Dynamics of two planets in the 3/2 mean-motion resonance: application to the planetary system of the pulsar PSR B

Celestial Mech Dyn Astr (2006) 94:381 397 DOI 10.1007/s10569-006-9002-4 ORIGINAL ARTICLE Dynamics of two planets in the 3/2 mean-motion resonance: application to the planetary system of the pulsar PSR B1257+12 N. Callegari Jr S. Ferraz-Mello T. A. Michtchenko Recieved: 29 July 2005 / Revised: 18 January 2006 / Accepted: 30 January 2006 / Published online: 13 June 2006 Springer Science+Business Media B.V. 2006 Abstract This paper considers the dynamics of two planets, as the planets B and C of the pulsar PSR B1257+12, near a 3/2 mean-motion resonance. A two-degrees-of-freedom model, in the framework of the general three-body planar problem, is used and the solutions are analyzed through surfaces of section and Fourier techniques in the full phase space of the system. Keywords Chaos Extra-solar planets Planetary systems Pulsar planets Resonance 1 Introduction The two outer planets orbiting the millisecond pulsar PSR B1257+12 were the first extra-solar planets ever discovered (Wolszczan and Frail, 1992). Some years after the discovery, another body was detected orbiting the same pulsar on an innermost orbit (Wolszczan, 1994). 1 In this paper, we denote the three planets with the letters A, B, C, in order of their distances to the pulsar. The long-term stability of the orbits of this planetary system was confirmed by several authors through long numerical integrations of the exact equations of motion (Malhotra et al., 1992; Quintana et al., 2002; Goździewski et al., 2005), and dynamical map studies (Ferraz-Mello and Michtchenko, 2002; Beaugé et al., 2005). The stability of a system for a given set of initial parameters is not enough to prove that the system is real. In the case of planets B and C, their mutual gravitational perturbation has been studied (Rasio et al., 1992; Konacki et al., 1999) for this task. Since the planets have mean motions nearly commensurable (close to the ratio 3/2), the mutual perturbation 1 On the possibility of existence of a fourth planet in distant orbit, see Wolszczan et al., 2000. N. Callegari Jr (B) DEMAC, UNESP, Av. 24A, Rio Claro, Brasil e-mail: calleg@rc.unesp.br S. Ferraz-Mello T. A. Michtchenko Instituto de Astronomia, Geofísica e Ciências Atmosféricas Universidade de São Paulo, Rua do Matão 1226, São Paulo, Brasil

382 N. Callegari Jr et al. are enhanced and can be detected in the periodic perturbations of the orbital elements of the planets (Malhotra, 1993; Peale, 1993; Wolszczan, 1994; Konacki and Maciejewski, 1996; Konacki and Wolszczan, 2003). In this paper we study the 3/2 planetary resonance, covering the whole phase space given by two free parameters H and A (the numerical value of energy and the parameter to measure the proximity of 3/2 near-commensurability, respectively). The main regimes of motion and the effects of the near-commensurability in the system were determined. This was done with the same averaged planar approximation in Hamiltonian form used in the study of the 2/1 planetary resonance (Callegari et al., 2004 hereafter denoted CMF 2004; see also Callegari et al., 2002; Callegari, 2003). This paper is divided as follows. The exact Hamiltonian and the model are given in Sections 2.1 and 2.2. Section 2.3 shows the set of initial condition used to map the phase space of the system. In Section 3 we study the dynamics of the pair B C through Fourier analysis of orbits and surfaces of section. The results for the dynamics of the 3/2 and 2/1 planetary resonances are very similar and the discussions in these sections are limited to the determining aspects of them. In the discussions included in Sections 3 and 4, the similarities and differences between two resonances are emphasized. 2 The model and the set of initial condition 2.1 Exact equations of motion Consider the system formed by a pair of planets with masses m 1 (inner planet) and m 2 (outer planet), orbiting a central star with mass m 0. The dynamics of this system can be studied using a Hamiltonian form when an adequate coordinate system is used. Here, we choose the canonical set of variables introduced by (Poincaré, 1897; Hori, 1985). These variables are defined by ( r i, p i ), i = 1, 2, where r i are the position vectors of the planets relative to the pulsar, and p i the momentum vectors of the bodies relative to center of mass of the system. In Poincaré canonical variables, the Hamiltonian of the problem is H = 2 i=1 ( p i 2 2β i µ i β i r i ) G m 1m 2 p1 p 2 +, (1) 12 m 0 where µ i = G(m 0 + m i ), β i = m 0m i m 0 +m i, 12 = r 1 r 2,andG = 4π 2 is the gravitational constant in units AU (astronomical unit), year and solar mass. The first term in the right-hand side of Equation (1) defines an unperturbed Keplerian motion of the planets around the pulsar. The second and third terms are the perturbation due to the interaction between the planets. The canonical equations of motion of the planets are d r i dt = H p i, d p i dt = H r i, i = 1, 2. (2) The initial conditions used to solve Equation (2) are calculated through classical formulae relating orbital elements and Poincaré canonical variables (Ferraz-Mello et al., 2004). Let us denote by a i, e i, i i, λ i, ϖ i, i, i = 1, 2, the semi-major axes, eccentricities, inclinations, mean longitudes, longitudes of pericenters and ascending nodes corresponding to the Keplerian part (1). The values of eccentricities and semi-major axes of planets B and C used in this work are given in the Appendix. In this work we consider only the planar case.

