# Popular Science Monthly/Volume 74/May 1909/The Closing of a Famous Astronomical Problem

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* THE CLOSING OF A FAMOUS ASTRONOMICAL PROBLEM
By Professor W. W. CAMPBELL
DIRECTOR OF THE LICK OBSERVATORY
THERE is perhaps no more striking illustration of the power of scientific method than that relating to the discovery of Neptune in 1846. The planet Uranus, until then the outermost known member of our solar system, refused to follow the path computed for it by mathematical astronomers. With the progress of time the discrepancies between its predicted and observed positions grew constantly larger until, in the early eighteen-forties, the discordance amounted to fully 75 seconds of arc. This is a small angle—not more than one twenty-fifth the angular diameter of our moon—yet a very large angle to refined astronomy, for a discrepancy of two seconds would have been detected with ease. The opinion gradually developed that Uranus was drawn from its natural course by the attractions of an undiscovered planet still farther from the sun than itself. Adams in 1843 and Le Verrier in 1845, independently, and each without knowledge of the other's plans, attacked the then extremely difficult problem of determining the approximate orbit, mass and position of an undiscovered body whose attractions should produce the perturbations observed. Regrettable and avoidable delays occurred in searching for the planet after Adams's results were communicated to the astronomer royal, in October, 1845. Le Verrier's results were communicated to the Berlin Observatory in September, 1846, with the request that a search be made. The disturbing planet, later named Neptune, was found on the first evening that it was looked for, less than one degree of arc from the position assigned by Le Verrier. If an energetic search had been made in England the year before, the planet would have been discovered within two degrees of the position assigned by Adams.
The above résumé of this unsurpassed achievement of the human mind forms a natural prelude to the present article, as it was the immediate forerunner of another problem, famous for half a century, which has now been brought to a satisfactory conclusion.
The determination of the orbit of the planet Mercury gave great difficulty to its investigators, principally from two causes:
1. Being the innermost known planet in our system, remaining always near the sun, and usually lost to view in the sun's glare, fairly ​accurate observations of its positions could be secured only when the planet was near its greatest angular distances from the sun and on the rare occasions when the planet passed between us and the sun's disk. Consequently, observations of the highest accuracy were few in number; and,
2. There were large discrepancies between Mercury's predicted and observed positions, certainly not due to the attractions of any known members of our solar system.
Le Verrier, of Neptunian fame, undertook a systematic investigation of Mercury's orbit, making use of all available observations. His results were derived and published in 1859. His work established that there were peculiarities in the planet's orbital motion which could not be due to the attractions of known masses of matter. Chief among the peculiarities was a slow rotation of the orbit itself. It is best described as a forward motion of the orbit's perihelion amounting to 38 seconds of arc per century.
Le Verrier announced that the outstanding differences between prediction and observation could be produced and explained by the disturbing attractions of an undiscovered planet closer to the sun than Mercury and revolving around the sun in an orbit lying nearly in the plane of Mercury's orbit. The mass of (the quantity of matter in) the hypothetical planet would depend upon its distance from Mercury: if half way between Mercury and the sun, its mass would be two thirds that of Mercury; if further from Mercury, the necessary mass would be greater; if nearer, smaller. A group or "ring" of small planets, instead of one large planet, would serve equally well, provided the total mass of the planetoids were of the same order of magnitude. Le Verrier did not say that such an undiscovered planet or ring of planetoids did exist, but simply that it would account for the observed anomalies. The accuracy of his computations, published in detail, could not be questioned. The recognition of his masterly skill, and the memory of his entirely similar discovery of Neptune, assisted in convincing astronomers quite generally that a planet or group of planets existed. The discovery of the disturbing mass became at once a noted problem.
A body traveling around the sun in a circular orbit whose radius is only one half Mercury's average solar distance would never be more than 12° from the sun as viewed by terrestrial observers. A search for it by ordinary methods would accordingly be fruitless. A body large enough to shine brilliantly on a dark-sky background would be hopelessly lost in the bright sky near the sun. Mercury itself, though running out between 20° and 30° from the sun every few weeks, is seldom seen by any save astronomers; and they know where to look for it in the twilight sky.
Two special methods of discovery were applicable: (1) To detect ​the planet projected upon the sun's disk when its orbital motion carried it between us and the sun; (2) to search for it when the sky background was darkened at the time of a total solar eclipse.
