Binary Disruption Research Articles

Populations of Stellar-mass Black Holes from Binary Systems

Abstract In large and complicated stellar systems like galaxies, it is difficult to predict the number and characteristics of a black hole (BH) population. Such populations may be modeled as an aggregation of homogeneous (i.e., having uniform star formation history and the same initial chemical composition) stellar populations. Using realistic evolutionary models, we predict the abundances and properties of BHs formed from binaries in these environments. We show that the BH population will be dominated by single BHs originating from binary disruptions and stellar mergers. Furthermore, we discuss how BH populations are influenced by such factors as initial parameters, metallicity, initial mass function, and natal kick models. As an example application of our results, we estimate that about 26 microlensing events happen every year in the direction of the Galactic Bulge due to BHs in a survey like OGLE-IV. Our results may be used to perform in-depth studies related to realistic BH populations, such as observational predictions for space survey missions like Gaia or Einstein Probe. We prepared a publicly available database with the raw data from our simulations to be used for more in-depth studies.

Liquid Densities and Excess Quantities for the Green Esterification Process

Liquid densities of deep eutectic solvents (DESs) such as choline chloride + glycerol {[ChCl][Gly]}, choline chloride + acetic acid {[ChCl][AA]}; ionic liquids (ILs), such as 1-ethyl-3-methylimidazolium hydrogen sulfate {[EMIM][HSO4]}, 1-ethyl-3-methylimidazolium ethyl sulfate {[EMIM][EtSO4]}, and 1-butyl-3-methylimidazolium acetate {[BMIM][Ac]}; and mixture of DESs + ILs as well as organic mixtures, such as n-butyl acetate + butanol and n-butyl acetate + acetic acid were measured at different temperatures from 293.15 to 343.15 K with an increase of 5 K. From this measured density, isobaric expansivity, excess molar volume, partial molar volume, excess partial molar volume, and apparent molar volume were calculated to understand and characterize the solution behavior as well as solvent–solvent interactions for the entire mole fraction at different temperatures. The excess molar properties showing positive deviation for DESs + ILs system, which indicates a creation of volume in the binary mixtures and disruption of H-bonds in the solution environment. Further, the sign and magnitude of all of the studied binary systems gave a good estimate of the attractive and disruptive interactions, chemical aggregation, H-bond formation, salt (choline chloride)–H (imidazolium) interaction, and salt (choline chloride)–O (acetate anion) interaction between those two components in the binary mixtures. The Redlich–Kister equation was used to fit the excess molar volume data and their corresponding regression coefficients (R2) were also found. The σ-profile and σ-potential for the compounds in the present study have been generated and analyzed by means of specific molecular interactions between all of the studied compounds using the conductor like screening model for real solvents model.

Discovery of an equal-mass ‘twin’ binary population reaching 1000 + au separations

ABSTRACT We use a homogeneous catalogue of 42 000 main-sequence wide binaries identified by Gaia to measure the mass ratio distribution, p(q), of binaries with primary masses 0.1 < M1/M⊙ < 2.5, mass ratios 0.1 ≲ q < 1, and separations $50 \lt s/{\rm au} \lt 50\, 000$. A well-understood selection function allows us to constrain p(q) in 35 independent bins of primary mass and separation, with hundreds to thousands of binaries in each bin. Our investigation reveals a sharp excess of equal-mass ‘twin’ binaries that is statistically significant out to separations of 1000–10 000 au, depending on primary mass. The excess is narrow: a steep increase in p(q) at 0.95 ≲ q < 1, with no significant excess at q ≲ 0.95. A range of tests confirm the signal is real, not a data artefact or selection effect. Combining the Gaia constraints with those from close binaries, we show that the twin excess decreases with increasing separation, but its width (q ≳ 0.95) is constant over $0.01 \lt a/{\rm au} \lt 10\, 000$. The wide twin population would be difficult to explain if the components of all wide binaries formed via core fragmentation, which is not expected to produce strongly correlated component masses. We conjecture that wide twins formed at closer separations (a ≲ 100 au), likely via accretion from circumbinary discs, and were subsequently widened by dynamical interactions in their birth environments. The separation-dependence of the twin excess then constrains the efficiency of dynamical widening and disruption of binaries in young clusters. We also constrain p(q) across 0.1 ≲ q < 1. Besides changes in the twin fraction, p(q) is independent of separation at fixed primary mass over $100 \lesssim s/{\rm au} \lt 50\, 000$. It is flatter than expected for random pairings from the initial mass function but more bottom-heavy for wide binaries than for binaries with a ≲100 au.

