Circumnuclear Media of Quiescent Supermassive Black Holes
We calculate steady-state, one-dimensional hydrodynamic profiles of hot gas in slowly accreting (“quiescent”) galactic nuclei for a range of central black hole masses , parametrized gas heating rates, and observationally-motivated stellar density profiles. Mass is supplied to the circumnuclear medium by stellar winds, while energy is injected primarily by stellar winds, supernovae, and black hole feedback. Analytic estimates are derived for the stagnation radius (where the radial velocity of the gas passes through zero) and the large scale gas inflow rate, , as a function of and the gas heating efficiency, the latter being related to the star-formation history. We assess the conditions under which radiative instabilities develop in the hydrostatic region near the stagnation radius, both in the case of a single burst of star formation and for the average star formation history predicted by cosmological simulations. By combining a sample of measured nuclear X-ray luminosities, , of nearby quiescent galactic nuclei with our results for we address whether the nuclei are consistent with accreting in a steady-state, thermally-stable manner for radiative efficiencies predicted for radiatively inefficiency accretion flows. We find thermally-stable accretion cannot explain the short average growth times of low mass black holes in the local Universe, which must instead result from gas being fed in from large radii, due either to gas inflows or thermal instabilities acting on larger, galactic scales. Our results have implications for attempts to constrain the occupation fraction of SMBHs in low mass galaxies using the mean correlation, as well as the predicted diversity of the circumnuclear densities encountered by relativistic outflows from tidal disruption events.
keywords:black hole physics – galaxies: active
Supermassive black holes (SMBHs) lurk in the centres of most, if not all nearby galaxies (see reviews by, e.g. Kormendy & Richstone 1995; Ferrarese & Ford 2005). However, only a few percent of these manifest themselves as luminous active galactic nuclei (AGN). Nearly quiescent SMBHs, such as those hosting low luminosity AGN, constitute a silent majority (e.g. Ho 2009).
Understanding why most SMBHs appear to be inactive requires characterizing their gaseous environments. Gas near the SMBH sphere of influence, hereafter denoted the ‘circumnuclear medium’ (CNM), controls the mass accretion rate, . The accretion rate in turn determines the SMBH luminosity and the feedback of its energy and momentum output on larger scales. Dense gas in the nucleus may lead to runaway cooling, resulting in bursty episodes of star formation and AGN activity (e.g. Ciotti & Ostriker 2007).
Knowledge of how depends on the SMBH mass, , and other properties of the nucleus informs key questions related to the co-evolution of SMBHs and their host galaxies with cosmic time (e.g. Kormendy & Ho 2013a; Heckman & Best 2014). In the low redshift Universe, SMBH growth is dominated by low mass black holes, (e.g. Heckman et al. 2004), a fact often attributed to the trend of ‘cosmic down-sizing’ resulting from hierarchical structure growth (e.g, Gallo et al. 2008). However, the physical processes by which typical low mass black holes accrete could in principle be distinct from those operating at higher SMBH masses, or those in AGN. Of key importance is whether SMBHs grow primarily by the accretion of gas fed in directly from galactic or extragalactic scales, or whether significant growth can result also from local stellar mass loss in the nuclear region.
A better understanding of what mechanisms regulate accretion onto quiescent SMBHs would shed new light on a variety of observations, such as the occupation fraction of SMBHs in low mass galaxies. Miller et al. (2015) use the average relationship between the nuclear X-ray luminosities, , of a sample of early type galaxies and their associated SMBH masses to tentatively infer that the SMBH occupation fraction becomes less than unity for galaxies with stellar masses (). This method relies on extrapolating a power-law fit of the distribution to low values of below the instrument detection threshold, an assumption which is questionable if different physical processes control the accretion rates onto the lowest mass SMBHs.
