The evolution of a supermassive binary black hole in an non-spherical nuclear star cluster

Alexander G. Polnarev; Pavel B. Ivanov

REVIEW 3 major objections 5 minor 69 references

The evolution of a supermassive binary black hole in an non-spherical nuclear star cluster

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that a quadrupole tidal potential with |m|=2 in a non-spherical nuclear star cluster, combined with Einstein apsidal precession, drives supermassive black-hole binaries to extreme eccentricity and can merge them within a…

desk verdict Good dynamics, bad arithmetic: the beta_E/10 eccentricity floor is real, but the few-Gyr SBBH merger timescale rests on a factor-10 error in Eq. (60). read the letter →

arxiv 2507.11684 v1 pith:U7MOJRC5 submitted 2025-07-15 astro-ph.GA gr-qc

Pavel B. Ivanov , Alexander G. Polnarev This is my paper

classification astro-ph.GAgr-qc

keywords supermassiveblackholebinariesnuclearstarclusterssecularorbitaldynamicsquadrupoletidalpotentialEinsteinapsidalprecessioneccentricityexcitationgravitational-wavecircularizationfinalparsecproblem

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper's central claim is that the quadrupole ($|m|=2$) tidal field of a non-spherical nuclear star cluster can slowly pump the eccentricity of a supermassive black hole binary to values extremely close to unity, even though no component of the binary's angular momentum is conserved. With Einstein apsidal precession included, the secular dynamics is shown to have a floor $\varepsilon_{\min}\approx \beta_E/10$ when $\beta_E>0.05$, and numerical solutions confirm that the averaged minimum of $\varepsilon=\sqrt{1-e^2}$ follows this scaling. Using observed nuclear-cluster scaling relations and a standard stellar density profile, the authors find that efficient circularization by gravitational-wave emission, with the circularization timescale shorter than an orbital period, can occur for mass ratios $q\gtrsim 0.04$ within a time of order or smaller than a few Gyr. If true, this gives a route to supermassive black hole mergers that does not require stellar hardening, and it predicts that some mergers should retain substantial eccentricity all the way down to the final plunge.

What carries the argument

The central object is the $|m|=2$ quadrupole term $V_2=\Omega^2\cos\nu\,(x^2-y^2)$ in the cluster tidal potential (eq. 2), the component that breaks conservation of every component of the binary's angular momentum and thereby allows eccentricity to reach values close to unity. The argument is carried by the dimensionless parameter $\beta_E=12(GM/(c^2a))(n_0/\Omega)^2$, the ratio of the Einstein apsidal-precession rate to the secular tidal rate; through the iterative map describing the evolution of the 'initial' inclination and nodal angle from cycle to cycle, and via Hamiltonian conservation, $\beta_E$ sets the limiting minimum of $\varepsilon=\sqrt{1-e^2}$ as $\beta_E/10$, the key relation that makes the astrophysical estimates possible.

What would settle it

Run a direct N-body simulation of a supermassive black hole binary with mass ratio $q=0.04$ embedded in a live triaxial nuclear star cluster with a Young density profile and no imposed symmetries, and compare the eccentricity extrema and the semi-major axis drift over roughly 100 secular timescales with the predictions of eqs. (60)–(62). If the semi-major axis changes by more than a factor of order unity, or if the minimum $\varepsilon$ does not approach $\beta_E/10$ when $\beta_E>0.05$, the central claim would be falsified; observationally, a sample of supermassive black hole mergers detected by future space-based detectors with $q\gtrsim 0.04$ that all show negligible eccentricity at frequencies above $10^{-5}$ Hz would cast doubt on the predicted eccentric signature.

Watch

Extended reading notes

Core claim

Within a non-spherical nuclear star cluster, once dynamical friction becomes inefficient because the enclosed stellar mass is below the secondary mass, the binary's secular evolution is governed by the cluster's quadrupole tidal potential. The paper studies the case where the $|m|=2$ harmonics dominate, so that no component of the orbital angular momentum is conserved; in the limit of small initial eccentricity and inclination the dynamics splits into cycles with long intervals at $i\approx 0$ and $i\approx \pi$, joined by 'sharp jumps' during which the eccentricity approaches unity. Einstein apsidal precession, acting mainly in these jumps, changes the effective initial orbital elements from cycle to cycle, and the conservation of the full Hamiltonian yields a lower bound $\varepsilon_{\min}=\beta_E/10$ on the minimal $\varepsilon=\sqrt{1-e^2}$. Numerical integrations over about 100 cycles show that the averaged minimal $\varepsilon$ is close to $\beta_E/10$ for $\beta_E>0.05$, and is roughly constant at about $5\cdot 10^{-3}$ for smaller $\beta_E$. Combined with the empirical relation between nuclear cluster size and primary black hole mass and a Young profile for the stellar density, this leads to the estimate that efficient gravitational-wave circularization occurs on timescales of a few Gyr or less for mass ratios $q\gtrsim 0.04$, with periastra possibly as small as a few gravitational radii, implying substantial eccentricity at merger or even direct collision from nearly parabolic orbits.

