A method for constructing the joint mass function of binary stars

The initial mass function (IMF) describes the distribution of stellar masses in a population of newly born stars and is amongst the most fundamental concepts in astrophysics. It is not only the direct result of the star formation process but it also explains the evolution of galaxies’ luminosities, metal yields, star-formation efficiencies, and supernova production rates. Because most stars exist in binary systems, however, a full statistical account of stellar mass requires not the IMF but rather the joint distribution of a binary population’s primary- and secondary-star masses. This joint distribution must respect the IMF of the stars from which the population has been assembled as well as the distribution of mass ratios that results from the assembly mechanism. Despite its importance, this joint distribution is known only in the case of random pairing. Here we present a method for constructing it in the general case. We also illustrate the use of our method by recovering the known result for random pairing and by finding the previously unknown result for uniform pairing.

💡 Research Summary

The paper addresses a fundamental gap in stellar population studies: while the initial mass function (IMF) describes the distribution of single-star masses, the majority of stars reside in binary systems, requiring a joint probability density function (JMF) for primary and secondary masses that simultaneously respects the IMF and the observed distribution of mass ratios. The authors develop a general, mathematically rigorous method to construct such a JMF for any prescribed IMF and any conditional mass‑ratio distribution (CMRF).

The framework begins by treating the masses of free (single) stars as independent, identically distributed (i.i.d.) continuous random variables M with PDF f_M (the IMF). Primary and secondary stars in binaries are also treated as i.i.d. random variables with PDFs f_{M1} (primary‑mass function, PMF) and f_{M2} (secondary‑mass function, SMF). The joint PDF of a binary is then f(M1,M2)=f_{M2|M1}(m2|m1)·f_{M1}(m1), where f_{M2|M1} is the conditional secondary‑mass distribution, directly related to the CMRF via a simple change of variables.

A key insight is that the overall IMF of the whole stellar population is a mixture of the primary and secondary distributions: f_M(m)=½