Stellar, Gas, and Maser Dynamical Masses

A dynamical black-hole mass is not read directly from a rotation curve or velocity dispersion. It is inferred by asking which gravitational potential, tracer distribution, observing geometry, and instrument model could have produced the measured positions and velocities. Stars, ionized or molecular gas, and water masers can constrain the same central point mass, yet their leading nuisance parameters differ: stellar anisotropy, gas pressure and noncircular motion, or maser-disk warp and distance. Agreement is therefore powerful only when the shared distance, center, and distributed mass are propagated rather than silently fixed.

Required background. Probability and Inference for Gravity supplies likelihoods, priors, nuisance-parameter marginalization, and posterior predictive checks.

Helpful background. Black-Hole Geometry and Mechanics supplies orbital invariants and relativistic scales; Relativistic Matter and Compact Objects supplies pressure support and matter models; and Black-Hole Astrophysics supplies the source and observational context.

One gravitational field, three tracer equations

In a spherical potential, write the enclosed gravitating mass as

$$M_{\rm enc}(r)=M_\bullet+M_\star(<r)+M_{\rm gas}(<r)+M_{\rm dm}(<r), \qquad v_c^2(r)=\frac{G M_{\rm enc}(r)}{r}.$$

For a flattened galaxy or a massive disk, instead compute the potential $\Phi$ and use $v_c^2=R\,\partial\Phi/\partial R$ in its equatorial plane; the spherical enclosed-mass formula is not exact there. The scientific target is $M_\bullet$, not $M_{\rm enc}$ at one unresolved radius. The nominal sphere of influence,

$$r_{\rm infl}=\frac{G M_\bullet}{\sigma_\star^2},$$

is a planning scale rather than a theorem: resolving it helps separate a point mass from a smooth stellar component, but unresolved data can still be informative if the light distribution, point-spread function, and orbital model are accurate. Conversely, nominally resolving $r_{\rm infl}$ does not cure a wrong mass-to-light profile or center.

For a spherical, steady, nonstreaming stellar tracer with luminosity density $\nu(r)$, radial dispersion $\sigma_r$, and one-component tangential dispersion $\sigma_t^2=(\sigma_\theta^2+\sigma_\phi^2)/2$, define $\beta=1-\sigma_t^2/\sigma_r^2$. The Jeans equation is

$$\frac{d(\nu\sigma_r^2)}{dr}+\frac{2\beta\nu\sigma_r^2}{r} =-\nu\frac{G M_{\rm enc}(r)}{r^2}.$$

The projected dispersion is then convolved with the PSF and aperture. A rise in dispersion can be reproduced partly by changing $\beta$, the stellar mass-to-light ratio, or the deprojection; orbit-superposition models replace the Jeans closure by a nonnegative orbit-weight distribution, but remain conditional on the potential and orbit library. The data and modelling hierarchy are laid out in Kormendy and Ho 2013, §§ 2.1–2.2.

For a thin, approximately circular gas disk, the radial Euler equation gives

$$v_c^2=v_\phi^2-\frac{r}{\rho}\frac{dP}{dr}, \qquad v_{\rm los}=v_\phi\sin i\cos\varphi+v_{\rm sys}.$$

Because pressure normally decreases outward, $v_\phi<v_c$; neglecting turbulent or thermal support biases the mass low. Shocks, bars, radial flow, disk thickness, dust obscuration, and an uncertain line-emissivity weighting can invalidate this axisymmetric reduction. The HST/STIS M87 analysis makes the convolution over slit, charge diffusion, line-spread function, and intrinsic surface brightness explicit in Walsh et al. 2013, §§ 2–3.

A geometrically thin maser disk approaches the cleanest limit. For a high-velocity spot near the disk midline,

$$(v_{\rm los}-v_{\rm sys})^2\simeq\frac{G M_\bullet}{D\theta}\sin^2 i,$$

so angular positions and Doppler speeds determine $M_\bullet/D$. Secular accelerations of systemic features supply the centripetal scale $G M_\bullet/(D\theta)^2$; together with proper motions or the warped-disk geometry, they break the mass–distance degeneracy. NGC 4258 illustrates the joint VLBI-position, velocity, acceleration, and proper-motion fit in Humphreys et al. 2013, §§ 2–4. A warp is information, not merely a correction, but maser self-gravity, eccentricity, blending, and incomplete azimuthal sampling must still be tested.

A shared forward model

Let $d_s$, $d_g$, and $d_m$ denote stellar, gas, and maser measurements. A defensible comparison uses

$$p(M_\bullet,D,\eta_s,\eta_g,\eta_m\mid d_s,d_g,d_m) \propto p(d_s\mid M_\bullet,D,\eta_s) p(d_g\mid M_\bullet,D,\eta_g) p(d_m\mid M_\bullet,D,\eta_m) p(M_\bullet,D,\eta_s,\eta_g,\eta_m),$$

where the nuisance vectors retain tracer-specific structure. The stellar vector includes anisotropy, deprojection, orbit weights, PSF, and mass-to-light gradients. The gas vector includes inclination, pressure, emissivity, noncircular flow, slit or beam convolution, and wavelength calibration. The maser vector includes warp coefficients, eccentricity, spot association, acceleration errors, self-gravity, and reference-frame registration. Distance, central position, systemic velocity, and sometimes astrometric calibration are shared; multiplying separately marginalized mass posteriors would count their priors more than once.

