Great Calculations: A Surprising Look Behind 50 Scientific Inquiries

by Colin Pask

  The calculations for the orbit of this comet became a challenge for the leading astronomers of the time. The most detailed calculations were made by Alexis-Claude Clairaut (see Wilson's appendix to Waff's paper and the book by Grier). Following Newton's work on planetary masses, Clairaut knew that Jupiter and Saturn directly cause perturbations to the comet's orbit, and they also cause variations in the sun's position, which then makes tiny changes in the comet's motion. The perturbations caused by Jupiter and Saturn were extremely difficult to handle, and this was one of the first times that extensive numerical integrations were performed. The extent and the complexity of the problem meant that a new approach to calculations was required, and Clairaut formed a team of calculators with the astronomer Joseph Jérôme Lefrançois de Lalande and Nicole-Reine Étable de Labrière Lepaute, the wife of a noted clockmaker. Clairaut divided the task of moving around the comet's orbit into a series of short steps so that each of them could manage one part of the calculation. For six months, the three of them calculated, often from morning till night. These truly were heroic calculations. Poor Madame Lepaute even missed out on a tribute to her written by Clairaut in his initial report—a jealous lady friend insisted that he remove it.

  Clairaut's calculations suggested that the comet would reach its perihelion (closest point to the sun) in mid-April 1759, although there was some uncertainty, perhaps as much as a month, associated with that. As it happened, the comet reached its perihelion on March 13th. Given the complexity of the problem and incomplete knowledge of the solar system at the time, that was surely a remarkable agreement between theory and experiment. Of course there were great rivalries leading people like Jean Baptiste le Rond d'Alembert to belittle Clairaut's work (see the article by Waff), but today we can only wonder at the dedication and persistence shown by Clairaut and his team.

  The validity of Newton's theory of universal gravitation was superbly demonstrated with this remarkable meeting of theory and observation. This is how it was announced on April 25, 1759, at the public assembly of the Paris Academy of Sciences:

  The Universe sees this year the most satisfying phenomenon that Astronomy has ever offered us; unique event up to this day, it changes our doubts into certainty, and our hypotheses into demonstrations.7

  It was at that meeting that the comet was also named Halley's Comet. The appearance of a comet exactly as predicted was surely the most wonderful and convincing evidence that Newton had truly answered Kepler's call and had found the underlying theory for astronomy. The epic struggles involved gave rise to my calculation 17, predicting the return of Halley's Comet.

  There have been three appearances of Halley's Comet since that wonderful event in 1759; it was carefully tracked in 1835, 1910, and 1986. The calculations for the 1835 return halved the error in Clairaut's great work, and by 1986 the error was measured in hours. (The stories behind all the return calculations are well told by Grier.) My own sense of wonder was still there in 1986 when I too saw a returning Halley's Comet and marveled that a theory could predict a long-term celestial event with such amazing accuracy.

  6.4 A SECOND CONFIRMING TRIUMPH

  In section 5.2, we saw Kepler struggling to explain why there were just the six planets: Mercury, Venus, Earth, Mars, Jupiter, and Saturn. He was not to know that in 1781 the game would be changed by a musician and amateur astronomer.

  William Herschel (1738–1822) was a German musician who moved to England in 1757. Apart from music, his interests included languages, mathematics, and astronomy. He built his own telescopes and established an observatory with the help of his sister Caroline. On March 13, 1781, Herschel observed what he thought was a curious nebulous star or maybe a comet. Future observations showed that its position changed and so it was deemed to be a comet, although one with strange features such as no tail. Observational evidence by Herschel and other astronomers finally led to the opinion that Herschel had stumbled on a seventh planet.

  There was some debate about a name for the planet. Herschel favored naming it after King George III, and the French wanted it named Herschel, after its discoverer. Eventually mythology prevailed, and the name Uranus was chosen. (Uranus was the god of the sky, husband of earth, father of Saturn, and so on.)

