = α 1 − 1 − e2 > 0 (43) ∂γ c Thus C (α, γ) strictly increases with γ. In the same 3.2. Critical AMD Cc(α, γ) c way We have thus demonstrated that for a given pair of    2  ratios of semi-major axes and masses, (α, γ), there is ∂Cc(α, γ) γ  1 + ec  = √ 1 −  < 0 (44) always a unique critical value Cc(α, γ) of the relative  q  0 ∂α 2 α  2  AMD C = C/Λ which defines the AMD-stability. 1 − ec The system of two planets is AMD-stable if and and Cc(α, γ) decreases with α. only if Now that the general behaviour of the critical AMD C (α, γ) is known, we can specify its explicit C c expression in a few special cases that are displayed C = 0 < Cc(α, γ) . (39) Λ in Table 1. The computations and the higher order developments can be found in Appendix C. The value of the critical AMD Cc(α, γ) is ob- A development of Cc(α, γ) for α −→ 1 can also tained by computing first the critical eccentricity be obtained (see Appendix C). With η = 1 − α, we ec(α, γ) which is the unique solution of the colli- have sional equation (35) in the interval [0, 1]. The criti- 2 0 γ η 3 cal AMD is then Cc(α, γ) = C (ec, ec) (Eq.29), where Cc(α, γ) ∼ + O(η ) . (45) 0 γ + 1 2 the critical value ec is obtained from ec through Eq. 28. It is important to note that the critical We now present two applications of the AMD- AMD, and thus the AMD-stability condition, de- stability, the formation of planetary systems, and a pends only on (α, γ). classification of observed planetary systems.

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Table 1. Special values of Cc(α, γ). The detail of the computations is provided in Appendix C.

0 γ α ec(α, γ) ec(α, γ) Cc((α, γ) 1 − α α(1 − α) √ √ γ → 0 α < 1/2 1 − 2 γ2 1 − 2α + 2 γ2 1 − 2 α(1 − α) + αγ (1 − 2α)2 (1 − 2α)2 γ γ → 0 α = 1/2 1 − (4γ)2/3 21/3γ2/3 √ √ 2 q e0 αe0γ √ 1 γ → 0 α > 1/2 e0 − p γ √ α − 2 − γ α(2α − 1) 2α − 1 α √ 1 1 − α 1 α(1 − α) √ α(1 − α)2 1 γ → +∞ 0 < α < 1 √ 1 − α − √ 1 − α(2 − α) − γ 2 − α γ 2 − α 2 − α γ √ q √ 1 − 1 − α + α2 √ √ α − 2 + 2 1 − α + α2 1 0 ≤ α ≤ 1 1 − α + α2 − α 1 + α − √ α α √ q √ − α 1 − 2α + 2 1 − α + α2 √ 1 − α 1 − α √ α 0 ≤ α ≤ 1 (1 − α)2 1 + α 1 + α

momentum in the barycentric reference frame. We 1 make an additional assumption here: between col- 0.8 lisions, the evolution of the orbits is similar to the evolution they would have in the averaged system in C (α, √α) 0.6 c the presence of chaotic behaviour. The orbital pa- C (α, 1) rameters will thus evolve in a limited manner, with C c 0.4 Cc(α, + ) a stochastic diffusion process, bounded by the con- ∞ servation of the total AMD. As shown in section 2.3, 0.2 during a collision the AMD decreases (eq. 24). We Cc(α, 0) assume the collisions to be totally inelastic with 0 perfect merging. Indeed, Kokubo & Genda (2010) 0 0.2 0.4 0.6 0.8 1 and Chambers (2013) have shown that the detailed α mechanisms of collisions, such as the possibilities of hit-and-run or fragmentation of embryos, barely Fig. 2. Values of Cc(α, γ) vs. α for the different γ values for which an explicit expression of Cc(α, γ) is obtained. change the final architecture of simulated systems. The total AMD of the system will thus be constant between collisions, and will decrease during collisions. On the other hand,√ the AMD for a particle 4. AMD model of planet formation 1 2 is of the order of 2 m ae . As the mass of the par- Once the disc has been depleted, the last phase ticle increases, its excursion in eccentricity will be of terrestrial planet formation begins with a disc more limited, and fewer collisions will be possible. composed of planetary embryos and planetesimals The collisions will stop when the total AMD of the (Morbidelli et al. 2012). In numerical simulations system is too small to allow for planetary collisions. of this phase, the AMD has been used as a statis- In the accretion process, we consider a planetes- tical measure for comparison with the inner solar imal of semi-major axis a and its immediate neigh- system (Chambers 2001). In this part, we recall the bour, defined as the planetesimal with semi-major axis a0, such that there is no other planetesimal with very simple model of embryo accretion described in 0 Laskar (2000) interpreting the N-body dynamics as semi-major axis between a and a . In this case we AMD-exchange. can assume that α is close to 1 and, as explained in Appendix C, we use an approximation of the critical AMD value Cc(α, γ): 4.1. Collisional evolution !2 So far we have not made special simplifications, δa C (α, γ) = k(γ) , (46) and the model simply preserves the mass and the c1 a Article number, page 6 of 18 Laskar & Petit: AMD-stability

0 0 where δa = a − a and k(γ) = γ/(2(γ + 1)) . where δn = 1 is the increment from planet a to a . By integration, this diﬀerence relation becomes for p − 3 , 4.2. AMD-stable planetary distribution , 2

