History topic: Mathematical discovery of planets

The first planet to be discovered was Uranus by William and Caroline Herschel on 13 March 1781. It was discovered by the fact that it showed a disk when viewed through even a fairly low powered telescope. The only other planets which have been discovered are Neptune and Pluto. These were predicted using ingenious mathematical arguments based on Newton's laws of gravitation and then observed near their predicted locations.

In fact Neptune could have been discovered without the mathematical arguments. It very nearly was discovered by Galileo, the first person who could possibly have discovered a new planet. Galileo turned his telescope on the planets and was immediately fascinated by the system of Jupiter and its moons which he observed. While he was observing the Jupiter system on 28 December 1612 he recorded Neptune as an 8th magnitude star. Just over a month later on 27 January 1613, he recorded two stars in his field of view. One was Neptune and the other a genuine star. Remarkably, Galileo observed the pair again the following night when he noted that the two stars appeared to be further apart. How close he was at that point to discovering that one of the stars was the planet Neptune. See [1].

Neptune was to be recorded several more times, without being recognised as a planet, over the following years. Lalande (1732-1807), a French astronomer whose tables of the planetary positions were the most accurate until the 19th Century, recorded Neptune on the 8th and 10th of May 1795 without recognising that it was not a star. John Herschel, who we shall see in a moment was to be involved with the discovery of Neptune, recorded Neptune on 14 July 1830 believing it to be a star.

Von Lamont (1805-1879), a Scottish born astronomer who lived most of his life in Munich, is famed for his determination of the orbits of moons of Saturn and Uranus, and also for discovering the periodic fluctuation of the Earth's magnetic field. He recorded Neptune on at least three occasions, namely on 25 October 1845, 7 September 1846 and 11 September 1846. As a highly skilled observer, one might have expected that von Lamont would have recognised the motion of Neptune over the four day period. However he failed to do so, only days before Neptune was discovered.

The discovery of Neptune however did not come from any of these chance observations. Rather it came from a mathematical analysis of the deviation of Uranus from its predicted orbit. Delambre computed tables of planetary positions Tables du Soleil, de Jupiter, de Saturne, d'Uranus et des satellites de Jupiter published in 1792. However discrepancies began to arise in the predicted position of Uranus. Bouvard (1767-1843), a French astronomer who was director of the Paris Observatory, had already published accurate tables of the orbits of Jupiter and Saturn in 1808 and he now undertook to produce a corrected version of Delambre's tables for Uranus. However he could not make all the observations fit, even after taking the perturbations of the other planets into account. He published his new tables of Uranus in 1821 but wrote

... I leave it to the future the task of discovering whether the difficulty of reconciling [the data] is connected with the ancient observations, or whether it depends on some foreign and unperceived cause which may have been acting upon the planet.
Although Bouvard had used the latest data to determine the orbit of Uranus, it soon became apparent that it was deviating from the position given in his tables. On 3 July 1841 Adams, while still an undergraduate at Cambridge, wrote
Formed a design in the beginning of this week, of investigating, as soon as possible after taking my degree, the irregularities of the motion of Uranus, which are yet unaccounted for; in order to find out whether they may be attributed to the action of an undiscovered planet beyond it; and if possible thence to determine the elements of its orbit, etc.. approximately, which would probably lead to its discovery.
Not everyone however attributed the problems of Uranus's orbit to an unknown planet beyond it. Airy, the Astronomer Royal, believed in another popular theory, namely that the inverse square law of gravitation began to break down over large distances. However after Adams had made an initial investigation of the effect that an undiscovered planet might have on Uranus, he was greatly encouraged in his belief that he was on the right track, and he obtained from Airy the Greenwich data on Uranus in February 1844.

In June 1845 Arago, the director of the Paris observatory, persuaded Le Verrier to start work on the problem of Uranus's orbit. Le Verrier quickly decided to devote himself fully to the problem and set aside the work he had been doing on comets. Neither Le Verrier nor Adams knew that the other was working on the problem.

