“The famous outing to the White Mountains was often the subject of much amusement at the dinner parties when Dr. Lowell and Judge Doe were both there. In later years that famous expedition seemed to be an inexhaustible source of fun—the voracious mosquitoes, the discomforts of a camp and beds under water, atrocious coffee, and so on!!
“And this reminds me of many dinner parties on Dr. Lowell’s and Judge Doe’s birthdays. These were jolly gatherings, and the brilliant repartee passing between Dr. Lowell and the Judge was a great delight to those who were present.
“Many things about the place often remind me of the intensely busy days before Dr. Lowell passed away. There were several excursions for his tree studies, to Sycamore Canyon, an arduous trip, and to other localities near Flagstaff for further studies of different species of junipers in their native habitat. The specimens were carefully sorted and packed for Professor Sargent of the Arnold Arboretum. Then I remember helping him plant many bulbs on the last two days before he was fatally stricken. The squills he planted at that time in the little bed under the oak tree near the entrance of the B. M. return every spring.”[34]
Ever inquiring, ever fertile, his mind turned to seek the explanation of divers astronomical phenomena. In 1912, for example, under the title “Precession and the Pyramids,” we find him discussing in the Popular Science Monthly the pyramid of Cheops as an astronomical observatory, with its relation to the position of the star then nearest to the North Pole, its lines of light and shadow, in a great gallery constructed with the object of recording the exact changes in the seasons.
But leaving aside these lesser interests, and the unbroken systematic observation of the planets, his attention in the later years of his life was chiefly occupied by two subjects, not unconnected, but which may be described separately. They are, first, the influence over each other’s position and orbits of two bodies, both revolving about a far larger one; and, second, the search for an outer planet beyond the path of Neptune. Each of these studies involved the use of mathematics with expanding series of equations which no one had better attempt to follow unless he is fresh and fluent in such forms of expression. For accurate and quantitative results they are absolutely essential, but an impression of what he was striving to do may be given without them.
Two bodies revolving about a common centre at different distances, and therefore different rates of revolution, will sometimes be on the same side of the central body, and thus nearer together; sometimes on opposite sides, when they will be much farther apart. Now it is clear that the attraction of gravity, being inversely as the square of the distance, will be greatest when they are nearest together; and if this happens at the same point in their orbits every time they approach each other the effect will be cumulative, and in the aggregate much larger than if they approach at different parts of their orbits and hence pull each other sometimes in one direction and sometimes in another. To use a homely, and not altogether apt, illustration: If a man, starting from his front door, walk every day across his front lawn in the same track he will soon make a beaten path and wear the grass away. If, instead, he walk by this path only every other day and on the alternate days by another, he will make two paths, neither of which will be so much worn. If he walk by three tracks in succession the paths will be still less worn; and if he never walk twice in the same place the effect on the grass will be imperceptible.
Now, if the period taken by the outer body to complete its orbit be just twice as long as that taken by the inner, they will not come close together again until the outer one has gone round once to the inner one’s twice, and they will always approach at the same point in their orbits. Hence the effects on each other will be greatest. If the outer one take just two turns while the inner takes three they will approach again only at the same point, but less frequently; so that the pull will be always the same, but repeated less often. This will be clearly true whenever the rates of the revolution differ by unity: e.g., 1 to 2, 2 to 3, 3 to 4, 4 to 5, etc.
Take another case where the periods differ by two; for example, where the inner body revolves about the central one three times while the outer one does so once; in that case the inner one will catch up with the outer when the latter has completed half a revolution and the inner one and a half; and again when the outer has completed one whole revolution and the inner three. In this case there will be two strong pulls on opposite sides of the orbits, and, as these pulls are not the same, the total effect will be less than if there were only one pull in one direction. This is true whenever the periods of revolution differ by two, e.g., 1 to 3, 3 to 5, 5 to 7. If the periods differ by three the two bodies will approach three times,—once at the starting point, then one third way round, and again two thirds way round, before they reach the starting point; three different pulls clearly less effective.
In cases like these, where the two bodies approach in only a limited number of places in their orbits the two periods of revolution are called commensurate, because their ratio is expressed by a simple fraction. The effect is greater as the number of such places in the orbit is less, and as the number of revolutions before they approach is less. But it is clearly greater than when the two bodies approach always at different places in their orbits, never again where they have done so before. This is when the two periods are incommensurate, so that their ratio cannot be expressed by any vulgar fraction. One other point must be noticed. The commensurate orbit, and hence the distance from the Sun, and the period of revolution, of the smaller and therefore most affected body, may not be far from a distance where the orbits would be incommensurate. To take the most completely incommensurate ratio known to science, that of the diameter of a circle to the circumference, which has been carried out to seven hundred decimal places without repetition of the figures. This is expressed by the decimal fraction .314159 etc. and yet this differs from the simple commensurate 1/3 or .333333 etc. by only about five per cent.; so that a smaller body may have to be pulled by the larger, only a very short way before it reaches a point where it will be seriously affected no more.