Load-bearing premise

The estimate that circularization happens within a few Gyr rests on the assumption that after dynamical friction stalls (eq. 57), the stars inside the binary orbit are quickly dispersed, no significant inflow replaces them, and the cluster's quadrupole potential remains fixed for approximately 100 eccentricity cycles; if the semi-major axis keeps evolving during the secular cycles, the $\beta_E$-scaling and the few-Gyr event rate do not follow.

Editorial extensions

If this is right

SBBH mergers can be produced in a few Gyr without a dense cusp or stellar hardening, giving a new route through the final parsec problem in galaxies with a non-spherical nuclear star cluster.
The mechanism predicts a distinctive gravitational-wave signature: for certain parameters, eccentricity remains of order 0.1–0.3 at frequencies around $10^{-5}$ Hz (the band of future space-based detectors), distinguishing these events from mergers produced by standard hardening.
For the largest achievable eccentricities, periastron can be as small as a few gravitational radii, so some events should appear as near-parabolic direct collisions or high-eccentricity plunges rather than quasi-circular inspirals.
The same secular dynamics applies to other settings, e.g. a binary star or proto-planetary system embedded in a massive deformed gas cloud, where the $|m|=2$ tidal term would similarly pump eccentricity.

Reading between the lines

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

Beyond the paper: the assumption of a fixed cluster potential and stalled semi-major axis is the paper's own caveat; in a live cluster that replenishes stars inside the orbit, the few-Gyr estimate and the $q\gtrsim 0.04$ threshold would need to be re-evaluated, possibly upward or downward.
Beyond the paper: the $\beta_E/10$ floor suggests a scaling law for the maximum eccentricity, $1-e_{\max}\sim (\beta_E/10)^2$, that could be tested against a suite of scattering or N-body experiments with varying cluster flattening and binary mass ratio.
Beyond the paper: because the $|m|=2$ term does not conserve angular momentum, the mechanism also changes the orbital plane; if real, it would produce a correlation between the merger orientation and the cluster's symmetry plane, which could in principle be probed with a statistical sample of future space-based gravitational-wave detections.
Beyond the paper: the iterative-map treatment of Einstein precession could be extended to include Lense-Thirring precession (neglected here on a stated inequality), which would introduce a dependence on the primary spin and alter the phase evolution of the final eccentric gravitational-wave signal.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Eliot — quick read of Ivanov & Polnarev. The dynamical core is better than I expected; the astrophysical wrapper is worse than the abstract suggests.

What's actually new: they give an asymptotic analytic solution for the m=2 quadrupole secular dynamics in the small-i0 limit, with a clean description of the sharp-jump structure, and they fold Einstein apsidal precession into an iterative map. The beta_E/10 floor on the minimum epsilon comes from Hamiltonian conservation and is confirmed numerically for beta_E > 0.05. That is a solid, checkable result, and the numerics in Figs. 5-7 look consistent. It's also a fair extension of the stellar-mass binary work by Petrovich & Antonini and Hamilton & Rafikov; the SBBH context is new.

Now the problem. I checked the Section 5 scalings myself. For q=0.04, mu=0.1, M8=1, their Eq. (57) gives a~2.7 pc, so n0~1.5e-4 yr^-1. With r_infl=23 pc, Eq. (59) gives Omega^2~3.7e-12 yr^-2. Then t_end~1e3 n0/Omega^2~4e10 yr, a factor of ~8-10 above the 4.9e9 yr quoted in Eq. (60). Eq. (58) looks off by a factor of ~3 in n0 as well. So the 'few Gyr for q>~0.04' headline is not supported by the paper's own formulas. You need either larger mu, a smaller r_infl, or a larger q to get the timescale down. The q>0.04 threshold really refers to the t_GW < P_orb condition, not to a few-Gyr event time.

