Astrophysical Black-Hole Parameters

An astrophysical black-hole parameter is an estimand in a forward model, not a raw detector reading. Its meaning requires calibrated data, a likelihood or confidence construction, nuisance parameters, selection effects, and model alternatives. This card fixes the stable dictionary; numerical estimates belong to a dated empirical record.

From detector output to a parameter

Let $d$ denote calibrated data, $\vartheta$ the parameters of interest, $\nu$ nuisance parameters, $\mathcal M$ a model family, and $\mathcal C$ the calibration state. A Bayesian analysis has the schematic form

$$p(\vartheta,\nu\mid d,\mathcal M,\mathcal C) \propto p(d\mid\vartheta,\nu,\mathcal M,\mathcal C) p(\vartheta,\nu\mid\mathcal M).$$

A frequentist likelihood analysis uses the same forward map even when it does not assign parameter priors. In either case, the reported quantity must say whether it is a likelihood estimate, posterior summary, confidence set, upper limit, or population-level parameter.

Keep five layers separate:

1. Raw data are detector samples, counts, voltages, or correlations with quality flags.
2. Calibrated observables have stated instrument response, standards, units, and calibration uncertainty.
3. Derived summaries include spectra, light curves, line profiles, reconstructed images, and orbital elements. Their reduction choices already introduce covariance and nuisance dependence.
4. Model-inferred parameters are values of $\vartheta$ supported by a likelihood or estimating equation.
5. Population parameters additionally require a sampling and selection model; they are not obtained by treating individual estimates as an unbiased catalog.

Mass, spin, orientation, and distance

For a compact object of mass $M$ and angular momentum $J$, the dimensionless spin in the site’s $c=1$ units is

$$a_{\ast}=\frac{J}{GM^2}.$$

Restoring SI units gives $a_{\ast}=cJ/(GM^2)$. Its sign requires an orientation convention, and the familiar Kerr bound is a statement about the assumed Kerr geometry, not a prior-free property of arbitrary compact-object models.

Estimand	Representative data and forward model	Principal degeneracy and check
$M$	Orbital periods, tracer velocities, timing, lensing, or calibrated gravitational angular scales	Inclination, distance, tracer dynamics, extended mass, and model family; predict an unused velocity or timing observable
$a_{\ast}$	Relativistic reflection, continuum spectra, timing, images, or gravitational waves	Mass, inclination, disk inner edge, atmosphere, emissivity, ionization, variability, and calibration; compare methods with shared nuisances retained
orientation	Projected orbital, disk, jet, polarization, or waveform geometry	Reflection symmetries, precession, thickness, and beaming; test another position angle or epoch
$D$	Parallax, standardizable source, redshift model, orbital geometry, or an external distance ladder	Calibration, peculiar motion, extinction, and environmental model; propagate $D$ into every dimensional parameter

In a single-lined binary with black-hole mass $M$, companion mass $M_2$, orbital period $P$, companion radial-velocity semi-amplitude $K_2$, eccentricity $e$, and orbital inclination $i$, Keplerian dynamics gives

$$f(M)=\frac{P K_2^3(1-e^2)^{3/2}}{2\pi G} =\frac{M^3\sin^3 i}{(M+M_2)^2}.$$

The mass function is a measured combination and a lower bound under the stated binary model, not by itself a posterior for $M$. Irradiation, veiling, tracer motion, the mass ratio, and especially inclination enter the inference Casares and Jonker 2014, §§ 2–2.1 and 3–5. The full derivation and its failure modes belong with dynamical mass measurements.

“Inclination” must name its axis: the binary orbital angular momentum, inner disk, outer disk, jet, and black-hole spin axis need not be aligned. Spin inferences from spectra therefore share mass, distance, inclination, atmosphere, inner-edge, density, ionization, emissivity, variability, and calibration nuisances Reynolds 2021, §§ 2–6. Use the mass–spin consistency treatment before combining nominally independent results.

Accretion, luminosity, efficiency, and jets

There is no unique unqualified “accretion rate.” Distinguish the supplied mass rate at an outer boundary, the inflow mass rate through a specified radius, the horizon mass rate crossing the horizon, and the retained mass rate that changes the hole’s mass after radiative, mechanical, and rest-mass energy bookkeeping. Winds, storage, and time dependence can make all four different.

