Empty Loss Cone Research Articles

Repeating partial disruptions and two-body relaxation

Two-body relaxation may drive stars onto near-radial orbits around a massive black hole, resulting in a tidal disruption event (TDE). In some circumstances, stars are unlikely to undergo a single terminal disruption, but rather to have a sequence of many grazing encounters with the black hole. It has long been unclear what is the physical outcome of this sequence: each of these encounters can only liberate a small amount of stellar mass, but may significantly alter the orbit of the star. We study the phenomenon of repeating partial tidal disruptions (pTDEs) by building a semi-analytical model that accounts for mass loss and tidal excitations. In the empty loss cone regime, where two-body relaxation is weak, we estimate the number of consecutive partial disruption that a star can undergo, on average, before being significantly affected by two-body encounters. We find that in this empty loss cone regime, a star will be destroyed in a sequence of weak pTDEs, possibly explaining the tension between the low observed TDE rate and its higher theoretical estimates.

Partial stellar tidal disruption events and their rates

ABSTRACT Tidal disruption events (TDEs) of stars operated by massive black holes (MBHs) will be detected in thousands by upcoming facilities such as the Vera Rubin Observatory. In this work, we assess the rates of standard total TDEs, destroying the entire star, and partial TDEs, in which a stellar remnant survives the interaction, by solving 1D Fokker–Planck equations. Our rate estimates are based on a novel definition of the loss cone whose size is commensurate with the largest radius at which partial disruptions can occur, as motivated by relativistic hydrodynamical simulations. Our novel approach unveils two important results. First, partial TDEs can be more abundant than total disruptions by a factor of a few to a few tens. Second, the rates of complete stellar disruptions can be overestimated by a factor of a few to a few tens if one neglects partial TDEs, as we find that many of the events classified as total disruptions in the standard framework are in fact partial TDEs. Accounting for partial TDEs is particularly relevant for galaxies harbouring a nuclear stellar cluster featuring many events coming from the empty loss cone. Based on these findings, we stress that partial disruptions should be considered when constraining the luminosity function of TDE flares; accounting for this may reconcile the theoretically estimated TDE rates with the observed ones.

Dynamics of tidal disruption events: statistical properties

ABSTRACT We study the correlations between the dynamical history of a star that eventually experiences a tidal disruption event (TDE) and the properties of the TDE itself. As expected, we find that (i) in empty loss cone at the radius of influence, most TDEs are non-violent TDEs; i.e. the star’s periastron is only a bit smaller than the tidal radius. These stars had experienced strong tidal interactions with the supermassive black hole (SMBH), just slightly outside the tidal radius. (ii) In full loss cone at the radius of influence, most TDEs are violent; i.e. the star’s periastron is significantly smaller than the tidal radius. These stars did not previously encounter strong tidal effects. Nevertheless, we demonstrate that even in extremely empty or extremely full loss cone a finite fraction of order ${\sim} 10{{\ \rm per\ cent}}$ of the events are non-typical. This is due to two surprising effects: (i) In an empty loss cone, a logarithmic fraction of the events are associated with stars that entered the loss cone via strong scatterings. Those events are usually violent penetrating deep into the loss cone. (ii) In full loss cone, orbits with semimajor axis smaller than the radius of influence, where the loss cone is marginally empty, dominate the TDE rate. As a result, a constant fraction of TDEs involve stars in empty loss cone orbits, which have typically experienced strong interactions with the SMBH in previous orbits.

Tidal Disruptions of Main-sequence Stars. I. Observable Quantities and Their Dependence on Stellar and Black Hole Mass

Abstract This paper introduces a series of papers presenting a quantitative theory for the tidal disruption of main-sequence stars by supermassive black holes. Using fully general relativistic hydrodynamics simulations and MESA-model initial conditions, we explore the pericenter-dependence of tidal disruption properties for eight stellar masses ( ) and six black hole masses ( ). We present here the results most relevant to observations. The effects of internal stellar structure and relativity decouple for both the disruption cross section and the characteristic energy width of the debris. Moreover, the full disruption cross section is almost independent of M ⋆ for M ⋆/M ⊙ ≲ 3. Independent of M ⋆, relativistic effects increase the critical pericenter distance for full disruption events by up to a factor of ∼3 relative to the Newtonian prediction. The probability of a direct capture is also independent of M ⋆; at M BH/M ⊙ ≃ 5 × 106 this probability is equal to the probability of a complete disruption. The breadth of the debris energy distribution ΔE can differ from the standard estimate by factors of 0.35 − 2, depending on M ⋆ and M BH, implying a corresponding change (∝(ΔE)−3/2) in the characteristic mass-return timescale. We provide analytic forms, suitable for use in both event rate estimates and parameter inference, to describe all these trends. For partial disruptions, we find a nearly universal relation between the star’s angular momentum and the fraction of M ⋆ remaining. Within the “empty loss-cone” regime, partial disruptions must precede full disruptions. These partial disruptions can drastically affect the rate and appearance of subsequent total disruptions.

