Inferring Black-Hole Properties from Tidal Events

A tidal disruption event can reveal a previously dormant black hole, but it does not provide a model-free mass or spin measurement. Timing constrains a combination of black-hole mass and stellar structure; spectra and luminosities add emission geometry and radiative efficiency; the high-mass event rate adds capture physics and population selection. This page builds the inference in that order and keeps catalog facts separate from mutable physical claims. Generic stellar dynamics and exact geodesic derivations are outside scope.

Required background. Geodesics and Jacobi Fields supplies affine geodesics, deviation, and capture-orbit language.

Helpful background. Black-Hole Geometry and Mechanics supplies Kerr spin and horizon structure; Relativistic Matter and Compact Objects supplies stellar mass–radius relations; Black-Hole Astrophysics supplies survey and accretion context.

What the fallback clock actually measures

For a star of mass $m_*$ and radius $R_*$ disrupted by a black hole of mass $M$, the impulse estimate at the tidal radius gives

$$r_t=R_*\left(\frac{M}{m_*}\right)^{1/3}, \qquad \Delta\varepsilon\simeq\frac{GMR_*}{r_t^2}.$$

The most-bound debris returns after one Kepler period,

$$t_{\min}=\frac{2\pi GM}{(2\Delta\varepsilon)^{3/2}} =\frac{\pi}{\sqrt{2G}}\frac{M^{1/2}R_*^{3/2}}{m_*}.$$

The familiar $t_{\min}\propto M^{1/2}$ scaling is conditional on $m_*$, $R_*$, a nearly parabolic orbit, and energy freezing near $r_t$. Freezing the spread at pericenter would instead give $\Delta\varepsilon\propto\beta^2$ and $t_{\min}\propto\beta^{-3}$. Hydrodynamic results do not support using that strong $\beta$ dependence indiscriminately for full disruptions. Moreover, the observed optical rise can trace diffusion, stream collision, or reprocessing rather than first return. Gezari 2021, §3.1, Eqs. 1–5 gives the derivation and the observed timing tension.

An observed blackbody adds

$$L_{\rm bol}=4\pi R_{\rm bb}^2\sigma_{\rm SB}T_{\rm bb}^4,$$

but $R_{\rm bb}$ is a photospheric radius, not automatically the ISCO, circularization radius, or stream intersection. A host velocity dispersion or bulge relation can supply an external mass prior; it must not be counted as an independent confirmation after being used to tune the transient model.

Capture provides a population constraint

The ratio

$$\frac{r_t}{R_g} =\frac{R_*}{G}\,M^{-2/3}m_*^{-1/3}, \qquad R_g=GM,$$

decreases with black-hole mass. Near the Hills scale, disruption competes with direct capture. Kerr spin changes the capture boundary differently for prograde, retrograde, and inclined orbits, so events above the nonspinning solar-star threshold can carry population-level spin information. They do not uniquely determine $a_*$: a larger star, uncertain host mass, inclination, loss-cone anisotropy, or misclassification can mimic the effect. Kesden 2012, §§II–IV, Eqs. 2–11 and Figs. 3–4 computes relativistic tidal and capture outcomes; Gezari 2021, §4.3, Eq. 16 and Figs. 8–9 reviews observational mass and spin proposals.

Dataset, calibration, and selection

The frozen population input is TDECat, CDS VizieR J/A+A/707/A171, inserted on 2026-03-12 and retrieved for this contract on 2026-08-18. Table tdecat has 134 rows—131 optical TDEs and three jetted events—discovered through the end of 2024; table1 separately has 24 candidates. This is a comprehensive compilation of known events, not a survey-complete sample. The main catalog requires a photometric-survey detection and a flare-time classification spectrum, with stated exceptions for featureless and three established jetted events. Unclassified detections and incompletely explored infrared/X-ray classes make the selection function heterogeneous. See Langis et al. 2026, §§2–2.2 and Appendix A.

