Nuclear Star Cluster Research Articles

Upper Limits on the Presence of Central Massive Black Holes in Two Ultra-compact Dwarf Galaxies in Centaurus A

Abstract The recent discovery of massive black holes (BHs) in the centers of high-mass ultra-compact dwarf galaxies (UCDs) suggests that at least some are the stripped nuclear star clusters of dwarf galaxies. We present the first study that investigates whether such massive BHs, and therefore stripped nuclei, also exist in low-mass (M < 107 M ⊙) UCDs. We constrain the BH masses of two UCDs located in Centaurus A (UCD 320 and UCD 330) using Jeans modeling of the resolved stellar kinematics from adaptive optics data obtained with the SINFONI integral field spectrograph at the Very Large Telescope (VLT/SINFONI). No massive BHs are found in either UCD. We find a 3σ upper limit on the central BH mass in UCD 330 of M • < 1.0 × 105 M ⊙, which corresponds to 1.7% of the total mass. This excludes a high-mass fraction BH and would only allow low-mass BHs similar to those claimed to be detected in Local Group globular clusters. For UCD 320, poorer data quality results in a less constraining 3σ upper limit of M • < 1 × 106 M ⊙, which is equal to 37.7% of the total mass. The dynamical mass-to-light ratios of UCD 320 and UCD 330 are not inflated compared to predictions from stellar population models. The non-detection of BHs in these low-mass UCDs is consistent with the idea that elevated dynamical mass-to-light ratios do indicate the presence of a substantial BH. Although no massive BHs are detected, these systems could still be stripped nuclei. The strong rotation (v/σ of 0.3–0.4) in both UCDs and the two-component light profile in UCD 330 support the idea that these UCDs may be stripped nuclei of low-mass galaxies whose BH occupation fraction is not yet known.

Gravitational wave sources from inspiralling globular clusters in the Galactic Centre and similar environments

We model the inspiral of globular clusters (GCs) towards a galactic nucleus harboring a supermassive black hole (SMBH), a leading scenario for the formation of nuclear star clusters. We consider the case of GCs containing either an intermediate-mass black hole (IMBH) or a population of stellar mass black holes (BHs), and study the formation of gravitational wave (GW) sources. We perform direct summation $N$-body simulations of the infall of GCs with different orbital eccentricities in the live background of a galaxy with either a shallow or steep density profile. We find that the GC acts as an efficient carrier for the IMBH, facilitating the formation of a bound pair. The hardening and evolution of the binary depends sensitively on the galaxy's density profile. If the host galaxy has a shallow profile the hardening is too slow to allow for coalescence within a Hubble time, unless the initial cluster orbit is highly eccentric. If the galaxy hosts a nuclear star cluster, the hardening leads to coalescence by emission of GWs within $3-4$ Gyr. In this case, we find a IMBH-SMBH merger rate of $\Gamma_{\rm IMBH-SMBH} = 2.8\times 10^{-3}$ yr$^{-1}$ Gpc$^{-3}$. If the GC hosts a population of stellar BHs, these are deposited close enough to the SMBH to form extreme-mass-ratio-inspirals with a merger rate of $\Gamma_{\rm EMRI} = 0.25$ yr$^{-1}$ Gpc$^{-3}$. Finally, the SMBH tidal field can boost the coalescence of stellar black hole binaries delivered from the infalling GCs. The merger rate for this merging channel is $\Gamma_{\rm BHB} = 0.4-4$ yr$^{-1}$ Gpc$^{-3}$.

Black Hole Mergers in Galactic Nuclei Induced by the Eccentric Kozai–Lidov Effect

Abstract Nuclear star clusters around a central massive black hole (MBH) are expected to be abundant in stellar black hole (BH) remnants and BH–BH binaries. These binaries form a hierarchical triple system with the central MBH, and gravitational perturbations from the MBH can cause high-eccentricity excitation in the BH–BH binary orbit. During this process, the eccentricity may approach unity, and the pericenter distance may become sufficiently small so that gravitational-wave emission drives the BH–BH binary to merge. In this work, we construct a simple proof-of-concept model for this process, and specifically, we study the eccentric Kozai–Lidov mechanism in unequal-mass, soft BH–BH binaries. Our model is based on a set of Monte Carlo simulations for BH–BH binaries in galactic nuclei, taking into account quadrupole- and octupole-level secular perturbations, general relativistic precession, and gravitational-wave emission. For a typical steady-state number of BH–BH binaries, our model predicts a total merger rate of ∼1–3 −3 −1, depending on the assumed density profile in the nucleus. Thus, our mechanism could potentially compete with other dynamical formation processes for merging BH–BH binaries, such as the interactions of stellar BHs in globular clusters or in nuclear star clusters without an MBH.