By September 1845 Adams had made a more detailed study of the problem and deduced an orbit for the perturbing planet. As well as the orbit he had calculated the mass of the planet and its position on 1 October 1845. He sent has predictions to James Challis, the director of the Cambridge Observatory.

Adams was breaking new mathematical ground here. Newton's theory of gravitation had been used many times to calculate the effects of bodies on one another but never had it been used to predict the position of a body from observations of the effects of its gravity on other bodies. Nevertheless Adams was very confident in his theory and referred to the "new planet". An attempt by Adams to give Airy information on the "new planet" failed when Adams visited Greenwich on 23 September on his way between his home in Laneast, Cornwall and Cambridge since Airy was in France at the time.

On 21 October 1845 Adams made a second attempt to visit Airy on his way to Cambridge. He was told that Airy was in London but would be back soon. Adams returned in the afternoon but Airy was at dinner. In fact Airy had the rather unusual habit of eating dinner at 3.30 every afternoon so Adams was turned away. Adams left a manuscript with his research on the orbit of Uranus in which he showed that, given the orbit which he proposed for the new planet, the errors in Uranus's position were extremely small.

In fact Airy was clearly interested in Adams' work for, on 5 November, he wrote to Adams asking what appeared to be a somewhat technical question. He wanted to know whether the postulated "new planet" explained not only the discrepancies in the longitude of Uranus but also the discrepancies in its radius vector. This question was designed to try to distinguish between the "new planet" theory and the "failure of the inverse square law" theory. However Adams, already cross at what he saw was Airy's refusal to see him, made no answer. He decided to search for the new planet himself:-

I could not expect, however, that practical astronomers, who were already fully occupied with important labours, would feel as much confidence in the results of my investigation, as I myself did; and I therefore had our instruments put in order, with the express purpose, if no one else took up the subject, of undertaking the search for the planet myself, with the small means afforded by our observatory at St John's.
On 10 November Le Verrier published his first paper on his investigations. In it he showed that the perturbations on the orbit of Uranus due to Jupiter and Saturn could not explain the observations. On 1 June 1846 Le Verrier published a second paper in which he showed that a variety of other possible causes could not explain the orbit of Uranus, and deduced that the only possible cause could be a planet further from the Sun than Uranus. He gave some details of a possible orbit of the "new planet" with a predicted position for the beginning of 1847. Le Verrier approached the Paris Observatory to search for the planet but after a very brief search they lost interest.

On 23 June the results of Le Verrier's paper reached Airy who immediately saw that Le Verrier's prediction and Adams prediction for the position of the "new planet" were almost identical. Three days later he wrote to Le Verrier asking the same question about the radius vector as he had asked Adams. Strangely Airy, who now knew that both Adams and Le Verrier had come to almost identical solutions to the same problem, did not tell either of them about the other, nor did he tell Le Verrier of his plans to begin a search. Le Verrier replied to Airy's questions convincing him that the deviations were indeed due to a "new planet". On 29 June Airy met with Challis and John Herschel in Greenwich and told them of

...the extreme probability of now discovering a new planet in a very short time, provided the powers of one observatory could be directed to search for it.
On 9 July Airy asked Challis to begin a search at the Cambridge Observatory. He was somewhat reluctant: as he later wrote:-
It was so novel a thing to undertake observations in reliance upon merely theoretical deductions; and that while much labour was certain, success appeared very doubtful.
Despite his reservations, Challis began the search on 29 July 1846, recording stars in the area of Adams's prediction. He observed on the nights of 29, 30 July, 4, 12 August and recorded the results. He checked out his methods by comparing the first 39 stars recorded on 12 August and checking that they appeared on his 30 July records. If he had continued his comparison he would have discovered the "new planet" which he had recorded on 12 August but which had not been in the search area on 30 July. He had also recorded the planet on 4 August but he never compared these records.

Later in August John Herschel visited an amateur astronomer William Dawes and told him of the "new planet" but, since Dawes had only a small telescope, he thought it not worth searching.

