Roche-lobe overflow

Roche-lobe overflow is mass transfer that begins when a donor star fills its Roche lobe. Gas near the inner Lagrange point can pass through the saddle in the rotating-frame effective potential toward the companion. Its orbital angular momentum can prevent direct radial infall and instead produce an accretion disk, depending on the geometry.

The process couples stellar evolution to orbital evolution. Losing mass changes the donor's stellar radius, while transferred or escaping matter changes the Roche lobe. Dynamical stability of binary mass transfer depends on which response dominates. Stable transfer can gradually strip an envelope; unstable interaction can lead to a common envelope. Case A mass transfer and Case B mass transfer distinguish the donor's nuclear state when overflow starts.

In nonconservative binary evolution, matter can escape the system and/or external processes change its orbital angular momentum. Escaping overflow, a stellar wind and a common envelope can invalidate the fixed-total-mass and period-product invariants of conservative mass transfer.

Let $x=\log (R/R_L)$, $q=M_1/M_2$, ${\dot{M}}_2=-f{\dot{M}}_1=M_{1}x/t_{\mathrm{dyn}}$, and $d\log R/dt=n{\dot{M}}_1/M_1+t_{\mathrm{nuc}}^{-1}$. Under the binary period invariant for a constant wind angular-momentum factor and $R_L=0.46a(M_1/M{)}^{1/3}$, the restoring coefficient is $A(q)=n+5/3-2fq-2(3\lambda +1)(1-f)q/[3(1+q)]$. Locally freezing the masses gives an attracting overfill $x_{\ast }=f{t}_{\mathrm{dyn}}/[A(q)t_{\mathrm{nuc}}]$ precisely when $A(q)>0$. If $A$ is of order one and $t_{\mathrm{dyn}}\ll t_{\mathrm{nuc}}$, transfer is driven slowly by nuclear expansion. A negative $A$ gives positive feedback on the short overfill timescale; detailed outcomes then exceed this linear contact model.

If an escaping stellar wind carries specific orbital angular momentum $\lambda J/M$ with constant $\lambda$, then $\dot{J}/J=\lambda \dot{M}/M$ and $J\propto M^{\lambda }$. Combining this with circular-binary orbital angular momentum gives the displayed period invariant. It is a different prescription from donor-wind angular-momentum loss, whose specific angular momentum generally varies with mass ratio.

An isotropic escaping stellar wind has the donor star's mean specific orbital angular momentum. With ${\dot{M}}_d<0$, retained fraction $f$, negligible spin and no additional wind torque, $\dot{J}=(1-f){\dot{M}}_d{a}_d^2\Omega$. The retained matter redistributes angular momentum internally. Unlike isotropic re-emission from a binary star, the escaping material originates at the donor rather than the accretor. With no retention this reduces to Jeans-mode mass loss.

In a circular binary star let the donor star lose mass ${\dot{M}}_1<0$, with constant retained fraction $f$ and ${\dot{M}}_2=-f{\dot{M}}_1$. A donor star stellar wind escaping with the donor star's orbital specific angular momentum has $\dot{J}/J=(1-f){\dot{M}}_1{M}_2/[M_1(M_1+M_2)]$. Differentiating $J=G^{2/3}(P_{\mathrm{orb}}/2\pi {)}^{1/3}{M}_1{M}_2(M_1+M_2{)}^{-1/3}$ and using this loss law givesFor constant $f$ this integrates to $P_{\mathrm{orb}}{M}_1^{3f}{M}_2^3(M_1+M_2{)}^2=\mathrm{constant}$. If $f$ varies, the differential relation remains valid but the first factor must be replaced by $\exp (3\int fd\log M_1)$; treating a time-dependent exponent as a fixed power introduces an extra $\dot{f}\log M_1$ term.

Under donor-wind angular-momentum loss and the Roche lobe approximation $R_L=0.46a(M_d/M{)}^{1/3}$, the Roche-lobe radius response exponent is affine in retained fraction: ${\zeta }_L=(1-f){\zeta }_0+f{\zeta }_1$, where $q=M_d/M_a$, ${\zeta }_0=1/3-4q/[3(1+q)]$ and ${\zeta }_1=2q-5/3$. Differentiate $\log R_L=\log a+(\log M_d-\log M)/3+\mathrm{constant}$ and the circular-binary orbital angular momentum to obtain this expression.

