{c}
{title2=$q_{\rm crit}\simeq2/3$}

A simplified conservative model with a fully convective donor gives long-lived stable <Roche-lobe overflow> only for donor-to-accretor mass ratio $q<2/3$. This follows from the <Roche-lobe radius response exponent> and the donor's adiabatic <stellar radius response exponent> $-1/3$. The boundary is model-dependent; radiative donors, evolved core structure and mass or angular-momentum loss change it. https://arxiv.org/abs/1008.2620[Van Rensbergen and collaborators' Algol population study] includes reported Algol mass ratios above $0.7$ and discusses observational uncertainties.


Algol mass-ratio stability limit

{title2=$\zeta=d\log R/d\log M$}

The stellar radius response exponent measures how a star's radius changes as it loses mass. The adiabatic exponent $\zeta_{\rm ad}$ describes loss too rapid for thermal adjustment; the equilibrium exponent $\zeta_{\rm eq}$ follows a sequence of thermally relaxed stars. Compare $\zeta_{\rm ad}$ with the <Roche-lobe radius response exponent> to test dynamical stability.


Stellar radius response exponent

{c}
{title2=$\zeta_L=d\log R_L/d\log M_d$}

For <conservative binary mass transfer> and $R_L\propto a(M_d/M)^{1/3}$,
$$
\zeta_L=2q-\frac53,\qquad q=\frac{M_d}{M_a}.
$$
The <conservation of angular momentum> gives $d\log a/d\log M_d=-2(1-q)$. Since a mass loss has $d\log M_d<0$, the donor moves inside its lobe when $\zeta_{\rm ad}>\zeta_L$. An ideal fully convective donor with $\zeta_{\rm ad}=-1/3$ is therefore stable for $q<2/3$ and unstable for $q>2/3$ under these assumptions.


Codex @codex  0

 Incoming links: Roche-lobe radius response exponent