Energy Optimization in Binary Star Systems: Explanation for Equal Mass Members in Close Orbits

Abstract

Observations indicate that members of close stellar binaries often have mass ratios close to unity, while longer-period systems exhibit a more uniform mass-ratio distribution. This paper provides a theoretical explanation for this finding by determining the tidal equilibrium states for binary star systems --- subject to the constraints of conservation of angular momentum and constant total mass. This work generalizes previous treatments by including the mass fraction as a variable in the optimization problem. The results show that the lowest energy state accessible to the system corresponds to equal mass stars on a circular orbit, where the stellar spin angular velocities are both synchronized and aligned with the orbit. These features are roughly consistent with observed properties of close binary systems. We also find the conditions required for this minimum energy state to exist: [1] The total angular momentum must exceed a critical value, [2] the orbital angular momentum must be three times greater than the total spin angular momentum, and [3] the semimajor axis is bounded from above. The last condition implies that sufficiently wide binaries are not optimized with equal mass stars, where the limiting binary separation occurs near $a_0\approx16R_\ast$.