Hypervelocity Stars from a Supermassive Black Hole–Intermediate-mass Black Hole Binary

Abstract In this paper we consider a scenario in which the currently observed hypervelocity stars in our Galaxy have been ejected from the Galactic center as a result of dynamical interactions with an intermediate-mass black hole (IMBH) orbiting the central supermassive black hole (SMBH). By performing three-body scattering experiments, we calculate the distribution of the ejected stars’ velocities given various parameters of the IMBH–SMBH binary: IMBH mass, semimajor axis, and eccentricity. We also calculate the rates of change of the BH binary orbital elements due to those stellar ejections. One of our new findings is that the ejection rate depends (although mildly) on the rotation of the stellar nucleus (its total angular momentum). We also compare the ejection velocity distribution with that produced by the Hills mechanism (stellar binary disruption) and find that the latter produces faster stars on average. Also, the IMBH mechanism produces an ejection velocity distribution that is flattened toward the BH binary plane, while the Hills mechanism produces a spherically symmetric one. The results of this paper will allow us in the future to model the ejection of stars by an evolving BH binary and compare both models with Gaia observations, for a wide variety of environments (galactic nuclei, globular clusters, the Large Magellanic Clouds, etc.).

Black hole mergers induced by tidal encounters with a galactic centre black hole

We discuss the properties of stellar mass black hole (BH) mergers induced by tidal encounters with a massive BH at galactic centres or potentially in dense star clusters. The tidal disruption of stellar binaries by a massive BH is known to produce hypervelocity stars. However, such a tidal encounter does not always lead to the break-up of binaries. Since surviving binaries tend to become hard and eccentric, this process can produce BH mergers in principle. For initially circular binaries, we show that the gravitational wave (GW) merger times become shorter by a factor of more than $10^{2}$ ($10^5$) in $10\%$ ($1\%$) of the surviving cases. This reduction is primarily due to the growth in binary's eccentricity at the tidal encounter. We also investigate the effective spins of the survivors, assuming that BH spins are initially aligned with the binary orbital angular momentum. We find that binary orientations can flip in the opposite direction at the tidal encounter. For the survivors with large merger time reduction factors, the effective spin distribution is rather flat. We estimate the merger rate due to the tidal encounter channel to be $\sim 0.6\ \textrm{Gpc}^{-3}\textrm{yr}^{-1}$. This mechanism is unlikely to be the dominant formation channel of BH mergers. However, the current and near-future GW observatories are expected to detect an enormous number of BH mergers. If mergers are found in the vicinity of massive BHs (e.g. the detection of GW lensing echoes or preceding extreme-mass-ratio bursts), this mechanism would provide a possible explanation for their origin.

The Structural Dimensions of Race: Lock Ups, Systemic Chokeholds, and Binary Disruptions

Disrupting traditional conceptions of structural inequality, state decision making power, and the presumption of Black criminality, this Essay explores the doctrinal and policy implications of James Forman, Jr.’s Pulitzer Prize winning book, Locking Up Our Own, and Paul Butler’s evocative and transformative book, Chokehold. While both books grapple with how to dismantle the structural components of mass incarceration, state legitimized police violence against Black bodies, and how policy functions to reify oppressive state power, the approaches espoused by Forman and Butler are analytically distinct. Forman locates his analysis in the dynamics of decision-making power when African American officials wield power to combat crime with unintended consequences. He argues for incremental change focusing on discrete aspects of the system. By contrast, Butler offers a full conceptual attack on the oppressive machinery of mass incarceration—he seeks to break the grip of the systemic chokehold that threatens to strangle the life prospects of communities of color. Both books disrupt binary conceptions of the criminal justice system in the wake of Michelle Alexander’s The New Jim Crow and its progeny.