An Ultradeep Chandra Catalog of X-Ray Point Sources in the Galactic Center Star Cluster

Abstract We present an updated catalog of X-ray point sources in the inner 500″ (∼20 pc) of the Galactic center (GC), where the nuclear star cluster (NSC) stands, based on a total of ∼4.5 Ms of Chandra observations taken from 1999 September to 2013 April. This ultradeep data set offers unprecedented sensitivity for detecting X-ray sources in the GC, down to an intrinsic 2–10 keV luminosity of 1.0 × 1031 erg s−1. A total of 3619 sources are detected in the 2–8 keV band, among which ∼3500 are probable GC sources and ∼1300 are new identifications. The GC sources collectively account for ∼20% of the total 2–8 keV flux from the inner 250″ region where detection sensitivity is the greatest. Taking advantage of this unprecedented sample of faint X-ray sources that primarily traces the old stellar populations in the NSC, we revisit global source properties, including long-term variability, cumulative spectra, luminosity function, and spatial distribution. Based on the equivalent width and relative strength of the iron lines, we suggest that in addition to the arguably predominant population of magnetic cataclysmic variables (CVs), nonmagnetic CVs contribute substantially to the detected sources, especially in the lower-luminosity group. On the other hand, the X-ray sources have a radial distribution closely following the stellar mass distribution in the NSC, but much flatter than that of the known X-ray transients, which are presumably low-mass X-ray binaries (LMXBs) caught in outburst. This, together with the very modest long-term variability of the detected sources, strongly suggests that quiescent LMXBs are a minor (less than a few percent) population.

Deformation of the Galactic Centre stellar cusp due to the gravity of a growing gas disc

The nuclear star cluster surrounding the massive black hole at the Galactic Centre consists of young and old stars, with most of the stellar mass in an extended, cuspy distribution of old stars. The compact cluster of young stars was probably born in situ in a massive accretion disc around the black hole. We investigate the effect of the growing gravity of the disc on the orbits of the old stars, using an integrable model of the deformation of a spherical star cluster with anisotropic velocity dispersions. A formula for the perturbed phase space distribution function is derived using linear theory, and new density and surface density profiles are computed. The cusp undergoes a spheroidal deformation with the flattening increasing strongly at smaller distances from the black hole; the intrinsic axis ratio ~ 0.8 at ~ 0.15 pc. Stellar orbits are deformed such that they spend more time near the disc plane and sample the dense inner parts of the disc; this could result in enhanced stripping of the envelopes of red giant stars. Linear theory accounts only for orbits whose apsides circulate. The non-linear theory of adiabatic capture into resonance is needed to understand orbits whose apsides librate. The mechanism is a generic dynamical process, and it may be common in galactic nuclei.

Super-solar Metallicity Stars in the Galactic Center Nuclear Star Cluster: Unusual Sc, V, and Y Abundances

Abstract We present adaptive-optics assisted near-infrared high-spectral-resolution observations of late-type giants in the nuclear star cluster of the Milky Way. The metallicity and elemental abundance measurements of these stars offer us an opportunity to understand the formation and evolution of the nuclear star cluster. In addition, their proximity to the supermassive black hole (∼0.5 pc) offers a unique probe of the star formation and chemical enrichment in this extreme environment. We observed two stars identified by medium spectral-resolution observations as potentially having very high metallicities. We use spectral-template fitting with the PHOENIX grid and Bayesian inference to simultaneously constrain the overall metallicity, [M/H], alpha-element abundance [α/Fe], effective temperature, and surface gravity of these stars. We find that one of the stars has very high metallicity ([M/H] > 0.6) and the other is slightly above solar metallicity. Both Galactic center stars have lines from scandium (Sc), vanadium (V), and yttrium (Y) that are much stronger than allowed by the PHOENIX grid. We find, using the spectral synthesis code Spectroscopy Made Easy, that [Sc/Fe] may be an order of magnitude above solar. For comparison, we also observed an empirical calibrator in NGC 6791, the highest metallicity cluster known ([M/H] ∼ 0.4). Most lines are well matched between the calibrator and the Galactic center stars, except for Sc, V, and Y, which confirms that their abundances must be anomalously high in these stars. These unusual abundances, which may be a unique signature of nuclear star clusters, offer an opportunity to test models of chemical enrichment in this region.