  Even more debate ensued when astronomers began calculating the orbit of the new planet. Gradually a set of observations was established, and a search through the astronomical records revealed that in fact Uranus had been observed by John Flamsteed in 1690, 1712, and 1715; by Pierre Charles Lemonnier in 1750, in 1764, twice in 1768, and six times in 1769; by James Bradley in 1756; and by Tobias Mayer in 1756. Thus several reliable, accurate sightings of Uranus were available, although, of course, none of the observers realized at the time that they were recording the position of a seventh planet.

  A problem arose: the tables calculated on the basis of the recent observations did not fit those older ones. In 1820, a former assistant to the great Pierre-Simon Laplace, Alexis Bouvard (1767–1843), produced comprehensive tables and suggested that the older observations had large errors attached to them. However, it was later discovered that the suggested errors were up to ten times larger than those experienced observers were believed to have made. Even worse, within a few years, there were new observations of Uranus that did not fit in with Bouvard's tables.

  Astronomers casting around for explanations of Uranus's strange behavior came up with five possibilities. The first three—the Cartesian hypothesis of a cosmic fluid, the existence of a massive satellite around Uranus, and the occurrence of a catastrophic effect like the collision with a comet—were all easily discredited. The fourth possibility was more scientific and interesting: Did the law of gravity (see equation (6.1)) change at the very great distance from the sun where Uranus was located? This possibility had been canvassed before when calculations failed to match observations, but each time it was found that the calculations needed amending or correcting, and Newton's law of gravity reigned supreme.

  It was the fifth possibility that seemed to be the answer: maybe there was another massive planet beyond Uranus, and it was perturbing the new planet's orbit so that the standard orbit theory used when constructing tables needed correction. (Clairaut had speculated about very distant planets when he considered effects on Halley's Comet.) Here was one of the great challenges to mathematicians working in astronomy.

  (To read more about this episode and its aftermath consult the references to Grosser, Hanson, Linton, Morando, and Smith.)

  6.4.1 Aspects of the Calculations

  It is worth pausing for a moment to consider the mathematical problems involved. First, there was the question of tables for Uranus; this required finding orbital parameters to fit observations and then computing new data to guide future observations. This was a critical step in revealing the new physical problem—observations just did not fit the conventionally calculated orbit.

  If the theoretical tables and observational data were accepted and the discrepancies were due to the influence of an unknown planet, how was that planet to be located? Newton had already explained in his Principia that even though planets were much less massive than the sun, their effects in perturbing each other's orbits could be substantial. Thus the branch of applied mathematics known as perturbation theory was to be used (see Linton).

  However, for the case of Uranus, there was a twist to the problem: the orbit perturbations for Uranus were not due to some known planet (although those due to Jupiter and Saturn had been considered), but were instead to be used to find the properties of a new planet. Instead of asking what perturbations a particular planet caused, the perturbations were now assumed, and the planet causing them was to be found. This is another classic inverse problem (as discussed in section 4.4.5), with all the difficulties associated with such problems.

  Assumptions would need to be made about the plane of the unknown planet's orbit and its mass, as well as about some geometrical details. One way to start was to use the curious Titius-Bode law, which is perhaps be
tter characterized as an empirical formula since it has no real scientific basis. This “law” describes orbit sizes according to

  distance from the sun = 4 + 3 × 2n.

  The earth's distance is taken as 10, corresponding to n = 1. Venus has n = 0, and for Mercury the distance is taken as 4. Mars, Jupiter, Saturn, and Uranus correspond to n = 2, 4, 5, and 6, respectively. (The gap at n = 3 was filled by the discovery of asteroids.) The predicted values are surprisingly accurate (the error for Uranus is about 2 percent), and so n = 7 could be used to suggest a possible orbit size for the unknown perturbing planet.

  Finding the origin of the variations in Uranus's orbit was no easy problem, and it became the outstanding mathematical challenge of the time.

