Future gravitational-wave detectors, especially the Laser Interferometer Space Antenna (LISA), will be sensitive to black hole binaries formed in astrophysical environments that promote large eccentricities and spin-induced orbital precession. Approximate models of gravitational waves that include both effects have only recently begun to be developed. The Efficient Fully Precessing Eccentric (EFPE) family is one such model, covering the inspiral stage with small-eccentricity-expanded gravitational-wave amplitudes accurate for initial time eccentricities $e < 0.3$ at 4 years before reaching an orbital frequency of 1 Hz. In this work, we extend this model to cover a larger range of initial eccentricities. The new EFPE for moderate eccentricities (EFPE_ME) model is able to accurately represent the leading-order gravitational-wave amplitudes to $e \leq 0.8$. Comparing the EFPE and the EFPE_ME models in the LISA band, however, reveals that there is no significant difference when the eccentricity at 4 years before merger, $e_0$, is less than or equal to 0.5, as radiation reaction circularizes supermassive black hole binaries too quickly. This suggests that the EFPE model may have a larger regime of validity in eccentricity space than previously thought, making it suitable for some inspiral parameter estimation with LISA data. On the other hand, for systems with $e_0 > 0.5$, the deviations between the models are significant, particularly for binaries with total masses below $10^5\, \mathrm{M}_{\odot}$. This suggests that the EFPE_ME model will be crucial to avoid systematic bias in parameter estimation with LISA in the future, once this model has been hybridized to include the merger and ringdown and the computation of the amplitudes is optimized.
We derive explicit expressions for the multi-detector F-statistic metric applied to short-duration non-precessing inspiral signals. This is required for template bank production associated with coherent searches for short-duration non-precessing inspiral signals in gravitational-wave data from a network of detectors. We compare the metric's performance with explicit overlap calculations for all relevant dimensions of parameter space and find the metric accurately predicts the loss of detection statistic above overlaps of 95%. We also show the effect that neglecting the variations of the detect…
Precessing black hole-neutron star (BH-NS) binaries produce a rich gravitational wave signal, encoding the binary's nature and inspiral kinematics. Using the lalinference\_mcmc Markov-chain Monte Carlo parameter estimation code, we use two fiducial examples to illustrate how the geometry and kinematics are encoded into the modulated gravitational wave signal, using coordinates well-adapted to precession. Even for precessing binaries, we show the performance of detailed parameter estimation can be estimated by "effective" estimates: comparisons of a prototype signal with its nearest neighbors, …
It is customary to use a precessing convention, based on Newtonian orbital angular momentum ${\bf L}_{\rm N}$, to model inspiral gravitational waves from generic spinning compact binaries. A key feature of such a precessing convention is its ability to remove all spin precession induced modulations from the orbital phase evolution. However, this convention usually employs a post-Newtonian (PN) accurate precessional equation, appropriate for the PN accurate orbital angular momentum ${\bf L}$, to evolve the ${\bf L}_{\rm N}$-based precessing source frame. This motivated us to develop inspiral wa…
The detection and characterization of gravitational wave signals from compact binary coalescence events relies on accurate waveform templates in the frequency domain. The stationary phase approximation (SPA) can be used to compute closed-form frequency-domain waveforms for non-precessing, quasi-circular binary inspirals. However, until now, no fast frequency-domain waveforms have existed for generic, spin-precessing quasi-circular compact binary inspirals. Templates for these systems have had to be computed via a discrete Fourier transform of finely sampled time-domain waveforms, which is far …
