# Popular Science Monthly/Volume 15/August 1879/The Brightness and Distribution of the Fixed Stars

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* THE BRIGHTNESS AND DISTRIBUTION OF THE FIXED STARS.
By HENRY FARQUHAR.
THOSE who view and admire the starry canopy above us—so fittingly associated, in the oft-quoted language of a great philosopher, with the moral nature of man—can hardly fail to remark how largely their pleasure in the grand prospect is due to the endless variety in its brilliancy. Just as the magnificence of mundane potentates is fully brought out only by the presence of a long train of inferiors more modestly arrayed, so Sirius and Capella would be less splendid had they not a multitude of lesser luminaries to heighten their glory by contrast. And how many hundreds of twinkling points, almost lost in the wide abyss, are there for every star of highest rank! In the proportion of common soldiers to captains, and of captains to corps commanders, this silent host of heaven is not unlike the less stately armies that tread earth instead of ether. And if astronomers have hitherto interested themselves less in questions of precedence and seniority than in the particular spot on the field occupied by each individual in the great array when drawn up for review; if, dropping the figure, differences of luster and the number of stars of the various grades have occupied less of their attention than the comparatively dry details of right-ascension and declination, with all the refinements of precession, nutation, aberration, proper motion, parallax, refraction, etc., affecting these—they are now making some amends for their neglect. The methodical study of stellar brightness belongs almost entirely, however, to the present century, Sir W. Herschel's first paper calling attention to the importance of the subject having appeared in ​the publication quaintly termed "Philosophical Transactions" of the London Royal Society in 1796.
Sir W. Herschel here mentions the number of variable stars, constantly increasing under new discoveries, very naturally predicts that closer observation will be likely to show variability in objects previously unsuspected, and recommends that careful comparisons be made from time to time between neighboring stars all over the heavens, so that any change occurring may be at once detected. The original comparisons accompanying this paper have been of but little use, however; they are interesting chiefly as having been the first attempt to introduce scientific methods into this unexplored territory of the astronomical realm. They were made without the aid of any instrument, and consisted of such indefinite statements as—"Star No. 7 about equal to No. 4, and just perceptibly fainter, or decidedly brighter, than No. 12." The difference of brightness which Herschel considered as "just perceptible" seems to have been from one fifth to one fourth of a magnitude.
That his least appreciable difference should have some constant relation to the traditional "magnitude" was to have been expected, bearing in mind what this oldest and most universal scale of reference was intended to express. The fixed stars were assigned to classes of brightness, we learn, before the Christian era; and the very term "magnitudes," used from the first to designate these classes, shows the state of knowledge under which the study had its origin, for, as we now know, the apparently greater size of the brighter stars is due only to imperfections of the eye. All visible stars—all that existed, that is, for the early astronomers of the Mediterranean—were included in six magnitudes, the first containing the dozen or score of brightest stars in the heavens, the second perhaps twice as many ranking next to these, and so on out in gradually increasing circles. The work of the ancients has in this case been well preserved, no modern innovator having been found bold enough to disturb this time-honored system of reckoning. Still, as in the days of the "Father of Astronomy," the two chief stars of Orion serve as examples of the first magnitude, while his Belt and the Dipper in the northern sky furnish types of the second order. But, while astronomy was yet in its infancy, observers had noticed that the stars were not assorted into well-defined orders, in which all the individuals were equally bright; and so, in assigning to a star its magnitude, they would often add that it was "smaller" or "larger" than the mean of that magnitude. They thus practically trebled the number of their classes. The same division into thirds of a magnitude is still employed by those who judge of brightness by eye estimates, though some are content with dividing into halves, and some undertake to be exact to tenths. Now, even though no scientific precision was attained, or even thought of, in this original apportionment of visible stars among the six magnitudes, to which all later estimates ​are adjusted, we would expect to find one principle underlying such a classification, making it that of greatest convenience. It must be just as easy to tell a fourth magnitude star, for instance, from a fifth magnitude as from a third, and there must be as little doubt in distinguishing between the fifth and sixth magnitudes as between the third and second. The numbers expressing magnitudes, then, must actually represent a scale of equal differences as measured by the sensibility of the eye. When an astronomer pays attention to differences of luster, measured also by the sensibility of the eye, but closer than the founders of the science cared to notice, he naturally finds that he can distinguish the same number of intermediate grades between two adjoining magnitudes, whether faint or bright. Herschels estimates, having been of this character, are, as we have seen, subject to the same condition.
