Astronomy & Astrophysics manuscript no. mmramdAA edit c ESO 2018 October 4, 2018
AMD-stability in the presence of first-order mean motion resonances
A. C. Petit, J. Laskar and G. Bou´e
ASD/IMCCE, CNRS-UMR8028, Observatoire de Paris, PSL Research University, UPMC, 77 Avenue Denfert-Rochereau, 75014 Paris, France e-mail: [email protected] Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
The AMD-stability criterion allows to discriminate between a-priori stable planetary systems and systems for which the stability is not granted and needs further investigations. AMD-stability is based on the conservation of the Angular Momentum Deficit (AMD) in the averaged system at all orders of averaging. While the AMD criterion is rigorous, the conservation of the AMD is only granted in absence of mean-motion resonances (MMR). Here we extend the AMD-stability criterion to take into account mean-motion resonances, and more specifically the overlap of first-order MMR. If the MMR islands overlap, the system will experience generalized chaos leading to instability. The Hamiltonian of two massive planets on coplanar quasi-circular orbits can be reduced to an integrable one degree of freedom problem for period ratios close to a first-order MMR. We use the reduced Hamiltonian to derive a new overlap criterion for first-order MMR. This stability criterion unifies the previous criteria proposed in the literature and admits the criteria obtained for initially circular and eccentric orbits as limit cases. We then improve the definition of AMD-stability to take into account the short term chaos generated by MMR overlap. We analyze the outcome of this improved definition of AMD-stability on selected multi-planet systems from the Extrasolar Planets Encyclopædia. Key words. Celestial mechanics - Planets and satellites: general - Planets and satellites: dynamical evolution and stability
1. Introduction MMR can be dynamically stable, chaotic behavior may result from the overlap of adjacent MMR, lead- The AMD-stability criterion (Laskar 2000; Laskar ing to a possible increase of the AMD and eventu- & Petit 2017) allows to discriminate between a- ally to close encounters, collisions or ejections. For priori stable planetary systems and systems need- systems with small orbital separations, averaging ing an in-depth dynamical analysis to ensure their over the mean anomalies is thus impossible due to stability. The AMD-stability is based on the con- the contribution of the first-order MMR terms. For servation of the angular momentum deficit (AMD, example, two planets in circular orbits very close to Laskar 1997) in the secular system at all orders of each other are AMD-stable, however the dynamics averaging (Laskar 2000; Laskar & Petit 2017). In- of this system cannot be approximated by the secu- deed, the conservation of the AMD fixes an upper lar dynamics. We thus need to modify the notion of bound to the eccentricities. Since the semi-major AMD-stability in order to take into account those axes are constant in the secular approximation, a configurations. low enough AMD forbids collisions between plan- ets. The AMD-stability criterion has been used to In studies of planetary systems architecture, a classify planetary systems based on the stability of minimal distance based on the Hill radius (Marchal their secular dynamics (Laskar & Petit 2017). & Bozis 1982) is often used as a criterion of sta- arXiv:1705.06756v2 [astro-ph.EP] 21 Jul 2017 However, while the analytical criterion devel- bility (Gladman 1993; Chambers et al. 1996; Smith oped in (Laskar & Petit 2017) does not depend on & Lissauer 2009; Pu & Wu 2015). However, Deck series expansions for small masses or spacing be- et al. (2013) suggested that stability criteria based tween the orbits, the secular hypothesis does not on the MMR overlap are more accurate in charac- hold for systems experiencing mean motion reso- terizing the instability of the three-body planetary nances (MMR). Although a system with planets in problem.
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Based on the considerations of Chirikov (1979) We follow here the reduction of the Hamiltonian for the overlap of resonant islands, Wisdom (1980) used in (Delisle et al. 2012, 2014). proposed a criterion of stability for the first-order MMR overlap in the context of the restricted cir- 2.1. Averaged Hamiltonian in the vicinity of a cular three-body problem. This stability criterion resonance defines a minimal distance between the orbits such that the first-order MMR overlap with one another. Let us consider two planets of masses m1 and m2 For orbits closer than this minimal distance, the orbiting a star of mass m0 in the plane. We denote MMR overlapping induces chaotic behavior even- the positions of the planets, ui, and the associated tually leading to the instability of the system. canonical momenta in the heliocentric frame, u˜ i. Wisdom showed that the width of the chaotic The Hamiltonian of the system is (Laskar & Robu- region in the circular restricted problem is pro- tel 1995) portional to the ratio of the planet mass to the 2 2 1 X ku˜ ik m0mi star mass to the power 2/7. Duncan et al. (1989) Hˆ = − G + confirmed numerically that orbits closer than the 2 m u i=1 i i Wisdom’s MMR overlap condition were indeed un- 1 ku˜ + u˜ k2 m m stable. More recently, another stability criterion 1 2 − G 1 2 (1) was proposed by Mustill & Wyatt (2012) to take 2 m0 ∆12 into account the planet’s eccentricity. Deck et al. where ∆12 = ku1 − u2k, and G is the constant of (2013) improved the two previous criteria by de- gravitation. Hˆ can be decomposed into a Keplerian veloping the resonant Hamiltonian for two massive, part Kˆ describing the motion of the planets if they coplanar, low-eccentricity planets and Ramos et al. εHˆ (2015) proposed a criterion of stability taking into had no masses and a perturbation part 1 due to account the second-order MMR in the restricted the influence of massive planets, three-body problem. X2 1 ku˜ k2 Gm m Kˆ = i − 0 i (2) While Deck’s criteria are in good agreement 2 m u with numerical simulations (Deck et al. 2013) and i=1 i i 2 can be applied to the three-body planetary prob- ˆ 1 ku˜ 1 + u˜ 2k Gm1m2 lem, the case of circular orbits is still treated sepa- εH1 = − . (3) 2 m0 ∆12 rately from the case of eccentric orbits. Indeed, the minimal distance imposed by the eccentric MMR The small parameter ε is defined as the ratio of the overlap stability criterion vanishes with eccentrici- planet masses over the star mass m + m ties and therefore cannot be applied to systems with ε = 1 2 . (4) small eccentricities. In this case, Mustill & Wyatt m0 (2012) and Deck et al. (2013) use the criterion de- Let us denote the angular momentum, veloped for circular orbits. A unified stability cri- X2 terion for first-order MMR overlap had yet to be ˆ proposed. G = ui ∧ u˜ i (5) In this paper, we propose in Section 2 a new i=1 derivation of the MMR overlap criterion based on which is simply the sum of the two planets Keple- the development of the three-body Hamiltonian by rian angular momentum. Gˆ is a first integral of the Delisle et al. (2012). We show in Section 3 how to system. obtain a unified criterion of stability working for Following (Laskar 1991), we express the Hamil- both initially circular and eccentric orbits. In Sec- tonian in terms of the Poincar´ecoordinates tion 4, we then use the defined stability criterion to Hˆ = Kˆ + εHˆ (Λˆ , xˆ , xˆ¯ ) limit the region where the dynamics can be consid- 1 i i i 2 2 3 2 ered to be secular and adapt the notion of AMD- X µ m X Y ¯ − i ˆ li ¯li ikiλi stability thanks to the new limit of the secular dy- = + ε Cl,l¯,k(Λ) xˆi xˆi e , (6) 2Λˆ 2 namics. Finally we study in Section 5 how the mod- i=1 i l,l¯∈N2 i=1 ification of the AMD-stability definition affects the k∈Z2 classification proposed in (Laskar & Petit 2017). where µ = Gm0 and for i = 1, 2, √ Λˆ i = mi µai q 2. The resonant Hamiltonian ˆ ˆ 2 Gi = Λi 1 − ei
The problem of two planets close to a first-order Cˆi = Λˆ i − Gˆi MMR on nearly circular and coplanar orbits can be q −i$i reduced to a one-degree-of-freedom system through xˆi = Cˆie a sequence of canonical transformations (Wisdom 1986; Henrard et al. 1986; Delisle et al. 2012, 2014). λi = Mi + $i.