Calibration must follow the contributing instrument, not a single catalog-wide zero point. Optical/IR data mix LINEAR, CRTS, ZTF, iPTF, SDSS, ASAS-SN, ATLAS, NEOWISE, Pan-STARRS, and Gaia; some magnitudes are AB, some Vega converted with stated offsets, while Gaia uses its internal system. Swift UV photometry is extinction corrected but not host subtracted. X-ray products combine Swift, Chandra, XMM-Newton, 4XMM-DR13, and eRASS1; the catalog treats individual observations with signal-to-noise at least three as detections, so stacking changes the selection. Spectra come from TNS or WISeREP. Exact apertures, backgrounds, conversions, and response handling are in Langis et al. 2026, §§3.1–3.4 and Appendix B.

Results and limitations through 2026-08-18

Results after 2026-08-18 are outside this evidence snapshot. The origin of optical TDE emission and the reliability of transient-only black-hole mass inference remain unresolved.

Evidence axis	Durable result	Inferred quantity	Scope and limitations
TDECat VizieR tables and ReadMe	134 main rows, 24 candidates, identifiers, redshifts, classes, and release provenance	Population membership and descriptive distributions	Literature compilation is not rate-complete; discovery and classification selection vary by survey
Multi-band light curves	Calibrated epochs, fluxes or limits, cadence, reported rise/decay blocks	$M$, $m_*$, $\beta$, disruption time, viscous delay	Fallback, circularization, diffusion, and reprocessing clocks can be degenerate; gaps censor peaks
Host spectroscopy and imaging	Redshift, stellar population, velocity dispersion or photometric mass when available	External prior on $M$ and nuclear environment	Scaling-relation scatter, aperture, bulge decomposition, and post-starburst selection are systematic
High-mass cutoff and capture	Few candidates occupy the regime where relativistic capture matters	Distribution of $a_*$ and stellar radii	Requires survey efficiency, mass function, orbital inclinations, stellar population, and classification purity
X-ray/UV/optical/IR spectra and timing	Temperatures, lines, lags, detections and instrument responses	Disk radius, inclination, absorption, circularization, spin	Reprocessing, obscuration, shocks, color correction, and non-simultaneity prevent a universal mapping

TDECat’s Bayesian-Blocks timing analysis used 103 flares and reported log-normal-compatible timescale distributions, but the parent sample itself is incomplete and events with poorly constrained rise or decay were excluded. Those descriptive results are useful checks, not a likelihood for a physical mass model. Langis et al. 2026, §4.1, Figs. 1–4 states the timing selection and uncertainty construction.

A likelihood that exposes the assumptions

For event $i$, let $D_i$ contain calibrated photometry, spectra, upper limits, host information, and observation times. Let $\theta_i$ contain source-frame $M$, $a_*$, $m_*$, $R_*$, $\beta$, inclination, disruption time, extinction, photosphere, efficiency, and delay parameters. For hyperparameters $\Phi$ describing the black-hole and stellar populations, a selection-corrected hierarchical posterior has the form

$$p(\Phi,\{\theta_i\}\mid\{D_i\},S=1) \propto p(\Phi)\prod_i \frac{p(D_i\mid\theta_i)\,p(\theta_i\mid\Phi)} {\alpha(\Phi)},$$

with

$$\alpha(\Phi)=\int p(S=1\mid\theta)\,p(\theta\mid\Phi)\,d\theta.$$

This expression conditions on the observed number $N$ of selected events. It assumes independent events and that the selection rule is fixed by the included data, or that $p(D_i\mid\theta_i)$ already includes the joint probability of the data and selection. If detection uses additional noisy measurements not included in $D_i$, that joint detection model must be supplied explicitly. An absolute event-rate analysis instead needs a Poisson count factor with the expected detected number, survey duration and exposure. A heterogeneous literature compilation does not provide a known $\alpha(\Phi)$ merely because its listed events are public.