Dynamics of two planets in the 3/2 mean-motion resonance 383 2.2 Model Here we apply the averaged planar planetary model developed in CMF 2004 to the case of the 3/2 mean-motion resonance. The model consists of a two-degrees-of-freedom Hamiltonian constructed in the framework of the general three-body problem. The classical Laplacian expansions of the Hamiltonian (Equation (1)) is used. This expansion is valid for low eccentricities and may be used to obtain the main features of the dynamics of a planetary system with the same characteristics as the planets of the pulsar PSR B1257+12. Additionally, in this paper, we study the secular and resonant orbital evolution of the system, and for this sake, we keep only the main resonant and secular terms in the expression of the disturbing function. The model does not include the effects of planet A. Planet A does not contribute to the critical 3/2 near resonant terms and its contribution to the secular terms is too small (its mass is two orders of magnitude smaller than that of the considered planets B and C). The resonant variables are I 1 = L 1 G 1, σ 1 = 3λ 2 2λ 1 ϖ 1 ; I 2 = L 2 G 2, σ 2 = 3λ 2 2λ 1 ϖ 2 ; J 1 = L 1 + 2(I 1 + I 2 ), λ 1 ; (3) J 2 = L 2 3(I 1 + I 2 ), λ 2. where L i = β i µi a i and G i = L i 1 ei 2. In terms of elliptic non-singular variables x i = 2I i cos σ i, y i = 2I i sin σ i, the abridged averaged Hamiltonian is given by H = H const + B(x 2 1 + y2 1 + x2 2 + y2 2 )2 +A(x1 2 + y2 1 + x2 2 + y2 2 ) +C(x1 2 + y2 1 ) + D(x2 2 + y2 2 ) + E(x 1x 2 + y 1 y 2 ) +Fx 1 + Ix 2 + R(x1 2 y2 1 ) +S(x2 2 y2 2 ) + T (x 1x 2 y 1 y 2 ) (4) where A, B, C, D, E, F, I, R, S, T are constant coefficients of the Hamiltonian; H const = 1.6874 10 3 is the constant part of Hamiltonian, which will be neglected in all calculations involving Equation (4). The coefficients are functions of the masses, semi-major axes and eccentricities, and their numerical values are given in Table A.1 in the Appendix. The averaged Hamiltonian is cyclic in λ 1 and λ 2 and, consequently, J 1 and J 2 are constants of motion whose values are given in Table A.2 in the Appendix. The initially system is, thus, reduced to a system with two degrees of freedom. The mean equations of motion are given by dx i dt 2.3 Plane of initial conditions J i = const. = H y i, dλ i dt dy i dt = H J i ; = H x i, i = 1, 2. (5) In this section, we present the set of initial conditions (eccentricities and critical angles: σ 1, σ 2 ). The eccentricities are taken in the interval 0 e i 0.07, while the initial critical

