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L.M. Malyshkin (Princeton University), R.M. Kulsrud (Princeton University, PPPL)
We consider two effects that reduce electron thermal conduction in the stochastic magnetic fields in clusters of galaxies. First, the parallel conduction along magnetic field lines may be reduced because the heat conducting electrons become trapped and detrapped between regions of strong magnetic field (magnetic mirrors). This problem reduces to a simple but realistic model for parallel diffusion of mono-energetic electrons based on the fact that when there is a reduction of diffusion, it is controlled by a subset of the mirrors, the principle mirrors. The reduction of parallel diffusion can be considered as equivalent to an enhancement of the pitch angle scattering rate. Therefore, in deriving the full perturbed electron-electron and electron-proton collision integral, we modify the pitch angle scattering term, and then find the effective parallel thermal conductivity as a function of the ratio of the magnetic field decorrelation length l0 to the electron mean free path at the electron thermal speed VT=\sqrt{2kT/me}.
The second effect is that conducting electrons have to travel along tangled magnetic field lines, and as a result, they have to go larger distances between hot and cold regions of space. This effect leads to further reduction of electron thermal conduction, which is given by the product of (l0/X0)2 and a function of a single parameter \beta=2(X0/l0)\,\langle B\rangle\big/\langle\delta B\rangle. Here, X0 is the cluster size (or the temperature gradient scale), \langle B\rangle is the mean magnetic field component (assumed to be homogeneous), and \langle\delta B\rangle is the averaged absolute value of the random field component. We discuss the application of our final results to clusters of galaxies.
Leonid Malyshkin would like to thank the Department of Astrophysical Sciences at Princeton University and Professor Bruce Draine for financial support.