Time Of Core Collapse Research Articles

Interacting supernovae from wide massive binary systems

Context. The features in the light curves and spectra of many Type I and Type II supernovae (SNe) can be understood by assuming an interaction of the SN ejecta with circumstellar matter (CSM) surrounding the progenitor star. This suggests that many massive stars may undergo various degrees of envelope stripping shortly before exploding, and may therefore produce a considerable diversity in their pre-explosion CSM properties. Aims. We explore a generic set of about 100 detailed massive binary evolution models in order to characterize the amount of envelope stripping and the expected CSM configurations. Methods. Our binary models were computed with the MESA stellar evolution code, considering an initial primary star mass of 12.6 M⊙ and secondaries with initial masses of between ∼12 M⊙ and ∼1.3 M⊙, and focus on initial orbital periods above ∼500 d. We compute these models up to the time of iron core collapse in the primary. Results. Our models exhibit varying degrees of stripping due to mass transfer, resulting in SN progenitor models ranging from fully stripped helium stars to stars that have not been stripped at all. We find that Roche lobe overflow often leads to incomplete stripping of the mass donor, resulting in a large variety of pre-SN envelope masses. In many of our models, the red supergiant (RSG) donor stars undergo core collapse during Roche lobe overflow, with mass transfer and therefore system mass-loss rates of up to 0.01 M⊙ yr−1 at that time. The corresponding CSM densities are similar to those inferred for Type IIn SNe, such as SN 1998S. In other cases, the mass transfer becomes unstable, leading to a common-envelope phase at such late time that the mass donor explodes before the common envelope is fully ejected or the system has merged. We argue that this may cause significant pre-SN variability, as witnessed for example in SN 2020tlf. Other models suggest a common-envelope ejection just centuries before core collapse, which may lead to the strongest interactions, as observed in superluminous Type IIn SNe, such as SN 1994W and SN 2006gy. Conclusions. Wide massive binaries exhibit properties that may not only explain the diverse envelope stripping inferred in Type Ib, IIb, IIL, and IIP SNe, but also offer a natural framework to understand a broad range of hydrogen-rich interacting SNe. On the other hand, the flash features observed in many Type IIP SNe, such as SN 2013fs, may indicate that RSG atmospheres are more extended than currently assumed; this could enhance the parameter space for wide binary interaction.

Uncertainties impact on the major FOMs for severe accidents in CANDU 6 nuclear power plant

In the nuclear safety studies, a new trend refers to the evaluation of uncertainties as a mandatory component of best-estimate safety analysis which is a modern and technically consistent approach being known as BEPU (Best Estimate Plus Uncertainty).The major objectives of this study consist in performing a study of uncertainties/sensitivities of the major analysis results for a generic CANDU 6 Nuclear Power Plant during Station Blackout (SBO) progression to understand and characterize the sources of uncertainties and their effects on the key figure-of-merits (FOMs) predictions in severe accidents (SA).The FOMs of interest are hydrogen mass generation and event timings such as the first fuel channel failure time, beginning of the core disassembly time, core collapse time and calandria vessel failure time. The outcomes of the study, will allow an improvement of capabilities and expertise to perform uncertainty and sensitivity analysis with severe accident codes for CANDU 6 Nuclear Power Plant.

Supernova Precursor Emission and the Origin of Pre-explosion Stellar Mass Loss

A growing number of core-collapse supernovae (SNe) that show evidence for interaction with dense circumstellar medium (CSM) are accompanied by “precursor” optical emission rising weeks to months prior to the explosion. The precursor luminosities greatly exceed the Eddington limit of the progenitor star, implying that they are accompanied by substantial mass loss. Here, we present a semi-analytic model for SN precursor light curves, which we apply to constrain the properties and mechanisms of the pre-explosion mass loss. We explore two limiting mass-loss scenarios: (1) an “eruption” arising from shock breakout following impulsive energy deposition below the stellar surface; and (2) a steady “wind,” due to sustained heating of the progenitor envelope. The eruption model, which resembles a scaled-down version of Type IIP SNe, can explain the luminosities and timescales of well-sampled precursors, for ejecta masses ∼ 0.1–1 M ⊙ and velocities ∼ 100–1000 km s−1. By contrast, the steady wind scenario cannot explain the highest precursor luminosities ≳ 1041 erg s−1, under the constraint that the total ejecta mass does not exceed the entire progenitor mass (though the less luminous SN 2020tlf precursor can be explained by a mass-loss rate ∼ 1 M ⊙ yr−1). However, shock interaction between the wind and pre-existing (earlier ejected) CSM may boost its radiative efficiency and mitigate this constraint. In both the eruption and wind scenarios, the precursor ejecta forms compact (≲1015 cm) optically thick CSM at the time of core collapse; though only directly observable via rapid post-explosion spectroscopy (≲ a few days before being overtaken by the SN ejecta), this material can boost the SN luminosity via shock interaction.