Needless to say, a crop of discoverers by the first method grew up without delay. The observer of greatest note was Lescarbault, a rural physician of France. Immediately following the publication of Le Verrier's conclusions, Lescarbault announced that he had observed the transit of an unknown planet across the sun's disk several months earlier. Le Verrier journeyed to Lescarbault's home, investigated all the circumstances of the observation, weighed the evidence and concluded that a real planet had been seen. In fact, so convinced of its reality were many scientific men that the name Vulcan was given to it. Older and later reported observations of the same character, to the number of twenty, were collected by Le Verrier, and those which seemed to be in harmony with each other were made the basis of an orbit. Vulcan was found to be about one third Mercury's distance from the sun, revolving once around the sun in between nineteen and twenty days. In some of the text-books on astronomy appearing in the sixties and seventies, Vulcan was assigned a place in the solar system as conspicuous and as secure as that of Mercury itself.
Now it is probable that every one of the twenty observations referred to was erroneous, though made in good faith. In essentially every case the observer was inexperienced, and used a telescope of insufficient power, or one unprovided with measuring apparatus suitable for determining whether or not the subject observed was in motion across the sun's disk. Even the observation of Lescarbault was in doubt when it later transpired that a Brazilian observer of considerable professional experience was at the same hour studying the region of the sun in question and saw only uniform normal solar surface. The situation was not without its humorous side. For example, a Mississippi Valley weather prophet who saw Vulcan crossing the sun's disk, said it was about "as large as a new [sic] silver half dollar"! Many of the observations no doubt referred to small sun spots which, with small telescopes, would look round.
Vulcan was searched for by visual observers at the principal eclipses of the sixties, seventies and eighties. Two noted astronomers at the eclipse of 1878, Watson and Swift, believed that they saw two new planets near the sun. However, the two seen by Watson did not agree with those seen by Swift, and still other astronomers at the same eclipse saw no strange bodies in the same regions. As the assigned locations depended upon the hasty readings of graduated circles, in which one can so easily make errors, in the press and excitement of eclipse conditions, the astronomical world quickly, and no doubt ​correctly, concluded that the objects seen were well-known neighboring stars.
The perfecting of dry-plate photography gave renewed interest to the search for Vulcan, both when passing over the solar surface and at times of eclipse. Although the sun has been photographed almost daily during the past twenty years, at one observatory or another, no experienced observer has seriously claimed that his plates recorded an unknown planet crossing the sun. Neither were eclipse searches more successful: the well-known bright stars lying nearly in the direction of the sun were photographed, but no strange bodies. Curiously enough, the optical principles governing the efficiency of cameras in this search were overlooked for many years, and faint objects near the sun—say stars fainter than the fourth magnitude—were not observable, because their images, though formed on the photographic plates, were overwhelmed and buried from sight in the general darkening of the photograph by the bright-sky background. It was not until 1900 that the elements of the problem of photographing faint bodies near the sun were comprehended. While preparing for the eclipse of that year, three astronomers, Professor W. H. Pickering, of Harvard College Observatory, and Messrs. Perrine and Campbell, of the Lick Observatory, independently arrived at the same simple conclusion that the focal lengths of the intramercurial-search cameras should be relatively long, in order to reduce the intensity of the sky exposure on the plates without reducing the intensity of the star images, and thus let the latter be seen on the negative. The principles involved are so simple as hardly to call for elucidation.
Let the two cameras have lenses of equal aperture, say 3 inches, of equal transparency and capable of covering equal angular fields of view, say a circle 10 degrees in diameter. Let one be of short focus, 21 inches, and the other of long focus, 135 inches. The powers of the two lenses to record stellar points on the sensitive plates in focus, under good atmospheric conditions, are not very unequal, for the two lenses collect equal quantities of light and condense the light into images of very nearly the same size. Both collect the same quantity of sky light, but the longer-focus camera spreads it (more thinly) over an area (135)$^2$/(21)$^2$ = 41 times the greater. It is evident that faint-star images hopelessly lost to view on the sky-blackened small plate may be seen with ease on the nearly clear glass of the large plate. "We may safely say that the large plate will show images of stars 3 or 3$^1$⁄$_2$ magnitudes fainter than the small plate. The same advantage exists for small intramercurial planets as for stars, provided the exposures do not exceed two or three minutes in length, as they seldom do at eclipses. In longer exposures on intramercurial planetoids the advantage would usually be lost, as their rapid (and unknown) motions would cause their ​images on the plate to move, slowly with short-focus and rapidly with long-focus cameras, thus drawing them out into trails. With the longer instrument here described, an eight-minute exposure would in general be no more effective than one of four minutes. The most successful instrument for the search in question must compromise between the advantage of long focus in reducing sky density, and the disadvantage of long focus in producing long trails. Shorter exposures, giving shorter trails, may be provided by increasing the diameter of the lens, but this in turn means greater unavoidable optical aberrations in the outer areas of the region photographed, which is a reduction in efficiency. In this as in all instruments, extensive experience and good judgment must combine to decide upon the best compromise-proportions.
