text stringlengths 128 2.05k |
|---|
Here [MATH] is the 4-velocity of the ISCO orbit. To ensure that the disk stays near the target temperature, the cooling rate should be rapid (i.e., [MATH] ), but drop to zero when the temperature falls below [MATH] . All of these criteria are satisfied by a cooling function with the form |
[EQUATION] where [MATH] and [MATH] is a constant of proportionality. Note that the term in the square brackets serves as a switch, so that |
[MATH] whenever [MATH] . The exponent [MATH] controls how rapidly the cooling rate grows when the temperature exceeds the target. We found that |
[MATH] cools the plasma efficiently while maintaining a stable evolution, and we set [MATH] . Only those fluid elements on bound orbits—where [MATH] —are cooled. |
In addition to controlling the vertical thickness of the disk, the cooling function provides a self-consistent way of comparing emission from the simulated disk with that expected in a standard NT model. When making this comparison, we use the angle-averaged fluid-frame luminosity per unit area (of an annulus located a... |
[EQUATION] where each component of the vector [MATH] represents the extent of a cell’s dimension as measured in the fluid element’s rest frame, and [MATH] is the orthonormal tetrad that transforms vectors in the Boyer-Lindquist coordinate frame to the local fluid frame (see Beckwith et al. ( 2008a for explicit expressi... |
[MATH] We also wish to calculate the radiated luminosity measured by a distant observer in order to include the effect of photon losses into the black hole. This is done by ray-tracing through the spacetime and integrating the radiative transfer equation along geodesics. Redshift factors from differences in inertial re... |
To briefly summarize our method, we trace a large number of rays from observers at infinity at 8 polar angles and integrate the transfer equation along each ray. Since we aim only at estimating the bolometric luminosity of the disk, and not at computing its spectrum, we assume that all radiated energy is emitted at a s... |
2.4 Coordinates, grid, and boundary conditions Equation ( ) is solved using finite volume techniques on a uniform grid in the so-called “Modified Kerr-Schild” (MKS) coordinate system described in Gammie et al. ( 2003 . It is based on the Kerr-Schild (KS) coordinate system that eliminates the coordinate singularity at t... |
[MATH] (Gammie et al., 2003 , which makes the poloidal cell scale at the axis about [MATH] times larger than that at the equator and allows us to resolve greater detail in the accretion disk than would be possible with the same number of equally spaced grid cells. |
The simulation reported here used [MATH] cells in the radial, poloidal, and azimuthal directions respectively, with [MATH] [MATH] [MATH] The radial extent is as large as the one used in KDP except our coordinates penetrate the horizon by five cells. We tested whether our polar angle discretization adequately resolved t... |
is adequate when used with the existing step size control method in HARM In the original MKS coordinate system, cells are placed all the way to the axis. We have introduced a new reflecting boundary condition that allows us to excise the coordinate singularity there. With the boundary placed at an angle of |
[MATH] from the axis (as in KDP), the excision enlarges the time step we can take, speeding up the evolution by about a factor of four relative to simulations using grids without the cut-out. The radial boundary conditions are the same as in the released version of HARM and we use periodic boundary conditions for the a... |
2.5 Algorithmic details The equations of motion are integrated using almost the same high-resolution shock-capturing methods as described in Gammie et al. ( 2003 . We still use HARM ’s Lax-Friedrichs numerical flux formula, as it is more diffusive than the HLL formula and seems to be stabler for our purposes. However, ... |
as our means of reconstructing values at cell faces. As in HARM , we use an MC (monotonized central-differenced) slope limiter. Parabolic reconstruction improves stability in low density regions where [MATH] , such as are found in the funnel |
(McKinney, 2006 We also use parabolic interpolation in the “Flux-CT” scheme of Tóth ( 2000 that preserves the divergence constraint. Originally, the electromotive forces (EMFs) at the cell faces were calculated by a second-order accurate two-point averaging procedure. This method failed to dissipate a cell-scale sawtoo... |
Even with the improvements described so far, stably evolving plasma whose total energy is dominated by magnetic and kinetic energies is difficult. In a conservative code like HARM3D , the critical step is deriving good primitive variables, [MATH] , from the conserved quantites. For instance, the magnetic field is typic... |