Constraining the Time of Gravitational-wave Emission from Core-collapse Supernovae

The advent of sensitive gravitational-wave (GW) detectors, coupled with wide-field, high-cadence optical time-domain surveys, raises the possibility of the first joint GW–electromagnetic detections of core-collapse supernovae (CCSNe). For targeted searches of GWs from CCSNe, optical observations can be used to increase the sensitivity of the search by restricting the relevant time interval, defined here as the GW search window (GSW). The extent of the GSW is a critical factor in determining the achievable false alarm probability for a triggered CCSN search. The ability to constrain the GSW from optical observations depends on how early a CCSN is detected, as well as the ability to model the early optical emission. Here we present several approaches to constrain the GSW, ranging in complexity from model-independent analytical fits of the early light curve, model-dependent fits of the rising or entire light curve, and a new data-driven approach using existing well-sampled CCSN light curves from Kepler and the Transiting Exoplanet Survey Satellite. We use these approaches to determine the time of core-collapse and its associated uncertainty (i.e., the GSW). We apply our methods to two Type II SNe that occurred during LIGO/Virgo Observing Run 3: SN 2019fcn and SN 2019ejj (both in the same galaxy at d = 15.7 Mpc). Our approach shortens the duration of the GSW and improves the robustness of the GSW compared to the techniques used in past GW CCSN searches.

Remnant masses of core collapse supernovae in the jittering jets explosion mechanism

ABSTRACT We conduct one-dimensional (1D) stellar evolution simulations of non-rotating stars with initial masses in the range of $11\!-\!48 \, \mathrm{M}_{\odot }$ to the time of core collapse and, using a criterion on the specific angular momentum fluctuations in the inner convective zones, estimate the masses of the neutron star (NS) remnants according to the jittering jets explosion mechanism. From the 1D simulations, we find that several convective zones with specific angular momentum fluctuations of $j_{\rm {conv}} \gtrsim 2.5 \times 10^{15} {\, \rm cm}^2 {\, \rm s}^{-1}$ develop near the edge of the iron core in all models. For this condition for explosion, we find the NS remnant masses to be in the range of $1.3\!-\!1.8 \, \mathrm{M}_\odot$, while if we require twice as large values, i.e. $j_{\rm {conv}} \gtrsim 5 \times 10^{15} {\, \rm cm}^2 {\, \rm s}^{-1}$, we find the NS remnant masses to be in the range of $1.4\!-\!2.8 \, \mathrm{M}_\odot$ (the upper values here might form black holes). Note that in general, the formation of black holes in the jittering jets explosion mechanism requires a rapidly rotating pre-collapse core, while we simulate non-rotating stars.

Introducing a new multi-particle collision method for the evolution of dense stellar systems

Context.In a previous paper we introduced a new method for simulating collisional gravitationalN-body systems with linear time scaling onN, based on the multi-particle collision (MPC) approach. This allows us to easily simulate globular clusters with a realistic number of stellar particles (105 − 106) in a matter of hours on a typical workstation.Aims.We evolve star clusters containing up to 106stars to core collapse and beyond. We quantify several aspects of core collapse over multiple realizations and different parameters while always resolving the cluster core with a realistic number of particles.Methods.We run a large set ofN-body simulations with our new code MPCDSS. The cluster mass function is a pure power law with no stellar evolution, allowing us to clearly measure the effects of the mass spectrum on core collapse.Results.Leading up to core collapse, we find a power-law relation between the size of the core and the time left to core collapse. Our simulations thus confirm the theoretical self-similar contraction picture but with a dependence on the slope of the mass function. The time of core collapse has a non-monotonic dependence on the slope, which is well fitted by a parabola. This also holds for the depth of core collapse and for the dynamical friction timescale of heavy particles. Cluster density profiles at core collapse show a broken-power-law structure, suggesting that central cusps are a genuine feature of collapsed cores. The core bounces back after collapse, with visible fluctuations, and the inner density slope evolves to an asymptotic value. The presence of an intermediate-mass black hole inhibits core collapse, making it much shallower, irrespective of the mass-function slope.Conclusions.We confirm and expand on several predictions of star cluster evolution before, during, and after core collapse. Such predictions were based on theoretical calculations or small-size directN-body simulations. Here we put them to the test in MPC simulations with a much larger number of particles, allowing us to resolve the collapsing core.