A Chandra Survey of Milky Way Globular Clusters. II. Testing the Hills–Heggie Law

Abstract Binary–single and binary–binary encounters play a pivotal role in the evolution of star clusters, as they may lead to the disruption or hardening of binaries, a novel prediction of the Hills–Heggie law. Based on our recent Chandra survey of Galactic globular clusters (GCs), we revisit the role of stellar dynamical interactions in GCs, focusing on main-sequence (MS) binary encounters as a potential formation channel of the observed X-ray sources in GCs. We show that the cumulative X-ray luminosity (L X), a proxy of the total number of X-ray-emitting binaries (primarily cataclysmic variables and coronally active binaries) in a given GC, is highly correlated with the MS binary encounter rate (Γ b ), as . We further test the Hills–Heggie law against the binary hardness ratio, defined as the relative number of X-ray-emitting hard binaries to MS binaries and approximated by , with L K being the GC K-band luminosity and f b the MS binary fraction. We demonstrate that the binary hardness ratio of most GCs is larger than that of the solar neighborbood stars, and exhibits a positive correlation with the cluster specific encounter rate (γ), as . We also find a strong correlation between the binary hardness ratio and cluster velocity dispersion (σ), with , which is consistent with the Hills–Heggie law. We discuss the role of binary encounters in the context of the Nuclear Star Cluster, arguing that the X-ray-emitting, close binaries detected therein could have been predominantly formed in GCs that later inspiralled to the Galactic center.

The MEGaN project II. Gravitational waves from intermediate-mass and binary black holes around a supermassive black hole

We investigate the evolution of intermediate-mass (IMBHs), stellar (BHs) and binary black holes (BHBs), deposited near a supermassive black hole (SMBH) by a population of massive star clusters. Stellar BHs rapidly segregate around the SMBH, driving the formation of extreme mass-ratio inspirals that coalesce at a rate $\Gamma= 0.02-0.2$ yr$^{-1}$ Gpc$^{-3}$ at redshift $z=0$. A few IMBHs orbiting the SMBH favour the formation of massive pairs that coalescence within a Hubble time, being the merger rate for this channel $\Gamma =0.03$ yr$^{-1}$ Gpc$^{-3}$. Recoiling kicks post-merger can eject the remnant from the galaxy centre, especially in dwarf galaxies. Our results suggest that this mechanism can lead to up to $10^5$ ejected SMBH within 1 Gpc. An IMBH co-existing with a few single and binary BHs in the same cluster can affect significantly their evolution, either driving binary disruption, yielding to intermediate-mass ratio inspirals (merger rate $\Gamma =9.5$ yr$^{-1}$ Gpc$^{-3}$), or boosting BHBs coalescence ($\Gamma =2-8$ yr$^{-1}$ Gpc$^{-3}$). In a few simulations, the SMBH boosts BHBs coalescence, leading this process to a merger rate $\Gamma =1$ yr$^{-1}$ Gpc$^{-3}$. We note that BHBs experiencing a merger in a galactic nucleus can be erroneously estimated $\sim 30\%$ heavier than it really is because of the Doppler shift of the wave frequency as caused by the rapid motion around the SMBH. All our simulations are carried out using an $N$-body code tailored to treat close encounters and post-Newtonian dynamics, that includes also the galaxy field and dynamical friction in the particles' equation of motion.

A reliable location model for heterogeneous systems under partial capacity losses

Due to the interdependency between multiple infrastructure systems, the performance of a facility may depend on the resources or supplies received from other facilities. However, cross-system interdependence has seldom been studied in the location design context, probably due to the lack of a concise model describing interdependence across heterogeneous systems. This paper proposes a new heterogeneous flow scheme to describe cross-system interdependence. This scheme has two features distinguished from existing models in describing an interdependent facility location problem. First, it is a simple linear model upon which a compact facility location model can be built. Secondly, it relaxes the need to maintain flow conservation between different systems and is suitable in describing heterogeneous systems that take in and output different resources or services. Built on this scheme, this paper proposes a reliable location design model for a nexus of interdependent infrastructure systems. This model aims to locate the optimal facility locations in multiple heterogeneous systems to balance the tradeoff between the facility investment and the expected nexus operation performance. Different from other reliable facility location models, this expected performance captures interdependence among heterogeneous systems due to the resource input-output relationships. The consideration of continuous partial capacity losses complements the reliable location literature that mainly focuses on binary disruptions. Two numerical examples are conducted for investigating features and applications of the proposed model. The results indicate that with a standard off-the-shelf integer programming solver, the proposed model is able to solve optimal facility location design for problem instances of realistic scales to the near-optimum solutions with optimality gap assurance. Sensitivity analyses of key parameters indicate that improving facility capacity and reducing interdependency between systems can mitigate impacts of facility capacity losses and reduce the overall system cost.