On 31 August Le Verrier published his third paper on the "new planet". This time he gave full details of the predicted orbit and the mass. He also deduced the angular diameter and wrote:-

It should be possible to see the new planet in good telescopes and also to distinguish it by the size of its disk. This is a very important point .... if a simple study of its physical appearance can replace the rigorous determination of the positions of all the stars, the search will proceed much more rapidly.
Adams wrote to Airy on 2 September giving a more through analysis of the problem. His first solution had depended on assuming a distance for the "new planet" of twice that of Uranus from the Sun. He was unhappy with this arbitrary part of his solution and he had redone the mathematical analysis finding a better estimate of the distance of the "new planet" by testing different distances against the observed perturbations of Uranus.

Dawes, although thinking (wrongly) that he could not observe the "new planet", suddenly had a thought. His friend William Lassell, another anateur astronomer and a brewer by trade, had just completed building a large telescope that would be able to record the disk of the planet. He wrote to Lassell giving him Adams's predicted position. However Lassell had sprained his ankle and was confined to bed. He read the letter which he gave to his maid who then promptly lost it. His ankle was sufficiently recovered on the next night and he looked in vain for the letter with the predicted position. His chance of fame had gone!

On 10 September John Herschel addressed a meeting of the British Association in Southampton. He spoke of the "new planet" saying:-

We see it as Columbus saw America from the shores of Spain. Its movements have been felt, trembling along the far-reaching line of our analysis, with a certainty hardly inferior to that of ocular demonstration.
Herschel was a very fine mathematician and clearly had a faith in the mathematical analysis which many astronomers failed to have. Adams intended to present a paper on his researches at the Southampton meeting. However Section A of the British Association ended its session one day before he expected and Adams arrived in Southampton too late to announce his predictions.

Le Verrier wrote to the German astronomer Galle on 18 September asking him to search for the "new planet" at his predicted location. Galle received the letter on 23 September and together with his assistant Heinrich d'Arrest began a search that night at the Royal Observatory in Berlin. D'Arrest suggested they use the latest star chart which had only just been produced. It took them less than 30 minutes to locate a star not on their map. Of course they knew that they had found the "new planet" but they confirmed it the following night by observing it had moved relative to the stars.

Galle wrote to Le Verrier on 25 September, saying:-

Monsieur, the planet of which you indicated the position really exists.
Le Verrier replied:-
I thank you for the alacrity with which you applied my instructions. We are thereby, thanks to you, definitely in possession of a new world.
On 29 September Le Verrier's paper of 31 August reached Challis. He observed that night, searching the predicted position for the disk of the planet. He noted that only one from 300 stars in the region appeared to show a disk. Of course he had observed the planet but, a cautious man by nature, he waited until he could confirm the result by showing the motion of the planet. He did not observe again before 1 October when The Times carried the announcement of Galle's discovery with the headline Le Verrier's Planet Found.

As soon as he read the headline, Herschel wrote to Lassell saying

Look out for satellites with all possible expedition.
Lassell began observing on 2 October and on 10 October he discovered Neptune's moon Triton.

It was on 3 October that Herschel wrote to the Athenaeum making public the role of Adams in the discovery of Neptune. The subsequent argument over the priority and naming of the planet is discussed in the article on Orbits and Gravitation. The full story of the contributions of Adams, Challis and Airy were published at the 13 November meeting of the Royal Astronomical Society. Challis and Airy each reported on the story of Adams's predictions and Adams himself published his memoir An explanation of the observed irregularities in the motion of Uranus, on the hypothesis of disturbances caused by a more distant planet; with a determination of the mass, orbit, and position of the disturbing body.

Once the orbit of Neptune was worked out sufficiently well, older records were searched to see if it had been recorded earlier. When Lalande's observations of Neptune on the 8th and 10th of May 1795 were discovered it was noted that Lalande had rejected the 8 May position and recorded a star in the 10 May position of Neptune, but marked it as doubtful. He never bothered to make a further observation to confirm the data which would have certainly resulted in his discovery of Neptune. When Airy learnt of this he wrote to Adams saying

Let no one after this blame Challis.
Once the orbit of Neptune was computed it was seen that both Adams and Le Verrier had been quite lucky with their predictions. Both had predicted positions which were very close to the actual position but both had predicted orbits which meant that Neptune would only be close to its predicted position around 1840-1850 while at other times (it takes about 165 years to complete one orbit and has not yet completed one since its discovery) it would be far from the positions predicted by both Adams and Le Verrier.

