Here was the situation because I know there are some grammar-minded people here.
"His father, the sun god Helios, used the sun as his chariot."
My suggestion: hyphenate "sun-god" because it is an adjective (describing Helios).
But, no, it's not an adjective. It's not describing Helios. It's identifying him, I suppose. So I guess the rule according to the Chicago Manual Style is just keep it "sun god" but perhaps in this case, put a comma after "god."
(And this has been another chapter of First World Problems).