ResearchPreprint
Finite-time transport in the circular restricted three-body problem (CR3BP) can change qualitatively under perturbations to initial conditions, making robustness difficult to characterize using smooth trajectory sensitivities alone. This work investigates whether geometric distance to an outcome-change boundary can provide a usefulmeasure of transport robustness for a two dimensional family of invariant-manifold seeds generated from planar L1 Lyapunov orbits in the Earth—Moon CR3BP. A 160 × 33 parameter atlas comprising 5280 initial conditions was classified by first-passage outcome, and adaptive refinement produced 484 numerical samples of the resultingoutcome boundaries. To avoid arbitrary Euclidean distance in the family parameters, the seed-state map was used to induce a local pullback metric, defining a family-constrained boundary-distance diagnostic ρη. The metric construction was numerically consistent, and ρη revealed strongly nonuniform proximity to the resolved transportinterfaces. Its ordering, however, depended on the prescribed state weighting, with Spearman correlations ranging from approximately 0.794 to 0.999 across the tested metrics. Direct perturbation tests also rejected a guaranteed-radius interpretation: one outcome change occurred at approximately 0.5ρη, while finite directional searches at five representative points recovered no independent minimum flip distance. These results support ρη as a family- constrained geometric diagnostic of proximity to resolved finite-time outcome boundaries, but not as a certifiedoutcome-preserving radius or validated transport condition number. A stronger robustness measure requires converged boundary resolution, physically specified uncertainty geometry, and direct computation of minimum outcome-changing perturbations.
ResearchPreprint
Orbital debris has become one of the most significant challenges facing the long-term sustainability of Low Earth Orbit (LEO). As the number of satellites continues to increase, so does the likelihood of collisions capable of generating large quantities of debris, potentially triggering a self-sustaining chain reaction known as Kessler Syndrome. This study develops a stochastic two-shell Monte Carlo model to investigate whether a critical orbital density exists beyond which such debris cascades become increasingly likely. The model represents the orbital environment using two concentric shells and incorporates stochastic collisions, debris fragmentation, atmospheric decay, active satellite failure, and collision avoidance. A systematic density sweep was performed using 200 independent Monte Carlo simulations for each normalized orbital density. The results reveal a rapid transition from stable orbital evolution to near-certain cascade formation over a narrow range of densities, with the critical normalized density multiplier estimated to be approximately $alpha_c approx 0.093$. Near this threshold, simulations with identical initial conditions frequently produced different long-term outcomes, highlighting the importance of stochastic collision dynamics in determining orbital stability. Although the model adopts several simplifying assumptions, it reproduces the fundamental feedback mechanism underlying the Kessler hypothesis and provides a computationally efficient framework for investigating the probabilistic onset of orbital debris cascades.
ResearchPreprint
Fin geometry plays a critical role in determining the aerodynamic stability andperformance of rockets, yet excessive fin area may reduce efficiency through added drag. In this study I investigated the effect of fin geometry on rocket flight characteristics using ten simulated variants analyzed in OpenRocket under controlled launch conditions with four wind levels in a hypothetical situation where we are in Guadalajara, Spain. Metrics including apogee altitude, maximum velocity, acceleration, and time to apogee were compared across designs, revealing measurable tradeoffs between stability margin and ascent performance. The results indicate that moderate fin configurations can outperform both oversized and undersized designs by balancing stability with reduced aerodynamic losses.
