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According to historians both Adams and Leverrier used Bode's law to guess the distance to Neptune, which led to a vast overestimation of its orbital period (Adams - 227 years, Le Verrier - 218 years, actual - 165 years). Apparently, their estimate of Neptune's mass was also largely off. Nonetheless, both came up with a position in the sky close to each other's, and to the actual one:"The only actual value they were close to, if one looks at the table, is the location in the sky it would be found. It is possible that both men were simply lucky with this. We shall never truly know."

This is somewhat mystifying. The first thought is that their errors in distance and mass "compensated" each other, although I am not familiar with this type of calculation to tell if this makes sense. But it should be possible to tell if they were "simply lucky" or not. For instance, is the prediction of Neptune's position in the sky based on the parameters they assumed correct only for a short time period or more broadly? Is it the case that visible orbital deviations of Uranus only determine some function of Neptune's distance and mass, but not each value separately? And was their value close to the actual one? Did anybody look into these kinds of questions? Is there a "reason" for the correct prediction despite the erroneous estimates, or were they indeed "simply lucky"?

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    $\begingroup$ Nice question! Some might argue that this is better suited to the history of science/math SE, but I'd argue that this question really focusses on the physics (motivated by a bit of history). $\endgroup$ – Kyle Oman Aug 24 '15 at 23:35
  • $\begingroup$ I doubt very much that they were 'simply lucky', as it would require extraordinary luck to find where Neptune was. The error on orbital period was the more important one but one has to assume it was still possible to estimate position without accurate knowledge of orbital period. $\endgroup$ – Gert Aug 25 '15 at 0:50
  • $\begingroup$ Of course, Pluto (and to a lesser degree, Ceres) were both found VERY coincidentally, especially Pluto because of its wide orbit, relatively close solar distance (at the time) and low brightness significantly lowered the odds of seeing it there at that very time.... $\endgroup$ – Damon Blevins Aug 25 '15 at 3:06
  • $\begingroup$ Maybe this is better migrated to Astronomy? $\endgroup$ – Anixx Aug 25 '15 at 5:57
  • $\begingroup$ No....... People don't answer worth a crap there. It takes FOREVER to get an answer in that website. I submitted a question months and months and months ago that never got answered $\endgroup$ – Damon Blevins Aug 30 '15 at 5:59
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Although this may not be what you're looking for... They weren't "simply lucky." In fact, they didn't use Bode's law at all- they used calculations based on Neptune's supposed gravitational effect on Uranus. In fact, had the two used Bode's law, they would never have found Neptune, as the Bode "law" would predict a completely different location. (This is when the "law" became discredited)

enter image description here

Astronomers noticed that the then-newly discovered planet, Uranus was appearing at its predicted positions later than what would be expected- this meant something should be tugging on it from behind, the opposite of its direction of motion. From the angle of this apparent slowing, astronomers could tell what angle the perturbing "bully" planet was from- applying this over multiple places in the Uranian orbit, and newtons laws of gravity, enabled them to predict where this new planet was at multiple given times, thus being able to predict its orbit. Then, they predicted where it should be at a certain time and looked for it- there it was!

Thus, it was not sheer coincidence and luck that this planet was discovered; rather, their math was what led them to it. Note that the perturbations could be attributed to bodies of different mass at different distances from Uranus -I take it that's the reason for the two astronomers' miscalculation of

Neptune's orbital semi-major axis. NOTE- Please don't downvote it if it's not what you were looking for, I will try to fix that if I haven't.

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    $\begingroup$ This is a nice answer, but I think references are in order! $\endgroup$ – march Aug 25 '15 at 4:59
  • $\begingroup$ I literally got all,that from prior knowledge... Minus the picture, obviously.... csep10.phys.utk.edu/astr161/lect/history/perturbations.html also has some good info on the topic, and the picture as well. $\endgroup$ – Damon Blevins Aug 25 '15 at 11:17
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    $\begingroup$ Historical accounts are fairly unanimous on Leverrier using distance estimates from Bode's law. And generally the discrepancy between Leverrier's assigned parameters for Neptune and the actual values was so great that some contemporary astronomers even claimed that what was discovered was not what Leverrier predicted. So the puzzle remains. The idea about relative angular positions is interesting, but Newton's law requires estimates for masses and distances in any calculation of disturbances, so it would not be enough for this kind of precise prediction. $\endgroup$ – Conifold Aug 25 '15 at 17:55
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I suspect that they were lucky that their predictions agreed with reality so closely, but any prediction was going to have Neptune roughly (perhaps very roughly) in the same direction as Uranus, during the times when it affects Uranus the most. So I suspect their calculations meaningfully ruled out large swathes of sky, which improved odds of finding it.

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