Study of the temporal distribution density of the shot sheaf span to create a simulation model of the optical sensor signal

When studying the space path of a cloud of pellets from a shotgun and evaluating its parameters, it is advisable to have a simulation model of the optical sensor signal when a shot sheaf crosses the light screen. To create such a model, you need to get the relationship between the scale parameter in the Rayleigh distribution and the time span of the shot sheaf. Studies of the temporal distribution density of the shot sheaf span are performed, and graphs of the distribution density of the span for near-real situations are constructed. A linear dependence of the mathematical expectation of the span distribution density on the scale parameter in the Rayleigh distribution is established. A simplified expression is obtained for calculating the mathematical expectation of the time span of the shot sheaf. Statistical modeling confirmed the possibility of the practical use of the proposed formulas, including with a large number of pellets (up to 1000). The dependence of the mathematical expectation of the span on the number of pellets is investigated, its approximation by various functions is carried out, and the approximation errors are estimated. The research conducted allows us to create a simulation model of the optical sensor signal when the shot sheaf crosses the light screen on the basis of empirical data on the real-signal duration (with averaged measurement data for several shots being helpful).