You can be non-specific about the orbit (save that one is being performed) or qualify it, and the qualification can take various forms.
- "It orbits." "Where does it orbit?" "Around the Sun."
- "It orbits." "Around what does it orbit?" "The Sun."
In the latter, the questioner supplies the (default?) relationship to qualify with, while the answer to the first question really needs the qualifier[1] and you could also qualify in many other ways, for different things: "Across the Earth's path." / "Through a solar flare." / "Beyond the Kuiper Belt." All of which can be directly pressed into service, from the start, as "It orbits <in this manner>." For each context.
I'd defer to an authority on this (I just speak the language, perhaps better than some; but I'm sure you could Summon Bigger Fish on that issue), and probably appreciate one who knows how this might differ under US and British standards (there's plenty of verb uses that rightfully differ between the two standards - "How are you?" "I'm good..."/"I'm well...")
[1] "That particular spaceplane will need extra fuel in its tanks, as it may have to wait for final permission to land. Until then, it orbits." "Where will that spaceplane orbit?" "Titan. In a holding pattern around Minerva Crater Spaceport." - It orbits
at Titan.
In Titan's atmosphere.
Circling a point/region on its surface. For reasons best known only to 'The Future' it does not orbit Titan itself, though the first bit of reply is correct ("Where does it orbit?" "Titan." "It orbits Titan?" "No, it circles the spaceport in a holding pattern..." [2]), yet it is viable to suggest that a (space-)plane's holding pattern/'orbit' is in a place (Greater Los Angelese metropolitan area) rather than around one (LAX). Could one similarly orbit
in the Sun? Possibly. I mean, right now "Where does it orbit?"->"The Sun" seems to be unambiguous for "Around the Sun" but... And "It orbits so close to the Sun that it is effectively in it..." is sort of a current thing.
[2] At this point I'm not going to predict how aeronautical and astronautical terminology is going to be resolved for advanced interplanetary aerospace transportation systems, especially when a veteran spaceport groundcrew supervisor is clearly having to initiate a rookie into job-specific descriptions that have been honed into what should be contextually understandable if not for the greenhorn being so new to the job and/or the grizzled old-timer forgets how much he has to spell things out to the raw and uninitiated.