3.2. Analytic modeling
Analytic `toy' models are another useful way to model the principal features of mechanical heating by AGNs. We consider the pdV work done by a spherically symmetric ensemble of bubbles rising subsonically through the pressure gradient. Since the timescale for the bubbles to cross the cluster (of order the free-fall time) is much shorter than the cooling timescale, the flux of bubble energy through the ICM approaches a steady state, implying that details of the energy injection process - such as the number flux of bubbles (e.g., one big one or many small ones), the bubble size, filling factor, and rate of rise - do not affect the mean heating rate. If we assume that the pdV work is dissipated within a pressure scale height of where it is generated, we can devise an average volume heating rate for the ICM, as a function of radius (Begelman 2001):
In eq. (3.1), <Lb> is the time-averaged power output of the AGN going into bubbles, p(r) is the pressure inside the bubbles (and p0 is the pressure where the bubbles are formed), and the exponent 1/4 equals ( - 1) / for a relativistic plasma (the exponent would be 2/5 for a nonrelativistic gas). A major assumption of the model, that the pdV work is absorbed and converted to heat within a pressure scale height, will have to be assessed using numerical simulations and studies of the microphysics of cluster gas. We have also assumed that the energy is spread evenly over 4 sr, a likely consequence of buoyancy.
The most important property of the above effervescent heating rate is its proportionality to the pressure gradient (among other factors), since this determines the rate at which pdV work is done as the bubbles rise. Since thermal gas that suffers excess cooling will develop a slightly higher pressure gradient, the effervescent heating mechanism targets exactly those regions where cooling is strongest. Therefore, it has the potential to stabilize radiative cooling (Begelman 2001). This potential is borne out in 1D, time-dependent numerical simulations of a cooling core (Ruszkowski & Begelman 2002), which show that the flow settles down to a steady state that resembles observed clusters, for reasonable parameters and without fine-tuning the initial or boundary conditions. Although these models include conduction (at 23% of the Spitzer rate), which may be necessary for global stability (Kim & Narayan 2003), the heating is overwhelmingly dominated by the AGN at all radii. The mass inflow rate through the inner boundary, which determines the AGN feedback in these simulations, stabilizes to a reasonable value far below that predicted by cooling flow models.
According to the analytic models, a large fraction of the injected energy reaches radii much larger than the cooling radius. Roychowdhury et al. (2004) have used these models to study whether AGN heating can explain the excess entropy of cluster gas. We find that the distributed heating rate given by eq. (3.1) can reproduce simultaneously the luminosity-temperature correlations measured at two different radii (roughly 0.5 and 0.08 of the virial radius), for cluster temperatures between 1 and 10 keV. These results appear to require a high efficiency of kinetic energy production (KE) during the growth of the black hole as well as a black hole mass (or sum of black hole masses) roughly proportional to the virial mass of the cluster. If we assume MBH 0.0015Mbulge for each galactic bulge in the cluster (Häring & Rix 2004), and further assume that bulges comprise a fraction 0.01f-2 of the cluster's virial mass, then the entropy measurements require KE f-2 0.05. Realistically this represents an upper limit, since it was calculated assuming that all of the heating occurs at low redshift, i.e., after each cluster has fully formed. Since most black hole growth appears to have occurred by accretion during the `quasar era' at z ~ 2 - 4 (Yu & Tremaine 2002), the entropy we see today was probably injected into the lower-mass precursors of modern clusters, which had lower virial temperatures. According to the second law of thermodynamics, the heat input required to produce a given change in entropy is proportional to temperature; therefore, the required value of KE f-2 is probably smaller than the one stated above.