How would I go about finding the distribution of initial separations (i.e. the lengths between the centres of mass) of stars that make up binary systems. I am interested in neutron stars and stellar black holes.
This paper uses an assumption that
"The initial separation of a binary is chosen from a distribution that is assumed to be logarithmically flat: $\Gamma(A) \propto 1/A$."
but it neither defines $\Gamma$ or $A$, so yeah... helpful.
The paper is rather lacking in definitions, I'll give you that.
As Murphrid points out, $\Gamma$ is just the PDF for separation, and $A$ is the variable used for the separation itself. The constant of proportionality is set by requiring the total probability to be unity.
Of course, as defined $\Gamma$ is not normalizable - it diverges logarithmically for both $A \to 0$ and $A \to \infty$. Presumably they put in a cutoff (much as they have a mass distribution $\Psi(M) \propto M^{-2.7}$, but they cut it off at $4~M_\odot$ and $100~M_\odot$), but I couldn't find any reference to the values.
The take-away message is that our understanding of the initial distributions of star properties still needs a lot of work. We can hardly claim to understand the distribution of initial masses beyond rough empirical measurements, and binary mass ratios (assumed to be uniform on $[0, 1]$ in this paper) and binary separations are at least as elusive. The assumptions in this paper were made so as to have something to feed into the simulations - they should by no means be taken to be the actual distributions of parameters.
