The way the circle is wider at the bottom reminds me more of a "star and crescent" than a star and circle.
Possible Louisiana or 7th Corps insignia...?
Some context on where it was found might help.
Edit: Oh and don't shine it ! At least not till you know what it is.... you can always clean more, but you can't clean less.....
Folks I’m not quite with this being a Texas ranger badge. The circle around the star isn’t even, looking more like a crescent and star which was a common design for cape pins, some of which bore holes in them. Also a common design for corps badges. The piece is the right size for either of those uses.