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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.

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.

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.

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

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.

The Lagrange point lying between the components of a circular restricted three-body problem. The critical effective potential through this saddle defines their Roche lobes; gas passing through it can initiate Roche-lobe overflow.

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.

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.

For a circular binary with negligible spin, the two orbital radii about the center of mass are $a_1=a{M}_2/M$ and $a_2=a{M}_1/M$. Summing their orbital angular momenta givesUsing Kepler's third law, ${\Omega }^2=GM/a^3$, also gives $J=M_1{M}_2\sqrt{Ga/M}$. Under conservative binary mass transfer, $M$ and $J$ stay fixed, so $a\propto (M_1{M}_2{)}^{-2}$. Since $d\log M_2/d\log M_1=-q$, the Roche-lobe radius response exponent isFor the supplied giant response, luminosity stays fixed during the mass-loss perturbation and ${\zeta }_{\ast }=d\log R/d\log M_1=-0.27$. The change of the logarithmic overfill is $({\zeta }_{\ast }-{\zeta }_L)d\log M_1$. Because the donor loses mass, this reduces overfill and stabilizes transfer precisely when ${\zeta }_{\ast }>{\zeta }_L$. Thus the stated model requiresThis is a response criterion within the given radius law; a different adiabatic donor response would change the dynamical threshold.

The initially more massive star evolves first. If it waits until it has the supplied giant structure while still having $q>1$, conservative overflow is unstable by this criterion. If Roche-lobe overflow instead begins while the donor is on the main sequence or shortly after central hydrogen exhaustion, its largely radiative-envelope response can allow transfer before a deep convective giant envelope develops. Mass loss and accretion can then reverse the mass ratio before it reaches the giant state. This is the binary mass-ratio reversal route to an Algol binary, through Case A mass transfer or early case B mass transfer.

At lobe filling, the lobe approximation and Kepler's law give the pre-giant Roche-lobe-filling period limitIt increases with the donor's radius. Requiring $R_L<R_{\mathrm{BGB}}=1.68R_{\odot }(M_1/M_{\odot }{)}^{5/3}$ therefore yieldsThe companion mass cancels in this lobe approximation.

Since $P\propto a^{3/2}$ at fixed total mass, the separation scaling gives $\boxed{P(M_1{M}_2{)}^3=\text{constant}}$. Use solar-unit masses $m_i=M_i/M_{\odot }$ and $m=m_1+m_2$, and let $K=P(m_1{m}_2{)}^3$. ThenThe derivative vanishes at $m_1=5m/8$, and its derivative is $5/m_1^2+3/(m-m_1{)}^2>0$, while the ratio diverges at both endpoints. Hence this is the global minimum, the conservative pre-giant period-ratio minimum.

For the stated Algol parameters, $m=4$ and $K=3(1\cdot 3{)}^3=81$ days. At the minimum $m_1=2.5,m_2=1.5$, the period is $81/(2.5\cdot 1.5{)}^3=1.536$ days and $P/m_1^2=0.24576$ days, well below $P_0=0.8$ days. This is a possible initially more-massive donor configuration with overflow before the giant branch. The current $q=1/3$ is also below the giant stability threshold. Thus Algol is compatible with the proposed conservative early-overflow route, although this necessary geometric test alone does not prove its full evolutionary history.

For RT Lac, $m=2.4$ and $K=5.1(0.9\cdot 1.5{)}^3$. Its minimum occurs at $m_1=1.5,m_2=0.9$, where the mass product equals its current value, so $P=5.1$ days andNo conservative antecedent on this orbit can begin overflow before the giant branch. Beginning with an initially more massive giant instead encounters the instability just derived. RT Lac cannot follow the stated coeval, conservative Algol-formation model. A plausible alternative is rapid initial transfer with substantial mass and angular-momentum loss, via an outflow, strong wind or envelope-ejection episode. Such nonconservative binary mass transfer changes both total mass and $J$ and removes the period-product constraint. Its present mass ratio $0.9/1.5=0.6$ permits stable current giant transfer. The given data do not determine a unique mass-loss history or establish a specific ejection mechanism.

The elements have several complementary origins: primordial light nuclei, stellar and explosive fusion products, and nuclei built by neutron capture. Chemical enrichment also requires their ejection, rather than merely their production inside a star. Nucleosynthesis and transport into the interstellar medium must therefore be considered together.