Supplying angular momentum to the jittering jets explosion mechanism using inner convection layers

ABSTRACT We conduct one-dimensional stellar evolution simulations in the mass range 13–20 M⊙ to late core collapse times and find that an inner vigorous convective zone with large specific angular momentum fluctuations appears at the edge of the iron core during the collapse. The compression of this zone during the collapse increases the luminosity there and the convective velocities, such that the specific angular momentum fluctuations are of the order of $j_{\rm conv} \simeq 5 \times 10^{15} {~\rm cm}^2 {~\rm s}^{-1}$. If we consider that three-dimensional simulations show convective velocities that are three to four times larger than what the mixing length theory gives, and that the spiral standing accretion shock instability in the post-shock region of the stalled shock at a radius of ${\simeq}100 {~\rm km}$ amplifies perturbations, we conclude that the fluctuations that develop during core collapse are likely to lead to stochastic (intermittent) accretion discs around the newly born neutron star. In reaching this conclusion, we also make two basic assumptions with uncertainties that we discuss. Such intermittent discs can launch jets that explode the star in the frame of the jittering jets explosion mechanism.

Fast Multipole Methods for N-body Simulations of Collisional Star Systems

Abstract Direct N-body simulations of star clusters are accurate but expensive, largely due to the numerous  ( N 2 ) pairwise force calculations. To solve the post-million-body problem, it will be necessary to use approximate force solvers, such as tree codes. In this work, we adapt a tree-based, optimized Fast Multipole Method (FMM) to the collisional N-body problem. The use of a rotation-accelerated translation operator and an error-controlled cell opening criterion leads to a code that can be tuned to arbitrary accuracy. We demonstrate that our code, Taichi, can be as accurate as direct summation when N > 104. This opens up the possibility of performing large-N, star-by-star simulations of massive stellar clusters and would permit large parameter space studies that would require years with the current generation of direct summation codes. Using a series of tests and idealized models, we show that Taichi can accurately model collisional effects, such as dynamical friction and the core-collapse time of idealized clusters, producing results in strong agreement with benchmarks from other collisional codes such as NBODY6++GPU or PeTar. Parallelized using OpenMP and AVX, Taichi is demonstrated to be more efficient than other CPU-based direct N-body codes for simulating large systems. With future improvements to the handling of close encounters and binary evolution, we clearly demonstrate the potential of an optimized FMM for the modeling of collisional stellar systems, opening the door to accurate simulations of massive globular clusters, super star clusters, and even galactic nuclei.

Precollapse Properties of Superluminous Supernovae and Long Gamma-Ray Burst Progenitor Models

Abstract We analyze the properties of 42 rapidly rotating, low-metallicity, quasi-chemically homogeneously evolving stellar models in the mass range between 4 and 45 at the time of core collapse. Such models were proposed as progenitors for both superluminous supernovae (SLSNe) and long-duration gamma-ray bursts (lGRBs) and the Type Ic-BL supernovae (SNe) that are associated with them. Our findings suggest that whether these models produce a magnetar-driven SLSN explosion or a near-critically rotating black hole is not a monotonic function of the initial mass. Rather, their explodability varies nonmonotonically depending on the late core evolution, once chemical homogeneity is broken. Using different explodability criteria, we find that our models have a clear preference to produce SLSNe at lower masses and lGRBs at higher masses, but we find several exceptions, expecting lGRBs to form from stars as low as 10 and SLSNe with progenitors as massive as 30 . In general, our models reproduce the predicted angular momenta, ejecta masses, and magnetic field strengths at core collapse inferred for SLSNe and lGRBs and suggest significant interaction with their circumstellar medium, particularly for explosions with low ejecta mass.