The idea that commensurateness affects the mutual attraction of bodies, and hence the perturbations in their orbits, especially of the smaller one, was not new; but Percival carried it farther, and to a greater degree of accuracy, by observation, by mathematics and in its applications. The most obvious example of its effects lay in the influence of Jupiter upon the distribution of the asteroids, that almost innumerable collection of small bodies revolving about the Sun between the orbits of Jupiter and Mars, of which some six hundred had been discovered. These are so small, compared with Jupiter, that, not only individually but in the aggregate, their influence upon it may be disregarded, and only its effect upon them be considered. In its immediate neighborhood the commensurate periods, Percival points out, come so close together (100 to 101, 99 to 100, etc.) that although occasions of approach would be infrequent they would be enough in time to disturb any bodies so near, until the planet had cleared out everything in its vicinity that did not, by revolving around it, become its own satellite.
Farther off Jupiter’s commensurate zones are less frequent, but where they occur the fragments revolving about the Sun would be so perturbed by the attraction of the planet as to be displaced, mainly, as Percival points out, to the sunward side. This has made gaps bare of such fragments, and between them incommensurate spaces where they could move freely in their solar orbits. Here they might have gathered in a nucleus and, collecting other fragments to it, form a small planet, were it not that the gaps were frequent enough to prevent nuclei of sufficient size arising anywhere. Thus the asteroids remained a host of little bodies revolving about the Sun, with gaps in their ranks—as he puts it “embryos of planets destined never to be born.”
The upper diagram in the plate opposite page 166 shows the distribution and relative densities of the asteroids, with the gaps at the commensurate points. The plate is taken from his “Memoir on Saturn’s Rings,”[35] and brings us to another study of commensurate periods with quite a different set of bodies obeying the same law. Indeed, among the planets observed at Flagstaff not the least interesting was Saturn, and its greatest peculiarity was its rings.
In Bulletin No. 32 of the Observatory (Nov. 24, 1907) Percival had written: “Laplace first showed that the rings could not be, as they appear, wide solid rings inasmuch as the strains due to the differing attraction of Saturn for the several parts must disrupt them. Peirce then proved that even a series of very narrow solid rings could not subsist and that the rings must be fluid. Finally Clerk-Maxwell showed that even this was not enough and that the rings to be stable must be made up of discrete particles, a swarm of meteorites in fact. But, if my memory serves me right, Clerk-Maxwell himself pointed out that even such a system could not eternally endure but was bound eventually to be forced both out and in, a part falling upon the surface of the planet, a part going to form a satellite farther away.
“Even before this Edward Roche in 1848 had shown that the rings must be composed of discrete particles, mere dust and ashes. He drew this conclusion from his investigations on the minimum distance at which a fluid satellite could revolve around its primary without being disrupted by tidal strains.
“The dissolution which Clerk-Maxwell foresaw can easily be proved to be inevitable if the particles composing the swarm are not at considerable distances from one another, which is certainly not the case with the rings as witnessed by the light they send us even allowing for their comminuted form. For a swarm of particles thus revolving round a primary are in stable equilibrium only in the absence of collisions. Now in a crowded company collisions due either to the mutual pulls of the particles or to the perturbations of the satellites must occur. At each collision although the moment of momentum remains the same, energy is lost unless the bodies be perfectly elastic, a condition not found in nature, the lost energy being converted into heat. In consequence some particles will be forced in toward the planet while others are driven out and eventually the ring system disappears.
“Now the interest of the observations at Flagstaff consists in their showing us this disintegration in process of taking place and furthermore in a way that brings before us an interesting case of celestial mechanics.”
He examines the rings mathematically, as the result of perturbations caused by the two nearest of the planet’s satellites, Mimas and Enceladus.
The effect is the same that occurs in the case of Jupiter and the asteroids, Saturn taking the place of the Sun, his satellites that of Jupiter, and the rings that of the asteroids. In spite of repetition it may be well to state in his own words the principle of commensurate periods and its application to the rings:[36]
“The same thing can be seen geometrically by considering that the two bodies have their greatest perturbing effect on one another when in conjunction and that if the periods of the two be commensurate they will come to conjunction over and over in these same points of the orbit and thus the disturbance produced by one on the other be cumulative. If the periods are not commensurate the conjunctions will take place in ever shifting positions and a certain compensation be effected in the outstanding results. In proportion as the ratio of periods is simple will the perturbation be potent. Thus with the ratio 1:2 the two bodies will approach closest only at one spot and always there until the perturbations induced themselves destroy the commensurability of period. With 1:3 they will approach at two different spots recurrently; with 1:4 at three, and so on....
