Stellar Capture and Extreme-Mass-Ratio Sources

Dense stellar nuclei feed massive black holes through several neighboring outcomes. A star or compact remnant can be tidally disrupted, cross the relativistic capture boundary in a prompt plunge, or lose orbital energy to gravitational waves fast enough to become an extreme mass-ratio inspiral (EMRI). The distinctions are phase-space and timescale distinctions, not labels assigned from pericenter alone. Generic stellar dynamics and exact geodesic derivations are outside scope.

Required background. Geodesics and Jacobi Fields supplies affine geodesics, deviation, and turning-point language.

Helpful background. Black-Hole Geometry and Mechanics supplies Kerr capture boundaries; Relativistic Matter and Compact Objects supplies compact-remnant populations; Black-Hole Astrophysics supplies nuclear-cluster and detector context.

Two boundaries and two clocks

At fixed orbital energy $E$, define the normalized angular momentum

$$\mathcal R=\frac{J^2}{J_c^2(E)}, \qquad \mathcal R_{\rm lc}=\frac{J_{\rm lc}^2(E)}{J_c^2(E)},$$

where $E$ and $J$ are specific orbital energy and angular momentum, and $J_c(E)$ belongs to the circular orbit of the same energy. This spherical, orbit-averaged description is most transparent in a nearly Keplerian potential dominated by the central hole; it is not an exact Kerr phase-space coordinate system. Two-body encounters diffuse objects in $\mathcal R$. The dimensionless refilling parameter has the schematic form

$$q(E)\sim \frac{P(E)\,\overline\mu(E)}{\mathcal R_{\rm lc}(E)},$$

with orbital period $P$ and orbit-averaged angular-momentum diffusion coefficient $\overline\mu$. For $q\ll1$, the loss cone is empty: the distribution is depleted near its edge and flux depends on relaxation. For $q\gg1$, individual angular-momentum steps exceed the cone size, the local full-loss-cone or pinhole regime. At fixed total population in an energy interval, however, the capture flux approaches its isotropic, area-limited value only for the stronger ordering $q\gg\ln(1/\mathcal R_{\rm lc})$. The logarithm comes from depletion over the surrounding angular-momentum distribution; local mixing across the cone and saturation of the integrated rate are different statements. See the derivation in Loss Cones and Capture Rates and Stone et al. 2020, §§3.1–3.3, Eqs. 13–17.

For a compact remnant, crossing the capture separatrix produces a prompt plunge unless gravitational radiation decouples the orbit from scattering first. A useful local criterion is

$$t_{\rm GW}(a,e)<t_J(a,e), \qquad t_J\sim\mathcal R\,t_r,$$

where $t_r$ is the energy-relaxation time. The factor $\mathcal R\ll1$ matters: a highly eccentric orbit can change its angular momentum much faster than its energy. An EMRI must satisfy the inequality over enough cycles while remaining outside direct capture; otherwise it is a plunge that contributes no long, phase-coherent signal. Relativistic precession can quench coherent torques, mass segregation changes the compact-object supply, and spin shifts the capture boundary. Stone et al. 2020, §§3.5–3.6 and §6.3 explains why an EMRI rate cannot be obtained by multiplying a stellar TDE rate by a remnant fraction.

For a Kepler ellipse with semimajor axis $a$ and eccentricity $e$, $J^2=GMa(1-e^2)$ and $J_c^2=GMa$, so $\mathcal R=1-e^2$. In this estimate $t_{\rm GW}$ must be a declared local evolution time, such as $a/\lvert\dot a\rvert$, evaluated along the orbit’s inspiral history; it is not automatically the total time to merger. The timescale criterion identifies a promising region of phase space, not a deterministic classification of every stochastic scattering history.

Source data available through 2026-08-18

The named nuclear-cluster product is GALACTICNUCLEUS first data release, CDS VizieR J/A+A/631/A20, as served on 2026-08-18. It contains the HAWK-I $JHK_s$ point-source catalogues from 49 Galactic-centre and inner-bulge pointings, including corrected table versions recorded by CDS. The survey reports about $3.3\times10^6$ stars over $0.3\,\mathrm{deg}^2$, with $0.2''$ resolution. It is a tracer catalog, not a direct catalog of stellar-mass black holes.

Calibration is part of the likelihood, not a footnote. Astrometry was tied to VVV, photometric zero points were recomputed against SIRIUS as pointings were combined, and the quoted zero-point systematic is at most about $0.04$ mag. Catalogue acceptance required detection in all three sub-images; artificial-star tests quantify crowding completeness. The $K_s$ photometry becomes unreliable for bright sources near $K_s\simeq11.5$ because of saturation, while the three-sub-image requirement can omit roughly 30–40% of sources in the final holographic products. These boundaries and the 25-column schema are documented in Nogueras-Lara et al. 2019, §§3–4.4, Tables 1–2 and Figs. 4–7, with limitations in §6.

