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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.

For Roche-lobe overflow, $R_2=R_L$. Cubing the Roche lobe approximation and using the donor's mass-radius relation givesKepler's third law therefore yields the Roche-lobe period-mass relation for a linear donor radius law:The component distances from the centre of mass are $a_1=a{M}_2/M$ and $a_2=a{M}_1/M$. Summing their angular momenta gives the circular-binary orbital angular momentumDuring conservative binary mass transfer, $\dot{M}=0$ and ${\dot{M}}_1=-{\dot{M}}_2$. Logarithmic differentiation at constant angular momentum gives $\dot{a}/a=-2(1-q){\dot{M}}_2/M_2$. Thus the Roche-lobe radius response exponent givesTheir difference is $d\log (R_L/R_2)/dt=(2q-8/3){\dot{M}}_2/M_2$. Since the donor loses mass, this is positive for $q<4/3$: the donor shrinks relative to its Roche lobe, so transfer shuts off without an external driver. The Roche lobe itself expands only for $q<5/6$; for $5/6<q<4/3$ it shrinks more slowly than the donor.

Gravitational-wave emission from a binary system provides such a driver: the orbit emits gravitational waves carrying energy and angular momentum, and the shrinking Roche lobe can maintain contact. Allowing $\dot{J}<0$, the binary mass-transfer contact equation becomesIt gives a negative transfer rate for $q<4/3$.

For the classical nova, assume isotropic re-emission from a binary star: ejecta carry the accretor's specific angular momentum $j_1=a_1^2\Omega$. The eruption lasts many orbital periods, so the orbit responds approximately adiabatically and remains nearly circular; an instantaneous asymmetric kick would be a different model. To first order,Differentiating the circular-binary orbital angular momentum formula during the ejection givesThe accompanying Roche lobe change is thereforeThe donor does not change mass in the eruption and moves inside its expanded Roche lobe: this is nova-induced binary detachment.

Put $\Gamma =-\dot{J}>0$ for the constant external loss. During detachment the component masses are fixed, so the Roche lobe shrinks at fractional rate $2\Gamma /J$. Closing the fractional gap $4\delta m/(3M)$ takes $t_d=2J\delta m/(3M\Gamma )$. During the subsequent semidetached binary phase, the binary mass-transfer contact equation gives $|{\dot{M}}_2|=3M_2\Gamma /[J(4-3q)]$, so accumulating the next eruption's mass takes $t_s=J(4-3q)\delta m/(3M_2\Gamma )$. ConsequentlyBoth times are evaluated to first order in $\delta m/M$, with the same equilibrium donor mass-radius relation; changes of $q,J$ and the transfer rate within one cycle affect higher orders.

A cataclysmic variable is a close semidetached binary in which a white dwarf accretes from a cool, usually low-mass donor star filling its Roche lobe. In an ordinary hydrogen-rich system the donor star is often near the lower main sequence. In a nonmagnetic cataclysmic variable (a disk-fed CV), the white dwarf's magnetic field is too weak to control the flow over the disk. This leaves a characteristic disk-fed geometry, summarized in the original schematic below. NASA's introduction to cataclysmic variables describes the basic components.

Roche-lobe overflow passes through the inner Lagrange point $L_1$. A nearly ballistic stream bends in the rotating frame and strikes the outer accretion disk, producing an accretion-disk stream-impact hot spot. Its retained angular momentum prevents direct radial infall. Viscous evolution of an accretion disk transports angular momentum outward while gas moves inward through a nearly Keplerian accretion disk. Close to the white dwarf, gas slows from orbital rotation toward the stellar rotation in an accretion-disk boundary layer. The disk, accretion-disk stream-impact hot spot, accretion-disk boundary layer, white dwarf and donor star all contribute to the spectrum and, where the orbital inclination permits, to the eclipses of an eclipsing binary. In particular, “nonmagnetic” describes the accretor's control of the flow; it does not imply that the donor star cannot sustain a magnetic field.

The gravitational power available from accretion is approximately $L_{\mathrm{acc}}=G{M}_{\mathrm{WD}}{\dot{M}}_{\mathrm{acc}}/R_{\mathrm{WD}}$. For a slowly rotating white dwarf and a thin steady Keplerian accretion disk, the specific energy changes from approximately zero far out to $-G{M}_{\mathrm{WD}}/(2R_{\mathrm{WD}})$ at the inner disk. Roughly half the available power is radiated by the accretion disk; the rest is released in the accretion-disk boundary layer as the orbital kinetic energy is dissipated. Stellar rotation and departures from a steady thin disk change this partition.

A classical nova has a different energy source. Transferred hydrogen accumulates on the white dwarf; compression heats the base of its envelope until temperature-sensitive hydrogen burning accelerates. Electron degeneracy pressure initially weakens the expansion response to heating, helping a thermonuclear runaway develop. The envelope subsequently expands and ejects material, producing a large optical outburst followed by a decline as the ejecta expand and residual burning ends. The white dwarf usually survives, so continued accretion can build another fuel layer. A rough recurrence scale is the ignition-envelope mass divided by the mean accretion rate; both this mass and the rate vary strongly among systems. The event is an envelope eruption, and the retained fraction is not automatically unity. Starrfield, Iliadis and Hix's nova calculations explains this nuclear mechanism. A classical nova is powered by unstable nuclear burning on the white dwarf.