The frozen-a assumption is also asserted rather than defended, and the no-inflow argument is hand-wavy. If the cluster replenishes or the potential evolves over ~100 eccentricity cycles, the whole t_end estimate moves. So I'd treat the timescale and event-rate numbers as order-of-magnitude illustrations, not robust predictions.

Bottom line: the mechanism is interesting and the analytic work deserves a serious referee. But the authors need to fix the arithmetic and tone down the event-rate claims. I'd send it out, with a clear request to redo Section 5 carefully.

Referee Report

3 major / 5 minor

Summary. The paper studies the secular evolution of a supermassive black hole binary (SBBH) embedded in a non-spherical nuclear star cluster (NSC), focusing on the quadrupole term with azimuthal number |m|=2 together with Einstein apsidal precession. It constructs an asymptotic analytic solution for small initial eccentricity and inclination, describes the slow evolution of the system as an iterative map, and derives a lower bound on the dimensionless pericentre parameter ε_min ≈ β_E/10 for β_E>0.05, which is verified numerically. The final section uses empirical NSC relations to estimate that, for mass ratios q≳0.04, gravitational-wave circularization can operate on timescales of order a few Gyr and can leave substantial eccentricity at merger. The central dynamical claim is the m=2 quadrupole can drive near-unity eccentricities and rapid GW-driven circularization without stellar hardening.

Significance. If the mechanism and the astrophysical estimates are correct, this is a distinctive alternative to the classical 'final parsec problem' and it produces a falsifiable prediction: SBBH mergers with non-negligible eccentricity down to the final stages, observable through the GW waveform (Eqs. 69–72). The paper contains genuine analytic progress: the asymptotic solution in Section III is nontrivial, the β_E/10 bound follows from Hamiltonian conservation, and the comparison of analytic results with numerical integrations in Figs. 2, 4, 6 and 10 is convincing in the tested regimes. The external inputs (Hamilton–Rafikov averaged potential, Peters GW decay, empirical NSC relations) are not fitted to the predicted events, so there is no obvious circularity. The main weakness is not the orbital dynamics itself but the quantitative astrophysical application, which contains an internal arithmetic/scaling inconsistency in the few-Gyr timescale estimate.

major comments (3)

[Section V, Eqs. (57)–(60)] The quoted event time is inconsistent with the paper's own scaling relations. Using Eq. (57) directly with nominal parameters (α_r=α_s=1, q_-2=4, M_8=1) gives a≈2.8 pc, and with GM_☉≈4.5×10^-7 pc^3 yr^-2, n_0≈1.45×10^-4 yr^-1. Eq. (59) with μ=0.1 and r_infl=23 pc gives Ω²≈3.7×10^-12 yr^-2, so Eq. (53) yields t_end≈10^3 n_0/Ω²≈3.9×10^10 yr, whereas Eq. (60) gives 1.2×10^10 yr for the same inputs. Moreover, combining Eqs. (57)–(59) gives t_end∝n_0/Ω²∝α_r^{3/2} α_s^{-1} M_8^{-0.475}, while Eq. (60) has α_s^{-1} replaced by α_s^{+1} and M_8^{+0.025}. The numerical coefficient of Eq. (58) is also about a factor 3 too large relative to Eq. (57). Since Eq. (60) is the basis of the abstract's 'a few Gyr' claim, the headline event-time estimate is currently unsupported.
[Section V, after Eq. (57); Section VI] The assumption that the evolution of the semi-major axis stops once a is given by Eq. (57), with the NSC quadrupole potential (Eq. 2) remaining fixed for ~100 secular cycles, is load-bearing but is not backed by any quantitative estimate. Dynamical friction and stellar replenishment from outside the orbit, or a change in the cluster's non-spherical mass distribution due to scattering by the secondary, would alter a, n_0, t_* and β_E (Eqs. 11, 17, 53). The paper states that 'there is no significant inflow of stars' and that the stars inside the orbit are 'quickly dispersed', but no timescale for replenishment or for the resulting change of the quadrupole field is provided. A quantitative statement about the validity time of the frozen-a, frozen-quadrupole approximation is needed before the few-Gyr scenario can be accepted.
[Section V, Eqs. (64)–(67); Conclusions] The paper connects the mass-ratio threshold q≳0.04 with a few-Gyr event time, but the threshold is derived from the GW circularization condition t_GW<P_orb in Eq. (64), not from the condition t_end<few Gyr. Even using the paper's own Eq. (60), q=0.04 gives t_end≈1.2×10^10 yr, which is already longer than a few Gyr; with the corrected scaling from the first major comment, the discrepancy is larger. The abstract and conclusions therefore conflate two different conditions. The authors should either revise the timescale claim or clearly specify the parameter range (e.g., α_r significantly smaller than unity or larger q) for which t_end is actually a few Gyr.