Habitable Evaporated Cores and the Occurrence of Panspermia Near the Galactic Center

Abstract Black holes growing via the accretion of gas emit radiation that can photoevaporate the atmospheres of nearby planets. Here, we couple planetary structural evolution models of sub-Neptune-mass planets to the growth of the Milky Way’s central supermassive black hole, Sgr A*, and investigate how planetary evolution is influenced by quasar activity. We find that, out to ∼20 pc from Sgr A*, the XUV flux emitted during its quasar phase can remove several percent of a planet’s H/He envelope by mass; in many cases, this removal results in bare rocky cores, many of which are situated in the habitable zones of G-type stars. Near the Galactic Center, the erosion of sub-Neptune-sized planets may be one of the most prevalent channels by which terrestrial super-Earths are created. As such, the planet population demographics may be quite different close to Sgr A* than in the galactic outskirts. The high stellar densities in this region (about seven orders of magnitude greater than the solar neighborhood) imply that the distance between neighboring rocky worlds is short (500–5000 au). The proximity between potentially habitable terrestrial planets may enable the onset of widespread interstellar panspermia near the nuclei of our galaxy. More generally, we predict these phenomena to be ubiquitous for planets in nuclear star clusters and ultra-compact dwarfs. Globular clusters, on the other hand, are less affected by the central black holes.

Random forest classification of stars in the Galactic Centre

Near-infrared high-angular resolution imaging observations of the Milky Way's nuclear star cluster have revealed all luminous members of the existing stellar population within the central parsec. Generally, these stars are either evolved late-type giants or massive young, early-type stars. We revisit the problem of stellar classification based on intermediate-band photometry in the K-band, with the primary aim of identifying faint early-type candidate stars in the extended vicinity of the central massive black hole. A random forest classifier, trained on a subsample of spectroscopically identified stars, performs similarly well as competitive methods (F1=0.85), without involving any model of stellar spectral energy distributions. Advantages of using such a machine-trained classifier are a minimum of required calibration effort, a predictive accuracy expected to improve as more training data becomes available, and the ease of application to future, larger data sets. By applying this classifier to archive data, we are also able to reproduce the results of previous studies of the spatial distribution and the K-band luminosity function of both the early- and late-type stars.

The combined effect of AGN and supernovae feedback in launching massive molecular outflows in high-redshift galaxies

We have recently improved our model of active galactic nucleus (AGN) by attaching the supermassive black hole (SMBH) to a massive nuclear star cluster (NSC). Here we study the effects of this new model in massive, gas-rich galaxies with several simulations of different feedback recipes with the hydrodynamics code RAMSES. These simulations are compared to a reference simulation without any feedback, in which the cooling halo gas is quickly consumed in a burst of star formation. In the presence of strong supernovae (SN) feedback, we observe the formation of a galactic fountain that regulates star formation over a longer period, but without halting it. If only AGN feedback is considered, as soon as the SMBH reaches a critical mass, strong outflows of hot gas are launched and prevent the cooling halo gas from reaching the disk, thus efficiently halting star formation, leading to the so-called "quenching". If both feedback mechanisms act in tandem, we observe a non-linear coupling, in the sense that the dense gas in the supernovae-powered galactic fountain is propelled by the hot outflow powered by the AGN at much larger radii than without AGN. We argue that these particular outflows are able to unbind dense gas from the galactic halo, thanks to the combined effect of SN and AGN feedback. We speculate that this mechanism occurs at the end of the fast growing phase of SMBH, and is at the origin of the dense molecular outflows observed in many massive high-redshift galaxies.

Disrupted globular clusters and the gamma-ray excess in the Galactic Centre

The Fermi Large Area Telescope has provided the most detailed view toward the Galactic Centre (GC) in high-energy gamma rays. Besides the interstellar emission and point-source contributions, the data suggest a residual diffuse gamma-ray excess. The similarity of its spatial distribution with the expected profile of dark matter has led to claims that this may be evidence for dark matter particle annihilation. Here, we investigate an alternative explanation that the signal originates from millisecond pulsars (MSPs) formed in dense globular clusters and deposited at the GC as a consequence of cluster inspiral and tidal disruption. We use a semi-analytical model to calculate the formation, migration, and disruption of globular clusters in the Galaxy. Our model reproduces the mass of the nuclear star cluster and the present-day radial and mass distribution of globular clusters. For the first time, we calculate the evolution of MSPs from disrupted globular clusters throughout the age of the Galaxy and consistently include the effect of the MSP spin-down due to magnetic-dipole breaking. The final gamma-ray amplitude and spatial distribution are in good agreement with the Fermi observations and provide a natural astrophysical explanation for the GC excess.

