Issue 
A&A
Volume 607, November 2017



Article Number  A35  
Number of page(s)  17  
Section  Planets and planetary systems  
DOI  https://doi.org/10.1051/00046361/201731196  
Published online  03 November 2017 
AMDstability in the presence of firstorder mean motion resonances
ASD/IMCCE, CNRSUMR 8028, Observatoire de Paris, PSL Research University, UPMC, 77 avenue DenfertRochereau, 75014 Paris, France
email: antoine.petit@obspm.fr
Received: 18 May 2017
Accepted: 10 July 2017
The angular momentum deficit (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. AMDstability is based on the conservation of the 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 meanmotion resonances (MMR). Here we extend the AMDstability criterion to take into account meanmotion resonances, and more specifically the overlap of firstorder MMR. If the MMR islands overlap, the system will experience generalized chaos leading to instability. The Hamiltonian of two massive planets on coplanar quasicircular orbits can be reduced to an integrable one degree of freedom problem for period ratios close to a firstorder MMR. We use the reduced Hamiltonian to derive a new overlap criterion for firstorder 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 AMDstability to take into account the short term chaos generated by MMR overlap. We analyze the outcome of this improved definition of AMDstability on selected multiplanet systems from the Extrasolar Planets Encyclopædia.
Key words: celestial mechanics / planets and satellites: general / planets and satellites: dynamical evolution and stability / planets and satellites: atmospheres
© ESO, 2017
1. Introduction
The angular momentum deficit (AMD)stability criterion (Laskar 2000; Laskar & Petit 2017) allows to discriminate between apriori stable planetary systems and systems needing an indepth dynamical analysis to ensure their stability. The AMDstability is based on the conservation of the angular momentum deficit (AMD, Laskar 1997) in the secular system at all orders of averaging (Laskar 2000; Laskar & Petit 2017). Indeed, the conservation of the AMD fixes an upper bound to the eccentricities. Since the semimajor axes are constant in the secular approximation, a low enough AMD forbids collisions between planets. The AMDstability criterion has been used to classify planetary systems based on the stability of their secular dynamics (Laskar & Petit 2017).
However, while the analytical criterion developed in (Laskar & Petit 2017) does not depend on series expansions for small masses or spacing between the orbits, the secular hypothesis does not hold for systems experiencing mean motion resonances (MMR). Although a system with planets in MMR can be dynamically stable, chaotic behavior may result from the overlap of adjacent MMR, leading to a possible increase of the AMD and eventually to close encounters, collisions or ejections. For systems with small orbital separations, averaging over the mean anomalies is thus impossible due to the contribution of the firstorder MMR terms. For example, two planets in circular orbits very close to each other are AMDstable, however the dynamics of this system cannot be approximated by the secular dynamics. We thus need to modify the notion of AMDstability in order to take into account those configurations.
In studies of planetary systems architecture, a minimal distance based on the Hill radius (Marchal & Bozis 1982) is often used as a criterion of stability (Gladman 1993; Chambers et al. 1996; Smith & Lissauer 2009; Pu & Wu 2015). However, Deck et al. (2013) suggested that stability criteria based on the MMR overlap are more accurate in characterizing the instability of the threebody planetary problem.
Based on the considerations of Chirikov (1979) for the overlap of resonant islands, Wisdom (1980) proposed a criterion of stability for the firstorder MMR overlap in the context of the restricted circular threebody problem. This stability criterion defines a minimal distance between the orbits such that the firstorder MMR overlap with one another. For orbits closer than this minimal distance, the MMR overlapping induces chaotic behavior eventually leading to the instability of the system.
Wisdom showed that the width of the chaotic region in the circular restricted problem is proportional to the ratio of the planet mass to the star mass to the power 2 / 7. Duncan et al. (1989) confirmed numerically that orbits closer than the Wisdom’s MMR overlap condition were indeed unstable. More recently, another stability criterion was proposed by Mustill & Wyatt (2012) to take into account the planet’s eccentricity. Deck et al. (2013) improved the two previous criteria by developing the resonant Hamiltonian for two massive, coplanar, loweccentricity planets and Ramos et al. (2015) proposed a criterion of stability taking into account the secondorder MMR in the restricted threebody problem.
While Deck’s criteria are in good agreement with numerical simulations (Deck et al. 2013) and can be applied to the threebody planetary problem, the case of circular orbits is still treated separately from the case of eccentric orbits. Indeed, the minimal distance imposed by the eccentric MMR overlap stability criterion vanishes with eccentricities and therefore cannot be applied to systems with small eccentricities. In this case, Mustill & Wyatt (2012) and Deck et al. (2013) use the criterion developed for circular orbits. A unified stability criterion for firstorder MMR overlap had yet to be proposed.
In this paper, we propose in Sect. 2 a new derivation of the MMR overlap criterion based on the development of the threebody Hamiltonian by Delisle et al. (2012). We show in Sect. 3 how to obtain a unified criterion of stability working for both initially circular and eccentric orbits. In Sect. 4, we then use the defined stability criterion to limit the region where the dynamics can be considered to be secular and adapt the notion of AMDstability thanks to the new limit of the secular dynamics. Finally we study in Sect. 5 how the modification of the AMDstability definition affects the classification proposed in Laskar & Petit (2017).
2. The resonant Hamiltonian
The problem of two planets close to a firstorder MMR on nearly circular and coplanar orbits can be reduced to a onedegreeoffreedom system through a sequence of canonical transformations (Wisdom 1986; Henrard et al. 1986; Delisle et al. 2012, 2014). We follow here the reduction of the Hamiltonian used in Delisle et al. (2012, 2014).
