 Synchronous fission model for binary formation

In this idealized model, a critically rotating spherical cloud splits into two equal-mass homologous spheres of radius $a/2$ that share the orbital frequency of their touching circular binary. Their combined orbital and spin inertia is $(1+\alpha )M{a}^2/4$. Equating post-fission angular momentum to $\alpha M\sqrt{GM{R}_c}$ gives the displayed separation relation. A compact outcome $a<R_c$ requires $0<\alpha <1/3$. The model is an angular-momentum budget, not a hydrodynamic proof that stellar fission occurs.