The system of comparisons introduced by Herschel was not followed by later astronomers. Determinations of brightness in which accuracy is sought are now made by means of instruments constructed expressly for the purpose. These instruments, called photometers—measurers of light, that is, their office being to show the amount of light that one star gives as compared with others—add nothing to the discriminating power of the eye, it should be stated. In deciding a question as to which is the brighter of two stars, situated sufficiently near together, no appliance yet invented can assist. But they have these three advantages: they facilitate comparison between faint stars, they furnish a means of comparing distant stars as though side by side, and they give results in a numerical form. That is to say, we get by means of them a definite difference, which may be expressed as a fraction of a magnitude. The magnitude, we see, is no longer regarded as a class, but as a fixed point on a continuous scale; a striking example of that progress of science in all its branches from a qualitative to a quantitative stage, on which philosophers delight so to insist.
But how are measures of light to give us fractions of a magnitude? How can the vague, qualitative relation, the brighter the light the higher the magnitude, become an exact and quantitative one? The discussion of this question may be of use by showing that, even with matters of so uncertain a nature, science does not proceed by guesswork. It is a general law, that the human senses measure ratios and not differences. If I am carrying a small weight, for example, and the addition of an ounce is required to make the burden perceptibly heavier to me, two ounces will have to be added in order that I may notice a difference when I carry twice the weight, and a whole pound when I carry sixteen times the weight. Similarly with the other senses, and, in no slight degree, with the emotions as well. Sensibility to grief and joy, as the experience of every one will attest, becomes feebler with an increase of the amount sustained. So, a faint sound can be heard only in comparative silence, and our footsteps surprise us by their resounding din on the floor of an empty hall, though no louder, ​as reflection easily assures us, than when the hall is filled with a bustling multitude. So, though the stars give us their whole light in the daytime, our eye, with the stimulus of an illuminated atmosphere, fails to discover them. This law, first stated by Fechner, is, in mathematical language, the excitement of a nerve varies in arithmetical progression as the exciting cause varies in geometrical progression, or degrees of sensation correspond to logarithms of the quantities perceived. Since, as just shown, the scale of magnitudes is that of equal differences in sensation, it must be at the same time that of equal ratios of light. We must thus have a constant light-ratio between each magnitude and the one next it, and these magnitudes must be logarithms of the quantities of light given, this ratio being taken as the base of our system. In fact, one of the first discoveries in photometry was that such a ratio actually exists; that, for example, if each star rated as third magnitude by good observers gives as much light as 212 stars of the fourth magnitude, a star of the fourth equals 212 of the fifth, and so on. Here was a practical confirmation of the character ascribed to ancient estimates of magnitude, and, at the same time, of Fechner's law.
This relation affords us the means of substituting exact measurement for estimates on an ill-defined scale by different observers, among whom a perfect agreement as to standard is out of the question. The idea that each observer has of the meaning of second or fifth magnitude is derived entirely from tradition and confirmed by habit, very much as are his notions of the significance of ordinary adjectives of degree—the only precaution observed being to alter the estimates of antiquity as little as possible, a vague limitation at best. Measures with the photometer depend no less on estimates with the eye, but the determination in them, as to the exact agreement of two lights, is subject to far less uncertainty.