“We see, then, that perturbations, which in this case will result in collisions, must be greatest on those particles which have periods commensurate with those of the satellites. But inasmuch as there are many particles in any cross-section of the ring there must be a component of motion in any collision tending to throw the colliding particles out of the plane of the ring, either above or below it.
“Considering, now, those points where commensurability exists between the periods of particle and satellite we find these in the order of their potency:
| With Mimas, | 1:2 |
| 1:3 | |
| 1:4 | |
| With Enceladus, | 1:3 |
2:3 of Mimas and 1:2; 2:3 of Enceladus falling outside the ring system. 1:2 of Mimas and 1:3 of Enceladus fall in Cassini’s division, which separates ring A from ring B.... 1:3 of Mimas’ period falls at the boundary of ring B and ring C at 1:50 radii of Saturn from the centre.”
In the following years this supposition was reinforced by the discovery of six new divisions in the rings. Three of them were in ring A and three in ring B, two of them in each case seen by Percival for the first time. This led to very careful measurements of Saturn’s ball and rings in 1913-14 and again in 1915; recorded in Bulletins 66 and 68 of the Observatory. Careful allowance was made for irradiation, and the results checked by having two sets of measurements, one made by Percival, the other by Mr. E. C. Slipher. The observations were, of course, made when the rings were so tilted to the Earth as to show very widely, the tilt on March 21, 1915, showing them at their widest for fifteen years.
But unfortunately, as it seemed, the divisions in the rings did not come quite where the commensurate ratios with the two nearest satellites should place them. They came in the right order and nearly where they ought to be, but always a little farther from Saturn. It occurred to Percival that this might be due to an error in the calculation of the motion of the rings, that if the attraction of Saturn were slightly more than had been supposed the revolutions of all parts of the rings would be slightly faster, and the places in them where the periods would be commensurate with the satellites would be slightly farther out, that is where the divisions actually occur. Everyone knows that the earth is not a perfect sphere but slightly elliptical, or oblate, contracted from pole to pole and enlarged at the equator; and the same is even more true of Saturn on account of its greater velocity of rotation. Now its attraction on bodies as near it as the rings, and to a less extent on its satellites, is a little greater than it would be if it were a perfect uniform sphere; and it would be greater still if it were not uniform throughout, but composed of layers increasing in density, in rapidity of rotation, and hence in oblateness, toward the centre. Percival made, therefore, a highly intricate calculation on what the attraction of such a body would be (“Observatory Memoir on Saturn’s Rings,” Sept. 7, 1915), and found that it accounted almost exactly for the discrepancy between the points of computed commensurateness and the observed divisions in the rings. Such a constitution of Saturn is by no means improbable in view of its still fluid condition and the process of contraction that it is undergoing. He found it noteworthy that a study of the perturbations of the rings by the satellites should bring to light the invisible constitution of the planet itself:
“Small discrepancies are often big with meaning. Just as the more accurate determination of the nitrogen content of the air led Sir William Ramsay to the discovery of argon; so these residuals between the computed and the observed features of Saturn’s rings seem to lead to a new conception of Saturn’s internal constitution. That the mere position of his rings should reveal something within him which we cannot see may well appear as singular as it is significant.” (p. 5); and he concludes: (pp. 20-22).
“All this indicates that Saturn has not yet settled down to a uniform rotation. Not only in the spots we see is the rate different for different spots but from this investigation it would appear that the speed of its spin increases as one sinks from surface to centre.[37]
“The subject of this memoir is of course two-fold: first, the observed discrepancy, and second, the theory to account for it. The former demands explanation and the latter seems the only way to satisfy it. From the positions of the divisions in its rings we are thus led to believe that Saturn is actually rotating in layers with different velocities, the inside ones turning the faster. If these layers were two only, or substantially two, this would result in Saturn’s being composed of a very oblate kernel surrounded by a less oblate husk of cloud.”
ASTEROIDS and SATURN’S RINGS
The divisions so made in Saturn’s rings by its satellites may be seen in the lower of the two diagrams opposite; the three fractions followed by an E indicating the divisions caused by Enceladus, the rest those caused by Mimas. The upper diagram represents, as already remarked, the similar effects by Jupiter on the asteroids. A slight inspection shows their coincidence.
