2.3 The double crossing case

We have extended the theory to have the possibility of efficiently computing the secular evolution around a double node crossing, that is a simultaneous crossing both at ascending and descending node with the same planet. In fact we have detected cases with almost exact double crossings in more than one case of real asteroids.

There is even a real asteroid whose initial conditions are almost double crossing at the present time: the recently discovered 1999AN10, whose short period dynamics has been intensively studied [Milani et al. 1999]. This has important implications, in particular in the persistence of impact risk for an extended time span (several centuries).

Here we present briefly the changes to our theory that allows the extension to the double crossing case.

Because of the double periodic nature of the true distance function $D$, the integration domain covering the torus $T$ can be shifted arbitrarily. However, it is essential to make sure that the singularities of collision do not occur on the boundary of the integration domain, otherwise a ``phantom'' singularity would appear.

A double crossing can occur only for $\omega=\pi/2, 3\pi/2$, that is for $\cos\omega=0$. The eccentric anomaly $u$ corresponding to the passage at the nodes satisfies the equations

$$\cos u = \frac{e\pm \cos\omega}{1\pm e\cos\omega}$$

and for $\omega=\pi/2, 3\pi/2$ this implies $\cos u=e<1$, which is incompatible with $u=0$, which corresponds to $\ell=0$. Thus the interval $[0, 2\pi]$ for $\ell$ is such that the singularity of collision would not occur on the boundary. For $\ell'$, if the origin of the planetary orbital phase is taken to be the ascending node of the asteroid orbits, collisions can only occur at $\ell'=0,\pi$. Thus the interval $[-\pi/2, 3\pi/2]$ for $\ell'$ does not have singularities on the boundary.

In a neighborhood of a double crossing the approximated inverse distance near the ascending node $1/d$ is used along with the other one $1/d_1$, applicable near the descending node. The difference $1/D-1/d-1/d_1$ is shown to be limited and with integrable derivatives. Note that this is true only if ``phantom'' singularities are avoided, because $1/d$ and $1/d_1$ are not periodic functions. Then the computations can proceed as in the single crossing case.