The gas density in galactic nuclei also influences the emission from stellar tidal disruption events (TDEs), such as the high energy transient Swift J1644+57 (Bloom et al. 2011, Burrows et al. 2011; Levan et al. 2011; Zauderer et al. 2011). This event was powered by an impulsive relativistic jet, which produced synchrotron radio emission as the jet material decelerated from shock interaction with the CNM of the previously quiescent SMBH (Giannios & Metzger 2011; Zauderer et al. 2011). Modeling of J1644+57 showed that the CNM density was much lower than that measured surrounding Sgr A on a similar radial scale (Berger et al. 2012; Metzger et al. 2012). However, a TDE jet which encounters a denser CNM would be decelerated more rapidly, producing different radio emission than in J1644+57. Variations in the properties of the CNM could help explain why most TDEs appear to be radio quiet (e.g. Bower et al. 2013; van Velzen et al. 2013).
Gas comprising the CNM of quiescent (non-AGN) galaxies can in principle originate from several sources: (1) wind mass loss from predominantly evolved stars; (2) stellar binary collisions; and (3) unbound debris from a recent TDE. Stellar wind mass loss is probably the dominant source insofar as collisions are relevant only in extremely dense stellar environments for very young stellar populations (Rubin & Loeb 2011), while MacLeod et al. (2013) find that TDEs contribute subdominantly to the time-averaged accretion rate of quiescent SMBHs.
The gas inflow rate on large scales is much easier to constrain both observationally and theoretically than the black hole (horizon scale) accretion rate. Ho (2009) determines the inflow rates in a sample of early-type galaxies by using X-ray observations to determine the Bondi accretion rate, and also by using estimated mass loss rates of evolved stars. Both methods lead him to conclude that the available gas reservoir is more than sufficient to power the observed low-luminosity AGN, assuming the standard 10 per cent radiative efficiency for thin disc accretion. Several lines of evidence now suggest that low-luminosity AGN result from accretion proceeding in a radiatively inefficient mode (Yuan & Narayan 2014), due either to the advection of gravitationally-released energy across the SMBH horizon (e.g. Narayan & Yi 1995) or due to disc outflows, which reduce the efficiency with which the inflowing gas ultimately reaches the SMBH (e.g. Blandford & Begelman 1999; Li et al. 2013).
Another approach to determine the inflow rates, which we adopt in this paper, is to directly calculate the density, velocity and temperature profiles of the CNM using a physically motivated hydrodynamic model. Mass is injected into the nuclear environment via stellar winds, while energy is input from several sources including stellar winds, supernovae (SNe), and AGN feedback (Quataert 2004; De Colle et al. 2012; Shcherbakov et al. 2014). Unlike previous works, which focused primarily on modeling individual galaxies, here we model the CNM properties across a representative range of galaxy properties, including different SMBH masses, stellar density profiles, and star formation histories (SFHs).
Previous studies, employing multi-dimensional numerical hydrodynamics and including variety of (parametrized) physical effects, have focused on massive elliptical galaxies (e.g. Ciotti & Ostriker 2007; Ciotti et al. 2010). These works show the periodic development of cooling instabilities on galactic scales, which temporarily increase the gas inflow rate towards the nucleus until feedback becomes strong enough to shut off the flow and halt SMBH growth.
In this paper we focus on time-independent models, in which the nuclear gas receives sufficient heating to render radiative cooling negligible. This approach allows us to systematically explore the relevant parameter space and to derive analytic expressions that prove useful in determining under what conditions cooling instabilities manifest in the nuclear region across the expected range of galaxy properties, or whether other (non-AGN) forms of feedback can produce a prolonged state of steady, thermally stable accretion. Even if cooling instabilities develop on galactic scales over longer Gyr time-scales, we aim to explore whether a quasi steady-state may exist between these inflow events on smaller radial scales comparable to the sphere of influence.
In the presence of strong heating, one-dimensional steady-state flow is characterized by an inflow-outflow structure, with a critical radius known as the“stagnation radius” where the radial velocity passes through zero. Mass loss from stars interior to the stagnation radius is accreted, while that outside is unbound in an outflow from the nucleus. The stagnation radius, rather than the Bondi radius, thus controls the inflow rate (although we will show that usually resides near the nominal Bondi radius). When heating is sufficiently weak, however, the stagnation radius may move to much larger radii or not exist at all, significantly increasing the inflow rate the SMBH, i.e. a “cooling flow”. However, the hydrostatic nature of gas near the stagnation radius also renders the CNM at this location particularly susceptible to local thermal instabilities, the outcome of which could well be distinct from the development of a cooling flow.
