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For Roche-lobe overflow in the stipulated stellar thermal equilibrium, $R_2=R_L$. Using the mass-radius relation and Kepler's third law,The total mass cancels, givingHere $R_{\odot }$ is the solar radius. This is the Roche-lobe-filling period-density relation specialized to $R_2\propto M_2$.

Neglecting spin, the angular momentum of a circular orbit is $J=M_1{M}_2\sqrt{Ga/M}$. Between classical novae the transfer conserves total mass, so $\dot{M}=0$ and ${\dot{M}}_1=-{\dot{M}}_2$. Its logarithmic derivative givesIn particular, for conservation of angular momentum,The source PDF has $\dot{J}=0$ here; the TeX's $J=0$ is a transcription defect. For $q>4/3$, the Roche-lobe radius response exponent exceeds the equilibrium stellar radius response exponent $1$. Since ${\dot{M}}_2<0$, the Roche lobe then shrinks faster than the donor star; the overfill increases and drives more transfer, giving positive feedback. Equality at $q=4/3$ is marginal in this linear response test.

Two qualifications matter. First, $q>4/3$ is a formal extrapolation outside the stated $q<1$ range, and the supplied Roche lobe approximation need not remain accurate there. Second, true dynamical stability of binary mass transfer compares ${\zeta }_L$ with the adiabatic stellar radius response exponent ${\zeta }_{\mathrm{ad}}$, not ${\zeta }_{\mathrm{eq}}=1$. The stellar thermal equilibrium argument supplies the displayed feedback threshold under the imposed radius law, rather than a necessary and sufficient physical dynamical threshold. For example, the polytropic mass-radius relation for a fully convective adiabatic stellar polytrope with index $3/2$ gives ${\zeta }_{\mathrm{ad}}=-1/3$; within this same Roche lobe approximation its dynamical threshold is $q=2/3$. Thus $q<4/3$ by itself does not guarantee dynamical stability.

Gravitational-wave emission from a binary system removes orbital energy and angular momentum. Magnetic braking of a binary star provides another important loss: a magnetized stellar wind carries away donor spin, while tidal locking couples that spin to the orbit. Tides alone redistribute angular momentum and are not an external sink. Matter expelled from the system can also carry orbital angular momentum, although appreciable continuous mass loss would require nonconservative modifications of the preceding equations. Setting ${\dot{R}}_L/R_L={\dot{R}}_2/R_2$ gives the binary mass-transfer contact equationFor $q<4/3$, a negative external $\dot{J}$ therefore sustains negative ${\dot{M}}_2$ along the stipulated stellar thermal equilibrium contact sequence, provided the system is otherwise stable and can remain thermally relaxed.

During the classical nova, the ejecta carry the white dwarf's specific angular momentum, not zero angular momentum. With ${\Omega }^2=GM/a^3$ and the accretor's distance from the centre of mass $a_1=a{M}_2/M$, isotropic escape givesThe slow ejection relative to the orbital period permits the adiabatic circular orbit approximation; the donor star does not transfer appreciable additional mass during it. Take $\delta M_1=-\delta m$, $\delta M_2=0$ and $\delta M=-\delta m$. Differentiating $J=M_1{M}_2\sqrt{Ga/M}$ givesSince $M/M_1=1+q$, this yieldsThese are first-order formulae with errors of order $(\delta m/M{)}^2$. The orbital widening agrees with Jeans-mode mass loss. The donor star's radius is unchanged to this order by the assumed ejection, so its Roche lobe expands away from it, causing nova-induced binary detachment. This idealization neglects changes in the donor from irradiation or interaction with ejecta; those are not supplied in the model.

Let $\Gamma =-\dot{J}>0$ be the fixed external loss rate. During the detached binary phase both component masses are fixed, and ${\dot{R}}_L/R_L=2\dot{J}/J=-2\Gamma /J$. Closing the fractional gap $4\delta m/(3M)$ takesIn the subsequent semidetached binary, the binary mass-transfer contact equation gives $-{\dot{M}}_2=3M_2\Gamma /[J(4-3q)]$. Accumulating a fresh layer of mass $\delta m$ therefore takesThroughout these first-order estimates $J,M,q$ can be evaluated at the start of the cycle: their fractional changes during it are of order $\delta m/M$. The limit $\Gamma =0$ is excluded, since no finite reconnection time follows without an external shrinkage mechanism.

