My usual position is that questions that are well suited to either site stay where the OP puts them. This does have the side effect of creating duplicate question on the two sites.
But I will suggest some principles:
- Questions about particular astronomical objects (i.e. the X ray source in Cygnus as opposed to a general question about black hole emissions which could go to either site) belong on Astronomy.
- Questions about astronomical observations, instruments (again "How telescopes work?" is good on either, "What are the current limits on angular resolution of ground based optical telescopes" is astronomy specific) , and data sets belong on Astronomy. Likewise for questions about getting telescope time or other cultural issues (if you are accepting those).
- Questions where the answer is in some sense wrapped up in the politics of the discipline (i.e. "What qualifies a body as a planet and why are these conditions important enough to justify demoting Pluto?") got to Astronomy. {*}
I see that there is a "What should we do with questions which are really physics questions" entry on meta.astronomy, and it already hosts some suggests for the kinds of questions that should be moved here. That link pointed at a question on the meta of the first [astronomy.se] which dies in beta. The same numbered meta question on the current [astronomy.se] seems to be about naming the chatroom.
{*} Aside: I was lucky enough to talk to Clive Tombaugh a couple of times during the last years of his life. Seemed like a nice guy.