### H. Appendices

 Equations Used The following equations are integral to calculating the outcome of interactions between particles and a material and are essential to understanding the interaction between the atmosphere and a particle. These equations describe the phenomena accurately and are essential to describing several particle characteristics. Bethe-Block Stopping Formula β = v/c, γ is its Lorentz factor, z is the charge of the ionizing particle in units of e. κ1 = 0.153287 MeV g −1 cm2 and κ2 = 9.386417 are derived from values for dry air Deflection in Earth's Magnetic Field This is a description of particle deflection in Earth's magnetic field, where l = length, z = charge, B vector = magnetic field vector, p vector = particle momentum Time of Flight At the first interaction of the primary in the atmosphere, the timing of the shower is started. The time interval dt is the time elapsed as the particle moves along its path; dt is calculated by dividing the path length l by the average particle velocity, where B ave is the arithmetic mean of the particle at the beginning and end of the trajectory. Mean Path Describes the interaction between muons and a material, in this case air. The mean free path for these interactions is given by the equation above where m air = 14.54 is the average atomic weight of air in g/mol and λ tni is given in g/mol and λ tni is given in g/cm2 Probability of Material Traverse Describes the probability of a muon to traverse an atmospheric layer of thickness λ without corresponding interaction.**Note: 'tni' as used in these equations represents the initial value for the associated factor.