= Solution
\b[<mass> from motion.] A resolved <Keplerian orbit> gives $M=4\pi^2a^3/(GP_{\rm orb}^2)$ when the orbiting object has negligible <mass>. At the Galactic centre, repeated infrared <astrometry> and <spectroscopy> measure sky motion, acceleration and line-of-sight velocity of <stars> around <Sagittarius A*>. Resolving a compact enclosed <mass> and fitting complete orbits avoids reliance on a single projected velocity. Source confusion, reference-frame stability, distance, inclination and possible extended <mass> must be included in the fit.<Measuring Distance and Properties of the Milky Way's Central Supermassive Black Hole with Stellar Orbits by Andrea M. Ghez and others>[Abstract and orbital-model/systematic-error discussion]{ref}
For an unresolved stellar-mass binary in a circular orbit, the donor's velocity semiamplitude $K_2$ and <orbital period> give the <binary mass function>
$$
f(M_{\rm BH})=\frac{P_{\rm orb}K_2^3}{2\pi G}
=\frac{M_{\rm BH}^3\sin^3i}{(M_{\rm BH}+M_2)^2}.
$$
It provides a lower <mass> bound; donor <mass> and <orbital inclination> are needed for an actual <mass> estimate. For an eccentric orbit, multiply the observational combination $P_{\rm orb}K_2^3/(2\pi G)$ by $(1-e^2)^{3/2}$ when evaluating the <binary mass function>. Resolved gas or maser <rotation> and stellar-dynamical models also measure <masses>, provided <pressure> support, noncircular motion and <line-of-sight projection> are modeled. Resolving the <black-hole sphere of influence> improves separation of central <mass> from the host <galaxy>.
For active nuclei, <reverberation mapping> measures a lag $\tau$ between variable continuum and broad-line response. With a line velocity scale $\Delta v$,
$$
M_{\rm BH}=f_{\rm geom}\frac{c\tau\,\Delta v^2}{G}.
$$
The factor $f_{\rm geom}$ accounts for geometry and kinematics and is not known simply from the lag. Cadence, line responsivity, inclination and winds require control. Rapid variability demonstrates compactness but is not by itself a unique <mass> measurement.
\b[<black-hole spin> from strong-field emission.] Define $a_*=cJ/(GM^2)$. Large-radius Keplerian motion is principally a <mass> probe and usually insensitive to <black-hole spin>. In the Kerr model, the <radius> and <binding energy> of the <innermost stable circular orbit> vary with $a_*$ and orbital orientation, making inner-disk <radiation> useful.
<black-hole continuum fitting> infers an inner emitting <radius> from the disk's thermal spectrum and flux normalization, then identifies it with the <innermost stable circular orbit> in a relativistic thin-disk model. Independent <mass>, distance and inclination, together with atmospheric spectral hardening, <absorption> and contamination, are needed. Spectra should be selected in a disk-dominated regime where truncation and strong <advection> are limited. A 2011 Cygnus X-1 analysis combined such spectra with independently constrained binary parameters and reported a near-extreme <black-hole spin> conditional on <black-hole spin>–orbit alignment and its disk model.<The Extreme Spin of the Black Hole in Cygnus X-1 by Lijun Gou and others>[Abstract, thermal-continuum fit and dominant mass/distance/inclination/flux-calibration uncertainties]{ref}
<black-hole reflection spectroscopy> instead models a relativistically broadened iron-line and reflection continuum: orbital <Doppler shifts> and beaming broaden the profile, while <gravitational redshift> contributes to the extended red wing. Its innermost emitting <radius> can constrain <black-hole spin> if reflection extends to the stable inner edge. Ionization, abundance, <absorption>, coronal illumination, disk thickness and possible truncation are competing parameters; a broad line is not a direct <black-hole spin> reading independent of these assumptions. Timing or precession models can add information but introduce their own dynamical identification.
\b[<mass> is principally inferred from gravitating dynamics; <black-hole spin> is inferred from strong-field effects with a specified spacetime and <accretion> model.] Agreement between independent methods is especially valuable because disk inner <radius>, inclination and radiation-transfer assumptions can otherwise mimic a change of <black-hole spin>.
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