Approximation of equal-volume radii through lagrange points

Accurate Roche-lobe radii are vital for mass transfer and binary evolution modeling. While the formula is widely used for its simplicity and 1% accuracy, observations reveal systematic underestimations in certain mass-ratio ranges—a discrepancy expected within the standard Roche framework. Retaining these standard assumptions, we refined the numerical accuracy using Eggleton's formula as a baseline. Crucially, because Eggleton's model only constrains the inner critical surface (rm L_ 1983ApJ...268..368E 1 ), it cannot accommodate over-contact binaries that require outer boundary modeling. By extending numerical integrations to the outer critical surface, we derived a more precise formula for the outer Roche-lobe radii. This provides a more realistic physical representation and is a key improvement in this work. Beyond determining the inner and outer Roche-lobe radii, we also established more precise fitting formulae for the positions of ̊m L_ 1 , ̊m L_ 2 , and ̊m L_ 3 . This study aimed to construct volume-equivalent radii formulae for the equipotential surfaces passing through these three Lagrangian points, providing a more comprehensive tool for binary evolution modeling. Furthermore, observational data of contact binaries covering mass ratios q= from 0 to 20 were utilized for validation. M_ 2 M_ 1 In this work, we defined a spherical coordinate system centered on the primary star (rm M_ 1 ) and derived analytical approximations for the positions of the three collinear Lagrangian points (rm L_ 1 , rm L_ 2 , and rm L_ 3 ). Furthermore, we proposed a new method to calculate the volume-equivalent radii for rm L_ 2 and rm L_ 3 . By using the lowest points along the rm L_ 2 and rm L_ 3 equipotential surfaces as boundaries, this approach divides the peanut-shaped structure into two components, thereby yielding the corresponding volume-equivalent radii rm R_ L2 and rm R_ L3 The improved Roche-lobe radii formulae in this study achieve errors łe 0.5% across all mass ratios q.The research results provide a more precise theoretical framework for simulating mass overflow and mass loss through ̊m L_ 1 , ̊m L_ 2 and ̊m L_ 3 . Determining the positions of the Lagrangian points aims to calculate the Roche potential and investigate mass transfer and loss. These new formulae are applicable to any q with higher precision, and rm L_ 2 , and rm L_ 3 facilitates the analysis of mass loss through the outer Lagrangian points.