Why should he?
The Sun is not part of a fixed constellation. It is also well known by its traditional name, so why would you want to rename it? Bayer provided alternate names for the bright stars based on a unified system, rather than the collection of ancient, multi-cultural names that some stars had acquired. Tanks to this system of naming stars, and other catalog systems that followed, astronomers can speak meaningfully about stars across national boundaries and language groups. However, if you are writing poetry or in a style other than science fiction, it is always a good idea to use the common names for stars within the language you are using!
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