The sun is a public good, is it not?
It seems to me that claiming ownership of the sun is absolutely pointless, and will remain so for all foreseeable futures.
Quite simply, laying claim to the sun is silly. Sillier by far than laying claim to, say, the moon, or Titan, or Venus or Mars, or Ceres.
The sun is, by far, the vastest thing in the solar system. Additionally, unlike the physical resource wealth of other locations, which would be more potentially worth laying claim to (my money's on Titan as being the most practical to lay claim and extract resources/set up colonies on,) the sun's use is in the energy it beams out into space.
Unfortunately for anyone who would like to claim the Sun, it meets the defining characteristics of a public good - it is non-excludable in that you cannot reasonably or even unreasonably restrict access to the energy being emitted from it, and it is non-rivalrous in that use by one party does not in any meaningful way impact the use of that energy by another party. You can't stop someone from putting up solar panels, or deploying a solar sail, or benefiting from the effects of solar energy reaching their home, such as photosynthetic plants growing, or what-not. Not without going and shooting them, which would be a tad difficult if you're set up on near-sol asteroid exulting over your ownership of the majestic fire god and they're half-way to the Oort Cloud.
It would be like trying to sell the Earth's atmosphere. Not canned bits of it, just the atmosphere of the Earth, in part or in whole. It's laughable, and any attempt to actually use force to enforce your claim would be seen as the efforts of a madman.
On the topic of laying claim to portions of other bodies in the solar system, the entire question is moot. No matter what some 50/100/150-year-old document may say, ownership will belong to the first party to get out there and lay claim to it, with the only way to contest said claims being the old-school way: to go there and fight them for it.