The dynamical evolution of massive black hole binaries - I. Hardening in a fixed stellar background

Gerald D. Quinlan

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The dynamical evolution of massive black hole binaries - I. Hardening in a fixed stellar background

Gerald D. Quinlan

Astrophysics · Submitted 17 Jan 1996 · 31 references

Pith's verdict: The standard reference for the hard-binary hardening rate and for a hardness criterion grounded in the measured shape of H1(v); the eccentricity conclusion is the one place the paper outruns its data.

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CitationGerald D. Quinlan. "The dynamical evolution of massive black hole binaries - I. Hardening in a fixed stellar background." arXiv:astro-ph/9601092 (1996). https://pith.science/paper/YSPLUUBC

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This is the paper people cite for the hard-binary hardening normalization — H0 around 18 for equal masses, nearly flat in mass ratio — and, more usefully, for replacing the hand-waving definition of 'hard' with one tied to the measured shape of H1(v). That is the takeaway. The abstract's hardness criterion (abstract:s4) is the real contribution, and it is not a fitted constant: H1 goes flat below v roughly w and falls as 1/v^2 above it, and the capture cross section collapses at the same velocity. Two independent measurements with the same mass scaling. That holds up. The Chandrasekhar comparison is also right on its own equations — distant encounters move the center of mass, not the semimajor axis — and the demonstration that the friction route gives nonsense for H1 is worth having in print. The cometary cross-reference (Everhart's A/B/C parts, the N^-1/2 survival law) is a genuine connection, not decoration. The numerics are described well enough to reproduce: dopri8 at 1e-9, Jacobi conserved to 1e-6, Sobol sampling, adaptive allocation across impact-parameter bins, an explicit systematic-versus-statistical error budget, and a stated Keplerian shortcut beyond rk. The chaotic-orbit caveat is handled honestly. The soft spot that matters is the eccentricity extrapolation behind abstract:s2. K1 is measured only at m1/m2 = 1 and 16, and section 3.3 extends it by assuming w is the only mass-ratio dependence. But that same section reports the hard-binary maximum K1 is about 40% larger at r = 16 than at r = 1, at fixed v/w approaching zero. That is a counterexample to the stated assumption, and the interpolation simply adopts the r = 16 curve everywhere, leaving K at r = 64 and 256 unconstrained. In practice the growth rates are small, max around 0.2 to 0.3, and the qualitative conclusion probably survives a factor-of-two error in K. But the abstract states it with no mass-ratio qualifier, and it should have one. Second, the fixed uniform core: the author says plainly that the quoted merger times are too short and that loss-cone depletion is deferred to paper II. That is acknowledged, not hidden. Worth noting that the classifier's 'limitations: none found' is wrong — the paper states several. Read it if you compute BH binary merger timescales or gravitational-wave event rates, or if you are tempted to use Chandrasekhar's formula on a binary. Send it out.

Referee report

Significance

This is a careful, model-independent tabulation that supersedes earlier restricted three-body work on massive binaries (Roos 1981; Hills 1983a; Mikkola & Valtonen 1992) and corrects a long-standing misuse of Chandrasekhar's dynamical-friction formula for binary hardening, where distant encounters perturb the center of mass but not the semimajor axis. Several features deserve explicit credit: the capture definition is reproducible rather than program-dependent (a real improvement on Hills' step-count criterion); the error budget separates systematic from statistical contributions and states where each dominates; the quasi-random sampling is used deliberately to beat N^(-1/2) convergence; and the results are checked against independent analytic limits (Gould's impulse approximation at high v, the N^(-1/2) survival scaling of Everhart) rather than presented in isolation. The hardness criterion is a falsifiable prediction with a specific mass-ratio scaling, and it is corroborated by two measurements that were made independently — the bend in H1(v) (Table 1) and the transition velocity c2 Vbin in the capture cross section (Table 2) — which scale with m2/M12 in the same way. That cross-c

Major objections (3)