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Here, Mi corresponds to the mean anomaly, $i to been used extensively in the study of the first-order the longitude of the pericenter, ai to the semi-major MMR dynamics. axis and ei to the eccentricity of the Keplerian orbit Expressed with the variables (Λˆ , λ, xˆ, xˆ¯), the of the planet i. Gˆi is the Keplerian angular momen- Hamiltonian can be written tum of planet i. We use the set of symplectic coordi- 2 2 3 ˆ ˆ X µ mi nates of the problem (Λi, λi, Ci, −$), or the canoni- Hˆ = − + av 2 cally associated variables (Λˆ i, λi, xˆi, −ixˆ¯i). The coeffi- 2Λˆ ˆ i=1 i cients Cl,l¯,k depend on Λ and the masses of the bod- X ¯ ¯ ˆ l1 ¯l1 l2 ¯l2 i j((p+1)λ2−pλ1) ies. They are linear combinations of Laplace coeffi- ε Cl,l¯, j(Λ)x ˆ1 xˆ1 xˆ2 xˆ2 e , (11) cients (Laskar & Robutel 1995). As a consequence l,l¯∈N2 of angular momentum conservation, the d’Alembert j∈Z rule gives a relation on the indices of the non-zero where we dropped the terms of order ε2. Cl,l¯,k coefficients
X2 2.1.2. Poincare-like complex coordinates k − l + l¯ = 0. (7) i i i Delisle et al. (2012) used a change of the angular co- i=1 ordinates in order to remove the exponential in the ˆ ˆ We study here a system with periods close to second term of eq. (11) and use G and Γ as actions. the first-order MMR p : p + 1 with p ∈ N∗. The new set of angles (θΓ, θG, σ1, σ2) is defined as For periods close to this configuration, we have 2 3 ˆ 3 −pn1 + (p + 1)n2 ' 0, where ni = µ m /Λ is the Ke- θΓ p −p 0 0 λ1 i i plerian mean motion of the planet i. θG −p p + 1 0 0 λ2 = · . (12) σ1 −p p + 1 1 0 −$1 σ −p p + 1 0 1 −$ 2.1.1. Averaging over non-resonant mean-motions 2 2 The conjugated actions are Due to the p : p + 1 resonance, we cannot average on both mean anomalies independently. Therefore, Γˆ p+1 Λˆ p 1 0 0 1 there is no conservation of Λˆ as in the secular prob- ˆ ˆ i G 1 1 −1 −1 Λ2 lem. However, the partial averaging over one of the = · . (13) Cˆ1 0 0 1 0 Cˆ 1 mean anomaly gives another first integral. Follow- Cˆ 0 0 0 1 Cˆ ing (Delisle et al. 2012), we consider the equivalent 2 2 ˆ ˆ p iσ set of coordinates (Λi, Mi, Gi, $i), and make the fol- We define Xˆi = Cˆie i , the complex coordinates iθ lowing change of angles associated to (Cˆi, σi). Since we have Xˆi = xˆie G , the ! ! ! terms of the perturbation in (11) can be written σ −p p + 1 M1 = . (8) 2 2 M2 0 1 M2 Y l l¯ i jθ Y l l¯ i(−l +l¯ + j)θ i ¯ i G X i X¯ i i i G xˆi xˆi e = ˆi ˆi e The actions associated to these angles are i=1 i=1 2 Y l l¯ ! 1 ! −1 ˆ X i X¯ i Γˆ − 0 Λˆ Λ1 = ˆi ˆi ; (14) 1 = p 1 = p . ˆ p+1 ˆ p+1 ˆ ˆ (9) i=1 Γ p 1 Λ2 p Λ1 + Λ2 the last equality resulting from the d’Alembert We can now average the Hamiltonian over M2 using rule (7). Γˆ and Gˆ are conserved and the aver- a change of variables close to the identity given by aged Hamiltonian no longer depends on the angles 2 the Lie series method. Up to terms of orders ε , we θΓ and θG can kill all the terms with indices not of the form 2 2 3 2 X µ m X Y ¯ Cl,l¯,− jp, j(p+1). In order to keep the notations light, we ˆ i ˆ li ¯li Hav = − + ε Cl,l¯, j(Λ) Xˆ Xˆ . (15) do not change the name of the variables after the 2Λˆ 2 i i i=1 i l,l¯∈N2 i=1 averaging. We also designate the remaining coef- j∈Z ficients Cl,l¯,− jp, j(p+1) by the lighter expression Cl,l¯, j. Since M2 does not appear explicitly in the remain- Λˆ 1 and Λˆ 2 can be expressed as functions of the new ing terms, variables and we have ˆ ˆ ˆ ˆ p + 1 Λ1 = −p(C + G − Γ) (16) Γˆ = Λˆ 1 + Λˆ 2 (10) p Λˆ 2 = (p + 1)(Cˆ + Gˆ) − pΓˆ, (17) is a first integral of the averaged Hamiltonian. where Cˆ = Cˆ1 + Cˆ2 is the total AMD of the system. The parameter pΓˆ is often designed as the spac- Up to the value of the first integrals Γˆ and Gˆ, the ing parameter (Michtchenko et al. 2008) and has system now has two effective degrees of freedom.
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2.2. Computation of the perturbation coefficients Using the expression of α at the resonance p : p + 1, We now truncate the perturbation, keeping only !2/3 the leading-order terms. Since we consider the first- p order MMR, the Hamiltonian contains some linear α = , (27) 0 p + 1 terms in Xi. Therefore the secular terms are ne- glected since they are at least quadratic. Moreover, we can give the asymptotic development of the co- the restriction to the planar problem is justified efficients r1 and r2 for p → +∞ (see Appendix A.1). since the inclination terms are at least of order two. The equivalent is We follow the method described in Laskar (1991) and Laskar & Robutel (1995) to determine K1(2/3) + 2K0(2/3) ˆ −r1 ∼ r2 ∼ (p + 1). (28) the expression of the perturbation H1. The details π of the computation are given in Appendix A. Since we compute an expression at first order in eccentric- where Kν(x) is the modified Bessel function of the ities and ε, the semi major axis and in particular second kind. We note r the numerical factor of the their ratio, equivalent (28), we have
a K1(2/3) + 2K0(2/3) α = 1 , (18) r = = 0.80199. (29) a2 π
are evaluated at the resonance. At the first order, For the resonant coefficients r1 and r2, Deck et al. ˆ (2013) used the expressions fp (α) and fp (α) the perturbation term H1 has for expression +1,27 +1,31 given in (Murray & Dermott 1999, pp. 539-556). The expressions (23) and (24) are similar to ˆ εH1 = Rˆ1(Xˆ1 + Xˆ¯1) + Rˆ2(Xˆ2 + Xˆ¯2), (19) fp+1,27(α) and fp+1,31(α) up to algebraic transforma- tions using the relations between Laplace coeffi- where cients (Laskar & Robutel 1995). In their compu- s tations, Deck et al. used a numerical fit of the co- γ µ2m3 1 2 Rˆ = −ε 2 r (α) (20) efficients for p = 2 to 150 and obtained 1 1 + γ ˆ 2 2 ˆ 1 Λ2 Λ1 − fp+1,27 ∼ fp+1,31 ∼ 0.802p. (30) and s We obtain the same numerical factor r through the γ µ2m3 1 2 Rˆ = −ε 2 r (α) (21) analytical development of the functions r1 and r2. 2 1 + γ ˆ 2 2 ˆ 2 Λ2 Λ2 (22) 2.3. Renormalization
with γ = m1/m2, So far, the Hamiltonian has two degrees of freedom (Xˆ1, Xˆ¯1, Xˆ2, Xˆ¯2) and depends on two parameters Gˆ and α (p) (p+1) (p+2) ˆ ˆ r1(α) = − 3b (α) − 2αb (α) − b (α) , (23) Γ. As shown in (Delisle et al. 2012), the constant Γ 4 3/2 3/2 3/2 can be used to scale the actions, the Hamiltonian and and the time without modifying the dynamics. We α (p−1) (p) (p+1) define r2(α) = 3b3/2 (α) − 2αb3/2(α) − b3/2 (α) 4 ˆ ˆ 1 Λi = Λi/Γ, + b(p) (α). (24) 2 1/2 G = Gˆ/Γˆ, (k) C Cˆ /ˆ, In the two previous expressions, b (α) are the i = i Γ s p Laplace coefficients that can be expressed as Xi = Xˆi/ Γˆ, Z π 1 cos(kφ) H = Γˆ 2Hˆ , b(k)(α) = dφ s 2s (25) π −π 1 − 2α cos φ + α t = tˆ/Γˆ 3. k > 0 k = 0 1/2 for . For , a factor has to be added in With this change of variables, the new Hamiltonian the second-hand member of (25). no longer depends on Γˆ. p = 1 For , it should be noted that a contribution The shape of the phase space is now only de- from the kinetic part should be added (Appendix A pendent on the first integral G. However, G does and Delisle et al. 2012) not vanish for the configuration around which the s Hamiltonian is developed: the case of two resonant µ2m2m2 1 2 2 planets on circular orbits. To be able to develop the H1,i = (Xˆ2 + Xˆ¯2). (26) 2m0Λˆ 1Λˆ 2 Λˆ 2 Keplerian part in power of the system’s parameter, Article number, page 4 of 19 Petit, Laskar & Bou´e:AMD-stability we define ∆G = G0 −G, the difference in angular mo- thanks to the relations (35). We develop the Keple- mentum between the circular resonant system and rian part to the second order in (C − ∆G) since the the actual configuration. We have first order vanishes (see Appendix B). The compu- tation of the second-order coefficient gives G0 = Λ1,0 + Λ2,0, (31) 5 1 3 2 3 (γ + α0) 2 where Λ and Λ are the value of Λ and Λ at K2 = − µ m (p + 1) . (39) 1,0 2,0 1 2 2 2 2 γα4 resonance. By definition, we have 0 !1/3 We drop the constant part of the Hamiltonian and Λ1,0 p √ = γ = γ α0. (32) obtain the following expression Λ2,0 p + 1 K p H = 2 (I + I − ∆G)2 + 2R I cos(θ ). (40) Moreover, we can express Λ1,0 as a function of the 2 1 2 1 1 ratios α0 and γ, We again change the time scale by dividing the ! Λˆ 1,0 1 p γ Hamiltonian by −K2 and multiplying the time by Λ1,0 = = = . (33) p+1 Λ2,0 this factor. We define Γˆ 0 + p + 1 γ + α0 p Λ1,0 √ 2R Similarly, Λ2,0 can be expressed as χ = − (41) K2 α0 Λ2,0 = . (34) and after simplification, α0 + γ 1 ε(γα )3/2 r (α ) G ∆G χ 0 2 0 f p Since 0 is constant, is also a first integral of = 2 2 ( ) (42) H. From now on, we consider ∆G as a parameter of 3 (1 + γ)(α0 + γ) (p + 1) 3/2 the two-degrees-of-freedom (X , X ) Hamiltonian H. r εγ 1 1 2 = + O((p + 1)−2), (43) The Keplerian part depends on the coordinates Xi 3 (1 + γ)3 p + 1 through the dependence of Λi in C. −1 Λ1 and Λ2 can be expressed as functions of the where r was defined in (29) and f (p) = 1 + O(p ) is Hamiltonian coordinates and their value at the res- a function of p and γ onance, s !2 α0 p + 1 r1 Λ1 = Λ1,0 − p(C − ∆G) f (p) = 1 − 1 − . (44) α0 + γ p r2 Λ2 = Λ2,0 + (p + 1)(C − ∆G). (35) At this point the Hamiltonian can be written