In order to treat the positive pressure problem and correct for other unphysical states that may arise (e.g. [MATH] ), we have completely redesigned HARM ’s recovery procedure. The most significant change is the inclusion of the conservation of entropy equation |
[EQUATION] where [EQUATION] Following a method similar to that of Balsara & Spicer ( 1999 , we integrate equation ( 19 ) in parallel with ( ). Whenever the standard primitive variable method fails to converge, [MATH] is unphysical, or |
[MATH] , we use a new inversion method which is identical to the standard one except the total energy equation is replaced by equation ( 20 ). Even though this inversion method is guaranteed to yield a positive pressure, it can either fail to converge to a solution or yield an unphysical [MATH] . If either happens, we ... |
We note that using equation ( 19 leads to a method that no longer conserves total energy to round-off error, but the impact of these departures from strict conservation is limited. The entropy equation is substituted for the energy equation only where the fluid is very strongly magnetically-dominated, and only when no ... |
We have verified that our new code is second-order accurate for smooth solutions and satisfactorily passes the tests described in Gammie et al. ( 2003 A quantitative comparison of our code’s performance to that of GRMHD |
will be left for future work. Results Our initial condition is a torus of gas in hydrostatic equilibrium, entirely contained within the simulated volume; our goal is to present results characteristic of an accretion flow with a fixed aspect ratio in a long-term equilibrium. Before quoting results directly from the simu... |
3.1 Scale-height regulation We set the parameters of our cooling function so that the ratio of the sound speed to the local orbital speed would produce a disk with a constant aspect ratio [MATH] . In Figure , we show how well the temperature was held to [MATH] by comparing the time-averaged volume-weighted temperature ... |
How well our temperature-regulation led to a disk aspect ratio matching the goal value of 0.13 can be seen in Figure . The actual [MATH] |
was slightly above the goal ( [MATH] ) through most of the simulation volume, but with a tendency to diminish inward inside [MATH] . At [MATH] |
[MATH] ; by the time the flow reaches the ISCO, it is only [MATH] . Comparison with the curve showing how the scale-height changes as a result of including the relativistic correction to the vertical gravity (as discussed in § 2.3 ) demonstrates that this thinning at small radius can be largely attributed to neglect of... |
Because our cooling function has a target temperature depending only on radius, at any particular radius the gas in the main body of the disk is nearly isothermal, and the density profile is therefore close to Gaussian (Fig. ). At higher altitudes above the midplane, the density falls slower than the Gaussian, presumab... |
We chose a value of [MATH] small enough that a key approximation of the NT theory could be approximately replicated in the simulation: the prompt radiation of dissipated heat. However, if the disk is to have a finite thickness, it cannot radiate all its heat. The parameters we chose for the cooling function yielded an ... |
[MATH] . Thus, this toy-model does assure that the majority of the dissipated heat is radiated. 3.2 Inflow equilibrium If the accretion flow were in a strict steady-state, the local (i.e., shell-integrated) mass accretion rate [MATH] would be the same at all radii at all times and the mass interior to a given radius wo... |
[MATH] However, for the purposes of estimating the radiative efficiency, we require a tighter definition of inflow equilibrium. This is because we wish to contrast the computed radiation rate with the NT rate at an accuracy of a few percent or better. In the NT model, |
[MATH] of the total light is emitted between [MATH] and [MATH] where our simulation shows significant departures from inflow equilibrium; a further [MATH] comes from outside [MATH] , where our simulation is not an accretion flow and we do not compute the luminosity at all. For these reasons, when we contrast the NT lum... |
3.3 Explicit radiative efficiency The fluid-frame emissivity [MATH] found in the simulation and the NT prediction for this quantity are displayed in Figure . In the leftmost panel, we show how they compare when the NT emissivity uses the time-averaged accretion rate at the horizon over the same interval for which the s... |