[Abstract (s2) vs. §4.3 and Fig. 8] The abstract states without qualification that 'the eccentricity growth is found to be unimportant unless the binary forms with a large eccentricity,' but §4.3 and Fig. 8 report that for e0 = 0.9 the merger time is reduced 'by about a factor of 20.' Since the merger time is one of the paper's stated motivations (§1: predicting the black hole merger rate and the detectability of the resulting gravitational waves), a factor of 20 is not naturally described as unimportant. The qualitative statement is defensible for e0 <= 0.3, where the eccentricity indeed barely grows, but the abstract should either restrict the claim to that range or state the merger-time sensitivity explicitly. As written, a reader who takes the abstract at face value would underestimate the systematic uncertainty on merger rates by more than an order of magnitude.
[§3.3, Table 4, Fig. 6] K1 is measured only at m1/m2 = 1 and 16, and is extended to all other mass ratios 'by assuming that the velocity w ... is the only important dependence on the mass ratio.' The paper's own §3.3 reports that the maximum K1 for m1/m2 = 16 is 'about 40% larger' than for m1/m2 = 1 in the hard-binary limit, where v/w -> 0 for both mass ratios. That is a mass-ratio dependence at fixed v/w, i.e. a direct counterexample to the stated scaling assumption, and it is not a small one. There are no measurements at m1/m2 = 64 or 256, so there is no evidence that the curve has saturated at 16. Because K enters de/d ln(1/a) (Eq. 4) and is integrated over several e-foldings, an underestimated K at extreme mass ratios could make the abstract's eccentricity conclusion fail for exactly the systems (m1 >> m2) where the rest of the paper's results are most distinctive. I recommend either restricting the abstract's eccentricity claim to the mass ratios actually measured, or quantifying how much K would have to grow at m1/m2 = 64–256 before the conclusion changes.
[§3.1.1, Eq. (17) and Table 1] The parameter w is load-bearing for the paper's headline claim: the abstract's (1 + m1/m2)^(1/2) scaling is Eq. (17) rewritten, and the hardness criterion is 'sigma << w.' Table 1 nevertheless quotes w/Vbin to four significant figures with no uncertainty at all, and the text states of the fit in Eq. (16) that 'it does not fit so well when H1 first starts to decrease; those data points were given little weight in the fitting procedure' — that is, the fit downweights precisely the bend region that determines w. The quoted prefactor 0.85 is a compromise across five mass ratios whose individual values deviate from it by up to about 12% (e.g. 0.4263 versus 0.85 sqrt(1/5) = 0.380 at m1/m2 = 4). Please attach uncertainties to the Table 1 entries and show explicitly that the (1 + m1/m2)^(1/2) scaling survives them. The independent corroboration from the capture-cross-section transition velocity c2 Vbin (Table 2) mitigates this concern substantially, and I would be satisfied with a short quantitative statement rather than new experiments.

Minor objections (8)

[§2.1, Eq. (5)] Eq. (5) as printed, d/dt(1/a) = G rho v H1, is dimensionally inconsistent with Eqs. (2), (6), (11), and (30): with H1 dimensionless (H0 = 17.97) and H dimensionless (H ~ 16 in Eq. 30), the velocity factor in Eq. (5) must appear in the denominator. The rest of the normalization chain is self-consistent, so this looks like a typographical slip, but since Eq. (5) is the defining equation for H1 it should be corrected.
[§3.1.1, Eqs. (19)–(20)] The text states that for an equal-mass binary 'the coefficients of the log terms differ by about 30%.' Substituting H0 = 17.97, w = 0.5675 Vbin, and beta = 2.40 into Eq. (19), and mu/M12 = 1/2 into Eq. (20), gives coefficients of roughly 2.3 (Vbin/sigma)^2 and 6.7 (Vbin/sigma)^2 respectively, a ratio closer to three than to 1.3. Please show the comparison explicitly, or state which normalization is being compared, since this agreement is used to validate the measurement against an independent analytic result.
[§3.2, Eq. (24) and Table 3] The text describes the Table 3 parameters as 'erratic' and notes that 'the velocity at the bend in the curves in panel (b) is not fit well by j2 Vbin'; for m1/m2 = 256 the parameters (j2 = 0.0214, j3 = 3.13, j4 = 0.53) are qualitatively unlike those at the other four mass ratios. Since Eq. (24) is integrated in §4 to obtain Mej and hence the open circles in Fig. 7 that mark where the fixed-background approximation breaks down, it would help to show fit residuals or to state how much Mej changes if the m1/m2 = 256 fit is replaced by direct interpolation of the measured points.
[§4.4, Eq. (37)] The origin of the numerical factor 10 and of the (r/rc)^2 scaling in Eq. (37) is not explained, and the expression does not reduce to the uniform-core value (4 pi/3) rho r^3 in the limit r << rc. A sentence of derivation, or a statement that this is a deliberately conservative estimate, would let the reader judge how much weight the open circles in Fig. 7 can bear.
[References] Everhart (1969) is listed as Astron. J. 73, 1039–1052, identical to Everhart (1968); the 1969 paper ('Close encounters of comets and planets') is Astron. J. 74, 735. Please correct the volume and pages.
[Fig. 4 caption] The caption defines the dotted lines as the cross section 'to remain bound with energy <= -vf^2/2,' which mixes a velocity threshold with an energy threshold without stating the sign convention for E*. Please state explicitly that the dotted curves show the bound (negative-energy) tail and how vf maps onto E*.
[§4.1] The integrations start at Vbin = sigma/5, described in the abstract as 'soon after the black holes become bound.' Some justification for that particular starting hardness, and a statement of how much the quoted merger times depend on it, would strengthen the connection between the abstract's framing and the actual initial condition.
[Appendix A] The appendix sets K1(e, v) = 0 for v > 2 Vbin and calls this 'some arbitrariness at high velocity' that 'does not matter for our applications.' Since K at high sigma is where the Chandrasekhar comparison in panel (f) is most discrepant, a one-line statement of the sensitivity of the plotted K to that cutoff would be useful.

