Mass Functions in Compact Binaries

The compact-binary mass function is the most model-independent mass information available from a single-lined spectroscopic orbit. It combines the period, radial-velocity semiamplitude, and eccentricity into a strict lower bound on the unseen object’s mass. It is not itself that mass: inclination, the companion-to-compact-object mass ratio, and line-formation systematics enter before a full inference is possible. Data-dependent statements on this page use the evidence cutoff 2026-08-18.

Required background. Probability and inference for gravity supplies joint likelihoods, priors, covariance, and selection effects.

Helpful background. Black-Hole Geometry clarifies what a mass estimate does not establish about a horizon; Relativistic Matter and Compact Objects supplies compact-star alternatives; Gravitational-Wave Astronomy gives the complementary mass inference from detector data.

Deriving the spectroscopic mass function

Let $M_1$ be the unseen compact-object mass and $M_2$ the visible donor mass. Define

$$q=\frac{M_2}{M_1}, \qquad M_{\rm tot}=M_1+M_2.$$

For a Keplerian binary with relative semimajor axis $a$ and period $P$,

$$P^2=\frac{4\pi^2a^3}{G(M_1+M_2)}.$$

The donor’s semimajor axis about the center of mass is

$$a_2=a\frac{M_1}{M_1+M_2}=\frac{a}{1+q}.$$

If the orbital inclination is $i=90^\circ$ for an edge-on orbit, the measured radial-velocity semiamplitude is

$$K_2=\frac{2\pi a_2\sin i}{P\sqrt{1-e^2}}.$$

Cubing this relation, eliminating $a$ with Kepler’s law, and gathering the directly spectroscopic quantities gives

$$\begin{aligned} f(M_1) &\equiv \frac{P K_2^3}{2\pi G}(1-e^2)^{3/2} \\ &=\frac{M_1^3\sin^3 i}{(M_1+M_2)^2} =\frac{M_1\sin^3 i}{(1+q)^2}. \end{aligned}$$

The eccentricity factor matters: the familiar circular expression is not valid merely because a sparse velocity curve looks sinusoidal. The derivation also assumes that the fitted absorption lines track the donor’s center of mass. Irradiation can suppress lines on one hemisphere, winds can distort them, and disk emission can contaminate the spectrum. Casares and Jonker 2014, §§ 2–2.1 and eq. (1) gives the circular case and its observational qualifications; the derivation above retains eccentricity explicitly.

Why it is a lower bound

Because $0<\sin i\le1$ and $(1+q)^2\ge1$ for positive masses,

$$f(M_1)=M_1\frac{\sin^3 i}{(1+q)^2}\le M_1.$$

Equality would require both $i=90^\circ$ and $q=0$: an exactly edge-on orbit and a massless donor. Real systems therefore have $M_1>f(M_1)$, apart from measurement uncertainty. This lower-bound logic is robust within the single-lined Keplerian model and is why a sufficiently large $f(M_1)$ can be more compelling than a larger but inclination-dominated point estimate.

The inequality does not license a population conclusion without a selection model. Edge-on systems are more likely to eclipse or dip and may be easier to constrain, while obscuration can remove some high-inclination systems. Systems with weak donor lines may be absent from a spectroscopic sample. An observed distribution of $f(M_1)$ is therefore neither the intrinsic mass distribution nor a randomly projected version of it.

From the bound to a mass posterior

The remaining quantities come from different observables and assumptions.

Period, velocity amplitude, and eccentricity. A time series of calibrated spectra supplies velocities after wavelength calibration, barycentric correction, template matching, and a choice of accepted lines. Phase coverage and the ephemeris determine how well $P$, $K_2$, and $e$ can be separated. The likelihood should retain their covariance.

Mass ratio. Rotational broadening can constrain $q$ if the donor fills its Roche lobe and rotates synchronously. Emission-line or disk-velocity methods introduce stronger assumptions because the emitting gas need not follow either stellar center of mass. A prior from the donor’s spectral type is conditional on its evolutionary and irradiation state.

Inclination. Ellipsoidal modulation, eclipses, polarimetry, astrometry, or a resolved jet may constrain $i$. A light curve mixes donor shape, limb and gravity darkening, irradiation, spots, and nonstellar light. A jet direction equals the orbital angular momentum only under an alignment assumption. Since $M_1\propto\sin^{-3}i$, inclination systematics are amplified.

