'When a spherical mass of incandescent gas contracts through the loss of its heat by radiation into space, its temperature continually becomes higher as long as the gaseous condition is retained.'
To put it in another way: if the compression was caused by external force and no heat was lost, the globe would get hotter by a calculable amount for each unit of contraction. But the heat lost in causing a similar amount of contraction is so little more than the increase of heat produced by contraction, that the slightly diminished total heat in a smaller bulk causes the temperature of the mass to increase.
But if, as there is reason to believe, the various types of stars differ also in chemical constitution, some consisting mainly of the more permanent gases, while in others the various metallic and non-metallic elements are present in very different proportions, there should really be a classification by constitution as well as by temperature, and the course of evolution of the differently constituted groups may be to some extent dissimilar.
With this limitation the process of evolution and decay of sun through a cycle of increasing and decreasing temperature, as suggested by Sir Norman Lockyer, is clear and suggestive. During the ascending series the star is growing both in mass and heat, by the continual accretion of meteoritic matter either drawn to it by gravitation or falling towards it through the proper motions of independent masses. This goes on till all the matter for some distance around the star has been utilised, and a maximum of size, heat, and brilliancy attained. Then the loss of heat by radiation is no longer compensated by the influx of fresh matter, and a slow contraction occurs accompanied by a slightly increased temperature. But owing to the more stable conditions continuous envelopes of metals in the gaseous state are formed, which check the loss of heat and reduce the brilliancy of colour; whence it follows that bodies like our sun may be really hotter than the most brilliant white stars, though not giving out quite so much heat. The loss of heat is therefore reduced; and this may serve to account for the undoubted fact that during the enormous epochs of geological time there has been very little diminution in the amount of heat we have received from the sun.
On the general question of the meteoritic hypothesis one of our first mathematicians, Professor George Darwin, has thus expressed his views: 'The conception of the growth of the planetary bodies by the aggregation of meteorites is a good one, and perhaps seems more probable than the hypothesis that the whole solar system was gaseous.' I may add, that one of the chief objections made to it, that meteorites are too complex to be supposed to be the primitive matter out of which suns and worlds have been made, does not seem to me valid. The primitive matter, whatever it was, may have been used up again and again, and if collisions of large solid globes ever occur—and it is assumed by most astronomers that they must sometimes occur—then meteoric particles of all sizes would be produced which might exhibit any complexity of mineral constitution. The material universe has probably been in existence long enough for all the primitive elements to have been again and again combined into the minerals found upon the earth and many others. It cannot be too often repeated that no explanation—no theory—can ever take us to the beginning of things, but only one or two steps at a time into the dim past, which may enable us to comprehend, however imperfectly, the processes by which the world, or the universe, as it is, has been developed out of some earlier and simpler condition.
CHAPTER VII
ARE THE STARS INFINITE IN NUMBER?
Most of the critics of my first short discussion of this subject laid great stress upon the impossibility of proving that the universe, a part of which we see, is not infinite; and a well-known astronomer declared that unless it can be demonstrated that our universe is finite the entire argument founded upon our position within it fall to the ground. I had laid myself open to this objection by rather incautiously admitting that if the preponderance of evidence pointed in this direction any inquiry as to our place in the universe would be useless, because as regards infinity there can be no difference of position. But this statement is by no means exact, and even in an infinite universe of matter containing an infinite number of stars, such as those we see, there might well be such infinite diversities of distribution and arrangement as would give to certain positions all the advantages which I submit we actually possess. Supposing, for example, that beyond the vast ring of the Milky Way the stars rapidly decrease in number in all directions for a distance of a hundred or a thousand times the diameter of that ring, and that then for an equal distance they slowly increase again and become aggregated into systems or universes totally distinct from ours in form and structure, and so remote that they can influence us in no way whatever. Then, I maintain, our position within our own stellar universe might have exactly the same importance, and be equally suggestive, as if ours were the only material universe in existence—as if the apparent diminution in the number of stars (which is an observed fact) indicated a continuous diminution, leading at some unknown distance to entire absence of luminous—that is, of active, energy-emitting aggregations of matter.1 As to whether there are such other material universes or not I offer no opinion, and have no belief one way or the other. I consider all speculations as to what may or may not exist in infinite space to be utterly valueless. I have limited my inquiries strictly to the evidence accumulated by modern astronomers, and to direct inferences and logical deductions from that evidence. Yet, to my great surprise, my chief critic declares that 'Dr. Wallace's underlying error is, indeed, that he has reasoned from the area which we can embrace with our limited perceptions to the infinite beyond our mental or intellectual grasp.' I have distinctly not done this, but many astronomers have done so. The late Richard Proctor not only continually discussed the question of infinite matter as well as infinite space, but also argued, from the supposed attributes of the Deity, for the necessity of holding this material universe to be infinite, and the last chapter of his Other Worlds than Ours is mainly devoted to such speculations. In a later work, Our Place among Infinities, he says that 'the teachings of science bring us into the presence of the unquestionable infinities of time and of space, and the presumable infinities of matter and of operation—hence therefore into the presence of infinity of energy. But science teaches us nothing about these infinities as such. They remain none the less inconceivable, however clearly we may be taught to recognise their reality.' All this is very reasonable, and the last sentence is particularly important. Nevertheless, many writers allow their reasonings from facts to be influenced by these ideas of infinity. In Proctor's posthumous work, Old and New Astronomy, the late Mr. Ranyard, who edited it, writes: 'If we reject as abhorrent to our minds the supposition that the universe is not infinite, we are thrown back on one of two alternatives—either the ether which transmits the light of the stars to us is not perfectly elastic, or a large proportion of the light of the stars is obliterated by dark bodies.' Here we have a well-informed astronomer allowing his abhorrence of the idea of a finite universe to affect his reasoning on the actual phenomena we can observe—doing in fact exactly what my critic erroneously accuses me of doing. But setting aside all ideas and prepossessions of the kind here indicated, let us see what are the actual facts revealed by the best instruments of modern astronomy, and what are the natural and logical inferences from those facts.