We present the first analytical inspiral-merger-ringdown gravitational waveforms from binary black holes (BBHs) with non-precessing spins, that is based on a description of the late-inspiral, merger and ringdown in full general relativity. By matching a post-Newtonian description of the inspiral to a set of numerical-relativity simulations, we obtain a waveform family with a conveniently small number of physical parameters. These waveforms will allow us to detect a larger parameter space of BBH coalescence, including a considerable fraction of precessing binaries in the comparable-mass regime,…
We describe a general procedure to generate spinning, precessing waveforms that include inspiral, merger and ringdown stages in the effective-one-body (EOB) approach. The procedure uses a precessing frame in which precession-induced amplitude and phase modulations are minimized, and an inertial frame, aligned with the spin of the final black hole, in which we carry out the matching of the inspiral-plunge to merger-ringdown waveforms. As a first application, we build spinning, precessing EOB waveforms for the gravitational modes l=2 such that in the nonprecessing limit those waveforms agree wit…
One of the greatest theoretical challenges in the build-up to the era of second-generation gravitational-wave detectors is the modeling of generic binary waveforms. We introduce an approximation that has the potential to significantly simplify this problem. We show that generic precessing-binary inspiral waveforms (covering a seven-dimensional space of intrinsic parameters) can be mapped to a two-dimensional space of non-precessing binaries, characterized by the mass ratio and a single effective total spin. The mapping consists of a time-dependent rotation of the waveforms into the quadrupole-…
We construct a new global, fully analytic, approximate spacetime which accurately describes the dynamics of non-precessing, spinning black hole binaries during the inspiral phase of the relativistic merger process. This approximate solution of the vacuum Einstein's equations can be obtained by asymptotically matching perturbed Kerr solutions near the two black holes to a post-Newtonian metric valid far from the two black holes. This metric is then matched to a post-Minkowskian metric even farther out in the wave zone. The procedure of asymptotic matching is generalized to be valid on all spati…
Biased parameter inference of eccentric, spin-precessing binary black holes
Cardiff University（卡迪夫大学）；Indian Institute of Technology Madras（马德拉斯印度理工学院）；Penn State University（宾夕法尼亚州立大学）；Tata Institute of Fundamental Research（塔塔基础研究所）
Multiple approaches are required to study the evolution of black-hole binaries. While the post-Newtonian approximation is sufficient to describe the early inspiral (even from infinitely large orbital separation), only numerical relativity can capture the full complexity of the dynamics near merger. We combine multi-timescale post-Newtonian integrations with numerical-relativity surrogate models, thus mapping the entire history of the binary from its asymptotic configuration at past-time infinity to the post-merger remnant. This approach naturally allows us to assess the impact of the precessio…
We study the performance of various analytical frequency-domain templates for detection and parameter estimation of gravitational waves from spin-precessing, quasi-circular, compact binary inspirals. We begin by assessing the extent to which non-spinning, spin-aligned, and the new (analytical, frequency-domain, small-spin) double-precessing frequency-domain templates can be used to detect signals from such systems. For effective, dimensionless spin values above $0.2$, the use of non-spinning or spin-aligned templates for detection purposes will result in a loss of up to $30%$ of all events, wh…
In this paper, we derive a set of equations of motions for binaries on eccentric orbits undergoing spin-induced precession that can efficiently be integrated on the radiation-reaction timescale. We find a family of solutions with a computation cost improved by a factor $10$ - $50$ down to $\sim 10$ ms per waveform evaluation compared to waveforms obtained by directly integrating the precession equations, that maintain a mismatch of the order $10^{-4}$ - $10^{-6}$ for waveforms lasting a million orbital cycles and a thousand spin-induced precession cycles. We express it in terms of parameters t…
I present an analytical fit to the duration of the inspiral of two point masses driven by gravitational-wave emission. The fit is accurate to within 3% over the entire range of initial eccentricities from 0 to 0.99999.