Hydrogen and much of the helium are primordial. During Big Bang nucleosynthesis, a cooling, expanding plasma assembled surviving neutrons and protons into deuterium, helium-3 and especially helium-4, with small abundances of lithium-7 or its beryllium-7 progenitor. Unstable nuclei at mass numbers five and eight, together with the rapidly falling density and short available time, obstructed sustained production of heavier nuclei. The carbon, oxygen and heavier material in later generations consequently require other sites. Cosmic rays fragment heavier interstellar nuclei and contribute importantly to the light elements lithium, beryllium and boron, which ordinary stellar burning tends to destroy.

Burning stages depend on initial stellar mass. Stellar nuclear fusion can release energy when the products have greater nuclear binding energy than the reactants. The resulting stellar thermostat links nuclear burning to hydrostatic equilibrium: when fuel exhaustion lowers heating, contraction raises the central temperature until another fuel can ignite, if the core becomes hot enough. Higher nuclear charges bring larger Coulomb barriers, so later charged-particle burning generally requires progressively higher temperatures. Whether the core reaches them depends on its mass, electron degeneracy pressure, mixing and envelope loss.

An approximately solar-composition $1M_{\odot }$ star first spends most of its life on the main sequence, converting hydrogen into helium mainly through the proton–proton chain. The net conversion is four protons into one helium-4 nucleus, with positrons and neutrinos; the small CNO cycle contribution redistributes CNO catalyst abundances but is not a net source of carbon at this stage. Exhaustion of central hydrogen leaves a helium-rich core. A surrounding hydrogen-burning shell continues to add helium, while the core contracts and the envelope expands into a red giant.

The low-mass helium core becomes supported substantially by electron degeneracy pressure. At roughly $10^8\mathrm{K}$, the Triple-alpha process ignites, first making transient beryllium-8 and then carbon-12 through the Hoyle state. Because degeneracy initially weakens the usual expansion response, this ignition is a helium flash. It heats and lifts the degeneracy of the core, leading to a stable phase of core helium burning. The Triple-alpha process makes carbon; competing carbon-12 alpha capture makes oxygen:After central helium exhaustion, the star develops a carbon-oxygen core, a helium-burning shell and a more external hydrogen-burning shell. On the Asymptotic giant branch, recurrent thermal pulses of an asymptotic-giant-branch star arise from unstable helium-shell burning. Stellar dredge-up can bring processed material toward the surface, with its efficiency depending on mass and composition. The eventual stellar wind removes the envelope; the hot exposed core can illuminate a planetary nebula, and the remnant becomes a carbon-oxygen white dwarf. A usual isolated $1M_{\odot }$ star does not ignite carbon burning. In particular, an element synthesized in its core need not all escape: much of the carbon and oxygen remains locked in the white dwarf.

An approximately $5M_{\odot }$ star burns central hydrogen predominantly through the CNO cycle, usually in a convective core. The CNO nuclei act as catalysts; the slow nitrogen-14 proton-capture step tends to accumulate nitrogen-14 while converting hydrogen into helium. After central fuel exhaustion, a hydrogen-burning shell surrounds the contracting helium core. Unlike the $1M_{\odot }$ case, core helium burning normally begins before strong electron degeneracy, without a classical helium flash. The same Triple-alpha process and carbon-12 alpha capture create a carbon-oxygen core.

The $5M_{\odot }$ star then enters the Asymptotic giant branch, with hydrogen and helium burning in shells. Repeated thermal pulses and stellar dredge-up can enrich its envelope with freshly made carbon and s-process nuclei. Hot-bottom burning can occur when the base of the convective envelope is sufficiently hot: the CNO cycle processes dredged-up carbon into nitrogen, so a massive AGB star need not become carbon-rich despite making carbon below its envelope. Helium-shell reactions on neon-22 can also provide neutrons for the s-process. For an ordinary $5M_{\odot }$ model, strong envelope loss generally precedes carbon ignition, leaving a carbon-oxygen white dwarf. The exact threshold for carbon ignition and the size of these yields depend on metallicity, mass loss and mixing; they are not universal integers fixed solely by initial mass.

