When I pray, I am also worshiping.
Well, if you only pray to someone deserving divine honors and worship, that would be appropriate.
In our usage, which is admittedly equivocal, "pray" has several meanings and shades of meaning. The most basic is "request." I can certainly make a request of someone, even of someone you think to be dead, without worshipping (in the divine sense) him. Ditto for "converse" or even "commune".
Now I hear the mocking howls in the background, but mockery is not argument. The technical term for the attitude with which Catholics approach or regard Mary is "hyperdulia" as distinct from "latria". English usage is so varied that one will always be able to find an exception to any forced rigor of usage of "worship" and "venerate", but while we do have a "hyper respect" for Mary most of us know what the Church teaches and do not think of Mary as Divine.
Most Catholics are not as fearful as some Protestants and we trust that God understands what some Protestants will not allow, namely: that one can kneel in front of a statue or painting and not be committing "latria."
So, they disagree. I can live with that.
So, you can pray to the "god of the sun" but yet not worship it?
I wouldn't pray to a "god of the sun," because I don't believe one exists.
Why on earth would you pray to someone whom you do not worship?
For the many of the same reasons I would communicate with any personal entity, but mostly to ask for prayers.
latria....I had NO idea what that word meant, so I looked it up. This is what I found in the Catholic encyclopedia...."Latria (latreia) in classical Greek originally meant "the state of a hired servant" (Aesch., "Prom.", 966), and so service generally. It is used especially for Divine service (Plato, "Apol.", 23 B). In Christian literature it came to have a technical sense for the supreme honour due to His servants, the angels and saints. This latter was styled "dulia". Etymologically, however, there is no reason why latria should be preferred to designate supreme honour; and indeed the two words were often used indiscriminately. The distinction is due to St. Augustine, who says: "Latria . . . ea dicitur servitus quae pertinet ad colendum Deum" (City of God X.1). (See ADORATION; WORSHIP.).... http://www.newadvent.org/cathen/09036a.htm
""the state of a hired servant".....I would take this as God's servant....not yours. Certainly not one to be prayed to.