Blue straggler stars beyond the Milky Way: a non-segregated population in the Large Magellanic Cloud cluster NGC 2213

Using the high-resolution observations obtained by the Hubble Space Telescope, we analyzed the blue straggler stars (BSSs) in the Large Magellanic Cloud cluster NGC 2213. We found that the radial distribution of BSSs is consistent with that of the normal giant stars in NGC 2213, showing no evidence of mass segregation. However, an analytic calculation carried out for these BSSs shows that they are already dynamically old, because the estimated half-mass relaxation time for these BSSs is significantly shorter than the isochronal age of the cluster. We also performed direct N-body simulations for a NGC 2213-like cluster to understand the dynamical processes that lead to this none-segregated radial distribution of BSSs. Our numerical simulation shows that the presence of black hole subsystems inside the cluster centre can significantly affect the dynamical evolution of BSSs. The combined effects of the delayed segregation, binary disruption and exchange interactions of BSS progenitor binaries may result in this none-segregated radial distribution of BSSs in NGC 2213.

Tidal breakup of triple stars in the Galactic Centre

The last decade has seen the detection of fast moving stars in the Galactic halo, the so-called hypervelocity stars (HVSs). While the bulk of this population is likely the result of a close encounter between a stellar binary and the supermassive black hole (MBH) in the Galactic Centre (GC), other mechanims may contribute fast stars to the sample. Few observed HVSs show apparent ages which are shorter than the flight time from the GC, thereby making the binary disruption scenario unlikely. These stars may be the result of the breakup of a stellar triple in the GC which led to the ejection of a hypervelocity binary (HVB). If such binary evolves into a blue straggler star due to internal processes after ejection, a rejuvenation is possible that make the star appear younger once detected in the halo. A triple disruption may also be responsible for the presence of HVBs, of which one candidate has now been observed. We present a numerical study of triple disruptions by the MBH in the GC and find that the most likely outcomes are the production of single HVSs and single/binary stars bound to the MBH, while the production of HVBs has a probability $\lesssim 1\%$ regardless of the initial parameters. Assuming a triple fraction of $\approx 10\%$ results in an ejection rate of $\lesssim 1\ \mathrm{Gyr}^{-1}$, insufficient to explain the sample of HVSs with lifetimes shorter than their flight time. We conclude that alternative mechanisms are responsible for the origin of such objects and HVBs in general.

Steeper Stellar Cusps in Galactic Centers from Binary Disruption

Abstract The relaxed distribution of stars around a massive black hole is known to follow a cusp profile, , with a characteristic slope . This follows from energy conservation and a scattering rate given by two-body encounters. However, we show that the injection of stars close to the black hole, i.e., a source term in the standard cusp picture, modifies this profile. In the steady-state configuration, the cusp develops a central region with a typical slope in which stars diffuse outward. Binary disruption by the intense tidal field of the massive black hole is among the phenomena that take place in the Galactic Center (GC). In such a disruption, one of the binary members remains bound to the black hole, thus providing a source term of stars close to the black hole. Assuming a binary fraction of 0.1 and an orbital circularization efficiency of 0.35, we show that this source is strong enough to modify the cusp profile within pc of the GC. If the binary fraction at the influence radius is of order unity and the orbits of all captured stars are efficiently circularized, the steeper cusp extends almost as far as the radius of influence of the black hole.

Infalling Young Clusters in the Galactic Centre: implications for IMBHs and young stellar populations

The central parsec of the Milky Way hosts two puzzlingly young stellar populations, a tight isotropic distribution of B stars around SgrA* (the S-stars) and a disk of OB stars extending to ~0.5pc. Using a modified version of Sverre Aarseth's direct summation code NBODY6 we explore the scenario in which a young star cluster migrates to the Galactic Centre within the lifetime of the OB disk population via dynamical friction. We find that star clusters massive and dense enough to reach the central parsec form a very massive star via physical collisions on a mass segregation timescale. We follow the evolution of the merger product using the most up to date, yet conservative, mass loss recipes for very massive stars. Over a large range of initial conditions, we find that the very massive star expels most of its mass via a strong stellar wind, eventually collapsing to form a black hole of mass 20 - 400 M_Sun, incapable of bringing massive stars to the Galactic Centre. No massive intermediate mass black hole can form in this scenario. The presence of a star cluster in the central ~10 pc within the last 15 Myr would also leave a ~2 pc ring of massive stars, which is not currently observed. Thus, we conclude that the star cluster migration model is highly unlikely to be the origin of either young population, and in-situ formation models or binary disruptions are favoured.

