Retrograde resonance in the planar three-body problem

Transcription

1 Noname manuscript No. (will be inserted b the editor) Retrograde resonance in the planar three-bod problem M. H. M. Morais F. Namouni Received: date / Accepted: date Abstract We continue the investigation of the dnamics of retrograde resonances initiated in Morais and Giuppone (22). After deriving a procedure to deduce the retrograde resonance terms from the standard expansion of the three-dimensional disturbing function, we concentrate on the planar problem and construct surfaces of section that explore phase-space in the vicinit of the main retrograde resonances (2/-, /- and /-2). In the case of the /- resonance for which the standard expansion is not adequate to describe the dnamics, we develop a semi-analtic model based on numerical averaging of the unexpanded disturbing function, and show that the predicted libration modes are in agreement with the behavior seen in the surfaces of section. Kewords Resonance. Three-bod problem. urface of section. Introduction The discover of extrasolar planets that orbit their host stars in the direction opposite the star s rotation has renewed interest in the dnamics of retrograde motion in gravitational sstems (Triaud et al, 2). In the solar sstem, retrograde motion is confined to smaller bodies such as satellites of the outer planets and long period comets. Understanding the structure of retrograde motion and in particular retrograde resonances will help elucidate the origin and evolution of the observed sstems. Gaon and Bois (28) performed numerical integrations of sstems with planets moving in opposite directions and observed that a retrograde resonance in two-planet sstems is more stable than the equivalent prograde resonance confirming the idea that when two bodies orbit in different directions, encounters occur at a higher relative velocit during a shorter time and mutual perturbations are therefore weaker. In Gaon et al (29) the authors obtained the 2/ retrograde resonance disturbing function, and identified retrograde resonance angles. However, the were unable to M. H. M. Morais Department of Phsics & I3N, Universit of Aveiro, Campus de antiago, Aveiro, Portugal helena.morais@ua.pt F. Namouni Université de Nice, CNR, Observatoire de la Côte d Azur, BP 4229, 634 Nice, France

2 2 M. H. M. Morais, F. Namouni find initial conditions that correspond to libration in retrograde resonance, and did not identif the theoretical reason for the observed enhanced stabilit of retrograde resonances. In Morais and Giuppone (22) (hereafter Paper I) we compared the stabilit of prograde and retrograde planets within a binar sstem. We observed that retrograde planets remain stable nearer to the secondar star than prograde planets. We showed that instabilit is caused b single mean motion resonances (MMRs) and the possible overlap of adjacent pairs. We used a standard expansion of the disturbing function for the planar circular restricted three-bod problem (CR3BP) to obtain the retrograde resonance terms, and we explained how these terms show wh retrograde resonances are more stable than prograde resonances. Indeed, the magnitude of the p/q resonance terms is proportional to a power of the eccentricit, which at the lowest order, is e p+q in the retrograde case and e p q in the prograde case. In this paper, we continue our investigation of retrograde motion in the three-bod problem b sstematicall studing the structure of the phase space near the main retrograde resonances. We concentrate on the planar problem and examine in detail motion near the 2/, / and /2 retrograde resonances. In section 2, we explain how to obtain the retrograde resonance terms from an expansion in Laplace coefficients of the three-dimensional disturbing function. We also show that, in the planar problem, these coincide with the retrograde resonance terms obtained in Paper I. ince Laplace coefficients diverge when the semi major axes ratio is close to unit, in section 3, we develop a semi-analtical model for the co-orbital / retrograde resonance based on numerical averaging of the unexpanded disturbing function. In section 4, we present our numerical approach and results, and describe how retrograde motion phase-space is structured and where stable motion is possible. ection 5 contains a discussion of our results. 2 Differences between prograde and retrograde resonance The encounter of two bodies in a retrograde configuration (orbiting in opposite directions) occurs at a higher relative velocit during a shorter time than in a prograde configuration. This implies that mutual perturbations are generall weaker for retrograde MMRs. In Paper I, we studied retrograde MMRs analticall in the context of the planar circular restricted three-bod problem and compared their relative strength and stabilit to prograde resonances. Here, we will explain how we obtain the slow terms of the disturbing function for a p/q retrograde MMR in the three-dimensional CR3BP. 2. Disturbing function Consider a test particle that moves under the gravitational effect of a binar composed of a primar with mass M and a secondar with mass m M. The motion of m with respect to M is a circular orbit of radius a = and longitude angle λ. The reference plane is defined b the binar s orbit. The test particle s osculating Keplerian orbit with respect to M has semi-major axis a, eccentricit e, inclination I, true anomal f, argument of pericentre ω, and longitude of ascending node Ω. The disturbing function reads: R = G m(/ r cos ψ) () where r is the radius of test particle, ψ is the angle between the radius vectors of the binar and the test particle, 2 = + r 2 2 r cos ψ, and cos ψ = cos(ω λ ) cos(ω + f) sin(ω λ ) sin(ω + f) cos I. (2)

