Binary star

A binary star is a gravitationally bound pair orbiting their common centre of mass.

An X-ray binary has a compact mass-gaining star or black hole accreting from a companion, so that much of its emitted power appears as X-rays. Gravitational binding energy supplies an accretion luminosity of order $GM\dot{M}/R$ when there is a surface or an appropriate inner disk radius. The donor star may feed an accretion disk through Roche-lobe overflow or a stellar wind.

A Type I X-ray burst is a transient thermonuclear runaway in accreted hydrogen or helium on a neutron star surface. It rises rapidly and cools after the fuel burns. A black hole has no material surface on which the same accumulated surface-fuel event can occur. This is distinct from an accretion-powered X-ray pulse or an accretion instability.

An accreting X-ray pulsar contains a magnetized neutron star whose magnetic field channels the accreting gas toward its magnetic poles. Rotation carries the bright accretion regions through the observer's view and produces pulses. The input angular momentum can change the spin period; this differs from the rotational energy source of an isolated pulsar.

A high-mass X-ray binary has a massive donor star supplying an accreting neutron star or black hole. A strong stellar wind often feeds it, although Roche-lobe overflow and donor equatorial outflows also occur. The short lifetime of the donor associates these systems with young stellar populations.

In a low-mass X-ray binary, the donor star is relatively low mass. Roche-lobe overflow usually feeds an accretion disk around a neutron star or black hole. The low-mass label concerns the donor rather than the total binary mass. Accumulated fuel on a neutron star can produce a Type I X-ray burst.

Coalescence of a binary containing two neutron stars can eject neutron-rich material and power a kilonova. Rapid neutron-capture process reactions in the ejecta produce heavy elements. This connects compact-binary dynamics to chemical enrichment.

An optical and infrared transient powered by radioactive decay in material ejected from a compact-object merger. Its evolving spectrum and light curve probe ejecta composition and rapid neutron-capture process production.

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.

For a circular Kepler orbit, neglecting stellar spin, the orbital angular momentum of a binary star is $J=\mu a^2\Omega =\mu \sqrt{GMa}$, where $M$ is total mass, $\mu$ is the reduced mass, and $a$ is separation. Kepler's third law also gives $J=G^{2/3}{P}^{1/3}{M}_1{M}_2/[(2\pi {)}^{1/3}{M}^{1/3}]$. Distances from the center of mass are $a{M}_2/M$ and $a{M}_1/M$; adding their individual orbital angular momenta proves the formula.

A binary binary mass ratio is the quotient of its component masses. For mass-transfer stability, specify the donor-to-accretor convention $q=M_d/M_a$ explicitly, since the inverse convention also occurs. Mass-ratio reversal changes which component is more massive.

A detached binary has both components lying inside their respective Roche lobes, so neither currently transfers matter by Roche-lobe overflow. A detached system may still exchange matter through a stellar wind.

A coeval binary has components formed at the same time. A population of such binary stars can still have a distribution of system ages. If component masses have independence and the state at age $t$ is determined separately for each component, the states have conditional independence given $t$. If $g(t)$ and $w(t)$ are the conditional probabilities of two states, the unconditional unordered pair fractions are $\mathbb{E}[g(t{)}^2]$ and $2\mathbb{E}[g(t)w(t)]$, not generally products of marginal state fractions.

Suppose binary ages have uniform coordinate $y\in [0,1]$ and independently sampled component masses give disjoint conditional state fractions $g(y)=ay$ and $w(y)=by$. Integrating the conditional pair probabilities gives $P(GG)=a^2/3$, $P(GW)=2ab/3$, and $P(WW)=b^2/3$. The fraction with at least one $G$ is $a-a^2/3$, and the fraction of those systems with a second evolved component is $(a^2+2ab)/(3a-a^2)$. Components share their age, so their unconditional states are correlated even when their birth masses have independence.

In an instantaneous burst all binary components have a fixed common age. If the initial component masses are independent, the probability of two red giants is the square of the individual giant probability at that age. This need not equal the age-averaged pair fraction in a continuously forming population.

If $g(t)$ is the conditional giant fraction and $w(t)$ a second evolved-state fraction at common age $t$, independent component masses give the fraction of giant-containing systems with another evolved component as $\mathbb{E}[g^2+2gw]/\mathbb{E}[2g-g^2]$. Averaging must follow conditioning on age, rather than multiplying marginal state probabilities.

A semidetached binary has one component filling its Roche lobe while the other remains inside its own lobe. Algol binaries and most ordinary cataclysmic variables are examples.