2.4. Integrable Hamiltonian 1 2 p H = − (I1 + I2 − ∆G) + χ 2I1 cos(θ1) (45) The system can be made integrable by a rotation 2 of the coordinates Xi (Sessin & Ferraz-Mello 1984; and has almost its final form. We divide the actions Henrard et al. 1986; Delisle et al. 2014). We intro- and the time by χ2/3 and the Hamiltonian by χ4/3 duce R and φ such that and we obtain 1 √ R1 = R cos(φ) and R2 = R sin(φ). (36) H = − (I − I )2 − 2I cos(θ ), (46) A 2 0 1 2 2 2 We have R = R1 + R2 and tan(φ) = R2/R1. If we note R the rotation of angle φ we define y such that where φ P X R −2/3 −2/3 = φy. We still have C = yiy¯i so the only change I = χ I1 and I0 = χ (∆G − I2). (47) in the Hamiltonian is the perturbation term
H = K(C, ∆G) + R(y1 + y¯1) 2.5. Andoyer Hamiltonian p = K(C, ∆G) + 2R I1 cos(θ1), (37) We now perform a polar to Cartesian change of co- ordinates with where (I, θ) are the action-angle coordinates associ- √ ated to y. With these coordinates, I2 is a first inte- X = − 2I cos(θ1), gral. R has for expression √ Y = 2I sin(θ1). (48)
2 3 2 ! We change the sign of X in order to have the same εγ µ m r (α )2 r (α )2 R2 = 2 1 0 + 2 0 . (38) orientation as (Deck et al. 2013). Doing so, the γ 2 1 + Λ2,0 2Λ1,0 2Λ2,0 Hamiltonian becomes !2 We now develop the Keplerian part around the 1 1 H = − (X2 + Y2) − I − X. (49) circular resonant configuration in series of (C − ∆G) A 2 2 0 Article number, page 5 of 19 A&A proofs: manuscript no. mmramdAA edit
single resonant term the system is integrable, over- lapping resonant islands will lead to chaotic motion (Chirikov 1979). Wisdom (1980) first applied the resonance over- 0 = 3 lap criterion to the first-order MMR and found, in I the case of the restricted three-body problem with a circular planet, that the overlap occurs for 1 − α < 1.3ε2/7. (53) Through numerical simulations, (Duncan et al. X X2 1989) confirmed Wisdom’s expression up to the nu- 1 2/7 X3 merical coefficient (1−α < 1.5 ε ). A similar crite- rion was then developed by Mustill & Wyatt (2012) for an eccentric planet, they found that for an ec- centricity above 0.2 ε3/7, the overlap region satis- fies the criterion 1 − α < 1.8(ε e)1/5. Deck et al. (2013) adapted those two criteria to the case of two massive planets, finding little difference up to the numerical coefficients. However, they treat two dif- ferent situations; the case of orbits initially circular and the case of two eccentric orbits. As in (Mustill Fig. 1. Hamiltonian HA (49) represented with the sad- & Wyatt 2012), the eccentric criterion proposed can dle point and the separatrices in red. 3/7 be used for eccentricities verifying e1 +e2 & 1.33 ε . We show here that the two Deck’s criteria can be obtained as the limit cases of a general expression. We recognize the second fundamental model of res- onance (Henrard & Lemaitre 1983). This Hamilto- 3.1. Width of the libration area nian is also called an Andoyer Hamiltonian (Ferraz- Mello 2007). We show in Figure 1 the level curves Using the same approach as (Wisdom 1980; Deck of the Hamiltonian HA for I0 = 3. et al. 2013), we have to express the width of the The fixed points of the Hamiltonian satisfy the resonant island as a function of the orbital param- equations eters and compare it with the distance between the ! two adjacent centers of MMR. 1 2 2 In the (X, Y) plane, the center of the resonance X˙ = Y (X + Y ) − I0 = 0 (50) 2 is located at the point of coordinates (X1, 0). The ! width of the libration area is defined as the distance 1 2 2 Y˙ = −X (X + Y ) − I0 − 1 = 0, (51) between the two separatrices on the Y = 0 axis. It 2 is indeed the direction where the resonant island is which have for solutions Y = 0 and the real roots of the widest. ∗ ∗ the cubic equation in X We note X1, X2 the abscissas of the intersections between the separatrices and the Y = 0 axis. Re- 3 − I ∗ ∗ X 2 0X + 2 = 0. (52) lations between X1, X2, and X3 can be derived (see ∗ Appendix C.1) and we obtain the expressions of X1 ∗ Equation (52) has three solutions (Deck et al. 2013) and X as functions of X3 (Ferraz-Mello 2007; Deck I3 − I 2 if its determinant ∆ = 32( 0 27/8) > 0, i.e. 0 > et al. 2013). We have 3/2. In this case, we note these roots X1 < X2 < X . X and X are elliptic fixed points while X is a 2 3 1 2 3 ∗ − − hyperbolic one. X1 = X3 √ , (54) X3 ∗ 2 X2 = −X3 + √ . (55) 3. Overlap criterion X3 As seen in the previous section, the motion of two The width of the libration zone δX depends solely planets near a first-order MMR can be reduced to on the value of X3, an integrable system for small eccentricities and planet masses. However, if two independent combi- 4 δX = √ . nations of frequencies are close to zero at the same (56) X3 time, the previous reduction is not valid anymore. Indeed, we must then keep, in the averaging, the In order to study the overlap of resonance is- terms corresponding to both resonances. While for a lands, we need the width of the resonance in terms
Article number, page 6 of 19 Petit, Laskar & Bou´e:AMD-stability of α. Let us invert the previous change of variables Cmin(p) as the minimal value of I1 to enter the res- in order to express the variation of α in terms of the onant island given ∆G − I2. Two cases must be dis- variation of X. In this subsection, for any function cussed: Q(X), we note – The point I1 = 0 is already in the libration zone ∗ ∗ δQ = |Q(X1) − Q(X2)|. (57) and then Cmin = 0, – The point I = 0 is in the inner circulation zone δI 1 The computation of (48) is straightforward from and then we have the computation of δX !2 χ2/3 2 X∗2 X∗2 C = I (X∗) = X − √ . (62) 1 2 min 1 2 3 δI = − 2 X3 2 2 In the second case, we have an implicit expression 1 ∗ ∗ ∗ ∗ = X + X X − X of X3 depending on Cmin 2 2 1 2 1 −1/3 p 2 = X3δX χ 2Cmin = X3 − √ , (63) p X3 δI = 4 X3. (58) where χ was defined in (41). In other words, there 2/3 I We then directly deduce δI1 = χ δ from (47). is a one-to-one correspondence between Cmin (62) Since I2 and ∆G are first integrals, the variation of and the Hamiltonian parameter I0 for Cmin > 0. Λi only depends on δI1. And finally, since we have The shape of the resonance island is completely de- !2 scribed by Cmin. −1 Λ1 α = γ , (59) We can also use the definition of Cmin to give an Λ2 expression depending on the system parameters α can be developed to the first order in (C − ∆G) C = I = u u¯ thanks to (35). This development gives min 1 1 1 r r 2 2 ! R1 Λ1,0 R2 Λ2,0 2(α0 + γ) 2/3 = X1 + X2 α = α0 1 − (p + 1)(I1 − χ I0) . (60) R 2 R 2 γα0 s 2 The width of the resonance in terms of α is then 2 R1 Λ1,0 R2 Λ2,0 = X1 − X2 directly related to X3 through R 2 R Λ 1 1,0 r2/3 8 2/3 1/3 p 2/3 1/3 α0γ δα = α0 ε (p+1) X3 +o(ε (p+1) ). (61) ' c2 c2 − c c $ , 2/3 2 ( 1 + 2 2 1 2 cos ∆ ) (64) 3 2(α0 + γ) The computation of the width of resonance is thus √ r q reduced to the computation of the root X3 as a func- where c = 2 1 − 1 − e2 = |X |. We note tion of the parameters. It should also be remarked i i i that at the first order, the width of resonance does 2 2 − not depend on the mass ratio γ. cmin = c1 + c2 2c1c2 cos ∆$, (65) the reduced minimal AMD. We can use the√ expres- −1/3 3.2. Minimal AMD of a resonance sion (64) to compute the quantity χ 2Cmin ap- pearing in equation (63) We are now interested in the overlap of adjacent res- onant islands. Planets trapped in the chaotic zone p 31/3 (p + 1)1/3 √ χ−1/3 2C ' c + o(p1/3). (66) created by the overlap will experience variations of min r1/3 ε1/3 min their actions eventually leading to collisions. For a configuration close to a given resonance The function Cmin(X3) (Eq. 62) is plotted in Fig- p : p + 1, the AMD can evolve toward higher values ure 2 with the two approximations used by Deck et al. (2013) to obtain the width of the resonance. if the original value places the system in a configura- 2/3 tion above the inner separatrix, eventually leading For Cmin χ or Cmin close to zero, the relation the planets to collision or chaotic motion in case can be simplified and we obtain of MMR overlap. On the other hand, if the initial p X ∼ χ−1/3 2C (67) AMD of the planets forces them to remain in the 3 min 2 p inner circulation region of the overlapped MMR is- X = 22/3 + χ−1/3 2C + O(χ−2/3C ). (68) lands, the system will remain stable in regards to 3 3 min min this criterion. Since C = I + I , and I is a first in- 1 2 2 We can use the developments (67) and (68) in order tegral, we define the minimal AMD of a resonance1 to compute the width of the resonance in these two 1 We summarize the notations of the various AMD ex- cases (see Appendix C). It should be noted as well 2/3 pressions used in this paper in Table 1. that for Cmin = 0, we have X3 = 2 .