384 N. Callegari Jr et al. Fig. 1 Level curves of the Hamiltonian given by Equation (4) on the initial plane (e 1, e 2 ). Positive and negative values of e i correspond to σ i = 0andπ, respectively. The other initial values are y i e i sin σ i = 0. Energy levels indicated by letters a, b,...,i will be used in the calculations of Section 3. The black points indicated by r 1, r 2, r 3, r 4 correspond to the four real roots of the quartic Hamiltonian (4), considering the level of energy H = 4.13 10 8,ande 1 = 0.018567 angles are fixed at σ i = 0orπ, i = 1, 2, i.e., at conjunctions occurring on the common apsidal line of aligned ( ϖ = ϖ 1 ϖ 2 = 0), or anti-aligned orbits ( ϖ = π). The set of initial conditions can thus be studied plotting the level curves of the function H = f (x 1 (e 1 ), x 2 (e 2 )) (Fig. 1). We can note in Fig. 1 the presence of three points where the gradient of H = f (e 1, e 2 )is zero: the saddle point PII, the point O corresponding to a local minimum of energy close to the origin (H min ), and the point PII + corresponding to a maximum of energy H(PII + ). According to these points, the plane of initial conditions can be divided in three distinct regions: H < H min (dashed region), H min H < H(PII ) (white regions), and H(PII ) H H(PII + ) (gray region). The gray region includes the energy levels which should reveal the main features of the 3/2 resonance. The white region is the near-resonance zone, where the dynamics is dominated by secular interactions (Section 3.2), but where resonant effects are also significant. The dashed region is the region far from resonance, where the effects of resonance are negligible. Considering σ i = 0orπ, the averaged Hamiltonian H (4) becomes a quartic polynomial with respect to x i. Figure 1 shows four points (r 1, r 2, r 3 and r 4 ), which are real roots of Hamiltonian for e 1 = 0.018567 and the energy level H = 4.13 10 8 (energy level cor-

Dynamics of two planets in the 3/2 mean-motion resonance 385 responding to the actual configuration of the planets B and C see below). In general, for agivene 1, there may exist up to four real roots of the eccentricity of the outer planet, e 2. The roots r 1 and r 2 correspond to σ 2 = π (e 2 cos σ 2 < 0), while roots r 3 and r 4 correspond to σ 2 = 0(e 2 cos σ 2 > 0). The pairs (r 2,r 3 )and(r 1,r 4 ) will be called inner and outer pair of roots, respectively. In this work, we always use the inner root r 2. However, the complete study of the dynamics of the system (4) must be done analyzing all four roots (CMF 2004; see also Section 3 of Tittemore and Wisdom, 1988). The initial condition indicated by letter r 2 in Fig. 1 corresponds to the current eccentricities of the planets (Table A.2 in the Appendix) and σ 1 = 0, σ 2 = π (energy level H = 4.13 10 8 ). We can associate the actual position of the system in Fig. 1 with the point r 2. We have chosen the anti-aligned configuration in the fourth quadrant based on the fact that the angle ϖ is oscillating around π (Goździewski et al., 2005; see also discussion in Section 3.1). 3 Dynamics of the 3/2 planetary resonance In this section, we study the dynamics of the system given by the Hamiltonian (4) through spectral analyses and the method of surfaces of section. We begin showing, in Fig. 2, a dynamical map based on the spectral number (Michtchenko and Ferraz-Mello, 2001a, b; for a detailed description see Ferraz-Mello et al., 2005). The map was constructed integrating the averaged equations of motion (two last equations of Equations (5)), for initial conditions in a grid of 201 201 points in the plane shown in Fig. 1. For each integration, the solutions were Fourier analyzed. The spectral number N, plotted in the map, was defined as the number of significant spectral peaks (e.g., more than 10% of the largest peak) of the Fourier transform of the variable x 2. The spectrum of one solution generally consists of fundamental frequencies, their harmonics and linear combinations. The number of fundamental frequencies is defined by the number of degrees of freedom of the system; since the Hamiltonian (4) has two degrees of freedom, we have two fundamental frequencies: the secular and resonant frequencies. The white strips in Fig. 2 correspond to the immediate neighborhood of periodic orbits. In these strips, the amplitude associated to one of the frequencies tends to zero and only one peak is seen in the spectrum (N = 1). Going away from initial conditions associated to the periodic orbits, we find at least the peaks corresponding to the two fundamental frequencies. Other peaks associated to the harmonics of the two frequencies and their linear combinations appear in the spectrum, increasing the number N. The gray-scale used to represent the values of spectral number in the plane of initial condition allows us to distinguish the chaotic and regular regions of the phase space. In Fig. 2, light regions (small N) correspond to regular (periodic or quasi-periodic) orbits, while darker regions (large N) indicate non-harmonic or chaotic motion associated to the separatrices of the problem. We have divided the analysis of the study of the dynamical system (4) into three parts, depending on the initial conditions: the secular domain, the resonant domain, and the transition between them. 3.1 Secular modes In this section we study the dynamics of the system with initial conditions far from resonance and in the near-resonance zone. The dynamics in these regions is dominated by secular

