Stellar-Mass X-Ray Binaries

A stellar-mass black-hole X-ray binary is a coupled system: an evolving donor supplies gas, the orbit fixes the available angular momentum and dynamical mass, the accretion flow selects a radiative state, and part of the power may leave in a jet or wind. A self-consistent interpretation must use the same distance, inclination, masses, and observing epoch wherever they are genuinely shared. It must also leave donor distortion, disk alignment, absorption, and state history free where the data do not force them to agree.

Helpful background. Black-Hole Geometry supplies the compact-object length and efficiency scales; Relativistic Matter and Compact Objects supplies stellar remnants and radiation; Black-Hole Astrophysics supplies the source-to-data inference chain; Two-Body Dynamics supplies orbital evolution; Gravitational-Wave Astronomy supplies compact-binary population inference.

Orbit, donor, and mass supply

Let $M_1$ be the black-hole mass, $M_2$ the donor mass, and $q=M_2/M_1$. A single-lined orbit gives

$$f(M_1)= \frac{PK_2^3}{2\pi G}(1-e^2)^{3/2} =\frac{M_1\sin^3 i}{(1+q)^2}.$$

The left side is close to the data; the right side introduces inclination and mass ratio. Ellipsoidal light curves, rotational broadening, eclipses, and a donor atmosphere can constrain those quantities, but disk light, irradiation, asynchronous rotation, Roche distortion, and spots can bias them. A compact-object mass should therefore come from a joint orbital likelihood, not from inserting separately quoted best fits.

Gas reaches the black hole through Roche-lobe overflow or capture from a stellar wind. For a circular binary, the donor’s approximate Roche-lobe radius is

$$\frac{R_{L,2}}{a}\simeq \frac{0.49q^{2/3}} {0.6q^{2/3}+\ln(1+q^{1/3})}.$$

This is the volume-equivalent radius in the circular, corotating Roche potential, rather than the distance to the inner Lagrange point Eggleton 1983, equation (2), p. 368. Roche contact couples the donor’s radius response to orbital angular momentum and mass transfer. Wind capture depends additionally on wind speed, ionization, and orbital phase. A steady mass budget is

$$\dot M_{\rm donor} =\dot M_{\rm lost} +\dot M_{\rm disk}, \qquad \dot M_{\rm disk} =\dot M_H+\dot M_{\rm out}.$$

All rates here are positive magnitudes. The donor loss rate is not the horizon accretion rate. In an evolving disk, the second equality also includes $dM_{\rm disk,stored}/dt$ on its right; outbursts can drain matter accumulated much earlier. This distinction is essential in wind-fed systems and in disk states with strong outflow.

Accretion state and jet coupling

An X-ray state is a reproducible location in spectral–timing space. A hard state commonly has a Comptonized spectrum, strong broad-band variability, and compact radio emission; a thermal-dominant state has a stronger disk component and weaker rapid variability; intermediate states connect them and often host rapid changes. These associations constrain geometry but do not uniquely identify a truncation radius, corona, or jet-launching mechanism.

For a control volume containing the flow, a time-dependent total-energy budget is

$$\dot E_{\rm supply} =\dot E_H+L_{\rm rad} +\dot E_{\rm jet}+\dot E_{\rm wind} +\frac{dE_{\rm flow}}{dt}.$$

Every $\dot E$ includes rest mass and uses the same energy and time normalization; $\dot E_H$ is positive into the hole and includes advected matter and electromagnetic extraction. Spin power must not be added again when already included there. $L_{\rm rad}$ is the total escaping radiation: disk photons scattered into the Compton continuum are counted once. The storage derivative cannot be dropped merely because several instantaneous luminosities are available. Radio flux is not jet kinetic power without a synchrotron, opacity, geometry, and distance model, and a band-limited X-ray luminosity is not bolometric power. A state transition can change this partition even while donor supply varies slowly.

Long-term Cyg X-1 monitoring illustrates both the connection and its ambiguity. Its 15 GHz flux correlates more directly with the Comptonized X-ray component than with a fixed soft-X-ray band, and the inferred correlation slope changes with radio free–free absorption in the donor wind. Zdziarski et al. 2011, §§ 3–4 shows why a band-specific radio/X-ray correlation cannot be promoted directly to a universal accretion–jet law.

A reproducible Cyg X-1 data model

Use two named datasets. The dynamical data release is the radial-velocity, $U\!BV$ light-curve, rotational, radius, and parallax-constrained model assembled by Orosz et al. 2011, §§ 2–4 and Tables 1–2. The state dataset is the simultaneous daily RXTE/ASM and Swift/BAT monitoring with 15 GHz Ryle/AMI measurements analyzed by Zdziarski et al. 2011, § 2. Freeze the final RXTE archive products, mission response and background calibration, radio flux scale, barycentric times, and the optical wavelength and photometric zero points at the evidence cutoff.

The selection must record orbital phases and spectral lines used for velocities, eclipse and irradiation masks, disk-light treatment, X-ray good-time and energy bands, the state classifier, radio nondetections, flare policy, and the simultaneity window. Because Cyg X-1’s focused wind absorbs radio emission with orbital phase, either forward-model that modulation or select and report a phase rule.

