In fairness to Paddoboy, it appears to me that he and many of the other pro-science posters are giving chinglu the benefit of the doubt, allowing that it's possible the traveling twin can count the orbits at all. As I mentioned before, that's entirely hypothetical since chinglu never bothered to explain how it might be done. (Not that I've seen in rebuttals, that is. I have the worst of trolls like chinglu on ignore.) Lacking the formalization of his question, which chinglu simply can't do because he doesn't have the chops, there is no geometric framework given for even assuming the traveling twin sees the Earth orbit the sun at all. Therein lies the rub; and I sense that paddoboy recognizes this and is reluctant to give the answer you are expecting, since there simply is no answer other than it depends on how we declare that the traveling twin is able to count the orbits at all.
I gave what I think is the only plausible scenario, the one in which the twin travels in the plane of the Earth's orbit, and has Star Trek equipment onboard to count the number of eclipses observed as it zooms 5 eclipses away from Earth, then, as Spock advises chinglu that sensors detect a 6th eclipse, he stops the Enterprise on a dime and reverses at near light speed only to return 6 eclipses later, for a total of 12. Apparently this is the nonsensical scenario that enables chinglu to count orbits.
paddoboy is actually right not to answer, since the scenario wasn't laid out (or so I imagine since I'm reading through a filter). There is no answer, Lakon, to a problem that's incompletely specified, other than to invent hypotheticals and try to answer it as I have. But note: once we do this we introduce geometry. And with that, we restore the rules of Nature that chinglu keeps trying to undermine by "disappearing them" in diabolically conceived scenarios in which reference frames are ignored. That is, geometry is ignored. But that's an invalid construction from the get go. Relativity is counter intuitive for this reason alone. It is entirely a consequence of geometry. Throw that in the trash, and there is no scenario; it's all bogus machinations of a diseased mind.
In my scenario the geometry is restored, although it relies on sci-fi as much as any near light speed travel does. The frames of reference are clear enough to establish that the traveling twin experiences the eclipses in relative time (not Star Dates) that are 10/12 of a year elapsed time on the ship's clock. (I'm working under the assumption that the ship's clock reads 10 elapsed years after traveling 12 light years (rel. to Earth) round trip from an Earth station to an arbitrary point on the Earth's orbital plane, and back again.) For purposes of my scenario, we can assume the ship is using an atomic fountain clock that was calibrated on Earth just before departure and which magically remains calibrated for the next 12 Earth years. (A Vulcan engineer mind melds it into perpetual calibration.) Further I'm couching this as a purely velocity dependent scenario (ignoring GR altogether) although it's just as valid either way since we haven't reckoned the ship's speed.
Yes, the person in motion in this scenario counts 12 eclipses even though he ages only 10 years. And yes, the Earth appears to him to speed up in its orbit around the Sun, although that's not anything he can detect with his eye. He's simply got Spock at the sensor panel telling him so. And yes, every 304 days of ship-time or so, chinglu is shocked to discover that he has wasted Federation money: he should have stayed in school rather than hijacking the Enterprise in the vain attempt to disprove what is a matter of high school science.
Paddoboy is right - there is simply no direct answer to an absurd and incompletely formulated question, other than to declare it invalid.