In a paper presented to the American Academy in April, 1913, and printed in their Memoirs[38] Percival explained the “Origin of the Planets” by the same principle of commensurate periods. In addition to what has already been said about the places where these periods occur coming closer and closer together as an object nears the planet, so that it is enabled to draw neighboring small bodies into itself, he points out that in attracting any object outside of its own orbit a planet is acting from the same side as the Sun thereby increasing the Sun’s attraction, accelerating the motion of the particle and making it come sunward. Whereas on a particle inside its orbit the planet is acting against the Sun, thereby diminishing its attraction, slowing the motion of the particle and causing it to move outward. “Thus a body already formed tends to draw surrounding matter to itself by making that matter’s mean motion nearly synchronous with its own.” These two facts, the close—almost continuous—commensurate points, and the effects on the speed of revolution of particles outside and inside its own orbit, assist a nucleus once formed to sweep clear the space so far as its influence is predominant, drawing all matter there to itself, until it has attained its full size. “Any difference of density in a revolving nebula is thus a starting point for accumulation. So soon as two or three particles have gathered together they tend by increased mass to annex their neighbors. An embryo planet is thus formed. By the same principle it grows crescendo through an ever increasing sphere of influence until the commensurate points are too far apart to bridge by their oscillation the space between them.”
So much for the process of forming a planet; but what he was seeking was why the planets formed just where they did. For this purpose he worked out intricate mathematical formulae, based on those already known but more fully and exactly developed. These it is not necessary to follow, for the results may be set forth,—so far as possible in his own words. “Beyond a certain distance from the planet the commensurate-period swings no longer suffice to bridge the intervening space and the planet’s annexing power stops. This happens somewhat before a certain place is reached where three potent periodic ratios succeed each other—1:2, 2:5, 1:3. For here the distances between the periodic points is greatly increased....
“At this distance a new action sets in. Though the character of its occasioning be the same it produces a very different outcome. The greater swing of the particles at these commensurate points together with a temporary massing of some of them near it conduces to collisions and near approaches between them which must end in a certain permanent combining there. A nucleus of consolidation is thus formed. This attracts other particles to it, gaining force by what it feeds on, until out of the once diffused mass a new planet comes into being which in its turn gathers to itself the matter about it.
“A new planet tends to collect here: because the annexing power of the old has here ceased while at the same time the scattered constituents to compose it are here aided to combine by the very potent commensurability perturbations of its already formed neighbor.
“So soon as it has come into being another begins to be beyond it, called up in the same manner. It could not do so earlier because the most important deus ex machina in the matter, the perturbation of its predecessor, was lacking.
“So the process goes on, each planet acting as a sort of elder sister in bringing up the next.
“That such must have been the genesis of the several planets is evident when we consider that had each arisen of itself out of surrounding matter there would have been in celestial mechanics nothing to prevent their being situated in almost any relative positions other than the peculiar one in which they actually stand....
“It will be noticed that the several planets are not quite at the commensurate points. They are in fact all just inside them.... Suppose now a particle or planet close to the commensurable point inside it. The mean motion in consequence of the above perturbation will be permanently increased, and therefore the major axis be permanently decreased. In other words, the particle or planet will be pushed sunward. If it be still where” the effect of the commensurateness is still felt “it will suffer another push, and so on until it has reached a place where the perturbation is no longer sensible.” He then goes on to show from his formulae that if the particle were just within the outer edge of the place where the perturbation began to be effective it would also be pushed sunward, and so across the commensurable point until it joined those previously displaced.
“We thus reach from theory two conclusions:
“1. All the planets were originally forced to form where the important and closely lying commensurable points 1:2, 2:5, or 1:3, and in one case 3:5, existed with their neighbors; which of these points it was being determined by the perturbations themselves.
“2. Each planet was at the same time pushed somewhat sunward by perturbation.”
He then calculates the mutual perturbations of the major axes of the outer planets taken in pairs and of Venus and the Earth.
“From them we note that:
“1. The inner planet is caeteris paribus more potent than the outer.
“2. The greater the mass of the disturber and, in certain cases, the greater the excentricity of either the disturber or the disturbed the greater the effect.”
As he points out, the effect of each component of the pair is masked by the simultaneous action of the other, and refers to the case of Jupiter and the asteroids, where the effect they have upon it is imperceptible, and we can see its effect upon them clearly.
