Detlef de Niem
Institute of Space Sensor Technology and Planetary Exploration - German Aerospace Center, Berlin,
Germany
An analytical formula for the scattering cross section for capture
of hyperbolic test bodies in the restricted problem of three bodies
is found, approximating the encounter with the secondary as a
two-body gravity assist.
Previous results for this problem by H. Newton
or Radzievsky and Tomanov treated the
interaction with the secondary using an impact parameter valid for
hyperbolic encounter from infinity.
Here, a conic matching at the Hill sphere of the secondary is
made instead, as recommended by Everhard, and an analytic
expression for the cross section corresponding to capture into an
elliptic orbit (around the central body) with semimajor axis smaller
than some limit is obtained. This cumulative capture cross section is
valid locally, only.
To find the capture rate corresponding to certain conditions
'at infinity' with respect to the central body, the cross section
is summed over the velocity distribution of incoming test bodies.
For an isotropic velocity distribution at infinity this was carried out,
in Pineault et al, using the cross section in the approximation by Radzievsky. This rate is compared with one obtained using the new
cross section.
Further, probability distributions of semimajor axes and Tisserand
parameters of captured bodies are derived.
Altough there are limits due to the two-body
approximation, these are shown to be not severe for test bodies
that are hyperbolic with respect to the central body. For
transitions between bound elliptic orbits
(e.g. for capture of Oort cloud comets) the present result for the
cross section is applicable even for small energy changes.
As an application, the probability of capture of near Earth asteroids due to
lunar gravity assist is compared to the rate of impacts on the Moon.
For moderate relative velocities of the asteroids with respect
to the orbit of the Earth, the capture rate becomes larger than the impact rate.