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Table 1. Summary of the diverse notations of AMD used in this paper.
Notation Description Equation C Total AMD of the system Cmin Minimal AMD to enter a resonance island (62) cmin Normalized minimal AMD (65) C Relative AMD (91) (0) Cc Critical AMD deduced from the collision condition (Laskar & Petit 2017) (1) Cc Critical AMD deduced from the MMR overlap (97) Cc Complete critical AMD (101)
α = (p/(p + 1))2/3. We develop α for p 1 10 0,p 0,p
!2/3 p α0,p = 8 p + 1 2 1 = 1 − − + O((p + 1)−3). (70) 3(p + 1) 9(p + 1)2 6 min C
2 Therefore, we have at second order in p √ 3 / 1
− 4 ∆α 2 1 χ . = 2 (71) α0,p 3 (p + 1)
2 We can use√ the implicit expression (63) of X3 as Implicit expression a function of cmin (Eq. 65) in order to derive an High ecc. approximation Circular approximation overlap criterion independent of approximations on 0 the value of Cmin. Equating the general width of res- 0 2 4 6 8 10 onance (61) with the distance between to adjacent X3 centers (71) and isolating X3 gives
4/3 Fig. 2. Relation (63) between X3 and Cmin (62) and 3 X ε−4/3 p −14/3. two different approximations. In red, the approximation 3 = 4/3 ( + 1) (72) used by Deck et al. (2013) for eccentric orbits and in 144r purple the constant evaluation used for circular orbits. We can inject this expression of X3 into (63), and using equation (66), √ 1 c = − 8rε(p + 1)2. (73) 3.3. Implicit overlap criterion min 48rε(p + 1)5
The overlap of MMR can be determined by finding Using the first order expression of (p + 1) as a func- the first integer p such that the sum of the half- tion of α, width of the resonances p : p + 1 and p + 1 : p + 2 is 1 3 larger than the distance between the respective cen- = (1 − α) (74) ters of these two resonances (Wisdom 1980; Deck p + 1 2 et al. 2013) we obtain an implicit expression of the overlap cri- terion ! √ 34(1 − α)5 32rε ∆α 1 δαp δαp+1 c = − . (75) . + , (69) min 29rε 9(1 − α)2 α0,p 2 α0,p α0,p+1
3.4. Overlap criterion for circular orbits where ∆α is the distance between the two centers The implicit expression (75) can be used to find the and δα corresponds to the width of the resonance k criteria proposed by Deck et al. (2013) for circular k : k + 1. and eccentric orbits. Let us first obtain the circular 2/3 Up to terms of order ε , the center of the reso- criterion by imposing cmin = 0 in equation (75) nance island p : p + 1 is located at the center of the resonance of the unperturbed Keplerian problem, 36(1 − α)7 = 214r2ε2. (76)
Article number, page 8 of 19 Petit, Laskar & Bou´e:AMD-stability
We can express 1 − α as a function of ε and we which leads to obtain √ 3/7 cmin 9.30ε . (85) 4r2/7 1 − α = ε2/7 = 1.46ε2/7. overlap 6/7 (77) It is worth noting that the low-eccentricity approx- 3 imation allows to cover the range of eccentricities The exponent 2/7 was first proposed by Wisdom where the criterion (79) is not applicable, since both (1980) and the numerical factor 1.46 is similar to boundaries depend on the same power of ε. the one found by Deck et al. (2013). We plot in Figure 3 the overlap criteria (75) for ε = 10−6, the two approximations (77) and (79) from (Deck et al. 2013), as well as the collision con- 3.5. Overlap criterion for high-eccentricity orbits dition used in (Laskar & Petit 2017) approximated α → 1 For large eccentricity, Deck et al. (2013) proposes for , √ a criterion based on the development (67) of equa- 1 − α ' e1 + e2 ' cmin. (86) tion (63). This criterion is obtained from (75) by ig- noring the second term of the right-hand side which We also plot the first MMR islands in order to show leads to the agreement between the proposed criterion and the actual intersections. We see that for high ec- 9 √ 4 5 2 rε cmin = 3 (1 − α) . (78) centricities, and large 1 − α, the system can verify the MMR overlap stability criterion while allowing Isolating 1 − α gives for collision between the planets. For small α, the MMR overlap criterion alone cannot account for the 29/5 1 − α = r1/5ε1/5c1/10 = 1.38ε1/5c1/10. (79) stability of the system. 34/5 min min This result is also similar to Deck’s one. For small 4. Critical AMD and MMR cmin, the criterion (79) is less restrictive than the criterion (77) obtained for circular orbits. The com- 4.1. Critical AMD in a context of resonance overlap parison of these two overlap criteria provides a min- In (Laskar & Petit 2017), we present the AMD- imal value of cmin for the validity of the eccentric stability criterion based on the conservation of criterion AMD. We assume the system dynamics to be sec- √ ular chaotic. As a consequence the averaged semi- c 3/7 min = 1.33ε . (80) major axis and the total averaged AMD are con- served. Moreover, in this approximation the dynam- 3.6. Overlap criterion for low-eccentricity orbits ics is limited to random AMD exchanges between planets with conservation of the total AMD. Based For smaller eccentricities,√ we can develop the on these assumptions, collisions between planets are equation (75) for small cmin and α close to possible only if the AMD of the system can be dis- 2/7 αcir = 1 − 1.46ε , the critical semi major axis ra- tributed such that the eccentricities of the planets tio for the circular overlap criterion (77). We have allow for collisions. Particularly, for each pair of ad- jacent planets, there exists a critical AMD, noted 2 9 2 √ 6 7 14 2 2 3 2 rε(1 − α) cmin = 3 (1 − α) − 2 r ε . (81) Cc(α, γ), such that for smaller AMD, collisions are forbidden. We develop the right-hand side at the first order in The critical AMD was determined thanks to the (αcir − α) and evaluate the left-hand side for α = αcir limit collision condition and after some simplifications obtain √ α(1 + e1) = 1 − e2. (87) 9 2 rε cmin αcir − α = . (82) However, in practice, the system may become un- 7 × 34 (1 − α )4 cir stable long before orbit intersections; in particular
We inject the expression of αcir into this equation the secular assumption does not hold if the sys- and obtain the following development of the overlap tem experiences chaos induced by MMR overlap. criterion for low eccentricity: We can, though, consider that if the islands do not overlap, the AMD is, on average, conserved on √ √ −2/3 2 cmin cmin timescales of order ε (i.e., of the order of the α − α = = 0.157 . (83) cir 7 × 34/7r1/7ε1/7 ε1/7 libration timescales). Therefore, the conservation, on average, of the AMD is ensured as long as the √This development remains valid for small enough system adheres to the above criteria for any dis- cmin if αcir − α 1 − αcir, which can be rewritten tribution of the AMD between planets. Based on the model of (Laskar & Petit 2017), we compute a −1/7 √ 2/7 0.157ε cmin 1.46ε , (84) critical AMD associated to the criterion (75).