A $20M_{\odot }$ star reaches much more advanced stages. It starts with convective-core CNO cycle hydrogen burning, followed by core helium burning. After central helium exhaustion, contraction can ignite carbon burning at temperatures approaching $10^9\mathrm{K}$. Carbon-fusion channels make products including neon, sodium and magnesium. Neon burning then uses photodisintegration of neon-20 to release oxygen-16 and an alpha particle, followed by capture of that alpha particle on another neon-20 nucleus; the combined network releases energy and produces magnesium-24. Oxygen burning makes silicon-group nuclei through channels such asAt still higher temperature, silicon burning rearranges nuclei by photodisintegration and captures of protons, neutrons and alpha particles. It is not principally direct fusion of two silicon nuclei. Networks approach quasi-equilibrium and then nuclear statistical equilibrium, producing an iron-group core whose precise mixture depends on the electron fraction in stellar matter. Outer shells retain different stages of burning, giving an approximately layered structure. Increasing stellar neutrino energy loss makes advanced burning stages much shorter than hydrogen burning.

Near the iron group, nuclear binding energy per nucleon is close to its maximum. Further fusion cannot provide the sustained energy source required to support the core. As the core grows, electron capture reduces electron-pressure support and photodisintegration absorbs energy, assisting gravitational collapse. A successful core-collapse supernova ejects some of the envelope and processed shells, and leaves a neutron star or black hole. The shock also causes explosive burning, changing the pre-existing abundances. A $20M_{\odot }$ initial mass does not uniquely fix the explosion or remnant: envelope loss, rotation, composition and fallback all matter, and some collapses fail to eject much material.

Ejection determines the enrichment of the interstellar medium. Low- and intermediate-mass stars return helium, CNO-processed matter, dredged-up carbon and s-process products through stellar winds and envelope expulsion. Massive-star winds can expose hydrogen- or helium-burning products before collapse. Successful core-collapse supernovae eject substantial oxygen and other alpha-capture products, together with some iron-group matter formed in hydrostatic and explosive burning. Material that falls back or remains in a stellar remnant is withheld from the interstellar medium. Subsequent mixing, cooling and star formation distribute the escaped material into new stars. The distinction between production and ejection is why an internal burning inventory alone is not a stellar yield.

Binary stars provide an additional major iron-group source. In a Type Ia supernova, a carbon-oxygen white dwarf in a binary undergoes thermonuclear disruption after an appropriate accretion or merger history. Explosive carbon and oxygen burning makes intermediate-mass and iron-group nuclei. Much initially synthesized nickel-56 decays through cobalt-56 into iron-56; this radioactive decay also powers the supernova light curve. Such explosions have delay times and yields different from those of massive-star collapse, and help explain why iron enrichment need not track oxygen enrichment exactly.

Neutron capture builds much of the material beyond iron. Charged-particle reactions encounter increasing Coulomb barriers as nuclear charge grows, but a neutral neutron does not encounter that barrier. Neutron capture raises mass number, and subsequent beta-minus decay can turn a neutron into a proton and raise atomic number. The relative capture and decay times select different paths through the nuclear chart.

In the slow neutron-capture process, most unstable intermediates decay before the next capture, so the path stays near stable nuclei. In Asymptotic giant branch stars, important neutron sources areThe carbon-13 source operates in suitable partially mixed layers, while the hotter helium-shell pulses can activate the neon-22 source. Captures on pre-existing seed nuclei produce elements such as strontium and barium and, under suitable conditions, lead. Stellar dredge-up and stellar winds return these products to the interstellar medium. Massive stars also have a weak s-process, notably using neon-22 during helium- and carbon-burning phases; it need not produce the same abundance range as the main AGB contribution.

In the rapid neutron-capture process, very high neutron densities permit many captures before typical beta-minus decays. The path moves to neutron-rich nuclei, which later decay toward stability, producing heavy nuclei including lanthanides and actinides. Neutron-rich ejecta from neutron star mergers provide an established site: a merger can eject dynamical material and disk outflows, and radioactive decay of the newly made nuclei powers a kilonova. Special neutron-rich outflows associated with some massive-star collapses are additional candidate sites. It would be unjustified to assume that every ordinary core-collapse supernova produces the full heavy r-process pattern; the required neutron richness and expansion history are restrictive.

The two neutron-capture processes do not make every isotope beyond iron. Some proton-rich isotopes bypass their main paths. Gamma-process nucleosynthesis, for example, uses photodisintegration of pre-existing heavy seed nuclei in hot explosive material. Other proton-rich reaction paths can matter in suitable environments. Thus “heavier than iron” describes a family of production mechanisms rather than a single further ordinary fusion stage.

