How large were the deviations of Uranus's positon from predicted which led to the discovery of Neptune?

How large were the deviations of Uranus's positon from predicted which led to the discovery of Neptune?

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After Uranus was discovered and its orbit calculated, its future orbit was calculated, and its future positions as seen from Earth were calculated.

And observers of Uranus began to notice that Uranus was deviating from its calculated positions. Eventually the possibility that an undiscovered planet was perturbing the orbit of Uranus was used by Adams and Le Verrier to calculate the orbit of Neptune and led to the discovery of Neptune.

So how large were the deviations in the apparent position of Uranus that were used to calculate the orbit and apparent position of Neptune?

I also asked this question at the History of Science and Mathematics This Site and got two answers so far.

The answer by Ben Crowell makes a rough calculation that the displacement should be on the order of ten arc seconds. It also has a plot of the variation in longitude of the observed and predicted positions of Uranus over time, which seem to differ by up to a few tens of arc seconds from predicted positions.

The answer by cktai states that according to his source the deviation between the calculated and the observed orbit was 30 arc seconds in 1835 and 70 arc seconds by 1840.

An arc second is one in 1,296,000, or 0.0000007, of a full circle.

At the average distance between the Sun and Uranus, about 19.22 Astronomical units or AU, a full circle is about 120.76271 AU. Since the orbital period of Uranus is 84.01 years, Uranus travels at an orbital speed of about 1.4374801 AU per year, or about 0.0039356 AU per day, or about 0.0001639 AU per hour, or about 0.0000027 AU per minute, or about 0.000000045 AU per second.

0.000000045 AU is 6.81433 kilometers, so the average orbital speed of Uranus should be about 6.81433 kilometers per second. The average orbital speed of Uranus is listed as 6.8 kilometers per second.

Earth is at an average distance of 1.0 Astronomical Units from the Sun, so the distance between Earth and Uranus varies between about 17.33 and 21.11 AU, and the average distance is about 19.22 AU.

So since there are 360 degrees in a full circle, at the average distance between Earth and Uranus a degree of arc would be about 0.33545 AU, and an arc minute would be about 0.00559 AU, and an arc second would be about 0.00009318 AU or 13,939.61 kilometers.

Being off position by 30 arc seconds would equal being off position by 418,188.38 kilometers, and being off position by 70 arc seconds would equal being off position by 975,772.88 kilometers. And that would be approximately the distance traveled by Uranus in about a day.

And I guess that gives some idea of the scale of the deviations which were considered a problem and which led to the discovery of Neptune.

Who Predicted the Existence of the Planet Neptune?

The tale of the discovery of Neptune, the eighth planet in the solar system (Figure 1) reads like a detective story.

Figure 1. Neptune as seen with the Hubble Space Telescope (from:

The planet Uranus was discovered by astronomer William Herschel in 1781. Almost immediately, astronomers started to see deviations of its orbit from predictions made by Newton's theory of gravitation. By the early 1840s, these deviations became substantial, leading a few astronomers to speculate that Uranus's orbit might be perturbed by the presence of an unseen planet (alternatively, Newtonian gravity would have to be modified at large separations). In some sense, this was the first suggestion for "dark matter" -- mass not detected through its light, but rather through its gravitational effects.

On June 1, 1846, the French mathematician Urbain Le Verrier published a calculation that explained the discrepancies by predicting the existence of a transuranian planet, and its expected location in the sky. Based on his prediction, the German astronomer Johann Galle and his student, Heinrich Louis d'Arrest, discovered Neptune just after midnight on the night of September 23, 1846. The new planet's celestial longitude was just about one degree away from Le Verrier's prediction! The excited Galle informed Le Verrier: "the planet whose place you have really exists."

If the story would have ended here, this would have simply been a fascinating story of how sciences progress through theoretical predictions and observational verifications. This is, however, where the conventional plot started to thicken. According to the commonly told version, as soon as news of the discovery reached England, George Biddell Airy, the British Astronomer Royal at the time, realized that he had seen a similar prediction in the fall of 1845 on a note left at his house by a little-known English mathematician named John Couch Adams.