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13 12 11 10 9 : : : : : 8 7 6 5 4 3 2 1 14 13 12 11 10 9: 8: 7: 6: 5: 4: 3: 2:
0.5 −
1.0 − m c √ 10
log 1.5 −
MMR overlap criterion 2.0 High eccentricity approximation − Circular approximation Collision condition
1.2 1.0 0.8 0.6 0.4 − − − − − log (1 α) 10 −
Fig. 3. Representation of the MMR overlap criteria. The dotted lines correspond to the criteria proposed by (Deck et al. 2013), and the collision curve is the approximation of the collision curve for α → 1. We represented in transparent green (p odd) and blue (p even) the first p : p + 1 MMR islands to show the agreement between the proposed overlap criterion and the actual intersections. In this figure, ε = 10−6.
(1) We consider a pair as AMD-stable if no distri- The critical AMD Cc associated to the overlap cri- bution of AMD between the two planets allows the terion (75) can be defined as the smallest value of overlap of MMR. A first remark is that no pair can relative AMD such that the conditions be considered as AMD-stable if α > αcir, because in this case, even the circular orbits lead to MMR E (c1, c2) = c1 + c2 = g(α, ε) (92) √ overlap. Let us write the criterion (75) as a function 1 2 2 (1) of α and ε; C (c1, c2) = γ αc1 + c2 = Cc (93) √ 2 c = g(α, ε), (88) min are verified by any couple (c1, c2). We represent this where configuration in Figure 4. As in (Laskar & Petit 34(1 − α)5 32rε 2017), the critical AMD is obtained through La- g(α, ε) = − α < α , grange multipliers 29rε 9(1 − α)2 cir ∇ ∝ ∇ = 0 α > αcir. (89) C E . (94) √ cmin depends on ∆$ and has a maximum for ∆$ = The tangency condition gives a relation between c1 π. Since the variation of ∆$ does not affect the and c2, AMD of the system, we fix ∆$ = π since it is the √ least-favorable configuration. Therefore we have γ αc1 = c2. (95) √ cmin = c1 + c2. (90) Replacing c2 in relation (92) gives the critical ex- pression of c and we immediately obtain the ex- We define the relative AMD of a pair of planets 1 pression of c2 C and express it as a function of the variables ci √ C 1 √ g(α, ε) γ αg(α, ε) 2 2 c = √ c = √ . C = = γ αc1 + c2 . (91) c,1 c,2 (96) Λ2 2 1 + γ α 1 + γ α Article number, page 10 of 19 Petit, Laskar & Bou´e:AMD-stability
3 α αcir − R
4
ε − 10 log 5 −
Collision 6 Eccentric MMR overlap − Circular MMR overlap
0.7 0.8 0.9 1.0 α
Fig. 5. Regions of application of the different criteria (0) Fig. 4. MMR overlap criterion represented in the (c1, c2) presented in this work. The purple region represents Cc plane. (1) is the smallest, in the green zone, Cc is the smallest and the circular overlap criterion is verified in the red zone. We see that the curve αR computed through a (0) (1) (1) development of Cc and Cc presents a good agreement The value of Cc is obtained by injecting the critical with the real limit between the green and the purple values cc,1 and cc,2 into the expression of C area. Here γ = 1. √ 2 (1) g(α, ε) γ α Cc (α, γ, ε) = √ . (97) 2 1 + γ α as a solution of the polynomial equation in (1 − α); 36(1 − α)7 − 3229rε(1 − α)3 − 214(rε)2 = 0. (99) 4.2. Comparison with the collision criterion While an exact analytical solution cannot be pro- It is then natural to compare the critical AMD (1) vided, a development in powers of ε gives the fol- Cc to the critical AMD Cc (denoted hereafter by lowing expression (0) Cc ) derived from the collision condition (87). If √ 4 1/4 1 3/4 α > αcir, the circular overlap criterion implies that 1 − αR = (2rε) + 2rε + O(ε ) (1) (1) 3 4 Cc = 0 and therefore Cc should be preferred to the √ (0) (1) . ε1/4 . ε ε3/4 . previous criterion Cc . However, Cc was obtained = 1 50 + 0 316 + O( ) (100) thanks to the assumption that α was close to 1. Par- It should be remarked that the first term can be ticularly, it makes no sense to talk about first-order directly obtained using Deck’s high-eccentricity ap- MMR overlap for α < 0.63 which corresponds to proximation. the center of the MMR 2:1. Therefore, the collision In Figure 5 we plot α and α and indicate criterion should be used for small α. We need then R cir which criterion is used in the areas delimited by the to find αR such that for α < αR, we should use the (0) curves. We specifically represented the region α > critical AMD Cc . Since we are close to 1, we use (0) αcir because we cannot treat this region in a similar a development of Cc presented in (Laskar & Petit manner to the remaining region since comparing 2017), and similarly, only keep the leading terms in (1) the relative AMD C to Cc does not provide any − (1) 1 α in Cc . The two expressions are information. We see that the curve αR is not exactly (0) (1) at the limit where Cc = Cc for higher ε due to the γ (1 − α)2 γ g(α, ε)2 C(0) = , C(1) = . (98) development of the critical AMDs for α → 1. We c 1 + γ 2 c 1 + γ 2 study the influence of γ on the difference between αR and the actual limit in Appendix D We observe that for α close to 1, the two expres- For stability analysis, we need to choose the sion have the same dependence on γ, therefore, αR smallest of the two critical AMD. For α < αR, the (0) (1) depends solely on ε. Simplifying Cc = Cc gives αR collisional criterion is better and the MMR overlap
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We can first observe that β(MMR) is not defined 1.0 (0) for α > αcir. Indeed, the conservation of the AMD Cc (α, γ) (1) cannot be guaranteed for orbits experiencing short- 0.8 Cc (α, γ, ε) term chaos. We use the modified definition of AMD-stability
) 0.6 in order to test its effects on the AMD-classification
α, γ proposed in (Laskar & Petit 2017). ( c
C 0.4
2.0 102 − 0.2 αR αcir
2.5 0.0 − 0.0 0.2 0.4 0.6 0.8 1.0 101 α 3.0 0.02 − c
αR αcir /C ) 0 ε ) 0 Λ 10 3.5 10
− C/ log α, γ 0.01 ( c = ( C β 4.0 − 0.00 1 0.80 0.82 0.84 0.86 0.88 0.90 10− α 4.5 −
Fig. 6. Representation of the two critical AMD pre- 2 (0) 5.0 10− sented in this paper. Cc in black is the collisional crite- − 0.0 0.2 0.4 0.6 0.8 1.0 (1) rion from (Laskar & Petit 2017), Cc in red is the critical α AMD derived from the MMR overlap criterion. In this plot, ε = 10−4 and γ = 1. Fig. 7. Pairs of adjacent planets represented in the α − ε plane. The color corresponds to the AMD-stability coefficient. We plotted the two limits αR corresponding criterion is used for α > αR. We thus define a piece- to the limit between the collision and the MMR-overlap- wise global critical AMD represented in Figure 6 based criterion and αcir corresponding to the MMR over- lap for circular orbits. (0) Cc(α, γ, ε) = Cc (α, γ) α < αR(ε, γ), (1) = Cc (α, γ, ε) α > αR(ε, γ). (101) 5.1. Sample and methodology 5. Effects of the MMR overlap on the We first briefly recall the methodology used in AMD-classification of planetary systems (Laskar & Petit 2017); to which we refer the reader for full details. We compute the AMD-stability co- In (Laskar & Petit 2017), we proposed a classifica- efficients for the systems taken from the Extrasolar tion of the planetary systems based on their AMD- Planets Encyclopædia2 with known periods, planet stability. A system is considered as AMD-stable if masses, eccentricities, and stellar mass. For each every adjacent pair of planets is AMD-stable. A pair pair of adjacent planets, ε was computed using the is considered as AMD-stable if its AMD-stability expression coefficient m1 + m2 C ε = , (103) β = < 1, (102) m0 0 (0) Λ Cc where m1 and m2 are the two planet masses and where C is the total AMD of the system, Λ0 is the m0, the star mass. The semi-major axis ratio was (0) derived from the period ratio and Kepler third law circular momentum of the outer planet and Cc is the critical AMD derived from the collision con- in order to reduce the uncertainty. dition. A similar AMD-coefficient can be defined The systems are assumed coplanar, however in using the global critical AMD defined in (101) in- order to take into account the contribution of the (0) real inclinations to the AMD, we define Cp, the stead of the collisional critical AMD Cc . Let us coplanar AMD of the system, defined as the AMD note β(MMR), the AMD-stability coefficient associ- ated to the critical AMD (101). 2 http://exoplanet.eu/
Article number, page 12 of 19 Petit, Laskar & Bou´e:AMD-stability of the same system if it was coplanar. We can com- (MMR) 2 1 0 1 2 pute coplanar AMD-stability coefficients βp us- 10− 10− 10 10 10 ing Cp instead of C, and we define the total AMD- AMD coefficient β = C /Cc (MMR) stability coefficients as β = 2βp . Doing so, we assume the equipartition of the AMD between the HD 5319 different degree of freedom of the system. We assume the uncertainties of the database HD 200964 quantities to be Gaussian. For the eccentricities, we use the same method as in the previous paper. The HD 33844 quantity e cos $ is assumed to be Gaussian with HD 128311 the mean, the value of the database and standard deviation, the database uncertainty. The quantity HD 47366 e sin $ is assumed to have a Gaussian distribution with zero mean and the same standard deviation. HD 45364 The distribution of eccentricity is then derived from HD 73526 these two distributions. We then propagate the uncertainties through HD 204313 the computations thanks to Monte-Carlo simula- tions of the original distributions. For each of the 101 102 103 104 systems, we drew 10,000 values of masses, periods Period (days) and eccentricities from the computed distributions. We then compute β(MMR) for each of these configu- Fig. 8. Architecture of the systems where the MMR rations and compute the 1-σ confidence interval. overlap criterion changes the AMD-stability. The color In (Laskar & Petit 2017), we studied 131 sys- corresponds to the value of the AMD-stability coeffi- tems but we did not find the stellar mass for 4 cient associated with the inner pair. For the innermost of these systems. They were, as a consequence, ex- planet, it corresponds to the star AMD-stability crite- cluded from this study. Moreover, the computation rion (Laskar & Petit 2017). The diameter of the circle of ε for the pairs of planets of the 127 remaining is proportional to the log of the mass of the planet. systems of the sample led in some cases to high planet-to-star mass ratios. We decide to exclude the systems such that αcir was smaller than the center of the resonance 2:1. We thus discard systems such that a pair of planets has 102 β(col) (MMR) −3 β ε > εlim = 8.20 × 10 . (104) ) 0 c (Λ As a result, we only consider in this study 111 sys- 101 tems that meet the above requirements. C/ =
A pair is considered stable if the 1-σ confi- β dence interval (84% of the simulated systems) of the AMD-stability coefficient β(MMR) is below 1. A 100 system is stable if all adjacent pairs are stable.
