Cygnus X-1

Cygnus X-1 is a persistent, wind-fed high-mass X-ray binary containing an optically visible massive star and an unseen compact companion. The compact object is dynamically established as a stellar-mass black hole: the orbital mass function, radio parallax, light-curve geometry, rotational broadening, and donor model jointly place it above viable neutron-star masses. Its often-quoted spin is a different kind of result—a posterior conditioned on accretion-disk and spacetime assumptions.

Cygnus X-1 as a wind-fed black-hole binary

Property	Working statement
Object class	Named observed system
System type	High-mass X-ray binary; focused stellar-wind accretion
Physical status	Dynamically established stellar-mass black hole
Direct measurements	Orbital period, optical radial velocities and line profiles, photometric modulation, radio parallax, X-ray counts and timing, radio flux and morphology
Model-dependent posteriors	Black-hole and donor masses, inclination, distance, wind geometry, accretion state, inner-disk radius, and spin
Evidence cutoff	20 August 2026

The system’s persistence and luminous donor make it observationally rich but not simple. Irradiation and a focused wind distort the donor spectrum; radio emission is absorbed through the wind; and the disk, orbital plane, and jet need not be perfectly aligned.

From orbital data to a black-hole mass

For donor radial-velocity semiamplitude $K_2$, period $P$, and eccentricity $e$,

$$f(M_{\rm BH})= \frac{PK_2^3}{2\pi G}(1-e^2)^{3/2} =\frac{M_{\rm BH}^3\sin^3 i}{(M_{\rm BH}+M_2)^2}.$$

The left side follows closely from direct spectroscopic observables. The individual masses require inclination $i$ and donor mass $M_2$. Those come from a joint model of the ellipsoidal light curve, rotational broadening, Roche geometry, radius and temperature, wind and irradiation, and distance. The very-long-baseline radio parallax is itself an angular fit with calibrator and orbital-motion systematics; it becomes a distance through parallax inversion and its likelihood.

The updated radio astrometry and binary model of Miller-Jones et al. 2021, Table 1 and Supplementary §§1.6–1.9 and Table S3 shifted the inferred distance and mass upward relative to earlier solutions. This is a useful lesson: the observed spectrum and sky position did not change into a “more massive” datum; a better parallax and joint nuisance model changed the posterior. Earlier joint dynamical work by Orosz et al. 2011, §§2–4 shows the covariance among distance, donor radius, inclination, and component masses.

Measured parameters

The table reports posterior medians and central $68\%$ ranges from the 2021 analysis unless a symmetric $1\sigma$ uncertainty is shown. Miller-Jones et al. label the luminous O-star as component 1, so their $K_1$ is the donor semiamplitude $K_2$ used in the mass-function equation above. “Fitted” means a parameter was adjusted in the stated forward model; it does not make that parameter a raw observable.

Parameter	Quoted result	Evidential status
VLBA parallax	$\varpi=458\pm35\ \mu{\rm as}$	One-dimensional astrometric fit perpendicular to the jet, used to reduce wind-driven radio-core shifts; optical orbital priors and astrometric error floors enter
Distance	$D_{\rm med}=2.22\ {\rm kpc}\;(68\%:\ 2.05\text{–}2.40\ {\rm kpc})$	Parallax-derived posterior using an exponentially decreasing space-density prior
Donor semiamplitude	$K_2\equiv K_{1,{\rm source}}=75.21\ {\rm km\,s^{-1}}\;(68\%:\ 74.80\text{–}75.63\ {\rm km\,s^{-1}})$	Fitted in the joint optical light-curve and radial-velocity model
Orbital inclination	$i=27.51^\circ\;(68\%:\ 26.94^\circ\text{–}28.28^\circ)$	Model posterior coupling Roche geometry, donor atmosphere, distance, irradiation, and rotation
Black-hole mass	$M_{\rm BH}=21.2\,M_\odot\;(68\%:\ 18.9\text{–}23.4\,M_\odot)$	Derived joint posterior, not a direct dynamical datum
Physical black-hole semimajor axis	$a_{\rm BH}=0.160\ {\rm au}\;(68\%:\ 0.147\text{–}0.173\ {\rm au})$	Derived from the binary solution
Angular black-hole semimajor axis	$a_{\rm BH,ang}=58\pm20\ \mu{\rm as}$	Separately fitted in the VLBA astrometric model, with covariance with parallax and orientation