3 Retrograde resonance in the planar three-bod problem 3 The first term of R (direct perturbation) is the gravitational force from the mass m on the test particle whereas the second term (indirect perturbation) comes from the reflex motion of the star under the influence of the mass m as the standard coordinate sstem is chosen to be centered on the star. The classic series of the disturbing function is expanded in powers of sin 2 (I/2). This is adequate for nearl coplanar prograde motion since sin 2 (I/2) but not for nearl coplanar retrograde motion since sin 2 (I/2). Therefore, for nearl coplanar orbits, we define β such that I = β or I = 8 β, respectivel, for prograde or retrograde motion. We ma therefore write: cos ψ = ( s 2 ) cos(f + ω ± (Ω λ )) + s 2 cos(f + ω (Ω λ )), (3) where s 2 = sin 2 (β/2), and the ± sign applies to prograde or retrograde cases, respectivel. Next, we write 2 = + r 2 2 r cos(f + ω ± (Ω λ )) 2rΨ, where Ψ is defined as: Ψ = cos ψ cos(f + ω ± (Ω λ )) = 2 s 2 sin(±(ω λ )) sin(ω + f). (4) Expanding the direct perturbation term in the vicinit of Ψ = (as s 2 ), we ma write: = ( ) i (2i)! (i!) 2 2 rψ 2 i+, (5) i= where 2 = + r 2 2 r cos(f + ω ± (Ω λ )). (2 i+) Finall, defining ɛ = r/a = O(e) and expanding around ɛ = : ( ) 2 i+ = + k! ɛk α k d k, (6) d k α ρ2 i+ with α = a/a, and k= ρ 2 i+ = ( + α2 2 α cos(f + ω ± (Ω λ ))) (i+/2), = 2 bj i+/2 (α) cos(j(f + ω ± (Ω λ ))), (7) j where b j i+/2 (α) are Laplace coefficients. For α <, the ma be expanded as convergent series in α (Ellis and Murra, 2). Therefore, for retrograde orbits (I 8 ) the disturbing function (Eq. 5) is expanded in powers of cos(i/2), whereas for prograde orbits (I ) it is expanded in powers of sin(i/2). Yokoama et al (25) studied the effect of Triton s retrograde orbit on the motion of Neptune s satellites. The used computer algebra to expand the disturbing function in powers of sin(i), and compared direct numerical integrations of the equations of motion with results obtained from the integration of Lagrange s equations using the retrograde disturbing function. The advantage of our approach is that it allows us to obtain the retrograde disturbing function directl from the well known prograde disturbing function, without the need for specific computer algebra. We will now explain that procedure in detail.

4 4 M. H. M. Morais, F. Namouni Examination of the angles in the expression of Ψ (4) and that of ρ (7) show that there are two was with which the expansion of the disturbing function for retrograde motion can be obtained from the expansion of the disturbing function for prograde motion. The two was are equivalent as the depend on whether one chooses to invert the motion of the inner bod or that of the outer one. Inverting the motion of the outer bod gives the first transformation: I = 8 I, λ = λ, ω = ω π and Ω = Ω π. In this case, the longitude of pericentre ϖ = ω + Ω is transformed into ϖ = ω Ω and the mean longitude λ = M + ω + Ω into λ = M + ω Ω where M is the mean anomal. Equivalentl, this means appling the generating function F = λ Λ + (λ + 2z)Λ + (g 2z)Γ (z π)z to the usual Poincaré action-angle variables. Inverting the motion of the inner bod gives the second possible transformation: I = 8 I, f = f, ω = π ω and Ω = π + Ω that ma be obtained with the generating function F 2 = λ Λ (λ+2z)λ (g 2z)Γ + (z π)z. We note however that these two transformations are passive in that the allow us onl to obtain the expression of the resonant arguments. Once the arguments are obtained formall, the assumption that λ > and λ > alwas holds. This is in contrast to the approach adopted in Paper I where an active transformation was used to stud the planar dnamical problem b choosing explicitl from the outset the convention λ > and λ <. A similar transformation has been used b aha and Tremaine (993) to analze long-term numerical integrations of the retrograde jovian satellites. 2.2 Resonant terms Now that we have shown how the expansion of the disturbing function for I = 8 β is obtained from that with I = β, we ma use the literal expansion of Ellis and Murra (2) valid for prograde motion and transform the relevant resonance terms to describe the corresponding retrograde resonance. We will use the first transformation described above with s = cos(i/2), λ = M + ω Ω and ϖ = ω Ω. The 2/ retrograde resonance terms are of tpe e 3 cos(λ 2 λ 3 ϖ ) [term 4D3. with j = 2], and e s 2 cos(λ 2 λ ϖ + 2 Ω) [term 4D3.5 with j = 2]. The /2 retrograde resonance has direct and indirect terms of tpe e 3 cos(2 λ λ 3 ϖ ) [terms 4D3.4 with j = 2 and 4I3.6], and e s 2 cos(2 λ λ ϖ +2 Ω) [terms 4D3. with j = 2 and 4I3.3]. The / retrograde resonance has direct and indirect terms. These are of tpe e 2 cos(λ λ 2 ϖ ) [terms 4D2. with j = and 4E2.2 ], and s 2 cos(λ λ + 2 Ω) [terms 4D2.4 with j = and 4E2.6]. However, the / resonance direct terms cannot be obtained from the literal expansion of the disturbing function (since Laplace coefficients diverge when α ). We develop a semi-analtic model for the co-orbital resonance in the next section. A similar analsis for an p/q retrograde resonance shows that there are resonant terms 2 e p+q 2 k s 2 k cos(qλ pλ (p + q 2 k)ϖ + 2 kω). (8) with k =,, 2,... and p + q 2 k. A retrograde orbit with inclination I > 9 can be obtained from a prograde orbit with inclination 8 I b inverting the direction of motion which implies a swap between ascending and descending nodes. 2 D Alembert rule is not obeed because the canonical transformations described in ect. 2. impl that angles for the test particle are measured in the opposite direction of the binar s motion.