A dwarf nova undergoes recurrent, shorter brightenings powered principally by enhanced gravitational accretion. The hydrogen-ionization disk instability creates cold, mostly neutral and hot, ionized branches of the disk's accretion-disk thermal S-curve, separated by unstable equilibria. In quiescence the cool disk stores matter because inward transport is slow. Once a critical surface density is reached, a heating transition puts the disk into a hotter, more state with higher effective viscosity: the inward accretion rate and luminosity rise and the disk drains. A cooling transition returns it to quiescence, completing the cycle. A persistent increase in the donor star's transfer rate is not required. Sufficiently high transfer rates can keep the disk on its hot stable branch, giving a nova-like variable rather than ordinary disk cycles. Lasota's disk-instability analysis and Hameury's disk-instability review develop this picture. A dwarf-nova outburst is a disk instability, not a white-dwarf thermonuclear explosion. The names classify mechanisms and need not identify permanently distinct binaries: a nova-producing binary can also possess an unstable disk between nuclear eruptions.

The cataclysmic-variable orbital-period distribution is not smooth. For ordinary hydrogen-rich cataclysmic variables, prominent features are a cataclysmic-variable period gap around two to three hours, a cataclysmic-variable period minimum near eighty minutes, and an accumulation near that minimum. These are population features rather than absolute exclusions. Selection effects matter: luminous high-accretion rate systems are easier to find than faint evolved systems. Helium-transferring binaries have a different period range and are not described by the hydrogen-rich minimum. Gänsicke and collaborators' period-minimum study documents the observed accumulation.

The Roche-lobe-filling period-density relation makes the orbital period a measure of donor structure. Combining $R_d\simeq 0.46a(M_d/M{)}^{1/3}$ with Kepler's third law givesAs the donor star loses mass, its stellar radius response exponent $\zeta =d\log R_d/d\log M_d$ impliesA donor with $\zeta >1/3$ evolves toward shorter orbital periods. When its effective response falls below $1/3$, continued mass loss instead lengthens the orbital period: this is the cataclysmic-variable period bounce. A very low-mass donor may be substellar and increasingly affected by electron degeneracy pressure; the ideal degenerate scaling $R_d\propto M_d^{-1/3}$ illustrates the reversal. The precise cataclysmic-variable period minimum depends on thermal disequilibrium and the strength of orbital angular momentum loss. Near a turning point $|\dot{P}|$ is small; for an approximately steady evolutionary flow of systems, the number per period interval scales as $N(P)\propto 1/|\dot{P}|$, explaining the accumulation. Knigge, Baraffe and Patterson's donor-based evolutionary study relates the donor star's response to these period features.

Long-term transfer is driven mainly by losses of orbital angular momentum, rather than by disk outbursts. Magnetic braking of a binary star removes the cool donor star's spin through a magnetized stellar wind. Tidal synchronization makes the orbit replenish that spin, so the wind extracts orbital angular momentum. This is usually the dominant standard driving mechanism above the cataclysmic-variable period gap. Gravitational-wave emission from a binary system supplies a baseline loss, especially important below the gap. For a weak-field, slowly moving circular orbit, the circular gravitational-wave inspiral givesBecause ordinary transfer from the lighter donor star tends to expand its Roche lobe if orbital angular momentum is conserved, an external loss is needed to sustain contact. In the conservative contact approximation, the binary mass-transfer contact equation readswhere the stellar radius response exponent must match the evolutionary timescale and additional donor expansion has been neglected. The positive denominator on the stable branch makes ${\dot{J}}_{\mathrm{ext}}<0$ drive ${\dot{M}}_d<0$. Nova ejecta can introduce additional nonconservative binary mass transfer. Knigge's evolutionary discussion describes the standard loss mechanisms and their limitations.

In the disrupted magnetic braking model, relatively rapid mass loss above the cataclysmic-variable period gap keeps the donor star inflated relative to stellar thermal equilibrium. When the donor approaches the fully convective star transition, the model postulates a substantial reduction in magnetic braking of a binary star. The donor star can contract within its Roche lobe, suppressing Roche-lobe overflow near the upper edge of the gap. Gravitational-wave emission from a binary system continues to shrink the detached binary; near the lower edge the Roche lobe again reaches the donor radius and transfer resumes. No mass transfer is needed during the detached crossing. The density relation predicts a radius ratio $(3/2{)}^{2/3}\simeq 1.31$ between contact at three and two hours if the masses remain nearly fixed, illustrating the required inflation before detachment. The torque reduction is a model ingredient, not a claim that all fully convective stars lose their magnetic fields. Zorotovic and collaborators' detached-binary study tests the predicted detached population in the gap.

The standard formation channel starts with an initially wider binary star containing two main sequence stars. The initially more massive component evolves first and becomes a giant. Unstable Roche-lobe overflow can engulf the companion in a common envelope. Drag causes inward orbital motion, releasing two-body orbital energy and transferring angular momentum to the envelope. If the envelope is expelled before merger, a close detached binary survives, containing the exposed core, which becomes a white dwarf, and the lower-mass companion. This explains how a binary becomes much tighter than the giant progenitor's radius would have allowed. The outcome depends on envelope binding and the efficiency of energy deposition, summarized approximately by the common-envelope energy formalism; ejection is not guaranteed. Ivanova and collaborators' common-envelope analysis discusses the relevant physics and uncertainties.

Subsequent magnetic braking of a binary star and gravitational-wave emission from a binary system shrink the detached binary until its companion fills its Roche lobe. Stable Roche-lobe overflow, for a suitable binary mass ratio and stellar radius response exponent, then creates a cataclysmic variable. Its later nuclear eruptions, disk cycles and secular orbital period evolution occur on different timescales. The formation sequence is a wide binary, envelope ejection, a close detached white-dwarf binary, and angular-momentum-driven contact.

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.

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.