For opacity per unit mass $\kappa$, hydrostatic electron-scattering balance in $c=1$ units defines

$$L_{\mathrm{Edd}}=\frac{4\pi GM}{\kappa}.$$

For ionized hydrogen with Thomson cross section $\sigma_T$, restoring constants gives $L_{\mathrm{Edd}}=4\pi GMm_p c/\sigma_T$. Composition changes the opacity. The Eddington ratio is $\lambda_{\mathrm{Edd}}=L_{\mathrm{bol}}/L_{\mathrm{Edd}}$. A dimensionless accretion rate is undefined until its reference rate is stated; $\dot M c^2/L_{\mathrm{Edd}}$ and $\dot M/\dot M_{\mathrm{Edd}}$ differ whenever the definition of $\dot M_{\mathrm{Edd}}$ builds in an efficiency.

The generic conversion efficiency is

$$\eta=\frac{L}{\dot{M}c^2}.$$

One must name both $L$—bolometric radiation, a band-limited luminosity, or mechanical power—and the surface at which $\dot M$ is evaluated. Disk thickness, stress at the inner edge, photon capture, advection, winds, and viewing transfer control the map from flow variables to luminosity Abramowicz and Fragile 2013, §§ 2–3 and 5–10. Definitions and limiting cases are developed under Eddington scales and efficiency.

Jet power is another inferred quantity. Horizon electromagnetic energy flux in the Blandford–Znajek process depends on horizon rotation and magnetic flux, but large-scale jet power is inferred after mass loading, collimation, propagation, dissipation, beaming, radiative conversion, and environmental work. A radio luminosity or cavity estimate is therefore not the horizon electromagnetic flux itself Blandford, Meier, and Readhead 2019, §§ 2–4. The physical separation is derived on the Blandford–Znajek process page.

Angular scales, images, and environment

Mass and distance define the gravitational angular scale

$$\theta_g=\frac{GM}{c^2D}.$$

Here $D$ is the angular-diameter distance, reducing to the ordinary Euclidean distance in the local asymptotically flat limit. The scale $\theta_g$ is not itself an image diameter. Interferometric measurements begin with calibrated visibilities and closure quantities. A reconstructed emission ring depends on sampling, calibration, reconstruction, variability, and plasma emissivity. The spacetime critical curve is a lensing feature predicted by a metric and observer configuration. In an ideal illumination model, the geometric shadow is the captured-ray region on the observer’s screen and its boundary is the critical curve; a “shadow diameter” must declare which shape statistic is being reduced to one number. A central brightness depression is an image feature. The horizon is a null boundary. These objects are not synonymous.

Converting image structure into $\theta_g$, $M$, or spin requires a declared geometric or simulation-based forward model, plus distance, orientation, electron thermodynamics, magnetization, variability, scattering, and calibration nuisances Event Horizon Telescope Collaboration 2019, §§ 3–6 and 8. The geometric distinctions are maintained in critical curves, shadows, and photon rings.

Environment parameters—density, temperature, magnetic flux, composition, radiation field, companion properties, or surrounding potential—must likewise name a radius, frame, averaging rule, and epoch. They are nuisance parameters for some questions and primary estimands for others.

Combining and checking inferences

Different labels do not make two estimates independent: they may share distance, inclination, calibration, simulations, selection, or theory. Combine their likelihoods with shared nuisances and covariance represented once; do not average published one-dimensional intervals as independent measurements.

A credible parameter claim should survive checks matched to its failure modes:

• reproduce the data summary and propagate calibration uncertainty;
• vary priors, nuisance models, cuts, and plausible model families;
• use injections, coverage, or posterior-predictive checks as appropriate;
• predict data withheld from the fit, another epoch, or another observable;
• compare methods only after translating definitions and retaining common nuisance parameters;
• route every numerical estimate, catalog statement, or ranking of method accuracy to a dated observational record.

The Black-Hole Astrophysics overview supplies the full physical setting. This stable card deliberately contains no object-specific estimate or universal accuracy claim.

References

• Abramowicz, Marek A., and P. Chris Fragile. “Foundations of Black Hole Accretion Disk Theory.” Living Reviews in Relativity 16 (2013): 1. DOI
• Blandford, Roger, David Meier, and Anthony Readhead. “Relativistic Jets from Active Galactic Nuclei.” Annual Review of Astronomy and Astrophysics 57 (2019): 467–509. DOI
• Casares, Jorge, and Peter G. Jonker. “Mass Measurements of Stellar and Intermediate-Mass Black Holes.” Space Science Reviews 183 (2014): 223–252. Open PDF
• Event Horizon Telescope Collaboration. “First M87 Event Horizon Telescope Results. VI. The Shadow and Mass of the Central Black Hole.” Astrophysical Journal Letters 875 (2019): L6. DOI
• Reynolds, Christopher S. “Observational Constraints on Black Hole Spin.” Annual Review of Astronomy and Astrophysics 59 (2021): 117–154. DOI