Update: since writing this I’ve learned a lot more about what happens on the really long routes, the ones that curve up over the Arctic. A great-circle flight like San Francisco → Doha passes within a few degrees of the North Pole, and prayer times get genuinely strange there: two sunsets on one flight, Fajr and sunrise fifteen minutes apart, and in polar winter no sunrise at all. The standard formulas simply break down, so the next version of Flight Salah follows the great-circle route and uses the “nearest normal latitude” rule that scholars apply in high-latitude regions.
I’ve written the whole thing up here: Praying on a long-haul flight: why prayer times follow the plane
The same explanation will be in the app (“How are these times calculated?” under any flight’s prayer list), so you can read it offline on the plane.