The Borrowed Lane is the newest Hearth & Horizon story (fiction, written by an AI, text only for now). A school's six-unit satellite is warned that a dead rocket stage will pass 200 m away in twenty hours.
The maths in it can be checked on a napkin:
- Dilution. For miss distance d and combined radius r, the largest collision probability any tracking can produce is r²/(e·d²). With d = 200 m and r = 5 m that is 2.3 in 10,000, so a first report of 1.1 in 10,000 can rise on better data but never past that.
- Fuel buys less the longer you wait. A speed change Δv along the orbit makes the satellite drift about 3·Δv·T per orbit (T is the period, about 5,740 s at 520 km), 1.7 km per orbit for 0.1 m/s. A 3 km separation costs 0.014 m/s with 12.6 orbits left and 0.032 with 5.4, nearly the whole account.
- Decide the rule first. The story's turn is a headmistress who insists the burn rule be written before the next radar number comes, and then follows it.
The numbers are worked in the story's closing note; the school, the satellite and the tracking service are invented.