5.4.3. Effects of the magnetic field
Charged particles gyrate around magnetic field lines on orbits with a radius (the gyroradius) of rg = (mv c/ZeB), where m is the particle mass, v is the component of its velocity perpendicular to the magnetic field, z is the particle charge, and B is the magnetic field strength. If v = (2 k Tg / m)1/2, which is the rms value in a thermal plasma, then
which is much smaller then any length scale of interest in clusters, and is also much smaller than the mean free path of particles due to collisions rg << e. Then, the effective mean free path for diffusion perpendicular to the magnetic field is only on the order of rg2 / e (Spitzer, 1956). Because they have larger gyroradii, the ions are most effective in transport processes perpendicular to the magnetic field. In practice, the gyroradii are so small that diffusion perpendicular to the magnetic field can be ignored in the intracluster gas.
Consider the effect of the magnetic field on thermal conduction, when the temperature gradient lies at an angle to the local magnetic field direction. Only the component of the gradient parallel to the field is effective in driving a heat flux, and only the the component of the resulting heat flux in the direction of the temperature gradient transports any net energy. If the conduction is unsaturated, the heat flux parallel to the thermal gradient is thus reduced by a factor of cos 2. If the conduction is saturated, the heat flux is independent of the temperature gradient; the appropriate factor is just cos (Cowie and McKee, 1977).
Observations suggest that the magnetic field in clusters may be tangled, so that the direction varies throughout the intracluster gas (Section 3.6). Let lB be the coherence length of the magnetic field, so that the field will typically have changed direction by 90° over this distance. Let lT be the temperature scale height (Section 5.4.2 above) and let e be the mean free path of electrons. Consider first the case where e << lT, so that the conduction is unsaturated. Then, if lB lT, the magnetic field is ordered over the scales of interest in the cluster, and the value of cos 2 depends on the geometry of the magnetic field. For example, for a cluster with a radial temperature gradient and a circumferential magnetic field, thermal conduction would be suppressed. Alternatively, if lT >> lB >> e, then the field direction can be treated as a random variable, and the heat flux is reduced by < cos2 > = 1 / 3. If the coherence length of the magnetic field is less than the mean free path lB << e, the conductivity depends on the topology of the magnetic field (i.e., whether it is connected over distances greater than lB). In general, the effective mean free path for diffusion will always be at least as small as lB, and could be as small as lB2 / e if the magnetic field is disconnected on the scale of lB. In this limit, the thermal conduction would be very significantly reduced.
The Faraday rotation observations (Section 3.6) suggest that lB e 20 kpc. Thus thermal conduction will probably be reduced by a factor of at least 1/3, and the conductivity time scale (equations 5.41 and 5.43) should be increased by at least this factor.