Hydrodynamic Simulations of Pre-supernova Outbursts in Red Supergiants: Asphericity and Mass Loss

Abstract The activity of a massive star approaching core-collapse can strongly affect the appearance of the star and its subsequent supernova. Late-phase convective nuclear burning generates waves that propagate toward the stellar surface, heating the envelope and potentially triggering mass loss. In this work, we improve on previous one-dimensional models by performing two-dimensional simulations of the pre-supernova mass ejection phase due to deposition of wave energy. Beginning with stellar evolutionary models of a 15 M ⊙ red supergiant star during core O-burning, we treat the rate and duration of energy deposition as model parameters and examine the mass-loss dependence and the pre-explosion morphology accordingly. Unlike one-dimensional models, density inversions due to wave heating are smoothed by Rayleigh–Taylor instabilities, and the primary effect of wave heating is to radially expand the star’s hydrogen envelope. For low heating rates with long durations, the expansion is nearly homologous, whereas high but short-lived heating can generate a shock that drives envelope expansion and results in a qualitatively different density profile at the time of core-collapse. Asymmetries are fairly small, and large amounts of mass loss are unlikely unless the wave heating exceeds expectations. We discuss implications for pre-supernova stellar variability and supernovae light curves.

Radiation hydrodynamical simulations of eruptive mass loss from progenitors of Type Ibn/IIn supernovae

Context. Observations suggest that some massive stars experience violent and eruptive mass loss associated with significant brightening that cannot be explained by hydrostatic stellar models. This event seemingly forms dense circumstellar matter (CSM). The mechanism of eruptive mass loss has not been fully explained. We focus on the fact that the timescale of nuclear burning gets shorter than the dynamical timescale of the envelope a few years before core collapse for some massive stars. Aims. To reveal the properties of the eruptive mass loss, we investigate its relation to the energy injection at the bottom of the envelope supplied by nuclear burning taking place inside the core. In this study, we do not specify the actual mechanism for transporting energy from the site of nuclear burning to the bottom of the envelope. Instead, we parameterize the amount of injected energy and the injection time and try to extract information on these parameters from comparisons with observations. Methods. We carried out 1D radiation hydrodynamical simulations for progenitors of red, yellow, and blue supergiants, and Wolf–Rayet stars. We calculated the evolution of the progenitors with a public stellar evolution code. Results. We obtain the light curve associated with the eruption, the amount of ejected mass, and the CSM distribution at the time of core-collapse. Conclusions. The energy injection at the bottom of the envelope of a massive star within a period shorter than the dynamical timescale of the envelope could reproduce some observed optical outbursts prior to the core-collapse and form the CSM, which can power an interaction supernova classified as Type IIn.

Formation of binary black holes in star clusters as gravitational wave sources

Direct detections of gravitational wave suggest that ~ 30 solar mass binary black holes (BBHs) commonly exist in the Universe. One possible formation scenario of such BBHs is dynamical three-body encounters in a dense core of globular clusters, which consist of millions of stars. Compared to globular clusters, open clusters are less dense and less massive but more populous. Because of their shallow potential, they have not been considered to be a formation site of merging BBHs. However, we found a new channel for the formation of BBHs, which is dominant in open clusters. We performed direct N-body simulations of open clusters with a mass of 2500 solar mass for metallicity of Z/Z ⊙ = 0.1. The core-collapse time of open clusters is much shorter than that of globular clusters. Therefore, massive main-sequence stars can form binaries before they evolve to BHs. These main-sequence binaries experience mass transfer evolution, and some of them evolve to BBHs merging within the Hubble time. From our simulations, we estimated the merger rates of BBHs originated from open clusters.

Constraining Dissipative Dark Matter Self-Interactions.

We study the gravothermal evolution of dark matter halos in the presence of dissipative dark matter self-interactions. Dissipative interactions are present in many particle-physics realizations of the dark-sector paradigm and can significantly accelerate the gravothermal collapse of halos compared to purely elastic dark matter self-interactions. This is the case even when the dissipative interaction timescale is longer than the free-fall time of the halo. Using a semianalytical fluid model calibrated with isolated and cosmological N-body simulations, we calculate the evolution of the halo properties-including its density profile and velocity dispersion profile-as well as the core-collapse time as a function of the particle model parameters that describe the interactions. A key property is that the inner density profile at late times becomes cuspy again. Using 18 dwarf galaxies that exhibit a corelike dark matter density profile, we derive constraints on the strength of the dissipative interactions and the energy loss per collision.