In this section, we search for the laws followed by  1/3 2p+3 2p+3 2p + 3 4C˜ the planetary distribution of a model formed fol- 6   a 6 = a +   n . (54) lowing the above assumptions. We thus start from 0 6  ζ  an arbitrary distribution of mass of planetesimals 3 ρ(a), and let the system evolve under the previous For p = − 2 , we obtain rules. We search for the condition of AMD-stable 1/3 −1/3 planetary systems, obtained by random accretion log(a) = log(a0) + (4C˜) ζ n. (55) of planetesimals. This condition requires that the final AMD value cannot allow for planetary crossing In particular, for p = 0 (constant distribution among the planets. Let C be the value of the AMD and ρ(a) = ζ), from (53), we have at the end of the accretion process. If we consider √  ˜ 1/3 the planetesimal of semi-major axis a, its mass will √  C  a = a0 +   n . (56) continue to increase by accretion with a body of 2ζ  semi-major axis a0 > a, as long as !2 For the masses, from (51), we have for large n C δa (or if a0 is small ) C = ≥ Cc = k (47) Λ0 1 a m(n) ∼ (2C˜2ζ)1/3n . (57) The initial linear density of mass is ρ(a). As a0 is the closest neighbour to a, we can assume that For p = −3/2, we find a power law similar to the 0 all the planetesimals initially between a and a have Titius-Bode law for planetary distribution2. The been absorbed by the two bodies of mass m(a) and different expressions deduced from this model of 0 m(a ). At first order with respect to δa/a, we have planetary accretion are listed in Table 2. thus The previous analytical results were tested on 0 a numerical model of our accretion scheme (Laskar m(a ) ∼ m(a) ∼ ρ(a)δa, (48) 2000). The model was designed to fulfil the condi- and from (47) at the limiting case we have tions (12,13) of section 2.3. Five thousand simula- !2 tions were started with a large number of orbits (10 C˜ δa √ = k , (49) 000) and followed in order to look for orbit intersec- δaρ(a) a a tions. When an intersection occurs, the two bodies ˜ √ merge into a new one whose orbital parameters are where C = C/ µ. We have thus determined by the collisional equations (12,13) (see C˜ 1/3 Appendix A). Between collisions, the orbits do not   1/2 −1/3 δa =   a ρ , (50) evolve, apart from a diffusion of their eccentricities, k which fulfils the condition of conservation of the to- and from (48) tal AMD. This is roughly what would occur in a chaotic secular motion.  1/3 C˜  The main parameter of these simulations is the m(a) =   a1/2ρ2/3. (51)  k  final AMD value, C. Because the AMD decreases during collisions, and in order to obtain final systems with a given value C of the AMD, the eccen- 4.3. Scaling laws with initial mass distribution tricities were increased by a small amount in order ρ(a) ∝ ap. to raise the AMD to the desired final value. This is justified as close encounters can also increase the Using the previous relations, we can compute the AMD value. Indeed, N-body simulations (Cham- resulting systems for various initial mass distribu- bers 2001; Raymond et al. 2006) present a first tion, in particular for ρ(a) = ζap. From equation phase of AMD increase at the beginning of the sim- (51), we obtain for two consecutive planets ulations before the AMD decreases and converges !2/3 m ρ(a) 1 2 to its final value. These simulations were extremely 1/2 2 + 3 p 0 = α 0 = α (52) m ρ(a ) 2 The value p = −3/2 corresponds to a surface density proportional to r−5/2, which is different from the and from Eq. (46), as limα−→1 γ = 1, we thus have −3/2 surface density exponent of the minimum mass 1 k(γ) = in all cases. Relation (50) can be rewritten solar nebula (Weidenschilling 1977). For the solar sys- 4 tem, (Laskar 2000) found p = 0 to be the best fitting

p 1 value; it corresponds to a surface density proportional − 1/3 −1/3 −1 δaa 3 2 = (4C˜) ζ δn , (53) to r .

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Table 2. Planetary distribution corresponding to diﬀerent initial mass distribution.

p a(n) m(a) m(n)

−1 3 !1/3 ! 2p+3 !2− 2p+3 2p+3 2p+3 ˜ 1/3 4p+3 ˜ 6 2p + 3 4C 2 4C 2p + 3 p −3/2 a 6 = a + n 4C˜ζ a 6 m(n) ∼ 4C˜ n , 0 6 ζ ζ 6 !1/3 !1/3 3 4C˜ C˜ − log(a) = log(a ) + n (4C˜ζ2)1/3a−1/2 log(m) ∼ − n 2 0 ζ 2ζ √ √ n 0 a = a + (4C˜/ζ)1/3 (4C˜ζ2)1/3 a1/2 m ∼ (2C˜2ζ)1/3n 0 2

rapid to integrate as the orbital motion of the orbits was not integrated. Instead, we looked here for colli- 1.0 sions of ellipses which fulfil the conservation of mass and of linear momentum. These simulations were thus started with a large number of initial bodies 0.8 (10 000) and continued until their final evolution. The different runs resulted in different numbers of 0.6 planets, which ranged from four to nine, but the averaged values fitted very well with the analytical results of Table 2 (Laskar 2000). 0.4

Sample studied Cumulative distribution 5. AMD-classification of planetary systems 0.2 AMD-stable systems AMD-unstable systems Here we show how the AMD-stability can be used Exoplanet.eu catalog as a criterion to derive a classification of planetary 0.0 100 101 102 103 systems. In section 3.2 we saw that in the secular Period ratio approximation, the stability of a pair of planets is determined by the computation of Fig. 3. Cumulative distributions of the period ratios of adjacent planets in the sample studied here, of the C C AMD-stable systems (both weak and strong), AMD- β = = 0 . (58) Cc Λ Cc unstable systems, and for all the systems referenced in the exoplanet.eu database. We call β the AMD-stability coefficient. For pairs of planets, β < 1 means that collisions are not possible. The pairs of planets are then called AMD- stable. We naturally extend this definition to mul- axes, and eccentricities for all their planets. Since tiple planet systems. A system is AMD-stable if the number of systems with known mutual inclina- every adjacent pair is AMD-stable3. We can also tions is too small, we assumed the systems to be define an AMD-stability coefficient regarding the almost coplanar. This claim is supported by previous statistical studies that constrain the observed collision with the star. We define βS , the AMD- stability coefficient of the pair formed by the star mutual inclinations distribution (Fang & Margot and the innermost planet. For this pair, we have 2012; Lissauer et al. 2011; Fabrycky et al. 2014; and Figueira et al. 2012). For some systems where the α = 0 and thus Cc = 1. With this simplification β C/ uncertainties were not provided, we consulted the S = Λ, where Λ is the circular momentum of 4 the innermost planet. original papers or the Exoplanet Orbit Database (Wright et al. 2011). We compare the cumulative distribution of 5.1. Sample studied and methods of computation the adjacent planets’ period ratios in our sam- We have studied the AMD-stability of some systems ple and that of all the multiplanetary systems in referenced in the exoplanet.eu catalogue (Schneider the database exoplanet.eu (see Fig. 3). The sam- et al. 2011). From the catalogue, we selected 131 ple is biased toward higher period ratios. Indeed, systems that have measured masses, semi-major most of the multiplanetary systems in the database come from the Kepler data. Since these systems are 3 This is equivalent to require that all pairs are AMD- stable. 4 http://exoplanets.org/

