# How Was Neptune Discovered? The Story of Mathematical Competition and an Unresolved Controversy

Leonard Kelley holds a bachelor's in physics with a minor in mathematics. He loves the academic world and strives to constantly explore it.

## Something Is Wrong With Planet Uranus

When Uranus was discovered, it was a major accomplishment for the scientific community. Never before has a planet been discovered by anyone, for all the planets up to that point could be seen without any telescopic aid.

Once found, Uranus was studied heavily. Astronomer Alexis Bouvard even compiled tables of various positions of Uranus in its orbit to extrapolate its orbit. People began to notice that when Kepler’s planetary laws (three rules that all orbiting bodies follow) were applied to Uranus, it had discrepancies that were not within observational error such as gravitational perturbations from the other solar system objects.

In 1821, Bouvard commented in his book Tables for Uranus that, “… The difficulty of harmonizing the two systems is really due to the inaccuracy of early observations or whether it is due to some strange and at present unknown force acting upon the planet and influencing its movement” (Airy 123, Moreux 153).

Many ideas explained this, including the notion that gravity may work differently in that region of space (Lyttleton 216). In 1829, a scientist named Harrison supposed that not one but two planets must be affecting Uranus’ orbit (Moreux 153). However, the general consensus was that a single missing planet must exist past Uranus and pull on it with its gravity (Lyttleton 216).

## What Were the Unknowns?

When searching for a new planet, there are many quantities to solve for. You will need to find the planet's mass (mn) and its average distance from the sun (dn), which would involve knowing its semi-major axis and semi-minor axis (since all planetary bodies orbit in some form of an ellipse). This would give us its eccentricity (en).

We also do not know if the planet orbits our plane, but since all the planets orbit within +-4 degrees of the ecliptic, it is a safe assumption that an unknown planet will also (Lyttleton 218).

## Initial Wanderings

George Airy, who was the Astronomical Royal of Britain and a central figure in this story, was first brought into this search by the Reverend T.J. Hussey in a letter from November 17, 1834. He mentions in his letter how he has heard of a possible planet beyond Uranus and looked for it using a reflector telescope, but to no avail. He presented the idea of using mathematics as a tool in the search but admitted to Airy that he would not be of much help in that regard.

On November 23, Airy writes back to the reverend and admits he, too, has been preoccupied with a possible planet. He had observed that Uranus' orbit deviated the most in 1750 and 1834 when it would be at the same point. This was strong evidence of an object pulling on the planet, but Airy felt that until more observations were made no mathematical tools would be of help (Airy 124).

The return of Halley's comet in 1835 also sparked interest in the search for the eight planet. After 76 years, scientists had the orbit figured and and awaited to see it.

The problem was, it came a day late.

Calculations were quickly made and based off the deviation, it pointed to a trans-Uranus object at 38 AU. With so many heavenly bodies not working as they were predicted to, the Royal Academy of Sciences in 1842 offered a cash prize to anyone who could find the missing planet (Weintraub 111).

## John Couch Adams and His Method

Adams, a British astronomer, was an undergraduate student when he began his search for the missing planet in 1841. He had compiled himself additional observational errors in Uranus’ orbit. Starting in 1843, he began his calculations for the unknowns mentioned before, and by September 1845, he finally finished (Lyttleton 219).

Among the tools he used to solve for Neptune’s orbit was a false correlation known as Bode’s Law which noted that the distance from Saturn to the Sun was twice the distance from Jupiter to the Sun and that the distance from Uranus to the Sun was twice the distance from Saturn to the Sun, and so on. Essentially, it states that the distance from a planet to the Sun is twice the distance from the previous planet to the Sun. As it turns out, Bode’s Law fails to place Mercury correctly, and it requires a planet to be placed between Mars and Jupiter if the pattern is to hold. Bode’s Law will ultimately fail on Neptune also (217).

Along with using Bode’s Law, Adams also used a circular orbit as his initial attempt at a solution. He knew that it would not be correct, but it was a good starting point to compare it with the observational data and refine it to a more elliptical orbit as he iterated more solutions. Another technique involved taking all the gravitational perturbations that the other planets imparted on Uranus away would help reveal the missing component provided by the missing planet (Moreux 158, Jones 8-10).

As he worked on these calculations, Adams needed data from past observations, and he contacted Challis, who was in charge of the observatory at Cambridge. In a letter dated February 13, 1844, Challis writes to Airy about Adam’s finished work and the desire of Adams for the errors in the “geocentric longitudes” and “heliocentric longitudes” of Uranus from 1818 to 1826. Airy does even better and sends data from 1754 to 1830 as well as notes on any discrepancies there may be from other published material that existed at the time (Airy 129, Jones 12).