The solar system did not, however, behave as expected. Neptune did not follow the orbit computed, even after taking the gravitational attraction of all the other known planets into account. To a lesser extent neither did Uranus and Saturn. Percival Lowell (1855-1916), an American astronomer, was interested in Mars. He built a private observatory at Flagstaff, Arizona specifically to study the planet. He began a mathematical analysis of the orbit of Uranus which was known more accurately than that of Neptune and yet failed to follow its predicted path. In 1905 Lowell completed his analysis of the data and predicted the existence of a planet beyond Neptune which was responsible for the perturbations.

By 1905, of course, astronomical observations had greatly improved due to photography. A search was begun at the Flagstaff Observatory in 1915 and for two years they photographed the area of the sky in which "Planet X", as Lowell called it, was predicted. Nothing was found. Lowell redid his mathematical analysis and, between 1914 and 1916, he again photographed the area of the sky where his predictions showed that Planet X would lie. In fact there are images of Pluto (Lowell's Planet X) on these plates but they are faint and were not recognised.

Lowell presented has paper Memoir on a Trans-Neptunian Planet to the American Academy on 13 January 1915. He wrote to his chief observer at Flagstaff:-

I am giving my work before the Academy on January 13. It would be thoughtful of you to announce discovery at the same time.
No discovery was made however and, as Lowell's brother later wrote:-
That X was not found was the sharpest disappointment of his life.
Another American astronomer, William Henry Pickering (1858-1938), actually constructed Lowell's Flagstaff Observatory in 1894. He moved to the Harvard College Observatory and, in 1919, he also predicted a position of a trans-Neptunian planet using the discrepancies in both the orbits of Uranus and Neptune as data. A search of photographs taken at the Mount Wilson Observatory failed to find the planet at the position predicted by Pickering.

In January 1929 Clyde W Tombaugh (1906- ) joined the staff at Flagstaff, with the task of finding Planet X. He used a new technique, namely comparing two plates taken some time apart by "blinking", that is shining a light successively through one plate and then the other so that objects on both plates remained steady while an object on one plate but not the other blinked. Tombaugh wrote:-

... on the afternoon of February 18, 1930, I suddenly came upon the images of Pluto! The experience was an intense thrill, because the nature of the object was apparent at first sight.
The planet was photographed every night from then on to confirm the observation and on 13 March 1930, the 75th anniversary of Lowell's birth and the 149th anniversary of Uranus's discovery (it is a remarkable coincidence that these should be the same day), an announcement was made from Flagstaff. In May 1930 the Flagstaff Observatory proposed the name Pluto for Planet X. They proposed a symbol consisting of interlocking letters P and L. It is interesting after the arguments about the naming of Neptune, that they managed to work Percival Lowell's initials into the planet name in such a major way.

This looked like another fantastic piece of mathematical theory by Lowell. However Brown reviewed the data which Lowell had used and showed that there was no way that he could have made the correct prediction based on the data. Russell, a leading American astronomer, wrote:-

The question arises ... why is there an actual planet moving in an orbit which is so uncannily like the one predicted?... There seems no escape from the conclusion that this is a matter of chance. That so close a set of chance coincidences should occur is almost incredible, but the evidence employed by Brown permits of no other conclusion...
The mass of Pluto is now known accurately since a satellite Charon has been discovered. Pluto's mass is 0.002 Earth masses, while Lowell required Planet X to have seven Earth masses to produce the effects on the other planets. The mystery remains as to how Lowell was able to predict the orbit so accurately and the mystery also remains as to why the outer planets fail to keep to their correct orbits.

Article by: J J O'Connor and E F Robertson

September 1996

MacTutor History of Mathematics