Article number, page 8 of 18 Laskar & Petit: AMD-stability mostly tightly packed ones, their period ratios are rather small. However, the majority of them do not 0.05 have measured eccentricities and are consequently 8 Planet systems excluded from this study. Our sample thus mostly 6 Planet systems 0.04 5 Planet systems contains systems detected by radial velocities (RV) 4 Planet systems methods that have, on average, higher period ratios. 3 Planet systems

) 0.03 Since all the AMD computations are done with 2 Planet systems β the relative quantities α and γ, we can use equiva- ( lent quantities that are measured more precisely in PDF 0.02 observations than the masses and semi-major axis. We used the period ratios elevated to the power 3/2 instead of the semi-major axis ratios, and the 0.01 minimum mass m sin(i) for RV system. This is not γ a problem for the computation of ; if we assume 0.00 3 2 1 0 1 2 3 that the systems are close to coplanarity, then 10− 10− 10− 10 10 10 10 m m sin(i) m sin(i) β = C /Cc γ = = ' . (59) m0 m0 sin(i) m0 sin(i0) Fig. 4. Probability distribution function of β for the Even though we assume the systems to be copla- sample studied. The systems are grouped by multiplic- nar, we want to take into account the contribution ity. The vertical line β = 1 marks the separation between of mutual inclinations to the AMD. Since we only AMD-stable and AMD-unstable pairs. have access to the eccentricities, we define the coplanar AMD of a system Cp, as the AMD of the same system if it was coplanar. We can also define For masses (or m sin(i) if no masses are provided) 0 and periods, we assume a Gaussian uncertainty cen- βp = Cp/(Λ Cc) , (60) tred on the value referenced in the database and which is the coplanar AMD-stability coefficient. with standard deviation, the half width of the con- Motivated by both theoretical arguments on chaotic fidence interval. The distributions are truncated to diffusion in the secular dynamics (Laskar 1994, 0. 2008) and observed correlations in statistical distri- For eccentricity distributions, the previous butions (Xie et al. 2016), we assume that the AMD method does not provide satisfying results. Most contribution of mutual inclinations is equal to that of the Gaussian distributions constructed with the of the eccentricities. This hypothesis is equivalent mean value and confidence interval given in the to assuming the average equipartition of the AMD catalogue make negative eccentricities probable (in among the secular degrees of freedom for a chaotic the case of almost circular planets with a large system. The total AMD is thus twice the measured upper bound on the eccentricity). One solution is AMD, and in this study we use to assume that the rectangular eccentricity coordinates (e cos ω, e sin ω) are Gaussian. Since the aver- β = 2βp . (61) age value of ω has no importance in the computation of the eccentricity distribution, we assume We can also define a coplanar AMD-stability co- it to be 0. Therefore, e sin ω has 0 mean. We de- efficient associated with the star, and similarly we fine the distribution of e˜ = e cos ω as a Gaussian set β = 2β . We then compute the coefficients β S S p S distribution with the mean value, the value refer- and β for each pair and the associated uncertainty enced in the catalogue, and standard deviation, the distributions. We list the results of the analysis in half-width of the confidence interval. If we assume Table 3. In the considered dataset, 70 systems are e sin ω has the same standard deviation as e cos ω, AMD-stable. The majority of the highest multiplic- we have e sin ω = e˜ − he˜i. The distribution of e is ity systems are AMD-unstable. In Figure 4 we plot then deduced from that of e˜ using the probability distribution of β for the considered p systems. e = e˜2 + (˜e − he˜i)2 . (62) Using the Gaussian assumption means that 5.2. Propagation of uncertainties some masses or periods can take values close to 0 with a small probability. This causes the distribu- The uncertainties are propagated using Monte tions of α or γ to diverge if it happens that a0 or m0 Carlo simulations of the distributions. After deter- can take values close to 0. To address this issue, a mining the distributions from the input quantities linear expansion around the mean value is used for (m, a, e), we generate 10,000 values for each of these the quotients, for example for α, parameters. We then compute the derived quanti- ! ties (α, γ, C , β. . . ) in these 10,000 cases and the as- a a ∆a0 c α = = 1 − , (63) sociated distributions. a0 ha0i ha0i Article number, page 9 of 18 A&A proofs: manuscript no. 30022 hk rev

Table 3. Result of the analysis split in function of the multiplicity of the system

Multiplicity Strong stable Weak stable Unstable Total 2 42 21 34 97 3 4 1 17 22 4+ 2 0 10 12 Total 48 22 61 131

