Equal-volume radii approx. through Lagrange points

Beyond determining the inner and outer Roche-lobe radii, we also established more precise fitting formulae for the positions of L1, L2, and L3. 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=M2/M1 from 0 to 20 were utilized for validation.

In this work, we defined a spherical coordinate system centered on the primary star (M1) and derived analytical approximations for the positions of the three collinear Lagrangian points (L1, L2, and L3). Furthermore, we proposed a new method to calculate the volume-equivalent radii for L2 and L3. By using the lowest points along the L2 and L3 equipotential surfaces as boundaries, this approach divides the peanut-shaped structure into two components, thereby yielding the corresponding volume-equivalent radii R_L2_ and R_L3_.

The improved Roche-lobe radii formulae in this study achieve errors <=0.5% across all mass ratios q. The research results provide a more precise theoretical framework for simulating mass overflow and mass loss through L1, L2 and L3. 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 L2, and L3 facilitates the analysis of mass loss through the outer Lagrangian points.