RT Lac is a named interacting binary used in studies of binary star evolution. Conservative-progenitor tests and nonconservative mass transfer provide ways to interpret a stripped, mass-losing component.

An eclipsing binary has an orbital orientation that lets one star pass in front of the other as seen by the observer, producing periodic changes in received radiative flux.

An Algol variable is an eclipsing binary with well-separated eclipse minima and relatively little variation between eclipses. This light-curve classification includes detached systems as well as interacting Algol binaries; it does not by itself imply Roche-lobe overflow.

A donor star loses matter to its companion through Roche-lobe overflow or a stellar wind.

A mass-gaining star accretes material from a companion. Its changing mass and composition can alter its main sequence location and cause stellar rejuvenation.

For a single-lined circular spectroscopic binary whose visible component has radial-velocity amplitude $K_1$ and orbital period $P$, the binary mass function is

Randomly oriented binary angular-momentum vectors are uniform in $\cos i$. After identifying $i$ with $\pi -i$, their normalized inclination density on $0\le i\le \pi /2$ is $p(i)=\sin i$.

A recycled pulsar is a neutron star spun up by accretion from a binary companion. Accreted angular momentum can reduce its spin period to milliseconds, while donor-envelope removal often leaves a white-dwarf companion.

A binary millisecond pulsar is a rapidly rotating recycled neutron star that remains in a binary, commonly with the white-dwarf remnant of its donor.

Wind mass transfer occurs when one star accretes part of a companion's stellar wind without the donor necessarily filling its Roche lobe.

For a fast isotropic cold wind, the mass density at a companion at separation $a$ is $(-{\dot{M}}_d)/(4\pi a^2{v}_w)$. The ballistic capture model gives ${\dot{M}}_a/(-{\dot{M}}_d)=G^2{M}_a^2/(a^2{v}_w^4)$. Fast relative to orbital motion also makes the capture radius small compared with the separation.

Wind Roche-lobe overflow occurs when a slow dense wind nearly fills the donor's Roche lobe and is gravitationally focused through the inner Lagrange region. It can transfer a much larger fraction of the outflow than a uniform fast-wind Bondi--Hoyle estimate.

A tidally enhanced stellar wind is a model in which a companion increases a giant's wind mass loss as the giant approaches its Roche lobe. Removing envelope mass before contact can reduce or reverse the mass ratio and make subsequent Roche-lobe overflow more stable. This is one proposed route to wide systems with Algol-like reversed evolutionary ordering. Van der Sluys's compact-binary study, section 5.7.1 discusses the enhanced-wind scenario and its effect on mass-transfer stability.

In Jeans-mode mass loss, an isotropic wind escapes from one binary component carrying that star's specific orbital angular momentum. If the companion accretes negligibly, a circular binary obeys $aM=\text{constant}$ and $P{M}^2=\text{constant}$.

A barium star is a main-sequence or giant star enriched in carbon and slow-neutron-capture elements despite being too unevolved to have made them itself. It was polluted by an asymptotic-giant-branch companion that is now a white dwarf.

An S-type star has strong zirconium-oxide bands and enhanced slow-neutron-capture elements. Intrinsic S-type stars are thermally pulsing asymptotic giants, while extrinsic S-type stars were polluted by a former asymptotic-giant-branch companion.

A cataclysmic variable is a close binary in which a Roche-lobe-filling low-mass star transfers matter to a white dwarf. Angular-momentum loss drives long-term transfer, while unstable burning of accumulated hydrogen produces classical novae.

A cataclysmic variable whose persistent luminous disk resembles a dwarf nova in outburst. A sufficiently high transfer rate can keep its accretion disk on the hot stable branch of the hydrogen-ionization disk instability, avoiding ordinary dwarf-nova cycles.

A cataclysmic variable with recurrent disk outbursts associated with the hydrogen-ionization disk instability. Enhanced accretion powers the bright phase; the event does not require thermonuclear runaway on the white dwarf.

A cataclysmic variable whose white dwarf magnetic field does not dominate the large-scale accretion flow. It normally supports an accretion disk fed through Roche-lobe overflow. The term does not rule out a magnetized donor star or magnetic braking of a binary star.

The bright region where the stream from Roche-lobe overflow strikes an accretion disk. Dissipation of relative kinetic energy produces a localized source of light whose visibility changes around the orbital period.

Ordinary hydrogen-rich cataclysmic variables occur on both sides of a cataclysmic-variable period gap at approximately two to three hours, with a cataclysmic-variable period minimum near 82 minutes. Selection effects change the observed proportions. Knigge, Baraffe and Patterson's donor-based evolutionary study summarizes these empirical landmarks.