A defensible star-count model bins projected radius $R$, magnitude $K_s$, and, where usable, color. If $n_i$ is the count in bin $i$,

$$n_i\sim {\rm Poisson}(\lambda_i), \qquad \lambda_i=\int_i C(K_s,R)\,\Sigma(K_s,R\mid\vartheta)\,dA\,dK_s+B_i,$$

where $C$ is injection–recovery completeness, $B_i$ is foreground/background contamination, and $\vartheta$ includes projected cusp slope, normalization, and extinction parameters. A posterior needs explicit priors on deprojection, mass-to-light conversion, star-formation history, and remnant retention. Poisson width is statistical; extinction-law choice, unresolved blends, age–metallicity degeneracy, anisotropy, mass segregation, and mapping luminous stars to dark remnants are systematic or model uncertainties.

Detector selection is a second experiment

For detector claims, use the adopted-mission LISA Definition Study Report ESA-SCI-DIR-RP-002, created 2024-02-12, rather than an unlabeled internet sensitivity curve. Its nominal duration is 4.5 yr with duty cycle above 82%, corresponding to at least 3.7 yr of data; its source examples and horizon calculations are sky averaged and apply stated signal-to-noise thresholds. The report places typical EMRI power across roughly $1$–$10$ mHz and emphasizes multi-year, many-harmonic signals. See ESA 2024, PDF, §2.2 and Fig. 2.2; §3.3.1, Fig. 3.6 and Tables 3.6–3.7, plus ESA 2024, PDF, §4.4 and §7.1, Fig. 7.1.

The detector selection function must inject source-frame masses, spin, eccentricity, inclination, sky position, distance and redshift through a specified waveform and time-dependent response, then apply the stated mission window and threshold. Detector-frame masses are $(1+z)$ times source-frame masses. Prompt plunges, direct captures, and adiabatic inspirals must be separate population components. Waveform mismatch, confusion foreground, nonstationary gaps, cosmology, and source evolution are systematic; Monte Carlo counting error is statistical.

Reproduction boundary and workflow

The public reproduction path has two honest layers:

1. Download the exact VizieR data product J/A+A/631/A20 and its ReadMe; record retrieval date and the per-table correction date. Select a declared sky region, exclude sentinel non-detections and saturated $K_s<11.5$ sources, retain the catalog’s measurement uncertainties, and apply the authors’ artificial-star completeness rather than a hard faint cut.
2. Fit the Poisson or hierarchical star-count model and publish the query, masks, extinction map, priors, posterior draws, and covariance. Report the luminous-density inference before any remnant conversion.
3. Draw a remnant population and orbital distribution, evolve the competition $t_{\rm GW}<t_J$ with a specified capture boundary, and then inject survivors through the 2024 LISA response and mission window. Publish both intrinsic and detected counts.

The GALACTICNUCLEUS release does not supply six-dimensional orbits, compact-remnant identities, a relaxation model, EMRI waveforms, or detector software. Consequently, it can reproduce projected tracer counts but cannot by itself reproduce an EMRI rate. Results after 2026-08-18 are outside this contract.

Diagnostic: plunge or inspiral?

At some $(a,e)$, suppose $t_r=10^9$ yr, $\mathcal R=10^{-4}$, and $t_{\rm GW}=3\times10^4$ yr. The pericenter remains outside the capture separatrix. Classify the orbit and state what would change the answer.

Solution

The angular-momentum scattering time is

$$t_J\sim\mathcal R t_r=10^{-4}\times10^9\ {\rm yr}=10^5\ {\rm yr}.$$

Since $t_{\rm GW}=3\times10^4$ yr is shorter, radiation reaction can decouple the object from scattering and the orbit is an EMRI candidate, not a prompt plunge. Using $t_r$ instead of $t_J$ would exaggerate the safety margin by $10^4$. The classification can still change if spin and inclination move the capture separatrix across the pericenter or if coherent torques shorten $t_J$. A physically successful inspiral may still fall outside the detector selection window; that changes detectability, not whether the source is intrinsically an EMRI.

References

• European Space Agency. 2024. LISA Definition Study Report. ESA-SCI-DIR-RP-002, 12 February 2024. Public PDF.
• Nogueras-Lara, Francisco, Rainer Schödel, Aurelia Teresa Gallego-Calvente, et al. 2019. “GALACTICNUCLEUS: A High-Angular-Resolution JHKs Imaging Survey of the Galactic Centre. II. First Data Release of the Catalogue and the Most Detailed CMDs of the GC.” Astronomy & Astrophysics 631, A20. https://doi.org/10.1051/0004-6361/201936584. CDS VizieR J/A+A/631/A20.
• Stone, Nicholas C., Eugene Vasiliev, Michael Kesden, Elena M. Rossi, Hagai B. Perets, and Pau Amaro-Seoane. 2020. “Rates of Stellar Tidal Disruption.” Space Science Reviews 216, 35. https://doi.org/10.1007/s11214-020-00651-4.