Is there a star that sets at midnight?

Answer 1

Yes. There are a great number of stars, and almost any star, I can calculate the date on which the star will set at midnight.

The problem is that as the Earth goes around the Sun and the apparent direction to each star shifts slightly as the Earth moves, all the stars appear to rise and set four minutes earlier every night. So if today Star X sets precisely at midnight, then tomorrow night Star X will set at 11:56 PM. There's a Star Z that will set tonight at 12:20AM. In five days, Star Z will set at midnight.