Binaries change both stellar evolution and the delivery of elements. Roche-lobe overflow, wind transfer and mergers can strip a donor, rejuvenate a mass-gaining star, alter core and envelope masses, and change the timing and character of an eventual explosion. Binary stripping can remove the hydrogen envelope of a massive core-collapse supernova progenitor. Accretion onto a white dwarf can cause a classical nova, ejecting hydrogen-burning products and particular isotopes while ordinarily leaving the white dwarf intact; this differs from the destructive Type Ia supernova. Compact binary stars can lose orbital energy through gravitational radiation and merge, providing the neutron star merger route to heavy r-process material. Consequently chemical enrichment cannot be inferred entirely from isolated-star tracks, even when a simplified population calculation treats binary components as evolving independently.

An X-ray binary contains a compact accretor supplied by a donor star. Most bright systems contain a neutron star or black hole. Gas arrives by Roche-lobe overflow, a captured stellar wind, or donor outflows, commonly forming an accretion disk. The primary energy source is the release of gravitational binding energy during accretion, with a characteristic accretion luminosityFor a neutron star, $R_{\mathrm{in}}$ can be near its material radius; for a black hole, radiation is produced by the flow outside the horizon. For $M\sim 1.4M_{\odot }$ and $R\sim 10\mathrm{km}$, $GM/(R{c}^2)\sim 0.2$, so accretion can release much more energy per unit mass than hydrogen burning. The hot inner flow, boundary layer or accretion column emits X-rays.

In a high-mass X-ray binary, a massive, relatively short-lived donor often supplies gas through a strong wind or an equatorial outflow; some systems instead undergo Roche-lobe overflow. These objects trace young stellar populations. In a low-mass X-ray binary, a lower-mass donor commonly fills its Roche lobe and feeds a disk. Such systems can persist in older populations. These mass labels refer to the donor, not the compact accretor.

A magnetized neutron star can channel accretion onto its poles, producing an accreting X-ray pulsar as the star rotates. Accretion transfers angular momentum, so changes in spin constrain the flow and magnetic coupling. A Type I X-ray burst has a different origin: accumulated hydrogen or helium undergoes unstable thermonuclear burning on a neutron star surface. It rises rapidly and then cools as the fuel is exhausted. A black hole has no material surface supporting the same surface-fuel mechanism.

Orbital motion and optical observations of the donor constrain masses, with inclination and donor-model uncertainties. A compact mass exceeding plausible stable neutron star masses supports a black hole interpretation. Rapid variability probes the small emitting region, while disk spectra and recurrent outbursts probe accretion and its instabilities. The Eddington luminosity,is obtained by balancing radiative acceleration $\kappa L/(4\pi r^{2}c)$ against $GM/r^2$; it is a useful luminosity scale under the assumed isotropic, opacity-controlled force law. Geometry and time dependence can complicate its application. X-ray binaries reveal compact stars through efficient gravitational accretion, with pulsations, bursts and orbital dynamics distinguishing their physical components.

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.

Let $M=M_1+M_2$. The centre of mass condition gives $a_1=a{M}_2/M$, $a_2=a{M}_1/M$. For circular motion the component angular momenta are $J_i=M_i{a}_i^2\Omega$, soUsing Kepler's third law, $a^3{\Omega }^2=GM$, and $\Omega =2\pi /P_{\mathrm{orb}}$ givesThis is the circular-binary orbital angular momentum. Mass received by the companion redistributes angular momentum within the binary star; only the escaping stellar wind removes it under the prescribed model.

The stellar wind from the donor star has specific angular momentumHere ${\dot{M}}_1<0$, ${\dot{M}}_2=-f{\dot{M}}_1$ and $\dot{M}=(1-f){\dot{M}}_1$. Thus donor-wind angular-momentum loss givesTaking the logarithmic derivative of the period expression for $J$ givesUse $M_2/(M_{1}M)=1/M_1-1/M$ to simplify this toFor constant $f$, integration therefore givesThis is the donor-wind period invariant with fixed retention fraction. If $f$ changes during the evolution, the differential equation remains valid but the integrated invariant is instead $P_{\mathrm{orb}}{M}_2^3{M}^2\exp (3\int fd\log M_1)=\mathrm{constant}$. The fixed-power expression cannot be used with a time-dependent exponent without this modification.