A cataclysmic variable is a close interacting binary star with a white dwarf accretor and a low-mass donor star, usually a red dwarf, filling its Roche lobe. Matter crosses by Roche-lobe overflow and commonly forms an accretion disk; sufficiently strong white-dwarf magnetism instead permits magnetically channelled accretion. Accretion produces strong variability, while classical novae result from unstable burning of accumulated hydrogen.

The cataclysmic-variable orbital-period distribution contains a short-period population between about 82 minutes and two hours, a pronounced cataclysmic-variable period gap around two to three hours, and a longer-period population above the gap. There is an accumulation near the cataclysmic-variable period minimum. These describe ordinary hydrogen-rich systems; evolved donors and helium-rich systems can have shorter periods. Selection by luminosity and outburst activity affects the observed counts, so $N(P)$ is not simply the intrinsic distribution.

For the contact red dwarf, $R_2=R_L\propto a(M_2/M{)}^{1/3}$. Using its supplied thermal-equilibrium mass-radius relation, $R_2\propto M_2$, gives $a\propto M_2^{2/3}{M}^{1/3}$. Kepler's third law then givesindependent of $M_1$. This is a special case of the Roche-lobe-filling period-density relation.

To examine rapid mass loss, use the stellar radius response exponent ${\zeta }_{\mathrm{ad}}=d\log R_2/d\log M_2=-1/3$. During conservative binary mass transfer, $M$ and $J=M_1{M}_2\sqrt{Ga/M}$ are fixed, and $d{M}_1=-d{M}_2$. ThereforeThe change in overfill is $d\log (R_2/R_L)=({\zeta }_{\mathrm{ad}}-{\zeta }_L)d\log M_2$. Since $d\log M_2<0$, overfill grows, and mass loss runs away, when ${\zeta }_{\mathrm{ad}}<{\zeta }_L$. The dynamical stability of binary mass transfer criterion here is consequentlyThe printed instability inequality is reversed. With the defined $q=M_2/M_1$, the specified radius laws imply the inequality above; $q<q_{\mathrm{crit}}$ is the stable side. Equality is marginal, and the supplied Roche-radius approximation restricts this calculation to its stated mass-ratio range.

Long-term contact is maintained by angular momentum transport out of the orbit. Gravitational-wave emission from a binary system provides one loss mechanism. Magnetic braking of a binary star uses the donor's stellar wind: its magnetic field enforces approximate corotation out to the Alfvén radius, so the wind carries much more specific angular momentum than a nonmagnetic surface outflow. At that radius, roughlyThe geometry contributes order-one factors to the torque. Even when ${\dot{M}}_w\ll |{\dot{M}}_2|$, the large lever arm can remove substantial spin angular momentum. Tidal locking couples donor spin to orbital motion, so the wind torque ultimately brakes the orbit. In practice classical novae can also eject matter; the conservative calculations here isolate the stated idealization.

For the prescribed $\dot{J}/J=-\alpha$, now use the supplied equilibrium law $R_2\propto M_2$ for the slowly evolving donor. Contact implies $\dot{a}/a=(2/3){\dot{M}}_2/M_2$, assuming negligible net wind mass loss. Taking the logarithmic derivative of $J$ givesSince $P\propto M_2$ along this equilibrium sequence,For the dynamically stable mass ratios, this is negative. The result uses thermal equilibrium as well as slow hydrostatic evolution; evolution slow compared only with the dynamical time does not by itself guarantee the radius law $R_2\propto M_2$.

In the disrupted magnetic braking model, the donor above the gap loses mass fast enough to be inflated relative to thermal equilibrium. When the donor becomes a fully convective star, the model assumes a substantial reduction in magnetic braking. The red dwarf then contracts toward equilibrium on its Kelvin-Helmholtz cooling time, becoming smaller than its Roche lobe; accretion largely stops. Gravitational-wave emission from a binary system continues to reduce the separation, carrying the detached binary through the gap until contact is restored near its lower edge.

