The reason the orbital perturbations were much stronger than Le Verrier and Adams thought is related to the fact that the 2:1 resonance is not exact. (The period of Neptune differs from twice that of Uranus by 2 percent.) Though Le Verrier and Adams worried about the 2:1 resonance destabilizing their calculations, they couldn’t anticipate a side-effect of the actual near-resonance. Namely, the orbital perturbation Uranus experiences undergoes “beats,” slow variations in amplitude that occur when two objects are very nearly, but not quite, in resonance, like strings of a musical instrument slightly out of tune.
In light of this analysis, it turns out Le Verrier and Adams’ incomplete understanding of perturbation theory led them to make two mistakes. First, they made the deviations symmetric around the 1822 Uranus-Neptune conjunction, which was incorrect. The deviation from the mean motion that they thought was a maximum during this period was actually a minimum. Second, they entirely missed another possible location for Neptune, 180 degrees out of phase from the first, on the opposite side of the Sun! That they chose the position they did, which happened to be near the planet at the time, was indeed a happy accident.
Nevertheless, their efforts represented a significant accomplishment. According to CUHK physicist Kenneth Young, who co-authored the 1990 paper, “Le Verrier’s and Adams’ calculations were valid within the limitations of the theory they used.” In fact, Young adds, “even a search with many fewer parameters than those actually involved would have been a major computational endeavor in the days before electronic computers.”
So, the fulfillment of their prediction in d’Arrest’s exclamation, “That star is not on the map!” proves not to have been, as expected by later investigators, a model for future planetary discovery. Instead, it was a freak event, not likely to be repeated again in the history of astronomy.