5 Retrograde resonance in the planar three-bod problem 5 If β then s, hence the term with k = is dominant. ince we restrict our stud to planar retrograde resonance (s = ), onl the term e p+q cos φ remains, where φ = qλ pλ (p + q)ϖ. (9) We thus recover the retrograde resonant angle from Paper I (with the expected change of sign for the term in λ ). Following Paper I, we use the notation p/-q resonance when referring to a p/q retrograde resonance. 3 A model for co-orbital resonance Consider a test particle in the co-orbital region of the secondar ( a ). The disturbing function ma be expressed using the natural angles: the fast epicclic motion represented b the mean longitude λ of the particle and the guiding centre phase represented b the relative mean longitude τ = λ λ. To obtain the resonant Hamiltonian, the disturbing function, R (Eq. ), is averaged with respect to the fast angle λ. The corresponding function is the ponderomotive potential = R used in our previous work on the co-orbital resonance (Namouni, 999; Namouni et al, 999). When the relative longitude τ is introduced, we ma write: cos ψ = 2 ( + cos I) cos(f M + τ) + ( cos I) cos(f + M τ + 2ω), () 2 and the ponderomotive potential is given as: = 2π( e 2 ) /2 2π R r 2 df, () where r = a( e 2 )/( + e cos f), and the average over the mean anomal M has been replaced b an average over f using the conservation of angular momentum. The mean anomal M is related to the true anomal b the eccentric anomal tan(e/2) = ( e) /2 ( + e) /2 tan(f/2) and Kepler s equation M = E e sin E. The expansion-free expression of ψ gives the natural resonant angles for planar motion. For prograde motion (cos I = ), libration occurs around φ = τ whereas for retrograde motion (cos I = ), libration occurs around φ = τ 2ω 3. Figure (st and 2nd rows) shows the shape of the potential as a function of the resonant argument φ = τ 2ω for planar retrograde motion where a =.. At low eccentricit, libration occurs onl around φ = 8 and the potential is quite shallow. This explains wh in the next section we observe that low eccentricit libration orbits for relativel large mass ratios (e.g. µ =.) are difficult to set up as the larger the mass ratio the stronger the mutual perturbations, the more destructive the close encounters. We shall show that such librations are quite stable at smaller mass ratios. As the eccentricit is increased, the collision boundar appears and librations ma occur around or 8. The extent of the libration amplitude depends on eccentricit, for. e.7, libration around zero has the largest amplitude. Libration around 8 regains some importance as e approaches unit. As the ponderomotive potential is derived for three-dimensional orbits, it is instructive to see how the introduction of a small inclination modifies the dnamics as realistic orbits in the planetar three-bod problem never lie exactl on the same plane. Moreover, in the co-orbital resonance, inclination is 3 Here, we define λ = M + ω + Ω. If we define λ = M + ϖ with ϖ = ω Ω then the retrograde resonant angle is φ = λ λ 2 ϖ in agreement with the conclusions of the previous section.

6 6 M. H. M. Morais, F. Namouni known to mitigate collisional encounters and facilitate stable orbital transitions (Namouni, 999; Namouni et al, 999). Figure (3rd and 4th rows) shows how is modified when the retrograde orbits have a mutual inclination of and ω is set to zero. As expected, collision singularities are absent. There appears a bifurcation near e =.6 where librations around and 8 have comparable amplitudes and ma be associated with the same energ level. The remaining features of the planar problem are present: the potential s shallowness for low eccentricit and the dominance of libration around zero for more eccentric orbits. We remark that the potential s amplitude and bifurcation modes depend on the relative semimajor axis. Figure onl illustrates the similarities and differences with the planar problem. We also note that in the full three dimensional problem, the time evolution of the argument of pericenter modifies the potential s shape and equilibria whereas for planar orbits, the potential depends on the combined phase φ = τ 2ω. We shall present the stud of the three-dimensional retrograde co-orbital resonance elsewhere. 4 Retrograde resonance phase space Poincaré surfaces of section are useful tools to stud the phase space structure in the threebod problem. In what follows, we define how we set up surfaces of section for the phase space of the 2/-, /- and /-2 resonances. We then examine the tpes of orbits involved, as well as their potential stabilit. 4. urface of section construction In the barcentric rotating frame, the planar circular restricted three-bod problem has two degrees of freedom (x, ) and one integral of motion, the Jacobi constant (Murra and Dermott, 999): C = x (ẋ 2 + ẏ 2 2 ( µ) ) µ, (2) r r 2 where µ < is the mass ratio of the secondar and the primar, r 2 = (x + µ) and r2 2 = (x +µ) Orbits therefore lie on a 3D subspace C(x,, ẋ, ẏ) = C embedded in the 4D phase space. Points of an orbit that intersects a given surface, e.g. =, in a given direction, e.g. ẏ >, lie on a 2D surface of section (x, ẋ). An order k resonance corresponds to a set of k islands on the surface of section (Winter and Murra, 997a,b). Here, we prefer to define the surface of section b ẋ =, allowing us to follow orbit intersections in the (x, )-plane. We choose a mass ratio µ =. that is small enough for perturbation theor to appl and Keplerian osculating elements to be used. These elements (a, e, ϖ) var on a longer scale than the orbital period. We then set at t =, ẋ =, so that: ẏ = ± x ( µ) x + µ + 2 µ (x + µ) C. (3) The transformation from barcentric variables (x,, ẋ, ẏ ) to astrocentric (centered on the primar) variables (x,, ẋ, ẏ ) is given as: x = x + µ ẋ = ẋ (4) = ẏ = ẏ + x (5)