2.1. Averaged Hamiltonian in the vicinity of a resonance
Let us consider two planets of masses m_{1} and m_{2} orbiting a star of mass m_{0} in the plane. We denote the positions of the planets, u_{i}, and the associated canonical momenta in the heliocentric frame, . The Hamiltonian of the system is (Laskar & Robutel 1995) (1)where Δ_{12} = ∥ u_{1}−u_{2} ∥, and is the constant of gravitation. can be decomposed into a Keplerian part describing the motion of the planets if they had no masses and a perturbation part due to the influence of massive planets, The small parameter ε is defined as the ratio of the planet masses over the star mass (4)Let us denote the angular momentum, (5)which is simply the sum of the two planets Keplerian angular momentum. Ĝ is a first integral of the system.
Following Laskar (1991), we express the Hamiltonian in terms of the Poincaré coordinates (6)where and for i = 1,2, Here, M_{i} corresponds to the mean anomaly, ϖ_{i} to the longitude of the pericenter, a_{i} to the semimajor axis and e_{i} to the eccentricity of the Keplerian orbit of the planet i. Ĝ_{i} is the Keplerian angular momentum of planet i. We use the set of symplectic coordinates of the problem , or the canonically associated variables . The coefficients depend on and the masses of the bodies. They are linear combinations of Laplace coefficients (Laskar & Robutel 1995). As a consequence of angular momentum conservation, the d’Alembert rule gives a relation on the indices of the nonzero coefficients (7)We study here a system with periods close to the firstorder MMR p:p + 1 with p ∈ N^{∗}. For periods close to this configuration, we have −pn_{1} + (p + 1)n_{2} ≃ 0, where is the Keplerian mean motion of the planet i.
2.1.1. Averaging over nonresonant meanmotions
Due to the p:p + 1 resonance, we cannot average on both mean anomalies independently. Therefore, there is no conservation of as in the secular problem. However, the partial averaging over one of the mean anomaly gives another first integral. Following (Delisle et al. 2012), we consider the equivalent set of coordinates , and make the following change of angles (8)The actions associated to these angles are (9)We can now average the Hamiltonian over M_{2} using a change of variables close to the identity given by the Lie series method. Up to terms of orders ε^{2}, we can kill all the terms with indices not of the form . In order to keep the notations light, we do not change the name of the variables after the averaging. We also designate the remaining coefficients by the lighter expression . Since M_{2} does not appear explicitly in the remaining terms, (10)is a first integral of the averaged Hamiltonian. The parameter is often designed as the spacing parameter (Michtchenko et al. 2008) and has been used extensively in the study of the firstorder MMR dynamics.
Expressed with the variables , the Hamiltonian can be written (11)where we dropped the terms of order ε^{2}.
2.1.2. Poincarelike complex coordinates
Delisle et al. (2012) used a change of the angular coordinates in order to remove the exponential in the second term of Eq. (11) and use Ĝ and as actions. The new set of angles (θ_{Γ},θ_{G},σ_{1},σ_{2}) is defined as (12)The conjugated actions are (13)We define , the complex coordinates associated to (Ĉ_{i},σ_{i}). Since we have , the terms of the perturbation in Eq. (11) can be written (14)the last equality resulting from the d’Alembert rule (7). and Ĝ are conserved and the averaged Hamiltonian no longer depends on the angles θ_{Γ} and θ_{G}(15)and can be expressed as functions of the new variables and we have where Ĉ = Ĉ_{1} + Ĉ_{2} is the total AMD of the system. Up to the value of the first integrals and Ĝ, the system now has two effective degrees of freedom.
2.2. Computation of the perturbation coefficients
We now truncate the perturbation, keeping only the leadingorder terms. Since we consider the firstorder MMR, the Hamiltonian contains some linear terms in X_{i}. Therefore the secular terms are neglected since they are at least quadratic. Moreover, the restriction to the planar problem is justified since the inclination terms are at least of order two.
We follow the method described in Laskar (1991) and Laskar & Robutel (1995) to determine the expression of the perturbation . The details of the computation are given in Appendix A. Since we compute an expression at first order in eccentricities and ε, the semi major axis and in particular their ratio, (18)are evaluated at the resonance. At the first order, the perturbation term has for expression (19)where (20)and (21)with γ = m_{1}/m_{2}, In the two previous expressions, are the Laplace coefficients that can be expressed as (24)for k> 0. For k = 0, a 1 / 2 factor has to be added in the secondhand member of (24).
For p = 1, it should be noted that a contribution from the kinetic part should be added (Appendix A and Delisle et al. 2012) (25)Using the expression of α at the resonance p:p + 1, (26)we can give the asymptotic development of the coefficients r_{1} and r_{2} for p → + ∞ (see Appendix A.1). The equivalent is (27)where K_{ν}(x) is the modified Bessel function of the second kind. We note r the numerical factor of the equivalent (27), we have (28)For the resonant coefficients r_{1} and r_{2}, Deck et al. (2013) used the expressions f_{p + 1,27}(α) and f_{p + 1,31}(α) given in Murray & Dermott (1999, pp. 539−556). The expressions (22) and (23) are similar to f_{p + 1,27}(α) and f_{p + 1,31}(α) up to algebraic transformations using the relations between Laplace coefficients (Laskar & Robutel 1995). In their computations, Deck et al. used a numerical fit of the coefficients for p = 2 to 150 and obtained (29)We obtain the same numerical factor r through the analytical development of the functions r_{1} and r_{2}.
2.3. Renormalization
So far, the Hamiltonian has two degrees of freedom and depends on two parameters Ĝ and . As shown in Delisle et al. (2012), the constant can be used to scale the actions, the Hamiltonian and the time without modifying the dynamics. We define With this change of variables, the new Hamiltonian no longer depends on .
