The history of Le Verrier’s mathematical discovery of Neptune is my favorite story, period. It’s literally got everything you’d want in a good novella - differential equations, integrals, telescopes, intrigue, you name it. Rather than try to rehash it here without doing it justice, I’ll point the interested reader to an excellent 2016 article by Davor Krajnović called “The contrivance of Neptune.” Here, I only want to call attention to what Le Verrier (and Adams) got right and what they got less right.
It’s widely known that Neptune was discovered “with the tip of a pen.” Indeed, Le Verrier was able to derive Neptune’s location on the sky from orbital anomalies of Uranus with exquisite accuracy, such that Galle and D’Arrest’s observational campaign to discover this elusive planet took less than a single night. What is somewhat less well known is that Le Verrier and Adams’ calculations of Neptune’s orbit and mass were not as precise. The figure below shows the true orbits and locations of Uranus (gray) and Neptune (black) between 1830 and 1860, as well as the predicted orbits of Neptune (in purple).
Notice that the inferred semi-major axis of Neptune was about 40 (rather than 30) AU and the derived mass (36 and 50 Earth masses for Le Verrier and Adams, respectively) also significantly exceeded that of Neptune. In light of the fact that the discovery of Neptune represents the only successful mathematical prediction of a planet to date, this level of uncertainty sets the gold standard for dynamically motivated planetary predictions. In other words, if we get Planet Nine to a similar level of precision, I’ll be satisfied. It is also useful to point out that the most significant quantity in perturbing the orbit of Uranus was the anomalous acceleration in the radial direction produced by the new body - GM/r^2 - a ratio that was calculated to higher accuracy than the individual values of mass and semi-major axis. As I will highlight later, the general framework of the Planet Nine hypothesis is characterized by comparable degeneracies between P9’s mass and orbital parameters.
Following Le Verrier’s mathematical discovery of Neptune, the planet prediction business didn’t simply get a boost - it exploded. Jacques Babinet (1848), David P. Todd (1877), George Forbes (1880), Camille Flammarion (1884), William Pickering (1909-1932) all took turns predicting trans-Neptunian planets that later turned out to not be there. But no planetary prediction is quite as emblematic as Percival Lowell’s hypothesized “Planet X.” Briefly, the story goes as follows: despite the addition of Neptune to the solar system’s ledger of planets, small apparent discrepancies in the orbits of the giant planets remained, and pointed to the existence of a ~7 Earth mass planet beyond Neptune. The search continued well past Lowell’s death, and in 1930, a bright moving object was discovered by Clyde Tombaugh in the approximate location on the sky where Planet X was envisioned to be. Because Planet X was the object of the original search, the newly found body was initially considered to be the long-sought-after Planet X.
Immediately, however, there was a problem. Planet X was supposed to be like Neptune, but Tombaugh’s new planet appeared dim and point-like, and therefore much much smaller. It soon became clear that the new member of the solar system could not be THE Planet X. The object was subsequently named Pluto, and its estimated mass steadily declined for the next 5 decades.
Although Planet X - as originally formulated by Lowell - does not exist, the discovery of Pluto turned out to be the tip of an extraordinary trans-Neptunian iceberg called the Kuiper belt. The mapping and subsequent characterization of the Kuiper belt in the ‘90s and the ‘00s, generated a new wave of planetary proposals — check out Brunini & Melita (2002), Gladman & Chan (2006), Gomes et al. (2006), Lykawka and Mukai (2008), Trujillo and Sheppard (2014), Volk and Malhotra (2017) and many others that are referenced therein. All of these hypothetical planets were invoked to explain different observational puzzles, and attempt to do so through individual dynamical mechanisms. Stepping away from specific predictions, however, it is worthwhile to examine the question of where still-undetected planets can hide in the solar system, from a completely model-independent perspective. As it turns out, the combination of ephemerides, orbital stability, and definition of a planet alone leave only a limited parameter space where additional solar system planets can hide (shown as the shaded region on the plot below):
Remarkably, Planet Nine falls right in the center of that region.