Thermal equilibrium under the influence of gravitation

Gas clouds under the influence of gravitation in thermodynamic equilibrium cannot be isothermal due to the Dufour effect, the energy flux induced by density gradients. In galaxy clusters this effect may be responsible for most of the ”cooling flows” instead of radiative cooling of the gas. Recent observations of galaxy clusters with high spatial resolution have shown that in most of them the intracluster plasma is far from being isothermal, exhibiting strong decrease of the plasma temperature in the vicinity of individual large galaxies. This effect is commonly attributed to ”cooling flows”, that means, increasing radiative energy loss with increasing density of the gas, which is gravitationally attracted by the galaxies. But several discussions (Voigt et al. [1], Markevitch et al. [2]) have shown that most of the temperature gradients would be washed out by thermal conduction, if the thermal conductivity of the plasma were of the magnitude given in the classical book by Spitzer [3]. In a paper of Loeb [4] it has been even demonstrated that complete clusters would have cooled off, if thermal conductivity were at the Spitzer value. It has been proposed that the action of magnetic fields reduces the thermal conductivity. But apart from the fact that no explanation exists for the generation of ordered magnetic fields of the required magnitude, the heat flux reduction works only perpendicular to the magnetic field lines. Thus no reasonable field geometry is imaginable, which would reduce the overall flux in a radially symmetric matter distribution to the required extent. In all these discussions it is implicitly assumed that in thermodynamic equilibrium conduction should lead to a uniform temperature field, though it is well known that density gradients may cause a flux of energy, an effect known from the textbooks as ”Dufour effect”. This effect is complementary to the more familiar thermal diffusion, where concentration gradients are established as a consequence of a temperature gradient. Speaking of equilibrium means that the net fluxes of all properties determining the state of a system are locally balanced. Normally in a gas we associate this with constant pressure, density and temperature. But in a system, which is