The shape of the phase space is now only dependent on the first integral G. However, G does not vanish for the configuration around which the Hamiltonian is developed: the case of two resonant planets on circular orbits. To be able to develop the Keplerian part in power of the system’s parameter, we define ΔG = G_{0}−G, the difference in angular momentum between the circular resonant system and the actual configuration. We have (30)where Λ_{1,0} and Λ_{2,0} are the value of Λ_{1} and Λ_{2} at resonance. By definition, we have (31)Moreover, we can express Λ_{1,0} as a function of the ratios α_{0} and γ, (32)Similarly, Λ_{2,0} can be expressed as (33)Since G_{0} is constant, ΔG is also a first integral of ℋ. From now on, we consider ΔG as a parameter of the twodegreesoffreedom (X_{1},X_{2}) Hamiltonian ℋ. The Keplerian part depends on the coordinates X_{i} through the dependence of Λ_{i} in C.
Λ_{1}and Λ_{2} can be expressed as functions of the Hamiltonian coordinates and their value at the resonance, (34)
2.4. Integrable Hamiltonian
The system can be made integrable by a rotation of the coordinates X_{i} (Sessin & FerrazMello 1984; Henrard et al. 1986; Delisle et al. 2014). We introduce R and φ such that (35)We have and tan(φ) = R_{2}/R_{1}. If we note ℛ_{φ} the rotation of angle φ we define y such that X= ℛ_{φ}y. We still have so the only change in the Hamiltonian is the perturbation term (36)where (I,θ) are the actionangle coordinates associated to y. With these coordinates, I_{2} is a first integral. R has for expression (37)We now develop the Keplerian part around the circular resonant configuration in series of (C−ΔG) thanks to the relations (34). We develop the Keplerian part to the second order in (C−ΔG) since the first order vanishes (see Appendix B). The computation of the secondorder coefficient gives (38)We drop the constant part of the Hamiltonian and obtain the following expression (39)We again change the time scale by dividing the Hamiltonian by and multiplying the time by this factor. We define (40)and after simplification, where r was defined in (28) and f(p) = 1 + O(p^{1}) is a function of p and γ(43)At this point the Hamiltonian can be written (44)and has almost its final form. We divide the actions and the time by χ^{2 / 3} and the Hamiltonian by χ^{4 / 3} and we obtain (45)where (46)
2.5. Andoyer Hamiltonian
We now perform a polar to Cartesian change of coordinates with (47)We change the sign of X in order to have the same orientation as (Deck et al. 2013). Doing so, the Hamiltonian becomes (48)We recognize the second fundamental model of resonance (Henrard & Lemaitre 1983). This Hamiltonian is also called an Andoyer Hamiltonian (FerrazMello 2007). We show in Fig. 1 the level curves of the Hamiltonian ℋ_{A} for ℐ_{0} = 3.
The fixed points of the Hamiltonian satisfy the equations which have for solutions Y = 0 and the real roots of the cubic equation in X(51)Equation (51) has three solutions (Deck et al. 2013) if its determinant , i.e. ℐ_{0}> 3 / 2. In this case, we note these roots X_{1}<X_{2}<X_{3}. X_{1} and X_{2} are elliptic fixed points while X_{3} is a hyperbolic one.
3. Overlap criterion
As seen in the previous section, the motion of two planets near a firstorder MMR can be reduced to an integrable system for small eccentricities and planet masses. However, if two independent combinations of frequencies are close to zero at the same time, the previous reduction is not valid anymore. Indeed, we must then keep, in the averaging, the terms corresponding to both resonances. While for a single resonant term the system is integrable, overlapping resonant islands will lead to chaotic motion (Chirikov 1979).
Wisdom (1980) first applied the resonance overlap criterion to the firstorder MMR and found, in the case of the restricted threebody problem with a circular planet, that the overlap occurs for (52)Through numerical simulations, (Duncan et al. 1989) confirmed Wisdom’s expression up to the numerical coefficient {(1−α< 1.5 ε^{2 / 7})}. A similar criterion was then developed by Mustill & Wyatt (2012) for an eccentric planet, they found that for an eccentricity above 0.2 ε^{3 / 7}, the overlap region satisfies the criterion 1−α< 1.8(εe)^{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 different situations; the case of orbits initially circular and the case of two eccentric orbits. As in Mustill & Wyatt (2012), the eccentric criterion proposed can be used for eccentricities verifying e_{1} + e_{2} ≳ 1.33 ε^{3 / 7}. We show here that the two Deck’s criteria can be obtained as the limit cases of a general expression.
Summary of the diverse notations of AMD used in this paper.
3.1. Width of the libration area
Using the same approach as Wisdom (1980), Deck et al. (2013), we have to express the width of the resonant island as a function of the orbital parameters and compare it with the distance between the two adjacent centers of MMR.
In the (X,Y) plane, the center of the resonance is located at the point of coordinates (X_{1},0). The width of the libration area is defined as the distance between the two separatrices on the Y = 0 axis. It is indeed the direction where the resonant island is the widest.
We note the abscissas of the intersections between the separatrices and the Y = 0 axis. Relations between , and X_{3} can be derived (see Appendix C.1) and we obtain the expressions of and as functions of X_{3} (FerrazMello 2007; Deck et al. 2013). We have The width of the libration zone δX depends solely on the value of X_{3}, (55)In order to study the overlap of resonance islands, we need the width of the resonance in terms of α. Let us invert the previous change of variables in order to express the variation of α in terms of the variation of X. In this subsection, for any function Q(X), we note (56)The computation of δℐ (47) is straightforward from the computation of δX(57)We then directly deduce δI_{1} = χ^{2 / 3}δℐ from Eq. (46). Since I_{2} and ΔG are first integrals, the variation of Λ_{i} only depends on δI_{1}. And finally, since we have (58)αcan be developed to the first order in (C−ΔG) thanks to Eq. (34). This development gives (59)The width of the resonance in terms of α is then directly related to X_{3} through (60)The computation of the width of resonance is thus reduced to the computation of the root X_{3} as a function of the parameters. It should also be remarked that at the first order, the width of resonance does not depend on the mass ratio γ.