Satellite capture mechanism in a sun-planet-binary four-body system

This paper studies the binary disruption problem and asteroid capture mechanism in a sun-planet-binary four-body system. Firstly, the binary disruption condition is studied and the result shows that the binary is always disrupted at the perigee of their orbit instantaneously. Secondly, an analytic expression to describe the energy exchange between the binary is derived based on the instantaneous disruption hypothesis. The analytic result is validated through numerical integration. We obtain the energy exchange in encounters simultaneously by the analytic expression and numerical integration. The maximum deviation of this two results is always less than 25% and the mean deviation is about 8.69%. The analytic expression can give us an intuitive description of the energy exchange between the binary. It indicates that the energy change depends on the hyperbolic shape of the binary orbit with respect to the planet, the masses of planet and the primary member of the binary, the binary phase at perigee. We can illustrate the capture/escape processes and give the capture/escape region of the binary clearly by numerical simulation. We analyse the influence of some critical factors to the capture region finally.

RUNAWAY DWARF CARBON STARS AS CANDIDATE SUPERNOVA EJECTA

ABSTRACT The dwarf carbon (dC) star SDSS J112801.67+004034.6 has an unusually high radial velocity, 531 ± 4 km s−1. We present proper motion and new spectroscopic observations which imply a large Galactic rest frame velocity, 425 ± 9 km s−1. Several other SDSS dC stars are also inferred to have very high galactocentric velocities, again each based on both high heliocentric radial velocity and also confidently detected proper motions. Extreme velocities and the presence of C 2 bands in the spectra of dwarf stars are both rare. Passage near the Galactic center can accelerate stars to such extreme velocities, but the large orbital angular momentum of SDSS J1128 precludes this explanation. Ejection from a supernova in a binary system or disruption of a binary by other stars are possibilities, particularly as dC stars are thought to obtain their photospheric C 2 via mass transfer from an evolved companion.

The dynamical importance of binary systems in young massive star clusters

AbstractCharacterization of the binary fractions in star clusters is of fundamental importance for many fields in astrophysics. Observations indicate that the majority of stars are found in binary systems, while most stars with masses greater than 0.5M⊙ are formed in star clusters. In addition, since binaries are on average more massive than single stars, in resolved star clusters these systems are thought to be good tracers of (dynamical) mass segregation. Over time, dynamical evolution through two-body relaxation will cause the most massive objects to migrate to the cluster center, while the relatively lower-mass objects remain in or migrate to orbits at greater radii. This process will globally dominate a cluster’s stellar distribution. However, close encounters involving binary systems may disrupt ‘soft’ binaries. This process will occur more frequently in a cluster’s central, dense region than in its periphery, which may mask the effects of mass segregation. Using high resolution Hubble Space Telescope observations, combined with sophisticated N-body simulations, we investigate the radial distributions of the main-sequence binary fractions in massive young Large Magellanic Cloud star clusters. We show that binary disruption may play an important role on very short timescales, depending on the environmental conditions in the cluster cores. This may lead to radial binary fractions that initially decline in the cluster centers, which is contrary to the effects expected from dynamical mass segregation.

THE FASTEST UNBOUND STARS IN THE UNIVERSE

The discovery of hypervelocity stars (HVSs) leaving our galaxy with speeds of nearly 103 km s−1 has provided strong evidence of the existence of a massive compact object at the galaxy's center. HVSs ejected via the disruption of stellar binaries can occasionally yield a star with km s−1; here we show that this mechanism can be extended to massive black hole (MBH) mergers, where the secondary star is replaced by a MBH with mass . We find that stars that are originally bound to the secondary MBH are frequently ejected with km s−1, and occasionally with velocities ∼105 km s−1 (one third the speed of light). For this reason we refer to stars ejected from these systems as "semi-relativistic" hypervelocity stars (SHSs). Bound to no galaxy, the velocities of these stars are so great that they can cross a significant fraction of the observable universe in the time since their ejection (several Gpc). We demonstrate that if a significant fraction of MBH mergers undergoes a phase in which their orbital eccentricity is ≳0.5 and their periapse distance is tens of the primary's Schwarzschild radius, the space density of fast-moving ( km s−1) SHSs may be as large as 103 Mpc−3. Hundreds of SHSs will be giant stars that can be detected by future all-sky infrared surveys such as WFIRST or Euclid and proper motion surveys such as LSST, with spectroscopic follow-up being possible with the James Webb Space Telescope.