with ∆a0 = a0 − ha0i. resonances, etc. In all these cases, an in-depth dynamical study is necessary to determine the short- or long-term stability of these systems. In the fol- 5.3. AMD-stable systems lowing sections, we detail how the AMD-stability As said above, we consider AMD-stable a system and AMD driven dynamics can help to understand where collisions between planets are impossible be- these systems. cause of the dynamics ruled by AMD (max β < 1). In addition, if the AMD-stability coefficient of the 5.4.1. Hierarchically AMD-stable systems (solar star βS < 1 (resp. βS > 1), the system is defined as system-like) strong AMD-stable (resp. weak AMD-stable). We first consider the solar system. Owing to the 5.3.1. Strong AMD-stable systems large amount of AMD created by the giant planets, the inner system is not AMD-stable (Laskar 1997). The strong AMD-stable systems can be considered However, long-term secular and direct integrations dynamically stable in the long term. In Figure 5, have shown that the inner system has a probabil- we plot the architecture of the strong AMD-stable ity of becoming unstable of only 1% over 5 Gyr systems. If the system is out of the mean motion res- (Laskar 2008; Laskar & Gastineau 2009; Batygin onance (MMR) islands, the AMD will not increase et al. 2015). Indeed, the chaotic dynamics is mainly and therefore no collision between planets or with restricted to the inner system, while the outer sys- the star can occur. We can see in Figure 3 that the tem is mostly quasi-periodic. However, when AMD AMD-stable systems have period ratios on average exchange occurs between the outer and inner sys- larger than those from the sample. tems, the inner system becomes highly unstable and can lose one or several planets. In Figure 8 we plot the AMD and the circu- 5.3.2. Weak AMD-stable systems lar momenta of the solar system planets. We see As defined above, in a weak AMD-stable system, that the inner system planets (resp. the outer plan- no planetary collisions can occur, but the innermost ets) have comparable AMD values. Laskar (2008) planet can increase its eccentricity up to 1 and col- showed the absence of exchange between the two lide with the star. We separate these systems from parts of the system. Moreover, the spacing of the the strong AMD-stable ones because the system can planets follows surprisingly well the distribution still lose a planet only by AMD exchange. How- laws mentioned in section 4 if we consider the two ever, the remaining system will not be affected by parts of the system separately (Laskar 2000). the destruction of the inner planet. In Figure 6, we We see that given an analysis of the secular dy- plot their architecture. In these systems, the inner namics, the tools developed above can still help to planet is much closer to the star than the others. understand how an a priori unstable system can still exist in its current state. The case of AMD-unstable systems with an AMD-stable part is not restricted 5.4. AMD-unstable systems to the solar system. If we look for systems where The AMD-unstable systems have at least one un- the outer part is AMD-stable while the whole sys- stable planet pair, but as we show in Figure 7 where tem is AMD-unstable, we find four similar systems we plot the architecture of these systems, this cate- in our sample as shown in Figure 9. We call these gory is not homogeneous. It gathers high multiplic- systems hierarchic AMD-stable systems. ity systems where planets are too close to each other for the criterion to be valid, multiscale systems like 5.4.2. Resonant systems the solar system where the inner system is AMD- unstable owing to its small mass in comparison to As shown by the cumulative distributions plotted the outer part, systems experiencing mean motion in Figure 3, the AMD-unstable systems have period

Article number, page 10 of 18 Laskar & Petit: AMD-stability

10 2 10 1 100 101 102 − − HD 85390 AMD coefﬁcient β = C /Cc HD 113538

HD 147018 HD 1605

HD 143761 nu Oph

HD 11964 HD 154857

Kepler-289 eta Cet

Kepler-46 HD 108874

HD 9446 HD 4732

HIP 65407 HD 159868

HD 117618 HD 142

K2-3 HD 12661

HD 13908 HD 134987

HD 20794 HD 169830

XO-2S HD 60532

GJ 682 TYC+1422-614-1

Kepler-79 HD 207832

HD 159243 HD 147873

Kepler-454 HD 177830

HD 69830 HIP 67851

HD 96700 HD 164922

HD 109271 HD 163607

HIP 14810 GJ 785

HD 1461 HD 168443

GJ 876 HD 37605

CoRoT-7 Kepler-51

1 0 1 2 3 4 1 2 3 4 10− 10 10 10 10 10 10 10 10 10 Period (days) Period (days)

Fig. 5. Architecture of the strong AMD-stable systems. Each planet is represented by a circle whose size is proportional to log10(m), where m is the mass of the planet. The colour represents the AMD-stability coefficient of the inner pair associated with the planet (in particular, the first planet is represented with the AMD-stability coefficient associated with the star βS ). This means that a red planet can collide with its inner neighbour ratios that are lower than the AMD-stable systems. 5.4.3. Concentration around MMR For example, in our sample 66% of the adjacent pairs of the AMD-unstable systems have a period Here we want to test whether the unstable systems ratio below 4, whereas this proportion is only 33% have been drawn randomly from the exoplanet.eu for the AMD-stable systems. For period ratios close catalogue or if the period ratios of the pairs of adja- to small integer ratios, the MMR can rule a great cent planets are statistically closer to small integer part of the dynamics. Particularly, if pairs of planets ratios. We denote H0, the hypothesis that the pe- are trapped in a large resonance island, the system riod ratios of the unstable systems have been drawn could be dynamically stable even if it is not AMD- randomly from the catalogue distribution. We con- stable. sider only the period ratios lower than 4 because the higher ones are not significant for a study of Individual dynamical studies are necessary in the MMR. We plot in figure 10 the cumulative dis- order to claim that a system is stable due to a tribution of the period ratios of the AMD-unstable MMR. We note, however, that the AMD-unstable systems, as well as the cumulative distribution of systems considered here are statistically constituted the period ratios of all the systems of exoplanet.eu. of more resonant pairs than a typical sample in the We call Ru the set of period ratios of the AMD- catalogue. unstable planets.

Article number, page 11 of 18 A&A proofs: manuscript no. 30022 hk rev

which means studying the ﬁne structure of the

2 1 0 1 2 distribution. We propose here another method for 10− 10− 10 10 10 AMD coefﬁcient β = C /Cc demonstrating this. Let us denote the probability density of the catalogue period ratios f and the associated cumu- HIP 5158 lative distribution F . Let us consider an interval HD 75784 Ix = (x, x + ∆x), if we assume H0, the probability for a ratio r to be in Ix is HD 11506 Z x+∆x HD 110014 0 0 P(r ∈ Ix|H0) = f (r )dr x Kepler-419 = F (x + ∆x) − F (x) = ∆F (x) . (64) HD 74156 Therefore, the probability that more than k GJ 832 0 pairs out of N pairs drawn randomly from the dis- HD 190360 tribution F are in Ix is

HD 38529 N ! X N P(x, k |H ) = (1 − ∆F )N−k∆F k. (65) HD 87646A 0 0 k k=k0 KELT-6

HD 217107 This is the probability of a binomial trial. From now on, N designates the number of period ratios in Ru. BD-08 2823 For a given ∆x, we can compute the probabil- GJ 649 ity P(x, Ru) = P(x, ku(x)), where ku(x) is the number of pairs from Ru in Ix. This probability measures HD 47186 the likelihood of a concentration of elements of Ru HD 215497 around x, assuming they were drawn from F . We choose ∆x = 0.05 and plot the function P(x, Ru) as BD-06 1339 well as ku in Figure 11. We observe that the con-