3.2. Minimal AMD of a resonance
We are now interested in the overlap of adjacent resonant islands. Planets trapped in the chaotic zone created by the overlap will experience variations of their actions eventually leading to collisions.
For a configuration close to a given resonance p:p + 1, the AMD can evolve toward higher values if the original value places the system in a configuration above the inner separatrix, eventually leading the planets to collision or chaotic motion in case of MMR overlap. On the other hand, if the initial AMD of the planets forces them to remain in the inner circulation region of the overlapped MMR islands, the system will remain stable in regards to this criterion. Since C = I_{1} + I_{2}, and I_{2} is a first integral, we define the minimal AMD of a resonance^{1}C_{min}(p) as the minimal value of I_{1} to enter the resonant island given ΔG−I_{2}. Two cases must be discussed:

the point I_{1} = 0 is already in the libration zone and then C_{min} = 0;

the point I_{1} = 0 is in the inner circulation zone and then we have
(61)In the second case, we have an implicit expression of X_{3} depending on C_{min}(62)where χ was defined in Eq. (40). In other words, there is a onetoone correspondence between C_{min} Eq. (61) and the Hamiltonian parameter ℐ_{0} for C_{min}> 0. The shape of the resonance island is completely described by C_{min}.
We can also use the definition of C_{min} to give an expression depending on the system parameters (63)where . We note (64)the reduced minimal AMD. We can use the expression (63) to compute the quantity appearing in Eq. (62) (65)The function C_{min}(X_{3}) (Eq. (61)) is plotted in Fig. 2 with the two approximations used by Deck et al. (2013) to obtain the width of the resonance. For C_{min} ≫ χ^{2 / 3} or C_{min} close to zero, the relation can be simplified and we obtain We can use the developments (66) and (67) in order to compute the width of the resonance in these two cases (see Appendix C). It should be noted as well that for C_{min} = 0, we have X_{3} = 2^{2 / 3}.
Fig. 2
Relation Eq. (62) between X_{3} and C_{min} Eq. (61) and two different approximations. In red, the approximation used by Deck et al. (2013) for eccentric orbits and in purple the constant evaluation used for circular orbits. 
3.3. Implicit overlap criterion
The overlap of MMR can be determined by finding the first integer p such that the sum of the halfwidth of the resonances p:p + 1 and p + 1:p + 2 is larger than the distance between the respective centers of these two resonances (Wisdom 1980; Deck et al. 2013) (68)where Δα is the distance between the two centers and δα_{k} corresponds to the width of the resonance k:k + 1.
Up to terms of order ε^{2 / 3}, the center of the resonance island p:p + 1 is located at the center of the resonance of the unperturbed Keplerian problem, α_{0,p} = (p/ (p + 1))^{2 / 3}. We develop α_{0,p} for p ≫ 1(69)Therefore, we have at second order in p(70)We can use the implicit expression (62) of X_{3} as a function of (Eq. (64)) in order to derive an overlap criterion independent of approximations on the value of C_{min}. Equating the general width of resonance (60) with the distance between to adjacent centers (70) and isolating X_{3} gives (71)We can inject this expression of X_{3} into (62), and using Eq. (65), (72)Using the first order expression of (p + 1) as a function of α, (73)we obtain an implicit expression of the overlap criterion (74)
3.4. Overlap criterion for circular orbits
The implicit expression (74) can be used to find the criteria proposed by Deck et al. (2013) for circular and eccentric orbits. Let us first obtain the circular criterion by imposing c_{min} = 0 in Eq. (74) (75)We can express 1−α as a function of ε and we obtain (76)The exponent 2/7 was first proposed by Wisdom (1980) and the numerical factor 1.46 is similar to the one found by Deck et al. (2013).
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}. 
3.5. Overlap criterion for higheccentricity orbits
For large eccentricity, Deck et al. (2013) proposes a criterion based on the development (66) of Eq. (62). This criterion is obtained from Eq. (74) by ignoring the second term of the righthand side which leads to (77)Isolating 1−α gives (78)This result is also similar to Deck’s one. For small c_{min}, the criterion (78) is less restrictive than the criterion (76) obtained for circular orbits. The comparison of these two overlap criteria provides a minimal value of c_{min} for the validity of the eccentric criterion (79)
3.6. Overlap criterion for loweccentricity orbits
For smaller eccentricities, we can develop Eq. (74) for small and α close to α_{cir} = 1−1.46ε^{2 / 7}, the critical semi major axis ratio for the circular overlap criterion (76). We have (80)We develop the righthand side at the first order in (α_{cir}−α) and evaluate the lefthand side for α = α_{cir} and after some simplifications obtain (81)We inject the expression of α_{cir} into this equation and obtain the following development of the overlap criterion for low eccentricity: (82)This development remains valid for small enough if α_{cir}−α ≪ 1−α_{cir}, which can be rewritten (83)which leads to (84)It is worth noting that the loweccentricity approximation allows to cover the range of eccentricities where the criterion (78) is not applicable, since both boundaries depend on the same power of ε.
We plot in Fig. 3 the overlap criteria (74) for ε = 10^{6}, the two approximations (76) and (78) from Deck et al. (2013), as well as the collision condition used in Laskar & Petit (2017) approximated for α → 1, (85)We also plot the first MMR islands in order to show the agreement between the proposed criterion and the actual intersections. We see that for high eccentricities, and large 1−α, the system can verify the MMR overlap stability criterion while allowing for collision between the planets. For small α, the MMR overlap criterion alone cannot account for the stability of the system.
