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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 an equatorial circular orbit in the Paczyński-Wiita potential, radial force balance givesIts specific angular momentum is $j=R^2\Omega =\sqrt{GM}{R}^{3/2}/(R-R_S)$. Perturbing the circular radius at fixed $j$ gives the radial epicyclic frequency squared ${\kappa }^2=R^{-3}d(j^2)/dR$. ThereforeCircular motion is radially stable for $R>3R_S$, marginal at $3R_S$, and unstable between $R_S$ and $3R_S$. Thus the inner edge of the nearly circular thin accretion disk is $R_{\mathrm{in}}=3R_S$, joining a plunging region of a black-hole accretion disk inside the innermost stable circular orbit. This conclusion concerns the circular thin-disk model, not an assertion that all gas disappears inside this radius.

Take the inward mass accretion rate $\dot{M}$ as positive, and write the positive outward viscous torque in an accretion disk asSteady conservation of angular momentum makes the net inward flux $C=\dot{M}j-\mathcal{G}$ independent of radius. Impose a zero-torque inner boundary condition at $3R_S$, appropriate to an approximately stress-free transition to plunging motion. ThenThis is the steady accretion disk with arbitrary rotation law angular-momentum balance.

The local viscous dissipation integrated through the full disk thickness is $D=\nu \Sigma (R{\Omega }^{\prime }{)}^2=-\mathcal{G}{\Omega }^{\prime }/(2\pi R)$. It is the power per unit planar area summed over both radiating faces. Sincesubstitution gives the two-face dissipation of a Paczyński-Wiita disk:In particular $D$ vanishes at the stress-free inner edge and is nonnegative outside it because $j$ increases there. A one-face radiative flux is $D/2$; the total accretion luminosity is therefore $L={\int }_{3R_S}^{\infty }2\pi RDdR$, with no additional factor of two.

For a direct evaluation put $x=R/R_S$. ThenThe first integral is $7/4$, using $(3x-1)/(x-1{)}^3=3/(x-1{)}^2+2/(x-1{)}^3$. For the second,so it equals $2/[\sqrt{3}(3-1)]=1/\sqrt{3}$. HenceThis is the radiative efficiency of black-hole accretion within this pseudo-Newtonian model.

Independently, the specific orbital energy of a circular particle isThus $E(\infty )=0$ and $E(3R_S)=-GM/(8R_S)=-c^2/16$. The energy released per unit accreted mass is exactly $c^2/16$, confirming the integrated accretion luminosity. The Paczyński-Wiita circular orbit reproduces the Schwarzschild inner radius but gives an approximate binding energy; the efficiency derived here is for the stated potential.

The helium mass fraction is $Y=1-X-Z$. For the two compositions, $Y_1=0.28$ and $Y_2=0.18$. The mean molecular weight is thereforeAt fixed mass and with the same prescribed opacity coefficient, part (a) gives $T_e\propto {\mu }^{1/2}{R}^{1/8}$ and $L\propto {\mu }^2{R}^{5/2}$. Eliminating radius givesThis is the composition shift of a convective pre-main-sequence track. The higher-helium mass fraction case $X_1=0.7$ is hotter at a given luminosity, lying to the left of the $X_2=0.8$ case in a Hertzsprung-Russell diagram. Its fixed-luminosity temperature ratio is $(349/324{)}^{2/5}\simeq 1.030$.

Both stars contract, so both $L$ and $T_e$ decrease in this particular opacity model. Their nearly vertical Hayashi tracks therefore run downward and slightly to the right, with arrows in that direction:

The plotted luminosity and effective temperature are normalized to arbitrary common references; no opacity normalization or absolute age was specified. The comparison holds the given opacity coefficient fixed.

These paths cease to apply when their assumptions fail. Deuterium-burning energy-generation rate can temporarily slow contraction; sustained hydrogen burning ends the purely gravitational phase. Formation of a radiative core breaks the fully convective adiabat and can lead to a Henyey track toward hotter effective temperature. Partial ionization, a changing atmospheric opacity, and electron degeneracy pressure alter the equation of state or boundary relation. Accretion luminosity, changing mass, and magnetic inhibition of convection can also modify a young star's path.

The specific orbital energy released in a black-hole accretion flow reaching an inner radius $r_{\mathrm{in}}$ is of order $GM/r_{\mathrm{in}}$. For a circular Keplerian orbit, the binding energy is more precisely $GM/(2r_{\mathrm{in}})$, with the other half of the potential-energy decrease retained as orbital kinetic energy. Multiplying by the mass accretion rate gives the accretion luminosity scaleFor an inner radius of order the Schwarzschild radius, $r_{\mathrm{in}}\sim 2GM/c^2$, this becomesThis is an energy-release scaling, not unit radiative efficiency of black-hole accretion. The actual inner orbit, stresses and advected energy determine the coefficient and the emitted luminosity; the innermost stable circular orbit of a Schwarzschild black hole lies outside its horizon.

Equate the accretion luminosity to the surface luminosity:Using $R\propto M^3{T}_{\mathrm{eff}}^{-24}$ from part i givesSubstitution into $L\propto M^6{T}_{\mathrm{eff}}^{-44}$ yields the accreting Hayashi track

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.