Tidal Disruptions of Main-sequence Stars. V. The Varieties of Disruptions

Abstract Tidal disruption events (TDEs) are generally imagined as the complete disruption of a star when it passes close to a supermassive black hole (SMBH). Relativistic apsidal precession is thought to quickly “circularize” the bound debris, forming a compact accretion disk, which then emits a flare of standardized light curve and spectrum. We show here that this picture holds in only a minority of cases. TDEs are more diverse and can be grouped into several categories distinguished by stellar pericenter distance r p ; we estimate the relative frequency of these categories. Rapid circularization is rare both because it requires ( ) and because most events with lead to direct capture. For larger pericenter distances, (for M BH = 106 M ⊙), main-sequence stars with M * ≲ 3 are completely disrupted, but the bound debris orbits are highly eccentric and possess semimajor axes ∼100× the scale of the expected compact disk. Partial disruptions with fractional mass loss ≳10% occur with a rate similar to that of total disruptions; for fractional mass loss ≳50%, the rate is ≈1/3 as large. Partial disruptions—which must precede total disruptions when the stars’ angular momenta evolve in the “empty loss-cone” regime—change the orbital energy by factors ≳O(1). Partial disruption remnants are in general far from thermal equilibrium. Depending on its orbital energy and conditions within the stellar cluster surrounding the SMBH, a remnant may return after ∼O(100)–O(1000) yr and be fully disrupted, or it may rejoin the stellar cluster.

Shock breakouts from tidal disruption events

Tidal disruption events of stars by supermassive black holes have so far been discovered months to years after the fact. In this paper we explore the short, faint and hard burst of radiation is emitted at maximum compression, as a result of shock breakout. The detection of this burst can be used to capture tidal disruption events in real time. We verify that shock breakout from main sequence stars produces radiation in the X-ray range, but find that it is difficult to detect using all sky X-ray surveying telescopes. In the case of shock breakout from red giants, most of the radiation is emitted in the UV and visible range, which is significantly easier to detect. A similar burst of UV/optical radiation will also be emitted by stars puffed by tidal heating from a previous passage close to the central black hole. This radiation can be detected by surveys like ZTF and LSST. We calculate detection rates for different types of galactic nuclei. For the case of a very full loss cone we predict a detection rate of once per month with LSST, whereas for the case of a very empty loss cone we predict a rate of once per year with LSST. Evidence from a recent tidal disruption event, ASASSN-14li, seems to favour a very full loss cone, in which case LSST is expected to detect one such event every month.

Isotopic effect in microstability of electrostatic oscillations in magnetic mirror traps

It has been long known that Drift-Cyclotron Loss-Cone (DCLC) and Double-Humped (DH) oscillations can be unstable in a mirror trap because of the presence of an empty loss cone in the distribution of hot ions and that the instability can be suppressed by the addition of a small amount of warm ions with isotropic distribution. Unfortunately, previous analyses have been limited almost exclusively to the case when the populations of both hot and warm ions consist of the same single isotope. In the present paper, we studied the stability of DCLC and DH modes in a multispecies plasma with two different isotopes of hydrogen confined in a mirror trap. We found that both DCLC and DH microinstabilities can be suppressed by the addition of warm ions with isotropic Maxwellian distribution provided that the concentrations of different isotopes in the warm ion population are proportional to those in the population of hot ions. We came to the conclusion that overlapping of the full set of cyclotron harmonics of hot ions by the set of cyclotron harmonics of the warm ion mixture is essential for effective suppressing of the instability of DCLC and DH modes.

Electrostatic instabilities in a mirror trap revisited

The conditions for the stabilization of the Drift-Cyclotron Loss-Cone (DCLC) and Double-Humped (DH) microinstabilities in a mirror trap are critically revisited assuming the plasma is confined in the kinetic regime, which is characterized by an empty loss cone. The temperature of warm ions, necessary for stabilization of the DH instability, is calculated. The fraction of warm ions necessary to stabilize the DCLC instability at a given radial density gradient is calculated. Assuming the wavelength is much shorter than the Larmor radius, a simple criterion for the stability of drift-cyclotron loss-cone oscillations is derived whose accuracy is verified by comparison with the solution of the exact dispersion equation and with known experimental data obtained in the past decades in PR-6, 2XII, 2XIIB, TMX, and TMX-U devices for plasma confinement.

