When a spacecraft returns to Earth, it hits the atmosphere at thousands of miles per hour. Engineers and defense researchers need a guaranteed safety map showing everywhere the vehicle could possibly land. Supercomputers cannot brute-force the exact flight path. A vehicle reentering through the atmosphere is highly chaotic—affected by shifting winds, extreme heat, and vibrations—exhausting even the most capable of compute clusters. Instead of a mathematical guarantee, they are reduced to an educated guess.
This research proposes a framework to solve this problem without relying on numerical simulations. Instead of tracking every chaotic detail, we use fundamental physics to draw an absolute outer boundary containing every possible place the vehicle can go.
To do this, we partition the flight into three regimes:
The hardest parts of the flight to calculate are the bubbles. These dips happen so fast compared to the rest of the journey that standard equations fail to track them entirely, losing the exact moments when the spacecraft violently slows down.
To address this, we dilate the timeline so those split-second atmospheric dips can be isolated. Using this stretched "slow time," we track the energy dissipated during these events. Even if a dip takes almost zero time, we maintain a strict account of the energy lost.
Because the spacecraft has a finite amount of energy, it can only hit a limited number of atmospheric bumps before it runs out of speed and is forced to land. By tracking this sudden energy loss on a rescaled timeline, we mathematically bound exactly how far the ship can travel.
Formally, this is achieved by casting the flight dynamics as a linear problem on occupation measures, applying Lions' concentration-compactness principle in the thin-atmosphere limit to extract the profile decomposition without losing the boundary-layer events. The ultimate goal is to tie these pieces together to obtain a guaranteed boundary map of where these returning vehicles can land.