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

Abstract

The cube method reduces conjunction screening in orbital debris simulations to $\mathcal{O}(N)$ 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 $\mathcal{O}(N)$ complexity. In an isotropic convergence benchmark at $L=50$ km, DC reduces the blindness rate from ${\beta }_{\mathrm{Cube}}=7.09\%$ to ${\beta }_{\mathrm{DC}}=0.55\%$, and a synchronized experiment separates the temporal component from the geometric one and leaves DC below $0.01\%$ against $0.80\%$ for the cube. Recovering these pairs changes which pairs are evaluated and leaves the probability formula untouched, so it does not improve calibration by itself, and both uncorrected methods remain substantially miscalibrated, with a reliability slope of $m=1.1314$ for the cube and $m=1.0996$ for DC. The formula applies the full cell volume to every evaluated pair irrespective of separation, which is the same bias characterized in the no-time-counter scheme of Direct Simulation Monte Carlo. Two corrections are derived from the geometry of the uniform cell alone: a power-law form referenced to the Robbins mean separation $\bar{d}=0.6617L$ and a Gaussian form using both that mean and the analytically derived standard deviation $\sigma =0.2494L$, with no quantity fitted to simulation output. The power-law correction moves the reliability slope to $0.9973$ at $k=1$ and $0.9429$ at $k=2$, and the Gaussian correction reaches $0.9806$ with an intercept of $0.975$, the only configuration within $3\%$ of unity on both. Both corrections have been implemented in MOCAT-MC.

arXiv comment21 pages, 7 figures, Submitted and presented at the 2026 AAS/AIAA Astrodynamics Specialist Conference, Whistler, British Columbia, July 26-30