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: [email protected]
S. Ferraz-Mello · T. A. Michtchenko
Instituto de Astronomia, Geofísica e Ciências Atmosféricas
Universidade de São Paulo, Rua do Matão 1226,
São Paulo, Brasil
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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 ( ri , pi ), i = 1, 2, where ri are the position vectors of the planets relative to the
→
pulsar, and pi the momentum vectors of the bodies relative to center of mass of the system.
In Poincaré canonical variables, the Hamiltonian of the problem is
→
→ →
2
µi βi
p1 · p2
m1m2
| pi |2
H=
− →
+
,
(1)
−G
2βi
m0
12
| ri |
i=1
→
→
mi
, 12 = | r1 − r2 |, and G = 4π 2 is the gravitational
where µi = G(m 0 + m i ), βi = mm00+m
i
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 ri
∂H
= →,
dt
∂ pi
→
d pi
∂H
= − → , i = 1, 2.
dt
∂ ri
(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 ai , ei , i i , λi , 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
I1 = L 1 − G 1 , σ1 = 3λ2 − 2λ1 − 1 ;
I2 = L 2 − G 2 , σ2 = 3λ2 − 2λ1 − 2 ;
J1 = L 1 + 2(I1 + I2 ), λ1 ;
(3)
J2 = L 2 − 3(I1 + I2 ), λ2 .
√
where L i = βi µi ai and G i = L i 1 − ei2 .
√
√
In terms of elliptic non-singular variables xi = 2Ii cos σi , yi = 2Ii sin σi , the abridged
averaged Hamiltonian is given by
H = Hconst + B(x12 + y12 + x22 + y22 )2
+A(x12 + y12 + x22 + y22 )
+C(x12 + y12 ) + D(x22 + y22 ) + E(x1 x2 + y1 y2 )
+F x1 + I x2 + R(x12 − y12 )
+S(x22 − y22 ) + T (x1 x2 − y1 y2 )
(4)
where A, B, C, D, E, F, I, R, S, T are constant coefficients of the Hamiltonian; Hconst =
−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 λ1 and λ2 and, consequently, J1 and J2 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
Ji = const.
d xi
∂H
=−
,
dt
∂ yi
dλi
∂H
;
=
dt
∂ Ji
∂H
dyi
=
, i = 1, 2.
dt
∂ xi
(5)
2.3 Plane of initial conditions
In this section, we present the set of initial conditions (eccentricities and critical angles:
σ1 , σ2 ). The eccentricities are taken in the interval 0 ≤ ei ≤ 0.07, while the initial critical
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N. Callegari Jr et al.
Fig. 1 Level curves of the Hamiltonian given by Equation (4) on the initial plane (e1 , e2 ). Positive and negative values of ei correspond to σi = 0 and π , respectively. The other initial values are yi ∝ ei sin σi = 0.
Energy levels indicated by letters a, b, . . . , i will be used in the calculations of Section 3. The black points
indicated by r1 , r2 , r3 , r4 correspond to the four real roots of the quartic Hamiltonian (4), considering the
level of energy H = 4.13 × 10−8 , and e1 = 0.018567
angles are fixed at σi = 0 or π, i = 1, 2, i.e., at conjunctions occurring on the common apsidal line of aligned ( = 1 −2 = 0), or anti-aligned orbits ( = π). The set of initial
conditions can thus be studied plotting the level curves of the function H = f (x 1 (e1 ), x2 (e2 ))
(Fig. 1).
We can note in Fig. 1 the presence of three points where the gradient of H = f (e1 , e2 ) is
zero: the saddle point PII− , the point O corresponding to a local minimum of energy close
to the origin (Hmin ), 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 < Hmin (dashed region), Hmin ≤ 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 σi = 0 or π, the averaged Hamiltonian H (4) becomes a quartic polynomial
with respect to xi . Figure 1 shows four points (r1 , r2 , r3 and r4 ), which are real roots of
Hamiltonian for e1 = 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
a given e1 , there may exist up to four real roots of the eccentricity of the outer planet, e2 .
The roots r1 and r2 correspond to σ2 = π (e2 cos σ2 < 0), while roots r3 and r4 correspond
to σ2 = 0 (e2 cos σ2 > 0). The pairs (r2 ,r3 ) and (r1 ,r4 ) will be called inner and outer pair
of roots, respectively. In this work, we always use the inner root r2 . 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 r2 in Fig. 1 corresponds to the current eccentricities of the planets (Table A.2 in the Appendix) and σ1 = 0, σ2 = π (energy level
H = 4.13 × 10−8 ). We can associate the actual position of the system in Fig. 1 with the
point r2 . We have chosen the anti-aligned configuration in the fourth quadrant based on the
fact that the angle 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 x2 .
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.
