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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
On convergence domains of expansions of disturbing
function of the planetary three--body problem in
powers of eccentricities
V. G. Sokolov
Pulkovo Observatory, St. Petersburg, Russia
The problem of convergence of the expansion of the disturbing function R in
power series has long history [1--4]. The most intriguing point in these works is the
fact that their results are different even for the plane case. Meanwhile as noted by
Ferraz­Mello [5] ``the importance of this subject in current research is increasing
because of the possibilities open for the analytical study of the motion of planets
and asteroids by new averaging techniques and by computational facilities for
algebraic manipulation''. He has analysed the results of [2, 4] in [5] and has
shown that the convergence conditions by Sundman [4] are indeed necessary and
sufficient, but that ones of Silva [2] are only sufficient. Using the steepest descent
methods Ferraz­Mello has extended the Sundman criterion to the case of mutually
inclined planetary orbits. He has established that the expansion of R for the
action of Jupiter on an asteroid diverges for considerable number of asteroids in
the outer region of the main belt, and hence their motions cannot be studied
using expansion of this kind.
The above--mentioned authors have investigated the expansion of the princi­
pal part of R, \Delta \Gamma1 , in power series with coefficients in terms of the mean anomaly
l as an independent variable of the equations of perturbed motion. However, as it
is known [6], the use of the eccentric, u, or true, f , anomaly instead of l may give
some advantages because, for instance, it leads to the more rapid convergence of
series in the two--body problem and allows to represent the coordinates of the per­
turbed planet through the elementary functions. The problem of the convergence
of the \Delta \Gamma1 expansions in powers of the eccentricities with coefficients as functions
of u and f has been considered in [7, 8], but only the divergence conditions have
been given in them.
In the present paper a general method for determining the convergence do­
mains of the R expansions in powers of the eccentricities with coefficients in terms
of any anomaly, l, u or f , is suggested. The method is based on the general prop­
erties of the analytical functions [9]. The planetary configurations corresponding
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to the greatest lower bound of the convergence domains have been found. It has
been shown that the use of l and f gives, respectively, the least and the largest
convergence domains for \Delta \Gamma1 , but at the same time it gives the largest and the
least ones for the indirect part of R. Advantages of using each anomaly are dis­
cussed.
References
1. Poicar'e H. Sur le d'eveloppement de la fonction perturbatrice. Bull.Astron.,
1898, 15, 449--464.
2. Silva G. Sur les limities de convergence du d'eveloppement de la fonction
perturbatrice, Bull.Astron., 1909, 26, 49--75, 97--114.
3. Von Zeipel H. Sur les limites de convergence des coefficients du
d'eveloppements de la fonction perturbatrice. Arkiv Matem. Astron. Fysik,
1911, 6, No. 33.
4. Sundman K. F. Sur les conditions n'ecessaires et suffisantes pour la conver­
gence du d'eveloppement de la fonction perturbatrice dans le mouvement plan.
¨
Ofversigt Finska Vetenskaps­Soc. F¨orh., 1916, 58A, No. 24.
5. Ferraz--Mello S. The Convergence Domain of the Laplacian Expansion of the
Disturbing Function. Cel. Mech. & Dyn. Astron., 1994, 58, 37--52.
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7. Samoilova--Yakhontova N. S. On convergence of the expansions of the per­
turbing function in powers of eccentricities, Bull. ITA, 1939, 45, 184--188.
8. Banachiewicz T. ¨
Uber die Anwendbarkeit der Gylden­Brendelschen
St¨orungstheorie auf die jupiternahen Planetoiden. Circulaire de l'Observatoire
de Cracovie, 1926, No. 22, 1--12.
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