A contribution to the theory of the equatorial noise generation in the Earth’s magnetosphere

This work is devoted to the study of the generation of the equatorial noise—electromagnetic emission below the LHR frequency observed near the equatorial plane of the magnetosphere at distances of ~4RE. According to accepted views, the generation of the equatorial noise is related to the instability of ring current protons. In this work, a logarithmic distribution of energetic protons over the magnetic moment with an empty loss cone is proposed, and arguments for the formation of such a distribution are presented. The main result of the work is the calculation and analysis of the instability growth rate of waves forming the equatorial noise. The growth rate obtained in this work significantly differs from that encountered in the literature.

Interactions between multiple supermassive black holes in galactic nuclei: a solution to the final parsec problem

Using few-body simulations, we investigate the evolution of supermassive black holes (SMBHs) in galaxies ($M_{\star}=10^{10}-10^{12}{\rm M}_{\odot}$ at $z=0$) at $0<z<4$. Following galaxy merger trees from the Millennium simulation, we model BH mergers with two extreme binary decay scenarios for the `hard binary' stage: a full or an empty loss cone. These two models should bracket the true evolution, and allow us to separately explore the role of dynamical friction and that of multi-body BH interactions on BH mergers. Using the computed merger rates, we infer the stochastic gravitational wave background (GWB). Our dynamical approach is a first attempt to study the dynamical evolution of multiple SMBHs in the host galaxies undergoing mergers with various mass ratios ($10^{-4} < q_{\star} < 1$). Our main result demonstrates that SMBH binaries are able to merge in both scenarios. In the empty loss cone case, we find that BHs merge via multi-body interactions, avoiding the `final parsec' problem, and entering the PTA band with substantial orbital eccentricity. Our full loss cone treatment, albeit more approximate, suggests that the eccentricity becomes even higher when GWs become dominant, leading to rapid coalescences (binary lifetime $\lesssim1 {\rm ~Gyr}$). Despite the lower merger rates in the empty loss cone case, due to their higher mass ratios and lower redshifts, the GWB in the full/empty loss cone models are comparable ($0.70\times10^{-15}$ and $0.53\times10^{-15}$ at a frequency of $1~{\rm yr}^{-1}$, respectively). Finally, we compute the effects of high eccentricities on the GWB spectrum.

How empty is an empty loss cone?

We consider two body relaxation in a spherical system with a loss cone. Considering two-dimensional angular momentum space, we focus on "empty loss cone" systems, where the typical scattering during a dynamical time $j_{d}$ is smaller than the size of the loss cone $j_{\rm lc}$. As a result, the occupation number within the loss cone is significantly smaller than outside. Classical diffusive treatment of this regime predict exponentially small occupation number deep in the loss cone. We revisit this classical derivation of occupancy distribution of objects in the empty loss cone regime. We emphasize the role of the rare large scatterings and show that the occupancy does not decay exponentially within the loss cone, but it is rather flat, with a typical value $\sim [(j_d/j_{\rm lc})]^2\ln^{-2}(j_{\rm lc}/j_{\min})$ compared to the occupation in circular angular momentum (where $j_{\min}$ is the smallest possible scattering). Implication are that although the loss cone for tidal break of Giants or binaries is typically empty, tidal events which occurs significantly inside the loss cone ($\beta\gtrsim 2$), are almost as common as those with $\beta\cong 1$ where $\beta$ is the ratio between the tidal radius and the periastron. The probability for event with penetration factor $>\beta$ decreases only as $\beta^{-1}$ rather than exponentially. This effect has no influence on events characterized by full loss cone, such as tidal disruption event of $\sim 1m_\odot$ main sequence star.

Stellar dynamics around a massive black hole – III. Resonant relaxation of razor-thin axisymmetric discs

We study the Resonant Relaxation (RR) of an axisymmetric low mass (or Keplerian) stellar disc orbiting a more massive black hole (MBH). Our recent work on the general kinetic theory of RR is simplified in the standard manner by ignoring the effects of `gravitational polarization', and applied to a zero-thickness, flat, axisymmetric disc. The wake of a stellar orbit is expressed in terms of the angular momenta exchanged with other orbits, and used to derive a kinetic equation for RR under the combined actions of self-gravity, 1 PN and 1.5 PN relativistic effects of the MBH and an arbitrary external axisymmetric potential. This is a Fokker-Planck equation for the stellar distribution function (DF), wherein the diffusion coefficients are given self-consistently in terms of contributions from apsidal resonances between pairs of stellar orbits. The physical kinetics is studied for the two main cases of interest. (1) `Lossless' discs in which the MBH is not a sink of stars, and disc mass, angular momentum and energy are conserved: we prove that general H-functions can increase or decrease during RR, but the Boltzmann entropy is (essentially) unique in being a non-decreasing function of time. Therefore secular thermal equilibria are maximum entropy states, with DFs of the Boltzmann form; the two-Ring correlation function at equilibrium is computed. (2) Discs that lose stars to the MBH through an `empty loss-cone': we derive expressions for the MBH feeding rates of mass, angular momentum and energy in terms of the diffusive flux at the loss-cone boundary.