5.2. Results HD 5319 HD 47366HD 73526HD 45364HD 33844 Figure 7 shows the planet pairs of the considered HD 128311HD 200964HD 204313 systems in a plane α-ε. The color associated to each point is the AMD-stability coefficient of the pair. Fig. 9. AMD-stability coefficient of the pairs affected (col) The values chosen for the plot correspond for all by the change of criterion. β corresponds to the co- quantities to the median. We remark that very few efficient computed with the collisional critical AMD, and β(MMR) refers to the one computed with the MMR systems are concerned by the change of the critical 3 overlap critical AMD. The triangles represent the pairs AMD, indeed, only eight systems have a pair of (MMR) (1) (0) where β goes to infinity. planets such that Cc < Cc . The 111 considered systems contain 162 planet pairs plotted in Fig- ure 7. This means that less than 5% of the pairs are in a configuration leading to MMR overlap. We plot in Figure 8, the architecture of these 3 It should be noted that for one of the systems, the eight systems and give in Table E.1 the values of MMR overlap criterion was preferred in 16% of the the AMD-stability coefficients. For each of these Monte Carlo simulations. systems, the pair verifying the MMR overlap crite-
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rion was already considered AMD-unstable by the expression (75), valid for all cases. The previous cir- criterion based on the collision. cular (77) and eccentric (79) criteria an then be de- In order to show this, we plot in Figure 9 rived from (75) as particular approximations. More- the AMD-stability coefficients computed with both over, we show that expression (75) can be used to critical AMD. We see that the pairs affected by the directly take into account the first-order MMR in change of criterion were already considered AMD- the notion of AMD-stability. unstable in the purely secular dynamics. However, With this work on first-order MMR, we im- while those pairs have a collisional AMD-coefficient prove the AMD-stability definition by addressing β between 1 and 10, the global AMD-stability co- the problem of the minimal distance between close efficient is increased by roughly an order of magni- orbits. For semi-major axis ratios α above a given tude for the four pairs with α between αR and αcir. threshold αcir (77), that is, αcir < α < 1, the system The AMD-coefficient is not defined for the three is considered unstable whichever value the AMD pairs verifying the circular MMR overlap criterion. may take given that even two circular orbits sat- The pair HD 47366 b/c does not see a significant isfy the MMR overlap criterion. At wider separa- change of its AMD-stability coefficient due to the tions, circular orbits are stable but as eccentricities (0) (1) small number of cases where Cc > Cc . increase two outcomes may happen: Either the sys- We identify three systems, HD 200964, tem enters a region of MMR overlap or the colli- HD204313 and HD 5319, that satisfy the circular sion condition is reached. The system is said to be overlapping criterion. As already explained in AMD-unstable as soon as any of these conditions (Laskar & Petit 2017), AMD-unstable planetary is reached. Above a second threshold, αR < α < αcir systems may not be dynamically unstable. How- (Eq. 100) the AMD-stability is governed by MMR ever, it should be noted that the period ratios of overlap while for wider separations (α < αR) we re- the AMD-unstable planet pairs are very close to trieve the critical AMD defined in (Laskar & Petit particular MMR. 2017) which only depends on the collision condition. Indeed, we have In order to improve the AMD-stability defini- tion for the collision region, we could even take HD 200964 Tc into account the non-secular dynamics induced = 1.344 ' 4/3, (105) T HD 200964 by higher-order MMR and close-encounter conse- b quences on the AMD. To study this requires more T HD 204313 d . ' / , elaborated analytical considerations than those pre- HD 204313 = 1 399 7 5 (106) Tc sented here that are restricted to the first-order T HD 5319 MMR; this will be the goal of future work. c = 1.313 ' 4/3. HD 5319 (107) We show in Section 5 that very few systems Tb satisfy the circular MMR overlap criterion. More- over, the presence of systems satisfying this crite- The AMD-instability of those systems strongly sug- rion strongly suggests that they are protected by a gests that they are indeed into a resonance which particular MMR. In this case, the AMD-instability stabilizes their dynamics. is a simple tool suggesting unobvious dynamical properties. 6. Conclusions As shown in Laskar & Petit (2017), the notion of References AMD-stability is a powerful tool to characterize the Chambers, J., Wetherill, G., & Boss, A. 1996, Icarus, 119, stability of planetary systems. In this framework, 261 the dynamics of a system is reduced to the AMD Chirikov, B. V. 1979, Physics Reports, 52, 263 transfers allowed by the secular evolution. Deck, K. M., Payne, M., & Holman, M. J. 2013, The Astro- However, we need to ensure that the system physical Journal, 774, 129 Delisle, J.-B., Laskar, J., & Correia, A. C. M. 2014, Astron- dynamics can be averaged over its mean motions. omy & Astrophysics, 566, A137 While a system can remain stable and the AMD Delisle, J.-B., Laskar, J., Correia, A. C. M., & Bou´e, G. 2012, or semi-major axis can be averaged over timescales Astronomy & Astrophysics, 546, A71 longer than the libration period in presence of Duncan, M., Quinn, T., & Tremaine, S. 1989, Icarus, 82, 402 MMR, the system stability and particularly the Ferraz-Mello, S. 2007, Astrophysics and Space Science Li- brary, Vol. 345, Canonical Perturbation Theories (New conservation of the AMD is no longer guaranteed York, NY: Springer New York) if the system experiences MMR overlap. In this pa- Gladman, B. 1993, Icarus, 106, 247 per, we use the MMR overlap criterion as a condi- Henrard, J. & Lemaitre, A. 1983, Celestial Mechanics, 30, tion to delimit the zone of the phase space where 197 the dynamics can be considered as secular. Henrard, J., Lemaitre, A., Milani, A., & Murray, C. D. 1986, Celestial Mechanics, 38, 335 We refine the criteria proposed by (Wisdom Laskar, J. 1991, in NATO ASI Series, Vol. 272, Predictability, 1980; Mustill & Wyatt 2012; Deck et al. 2013) and Stability, and Chaos in N-Body Dynamical Systems SE - demonstrate that it is possible to obtain a global 7, ed. A. Roy (Springer US), 93–114