[MATH] (notation as in Krolik ( 1999a ); we adjust this to [MATH] , with [MATH] the time-averaged accretion rate at radius [MATH] in the simulation. By doing so, we compare the two radiation models in a way that factors out any contrasts due solely to fluctuations in the accretion rate. The adjusted version is shown in... |
As this pair of figures shows, the two models coincide closely in the main disk body, but contrast sharply near and within the ISCO, which is at |
[MATH] for this spin. Because the NT model is founded on energy and angular momentum conservation in a time-steady disk, this coincidence is no surprise where the influence of the NT no-stress boundary condition is small. The principal departure between the two, a systematic offset in which the simulation curve lies [M... |
At small radius, however, the fluid-frame surface brightness of the simulation differs substantially from the NT model. At [MATH] the simulation surface brightness is greater by [MATH] ; at the ISCO, although the NT model would predict no radiation, the surface brightness indicated by the simulation is roughly the same... |
Another way to characterize the contrast between the simulation results and the NT model is through the intermediary of another analytic model. In the model of Agol & Krolik ( 2000 , it is supposed that a finite stress is exerted at the ISCO, but all other assumptions follow those of NT. This model (which we will abbre... |
Only some of this radiation reaches infinity, and any that does arrives with a significant Doppler shift, most often toward the red. Using the techniques described in § 2.3 and Appendix A, we computed the luminosity received at infinity per unit radial coordinate [MATH] , which is shown in Figure . Like the emissivity ... |
to [MATH] . Although the fluid-frame emissivity extends farther inward, its efficiency in creating luminosity at infinity is cut off by a combination of increasing redshift and probability of photon capture by the black hole. |
At larger radii, departures from inflow equilibrium become significant. To compute accurately [MATH] , the data of our simulation must be both adjusted so as to correspond to true inflow equilibrium and supplemented by an extension to larger radius to account for the substantial radiation from radii larger than |
[MATH] . Because the time-averaged fluid-frame emissivity in the simulation tracks the NT model so closely for [MATH] , we define the simulation luminosity as its [MATH] integrated from the horizon to [MATH] plus the NT luminosity at the mean accretion rate from [MATH] outward. |
Given that definition, we find that the efficiency with which this simulation generated light reaching infinity, averaged from 7000– [MATH] , was 0.151. This number is [MATH] greater than the NT figure, which is 0.143 after allowing for photon capture. |
3.4 Extrapolating to the complete radiation limit As discussed in the Introduction, our principal goal in this initial simulation was to achieve as close a test as possible of the effect of the ISCO stress boundary condition on the radiative efficiency. We must now evaluate the degree to which our only approximate repl... |
We have already seen that our radiation rate closely tracks the NT radiation rate in the main disk body, but is about [MATH] lower. Thus, to extrapolate to complete radiation, the emission from this portion of the flow should be increased by this amount. Near the ISCO, where the effects of the stress boundary condition... |
[MATH] orbits. At the same place, the inflow rate is [MATH] . Thus, the photon diffusion time near the ISCO in a real disk should be shorter than the inflow time—and shorter than our toy-model cooling time [MATH] —for all accretion rates below Eddington. A second standard of comparison may be derived from the magnitude... |
at [MATH] . That the retained heat is [MATH] of the binding energy there is consistent with the fact that [MATH] of the heat dissipated in the main disk body is left unradiated. Combining these two arguments, we might expect that in the limit of truly complete radiation of dissipated heat, the efficiency could have bee... |
[MATH] above the classical number as adjusted for photon capture. Additional heat is created in the plunging region (the mean accreted specific enthalpy rises from 1.02 at the ISCO to [MATH] |
at the horizon), but, as we have already seen, the fraction of photons escaping from regions so close to the horizon to infinity is relatively small, so only a small part of the additional 0.01 in rest-mass equivalent is likely to reach distant observers. |
We might also ask what effect truly radiating all the heat would have on electromagnetic energy fluxes. To approach this question we begin by considering it from the point of view of the classical (NT) theory of accretion, where much attention is paid to the [MATH] [MATH] component of the stress tensor [MATH] , but lit... |
