ThreeBody Expansion

This is a joint work with U. Locatelli and A. Giorgilli.
The expansion of the Hamiltonian of the threebody is a common topic in Celestial Mechanics.
The program threebody_plane.c
(see main) computes the expansion of the Hamiltonian of the threebody in Poincare' canonical heliocentric variables.
For sake of completeness, we report here a brief description of the main issues. All the details can be found in any books about Celestial Mechanics (see the References).
The orbital elements describe uniquely a specific orbit. These elements are the ones generally considered in classical twobody systems, where a Kepler orbit is used.
The traditional orbital elements are the six Keplerian elements, after J. Kepler and his laws of planetary motion,
The mean anomaly, , is a parameter relating position and time for a body moving in a Kepler orbit. It is based on the fact that equal areas are swept at the focus in equal intervals of time,
The relation between the mean anomaly and the true anomaly is not trivial. To transform an expansion in the true anomaly, into one in the mean anomaly (needed later), we use an algorithm by J. Henrard (see [1]).
The idea is to trasnform a function
into
The algorithm by Henrard uses the Lie transforms method, and the generator is
The transformed function is computed by the recursive formula
where and .
It also useful to define the quantity
that is related to the modulus of the secular Poincare' canonical heliocentric variables
The Hamiltonian of the threebody problem can be written as
where is the unperturbed part and is the perturbation. Moreover, the perturbation can be written as the sum of the potential energy, , and the kinetic energy, ,
The expansion is simple,
For small values of the eccentricity and inclinations, we can introduce the small quantity
We can now write
where is defined by
and the symbols are the Pochhammer symbols, defined as
This expansion is easy to compute, except for the terms . To complete the scheme we use the expansion in Laplace coefficients,
The Laplace coefficients can be computed as
The hypergeometric function can be computed as
and moreover we have the following relation
Binomial expansion
Trigonometric expansion
from which we have