Adams had indeed been doing calculations for several years concerning the potential location of a new planet, and he briefly communicated his results to Airy and to James Challis, director of the Cambridge Observatory. However, the fact is, that the materials Adams provided to Challis and Airy were insufficient to convince the two to initiate an aggressive observational search. Only after Le Verrier's prediction became known in England did Challis make an unsuccessful attempt to search for the planet suggested by Adams' calculations. Nevertheless, following the announcement of the discovery of Neptune, a controversy arose between the English and French astronomical communities, with the English astronomers claiming that Le Verrier and Adams should share the credit for the prediction. The French initially reacted with suspicion, and the magazine L'Illustration even published a cartoon depicting Adams "discovering" Neptune in Le Verrier's manuscript (Figure 2). The controversy eventually subsided after Airy presented certain documents which supposedly demonstrated that Adams had indeed produced predictions deviating from those of Le Verrier by only about one degree. The emerging consensus was, therefore, that Le Verrier and Adams should both be credited with predicting Neptune.

Figure 2. A cartoon that appeared in the November 7, 1846 issue of L'Illustration, showing Adams looking for Neptune in vain, and then "discovering" it in Le Verrier's notebook. From:

Amazingly, the story did not end there. Modern science historians, such as Neptune scholar Dennis Rawlins and astronomy historian Robert Smith, discovered that it was difficult to examine the details of the discovery of Neptune since the "Neptune file" containing Adams' correspondence with Airy had gone missing from the library in the 1960s. In 1999, following the death of astronomer Olin Eggen, the file was discovered in his office in Chile, even though Eggen had previously denied having it. Eggen had served as chief assistant to the Astronomer Royal in Britain in the early 1960s, and apparently he was given the file -- or he "borrowed" it (as well as other rare books) -- for some research he was doing on Airy's work. He then took the file with him to Australia and eventually to Chile.

Historian of science Nicholas Kollerstrom and his colleagues William Sheehan and Craig Woff have examined the file in detail, and in their view "the achievement [of predicting Neptune] was Le Verrier's alone." They based their conclusion on two main points. First, in successive calculations, Adams kept changing his prediction for the location of the putative new planet, once by as much as twenty degrees. Second, even though there is no doubt that Adams' calculations were similar in nature, and at some level even in accuracy, to those of Le Verrier (both miscalculated Neptune's distance from the Sun), Adams failed to convince his contemporaries to engage in an extensive search. Did, therefore, (in Kollerstrom's words) "the Brits steal Neptune"? Maybe this is too strong a statement, but they certainly presented the evidence in a way that boosted Adams' (and thereby Britain's) claim for credit.

Our Obsession With Hidden Planets Didn't Start With Planet Nine

A view from the hypothetical Planet Nine back toward Earth.

Caltech/R. Hurt (IPAC) via NASA

For most of modern astronomy, we've been convinced that there's more out there at the edges of our solar system than we could see. Sometimes planets don’t orbit the Sun in quite the way physics predicts, which means there could be some other object exerting a gravitational influence, and astronomers just haven’t found it yet. Or we could simply be a little off in our estimates of a planet’s mass, or not accounting for the complexities of interactions between smaller objects.

Both have happened several times over the last 250 years. Here’s a look:

An Eighth Planet

When astronomer Sir William Herschel discovered Uranus in 1781, it didn’t take long at all for astronomers to decide there must be more planets out there. By the late 1700s, astronomers had a pretty good grasp of orbital mechanics. Isaac Newton had published his laws of gravitation – which described how a planet’s movement is determined by the gravity of the Sun, as well as other planets – more than a century earlier, and so far the universe seemed to be playing by his rules. But Uranus didn’t quite follow the orbit Newton’s equations predicted, and its deviations seemed to hint that astronomers were missing a variable in their calculations: a whole planet.

In the early 1830s, British astronomer Thomas Hussey and French astronomer Alexis Bouvard began to consider the idea that there must be another planet out there whose gravity was perturbing Uranus' orbit. French mathematician Urbain LeVerrier was having the very same notion, and he calculated the mass of the planet and the orbit it would have to follow in order to cause the strangeness in Uranus’ orbit. And in 1846, German astronomer Johann Gottfried Galle discovered Neptune in exactly the spot LeVerrier had predicted.