minor comments (5)

[Title] The title reads 'in an non-spherical nuclear star cluster'; it should be 'in a non-spherical nuclear star cluster'.
[Section IV, paragraph 2] The line 'We also set π=π/2 throughout this Section' should be 'We also set ν=π/2', since ν is the mixing angle in Eq. (2).
[Section V, Eq. (58)] The coefficient 1.8×10^-3 in Eq. (58) is about a factor 3 larger than the value obtained by substituting Eq. (57) into n_0=sqrt(GM/a^3); this likely explains part of the inconsistency in Eq. (60) and should be corrected together with the exponents.
[Section IV.B, Eq. (54)] The numerical floor ε_min≈5×10^-3 for β_E<0.05 is adopted from Fig. 6, but that figure is computed for e_0=i_0=0.1; the applicability of this floor to other initial inclinations, and in particular to the large-i_0 cases invoked in the conclusions, should be stated more carefully.
[Section V, Eq. (66)] Eq. (66) assumes the regime β_E>0.05, but Eq. (62) gives β_E<0.05 for q≳0.06 at nominal parameters; in that regime the floor ε_min≈5×10^-3 from Eq. (54) should be used rather than ε≈β_E/10. The periastron estimate Eq. (69) therefore needs to be restricted to the stated regime or modified.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the secular dynamics, epsilon_min bound, and event-rate scalings are derived from external averaged Hamiltonians, Hamiltonian conservation, and standard GW loss; self-citations are background only.

full rationale

The claimed derivation chain is self-contained: the secular equations start from the averaged quadrupole Hamiltonian of Hamilton and Rafikov [35] (external to this paper), the Einstein apsidal precession is the standard 1PN rate, and the final event criteria combine Peters' GW-averaged flux with empirical NSC-size and Young-profile relations [38,39]. The key bound epsilon_min approximately equal to beta_E/10 is obtained by conservation of the full Hamiltonian (Eq. 52 plus the bound of 10 on the tidal term) rather than fitted to the event rate; the companion numerical runs independently reproduce Eq. (51)/(54). Section V's parameters (a from M_st(a)=M_s, Omega from a mu-scaled NSC quadrupole, q threshold from t_GW < P_orb) are stated assumptions with free parameters, not retrofits to a target outcome. The several self-citations ([33], [31], [60]) are used for background, for eliminating m=1 terms, and for comparing the standard ZLK case; none supplies the load-bearing averaging or the GW-loss law. The stated Section V freeze-a approximation (stars dispersed, no inflow) is an explicit simplifying assumption rather than a definitional circularity. A possible order-of-magnitude arithmetic issue in Eq. (60) (t_end approximately 1.2 times 10^10 yr rather than a few Gyr at nominal q=0.04, mu=0.1, M8=1) is a correctness risk, not a circular step. No fitted quantity is renamed as a prediction, and no self-citation supplies the central theorem.

Assumptions & free parameters 8 free parameters · 6 assumptions · 0 invented entities

Central claim rests on a sequence of modeling choices: a smooth quadrupole NSC potential with m=2 dominance, a binary whose semi-major axis is frozen because nearby stars are dispersed, neglect of all relativistic effects except Einstein precession, and empirical NSC scaling relations. None of these are fitted to the target result, so the circularity burden is low even though the astrophysical predictions inherit the uncertainty of each choice.