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The paper

The paper claims

Scattering stars off a pair of massive black holes one at a time, Quinlan measures how fast the pair's orbit shrinks, how much stellar mass it throws out, and how its shape changes, and concludes that a black hole binary should not be class

The reading

Quinlan asks how a pair of massive black holes at the centre of a galaxy spirals inward, and what sets the rate. He answers by scattering individual stars off the pair, up to a million orbits per measurement, and boiling the results into three numbers: the hardening rate (how fast the separation shrinks), the ejection rate (how much stellar mass is thrown out), and the eccentricity growth rate. The headline result is a redefinition: a black hole binary should not be called hard, meaning it shrinks at a steady rate, until its orbital velocity exceeds the stellar velocity dispersion by a factor of $[1+m_1/m_2]^{1/2}$, not merely by order unity. If that is right, the usual assumption that any bound pair of massive black holes is automatically hard is wrong, and the time a binary spends stalled before gravitational waves take over is longer than commonly assumed.

What carries the argument

The load-bearing object is the dimensionless hardening rate $H_1(v)$, defined so that $d(1/a)/dt = G rho v H_1$, and its two-parameter fit $H_1 = H_0/[1+(v/w)^4]^{1/2}$, where $H_0$ is the flat low-velocity level and $w$ is the velocity at which the binary stops capturing stars. The fit does the work of turning a table of scattering outcomes into a single statement about when a binary is hard, and the same $w$ is found to control the capture cross section and the peak of the ejection-velocity distribution. The mechanism behind the flat plateau is a cancellation: for a fast star, energy gains and energy losses almost exactly cancel, removing four powers of $v$ from the rate, but below $w$ the stars that lose energy are captured and later expelled with a net gain, so the cancellation is broken and the rate stays constant. The paper's second piece of machinery is the pair of dimensionless changes $C$ (energy) and $B$ (angular momentum) used to build both $H_1$ and the eccentricity growth rate $K_1$ from the same scattering runs.

