Motion of parallel-incoming particles in gravitational field

[Aaronaut]

Question:
There is a large parallel beam of incoming particles with mass m and uniform velocity v0 (v0≪c) in the presence of a gravitational field of a (spherical) planet with mass M and radius R. (without GR) The question is what fraction of the particles will eventually arrive at the planet.
Relevant formulas/attempt to solve
The conservation of Energy implies that
v20=v2−γMR,
which would give the norm of the velocity at the moment of impact. Furthermore, for the tangential component of the velocity could be at least in principle calculated using the conservation of angular momentum,
rv0,t=Rvt,
where r is the original distance vector of the particle. However, there is the problem, that since the case of a far-away particle is considered, r→∞, the left-hand expression is hard to evaluate. In addition, these equations still do not, at least from my point of view, contain enough information to calculate the fraction of particles arriving.

 

[haruspex]

Not sure I'm interpreting the question correctly. Are we to assume that the diameter of beam exceeds that of the planet, so it's a question of which particles would have a perigee closer than the radius of the planet? If so, I agree there's not enough information. We need to know the diameter of the beam.

 

[Aaronaut]

I think the problem is to determine the radius of the particle beam, because it should be proportional to the number of particles.