27,000 years is the number reported in the article.
Even assuming observation from above or below the orbital plain, an orbit that large would be difficult to discern from a straight line over a short period of observation. So your surmise concerning fitting observed data to theoretical orbit calculations makes sense. Once, that is, you believe you are actually dealing with a planet in orbit.
Small as it appears, how big would that planet have to be to register as an image of (?) pixels from 1200 light years away?
Good question. In professional telescopes and other professional optics pixel pitch (pixels per millimeter) are matched to optical resolution. The resolving power of an optical system is in radians is:
ds = 1.22 x wavelength / aperture
where:
ds = resolving power in radiansFor the Hubble Space telescope that works out to about 2.29 x 10^-7 radians. The resolution at 1200 light years would be about 2.6 billion kilometers or 17 AU. For two objects to be resolvable to the HST at that distance they would have to about 17 times the distance between the earth and sun apart.
1.22 = empirical constant for a circular aperture
wavelength = optical wavelenth, use 550 nanometers for visible light
aperture = objective diameter in the same units as wavelength.
At the distance to Pluto (40 AU) the resolution of the HST is about 1375 km about half the diameter of Pluto, 2377 km. Pluto would occupy four pixels, two horizontally, two vertically, in the HST.