2. A generalized averaging principle

Let us consider, for simplicity, a restricted 3-body problem: Sun-Earth-asteroid, in which the Earth is on a circular orbit with radius $a'$ and the asteroid is on a Keplerian elliptic orbit with semimajor axis $a$.

The Hamiltonian has the form $H = H_0 - R$ where $H_0$ is constant 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, $\mu ={m_\oplus\over m_\odot}$, while $\vec x$ and $\vec x'$ are the asteroid and the Earth position in a Heliocentric reference frame, expressed as function of the Delaunay variables $(\ell,g,z,L,G,Z)$ and $(\ell',g',z',L',G',Z')$.

If there is an intersection between the two orbits, the perturbing function $R$ has a first order polar singularity, so we can compute its average over the fast variables $\ell, \ell'$ because it is an improper convergent integral:

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

On the other hand, if we try to study the qualitative behavior of the solutions of Hamilton's equations by the classical averaging principle, we have to average the right hand side of the equations of motion, that is the derivatives of $R$, which have second order polar singularities; so these integrals are absolutely divergent.

We have previously proved (see Gronchi and Milani 1998) that is possible to give a generalized definition of averaged solutions for planet-crossing orbits by showing that Hamilton's equations built with the averaged perturbing function $\overline{R}$, that is

$$\cases{ \dot{\overline g} = -{\partial \overline{R} \over \... ...verline Z} = {\partial \overline{R} \over \partial z} = 0 \cr}$$ (1)

have unique piecewise smooth solutions which could be considered as representative of the solutions of the averaged equations when the latter have no meaning. This is a mathematical generalization of the classical averaging principle; in fact, if the two orbits do not intersect each other, we can show by the theorem of differentiation under the integral sign that the classical averaged equations correspond to Eqs.(1).

Let us describe briefly this situation in the $(e\cos \omega, e\sin \omega)$ plane: we have the conservation of the $z$ component of the angular momentum, that is of the action variable $Z$; so, for a given value of $Z$ we have a maximum value of $e$ (say $e_{max})$, corresponding to inclination zero.

  

Figure: Kozai domain in the $(e\cos \omega, e\sin \omega)$ plane and the node crossing lines.

The Kozai domain, where the averaged solutions can evolve, is represented by $W = \{0 < e < e_{max}\}$ in this plane; in Figure 1 we have plotted also the ascending node crossing line and the descending one ( $d^+_{nod}=0$ and $d^-_{nod}=0$respectively). If we call $W_i$ the interior of the connected components in which $W$ is divided by the node crossing lines we already know, as classical results, that $\overline{R}$ is continuous in $W$and differentiable in each $W_i$.

  

Figure: Level lines of the averaged Hamiltonian of an asteroid with $a$ = 7 AU.

Using a Wetherill approximation (see Wetherill 1967) and Kantorovich's method of singularity extraction we have shown that $\overline{R}$ is differentiable in the closure of each $W_i$ in $W$ and, more important, that we can define piecewise smooth solutions of Hamilton's equations built with $\overline{R}$ simply by extending to open regions including the node crossing lines the expressions of the derivatives in the right-hand sides of the equations in a smooth, but two-valued, way (see Gronchi and Milani 1998).

We can see in Figure 2 the level lines of the averaged Hamiltonian of a Jupiter crosser asteroid (or ``Centaur'') with semimajor axis $a$ = 7 AU; these lines have a corner when they go through a node crossing line and the presence of these corners has been detected analytically, by isolating, from the averaged Hamiltonian, the term that is responsible.

This term turns out to be the distance between the two straight lines that, according to Wetherill's approach, approximate the orbit of the Earth and of the asteroid in a neighborhood of the intersection points; thus it behaves like an absolute value function.