This explanation requires strong braking immediately above the gap: the donor must already be out of equilibrium and inflated, so there is room to contract and detach when braking drops. With nearly fixed component masses during detachment, $R_L\propto a\propto P^{2/3}$, giving the illustrative inflation factorA weak braking torque that kept the donor in thermal equilibrium just above the gap would not produce this detachment. The model requires a decrease in the torque, not disappearance of the donor's magnetic field.

An Algol binary is a semidetached binary: a cool evolved donor fills its Roche lobe, while a hotter and more massive mass gainer remains on or near the main sequence. Roche-lobe overflow feeds the companion through a stream, sometimes forming an accretion disk. A suitably inclined system is an eclipsing binary. The prototype's periodic dimming led John Goodricke to propose an occulting companion in 1783; his original observations record an early eclipse interpretation.

The Algol paradox arises if the present masses are assumed to have been constant: the lower-mass star is the more evolved one, even though coeval isolated stars of higher mass normally exhaust core fuel sooner. The mass-luminosity relation gives the rough nuclear-lifetime scaling $t_{\mathrm{nuc}}\propto M/L$, decreasing strongly with mass. The resolution is binary mass-ratio reversal. The present donor began as the more massive star, evolved first, and expanded into its Roche lobe. Transferring much of its envelope made it less massive and made its initially less massive companion the present mass gainer. Their current masses therefore do not reveal their original evolutionary ordering.

On a Hertzsprung-Russell diagram, both components start on the zero-age main sequence. The initially more massive donor leaves the main sequence first, moving toward lower effective temperature and higher luminosity as a subgiant or red giant. During envelope stripping, it remains oversized and overluminous for its decreasing mass because its evolved core continues to supply energy. The accretor gains mass, moves to higher effective temperature and luminosity, and can undergo stellar rejuvenation if fresh hydrogen mixes into its core. After substantial envelope removal, the donor contracts to a hot stripped star; a sufficiently low-mass helium core ultimately becomes a helium white dwarf. The schematic below separates the two identities through the transfer episode rather than relabelling them when their masses cross.

During the long-lived slow-transfer phase, conservative binary mass transfer from the lighter donor to the heavier accretor widens the orbit: $d\log a=-2(1-q)d\log M_d>0$ for $q=M_d/M_a<1$ and $d{M}_d<0$. Transfer eventually stops when the donor's shrinking envelope can no longer maintain contact. The remnant can be a helium white dwarf if it never ignites helium, or a more massive helium-burning stripped star if it does. Later the mass gainer also leaves the main sequence. Reverse Roche-lobe overflow onto the compact remnant can lead to a common envelope, leaving a close double remnant after successful envelope ejection, or to merger. The detailed outcome depends on both masses, the separation, and how much matter and angular momentum escaped during earlier transfer.

The approximate mass-ratio boundary in the question has a stability interpretation. For an ideal fully convective donor star with adiabatic stellar radius response exponent ${\zeta }_{\mathrm{ad}}=-1/3$, the conservative Roche-radius approximation gives ${\zeta }_L=2q-5/3$. Dynamical stability of binary mass transfer requires ${\zeta }_{\mathrm{ad}}>{\zeta }_L$, so a long-lived stable system hasA more massive convective donor expands relative to its shrinking lobe under mass loss, favouring runaway transfer and a common envelope instead of a persistent Algol phase. This explains the approximate Algol mass-ratio stability limit in that model. It is not a universal observational boundary: a donor with a radiative envelope or a substantial evolved core has a different adiabatic response, and nonconservative loss changes the Roche-lobe response. Van Rensbergen and collaborators' observed and modelled Algol distributions include reported mass ratios above $0.7$ and discuss uncertainties in their determination. The literal claim that all Algols obey the same cutoff is therefore too strong.

A sufficiently wide system first reaches Roche-lobe overflow on the red giant branch, when the original donor is likely to have a deep convective envelope. Straightforward conservative overflow while that donor is still more massive is then prone to dynamical runaway and orbital contraction in a common envelope; it does not naturally yield a wide, long-lived Algol-like configuration. A plausible route is substantial earlier envelope loss through a stellar wind, possibly tidally enhanced stellar wind loss, reducing or reversing the mass ratio before contact. Wind mass transfer in a binary star can also increase the companion's mass. The lower donor mass, reduced envelope and larger core fraction make later transfer easier to stabilize. Alternatively, a detached pair with the same reversed evolutionary appearance may be interacting only through a wind and need never have undergone overflow. Its current width alone does not uniquely determine the initial orbit, but it indicates that prior mass loss or a more general nonconservative history must be considered, rather than applying the simple conservative convective-donor picture unchanged.