386 N. Callegari Jr et al. e 2 0.07 0.05 0.03 0.01 0-0.01-0.03-0.05 MII- level a level a SECULAR DOMAIN SECULAR DOMAIN MI- MII- MII+ MI- -0.07-0.07-0.05-0.03-0.01 0 0.01 0.03 0.05 0.07 e 1 MII+ r RIV MI+ A B E D C Fig. 2 Gray scale plot of the Spectral Number N in the same plane of Fig. 1. Some level curves of Fig. 1 are also plotted in Fig. 2. MI+,MI,MII+,MII, RIV are the loci of periodic orbits associated to secular modes and resonant regime RIV. Point r 2 represents the actual position of the pair B C. Point A is an initial condition near the periodic orbit of MII shown in Fig. 3(a). Point B is an initial condition near the transition between near-resonance and resonance domain (Section 3.2). Points C I (white points) are a sequence of initial conditions inside the resonance zone near the periodic orbit of Regime III of motion. Points A I correspond to fixed points of surfaces of sections (Section 3.3). The corresponding energy levels and coordinates are given in Table 1 2 PII+ I H G F MI+ N 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Table 1 Energies and initial conditions of the points A,...,I of Fig. 2. These points were obtained from the surfaces of section in the energy levels a,...,i, shown in Fig. 1 Point Energy ( 10 8 ) e 1 cos σ 1 e 2 cos σ 2 A a) 7.0000 0.0315 0.0333 B b) 7.31004 0.03200 0.03674 C c) 7.3280 0.0342 0.0350 D d) 7.4095 0.0415 0.0276 E e) 7.4230 0.0450 0.0222 F f) 7.4380 0.0500 0.0087 G g) 7.4530 0.0513 +0.0017 H h) 7.5036 0.0448 +0.0288 I i) 7.5100 0.0421 +0.0334 interactions, and the motion is a composition of two secular modes of motion (Modes I and II): one where ϖ = 0 and another where ϖ =±π. These periodic orbits lie along the two white strips going across the origin in Fig. 2. The secular dynamics can be understood by plotting the surfaces of section and analyzing the dynamical power spectra of some orbits from near-resonant domain. First of all,

Dynamics of two planets in the 3/2 mean-motion resonance 387 we define the surfaces of sections and the spectral techniques used in this paper. The inner planet surface of section is defined by the condition y 2 = 0 and is represented on the plane (e 1 cos ϖ e 1 sin ϖ ); the outer planet section is defined by the condition y 1 = 0and is represented on the plane (e 2 cos ϖ e 2 sin ϖ ). Since the Hamiltonian (4) is quartic with respect to the variables x i, it is necessary to fix two conditions to construct the surfaces of sections. These two conditions depend on the chosen root; since we have chosen the inner root r 2, these conditions are dy 2 < 0andx 2 < 0(σ 2 = π) in the inner planet section and dy 1 dt dt > 0andx 1 > 0(σ 1 = 0) in the outer planet sections. In other words, the inner sections correspond to conjunctions occurring with the outer planet located at apocenter (σ 2 = π), and the outer sections correspond to conjunctions occurring with the inner planet located at pericenter (σ 1 = 0) (Callegari, 2003). We also construct the dynamic power spectra (or spectral maps) for the set of solutions with initial condition e 1 sin ϖ = 0 in the inner planet section. For each initial condition, the orbit is Fourier analyzed with a FFT and maps are constructed showing, for each initial condition, the main frequencies present in the oscillation of the x 2 variable whose associated peaks in the spectrum have amplitudes greater than 1% of largest peak. In the surfaces of section of Fig. 3(a) (top and middle), the energy level is H = 7.0 10 8 (indicated by letter a and dashed lines in Fig. 1 see Table 2). Two periodic solutions appear in the sections as fixed points, which are indicated as MI and MII. Inside the domains around Modes I and II, the angle ϖ oscillates about 0 and π respectively. The curves around the fixed points (and between them) are quasi-periodic solutions in which the angle ϖ performs a direct circulation, and the motion is a linear composition of the two main modes. The direct or retrograde direction of motion of ϖ of these quasi-periodic orbits depend on the mass configuration of the system: when the mass of the inner planet is larger then the outer one, the motion is direct; otherwise, the motion is retrograde. For example: in the case of the pair Jupiter-Saturn, the corresponding motion of ϖ is direct, while in the Uranus- Neptune system the motion of ϖ is retrograde (Michtchenko and Ferraz-Mello, 2001a; CMF 2004). We note that the direction of motion of the angle ϖ is independent of the root chosen to study the dynamics of the system (in the near-resonance zone). On the other hand, the motion of the critical angles σ 1, σ 2 depends on the initial condition and can be direct (C D ) or retrograde (C R ) circulations. The direct case corresponds to the inner roots, while the retrograde corresponds to the outer roots. In the case of Fig. 3(a), σ 1 and σ 2 are in direct circulation in both modes (denoted by C D ). We note also that for the position of the planets B and C given by the inner root (point r 2 in Figs. 1 and 2), the conjunction line of planets B and C is circulating in direct sense. This is in agreement with the numerical solutions of Equation (2), considering the actual set of initial conditions (see Appendix). In the near-resonance zone, all motions are regular. That is, there is no separatrix associated to the frontier between Modes I and II, and the passage between them is continuous. A consequence of this fact is shown in the dynamic power spectra of Fig. 3(a) (bottom) where the main frequencies vary continuously along the x-axis e 1 cos ϖ. To better show the main lines of the spectrum it was split into two panels with different vertical scales. The frequency associated with the circulation of the critical angles (resonant frequency) is shown in panel A, together with its higher harmonics, while the secular frequency is shown in panel B. Near the center of Mode I, the resonant frequency is around 1/14.9 year 1, and the secular frequency is about 1/2677.3 year 1. Around the center of Mode II the resonant frequency is approximately 1/13.9 year 1, and the secular frequency is approximately 1/2600.0 year 1. We can see that the amplitude associated to the secular frequency tends to zero for initial conditions near the fixed points.

