Conjunction Risk 2D: Monte Carlo Sampler & Pc-vs-Horizon Curve

Orbit-determination centers condense every close approach into one number, P_c, computed from a closed-form Gaussian-overlap formula. This 2D companion checks that formula the way a statistician would: by drawing thousands of random debris positions from the same tracking-uncertainty distribution and directly counting how many actually fall inside the hard-body threat circle. Watch the Monte Carlo estimate converge toward the analytic value as samples accumulate, then sweep the propagation horizon on a log-probability curve to see why operators sometimes wait before maneuvering — more lead time both grows the position uncertainty and gives a tiny radial burn more time to work, and the two effects compound to push P_c below the standard 10⁻⁴ threshold.