This paper is organized as follows. In we describe our model, including the sample of galaxy properties used in our analysis () and our numerical procedure for calculating the steady-state hydrodynamic profile of the CNM (). In we describe our analytic results, which are justified via the numerical solutions we present in . We move from a general and parametrized treatment of heating to a physically motivated one in §5, where we consider a range of physical processes that can inject energy into the CNM. In we discuss the implications of our results for topics which include the nuclear X-ray luminosities of quiescent black holes, jetted TDEs, and the growth of SMBHs in the local Universe. In we summarize our conclusions. Table LABEL:table:definitions provides the definitions of commonly used variables. Appendix A provides useful analytic results for the stagnation radius, while Appendix B provides the details of our method for calculating stellar wind heating and mass input.
|Black hole mass|
|Total heating parameter, including minimum heating rate from stellar and black hole velocity dispersion|
|Total heating parameter, excluding minimum heating rate from velocity dispersion|
|Stellar velocity dispersion (assumed to be radially constant).|
|: Dynamical time at large radii (where stellar potential dominates)|
|: Free fall time (where the black hole potential dominates)|
|Alternative heating parameter, (eq. )|
|Stagnation radius, where gas radial velocity goes to zero|
|Radius of sphere of influence (eq. )|
|Outer break radius of stellar density profile|
|Radius interior to which SN Ia are infrequent compared to the dynamical time-scale (eq. )|
|3D radial stellar density profile|
|Gas density of CNM|
|Total enclosed stellar mass inside radius|
|Total enclosed mass inside radius (SMBH + stars)|
|Mass source term due to stellar winds, (eq. )|
|Parameter setting normalization of mass input from stellar winds (eq. )|
|Age of stellar population, in case of a single burst of star formation|
|Power-law slope of radial stellar surface brightness profile interior to the break radius|
|Power-law slope of the 3D stellar density profile inside of the break radius, .|
|Large scale inflow rate (not necessarily equal to the SMBH accretion rate)|
|SMBH accretion rate, , where accounts for outflows from the accretion disc on small scales.|
|Maximum accretion rate as limited by SN Ia (eq. )|
|Equilibrium inflow rate set by Compton heating acting alone (eq. )|
|Maximum accretion rate for thermally stable accretion (eq. )|
|Ratio of external heating (stellar winds, SN Ia, MSPs) to radiative cooling (eq. )|
|Gas density power-law slope at the stagnation radius (eq. )|
2.1 Galaxy models
Lauer et al. (2007) use Hubble Space Telescope WFPC2 imaging to measure the radial surface brightness profiles for hundreds of nearby early type galaxies. The measured profile is well fit by a “Nuker” law parameterization:
i.e., a broken power law that transitions from an inner power law slope, , to an outer power law slope, , at a break radius, . If the stellar population is spherically symmetric, then this corresponds to a 3D stellar density for and for . We can write the deprojected stellar density approximately (formally, this is the limit) as
where is the stellar density at the radius of
the black hole sphere of influence
This may be of questionable validity for low mass black holes (e.g., Greene et al. 2010; Kormendy & Ho 2013b). Also, several of the black hole masses used in McConnell et al. (2011) were underestimated (Kormendy & Ho, 2013b). However, our results are not overly sensitive to the exact form of the relationship that we use.
A galaxy model is fully specified by four parameters: , , , and . We compute models for three different black hole masses, , , . The distribution of in the Lauer et al. (2007) sample is bimodal, with a concentration of “core” galaxies with and a concentration of “cusp” galaxies with . We bracket these possibilities by considering models with and .
We fix but find that our results are not overly sensitive
to the properties of the gas flow on radial scales . The
presence of the break radius is, however, necessary to obtain
a converged steady state for some regions of our parameter
Finally, we note that Lauer et al. 2005 find that of cusp galaxies and of core galaxies have (generally unresolved) emission in excess of the inward extrapolation of the Nuker law. Indeed, some low mass galaxies with possess nuclear star clusters (Graham & Spitler 2009), which are not accounted for by our simple parametrization of the stellar density. In such cases gas and energy injection could be dominated by the cluster itself, i.e. concentrated within its own pc-scale “break radius” which is much smaller than the outer break in the older stellar population on much larger scales. Although our analysis does not account for such an inner break, we note that for high heating rates the stagnation radius and concomitant inflow rate are not sensitive to the break radius.