The distribution of stars around the Milky Way’s central black hole

Context. The existence of dynamically relaxed stellar density cusps in dense clusters around massive black holes is a long-standing prediction of stellar dynamics, but it has so far escaped unambiguous observational confirmation. Aims. In this paper we aim to revisit the problem of inferring the innermost structure of the Milky Way’s nuclear star cluster via star counts, to clarify whether it displays a core or a cusp around the central black hole. Methods. We used judiciously selected adaptive optics assisted high angular resolution images obtained with the NACO instrument at the ESO VLT. Through image stacking and improved point spread function fitting we pushed the completeness limit about one magnitude deeper than in previous, comparable work. Crowding and extinction corrections were derived and applied to the surface density estimates. Known young, and therefore dynamically not relaxed stars, are excluded from the analysis. Contrary to previous work, we analyse the stellar density in well-defined magnitude ranges in order to be able to constrain stellar masses and ages. Results. We focus on giant stars, with observed magnitudes K = 12.5−16, and on stars with observed magnitudes K ≈ 18, which may have similar mean ages and masses than the former. The giants display a core-like surface density profile within a projected radius R ≤ 0.3 pc of the central black hole, in agreement with previous studies, but their 3D density distribution is not inconsistent with a shallow cusp if we take into account the extent of the entire cluster, beyond the radius of influence of the central black hole. The surface density of the fainter stars can be described well by a single power-law at R < 2 pc. The cusp-like profile of the faint stars persists even if we take into account the possible contamination of stars in this brightness range by young pre-main sequence stars. The data are inconsistent with a core-profile for the faint stars. Finally, we show that a 3D Nuker law provides a good description of the cluster structure. Conclusions. We conclude that the observed density of the faintest stars detectable with reasonable completeness at the Galactic centre, is consistent with the existence of a stellar cusp around the Milky Way’s central black hole, Sagittarius A*. This cusp is well developed inside the influence radius of Sagittarius A* and can be described by a single three-dimensional power-law with an exponent γ = 1.43 ± 0.02 ± 0.1sys. This corroborates existing conclusions from Nbody simulations performed in a companion paper. An important caveat is that the faint stars analysed here may be contaminated significantly by dynamically unrelaxed stars that formed about 100 Myr ago. The apparent lack of giants at projected distances of R ≲ 0.3 pc (R ≲ 8′′) of the massive black hole may indicate that some mechanism may have altered their distribution or intrinsic luminosity. We roughly estimate the number of possibly missing giants to about 100.

Rapid Formation of Black Holes in Galaxies: A Self-limiting Growth Mechanism

Abstract We present high-quality fluid dynamical simulations of isothermal gas flows in a rotating barred potential. We show that a large quantity of gas is driven right into the nucleus of a galaxy when the model lacks a central mass concentration, but the inflow stalls at a nuclear ring in comparison simulations that include a central massive object. The radius of the nuclear gas ring increases linearly with the mass of the central object. We argue that bars drive gas right into the nucleus in the early stages of disk galaxy formation, where a nuclear star cluster and perhaps a massive black hole could be created. The process is self-limiting, however, because inflow stalls at a nuclear ring once the mass of gas and stars in the nucleus exceeds ∼1% of the disk mass, which shuts off rapid growth of the black hole. We briefly discuss the relevance of these results to the seeding of massive black holes in galaxies, the merger model for quasar evolution, and the existence of massive black holes in disk galaxies that lack a significant classical bulge.

Detailed Abundances for the Old Population near the Galactic Center. I. Metallicity Distribution of the Nuclear Star Cluster

Abstract We report the first high spectral resolution study of 17 M giants kinematically confirmed to lie within a few parsecs of the Galactic center, using spectroscopy from Keck/NIRSPEC and a new line list for the infrared K band. We consider their luminosities and kinematics, which classify these stars as members of the older stellar population and the central cluster. We find a median metallicity of and a large spread from approximately −0.3 to (quartiles). We find that the highest metallicities are , with most of the stars being at or below the solar iron abundance. The abundances and the abundance distribution strongly resemble those of the Galactic bulge rather than the disk or halo; in our small sample we find no statistical evidence for a dependence of velocity dispersion on metallicity.