4. Critical AMD and MMR
4.1. Critical AMD in a context of resonance overlap
In Laskar & Petit (2017), we present the AMDstability criterion based on the conservation of AMD. We assume the system dynamics to be secular chaotic. As a consequence the averaged semimajor axis and the total averaged AMD are conserved. Moreover, in this approximation the dynamics is limited to random AMD exchanges between planets with conservation of the total AMD. Based on these assumptions, collisions between planets are possible only if the AMD of the system can be distributed such that the eccentricities of the planets allow for collisions. Particularly, for each pair of adjacent planets, there exists a critical AMD, noted C_{c}(α,γ), such that for smaller AMD, collisions are forbidden.
The critical AMD was determined thanks to the limit collision condition (86)However, in practice, the system may become unstable long before orbit intersections; in particular the secular assumption does not hold if the system experiences chaos induced by MMR overlap. We can, though, consider that if the islands do not overlap, the AMD is, on average, conserved on timescales of order ε^{− 2 / 3} (i.e., of the order of the libration timescales). Therefore, the conservation, on average, of the AMD is ensured as long as the system adheres to the above criteria for any distribution of the AMD between planets. Based on the model of Laskar & Petit (2017), we compute a critical AMD associated to the criterion (74).
We consider a pair as AMDstable if no distribution of AMD between the two planets allows the overlap of MMR. A first remark is that no pair can be considered as AMDstable if α>α_{cir}, because in this case, even the circular orbits lead to MMR overlap. Let us write the criterion (74) as a function of α and ε; (87)where (88)depends on Δϖ and has a maximum for Δϖ = π. Since the variation of Δϖ does not affect the AMD of the system, we fix Δϖ = π since it is the leastfavorable configuration. Therefore we have (89)We define the relative AMD of a pair of planets C and express it as a function of the variables c_{i}(90)The critical AMD associated to the overlap criterion (74) can be defined as the smallest value of relative AMD such that the conditions are verified by any couple (c_{1},c_{2}). We represent this configuration in Fig. 4.
Fig. 4
MMR overlap criterion represented in the (c_{1},c_{2}) plane. 
As in Laskar & Petit (2017), the critical AMD is obtained through Lagrange multipliers (93)The tangency condition gives a relation between c_{1} and c_{2}, (94)Replacing c_{2} in relation (91) gives the critical expression of c_{1} and we immediately obtain the expression of c_{2}(95)The value of is obtained by injecting the critical values c_{c,1} and c_{c,2} into the expression of C(96)
Fig. 5
Regions of application of the different criteria presented in this work. The purple region represents is the smallest, in the green zone, is the smallest and the circular overlap criterion is verified in the red zone. We see that the curve α_{R} computed through a development of and presents a good agreement with the real limit between the green and the purple area. Here γ = 1. 
4.2. Comparison with the collision criterion
It is then natural to compare the critical AMD to the critical AMD C_{c} (denoted hereafter by ) derived from the collision condition (86). If α>α_{cir}, the circular overlap criterion implies that and therefore should be preferred to the previous criterion . However, was obtained thanks to the assumption that α was close to 1. Particularly, it makes no sense to talk about firstorder MMR overlap for α< 0.63 which corresponds to the center of the MMR 2:1. Therefore, the collision criterion should be used for small α. We need then to find α_{R} such that for α<α_{R}, we should use the critical AMD . Since we are close to 1, we use a development of presented in Laskar & Petit (2017), and similarly, only keep the leading terms in 1−α in . The two expressions are (97)We observe that for α close to 1, the two expression have the same dependence on γ, therefore, α_{R} depends solely on ε. Simplifying gives α_{R} as a solution of the polynomial equation in (1−α); (98)While an exact analytical solution cannot be provided, a development in powers of ε gives the following expression (99)It should be remarked that the first term can be directly obtained using Deck’s higheccentricity approximation.
In Fig. 5 we plot α_{R} and α_{cir} and indicate which criterion is used in the areas delimited by the curves. We specifically represented the region α>α_{cir} because we cannot treat this region in a similar manner to the remaining region since comparing the relative AMD C to does not provide any information. We see that the curve α_{R} is not exactly at the limit where for higher ε due to the development of the critical AMDs for α → 1. We study the influence of γ on the difference between α_{R} and the actual limit in Appendix D
For stability analysis, we need to choose the smallest of the two critical AMD. For α<α_{R}, the collisional criterion is better and the MMR overlap criterion is used for α>α_{R}. We thus define a piecewise global critical AMD represented in Fig. 6(100)
Fig. 6
Representation of the two critical AMD presented in this paper. in black is the collisional criterion from Laskar & Petit (2017), in red is the critical AMD derived from the MMR overlap criterion. In this plot, ε = 10^{4} and γ = 1. 
5. Effects of the MMR overlap on the AMDclassification of planetary systems
In Laskar & Petit (2017), we proposed a classification of the planetary systems based on their AMDstability. A system is considered as AMDstable if every adjacent pair of planets is AMDstable. A pair is considered as AMDstable if its AMDstability coefficient (101)where C is the total AMD of the system, Λ′ is the circular momentum of the outer planet and is the critical AMD derived from the collision condition. A similar AMDcoefficient can be defined using the global critical AMD defined in (100) instead of the collisional critical AMD . Let us note β^{(MMR)}, the AMDstability coefficient associated to the critical AMD (100).
We can first observe that β^{(MMR)} is not defined for α>α_{cir}. Indeed, the conservation of the AMD cannot be guaranteed for orbits experiencing shortterm chaos.
We use the modified definition of AMDstability in order to test its effects on the AMDclassification proposed in Laskar & Petit (2017).
5.1. Sample and methodology
We first briefly recall the methodology used in Laskar & Petit (2017); to which we refer the reader for full details. We compute the AMDstability coefficients for the systems taken from the Extrasolar Planets Encyclopaedia^{2} with known periods, planet masses, eccentricities, and stellar mass. For each pair of adjacent planets, ε was computed using the expression (102)where m_{1} and m_{2} are the two planet masses and m_{0}, the star mass. The semimajor axis ratio was derived from the period ratio and Kepler third law in order to reduce the uncertainty.
