[DuncanWaring]

anti-aircraft missiles continue to burn fuel in their primary rocket motors throughout the flight envelope.

Not all missiles.

I don't know about the "PAC-3" PATRIOT missiles, but the original version had a motor that burned for about six seconds. On a long-range engagement, the flight could last several minutes.

[Rinnwald]

A more likely contender would be a U.S. Navy Standard missile. Burn time is listed as 6 seconds too, for the BOOSTER. Primary motor burn is about 260 seconds.

Either way, a nighttime missile launch is not a stealthy event.