Photometers agree in this particular, whatever their differences in mechanical construction. Seidel, of Munich, who was twelve years in comparing the light of but 208 fixed stars, used an apparatus where two stars seen through a telescope with divided object-glass, each out of focus, were made of the same brightness to the eye by diffusing or concentrating the light of one of them, its half of the object-glass being drawn out or in. The stars thus appeared as two disks, of different sizes but equally bright, and the amount of light given by each was taken as proportionate to the area covered by its disk. The same Dr. Zöllner who has lately become so conspicuous in "spiritualist" investigations, invented a much more convenient style of photometer, with which he made some interesting researches into the comparative light of the planets. Other astronomers, European and American, have also used it. With one of these instruments, belonging to the observatory of Harvard University, Mr. Peirce finished, a few years ago, perhaps the most extensive and methodical photometric work that ​has yet been done. His measures included the visible stars, about five hundred in all, of the zone between 40° and 50° of north declination—those passing overhead in the Northern United States and Canada. The light of a kerosene lamp, in Zöllner's photometer, shines through a small round hole in a thin metallic plate, so as to form an imitation of a star, slightly brighter than the real stars with which it is compared. The light of this artificial star, having been polarized by passing through one Nicol prism, is partially cut off by turning another. The proportion of light so cut off depends on the angle of the second prism from parallelism with the first. Having thus found the amounts of light given by two or more stars, as compared with a fixed light, their differences of magnitude are calculated by applying the rule involving logarithms, alluded to above.
Owing to the labor involved in making any large number of photometric comparisons, the less accurate but more convenient method of eye-estimates has not yet been entirely superseded. It becomes necessary, then, to find some way of reducing different observers to one uniform scale, in order to have their work available for determination of variability and questions of distribution. Mr. Peirce, in his "Photometric Researches," recently published by the Harvard College Observatory, has shown that this may be done by the simple process of counting. When we find in any catalogue a star recorded as of magnitude 413, say, though we can not tell exactly what degree of brightness this figure denotes, we yet know something definite, namely, that this observer classes the star in question as fainter than those he calls 4 and brighter than those he calls 423. And we know something more: if in the northern hemisphere—a limited part of the heavens must be taken for the purpose, few catalogues being complete in southern stars—he classes 200 stars in all as brighter than 413, while he calls 25 stars 413 exactly, he means to tell us that his 413 magnitude stars would all fall between 200 and 225 on a list of northern stars arranged in order of brightness. It is by considering the order which stars would follow when so arranged, leaving entirely out of view the numbers by which their magnitudes are expressed, that Mr. Peirce brings all observers to a single standard of reference; for he is justified in assuming that each of them attaches the same idea of brightness to the 50th or 150th star in his order, as an assumption of some such nature must be made to have their estimates of any service at all. To reduce to magnitudes these numbers expressing order of arrangement, we have to notice that equal ratios among them correspond to equal differences in magnitude. If we take a good catalogue and find the number of stars in it brighter than 2·0, and add to this number successively the number between 2·0 and 3·0, 3·0 and 4·0, and so on, we shall find that our series of numbers increases geometrically, the common ratio being nearly 338. This is a remarkable fact, but it is not difficult to account for, on the supposition that the stars are uniformly ​scattered throughout space, or that portion of space in which visible stars are situated. In this case, the number of them out to any distance from our solar system must vary as the cube of that distance, while their light, supposing no important variations in real size and brightness among them, is inversely proportionate, in the mean, to the square of their distance. And since we have a constant ratio of light between each magnitude and the next, we must accordingly have a constant ratio of mean distance, equal to the square root of this ratio inverted, and a constant ratio of number, equal to the cube of the ratio of distance. Mr. Peirce adopted the ratio 338. While introducing no perceptible change in the traditional magnitude-scale, except to rid it of irregularities, this number has the convenience of being exactly the cube of 112. Considering differences in brightness as due exclusively to differences in distance, we may conclude that a star of the second magnitude, for instance, is just half as far again from us as one of the first, and two thirds as far as one of the third. The magnitude of any star, then, is to be regarded as a logarithm of the number expressing its ordinal rank, 338 being the base of the system. We may thus find to what magnitudes the ordinal numbers, 200 and 225 in the example given, correspond, and take these as the superior and inferior limits of our observer's magnitude 413. The probable corrected magnitude may be considered as half way between these limits, and we can not be more exact than this in our reduction, because his discrimination has not been close enough to admit of it.
There are, it will thus be seen, three ways of stating the rank of the stars: by magnitudes or other devices to express differences of visual sensibility, by quantities of light, and by positions on a list arranged in order of decreasing luster. These three are reduced to one, through Fechner's law connecting the first two, and the hypothesis of equable distribution connecting the second and third.
