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 make a limited number of atmospheric dips before it runs out of speed and is forced to land. That budget is what turns an open-ended problem into a finite one: it bounds how many dips there can be and how long each one lasts, and so bounds how far the vehicle can travel.
The open question was whether the three-part picture is a genuine bound or only a convenient description. It is a bound, and it is now proved — for a reduced model that keeps the concentration and discards everything else. Everywhere the vehicle can reach is contained in what the skeleton-and-bubble system can reach, widened by a residual whose size is written down in closed form and shrinks as the atmosphere thins. The bound holds uniformly over every steering history and every admissible model of the aerodynamics, which is what separates a certificate from an estimate.
The proof is deliberately elementary: an energy identity integrated once, a single change of variable that makes the fast problem independent of the small parameter, and Gronwall's inequality. Occupation measures enter after the limit rather than before it — once the reduced system is in hand it is a low-dimensional problem with one guard and one jump, which is the form the occupation-measure relaxation was built for. Lions' concentration-compactness principle is expected to be needed for the companion statement at the level of measures, not for the bound itself.
Carrying this to a full entry vehicle needs three further results. The critical one is an averaging theorem with explicit constants that stays valid where the vehicle's attitude and structural frequencies cross. That is the work an advisor is wanted for.