A New Fokker–Planck Approach for the Relaxation-driven Evolution of Galactic Nuclei

Abstract We present an approach for simulating the collisional evolution of spherical isotropic stellar systems based on the one-dimensional Fokker–Planck equation. A novel aspect is that we use the phase volume as the argument of the distribution function instead of the traditionally used energy, which facilitates the solution. The publicly available code PhaseFlow implements a high-accuracy finite-element method for the Fokker–Planck equation, and can handle multiple-component systems, optionally with the central black hole and taking into account loss-cone effects and star formation. We discuss the energy balance in the general setting, and in application to the Bahcall–Wolf cusp around a central black hole, for which we derive a perturbative solution. We stress that the cusp is not a steady-state structure, but rather evolves in amplitude while retaining an approximately density profile. Finally, we apply the method to the nuclear star cluster of the milky Way, and illustrate a possible evolutionary scenario in which a two-component system of lighter main-sequence stars and stellar-mass black holes develops a Bahcall–Wolf cusp in the heavier component and a weaker cusp in the lighter, visible component, over the period of several Gyr. The present-day density profile is consistent with the recently detected mild cusp inside the central parsec, and is weakly sensitive to initial conditions.

Probing stellar binary black hole formation in galactic nuclei via the imprint of their center of mass acceleration on their gravitational wave signal

Multi-frequency gravitational wave (GW) observations are useful probes of the formation processes of coalescing stellar-mass binary black holes (BBHs). We discuss the phase drift in the GW inspiral waveform of the merging BBH caused by its center-of-mass acceleration. The acceleration strongly depends on the location where a BBH forms within a galaxy, allowing observations of the early inspiral phase of LIGO-like BBH mergers by the Laser Interferometer Space Antenna (LISA) to test the formation mechanism. In particular, BBHs formed in dense nuclear star clusters or via compact accretion disks around a nuclear supermassive black hole in active galactic nuclei would suffer strong acceleration, and produce large phase drifts measurable by LISA. The host galaxies of the coalescing BBHs in these scenarios can also be uniquely identified in the LISA error volume, without electromagnetic counterparts. A non-detection of phase drifts would rule out or constrain the contribution of the nuclear formation channels to the stellar-mass BBH population.

Probing the formation history of the nuclear star cluster at the Galactic Centre with millisecond pulsars

The origin of the Nuclear Star Cluster in the centre of our Galaxy is still unknown. One possibility is that it formed after the disruption of stellar clusters that spiralled into the Galactic Centre due to dynamical friction. We trace the formation of the Nuclear Star Cluster around the central black hole, using state-of-the-art N-body simulations, and follow the dynamics of the neutron stars born in the clusters. We then estimate the number of Millisecond Pulsars (MSPs) that are released in the Nuclear Star Cluster, during its formation. The assembly and tidal dismemberment of globular clusters lead to a population of MSPs distributed over a radius of about 20 pc, with a peak near 3 pc. No clustering is found on the sub-parsec scale. We simulate the detectability of this population with future radio telescopes like the MeerKAT radio telescope and SKA1, and find that about of order ten MSPs can be observed over this large volume, with a paucity of MSPs within the central parsec. This helps discriminating this scenario from the in-situ formation model for the Nuclear Star Cluster that would predict an over abundance of MSPs closer to the black hole. We then discuss the potential contribution of our MSP population to the gamma-ray excess at the Galactic Centre.

Greatly Enhanced Merger Rates of Compact-object Binaries in Non-spherical Nuclear Star Clusters

