Hop on the Ark Turret and kill all incoming UFO's and the Floating Mines. Once you do that and the Mystery Ship arrives you will notice on the sides and middle, these disks coming in and out of the ships. There are 4 on each side and 4 in the middle you need to destroy them all with either with your rocket guns or a Fighter. Once you do that you will see a cut scene indicating you need to destroy the inner core. Go to the marker pull out the wires and shoot the inner core on both sides to defeat the ship.