For the Roche lobe, write $q=M_1/M_2$. The equivalent separation expression $J=M_1{M}_2\sqrt{Ga/M}$ givesTaking the logarithmic derivative of $R_L=0.46a(M_1/M{)}^{1/3}$ then adds ${\dot{M}}_1/(3M_1)-\dot{M}/(3M)$, yieldingThis donor-wind Roche-lobe response uses only the instantaneous value of $f$ and thus still holds if it varies.

Define $A(q)=2q+4q/[3(1+q)]-2$ and $B(q)=1/3-4q/[3(1+q)]$. The Roche-lobe radius response exponent is ${\zeta }_L=B+Af$, whereas the donor star's stellar radius response exponent is ${\zeta }_{\ast }=-n$. Keeping $R_1=R_L$ requires the two logarithmic derivatives of radius to agree, soFor $A\ne 0$, this determinesA physical mixture of Roche-lobe overflow and stellar wind from the donor star needs $0<f<1$. Equivalently, $-n$ must lie strictly between the pure-wind response $B$ and the conservative response $B+A=2q-5/3$. The values $f=0$ and $1$ are valid limiting prescriptions, respectively pure stellar wind and conservative mass transfer, rather than mixtures. If $f_{\mathrm{req}}$ lies outside $[0,1]$, no allowed division of the lost mass can maintain contact under this radius law and angular-momentum-loss prescription. This is feasibility of donor-wind binary contact.

The likely direction away from contact follows without assigning a nonphysical fraction. For any actual $f$, let $\Delta =\log (R_1/R_L)$. ThenSince ${\dot{M}}_1<0$, a positive value of $A(f_{\mathrm{req}}-f)$ makes the donor star move inside its lobe and detach; a negative value increases overfill and promotes stronger transfer. Thus if $A>0$, $f_{\mathrm{req}}<0$ gives increasing overfill and $f_{\mathrm{req}}>1$ gives detachment; if $A<0$, these outcomes are reversed. A runaway or a new equilibrium would require the donor star's response or the angular-momentum-loss model to change, possibly with additional driving. The contact calculation alone does not prove a particular nonlinear endpoint.

There is also a genuine exceptional denominator: $A=0$ at $q=(\sqrt{10}-1)/3$. At this ratio both limiting lobe responses coincide. Contact is possible only if $-n=B$, namely $n=(\sqrt{10}-2)/(\sqrt{10}+2)$; then every $f$ gives the same instantaneous radius response. Otherwise no fraction works, and the sign of $(-n-B){\dot{M}}_1/M_1$ determines detachment or increasing overfill.

Treat the stars as point masses for the orbital motion, neglecting their spin angular momenta. Let $M=M_1+M_2$. Their distances from the centre of mass are $a_1=a{M}_2/M$ and $a_2=a{M}_1/M$. The sum of their orbital angular momenta isThus the circular-binary orbital angular momentum is

For conservative binary mass transfer, both $M$ and $J$ are fixed. In terms of the binary mass ratio $q=M_1/M_2$, $M_1=Mq/(1+q)$ and $M_2=M/(1+q)$, henceMass conservation also gives $d{M}_2=-d{M}_1$, soDifferentiating $J\propto M_1{M}_2{a}^{1/2}$ at fixed $M,J$ gives $d\log a/d\log M_1=2(q-1)$. Therefore the stipulated Roche lobe law has the Roche-lobe radius response exponent

Define the overfill by $\Delta =\log (R_1/R_L)$. On a time scale short compared with the slow change of $K$, the donor's response is $d\log R_1=\beta d\log M_1$. ThusDuring mass loss $d\log M_1<0$. If ${\zeta }_L>\beta$, mass loss increases the overfill and therefore drives more mass loss: this is positive feedback, giving more rapid Roche-lobe overflow. If ${\zeta }_L<\beta$, mass loss decreases the overfill and can balance the slow expansion due to stellar evolution. More explicitly, with $k_{\mathrm{ev}}=d\log K/dt>0$, contact requiresA slow negative transfer rate can satisfy this only on the stable side $\beta >{\zeta }_L$. The rapid-transfer condition is thereforeIts time scale depends on the physical donor response represented by $\beta$; it need not be a dynamical time scale if $\beta$ is a thermal-equilibrium mass-radius exponent.