Thus he shows that a new planet would naturally arise near to a point where its orbit would be commensurate with that of the older one next to it. But the particular commensurate fraction in each case is not so certain. In general it would depend upon the ratio of the two pulls to each other, for if “the action of the more potent planet greatly exceeds the other’s it sweeps to itself particles farther away than would otherwise be possible”; if it does not so greatly exceed it would not sweep them from so far and hence allow the other planet to form nearer. Now of the four commensurate ratios mentioned, near which a planet may form its neighbor, that of 3:5 means that the two planets are relatively nearest together, for the inner one makes only five revolutions while the outer makes three, that is the inner one revolves around the Sun less than twice as fast as the outer one. The ratio 1:2 means that the inner one revolves just twice as fast as the outer; while 2:5 means that it revolves twice and a half as fast, and 1:3 that it does so three times as fast. Thus the nearer equal the pulls of any pair of forming planets the larger the fraction and the nearer the relative distance between them. Relative, mind, for as we go away from the Sun all the dimensions increase and the actual distances between the planets among the rest.
Venus is smaller than the Earth, but her interior position gives her an advantage more than enough to make up for this, with the result that the pulls of the two are more nearly equal than those of any other pair, the commensurate ratio being 3:5. The next nearest equality of pull is between Uranus and Neptune, where the commensurate ratio is 1:2; the next between Jupiter and Saturn, and Venus and Mercury, where it is 2:5; the least equality being between Saturn and Uranus, where it is only 1:3. Mars seems exceptional for, as Percival says, from the mutual pulls we should expect its ratio with the Earth to be 1:3 instead of 1:2 as it is, and he suggests as the explanation, “the continued action of the gigantic Jupiter in this territory, or it may be that a second origin of condensation started with the Earth while Jupiter fashioned the outer planets.”
He brings the Memoir to an end with the following summary:
“From the foregoing some interesting deductions are possible:
“1. The planets grew out of scattered material. For had they arisen from already more or less complete nuclei these could not have borne to one another the general comensurate relation of mean motions existent to-day.
“2. Each brought the next one into being by the perturbation it induced in the scattered material at a definite distance from it.
“3. Jupiter was the starting point, certainly as regards the major planets; and is the only one among them that could have had a nucleus at the start, though that, too, may equally have been lacking.
“4. After this was formed Saturn, then Uranus, and then Neptune.” (This he shows from the densities of these planets.)
“5. The asteroids point unmistakably to such a genesis, missed in the making.
“6. The inner planets betray inter se the action of the same law, and dovetail into the major ones through the 2:5 relation between Mars and the asteroids.
“We thus close with the law we enunciated: Each planet has formed the next in the series at one of the adjacent commensurable-period points, corresponding to 1:2, 2:5, 1:3, and in one instance 3:5, of its mean motion, each then displacing the other slightly sunward, thus making of the solar system an articulated whole, an inorganic organism, which not only evolved but evolved in a definite order, the steps of which celestial mechanics enables us to retrace.
“The above planetary law may perhaps be likened to Mendelief’s law for the elements. It, too, admits of prediction. Thus in conclusion I venture to forecast that when the nearest trans-Neptunian planet is detected it will be found to have a major axis of very approximately 47.5 astronomical units, and from its position a mass comparable with that of Neptune, though probably less; while, if it follows a feature of the satellite systems which I have pointed out elsewhere, its excentricity should be considerable, with an inclination to match.”
The last paragraph we shall have reason to recall again.
This paper on the “Origin of the Planets” has been called the most speculative of Percival’s astronomical studies, and so it is; but it fascinated him, and is interesting not more in itself, than as an illustration of the inquiring and imaginative trend of his mind and of the ease with which intricate mathematical work came to the aid of an idea.
Meanwhile his reputation was growing in Europe. At the end of 1909 he is asked to send to the German National Museum in Munich some transparencies of his fundamental work on Mars and other planets with Dr. Slipher’s star spectra, and Dr. Max Wolf of Heidelberg who writes the letter adds: “I believe there is no American astronomer, except yours, [sic] invited till now to do so.” A year later the firm in Jena which had just published a translation of his “Soul of the Far East” wants to do the same for “Mars as the Abode of Life.” In August 1914 he writes to authorize a second French edition of this last book which had been published with the title “Evolution des Mondes.” Every other year, he took a vacation of a few weeks in Europe to visit his astronomic friends, and to speak at their societies. We have seen how he did so after his marriage in 1908. He went with Mrs. Lowell again in the spring of 1910, giving lectures before the Société Astronomique in Paris, and the Royal Institution in London, and once more, two years later, when we find him entertained and speaking before several scientific bodies in both Paris and London. That autumn he was confined to the house by illness; and although he improved and went to Flagstaff in March, he writes of himself in August 1913 as “personally still on the retired list.” In the spring it was thought wise for him to take another vacation abroad; and since his wife was recovering from an operation he went alone. He saw his old friends in France and England and enjoyed their hospitality; but he did not feel well, and save for showing at the Bureau des Longitudes “some of our latest discoveries” he seems to have made no addresses. He sailed back on the Mauretania on August 1, just before England declared war, and four days later she was instructed to run to Halifax, which she did, reaching it the following day.
