A method for constructing the joint mass function of binary stars

TL;DR

The paper addresses the challenge of constructing the joint mass function (JMF) for binary stars from the single-star mass function (IMF) and a conditional mass-ratio function (CMRF). It casts the problem as an inhomogeneous Fredholm integral equation of the second kind for the primary-mass function (PMF), with the JMF obtained via $f_{(M_1,M_2)}(m_1,m_2)=f_{M_2|M_1}(m_2|m_1)f_{M_1}(m_1)$. The authors demonstrate the method by recovering the known random-pairing result and by solving for uniform pairing (requiring Nyström numerical methods), while outlining boundary behavior for main-sequence binaries. They discuss how the choice of minimum and maximum stellar masses and the simplifications in pairing schemes influence the PMF and JMF, underscoring the need for more realistic MFs and CSMFs to enable accurate IMF inference and binary-population synthesis.

Abstract

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.

Figures (4)

• Figure 1: The primary-mass function, $f_{M_{1}}$ (Equation \ref{['eq:primary_mass_pdf_random']}), and secondary-mass function, $f_{M_{2}}$ (Equation \ref{['eq:secondary_mass_pdf_random']}), alongside the mass function, $f_{M}$ (Equation \ref{['eq:kroupa_imf']}). Here we have used the IMF of kroupa2001 and random pairing (Section \ref{['sec:orgcc3815a']}).
• Figure 2: The joint mass function, $f_{(M_{1}, M_{2})}$ (Equation \ref{['eq:joint_mass_function_random']}), constructed using the IMF of kroupa2001 and random pairing (Section \ref{['sec:orgcc3815a']}, Figure \ref{['fig:org91e602c']}). It is non-zero for $m_{2} \le m_{1}$. Low-mass pairs are most probable and high-mass pairs least probable.
• Figure 3: The primary-mass function, $f_{M_{1}}$, and secondary-mass function, $f_{M_{2}}$, alongside the mass function, $f_{M}$ (Equation \ref{['eq:kroupa_imf']}). Here we have used the IMF of kroupa2001 and uniform pairing (Section \ref{['sec:org96ac7dd']}).
• Figure 4: The joint mass function, $f_{(M_{1}, M_{2})}$, constructed using the IMF of kroupa2001 and uniform pairing (Section \ref{['sec:org96ac7dd']}, Figure \ref{['fig:org1e8a02d']}). It is non-zero for $m_{2} \le m_{1}$ and $m_{2} \ge q_{\min}m_{1}$ where $q_{\min} = 0.1$. As is the case for random pairing (Figure \ref{['fig:org0ab2b78']}), low-mass pairs are most probable and high-mass pairs least probable.