My first thought was, "Bird strike!" and I said the same to my student. As the reality of our situation set in, I gently tested right aileron, left aileron and back to wings level when I realized I was holding almost full right rudder. In my mind, I instantly replayed the loud thump, wondering how bad the vertical stabilizer or rudder was damaged. "How big of a bird did we hit? Was it a bird or maybe another aircraft from behind us?" I remember a split-second or maybe several seconds early on when I was really scared and thought, "This is it!" When our aircraft didn't begin tumbling from the sky out of control, I was able to compartmentalize my fears and turn my entire focus to flying the airplane. It was natural for me to focus on flying the airplane first and foremost.