## Airy and His Mistake

In a letter dated September 22, 1845, Challis writes to Airy about Adams’ finished work and his desire to meet with Challis and Airy to discuss them. Airy responds September 29 that such a meeting would be a great idea and that Adams should write to Airy to settle the date. Ironically, Adams sent where the possible location should have been for a missing planet if you looked on October 1, 1845. Backtracking what we know now, if Challis had looked, he would have found Neptune only 2 degrees from the expected location (Airy 129, Jones 13)!

On October 21, 1845, Adams sent his work to Airy in hopes that he would assist him in the search for Neptune. Adams did not seem to have enough conviction in his work to officially submit it for publication and would eventually revise his work several times. Adams was foremost a mathematician and an astronomer second. He may have wanted his work in more capable hands before taking the plunge of making his work official. (Rawlins 116).

Officially, Airy does not fully appreciate what he has received. He feels that certain portions of Adams’s work are assumed numbers when in reality, Adams had made hard calculations over those elements. Airy was also more focused on how Adams’s work could help solve a problem with the radius vector of Uranus, or the distance problem that helped spark the quest for a new planet in the first place, than on the implications of Adam’s work.

He felt that gravity might work differently around there and so wanted Adams to see if it could solve that problem because, to Airy, the work Adams submitted could be separated from the vector dilemma and still be valid, so why not see if a correlation existed. He writes back to Adams on November 5 expressing this (Lyttleton 221-2, Airy 130).

Finally, he also mentions in his letter to Adams that he has concerns about if the data takes into account recently measured errors in the orbits of Jupiter and Saturn because of gravitational tugging between them all. Naturally, not having his request met and instead being dealt all these comments and questions made Adams mad, though he would respond to Airy a full year later (November 18, 1845), stating that he was trying to resolve a distance calculation to ensure Airy’s queries were resolved.

He also points out that the radius vector problem is merely a result of angular momentum mistakes taken from Uranus that once considered render the problem obsolete. Finally, Adams also wanted to ensure Airy that his work was indeed his own, found as a result of rigorous calculations, and so confidence should be placed in his work (despite his lack of publication) (Lyttleton 222-3, Jones 18-21).

## Enter Le Verrier

Around this same time, an astronomer named Arago, director of the Paris Observatory, encourages a young French astronomer named Urbain Le Verrier to find this missing planet (Moreux 153). Unaware of Adams and his work, Le Verrier did use some similar techniques as Adams. He, too, felt that Bode’s Law was an acceptable tool to find the distance to Neptune from the Sun. He also made similar conclusions about the plane of the orbit as well as the maximum number of degrees it could be above/below the ecliptic (155).

Le Verrier did many different calculations from Adams. He started by tracing out Uranus’ 84-year orbit and taking into account all known influences, including gravitational pulls from Saturn and Jupiter. To help determine this orbit, Le Verrier needed to know the elements of an elliptical orbit that would best match. He also needed to know what his uncertainty values were for each of those values calculated (Lyttleton 231). Also, using this model, original measurements of Uranus, and current (at the time) measurements of Uranus, he made a calculation for the mass of Neptune, which he felt would be smaller than Uranus (Moreux 154).

To get a feel as to how grueling the calculations both men worked on were, consider the following: During one portion of his work, Le Verrier was presented with 40 possible solutions to a particular value, based on unknowns such as satellites of Uranus, the range of Uranus’ orbit, different space physics, or alterations to gravity. He solved for each value and then determined which one was the best fit to his data (Lyttleton 232, Levenson 36-7).

Also, consider this: The Theory of the Perbutations, which contains some of Le Verrier and Adams calculations, has said values for Jupiter, Saturn, Uranus, and Neptune properties. This encompasses five volumes and totals about 2,300 pages. The calculations behind the book's values take up about 3-4 times as much space (Moreux 156).

## The Hunt Is On, The Conspiracy Plotted

Le Verrier publishes his first set of calculations on November 10, 1845, and later his second set on June 1, 1846, in Comptes Rendus. Interestingly, between these publications, Airy reads of Le Verrier’s work in December 1845 and remarks on his ability to incorporate Jupiter and Saturn’s perturbations on Uranus, thus reducing errors in his work. With Adam’s work in tow, he notes the similarities with Le Verrier’s and is further persuaded by the mounting evidence that surrounds him.