  6.4.2 Victory to Leverrier

  Urbain-Jean-Joseph Leverrier (1811–1877) was a highly talented French mathematician and astronomer who gained experience in perturbation theory early in his career. He published three papers on his work on the orbit of Uranus. In the first two (November 1845 and June 1846), he discussed the perturbations due to Jupiter and Saturn and the remaining discrepancy with Bouvard's tables and made his conclusion:

  I have demonstrated a formal incompatibility between the observations of Uranus and the hypothesis that this planet is subject only to the actions of the Sun and of other planets in accordance with the principle of universal gravitation.8

  In his third paper (August 1846) Leverrier announced the new, perturbing planet's mass and orbital elements. It was now up to the observational astronomers. There seems to have been little response in France and Leverrier communicated with his friend Johann Gottfried Galle at the Berlin Observatory. Galle, with the assistance of an astronomy student, Heinrich d'Arrest, began the search on September 23rd, the day he received Leverrier's letter. Searching the sky in the area nominated by Leverrier and comparing star charts, it took Galle and d'Arrest about half an hour to locate the eighth planet in the solar system. Other astronomers later confirmed the sighting, and a disc was observed, rather than a twinkling point of light, verifying that the object was indeed a planet.

  Here was an enormous triumph for French science (and German astronomers) and another victory for Newton's theory of gravitation. After the usual debate, the planet was given the name Neptune. Leverrier was showered with honors, and, in 1854, he became head of the Paris Observatory.

  6.4.3 Another Side to the Story

  Leverrier and Galle's discovery of Neptune is one of science's great stories, but there is another series of events that make it also one of the most fascinating and sensational events in the history of science.

  John Couch Adams (1819–1892) graduated from Cambridge University as the Senior Wrangler (top student) in the mathematics class of 1843. He had learned about the problems generated by Uranus and the unknown planet hypothesis and resolved to work on a perturbation theory for predicting the nature of the unknown planet as soon as he had graduated from Cambridge and become a fellow of St John's College. Adams was helped by James Challis, the Plumian Professor of Astronomy at Cambridge and head of the university's observatory. After working over the summers of 1843 and 1845, Adams had determined the characteristics required of the new planet and predicted its position.

  But now comes the big difference between Adams and Leverrier: Adams did not publish his findings. Knowing Adams's results, Challis corresponded with Astronomer Royal George Biddell Airy, and, at Challis's suggestion, Adams famously went to call on Airy in September only to find him away in France. Adams tried to see Airy again on October 21st, first to find him out and then later in the day to be told Airy could not be disturbed. Adams left a note (Grosser gives the details), but Airy was doubtful about the whole business, and no decisive action was taken. In fact Airy had written that

  With respect to the errors of the tables of Uranus…if it be the effect of any unseen body, it will be nearly impossible to ever find out its place.9

  In short, Airy and Challis procrastinated while Leverrier and Galle gained the glory.

  There is a long and complex story involving Adams, Challis, and Airy, which resulted in little action being taken to use Adams's predictions in a search for the unknown planet—particularly until Airy received a letter from Leverrier that showed him Adams was indeed on the right track. Poor Adams apparently prepared a paper to be read at the Southampton British Association for the Advancement of Science in September 1845, but he arrived a day too late for the physical-sciences sessions. The result was that England lost the glory to France, and there was an almighty row in England when all was revealed. Airy was scorned, and the effects lasted until his death in 1890 when thoughts of burial in Westminster Abbey were overturned by memories of his part in the Neptune debacle.

  There was a further aspect to this story. Eventually Adams's work was made known, and it was suggested that he should share the credit with Leverrier. It is not hard to imagine the uproar and the vicious exchanges in the now-declining relationship between Britain and France. Figure 6.4 is a French cartoon showing how Adams discovered Neptune by spying on Leverrier across the English Channel. (See references, especially Smith and Grosser, for the full story.) Scientists and scientific societies at the highest level became embroiled in the controversy. Eventually the whole matter was resolved amicably; Adams was also honored, and Leverrier and Adams even became friends.