2.2 Hydrodynamic Equations
Following Quataert (2004, see also Holzer & Axford 1970; De Colle et al. 2012; Shcherbakov et al. 2014), we calculate the density , temperature , and radial velocity of the CNM for each galaxy model by solving the equations of one-dimensional, time-dependent hydrodynamics,
where and are the pressure and specific entropy, respectively, and is the enclosed mass (we neglect dark matter contributions). We adopt an ideal gas equation of state with with and . The source term in equation (5),
represents mass input from stellar winds, which we parametrize in
terms of the fraction of the stellar density being
recycled into gas on the Hubble time yr.
To good approximation at
time following an impulsive starburst (e.g.,
Ciotti et al. 1991)
Source terms also appear in the momentum and entropy equations (eqs.  and ) because the isotropic injection of mass represents, in the SMBH rest frame, a source of momentum and energy relative to the mean flow. Physically, these result from the mismatch between the properties of virialized gas injected by stellar winds and the mean background flow. The term is important because it acts to stabilize the flow against runaway cooling ().
The term in the entropy equation accounts for external heating sources (e.g., Shcherbakov et al. 2014), where
is the stellar velocity dispersion. This accounts for the minimal amount of shock heating from stellar winds due to the random motion of stars in the SMBH potential. We take to be constant and use , where km/s is the velocity dispersion from the McConnell et al. (2011) relation. The second term, , parametrizes additional sources of energy input, including faster winds from young stars, millisecond pulsars (MSPs), supernovae, AGN feedback, etc (). We assume that is constant with radius, i.e. that the volumetric heating rate is proportional to the local stellar density.
Our model does not take into account complications such as more complicated geometries or the discrete nature of real stars (Cuadra et al., 2006; Cuadra et al., 2008). In the case of Sgr A* these effects reduce the time-averaged inflow rate to – an order of magnitude less than a 1D spherical model (Cuadra et al., 2006). Motions of individual stars can produce order of magnitude spikes in the accretion rate (Cuadra et al., 2008).
To isolate the physics of interest, our baseline calculations neglects three potentially important effects: heat conduction, radiative cooling, and rotation. Heat conduction results in an an additional heating term in equation (7),
where (Dalton & Balbus 1993) is the conductivity and is the classical Spitzer (1962) value ( in cgs units). The flux limiter saturates the conductive flux if the mean free path for electron coulomb scattering exceeds the temperature length scale, where is the isothermal sound speed and is an uncertain dimensionless constant (we adopt ). Even a weak magnetic field that is oriented perpendicular to the flow could suppress the conductivity by reducing the electron mean free path. However, for radially-decreasing temperature profiles of interest, the flow is susceptible to the magneto-thermal instability (Balbus 2001), the non-linear evolution of which results in a radially-directed field geometry (Parrish & Stone 2007). In we show that neglecting conductivity results in at most order-unity errors in the key properties of the solutions.
Radiative cooling contributes an additional term to equation (7), of the form
where and is the cooling function. We neglect radiative cooling in our baseline calculations, despite the fact that this is not justified when the wind heating is low or if the mass return rate is high. Once radiative cooling becomes comparable to other sources of heating and cooling, its presence can lead to thermal instability (e.g. Gaspari et al. 2012b, McCourt et al. 2012, Li & Bryan 2014a) that cannot be accounted for by our 1D time-independent model. Our goal is to use solutions which neglect radiation to determine over what range of conditions cooling instabilities will develop ().
Equations (5)-(7) are solved using a sixth
order finite difference scheme with a third order Runge-Kutta scheme
for time integration and artificial viscosity terms in the velocity
and entropy equations for numerical stability
Our solutions can be scaled to any value of the mass input parameter, , since the mass and energy source terms scale linearly with or ; however, the precise value of must be specified when cooling or thermal conduction are included. We check the accuracy of our numerical solutions by confirming that mass is conserved across the grid, in addition to the integral constraint on the energy (Bernoulli integral).
