5.7.5. Cooling flow models with star formation
If cooling gas is being converted into stars in cooling flows, then terms representing the loss of gas should be added to the equations for the flow (equations 5.101). Regardless of the ultimate fate of the cooling gas, the fact that the gas is thermally unstable means that it is not reasonable to treat the cooling flow as homogeneous. In homogeneous models for cooling flows, the gas remains hot enough to emit X-rays until it is within the sonic radius (Section 5.7.2). However, in an inhomogeneous flow, thermal instability will cause denser lumps of gas to cool below X-ray emitting temperatures while the more diffuse gas is still quite hot. Thus, even if star formation were not occurring, thermal instabilities would reduce the amount of hot gas as it moves towards the center of the galaxy.
If the mass flow rate 
of hot gas decreases with distance from the center of
the flow, this should result in an observable reduction in the amount of
X-ray emission near the center of the flow. Such a reduction does appear
to be required
by the X-ray surface brightness data of cooling flow clusters. Using the
semi-empirical method to determine
from X-ray surface
brightness profiles (Section 5.7.1),
Fabian et al.
(1984b)
and Stewart et
al. (1984a)
found that the cooling rate
increased with radius
in M87/Virgo and NGC1275/Perseus. Unfortunately, their method of determining
is
inconsistent if is
not constant
(White and Sarazin,
1987a,
b).
First, their particular form of the energy equation (last of equations
5.101) requires that
be constant. Second, they assumed that only the hot diffuse gas
contributed to the X-ray emission. But the gas being removed from the
flow by thermal instabilities cools radiatively, producing X-ray emission.
Thus the X-ray emissivity should include a term proportional to the
rate of gas loss through thermal instabilities. Semi-empirical methods
to determine (r)
including more consistent treatments of mass loss have been given by
Fabian et al. (1985,
1986a)
and White and Sarazin (1987a,
b).
These studies indicate that
(r)
increases with increasing r
in the best studied cooling flow clusters. Fabian et al. argue
that the variation of
(r) is well
represented by
(r)
r, while White and
Sarazin contend that the variation of
(r) is too
sensitive to the assumed form of the gravitational potential to allow
any strong statements to be made.
If the gas in cooling flows is inhomogeneous, it is much more difficult to
model the dynamics of the flows. In principle, in a correct treatment of
the flow the gas would be represented by a continuous range of densities
g.
Rather than giving a single set of thermodynamic variables, say
g,
Tg, and v, as a function of
position r, one should specify a distribution of densities. For
example,
f(g , r)
might be the fraction of the volume or mass in the flow at r
which is in the form of gas at a density
g.
Correspondingly, Tg(g , r) and
vg(g , r)
would be the mean temperature and velocity of gas having a density
g
located at r. Obviously, this would vastly increase the
complexity of the hydrodynamical modeling of the flow. For one thing,
there is no clear physical
argument which specifies the boundary conditions on the inhomogeneities (for
example, the value of
f(g) at the cooling radius
rcool. Perturbations in
the flow probably cool isobarically until they are too cool to produce a
significant amount of X-ray emission (Tg
106
K), so that it is probably
reasonable to assume that all density phases have the same pressure at each
position. Then the temperature is just
Tg(g , r) = P(r)
/ g.
However,
the possibility that the different densities phases would have different
velocities still is an enormous complication, since the hydrodynamical
interactions between lumps of differing density and velocity would be
extremely complex.
Two opposite approximations have been made to deal with this problem.
First, Fabian et al.
(1985,
1986a)
assumed that all the density phases
comove, so that both v and P are functions only of
position r and not of density
g.
As noted above (Section 5.7.3),
the fastest growing linear
perturbations (
g
/ g)
<< 1 in a homogeneous flow do comove, which
supports this idea. However, one might expect that once perturbations
grow nonlinearly
(
g
/ g)
>> 1 they might drop out of the flow. This led
White and Sarazin (1987a,
b)
to an opposite approximation. They argued
that the isobaric cooling time decreases quite rapidly with decreasing
temperature. Thus once a lump has cooled significantly below the average
temperature, it will cool below X-ray emitting temperatures rapidly. As
the density of the lump increases, its surface area will decrease and it
can decouple from the flow and fall ballistically. Since the flow time is
determined by the cooling time of the diffuse gas, the cooling and
decoupling of
higher density lumps can occur before the flow has moved inwards by a
significant amount. In this limit, the cooling of dense lumps of gas can be
treated as a local sink for the diffuse gas, and the flow equations revert
to equations 5.101 with loss terms. Numerical models for cooling flow
including loss terms which are proportional to either the cooling time
or the growth time of thermal instabilities have been given by
White and Sarazin (1987a,
b).
These inhomogeneous models for cooling flows can be used to predict the
X-ray surface brightness profiles and spectral variations of cooling flows
(White and Sarazin,
1987b).
If clumps of gas cool rapidly from X-ray emitting
temperatures to Tg
104 K,
they can also predict the surface brightness of
optical line emission. If these cooling condensates form stars quickly
or if the cool
lumps are decoupled from the flow and falling ballistically (and form stars
eventually), these models can give the predicted distribution of these
stars. In Section 5.7.4,
attempts to detect the presence of a younger stellar population due
to accretion in the optical spectra of central galaxies in cooling flows
were
discussed. It is also very important to study the spatial distribution
of this population. As discussed in
Section 2.10.1, central dominant
galaxies in
clusters appear to be composed of an extended giant elliptical interior,
surrounded in the case of rich cluster cDs by a very extended
halo. These cDs may also have dark, missing mass haloes
(Section 5.8.1). Which of these
components could be the result of accretion-driven star formation? If
accretion produces the giant elliptical interiors, why do these resemble
the stellar distributions in other nonaccreting giant ellipticals? Giant
ellipticals have light distributions that are reasonably fit by de
Vaucouleurs
or Hubble profiles (Section 2.10.1),
and recent numerical studies have
suggested that these form naturally in violent relaxation
(Section 2.9.2).
Accretion-driven star formation is a slow process; will it give a similar
distribution?
As discussed above, semi-empirical determinations of the hot gas inflow
rates by
Fabian et al. (1985,
1986a)
are consistent with
r, although
White and Sarazin (1986a,b)
have argued that
(r) is
extremely uncertain. If the cooling
lumps form stars rapidly and if the orbits of the newly formed stars are not
affected by the galaxy potential, this might imply that the density of
the new stars varied as
*
r-2. This is much flatter than the density
distribution of the luminous stars in elliptical galaxies. This is just
the density distribution of an isothermal sphere, and is similar to the
density distributions
inferred for the missing mass haloes of spiral galaxies
(Section 2.8). This led
Fabian et al. (1986a,
b)
to suggest that the missing mass is very low mass
stars formed in cooling flows; this requires that the initial mass function
for star formation in cooling flows produce mainly very low mass stars
(M*
0.1M
).
On the other hand, White and Sarazin (1986a,b) found that the predicted stellar distributions in their cooling flow models with star formation were very similar to those of the light from giant elliptical galaxies. In calculating the stellar distributions, they self-consistently included the effects of the stellar orbits in the galaxy and cluster gravitational potential.
Accretion-driven star formation may result in different stellar orbits in cD galaxies than in nonaccreting giant ellipticals. If the flows have little angular momentum (see above) and are radial, the resulting stellar orbits may be radial. If the processes of clumping and star formation impart significant random velocities to the star forming regions, the orbits might be isotropic. If the flow stagnates and forms a disk, stellar disks may be found within cD galaxies.