Professor Pickering, of Harvard, and Mr. Abbot, of the Smithsonian Institution, used such cameras at the total solar eclipse of 1900. The latter observer was favored with good conditions, in North Carolina, and he secured one photograph of a considerable area surrounding the eclipsed sun. Quite a number of the stars known to exist in this region were photographed; but in the absence of a duplicate photograph of the same region, he could not decide whether certain apparent images on the plate were due to unknown planets, or were defects such as always exist in photographic films.
At the eclipse of 1901, in Sumatra, Mr. Abbot, of the Smithsonian Expedition, and Mr. Perrine, in charge of the Crocker Expedition from the Lick Observatory, were prepared, with four cameras each, to secure duplicate photographs covering a large area extending east and west from the sun. Conditions were unfortunately against the success of Mr. Abbot's plans, but thin clouds at the time of the eclipse let 25 per cent, of the light come through to Mr. Perrine's photographic plates. The area covered in duplicate was 6° x 38°, extending along the direction of the sun's equator, with the sun in the center of the region. The plates recorded 170 well-known stars; and all apparent images not of ordinary stars were proved by the duplicate plates to be defects in the films. In two thirds of the area stars down to the eighth magnitude and many fainter ones were recorded; and in one third the area, covered with thicker clouds, stars were recorded down to the fifth and sixth magnitudes.
At the eclipse of 1905 Mr, Crocker made it possible for me to organize expeditions to Labrador, Spain and Egypt, each equipped with four intramercurial cameras, in addition to apparatus for other lines of research. The details of the twelve cameras were planned by Dr. Perrine, the instruments were constructed under his supervision, and any photographic plates obtained with them at the three stations were to be assigned to him to examine for possible intramercurial-planet images. The Labrador group of four cameras, mounted at the ​station by Dr. Curtis, made no contribution because of the severe storm conditions prevailing at the time of totality. The Egyptian cameras, mounted by Professor Hussey, recorded a considerable number of stars, but the sky, though clear in the usual sense, was full of dust, and the sun and the surrounding region covered by the search were at a low altitude. The Spanish cameras, photographing through clouds which permitted only 20 or 30 per cent, of the light to pass, recorded 55 stars down to about the seventh and eighth magnitudes. All suspected images not occupying the positions of known stars were proved to be defects in the films.
The eclipse of 1908 in the South Seas was utilized by the Crocker Expedition to cover a region extending east and west along the sun's equator with duplicate exposures. Notwithstanding interference from rain and clouds at the beginning of totality, clear sky prevailed during the last two thirds of the four critical minutes. Dr. Perrine finds more than 500 images of well-known stars on the plates, and no images of unknown bodies. Stars are recorded down to nearly the ninth visual magnitude.
It is not absolutely certain that intramercurial planets, revolving around the sun in elliptical orbits would be seen in projection entirely within the area 9° x 29° lying along the solar equator and equally east and west of the sun's center, yet there are exceedingly strong reasons to believe such would be the case. The eight large planets and the 650 ± minor planets in our system revolve around the sun in the same direction and, excepting a small proportion of the asteroids, so nearly in the sun's equatorial plane that the parts of their orbit planes lying within the limits for intramercurial planets would be projected upon the photographed area. The central plane of the zodiacal light differs little from the sun's equatorial plane. It is certain, also, that any intramercurial planets originally moving in planes inclined at large angles to Mercury's orbit plane would gradually be compelled by the attractions of Mercury and the other major planets to move in planes inclined at small angles to the ecliptic. The coincidence of the satellite planes in the systems of Jupiter and Saturn, and no doubt of Uranus and Neptune also, with the equatorial planes of these planets is another analogy of some weight. Admitting, for completeness, the hypothesis of an extensive system of small planets moving in planes making a variety of angles with the ecliptic and sun's equator, some would certainly have been caught in the region photographed. A single planet, or a half dozen planets, massive enough to meet the requirements, moving in any orbit planes would no doubt have been discovered a generation ago. In view of these facts, there is little reason to fear that any planets effective in disturbing Mercury's motions were north or south of the regions covered by photography.