[MATH] . As Beckwith et al. ( 2008a pointed out, this assumption is consistent with the sort of stress NT had in mind, i.e., ordinary viscosity, but not necessarily with other physical stress mechanisms. In particular, it is inconsistent with MRI-driven MHD turbulence: the electromagnetic stress tensor contains a term ... |
not orthogonal to the four-velocity; in addition, the turbulence entails another (generally rather smaller) contribution to the stress tensor |
[MATH] , where [MATH] is the fluctuating part of the four-velocity. Described in more qualitative terms, the classical theory accounts for the energy flow due to the work done by the stress, but not the energy flow due to the advection, by the mean flow, of an energy density associated with the stress mechanism. |
As numerous numerical studies of the MRI-driven turbulence have shown, the fluid-frame ratio [MATH] –0.3 in the disk body, rising by factors of a few in the plunging region (e.g. Hawley & Krolik ( 2002 ). At the order of magnitude level, the ratio of the advected magnetic energy flux to the magnetic work is [MATH] , wh... |
To complete our extrapolation to complete radiation therefore means that we need to determine how [MATH] near the ISCO might change when that limit is taken at fixed accretion rate. Fixing the accretion rate means that the vertically-integrated stress does not change, and we do not need to estimate how the stress would... |
Summary and Implications Global disk simulations have for many years focused on dynamical effects, i.e., angular momentum transport leading to inflow. To link them to observations, however, requires including considerations of thermodynamics, for the energy to radiate photons is drawn from the thermal energy of the gas... |
In this first application of our new technique, we have found that a disk with [MATH] accreting onto a black hole with spin parameter [MATH] carries thermal and magnetic energy past the ISCO at a rate [MATH] per unit rest-mass, while producing radiation that reaches infinity at a rate [MATH] per unit rest-mass. These n... |
These results have implications for the spectral shape of the emitted radiation. Generically, the effect of the continuing stresses is to move the radius of peak emission inward and raise the fluid-frame effective temperature at that location. For example, in this instance the maximum in [MATH] (after allowing for phot... |
is about [MATH] greater ( [MATH] higher effective temperature) in the simulation data than in the NT model. In terms of the radiation edge terminology introduced by Krolik & Hawley ( 2002 and Beckwith et al. ( 2008a , we find that |
[MATH] of the radiation reaching infinity is produced outside [MATH] in contrast to [MATH] in the NT model. In a previous study, Beckwith et al. ( 2008a used the stress distributions observed in an ensemble of disk simulations to estimate the dissipation that might be associated with those stresses, and from this the a... |
Applying Beckwith et al.’s expression to our data leads to a prediction for the dissipation rate very similar to theirs . The fact that our radiation rate is considerably less than this prediction suggests that the simple ansatz used by Beckwith et al. to directly compute dissipation from stress and equate dissipation ... |
Framed in the context of predictions for real accretion flows in Nature, these questions emphasize the importance of realistic dissipation and radiation physics for obtaining more accurate accounts of radiation associated with accretion. In the vicinity of the ISCO, where the energy available for release is largest, on... |
In sum, we have shown that by use of a toy-model optically-thin cooling function, it is possible both to control the thickness of the accretion flow and to tally (approximately) the rate at which radiation can be produced by dissipation in the flow. At relatively large radii, where the inflow time is long compared to t... |
Having demonstrated the technical feasibility of this approach, we will next employ it to explore more fully how accretion onto black holes depends on disk thickness and on black hole rotation rate. In this context, we point out that although there is a standard notation for describing black hole rotation (the spin par... |
Lastly, we remark that in this paper we have set aside the fact that photons are not the only form in which energy can be sent to infinity from the vicinity of black holes. Accreting black holes are also capable of driving mass motions, often relativistic, that can carry significant power in Poynting flux. Simulational... |