Extended reading notes

Core claim

The paper's central claim is that the hardening rate of a massive black hole binary is flat with respect to the incident stellar velocity $v$ below a threshold $w$, and falls as $1/v^2$ above it, with $w = 0.85 (G m_2/a)^{1/2}$. The flat level is $H_0$ in the range 18 to 23, almost independent of the mass ratio, so all hard binaries of this kind shrink at essentially the same fractional rate. The threshold has a physical reading: below $v = w$ the binary captures stars into bound orbits, and above it, it cannot. Since the hardening rate is constant only when $v < w$, and $v$ is set by the background dispersion, a binary is hard only when $V_{bin}/sigma$ exceeds roughly $[1+m_1/m_2]^{1/2}$. The paper supports this with three independent measurements: the hardening curve itself, the capture cross section, which collapses at the same velocity, and the distribution of ejection velocities, whose peak sits near $1.75w$. It also shows that applying Chandrasekhar's dynamical-friction formula to the semimajor axis is a category error, because distant encounters move the binary's centre of mass but leave the separation alone; the formula gives the nonsensical result that the hardening rate vanishes at zero stellar velocity. The measured eccentricity growth is modest, with a maximum near $K_1 = 0.2$ at $e = 0.7$ for an equal-mass binary, and the paper concludes eccentricity growth matters only if the binary forms with $e_0 > 0.3$. Applied to uniform galactic cores, the binary passes through three stages, and in most of the models shown it runs out of stars on intersecting orbits before it has ejected even a tenth of the mass needed to merge, so the quoted merger times are too short.

Load-bearing premise

The conclusion that eccentricity growth can be ignored for mass ratios other than 1 and 16 rests on assuming that the single velocity $w$ is the only way the mass ratio enters the growth rate, an interpolation adopted because that rate is the hardest quantity to measure.

Editorial extensions

If this is right

Because the flat hardening level $H_0$ is nearly the same for all mass ratios, the shrinkage timescale of a hard binary is set by the available stellar density rather than by the mass ratio, which simplifies estimates of black hole merger rates.
A hard binary does not need to eject its own mass in stars to shrink by a factor of $e$; the mean energy change per encounter is of order unity, so ejection and hardening are related but not equivalent.
A black hole of mass $m$ merging with $N$ holes of mass $m/N$ ejects more stellar mass than one merging with a single hole of mass $m$, so core scouring in merged galaxies should depend on the mass spectrum of the holes and not just on their total mass.
In the uniform-core models the binary typically depletes the stars on intersecting orbits well before it can merge, so the fixed-background merger times are lower bounds and some binaries would stall for the age of the galaxy.
Unless the binary forms with $e_0 > 0.3$, the orbit circularises before gravitational radiation dominates, so the final inspiral is circular; a large initial eccentricity can shorten the merger time by roughly a factor of twenty.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The redefinition of hard is really a statement about which stars count: only those the binary can capture contribute, which suggests the effective loss cone for a massive binary is narrower than the standard one by a factor of order $1+m_1/m_2$, a geometric prediction that N-body models could check directly.
The $N^{-1/2}$ survival law borrowed from cometary scattering implies that captured stars random-walk in energy at nearly fixed perihelion, so the mass ejected per e-folding should depend mainly on $w$ and hardly at all on the details of the orbit.
If ejection really is more efficient for very unequal mass ratios, then the observed trend of lower central densities in large ellipticals should correlate with the mass ratio of the holes that merged, not only with the total mass.
Since the paper's own models make merger times too short, the gravitational-wave merger rate from this channel in a realistic, self-consistent galaxy is probably lower than fixed-background estimates, which is in principle checkable against the shape of the low-frequency background.

Original abstract

The stellar ejection rate and the rates of change of the binary semimajor axis and eccentricity are derived from scattering experiments for the restricted three-body problem. They are used to study the evolution of binaries in simple models for galactic nuclei, starting soon after the black holes become bound and continuing until the evolution is dominated by the emission of gravitational radiation, or until the ejected mass is too large for the galaxy to be considered fixed. The eccentricity growth is found to be unimportant unless the binary forms with a large eccentricity. The scattering results are compared with predictions from Chandrasekhar's dynamical-friction formula and with previous work on the capture and scattering of comets by planetary systems. They suggest that a binary with masses $m_1\geq m_2$ should not be considered hard until its orbital velocity exceeds the background velocity dispersion by a factor that scales as $(1+m_1/m_2)^{1/2}$.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The stellar ejection rate and the rates of change of the binary semimajor axis and eccentricity are derived from scattering experiments for the restricted three-body problem. They are used to study the evolution of binaries in simple models for galactic nuclei, starting soon after the black holes become bound and continuing until the evolution is dominated by the emission of gravitational radiation, or until the ejected mass is too large for the galaxy to be considered fixed. The eccentricity growth is found to be unimportant unless the binary forms with a large eccentricity. The scattering results are compared with predictions from Chandrasekhar's dynamical-friction formula and with previous work on the capture and scattering of comets by planetary systems. They suggest that a binary with masses $m_1\geq m_2$ should not be considered hard until its orbital velocity exceeds the background velocity dispersion by a factor that scales as $(1+m_1/m_2)^{1/2}$.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The stellar ejection rate and the rates of change of the binary semimajor axis and eccentricity are derived from scattering experiments for the restricted three-body problem.	method 0.92	0.88	0.04	asserted 1.00
s1	They are used to study the evolution of binaries in simple models for galactic nuclei, starting soon after the black holes become bound and continuing until the evolution is dominated by the emission of gravitational radiation, or until the ejected mass is too large for the galaxy to be considered fixed.	method 0.53	0.85	0.07	asserted 0.97
s2	The eccentricity growth is found to be unimportant unless the binary forms with a large eccentricity.	result 0.99	0.94	0.16	asserted 1.00
s3	The scattering results are compared with predictions from Chandrasekhar's dynamical-friction formula and with previous work on the capture and scattering of comets by planetary systems.	method 0.63	0.74	0.04	hedged possibility 0.56
s4	They suggest that a binary with masses $m_1\geq m_2$ should not be considered hard until its orbital velocity exceeds the background velocity dispersion by a factor that scales as $(1+m_1/m_2)^{1/2}$.	interpretation 0.51	0.79	0.41	hedged possibility 0.97