free parameters (8)

quadrupole amplitude mu (degree of non-sphericity) = 0.1 assumed in estimates
Eq. (59) parametrizes Omega through mu; final q > 0.04 and few-Gyr estimates use mu = 0.1 ('non-spherical component order of ten per cent').
initial eccentricity e0 = 0.1 in numerical runs
The small-e0 expansion underlies the asymptotic solution; all runs set e0 = 0.1.
initial inclination i0 = 0.05 to 1.6 in numerical runs; analytic limit i0 to 0
epsilon_min scales as i0 / e0^(1/5) in eq. (44); event fraction depends on the distribution of i0.
initial nodal angle omega0 = uniform in (-pi/2, pi/2) for runs
epsilon_min depends on sin and cos of omega0 (eq. 45); the iterative map dynamics depends on omega0.
mass ratio q = constrained q >~ 0.04
Eq. (67) yields this threshold for efficient circularization; q is a model input.
primary mass M8 = nominal M8 = 1 in estimates
Scaling with M8 powers enters eqs. (55)-(68); not fitted to data.
NSC scale factors alpha_r and alpha_s = order unity assumed
alpha_r relates r_infl to NSC size (eq. 55), alpha_s sets M_st(r=a) = alpha_s M_s (eq. 57); both are assumed of order unity.
mixing angle nu = pi/2 in main analysis (m=2 only)
Sets relative strength of m=0 and m=2 terms in eq. (2); Appendix A checks nu not equal to pi/2 numerically.

assumptions (6)

standard math The averaged secular Hamiltonian for the quadrupole tidal potential (eqs. 8-9) is adopted from Hamilton and Rafikov [35] without re-derivation.
The central equations of motion (12)-(15) follow from this Hamiltonian; if it is incorrect, the analytic results change.
domain assumption The NSC potential is represented by a static quadrupole of the form Omega^2 [sin(nu)(2z^2 - x^2 - y^2) + cos(nu)(x^2 - y^2)] (eq. 2) with Omega much less than n0.
Relies on the Introduction's essential assumption that stellar density deviates significantly from uniform elliptical shells so the potential is non-trivial at small radii.
domain assumption After dynamical friction stalls, the stars in the secondary's orbit are quickly dispersed and no significant inflow replenishes them, freezing the semi-major axis at eq. (57).
Stated in Section V; without this, a continues to shrink and the secular timescale t* is not constant.
domain assumption All relativistic corrections except Einstein apsidal precession are neglected; Lense-Thirring is dropped using inequality (20).
Section II; at periastra near r_mb higher post-Newtonian terms matter, though the paper says full GR is needed for quantitative conclusions.
domain assumption The analytic small-inclination solution extends to large initial inclinations because the system spends long transients with small minimal i; this is shown numerically, not analytically (Fig. 7).
Required for astrophysical applicability, since a uniform distribution of initial inclinations mostly gives large i0.
domain assumption The empirical r_infl-M_* relation (eq. 55 from [38], early-type galaxies) and the Young r^(-3/2) density profile (eq. 56 from [39]) describe the NSC of interest.
Supplies the normalization for all timescale and mass-ratio estimates in Section V.

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Pith. "Pith review of The evolution of a supermassive binary black hole in an non-spherical nuclear star cluster." pith.science (2026). https://pith.science/paper/U7MOJRC5

@misc{pith2026250711684,
  author       = {Pith},
  title        = {Pith review of: The evolution of a supermassive binary black hole in an non-spherical nuclear star cluster},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U7MOJRC5}},
  note         = {Machine review of arXiv:2507.11684}
}

abstract

(abridged) We consider a secular orbital evolution of a supermassive binary black hole (SBBH) with unequal masses $M_p$ and $M_s < M_p$ in a central part of a non-spherical nuclear star cluster (NSC). When the mass of NSC inside the orbit is smaller than $M_{s}$ dynamical friction becomes inefficient. The subsequent orbital evolution of SBBH is largely governed by perturbing tidal potential of NSC arising from its non-sphericity. When the perturbing potential is mainly determined by quadrupole harmonics with azimuthal number $|m|=2$ the secular dynamics of the SBBH does not conserve any components of the angular momentum and can lead to the formation of highly eccentric orbits. Such orbits can experience an efficient circularization due to emission of gravitational waves (GW). In this Paper we consider this situation in some detail. We study analytically and numerically the orbital evolution taking into account the important effect of Einstein apsidal precession and estimate the largest possible value of $e$, which can be obtained. We then estimate a possibility of fast orbital circularization through emission of gravitation waves on a highly eccentric orbit, with circularization timescale of the order of the orbital period. We find that our mechanism could result in such events on a time scale of the order of or smaller than a few Gyr. It is stressed that for particular values of the model parameters such events may sometimes be distinguished from the ones expected in more standard scenarios, since in our case the eccentricity may remain substantial all the way down to the final merger. It is also noted that our results can be applied to other astrophysical settings, e.g. to study the orbital evolution of a binary star or a proto planetary system inside a massive deformed gas cloud.

