Shooting stars on schedule: synchronising re-entry of particles launched from a satellite

The ALE-3 mission plans to create artificial “shooting stars” in the upper atmosphere by launching pellets sequentially from a satellite in low Earth orbit. How does one arrange for all pellets to re-enter over the intended location simultaneously? Starting with conservation of energy and angular momentum, I derive a version of Kepler’s equation that gives time as a function of radial distance in an orbit, which can be used to find the transfer time from the pellet’s ejection to its re-entry as a function of its launch velocity. I show that for a given pellet ejection speed, there is a launch angle from the satellite that results in the fastest transfer time. I use these results to determine the pellets’ launch times and velocities for simultaneous arrival at a desired re-entry point. These results can be applied to de-orbiting any set of objects launched from a satellite, and use concepts that can be covered in an advanced undergraduate course in physics or aerospace engineering. Supplementary materials are provided online.