For Roche-lobe overflow feedback with a constant wind torque factor, retain mass conservation to the stated approximation but allow the prescribed angular-momentum loss. Since $d\log J=fd\log M_1$, differentiation givesThe new Roche-lobe radius response exponent isThe same overfill calculation givesA loss of angular momentum has $f>0$ because ${\dot{M}}_1<0$, and it lowers the critical ratio by making the orbit contract more strongly.

For the specified solar-mass donor exponent $\beta =1/2$, the answer isWithout the wind torque this becomes $q>22/23\simeq 0.957$, so donors slightly less massive than their companions can already lie on the rapid-transfer side of this model. For $0<f<11/10$ the lower threshold is $(22-20f)/23$, and for $f\ge 11/10$ every positive $q$ satisfies the strict inequality. There is no finite upper bound on $q$ from this criterion. Since the paper does not specify $f$, the wind case has a conditional range rather than a unique numerical cutoff.

To obtain the specified expansion, use the usual nova-ejection prescription: the escaping material is expelled isotropically from the white dwarf and carries its specific orbital angular momentum, with no additional torque. This is isotropic re-emission from a binary star. Circularity alone does not determine the angular momentum carried by the ejecta; the prescription is an additional physical assumption.

The accretor's orbital radius is $a_1=a{M}_2/M$, so its specific angular momentum satisfiesDuring ejection, take $\delta M_1=-\delta m$, $\delta M_2=0$ and $\delta M=-\delta m$, where $\delta m>0$. ConsequentlyDifferentiate $J=M_1{M}_2\sqrt{Ga/M}$ to first order:Using $M/M_1=1+q$ yieldsThe same lobe law now has changing total mass but unchanged donor mass, soThe donor radius is unchanged on the short eruption timescale, whereas its lobe expands. It therefore falls inside the lobe and Roche-lobe overflow stops in this idealized model. This is nova-induced binary detachment.

The need for the ejection prescription can also be seen algebraically. If the ejecta instead have mean specific angular momentum $j_{\mathrm{ej}}={\eta }_{J}J/M$, the same circular-orbit differentiation givesThe requested coefficient is obtained only when ${\eta }_J=q$. Other prescriptions can produce a different expansion or a contraction; the stated circularity condition does not rule them out. The ejection may be short compared with the transfer time while remaining slow compared with an orbit, consistent with circularity. If truly impulsive, an eccentricity is generally excited unless an additional circularization assumption is made.

Now put $\Gamma =-\dot{J}>0$ for the constant external torque. Keep the masses, mass ratio and torque coefficients fixed to leading order over one small-$\delta m$ cycle; corrections in those coefficients affect the calculated times only at the next order. During detachment there is no transfer, so fixed component masses giveThe fractional lobe gap $4\delta m/(3M)$ closes in timeDuring the semi-detached accumulation phase, use the same external torque and the equilibrium donor contact equation derived in (a):Accumulating the next layer therefore takesDivision cancels the layer mass, torque and angular momentum:The result uses $q<4/3$ for a positive steady accumulation rate, unchanged donor structure during the eruption and the same external torque in the two phases. Here the given $q<1$ ensures a positive denominator. It describes an idealized nova cycle; extra frictional loss, irradiation-driven donor changes or a different ejection angular momentum would change the ratio.

Stellar nucleosynthesis and the subsequent return of matter explain most elements heavier than the primordial hydrogen and helium. Big Bang nucleosynthesis supplied predominantly those two elements, with small light-isotope abundances; stars do not make every element from an initially metal-free mixture. Different stellar masses reach different central temperatures and leave different remnants. The mass comparisons below describe representative isolated, near-solar-composition stars; mass loss and binary interaction can alter the final core. A nucleus contributes to later generations only if it is eventually expelled into the interstellar medium, rather than remaining locked in a remnant.

A $1M_{\odot }$ star spends its main sequence life converting hydrogen to helium, predominantly through the proton–proton chain. After core hydrogen burning ends, a hydrogen-burning shell supplies an expanding red giant envelope and builds an electron-degenerate helium core. First dredge-up exposes material whose CNO cycle has redistributed the original carbon and nitrogen abundances. The helium flash starts core helium burning: the Triple-alpha process makes carbon, and carbon-12 alpha capture makes oxygen. Once core helium is exhausted, the Asymptotic giant branch has a carbon-oxygen core with helium-burning shell and hydrogen-burning shell. Shell instability causes AGB thermal pulses; subsequent third dredge-up, where efficient enough, can bring carbon and neutron-capture products into the envelope. Its efficiency at such low mass depends on composition and mixing. The star does not ordinarily ignite core carbon burning. Stellar winds remove the envelope, a hot remnant may illuminate a planetary nebula, and a carbon-oxygen white dwarf retains much of the core's synthesized matter.

