Err, all spartans were rich. (for a given value of "Rich") that's what comes form being a tiny upper-class ruling a large portion of slaves, serfs and conquered peoples.
The Spartiates were all upper-class, and all greek hoplites were at least middle-classed. The reason Athens was comparatively democratic even for greek city-states was because ofit's naval status: Rowers didn't have to afford the expensive armour the way more land-based militaries did, and service guarantees citizenship and all that.