2. Averaging theory for planet crossing orbits

The model employed in this theory is a N-body problem restricted and circular, that is the asteroid is considered as a massless object and the Solar system major planets are on circular and coplanar orbits. Let us describe, for simplicity, the averaging theory for a 3 body problem: Sun-Earth-asteroid.

The Hamiltonian of the system has the form $H = H_0 - R$, where $H_0$ is the 2-body Hamiltonian and $R$ is the perturbing function

$$R = k^2\mu \biggl[ {1 \over \vert\vec x - \vec x'\vert} - {(\vec x \cdot \vec x') \over x'^3 }\biggr]$$

($k$ is Gauss' gravitational constant, $\mu ={m_\oplus/ m_\odot}$is the ratio of the mass of the Earth and the mass of the Sun, $\vec x$and $\vec x'$ are the positions of the asteroid and the Earth in a Heliocentric reference frame). In the following we shall exploit the well known fact that $R$ can also be written as a function of Delaunay variables $(\ell,g,z,L,G,Z)$ and $(\ell',g',z',L',G',Z')$, of the asteroid and of the Earth, respectively.

If the two orbits intersect, then the perturbing function has a first order polar singularity given by the direct perturbation term ${1 \vert\vec x - \vec x'\vert}$; however the integral average of the perturbing function over the fast angle variables $\ell, \ell'$

$$\overline{R} ={1\over (2\pi)^2}\int_0^{2\pi}\int_0^{2\pi} R\, d\ell d\ell' \$$ (1)

is a convergent improper integral.

By a classical result the integral average of the indirect perturbation term

$$k^2\mu\; {(\vec x \cdot \vec x') \over x'^3 }$$

is zero; therefore only the direct perturbations plays a role in this theory.