Figures

Figures reproduced from arXiv: 2507.11684 by the authors.

**Figure 2.** Top panel. The numerical dependencies of [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** The dependencies of the ratio Ci/Ci,0 and i on time shown as the solid line and dashed lines, respectively. The initial values of i, e and ω are the same as the ones used in [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: The numerically obtained dependency of ǫ on time shown as the solid line and the analytic expression given by eq. (46), where the dashed line corresponds to κ = −, while the dotted line corresponds to κ = +. The initial values of i, e and ω are the same as the ones use…

**Figure 5.** Figure 5: Top panel. The minimal values of ǫmin found numerically over the calculation time corresponding to 102 cycles (∼ 400τ∗) as functions of the initial nodal angle ω0. e0 = i0 = 0.1 for all runs. Different curves correspond to different values of βE with the horizontal lin…

**Figure 6.** Figure 6: Top panel. The value of ǫmin averaged over the initial nodal angle ω0 as a function of βE is shown as a solid line. e0 = i0 = 0.1 for the shown case. The dashed line represents the theoretical value (51). Bottom panel. The same quantity calculated numerically as a func…

**Figure 7.** Figure 7: Top panel. The value of ǫmin as a function of time numerically obtained for the calculation with e0 = βE = 0.1, i0 = π/4 and ω0 = π/3. We show the whole time period t < tend. Bottom panel. e (solid line), i (dashed line) and sin(i) (dotted line) as functions of time sh…

**Figure 8.** Figure 8: Top panel. We show the dependency (70) calculated fo [PITH_FULL_IMAGE:figures/full_fig_p021_8.png]

**Figure 9.** Figure 9: The numerically obtained dependencies of [PITH_FULL_IMAGE:figures/full_fig_p023_9.png]

**Figure 10.** Figure 10: We show tan(ψ2) numerically obtained using eqns (B3), (B4), (B6) and (B7) after one iteration as a function of ω0 together with the corresponding results obtained by numerical solution of our equations of motion for the time period τ = 4τ∗. e0 = 0.1, i0 = 0.1 for all …

**Figure 11.** Figure 11: Top panel. The result of comparison of the map defin [PITH_FULL_IMAGE:figures/full_fig_p025_11.png]

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Works this paper leans on

69 extracted references · 53 canonical work pages

[1]

For that we use eq

The evolution of inclination in the linear regime When the inclination i is either small or close to π, we can use the solution described above to determine its evolution. For that we use eq. (13) setting there cos( i) = ±1, and sin( i) = i+, where i+ ≡ i when i ≪ 1, or sin(i) = −i−, where i− =i −π when i ≈π, to obtain i−1 ± di± dτ = − 2i ǫ (5e2 sin(2(̟ ±...

work page
[2]

(40) In order to check eq. (40) we show the ratio Ci/Ci,0, where Ci is evaluated for numerically calculated values of i, e and ω, and Ci,0 is a value of this quantity calculated for the initial values of these var iables, Ci,0 ≈i0 √ | cos(2ω)|, in Fig. 3. One can see that Ci is indeed close to a constant when i is either close to zero or π. Moreover, when...

work page
[3]

Therefore, we consid er below only the case α(τ = 0) = 0

It also follows from eqns (26) that they are invariant with respect to the change α →α +π, τ → −τ , and, therefore, the case α(τ ) = 0 and α(τ = 0) = π can be obtained from each other by the change of the sign of time. Therefore, we consid er below only the case α(τ = 0) = 0. In this case we expect that cos( α) > 0 in the course of the whole evolution and...

work page
[4]

We conclude that relatively large mass ratios q >∼ 0.04 are preferable

In Section 5 we estimate parameters of SBBHs and NSCs required for eﬃcient orbital circularization due to GWs emission. We conclude that relatively large mass ratios q >∼ 0.04 are preferable. We also provide an upper estimate of a 4 typical time elapsed before such circularization takes place

work page
[5]

Note that a reader who is interested in astrophysical applications o nly can omit Section 2-4 and go directly to Section 5, which is written in a self-consistent way