A $5M_{\odot }$ star generally uses the more temperature-sensitive CNO cycle for main-sequence hydrogen burning, with a substantial convective core. Its hotter helium core ignites core helium burning without the low-mass degeneracy-driven helium flash, producing carbon and oxygen. After core helium exhaustion it develops double-shell Asymptotic giant branch burning and a degenerate carbon-oxygen core. Second dredge-up and repeated third dredge-up alter the envelope composition. In sufficiently hot models, hot-bottom burning at the base of the convective envelope converts dredged-up carbon into nitrogen through the CNO cycle; consequently a carbon-producing shell need not produce a carbon-rich final envelope. Hot helium-shell pulses can activate the ${}^{22}\mathrm{Ne}(\alpha ,n){}^{25}\mathrm{Mg}$ neutron source. Strong stellar winds expel these processed layers. A conventional isolated five-solar-mass model ends as a carbon-oxygen white dwarf, below the mass range that normally develops sustained core carbon burning; mass loss and mixing govern the exact boundaries between outcomes.

A $32M_{\odot }$ star has core CNO cycle hydrogen burning, followed by core helium burning. The core then contracts through carbon burning, neon burning, oxygen burning and silicon burning, with active shells outside the successive exhausted cores. Carbon burning produces chiefly nuclei such as neon and magnesium. Neon burning combines photodisintegration and alpha capture; oxygen burning makes silicon-group nuclei. Silicon burning is a network of captures and photodisintegration approaching nuclear statistical equilibrium, rather than simply fusion of pairs of silicon nuclei. It produces an iron-group mixture governed by temperature, density and weak reactions. The maximum in nuclear binding energy per nucleon near the iron group prevents continued exothermic fusion from supplying ordinary hydrostatic support. Stellar neutrino energy loss accelerates the advanced stages; electron capture and photodisintegration help remove support during iron-core collapse.

The massive star can return products both before and during collapse. Radiation-driven stellar winds may expose and expel layers enriched in helium, nitrogen, carbon and oxygen. A successful core-collapse supernova ejects additional hydrostatic products and creates further nuclei by shock-driven explosive burning, including alpha-rich and iron-group products. The nickel-56 decay chain converts radioactive nickel into iron through cobalt. Initial mass alone does not fix the explosion or remnant: substantial fallback or a failed supernova can leave a black hole and retain much of the core material. Thus the possible core-collapse supernova contribution is not a guarantee that a thirty-two-solar-mass star ejects every element it made.

For nuclei well beyond the iron group, neutron capture is a principal assembly mechanism, avoiding the large Coulomb barrier for charged-particle fusion. Beta-minus decay converts captured neutrons into protons and raises the atomic number. In the s-process, captures are slower than relevant beta decays, so the sequence usually follows nuclei near stability. The main site is the Asymptotic giant branch: ${}^{13}\mathrm{C}(\alpha ,n){}^{16}\mathrm{O}$ is an important neutron source in low-mass intershell material, while ${}^{22}\mathrm{Ne}(\alpha ,n){}^{25}\mathrm{Mg}$ becomes important in hotter pulses and in massive-star helium/carbon burning. Third dredge-up and stellar winds make these products available to the interstellar medium. Barium and lead illustrate heavy products; the detailed yield depends on seed abundance, neutron exposure and branching at unstable nuclei.

In the r-process, neutron captures outrun beta decays, building very neutron-rich nuclei before they decay toward stable heavy isotopes. This needs much larger neutron densities than the s-process and can produce very heavy nuclei, including uranium. Neutron-rich ejecta from neutron star mergers provide an established site: observed kilonova emission and spectroscopy support heavy-element synthesis in such ejecta. Rare magnetically driven stellar-collapse outflows are additional candidate sites. An ordinary core-collapse supernova cannot automatically be assumed to supply a complete heavy r-process pattern.

Not every heavy isotope belongs to the two capture sequences. Gamma-process nucleosynthesis in hot supernova material uses photodisintegration on pre-existing heavy seed nuclei through neutron, proton and alpha-emission channels, helping produce proton-rich isotopes bypassed by the main capture paths. This is distinct from simply using charged-particle fusion to build every element beyond the iron group.

