Gravitational focusing
The concept of gravitational focusing describes how the gravitational attraction between two objects increases the probability that they will collide. Without gravitational force, the likelihood of a collision would depend on the cross-sectional area of the two objects. However, the presence of gravity can cause objects that would have otherwise missed each other to be drawn together, effectively increasing the size of their cross-sectional area.
Assuming two bodies having spherical symmetry, a collision will occur if the minimum separation between the two centres is less than the sum of the two radii. Because of the conservation of angular momentum, we have the following relationship between the relative speed when the separation equals this sum, and the relative speed when the objects are very far apart :
where is the minimum separation that would occur if the two bodies were not attracted one to the other. This means that a collision will occur not only when but when
and the cross-sectional area is increased by the square of the ratio, so the probability of collision is increased by a factor of However, by the conservation of energy we have
where is the escape velocity. This gives the increase in probability of a collision as a factor of When neither body can be treated as having a negligible mass, the escape velocity is given by:
When the second body is of negligible size and mass, we have:
where is the average density of the large body.
The equation of conservation of energy can be developed into
where is the minimum separation between the centres and is the total mass.
Instead of using this to find for a given we can solve for given and
When is less than the combined radii of the bodies, there will be a collision.
The eccentricity of the hyperbolic trajectory is:
When there is no collision, the trajectories turn by in the centre-of-mass fame of reference. The relative velocity vector changes by the velocity of the lighter body changing more and of the more massive body less. The relative speed goes asymptotically back down toward