Are the Stars Infinite in Number?
The views of those astronomers who have paid attention to this subject are, on the whole, in favour of the view that the stellar universe is limited in extent and the stars therefore limited in number. A few quotations will best exhibit their opinions on this question, with some of the facts and observations on which they are founded.
Miss A.M. Clerke, in her admirable volume, The System of the Stars, says: 'The sidereal world presents us, to all appearance, with a finite system.... The probability amounts almost to certainty that star-strewn space is of measurable dimensions. For from innumerable stars a limitless sum-total of radiations should be derived, by which darkness would be banished from our skies; and the "intense inane," glowing with the mingled beams of suns individually indistinguishable, would bewilder our feeble senses with its monotonous splendour.... Unless, that is to say, light suffer some degree of enfeeblement in space.... But there is not a particle of evidence that any such toll is exacted; contrary indications are strong; and the assertion that its payment is inevitable depends upon analogies which may be wholly visionary. We are then, for the present, entitled to disregard the problematical effect of a more than dubious cause.'
Professor Simon Newcomb, one of the first of American mathematicians and astronomers, arrives at a similar conclusion in his most recent volume, The Stars (1902). He says, in his conclusions at the end of the work: 'That collection of stars which we call the universe is limited in extent. The smallest stars that we see with the most powerful telescopes are not, for the most part, more distant than those a grade brighter, but are mostly stars of less luminosity situate in the same regions' (p. 319). And on page 229 of the same work he gives reasons for this conclusion, as follows: 'There is a law of optics which throws some light on the question. Suppose the stars to be scattered through infinite space so that every great portion of space is, in the general average, equally rich in stars. Then at some great distance we describe a sphere having its centre in our sun. Outside this sphere describe another one of a greater radius, and beyond this other spheres at equal distances apart indefinitely. Thus we shall have an endless succession of spherical shells, each of the same thickness. The volume of each of these shells will be nearly proportional to the squares of the diameters of the spheres which bound it. Hence each of the regions will contain a number of stars increasing as the square of the radius of the region. Since the amount of light we receive from each star is as the inverse square of its distance, it follows that the sum total of the light received from each of these spherical shells will be equal. Thus as we add sphere after sphere we add equal amounts of light without limit. The result would be that if the system of stars extended out indefinitely the whole heavens would be filled with a blaze of light as bright as the sun.'
But the whole light given us by the stars is variously estimated at from one-fortieth to one-twentieth or, as an extreme limit, to one-tenth of moonlight, while the sun gives as much light as 300,000 full moons, so that starlight is only equivalent at a fair estimate to the six-millionth part of sunlight. Keeping this in mind, the possible causes of the extinction of almost the whole of the light of the stars (if they are infinite in number and distributed, on the average, as thickly beyond the Milky Way as they are up to its outer boundary) are absurdly inadequate. These causes are (1) the loss of light in passing through the ether, and (2) the stoppage of light by dark stars or diffused meteoritic dust. As to the first, it is generally admitted that there is not a particle of evidence of its existence. There is, however, some distinct evidence that, if it exists, it is so very small in amount that it would not produce a perceptible effect for any distances less remote than hundreds or perhaps thousands of times as far as the farthest limits of the Milky Way are from us. This is indicated by the fact that the brightest stars are not always, or even generally, the nearest to us, as is shown both by their small proper motions and the absence of measurable parallax. Mr. Gore states that out of twenty-five stars, with proper motions of more than two seconds annually, only two are above the third magnitude. Many first magnitude stars, including Canopus, the second brightest star in the heavens, are so remote that no parallax can be found, notwithstanding repeated efforts. They must therefore be much farther off than many small and telescopic stars, and perhaps as far as the Milky Way, in which so many brilliant stars are found; whereas if any considerable amount of light were lost in passing that distance we should find but few stars of the first two or three magnitudes that were very remote from us. Of the twenty-three stars of the first magnitude, only ten have been found to have parallaxes of more than one-twentieth of a second, while five range from that small amount down to one or two hundredths of a second, and there are two with no ascertainable parallax. Again, there are 309 stars brighter than magnitude 3.5, yet only thirty-one of these have proper motions of more than 100" a century, and of these only eighteen have parallaxes of more than one-twentieth of a second. These figures are from tables given in Professor Newcomb's book, and they have very great significance, since they indicate that the brightest stars are not the nearest to us. More than this, they show that out of the seventy-two stars whose distance has been measured with some approach to certainty, only twenty-three (having a parallax of more than one-fiftieth of a second) are of greater magnitudes than 3.5, while no less than forty-nine are smaller stars down to the eighth or ninth magnitude, and these are on the average much nearer to us than the brighter stars!