The component distances from the centre of mass are $a_1=a{M}_2/M$ and $a_2=a{M}_1/M$, where $M=M_1+M_2$. Their orbital angular momenta add to

Let $W>0$ be the isotropic wind-loss magnitude, so ${\dot{M}}_1=-W$ and ${\dot{M}}_2=0$. The wind carries star 1's specific orbital angular momentum $j_1=a_1^2\Omega$, and henceUsing Kepler third law to write $J=M_1{M}_2\sqrt{Ga/M}$ and differentiating showsSince $P^2\propto a^3/M$, it follows thatThis is Jeans-mode mass loss.

Now allow transfer to star 2 at rate ${\dot{M}}_2>0$ while the wind continues. ThenThe same conservation of angular momentum calculation givesThe donor response $R\propto M_1^{-n}$ iswhile

Before contact, put ${\dot{M}}_2=0$. The wind drives the star farther into its Roche lobe when $\dot{\log }(R/R_L)>0$. With $q=M_1/M_2$, this condition reduces toIf the inequality is reversed, the Roche lobe expands relative to the donor, so wind loss detaches the star and no wind-driven Roche-lobe transfer is sustained; later nuclear expansion may restore contact.

During stable contact, impose $\dot{R}/R={\dot{R}}_L/R_L$ and solve for the transfer rate. Straightforward algebra givesIn the paper's signed notation $W=-\dot{M}$, this is exactlyIfthe stationary response has the wrong sign: transfer enlarges the overfill rather than removing it. Dynamical stability of binary mass transfer is lost, leading to runaway transfer and usually a common envelope or merger.

An intrinsic S-type star is a thermally pulsing AGB star whose third dredge-up exposes carbon and products of the slow neutron-capture process. An extrinsic S-type star has similar surface pollution but no current internal source; it received the material from a former AGB companion. A barium star is the warmer main-sequence or giant counterpart, recognized particularly through strong barium and other slow-neutron-capture elements.

This interpretation predicts a white dwarf companion, the remnant of the former AGB donor. Barium and extrinsic S-type stars are indeed binaries, often with periods long enough that the donor could not have undergone ordinary Roche-lobe overflow. Their abundance patterns, white-dwarf companions, and wide or eccentric orbits therefore point to wind mass transfer in a binary star.

The Bondi–Hoyle accretion estimate treats the companion as moving through a locally uniform wind with relative speed $v_{\mathrm{rel}}$, sound speed $c_s$, and density ${\rho }_w$. Gravity focuses gas from the accretion radiusgivingFor a roughly spherical donor wind, ${\rho }_w\simeq {\dot{M}}_w/(4\pi a^2{v}_w)$ and $v_{\mathrm{rel}}$ combines wind and orbital velocities. This supplies an order-of-magnitude accreted fraction; wind acceleration, density gradients, orbital deflection, and Wind Roche-lobe overflow can substantially change it.

For initial masses $5M_{\odot }$ and $0.7M_{\odot }$, increasing the initial period gives three broad channels:

The transition periods are set by the primary's maximum radius relative to its Roche lobe, and their exact values depend on mass-transfer efficiency, wind speed, and common-envelope energy formalism.

In a circular binary star, the distances from the centre of mass are $a_1=a{M}_2/M$ and $a_2=a{M}_1/M$. Summing the two orbital angular momenta givesEquivalently, $J=\mu \sqrt{GMa}$ by Kepler third law.

First consider a rapid conservative perturbation. Both $M=M_1+M_2$ and $J$ are fixed, while $d{M}_1=-d{M}_2$. With $q=M_2/M_1$,soThe Roche lobe formula then gives its mass-radius exponentAfter mass loss, dynamical stability of binary mass transfer requires the donor to shrink relative to its lobe. Since $d\log M_2<0$, this means ${\zeta }_{\mathrm{ad}}>{\zeta }_L$, or

For stable secular conservative binary mass transfer, put $m={\dot{M}}_2/M_2$. The angular-momentum and contact conditions areElimination of $\dot{a}/a$ givesHere magnetic braking of a binary star removes orbital angular momentum, while the donor's expansion maintains Roche-lobe overflow.