The coalescence of compact objects is one of the most promising sources of gravitational waves for ground-based interferometric detectors, such as advanced LIGO and Virgo. Generically, com- pact objects in binaries are expected to be spinning with spin angular momenta misaligned with the orbital angular momentum, causing the orbital plane to precess. This precession adds rich structure to the gravitational waves, introducing such complexity that an analytic closed-form description has been unavailable until now. We here construct the first closed-form frequency- domain gravitational waveforms …
We study precession dynamics of generic binary black holes in eccentric orbits using an effective potential based formalism derived in [M. Kesden et al., PRL {\bf 114}, 081103 (2015)]. This effective potential is used to classify binary black holes into three mutually exclusive spin morphologies. During the inspiral phase, binaries make transitions from one morphology to others. We evolve a population of binary black holes from an initial separation of $1000\mathbf{M}$ to a final separation of $10\mathbf{M}$ using post-Newtonian accurate evolution equations. We find that, given suitable initia…
Neutron star (NS) binaries formed dynamically may have significant eccentricities while emitting gravitational waves (GWs) in the LIGO/VIRGO band. We study tidal effects in such eccentric inspiralling NS binaries using a set of hybrid equations. The NS is modelled as a compressible ellipsoid, which can deform nonlinearly due to tidal forces, while the orbit evolution is treated with the post-Newtonian (PN) theory up to 2.5-PN order. We find that in general, the tidal interaction can accelerate the inspiral, and cause orbital frequency and phase shifts. For circular inspirals, our calculations …
We construct an inspiral-merger-ringdown eccentric gravitational-wave (GW) model for binary black holes with non-precessing spins within the effective-one-body formalism. This waveform model, SEOBNRv4EHM, extends the accurate quasi-circular SEOBNRv4HM model to eccentric binaries by including recently computed eccentric corrections up to 2PN order in the gravitational waveform modes, notably the $(l,|m|)=(2,2),(2,1),(3,3),(4,4),(5,5)$ multipoles. The waveform model reproduces the zero eccentricity limit with an accuracy comparable to the underlying quasi-circular model, with the unfaithfulness …
A precessing source frame, constructed using the Newtonian orbital angular momentum ${\bf L_{\rm N}}$, can be invoked to model inspiral gravitational waves from generic spinning compact binaries. An attractive feature of such a precessing convention is its ability to remove all spin precession induced modulations from the orbital phase evolution. However, this convention usually employs a post-Newtonian (PN) accurate precessional equation, appropriate for the PN accurate orbital angular momentum ${\bf L}$, to evolve the ${\bf L_{\rm N}}$-based precessing source frame. This influenced us to dev…
In this paper, we estimate the eccentricity of the 10 BBHs in GWTC-1 by using the inspiral-only BBH waveform template EccentricFD. Firstly, we test our method with simulated eccentric BBHs. Afterwards we apply the method to the real BBH gravitational wave data. We find that the BBHs in GWTC-1 except GW151226, GW170608 and GW170729 admit very small eccentricity. Their upper limits on eccentricity range from 0.033 to 0.084 with 90% credible interval at the reference frequency 10 Hz. For GW151226, GW170608 and GW170729, the upper limits are higher than 0.1. The relatively large eccentricity of GW…
We investigate the non-adiabatic dynamics of spinning black hole binaries by using an analytical Hamiltonian completed with a radiation-reaction force, containing spin couplings, which matches the known rates of energy and angular momentum losses on quasi-circular orbits. We consider both a straightforward post-Newtonian-expanded Hamiltonian (including spin-dependent terms), and a version of the resummed post-Newtonian Hamiltonian defined by the Effective One-Body approach. We focus on the influence of spin terms onto the dynamics and waveforms. We evaluate the energy and angular momentum rele…
Binary black holes which are both eccentric and undergo precession remain unexplored in numerical simulations. We present simulations of such systems which cover about 50 orbits at comparatively high mass ratios 5 and 7. The configurations correspond to the generic motion of a nonspinning body in a Kerr spacetime, and are chosen to study the transition from finite mass-ratio inspirals to point particle motion in Kerr. We develop techniques to extract analogs of the three fundamental frequencies of Kerr geodesics, compare our frequencies to those of Kerr, and show that the differences are consi…
Relativistic spin-orbit and spin-spin couplings has been shown to modify the gravitational waveforms expected from inspiraling binaries with a black hole and a neutron star. As a result inspiral signals may be missed due to significant losses in signal-to-noise ratio, if precession effects are ignored in gravitational-wave searches. We examine the sensitivity of the anticipated loss of signal-to-noise ratio on two factors: the accuracy of the precessing waveforms adopted as the true signals and the expected distributions of spin-orbit tilt angles, given the current understanding of their physi…
