The velocity is typically of order
500-1000 km/s for collision between MS star, or a MS star and a more
A collision occurs when the centers of the stars are closer to each
other than d = eta*(R_1+R_2). If we neglect tidal deformations (OK for
high speed collisions), eta=1 for genuine collision. At small relative
velocities, tidal captures may occur, which corresponds to eta>1.
Note that the fate of tidally formed binaries is a complicated problem
but it is likely that, in many cases, the stars will eventually merge.
Hence, tidal captures should always be included when one discuss
collisions (unless the velocity dispersion is high enough to suppress
The cross-section for encounters with closest approach smaller or equal
to d (neglectig tidal perturbations of the Keplerian trajectories) is:
The second term in the large brackets is
the "gravitational focusing", which. It dominates if V_rel <<
V_ast (for eta=1), which is the case in globular clusters (velocity
dispersions of a 10-20 km/s) or open clusters (1 km/s or less). In this
Note that, the cross section scales
linearly with the closest approach distance so that all values of d_min,
from 0 to eta*(R_1+R_2) have equal probability. This is not so if V_rel
>> V_ast, which is the case in the center-most part of galactic
nuclei, where a massive black hole impose very high velocities. In this
case, the cross section is simply geometrical and the probability of
having closest approach d_min is proportional to d_min. Hence, (near)
haed-on collisions are very unlikely.
The regime with V_rel << V_ast is refered as parabolic.
V_rel play little role in the hydrodynamics of genuine collisions (with
contact at first peri-center passage), but it is crucial to determine
the critical d_min (> R_1+R_2) for tidal captures. Such
collisions are only mildly supersonic as the thermal velocity of the
stellar gas is of order V_ast which is also approximately the relative
velocity at contact.