  Figure 6.4. French cartoon showing Adams discovering Neptune by spying on Leverrier's work. From L'Illustration (November 7, 1846).

  6.4.4 Planet Neptune

  The discovery of a new planet by means of calculations was a triumph for mathematical astronomers and provided yet more evidence for the validity of Newton's universal theory of gravity. An essential addition to my list is calculation 18, the discovery of Neptune.

  Before leaving this topic, I turn to a question many readers may be asking: Given the difficulty of the calculations, how good were Leverrier's and Adams's results for the orbit of the unknown planet Neptune? The answer is summarized in figure 6.5.

  Figure 6.5. The orbits of Uranus and Neptune, and the predictions of Leverrier and Adams. Reprinted with permission, © Cambridge University Press, from C. M. Linton, From Eudoxus to Einstein: A Mathematical History of Astronomy (Cambridge: Cambridge University Press, 2004).

  Clearly the predicted orbits are reasonable (given the technical difficulties) for the years around 1840, and both indicate roughly where Neptune should be found. For the rest of the orbit the results are not too good. However, we must remember that Neptune takes almost 165 years to make one revolution in its orbit, so maybe we should expect the results to be reasonable only in the time period covered in the discovery story.

  There is one final twist! The nineteenth century saw the growth of astronomy in America, and, naturally, interest in the discovery of Neptune was as great there as elsewhere. Analysis by Sears Cook Walker, supported by Harvard mathematician Benjamin Peirce, gave rise to the American view that the discovery of Neptune was a “happy accident” and the orbital parameters were quite wrong. Needless to say, there was a new row, this time with America confronting a united Europe. To delve further I recommend the fascinating paper by Kent and the book by Grosser.

  6.5 THE SIZE OF THE SOLAR SYSTEM

  Over the centuries, accurate measurements were made of the position of the moon and of the planets and the periods of their motions. The fixed stars provide a background for precise angular measurements. However, determining the absolute size of the solar system proved to be extremely difficult, and a great variety of mostly quite inaccurate values were obtained (see Van Helden's book for a detailed historical account). A major step forward came with the discovery of Kepler's third law (see the previous chapter) which tells us that a planet's period T and orbit size a satisfy

  If we measure T in earth years and a in terms of the Earth-sun distance (the astronomical unit, AU), then equation (6.2 (a)) for the Earth tells us that the constant is one. Thus for any other planet

  a = T2/3 AU if T is
measured in Earth years. (6.2 (b))

  For example, the period of Venus is 0.614 Earth years so it has an orbit size of 0.6142/3 = 0.72AU.

  The relative size of the solar system is now established, but the absolute size is still unknown. How do we find the value of an astronomical unit, AU, the Earth-sun distance?

  Many people, including the eminent astronomers Giovanni Cassini and John Flamsteed, relied on a measurement of the parallax of Mars, but there are difficulties in that method, and the result for the AU was not reliable.

  In the second century BCE, Hipparchus had used data from a solar eclipse to find a reasonable value for the Earth-moon distance in terms of the earth's radius (see Linton and Van Helden), but his value for the Earth-sun distance was quite inaccurate. An eclipse of the sun occurs when the moon passes between the earth and the sun, and after Copernicus's change to a heliocentric view of the solar system, it was apparent that two planets, the interior planets, also intervene between the earth and the sun. Could that be used to measure the AU?

  (An aside on notation: astronomers really wished to measure the angle subtended at the sun by the earth's diameter, and half that angle is called the solar parallax. Call it β. Since β is small, to a good approximation, the Earth-sun distance, one AU, is given by Re /β if β is measured in radians, or 180Re/πβ if β is measured in degrees. The modern value of β is 8.797". Note that this is a small angle and thus presents measurement difficulties.)