For a Roche lobe-filling donor with local radius response $R_2\propto M_2^{\zeta }$, the Roche-lobe-filling period-density relation gives $P\propto M_2^{(3\zeta -1)/2}$. As mass is lost, the period falls for $\zeta >1/3$ and rises for $\zeta <1/3$, with a minimum at the transition. The reversal is called a period bounce. Slow evolution near the turning point produces an accumulation in the period distribution: for a steady flow of systems along one evolutionary branch, $N(P)\propto 1/|\dot{P}|$.

The reversal from decreasing to increasing orbital period during continuing donor mass loss. The Roche-lobe-filling period-density relation gives $d\log P/d\log M_d=(3\zeta -1)/2$, where $\zeta$ is the effective stellar radius response exponent along the mass-losing sequence. A crossing from $\zeta >1/3$ to $\zeta <1/3$ gives a local cataclysmic-variable period minimum.

The cataclysmic-variable period gap is the observed deficit of accreting, ordinary hydrogen-rich cataclysmic variables with periods around two to three hours. The disrupted magnetic braking model interprets it as a detached phase after the donor contracts inside its Roche lobe.

A magnetized wind removes spin angular momentum from a cool donor star. Tidal synchronization replenishes that spin from the orbit, so the wind indirectly removes orbital angular momentum and drives the binary toward shorter separation.

In this model, strong magnetic braking of a binary star inflates the mass-losing donor above the cataclysmic-variable period gap. A reduction in the torque near the fully convective star transition allows thermal contraction and detachment. Gravitational-wave emission from a binary system continues to shrink the orbit until contact returns below the gap. The model requires a braking decrease and a previously inflated donor, not disappearance of stellar magnetism. Knigge's review describes this evolutionary picture and observational tests.

In isotropic re-emission, transferred mass is expelled from the vicinity of the accretor and carries the accretor's specific orbital angular momentum. The total binary mass and orbital angular momentum then decrease even though the accretor retains no net mass.

Asymmetric ejecta or neutrino emission can impart a natal kick to a compact remnant. The impulsive change in relative velocity and total mass can alter the binary's semimajor axis and eccentricity or unbind the system.

For an impulsive supernova kick in a binary star initially in a circular Kepler orbit, let $\beta$ be the remaining total mass fraction and $\alpha$ the kick speed divided by the original relative speed. The displayed post-kick specific orbital energy is negative for a bound pair and positive for escape. Over all directions, the velocity factor ranges from $(1-\alpha {)}^2$ to $(1+\alpha {)}^2$. Some direction leaves a bound pair exactly when $2\beta >(1-\alpha {)}^2$; some direction permits positive-energy escape exactly when $2\beta <(1+\alpha {)}^2$. Equalities are marginal parabolic Kepler orbits. The circular initial orbit and impulsive approximation are essential.

A disrupted binary star with positive post-impulse specific orbital energy ${\epsilon }^{\prime }$ has relative speed $V=\sqrt{2{\epsilon }^{\prime }}$ at infinite separation, because the Newtonian gravitational potential energy tends to zero. With the notation of the kick-direction binary survival criterion, $V=v\sqrt{1+2\alpha \cos \psi +{\alpha }^2-2\beta }$. This is not a systemic recoil velocity. A marginal parabolic Kepler orbit has $V=0$ at infinity.

An accelerating binary mass quadrupole emits gravitational waves. The resulting energy and angular-momentum loss shrinks and circularizes a detached compact binary, with the evolution accelerating sharply as its separation decreases.

For fixed component masses, total mass $M=M_1+M_2$ and reduced mass $\mu =M_1{M}_2/M$, in the weak-field slow-motion circular orbits, $E=-GM\mu /(2a)$ and $J=\mu \sqrt{GMa}$ obey $dE/dJ=\Omega$. The gravitational-wave energy and angular-momentum balance makes the secular loss tangent to this circular family. Substituting $v^2=GM/a$ into the instantaneous quadrupole luminosity of a Kepler binary and differentiating $E$ gives the displayed inspiral equation. Integration givesThe orbit is quasicircular, with small radial drift. The formal coalescence time is not a valid extension of the weak-field point-mass model into the final strong-field or stellar-contact phase.

The equipotential stellar-structure approximation replaces distorted closed equipotentials in a synchronized binary by one-dimensional shells labeled by enclosed volume and mass.

The volume-equivalent radius of a closed surface enclosing volume $V$ is defined by $V=4\pi r^{\prime 3}/3$.

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.