Spitzer & Saslaw Collision model

In a pioneering work about the collisional dynamics of dense proto-galactic nuclei, Spitzer & Saslaw (1966) introduced a simple, semi-analytical model to determine the outcome (mass and energy loss) of high speed stellar collisions. This approach has been applied by David, Durisen & Cohn (1987a,1987b) and extended to unequal mass collisions by  Sanders (1970) and by Murphy, Cohn & Durisen (1991).

Fig 1 from MCD91Figure taken from Murphy, Cohn & Durisen (1991).

The computation preceeds as follows. The colliding stars are decomposed into a lrage number of sticks, parallel to the direction of the relative motion. It is assumed that, from the time of first contact, the collision proceeds on a straight trajectory. In the overlapping regions of the stars, one considers conservation of momentum and energy (including thermal energy) in individual encounters between one stick from the fisrt star and one stick from the other (in black on the figure). No transverse energy or momentum transfer is taken into account.  If,  after the collision, the total specific energy of a stick  relative to its parent star is higher than the binding energy, G*M_star/R_star, this element of mass is considered lost from the parent star (of mass M_star and radius R_star).

Although quite crude, this model gives results for the mass lost in surprisingly good agreement with SPH simulations when the relative velocity at infinity is larger than the stellar escape velocity and the impact parameter is relatively large, d_min>0.4*(R_1+R_2).

Figure: comparison of the fractional mass loss between SPH simulations by Freitag (Freitag 2000, Freitag & Benz 2002) and the Spitzer & Saslaw semi-analytical model applied to the same stellar models.

This regime corresponds to most likely collisions in the centre-most parts of a galactic nucleus where a central massive black hole imposes high velocities. Future work, in particular detailed comparisons with  hydrodynamical simulations, will tell us if it can be used to determine the entropy of gas elements in order to predict the post-collisional stellar structure, in conjonction with some fluid sorting algorithm, and if an extension of this method can be devised for the regime of intermediate velocities (V_rel of order of V_star) and/or small impact parameters.