Meanwhile, it looked like the discovery of Neptune still failed to completely explain Uranus’ orbit. In 1848, two years after the discovery of Neptune, French physicist Jacques Babinet published a paper arguing that LeVerrier’s predicted orbit didn't match Neptune's actual orbit. It was close enough to enable astronomers to find the ice giant, but not, Babinet claimed, quite right to cause Uranus’s perturbed orbit. Babinet concluded that another giant planet, which he took the liberty of naming Hyperion, must be out there waiting to be discovered. Hyperion would be 12 times as massive as Earth and orbit the Sun once every 336 years at a distance 48 times further than Earth's orbit.

In a sharp rebuttal, LeVerrier replied that astronomrs just didn’t have enough information to make such a detailed claim. But although LeVerrier thought Babinet was jumping to conclusions, he also suspected the existence of a ninth planet. Astronomers just needed more time to study Neptune's orbit in order to figure out what unseen neighbor's gravity might be affecting it. It was a hotly debated idea and met with some very blunt criticism, but astronomers in several countries spent the second half of the 19th century searching the skies for a ninth planet.

Brahma and Vishnu

In the early 1880s, astronomers looking for evidence of a ninth planet noticed two comets, 1862 III and 1889 III, whose orbital paths appeared to show the invisible gravitational influence of a trans-Neptunian planet, which might have stretched their orbits into ellipses. Astronomer George Forbes studied the orbits of several more comets and calculated that it would take not just one but two hidden planets to shepherd the comets onto their observed orbital paths. The masses and orbits Forbes predicted actually sound a lot like an early version of the Planet Nine that astronomers are currently searching for.

And Forbes wasn't alone. In 1911, Indian astronomer Venkatesh P. Ketakar proposed that Neptune, Uranus, and two hypothetical planets he named Brahma and Vishnu were in an orbital resonance. When two objects’ orbital periods are related by a ratio of two whole numbers – Jupiter's moons Io, Europa, and Ganymede are in a 1:2:4 resonance, for instance – their gravitational interactions can have a greater effect than usual.

Pluto and Planet X

But instead of two large planets, astronomers found one very small one: a dwarf planet, in fact. In 1906, astronomer Percival Lowell, of Martian canal fame, began the search for Planet X, his preferred version of the unseen body whose gravity was influencing the orbits of Neptune and Uranus. Thirteen years after Lowell’s death, astronomer Cylde Tombaugh announced on March 13, 1930, that he’d discovered a planet exactly where Lowell predicted Planet X would be.

But in 1978, astronomer James Christy discovered Pluto’s moon Charon. Watching their orbits, astronomers were finally able to calculate Pluto’s mass, and it turned out to be tiny – much too small to account for the discrepancies in Neptune and Uranus’ orbits. It took another 11 years to find the real Planet X: an imprecise estimate of Neptune’s mass. Voyager 2 collected much more precise measurements in 1989 when it flew past the ice giants on its way out of the solar system, and Neptune’s and Uranus’ orbits suddenly made sense.

Planet Nine and Beyond

But now some astronomers say there’s something fishy about the extremely elliptical orbits of dwarf planets in the Kuiper Belt, the distant, icy outer reaches of the solar system. There’s a lively debate in progress about whether a giant rocky planet is lurking out there somewhere or if the Kuiper belt objects’ gravitational interactions are enough to explain their unconventional orbits. Planet Nine – or Hyperion, if you’re old-fashioned – may be real, or it may turn out to be just a figment of physicists’ calculations.

Either way, it’s unlikely that astronomers will stop looking for new worlds. Ad it’s been a fruitful search so far, even if we haven’t discovered an estranged super-Earth in the Kuiper Belt or another gas giant just beyond Neptune. The search for a hidden planet has led to other important discoveries: dwarf planets and their moons, new details about the planets we already know and love, and a better understanding of how the solar system formed.

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 8 th 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 ] .

You can see Galileo's notes about this observation at THIS LINK.

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 19 th Century, recorded Neptune on the 8 th and 10 th of May 1795 without recognising that it was not a star. John Herschel, whom 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

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:-

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

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:-

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 amateur 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:-

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.

