Sun and sun and Sol

I have it on good authority that the Sun (capitalized) is the correct English proper noun for the specific primary of the solar system we inhabit. "sun" refers to any star, or to various other meanings ("got too much sun"). "sun" can also be used when speaking casually about the specific star, but "Sun" is used to distinguish it from all other stars. "Sol" is mostly used in science-fiction.

Similarly, "Earth" is the planet to which science fiction writers refer as Sol III. "earth" is soil, and "the earth" is the planet in most contexts. Same goes for "Moon", "Universe", "Solar system", and "Asteroid Belt".

What is the authority?

While I would not quarrel with anyone who capitalized Earth or Sun anymore than one who capitalized G-d, the earth and the sun do not follow the normal pattern of proper names because they still use the article.

If "the" is supposed to be part of the name, then maybe you should write about The Sun.

Using the same pattern for "[The?] Solar system" (why not "Solar System") et.al. seems creative but dubious.

The Gliese Catalog (the most popular catalog of nearby stars used by astronomers) lists our star as "Sol".

Inneresting thought, jens... but Alexander the Great and Winnie the Pooh are counterexamples.