The systems are assumed coplanar, however in order to take into account the contribution of the real inclinations to the AMD, we define C_{p}, the coplanar AMD of the system, defined as the AMD of the same system if it was coplanar. We can compute coplanar AMDstability coefficients using C_{p} instead of C, and we define the total AMDstability coefficients as . Doing so, we assume the equipartition of the AMD between the different degree of freedom of the system.
We assume the uncertainties of the database quantities to be Gaussian. For the eccentricities, we use the same method as in the previous paper. The quantity ecosϖ is assumed to be Gaussian with the mean, the value of the database and standard deviation, the database uncertainty. The quantity esinϖ is assumed to have a Gaussian distribution with zero mean and the same standard deviation. The distribution of eccentricity is then derived from these two distributions.
We then propagate the uncertainties through the computations thanks to Monte Carlo simulations of the original distributions. For each of the systems, we drew 10 000 values of masses, periods and eccentricities from the computed distributions. We then compute β^{(MMR)} for each of these configurations and compute the 1σ confidence interval.
In Laskar & Petit (2017), we studied 131 systems but we did not find the stellar mass for 4 of these systems. They were, as a consequence, excluded from this study. Moreover, the computation of ε for the pairs of planets of the 127 remaining systems of the sample led in some cases to high planettostar mass ratios. We decide to exclude the systems such that α_{cir} was smaller than the center of the resonance 2:1. We thus discard systems such that a pair of planets has (103)As a result, we only consider in this study 111 systems that meet the above requirements.
A pair is considered stable if the 1σ confidence interval (84% of the simulated systems) of the AMDstability coefficient β^{(MMR)} is below 1. A system is stable if all adjacent pairs are stable.
5.2. Results
Fig. 7
Pairs of adjacent planets represented in the α−ε plane. The color corresponds to the AMDstability coefficient. We plotted the two limits α_{R} corresponding to the limit between the collision and the MMRoverlapbased criterion and α_{cir} corresponding to the MMR overlap for circular orbits. 
Fig. 8
Architecture of the systems where the MMR overlap criterion changes the AMDstability. The color corresponds to the value of the AMDstability coefficient associated with the inner pair. For the innermost planet, it corresponds to the star AMDstability criterion (Laskar & Petit 2017). The diameter of the circle is proportional to the log of the mass of the planet. 
Fig. 9
AMDstability coefficient of the pairs affected by the change of criterion. β^{(col)} corresponds to the coefficient computed with the collisional critical AMD, and β^{(MMR)} refers to the one computed with the MMR overlap critical AMD. The triangles represent the pairs where β^{(MMR)} goes to infinity. 
Figure 7 shows the planet pairs of the considered systems in a plane α–ε. The color associated to each point is the AMDstability coefficient of the pair. The values chosen for the plot correspond for all quantities to the median. We remark that very few systems are concerned by the change of the critical AMD, indeed, only eight systems^{3} have a pair of planets such that . The 111 considered systems contain 162 planet pairs plotted in Fig. 7. This means that less than 5% of the pairs are in a configuration leading to MMR overlap.
We plot in Fig. 8, the architecture of these eight systems and give in Table E.1 the values of the AMDstability coefficients. For each of these systems, the pair verifying the MMR overlap criterion was already considered AMDunstable by the criterion based on the collision.
In order to show this, we plot in Fig. 9 the AMDstability coefficients computed with both critical AMD. We see that the pairs affected by the change of criterion were already considered AMDunstable in the purely secular dynamics. However, while those pairs have a collisional AMDcoefficient β between 1 and 10, the global AMDstability coefficient is increased by roughly an order of magnitude for the four pairs with α between α_{R} and α_{cir}. The AMDcoefficient is not defined for the three pairs verifying the circular MMR overlap criterion. The pair HD 47366 b/c does not see a significant change of its AMDstability coefficient due to the small number of cases where .
We identify three systems, HD 200964, HD 204313 and HD 5319, that satisfy the circular overlapping criterion. As already explained in Laskar & Petit (2017), AMDunstable planetary systems may not be dynamically unstable. However, it should be noted that the period ratios of the AMDunstable planet pairs are very close to particular MMR.
Indeed, we have The AMDinstability of those systems strongly suggests that they are indeed into a resonance which stabilizes their dynamics.
6. Conclusions
As shown in Laskar & Petit (2017), the notion of AMDstability is a powerful tool to characterize the stability of planetary systems. In this framework, the dynamics of a system is reduced to the AMD transfers allowed by the secular evolution.
However, we need to ensure that the system dynamics can be averaged over its mean motions. While a system can remain stable and the AMD or semimajor axis can be averaged over timescales longer than the libration period in presence of MMR, the system stability and particularly the conservation of the AMD is no longer guaranteed if the system experiences MMR overlap. In this paper, we use the MMR overlap criterion as a condition to delimit the zone of the phase space where the dynamics can be considered as secular.
We refine the criteria proposed by Wisdom (1980), Mustill & Wyatt (2012), Deck et al. (2013) and demonstrate that it is possible to obtain a global expression (74), valid for all cases. The previous circular Eq. (76) and eccentric Eq. (78) criteria an then be derived from Eq. (74) as particular approximations. Moreover, we show that expression (74) can be used to directly take into account the firstorder MMR in the notion of AMDstability.
With this work on firstorder MMR, we improve the AMDstability definition by addressing the problem of the minimal distance between close orbits. For semimajor axis ratios α above a given threshold α_{cir} Eq. (76), that is, α_{cir}<α< 1, the system is considered unstable whichever value the AMD may take given that even two circular orbits satisfy the MMR overlap criterion. At wider separations, circular orbits are stable but as eccentricities increase two outcomes may happen: either the system enters a region of MMR overlap or the collision condition is reached. The system is said to be AMDunstable as soon as any of these conditions is reached. Above a second threshold, α_{R}<α<α_{cir} (Eq. (99)) the AMDstability is governed by MMR overlap while for wider separations (α<α_{R}) we retrieve the critical AMD defined in Laskar & Petit (2017) which only depends on the collision condition.
