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A binary mass function combines the observed orbital period $P$ and radial velocity semi-amplitude $K_1$ of a visible member of a spectroscopic binary. For a circular Kepler orbit, $K_1=2\pi a_1\sin i/P$, where $a_1$ is the visible star's orbital semimajor axis about the centre of mass and $i$ the orbital inclination. The centre of mass relation $a_1=a{M}_2/(M_1+M_2)$ and Kepler's third law $4{\pi }^2{a}^3=G(M_1+M_2)P^2$ eliminate the unobserved separation:It has units of mass. The right side is at most $M_2$, so $f(M)\le M_2$ is a strict observational lower bound on the unseen companion's mass within this model. If $M_1$ is independently estimated, setting $\sin i=1$ and solving the equation gives a stronger minimum companion mass. An unknown orbital inclination generally prevents a unique pair of stellar masses from being inferred from one velocity curve alone.

For an eccentric Kepler orbit with measured orbital eccentricity $e$, the radial-velocity semi-amplitude is $K_1=2\pi a_1\sin i/[P\sqrt{1-e^2}]$, so the observable expression becomes $f(M)=P{K}_1^3(1-e^2{)}^{3/2}/(2\pi G)$. A fitted semi-amplitude, rather than an arbitrary instantaneous radial velocity, is required. This is why an unseen companion can be constrained dynamically even when its own spectral lines are absent.

Doppler spectroscopy detects an exoplanet through the host star's changing Doppler shift. Both bodies orbit their shared astronomical barycenter: the star undergoes stellar reflex motion even when the exoplanet itself is unseen. Its velocity along the line of sight shifts its spectral lines; at speeds small compared with the speed of light, $\Delta \lambda /\lambda \simeq v_r/c$, with positive $v_r$ for recession. Repeated measurements can fit a Kepler orbit, rather than merely detect one shifted spectrum. Starspots and other changes in stellar spectral lines must be distinguished from actual orbital motion.

For one companion in an isolated two-body problem, the stellar radial-velocity semi-amplitude isHere $G$ is the gravitational constant, $P$ the orbital period, $e$ the orbital eccentricity, and $M_{\star },M_p$ the star and exoplanet masses. The orbital inclination $i$ is measured relative to the sky plane: $i=0$ is face-on and $i=90^{\circ }$ is an edge-on orbit. The exact binary mass function measured from the fitted orbit isWith known $M_{\star }$ and $M_p\ll M_{\star }$, this is usually reported as $M_p\sin i$. Taking $\sin i=1$ gives the minimum companion mass; a smaller orbital inclination allows a larger mass. A detected exoplanet transit constrains $i$ close to $90^{\circ }$ for a compact planet orbiting well outside the star's surface, substantially resolving this ambiguity. Combining Doppler spectroscopy and exoplanet transit photometry then supplies a mass and radius for exoplanet interior studies.

For a computed circular example, let $q=M_p/M_{\star }=0.001$ and choose a relative orbital speed $V_{\mathrm{rel}}=100\mathrm{km}{\mathrm{s}}^{-1}$: this is the speed of the planet relative to the star, not its speed about the astronomical barycenter. Conservation of momentum givesThe barycentric velocities point in opposite directions, so their speeds add to $V_{\mathrm{rel}}$. Here $e=0$ and $K=V_{\star }\sin i$: $K=99.9001\mathrm{m}{\mathrm{s}}^{-1}$ at $i=90^{\circ }$, $49.9500\mathrm{m}{\mathrm{s}}^{-1}$ at $i=30^{\circ }$, and zero at $i=0$. The star still orbits in the face-on case, but its orbital velocity has no line-of-sight component. The plotted circular signal is $v_r(t)-\gamma =K\sin (2\pi t/P)$ for a chosen phase origin and constant system velocity $\gamma$.

A repeating Doppler shift contains more than an amplitude. For an isolated two-body problem, let $\nu$ be the true anomaly, ${\omega }_{\star }$ the stellar argument of periapsis, $\tau$ the periapsis time and $\gamma$ the constant system line-of-sight velocity. The companion's argument of periapsis differs by $\pi$ from ${\omega }_{\star }$, so mixing the two conventions reverses the orbital signal. With positive velocity for recession, choose the stellar line-of-sight coordinate positive away from the observer. Kepler's equation supplies the clock:Here $M$ is the mean anomaly and $E$ the eccentric anomaly. The time coordinate is uniform in $M$, not in $\nu$: an eccentric Kepler orbit runs fastest near periapsis.

To derive the velocity law without guessing a sinusoid, write the stellar orbital coordinates as $X=a_{\star }(\cos E-e)$ and $Y=a_{\star }\sqrt{1-e^2}\sin E$. Project toDifferentiating $z$ and substituting the true anomaly identities givesThe constant $e\cos {\omega }_{\star }$ term makes the time average of the orbital velocity zero: $z$ returns to its original value after one orbital period. For $e=0$ this reduces to a sinusoid with an arbitrary phase origin; periapsis itself is then undefined. For $e>0$, the largest and smallest velocities are $\gamma +K(e\cos {\omega }_{\star }\pm 1)$, so $K$ is still half the peak-to-peak range, although the extrema need not be symmetric about $\gamma$.