388 N. Callegari Jr et al. Fig. 3 Top: Surfaces of section of the outer planet (Planet C). Middle: Surfaces of section of the inner planet (Planet B). Bottom: dynamical power spectra corresponding to the solutions crossing the axis sin ϖ = 0on the inner planet sections for energy levels a (left) and b (right). In the left-hand side the y-axis of dynamical power spectrum is in linear scale and is divided in two side parts (A and B), showing the resonant and secular frequencies, respectively. The fixed point MII corresponds to point A in Fig. 2 (the corresponding orbit is shown in Figure 4(a c)). The separatrix S1 corresponds to point C in Figure 2 (orbit shown in Figure 4(d f))

Dynamics of two planets in the 3/2 mean-motion resonance 389 Table 2 Energies used in the construction of the surfaces of section and dynamic power spectra, and the regimes of motion present in the corresponding sections Level/Figure Energy ( 10 8 ) Regimes of motion a/3a 7.0000 MI, MII b/3b 7.31004 MI, MII RIII c/5a 7.3280 MI, RIII d/5b 7.4380 MI, RIII, RIV e/5c 7.4530 MI, RIII, RIV f/6a 7.5036 RIII, RIV g/6b 7.5100 RIII, RIV, RV The spectra show that the difference between the numerical values of the fundamental frequencies around Modes I and II is very small. In the case of initial conditions corresponding to the pulsar planets (H = 4.13 10 8, not shown), the frequencies around Modes I and II were estimated from dynamic power spectrum with numerical simulations over more than 10 7 years. Typical values of resonant and secular frequencies around the actual position of the planets B and C (near Mode II), are given approximately by 1/5.7and1/5376 years 1, respectively. 3.2 The transition to the resonance The motion around Mode MII is stable as far we consider energy levels in the near-resonant domain. For energy values close to H = 7.31004 10 8, the system approaches the resonance and the fixed point associated to MII becomes unstable. As we have seen in Section 2, this energy level separates the representative plane in three regions, and on this level, the system is at the edge of the resonance domain. The regular behavior of the system for initial conditions around the fixed point of Mode II changes. These changes are not visible in the surfaces of section in the scale shown in Fig. 3(b) (top and middle), but are clearly seen in the dynamic power spectrum (Fig. 3(b) bottom): the smooth evolution of the main frequencies around the center of Mode II (Fig. 3(a)) is substituted by a discontinuity and a vertical line. This behavior is characteristic of chaotic trajectories close to a separatrix. We denote this separatrix as S1. In order to understand the entrance in the resonance zone (i.e., the formation of the separatrix), let us study some orbits located in the region of transition between the near-resonance and resonance domains. In Fig. 4, we show the transition of periodic orbit of Mode II for three initial conditions indicated in Fig. 2: i) an initial condition near the fixed point MII (point A in Fig. 2); ii) an initial condition near the chaotic separatrix S1 (point B in Fig. 2); and an initial condition inside the resonance zone (point C in Fig. 2). Case i: Fig. 4(a, b, c). Initial condition on the level a of Fig. 1. In this case, the critical angles σ 1, σ 2 circulate, and the angle ϖ oscillates around π with very small amplitude. Fig. 4(c) shows the surface of section corresponding to this solution showing its regular nature. This solution is very close to the center MII; note the very small value of proper eccentricity shown in the scale of Fig. 4(c). Case ii: Fig. 4(d, e, f). At this initial condition, the system is at the edge the 3/2 meanmotion resonance. The critical angles alternate between direct and retrograde circulation, as shown in Fig. 4(d, e). The chaotic nature of the orbit is clearly seen in the surface of section shown in Fig. 4(f). Case iii. Fig. 4(g, h, i). At this energy, the system is located inside the resonance zone; now MII ceases to exist and a new regime of motion appears where the critical angles librate (Fig. 4(g, h)) which is regular (Fig. 4(i)). The angle ϖ associated to the long-term