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Laskar, J. 1997, Astronomy and Astrophysics, 317, L75 V is at least of order one in eccentricity. We can Laskar, J. 2000, Physical Review Letters, 84, 3240 Laskar, J. & Petit, A. 2017, Astronomy and Astrophysics In therefore develop (A.5) for small V. We only keep press the terms of first order in eccentricity, Laskar, J. & Robutel, P. 1995, Celestial Mechanics & Dy- namical Astronomy, 62, 193 a2 a2 −1/2 1 a2 −3/2 2 Marchal, C. & Bozis, G. 1982, Celestial Mechanics, 26, 311 = A − VA + O(V ). (A.10) Michtchenko, T. A., Beaug´e,C., & Ferraz-Mello, S. 2008, ∆12 u2 2 u2 Monthly Notices of the Royal Astronomical Society, 387, 747 The well-known development of the circular Murray, C. D. & Dermott, S. F. 1999, Solar system dynamics coplanar motion A gives (e.g., Poincar´e1905) (Cambridge University Press), 592 Mustill, A. J. & Wyatt, M. C. 2012, Monthly Notices of the 1 X −s (k) ik(λ1−λ2) Royal Astronomical Society, 419, 3074 A = bs (α)e , (A.11) Poincar´e, H. 1905, Le¸cons de m´ecanique c´eleste, Tome I 2 k∈Z (Gauthier-Villars. Paris) Prˇsa,A., Harmanec, P., Torres, G., et al. 2016, The Astro- (k) nomical Journal, 152, 41 where bs (α) are the Laplace coefficients (25). Pu, B. & Wu, Y. 2015, The Astrophysical Journal, 807, 44 Because of the averaging over the non-resonant Ramos, X. S., Correa-Otto, J. A., & Beaug´e, C. 2015, Celes- tial Mechanics and Dynamical Astronomy, 123, 453 fast angles, the non-vanishing terms have a depen- Sessin, W. & Ferraz-Mello, S. 1984, Celestial Mechanics, 32, dence on λi of the form j ((p + 1)λ2 − pλ1). Since we 307 only keep the terms of first order in eccentricity, Smith, A. W. & Lissauer, J. J. 2009, Icarus, 201, 381 ± Wisdom, J. 1980, The Astronomical Journal, 85, 1122 the d’Alembert’s rule (7) imposes j = 1. Let us Wisdom, J. 1986, Celestial Mechanics, 38, 175 compute the first-order development of a2/u2 and V in terms of Poincar´evariables and combine these expressions with A−1/2 and A−3/2 in order to select Appendix A: Expression of the first-order the non-vanishing terms. iλi i(λ1−λ2) resonant Hamiltonian Let us denote zi = e and z = z1z¯2 = e . We use the method proposed in (Laskar 1991) and The researched terms are of the form (Laskar & Robutel 1995) to determine the expres- i((p+1)λ −pλ ) −p −(p+1) e 2 1 = z z = z z (A.12) ˆ ˆ 2 1 sion of the planetary perturbation H1. H1 can be −i((p+1)λ2−pλ1) p p+1 decomposed into a part from the gravitational po- e = z¯2z = z¯1z . (A.13) tential between planets Uˆ and a kinetic part Tˆ 1 1 Let us denote as s s εHˆ = Uˆ + Tˆ , (A.1) 2 2Cˆ 1 1 1 i −i$i −i$i 2 Xi = xˆi = e = eie + O(ei ), (A.14) with Λˆ i Λˆ i m m m µ2m3 a Uˆ −G 1 2 − 1 2 2 the first term in the development (A.10) gives 1 = = 2 (A.2) ∆12 m0 Λ ∆12 2 ! a 1 1 1 X u˜ 1 · u˜ 2 1 2 2 2 −1/2 (k) k 2 Tˆ = + (ku˜ k + ku˜ k ). A = 1 + X2z2 + X¯2z¯2 b (α)z +O(e ). 1 1 2 (A.3) u 2 2 2 1/2 2 m0 2m0 2 k∈Z The difficulty comes from the development of (A.15) a2/∆12 and its expression in terms of Poincar´evari- ables. We note S , the angle between u1 and u2. We The contributing term has for expression have 1 (p) ¯ 2 2 2 b1/2(α)(X2 + X2), (A.16) ∆12 = u1 + u2 − 2u1u2 cos S. (A.4) 4
Let us denote ρ = u1/u2, a2/∆12 can be rewritten p i((p+1)λ −pλ ) where Xi = Xˆi 2/Λˆ i = Xie 2 1 . a a −1/2 2 = 2 1 + ρ2 − 2ρ cos S For the computation of the second term of ∆12 u2 (A.10), the only contribution comes from V since a2 a /u ∼ 1. We define = (A + V)−1/2 , (A.5) 2 2 u2 U = X1z1 − X2z2 where we denote s s ˆ ˆ 2 2C1 i(λ −$ ) 2C2 i(λ −$ ) A = 1 + α − 2α cos(λ1 − λ2), (A.6) = e 1 1 − e 2 2 . (A.17) 2 Λˆ 1 Λˆ 2 V = α V2 + 2αV1, (A.7) ρ V can be expressed as a function of z, z¯, U and U¯ . V1 = cos(λ1 − λ2) − cos S, (A.8) α Indeed we have ρ 2 V = − 1. (A.9) ρ 1 2 α = 1 − (U + U¯ ) + O(e2) (A.18) α 2 Article number, page 15 of 19 A&A proofs: manuscript no. mmramdAA edit
and first-order terms must have an angular dependence j(−pλ +(p+1)λ ) ε 1 of the form 1 2 . At the first order in , cos S = z + z¯ + U(z − z¯) + U¯ (¯z − z) +O(e2), (A.19) such a term can only be present in the development 2 of the inner product u˜ 1 · u˜ 2. At the first order in where O(e2) corresponds to terms of total degree in eccentricities, we have (Laskar & Robutel 1995) eccentricities of at least 2. We deduce from these two last expressions that µ2m2m2 1 2 iω1 −iω2 2 u˜ 1 · u˜ 2 = <((e + X1)(e + X¯2)) + O(e ), 1 Λˆ Λˆ V = U(3¯z − z) + U¯ (3z − z¯) + O(e2), (A.20) 1 2 1 4 (A.30) 2 V2 = −(U + U¯ ) + O(e ). (A.21) where ω j is the true longitude of the planet j. The We can therefore write4 only term with the good angular dependence comes i(ω1−ω2) 1 from
After gathering the terms (A.16,A.23), we can Appendix A.1: Asymptotic expression of the resonant give the expression of the resonant Hamiltonian coefficients ˆ H = Kˆ + Rˆ1(Xˆ1 + Xˆ¯1) + Rˆ2(Xˆ2 + Xˆ¯2), (A.24) We present the method we used to obtain the an- alytic development of the coefficients r1 and r2 de- where fined in equations (A.28) and (A.29). Using the ex- s (k) 2 3 pression of bs (α), we have γ µ m2 1 2 Rˆ1 = −ε r1(α), (A.25) 1 + γ ˆ 2 2 ˆ "Z π Λ2 Λ1 α 3 cos(pφ) s r (α) = − dφ+ 2 3 1 4π (1 + α2 − 2α cos φ)3/2 γ µ m2 1 2 −π Rˆ = −ε r (α) (A.26) Z π 2 1 + γ ˆ 2 2 ˆ 2 −2α cos((p + 1)φ) Λ2 Λ2 φ 2 3/2 d + −π (1 + α − 2α cos φ) (A.27) # Z π − cos((p + 2)φ) dφ . (A.33) with γ = m /m , and 2 3/2 1 2 −π (1 + α − 2α cos φ) α r (α) = − 3b(p) (α) − 2αb(p+1)(α) − b(p+2)(α) , 1 4 3/2 3/2 3/2 We can rewrite this expression (A.28) " α α Z π (cos(φ) − α) cos((p + 1)φ) r (α) = 3b(p−1)(α) − 2αb(p) (α) − b(p+1)(α) r α − φ 2 3/2 3/2 3/2 1( ) = 2 3/2 d + 4 2π −π (1 + α − 2α cos φ) 1 Z π # b(p) α . 2 sin φ sin((p + 1)φ) + 1/2( ) (A.29) φ . 2 2 3/2 d (A.34) −π (1 + α − 2α cos φ) The kinetic part Tˆ1 has no contribution to the averaged resonant Hamiltonian for p > 1. Indeed, We make the change of variable φ = (1 − α)u in the as explained above, due to the d’Alembert rule, the integrals. Factoring (1 − α)3, the denominators in the integrals can be developed for α → 1 4 In Laskar & Robutel (1995) the first-order expres- ¯ ¯ sion of V is written W1 = (UZ + UZ) instead of !3/2 ¯ ¯ 1 − cos((1 − α)u) W1 = (UZ + UZ)/2. This misprint in equation (47) of (1 + α2 − 2α cos φ)3/2 = 1 + 2α (Laskar & Robutel 1995) is transmitted as well in equa- (1 − α)2 tion (51). It has no consequences in the results of the 3 2 3/2 paper. ' (1 − α) (1 + u ) . (A.35)
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2/3 Using the relation α0 = (p/(p + 1)) , the numera- The second-order term has for coefficient tors can be developed 1 3 γ3 p2 (p + 1)2 K = − µ2m3 + N − α u − α p − α u 2 2 4 4 1 = (cos((1 ) ) ) cos(( + 1)(1 ) ) 2 2 Λ1,0 Λ2,0 ! 2u ' − 2 2 (1 α) cos (A.36) 3 2 3 4 p (p + 1) 3 = − µ m (γ + α0) + 2 2 p 4 4 γ α0 N2 = 2 sin((1 − α)u) sin((p + 1)(1 − α)u) p+1 ! 2 2u α4 p+1 γ ' − α u . 3 0 p + 2(1 ) sin (A.37) − µ2m3 γ α 4 p 2 3 = 2( + 0) ( + 1) 4 2 γα0 5 Therefore, we deduce the equivalent of r for 1 3 (γ + α0) 1 K = − µ2m3 (p + 1)2. (B.4) p → +∞ 2 2 4 2 2 γα0
2u 2u 3(p + 1) Z +∞ cos + 2u sin r α ∼ − 3 3 u 1( ) 2 3/2 d Appendix C: Width of the resonance island 4π −∞ (1 + u ) K (2/3) + 2K (2/3) We detail in this Appendix the computation of ∼ − 1 0 (p + 1) (A.38) π the resonance island’s width (see also Ferraz-Mello ∼ 0.802(p + 1), (A.39) 2007, Appendix C). where Kν(x) is the modified Bessel function of the Appendix C.1: Coefficients-roots relations second kind. Similarly, we have r2 ∼ −r1 since the additional term is of lower order in p. We first explain how the width of the resonance can We can obtain the constant term of the devel- be related to the position of the saddle point on the X opment by using the second order expression of α0 -axis. The resonant island has a maximal width and developing the integrand to the next order in on the X-axis. Therefore we need to compute the (1 − α). We give here the numerical expressions of expression of the intersections of the separatrices the two developments with the X-axis. Let us note H3, the energy at the saddle point −1 H r1(α0) = −0.802(p + 1) − 0.199 + O(p ), (A.40) (X3, 0). Since the energy of the separatrices is 3 as −1 well, the two intersections of the separatrices with r2(α0) = 0.802(p + 1) + 0.421 + O(p ). (A.41) the X-axis are the solution of the equation