Yet amazingly, Airy is still concerned about the radius vector problem and does not appreciate the real meaning behind the work. Without revealing Adam’s work, Airy writes to Le Verrier on June 26, 1846, about the Uranus radius vector problem that still plagued him. Le Verrier writes back, explaining how his work solves that problem and still addresses the missing planet. Airy does not write back (Lyttleton 224, Airy 131-2, Jones 22-4)

It took him 11 months to complete his final calculations, but on August 31, 1846, Le Verrier makes his prediction before the Académie in France: Neptune would be at 326 degrees, 32’ on January 1, 1847 (155). The next day, September 1, 1846, Le Verrier publishes his findings in Comptes Rendus, a French scientific periodical. By this point, it had been seven months since Airy had received Adams’s work (Lyttleton 224, Levenson 38).

As it turns out, Airy had a secret search for Neptune started with Challis’ assistance. Since the expected location of Neptune was in a region the observatory had not cataloged before; Challis was not too hopeful about the odds of success. Why? One needs to find out what are stars, comets, asteroids, and so on before one can determine a planet so that the proper distinction can be made and you don’t falsely claim that a planet has been found (Lyttleton 225).

In a shocking twist of events, Airy began this hunt without revealing to either Adams or Le Verrier that he was using their work. He read Le Verrier’s work abound the 24 of June, months before its final publication, courtesy of a friend of Le Verrier, and held a meeting of the Board of Visitors of the Royal Observatory at Cambridge on June 29 where he points out many of the similarities of Adams and Le Verrier’s work.

Because of this similarity, he initiated the search, not because of the possible veracity of Adam’s initial submission. Airy mentions how if the task was distributed amongst observatories, then the likelihood of discovery would increase. General agreement was reached on the matter, but no game plan was set forth to move onward (Rawlins 117-8, Airy 133, Jones 25).

A few weeks later, on July 9, Airy writes to Challis asking for his assistance in the search. Challis had been present at the meeting and so knew of the agreement in the work of Adams and Le Verrier. As Challis admitted in a letter, “I can say, however, that this concurrent evidence of the reality of the disturbing body from two independent investigations, weighed strongly with me in coming to the determination of undertaking the observations in the face of the great amount of labor they might be expected to entail.”

Whether Airy was truly concerned about the radius vector problem is certainly doubtful in light of all of this and was most likely a cover for him to be clandestine in his operations. After all, he was consistently...inconsistent with his information distribution (Rawlins 121, Airy 133).

Airy was determined to be the one to find the new planet. He was so desperate to use the telescope at Cambridge that he was willing to pay Challis, who was not on board at first, a large sum of money. He was able to subtlety mention this payment in the July 9 letter, saying it would be for an assistant if needed be. He further states that Challis’ Northumberland Telescope was perfect because Airy’s location was bad based on where the sky needed to be observed.

No doubt that Airy was playing puppetmaster in this making of a conspiracy to be the finder, for many of his letters reveal his secret maneuverings around the people surrounding him. For a good example, look no further than a letter to Challis on November 13, 1846 (post-Neptune discovery): “The matter being one of delicacy, I will not compromise any one…All I as is will you allow me to publish your correspondence with me on this subject, or extracts from it taken at my discretion?” In fact, once Neptune was found, Airy destroyed many correspondences he had at the time. Several letters were sent between June 30 to July 21, and finally, on July 27, months before Le Verrier would publish his final work, their secrets now lost to time (Rawlins 118-20; Airy 135, 142; Jones 25).

With all of this nonsense, it is no surprise that Challis missed finding Neptune. Adam’s solution included a night sky span covering longitudes between 315 and 336 degrees. That is so much to look over. Also, Adams sent so many revisions to his work that portions of the search became redundant (Rawlins 120).

Rather than wait around for what he thought was further inaction, Adams kept busy. Though he certainly could have initiated the search himself, much less publish his calculations, he was busy revising his work as was Le Verrier. Adams claimed on September 2, 1846, in a letter to Airy just a few days after Le Verrier published his latest work on the calculations, that he had not initiated the search yet because he did not want to hunt for something that was not more determined to be correct.

Le Verrier would go on to publish a revised solution. Adams would not. Le Verrier’s new work reflects recent data from Uranus and other celestial objects, while Adams’s was more about tinkering about based on an idea rather than on observations. One of these was modifying Bode’s Law so that the distance was decreased by 1/30, and thus the eccentricity errors were reduced. All of this is further evidence of his lack of faith in his work (Rawlins 116-7, Airy 137).