7 Retrograde resonance in the planar three-bod problem 7 When ẋ = and = we have ẋ = and ẏ = ẏ + x. Hence, to set up prograde orbits we choose x > and ẏ >, or x < and ẏ <. However, to set up retrograde orbits we must have x > and ẏ < x, or x < and ẏ > x. B replacing = and ẏ = x into Eq. (3), we obtain a limit range on the Jacobi constant C < 2 ( µ) x + µ + 2 µ x + µ µ(µ + 2 x ), (6) such that, within this range of C, choosing x > and ẏ <, or x < and ẏ > ensures that the orbits are retrograde. When x < 3, this limit is C.7. We construct surfaces of section defined b ẋ = and ẏ ẏ > so that the initial condition lies on the surface of section. We var x between -3 and 3 with increments of.5, and between and.8 with increments of.. The Jacobi constant in the range.9 C.7 is incremented b.. We use Eq. (3) to obtain ẏ and choose ẏ > if x < and ẏ < if x >. This ensures that the initial conditions alwas correspond to retrograde orbits (as C.7). The equations of motion of the CR3BP were numericall integrated for up to 6 binar periods using a Bulirsch-toer algorithm with accurac 4. A selection of the surfaces of section will be discussed in ect The full set of surfaces of section can be seen as Online Resource 5. We show the level curves of constant C in (a, e) space for initial conditions at conjunction i.e. x > and = (Fig. 2 (a) and (b)) and for initial conditions at opposition i.e. x < and = (Fig. 2 (c) and (d)). The chosen range.9 C.7 spans semi-major axes between.5 and.5, thus includes the 2/-, /- and /-2 resonance regions. 4.2 Resonant angles The chosen initial conditions are λ = λ = (conjunction: = and x + µ > ) or λ = 8 and λ = (opposition: = and x + µ < ). The points on the surface of section (ẋ = ) with = correspond to the osculating orbits pericenter (λ = ϖ) or apocenter (λ = ϖ + 8 ), depending on the x and C values. tarting in conjunction and at pericenter (apocenter) corresponds to the resonant angle φ = q λ p λ (p + q) ϖ = ([p + q] 8 ). In this case, φ = (8 ) if p + q is even (odd). tarting at opposition and at pericenter (apocenter), φ = p 8 (q 8 ). Hence for opposition at pericenter, φ = (8 ) if p is even (odd) and at apocenter φ = (8 ) if q is even (odd). Therefore, even order resonant angles (such as that of the /- resonance) ma librate around for initial conditions at conjunction, or around 8 for initial conditions at opposition. Odd order resonant angles with p even (such as that of the 2/- resonance) ma librate around 8 for initial conditions at apocenter, or around for initial conditions at pericentre. Odd order resonant angles with an odd p (such as that of the /-2 resonance) ma librate around 8 for initial conditions at conjunction and apocenter, opposition and pericenter, or around for initial conditions at conjunction and pericentre, opposition and apocenter.