Abstract The Milky Way and a significant fraction of galaxies are observed to host a central massive black hole (MBH) embedded in a non-spherical nuclear star cluster. We study the secular orbital evolution of compact-object binaries in these environments and characterize the excitation of extremely large eccentricities that can lead to mergers by gravitational radiation. We find that the eccentricity excitation occurs most efficiently when the nodal precession timescale of the binary’s orbit around the MBH due to the non-spherical cluster becomes comparable (within a factor of ∼10) to the timescale on which the binary is torqued by the MBH due to the Lidov–Kozai (LK) mechanism. We show that in this regime the perturbations due to the cluster increase the fraction of systems that reach extreme eccentricities ( ) by a factor of ∼10–100 compared to the idealized case of a spherical cluster, increasing the merger rates of compact objects by a similar factor. We identify two main channels that lead to this extreme eccentricity excitation: (i) chaotic diffusion of the eccentricities due to resonance overlap; (ii) cluster-driven variations of the mutual inclinations between the binary orbit and its center-of-mass orbit around the MBH, which can intensify the LK oscillations. We estimate that our mechanism can produce BH–BH and BH–neutron star binary merger rates of up to and , respectively. Thus, we propose the cluster-enhanced LK mechanism as a new channel for the merger of compact-object binaries, competing with scenarios that invoke isolated binary evolution or dynamical formation in globular clusters.

Relaxation near Supermassive Black Holes Driven by Nuclear Spiral Arms: Anisotropic Hypervelocity Stars, S-stars, and Tidal Disruption Events

Abstract Nuclear spiral arms are small-scale transient spiral structures found in the centers of galaxies. Similarly to their galactic-scale counterparts, nuclear spiral arms can perturb the orbits of stars. In the case of the Galactic center (GC), these perturbations can affect the orbits of stars and binaries in a region extending to several hundred parsecs around the supermassive black hole (SMBH), causing diffusion in orbital energy and angular momentum. This diffusion process can drive stars and binaries to close approaches with the SMBH, disrupting single stars in tidal disruption events (TDEs), or disrupting binaries, leaving a star tightly bound to the SMBH and an unbound star escaping the galaxy, i.e., a hypervelocity star (HVS). Here, we consider diffusion by nuclear spiral arms in galactic nuclei, specifically the Milky Way GC. We determine nuclear-spiral-arm-driven diffusion rates using test-particle integrations and compute disruption rates. Our TDE rates are up to 20% higher compared to relaxation by single stars. For binaries, the enhancement is up to a factor of ∼100, and our rates are comparable to the observed numbers of HVSs and S-stars. Our scenario is complementary to relaxation driven by massive perturbers. In addition, our rates depend on the inclination of the binary with respect to the Galactic plane. Therefore, our scenario provides a novel potential source for the observed anisotropic distribution of HVSs. Nuclear spiral arms may also be important for accelerating the coalescence of binary SMBHs and for supplying nuclear star clusters with stars and gas.

On the relation between the mass of Compact Massive Objects and their host galaxies

Supermassive black holes and/or very dense stellar clusters are found in the central regions of galaxies. Nuclear star clusters are present mainly in faint galaxies while upermassive black holes are common in galaxies with masses $\geq 10^{10}$ M$_\odot $. In the intermediate galactic mass range both types of central massive objects (CMOs) are found. Here we present our collection of a huge set of nuclear star cluster and massive black hole data that enlarges significantly already existing data bases useful to investigate for correlations of their absolute magnitudes, velocity dispersions and masses with structural parameters of their host galaxies. In particular, we directed our attention to some differences between the correlations of nuclear star clusters and massive black holes as subsets of CMOs with hosting galaxies. In this context, the mass-velocity dispersion relation plays a relevant role because it seems the one that shows a clearer difference between the supermassive black holes and nuclear star clusters. The $M_{MBH}-{\sigma}$ has a slope of $5.19\pm 0.28$ while $M_{NSC}-{\sigma}$ has the much smaller slope of $1.84\pm 0.64$. The slopes of the CMO mass- host galaxy B magnitude of the two types of CMOs are indistinguishable within the errors while that of the NSC mass-host galaxy mass relation is significantly smaller than for supermassive black holes. Another important result is the clear depauperation of the NSC population in bright galaxy hosts, which reflects also in a clear flattening of the NSC mass vs host galaxy mass at high host masses.

Probabilistic eccentricity bifurcation for stars around shrinking massive black hole binaries

Based on the secular theory, we discuss the orbital evolution of stars in a nuclear star cluster to which a secondary massive black hole is infalling with vanishing eccentricity. We find that the eccentricities of the stars could show sharp transitions, depending strongly on their initial conditions. By examining the phase-space structure of an associated Hamiltonian, we show that these characteristic behaviours are partly due to a probabilistic bifurcation at a separatrix crossing, resulting from the retrograde apsidal precession by the cluster potential. We also show that separatrix crossings are closely related to realization of a large eccentricity and could be important for astrophysical phenomena such as tidal disruption events or gravitational wave emissions.