This work was supported by NSF grant AST-0507455 (JHK) and PHY-0205155 (JFH). We thank Charles Gammie for many enlightening discussions and an early version of the HAM code. Exploratory simulations were performed on the DataStar cluster at SDSC and the Woodhen cluster at Princeton University, while the TeraGrid T3 and ... |
Appendix A The Radiative Transfer Calculation Our method for calculating the radiative transfer closely follows the one described by Noble et al. ( 2007 . We have made many changes to the code, including the ability to use 3D simulation data, different emission models (such as the one explained here), and many optimiza... |
The algorithm integrates geodesics from the observer’s camera through the source domain—our simulation data. These geodesics point toward the camera and the future. A geodesic represents a path along which a bundle of photons travel. The Lagrangian form of the geodesic equations is used: |
[EQUATION] where [MATH] is the world-line of the photon bundle and [MATH] is the geodesic’s tangent vector parameterized by the affine parameter [MATH] |
Since there is no absorption or scattering, the radiative transfer equation takes the form [EQUATION] where [MATH] is the Lorentz invariant intensity, |
[MATH] is the specific intensity, [MATH] is the invariant emissivity, [MATH] is the emissivity, and [MATH] is the local frequency of the photon. For the purposes of calculating the bolometric luminosity, we consider only line emission. We can assume either constant emission frequency (e.g. Fe K [MATH] fluorescence) whe... |
We assume that the radiation is emitted isotropically, so [MATH] but we must also take into account the constraint that the fluid’s emission frequency is the blueshifted frequency at the observer: |
[EQUATION] where [MATH] , the redshift factor, is the ratio of the photon’s energy measured by the camera to the photon’s energy measured by the fluid: |
[EQUATION] Here, [MATH] is velocity of the camera which is assumed to be static in flatspace; this is a good approximation as we place the camera [MATH] away from the black hole. |
# Source: arxiv 0808.3150 # Title: Strong mass segregation around a massive black hole # Sections: all # Downloaded: 2026-03-02T07:58:34.243217+00:00 |
Strong mass segregation around a massive black hole Abstract We show that the mass-segregation solution for the steady state distribution of stars around a massive black hole (MBH) has two branches: the known weak segregation solution (Bahcall & Wolf, 1977 , and a newly discovered strong segregation solution, presented... |
mass-dependent cusps near the MBH, with indices [MATH] for the heavy stars and [MATH] for the light stars (i.e. [MATH] ). However, when the heavy stars are relatively rare ( [MATH] ), they scatter mostly on light stars, sink to the center by dynamical friction and settle into a much steeper cusp with [MATH] while the l... |
of main sequence and compact dwarfs, and [MATH] of stellar black holes (SBHs), and have [MATH] . We conclude that it is likely that many relaxed galactic nuclei are strongly segregated. We review indications of strong segregation in observations of the Galactic Center and in results of numeric simulations, and briefly ... |
Subject headings: Galaxy: kinematics and dynamics — stellar dynamics — black hole physics 1. Introduction Early theoretical studies of the dynamics and distribution of stars around a MBH (Peebles, 1972 ; Bahcall & Wolf, 1976 1977 ; Young, 1980 were triggered by the discovery of quasars (e.g. Matthews & Sandage, 1963 ; ... |
(Giacconi et al., 1972 , which were then thought to be accreting MBHs (e.g. Wyller 1970 ; Bahcall & Wolf 1976 ; see also Miller & Hamilton 2002 for a recent reevaluation of this possibility). The main motivations for these studies were the prospect of detecting MBHs by the observed stellar density profile and by tidal ... |
The renewed interest in this problem is driven by observations of stars closely orbiting the Galactic MBH (Eisenhauer et al., 2005 ; Ghez et al., 2005 and the accumulating data on their distribution and dynamics (Alexander, 1999 ; Genzel et al., 2000 2003 ; Schödel et al., 2003 2007 as well as by the prospects of detec... |
Hopman 2006 ). EMRI rates and properties strongly depend on the stellar density and the stellar dynamical processes within [MATH] |
of the MBH, where inspiraling sources originate (e.g. Hopman & Alexander, 2005 2006a Mass segregation occurs in dynamically relaxed systems. MBHs are naturally expected to lie in relaxed cores in scenarios where the MBH is formed by run-away mergers in the extreme central density following core collapse (Rees, 1984 , w... |
(e.g. Spitzer 1987 ; Quinlan 1996 ; Freitag et al. 2006b Likewise, the extreme mass ratio targets of the planned Laser Interferometer Space Antenna GW detector (LISA) are expected to originate in relaxed nuclei, since LISA design is sensitive to GW from MBHs with mass [MATH] The observed correlation between the MBH mas... |
