Synchronous fission model for binary formation ($a/R_c=16{\alpha }^2/(1+\alpha {)}^2$)

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.

If each daughter subsequently conserves its own spin while contracting at fixed outer separation, its next critical radius is $a/8$. A local repetition of the synchronous fission model for binary formation gives the displayed inner-to-outer separation ratio. Continuous tidal locking throughout the contraction would instead keep the spin frequency fixed and decrease its centrifugal-to-gravity ratio as the cube of its radius; the repeated fission would not follow. Spin-orbit torque assumptions must therefore be explicit.