But before accepting this hypothesis of equable distribution as part of our knowledge, we must see how well it agrees with the facts. Observation must determine if the "ratio of light" and the "ratio of number" have actually the mathematical relation given above. On the scale adopted by Mr. Peirce, as we have seen, the distance of a star should be two thirds that of one one magnitude fainter, and its light, by the law of the inverse square, 214 times as great. But the actual ratio of light between successive magnitudes is found by photometric measurement to be not far from 212; different observers varying from 2·3 to 2·8, but all giving values greater than the theory. By the fact, however, that the ratio thus found is constant or very nearly so for all grades of brightness, we are yet justified, notwithstanding the objection from its too high value, in determining magnitudes by counting, and so clearing individual estimates of much of their uncertainty and irregularity.
The conclusion seems unavoidable that a uniform distribution of ​stars does not hold even in the region of space immediately about our solar system. Since the ratio of number is smaller than it should be to correspond to that of light, the density in which the stars are aggregated—if the expression be permitted—must diminish as the distance increases. Our sun is therefore in a part of space more closely filled than are neighboring parts. This is perhaps the most interesting result to which the study of photometry leads us, because it seems so strange at first sight—and even more strange when we remember that the nearest of the other suns is distant from us more than three years' journey of light. Truly, astronomy is without a rival in its special mission, to contradict on every point the evidence of superficial observation. We would most naturally suppose our universe to be as we are told it appeared to a distinguished visitor, when at once
"The golden sun, in splendor likest heaven, Allured his eye; . . . . . . . where the great luminary, Aloof the vulgar constellations thick, That from his lordly eye keep distance due, Dispenses light from far."
But this appearance of standing aloof is wholly misleading, and, moreover, as our sun would rank by no means first among the fixed stars if placed at the distance of the nearest of them, and would sink below the third magnitude if removed as far as Sirius, its real insignificance in the stellar firmament is almost as striking as its supremacy in its own planetary system. The reflection is an interesting one, how lamentably the grandest of poems must have suffered had its author been compelled to regard the true proportions of the sidereal universe; but for the true lover of nature, it may be hoped, the glory of the Almighty handiwork will not be lessened through the disappearance from fancy of the universal sovereignty of the sun along the track made for it centuries ago by the vanished delusion that our earth was the unmoved center of all things.
It is very certain that an equable distribution could not hold throughout all space (for an infinite number of stars impartially scattered would, however vast the distances among them, give us a heaven shining like the sun in every part, with heat to correspond) unless, owing to the presence of innumerable dark bodies, or to a discontinuity in the luminiferous ether itself, as some physicists have suggested, light from remote distances is wholly or partly cut off before reaching us. But to this view, though it would agree with all the facts, that of a limitation of our firmament of stars, in extent and number, is generally preferred. That such a limitation exists we have other reasons for believing: prominent among these is the system of distribution which a census of the heavens brings to light. We could not expect ​an infinite universe of stars to show everywhere such a uniformity of plan.
How the fixed stars are actually distributed through space, an inquiry into which we are led by study of their number and brightness, it has been but recently found worth while to consider. So long as they were believed to be simply lights set in a hollow revolving structure that divided the waters beneath from the waters above; so long as the idea of the "firmament" retained the association with solidity that now only remains in the word; so long as the "spangled heavens, a shining frame" was a reality of opinion and not an unmeaning archaism of poetry—this question was never heard. No significance could attach to it, and it excited no curiosity. But when this solid celestial framework was broken up by the discovery of the earth's rotation, and its lights scattered afar on the deep ocean of unbounded space, when contemplation of the beautiful adjustments and proportions of our solar system had suggested the hope of discovering the same harmonies throughout the universe, it began to be asked if some of these far-distant orbs, or perhaps the mighty whole, our sun and its attendant planets included, were not connected in a system of similar character. Kant was one of the first to advance this idea. The elder Herschel, contenting himself with a working hypothesis to give form to his observations, supposed our firmament to be a mighty cluster of stars equally distributed within finite limits, so that the number visible in the field of his great reflector at any pointing would show the extent of occupied space in that direction; and he undertook to gauge the depths and discover the shape of this cluster by counting telescopic fields in different parts of the sky. The elder Struve considered the density of the stars as varying with the distance from the Milky Way, as does that of the atmosphere with its distance from the earth's surface; being equal in parallel plans. Argelander, of Bonn, relieved his laborious task of cataloguing over 300,000 northern stars, by investigation into the subject; Mr. Proctor has devoted to it numerous memoirs and popular lectures, and speaks of it as his chief incentive to the labor of constructing his set of twelve star maps; Mr. Peirce gives it considerable space in his "Photometric Researches." From these sources we have a few conjectures and a few facts.