Finally, binary stars change both production and delivery. Roche-lobe overflow and wind mass transfer in a binary star strip envelopes, change subsequent core growth, and expose processed layers. Mass transfer from an Asymptotic giant branch donor can enrich a surviving companion in carbon and s-process products, as in a barium star; this redistributes existing yields rather than creating them in the recipient. Accretion onto a white dwarf can trigger a classical nova with explosive hydrogen-burning products. Thermonuclear disruption of a carbon-oxygen white dwarf in a Type Ia supernova, through suitable accretion or the Double-degenerate Type Ia supernova scenario, supplies a major delayed iron-group contribution, including iron; it is not the same event as collapse of a massive iron core. Compact binaries containing two neutron stars can later merge and eject r-process material. Thus binaries affect stellar mass histories, enrich companions, open thermonuclear explosion channels, and create heavy-element-producing compact-remnant mergers.

The key distinction is between making nuclei and ejecting them: ordinary low/intermediate-mass stars mainly return processed envelopes, massive stars add advanced and explosive burning products when those layers escape, and neutron-capture environments supply much of the material beyond iron.

Take $\mathbf{r}={\mathbf{x}}_1-{\mathbf{x}}_2$, $r=|\mathbf{r}|$ and $M=M_1+M_2$. Newton's law of universal gravitation gives ${\ddot{\mathbf{x}}}_1=-G{M}_2\mathbf{r}/r^3$ and ${\ddot{\mathbf{x}}}_2=G{M}_1\mathbf{r}/r^3$. Subtracting yields the relative two-body equationThe reduced mass is $\mu =M_1{M}_2/M$. The orbital energy and angular momentum in the center of mass frame areDifferentiation givesThus both are conserved. The eccentricity vector isFor $\mathbf{h}=\mathbf{J}/\mu$, one has $\ddot{\mathbf{r}}\times \mathbf{h}/(GM)=\dot{\mathbf{r}}/r-\mathbf{r}(\mathbf{r}\cdot \dot{\mathbf{r}})/r^3=d(\mathbf{r}/r)/dt$, so $\dot{\mathbf{e}}=0$. In particular the orbital eccentricity is constant, withFor a bound Kepler orbit, $E=-GM\mu /(2a)$ and $J^2=GM{\mu }^{2}a(1-e^2)$.

For the far-field multipole expansion, set the center of mass to zero. Then ${\mathbf{x}}_1=(M_2/M)\mathbf{r}$ and ${\mathbf{x}}_2=-(M_1/M)\mathbf{r}$. At $x=|\mathbf{x}|\gg r$, expand each point-mass Newtonian gravitational potential:Repeated indices are summed. The dipole vanishes since ${\sum }_A{M}_A{\mathbf{x}}_A=0$, and ${\sum }_A{M}_A{x}_{Ai}{x}_{Aj}=\mu r_i{r}_j$. The second-order gravitational quadrupole potential of a point-mass binary is thereforeBoth tensors specified in the question are trace-free. Direct contraction givesso to the requested orderFor equal masses the cubic multipole also vanishes, but it need not vanish for unequal masses.

To evaluate the quadrupole formula, use $k=GM$, $\mathbf{v}=\dot{\mathbf{r}}$, $\mathbf{n}=\mathbf{r}/r$, $u=\dot{r}=\mathbf{n}\cdot \mathbf{v}$, and $\mathbf{w}=\mathbf{v}-u\mathbf{n}$. The acceleration and its derivative areThe third derivative of $r_i{r}_j$ is ${\dot{b}}_i{r}_j+r_i{\dot{b}}_j+3b_i{v}_j+3v_i{b}_j$. Its trace is $d^3(r^2)/d{t}^3=-2ku/r^2$. Substituting in $q_{ij}=\mu (3r_i{r}_j-r^2{\delta }_{ij})/2$ givesBecause $\mathbf{n}\cdot \mathbf{w}=0$, the cross tensor contraction is zero. The squared norms of the two bracketed pieces are $6u^2$ and $72w^2$, respectively. Thusand the instantaneous energy loss isThe parenthesis is $w^2+u^2/12\ge 0$, as required for positive radiated power. This is the instantaneous quadrupole luminosity of a Kepler binary with loss sign for the orbital energy.