This one’s a bit of a departure, but it appeals to my imagination. Given that almost every method astronomers use of locating exoplanets involve spotting a regular change in some measurement of its host star — light output, velocity, etc. — an independent-minded little planet like this one would have half the world’s astrophysicists tearing their hair out. Plus it would wreak havoc on the orbits of the other planets in its system, given their relative sizes. The illustration doesn’t reflect that fact, but hey, it’s a planet with eyeballs so I’m not going to nitpick.
The wallpaper reminded me of an excellent PhD TV episode, in which a team of CalTech astrophysicists explain the five methods of planet hunting. It’s layperson-friendly, but I should warn you that your mind will probably be blown.
(Click the image to orbate, and scroll down to find the size you want.)