This paper presents a post-Newtonian (PN) template family of gravitational waveforms from inspiralling compact binaries with non-precessing spins, where the spin effects are described by a single "reduced-spin" parameter. This template family, which reparametrizes all the spin-dependent PN terms in terms of the leading-order (1.5PN) spin-orbit coupling term \emph{in an approximate way}, has very high overlaps (fitting factor > 0.99) with non-precessing binaries with arbitrary mass ratios and spins. We also show that this template family is "effectual" for the detection of a significant fractio…
We analyze the inspiral dynamics of equal-mass precessing black-hole binaries using multi-timescale techniques. The orbit-averaged post-Newtonian evolutionary equations admit two constants of motion in the equal-mass limit, namely the magnitude of the total spin $S$ and the effective spin $ξ$. This feature makes the entire dynamics qualitatively different compared to the generic unequal-mass case, where only $ξ$ is constant while the variable $S$ parametrizes the precession dynamics. For fixed individual masses and spin magnitudes, an equal-mass black-hole inspiral is uniquely characterized by…
The space-based detector LISA may observe gravitational waves from the early inspiral of stellar-mass black hole binaries, some of which could have significant eccentricity. Current gravitational waveform templates are only valid for small orbital velocities (i.e., in a post-Newtonian expansion) and small initial eccentricity $e_0$ ("post-circular" expansion). We conventionally define $e_0$ as the eccentricity corresponding to an orbital frequency of $5 \text{ mHz}$, and we study the convergence properties of frequency-domain inspiral templates that are accurate up to 2PN and order $e_0^6$ in …
In a recent investigation of the effects of precession on the anticipated detection of gravitational-wave inspiral signals from compact object binaries with moderate total masses, we found that (i) if precession is ignored, the inspiral detection rate can decrease by almost a factor of 10, and (ii) previously proposed ``mimic'' templates cannot improve the detection rate significantly (by more than a factor of 2). In this paper we propose a new family of templates that can improve the detection rate by factors of 5--6 in cases where precession is most important. Our proposed method for these n…
Spherical harmonic modes of gravitational waveforms for inspiraling compact binaries in eccentric orbits from post-Newtonian (PN) theory accurate to third post-Newtonian order, and those extracted from numerical relativity (NR) simulations for binary black holes (BBHs) are compared. We combine results from the two approaches (PN and NR) to construct time-domain hybrid waveforms that describe the complete evolution of BBH mergers through inspiral-merger-ringdown (IMR) stages. These hybrids are then used in constructing a fully analytical dominant mode ($\ell$=2, $|m|$=2) eccentric IMR model. A …
We probe the ability of various types of post-Newtonian(PN)-accurate circular templates to capture inspiral gravitational-wave (GW) signals from compact binaries having tiny orbital eccentricities. The GW signals are constructed by adapting the phasing formalism, available in T. Damour, A. Gopakumar, and B. R. Iyer, [Phys. Rev. D 70, 064028 (2004)], employing the orbital energy and the time-eccentricity to describe the orbital evolution. Using the fitting factor estimates, relevant for the initial LIGO, we show that circular templates, based on the adiabatic TaylorT1, complete adiabatic Taylor…
Post-Newtonian inspiral waveform model for eccentric precessing binaries with higher-order modes and matter effects
Max Planck Institute for Gravitational Physics (Albert Einstein Institute)（马克斯·普朗克引力物理研究所（爱因斯坦研究所））；Instituto de Física Teórica UAM/CSIC, Universidad Autónoma de Madrid（马德里自治大学理论物理研究所）；School of Physics and Astronomy and Institute for Gravitational Wave Astronomy, University of Birmingham（伯明翰大学物理与天文学院及引力波天文研究所）；Department of Physics, University of Maryland, College Park（马里兰大学帕克分校物理系）
We present a new phenomenological gravitational waveform model for the inspiral and coalescence of non-precessing spinning black hole binaries. Our approach is based on a frequency domain matching of post-Newtonian inspiral waveforms with numerical relativity based binary black hole coalescence waveforms. We quantify the various possible sources of systematic errors that arise in matching post-Newtonian and numerical relativity waveforms, and we use a matching criteria based on minimizing these errors; we find that the dominant source of errors are those in the post-Newtonian waveforms near th…