For an original worked example, choose $M_{\star }=1M_{\odot }$, $P=10$ days, $K=50\mathrm{m}{\mathrm{s}}^{-1}$, $e=0.4$ and ${\omega }_{\star }=60^{\circ }$. Taking one day as $86400\mathrm{s}$ and $G=6.67430\times 10^{-11}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$, the binary mass function is $f=1.98270\times 10^{20}\mathrm{kg}$. Its units check directly: $[P{K}^3/G]=\mathrm{kg}$. With adopted solar mass $M_{\odot }=1.98847\times 10^{30}\mathrm{kg}$ and Jupiter mass $M_J=1.89813\times 10^{27}\mathrm{kg}$, solving the exact binary mass function gives

All three systems have the same chosen $P,K,e,{\omega }_{\star },\tau$ and therefore the same Doppler spectroscopy time series, despite different true masses. Exactly what remains fixed is $M_p\sin i/(M_{\star }+M_p{)}^{2/3}$; $M_p\sin i$ alone is only the small-companion approximation. The $90^{\circ }$ value is the minimum mass. For this example the extrema are $+60$ and $-40\mathrm{m}{\mathrm{s}}^{-1}$ relative to $\gamma$, and the periapsis value is $+35\mathrm{m}{\mathrm{s}}^{-1}$. This is a chosen eccentric system, not a reconstruction of 51 Pegasi b.

Doppler spectroscopy measures the line-of-sight component of a host star's stellar reflex motion, rather than the full companion mass. Let the measured orbital period, orbital eccentricity and stellar radial-velocity semi-amplitude be $P,e,K$, with host mass $M_{\star }$ and sky-plane orbital inclination $i$. The exact two-body relation isCubing and rearranging defines the observed binary mass functionFor fixed $M_{\star }$, the function $M_p^3/(M_{\star }+M_p{)}^2$ is strictly increasing: its logarithmic derivative is $3/M_p-2/(M_{\star }+M_p)>0$. Since $\sin i\le 1$, the smallest compatible companion mass occurs for an edge-on orbit, $i=\pi /2$.

Only when $M_p\ll M_{\star }$ does this simplify toA small inferred projected mass alone therefore does not exclude a more massive companion at small orbital inclination. An exoplanet transit constrains orbital inclination and helps remove this degeneracy. When the companion mass is appreciable, solve the exact binary mass function instead of inserting a total-mass correction on the wrong side of the approximate formula.

The most direct stellar mass determination uses orbital dynamics in a binary star. A resolved orbit and an independent distance give the relative semimajor axis $a$ from its angular size; measuring the period then gives the total mass by Kepler's third law:Individual barycentric orbital sizes, or the ratio of the two radial-velocity amplitudes, give $M_1/M_2=a_2/a_1=K_2/K_1$. A photocenter orbit alone needs the luminosity ratio to infer the barycentric orbit.

For a double-lined spectroscopic orbit, the line-of-sight amplitudes determine $a_i\sin i=K_{i}P\sqrt{1-e^2}/(2\pi )$. Combining them givesAn eclipsing binary light curve constrains the inclination and relative radii, turning these projected quantities into individual masses. Eclipse and orbit modeling must allow for eccentricity, surface brightness, limb darkening and possible tidal distortion.

If only one spectrum is measured, the binary mass function isWithout an inclination and an estimate of the visible star's mass it does not fix the unseen mass uniquely, but it provides useful constraints and lower limits.

Less direct estimates combine a spectroscopic surface gravity with a radius inferred from distance, bolometric flux and effective temperature: $R^2=L/(4\pi {\sigma }_{\mathrm{SB}}{T}_{\mathrm{eff}}^4)$ and $M=g{R}^2/G$. These depend on atmosphere modeling. Locating a star on an evolutionary track or using a calibrated main-sequence mass-luminosity relation also estimates mass, but assumes composition and evolutionary state; giants and mass-transfer products need not obey a single main-sequence relation. Dynamical binary masses provide the empirical calibration against which such model-dependent methods are tested.

Let $a_{\mathrm{ns}}$ be the neutron star's distance from the centre of mass. In a circular orbit,Combining this with Kepler third law,gives the binary mass functionSince $\sin i\le 1$ and $(M_{\mathrm{ns}}+M_{\mathrm{wd}}{)}^2>M_{\mathrm{wd}}^2$,

When a low-mass red giant undergoes stable Roche-lobe overflow, the neutron star receives matter with substantial specific angular momentum, usually through an accretion disk. The accretion torque spins it up and weakens its external magnetic field, producing a recycled pulsar with a millisecond period. Removal of the donor's envelope exposes its helium core as a low-mass white dwarf.

At detachment, the donor mass and core mass both become $M_{\mathrm{wd}}$. Its luminosity is fixed by the red-giant core-mass--luminosity relation, and the supplied radius law therefore makes its final radius $R_L$ a function only of $M_{\mathrm{wd}}$. Cubing the Roche-lobe relation givesEliminating $a$ with Kepler third law yieldsThis is the white-dwarf mass--orbital-period relation: dependence on the neutron-star mass cancels.

