From all these considerations, there are two problems: 1. Measurement technique (getting the data) 2. System dynamics calculations (analyzing the data)
For measurement, the article mentions both visual and spectroscopic (wavelength) measurements; overviews of these and other techniques are discussed in [1]. You are correct, distance from the measurement device makes things interesting, especially in the presence of atmospheric distortion, but there are lots of tricks that can improve the effective resolution (eg, speckle imaging, which uses a bunch of images taken in rapid succession [2]).
As for the orbital period, think of it like measurements of a line. You don't need every point, just two. If you have several measurements of a celestial body, you can plot them on one of the (MANY) celestial coordinate systems [3] and do some estimates. Longer periods mean more observations for better accuracy, but you don't need to wait for a complete orbital period to get a decent estimate.
Hope this at least gives some useful keywords for your searches!
[1] https://en.wikipedia.org/wiki/Binary_star#Methods_of_observa... [2] https://en.wikipedia.org/wiki/Speckle_imaging [3] http://csep10.phys.utk.edu/astr161/lect/time/coordinates.htm...
Although I doubt much at this distance is measured optically, but rather using estimates of gravitational forces, perhaps through measuring various "wobbles" within the system.
- Not a physicist.
For example how did they figure out that Pluto takes 247 Earth years per orbit? In our solar system I imagine they could measure it since it's close enough or see it in real time but even Pluto wasn't known for 248 years to see it go around.
With so many objects being so far away I thought it wouldn't really look like it moved from our perspective.