3 Analytic Results
We first describe analytic estimates of physical quantities, such as the stagnation radius and the mass inflow rate, the detailed derivation of which are given in Appendix A.
3.1 Flow Properties Near the Stagnation Radius
Continuity of the entropy derivative at the stagnation radius where requires that the temperature at this location be given by (eq. )
where km s) and is the density power-law slope at . Empirically, we find from our numerical solutions that
For high heating rates , the stagnation radius resides well inside the SMBH sphere of influence. In this case , such that equation (14) simplifies to
where we have used equation (13) to estimate separately for core (; ) and cusp (; ) galaxies. This expression is similar to that obtained by Volonteri et al. (2011) on more heuristic grounds (their eq. 6).
In the opposite limit of weak heating () the stagnation radius moves to large radii, approaching the break radius in the stellar density profile, implying that all of the interstellar medium (ISM) inside of is inflowing. In particular, we find that approaches for heating below a critical threshold
Condition (16) approximately corresponds to the requirement that the heating rate exceed the local escape speed at the break radius, . This result makes intuitive sense: gas is supplied to the nucleus by stars which are gravitationally bound to the black hole, so outflows are possible only if the specific heating rate significantly exceeds the specific gravitational binding energy.
3.2 Inflow Rate
The large scale inflow rate towards the SMBH is given by the total mass loss rate interior to the stagnation radius (eq. ),
where we have assumed by adopting equation (15) for . The resulting Eddington ratio is given by
where yr is the Eddington accretion rate, assuming a radiative efficiency of ten per cent. Note the sensitive dependence of the inflow rate on the wind heating rate. Equation (18) is the radial mass inflow rate on relatively large scales and does not account for outflows from the SMBH accretion disc (e.g. Blandford & Begelman 1999; Li et al. 2013), which may significantly reduce the fraction of that actually reaches the SMBH. Thus we distinguish between the large scale inflow rate, , and the accretion rate onto the black hole, .
The gas density at the stagnation radius, , is more challenging to estimate accurately. By using an alternative estimate of as the gaseous mass within the stagnation radius divided by the free-fall time ,
where is the Bondi radius, is the adiabatic sound speed, and is a parameter of order unity.
3.3 Heat Conduction
Our analytic derivations neglect the effects of heat conduction, an assumption we now check. The ratio of the magnitude of the conductive heating rate (eq. ) to the external heating rate at the stagnation radius is given by
where the stagnation radius is assume to reside well inside the Nuker break radius.
Equation (23) shows that, even when conduction is saturated, our neglecting of heat conduction near the stagnation radius results in at most an order unity correction for causal values of the saturation parameter . Our numerical experiments which include conductive heating confirm this (). We do not consider the possibility that the conduction of heat from the inner accretion flow can affect the flow on much larger scales (Johnson & Quataert, 2007).
3.4 Thermal Instability
Radiative cooling usually has its greatest impact near or external to the stagnation radius, where the gas resides in near hydrostatic balance. If radiative cooling becomes important, it can qualitatively alter key features of the accretion flow. Initially hydrostatic gas is thermally unstable if the cooling time is much less than the free-fall time , potentially resulting in the formation of a multi-phase medium (Gaspari et al. 2012b; Gaspari et al. 2013, 2015; Li & Bryan 2014b).
Even if the hot plasma of the CNM does not condense into cold clouds, the loss of pressure can temporarily increase the inflow and accretion rates by producing a large-scale cooling flow. When coupled to feed-back processes which result from such enhanced accretion, this can lead to time-dependent limit cycle behavior (e.g. Ciotti & Ostriker 2007; Ciotti et al. 2010; Yuan & Li 2011, Gan et al. 2014), which is also inconsistent with our assumption of a steady, single-phase flow.