A fast escaping Jeans-mode mass loss wind can drive an evolving donor into Roche-lobe overflow even while widening the orbit. This mechanism uses the star's changing radius; it is distinct from gravitational capture of a slow Wind Roche-lobe overflow stream.

Assume a slowly evolving circular orbit, no wind retention by the companion, $a(M_d+M_a)=\text{constant}$, and the approximate Roche lobe radius $R_L\propto a[M_d/(M_d+M_a){]}^{1/3}$. The accretor mass $M_a$ is fixed. With $q=M_d/M_a>0$,If the donor sequence obeys $R_d\propto M_d^{-n}$ locally, subtracting the two logarithmic responses givesSince ${\dot{M}}_d<0$, the overfill grows exactly when $3(1-n)q<1+3n$.

For $-1/3<n<1$, this is $q<(1+3n)/[3(1-n)]$. For $n\ge 1$, every positive $q$ satisfies the growth condition; dividing by $1-n$ would either be undefined or reverse the inequality. For $n\le -1/3$, no positive $q$ satisfies it. Equality is a marginal local response, not growth.

The criterion concerns increasing relative radius along the assumed mass-losing sequence. Reaching actual contact also depends on the initial underfill and how long the same sequence and wind prescription remain valid. Extra angular-momentum losses, wind capture or a changing radius-response exponent require a revised evolution calculation.

Let Roche-lobe overflow transfer mass conservatively within a circular binary while a separate fast wind escapes with the donor's specific orbital angular momentum. Then ${\dot{M}}_d=\dot{M}-{\dot{M}}_a$ and $\dot{a}/a=-\dot{M}/M+2{\dot{M}}_a(1/M_d-1/M_a)$. Differentiating the donor and Roche lobe radii and enforcing contact gives the displayed rate. It is positive and stabilizing when the wind drives overfill and $6q<5-3n$. When the latter inequality is reversed, transfer increases the overfill, producing runaway rather than a steady negative-transfer solution.

A donor star with stellar radius response exponent ${\zeta }_{\ast }$ can maintain exact contact in the donor-wind Roche-lobe response model for some retained fraction $0\le f\le 1$ precisely when ${\zeta }_{\ast }$ lies between ${\zeta }_0$ and ${\zeta }_1$. This follows because the attainable Roche-lobe radius response exponents form that closed interval. If its endpoints coincide, all fractions work when ${\zeta }_{\ast }$ equals the common value and none work otherwise. With negative donor mass-loss rate, $d\log (R_d/R_L)/dt=({\zeta }_{\ast }-{\zeta }_L){\dot{M}}_d/M_d$ distinguishes detachment from increasing overfill. Dynamical stability of binary mass transfer requires the adiabatic donor response.

For a slowly evolving circular orbit with constant retained fraction $f$ and donor-wind angular-momentum loss, logarithmic differentiation of the circular-binary orbital angular momentum gives $d\log P=-3fd\log M_d-3d\log M_a-2d\log (M_d+M_a)$. Integration gives the displayed invariant. At $f=1$ it becomes the period-product invariant for conservative mass transfer, while at $f=0$ it becomes $P{M}^2=\mathrm{constant}$. A varying retained fraction does not give this simple power-law invariant.

Case B mass transfer starts after core hydrogen exhaustion and before core helium ignition. It is subdivided by donor structure; early case B mass transfer occurs before a deep convective giant envelope develops.

Early case B overflow occurs in an early post-main-sequence donor, commonly crossing the Hertzsprung gap with a largely radiative envelope. A stabilizing mass-loss response can allow transfer at binary mass ratios that would destabilize a convective giant. This is an evolutionary-stage classification, distinct from recombination terminology.

Case A mass transfer starts while the donor star is on the main sequence, with core hydrogen burning. Its onset precedes giant-envelope formation; donor structure and the binary mass ratio still determine stability.

For a circular orbit, fixed total mass, transfer from $M_2$ to $M_1$, negligible spin and $R_L\propto a(M_2/M{)}^{1/3}$, the orbital angular momentum $J=M_1{M}_2\sqrt{Ga/M}$ givesIf the donor follows a radius sequence with stellar radius response exponent $\zeta$, has no additional radius change at fixed mass, and maintains contact, equating its radius derivative to this expression proves the displayed contact equation. Negative external $\dot{J}$ drives donor mass loss when $\zeta +5/3-2q>0$. The appropriate equilibrium or adiabatic response and its timescale must be specified separately; the equation alone is not a universal dynamical-stability criterion.