In order to improve the AMDstability definition for the collision region, we could even take into account the nonsecular dynamics induced by higherorder MMR and closeencounter consequences on the AMD. To study this requires more elaborated analytical considerations than those presented here that are restricted to the firstorder MMR; this will be the goal of future work.
We show in Sect. 5 that very few systems satisfy the circular MMR overlap criterion. Moreover, the presence of systems satisfying this criterion strongly suggests that they are protected by a particular MMR. In this case, the AMDinstability is a simple tool suggesting unobvious dynamical properties.
We summarize the notations of the various AMD expressions used in this paper in Table 1.
In Laskar & Robutel (1995) the firstorder expression of V is written instead of . This misprint in Eq. (47) of Laskar & Robutel (1995) is transmitted as well in Eq. (51). It has no consequences in the results of the paper.
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Appendix A: Expression of the firstorder resonant Hamiltonian
We use the method proposed in Laskar (1991) and Laskar & Robutel (1995) to determine the expression of the planetary perturbation . can be decomposed into a part from the gravitational potential between planets Û_{1} and a kinetic part as (A.1)with The difficulty comes from the development of a_{2}/ Δ_{12} and its expression in terms of Poincaré variables. We note S, the angle between u_{1} and u_{2}. We have (A.4)Let us denote ρ = u_{1}/u_{2}, a_{2}/ Δ_{12} can be rewritten (A.5)where we denote Vis at least of order one in eccentricity. We can therefore develop Eq. (A.5) for small V. We only keep the terms of first order in eccentricity, (A.10)The wellknown development of the circular coplanar motion A gives (e.g., Poincaré 1905) (A.11)where are the Laplace coefficients (24).
Because of the averaging over the nonresonant fast angles, the nonvanishing terms have a dependence on λ_{i} of the form j((p + 1)λ_{2}−pλ_{1}). Since we only keep the terms of first order in eccentricity, the d’Alembert’s rule (7) imposes j = ± 1. Let us compute the firstorder development of a_{2}/u_{2} and V in terms of Poincaré variables and combine these expressions with A^{− 1 / 2} and A^{− 3 / 2} in order to select the nonvanishing terms.
Let us denote z_{i} = e^{iλi} and . The researched terms are of the form Let us denote (A.14)the first term in the development (A.10) gives (A.15)The contributing term has for expression (A.16)where .
For the computation of the second term of (A.10), the only contribution comes from V since a_{2}/u_{2} ~ 1. We define (A.17)Vcan be expressed as a function of and . Indeed we have (A.18)and (A.19)where O(e^{2}) corresponds to terms of total degree in eccentricities of at least 2. We deduce from these two last expressions that We can therefore write^{4}(A.22)where . With this expression of V, it is easy to gather the corresponding terms and the second term in the development (A.10) gives the contributing term (A.23)After gathering the terms (A.16), (A.23), we can give the expression of the resonant Hamiltonian (A.24)where with γ = m_{1}/m_{2}, and The kinetic part has no contribution to the averaged resonant Hamiltonian for p> 1. Indeed, as explained above, due to the d’Alembert rule, the firstorder terms must have an angular dependence of the form j(−pλ_{1} + (p + 1)λ_{2}). At the first order in ε, such a term can only be present in the development of the inner product . At the first order in eccentricities, we have (Laskar & Robutel 1995) (A.30)where ω_{j} is the true longitude of the planet j. The only term with the good angular dependence comes from ℜe^{i(ω1−ω2)} since the other firstorder terms only depend on one mean longitude. The development of e^{i(ω1−ω2)} at the first order in eccentricities gives (A.31)Thus for p> 1, has no contribution to the averaged Hamiltonian, and for p = 1 we have (A.32)
Appendix A.1: Asymptotic expression of the resonant coefficients
We present the method we used to obtain the analytic development of the coefficients r_{1} and r_{2} defined in Eqs. (A.28) and (A.29). Using the expression of , we have (A.33)We can rewrite this expression (A.34)We make the change of variable φ = (1−α)u in the integrals. Factoring (1−α)^{3}, the denominators in the integrals can be developed for α → 1(A.35)Using the relation α_{0} = (p/ (p + 1))^{2 / 3}, the numerators can be developed Therefore, we deduce the equivalent of r_{1} for p → + ∞where K_{ν}(x) is the modified Bessel function of the second kind. Similarly, we have r_{2} ~ −r_{1} since the additional term is of lower order in p.
We can obtain the constant term of the development by using the second order expression of α_{0} and developing the integrand to the next order in (1−α). We give here the numerical expressions of the two developments
Appendix B: Development of the Keplerian part
We show here that the first order in (C−ΔG) of the Keplerian part vanishes and give the details of the computation for the second order. The Keplerian part can be written (B.1)Therefore, the first order in C−ΔG has for expression (B.2)since we have (B.3)The secondorder term has for coefficient (B.4)
Appendix C: Width of the resonance island
We detail in this Appendix the computation of the resonance island’s width (see also FerrazMello 2007, Appendix C).
Appendix C.1: Coefficientsroots relations
We first explain how the width of the resonance can be related to the position of the saddle point on the Xaxis. The resonant island has a maximal width on the Xaxis. Therefore we need to compute the expression of the intersections of the separatrices with the Xaxis.
