"Astronomy" is from the Greek meaning "star formation." It is the branch of science dedicated to studying bodies in outer space. Since astronomy is an investigation of God's creation, it is perfectly acceptable for Christians to study astronomy. Where we get caught up is when scientists draw conclusions about the data that contradict the Bible. The Bible tells us God formed the heavenly objects through His words. As Christians—particularly Christians interested in science—we need to understand that despite the fact astronomy has been studied for thousands of years, we still know very little. Even the latest interpretation is based on incomplete data. It is good to study God's creation, as long as we recognize that it is His creation. Study the stars, and trust that the more we learn, the more we will be pointed back to God.
The problem with a Christian studying astronomy isn't in the subject matter but in the culture of those who study the subject matter. Astronomy is one of the hardest of the sciences to understand, in part because the objects of study are so far away and not subject to human interaction. Scientific theories are validated either by experimentation or by discovering phenomena that had previously been predicted. We can't do experiments on a quasar. And considering the supposed time it takes for stars to change, it isn't easy to predict the phenomenon. Sometimes predictions do turn out to be true; earlier astronomers were proved to be right when they postulated that some stars wobble because of the gravitational pull of revolving planets, but sometimes they're wrong; science would indicate that at their supposed ages, the gas giants should have very weak magnetic fields—Saturn does, but Jupiter, Uranus, and Neptune have very strong fields. Overall, the study of astronomy is meant to point us to the Creator, who made us and the universe. This Creator loves us and died for us so we can be free and live with Him in heaven forever.