​Inasmuch as planets, shining by reflected light, do not act upon photographic plates so strongly as stars of the same visual magnitude, we may say that exposures which recorded stars down to the ninth magnitude should have recorded planets down to the eighth. From the known brightness, distance from the sun and approximate diameter of a few of the asteroids revolving in space between Mars's and Jupiter's orbits. Dr. Perrine has computed that an average eighth-magnitude intramercurial planet could scarcely be larger than thirty miles in diameter and that roughly a million such bodies, of great density, would be required to supply the disturbing effect observed in Mercury's orbit.
Taking all these points into consideration, 1 think we may say that the investigations by Perrine, forming a part of the work of the Crocker Eclipse Expeditions from Lick Observatory, have brought the observational side of the Intramercurial Problem, famous for a half century, definitely to a close. It is not contended that no such planets will be discovered in the future; in fact, it would not be surprising, nor in opposition to the opinions here expressed, if several such bodies should be found; but it is confidently believed that any such bodies would fail hopelessly to supply the great mass of material demanded by Le Terrier's theory, as Perrine pointed out in discussing the Sumatra observations of 1901.
On the occasion of a future eclipse of fairly long duration, occurring in the dry season, it might be well to repeat the observations, inasmuch as the instruments are in approximate readiness, and the observations at the three past eclipses were made through thin clouds twice, and with cloud-shortened exposures the third time. The cameras are capable of recording tenth-magnitude stars with three-minute exposures in clear sky. It will not be advisable to use these instruments at the eclipses of the next four years.
There are other chapters, on the theoretical side of the problem, to be entered here.
Professor Newcomb's researches on planetary motions extended much further than Le Verrier's. He found small terms in the motions of all the inner planetg—Mercury, Venus, Earth, Mars—which are not due to the disturbing attractions of any known masses of matter. The chief discrepancies, aside from the large one found in Mercury's motion by Le Verrier and confirmed by Newcomb, are in the perihelion of Mars, and in the nodes of Mercury and Venus. These outstanding residuals will be tabulated on a later page.
The attractions of any one planet or ring of small planets, sufficient to account for the excess motion of Mercury's perihelion, failed to account for the other discrepancies discovered by Newcomb for the ​four planets. Satisfactory causes were looked for in a possible ellipsoidal form of the sun, in a hypothetical ring of small planets between Mercury and Venus, in an assumed minute variation in the law of gravitation from the Newtonian inverse square of distances, and in other assumptions, but in vain. One hypothesis, that the finely divided material which gives rise to the zodiacal light (by reflecting the sun's rays) is the responsible disturbing mass, has been discussed several times since the days of Le Verrier and as many times rejected, with one exception.
The exception is Professor Seeliger's recently published investigation. with great skill and with entirely reasonable assumptions as to the form of space occupied by the zodiacal material, and as to the density of the distribution of the material in this space, he establishes that there is sufficient mass to account for the discrepancies in the motions of all the four planets.
The following table exhibits the results of Seeliger's theory in the first column of figures, and the actual results of observation as determined by Newcomb in the second column. The quantities in the third column, which bear the sign ±, are the "probable errors" assigned by Newcomb to his results; and, for the benefit of non-mathematical readers, we may explain that these "probable errors," deduced from the observations themselves, are indications of the uncertainties existing in the quantities to which they are attached. In this table e and i are respectively the eccentricity of the orbit and the inclination of the orbit plane to the ecliptic; and ΔΠ, ΔΩ, and Δi are respectively the changes, per century, in the longitude of perihelion, in the longitude of node and in the inclination of the orbit plane, unaccounted for by the attractions of known masses, as in the second column, and produced by the attractions of the zodiacal matter as computed by Seeliger. In the last column are the differences between the Seeliger and Newcomb numbers: in other words, a comparison of theory with actuality. These differences are small. All are within the probable errors in the third column; with one exception, far within these probable errors.