Conservative binary mass transfer moves material between two stars while retaining total binary mass and, in the usual approximation, orbital angular momentum. For a circular orbit with negligible spin angular momentum,At fixed $M$ and $J$, the separation obeys $a(M_d{M}_a{)}^2=\mathrm{constant}$. When an initially more massive donor star loses material, the product of component masses first increases and the orbit shrinks. After equal masses, continued transfer decreases that product and widens the orbit.

The period-product invariant for conservative mass transfer follows using Kepler's third law. Matter escaping the system, external torques or appreciable exchange with stellar spins changes the assumptions, so the simple invariant need not apply to nonconservative binary mass transfer.

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.

Dynamical stability of binary mass transfer asks whether an infinitesimal mass loss reduces or increases a donor's overfill of its Roche lobe. Define ${\zeta }_{\mathrm{ad}}=d\ln R_{\star }/d\ln M_d$ for the donor's rapid, approximately adiabatic response and ${\zeta }_L=d\ln R_L/d\ln M_d$ for the Roche lobe. ThenBecause mass loss has $d\ln M_d<0$, the overfill decreases when ${\zeta }_{\mathrm{ad}}>{\zeta }_L$ and grows in the opposite case.

The stellar radius response exponent depends on the donor's structure, while the Roche-lobe radius response exponent depends on orbital and mass-loss assumptions. There is consequently no universal critical binary mass ratio. Transfer that is dynamically stable can still evolve on a thermal or nuclear timescale.

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.

Stable envelope stripping of a low-mass red giant leaves a helium white dwarf. Because the giant radius is determined mainly by its core mass and fixes its Roche-lobe radius at detachment, the final binary orbital period is tightly related to the white-dwarf mass.

A common envelope is a phase of binary interaction in which the two stellar cores orbit within shared gas. It can arise when Roche-lobe overflow becomes unstable or when a companion is engulfed by an expanding giant. Motion through the gas transfers orbital energy and angular momentum to the envelope, causing the orbit to shrink.

If enough energy and momentum reach gas that can escape, envelope ejection can leave a much closer binary star; otherwise the cores can merge. The common-envelope energy formalism estimates the final separation by comparing released orbital energy with envelope binding energy, but its efficiency parameters compress complicated gas dynamics and energy transport. The outcome therefore depends on the donor's structure and the actual coupling to escaping material, rather than orbital energy being automatically available with unit efficiency.

The common-envelope energy formalism equates a fraction ${\alpha }_{\mathrm{CE}}$ of the released orbital energy to the envelope binding energy, often written $E_{\mathrm{bind}}=G{M}_d{M}_{\mathrm{env}}/(\lambda R_d)$. It estimates the post-envelope separation but compresses uncertain hydrodynamics into ${\alpha }_{\mathrm{CE}}$ and $\lambda$.

The double-degenerate scenario forms a close pair of carbon--oxygen white dwarfs, usually through mass transfer and one or more common-envelope phases. Gravitational radiation then shrinks the orbit until merger or unstable transfer triggers thermonuclear disruption.

An Algol binary is a semidetached binary in which an evolved, less massive donor star fills its Roche lobe while its more massive mass gainer remains near the main sequence. Binary mass-ratio reversal resolves the Algol paradox. Eclipsing members commonly have Algol variable light curves, but that observational class also includes detached systems.

The Algol paradox is the apparently inconsistent evolutionary ordering in an Algol binary: the less massive star is more evolved, whereas higher-mass isolated main sequence stars normally exhaust core fuel sooner. Binary mass-ratio reversal resolves it by identifying the present donor as the initially more massive star.

During binary mass-ratio reversal, an initially more massive donor star loses enough material that its companion becomes the more massive member. The stars retain their different prior nuclear evolution, resolving the Algol paradox. Conservative binary mass transfer shrinks the orbit before equal masses and widens it after reversal, because $a\propto (M_d{M}_a{)}^{-2}$ at fixed total mass and orbital angular momentum.

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. Van Rensbergen and collaborators' Algol population study includes reported Algol mass ratios above $0.7$ and discusses observational uncertainties.