The two orbit-size channels provide an internal scale check. Since $1\ {\rm au}$ at $1\ {\rm kpc}$ subtends $1\ {\rm mas}$, the physical binary solution predicts

$$\theta_{\rm BH} =1000\,\frac{a_{\rm BH}/{\rm au}}{D/{\rm kpc}}\ \mu{\rm as} =1000\,\frac{0.160}{2.22}\ \mu{\rm as} =72.1\ \mu{\rm as}.$$

Ignoring covariance only for a transparent uncertainty estimate gives $\sigma_\theta\simeq8.3\ \mu{\rm as}$, consistent with the authors’ quoted $73\pm8\ \mu{\rm as}$ and with the astrometric $58\pm20\ \mu{\rm as}$. Because distance and orbital parameters share data and covariance, this is a closure test, not a second independent black-hole mass measurement.

The detailed donor–orbit–accretion coupling is treated in Stellar-Mass X-Ray Binaries.

Accretion states, corona, and jet

Cygnus X-1 spends much time in a spectrally hard state, with strong Comptonized X-rays, broad-band variability, and a compact radio jet; it also enters softer, disk-dominated states. A hardness ratio is instrument specific. Physical comparisons should use response-folded spectra and timing products with an explicit state rule, absorption model, bolometric correction, and simultaneity window.

The donor’s wind affects both fuel supply and observation. Its density, velocity, ionization, and orbital geometry control capture and absorption; the donor mass-loss rate is not the rate crossing the horizon. Radio free–free absorption can change apparent radio/X-ray correlations with orbital phase. A jet-power estimate additionally requires distance, spectrum, opacity, geometry, particle content, and radiative efficiency.

Why spin is more conditional than mass

Continuum fitting maps a fitted inner-disk radius to the Kerr innermost stable circular orbit after adopting $M$, distance, inclination, spectral hardening, disk atmosphere, torque, thinness, and alignment. Reflection fitting maps relativistically broadened features through an illumination, density, ionization, abundance, absorption, and coronal-geometry model. The continuum analysis of Gou et al. 2014, §§4–6 is a concrete example of a selected Cygnus X-1 dataset favoring rapid Kerr rotation, but its quoted interval is a model posterior, not a directly timed rotation rate.

Reynolds 2021, §§2–4 details the shared and method-specific assumptions. Wind absorption, pile-up or response calibration, disk thickness, returning radiation, and a disk–orbit tilt can dominate over formal fit errors. The appropriate strong-gravity comparison belongs with X-Ray Spectral Tests of Strong Gravity, where the null spacetime and astrophysical alternatives can be varied together.

Controlled comparisons and limitations

A reproducible mass check first recovers the observed mass function, then adds the parallax and light-curve likelihood while retaining their covariance. A spin check fits the same state-selected count spectra with alternative absorption, atmosphere, reflection, coronal, and alignment models. Combining best-fit mass, distance, and inclination values from different analyses as independent Gaussian inputs is not equivalent.

Cygnus X-1 securely demonstrates a stellar-mass black hole in a high-mass binary. It does not by itself determine a natal spin, a universal wind-capture efficiency, or a unique disk–corona–jet geometry. Its non-eclipsing orientation, massive evolving donor, focused wind, and persistent selection also make it an unrepresentative draw from the dormant Galactic black-hole population.

An informative independent check compares the systemic velocity and orbital eccentricity with binary-evolution calculations, while allowing natal mass loss and kicks to vary. Even agreement would constrain a family of histories rather than uniquely reconstruct the collapse event.

References

• Gou, Lijun, et al. “The Extreme Spin of the Black Hole in Cygnus X-1.” Astrophysical Journal 790 (2014): 29. DOI.
• Miller-Jones, James C. A., et al. “Cygnus X-1 Contains a 21-Solar Mass Black Hole—Implications for Massive Star Winds.” Science 371 (2021): 1046–1049. DOI.
• Orosz, Jerome A., et al. “The Mass of the Black Hole in Cygnus X-1.” Astrophysical Journal 742 (2011): 84. DOI.
• Reynolds, Christopher S. “Observational Constraints on Black Hole Spin.” Annual Review of Astronomy and Astrophysics 59 (2021): 117–154. DOI.