Neutrino diffusion time-scale in NS

[zhermes]

I'm wondering how to calculate the neutrino diffusion time-scale from a neutron star---just to an order of magnitude or so accuracy.

Based on a 10km, 1.4 solar-mass star, and a cross section $$\sigma_{\mu n} \approx 10^{-42}$$ cm^2; I found the mean-free path to be
$$l \approx 10^{14}cm$$.

Then using
$$t \approx \frac{R^2}{l c} = 10^{-12} s$$
which is about 13 orders of magnitude too small... why?
The neutrinos are so relativistic that using the speed of light should be fine.

One clear problem is that in the limit that $$l \rightarrow \infty$$, the time-scale goes to zero----while it seems like it should simply approach R/c.

I was looking at a powerpoint (i.e. without thorough explanations), which showed a timescale
$$t_D = \frac{3}{\pi^2} \frac{\partial Y_L}{\partial Y_\nu} \frac{R^2}{c l_{\nu e}}$$
Can anyone explain this?

 

[LightHero]

The neutrino diffusion time-scale is defined as the time it takes for neutrinos to diffuse through a star. It depends on the mean free path, l_νe, which is the average distance a neutrino travels before interacting with an electron or other particle. The mean free path for a given neutrino energy is determined by its cross section, which is the probability of it interacting with a target particle. The equation you provided is a simplified version of the diffusion time-scale, where Y_L and Y_ν are the lepton and neutrino number densities, respectively. This equation assumes that the lepton and neutrino number densities remain constant during the diffusion process, which is usually a reasonable assumption. The equation gives the time-scale, t_D, in terms of the mean free path, l_νe, the radius of the star, R, and the speed of light, c. The numerical factor in front of the equation accounts for the fact that the diffusion timescale is proportional to the ratio of the change in the number densities of the lepton and neutrino number densities.To answer your question about why your calculation of the diffusion time-scale was too small, it's likely because you assumed that l_νe was equal to the radius of the star. However, since the mean free path is much larger than the radius of the star, the actual diffusion time-scale is much longer.