8 8 M. H. M. Morais, F. Namouni 4.3 Results of numerical integrations 4.3. Resonant configurations We examine a selection of orbits in the vicinit of the 2/-, /- and /-2 resonances shown in the frame rotating with the binar. These orbits are periodic when at exact resonance and quasi-periodic otherwise. Fig. (3) shows 2/- resonant orbits. The top left panel shows an orbit with C = +.6 that starts at conjunction or opposition ( = ), and pericenter (mode A). The top right panel shows an orbit with C = +.6 that starts at conjunction or opposition ( = ), and apocenter (mode B). Resonant libration occurs around in mode A and around 8 in mode B (Online Resource ). The low left panel shows an orbit with C = +.3 that starts at conjunction or opposition ( = ), and pericenter (mode A). The low right panel shows an orbit with C = +.3 that starts at conjunction or opposition ( = ), and apocenter (mode B). The latter orbit is ver close to collision with the secondar. Fig. (4) shows /- resonant orbits. The top left panel shows an orbit with C = +.6 that starts at x > and =, or x < and (mode I). Resonant libration occurs around and disruptive close encounters are avoided despite the high eccentricit (Online Resource 2 left: mode I). The top right panel shows an orbit with C = +.6 that starts at x < and = (mode II). The /- resonant angle librates around 8 (Online Resource 2 right: mode II) but the orbit is close to collision and becomes unstable when C <.6. Both orbits are described b the equilibria of the ponderomotive potential in Fig. (2nd row, rightmost panel). The mid left panel of Fig. (4) shows an orbit with C =.9 and moderate eccentricit that starts at x > and =, or x < and (mode I). The /- resonant angle φ librates around. The mid right panel of Fig. (4) shows an orbit with C =. that is nearl circular and has initiall x < and = (mode III). This crossing orbit is ver close to collision and we expect resonant libration around 8 (Online Resource 3 right: mode III). What happens is an interesting behavior best seen if we integrate a similar orbit for smaller mass ratios thus reducing the jitter due to close encounters. In Fig. (5, left panel), we plot in the (φ, e)-plane, a similar orbit but with µ = 4. The resonant argument alternates periodicall between libration and circulation in a state that is stable over long time scales. Observing libration around φ = 8 requires a finer search which becomes easier as the mass ratio is decreased (Fig. 5, right panel). The low left panel of Fig. (4) shows an orbit with C =.2 that has small eccentricit and starts at x > and =, or x < and (mode I). This crossing orbit is also close to collision and the resonant angle φ librates around (Online Resource 3 left: mode I). The low right panel of Fig. (4) shows an orbit with C =.2 that is nearl circular and starts at x > or x <, and = for C =.2. This is a non-crossing orbit just exterior to the secondar s orbit and the resonant angle circulates. Fig. (6) shows /-2 resonant orbits. The top left panel shows an orbit with C =.5 that starts at conjunction and pericenter, or opposition and apocenter (mode A). The top right panel shows an orbit with C =.5 that starts at conjunction and apocenter, or opposition and pericenter (mode B). Resonant libration occurs around in mode A and around 8 in mode B (Online Resource 4). When C =.5 both mode A and mode B orbits are close to collision with the secondar. The low left panel shows an orbit with C =.2 that starts at conjunction and pericenter, or opposition and apocenter (mode A). This is the onl stable configuration when C >.3. The low right panel shows an orbit with C =.8 that

9 Retrograde resonance in the planar three-bod problem 9 starts at conjunction and apocenter, or opposition and pericenter (mode B). This is the onl stable configuration when C = urfaces of section As seen in Figs. 3, 4, 6, an order k resonant orbit intersects the section ẋ = at 2k different points. However, owing to the constraint on the sign of ẏ, we onl see k of these intersections on the surface of section (x, ). Therefore, in the surfaces of section (Fig. 7) a set of k intersections on the left hand side (x < ) usuall represents the same configuration (in the snodic frame) as a set of k intersections on the right hand side (x > ). Fig. 7 shows a selection of surfaces of section with C =.6, C =.3, C =., C =.9, C =., C =.2, C =.5 and C =.8. The full set of surfaces of section (C between.7 and.9 at steps.) can be seen as Online Resource 5. At these values of C man initial conditions correspond to crossing orbits (see Fig. 2) which can onl be stable in resonance. Non-crossing small eccentricit orbits exist in the regions marked in green (left and right on Fig. 7). Other colors correspond to different libration modes as described below. Empt areas in the surfaces of section correspond to initial conditions that lead to collision or escape. When C =.6, we see nearl circular non-crossing orbits in the vicinit of the 3/- resonance (green) for initial conditions at opposition (left) or conjunction (right). Collision with the secondar occurs between the 3/- and 2/- resonances. We see islands of libration in the 2/- resonance. The 2/- resonant orbits that start at pericenter (magenta on left and right) correspond to the same configuration in the snodic frame (Fig. 3 top left panel: mode A) where the resonant angle librates around (Online Resource left). The 2/- resonant orbits that start at apocenter (blue on left and right) also correspond to the same configuration in the snodic frame (Fig. 3 top right panel: mode B) where the resonant angle librates around 8 (Online Resource right). When C =.6, we also see islands of libration in the /- resonance that correspond to orbits with ver high eccentricit values. The /- resonant orbits that start at conjunction (black on right) or with x < and (black on left) correspond to the same configuration in the snodic frame (Fig. 4 top left: mode I) where the resonant angle librates around (Online Resource 2 left). The /- resonant orbit that starts at opposition (red) is close to collision (Fig. 4 top right: mode II) and the resonant angle librates around 8 (Online Resource 2 right). When C =.3, we see orbits the vicinit of the 2/- resonance for initial conditions at opposition (left) or conjunction (right). The 2/- resonant angle can circulate for noncrossing orbits (green on left and right), it can librate in mode A (Fig. 3 low left) for initial conditions at pericenter (magenta on left and right), or it can librate in mode B (Fig. 3 low right) for initial conditions at apocenter (blue on left and right). Collision with the secondar occurs just outside the 2/- resonance separatrix. The /- resonant angle librates around (mode I) for initial conditions at conjunction (black on right) or with x < and (black on left). There are also islands of libration in the /-2 resonance for initial conditions at conjunction / pericenter (magenta on right) or opposition / apocenter (magenta on left). These correspond to the same configuration in the snodic frame where the resonant angle librates around (Fig. 6 mode A). When C =., we see nearl circular orbits in the vicinit of the 2/- resonance (green) for initial conditions at opposition (left) or conjunction (right). The 2/- resonant angle can onl circulate and all these orbits are non-crossing. Collision with the secondar occurs between the 2/- and 3/-2 resonances. The /- resonant angle librates around (mode I) for

