Stellar-Mass Black Holes

Stellar-mass black holes are compact remnants and merger products whose masses are broadly associated with stellar evolution. They are an established astrophysical population, but their intrinsic mass, spin, multiplicity, and natal-kick distributions are not read directly from any one catalog. The useful question is not merely which objects fall within a numerical interval; it is which formation and selection model makes the X-ray, astrometric, microlensing, and gravitational-wave samples mutually consistent.

Helpful background. Black-Hole Formation in Proto-Neutron-Star Evolution supplies the collapse endpoint used here.

Stellar-remnant black-hole population

Property	Working statement
Object class	Astrophysical class
Physical status	Established population inferred through dynamics, accretion, lensing candidates, and compact-binary mergers
Operational range	Approximately a few to hundreds of $M_\odot$; neither edge is universal
Principal channels	Single-star collapse, interacting-binary evolution, stellar collisions, and hierarchical compact-object mergers
Direct observables	Orbital periods and velocities, photometry, astrometry, spectra and timing, microlensing light curves, and detector strain
Inferred quantities	Remnant mass and spin, natal kick, merger rate, and population distribution
Evidence cutoff	20 August 2026

The lower boundary depends on the maximum neutron-star mass and on how confidently an unseen companion is classified. The upper boundary depends on winds, pair instability, metallicity, binary interactions, and whether second-generation merger remnants are included. A mass label must therefore state its purpose. “Stellar-mass” is a defensible formation-scale category, not a claim that every member formed in one isolated star.

Formation channels and controlled limits

In single and binary stellar evolution, the pre-collapse core is set by nuclear burning, mass loss, rotation, and envelope stripping. Collapse may produce an explosion with fallback or nearly direct black-hole formation. Lower metallicity generally weakens line-driven winds and can leave more massive cores, but the mapping is not one-dimensional: convection, reaction rates, rotation, and binary mass exchange matter. Pair-production softening can drive pulsations or complete disruption over model-dependent core-mass ranges. Mapelli 2021, §§1.2.1–1.2.6 reviews these dependencies.

Two controlled limits clarify the channel labels. In the isolated, noninteracting limit, the remnant follows a single-star prescription with its assumed winds and explosion engine. In the dense-cluster limit, exchanges and repeated mergers can assemble binaries dynamically and can produce remnants above a first-generation stellar boundary. Between them lie common-envelope evolution, stable mass transfer, chemically homogeneous evolution, and stellar collisions. No observed mass alone selects among these histories.

Natal kicks and mass loss alter binary survival, eccentricity, and systemic velocity. Spin is similarly nonunique: it can reflect core angular-momentum transport, tidal synchronization, accretion, or a previous merger. A present-day spin posterior is not a direct natal-spin measurement.

Dynamical inference and its nuisance parameters

For a single-lined binary with donor velocity amplitude $K_2$,

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

The left side is determined from orbital observables; the black-hole mass on the right is a posterior involving inclination $i$ and donor mass $M_2$. Ellipsoidal modulation, rotational broadening, eclipses, and atmosphere models help constrain them. Disk veiling, irradiation, Roche distortion, spots, asynchronous rotation, and correlated distance information can bias a plug-in estimate. Casares and Jonker 2014, §§2–2.1 gives the derivation and the principal failure modes.

X-ray spectra and timing diagnose an accretion flow, not the horizon directly. Continuum-fitting and reflection methods infer spin only after assuming a spacetime, an inner-disk geometry, inclination, spectral hardening or ionization structure, and calibration; Reynolds 2021, §§2–4 reviews the method-specific assumptions. These posteriors should not be merged with gravitational-wave spins as though the two samples or nuisance parameters were interchangeable.

Population observables and selection

An X-ray-binary census selects systems that transfer matter, radiate in the observed band, and receive follow-up sufficient for a dynamical solution. Transients are weighted by duty cycle; high extinction and donor brightness matter. Microlensing searches admit isolated candidates but infer lens mass jointly with parallax, finite-source effects, source motion, and the Galactic phase-space prior. Gravitational-wave catalogs select merging binaries through a mass-, spin-, redshift-, and orientation-dependent sensitive volume.

Consequently, apparent features such as a low-mass deficit, a high-mass taper, or a spin peak must be inferred hierarchically. Event likelihoods are convolved with a population distribution and divided by the selection efficiency; Mandel, Farr, and Gair 2019, §§ 2–4 derives this selection-aware framework. The same mass spectrum should be tested under alternative pairing, redshift-evolution, waveform, and catalog-threshold models, as in the dated LIGO–Virgo–KAGRA Collaboration 2026, §§ 2.2–3 and 5–7, Appendix B. The current interpretation framework belongs with Interpreting Binary Black-Hole Catalogs; detailed X-ray-binary coupling belongs with Stellar-Mass X-Ray Binaries.

What is established—and what is not

Secure dynamical binaries and gravitational-wave signals establish a stellar-scale black-hole population. They do not yet yield a selection-independent census of dormant Galactic remnants, a unique remnant-mass function, or a unique mixture of formation channels. The mass range itself is convention and evidence dependent, and proposed gaps should be reported with their model, credibility interval, and catalog version.

A useful cross-check predicts several observables together: masses, mass ratios, spins, eccentricities, merger redshifts, companion types, and host environments. Agreement with only a selected one-dimensional histogram can conceal compensating errors. Formation microphysics and the collapse threshold are developed in Proto-Neutron-Star Evolution and Black-Hole Formation.

References

• Casares, Jorge, and Peter G. Jonker. “Mass Measurements of Stellar and Intermediate-Mass Black Holes.” Space Science Reviews 183 (2014): 223–252. arXiv.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration. “GWTC-5.0: Population Properties of Merging Compact Binaries.” arXiv:2605.27226 (2026). arXiv.
• Mandel, Ilya, Will M. Farr, and Jonathan R. Gair. “Extracting Distribution Parameters from Multiple Uncertain Observations with Selection Biases.” Monthly Notices of the Royal Astronomical Society 486 (2019): 1086–1093. DOI.
• Mapelli, Michela. “Formation Channels of Single and Binary Stellar-Mass Black Holes.” In Handbook of Gravitational Wave Astronomy. Springer, 2021. arXiv.
• Reynolds, Christopher S. “Observational Constraints on Black Hole Spin.” Annual Review of Astronomy and Astrophysics 59 (2021): 117–154. DOI.