[MATH] (the [MATH] relation, Ferrarese & Merritt, 2000 ; Gebhardt et al., 2000 then implies that such nuclei are dynamically relaxed and very dense |
(Alexander, 2007 ; Merritt et al., 2007 . This can be seen by assuming for simplicity [MATH] (a higher value only reinforce these conclusions), and noting that the MBH radius of influence [MATH] |
encompasses a stellar mass of order [MATH] , so that the number of stars there is [MATH] , where [MATH] is the typical stellar mass, and the mean stellar density is [MATH] The “ [MATH] ” rate estimate of strong gravitational collisions then implies that [MATH] A more rigorous estimate shows that for the Galactic MBH ( ... |
Eisenhauer et al. 2005 ; Ghez et al. 2005 ), an archetype of LISA targets, [MATH] (the Hubble time) and [MATH] The density in the stellar cusp near the MBH is orders of magnitude higher still (see below). Since [MATH] , isolated nuclei with [MATH] are predicted to be relaxed. |
A single mass stellar system around a MBH is expected to relax to [MATH] cusp with [MATH] (Bahcall & Wolf, 1976 . This results from the fact that the gravitational orbital energy gained by the system when stars are destroyed near the MBH is conserved as it is shared and carried outward by the remaining stars, [MATH] |
(Binney & Tremaine, 1987 . When the system includes a spectrum of masses, [MATH] the approach toward equipartition by 2-body interactions decreases the specific kinetic energy of the high-mass stars, while that of the low-mass stars increases. As a result, the high-mass stars sink and concentrate in the center on the d... |
[MATH] while the low-mass stars float out (Spitzer, 1987 Bahcall & Wolf ( 1977 , hereafter BW77) approximated the mass segregation problem in the Fokker-Planck formalism, and solved for the steady state, angular momentum averaged stellar distribution functions (DFs) |
[MATH] . They found that near the MBH, the DFs can be approximated by power-laws, [MATH] , and that the stellar current into the MBH, [MATH] , is very small, which leads to a specific relation between the stellar mass and the logarithmic slope of the DF, [MATH] . In the Keplerian limit near the MBH, these DFs correspon... |
no longer holds, and the range of central concentrations far exceeds that of the BW77 solution. The Galactic Center (GC) provides to date the few available observations that directly bear on the question of mass segregation around a MBH. The over-abundance of X-ray transients in the central [MATH] pc of the GC was inte... |
(Mouawad et al., 2005 ; Gillessen et al., 2008 ; Ghez et al., 2008 , is still 10–100 times larger than predicted by approximate theoretical estimates (Morris, 1993 ; Miralda-Escudé & Gould, 2000 Fokker-Planck calculations (Hopman & Alexander, 2006b , or by the conservative drain limit (Alexander & Livio, 2004 , which p... |
Detailed analysis of observations of the GC (Levi, 2006 ; Schödel et al., 2007 suggests that the degree of segregation between the light and heavy stars is stronger than expected in the BW77 solution, although this interpretation of the data is not unique. Strong segregation in the GC is further supported by analytic (... |
(Freitag et al., 2006a , M. Freitag, priv. comm.) mass segregation results (Fig. ). These calculations do not take into account star formation and evolution, but instead assume for simplicity a non-evolving mass function that is based on a model of the GC’s present day mass mass function (PMF) as an old, continuously s... |
(Alexander & Sternberg, 1999 ; Alexander, 2005 , table 2.1) . In these theoretical models [MATH] while [MATH] at [MATH] pc. [MATH] -body simulations of stellar clusters with evolving stellar populations and a central intermediate mass black hole (IMBH) also provide hints of strong mass segregation (Baumgardt et al., 20... |
This paper is organized as follows. The Fokker-Planck formulation of the mass segregation problem, the choices of boundary conditions and the various simplifying approximations are described in § The physical meaning of the relaxational coupling parameter, [MATH] is discussed in § . The [MATH] parameters of different s... |
2. The Energy diffusion Fokker-Planck equation The MBH dominates the gravitational potential within its radius of influence, where the dynamics are approximately Keplerian. Following the approach introduced by Bahcall & Wolf ( 1976 1977 and its implementation by Hopman & Alexander ( 2006b , we calculate the approximate... |
[EQUATION] where [MATH] is the number density normalization of the reference star. Since the number density of unbound stars [EQUATION] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.