HD 187123 centrations around 3 and 2 are very unlikely with a probability of 2.3 × 10−3 and 2.9 × 10−3, respec- WASP-41 tively. Other peaks appear for x = 1.4 and around 3.8. However, P(x, Ru) is the probability of seeing a HAT-P-13 concentration around x precisely. It is not the prob- Pr 0211 ability of observing a concentration somewhere between 1 and 4. Kepler-10 To demonstrate that the concentrations around

1 0 1 2 3 4 5 2 and 3 are meaningful, we compare this result to 10− 10 10 10 10 10 10 F Period (days) samples drawn randomly from the distribution . We create 10,000 datasets Rα by drawing N points F Fig. 6. Architecture of the weak AMD-stable systems. from and compute P(x, Rα) for these datasets. Each planet is represented by a circle whose size is pro- Then, we record the two minimum values (at least distant by more than ∆x) and plot the distribution portional to log10(m), where m is the mass of the planet. The colour represents the AMD-stability coefficient of of these minima on Figure 12. From these simula- the inner pair associated with the planet (in particular, tions, we see that on average 17.2% of the samples the first planet is represented with the AMD-stability have a minimum as small as Ru. However, the pres- coefficient associated with the star βS ). ence of a secondary peak as significant as the second one of Ru has a probability of 1.3%. Moreover, the Ru concentrations are clearly situated around low integer ratios, which would not be the case in gen- We first test H0 via a Kolmogorov-Smirnov test eral for a randomly generated sample. (Lehmann & Romano 2006) between the sample Ru We thus demonstrated here that the AMD- and the period ratios of the catalogue. The test fails unstable systems period ratios are significantly con- to reject H0 with a p-value of about 9%. Therefore, centrated around low integer ratios. While we do we cannot reject H0 on the global shape of the dis- not prove that the pairs of planets close to these tribution of the AMD-unstable period ratios. ratios are actually in MMR, this result further mo- However, we want to determine if the AMD- tivates investigations toward the behaviour of these unstable pairs are close to small integer ratios, pairs in a context of secular chaotic dynamics.

Article number, page 12 of 18 Laskar & Petit: AMD-stability

10 2 10 1 100 101 102 − − NN Ser (AB) AMD coefﬁcient β = C /Cc HD 7449

HD 31527 47 Uma

HD 13808 HD 5319

HD 51608 HD 133131 A

HD 93385 HD 183263

HD 134606 HD 200964 HD 33844 HD 20003 HD 128311 HD 136352 24 Sex Kepler-56 HD 47366 Kepler-11 HD 67087 mu Ara HD 202206 HD 181433 HD 45364 GJ 163 HD 155358 HIP 57274 HD 73526 HD 215152 HD 92788 GJ 667 C HD 37124 Kepler-23 HD 65216 HD 10180 HD 102272 HD 39194 Kepler-87 Kepler-68 HD 141399 HD 7924 Sun Wolf 1061 HD 89744 ups And HD 52265 HD 40307 Kapteyn’s

61 Vir HD 204313

HD 125612 HD 3651

HD 219828 GJ 3293

GJ 676 A HD 20781

HD 134060 HD 21693

55 Cnc GJ 180

1 0 1 2 3 4 5 1 2 3 4 5 10− 10 10 10 10 10 10 10 10 10 10 10 Period (days) Period (days)

Fig. 7. Architecture of the AMD-unstable systems. Each planet is represented by a circle whose size is proportional to log10(m), where m is the mass of the planet. The colour represents the AMD-stability coefficient of the inner pair associated with the planet (in particular, the first planet is represented with the AMD-stability coefficient associated with the star βS ).

6. Conclusion stability of a system can be checked by the computation of the critical AMD Cc (eq. 35) and AMD- stability coefficients βi that depend only on the The angular momentum deficit (AMD) (Laskar eccentricities and ratios of semi-major axes and 1997, 2000) is a key parameter for the understand- masses (Eqs. 29, 35, 39, 58). This criterion thus does ing of the outcome of the formation processes of not depend on the degeneracy of the masses coming planetary systems (e.g. Chambers 2001; Morbidelli from radial velocity measures. Moreover, the uncer- et al. 2012; Tremaine 2015). We have shown here tainty on the relative inclinations can be compen- how AMD can be used to derive a well-defined stability criterion: the AMD-stability. The AMD-

Article number, page 13 of 18 A&A proofs: manuscript no. 30022 hk rev

109 1.0 Λ 108 k Ck 107 0.8 106

105 0.6

Momentum 10 0.4 103

2 10 Cumulative distribution 0.2 101 AMD-unstable systems Exoplanet.eu catalog 0.0 Earth Mars Venus Jupiter Saturn Uranus 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Mercury Neptune Period ratio Fig. 8. Circular momenta and AMD of the solar system Fig. 10. Cumulative period ratios of the AMD-unstable planets. The horizontal lines correspond to the respec- systems and of all systems in the catalogue. The vertical tive mean Λk and Ck for the inner and outer system. line represents low order MMR.