That was destined to be his last voyage, for although he seemed well again he was working above his strength. His time in these years was divided between Flagstaff, where his days and nights were spent in observing and calculating, and Boston, where the alternative was between calculations and business. He was always busy and when one summer he hired a house at Marblehead near to his cousins Mr. and Mrs. Guy Lowell he would frequently drop in to see them; and was charming when he did so; but could not spare the time to take a meal there, and never stayed more than five minutes.
We must now return to the last paragraph of his “Memoir on the Origin of the Planets,” where he suggests the probable distance of a body beyond Neptune. In fact he had long been interested in its existence and whereabouts. By 1905 his calculations had given him so much encouragement that the Observatory began to search for the outer planet, which he then expected would be like Neptune, low in density, large and bright, and therefore much more easily detected than it turned out to be. But the photographs taken in 1906, with a well planned routine search the next year revealed nothing, and he became distrustful of the data on which he was working. In March 1908, one finds in his letter-books from the office in Boston the first of a series of letters to Mr. William T. Garrigan of the Naval Observatory and Nautical Almanack about the residuals of Uranus—that is the residue in the perturbations of its normal orbit not accounted for by those due to the known planets. He suggests including later data than had hitherto been done; asks what elements other astronomers had taken into account in estimating the residuals; points out that for different periods they are made up on different theories in the publications of Greenwich Observatory, and that some curious facts appear from them. About his own calculation he writes on December 28, 1908: “The results so far are both interesting and promising.” He was hard at work on the calculations for such a planet, based upon the residuals of Uranus, and assisted by a corps of computers, with Miss Elizabeth Williams, now Mrs. George Hall Hamilton of the Observatory at Mandeville, Jamaica, at their head.
Before trying to explain the process by which he reached his results it may be well to give his own account of the discovery of Neptune by a similar method:[39]
“Neptune has proved a planet of surprises. Though its orbital revolution is performed direct, its rotation apparently takes place backward, in a plane tilted about 35° to its orbital course. Its satellite certainly travels in this retrograde manner. Then its appearance is unexpectedly bright, while its spectrum shows bands which as yet, for the most part, defy explanation, though they state positively the vast amount of its atmosphere and its very peculiar constitution. But first and not least of its surprises was its discovery,—a set of surprises, in fact. For after owing recognition to one of the most brilliant mathematical triumphs, it turned out not to be the planet expected.
“‘Neptune is much nearer the Sun than it ought to be,’ is the authoritative way in which a popular historian puts the intruding planet in its place. For the planet failed to justify theory by not fulfilling Bode’s law, which Leverrier and Adams, in pointing out the disturber of Uranus, assumed ‘as they could do no otherwise.’ Though not strictly correct, as not only did both geometers do otherwise, but neither did otherwise enough, the quotation may serve to bring Bode’s law into court, as it was at the bottom of one of the strangest and most generally misunderstood chapters in celestial mechanics.
“Very soon after Uranus was recognized as a planet, approximate ephemerides of its motion resulted in showing that it had several times previously been recorded as a fixed star. Bode himself discovered the first of these records, one by Mayer in 1756, and Bode and others found another made by Flamsteed in 1690. These observations enabled an elliptic orbit to be calculated which satisfied them all. Subsequently others were detected. Lemonnier discovered that he had himself not discovered it several times, cataloguing it as a fixed star. Flamsteed was spared a like mortification by being dead. For both these observers had recorded it two or more nights running, from which it would seem almost incredible not to have suspected its character from its change of place.
“Sixteen of these pre-discovery observations were found (there are now nineteen known), which with those made upon it since gave a series running back a hundred and thirty years, when Alexis Bouvard prepared his tables of the planet, the best up to that time, published in 1821. In doing so, however, he stated that he had been unable to find any orbit which would satisfy both the new and the old observations. He therefore rejected the old as untrustworthy, forgetting that they had been satisfied thirty years before, and based his tables solely on the new, leaving it to posterity, he said, to decide whether the old observations were faulty or whether some unknown influence had acted on the planet. He had hardly made this invidious distinction against the accuracy of the ancient observers when his own tables began to be out and grew seriously more so, so that within eleven years they quite failed to represent the planet.