Unexpectedly high formation rate of merging binary black holes in open clusters

AbstractGravitational wave direct detections suggest that 30 M⊙ binary black holes (BBHs) commonly exist in the universe. One possible formation scenario of such BBHs is dynamical three-body encounters in dense star clusters. We performed a series of direct N-body simulations with a mass of 2500 and 10000 M⊙ and found a new channel for the formation of BBHs which is dominant in open clusters. In open clusters, the core-collapse time is shorter than in globular clusters, and therefore massive main-sequence (MS) binaries can form before they evolve to BHs. These MS binaries experience common envelope evolution and evolve to hard BBHs, which can merge within the Hubble time. The number of BBH mergers per unit mass obtained from our simulations reached 20–50 % of that for globular clusters, assuming an initial cluster mass function. Thus, open clusters can be a dominant formation site of hard BBHs.

The hunt for self-similar core collapse

Context. Core collapse is a prominent evolutionary stage of self-gravitating systems. In an idealised collisionless approximation, the region around the cluster core evolves in a self-similar way prior to the core collapse. Thus, its radial density profile outside the core can be described by a power law, ρ ∝ r−α. Aims. We aim to find the characteristics of core collapse in N-body models. In such systems, a complete collapse is prevented by transferring the binding energy of the cluster to binary stars. The contraction is, therefore, more difficult to identify. Methods. We developed a method that identifies the core collapse in N-body models of star clusters based on the assumption of their homologous evolution. Results. We analysed different models (equal- and multi-mass), most of which exhibit patterns of homologous evolution, yet with significantly different values of α : the equal-mass models have α ≈ 2.3, which agrees with theoretical expectations, the multi-mass models have α ≈ 1.5 (yet with larger uncertainty). Furthermore, most models usually show sequences of separated homologous collapses with similar properties. Finally, we investigated a correlation between the time of core collapse and the time of formation of the first hard binary star. The binding energy of such a binary usually depends on the depth of the collapse in which it forms, for example from 100 kT to 104 kT in the smallest equal-mass to the largest multi-mass model, respectively. However, not all major hardenings of binaries happened during the core collapse. In the multi-mass models, we see large transfers of binding energy of ∼104 kT to binaries that occur on the crossing timescale and outside of the periods of the homologous collapses.

First targeted search for gravitational-wave bursts from core-collapse supernovae in data of first-generation laser interferometer detectors

We present results from a search for gravitational-wave bursts coincident with a set of two core-collapse supernovae observed between 2007 and 2011. We employ data from the Laser Interferometer Gravitational-wave Observatory (LIGO), the Virgo gravitational-wave observatory, and the GEO 600 gravitational-wave observatory. The targeted core-collapse supernovae were selected on the basis of (1) proximity (within approximately 15 Mpc), (2) tightness of observational constraints on the time of core collapse that defines the gravitational-wave search window, and (3) coincident operation of at least two interferometers at the time of core collapse. We find no plausible gravitational-wave candidates. We present the probability of detecting signals from both astrophysically well-motivated and more speculative gravitational-wave emission mechanisms as a function of distance from Earth, and discuss the implications for the detection of gravitational waves from core-collapse supernovae by the upgraded Advanced LIGO and Virgo detectors.

The role of stellar relaxation in the formation and evolution of the first massive black holes

We present calculations on the formation of massive black holes with 10^5 Msun at z > 6 that can be the seeds of supermassive black holes at z > 6. Under the assumption of compact star cluster formation in merging galaxies, star clusters in haloes of 10^8 ~ 10^9 Msun can undergo rapid core-collapse leading to the formation of very massive stars (VMSs) with ~1000 Msun which directly collapse into black holes with similar masses. Star clusters in halos of > 10^9 Msun experience type-II supernovae before the formation of VMSs due to long core-collapse time scales. We also model the subsequent growth of black holes via accretion of residual stars in clusters. 2-body relaxation efficiently re-fills the loss cones of stellar orbits at larger radii and resonant relaxation at small radii is the main driver for accretion of stars onto black holes. As a result, more than ninety percent of stars in the initial cluster are swallowed by the central black holes before z=6. Using dark matter merger trees we derive black hole mass functions at z=6-20. The mass function ranges from 10^3 to 10^5 Msun at z <~ 15. Major merging of galaxies of >~ 4*10^8 Msun at z ~ 20 successfully leads to the formation of >~ 10^5 Msun BHs by z >~ 10 which can be the potential seeds of supermassive black holes seen today.