0.07
MI+
SECULAR DOMAIN
MII-
N
level a
0.05
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
level a
SECULAR DOMAIN
MII-
0.03
I
MI+
H
e
2
0.01
G
0
-0.01
F
MII+
r2
MI-
-0.03
E
D
A
B C
PII+
V
RI
-0.05
MII+
MI-0.07
-0.07
-0.05
-0.03
-0.01 0 0.01
e
0.03
0.05
0.07
1
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 r2 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
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 )
e1 cos σ1
e2 cos σ2
A
B
C
D
E
F
G
H
I
a) 7.0000
b) 7.31004
c) 7.3280
d) 7.4095
e) 7.4230
f) 7.4380
g) 7.4530
h) 7.5036
i) 7.5100
0.0315
0.03200
0.0342
0.0415
0.0450
0.0500
0.0513
0.0448
0.0421
−0.0333
−0.03674
−0.0350
−0.0276
−0.0222
−0.0087
+0.0017
+0.0288
+0.0334
interactions, and the motion is a composition of two secular modes of motion (Modes I and
II): one where = 0 and another where = ±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 y2 = 0 and is represented on the plane
(e1 cos ×e1 sin ); the outer planet section is defined by the condition y1 = 0 and
is represented on the plane (e2 cos × e2 sin ). Since the Hamiltonian (4) is quartic
with respect to the variables xi , 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
2
root r2 , these conditions are dy
dt < 0 and x 2 < 0 (σ2 = π) in the inner planet section and
dy1
dt > 0 and x 1 > 0 (σ1 = 0) in the outer planet sections. In other words, the inner sections
correspond to conjunctions occurring with the outer planet located at apocenter (σ2 = π),
and the outer sections correspond to conjunctions occurring with the inner planet located at
pericenter (σ1 = 0) (Callegari, 2003).
We also construct the dynamic power spectra (or spectral maps) for the set of solutions
with initial condition e1 sin = 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 oscillates about 0 and π respectively. The curves around the
fixed points (and between them) are quasi-periodic solutions in which the angle performs a direct circulation, and the motion is a linear composition of the two main modes.
The direct or retrograde direction of motion of 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 is direct, while in the UranusNeptune system the motion of is retrograde (Michtchenko and Ferraz-Mello, 2001a;
CMF 2004). We note that the direction of motion of the angle 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 σ1 , σ2 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), σ1 and σ2 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 r2 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 e1 cos . 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.
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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 = 0 on
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
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
389
Level/Figure
Energy (×10−8 )
Regimes of motion
a/3a
b/3b
c/5a
d/5b
e/5c
f/6a
g/6b
7.0000
7.31004
7.3280
7.4380
7.4530
7.5036
7.5100
MI, MII
MI, MII→RIII
MI, RIII
MI, RIII, RIV
MI, RIII, RIV
RIII, RIV
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
107 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 1/5.7 and 1/5376 year s −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 σ1 , σ2 circulate, and the angle 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 associated to the long-term
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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 σ1 (panel a) and σ2 (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 σ1 (d) and σ2 (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 σ1 librates around 0 while σ2 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 r2 , 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
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N. Callegari Jr et al.
(σ1 librate around 0 and σ2 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 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 e2 cos 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 σ1 circulates and σ2 librate around
π in the 2/1 resonance (CMF 2004), in the 3/2 resonance both σ1 and σ2 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 r2 , 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
Fig. 6 The same as Fig. 3(b): (a) energy level h, (b) energy level i
393
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 ≈ 21 [3n 2 − 2n 1 ], where
n 1 , n 2 are the mean-motions.
Adopting A as a free parameter and fixing constant values of J1 and J2 , 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
FORBIDDEN ZONE
PII+
0
RESONANCE ZONE
H-Hsec (PII-)
PII-1E-8
NEAR-RESONANCE ZONE
-2E-8
SECULAR DOMAIN
-3E-8
O
-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.
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 (σ1 librates around 0 and σ2 librates around π), with 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 , 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
B
C
D
E
F
I
R
S
T
8.196906 × 10−1
−2229522.4
1.135716 × 10−3
1.152018 × 10−3
4.806767 × 10−4
3.605255 × 10−6
−4.310939 × 10−6
−5.632568 × 10−8
−1.133324 × 10−8
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 Ji . 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)
Semi-major axis (AU)
Orbital Period (d)
Eccentricity
Eccentricity
Mass (M Ear th )
Ji
0.359500 (P)
0.359512 (J)
66.5419(1) (J)
0.018567 (P)
0.0186(2) (J)
4.3
5.762639 × 10−5
0.466025 (P)
0.466042 (J)
98.2114(2) (J)
0.025238 (P)
0.0252(2) (J)
3.9
5.936144 × 10−5
Dynamics of two planets in the 3/2 mean-motion resonance
397
corrections from astrocentric to Poincaré were considered since they may, in some cases,
become large due to the fact that in Poincaré system, the orbital elements are not osculating (see Ferraz-Mello et al., 2004). The corrections due to the transformation from Jacobi
coordinates to astrocentric are smaller than the data uncertainties and were not considered.
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