Compact object mergers: observations of supermassive binary black holes and stellar tidal disruption events

AbstractThe capture and disruption of stars by supermassive black holes (SMBHs), and the formation and coalescence of binaries, are inevitable consequences of the presence of SMBHs at the cores of galaxies. Pairs of active galactic nuclei (AGN) and binary SMBHs are important stages in the evolution of galaxy mergers, and an intense search for these systems is currently ongoing. In the early and advanced stages of galaxy merging, observations of the triggering of accretion onto one or both BHs inform us about feedback processes and BH growth. Identification of the compact binary SMBHs at parsec and sub-parsec scales provides us with important constraints on the interaction processes that govern the shrinkage of the binary beyond the “final parsec”. Coalescing binary SMBHs are among the most powerful sources of gravitational waves (GWs) in the universe. Stellar tidal disruption events (TDEs) appear as luminous, transient, accretion flares when part of the stellar material is accreted by the SMBH. About 30 events have been identified by multi-wavelength observations by now, and they will be detected in the thousands in future ground-based or space-based transient surveys. The study of TDEs provides us with a variety of new astrophysical tools and applications, related to fundamental physics or astrophysics. Here, we provide a review of the current status of observations of SMBH pairs and binaries, and TDEs, and discuss astrophysical implications.

Revisiting the universality of (multiple) star formation in present-day star formation regions

Populations of multiple stars inside clustered regions are known to change through dynamical interactions. The efficiency of binary disruption is thought to be determined by stellar density. King and collaborators recently investigated the multiplicity properties in young star forming regions and in the Galactic field. They concluded that stellar density-dependent modification of a universal initial binary population (the standard or null hypothesis model) cannot explain the observations. We re-visit their results, analyzing the data within the framework of different model assumptions, namely non-universality without dynamical modification and universality with dynamics. We illustrate that the standard model does account for all known populations if regions were significantly denser in the past. Some of the effects of using present-day cluster properties as proxies for their past values are emphasized and that the degeneracy between age and density of a star forming region can not be omitted when interpreting multiplicity data. A new analysis of the Corona Australis region is performed within the standard model. It is found that this region is likely as unevolved as Taurus and an initial density of $\approx190M_{\odot}\;pc^{-3}$ is required to produce the presently observed binary population, which is close to its present-day density.

The disruption of close binaries in the gravitational field of a supermassive black hole and the formation of hypervelocity stars

The formation of hypervelocity stars due to the dynamical capture of one component of a closebinary system by the gravitational field of a supermassive black hole (SMBH) is modeled. The mass of the black hole was varied between 106 and 109M⊙. In the model, the problem was considered first as a three-body problem (stage I) and then as an N-body problem (stage II). In the first stage, the effect of the inclination of the internal close-binary orbit (the motion of the components about the center of mass of the binary system) relative to the plane of the external orbit (the motion of the close binary around the SMBH) on the velocity with which one of the binary components is ejected was assessed. The initial binary orbits were generated randomly, with 10 000 orbits considered for each external orbit with a fixed pericenter distance rp. Analysis of the results obtained in the first stage of the modeling enables determination of the binary-orbit orientations that are the most favorable for high-velocity ejection, and estimation of the largest possible ejection velocities Vmax. The boundaries of the region of stellar disruption derived from the balance of tidal forces and self-gravitation are discussed using Vmax-rp plots, which generalize the results of the first stage of the modeling. Since a point-mass representation does not enable predictions about the survival of stars during close passages by a SMBH, there is the need for a second stage of the modeling, in which the tidal influence of the SMBH is considered. An approach treating a star like a structured finite object containing N bodies (N = 4000) enables the derivation of more accurate limits for the zone of efficient acceleration of hypervelocity stars and the formulation of conditions for the tidal disruption of stars.