You can see a diagram showing the predicted and actual positions at THIS LINK.

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

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

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 8 th and 10 th 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

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.

A cartoon about the discovery of Neptune is at THIS LINK

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:-

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:-

The planet was photographed every night from then on to confirm the observation and on 13 March 1930 , the 75 th anniversary of Lowell's birth and the 149 th 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:-

Disclaimer: The following material is being kept online for archival purposes.

In 1821 Alexis Bouvard calculated and published the predicted motion of Uranus, but the planet deviated from the predicted orbit.

Two mathematicians tried independently to calculate the position of an unknown planet whose attraction might have led to the discrepancy: John Couch Adams , a Cambridge graduate trained in mathematical astronomy, and an astronomer at the Paris observatory, Urbain Le Verrier .

Adams made the first predictions, in 1845, but his communications with astronomers (and sometimes, his relations with them) were not consistent. He guessed the distance of Neptune from the Titius-Bode law and several times predicted where in the sky Neptune could be found. The Bode-Titius law unfortunately is a poor guide here--it suggested 38 AU, whereas the observed mean distance is near 30 AU. James Challis, of the Cambridge observatory, actually searched for the planet but could not find it. Perhaps Adams got less attention because he was young, not well established and a bit of a loner.

Le Verrier was a graduate of the Ecole Polytechnique and presented his prediction only in 1846. It was more accurate than the one by Adams, but again, astronomers in his own country were not keen to search for the planet. LeVerrier therefore sent his prediction to Johann Gottfried Galle , head of the Berlin observatory, who at once started looking. A student with Galle, Heinrich Louis d'Arrest (who apparently also lodged at the observatory), pointed out a sky map existing of the same region, and by comparing it to actual observations, Neptune was located that very same night.

A few weeks later William Lassell discovered a moon orbiting Neptune, and it was named Triton (after the mythological son of Poseidon-Neptune). It is quite big--diameter 2700 km. The orbits of planets and of almost all other moons, as well as the rotation of planets and of the Sun, are counter-clockwise when viewed from the northern side of the ecliptic. Triton however orbits in the opposite direction (as do 4 small moons of Jupiter and one of Saturn). It is therefore widely believed to be a captured minor planet, probably from the Kuiper belt lying beyond Neptune

Neptune itself is a gas planet very much like Uranus, with radius 3.83 times Earth's and a mean density 1.64 times that of water. Its visible disk contains mostly hydrogen and its rotation period is 16.11 hours. One big difference is the rotation axis, which is close to the perpendicuar to the ecliptic, making an angle 28.32 degrees, not too far from Earth's 23.5 degrees.

Voyager 2 passed Neptune on 25 August 1989 and observed a magnetic field, with "magnetic moment" about 25 times Earth's (for Uranus the number is about 50), but as with Uranus, the magnetic axis was also inclined steeply to the rotation axis, by 47°, making its precess rapidly around a cone.

Voyager's sensors observed Neptune to be blue like Uranus, but a different and much more intense shade of blue, and while the atmosphere of Uranus was bland, this one contained a large white cloud and a large dark spot near it, south of the equator. The Hubble telescope in 1994 no longer found that spot, but instead another one appeared north of the equator, about as big.

In addition to Triton, Neptune has 12 small moons and a narrow ring structure, which seem to change over time

Two last notes. First , modern scholars examining Galileo's drawing based on his telescope observations have noted that he did, in fact, observe Neptune, but marked it down as another distant star. The planet's motion was too slow to observe the difference between the two observations, and Galileo's telescope did not have enough resolution to make out the planetary disk.

Second , LeVerrier also calculated the motion of the planet Mercury and found a discrepancy, which renewed the search for an unknown planet "Vulcan" inside the orbit of Mercury. Such a planet was sought previously and not found, and it was never seen in total eclipses, yet the rotation of Mercury's perihelion (point closest to the Sun) was observed to exceed the calculated effect of all planets by 43 seconds of arc per century. This was explained in 1915 by Einstein's general theory of relativity,and is considered one of the crucial tests of that theory.

