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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
On the orbits of extrasolar planets
L. L. Sokolov
Astronomical Institute, Saint­Petersburg University, St. Petersburg, Russia
The discovery of planetary systems around alien stars is an outstanding
achievement of science. Many new problems, connected with extrasolar planets,
arise in astronomy. Main problems concern their formation. Discovered planetary
systems differ sufficiently from the Solar system, and known theories of planet
formation fail. Here we restricted ourselves to some dynamical problems. Many
massive planets contrary to Jupiter have large eccentricities of orbits. We can­
not observe small planets like the Earth. It is interesting to investigate possible
orbits of such small planets. How `Jupiter' with large orbital eccentricity affects
the orbital evolution of other possible planets and stability of the system? The
interesting problem is possible existence in this case of stable quasi--circular or­
bits similar to the Solar planet orbits. Another problem is to study the conditions
and boundaries of stable, regular motions of small mass planet in systems having
`Jupiter' with large orbital eccentricity as well as properties of regular, stable,
and irregular, chaotic motions. It may be that extrasolar planetary systems are
less stable than the Solar system. For example, in [1] chaotic properties of planets
motions in the system AE And are discussed.
We have investigated possible motion of small planet within the limits of
restricted planar elliptical three--body problem by using numerical integration and
analytical methods. Numerical integration is performed using codes developed by
Krogh [2]. Main usual properties of regular motions of small mass planet obtained
numerically are as follows: the semi--maior axis a is constant; in Lagrange variables
e cos g; e sin g the trajectories are approximately circular (e is the eccentricity, g
is the pericenter argument); center of circles is located on abscissa axis.
This picture can be described analytically using classical Laplace--Lagrange
theory of secular perturbations. After averaging over two fast variables we have
a = const and —
x = \GammaK 1 y, —
y = K 1 x \Gamma K 2 e p . Here x = e cos g; y = e sin g are
Lagrange variables,
K 1 = mn
ъ
''
1 + ff 2
(1 \Gamma ff 2 ) 2
E(ff) \Gamma
1
1 \Gamma ff 2
F(ff)
#
;
161

K 2
= mn
ъff
''
2(1 \Gamma ff 2 + ff 4 )
(1 \Gamma ff 2 ) 2
E(ff) \Gamma
2 \Gamma ff 2
1 \Gamma ff 2
F(ff)
#
;
m is mass of `Jupiter', its parameters denoted by index p, n = a \Gamma3=2 is mean
motion of small planet, ff = a p =a ! 1, the mass of the star and gravitational
constant are equal to unity, E and F are elliptic integrals. We have the integral
of motion
y 2 + (x \Gamma e p K 2 =K 1
) 2 = const:
After simplification we derive K 2 =K 1 ъ 9ff=8. We have circles with center on ab­
scissa axis x = 9ffe p =8. This result agrees with numerical data, accuracy is about
10%. For example, initially circular orbit reaches eccentricity e max = 9ffe p =4 at
most. The orbit with initial values e = 9ffe p =8; g = 0 has no evolution. The same
picture corresponds to the case ff ? 1. The evolution of distance between orbits
is investigated.
If the motion is irregular (chaotic) in the case ff ! 1, than the values a and e
usually increase consistently under condition a(1 \Gamma e) ъ const. Initially circular
orbit transforms to that similar to the case of long--periodic comets.
The regular eccentricity evolution described above demonstrates that stable
quasi--circular orbits of small planet are unusual. However, such rare orbits are
found and investigated.
Regions of regular and chaotic motions are separated and described for dif­
ferent values of mass and orbital eccentricity of large planet. The instability of
chaotic trajectories is investigated.
This work was supported by grant of Leading Scientific School N 00­15­96775,
grant of Russian Foundation for Basic Research N 02­02­17516, and grant of the
Program `Universities of Russia'.
References
1. Laughlin G., Adams F. C. Stability and chaos in the AE Andromedae planetary
system. Astrophysical Journal 1999, 526, 881--889.
2. Krogh F. T. Changing step size in the integration of differential equations
using modified divided differences. Lecture Notes in Mathematics, 1974, 362,
22--71.
162