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The range and energy loss tables have different maximum values for electrons,
muons, and protons so we have to dynamically set the maximum energy of the fit
in order to avoid a GSL interpolation error.
This commit adds {electron,muon,proton}_get_max_energy() functions to return
the maximum energy in the tables and that is then used to set the maximum value
in the fit.
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This commit updates the likelihood function to take into account Cerenkov light
produced from delta rays produced by muons. The angular distribution of this
light is currently assumed to be constant along the track and parameterized in
the same way as the Cerenkov light from an electromagnetic shower. Currently I
assume the light is produced uniformly along the track which isn't exactly
correct, but should be good enough.
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To characterize the angular distribution of photons from an electromagnetic
shower I came up with the following functional form:
f(cos_theta) ~ exp(-abs(cos_theta-mu)^alpha/beta)
and fit this to data simulated using RAT-PAC at several different energies. I
then fit the alpha and beta coefficients as a function of energy to the
functional form:
alpha = c0 + c1/log(c2*T0 + c3)
beta = c0 + c1/log(c2*T0 + c3).
where T0 is the initial energy of the electron in MeV and c0, c1, c2, and c3
are parameters which I fit.
The longitudinal distribution of the photons generated from an electromagnetic
shower is described by a gamma distribution:
f(x) = x**(a-1)*exp(-x/b)/(Gamma(a)*b**a).
This parameterization comes from the PDG "Passage of particles through matter"
section 32.5. I also fit the data from my RAT-PAC simulation, but currently I
am not using it, and instead using a simpler form to calculate the coefficients
from the PDG (although I estimated the b parameter from the RAT-PAC data).
I also sped up the calculation of the solid angle by making a lookup table
since it was taking a significant fraction of the time to compute the
likelihood function.
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