The likelihood should retain nondetections and calibration nuisance parameters; priors must distinguish source and cosmological frames. Statistical intervals come from finite signal-to-noise and sample size. Systematics include host subtraction, bolometric correction, stellar evolution, energy-freezing prescription, emission model, survey completeness, and catalog classification. If a host mass estimate already entered the prior, it cannot subsequently serve as independent validation of the inferred mass.

Fallback-powered disk. Luminosity is tied closely to return or accretion rate. It uses the clock efficiently but must explain weak rise-time–mass trends and color evolution.

Reprocessing envelope or wind. Inner radiation is absorbed and re-emitted at a large photosphere. It explains cool, nearly constant optical temperatures but adds geometry, opacity, and time-delay freedom.

Circularization shocks. Stream intersections power much of the optical emission before a compact disk forms. Timing depends on apsidal/nodal precession and stream width, not just $t_{\min}$.

Prompt disk or jet. X-rays or relativistic outflow probe smaller radii, but viewing angle, absorption, spectral hardening, and jet selection dominate. These are competing positions, not interchangeable templates with one shared “mass” parameter.

Reproducing the result and testing its robustness

To reproduce the catalog layer, download the VizieR ReadMe plus both tables, record the 2026-03-12 insertion, verify 134 and 24 rows, and keep candidates separate. Pin a TDECat GitHub commit before using the larger photometry tree or application; a moving default branch is not a version. Preserve native flux systems and instrument metadata, reproduce the authors’ survey-specific conversions, and encode limits rather than dropping them.

To reproduce inference, publish the event list, discovery/classification selection, calibration files, host subtraction, cosmology, source-frame conversion, likelihood, priors, sampler settings, posterior draws, and injection–recovery estimate of $p(S=1\mid\theta)$. A rerun without the private alert stream can validate catalog-conditioned inference but cannot recover the population of missed or unclassified events; that is the truthful reproduction boundary.

What could change the conclusion. The interpretation could change when a public survey-defined TDE sample releases end-to-end injection–recovery efficiency, when independent transient and dynamical masses agree across emission models, or when relativistic population inference separates spin from stellar radius and inclination under calibrated classification contamination.

Diagnostic: choose the clock before fitting mass

Two otherwise identical models fit an observed 60-day optical rise. Model A identifies it with $t_{\min}$ for a solar-type star. Model B sets $t_{\rm rise}=t_{\min}+45$ d for circularization and diffusion. What mass ratio follows, and what is the sound inference choice?

Solution

With fixed stellar properties, $t_{\min}\propto M^{1/2}$. Model A uses $t_{\min,A}=60$ d, while Model B uses $t_{\min,B}=15$ d. Therefore

$$\frac{M_A}{M_B} =\left(\frac{t_{\min,A}}{t_{\min,B}}\right)^2 =\left(\frac{60}{15}\right)^2=16.$$

The same light curve permits a sixteenfold mass shift before changing the star or $\beta$. The sound choice is not to average the two masses: fit a model indicator or model-averaged hierarchy, constrain delay with multi-band timing, and compare with a host-based mass held out from the transient likelihood.

References

• Gezari, Suvi. 2021. “Tidal Disruption Events.” Annual Review of Astronomy and Astrophysics 59, 21–58. https://doi.org/10.1146/annurev-astro-111720-030029.
• Kesden, Michael. 2012. “Tidal-Disruption Rate of Stars by Spinning Supermassive Black Holes.” Physical Review D 85, 024037. https://doi.org/10.1103/PhysRevD.85.024037.
• Langis, D. A., I. Liodakis, K. I. I. Koljonen, et al. 2026. “Repeating Flares, X-Ray Outbursts and Delayed Infrared Emission: A Comprehensive Compilation of Optical Tidal Disruption Events—TDECat.” Astronomy & Astrophysics 707, A171. https://doi.org/10.1051/0004-6361/202554882. CDS VizieR J/A+A/707/A171.