**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):

(3) |

In eq. (3.1), <*L*_{b}> is the time-averaged power
output of the AGN going into bubbles, *p*(*r*) is the
pressure inside the bubbles (and *p*_{0} 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
*M*_{BH}
0.0015*M*_{bulge} for each galactic bulge in the cluster
(Häring & Rix
2004),
and further assume that bulges comprise a fraction
0.01*f*_{-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.