390 N. Callegari Jr et al. Fig. 4 Orbits (top and middle panels) and surfaces of section (bottom panels) corresponding to initial conditions A, B and C (see Table 1 and Fig. 2). First column: The initial condition (A) is close to fixed point MII; the critical angles σ 1 (panel a) and σ 2 (b) are in direct circulation; the points indicate the initial conditions, and the arrows the direction of motion. Second column: Orbit with initial condition B. It lies near the separatrix S1; now the critical angles σ 1 (d) and σ 2 (e) are alternating between direct and retrograde circulation. Numbers 1, 2 and 4 correspond to direct circulation while number 3 corresponds to retrograde motion. The surface of section (panel f) shows the chaotic behavior in the transition. Third column: Orbit with initial condition C. It lies near the periodic orbit of Regime RIII; now the critical angle σ 1 librates around 0 while σ 2 librates around π evolution of system oscillates around π. This new regime of motion is called Regime III of motion. 3.3 Inside the resonance We consider now the regimes of motion in which the critical angles are in libration. In the surfaces of section corresponding to the inner root r 2, the center of RIII is located near the border of the resonance, inside the black strip separating secular and resonant domains in

Dynamics of two planets in the 3/2 mean-motion resonance 391 Fig. 5 The same as Fig. 3(b) for the energy levels c (left), f (middle) and g (right). The initial conditions of RIII in figures a), b) and c) are given by the points C, F and G in Table 1 and Figure 2, respectively Fig. 2. Since the resolution of the figure is small (in spite of representing more than 40,000 initial conditions), the white line corresponding to the periodic orbit at the center of RIII cannot be seen clearly. The points C to I, in Fig. 2, show the initial conditions of fixed points of RIII for several values of the energy. Typical examples of surfaces of section and dynamic power spectra in resonance zone are given in Fig. 5 corresponding to the energy levels c, f and g. The fixed points and the separatrix (S1) are shown. We can see that the center of RIII moves towards the origin in the outer planet section, and towards the border in the inner planet section. Fig. 5(b) shows the rise of a new separatrix (S2) inside the domain where the critical angles are librating. This separatrix splits the resonance domain in two parts. One, which is the continuation of RIII and a new one, RIV, which is a new regime of motion. The fixed points associated to the periodic orbits of this new regime of motion appear in Fig. 2 as a white strip in the middle of the resonance zone. In this regime, conjunction remain librating around the pericenter of the inner planet orbit and the apocenter of the outer planet orbit

