A collisional model for asteroids and their debris is presented. The model describes the evolution of a system of particles undergoing inelastic collisions and fragmentation, leading to a population index distribution of the form $ Am^{-\alpha}dm $, where $ A $ and $ \alpha $ are constants. The population index $ \alpha $ is theoretically derived as 1.837 for asteroids and their debris, consistent with empirical observations. The model shows that catastrophic collisions are the dominant physical process determining particle lifetimes and the distribution of particle sizes, especially for larger particles where radiation effects are negligible. The lifetime of the largest asteroids is comparable to the age of the solar system, suggesting some may have survived since their formation, while smaller ones are likely collisional fragments. The model is applied to estimate the number density and statistical properties of debris in the asteroidal belt. Observational evidence supports a power law distribution for asteroid masses, with the population index $ \alpha $ ranging from 1.8 to 2.34. The model's results align with empirical fits to observed asteroid data, showing that the theoretical value of $ \alpha $ for a steady-state distribution is within the margin of error of empirical observations. The model also accounts for particle creation and destruction rates, mass loss due to radiation, particle lifetimes, and erosion rates. The model's solution for asteroids and their debris shows that the population index $ \alpha $ is approximately 1.837, consistent with both theoretical and empirical data. The distribution of particle sizes is influenced by the balance between particle creation and catastrophic collisions, with erosion having a minor effect. The model's results are insensitive to material parameters and collisional velocities, indicating that the population index $ \alpha $ is primarily determined by the balance of these processes. The stability of the solution is confirmed by analyzing the effects of small deviations from the steady-state value of $ \alpha $, showing that the distribution remains stable under these conditions. The model's predictions are consistent with observed data, supporting the conclusion that the population index $ \alpha $ for asteroids and their debris is approximately 1.837.A collisional model for asteroids and their debris is presented. The model describes the evolution of a system of particles undergoing inelastic collisions and fragmentation, leading to a population index distribution of the form $ Am^{-\alpha}dm $, where $ A $ and $ \alpha $ are constants. The population index $ \alpha $ is theoretically derived as 1.837 for asteroids and their debris, consistent with empirical observations. The model shows that catastrophic collisions are the dominant physical process determining particle lifetimes and the distribution of particle sizes, especially for larger particles where radiation effects are negligible. The lifetime of the largest asteroids is comparable to the age of the solar system, suggesting some may have survived since their formation, while smaller ones are likely collisional fragments. The model is applied to estimate the number density and statistical properties of debris in the asteroidal belt. Observational evidence supports a power law distribution for asteroid masses, with the population index $ \alpha $ ranging from 1.8 to 2.34. The model's results align with empirical fits to observed asteroid data, showing that the theoretical value of $ \alpha $ for a steady-state distribution is within the margin of error of empirical observations. The model also accounts for particle creation and destruction rates, mass loss due to radiation, particle lifetimes, and erosion rates. The model's solution for asteroids and their debris shows that the population index $ \alpha $ is approximately 1.837, consistent with both theoretical and empirical data. The distribution of particle sizes is influenced by the balance between particle creation and catastrophic collisions, with erosion having a minor effect. The model's results are insensitive to material parameters and collisional velocities, indicating that the population index $ \alpha $ is primarily determined by the balance of these processes. The stability of the solution is confirmed by analyzing the effects of small deviations from the steady-state value of $ \alpha $, showing that the distribution remains stable under these conditions. The model's predictions are consistent with observed data, supporting the conclusion that the population index $ \alpha $ for asteroids and their debris is approximately 1.837.