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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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To calculate the expected number of photons from reflected light we now
integrate over the track and use the PMT response table to calculate what
fraction of the light is reflected. Previously we were just using a constant
fraction of the total detected light which was faster since we only had to
integrate over the track once, but this should be more accurate.
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This commit updates the calculation of the muon kinetic energy as a function of
distance along the track. Previously I was using an approximation from the PDG,
but it doesn't seem to be very accurate and won't generalize to the case of
electrons. The kinetic energy is now calculated using the tabulated values of
dE/dx as a function of energy.
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The RMS scattering angle calculation comes from Equation 33.15 in the PDG
article on the passage of particles through matter. It's not entirely obvious
if this equation is correct for a long track. It seems like it should be
integrated along the track to add up the contributions at different energies,
but it's not obvious how to do that with the log term.
In any case, the way I was previously calculating it (by using the momentum and
velocity at each point along the track) was definitely wrong.
I will try this out and perhaps try to integrate it later.
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