Cooling instability can, however, be prevented if destabilizing radiative cooling () is overwhelmed by other sources of cooling, namely the stabilizing term in the entropy equation (eq. ). Neglecting radiative cooling to first order, this term is balanced at the stagnation radius by the external heating term, . One can therefore assess thermal stability by comparing the ratio of external heating to radiative cooling (eq. ),
where we have used equations (20) and (24) for the gas and stellar densities at the stagnation radius, respectively. We have approximated the cooling function for K as erg cms, which assumes solar metallicity gas (Draine 2011; his Fig. 34.1).
To within a constant of order unity, equation (25) also equals the ratio of the gas cooling time-scale to the free-fall time at the stagnation radius. This equivalence can be derived using the equality
that results by combining equations (17),(19), and (24). Sharma et al. (2012) argue cooling instability develops in a initially hydrostatic atmosphere if , so equation (25) represents a good proxy for instability in this case as well.
Based on our numerical results (§4) and the work of Sharma et al. (2012) we define thermally stable flows according to the criterion () being satisfied near the stagnation radius. This condition translates into a critical minimum heating rate
Equations (25) and (27) are derived using expressions for the stagnation radius and gas density in the high heating limit of (eq. ). However, for , the stagnation radius diverges to the break radius . The quasi-hydrostatic structure that results in this case greatly increases the gas density, which in practice renders the flow susceptible to thermal runaway, even if according to equation (27). In other words, the true condition for thermal instability can be written
Figure 1 shows for different mass input parameters , as well as the [-independent] criterion, shown separately for cusp and core galaxies. Based on the Lauer et al. (2007) sample we take pc and (240 pc) and thus for cores (cusps). We see that the for high and low the criterion is more stringent, while for low and high , the criterion is more stringent.
Note that since the SMBH accretion rate cannot exceed the large scale inflow rate, also represents the maximum thermally stable accretion rate.
What we describe above as “thermal instability” may in practice simply indicate an abrupt transition from a steady inflow-outflow solution to a global cooling flow, as opposed a true thermal instability. In the former case the stagnation radius diverges to large radii, increasing the density in the inner parts of the flow, which increases cooling and creates a large inflow of cold gas towards the nucleus. A true thermal instability would likely result in a portion of the hot ISM condensing into cold clouds, a situation which may or may not be present in a cooling flow. In this paper we do not distinguish between these possibilities, although both are likely present at some level. Finally, note that if the CNM were to “regulate” itself to a state of local marginal thermal instability (as has previously been invoked on cluster scales; e.g., Voit et al. 2015), then equation (29) might naturally reflect the characteristic mass fall-out rate and concomitant star formation rate.
3.5 Angular Momentum
Our spherically symmetric model neglects the effects of angular momentum on the gas evolution. However, all galaxies possess some net rotation, resulting in centrifugal forces becoming important at some radius . Here is the stellar specific angular momentum near the stagnation radius, from which most of the accreted mass originates, where is the stellar azimuthal velocity.
Emsellem et al. (2007) use two-dimensional kinematic data to measure the ratio of ordered to random motion in a sample of early type galaxies, which they quantify at each galactic radius by the parameter
where is the velocity dispersion and the brackets indicate a luminosity-weighted average. The circularization radius of the accretion flow can be written in terms of as
where we have used the definition and in the second inequality have assumed that , a condition which is satisfied for the thermally-stable solutions of interest.
Emsellem et al. (2007) (their Fig. 2) find that is generally on radial scales 10 per cent of the galaxy half-light radius and that decreases with decreasing interior to this point. From equation (31) we thus conclude that . For the low inflow rates considered the gas () would be unable to cool on a dynamical time and would likely drive equatorial and polar outflows (Li et al., 2013). Our model cannot capture such two dimensional structures, but would still be relevant for intermediate polar angles where the gas is inflowing.
4 Numerical Results
Break radius of stellar density profile. as fixed in our numerical runs. Inflow rate in Eddington units, normalized to a stellar mass input parameter . Ratio of wind heating rate to radiative cooling rate at the stagnation radius. Calculated with radiative cooling included, assuming mass loss parameter . Calculated with radiative cooling and conductivity included, assuming mass loss parameter and conductivity saturation parameter . Solutions including radiative cooling were performed for cusp galaxies with km s and core galaxies with km s for = 0.02, 0.2, and 0.6. Values of resulting in thermally unstable solutions are marked in the final column. Solutions found to be thermally unstable for all are denoted as TI.