We summarize and ﬁnally discuss our results in Section 6. Note that a reader who is interested in astrophysical applications o nly can omit Section 2-4 and go directly to Section 5, which is written in a self-consistent way. II. THE SECULAR EVOLUTION OF SBBHS IN THE PRESENCE OF QUADRUP OLE PER TURBING POTENTIAL Let us consider the orbit of the secondary b...

work page
[6]

A minimal value of ǫ∗ obtained from a numerical analysis of our system In order to check the analytic approach discussed above we perfo rm a set of numerical runs with diﬀerent initial parameters. The initial eccentricity e0 is set to be equal to 0 .1 for all runs, but we consider several initial inclinations i0 in the range 0.05 − 1.6, the parameter βE i...

work page
[7]

However, our numerical results suggest the values of ǫ as small as the ones corresponding to small i0 can be obtained for i0 as large as π/2 when the suﬃciently large evolution time is considered, see Fig. 6. A possible explanation of this r esult is that over a large time there are transient periods of the evolution with suﬃciently small minimal values o...

work page
[8]

(B9) The condition (B9) tells again that our approach is invalid when ω0 ≈π/4, and, accordingly, ψ0 ≈ 0. Assuming that this condition holds we can extend the map (B8) to man y cycles by reducing it to ordinary diﬀerential equations according to the rule ∆ ψ → dψ0 dn and ∆ i → di0 dn , where n is a number of cycles [58]. Proceeding in this way we see that ...

work page

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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For that we use eq

The evolution of inclination in the linear regime When the inclination i is either small or close to π, we can use the solution described above to determine its evolution. For that we use eq. (13) setting there cos( i) = ±1, and sin( i) = i+, where i+ ≡ i when i ≪ 1, or sin(i) = −i−, where i− =i −π when i ≈π, to obtain i−1 ± di± dτ = − 2i ǫ (5e2 sin(2(̟ ±...

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(40) In order to check eq. (40) we show the ratio Ci/Ci,0, where Ci is evaluated for numerically calculated values of i, e and ω, and Ci,0 is a value of this quantity calculated for the initial values of these var iables, Ci,0 ≈i0 √ | cos(2ω)|, in Fig. 3. One can see that Ci is indeed close to a constant when i is either close to zero or π. Moreover, when...

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Therefore, we consid er below only the case α(τ = 0) = 0

It also follows from eqns (26) that they are invariant with respect to the change α →α +π, τ → −τ , and, therefore, the case α(τ ) = 0 and α(τ = 0) = π can be obtained from each other by the change of the sign of time. Therefore, we consid er below only the case α(τ = 0) = 0. In this case we expect that cos( α) > 0 in the course of the whole evolution and...

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We conclude that relatively large mass ratios q >∼ 0.04 are preferable

In Section 5 we estimate parameters of SBBHs and NSCs required for eﬃcient orbital circularization due to GWs emission. We conclude that relatively large mass ratios q >∼ 0.04 are preferable. We also provide an upper estimate of a 4 typical time elapsed before such circularization takes place

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Note that a reader who is interested in astrophysical applications o nly can omit Section 2-4 and go directly to Section 5, which is written in a self-consistent way

We summarize and ﬁnally discuss our results in Section 6. Note that a reader who is interested in astrophysical applications o nly can omit Section 2-4 and go directly to Section 5, which is written in a self-consistent way. II. THE SECULAR EVOLUTION OF SBBHS IN THE PRESENCE OF QUADRUP OLE PER TURBING POTENTIAL Let us consider the orbit of the secondary b...

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A minimal value of ǫ∗ obtained from a numerical analysis of our system In order to check the analytic approach discussed above we perfo rm a set of numerical runs with diﬀerent initial parameters. The initial eccentricity e0 is set to be equal to 0 .1 for all runs, but we consider several initial inclinations i0 in the range 0.05 − 1.6, the parameter βE i...

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However, our numerical results suggest the values of ǫ as small as the ones corresponding to small i0 can be obtained for i0 as large as π/2 when the suﬃciently large evolution time is considered, see Fig. 6. A possible explanation of this r esult is that over a large time there are transient periods of the evolution with suﬃciently small minimal values o...

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(B9) The condition (B9) tells again that our approach is invalid when ω0 ≈π/4, and, accordingly, ψ0 ≈ 0. Assuming that this condition holds we can extend the map (B8) to man y cycles by reducing it to ordinary diﬀerential equations according to the rule ∆ ψ → dψ0 dn and ∆ i → di0 dn , where n is a number of cycles [58]. Proceeding in this way we see that ...

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