Close Binary: Roche Lobes and Conservative Mass Transfer

A close-binary Roche-geometry and mass-transfer playground. The corotating Roche potential gives the five Lagrange points and the figure-eight critical equipotential through the inner point L1; the Eggleton (1983) formula gives the volume-equivalent Roche-lobe radius. Conservative mass transfer holds the total mass and the orbital angular momentum fixed, so the separation scales as a ~ (M1 M2)^-2 and the orbit shrinks while the donor is the more massive star, reaches a minimum at q = 1, then widens. Panel A is the corotating frame with both lobes, the Lagrange points and, on overflow, the L1 stream and accretion disk; Panel B is the separation and period under transfer with the shrink-then-widen turning point; Panel C is the stability map, the Roche-lobe response exponent against stiff and soft donors, classifying detached, stable transfer and common envelope. Conservative transfer holds total mass and orbital angular momentum fixed, so the orbit shrinks while the donor is the more massive star and widens past q = 1, with the donor-versus-Roche-lobe radius response setting whether transfer is stable or runs away.