The local sensitivity follows by differentiating the mass relation:

$$d\ln M_1=d\ln f+\frac{2\,dq}{1+q}-3\cot i\,di,$$

with $di$ measured in radians. At $i=30^\circ$, a $1^\circ$ upward shift in the assumed inclination lowers the inferred mass by about $9\%$ to first order; at $i=80^\circ$, the same shift changes it by only about $0.9\%$. This is a sensitivity calculation, not an error-combination rule: correlations must still be retained. For randomly oriented orbital axes the unselected inclination density is $p(i)=\sin i$ on $0\le i\le\pi/2$, equivalently uniform in $\cos i$; an observed binary sample need not retain that distribution.

A coherent analysis uses a joint model,

$$p(M_1,q,i,\eta\mid d,S) \propto p(d_{\rm RV},d_{\rm line},d_{\rm phot}\mid M_1,q,i,\eta,S) p(M_1,q,i,\eta\mid S),$$

where $\eta$ denotes line, atmosphere, irradiation, disk-light, and calibration nuisance parameters. The selection event $S$ records which phases, lines, photometric intervals, and systems entered the analysis. Multiplying a mass-function interval, an inclination interval, and a mass-ratio interval as if they were independent can count shared spectra or assumptions twice.

The Cygnus X-1 analysis in Orosz et al. 2011, §§ 2–4 illustrates an orbital, light-curve, and stellar constraint solved together. Its value here is methodological, not as an immutable mass number. A reproducible application should name the velocity and photometric data release or machine-readable tables, calibration and quality cuts, atmosphere and Roche models, priors, and software. First reconstruct the one-dimensional mass-function likelihood; then add $q$ and $i$ and report how alternative irradiation and disk-light models shift the result.

Statistical uncertainty describes the finite spectra and photometry conditional on this model. Systematic and model dependence includes wavelength zero point, template mismatch, phase-dependent line formation, asynchronous rotation, eccentricity, Roche geometry, disk veiling, spots, inclination prior, and sample selection. A small formal error on $K_2$ cannot compensate for a biased line tracer.

Common pitfalls

Calling $f(M_1)$ the black-hole mass. It is a lower bound. The factor $(1+q)^2/\sin^3i$ turns it into a conditional mass estimate.

Dropping eccentricity without testing it. The factor $(1-e^2)^{3/2}$ belongs to the spectroscopic relation. A circular prior should be justified by tidal evolution or the data, not convenience.

Using the wrong star’s velocity. The formula above uses the visible donor’s $K_2$. Reversing component labels changes both the mass ratio and which mass is bounded.

Treating line centroids as center-of-mass velocities. Irradiation, winds, and disk contamination can move the effective line-forming region. Phase-resolved residuals and alternative line masks are essential checks.

Compute the bound and the inferred mass

A binary has $P=6.47$ days, $K_2=208\ {\rm km\,s^{-1}}$, and negligible measured eccentricity. Use

$$\frac{f(M_1)}{M_\odot} =1.0385\times10^{-7}P_{\rm d}K_{\rm km/s}^3$$

to find the mass function. Then take $q=0.12$ and $i=70^\circ$ and infer $M_1$.

Solution

The spectroscopic bound is

$$\frac{f(M_1)}{M_\odot} =1.0385\times10^{-7}(6.47)(208)^3 \simeq6.05.$$

Thus $M_1\ge6.05M_\odot$ within the orbital model even before $q$ and $i$ are supplied. With the additional constraints,

$$M_1=f(M_1)\frac{(1+q)^2}{\sin^3i} =6.05M_\odot\frac{1.12^2}{\sin^3 70^\circ} \simeq9.15M_\odot.$$

The second number is more informative but less assumption-light. A complete answer would propagate the joint posterior of $P$, $K_2$, $e$, $q$, $i$, and the shared spectral and light-curve nuisance parameters.

References

• Casares, Jorge, and Peter G. Jonker. “Mass Measurements of Stellar and Intermediate-Mass Black Holes.” Space Science Reviews 183 (2014): 223–252. DOI.
• Orosz, Jerome A., et al. “The Mass of the Black Hole in Cygnus X-1.” Astrophysical Journal 742 (2011): 84. DOI.