Our numerical results, summarized in Table LABEL:table:models, allow us to study a range of CNM properties and to assess the validity of the analytic estimates from the previous section.
Figure 2 shows profiles of the density , temperature , and radial velocity , for the cusp () solutions within our grid. As expected, the gas density increases towards the SMBH with , i.e. shallower than the power law for Bondi accretion. This power law behavior does not extend through all radii, however, as the gas density profile has a break coincident with the location of the break in the stellar light profile ( pc). The temperature profile is relatively flat at large radii, but increases as interior to the sphere of influence, where , somewhat shallower than expected for virialized gas within the black hole sphere of influence. The inwardly directed velocity increases towards the hole with a profile that is somewhat steeper than the local free-fall velocity . The flow near the stagnation radius is subsonic, but becomes supersonic at two critical points. The inner one at is artificially imposed for numerical stability, although we have verified that moving the inner boundary has a small effect on the solution properties near the stagnation radius. The outer one is located near the break radius, and is caused by the transition to a steeper stellar density profile exterior to the outer Nuker break radius.
Figure 3 shows our calculation of the stagnation radius as a function of the wind heating parameter , with different colors showing different values of . Cusp and core galaxies are marked with square and triangles, respectively. Shown for comparison are our analytic results (eq. ) with solid and dashed lines for cusp and core galaxies, respectively, calculated assuming and , respectively.
Our analytic estimates accurately reproduce the numerical results in the high heating limit (; ). However, for low heating the stagnation radius diverges above the analytic estimate, approaching the stellar break radius . This divergence occurs approximately when (eq. ). Physically this occurs because the heating rate is insufficient to unbind the gas from the stellar potential. Thus, for small values of the heating rate (small ) the location of the stagnation radius will vary strongly with the break radius. This explains the behavior of the three vertically aligned green squares. These are three cusp () galaxies with and km s but with different break radii ( 25, 50, and 100 pc from top to bottom). This divergence of the stagnation radius to large radii occurs at a higher value of in core galaxies (), explaining the behavior of the two core galaxies shown in Fig. 3 as vertically aligned orange triangles.
Figure 4 shows the inflow rate for a sample of our numerical solutions for different values of 300, 600 and 1200 km s, and for both core (=0.1) and cusp galaxies (=0.8). Shown for comparison is our simple analytic estimate of from equation (18). For high wind velocities () the stagnation radius lies well inside the black hole sphere of influence and our analytic estimate provides a good fit to the numerical results. However, for low wind velocities and/or high (large ), the numerical accretion rate considerably exceeds the simple analytic estimate as the stagnation radius diverges to large radii (Fig. 3).
Fig. 5 shows the ratio of wind heating to radiative cooling, (eq.  and surrounding discussion) as a function of radius for and 1200 km/s, , and . Radiative cooling is calculated using the cooling function of Draine (2011) for solar metallicity. For high heating rates of and km s cooling is unimportant across all radii, while for km s we see that and can be less than unity, depending on the wind mass loss parameter . Because at the stagnation radius is within a factor of two of its minimum across the entire grid, the value of is a global diagnostic of thermal instability for cusp galaxies. For core galaxies, the minimum value for the ratio of wind heating to radiative cooling may be orders of magnitude less than the value at the stagnation radius. However, we find that the criterion (see eq. (16)) combined with the ratio of wind heating to radiative cooling at still gives a reasonable diagnostic of thermal stability for core galaxies.
Also note that for and km s we have near the stagnation radius, but these ratios diverge from each other at low heating ( km s). This results because equations (19), (20) underestimate the true gas density in the case of subsonic flow (weak heating).
Although most of our solutions neglect thermal conduction and radiative cooling, these effects are explored explicitly in a subset of simulations. Dagger symbols in Table LABEL:table:models correspond to solutions for which we turned on radiative cooling. For cases which are far from being thermal instability when cooling is neglected (e.g.,