METAMORPHOSIS OF SN 2014C: DELAYED INTERACTION BETWEEN A HYDROGEN POOR CORE-COLLAPSE SUPERNOVA AND A NEARBY CIRCUMSTELLAR SHELL

We present optical observations of supernova SN 2014C, which underwent an unprecedented slow metamorphosis from H-poor type Ib to H-rich type IIn over the course of one year. The observed spectroscopic evolution is consistent with the supernova having exploded in a cavity before encountering a massive shell of the progenitor star's stripped hydrogen envelope. Possible origins for the circumstellar shell include a brief Wolf-Rayet fast wind phase that overtook a slower red supergiant wind, eruptive ejection, or confinement of circumstellar material by external influences of neighboring stars. An extended high velocity Halpha absorption feature seen in near-maximum light spectra implies that the progenitor star was not completely stripped of hydrogen at the time of core collapse. Archival pre-explosion Subaru Telescope Suprime-Cam and Hubble Space Telescope Wide Field Planetary Camera 2 images of the region obtained in 2009 show a coincident source that is most likely a compact massive star cluster in NGC 7331 that hosted the progenitor system. By comparing the emission properties of the source with stellar population models that incorporate interacting binary stars we estimate the age of the host cluster to be 30 - 300 Myr, and favor ages closer to 30 Myr in light of relatively strong Halpha emission. SN 2014C is the best-observed member of a class of core-collapse supernovae that fill the gap between events that interact strongly with dense, nearby environments immediately after explosion and those that never show signs of interaction. Better understanding of the frequency and nature of this intermediate population can contribute valuable information about the poorly understood final stages of stellar evolution.

LUMINOUS BLUE VARIABLES AND SUPERLUMINOUS SUPERNOVAE FROM BINARY MERGERS

Evidence suggests that the direct progenitor stars of some core-collapse supernovae (CCSNe) are luminous blue variables (LBVs), perhaps including some `superluminous supernovae' (SLSNe). We examine models in which massive stars gain mass soon after the end of core hydrogen burning. These are mainly intended to represent mergers following a brief contact phase during early Case B mass transfer, but may also represent stars which gain mass in the Hertzsprung Gap or extremely late during the main-sequence phase for other reasons. The post-accretion stars spend their core helium-burning phase as blue supergiants (BSGs), and many examples are consistent with being LBVs at the time of core collapse. Other examples are yellow supergiants at explosion. We also investigate whether such post-accretion stars may explode successfully after core collapse. The final core properties of post-accretion models are broadly similar to those of single stars with the same initial mass as the pre-merger primary star. More surprisingly, when early Case B accretion does affect the final core properties, the effect appears likely to favour a successful SN explosion, i.e., to make the core properties more like those of a lower-mass single star. However, the detailed structures of these cores sometimes display qualitative differences to any single-star model we have calculated. The rate of appropriate binary mergers may match the rate of SNe with immediate LBV progenitors; for moderately optimistic assumptions we estimate that the progenitor birthrate is ~1% of the CCSN rate.

The moment of core collapse in star clusters with a mass function

Star clusters with multi-mass components dynamically evolve faster than those modeled with equal-mass components. Using a series of direct $N$-body simulations, we investigate the dynamical evolution of star clusters with mass functions, especially their core collapse time. Multi-mass clusters tend to behave like systems with a smaller number of particles, which we call the effective number of particles ($N_{eff}$) and for which $N_{eff} = M/m_{max}$ (here $M$ and $m_{max}$ are the total cluster mass and the mass of the most massive star in the cluster, respectively). We find that the time of core collapse is inversely proportional to the mass of the most massive star in the cluster and analytically confirm that this is because the core collapse of clusters with a mass function proceeds on the dynamical friction timescale of the most massive stars. As the mass of the most massive star increases, however, the core-collapse time, which is observed as a core bounce of the cluster core from the evolution of the core density or core radius, becomes ambiguous. We find that in that case the total binding energy of the hard binaries gives a good diagnosis for determining the moment of core collapses. Based on the results of our simulations, we argue that the core bounce becomes ambiguous when the mass of the most massive star exceeds 0.1\% of the total mass of the cluster.