I am an undergraduate student working on a project on neutrino oscillations. I am currently trying to code the probability of neutrino oscillations in matter (MSW effect). I am using the following formulae:
$$ P (\nu_{\alpha} \rightarrow \nu_{\beta}) = \sin^2 ( 2\theta_M ) \, \sin^2 \bigg(\frac{\Delta m^2_M \, x}{4E} \bigg), $$
where
$$ \sin^2 (2\theta_M) = \frac{\sin^2(2\theta)}{\sin^2(2\theta) + ( \cos(2\theta) - x)^2} $$
and
$$ \Delta m^2_M = \Delta m^2 \sqrt{\sin^2 (2\theta) + (\cos(2\theta) - x )^2} $$
with the vacuum mixing angle $\theta$ and $ x = \displaystyle{\frac{2\sqrt{2} \, G_F \, N_e \, E}{\Delta m^2}} $ in terms of the electron density $N_e$, neutrino energy $E$ and the mass difference squared in vacuum $\Delta m^2$. $G_F$ is the Fermi constant. The math is taken from here on page 16.
My questions are the following:
1) To my understanding, $x$ is dimensionless, but the argument of the second sine squared in the first equation I have written down (the transition probability) needs to be dimensionless as well. How is this possible? In the formula for the transition probability of the vacuum oscillations the $x$ is replaced by the distance $L$ which has units of meters (or km). This makes sense and in that case the argument of the sine squared is dimensionless. Shouldn't the matter oscillations also depend on the distance? This would give the right units.
2) I have tried to find some data on the electronic density of the Sun (not only the core, but also the other layers) and I couldn't find much. Is there any source you could recommend to look up this information? As I am considering solar neutrinos, I would ideally also need data on the Earth's atmosphere and various layers.
Any help would be very much appreciated!