On September 18, 1846, Le Verrier writes a letter to Dr. Galle, the director of the Berlin Observatory, about many topics, and as a post-script, mentions his calculations for Neptune (Moreux 156, Levenson 39). On September 23, Galle receives Le Verrier’s letter. The Berlin Observatory had recently compiled a map of the supposed region where Neptune would be located, so they would be able to tell what was a celestial object was and what was a planet (Lyttleton 225).

The same day he received the letter, Galle and his assistant d’Arrest begin the search at night. Within one hour of the search, a “star that is not on the map” as d’Arrest proclaimed was found a mere 52’ from its anticipated location (Moreux 157, Levenson 39). They took an extra night to confirm their discovery and formally announced it to the world on September 25 (Lyttleton 226).

When news reached Britain, Challis stopped his search. It was not noted until reviewing their work that Challis had observed Neptune several times during his hunt and had never realized it. As directed by Airy, Challis had conducted sweeps of the region in question on July 29, July 31, August 4, and August 12. In a letter from October 12, Challis tells Airy that he had an unnoticed find of the planet in early August. He continues on, saying how on August 12, he has noticed an 8th magnitude star that did not match with his July 31 observation of the same part of the sky. He had been busy completing a catalogue of comet observations and did not have the time to look over old results yet. He was too busy gathering data.

The additional insult to injury was the examination of the area on September 29 after Le Verrier had published a new set of results. Challis thought that he say a disc but was not sure. Altogether, Neptune had been observed twice in the first four days on the search and many more times throughout (Airy 143, Lyttleton 225, Jones 26-7).

Average Distance from Sun (AU)

36.2

37.2

30.07

Eccentricity

0.208

0.121

0.0086

Mass (10^24 kg)

212.74

298.22

103.06

Location (Degrees)

327.4

330.9

328.4

## Aftermath

For England, the message was quite clear: They missed out on a great, once-in-a-lifetime discovery. They had knowledge of this planet a full year before it was found, and now no credit would go to Adams, Airy, or Challis. Adams can hardly shoulder all the blame, for Challis had clearly missed the signs of Neptune and Airy has several offenses we can credit him with. Airy had the information at hand and tried to outmaneuver both men, only to come up empty-handed.

In an attempt to perhaps save his own skin, he publicly gives Le Verrier credit for the find, earning the scorn of Britains for the rest of his life. Despite this, Airy did manage to prevent Le Verrier from winning the Royal Astronomical Society Medal for his work, which would have meant that Adams’s work was not on par with Le Verrier. Adams became an inspiration for several generations of British mathematicians. Not at any point in his work did he learn of Le Verrier’s prior to the discovery.

Adams would acknowledge his mistake of not being bold with his work. In a letter from December 17, 1846, Adams wrote, “I fully allow that I have to blame myself severely in this matter…for having trusted to anyone but myself to make known the results at which I had arrived.” For Le Verrier, it ensured his place in French mathematical astronomy, a pedestal shared with Lagrange and Laplace (Lyttleton 226, Rawlins 117-8).

The world was excited by the find, for never before had mathematics predicted a natural object. This confidence in the results was lessened, however, when discrepancies were noticed in the calculated values and the actual ones (Lyttleton 227). For example, Adams calculated an orbital period of 227 years and Le Verrier found it to be 218 years using Kepler’s Third Law (Period squared is proportional to average distance cubed). The actual value of the orbit is 165 years. This discrepancy was not a result of using Kepler’s Third Law but because of using Bode’s Law for the average distance (229).

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 (233). Neptune, the last planet in our Solar System, proved to be the ultimate challenge in mathematical astronomy.

## Works Cited

Airy, Georges. Royal Astronomical Society Vol. 7 No. 9: 13 Nov. 1846. Print. 16 Nov. 2014.

Jones, Sir Harold Spencer. John Couch Adams and the Discovery of Neptune. Cambridge University Press: New York, 1947. Print. 8-10, 12-14, 18-27.

Levenson, Thomas. The Hunt for Vulcan. Pandin House: New York, 2015. Print. 36-9.

Lyttleton, Raymond Arthur. Mysteries of the Solar System. Oxford: Clarendon P., 1968. 216-33. Print.

Moreux, Théophile. "Uranus & Neptune." Astronomy To-day. Trans. C. F. Russell. New York: E.P. Dutton and, 1926. 153-58. Print.

Rawlins, Dennis. "The Neptune Conspiracy." DIO 2.3 (1992): 116-21. Print.

Weintraub, David A. Is Pluto a Planet? New Jersey: Princeton University Press, 2007: 111. Print.

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