What kind of paper the classifier reports this is: simulation study 0.78 · evidence from simulation or computation 0.89.

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sources read

arXiv listing abstract v1

not examined

full text
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references and citing sentences
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levels run

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Binary hardening rate versus the initial stellar velocity v: (a) shows three eccentricities for m1/m2 = 1; (b) and (c) show five mass ratios for e = 0, on linear and logarithmic scales. The lines are the fits to eq. (16). Roos (1981) and Hills (1983a) showed that the lowvelocity hardening rate H0 does not vary much with the mass ratio. But the velocity w does. The variation is fit by w ≃ 0.85r m2 M12 Vbin = 0.85r Gm2 a . (17) The physical significance of this mass dependence is the following: if v < w the binary can easily capture stars into bound orbits; if v > w it cannot. Th

**Figure 2.** The cross-section Σcap for a circular binary to capture an orbit with initial velocity v, plotted for five mass ratios on (a) linear and (b) logarithmic scales. The solid lines in (a) and (b) are the fits to eq. (22); the dashed line varies as (v/Vbin)−4 . Panels (c) and (d) show the cross section for an orbit to be captured for at least N revolutions for the same five mass ratios, plotted in (c) at the lowest v for each mass ratio, and in (d) at v = w. The dotted lines in (c) and (d) vary as N−1/2 . 1990 for further discussion). The N −1/2 scaling does not work as well if the…

**Figure 3.** Close-encounter cross sections for a hard, circular binary (v equals the lowest value in

**Figure 4.** Distribution of final velocities for a hard, circular binary (v equals the lowest value in

**Figure 5.** Mass ejection rate J for a circular binary, plotted on (a) linear and (b) logarithmic scales. The five lines are for mass ratios (increasing from right to left) m1/m2 = 1, 4, 16, 64, and 256. v/Vbin m1/m2 = 1 m1/m2 = 16 k0 k1 k2 k0 k1 k2 0.01 0.731 0.265 0.230 0.822 0.383 0.402 0.032 0.841 0.106 0.534 0.584 0.552 -0.140 0.1 0.724 0.053 0.275 0.381 0.202 -0.120 0.32 1.271 -0.198 0.445 0.739 -0.156 0.135 1.00 1.169 0.102 -0.022 1.221 -0.180 0.463

**Figure 6.** of Spitzer 1987) and is far more useful when the

**Figure 7.** Hardening time |a/a˙| and ejected mass Mej versus semimajor axis for BH binaries in two galaxy models: in model I (panels a–d), ρ = 103 and σ = 300; in model II (panels e and f), ρ = 106 and σ = 100. The initial eccentricity is 0.1. In (a) and (b), m1 = m2 = 108 (solid line), 107 (dotted), and 106 (dashed). In (c) and (d), m1 = 108 and m1/m2 = 1 (solid), 4, 16, 64, and 256 (dashed-dotted). Panels (e) and (f) are similar to (c) and (d) but use galaxy model II with m1 = 105 . The filled and open circles mark the points where a = ah (filled) and where the evolution could stall because…

**Figure 8.** for one of the binaries from

Figures reproduced from arXiv:astro-ph/9601092 by the authors.