Observations of transient gravitational wave (GW) events with non-negligible orbital eccentricity can be highly rewarding from astrophysical considerations. Ready-to-use fully analytic frequency domain inspiral GW templates are crucial ingredients to construct eccentric inspiral-merger-ringdown waveform families, required for the detection of such GW events. It turns out that a fully analytic, post-Newtonian (PN) accurate frequency domain inspiral template family, which uses certain post-circular approximation, may only be suitable to model events with initial eccentricities $e_0 \leq 0.2$.We …
The inspiral-merger-ringdown (IMR) consistency test checks the consistency of the final mass and final spin of a binary black hole merger remnant, independently inferred via the inspiral and merger-ringdown parts of the waveform. As binaries are expected to be nearly circularized when entering the frequency band of ground-based detectors, tests of general relativity (GR) currently employ quasicircular waveforms. We quantify the effect of residual orbital eccentricity on the IMR consistency test. We find that eccentricity causes a significant systematic bias in the inferred final mass and spin …
We study the orbital evolution of eccentric binary neutron stars. The orbit is described as a Quasi-Keplarian orbit with perturbations due to tidal couplings. We find that the tidal interaction between stars contributes to orbital precession, in addition to the Post-Newtonian procession. The coupling between the angular and radial motion of the binary also excites a series of harmonics in the stars' oscillation. In the small eccentricity limit, this coupling mainly gives rise to an additional orbital resonance, with the orbital frequency being one third of the f-mode frequency. For a binary wi…
We build two families of inspiral waveforms for precessing binaries on eccentric orbits in the Fourier domain. To achieve this, we use a small eccentricity expansion of the waveform amplitudes in order to separate the periastron precession timescale from the orbital timescale, and use a shifted uniform asymptotics transformation to compute the Fourier transform in the presence of spin-induced precession. We show that the resulting waveforms can yield a median faithfulness above 0.993 when compared to an equivalent time domain waveform with an initial eccentricity of $e_0 \approx 0.3$. We also …
We present new developments and comparisons of competing inspiral and waveform models for highly eccentric non-spinning extreme and intermediate mass-ratio inspirals (EMRIs and IMRIs). Starting from our high eccentricity self-force library, we apply the near-identity transform (NIT) technique to rapidly compute highly eccentric self-forced inspirals for the first time. Upon evaluating our approximate NIT results via comparison with full self-force inspirals, we couple our accurate and streamlined inspiral data to potential waveform generation schemes. We find that, although high eccentricity s…
[abridged] Although gravitational radiation causes inspiralling compact binaries to circularize, a variety of astrophysical scenarios suggest that binaries might have small but nonnegligible orbital eccentricities when they enter the low-frequency bands of ground and space-based gravitational-wave detectors. If not accounted for, even a small orbital eccentricity can cause a potentially significant systematic error in the mass parameters of an inspiralling binary. Gravitational-wave search templates typically rely on the quasi-circular approximation, which provides relatively simple expression…
We derive analytic expressions that provide Fourier domain gravitational wave (GW) response function for compact binaries inspiraling along moderately eccentric orbits. These expressions include amplitude corrections to the two GW polarization states that are accurate to the first post-Newtonian (PN) order. Additionally, our fully 3PN accurate GW phase evolution incorporates eccentricity effects up to sixth order at each PN order. Further, we develop a prescription to incorporate analytically the effects of 3PN accurate periastron advance in the GW phase evolution. This is how we provide a rea…
Constraining initial orbital eccentricity of inspiral-dominated gravitational-wave events with an analytic approximant
The Chinese University of Hong Kong（香港中文大学）；Tata Institute of Fundamental Research（塔塔基础研究所）；Nikhef—National Institute for Subatomic Physics（NIKHEF-国家亚原子物理研究所）；KU Leuven（荷语鲁汶大学）；The Inter-University Centre for Astronomy and Astrophysics（大学间天文学与天体物理学中心）
Universal phenomenological relations between spherical harmonic modes in non-precessing eccentric binary black hole merger waveforms
Kavli Institute for Theoretical Physics, University of California Santa Barbara（加州大学圣塔芭芭拉分校卡弗里理论物理研究所）；Theoretical AstroPhysics Including Relativity and Cosmology, California Institute of Technology（加州理工学院理论天体物理学（含相对论与宇宙学）部门）；Department of Physics, University of California at Santa Barbara（加州大学圣塔芭芭拉分校物理系）；International Centre for Theoretical Sciences, Tata Institute of Fundamental Research（印度塔塔基础研究院国际理论科学中心）
Many relevant applications in gravitational wave physics share a significant common problem: the seven-dimensional parameter space of gravitational waveforms from precessing compact binary inspirals and coalescences is large enough to prohibit covering the space of waveforms with sufficient density. We find that by using the reduced basis method together with a parametrization of waveforms based on their phase and precession, we can construct ultra-compact yet high-accuracy representations of this large space. As a demonstration, we show that less than $100$ judiciously chosen precessing inspi…