10 M. H. M. Morais, F. Namouni initial conditions at conjunction (black on right) or with x < and (black on left). The /-2 resonant angle librates around (mode A) for initial conditions at conjunction / pericenter (magenta on right) or opposition /apocenter (magenta on left). When C =.9, all initial conditions correspond to crossing orbits hence the are onl stable in resonance. The /- resonant angle librates around (mode I) for initial conditions at conjunction (black on right) or with x < and (black on left). Collision with the secondar occurs in the /- resonance region. The /-2 resonant angle librates around (mode A) for initial conditions at conjunction / pericenter (magenta on right) or opposition /apocenter (magenta on left). When C =., nearl circular orbits in the /- resonance starting at opposition (red on left) correspond to libration around 8 (Fig. 4 mid right: mode III) although that is not ver clear from the behavior of the resonant angle (Online Resource 3 right: mode III) due to the effect of repeated ver close encounters. These are crossing orbits which are ver close to the collision boundar. The nearl circular orbit starting at conjunction (red on right) correspond to the /- resonance separatrix. The /- resonant angle can also librate around (mode I) for initial conditions at conjunction (black on left) or with x < and (black on right). These are also crossing orbits close to the collision boundar. The /-2 resonant angle librates around (mode A) for initial conditions at conjunction / pericenter (magenta on right) or opposition /apocenter (magenta on left). When C =.2, we see nearl circular orbits in the vicinit of the /- resonance (green) for initial conditions at opposition (left) or conjunction (right). The /- resonant angle circulates and the orbits are just exterior to the secondar s orbit thus ver close to the collision boundar (Fig. 4 low right: mode III). The /- resonant angle can also librate around (Online Resource 3 left and Fig. 4 low left: mode I) for initial conditions at conjunction (black on right) or with x < and (black on left). These are crossing orbits close to the collision boundar. The /- resonance is no longer possible when C =.3 (see Online Resource 5). When C =.2, the /-2 resonant angle librates around (mode A) for initial conditions at conjunction / pericenter (magenta on right) or opposition /apocenter (magenta on left). When C =.3 (see Online Resource 5) the /-2 resonant angle can also librate around 8 (mode B) for initial conditions at conjunction / apocenter or opposition / pericenter. When C =.5, we see nearl circular orbits in the vicinit of the 2/-3 resonance (green) for initial conditions at opposition (left) or conjunction (right). Collision with the secondar occurs in the 2/-3 resonance region. There are islands of libration in the /-2 resonance. The /-2 resonant orbits that start at conjunction / pericenter (magenta on right) and opposition / apocenter (magenta on left) correspond to the same configuration in the snodic frame (Fig. 6 top left panel: mode A) where the resonant angle librates around (Online Resource 4 left). The /-2 resonant orbits that start at conjunction / apocenter (blue on right) and opposition / pericenter (blue on left) correspond to the same configuration in the snodic frame (Fig. 6 top right panel: mode B) where the resonant angle librates around 8 (Online Resource 4 right). When C =.8, there are nearl circular orbits in the vicinit of the /-2 resonance (green) for initial conditions at opposition (left) or conjunction (right). The /-2 resonant angle can circulate or it can librate around 8 (mode B) for initial condition at opposition / pericenter (blue on left) or conjunction / apocenter (blue on right). Collision with the secondar occurs just outside the /-2 resonance separatrix. The /-2 resonant angle can no longer librate around (mode A).