In a cataclysmic variable, hydrogen-rich material accumulates on a degenerate white dwarf. Degeneracy prevents initial expansion from regulating its temperature, so nuclear ignition produces the thin-shell instability and a classical nova. Nuclear burning of hydrogen to helium releases about $0.007m{c}^2$, whereas the binding energy at a white-dwarf surface is only of order $G{M}_{1}m/R_1$, typically a few $10^{-4}m{c}^2$. Even modest coupling can therefore eject all the newly accreted envelope without disrupting the white dwarf.

Finally suppose every transferred mass element is expelled by isotropic re-emission from a binary star. Then ${\dot{M}}_1=0$, $\dot{M}={\dot{M}}_2$, and expelled matter carries the white dwarf's specific angular momentum $j_1=(M_2/M{)}^2{a}^2\Omega$. HenceOn the other hand, logarithmic differentiation of $J=M_1{M}_2{M}^{-1/2}{G}^{1/2}{a}^{1/2}$ and of the Roche-lobe radius givesEliminating the separation produces

The Roche lobe of star 1 is the volume around it bounded by the critical closed equipotential passing through the inner Lagrange point $L_1$. Material on a lower potential surface remains confined to star 1; at the critical surface a path opens toward star 2.

Let $\mathbf{g}=-\nabla \Psi$ and take $d\mathbf{S}$ outward. For a nearly spherical interior surface, the divergence theorem givesThe self-gravity term contributes $-4\pi G{M}_1$. The companion lies outside $S$, so its potential is harmonic inside and contributes zero net flux. For the centrifugal term, ${\nabla }^2(-{\Omega }^2{s}^2/2)=-2{\Omega }^2$, so its acceleration has divergence $2{\Omega }^2$ and contributes $2{\Omega }^{2}V$. Using $V=4\pi r^3/3$ and $S=4\pi r^2$,The sign is the outward-normal component; the dominant self-gravity is inward and therefore negative.

At the critical Roche surface, $g=0$ at $L_1$, making $⟨g^{-1}⟩$ singular. Once star 1 overfills its Roche lobe, the relevant equipotentials are no longer nested closed surfaces belonging to that star: they open through the $L_1$ saddle and connect to star 2. Matter then undergoes Roche-lobe overflow with finite velocity, so advective momentum and energy transport replace hydrostatic and purely radiative equilibrium near the nozzle. Density, temperature, and composition need not remain uniform on an equipotential. The one-dimensional volume coordinate $r^{\prime }(\Psi )$ and its correction factors therefore cease to describe the three-dimensional mass-transfer flow.

For a circular binary star, the relative orbit has speed $v=\sqrt{GM/a}$. Its orbital angular momentum is thereforeLogarithmic differentiation, using $\dot{\mu }/\mu ={\dot{M}}_1/M_1+{\dot{M}}_2/M_2-\dot{M}/M$, givesFor conservative binary mass transfer, $\dot{M}=0$, ${\dot{J}}_{\mathrm{orb}}=0$, and ${\dot{M}}_2=-{\dot{M}}_1$. HenceSince a donor has ${\dot{M}}_1<0$, the orbit widens for $M_1<M_2$ and shrinks for $M_1>M_2$.

Write the Roche lobe radius as $R_L=aF(q)$, whereThenConservative transfer has $d\ln q/d\ln M_1=1+q$, so

The donor response $R_1\propto M_1^{-n}$ has ${\zeta }_{\ast }=d\ln R_1/d\ln M_1=-n$, whereasAfter $d\ln M_1<0$, overflow decreases only if ${\zeta }_{\ast }>{\zeta }_L$. The dynamical stability of binary mass transfer criterion is thusFor $q>1$, both terms make $n_{\mathrm{crit}}<0$, which is incompatible with $n>0$. Transfer from the more massive donor is therefore dynamically unstable: mass loss shrinks its Roche lobe while the donor expands. The resulting runaway commonly produces a common envelope, followed by envelope ejection into a tighter binary or by a stellar merger.

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 $P\propto M_2^{(3\zeta -1)/2}$.

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.

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.