Generation of multi-charged high current ion beams using the SMIS 37 gas-dynamic electron cyclotron resonance (ECR) ion source

A gas-dynamic ECR ion source (GaDIS) is distinguished by its ability to produce high current and high brightness beams of moderately charged ions. Contrary to a classical ECR ion source where the plasma confinement is determined by the slow electron scattering into an empty loss-cone, the higher density and lower electron temperature in a GaDIS plasma lead to an isotropic electron distribution with the confinement time determined by the prompt gas-dynamic flow losses. As a result, much higher ion fluxes are available; however a decrease in the confinement time of the GaDIS plasma lowers the ion charge state. The gas-dynamic ECR ion source concept has been successfully realized in the SMIS 37 experimental facility operated at the Institute of Applied Physics, Russia. The use of high-power (∼100kW) microwave (37.5GHz) radiation provides a dense plasma (∼1013cm−3) with a relatively low electron temperature (∼50–100eV) and allows for the generation of high current (∼1A/cm2) beams of multi-charged ions. In this work we report on the present status of the SMIS 37 ion source and discuss the advanced numerical modeling of ion beam extraction using the particle-in-cell code WARP.

Comparative analysis of low‐altitude ENA emissions in two substorms

AbstractWe report on the dynamics of low‐altitude energetic neutral atom (ENA) emissions during two substorms that occurred during the main phases of two storms: (1) a CIR‐driven storm on 11 October 2008 and (2) a coronal mass ejection (CME)‐driven storm on 5 April 2010. For both of these storms, we have complementary spacecraft and ground‐based observations. The dual‐spacecraft Two Wide‐angle Imaging Neutral‐atom Spectrometers (TWINS) mission obtained ENA images containing low‐altitude emissions (LAEs). Substorm dynamics is inferred from THEMIS all‐sky imagers. TWINS‐observed LAEs are compared with trapped/loss cone proton fluxes from the low‐altitude NOAA/MetOp spacecraft constellation. We find that the timing and intensity profiles of LAEs are different for the two selected events. For the 11 October 2008 event, the LAEs rise during substorm recovery phase and storm main phase. On 5 April 2010, the LAEs tend to peak near substorm onset. We argue that the different LAE behavior results from different pitch‐angle distributions (PADs) of the ion source population. Ion PADs are isotropic during substorm recovery phase for the 11 October 2008 event and have empty loss cone for the 5 April 2010 event. For both cases, LAE intensification marks the onset of activity in the magnetotail and precedes the large substorm onset. We conclude that the LAE production starts in the transition region between the magnetotail and ring current and may expand/move into the inner magnetosphere together with ring current formation.

Statistics of the longitudinal splitting of proton aurora during substorms

[1] The lower-latitude boundary of the proton aurora, i.e., the isotropic boundary (IB), marks the transition from a full downgoing loss cone on the poleward side to an empty downgoing loss cone on the equatorward side. A number of authors have correlated this boundary with the amount of stretching in the magnetic field using GOES spacecraft. In this paper, we use 264 substorm events from the IMAGE SI-12 global proton auroral imager to show that during substorms, the proton aurora splits longitudinally 48% of the time. We hypothesize that the splitting is a result of the azimuthal growth of the substorm current wedge and show that splitting is more likely during stronger substorms (lower AL).

Excitation of electron cyclotron harmonic waves in the inner Saturn magnetosphere within local plasma injections

Strong electron cyclotron harmonic (ECH) waves have been observed by Cassini associated with local plasma injection regions in the Saturnian magnetosphere. Using measured electron distributions from the Cassini plasma spectrometer, we calculated local growth rates and path‐integrated wave gain of ECH waves inside an injection event using the HOTRAY code. We showed that electrons with energy near a few eV have a collision frequency comparable to their bounce frequency; thus, they cannot have an empty loss cone. We then demonstrated that the growth of ECH waves inside the injection event can be driven by electron phase space density gradients associated with the loss cone distribution of injected electrons at energies between a few hundred eV and a few keV. This conclusion is contrary to previous results that the source of free energy for growth of ECH waves is provided by electrons near a few eV. Results in this work are helpful for understanding the generation of ECH waves and their roles in electron dynamics in the Saturnian magnetosphere.