Appendix B: Development of the Keplerian X4 I X2 I2 H (X, 0) = − + 0 − X = H + 0 = H˜ . (C.1) part A 8 2 3 2 3 We show here that the first order in (C − ∆G) of the This equation has three solutions X∗, X∗, and Keplerian part vanishes and give the details of the 1 2 X3 which has a multiplicity of 2. We can therefore computation for the second order. The Keplerian rewrite the equation as part can be written (X − X∗)(X − X∗)(X − X )2 = X4 − 4I X2 + 8X + 8H˜ . µ2m3 1 2 3 0 3 Kˆ − 1 = 2 (C.2) 2(Λ1,0 − p(C − ∆G)) µ2m3 We detail here the relations between the coef- − 2 . 2 (B.1) ficients and the roots of the polynomial equation 2(Λ2,0 + (p + 1)(C − ∆G)) (C.2). We have
Therefore, the first order in C − ∆G has for expres- ∗ ∗ sion X1 + X2 + 2X3 = 0 (C.3) ∗ ∗ ∗ ∗ 2 X X + 2X3(X + X ) + X = −2I0 (C.4) 2 3 3 3 1 2 1 2 3 µ m pγ Λ , K = − 2 2 0 − (p + 1) (C − ∆G) = 0, (B.2) X∗X∗X2 = 8H˜ = −X4 + 2I X2 − 8X . 1 3 3 1 2 3 3 3 0 3 3 Λ2,0 Λ1,0 (C.5)
∗ ∗ since we have From relation C.3, we have directly X1 + X2 = −2X3, and since !3 Λ1,0 3 p = γ . (B.3) ∗ ∗ ∗ ∗ 2 − ∗ − ∗ 2 Λ2,0 p + 1 4X1 X2 = (X1 + X2) (X1 X2) , (C.6) Article number, page 17 of 19 A&A proofs: manuscript no. mmramdAA edit
∗ ∗ 2 Therefore for low-eccentricity systems, we have we can express (X1 − X2) as a function of X3 thanks to the relations (C.4) and (C.5) 1/3 ! 4 δα δαc 1 (p + 1) √ ∗ ∗ ' c , |X1 − X2| = √ . (C.7) 1 + 2/3 1/3 1/3 min (C.14) X3 α0 α0 6 r ε ∗ ∗ We thus deduce the expressions of X1 and X2 as functions of X3 where δαc is the width of the resonance for initially circular orbits defined in (C.10). ∗ 2 X1 = −X3 − √ , (C.8) X3 ∗ 2 X2 = −X3 + √ . (C.9) Appendix D: Influence of γ on the limit αR X3 As explained in section 3.1, we obtain the width As can be seen in Figure 5, the solution αR of equa- of the resonance in terms of variation of α as a func- tion (99) is not the exact limit where the collision and the MMR criteria are equal. Indeed, equation tion of X3 (equation (61)). We can use this expres- (0) sion to obtain the width of the resonance for par- (99) is obtained after the development of Cc and (1) ticular cases detailed in the following subsections. Cc for α close to 1. Since at first order, both ex- pressions have the same dependence on γ, αR does not depend on γ. In order to study the dependence Appendix C.2: Width for initially circular orbits (0) (1) on γ of the limit αlim where Cc = Cc , we plot in In the case of initially circular orbits, the minimal Figure (D.1), for different values of ε, the quantity AMD to enter the resonance is 0. For Cmin = 0, the 2/3 equation (63) gives X3 = 2 as a solution and we αR(ε) − αlim(ε, γ) have δαR(ε, γ) = , (D.1) 1 − αR(ε) δα 8 × 21/3r2/3 ε2/3 p 1/3 = 2/3 ( + 1) α0 3 which gives the error made when approximating 2/3 1/3 = 4.18 ε (p + 1) . (C.10) αlim by αR. We see that all the curves have the same shape with an amplitude increasing with ε. For high We find here the same width of resonance as (Deck γ, α is very accurate even for the greatest values et al. 2013). R of ε. Moreover, the error is maximum for very small γ and always within a few percent. Appendix C.3: Width for highly eccentric orbits The amplitude of the error scales with 1 − α ∝ ε1/4 as we can see in the Figure D.2. We If we consider a system with C χ2/3, our for- R min plot in this Figure D.2 the quantity δα /ε1/4; we malism gives us the result first proposed by Mustill R see that the curves are almost similar, particularly & Wyatt (2012) and improved by Deck et al. (2013) for the smaller values of ε. for eccentric orbits. In this case, we can inject the approximation (67) of X3 in the expression (61) of δα and obtain √ 0.09 δα 8 r p 6 1/4 0.08 ε = 10− = √ ε(p + 1)cmin (C.11) α0 5 3 ) 0.07 ε = 10− p 1/4 R α 4 = 4.14 ε(p + 1)cmin. (C.12) 0.06 ε = 10− √ − 3
(1 ε = 10−
This result is also similar to Deck’s one, using cmin / 0.05 instead of σ (Deck et al. 2013, equation (25)). ) lim 0.04 α
− 0.03 R
Appendix C.4: Width for low eccentric orbits α ( 0.02 2/3 For Cmin χ , we propose here a new expres- 0.01 sion of the width of resonance thanks to the ex- 0.00 pression (68). This expression is an extension of 3 2 1 0 1 2 3 − − − log γ the circular√ result presented above (C.10). Let us 10 2/3 develop X3 for Cmin χ r (0) 1/3 √ Fig. D.1. Difference between the limit αlim where Cc p 2/3 2 (p + 1) (1) X c and Cc are equal and its approximation αR scaled by 3 = 2 + 2/3 1/3 1/3 min 3 r ε 1 − α versus γ for various values of ε. ! R 1 (p + 1)1/3 √ ' 21/3 1 + c . (C.13) 62/3r1/3 ε1/3 min
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Table E.1. AMD-stability coefficients computed for the systems affected by the MMR overlap criterion
N p 2 (MMR) Planet Period (d) Mass (ME ) Eccentricity he i β β N HD 128311 Mass: 0.84 M b 454.2 463.14 0.345 0.352 0.312 c 923.8 1032.46 0.230 0.244 3.200 27.931 N HD 200964 Mass: 1.44 M b 613.8 587.98 0.040 0.067 0.024 c 825 284.46 0.181 0.184 3.872 +∞ N HD 204313 Mass: 1.045 M c 34.905 17.58 0.155 0.184 16.664 b 2024.1 1360.31 0.095 0.095 0.110 0.110 d 2831.6 533.95 0.280 0.308 8.032 +∞ N HD 33844 Mass: 1.75 M b 551.4 622.94 0.150 0.180 0.084 c 916 556.20 0.130 0.189 2.939 22.676 N HD 45364 Mass: 0.82 M b 226.93 59.50 0.168 0.171 0.070 c 342.85 209.10 0.097 0.099 1.975 13.700 N HD 47366 Mass: 1.81 M b 363.3 556.20 0.089 0.138 0.146 c 684.7 591.16 0.278 0.292 2.896 2.896 N HD 5319 Mass: 1.56 M b 675 616.59 0.120 0.162 0.053 c 886 365.50 0.150 0.171 8.659 +∞ N HD 73526 Mass: 1.08 M b 188.9 715.11 0.290 0.293 0.200 c 379.1 715.11 0.280 0.289 3.391 9.922 N Note: Masses are given in terms of nominal terrestrial masses ME and stellar masses in terms of N nominal solar masses M as recommended by the IAU 2015 Resolution B3 (Prˇsaet al. 2016).
the Monte Carlo realizations were affected by the 0.5 change of critical AMD. Apart for the system 6 ε = 10− HD 47366 where 16% of the simulations used the 5 0.4 ε = 10− new criterion, the seven other systems used the crit- (1) 4 ical AMD C for almost all the realizations. For ε = 10− c
4 3 HD 204313, only the pair (b/d) is affected.
/ 0.3
1 ε = 10− p h 2i /ε In Table E.1, e corresponds to the mean R value of the squared eccentricity computed as ex-
δα 0.2 plained in section (5.1).
0.1
0.0 3 2 1 0 1 2 3 − − − log10 γ
1/4 Fig. D.2. δαR scaled by ε versus γ for various values of ε.
Appendix E: AMD-stability coefficients of the system affected by the MMR overlap criterion We report in Table E.1 the AMD-stability coef- ficients of the systems where more than 5% of Article number, page 19 of 19