0 ] x 2 1 0 1 2 10− 10− 10 10 10 + ∆ AMD coefﬁcient β = C /Cc

1 x, x − [ in ) u u Sun outer R x, R ( 2 P − Sun inner 10

log 10

3 HD 141399 outer − 5 : Number of pairs of

HD 141399 inner u k 4 − 1.0 1.5 2.0 2.5 3.0 3.5 4.0 HD 10180 outer Period ratio

HD 10180 inner Fig. 11. Red curve: Probability of observing more than GJ 676 A outer k0 pairs in the interval Ix given a probability density f . Green curve: k0 is the number of pairs in Ix for the GJ 676 A inner unstable systems Ru

55 Cnc outer

55 Cnc inner and the simple computation of the AMD-stability coeﬃcients.

1 0 1 2 3 4 5 We have also proposed here a classiﬁcation 10− 10 10 10 10 10 10 Period (days) of the planetary systems based on AMD-stability (Section 5). The strong AMD-stable systems are the Fig. 9. Examples of systems where the outer part is systems where no planetary collisions are possible, AMD-stable and the inner part becomes AMD-stable if and no collisions of the inner planet with its cen- considered alone. tral star, while the weak AMD-stable systems allow for the collision of the inner planet with the central star. The AMD-unstable systems are the systems for which the AMD-coeﬃcient does not pre- sated by assuming equipartition of the AMD be- vent the possibility of collisions. The solar system tween eccentricities and inclinations. is AMD-unstable, but it belongs to the subcategory AMD-stability will ensure that in the absence of of hierarchical AMD-stable systems that are AMD- mean motion resonances, the system is long-term unstable but become AMD-stable when they are stable. A rapid estimate of the stability of a sys- split into two parts (giant planets and terrestrial tem can thus be obtain by a short-term integration planets for the solar system) (Laskar 2000). Out of

Article number, page 14 of 18 Laskar & Petit: AMD-stability

ies. Note: The AMD-stability coeﬃcients of selected 0.06 planetary systems will be made available on the ex- 1st min. 3 oplanet.eu database. 2nd min. −

0.05 10 × 9 .

2 References

0.04 3 − Albouy, A. 2002, Lectures on the two-body problem (Prince- 10

0.03 × ton University Press) 3

. Batygin, K., Morbidelli, A., & Holman, M. J. 2015, The As- 2 trophysical Journal, 799, 120 0.02 Chambers, J. 2001, Icarus, 152, 205 Chambers, J. 2013, Icarus, 224, 43 Chambers, J., Wetherill, G., & Boss, A. 1996, Icarus, 119, 0.01 261 Probability distribution function Deck, K. M., Payne, M., & Holman, M. J. 2013, The Astro- physical Journal, 774, 129 0.00 4.0 3.5 3.0 2.5 2.0 1.5 1.0 Fabrycky, D. C., Lissauer, J. J., Ragozzine, D., et al. 2014, − − − − − − − The Astrophysical Journal, 790, 146 min(log P (x, R )) 10 α Fang, J. & Margot, J.-L. 2012, The Astrophysical Journal, 761, 92 Fang, J. & Margot, J.-L. 2013, The Astrophysical Journal, Fig. 12. Probability density function of the two first 767, 115 minimum values of P(x, Rα) from 10,000 samples Rα Figueira, P., Marmier, M., Boué,G., et al. 2012, Astronomy drawn from the F distribution. & Astrophysics, 541, A139 Gladman, B. 1993, Icarus, 106, 247 Graner, F. & Dubrulle, B. 1994, Astronomy and Astro- physics, 282, 262 the 131 studied systems from exoplanet.eu, we find Hernández-Mena, C. & Benet, L. 2011, Monthly Notices of the Royal Astronomical Society, 412, 95 48 strong AMD-stable, 22 weak AMD-stable, and Kokubo, E. & Genda, H. 2010, The Astrophysical Journal, 61 AMD-unstable systems, including 5 hierarchical 714, L21 AMD-stable systems. Laskar, J. 1994, Astronomy and Astrophysics, 287, L9 Laskar, J. 1997, Astronomy and Astrophysics, 317, L75 As for the solar system, the AMD-unstable sys- Laskar, J. 2000, Physical Review Letters, 84, 3240 tems are not necessarily unstable, but determin- Laskar, J. 2008, Icarus, 196, 1 Laskar, J. & Gastineau, M. 2009, Nature, 459, 817 ing their stability requires some further dynamical Laskar, J. & Robutel, P. 1995, Celestial Mechanics & Dy- analysis. Several mechanisms can stabilise AMD- namical Astronomy, 62, 193 unstable systems. The absence of secular chaotic in- Lehmann, E. L. & Romano, J. P. 2006, Testing statistical teractions between parts of the systems, like in the hypotheses (Springer Science & Business Media) Lissauer, J. J., Ragozzine, D., Fabrycky, D. C., et al. 2011, solar system case, or the presence of mean motion The Astrophysical Journal Supplement Series, 197, 8 resonances, protecting pairs of planets from colli- Marchal, C. & Bozis, G. 1982, Celestial Mechanics, 26, 311 sion. In this case, the AMD-stability classification Mayor, M., Marmier, M., Lovis, C., et al. 2011, eprint arXiv:1109.2497 is still useful in order to select the systems that Moorhead, A. V., Ford, E. B., Morehead, R. C., et al. 2011, require this more in-depth dynamical analysis. It The Astrophysical Journal Supplement Series, 197, 1 should also be noted that the discovery of additional Morbidelli, A., Lunine, J., O’Brien, D., Raymond, S., & planets in a system will require a revision of the Walsh, K. 2012, Annual Review of Earth and Planetary Sciences, 40, 251 computation of the AMD-stability of the system. Nieto, M. M. 1972, The Titius-Bode law of planetary dis- This additional planet will always increase the to- tances: its history and theory (Pergamon Press), 161 tal AMD, and thus the maximum AMD-coefficient Poincaré, H. 1905, Le¸cons de mécanique céleste, Tome I of the system, decreasing its AMD-stability unless (Gauthier-Villars. Paris) Pu, B. & Wu, Y. 2015, The Astrophysical Journal, 807, 44 it is split into two subsystems. Ramos, X. S., Correa-Otto, J. A., & Beaugé, C. 2015, Celes- In the present work, we have not taken into tial Mechanics and Dynamical Astronomy, 123, 453 account mean motion resonances (MMR) and the Raymond, S. N., Quinn, T., & Lunine, J. I. 2006, Icarus, 183, 265 chaotic behaviour resulting from their overlap. We Schneider, J., Dedieu, C., Le Sidaner, P., Savalle, R., & Zolo- aim to take these MMR into consideration in a tukhin, I. 2011, Astronomy & Astrophysics, 532, A79 forthcoming work. Indeed, the criterion regard- Shabram, M., Demory, B.-O., Cisewski, J., Ford, E. B., & Rogers, L. 2016, The Astrophysical Journal, 820, 93 ing MMR developed by Wisdom (1980) or more re- Smith, A. W. & Lissauer, J. J. 2009, Icarus, 201, 381 cently Deck et al. (2013) may help to improve our Tremaine, S. 2015, The Astrophysical Journal, 807, 157 stability criterion by considering the MMR chaotic Weidenschilling, S. J. 1977, Astrophysics and Space Science, zone as a limit for stability instead of the limit 51, 153 Winn, J. N. & Fabrycky, D. C. 2015, Annual Review of As- considered here that is given by the collisions of tronomy and Astrophysics, 53, 409 the orbits (Eqs. 35). The drawback will then most Wisdom, J. 1980, The Astronomical Journal, 85, 1122 probably be giving up the rigorous results that we Wright, J. T., Fakhouri, O., Marcy, G. W., et al. 2011, Pub- lications of the Astronomical Society of the Pacific, 123, have established in section 3, and allowing for ad- 412 ditional empirical studies. The present work will in Xie, J.-W., Dong, S., Zhu, Z., et al. 2016, Proceedings of the any case remain the backbone of these further stud- National Academy of Sciences, 113, 11431