For randomly oriented binaries, the isotropic binary inclination distribution isThe observed period gives $M_{\mathrm{wd}}$ from the preceding relation. If $M_{\mathrm{ns}}=1.35M_{\odot }$ is adopted, the measured mass function then givesso the inclination is inferred without astrometry.

Two plausible causes of an apparent shortage of edge-on systems are:

Scatter or bias in the core-mass--period relation can reinforce the second effect.

A donor star can feed a compact companion through Roche-lobe overflow near the inner Lagrange point, or through capture of a stellar wind. The transferred gas usually carries angular momentum; when its circularization radius exceeds the accretor's effective inner radius it forms an accretion disc. Outward angular momentum transport permits inward mass flow, while dissipation converts gravitational binding energy into heat and radiation. For a compact object with a surface $L\sim GM\dot{M}/R$; for a black hole write $L=\eta \dot{M}{c}^2$, with efficiency determined by the flow and inner boundary. Optically thick disc regions emit a sum of thermal radiation spectrum components; a hot optically thin accretion corona can inverse-Compton scatter softer photons into a harder X-ray component. A black hole has no material surface on which to radiate a stellar boundary layer.

An explicit example is the wind-fed black-hole binary Cygnus X-1. Its massive stellar companion supplies gas, and the accretion flow produces strong X-ray emission.^[ref] Compact-object evidence comes from orbital dynamics rather than X-ray brightness alone. The single-lined binary mass function constrains the unseen mass; inclination, donor properties and distance refine it. Jerome A. Orosz and collaborators combined optical photometry, radial velocities, rotation and a distance constraint to obtain a black-hole mass of $14.8\pm 1.0M_{\odot }$ in their 2011 analysis. The inference depends on the stellar and orbital model, and that historical value should not be confused with a latest measurement.^[ref]

mass from motion. A resolved Keplerian orbit gives $M=4{\pi }^2{a}^3/(G{P}_{\mathrm{orb}}^2)$ 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.^[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 functionIt 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 $P_{\mathrm{orb}}{K}_2^3/(2\pi G)$ 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 $\Delta v$,The factor $f_{\mathrm{geom}}$ 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.

black-hole spin from strong-field emission. Define $a_{\ast }=cJ/(G{M}^2)$. 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 $a_{\ast }$ 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.^[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.

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.

The strongest pre-2014 evidence combines large gravitating mass with extreme compactness. In an X-ray binary, companion-star radial velocities determine a binary mass function; orbital inclination, donor properties and distance complete the mass inference. An unseen component above conventional neutron-star masses is a black-hole candidate. For example, a 2011 Cygnus X-1 analysis combined optical photometry, spectroscopy and distance information to infer a compact-object mass about $14.8M_{\odot }$, with model and observational uncertainties.^[ref]

Near Sagittarius A*, infrared stellar astrometry and radial velocities trace nearly Keplerian orbits around a dark compact mass. The concentration of several million solar masses in a small volume strongly challenges an extended cluster of ordinary remnants; source confusion, reference-frame errors, distance and possible extended mass must be fitted.^[ref] Gas and maser dynamics provide related compact-mass tests in other nuclei.

accretion explains luminous X-rays, rapid variability and broad relativistic disk features. These constrain size and the strong-field environment, but emission alone is not unique to an event horizon. Lack of surface-like signatures is supporting, model-dependent evidence. Mass, compactness and consistent accretion phenomenology together support the black-hole interpretation; large-radius mass measurements alone do not directly demonstrate a horizon.

The semimajor axis $a_1$ of the pulsar's orbit about the binary barycenter, projected along the line of sight and divided by the speed of light. The orbital inclination $i$ is zero for a face-on orbit. Thus $x$ is a time, often quoted as the numerical number of light-seconds in $a_1\sin i$, not the true binary separation. The relative-orbit semimajor axis is instead $a=a_1+a_2$; conservation of momentum gives $a_1=a{m}_2/(m_1+m_2)$.

Combining Kepler's third law with that barycenter relation gives the timing form of the binary mass function, also valid for eccentric Kepler orbits:Choose $x=2\mathrm{s}$ and $P_b=8\mathrm{h}$ for an original example: $a_1\sin i=599,585\mathrm{km}$ and $f\simeq 0.0773M_{\odot }$, where $M_{\odot }$ is a solar mass. For a specified pulsar mass $m_1=1.4M_{\odot }$, the smallest allowed companion mass occurs at $\sin i=1$ and is approximately $0.698M_{\odot }$. A different assumed $m_1$ changes this bound. One measured $f$ does not determine both masses and orbital inclination.

Orbital dynamics gives the most direct stellar masses: a resolved binary orbit and distance determine the relative semimajor axis, and Kepler's third law gives the total mass. The barycentric size or radial-velocity amplitude ratio gives individual masses. An eclipsing binary inclination removes the sine-cubed projection factor from spectroscopic masses; a single-lined binary mass function alone does not uniquely determine both masses. Surface-gravity/radius and stellar-track estimates are model-dependent and are calibrated against dynamical binary measurements.