Beyond the Cube: Overlapping Grid Methods for Debris Collision Risk Assessment

The cube method reduces conjunction screening in orbital debris simulations to O(N) cost by evaluating only object pairs sharing the same grid cell at each snapshot, but systematically assigns zero collision probability to pairs separated by a cell boundary at that epoch, a failure known as boundary blindness. This paper introduces the Double Cube (DC) method, which recovers boundary-crossing conjunctions through a spatially shifted secondary grid using bin-index lookup alone, preserving O(N) complexity. Validated across 8,000 Monte Carlo seeds, DC reduces the blindness rate from beta Cube = 9.70% to beta DC = 4.21%; a synchronized experiment confirms the residual is temporal in origin by reaching exactly 0.00%. Removing blindness reveals a systematic per-pair overestimation in the cube formula that blind zero-probability assignments had been masking, suppressing the overall predicted collision rate below the true rate. Two independent corrections are derived and validated: a power-law correction motivated by the Direct Simulation Monte Carlo kinetic theory analogy reduces the calibration error from 12.9% to 1.9% at k = 1 and 4.0% at k = 2, bracketing perfect calibration from opposite sides; a parameter-free Gaussian correction derived from the pair-distance distribution geometry achieves a residual of 0.08%. Both corrections have been implemented in MOCAT-MC.