11 Retrograde resonance in the planar three-bod problem 4.4 Analtic model for 2/- and /-2 resonances The structure of the 2/- and /-2 resonances, at low to moderate eccentricities, can be described b the analtic model for 3rd order resonance presented in Murra and Dermott (999) and described in Paper I. The Hamiltonian is: H = δ 2 (X2 + Y 2 ) + 4 (X2 + Y 2 ) 2 2 X(X 2 3 Y 2 ) (7) where δ measures the proximit to exact resonance, applies to 2/- or /-2 resonances, X = R cos(φ/3), Y = R sin(φ/3), and R is a scaling factor specific for each resonance and dependent on the resonance coefficient and on the mass ratio µ (Murra and Dermott, 999). For the 2/- resonance φ = λ 2 λ 3 ϖ and R = γ e, where γ = (3µ f 82 ) and f 82 =.42 is the amplitude of term 4D3. (j = 2) when α =.623. For the /-2 resonance φ = 2 λ λ 3 ϖ and R = 2 2 γ e where γ = (3µ (f 85.5/α 2 )) where f 85.5/α 2 =.533 is the combined amplitude of the terms 4D3.4 (j = 2) and 4I3.6 when α =.623. A resonant orbit corresponds to a set of 3 stable equilibrium points of the Hamiltonian (Eq. 7). In Paper I we showed curves of constant Hamiltonian and location of equilibrium points for several values of the parameter δ. This is in agreement with the behavior observed in the surfaces of section near the 2/- and /-2 resonances (Fig. 7(b),(h)). In particular, when δ = (exact resonance) there is a bifurcation at the origin, and the 3 stable equilibrium points have R = 6 and φ =, ±2π/3 (2/- resonance) or φ = π, ±π/3 (/-2 resonance). Appling the scaling we see that, when µ =., the stable equilibrium points at the 2/- resonance have e =.5, while the stable equilibrium points at the /-2 resonance have e =.3. In Fig. 8 we show real trajectories in the vicinit of the 2/- and /-2 resonances obtained b numerical integration of the equations of motion with µ =. at Jacobi constant values (C) close to bifurcation at the origin. There is ver good agreement with the analtic model for the 2/- resonance (Fig. 8a) since the equilibrium points and separatrix appear at the correct locations which implies that the scaling is correct. However, for the /-2 resonance an orbit close to the stable equilibrium points in the surface of section correspond to a large amplitude libration orbit in Fig. 8b, possibl due to the vicinit of the separatrix at e =. It is also known that for exterior resonances it ma be necessar to include higher order terms in the analtic model in order to obtain the correct dnamics (Winter and Murra, 997b). What is important for our purposes is that the scaling is still approximatel correct since the orbit encircles the equilibrium points predicted b the analtic model. This test provides further assurance on the correct identification of the resonant terms in ect Discussion This article is the continuation of our work on retrograde resonances (Paper I). We identified the transformation that must be applied to the standard expansion of the three-dimensional disturbing function in order to obtain the relevant resonance terms for retrograde motion and analze them quantitativel. We explored the phase-space near the retrograde resonances /, 2/, and /2, b constructing surfaces of section for the planar circular restricted three-bod problem using a mass ratio.. The resonant term amplitude is of order e 2 for the /- resonance, while

12 2 M. H. M. Morais, F. Namouni those of the 2/- and /-2 resonances are of order e 3. Therefore, these are the strongest retrograde resonances. We saw that for low eccentricit non-crossing orbits, libration in the 2/- resonance occurs around (corresponds to starting at conjunction or opposition and pericenter) whereas libration in the /-2 resonance occurs around 8 (corresponds to starting at conjunction and apocenter, or opposition and pericenter). These are the most stable configurations for non-crossing resonant orbits, since the ensure that closest approach with the secondar occurs alwas at pericenter for the 2/- resonance, and alwas at apocenter for the /-2 resonance. The behavior of low eccentricit 2/- and /-2 resonant orbits is in reasonable agreement with an analtic model for 3rd order resonances based on the literal expansion of the disturbing function. However, this analtic model is not valid for resonant crossing orbits. We observed that moderate to large eccentricit crossing orbits in the 2/- resonance or moderate eccentricit crossing orbits in the /-2 resonance ma librate around or 8, while large eccentricit crossing orbits in the /-2 resonance ma librate onl around. Recalling that the literal expansion of the disturbing function is not adequate for coorbital motion, we developed a semi-analtic model for the / retrograde resonance based on numerical averaging of the full disturbing function and valid for large eccentricit and inclination. We saw that this model correctl explains the /- resonant modes, namel libration around and around 8. Whereas /- resonant libration around is quite stable and occurs for a wide range of eccentricities, libration around 8 occurs onl near e and e, hence it is located close to the collision separatrix with the secondar. The disruptive effect of these close encounters implies that /- libration around 8 is easier to setup for smaller mass ratios ( 4 ). We also expect that transitions between different / retrograde resonant modes are possible in the three-dimensional problem, in analog with what is described for prograde / resonant orbits (Namouni, 999; Namouni et al, 999). hortl after completing this theoretical stud of retrograde resonance we identified a set of Centaurs and Damocloids that are temporaril captured in retrograde resonance with Jupiter and aturn (Morais and Namouni, 23). Acknowledgments We thank both reviewers for helpful suggestions that improved the article s clarit. We acknowledge financial support from FCT-Portugal (PEst-C/CTM/LA25/2). The surfaces of section computations were performed on the Blafis cluster at the Universit of Aveiro. References Ellis KM, Murra CD (2) The Disturbing Function in olar stem Dnamics. Icarus47:29 44, DOI.6/icar Gaon J, Bois E (28) Are retrograde resonances possible in multi-planet sstems? Astron. Astrophs. 482: , DOI.5/4-636:27846, 8.89 Gaon J, Bois E, choll H (29) Dnamics of planets in retrograde mean motion resonance. Celestial Mechanics and Dnamical Astronom 3: , DOI.7/ s , Morais MHM, Giuppone CA (22) tabilit of prograde and retrograde planets in circular binar sstems. Mon. Not. R. Astron. oc. 424:52 64, DOI./j x,