A coherent likelihood has an orbital block for velocities, light curves, distance, and rotational broadening; a count- or flux-space X-ray block for disk and Comptonized emission plus timing; and a radio block with calibration, censoring, and wind absorption. Shared parameters include distance and orbital ephemeris. The disk inclination should equal the orbital inclination only under an explicit alignment hypothesis. Priors on donor atmosphere, wind, and jet geometry must remain visible.

Statistical uncertainty arises from measurement noise, finite sampling, and posterior covariance. Systematic and model uncertainty includes inclination, Roche geometry, irradiation, atmosphere templates, parallax, X-ray response and bolometric correction, state hysteresis, free–free absorption, radio calibration, and non-simultaneity. The reproduction pipeline should first recover the published mass function and state-resolved radio/X-ray tracks, then vary alignment, energy bands, state boundaries, and wind absorption. A shared-parameter fit is informative only if its posterior-predictive residuals remain acceptable in every data block.

The observational comparison includes results through 2026-08-18. Later astrometry, reprocessed high-energy responses, or a revised radio flux scale requires the joint model to be rerun.

Evolutionary interpretation

The present binary is one frame in a longer evolution. Donor nuclear expansion, wind loss, tides, magnetic braking in lower-mass systems, and nonconservative transfer change the orbit and spin supply. Stable transfer requires the donor’s radius response and Roche-lobe response to remain compatible; otherwise a thermal runaway or common envelope can occur.

Observed black-hole spin, orbital eccentricity, and systemic motion are conditional clues about prior mass transfer and collapse. They do not uniquely reconstruct natal mass loss, kicks, or internal angular-momentum transport. Population statements require selection-aware synthesis across transient duty cycles, extinction, companion type, and follow-up probability.

This page develops the astrophysical coupling among donor, orbit, accretion state, and outflow. For the underlying orbital dynamics and the interpretation of gravitational-wave data, see Two-Body Dynamics and Gravitational-Wave Astronomy.

Common pitfalls

Using the donor loss rate as $\dot M_H$. Capture, storage, disk winds, and jets can remove most of the supplied mass before it crosses the horizon.

Classifying a state from one hardness ratio. The boundary is instrument dependent and should be checked with spectra, variability, and hysteresis.

Multiplying posteriors with shared inputs. Distance, inclination, ephemeris, and calibration covariances must enter once in a joint model.

Close a binary budget

Take $P=5.6\,{\rm d}$, $K_2=75\,{\rm km\,s^{-1}}$, $e=0.02$, $q=1.4$, and $i=27^\circ$. Compute $f(M_1)$ and $M_1$. A wind loses $2\times10^{-6}M_\odot\,{\rm yr^{-1}}$, of which one percent is captured by the disk; half of that reaches the horizon. If $L_X=1.3\times10^{37}\,{\rm erg\,s^{-1}}$ and $P_{\rm jet}=5\times10^{36}\,{\rm erg\,s^{-1}}$, what fraction of $\dot M_Hc^2$ appears in these two channels?

Solution

Using

$$\frac{f}{M_\odot} =1.0385\times10^{-7} \left(\frac{P}{\rm d}\right) \left(\frac{K_2}{\rm km\,s^{-1}}\right)^3 (1-e^2)^{3/2},$$

gives $f(M_1)\simeq0.245M_\odot$. Therefore

$$M_1=f(M_1)\frac{(1+q)^2}{\sin^3i} \simeq0.245M_\odot \frac{2.4^2}{\sin^3 27^\circ} \simeq15.1M_\odot.$$

The disk captures $2\times10^{-8}M_\odot\,{\rm yr^{-1}}$ and the horizon receives $\dot M_H=10^{-8}M_\odot\,{\rm yr^{-1}}$. Since

$$\dot M_Hc^2\simeq5.67\times10^{38}\,{\rm erg\,s^{-1}},$$

the measured channels contain

$$\frac{L_X+P_{\rm jet}}{\dot M_Hc^2} =\frac{1.8\times10^{37}}{5.67\times10^{38}} \simeq0.032.$$

About $3.2\%$ appears in the stated X-ray and jet channels, taking $P_{\rm jet}$ to exclude jet rest-mass energy. Most of the horizon rest-mass energy normally increases the black-hole mass; it is not available binding energy that must all become radiation or heat. Additional bolometric radiation, wind power, storage, and uncertainty in capture and horizon rates must be included before deriving an accretion efficiency.

References

• Eggleton, Peter P. “Approximations to the Radii of Roche Lobes.” Astrophysical Journal 268 (1983): 368–369. DOI.
• Orosz, Jerome A., et al. “The Mass of the Black Hole in Cygnus X-1.” Astrophysical Journal 742 (2011): 84. DOI.
• Zdziarski, Andrzej A., Gerald K. Skinner, Guy G. Pooley, and Piotr Lubiński. “X-Ray Variability Patterns and Radio/X-Ray Correlations in Cyg X-1.” Monthly Notices of the Royal Astronomical Society 416 (2011): 1324–1339. DOI.