This hierarchy also clarifies what “the same mass” means. Each tracer samples a different radial and angular window. The model should predict the actual pixels, spectra, or spot catalog, not a deconvolved curve treated as error-free. Posterior predictive residuals should be inspected in position–velocity space, and a distributed dark component should be fitted as a competitor to a point mass whenever the innermost resolved scale permits it.

Evidence and reproduction boundary through 2026-08-18

Concrete reference data sets are the Gebhardt et al. Gemini/NIFS, SAURON, and VIRUS-P stellar kinematics for M87; the Walsh et al. HST/STIS ionized-gas spectra for M87; and the Humphreys et al. VLBI and single-dish NGC 4258 maser monitoring set. Their calibrated spectra, images, spot tables, or published machine-readable measurements are not interchangeable releases. A reproduction must record observing epochs, archive product version, PSF or synthesized beam, line-spread or channel response, astrometric and velocity zero points, distance convention, and reference frame.

Selection enters before the likelihood. Report the radial and angular footprint, tracer membership and quality cuts, masked asymmetric gas, detected-line threshold, maser-feature persistence rule, acceleration coverage, and nondetections. Statistical intervals conditional on a fixed deprojection or warp must not be presented as total uncertainty. The leading systematics differ: stellar anisotropy and mass-to-light gradients; gas pressure, beam smearing, and noncircular motion; maser warp, self-gravity, and spot association. Shared distance and centering errors correlate the methods.

A practical reproduction path is to retrieve the named observatory products or the papers’ machine-readable tables, rebuild one tracer likelihood at a time, inject its leading nuisance effect into synthetic data, and only then form the joint model with shared distance and reference-frame covariance. Exact recovery of the published posterior may be impossible where orbit libraries, reduction states, or covariance products were not released; that is a reproducibility boundary, not permission to substitute digitized plots. The stellar-versus-gas M87 discrepancy is a useful adversarial test because a fit can be statistically precise while remaining sensitive to tracer physics; see Gebhardt et al. 2011, §§ 2 and 4 and Walsh et al. 2013, §§ 3–4.

Limitations that survive better data

A point mass is a model class. Keplerian scaling over a finite interval shows a compact concentration, but the strongest size limit comes from the innermost tracer and the allowed distributed-mass profile. It does not by itself demonstrate an event horizon.

More angular resolution is not automatically more accuracy. A narrower PSF reduces beam smearing while exposing smaller-scale asymmetry, population gradients, or disk warps. Calibration and model discrepancy therefore need their own nuisance terms rather than being absorbed into a reduced $\chi^2$.

Tracer agreement is conditional. Agreement after fixing a common distance, inclination, or center can be manufactured by the shared assumption. A leave-one-tracer-out prediction is more discriminating: fit two tracers and predict the third in its native observable space.

Exercises

A maser disk has an edge-on high-velocity feature at angular radius $\theta=4.0\ {\rm mas}$ and speed $v=900\ {\rm km\,s^{-1}}$. A geometric distance analysis gives $D=7.6\ {\rm Mpc}$. Estimate $M_\bullet$ from $v^2=GM_\bullet/(D\theta)$. If $D$ shifts upward by $3\%$ while $v$ and $\theta$ are unchanged, how does the inferred mass move?

Solution

At $7.6\ {\rm Mpc}$, one milliarcsecond subtends $0.03684\ {\rm pc}$, so $r=D\theta=0.1474\ {\rm pc}$. Using $G=4.3009\times10^{-3}\ {\rm pc}\,M_\odot^{-1}({\rm km\,s^{-1}})^2$,

$$M_\bullet=\frac{v^2r}{G} =\frac{(900)^2(0.1474)}{4.3009\times10^{-3}}M_\odot \simeq2.78\times10^7 M_\odot.$$

The angular Kepler relation measures $M_\bullet/D=v^2\theta/G$, so $M_\bullet\propto D$. A $3\%$ larger distance therefore gives a $3\%$ larger mass, $2.86\times10^7M_\odot$, before any correlated change in the warped-disk solution. Treating the distance error as independent after using the same maser data to infer $D$ would double count information; the joint likelihood must retain their covariance.

References

• Gebhardt, Karl, et al. 2011. “The Black Hole Mass in M87 from Gemini/NIFS Adaptive Optics Observations.” The Astrophysical Journal 729:119. DOI.
• Humphreys, Elizabeth M. L., et al. 2013. “Toward a New Geometric Distance to the Active Galaxy NGC 4258. III. Final Results and the Hubble Constant.” The Astrophysical Journal 775:13. DOI.
• Kormendy, John, and Luis C. Ho. 2013. “Coevolution (Or Not) of Supermassive Black Holes and Host Galaxies.” Annual Review of Astronomy and Astrophysics 51:511–653. DOI.
• Walsh, Jonelle L., et al. 2013. “The M87 Black Hole Mass from Gas-Dynamical Models of Space Telescope Imaging Spectrograph Observations.” The Astrophysical Journal 770:86. DOI.