Next Stop (following "The Planets"): #9c The Discovery of the Solar System , from Copernicus to Galileo

What's ahead?

At 3:54 PM on the May 11, our moon will be as far from us as it will get this year at 252,595 miles. This is the opposite of a supermoon. The moon will appear smaller than normal, but you won&rsquot be able to see it anyway as it is also the new moon.

Planet Visibility Report: As May begins, both Saturn and Jupiter sit high in the southern sky in the predawn sky. Venus, Mercury, and Mars are all up immediately after sunset. While Venus and Mercury move higher for two weeks, Mercury then turns to dive back toward the sun while Venus moves ever higher in the evening sky. Mars stays roughly three hours behind the sun all month. New moon occurs on May 11, with the full moon following on May 26. That full moon also passes into Earth&rsquos shadow, creating a total lunar eclipse, but it starts at sunset for us in Oklahoma, so we miss virtually all of it.

The rediscovery of neptune

The methods used by Adams and Le Verrier for the prediction of the position of Neptune led to extremely extensive calculations, and the question still remains open whether any much simpler approarch is possible that could have led to comparably accurate prediction with much reduced effort of calculation. Such a method is presented. It turns out that the time of conjunction with Uranus can be derived with very little calculation using only elementary considerations applied solely to the (observed) discrepancies in heliocentric longitude. The instant so found is only about six months later than actual value. On the basis of a circular orbit for the unknown planet, its position at discovery (25 years later than conjunction) even using Bode's law for the size of orbit is then only some 13° away from the actual position, and in fact within the zone that Challis began to search, and so probably sufficient for ultimate discovery. However, the distance appropriate to a best-fit of the observations can also be found, and this makes a considerable improvement in the predicted position. By suitably combining the equations of condition, the number of unknowns associated with the necessary correction of the orbit of Uranus can be reduced from four to two, and with any assumed radius for the (circular) Neptune orbit only the mass of the planet is unknown. Thus the number of unknowns can be reduced to three (for each assumed mean distance) compared with eight in the original methods. Even with the extra work of calculation of the coefficients in the perturbations for a sufficient number of assumed mean distances, there results an enormous reduction in the amount of calculation. For the demonstration of the method, further economy of arithmetic has been achieved by making use only of part of the observational material that was actually available at the time (the so-called “modern” observations). Nevertheless the procedure succeeds in predicting a position of Neptune (in 1846) with the same accuracy (1°) as Le Verrier and more accurately than Adams ( 2 1 2 ° ).

How Einstein’s Relativity Saved The Solar System

“Learn from yesterday, live for today, hope for tomorrow. The important thing is not to stop questioning.” -Albert Einstein

For millennia, throughout most of human history, the planets and the Moon were the only keys to a changing Universe that we had. The stars and the Milky Way — night after night, year after year — appeared the same, or appeared to change so little, so rarely and so gradually that humanity never took note. A careful eye would note that the planets didn’t just “wander” from night-to-night, but moved in a predictable set of fashions, exhibiting both prograde and retrograde motions.

There were two main methods of explaining their apparent patterns in the sky:

  1. Either the planets moved in orbits given by equants, deferents and epicycles around the Earth,
  2. Or the planets moved around the Sun, with the Earth as just another one of those planets.

For nearly 2,000 years, that former interpretation was the one that held sway. But after Copernicus put forth the latter one in the 16th century, followed by the work of Galileo, Kepler, and finally Isaac Newton, the heliocentric model won out.

Newton’s advance was by far the biggest, because he not only set out to describe the behavior of these objects — that the planets moved about the Sun in ellipses, with the Sun at one focus — but because he added in a mechanism for that behavior: the law of universal gravitation. This law accounted for gravitation not only on Earth, but of all the heavenly bodies. It explained why moons orbit their parent planet, why comets recurred and were often perturbed by the other planets, why our world experiences tides, and why the planets don’t disturb one another and cause frequent ejections.

It also explained some more subtle effects, ones that took generations to notice.

If the Universe consisted only of two point masses — the Sun and a planet — the orbit of that planet would make a perfect, closed ellipse that returned the world to its starting location with each trip around the Sun. But in a Universe governed by Newtonian gravity, with a plethora of massive bodies in our Solar System, that ellipse will precess, or rotate slightly in its orbit. In the mid-1800s, orbital deviations of Uranus from its predicted motions led to the discovery of Neptune, as the outermost world’s gravitational influence accounted for the excess motion.

