Planet Neptune was originally predicted thanks to Uranus’ perturbed orbit
There was a time when astronomers assumed they knew all the planets in our solar system, based mainly on checking what was immediately visible in the sky. Uranus can be seen without instruments, but it’s dimness and slow orbit made people assume it was some kind of weird comet, or maybe a distant star. In 1781, William Herschel finally realized what he was seeing, but the orbit was still a bit weird. It wasn’t behaving as predicted by physics models, which meant either Newtonian models of gravity were wrong, or something was missing from the equation. Something big and close enough to Uranus to perturb its orbit.
Mapping with math
While it seemed bizarre to even suggest there were planets hiding out there after centuries of accepting what the naked eye can see as the whole story, astronomers started crunching numbers to make sense of Uranus’ orbit. It wasn’t an easy problem to crack, since it required figuring out the gravitational influence of something of an unknown mass and distance, then reverse-engineering all that into a testable prediction. John Couch Adams and Urbain Le Verrier, independently, both came up with the answer of an eighth planet, which Le Verrier managed to get astronomer Johann Galle to look for in September of 1846.
Galle spotted what was to become Neptune within one degree of Le Verrier’s prediction. It was an amazing test of Newtonian physics, although there may have been some opportune timing involved— we now know that the two sets of calculations both diverge from Neptune’s actual orbit substantially enough that if it had been in a different part of its orbit when they started looking, it would likely have been missed. Eventually, direct observations would have likely found it at some point (Galileo actually found it 1613 without knowing what he was seeing), but it was still a success for a method that’s still in use today.
The current hunt for the so-called Planet Nine started because astronomers noticed disturbances in the orbits of known Kuiper Belt objects. Mathematical models have helped us narrow down where in the sky to look, but at this point the distances are involved enough that only a few instruments are likely to be able to detect it. Planet Nine is expected to be significantly dimmer than Pluto, ensuring that it probably hasn’t been seen and misidentified as a comet already.
Source: Neptune Facts, Nine Planets