X-ray Binary Luminosity Calculator

An X-ray binary (XRB) is a two-star system in which one component is a compact object (neutron star or black hole) that accretes matter from its companion star. As material falls onto the compact object, gravitational potential energy is released as X-ray emission. XRBs are among the brightest X-ray sources in the sky and are divided by companion type: high-mass X-ray binaries (HMXBs) with OB supergiant donors, and low-mass X-ray binaries (LMXBs) with Roche-lobe-filling late-type companions.

Accretion luminosity is L = eta * Mdot * c^2, where eta is the radiative efficiency (fraction of rest-mass energy radiated), Mdot is the mass accretion rate in kg/s, and c is the speed of light. This is derived from the gravitational potential energy released as material falls from infinity onto the compact object surface or innermost stable orbit. For a neutron star of mass M and radius R, the Newtonian estimate gives eta ~ GM/(Rc^2) ~ 0.2 for R = 10 km.

The Eddington luminosity L_Edd = 4*pi*G*M*m_p*c / sigma_T is the maximum luminosity at which radiation pressure on free electrons exactly balances gravitational attraction. For spherically symmetric accretion of hydrogen plasma, L_Edd ~ 1.3 x 10^31 * (M/M_sun) watts. Sources exceeding L_Edd cannot maintain steady spherical accretion, though super-Eddington emission is observed in ULXs via beamed, disk-driven, or magnetically dominated flows.

Radiative efficiency eta is the fraction of accreted rest-mass energy (Mdot*c^2) that emerges as radiation rather than falling into the compact object or being advected. For Schwarzschild BHs (non-spinning), the ISCO is at 6 GM/c^2 and eta ~ 0.057. For maximally spinning Kerr BHs, the ISCO is at 1.24 GM/c^2 and eta ~ 0.42. Neutron stars with solid surfaces retain all infalling kinetic energy as heat, giving eta typically 0.15 to 0.25. Advection-dominated accretion flows (ADAFs) at low Mdot can have very low effective eta.

The Eddington accretion rate is the mass transfer rate at which accretion at efficiency eta produces exactly L_Edd: Mdot_Edd = L_Edd/(eta*c^2) = 4*pi*G*M*m_p/(eta*sigma_T*c). For a 10 M_sun BH with eta = 0.1, Mdot_Edd ~ 1.4 x 10^-7 M_sun/yr. This is a characteristic rate used to normalize observed accretion rates and compare across source types.

High-mass X-ray binaries (HMXBs) have companion stars more massive than about 10 M_sun, typically OB supergiants or Be stars. The compact object accretes from the stellar wind. Examples: Cygnus X-1 (BH + O supergiant), Vela X-1 (NS + B supergiant). Low-mass X-ray binaries (LMXBs) have companions below about 2 M_sun, typically K or M stars filling their Roche lobe. Accretion is through an accretion disk. Examples: Sco X-1, Her X-1, XTE J1550-564.

ULXs are off-nucleus X-ray point sources with luminosity above ~10^33 W (10^39 erg/s), exceeding the Eddington limit for a 10 M_sun BH. They are found in nearby galaxies and may represent super-Eddington accretion onto stellar-mass compact objects (with beamed, wind-confined emission), intermediate-mass black holes (100 to 100,000 M_sun), or highly magnetized neutron stars (pulsating ULXs, where the strong B-field modifies the Eddington limit).

In an LMXB, the donor star fills its Roche lobe and transfers mass through the L1 Lagrange point at a rate set by the companion's evolutionary state. The accretion disk channels this material onto the compact object. The X-ray luminosity directly tracks the mass transfer rate through L = eta*Mdot*c^2. Transient LMXBs show outbursts when disk instability triggers a sudden increase in Mdot, while persistent LMXBs (like Sco X-1) maintain steady accretion.

Key X-ray observatories include: Chandra (subarcsecond imaging, 0.5-10 keV), XMM-Newton (high throughput, 0.15-15 keV), NuSTAR (hard X-ray focusing, 3-79 keV), NICER (neutron star timing, 0.2-12 keV), and eROSITA (all-sky survey). Historical missions include UHURU (first systematic X-ray survey, discovered Cygnus X-1), EXOSAT, ROSAT, RXTE, and Swift. Together they have catalogued thousands of X-ray binaries in the Milky Way and nearby galaxies.

Both X-ray binaries and active galactic nuclei (AGN) are powered by accretion onto compact objects, but the scales differ by 10^6 to 10^9. A typical XRB has a stellar-mass BH (3 to 100 M_sun) and X-ray luminosity of 10^30 to 10^32 W. An AGN harbors a supermassive BH (10^6 to 10^10 M_sun) with bolometric luminosity up to 10^40 W. The same physics (L = eta*Mdot*c^2, L_Edd = 4pi*G*M*m_p*c/sigma_T) governs both, making XRBs useful laboratories for understanding AGN on human timescales.

Cygnus X-1 was discovered in 1964 and became the first source widely identified as a black hole (dynamical mass measurement via orbital motion of its companion HDE 226868 showed M > 3 M_sun). The BH mass is now measured at 21.2 M_sun. It is a HMXB accreting from the stellar wind of a 41 M_sun O-type supergiant at a rate of about 2 x 10^-6 M_sun/yr. In its hard state it radiates at about 2% Eddington as a bright X-ray source detectable with simple instruments.