Let us note ℋ_{3}, the energy at the saddle point (X_{3},0). Since the energy of the separatrices is ℋ_{3} as well, the two intersections of the separatrices with the Xaxis are the solution of the equation (C.1)This equation has three solutions , and X_{3} which has a multiplicity of 2. We can therefore rewrite the equation as (C.2)We detail here the relations between the coefficients and the roots of the polynomial Eq. (C.2). We have From relation (C.3), we have directly , and since (C.6)we can express as a function of X_{3} thanks to the relations (C.4) and (C.5) (C.7)We thus deduce the expressions of and as functions of X_{3}As explained in Sect. 3.1, we obtain the width of the resonance in terms of variation of α as a function of X_{3} (Eq. (60)). We can use this expression to obtain the width of the resonance for particular cases detailed in the following subsections.
Appendix C.2: Width for initially circular orbits
In the case of initially circular orbits, the minimal AMD to enter the resonance is 0. For C_{min} = 0, Eq. (62) gives X_{3} = 2^{2 / 3} as a solution and we have (C.10)We find here the same width of resonance as Deck et al. (2013).
Appendix C.3: Width for highly eccentric orbits
If we consider a system with C_{min} ≫ χ^{2 / 3}, our formalism gives us the result first proposed by Mustill & Wyatt (2012) and improved by Deck et al. (2013) for eccentric orbits. In this case, we can inject the approximation (66) of X_{3} in the expression (60) of δα and obtain This result is also similar to Deck’s one, using instead of σ (Deck et al. 2013, Eq. (25)).
Appendix C.4: Width for low eccentric orbits
For C_{min} ≪ χ^{2 / 3}, we propose here a new expression of the width of resonance thanks to the expression (67). This expression is an extension of the circular result presented above Eq. (C.10). Let us develop for C_{min} ≪ χ^{2 / 3}(C.13)Therefore for loweccentricity systems, we have (C.14)where δα_{c} is the width of the resonance for initially circular orbits defined in Eq. (C.10).
Appendix D: Influence of γ on the limit α_{R}
As can be seen in Fig. 5, the solution α_{R} of Eq. (98) is not the exact limit where the collision and the MMR criteria are equal. Indeed, Eq. (98) is obtained after the development of and for α close to 1. Since at first order, both expressions have the same dependence on γ, α_{R} does not depend on γ. In order to study the dependence on γ of the limit α_{lim} where , we plot in Fig. (D.1), for different values of ε, the quantity (D.1)which gives the error made when approximating α_{lim} by α_{R}. We see that all the curves have the same shape with an amplitude increasing with ε. For high γ, α_{R} is very accurate even for the greatest values of ε. Moreover, the error is maximum for very small γ and always within a few percent.
The amplitude of the error scales with 1−α_{R} ∝ ε^{1 / 4} as we can see in Fig. D.2. We plot in this Fig. D.2 the quantity δα_{R}/ε^{1 / 4}; we see that the curves are almost similar, particularly for the smaller values of ε.
Fig. D.1
Difference between the limit α_{lim} where and are equal and its approximation α_{R} scaled by 1−α_{R} versus γ for various values of ε. 
Fig. D.2
δα_{R} scaled by ε^{1 / 4} versus γ for various values of ε. 
Appendix E: AMDstability coefficients of the system affected by the MMR overlap criterion
We report in Table E.1 the AMDstability coefficients of the systems where more than 5% of the Monte Carlo realizations were affected by the change of critical AMD. Apart for the system HD 47366 where 16% of the simulations used the new criterion, the seven other systems used the critical AMD for almost all the realizations. For HD 204313, only the pair (b/d) is affected.
In Table E.1, corresponds to the mean value of the squared eccentricity computed as explained in Sect. (5.1).
AMDstability coefficients computed for the systems affected by the MMR overlap criterion
All Tables
AMDstability coefficients computed for the systems affected by the MMR overlap criterion
All Figures
Fig. 1
Hamiltonian ℋ_{A} (48) represented with the saddle point and the separatrices in red. 

In the text 
Fig. 2
Relation Eq. (62) between X_{3} and C_{min} Eq. (61) and two different approximations. In red, the approximation used by Deck et al. (2013) for eccentric orbits and in purple the constant evaluation used for circular orbits. 

In the text 
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}. 

In the text 
Fig. 4
MMR overlap criterion represented in the (c_{1},c_{2}) plane. 

In the text 
Fig. 5
Regions of application of the different criteria presented in this work. The purple region represents is the smallest, in the green zone, is the smallest and the circular overlap criterion is verified in the red zone. We see that the curve α_{R} computed through a development of and presents a good agreement with the real limit between the green and the purple area. Here γ = 1. 

In the text 
Fig. 6
Representation of the two critical AMD presented in this paper. in black is the collisional criterion from Laskar & Petit (2017), in red is the critical AMD derived from the MMR overlap criterion. In this plot, ε = 10^{4} and γ = 1. 

In the text 
Fig. 7
Pairs of adjacent planets represented in the α−ε plane. The color corresponds to the AMDstability coefficient. We plotted the two limits α_{R} corresponding to the limit between the collision and the MMRoverlapbased criterion and α_{cir} corresponding to the MMR overlap for circular orbits. 

In the text 
Fig. 8
Architecture of the systems where the MMR overlap criterion changes the AMDstability. The color corresponds to the value of the AMDstability coefficient associated with the inner pair. For the innermost planet, it corresponds to the star AMDstability criterion (Laskar & Petit 2017). The diameter of the circle is proportional to the log of the mass of the planet. 

In the text 
Fig. 9
AMDstability coefficient of the pairs affected by the change of criterion. β^{(col)} corresponds to the coefficient computed with the collisional critical AMD, and β^{(MMR)} refers to the one computed with the MMR overlap critical AMD. The triangles represent the pairs where β^{(MMR)} goes to infinity. 

In the text 
Fig. D.1
Difference between the limit α_{lim} where and are equal and its approximation α_{R} scaled by 1−α_{R} versus γ for various values of ε. 

In the text 
Fig. D.2
δα_{R} scaled by ε^{1 / 4} versus γ for various values of ε. 

In the text 
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