We can not ascribe this remarkable agreement between Newcomb's
| | | | Seeliger | Newcomb | Newcomb | N -S. |
| αΔΠ | $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \ \end{matrix}}\right.}$ | Mercury | +8.49 | +8.48 | ±0.43 | -0.01 |
| αΔΠ | $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \ \end{matrix}}\right.}$ | Venus | +0.05 | -0.05 | ±0.25 | -0.10 |
| αΔΠ | $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \ \end{matrix}}\right.}$ | Earth | +0.07 | +0.10 | ±0.13 | +0.03 |
| αΔΠ | $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \ \end{matrix}}\right.}$ | Mars | +0.59 | +0.75 | ±0.35 | +0.16 |
| sin iΔΩ | $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \ \end{matrix}}\right.}$ | Mercury | +0.65 | +0.61 | ±0.52 | -0.04 |
| sin iΔΩ | $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \ \end{matrix}}\right.}$ | Venus | +0.58 | +0.60 | ±0.17 | +0.02 |
| sin iΔΩ | $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \ \end{matrix}}\right.}$ | Mars | +0.23 | +0.03 | ±0.22 | -0.20 |
| Δi | $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \ \end{matrix}}\right.}$ | Mercury | +0.52 | +0.38 | ±0.80 | -0.14 |
| Δi | $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \ \end{matrix}}\right.}$ | Venus | +0.17 | +0.38 | ±0.33 | +0.21 |
| Δi | $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \ \end{matrix}}\right.}$ | Mars | +0.02 | -0.01 | ±0.20 | +0.01 |
​and Seeliger's results to fortunate chance, and it is scarcely possible to doubt that in the zodiacal-light materials lie the causes of the discrepancies referred to. I have little hesitation in venturing the opinion that Seeliger's investigation marks an epoch in the application of Newton's law of gravitation to the motions within the solar system. At one stroke he appears to have removed a group of discrepancies which served as bases for many inquiries as to the preciseness and sufficiency of the Great Law. With all respect to Seeliger's genius and labor, however, scientific caution will value confirmation of his results by other investigators.
Seeliger's assumptions as to the distribution and mass of the zodiacal material are of interest, especially when we recall that the zodiacal light within some 20 degrees of the sun is unobservable, on account of the glare, and that the brightness of the light is a poor index to the mass: a given quantity of matter, finely divided, would reflect sunlight more strongly than the same quantity existing in larger particles. For the mathematical development of the subject he assumed that the material is distributed throughout a space represented by a much-flattened ellipsoid of revolution whose center is at the sun's center, whose axis of revolution coincides more or less closely with the sun's axis, whose polar surfaces extend 20 or 30 degrees north and south of the sun (as viewed from the earth), whose equatorial regions extend considerably beyond the earth's orbit, and in which the density-distribution of materials decreases as a function both of the linear distance out from the sun and of the angular distance out from the equatorial plane of symmetry. According to these assumptions, surfaces of equal densities are concentric ellipsoidal surfaces, and the number of such ellipsoids can be increased or decreased according as the computer may desire to represent more or less closely any assumed law of density-variation within the one great spheroid. Practically, Seeliger found that the disturbing effects on the planets are almost independent of the law of distribution of the material, as related to distance from the sun, as far out as two thirds of the distance to Mercury. He made use of only two ellipsoids: One with equatorial radius 0.24 unit and polar radius 0.024, of uniform density; and the other with corresponding radii 1.20 and 0.24, of uniform but much smaller density. The total mean densities determined for his volumes, on the basis of unity as the mean density of the sun, are, respectively, $2.18\times 10^{-11}$ and $3.1\times 10^{-15}$; and the resulting combined mass of the two ellipsoids is $3.1\times l0^{-7}$ of the sun's mass, which is roughly twice the mass of Mercury. The corresponding density of mass-distribution is surprisingly low. In the inner and denser ellipsoid, the matter, if as dense as water, would occupy 1 part in 30,000,000,000 of the space; if as dense as the earth, only 1 part in 160,000,000,000. ​The reader should be cautioned against obtaining the impression that Seeliger's two ellipsoids represent the truth as to the law of distribution of density, for such is not the case. A very large number of ellipsoids, doubtless decreasing rapidly in density as one proceeds from the sun outward, would be required to represent the actual law. Seeliger found that the attractive effect of the mass inside of the ellipsoid with maximum radius 0.24 was essentially independent of the law of distribution; and for convenience in the computations he therefore assumed the density in the said ellipsoid to be uniform. A solution based upon a greater number of constituent ellipsoids would perhaps be a slight improvement.
The logic of Seeliger's work rests finally upon the reasonableness of his assumptions and deductions concerning the distribution and density of the zodiacal-light materials; and these are not out of harmony with the meager knowledge of the zodiacal light which we have obtained by direct observation.
Casco Bay and the Site of the Harpswell Laboratory.
* ↑ Le Verrier's discrepancy amounted to 38″, Newcomb's to 41″.
* ↑ The distance from the sun to the earth being 1.00.