Ehh, not really, Astrouc (though it may differ from one college to the next). I was an aerospace engineering student before I switched to Mechanical. Essentially:
Aerospace and its derivatives are a subset of mechanical engineering. That's why you don't see them at every university. They are specialized and the demand isn't all that high. Many universities will have aerospace as a "track" or minor of ME.
Aerospace and aeronautical engineering are essentially interchangeable, but the term "aeronautcal" is falling out of favor because of the evolution of aircraft. When you start dealing with supersonic/hypersonic flow (which is touched on briefly as an undergrad with "Gas Dynamics", which is essentially a supersonic flow course and included rocket/jet nozzles), "aerospace" really is a more accurate term.
What I think you may be thinking of, Astronuc, is "Astronautics" (I think that's what enigma is) - which for my school (US Naval Academy) was a "track" of aerospace engineering focusing on spacecraft . The Aero-Aero and Aero-Astro tracks coincided completely until, iirc, 2nd semester junior year.
So, for "aerospace", think anything up to and including a sub-orbital space-plane. Astronautics is anything above that, but with some treatment of how to get back home...