Recent studies of TWINS Lyman- α observations have reported an increase in geocoronal column brightness during geomagnetic storms, indicating enhanced exospheric hydrogen atom density ( N H ). This suggests a complex role of exospheric neutrals in determining storm-time magnetosphere dynamics and their energy release through charge-exchange processes. We developed a Model for Analyzing Terrestrial Exosphere (MATE) to investigate storm-time exospheric behaviors and their physical drivers. MATE traces test hydrogen atoms backward in time from locations in the exosphere to a nominal exobase altitude of 500 km, employing Newtonian mechanics with gravitational force. The model then calculates the phase-space densities (PSDs) of test hydrogen atoms at the exobase using the Maxwellian distribution with physics-based exobase conditions from the TIMEGCM upper atmosphere model. MATE maps PSDs at the exobase to the exosphere using Liouville’s Theorem under collisionless assumptions and derives N H by integrating the PSDs across velocity space. We conducted MATE simulation before, during, and after a minor geomagnetic storm from 12 to 18 June 2008, and compared the model results with N H estimates from the TWINS geocorona data. MATE reproduces storm-time density enhancements soon after the minimum Dst is reached, matching well with a general trend of TWINS N H estimates. The results suggest that upper atmospheric heating during a geomagnetic storm increases the number of ballistic and escaping hydrogen atoms entering the exosphere from the exobase, thereby boosting N H . However, the magnitude of modeled N H mismatches the TWINS N H estimates. The potential mechanisms of this density discrepancy include the physics excluded in the MATE model — such as neutral-neutral collisions, neutral-plasma charge exchange, solar radiation pressure, and photoionization — as well as the higher exobase hydrogen density of TIMEGCM compared to typical empirical values, which will be addressed in future.