Conservative binary mass transfer

Conservative binary mass transfer moves matter between the components while preserving total binary mass and orbital angular momentum.

Negligible spin and a circular orbit give $J=M_1{M}_2\sqrt{Ga/M}$. Fixed $M$ and $J$ imply $a(M_1{M}_2{)}^2$ is constant. Kepler's third law then gives the period-product invariant $P(M_1{M}_2{)}^3$.

At fixed total mass $M$ and period-product invariant, $P/M_d^2$ is proportional to $[M_d^5(M-M_d{)}^3{]}^{-1}$. Its unique global minimum occurs at donor mass $5M/8$ and accretor mass $3M/8$. It is useful for testing all possible conservative progenitors against a pre-overflow period bound.

Binary mass transfer is dynamically stable when the donor's radius decreases relative to its Roche-lobe radius after a small mass loss, reducing overflow instead of amplifying it.

With fixed total mass and orbital angular momentum and $R_L\propto a(M_d/M{)}^{1/3}$, the Roche-lobe radius response exponent is $2q-5/3$. A donor with stellar radius response exponent ${\zeta }_{\ast }$ is stable to infinitesimal mass loss when ${\zeta }_{\ast }>2q-5/3$, giving $q<({\zeta }_{\ast }+5/3)/2$. The threshold depends on the donor response.

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

For conservative binary mass transfer and $R_L\propto a(M_d/M{)}^{1/3}$,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 ${\zeta }_{\mathrm{ad}}>{\zeta }_L$. An ideal fully convective donor with ${\zeta }_{\mathrm{ad}}=-1/3$ is therefore stable for $q<2/3$ and unstable for $q>2/3$ under these assumptions.

Let $q=M_1/M_2$, hold total binary mass fixed, use $R_L\propto a{q}^{3/10}$, and impose $d\log J=fd\log M_1$. The circular-binary orbital angular momentum then gives $d\log a/d\log M_1=2(q-1+f)$, whereas mass conservation gives $d\log q/d\log M_1=1+q$. Thus ${\zeta }_L=(23q-17+20f)/10$. If the donor obeys $R_1=K{M}_1^{\beta }$ with slowly varying $K$, loss of donor mass increases its overfill when ${\zeta }_L>\beta$, or $q>(17-20f+10\beta )/23$. This is a feedback criterion on the time scale of the assumed donor response, not automatically a dynamical instability for every choice of $\beta$. For $\beta =1/2$, the threshold is $(22-20f)/23$; if $f\ge 11/10$, every positive mass ratio satisfies it.