Roche lobe

A Roche lobe is the region around one component of a circular binary inside the critical effective-potential surface through the inner Lagrange point.

If $R_2=R_L\simeq Ca(M_2/M{)}^{1/3}$, Kepler's third law givesThus a Roche lobe-filling star's period measures its inverse square-root mean mass density, largely independently of the companion mass in this approximation. A local mass-radius relation $R_2\propto M_2^{\zeta }$ gives $P\propto M_2^{(3\zeta -1)/2}$.

For contact with $R_d/R_{\odot }=M_d/M_{\odot }$ and $R_L/a=f(M_d/M{)}^{1/3}$, Kepler's third law gives $P_0=2\pi \sqrt{R_{\odot }^3/(f^{3}G{M}_{\odot })}$. The total binary mass cancels. The linear period relation depends on that particular equilibrium radius sequence; a donor with a different stellar radius response exponent follows a different mass-period relation.

For $R_L=Ca(M_d/M{)}^{1/3}$, the period when a donor reaches radius $R$ is $P=2\pi \sqrt{R^3/(C^{3}G{M}_d)}$. Requiring overflow before a specified evolutionary radius therefore bounds the initial period independently of companion mass in this lobe approximation. If that radius scales as $M_d^{5/3}$, the bound scales as $M_d^2$.

For fixed total solar-unit mass $m$ and conserved orbital angular momentum, conservative binary mass transfer preserves $K=P[m_d(m-m_d){]}^3$. Hence $P/m_d^2=K/[m_d^5(m-m_d{)}^3]$. Its logarithmic derivative is $-5/m_d+3/(m-m_d)$, vanishing at $5m/8$ with positive second derivative. The ratio diverges at both endpoints, so this is its global minimum. Comparing this minimum to a pre-giant bound $P_0$ tests whether any such conservative antecedent can overflow before the giant branch; satisfying it is necessary, not a proof of an entire evolutionary history.

Roche-lobe overflow begins when a star fills its Roche lobe and matter flows through the inner Lagrange point toward its companion.

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 a giant into Roche-lobe overflow because its radius changes as it loses mass, even though the wind itself is not substantially captured. This differs from slow Wind Roche-lobe overflow. For $R_d\propto M_d^{-n}$ and $R_L\propto a(M_d/M{)}^{1/3}$, wind alone gives $d\log (R_d/R_L)/dt=\dot{M}[3(1-n)q-(1+3n)]/[3M_d(1+q)]$, where $q=M_d/M_a$ and $\dot{M}<0$ is the escaping wind rate. The displayed inequality is exactly the condition that overfill increases.

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 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.

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 phase occurs when unstable binary interaction engulfs both stellar cores in one envelope. Orbital energy and angular momentum may eject the envelope, leaving a compact binary, or the cores may merge.

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.