The richest parts of the sky, in bright and faint stars alike, are almost all about the Milky Way. This stream of suffused light follows, with some irregularities, the course of a great circle; and toward the plane of this circle, passing not very far, perhaps, from the sun, stars at all distances appear to become more densely packed. The Milky Way itself is evidence of this for the faintest magnitudes; and Herschel's star-gauges, from which he inferred for our cluster the shape of a disk or lens, give the comparison in a numerical form. ​Argelander's gauges show the same concentration in telescopic stars brighter than the tenth magnitude, and it is even more plainly to be made out from Mr. Proctor's chart of his great catalogue. If parallel to the great circle of the Milky Way two small circles be passed, each at a distance from it of 30°, having between them a broad belt about the celestial sphere somewhat like the torrid zone on the earth's surface, we shall leave two spherical caps whose united area will exactly equal that of the belt—just one hemisphere. From Argelander's gauges it may be calculated that the number of stars inside these 30 circles is to that outside nearly as 212 to 1, for stars of the ninth magnitude, and about as 2 to 1 for the eighth, diminishing with brighter stars. This condensation increases without interruption, to the Milky Way itself. The law holds also with stars visible to the naked eye, though not so conspicuously; for these, Mr. Peirce found the same ratio to be only as 4 to 3. He was also surprised to see that the stars were very little more numerous in the track of the Milky Way than at a distance of 20° from it, the decrease in density appearing almost suddenly about 30°. But as we approach the sun, the rate of condensation becomes greater again. Of the twenty stars classed as first magnitude by the best observers, fifteen are within the 30° circles; and of the five outside, but two, Arcturus and one far southern star, are equal in brightness to the average of the twenty. We have no right, however, unless we are dealing generally with a very great number of stars, to take light as a reliable indication of distance. Of our twelve nearest neighbors yet recognized, being all that have a parallax greater than one sixth of a second, and distant from us less than twenty years' journey of light, four are telescopic stars, to which attention was attracted by their large proper motion. Ten stars out of these twelve, it should therefore be added, are either in the Milky Way or within 15 of it. The exceptions are two minute stars in Ursa Major.
Will these facts enable us to decide what is the actual form of the immense cluster of stars in which our sun holds so humble a rank? We may conclude from them, with safety, that the strongly marked and surprising concentration of brightest and nearest stars in the galactic plane is irreconcilable with a generally prevailing uniform distribution, and agrees hardly better with Struve's theory of condensation in parallel planes. For this theory, it will be seen, requires a more decided concentration with a greater distance, the planes of equal density appearing to approach the galactic circle and each other as do the parallel lines of a perspective drawing. We do see some tendency of this kind in telescopic magnitudes, so that we might suppose that Struve's theory began to express the facts at the distance of the faintest visible stars—unless it could be shown that the density of aggregation in the central plane also varies at different distances. In Mr. Peirce's opinion, photometric observations have proved that this density increases from the seventh to the ninth magnitude, and that ​therefore the idea of the Milky Way itself as a vast ring of closest aggregation, including a more sparsely filled region, thus giving to the whole cluster, could it be seen laterally from a sufficient distance, an appearance not unlike that of the annular nebula in Lyra, is well founded. He also finds reason for supposing a similar but independent arrangement of the brightest stars, in the peculiar localization of the nearest of them, and the sudden falling off in density at about 30° from the galactic circle, above remarked—other systems of condensation requiring gradual changes. The true figure, if his reasoning is to be trusted, would therefore be a small ring of maximum density near the center of a very large one. Such speculations are, however, it is hardly necessary to say, very uncertain.
