I would probably not say "Jim has a sun", because it suggests the existence of several suns, which doesn't seem appropriate in your context.
"Jim has the sun", on the other hand, might not imply for everyone that Jim was looking at the sun.
There are other options, e.g. "Jim takes the sun"; but again, that doesn't necessarily imply "looking".
Could you reverse it, e.g. "The sun holds Jim"?
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