Description
The statistical properties of a population of binary stars are determined in large part by the joint distribution of the masses of their components at their time of formation. These masses determine the binary stars' subsequent evolution. However, this joint distribution is known neither from first principles, since theory is insufficiently developed, nor from observation, since observational surveys instead report the distribution of the masses of the ionized components of the binary stars (the initial mass function) and the conditional distribution of the masses of the secondary stars given the mass of the primary star (the pairing function). Here we show that the joint initial mass function (joint IMF), the initial mass function (IMF), and the pairing function (PF) are related by a simple integral equation. This equation constrains the forms that the IMF and PF may take even in principle. Given valid IMF and PF it typically admits a unique solution for the joint IMF. Using this solution we may find the distribution of the masses of the primary stars (the primary initial mass function). In the high-mass limit this primary initial mass function (primary IMF) is twice the value of the IMF and in the low-mass limit vanishes, regardless of the PF. Knowledge of the primary IMF allows us to initialize population synthesis models exactly and, in particular, to exactly specify the number of supernova progenitors at initialization. We explore the consequences of this for the late-time supernova abundances and iron yields computed using these models.
