update likelihood function to fit electrons!

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.

1 files changed, 1 insertions, 0 deletions

diff --git a/src/misc.h b/src/misc.h
index 3a55c2f..94ccee7 100644
--- a/src/misc.h
+++ b/src/misc.h
@@ -26,5 +26,6 @@ double norm(double x, double mu, double sigma);
 double norm_cdf(double x, double mu, double sigma);
 double mean(const double *x, size_t n);
 double std(const double *x, size_t n);
+double gamma_pdf(double x, double k, double theta);
 
 #endif