References

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[30]

Xu, G. and Ostriker, J. P. 1994, Dynamics of massive black hol es as a possible candidate of galactic dark matter, Astrophys. J. 437, 184–193

View on Pith 1994
[31]

Yabushita, S. 1979, A statistical study of the evolution of t he orbits of long-period comets, Mon. Not. R. Astron. Soc. 187, 445–462. APPENDIX A: ECCENTRICITY GROWTH RATE FOR A MAXWELLIAN DISTRIBUTION The derivation of K from K1 is more cumbersome than the derivation of H from H1 because K1 is a function of two variables (e and v), because we do not have...

View on Pith 1979

Forward citations

Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Gaia's dark partners: A new signature of dissipative dark matter
astro-ph.GA 2026-11 conditional novelty 7.0 of 10

Dissipative dark matter produces slow-moving dark compact objects that efficiently exchange into stellar binaries, matching the observed abundance of Gaia BH1-like systems and enabling competitive constraints on dark ...
Post-Circular Expansion of Eccentric Binary Inspirals: Fourier-Domain Waveforms in the Stationary Phase Approximation
gr-qc 2009-06 accept novelty 6.0 of 10

A small-eccentricity expansion plus the stationary phase approximation yields closed-form frequency-domain gravitational waveforms for eccentric binaries, and the resulting harmonics raise detector mass reach by up to...
Eccentric binary black-hole mergers: The transition from inspiral to plunge in general relativity
gr-qc 2007-10 conditional novelty 6.0 of 10

Equal-mass black hole binaries transition from inspiral to plunge at orbital angular momentum L approximately 0.8 M squared, and orbits at that threshold produce the largest remnant Kerr parameter seen, j approximately 0.724.
On an empirical method to build near-equilibrium axisymmetric and triaxial galaxy models
astro-ph.GA 2026-07 conditional novelty 5.0 of 10

An improved adiabatic-squeezing method with an Ornstein-Uhlenbeck noise term lets modelers prescribe the final axial ratios of near-equilibrium triaxial N-body galaxy models.

How later papers used it

163 later papers cite it

Evidence: substantial. 163 of the citing papers share no author with it, and 36 of those carry a confident label from Pith's citation classifier.

Pith read 19 of the 276 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

8 later papers by other authors are listed below: 7 used its method or tool and 1 used its data or values.

Uses its method or tool 7 of 21 papers with this label

[equation] where $s(t)$ and $K$ denote the time-dependent hardening rate and the eccentricity growth rate of the binary, respectively, defined as [ref] ,

Scaling Relations for Binary Black Hole Merger Times from Cosmological Initial Conditions (2026) · Integration until coalescence · 0.93
Described the stellar-interaction evolution of the orbital elements as well described by this paper, using fits obtained by another work for the coefficients.

Eccentricity evolution of massive black hole binaries from formation to coalescence (2022) · 0.95 · paraphrased: the licence does not allow quoting
Determined the hardening rate of the binary following this paper, with a hard binary hardening at a constant rate.

Predicting the hypervelocity star population in Gaia (2017) · 0.98 · paraphrased: the licence does not allow quoting
Used scattering experiments analyzed with the method originally presented in this paper to calculate the ejected star velocities.

Hypervelocity Stars from a Supermassive Black Hole-Intermediate-mass Black Hole binary (2018) · 1.00 · paraphrased: the licence does not allow quoting
Following as well from [ref] , the total mass $ΔM_ej$ in stars ejected by the MBHB as it shrinks from separation $a$ to $a-Δa$ is modelled using the dimensionless mass ejection rate $J$:

Constraints on the Galactic Centre environment from \textit{Gaia} hypervelocity stars III: Insights on a possible companion to Sgr A* (2023) · Orbital decay of the MBHB · 0.93
Fit the binary hardening rate using this paper's model to confirm that the hardening process is modeled correctly in a cosmological simulation.

