Monte Carlo Simulation

In order to obtain an expected count rate for each detector position, a Monte Carlo simulation was used to numerically evaluate the integral

over the angular acceptance range D for a given detector configuration. The exponent k was treated as a parameter for this purpose; different distributions were generated for a spread of values of k.

The geometry of the simulation is shown below. The simulation worked by choosing random angles theta and phi. These were drawn from probability distributions equal to the expected marginal muon-flux distributions (i.e. theta was drawn from a

distribution, while phi was drawn from a uniform distribution on ). Then, a random position (x, y) was chosen on the top panel, and the intersection point on the ground plane was calculated to determine whether the trajectory would also intersect the bottom panel.The number of hits was counted as a proportion of total events generated.

(Schematic of the geometry of the Monte Carlo integration).

Practical Considerations

For efficiency, the integration was performed for all desired lower-panel positions simultaneously. Each time a trajectory was generated, it was checked against all bottom panel positions, and separate counters incremented for each lower panel that registered a hit.

The simulation was implemented in Python using efficient vector operations provided by the NumPy that automatically take advantage of the underlying hardware parallelism. It took about 7 seconds to generate and test 10⁷ events, close to 10 times faster than an earlier implementation in C++.

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