392 N. Callegari Jr et al. (σ 1 librate around 0 and σ 2 librate around π). For larger energy values, the domain of RIV increases in the surfaces of sections. The main difference between the regimes RIII and RIV is the direction of phase flow around their fixed point. The dynamic power spectrum shows that the frequency associated with the motion of ϖ becomes equal to zero at the two points corresponding to the intersections of the separatrix S2 with the x-axis. A true resonance happens on S2 (with one proper frequency passing through zero). For larger values of energy (H = 7.4530 10 8 ), new changes occur with the periodic orbit of RIII: it leaves the x axis in the inner planet surface of section and is, now, located near the boundary of the energy manifold (Fig. 5(c), middle); in the outer planet section, it crosses the origin (Fig. 5(c), top). This geometric change of RIII is related to the evolution of the periodic orbit of RIII in the plane of initial condition: the initially anti-aligned orbits now become aligned. Despite the changes in the left-hand side of the surfaces of section, the domain around Mode I remains almost unaltered showing that non-resonant secular regimes continues to exist for all considered energy levels. A similar behavior was observed in the 2/1 resonance (CMF 2004). However, the domain is decreasing in size and for larger values of energy, the Mode I also touches the border of resonance zone, and becomes unstable (Figs. 6(a) and 7(a) H = 7.5036 10 8 ). Physically, the domain of MI is engulfed by the chaotic separatrix S1. This behavior fixed point of MI touching the border of the resonance domain can also be seen in Fig. 2, near the positions indicated by MI+ and MI : we can see that the two branches of MI, present in the inner and outer parts of Fig. 2, are separated by resonance zone. For slightly larger values of energy, a new thin regime arises in the place of Mode I (Figs. 6(b) and 7(b)). As in the case of 2/1 resonance (CMF 2004), this regime is called RV. Fig. 7(b) also shows the new separatrix (S3) associated to this new regime. As we can see in Fig. 7(b) (top), the fixed point associates to RV does not lie on the e 2 cos ϖ axis. In the dynamic power spectrum shown in Fig. 6(b) (bottom), we can see the regime RV squeezed between separatrices S2 and S3. (The main difference between Regime V in the 2/1 and 3/2 planetary resonances is the motion of critical angles: while σ 1 circulates and σ 2 librate around π in the 2/1 resonance (CMF 2004), in the 3/2 resonance both σ 1 and σ 2 librate around π. This means that conjunctions can oscillate around apocenter of both planets.) For larger values of energy, Regimes RIII and RV do not survive in the inner root family of initial conditions r 2, since their domains decrease while the chaotic regions emanating from separatrices S2 and S3 dominate the corresponding region of phase space. We finish this section noting that the domain of regime RV is very thin as compared with the domain of regime RIV, and is present only in the border of resonance. Regime RIV occupies the largest domain in the phase space and is the main regime of motion inside the 3/2 resonance. 4 The phase space for other values of A All results described in previous sections refer to a Hamiltonian with coefficients calculated for fixed values of the semi-major axes and values of eccentricities given in Table A.2 (Appendix). In this section, we consider the problem of extending the results to systems with other values of A, different of that used in previous sections (A = 8.196906 10 1 ), without having to recalculate all solutions necessary to construct a figure like Fig. 2. For near-circular

Dynamics of two planets in the 3/2 mean-motion resonance 393 Fig. 6 The same as Fig. 3(b): (a) energy level h, (b) energy level i

394 N. Callegari Jr et al. Fig. 7 (a) Details of separatrix S1 and Regime RIII shown in Fig. 6(a) (top and bottom). (b) Details of Fig. 6(b) showing Regimes III and V and separatrices S2 and S3 associated to the corresponding regimes. For better resolution of Regime V, we take the scales of axes of Fig. 7(b) different from Fig. 7(a) orbits, the expression for the coefficient A is well approximated by A 1 2 [3n 2 2n 1 ],where n 1, n 2 are the mean-motions. Adopting A as a free parameter and fixing constant values of J 1 and J 2, the level curves of Hamiltonian (4) are dependent on the value of A; the position (and even the existence) of points O, PII and PII + depend on the adopted value of A. As pointed in Section 2, these three points can be used to locate the approximated domain of the main regions of the plane of initial conditions: resonance, near-resonance and secular zone (see Fig. 2). Therefore, the knowledge of the behavior of the critical points O, PII and PII + as a function of A may allow us to know the main changes in the shape of the resonance and near-resonance zones for different values of A. The other parameters given in Table A.1 in the Appendix are kept unchanged. (This analysis is equivalent to the analysis done in CMF 2004 taking δ = 2(A + C) as parameter. Both choices are equivalent.) Figure 8 shows the value of the energy levels corresponding to the points C, PII and PII + as a function of A. For the sake of having a better figure, we plotted the value of the energies in Fig. 8 with respect to their values at PII. The long-term dynamics of the planets B and C for a large set of initial conditions is resumed in Fig. 8. The region above the curves PII + (forbidden region) correspond to a value

