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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
Dynamics of resonant planets:
The resonances 2:1, 3:2 and 5:2
N. Callegari Jr., T. A. Michtchenko, S. Ferraz­Mello
University of S~ao Paulo, IAG­Dept. Astronomy, S~ao Paulo, Brazil
This paper considers the planar problem of two planets orbiting a star with pe­
riods close to a commensurability. The problem is stated in Hamiltonian form, in
heliocentric canonical variables, and is reduced to two degrees of freedom through
the elimination of all non--critical terms involving the mean longitudes. In the
adopted approximation, the reduced Hamiltonian includes the main resonant and
secular terms up to fourth order in the eccentricities. In the case of first--order
resonances (Callegari Jr. et al, 2002) the second--order critical terms are also kept
in the model. Numerical tests have shown good agreement between the solutions
of the model and the solutions of the correspondent exact equations of a two--
planet system, at least as far as the eccentricities are kept small. In the domains
studied here, the eccentricity of the planets generally remains below 0.05 in the
resonances of higher ff, reaching 0.1 only in those of lower ff (ff is the ratio of the
semimajor axes of the 2 planets.)
The dynamics of the resulting system can be studied through the construction
of surfaces of section for a large set of initial conditions (eccentricities and critical
angles). Besides, the FFT spectrum of all solutions is calculated and the number
of peaks above a suitably chosen limit is used to determine the complexity of
the solution (see Michtchenko and Ferraz­Mello 2001a). The results from the
FFT spectrum are complementary to the surfaces of section in the study of the
dynamics. As the Hamiltonian has two degree of freedom, two main independent
frequencies of the system are expected to be found in the Fourier spectrum, as
well as higher harmonics and beats of both frequencies. In this way, we may
class the solutions as regular when the FFT spectrum shows a small number of
harmonics (peaks) and chaotic when the structure of the FFT spectrum becomes
complex and the number of peaks exceeds a given limit.
These two techniques were applied to three real 2­planet systems whose mem­
bers have semimajor axes such that they are in resonance and are close to some
important near resonant systems:
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(a) the system Sun­Jupiter­Saturn, near the 5:2 resonance. (ff ¸ 0:54);
(b) the system Sun­Uranus­Neptune, where both planets orbit the Sun near the
2:1 resonance (ff ¸ 0:63);
(c) the system formed by the pulsar PSB B1257+12 and its planets B and C,
near the 3:2 resonance (ff ¸ 0:763).
In order to explain the results obtained on the dynamics of these systems
let us define the critical angles. Let – i
; $ i
; denote the mean longitudes and the
perihelion longitudes of the inner (i=1) and outer (i=2) planets, respectively. The
critical angles are oe i
= (p+1)– 2 \Gamma p– 1 \Gamma $ i
(i = 1; 2) (p = 1 for the 2:1 resonance,
p = 2 for the 3:2 resonance and p = 2=3 for the 5:2 resonance. The long period
(secular) variable is \Delta$ = oe 1 \Gamma oe 2 :
The systems are studied following the technique proposed by Tittemore and
Wisdom (1988) and uses two parameters: the energy and one parameter ffi mea­
suring of the distance of the system to the exact resonance (which is equal to zero
at the exact resonance). This parameter is defined in terms of constants appearing
in the expression of the Hamiltonian. It is roughly equal to 5n 2 \Gamma 2n 1 \Gamma —
$ 1 \Gamma 2 —
$ 2
in the study of the 5:2 resonance and (p + 1)n 2 \Gamma pn 1 \Gamma —
$ 1 in the case of the
first­order resonances.
In the 5:2 resonance it was possible to identify several different regimes of
motion. Outside the resonance, the solutions are the well­known secular ones. The
surfaces of sections show 2 periodic orbits, named I and II by Michtchenko and
Ferraz­Mello (2001b), where we have, respectively, ffi $ = 0 and ffi$ = ú. These
two periodic orbits correspond to four possible stable geometrical configurations
of the planets:
ffl Mode I: the perihelia of the inner and outer planets are aligned in the same
direction (parallel apsidal lines) (ffi$ = oe 1 \Gamma oe 2 = 0).
ffl Mode II: the perihelia of both planets are aligned in opposite directions
(anti­parallel apsidal lines), (ffi$ = ú).
In the lower energy surfaces of section, the solutions appear divided in two
modes (mode I and mode II, respectively) formed by motions over two seemingly
regular families of tori enclosing the 2 periodic orbits. The classical classification
of the motions according with the behavior of the angle ffi$ = oe 1 \Gamma oe 2 leads
to some difficulties well­known in the study of secular dynamics, because the
periodic orbits are not at e = 0. Then, solutions on tori enclosing the origin
and those on tori which do not enclose the origin are kinematically different,
notwithstanding the fact that a bifurcation between them does not exist. The
torus going across the origin, which separates the two kinematically different
motions, is not a true topological separatrix. (Regular tori in both sides of this
separatrix belong to a same regime of motion and form a continuous family.)