Latitude distribution of vertically precipitating energetic neutral atoms observed at low altitudes

Earlier modeling work of the latitude distribution of earthward traveling energetic neutral atoms (ENA) originating in the ring current have suggested that an isotropic ring current with an empty loss cone will deposit most ENA away from the magnetic equator, while a ring current with an equatorial pitch angle distribution peaked at 90° will deposit most ENA at the magnetic equator. We present the first direct low altitude measurements of energetic particles at low and equatorial latitudes supporting this model. The evolution from an isotropic ring current to a more anisotropic one is manifested in the latitude distribution of the low altitude energetic particles during a geomagnetic storm. In the growth and main phase of the storm the latitude distribution of the energetic particles is wide and often show secondary peaks away from the equator, while in the recovery phase the latitude distribution narrows and peaks at the magnetic equator.

Secular Stability and Instability in Stellar Systems Surrounding Massive Objects

We examine the stability of a low-mass stellar system surrounding a massive central object. Examples of such systems include the centers of galaxies or star clusters containing a massive black hole, and the Oort comet cloud. If the self-gravity of the stellar system is the dominant non-Keplerian force, such systems may be subject to slowly growing (secular) lopsided instabilities. Stability to secular modes is largely determined by the dependence of the distribution function F on angular momentum J. If dF/dJ 0, as is expected if there is a loss cone at low angular momentum, all spherical systems in which F=0 at J=0 (an empty loss cone) are only neutrally stable, and flattened, non-rotating systems are generally unstable. These results suggest that secular instabilities may dominate the structure and evolution of the stellar systems in the centers of galaxies.

Energetic ions as remote probes of X type neutral lines in the geomagnetic tail

Martin and Speiser (1988) have predicted ridges in the velocity space distribution function as a signature of the interaction of energetic ions with an X type neutral line in the geomagnetic tail. In this paper we study the properties of these ridges as the observation point is moved relative to the X line, as phase angle is varied, and the effect of the ridges on initial distributions with a loss cone. With the ridges, one can remotely sense not only the presence of an X line, but also, potentially, the distance from the X line, whether the observer is earthward or tailward of the X line, and the vertical position within the current sheet. For example, we find that for single particle dynamics in a current sheet with neutral line, the phase space ridge is predicted to be found throughout the current sheet if it is not destroyed by collective behavior. The ridge is predicted to be found for distributions plotted in ν⊥, ν∥ space and also in νx, νy, νz space. Additionally, valleys are found in νx, νy, νz space, which are also signatures of a neutral line, and which depend on the initial flowing distribution. An initially tailward flowing distribution causes asymmetries in distributions earthward versus tailward of the neutral line. These asymmetries are due to the fact that part of the distribution below (at smaller pitch angles than) the ridge comes from initially earthward (tailward) particles when the modeling point is earthward (tailward) of the neutral line. The accelerated beam is not predicted inside the current sheet at x = L (one separatrix distance earthward of the neutral line), but close to the sheet center a perpendicular bulk flow is predicted. Spatially, the ridge is found to move to larger pitch angles as the observation point approaches the neutral line in the plasma sheet boundary layer. As the observation point moves toward the current sheet center, the ridge stays at about the same pitch angle, but then tends to be diminished very near the center where a perpendicular bulk flow is found. Chaotic pitch angle scattering can fill even a relatively large (10°), initially empty loss cone. This loss cone filling is more complete when orbits are calculated in a thicker current sheet. Thus chaotic pitch angle scattering may be a dominant mechanism to produce nightside proton isotropy in the auroral zones, as suggested by Sergeev et al. (1983).

Tidal capture of stars by a massive black hole

The processes leading to tidal capture of stars by a massive black hole and the consequences of these processes in a dense stellar cluster are discussed in detail. When the amplitude of a tide and the subsequent oscillations are sufficiently large, the energy deposited in a star after periastron passage and formation of a bound orbit cannot be estimated directly using the linear theory of oscillations of a spherical star, but rather numerical estimates must be used. The evolution of a star after tidal capture is discussed. The maximum ratio R of the cross-section for tidal capture to that for tidal disruption is about 3 for real systems. For the case of a stellar system with an empty capture loss cone, even in the case when the impact parameter for tidal capture only slightly exceeds the impact parameter for direct tidal disruption, tidal capture would be much more important than tidal disruption.