“The discrepancies between theory and observation attracted the attention of the astronomic world, and the idea of another planet began to be in the air. The great Bessel was the first to state definitely his conviction in a popular lecture at Königsberg in 1840, and thereupon encouraged his talented assistant Flemming to begin reductions looking to its locating. Unfortunately, in the midst of his labors Flemming died, and shortly after Bessel himself, who had taken up the matter after Flemming’s death.
“Somewhat later Arago, then head of the Paris observatory, who had also been impressed with the existence of such a planet, requested one of his assistants, a remarkable young mathematician named Leverrier, to undertake its investigation. Leverrier, who had already evidenced his marked ability in celestial mechanics, proceeded to grapple with the problem in the most thorough manner. He began by looking into the perturbations of Uranus by Jupiter and Saturn. He started with Bouvard’s work, with the result of finding it very much the reverse of good. The farther he went, the more errors he found, until he was obliged to cast it aside entirely and recompute these perturbations himself. The catalogue of Bouvard’s errors he gave must have been an eye-opener generally, and it speaks for the ability and precision with which Leverrier conducted his investigation that neither Airy, Bessel, nor Adams had detected these errors, with the exception of one term noticed by Bessel and subsequently by Adams.[40] The result of this recalculation of his was to show the more clearly that the irregularities in the motion of Uranus could not be explained except by the existence of another planet exterior to him. He next set himself to locate this body. Influenced by Bode’s law, he began by assuming it to lie at twice Uranus’ distance from the Sun, and, expressing the observed discrepancies in longitude in equations, comprising the perturbations and possible errors in the elements of Uranus, proceeded to solve them. He could get no rational solution. He then gave the distance and the extreme observations a certain elasticity, and by this means was able to find a position for the disturber which sufficiently satisfied the conditions of the problem. Leverrier’s first memoir on the subject was presented to the French Academy on November 10, 1845, that giving the place of the disturbing planet on June 1, 1846. There is no evidence that the slightest search in consequence was made by anybody, with the possible exception of the Naval Observatory at Washington. On August 31 he presented his third paper, giving an orbit, mass, and more precise place for the unknown. Still no search followed. Taking advantage of the acknowledging of a memoir, Leverrier, in September, wrote to Dr. Galle in Berlin asking him to look for the planet. The letter reached Galle on the 23rd, and that very night he found a planet showing a disk just as Leverrier had foretold, and within 55′ of its predicted place.
“The planet had scarcely been found when, on October 1, a letter from Sir John Herschel appeared in the London Athenaeum announcing that a young Cambridge graduate, Mr. J. C. Adams, had been engaged on the same investigation as Leverrier, and with similar results. This was the first public announcement of Mr. Adams’ labors. It then appeared that he had started as early as 1843, and had communicated his results to Airy in October, 1845, a year before. Into the sad set of circumstances which prevented the brilliant young mathematician from reaping the fruit of what might have been his discovery, we need not go. It reflected no credit on any one concerned except Adams, who throughout his life maintained a dignified silence. Suffice it to say that Adams had found a place for the unknown within a few degrees of Leverrier’s; that he had communicated these results to Airy; that Airy had not considered them significant until Leverrier had published an almost identical place; that then Challis, the head of the Cambridge Observatory, had set to work to search for the planet but so routinely that he had actually mapped it several times without finding that he had done so, when word arrived of its discovery by Galle.
“But now came an even more interesting chapter in this whole strange story. Mr. Walker at Washington and Dr. Petersen of Altona independently came to the conclusion from a provisional circular orbit for the newcomer that Lalande had catalogued in the vicinity of its path. They therefore set to work to find out if any Lalande stars were missing. Dr. Petersen compared a chart directly with the heavens to the finding a star absent, which his calculations showed was about where Neptune should have been at the time. Walker found that Lalande could only have swept in the neighborhood of Neptune on the 8th and 10th of May, 1795. By assuming different eccentricities for Neptune’s orbit under two hypotheses for the place of its perihelion, he found a star catalogued on the latter date which sufficiently satisfied his computations. He predicted that on searching the sky this star would be found missing. On the next fine evening Professor Hubbard looked for it, and the star was gone. It had been Neptune.[41]
“This discovery enabled elliptic elements to be computed for it, when the surprising fact appeared that it was not moving in anything approaching the orbit either Leverrier or Adams had assigned. Instead of a mean distance of 36 astronomical units or more, the stranger was only at 30. The result so disconcerted Leverrier that he declared that ‘the small eccentricity which appeared to result from Mr. Walker’s computations would be incompatible with the nature of the perturbations of the planet Herschel,’ as he called Uranus. In other words, he expressly denied that Neptune was his planet. For the newcomer proceeded to follow the path Walker had computed. This was strikingly confirmed by Mauvais’ discovering that Lalande had observed the star on the 8th of May as well as on the 10th, but because the two places did not agree, he had rejected the first observation, and marked the second as doubtful, thus carefully avoiding a discovery that actually knocked at his door.
