fix how multiple Coulomb scattering is treated

Previously I had been assuming that a particle undergoing many small angle
Coulomb scatters had a track direction whose polar angle was a Gaussian.
However, this was just due to a misunderstanding of the PDG section "Multiple
scattering through small angles" in the "Passage of particles through matter"
article. In fact, what is described by a Gaussian is the polar angle projected
onto a plane. Therefore the distribution of the polar angle is actually:

(1/(sqrt(2*pi)*theta0**2))*theta*exp(-theta**2/(2*theta0))

This commit updates the code in scattering.c to correctly calculate the
probability that a photon is emitted at a particular angle. I also updated
test-likelihood.c to simulate a track correctly.

1 files changed, 1 insertions, 1 deletions

diff --git a/src/muon.c b/src/muon.c
index 9a539de..7012214 100644
--- a/src/muon.c
+++ b/src/muon.c
@@ -262,5 +262,5 @@ double get_expected_charge(double x, double T, double T0, double *pos, double *d
     theta0 = get_scattering_rms(x,p0,beta0,z,rho);
 
     /* FIXME: add angular response and scattering/absorption. */
-    return 2*omega*2*M_PI*FINE_STRUCTURE_CONSTANT*z*z*(1-(1/(beta*beta*n*n)))*get_probability(beta, cos_theta, theta0)/(sqrt(2*M_PI)*theta0);
+    return omega*FINE_STRUCTURE_CONSTANT*z*z*(1-(1/(beta*beta*n*n)))*get_probability(beta, cos_theta, theta0);
 }