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Appendix A: Intersection of planar orbits r In this section, we present an eﬃcient algorithm for a(1 + e) the computation of the intersection of two elliptical orbits in the plane, following (Albouy 2002). Let us consider an elliptical orbit deﬁned by (µ, r, r˙) and let G = r ∧ r˙ be the angular momentum per unit of mass. The Laplace vector a(1 e) r˙ ∧ G r − p P = − (A.1) µ r

is an integral of the motion with coordinates y (e cos ω, e sin ω). One has r2 = x2 + y2 G2 P · r = − r = p − r, (A.2) x µ

where p = a(1 − e2) is the parameter of the ellipse. Let r = (x, y) in the plane. We can consider the Fig. A.1. Elliptical orbit, as the intersection of the cone (x, y, r) ellipse in three-dimensional space as the in- r2 = x2 + y2 and the plane P · r + r = p . tersection of the cone

r2 = x2 + y2 (A.3) (Eqs. B.4, B.6, B.7), such that H0 is independent of λ. Then, K is an integral of H0, i.e. with the plane deﬁned by Eq. (A.2), that is {K, H0} = 0 (B.3) x(e cos ω) + y(e sin ω) + r = p (A.4) 2 3 The generator S = εS 1 + ε S 2 + ε S 3 + ··· is If we consider now two orbits O1 and O2. Their in- obtained formally through the identiﬁcation order tersection is easily obtained as the intersection of by order

the line of intersection of the two planes 0 H0 = H0 x(e1 cos ω1) + y(e1 sin ω1) + r = p1 H0 = H + {H , S } (A.5) 1 1 0 1 x(e2 cos ω2) + y(e2 sin ω2) + r = p2 (B.4) 0 1 H2 = {H0, S 2} + {{H0, S 1}, S 1} + {H1, S 1} 2 2 2 2 with the cone of equation r = x + y . Depending ... on the initial conditions, if O1 and O2 are distinct, we will get either 0, 1, or 2 solutions. For any function G(Λ, λ, J, φ), let 1 Z hGi = Gdmλ (B.5) Appendix B: AMD in the averaged λ (2π)m equations be the average of G over all the angles λk. For each In this section we show that the AMD conserves the n ≥ 1, the S n is obtained through the resolution of same form in the averaged planetary Hamiltonian an equation of the form at all orders. More generally, this is true for any in- H0 = {H , S } + R , (B.6) tegral of H which does not depend on the longitude n 0 n n

λk. Let where Rn belongs to L(H0, H1, S 1,..., S n−1), the Lie 0 algebra generated by (H0, H1, S 1,..., S n−1). Hn will H = H0(Λ) + εH1(Λ, λ, J, φ) (B.1) be the averaged part of Rn, hRniλ, and S n is obtained be a perturbed Hamiltonian system, and let by solving the homological equation K(Λ, J, φ) H(Λ, λ, J, φ) be a ﬁrst integral of (such that {H0, S n} = hRniλ − Rn . (B.7) {K, H} = 0, where {·, ·} is the usual Poisson bracket), and independent of λ. In addition, let We show by recurrence that {K, S n} = 0 for all n ≥ 1. First, we note that as {K, H0} = 0, we also 0 LS H = e H (B.2) have {K, H1} = 0. As K is independent of λk, we also have for all G be a formal averaging of H with respect to λ. If S (Λ, λ, J, φ) is a generator deﬁned as below h{K, G}iλ = {K, hGiλ} (B.8)

Article number, page 16 of 18 Laskar & Petit: AMD-stability

This can be seen by formal expansion of G in a α < 1/2 : We have then e0 > 1, and the only P ihk,λi Fourier series G = gke . We have thus hGiλ = possibility for ec(α, 0) is e1 = 1; as it is the only g0. Let us assume now that {K, S k} = 0 for all root of (C.2) which belongs to [0, 1], we have thus k ≤ n. As Rn+1 ∈ L(H0, H1, S 1,..., S n), we also have {K, R } = 0 {K, hR i } = 0 n+1 , and from (B.8) n+1 λ and lim ec(α, γ) = 1 (C.3) thus γ→0

{K, {H0, S n+1}} = 0 (B.9) We have then

0 lim ec(α, γ) = 1 − 2α ; Using the Jacobi identity, as {F, H0} = 0, we have γ→0 p lim Cc(α, γ) = 1 − 2 α(1 − α); (C.4) {H0, {K, S n+1}} = 0 (B.10) γ→0

The solution of the homological equation (B.7) is In order to study the behaviour of ec(α, γ) in the unique up to a term independent of λ. But as vicinity of γ = 0, we can diﬀerentiate K(ec(α, γ), γ) = 0 h{K, S n+1}iλ = 0, then the only possible solution for twice, which gives (B.10) is 2 dec d ec 4(α − 1) {K, S } = 0 . (B.11) (α, 0) = 0 ; (α, 0) = < 0 , n+1 dγ dγ2 (1 − 2α)2