Resolving the Complex Evolution of a Supermassive Black Hole Triplet in a Cosmological Simulation (2021) · 0.96 · paraphrased: the licence does not allow quoting
Computed the time-dependent hardening rate of the binary in all models as in this paper, from binary formation to the end of the N-body integrations.

Defeating stochasticity: coalescence timescales of massive black holes in galaxy mergers (2020) · 1.00 · paraphrased: the licence does not allow quoting

Uses its data or values 1 of 12 papers with this label

Assumed the dimensionless parameter value of 3 as in this paper.

The Cosmic Evolution of Binary Black Holes in Young, Globular and Nuclear Star Clusters: Rates, Masses, Spins and Mixing Fractions (2021) · 0.95 · paraphrased: the licence does not allow quoting

How the literature uses it

48 arXiv papers cite this in 87 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 64 sentences without a confident label

Uses its method or tool 4 papers

In Eq. [ref] for the $ρ_env$ we include the density of the binaries $ρ_PBH binaries$, as the great majority of PBH binaries are wide and act as two single objects on the evolution of the tight binaries. $H(r,t)$ is the hardening rate [ref] , which can be approximated by [ref] , [equation] The hard semi-major axis $a_h(r,t)$ is defined by, [equation] The velocity dispersion of single objects (PBHs) and binaries in the halo depends on time and position, [equation] where $M(r,t)$ is the halo mass contained within a sphere of radius $r$ from its center, at time $t$.

Simulating Binary Primordial Black Hole Mergers in Dark Matter Halos · The evolution of PBH binaries properties · 0.81
[equation] where $s(t)$ and $K$ denote the time-dependent hardening rate and the eccentricity growth rate of the binary, respectively, defined as [ref] ,

Scaling Relations for Binary Black Hole Merger Times from Cosmological Initial Conditions · Integration until coalescence · 0.93
We create a Monte Carlo setup similar to that of Ref. [ref] used to determine the hardening rate of massive black hole binaries due to dynamical drag from a dense stellar environment.

Amplified capture rate of dark matter in compact binaries and constraints on bosonic dark matter from GW170817 · Simulation setup and results · 0.99

Uses its data or values 4 papers

The minimum value of $k_0$ estimated by Ref. [ref] is $k_0 ∼0.75$.

The merger rate of primordial-black-hole binaries · Dynamical friction by dark matter particles · 1.00
Past surveys [ref] have showed that $H$ lies in the interval [15,20] and $K$ is in [0,0.2].

Black Holes merging with Low Mass Gap Objects inside Globular Clusters · Hard binaries · 0.97
where the constant $H≈16$ is the typical hardening rate for a hard binary [ref] , and $ρ$ represents the galaxy core density.

A Population Study for Searching Supermassive Binary Black Holes in Active Galactic Nuclei: Continuum Spectral Features and Periodic Variabilities · Orbital evolution of active SMBBHs · 0.88

Compares with it 1 paper

The hardening rate $H$ and eccentricity growth rate $K$ have been studied previously in the literature [ref] , so our results for these serve as a validation of our code.

Self-acceleration of Hardening Binaries · Evolution parameters for $V=0$ · 0.84

Cites it as support 1 paper

Such a large difference in the stellar densities, explains the rapid BH mergers, since larger stellar densities increase the rate of binary hardening [ref] .

Rapid sinking and efficient mergers of supermassive black holes in compact high-redshift galaxies · Black hole binaries · 0.96

Cites it as background 12 papers

The binary separation is typically $a∼1-10 pc$ at this second stage and the further shrinking of the binary orbit will then primarily proceed by the scattering of individual stars, residing in the so called `loss-cone' [ref] .

Rapid sinking and efficient mergers of supermassive black holes in compact high-redshift galaxies · Introduction · 0.96
In gas-poor systems, the binary shrinks mainly through repeated three-body interactions with background stars [ref] , and may be further accelerated by interactions with a third massive BH brought in by subsequent mergers [ref] .

Winds Against Alignment: AGN Feedback and the Spin Evolution of Massive Black Hole Binaries · Introduction · 0.90
Following its formation, the binary hardens through the ejection of surrounding stars via three-body (slingshot) interactions [ref] .

Scaling Relations for Binary Black Hole Merger Times from Cosmological Initial Conditions · Evolution and timescales of black hole binaries · 0.84

Discussion

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View on Pith 1979