Dynamics of two planets in the 3/2 mean-motion resonance 395 1E-8 0 FORBIDDEN ZONE RESONANCE ZONE PII+ PII- H-H sec (PII - ) -1E-8-2E-8 NEAR-RESONANCE ZONE SECULAR DOMAIN -3E-8-4E-8-0.4-0.2 0.0 0.2 0.4 0.6 0.8 1.0 A Fig. 8 The parameter plane. The main regimes of motion (RIII and RIV) are located in resonance zone, between curves PII + and PII ; direct and retrograde circulation of conjunctions are located in near-resonance zone, between PII and O; below O, only retrograde circulation is possible. A full circle represents the actual position of the planets B and C, where conjunctions circulate in direct direction of H larger than the maximum. In this region, no motion is possible, since all roots of the quartic polynomial become imaginary. The resonance domain is located between the curves PII + and PII. In this region, all resonant regimes of motion discussed in Section 3 (RIII, RIV and RV) are present. Since the limits of the regimes are located between the curves PII + and PII, we see that they may exist for A 0. The area between O and PII is the near-resonance region, where both retrograde (C R ) and direct (C D ) circulation of the critical angles are possible. The full circle inside the near-resonance region shows the position of the system formed by the pulsar planets B and C, whose dynamics is dominated by secular interactions. Below the curve O, only retrograde circulation (C R ) of the critical angles exist. O 5 Conclusions The dynamics of the 3/2 planetary resonance was studied with a planar averaged model in the low eccentricity domain, with the purpose of determining the long-term behavior of both resonant and secular interactions among planets in or near the 3/2 resonance. The results show that the dynamics of 3/2 planetary resonance is very similar to the dynamics of the 2/1 planetary resonance studied by CMF 2004 with several regimes of motion, where the conjunction line can oscillate in different ways, in several stable regimes of motion, or circulate in direct or retrograde sense. Regimes of libration and circulation are separated by chaotic separatrices. In the near-resonance zone, the system is not in 3/2 mean-motion resonance, and the critical angles circulate. At the edge of the resonance zone, critical angles alternate between

396 N. Callegari Jr et al. direct and retrograde motion along the branches of a separatrix. Inside the resonance zone, critical angles librate (σ 1 librates around 0 and σ 2 librates around π), with ϖ oscillating kinematically around π (RIII). For higher energies, the resonance zone splits into two parts (RIII and RIV) separated by the separatrix (S2) where a true secular resonance occurs. The Regime III, which begins with anti-aligned orbits, evolves continuously to aligned orbits. Around the maximum energy value, the domain of the Regime IV (where both, the critical angles and ϖ, librate) increases, becoming the main regime of motion inside 3/2 resonance. The above scenario may exist for a wide range of values of the elements and masses around those of the pulsar planets B and C (Callegari, 2003). Acknowledgements Nelson Callegari Jr. thanks Prof. T. Yokoyama, DEMAC/UNESP (Rio Claro, Brazil) and Department of Mathematics/UFSCar (São Carlos). The authors thank two anonymous referees and Prof. J. D. Hadjidemetriou for their interesting comments. This investigation was sponsored by FAPESP (Proc. 98/13593-8) and CNPq. Appendix This Appendix gives the numerical values of the coefficients of the Hamiltonian given by Equation (4) (Table A.1). In the calculation of the values listed in Table A.1, we have used the orbital elements and masses given in Table A.2. In the transformation to the relative Poincaré elements, only the Table A.1 Numerical values of the coefficients of the Hamiltonian (4) obtained with the expressions given in the Appendix of CMF 2004, when r = 2 Coefficient Numerical Value A 8.196906 10 1 B 2229522.4 C 1.135716 10 3 D 1.152018 10 3 E 4.806767 10 4 F 3.605255 10 6 I 4.310939 10 6 R 5.632568 10 8 S 1.133324 10 8 T 1.871631 10 7 Table A.2 Keplerian elements in Poincaré s (J) and Jacobi s systems of coordinates (P), masses and constants of motion J i. The masses, and the elements in Jacobi s system, were obtained from Konacki and Wolszczan (2003). The adopted mass for the pulsar is m 0 = 1.4m Sun. Figures in parentheses are uncertainties in the last digits indicated Quantity Planet B Planet C Semi-major axis (AU) 0.359500 (P) 0.466025 (P) Semi-major axis (AU) 0.359512 (J) 0.466042 (J) Orbital Period (d) 66.5419(1) (J) 98.2114(2) (J) Eccentricity 0.018567 (P) 0.025238 (P) Eccentricity 0.0186(2) (J) 0.0252(2) (J) Mass (M Earth ) 4.3 3.9 J i 5.762639 10 5 5.936144 10 5

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