Building upon several recent advances in the development of effective-one-body models for spin-aligned eccentric binaries with individual masses $(m_1,m_2)$ we introduce a new EOB waveform model that aims at describing inspiralling binaries in the large mass-ratio regime, $m_1\gg m_2$. The model exploits the current state-of-the-art TEOBResumS-Dali model for eccentric binaries, but the standard EOB potentials $(A,\bar{D},Q)$, informed by Numerical Relativity (NR) simulations, are replaced with the corresponding functions that are linear in the symmetric mass ratio $ν\equiv m_1 m_2/(m_1+m_2)^2$…
Unified remnant models for aligned-spin, precessing, and eccentric binary black hole mergers
Kavli Institute for Theoretical Physics, University of California Santa Barbara（加州大学圣塔芭芭拉分校卡弗里理论物理研究所）；Weinberg Institute, University of Texas at Austin（德克萨斯大学奥斯汀分校温伯格研究所）；Department of Physics and Institute for AI & Computational Research, University of Rhode Island（罗德岛大学物理与人工智能及计算研究院）；Department of Physics and Center for Scientific Computing & Data Science Research, University of Massachusetts, Dartmouth（马萨诸塞大学达特茅斯分校物理与科学计算及数据科学研究中心）
We introduce \texttt{TEOBResumSP}: an efficient, accurate hybrid scheme for generating gravitational waveforms from spin-precessing compact binaries. The precessing waveforms are generated via the established technique of Euler rotating the non-precessing \texttt{TEOBResumS} waveforms from a precessing frame to an inertial frame. We obtain the Euler angles by solving the post-Newtonian precession equations expanded to second post-Newtonian order. Current version of \texttt{TEOBResumSP} produces precessing waveforms through the inspiral phase up to the onset of the merger. We compare \texttt{TE…
Inspiraling compact binaries with non-negligible orbital eccentricities are plausible gravitational wave (GW) sources for the upcoming network of GW observatories. In this paper, we present two prescriptions to compute post-Newtonian (PN) accurate inspiral templates for such binaries. First, we adapt and extend the post-circular scheme of Yunes {\it et al.} [Phys. Rev. D 80, 084001 (2009)] to obtain a Fourier-domain inspiral approximant that incorporates the effects of PN-accurate orbital eccentricity evolution. This results in a fully analytic frequency-domain inspiral waveform with Newtonian…
ESIGMAHM: An Eccentric, Spinning inspiral-merger-ringdown waveform model with Higher Modes for the detection and characterization of binary black holes
We study binary systems in which a stellar mass compact object spirals into a massive black hole, known as extreme mass ratio inspirals, in scenarios with a new fundamental scalar field. Earlier work has shown that, in most interesting such scenarios and to leading order in the mass ratio, the massive black holes can be adequately approximated by the Kerr metric and the imprint of the scalar field on the waveform is fully controlled by the scalar charge of the stellar mass object. Here we use this drastic simplification in the inspiral modelling and consider eccentric equatorial orbits. We stu…
We construct hybrid binary black holes merger waveforms using analytical model waveforms for the early inspiral phase and numerical relativity waveforms for late inspiral to merger and post merger phases. To hybridize analytical and numerical waveforms, we first perform a 3-dimensional rotation to align the instantaneous orbital planes associated with the two waveforms at some fiducial frequency, we then find appropriate phase and time translations that maximize the overlap of the two waveforms in a hybridization interval. We discuss the accuracy and limitations for hybrids constructed by this…
Standard choices of quasi-circular orbit parameters for black-hole binary evolutions result in eccentric inspiral. We introduce a conceptually simple method, which is to integrate the post-Newtonian equations of motion through hundreds of orbits, and read off the values of the momenta at the separation at which we wish to start a fully general relativistic numerical evolution. For the particular case of non-spinning equal-mass inspiral with an initial coordinate separation of $D = 11M$ we show that this approach reduces the eccentricity by at least a factor of five to $e < 0.002$ as compared t…
The localization and distance inference of gravitational waves are two crucial factors for dark sirens as precise probes of cosmology, astrophysics, and fundamental physics. In this Letter, for the first time we investigate the parameter estimation of gravitational waves emitted by the eccentric compact binaries in the mid-frequency (0.1--10 Hz) band. Based on the configuration of one cluster of DECIGO (B-DECIGO), we simulate five types of typical compact binaries in GWTC-3 with component mass ranging from $\mathcal{O}(1\sim100)~M_{\odot}$. For each type of binaries, we assign discrete eccentr…