It is not easy to make out, from the general distribution of the stars, that our sun is in one direction rather than another from the center of the sidereal system, and there is even some doubt about the position which some astronomers give us, on the northern side of the plane of the Milky Way. Indeed, beyond the prevailing condensation toward this plane, it seems that no important general law governing star aggregation has yet been found. Mr. Proctor's services in calling attention to the grouping of certain portions of the heavens in subordinate systems having a common "star-drift," should not be overlooked; but his discovery that a large part of the southern hemisphere is particularly rich in stars can not be admitted for several reasons: 1. Behrmann's catalogue of southern stars, in which magnitudes were observed with particular care, shows nothing of the sort; 2. Mr. Proctor's own maps show nothing of the sort, for stars brighter than the sixth magnitude; and it is far less credible that an anomalous law of distribution holds over a wide area, affecting but this one order of brightness, than that those who observed this part of the heavens included more and fainter stars in their sixth magnitude than did northern observers; 3. Mr. Proctor's own maps show that the boundary of his "rich region" is the Tropic of Capricorn; and it is far less credible that an artificial circle should limit any law of distribution than that the whole difference is due to the fact that this tropic was also the northern boundary of La Caille's observations, the source, in all probability, whence the magnitudes of Mr. Proctor's stars were originally derived. Observers, in fact, are particularly likely to differ in estimating the extent of the sixth magnitude, for it seems to have been agreed by general consent that this magnitude shall include all stars to be seen with the unaided eye on the clearest nights, and ​differences of climate and of individual eye-sight affect this considerably. Argelander, and after him Heis, catalogued all the stars visible to their eyes; their numbers, for the whole northern hemisphere, where 2,350 and 3,936 respectively. Heis, must therefore, have seen stars at least four tenths of a magnitude fainter than Argelander's faintest. La Caille's eye must also have been sharper than the average; and, if Mr. Proctor had thought to apply the test of enumeration to the different magnitudes in different parts of the sky, this explanation would doubtless have occurred to him, and nothing have been heard of his remarkable "rich region." His observation is valuable, certainly; but only by showing the undeveloped state of the whole subject, and the precautions necessary before venturing conclusions on it.
The search for a common center, about which the uncounted millions of stars composing the galactic cluster may revolve, has tempted many investigators, but it can not be said as yet to have proved altogether successful. Mädler, by calculations from the proper motions of stars in different parts of the heavens, sought to locate it among the Pleiades; some later astronomers have preferred the Sword of Perseus; Mr. Maxwell Hall has just decided, and informed the Astronomical Society of England, that the universe turns about the South Elbow of Andromeda. The proof advanced is always incomplete, resting on assumptions not generally admitted; and when we remember that the gravitative force exerted by the fixed stars on one another is so small that to keep the nearest of them from falling to the sun, supposing no counterbalancing attractions, an angular velocity of but one second of arc in eighty years is needed; that the proper motions to be explained are often far larger than this; that the distance of the attracting center must be many times that of the nearest fixed star; and that the heavens give no sign of any preeminent body or group of bodies to which we may ascribe the enormous attractive power necessary to control these motions—the skepticism of many astronomers as to the universal center seems excusable.
It must be admitted, then, that but little of the true character of our sidereal system is known to us, and that all speculation upon it rests as yet on a very insecure foundation. But, as the sudden development of spectrum analysis has shown, matters of pure conjecture to-day may become entirely settled to-morrow; and it may reasonably be hoped that the secrets of this domain, if due interest be taken in them, will not much longer elude the search of scientific explorers.
* ↑ Professor Newcomb's account of these researches and speculations, in his "Popular Astronomy," pages 462-476, is full and interesting.
* ↑ "Photometric Researches," pages 175-178. The sun, it would seem, is to be considered as in a region of exceptional rarity as compared with other regions through which the galactic plane passes, and at the same time of exceptional density when the comparison includes stars remote from this plane.
* ↑ Most positively stated in a lecture before the Royal Institution, May, 1870; also in the introduction to his "Star Atlas."