But in the inner Solar System, the nearest planet to the Sun, Mercury, was experiencing a similar problem.

With detailed, accurate observations going back to the late 1500s (thanks to Tycho Brahe), we could measure how Mercury’s perihelion — its closest orbital point to the Sun — was advancing. The number we came up with was 5,600″ per century, which is incredibly slow: just over 1.5 degrees over a 100 year period! But of that, 5025″ of that came from the precession of Earth’s equinoxes, a well-known phenomenon, while 532″ was due to Newtonian gravity.

But 5025 + 532 does not equal 5600 it comes up short by a small but significant amount. The big question, of course, is why.

There were, of course, a number of possibilities put forth.

  1. Perhaps the data was wrong an error of less than one percent hardly seems like a reason to panic. And yet, the errors at the time were less than 0.2%, meaning the data was significant.
  2. Perhaps there was an extra inner planet, one even interior to Mercury. This explanation was put forth by Urbain Le Verrier, the scientist who predicted the existence of Neptune. Yet after an exhaustive search, including of modifications to the Sun’s corona, no planet was found.
  3. Or perhaps the Newtonian force law needed a slight tweak. Rather than an inverse square law, it’s conceivable that there was a tiny, extra force: perhaps instead of the power of 2, the power law was 2.0000000(something), an explanation put forth by Simon Newcomb and Asaph Hall.

But none of these panned out none of them were satisfying. Moreover, the latter option — despite being conceivable as the explanation for this orbit — didn’t give a predictive “a-ha” that one could use to look for something else to validate or falsify it.

But after Einstein’s special theory of relativity came out in 1905, Henri Poincare showed that the phenomena of length contraction and time dilation contributed a fraction — between 15–25% — of the needed amount towards the solution, dependent on the error. That, plus Minkowski’s formalization of space and time as not separate entities, but as a single structure bound together by their union, spacetime, led Einstein to develop the general theory of relativity.

On November 25, 1915, he presented his results, computing the spectacular figure that the contribution of the extra curvature of space predicted an additional precession of 43″ per century, exactly the right figure needed to explain this observation.

The shockwaves through the astronomy and physics communities were tremendous. Less than two months after this, Karl Schwarzschild found an exact solution, predicting the existence of black holes. The deflection of starlight and gravitational redshifts/blueshifts were realized as possible tests, and finally the solar eclipse of 1919 validated general relativity as superseding Newtonian gravity.

And in the century since, its predictions, from gravitational lensing to frame dragging to orbital decay and more, have all been validated. Never once have observations contradicted the theory, and today, we celebrate 100 years of Einstein’s greatest accomplishment. A century later, improved observations and understanding of the Solar System have validated the precession of Mercury’s perihelion down to the precision of hundredths of arc-seconds per century, with uncertainties continuing to drop on both the theoretical and the observational fronts.

Who knows what new discoveries, or what new, novel possibilities, the next 100 years will hold?

How did Le Verrier calculate Neptune's position?

In the Wikipdia article on Neptune the discovery is described as a mathematical achievement:

Subsequent observations revealed substantial deviations from the tables, leading Bouvard to hypothesize that an unknown body was perturbing the orbit through gravitational interaction. In 1843, John Couch Adams began work on the orbit of Uranus using the data he had. Via Cambridge Observatory director James Challis, he requested extra data from Sir George Airy, the Astronomer Royal, who supplied it in February 1844. Adams continued to work in 1845–46 and produced several different estimates of a new planet. [. ] In 1845–46, Urbain Le Verrier, independently of Adams, developed his own calculations [. ]. Neptune was discovered within 1° of where Le Verrier had predicted it to be, and about 12° from Adams' prediction.

Q1: Which method did Le Verrier employ to calculate Neptune's position with such accuracy?

Q2: How would it be done with today's tools?

Edit: Springer has the chapter in the answer available for download.

Edit 2: since the work of Le Verrier seems beyond the scope of a stackexchange question, is it possible to explain the general approach?

