There are 1,440 60-second increments in a day. My suggestion would be back up your database by copying stevefire.mdb somewhere safe (in case something horrible happens), then enter 1440 into the Min Inc box above the Auto Turns box on the system view, then click Auto Turns and leave it going overnight!
The aliens shouldn't really stay there pinging you for ages. They should really either blow you up or leave. If they haven't moved after you've run the game for a day, they're probably bugged out. Post in the bugs thread and keep that DB backup in case you need to email it to Steve.