13 Retrograde resonance in the planar three-bod problem 3 Morais MHM, Namouni F (23) Asteroids in retrograde resonance with Jupiter and aturn. MNRA Letters, in press, eprintarxiv:38.26 Murra CD, Dermott F (999) olar sstem dnamics. Cambridge Universit Press Namouni F (999) ecular Interactions of Coorbiting Objects. Icarus37:293 34, DOI.6/icar Namouni F, Christou AA, Murra CD (999) Coorbital Dnamics at Large Eccentricit and Inclination. Phsical Review Letters 83: , DOI.3/PhsRevLett aha P, Tremaine (993) The orbits of the retrograde Jovian satellites. Icarus6:549, DOI.6/icar Triaud AHMJ, Collier Cameron A, Queloz D, Anderson DR, Gillon M, Hebb L, Hellier C, Loeillet B, Maxted PFL, Maor M, Pepe F, Pollacco D, égransan D, malle B, Udr, West RG, Wheatle PJ (2) pin-orbit angle measurements for six southern transiting planets. New insights into the dnamical origins of hot Jupiters. Astron. Astrophs. 524:A25, DOI.5/4-636/24525, Winter OC, Murra CD (997a) Resonance and chaos. I. First-order interior resonances. Astron. Astrophs. 39:29 34 Winter OC, Murra CD (997b) Resonance and chaos. II. Exterior resonances and asmmetric libration. Astron. Astrophs. 328: Yokoama T, Do Nascimento C, antos MT (25) Inner satellites of Neptune: I The disturbing function. Advances in pace Research 36: , DOI.6/j.asr

14 4 M. H. M. Morais, F. Namouni I 8, e I 8, e.8,ω I 8, e.2,ω Τ 2Ω I 8, e Τ I 8, e.5,ω 2 Τ I 8, e Τ 2Ω I 7, e.,ω Τ I 7, e.6,ω Τ 2Ω I 7, e.65,ω Τ I 7, e.2,ω Τ I 7, e.8,ω Τ I 7, e.98,ω Τ Τ Τ Fig. Ponderomotive potential of the /- resonance as a function of the resonant angle. The relative semi-major axis is.. The st and 2nd rows illustrate the planar potential whereas the 3rd and 4th rows appl to three dimensional orbits with small inclination. For the latter, τ 2ω is no longer the onl possible resonant argument and is plotted as a function of τ for a fixed ω.

15 Retrograde resonance in the planar three-bod problem 5 e e (a) a (b) a e e (c) a (d) a Fig. 2 Level curves of C at values (from left to right).6,.3,,-.3,-.6,-.9,-.2,-.5,-.8. Initial condition at conjunction and pericenter (a), conjunction and apocenter (b), opposition and pericenter (c), opposition and apocenter (d). The magenta and blue lines locate collision with secondar at pericenter or apocenter. The red vertical lines show location of 2/-, /- and /-2 resonances.

16 6 M. H. M. Morais, F. Namouni Fig. 3 Orbits in 2/- resonance seen in snodic frame: C =.6 mode A (top left) and mode B (top right); C =.3 mode A (low left) and mode B (low right). A unit radius circle in blue helps identif the crossing orbits and non-crossing orbits.

17 Retrograde resonance in the planar three-bod problem Fig. 4 Orbits in /- resonance seen in snodic frame: C =.6 mode I (top left) and mode II (top right); C =.9 mode I (mid left); C =. mode III libration (mid right); C =.2 mode I (low left) and mode III circulation (low right). A unit radius circle in blue helps identif the crossing orbits and non-crossing orbits.

18 8 M. H. M. Morais, F. Namouni.9 µ = e-4.7 µ = e-5 e e φ φ Fig. 5 Co-orbital resonant libration at small eccentricit. The orbits initial conditions are M = 8, ϖ = and for µ = 5, a/a =., e =., whereas for µ = 4, a/a =., e =.8. For better visibilit, orbits are shown onl for periods Fig. 6 Orbits in /-2 resonance seen in snodic frame: C =.5 mode A (top left) and mode B (top right); C =.2 mode A (low left) and C =.8 mode B (low right). A unit radius circle in blue helps identif the crossing orbits and non-crossing orbits.

19 Retrograde resonance in the planar three-bod problem 9 (a) (b) (c) Fig. 7 urfaces of section for selected values of C. Primar is at (, ) and secondar is at (, ). Different colors correspond to different libration modes, or circulation (see text).

20 2 M. H. M. Morais, F. Namouni (d) (e) (f) Fig. 7 urfaces of section for selected values of C. Primar is at (, ) and secondar is at (, ). Different colors correspond to different libration modes, or circulation (see text).

21 Retrograde resonance in the planar three-bod problem 2 (g) (h) Fig. 7 urfaces of section for selected values of C. Primar is at (, ) and secondar is at (, ). Different colors correspond to different libration modes, or circulation (see text).

22 22 M. H. M. Morais, F. Namouni e sin(φ/3) e sin(φ/3) e cos(φ/3) (a) e cos(φ/3) (b) Fig. 8 (a) Trajectories in the vicinit of 2/- resonance obtained b numerical integration of the equations of motion with µ =. at C =.5: near exact resonance (can), separatrix (red and green) and outer circulation (magenta). (b) Trajectories in the vicinit of /-2 resonance obtained b numerical integration of the equations of motion with µ =. at C =.886: near exact resonance in surface of section (red). The equilibrium points predicted b the analtic model are marked b crosses.