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A A
B
B
Fig. 1. Two surfaces of section of the 5:2 resonance showing the preserved sec­
ular mode I (A) and the main resonant modes R ex and R in (B). Taken from
Michtchenko and Ferraz­Mello, 2001b.
The solutions in modes I and II are such that the angle ffi$ librate about 0 or ú
as far as these solutions are close to the periodic orbits. Solutions on wider tori
may show this angle circulating, but this difference is topologically meaningless.
Besides, for lower energies, we have to distinguish the solutions corresponding to
systems in both sides of the resonance. When the ratio of the semi­major axes
of Jupiter and Saturn is larger than the exact resonant value 0.54, the angles oe i
have retrograde motion (negative velocity) and when ff is smaller than 0.54, the
orbits of Jupiter and Saturn are closer one to another, the angles oe i
have direct
circulation (positive velocity).
As the energy increases, the system reaches the resonance domain and the
solutions of mode II located in the immediate neighborhood of the period orbit
starts librating (both angles oe i
librate -- a true bifurcation is reached). The new
regime of motion is named R ex (see the figure). The chaoticity of the motions sep­
arating librating and circulating orbits is clearly seen in the surfaces of section
(the thick curves separating the domains A and B), and confirmed by the FFT
of the solutions. For energies yet larger (right­hand side section in the figure), a
secular resonance associated with the relative motion of the perihelia is reached
creating a new bifurcation inside the libration region and the surface of section
shows a new center and a new saddle point. The separatrix emanating from the
saddle point encloses a new regime of motion (R in ), in the immediate neighbor­
hood of a new stable periodic orbit. The sections were done for a given value of
ffi (0.02), but the regimes of motion reproduce themselves without modifications
for other values of ffi ? 0.
In the case of the Uranus­Neptune­like system the situation is more involving.
Studies done with ffi = 0:0012 first show a behavior very similar to that of the res­
onance 5:2. Far of the resonance, the system is dominated by secular interactions
showing the two regimes of motion (modes I an II) around stable periodic orbits;
as the energy increases, the exact resonant separation of the two planetary orbits
is reached and the solutions of mode II located in the immediate neighborhood
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of the period orbit starts librating (mode R ex ); for energies yet larger, a secular
resonance associated with the relative motion of the perihelia is reached creating
a new bifurcation inside the libration region and the surface of section shows a
new center and a new saddle point. The separatrix emanating from the saddle
point envelopes a new regime of motion in the immediate neighborhood of the
new stable periodic orbit (mode R in ). However, from this point on, new regimes
of motion, not seen in the 5:2 resonance, appear. While in the 5:2 resonance larger
energies came to correspond to a deep resonance regime, very regular, in the 2:1
resonance new bifurcation appear in the surface of section. Another feature seen
in the surfaces of section of the 2:1 resonance is that the chaotic zone separating
the domains A and B in the figure grows and evolves in such a way to completely
envelop the domain A. Periodic orbits at the center of the resonant modes may
show either ffi $ = 0 or ffi$ = ú. (The two angles oe i
may be librating on the same
side or in opposite sides.) These centers migrate and in some cases they may
cross the origin and change by 180 ffi the libration center of one of the oe i
. The
regimes of motion characterized by the coupled libration of oe 1 and oe 2 around 0;
and libration of \Delta$ = ú have the same characteristics of the motions of the two
planets orbiting around the star GJ 876 (Marcy et al. 2001).
The results for 3:2 resonance are very similar to those for the 2:1 resonance.
The analysis of the dynamics of these systems was limited to the search of
symmetrical librations of the angles oe 1 and oe 2 , that is oscillations of these angles
around 0 or ú. Some numerical experiments have, however, shown that asymmet­
ric librations as found by Beaug'e (1994), in which the angles oe i
oscillate around
angles not commensurable with ú also exist.
References
1. Beaug'e C. Asymmetric Librations in Exterior Resonances. Cel. Mech. &
Dyn. Astron., 1994, 60, 225.
2. Callegari N. J., Michtchenko T., Ferraz­Mello S. Dynamics of two planets in
the 2:1 and 3:2 resonances, 2002, in preparation.
3. Marcy G. et al. A pair of resonant planets orbiting GJ 876. ApJ, 2001, 556,
296.
4. Michtchenko T., Ferraz­Mello S. Resonant Structure of the Solar System.
2001a, 122, 474.
5. Michtchenko T., Ferraz­Mello S. Modeling the 5:2 Mean­motion Resonance
in the Jupiter­Saturn Planetary System. Icarus, 2001b, 149, 357.
6. Tittemore W., Wisdom J. Tidal evolution of the Uranian satellites I. Pas­
sage of Ariel and Umbriel through the 5:3 Mean­Motion Commensurability.
Icarus, 1988, 74, 172.
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