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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
Expansion of the Hamiltonian of the planetary
three--body problem into Poisson series in all
elements using Poisson series processor PSP
E. D. Kuznetsov 1 , K. V. Kholshevnikov 2 , A. V. Greb 3
1 Astronomical Observatory of Urals State University, Ekaterinburg, Russia
2 Astronomical Institute of St. Petersburg State University, St. Petersburg,
Russia
3 Institute of Applied Astronomy, St. Petersburg, Russia
Investigation of evolution of a planetary system similar to our Solar one is one
of the most important problems of Celestial Mechanics. The first step to solve
it involves expansion of the corresponding Hamiltonian into Poisson series in all
elements. In this work we consider the case of two planets (the Sun -- Jupiter --
Saturn). We plan to pass to the general case in the nearest future.
Let m 0 , Їm 0 m 1 , Їm 0 m 2 be the masses of the Sun, Jupiter and Saturn, respec­
tively. Small parameter Ї is set equal to 10 \Gamma3 . Indices 1 and 2 for radius vectors,
coordinates and orbital elements correspond to Jupiter and Saturn. We use Ja­
cobian coordinates as best--fitting for our problem. As to osculating elements we
deal with two systems of them.
The first system. Positional elements (a \Gamma a 0 )=a 0 , e, sin(I=2) are small and
dimensionless. Angular elements ff = l +g+ fi = g + \Omega\Gamma fl
=\Omega are expressed in
terms of `broken' angles. Here a and a 0 are the semi--major axis and its average
values, e, I , l,
g,\Omega are eccentricity, inclination, mean anomaly, argument of
pericenter, and longitude of ascending node, respectively.
The second system realizes simplifications due to the homogeneity of the per­
turbation function with respect to the semi--major axes. On the other hand, it
has a deficiency, mixing a part of elements of all planets. Namely, we use z s ,
e s , sin(I s =2), ff s , fi s , fl s . Here for the first planet z 1 = ! 0
1
=! 1 \Gamma 1, for the s th
(s – 2) planet z s = ! 0
1
! s =(! 0
s ! 1 ) \Gamma 1, ! s = џ s a \Gamma3=2
s being the mean motions, ! 0
s
being constants close to the mean values of ! s , џ s being gravitational parame­
ters. In this system denominators arising in the process of analytical integration
of equations of motion are extremely simple.
Let represent Hamiltonian h as a sum of the unperturbed part h 0 and the
perturbed one Їh 1 : h = h 0 + Їh 1 : The first term depends on semi--major axes
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only h 0 = \GammaGm 0 m 1 =(2a 1 ) \Gamma Gm 0 m 2 =(2a 2 ) : The second term may be thought
of as a constant factor having dimension of velocity squared and a dimensionless
part h 2 : h 1 = (Gm 0 =a 0 )h 2 [1].
The disturbing function h 2 may be developed into Poisson series h 2 =
P
A kn x k cos ny in positional x and angular y elements. Coefficients A kn are
found using the Poisson series processor PSP [2]. To decrease round--off errors
the rational version of PSP is used.
To obtain the averaged Hamiltonian h up to the second order with respect
to the small parameter it is necessary to take into account terms up to the order
k = 6 for the positional elements and up to the multiplicity n = 13 for the angular
elements [3].
Four variants of the expansion are constructed, two for each of the osculating
elements system. One of the variants furnishes numerical values of A kn corre­
sponding to the system the Sun -- Jupiter -- Saturn. Other one furnishes their
literal expressions depending on parameters of the system. The expansions with
numerical data contain 55228 terms. The expansions with literal parameters con­
tain 182744 terms for the first elements system and 183227 terms for the second
one.
This work was partly supported by the RFBR grant No. 02­02­17516 and the
Leading Scientific School grant No. 00­15­96775.
References
1. Kholshevnikov K. V., Greb A. V., Kuznetsov E. D. Development of Hamil­
tonian of Planetary Problem in Poisson Series in All Elements (theory). As­
tronomicheskij Vestnik, 2001, 35, 267--272 (Solar System Research, 2001, 35,
243--248).
2. Ivanova T. V. Poisson Series Processor PSP. Preprint No. 64 ITA RAS, St. Pe­
tersburg, 1997 (in Russian).
3. Kholshevnikov K. V., Greb A. V., Kuznetsov E. D. The Expansion of the
Hamiltonian of the Two--Planetary Problem into a Poisson Series in All El­
ements: Estimation and Direct Calculation of Coefficients. Astronomicheskij
Vestnik, 2002, 36, 75--87 (Solar System Research, 2002, 36, 68--79).
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