rjanish
commented
7 years ago Yes, I agree that he is right to expect thermalization, and thermalization will occur, but the scales involved may be more subtle than just saying t_thermal^-1 ~ n \sigma with \sigma ~ \alpha^2 \MeV^2.
Therefore, I feel like any thermalization time scale should really be the mean free path to transfer O(1) of your initial energy
Yes, I agree. The \sigma that SR gives is the \sigma for an electron with E~MeV to have an hard coulomb scatter, but we know that the majority of coulomb energy transfer is done via soft scatters, so a better estimate of the thermalization scale is the electron's range calculated from dE/dx.
This parametrically longer time scale must necessarily be longer than the diffusion time scale, which is set by electron-electron scattering (conduction) which, comparing the Coulomb stopping powers I calculated, is definitely more dominant than electron-ion scattering.
Yes, I think this is exactly the right thing to compare: thermalization vs diffusion. We really don't care how long anything takes in the star, but we do care if by the time the electrons thermalize with ions they've diffused out to having T<MeV. The diffusion time scales ~ L^2, where as the thermalization time is independent of L (all this is at fixed T), so there is always some L at which thermalization happens first. This is something we ought to compute, as this L sets the minimum heating length for an electromagnetic processes. We can estimate this just by T_thermal = R/v_electron \approx R and T_diff = (diffusivity)^(-1) * L^2, and I imagine the results will give the same minimal L (to order 1) as we'd get using Paul's iterative formalism for heating lengths.
As for actually calculating this, I am not sure the numbers you give above make sense to me. My gut is saying that if we do this carefully we'll find that L < \lambda_T. The intuition of diffusion is a bit subtle: things that scatter a lot (low mfp) diffuse slowly. So the time scale for e-e diffusion is probably slow, not fast, if e-e dominates e-ion in scattering. The intuition is that, if e-e happens often, then if a cold e picks up energy from a high-energy e, it is like to just dump that energy back into a neighboring e and not carry the energy off to a distant region. If I remember right, diffusivity ~ v_th d_e, where d is the mean free path for e-e. So T_diff ~ L^2/(v_th d_e). Then if T_thermal ~ R_ion / v_th, I get that the threshold L is L ~ sqrt(d_e R_ion).
[Anyway, this isn't too important if we focus on hadron showers, is it? Though I would like to understand it regardless for my own benefit.]
vijayoct27
So I recently had a Skype conversation with SR in which he claims the following: 1) We ultimately want to deposit energy to nuclei in order to initiate fusion (we knew this).
2) If we instead dump energy predominantly into a hot electron gas, there should be a thermalization time scale after which the combined system (ions + electrons) reach a uniform temperature (I agree).
3) SR claims that if we have a hot gas of ~ few MeV electrons, then in a time scale of ~10^(-15) s, the ions should also heat up to this temperature since diffusion is sufficiently slow. His claim is based on two assumptions -the ions and electrons are "strongly-coupled", meaning you can't heat up one component without heating the other. -The thermalization time is set by a mean free path to scatter, which he estimates as ~ 1/n \sigma, where \sigma = \alpha^2/MeV^2.
I disagree with both of these points. For Coulomb collisions of MeV electrons off ions, the maximum energy transfer (Emax in the Coulomb log) is set by kinematics ~ eV. Therefore, I feel like any thermalization time scale should really be the mean free path to transfer O(1) of your initial energy, perhaps after 10^6 collisions. This parametrically longer time scale must necessarily be longer than the diffusion time scale, which is set by electron-electron scattering (conduction) which, comparing the Coulomb stopping powers I calculated, is definitely more dominant than electron-ion scattering. SR was right to suspect what he did, but I think it is incorrect. Thoughts?