Quote from the article above:

Jean-Baptiste Biot attempted to explain Le Verrier's methods in six papers in *Journal des Savants (October 1846, pp. 577–596 November 1846, pp. 641–664 December 1846, pp. 750–768 January 1847, pp. 18–35 February 1847, pp. 65–86 March 1847, pp. 182–187). Arrived at the third paper, he writes: “As I progress in the task I have undertaken, the difficulty of the subject seems to increase.”

Neptune Completes Its First Circuit Around The Sun Since Its Discovery in 1846

On July 12, 2011, Neptune arrived at the same location in space where it was discovered nearly 165 years ago. To commemorate the event, NASA's Hubble Space Telescope took these "anniversary pictures" of the blue-green giant planet.

Neptune is the most distant major planet in our solar system. German astronomer Johann Galle discovered the planet on September 23, 1846. At the time, the discovery doubled the size of the known solar system. The planet is 2.8 billion miles (4.5 billion kilometers) from the Sun, 30 times farther than Earth. Under the Sun's weak pull at that distance, Neptune plods along in its huge orbit, slowly completing one revolution approximately every 165 years.

These four Hubble images of Neptune were taken with the Wide Field Camera 3 on June 25-26, during the planet's 16-hour rotation. The snapshots were taken at roughly four-hour intervals, offering a full view of the planet. The images reveal high-altitude clouds in the northern and southern hemispheres. The clouds are composed of methane ice crystals.

The giant planet experiences seasons just as Earth does, because it is tilted 29 degrees, similar to Earth's 23-degree-tilt. Instead of lasting a few months, each of Neptune's seasons continues for about 40 years.

The snapshots show that Neptune has more clouds than a few years ago, when most of the clouds were in the southern hemisphere. These Hubble views reveal that the cloud activity is shifting to the northern hemisphere. It is early summer in the southern hemisphere and winter in the northern hemisphere.

In the Hubble images, absorption of red light by methane in Neptune's atmosphere gives the planet its distinctive aqua color. The clouds are tinted pink because they are reflecting near-infrared light.

A faint, dark band near the bottom of the southern hemisphere is probably caused by a decrease in the hazes in the atmosphere that scatter blue light. The band was imaged by NASA's Voyager 2 spacecraft in 1989, and may be tied to circumpolar circulation created by high-velocity winds in that region.

The temperature difference between Neptune's strong internal heat source and its frigid cloud tops, about minus 260 degrees Fahrenheit, might trigger instabilities in the atmosphere that drive large-scale weather changes.

Neptune has an intriguing history. It was Uranus that led astronomers to Neptune. Uranus, the seventh planet from the Sun, is Neptune's inner neighbor. British astronomer Sir William Herschel and his sister Caroline found Uranus in 1781, 55 years before Neptune was spotted. Shortly after the discovery, Herschel noticed that the orbit of Uranus did not match the predictions of Newton's theory of gravity. Studying Uranus in 1821, French astronomer Alexis Bouvard speculated that another planet was tugging on the giant planet, altering its motion.

Twenty years later, Urbain Le Verrier of France and John Couch Adams of England, who were mathematicians and astronomers, independently predicted the location of the mystery planet by measuring how the gravity of a hypothetical unseen object could affect Uranus's path. Le Verrier sent a note describing his predicted location of the new planet to the German astronomer Johann Gottfried Galle at the Berlin Observatory. Over the course of two nights in 1846, Galle found and identified Neptune as a planet, less than a degree from Le Verrier's predicted position. The discovery was hailed as a major success for Newton's theory of gravity and the understanding of the universe.

Galle was not the first to see Neptune. In December 1612, while observing Jupiter and its moons with his handmade telescope, astronomer Galileo Galilei recorded Neptune in his notebook, but as a star. More than a month later, in January 1613, he noted that the "star" appeared to have moved relative to other stars. But Galileo never identified Neptune as a planet, and apparently did not follow up those observations, so he failed to be credited with the discovery.

Neptune is not visible to the naked eye, but may be seen in binoculars or a small telescope. Today it can be found in the constellation Aquarius, close to the boundary with Capricorn.