What does Phoenix mean?
From Greek 'phoinix', the mythical bird that is consumed by fire and reborn from its own ashes. The fire link is through the rebirth-by-fire myth rather than a literal word for fire.
What does Aster mean?
From Greek 'aster' ('star'). It is also the name of the star-shaped daisy-family flower, which was itself named for the Greek word, giving the name a botanical as well as celestial sense.