**5.3.4. Heating by galaxy motions**

Although ongoing heating of the intracluster gas may not be necessary to
account for the observed gas temperatures, the estimates given above and
the history of the gas are sufficiently uncertain that one cannot rule
out ongoing heating as an important processes. One way in which
intracluster gas could be heated would be
through friction between the gas and the galaxies that are constantly moving
throughout the cluster
(Ruderman and Spiegel,
1971;
Hunt, 1971;
Yahil and Ostriker,
1973;
Schipper, 1974;
Livio *et al.*,
1978;
Rephaeli and Salpeter,
1980).
The calculation of the magnitude of this drag force and of the
consequent heating
of the intracluster gas is complicated by the following problems. First,
the motion of an average cluster galaxy through the intracluster medium
is likely to be just transonic *m*
1, where
*m*
*v/c*_{s} is the Mach number, *v*
is the galaxy velocity, and *c*_{s} =
1480(*T*_{g} / 10^{8} K)^{1/2} km/s is the
sound
speed in the gas. If equation (4.10) for the observed gas temperatures is
assumed, and the average galaxy velocity is taken to be
3^{1/2}
_{r}, then the
average Mach number is <*M*>
1.5. Thus the galaxy
motion cannot be
treated as being either highly supersonic (strong shocks, etc.) or very
subsonic (incompressible, etc.). In some cases shocks will be formed by the
motion, and in some cases no shocks form. Second, the mean free path
_{i} of
ions in the intracluster medium due to Coulomb collisions (equation
5.34) is similar to the radius of a galaxy *R*_{gal}
20 kpc
(Nulsen, 1982).
Thus it is unclear whether the intracluster gas should be treated as a
collisionless gas or as a fluid, and the role of transport processes such as
viscosity (Section 5.4.4) is uncertain.
For example, the Reynolds number of
the flow about an object of radius *r* is *Re*
3(*R* /
_{i})*m*
(equation 5.46
below), and thus is somewhat larger than unity if *r* =
*R*_{gal}. It is therefore
uncertain whether the flow will be laminar or turbulent. The magnetic
field can affect transport processes
(Section 5.4.3), but the coherence
length of the field *l*_{B} estimated from Faraday rotation
observations is also comparable to the size of a galaxy
(Section 3.6). Finally, the nature of the drag
force depends on whether the galaxy contains interstellar gas or not.
If the galaxy contains no gas, it affects the intracluster medium only
through its gravitational field. If the galaxy contains high density gas, it
can give the galaxy an effective surface. For example, a gasless galaxy
in supersonic motion probably will not produce a bow shock, while a
gas-filled galaxy may
(Ruderman and Spiegel,
1971;
Hunt, 1971;
Gisler, 1976).

It is convenient to write the rate of energy loss by the galaxy and the heating rate of the intracluster medium as

(5.28) |

where _{g} is the intracluster gas density and
*R*_{D} is the effective radius of the
galaxy for producing the drag force. First, assume that the intracluster
gas is collisionless. Then the drag is given by the dynamical friction
force of equation (2.34) and

(5.29) |

where *R*_{A}
2*Gm/v*^{2} is the accretion radius
(Ruderman and Spiegel, 1971),
*m* is the galaxy mass,
is given by
equation (2.29), and *x*
(5/6)^{1/2} *M*. By the
virial theorem applied to the galaxy, *R*_{A}
*R*_{gal}(_{*}
/ _{r})^{2},
and the accretion radius
is typically much smaller than the galaxy radius, since the galaxy velocity
dispersion is smaller than that of a cluster. In the limit of hypersonic
motion
*M* >> 1, the term in brackets reduces to unity. In this
limit,
Rephaeli and Salpeter
(1980)
have shown that equation (5.29) gives the drag force for any
value of the mean free path or viscosity if no gas is present in the galaxy.
The definition of
must be slightly
modified
(Ruderman and Speigel,
1971).

For a 10^{11}
*M*_{} galaxy
moving at 1000 km/s through intracluster gas with a
proton number density of 10^{-3} cm^{-3}, the rate of
heating of the gas is
10^{41}
erg/s. If the cluster contained 1000 such galaxies, the total heating
rate would be
10^{44}
erg/s. While not trivial, this heating rate is too small to heat the
intracluster gas in a Hubble time
(Schipper, 1974;
Rephaeli and Salpeter,
1980).
If the total mass of intracluster gas is
10^{14}
*M*_{} and
the gas temperature is
6 ×
10^{7} K, then the time required to heat the gas at this rate
would be 7 ×
10^{11} yr.

The drag force can be considerably increased if the galaxy contains gas or
magnetic fields that prevent the penetration of the galaxy by the
intracluster gas
(Ruderman and Spiegel,
1971;
Yahil and Ostriker,
1973;
Livio *et al.*,
1978;
Shaviv and Salpeter,
1982;
Gaetz *et al.*,
1987).
However, the main
effect of the drag force may be to remove the gas from the galaxy and
heat it to the temperature determined by the galaxy kinetic energy per
mass (Section 5.3.3). The stripping of gas
from a galaxy is shown in
Section 5.9 to be very efficient, and as
discussed in Sections 2.10.2 and
4.6,
galaxies in X-ray clusters are known to be very deficient in gas. It is
possible that galaxies may retain a core of gas produced by stellar mass
loss within the galaxy.
Gaetz *et al.*
(1987)
give useful analytic fitting
formulas for drag coefficients for galaxies with stellar mass loss, based on
two-dimensional hydrodynamic simulations.
Yahil and Ostriker
(1973) and
Livio *et al.*
(1978)
have suggested very large heating rates produced
by having high rates of gas output in galaxies. Livio *et al.* assume a
rather large cross section for galaxies and a rather low intracluster gas
density. Both of these papers argue that the heating rate due to galaxy
motions is so large that the intracluster gas is heated beyond the escape
temperature from the cluster, and a cluster wind results (see
Section 5.6).

There is a simple argument against the great importance of heating due to drag forces from the motions of galaxies. The mass associated with intracluster gas in a typical X-ray cluster is comparable to or greater than the mass associated with galaxies (Section 4.4.1). The average thermal velocities in the gas are comparable to or greater than the typical galaxy velocities (Sections 4.6, 5.3.2, and 5.5). Thus the total thermal energy in the gas is greater than or comparable to the total kinetic energy in the galaxies. It is difficult to believe that the galaxies heated the gas under these circumstances. If they did, then the massive galaxies would have lost most of their initial kinetic energy in the process; this should have produced quite extreme mass segregation, which is not observed (Section 2.7).