“Meanwhile Peirce had made a remarkable contribution to the whole subject. In a series of profound papers presented to the American Academy, he went into the matter more generally than either of the discoverers, to the startling conclusion ‘that the planet Neptune is not the planet to which geometrical analysis had directed the telescope, and that its discovery by Galle must be regarded as a happy accident.’[42] He first proved this by showing that Leverrier’s two fundamental propositions,—
“1. That the disturber’s mean distance must be between 35 and 37.9 astronomical units;
“2. That its mean longitude for January 1, 1800, must have been between 243° and 252°,—were incompatible with Neptune. Either alone might be reconciled with the observations, but not both.
“In justification of his assertion that the discovery was a happy accident, he showed that three solutions of the problem Leverrier had set himself were possible, all equally complete and decidedly different from each other, the positions of the supposed planet being 120° apart. Had Leverrier and Adams fallen upon either of the outer two, Neptune would not have been discovered.[43]
“He next showed that at 35.3 astronomical units, an important change takes place in the character of the perturbations because of the commensurability of period of a planet revolving there with that of Uranus. In consequence of which, a planet inside of this limit might equally account for the observed perturbations with the one outside of it supposed by Leverrier. This Neptune actually did. From not considering wide enough limits, Leverrier had found one solution, Neptune fulfilled the other.[44] And Bode’s law was responsible for this. Had Bode’s law not been taken originally as basis for the disturber’s distance, those two great geometers, Leverrier and Adams, might have looked inside.
“This more general solution, as Peirce was careful to state, does not detract from the honor due either to Leverrier or to Adams. Their masterly calculations, the difficulty of which no one who has not had some experience of the subject can appreciate, remain as an imperishable monument to both, as does also Peirce’s to him.”
The facts, that is what was done and written, are of course correct; but the conclusions drawn from them are highly controversial to the present day.
The calculations for finding an unknown planet by the perturbations it causes in the orbit of another are extremely difficult, the more so when the data are small and uncertain. For Percival they were very small because Neptune,—nearest to the unknown body,—had been discovered so short a time that its true orbit, apart from the disturbances therein caused by other planets, was by no means certain. In fact Percival tried to analyze its residuals, but they yielded no rational result. This left only what could be gleaned from Uranus after deducting the perturbations caused by Neptune, and that was small indeed. In 1845, when the calculations were made which revealed that planet, “the outstanding irregularities of Uranus had reached the relatively huge sum of 133″. To-day its residuals do not exceed 4.5″ at any point of its path.”
Then there are uncertainties depending on errors of observation, which may be estimated by the method of least squares of the differences between contemporary observations. Moreover there is the uncertainty that comes from not knowing how much of the observed motion is to be attributed to a normal orbit regulated by the Sun, and how much to the other planets, including the unknown. Its true motion under these influences can be ascertained only by observing it for a long time, and by taking periods sufficiently far apart to distinguish the continuing effects of the known bodies from those that flow from an unknown source. This was the ingenious method devised by Leverrier as a basis for his calculations, and he thereby got his residuals caused by the unknown planet in a form that could be handled.
Finally there was the uncertainty whether the residual perturbations, however accurately determined, were caused by one or more outer bodies. Of this Percival was, of course, well aware, and in fact, in his study of the comets associated with Jupiter he had pointed out that there probably was a planet far beyond the one for which he was now in search. But, as no one has ever been able to devise a formula for the mutual attraction of three bodies, he could calculate only for a single body that would account as nearly as possible for the whole of the residuals.
Thus he knew that his work was an approximation; near enough, he hoped, to lead to the discovery of the unknown.
The various elements in the longitude of a planet’s orbit, that is in the plane of the ecliptic, that are affected by and affect another, are:
a—The length of its major, or longest, axis.
n—Its mean motion, which depends on the distance from the Sun.
ε—The longitude at a given time, that is its place in its orbit.
e—The eccentricity of its orbit, that is how far it is from a circle.
ῶ—The place of its perihelion, that is the position of its nearest approach to the Sun.
(These last two determine the shape of the ellipse, and the direction of its longer axis with respect to that of the other planet.)
m—Its mass.