0 0 (C.5) In the same way, as H1 = hH1iλ, we have {K, H1} = h{K, H1}iλ = 0. Thus {K, {H0, S 1}} = 0, by the Ja- cobi identity, {H0, {K, S 1}} = 0, and as previously, thus ec(α, γ) decreases with respect to γ at γ = 0. {K, S 1} = 0. Our recurrence is thus complete and it The second order development of Cc gives follows immediately that {K, H0} = 0.

r √ p 2 1 3 Appendix C: Special values of C (α, γ) Cc(α, γ) = 1−2 α(1 − a)+γ α−γ − 1+O(γ ). c α This appendix provides the detailed computations (C.6) and proofs of the results of section 3.2

α > 1/2 : In this case, e0 < 1. As ec(α, γ) ∈]0, e0[, Appendix C.1: Asymptotic value of Cc(α, γ) for we have ec(α, 0) ∈ [0, e0], which gives as the unique γ → 0 possibility

We have shown that ec(α, γ) is a diﬀerentiable func- lim ec(α, γ) = e0 (C.7) tion of γ, which is monotonic (41) and bounded γ→0 lim (ec(a, γ) ∈ [0, 1]). The limit ec(α, 0) = γ→0 ec(α, γ) exists for all α ∈]0, 1]. If ec(α, γ) is a solution of with equation(35), it will also be a solution of the following cubic equation (in e), which is directly ob- 0 lim ec(α, γ) = 0 ; lim Cc(α, γ) = 0 . (C.8) tained from (35) by squaring, multiplication, and γ→0 γ→0 simpliﬁcation by α(1 + e): By setting γ = 0 in (41), we also obtain K(e, α, γ) = α(γ2 − α)e3 −(2 − α)(γ2 − α)e2 2 2 dec e0 −(1 − α )e + (1 − α) = 0 (α, 0) = − q < 0 . (C.9) dγ 3/2 2 (C.1) α 1 − e0

e α, γ γ γ → As c( ), is a continuous function of , when The development of Cc gives 0 the limits e0c(α) will satisfy the limit equation

K (e, α) = (1 − α − αe)2(1 − e) = 0 (C.2)  r  0  √ 1  γ2 (1 − α)2 C (α, γ) = γ  α − 2 − − +O(γ3). c  α 2α − 1 α with solutions e0 = 1/α − 1 and e1 = 1. Depending on α, several cases are treated: (C.10)

Article number, page 17 of 18 A&A proofs: manuscript no. 30022 hk rev

α = 1/2 : In this case, e0 = 1, and the only solution Appendix√ C.3: Study of Cc(α, γ) for γ = 1 and is γ = α. √ For γ = 1 or γ = α, we can also obtain simple lim ec(α, γ) = 1 (C.11) γ→0 expressions for Cc(α, γ). Indeed, If γ = 1, K(e, α, γ) factorises in (1 − α)(1 + e)(αe2 − 2e + 1 − α) and we and have a single solution for ec in the interval [0, 1], √ 0 1 − 1 − α + α2 e α, γ C α, γ . lim c( ) = 0 ; lim c( ) = 0 (C.12) ec(α, 1) = ; γ→0 γ→0 α (C.20) √ 2 2 0 2 Moreover, equation (C.1) becomes γ 2e (3 − e) = ec(α, 1) = 1 − α + α − α ; 3 (1 − e) . We obtain thus the asymptotic value for and ec(1/2, γ) when γ → 0 as q √ √ α − 2 + 2 1 − α + α2 1 2/3 Cc(α, 1) = α − √ 1 − ec( , γ) ∼ (4γ) . (C.13) α 2 (C.21) √ q √ 2 For α = 1/2, the development of Cc in γ contains +1 − α 1 − 2α + 2 1 − α + α √ non-polynomial terms in γ giving For γ = α, the cubic equation (C.1) reduces to √ K(α, α) = −(1 − α2)e + (1 − α)2 = 0 (C.22) γ −1/3 4/3 −4/3 2 8/3 Cc(1/2, γ) = √ −2 γ −2 γ +O(γ ) (C.14) with the single solution 2 √ √ 1 − α e (α, α) = e0 (α, α)) = (C.23) c c 1 + α Appendix C.2: Asymptotic value of Cc(α, γ) for and γ → +∞ √ √ 2 Cc(α, α) = (1 − α) (C.24) This case is more simple. If ec(α, γ) is a solution of Eq. (35), then it will also be a solution of eq. (C.1), and thus of Appendix C.4: Cc(α, γ) for α −→ 1 K(e, α, γ) Let us denote η = 1 − α. The equation C.1 can be = 0 (C.15) developed in η, γ2

2 2 e α, γ K(e, α, γ) = (γ − 1)e (e − 1) + As c( ) is monotonic and bounded, it has a limit when γ → +∞, which will verify the limit equation = ηe (γ2 − 1)(e − e2) − 2 + (C.15), when γ → +∞, that is = η2(1 − e − e2 − e3) = 0. (C.25) 2 K∞(α, e) = e (2 − α − αe) = 0 (C.16) The zeroth and ﬁrst orders of equation C.25 imply that ec must go to zero; moreover, it scales As 0 < α < 1, the only solution is e = 0, and thus with η. We write ec(η, γ) = κ(γ)η + o(η). We inject this expression in C.25 and keep the second order lim ec(α, γ) = 0 (C.17) in η γ→+∞ 2 − 2 − and (γ 1)κ + 2κ 1 = 0. (C.26) We keep the solution that is positive and con- 0 p tinuous in γ and we have lim ec(α, γ) = 1−α ; lim Cc(α, γ) = 1− α(2 − α) γ→+∞ γ→+∞ η (C.18) ec(η, γ) = + o(η). (C.27) γ + 1 and more precisely If we now compute Cc developed for α −→ 1 we have √ p 1 α(1 − α)2 C(α, γ) = 1 − α(2 − α) − 2 2 2γ 2 − α Cc(1 − η, γ) = k(γ)η + o(η ) (C.19) 1 γ p 2   = η2 + o(η2) (C.28) 1 α(2 − α)(1 − α)  1  2 γ + 1 + + O   γ2 2(2 − α)3 γ3  Article number, page 18 of 18