Taking the whole of the stars whose parallaxes are given by Professor Newcomb, we find that the average parallax of the thirty-one bright stars (from 3.5 magnitude up to Sirius) is 0.11 seconds; while that of the forty-one stars below 3.5 magnitude down to about 9.5, is 0.21 seconds, showing that they are, on the average, only half as far from us as the brighter stars. The same conclusion was reached by Mr. Thomas Lewis of the Greenwich Observatory in 1895, namely, that the stars from 2.70 magnitude down to about 8.40 magnitude have, on the average, double the parallaxes of the brighter stars. This very curious and unexpected fact, however it may be accounted for, is directly opposed to the idea of there being any loss of light by the more distant as compared with the nearer stars; for if there should be such a loss it would render the above phenomenon still more difficult of explanation, because it would tend to exaggerate it. The bright stars being on the whole farther away from us than the less bright down to the eighth and ninth magnitudes, it follows, if there is any loss of light, that the bright stars are really brighter than they appear to us, because, owing to their enormous distance some of their light has been lost before it reached us. Of course it may be said that this does not demonstrate that no light is lost in passing through space; but, on the other hand, it is exactly the opposite of what we should expect if the more distant stars were perceptibly dimmed by this cause, and it may be considered to prove that if there is any loss it is exceedingly small, and will not affect the question of the limits of our stellar system, which is all that we are dealing with.
This remarkable fact of the enormous remoteness of the majority of the brighter stars is equally effective as an argument against the loss of light by dark stars or cosmic dust, because, if the light is not appreciably diminished for stars which have less than the fiftieth of a second of parallax, it cannot greatly interfere with our estimates of the limits of our universe.
Both Mr. E.W. Maunder of the Greenwich Observatory and Professor W.W. Turner of Oxford lay great stress on these dark bodies, and the former quotes Sir Robert Ball as saying, 'the dark stars are incomparably more numerous than those that we can see … and to attempt to number the stars of our universe by those whose transitory brightness we can perceive would be like estimating the number of horseshoes in England by those which are red-hot.' But the proportion of dark stars (or nebulæ) to bright ones cannot be determined a priori, since it must depend upon the causes that heat the stars, and how frequently those causes come into action as compared with the life of a bright star. We do know, both from the stability of the light of the stars during the historic period and much more precisely by the enormous epochs during which our sun has supported life upon this earth—yet which must have been 'incomparably' less than its whole existence as a light-giver—that the life of most stars must be counted by hundreds or perhaps by thousands of millions of years. But we have no knowledge whatever of the rate at which true stars are born. The so-called 'new stars' which occasionally appear evidently belong to a different category. They blaze out suddenly and almost as suddenly fade away into obscurity or total invisibility. But the true stars probably go through their stages of origin, growth, maturity, and decay, with extreme slowness, so that it is not as yet possible for us to determine by observation when they are born or when they die. In this respect they correspond to species in the organic world. They would probably first be known to us as stars or minute nebulæ: at the extreme limit of telescopic vision or of photographic sensitiveness, and the growth of their luminosity might be so gradual as to require hundreds, perhaps thousands of years to be distinctly recognisable. Hence the argument derived from the fact that we have never witnessed the birth of a true permanent star, and that, therefore, such occurrences are very rare, is valueless. New stars may arise every year or every day without our recognising them; and if this is the case, the reservoir of dark bodies, whether in the form of large masses or of clouds of cosmic dust, so far from being incomparably greater than the whole of the visible stars and nebulæ, may quite possibly be only equal to it, or at most a few times greater; and in that case, considering the enormous distances that separate the stars (or star-systems) from each other, they would have no appreciable effect in shutting out from our view any considerable proportion of the luminous bodies constituting our stellar universe. It follows, that Professor Newcomb's argument as to the very small total light given by the stars has not been even weakened by any of the facts or arguments adduced against it.
Mr. W.H.S. Monck, in a letter to Knowledge (May 1903), puts the case very strongly so as to support my view. He says:—'The highest estimate that I have seen of the total light of the full moon is 1/300000 of that of the sun. Suppose that the dark bodies were a hundred and fifty thousand times as numerous as the bright ones. Then the whole sky ought to be as bright as the illuminated portion of the moon. Every one knows that this is not so. But it is said that the stars, though infinite, may only extend to infinity in particular directions, e.g. in that of the Galaxy. Be it so. Where, in the very brightest portion of the Galaxy, will we find a part equal in angular magnitude to the moon which affords us the same quantity of light? In the very brightest spot, the light probably does not amount to one hundredth part that of the full moon.' It follows that, even if dark stars were fifteen million times as numerous as the bright ones, Professor Newcomb's argument would still apply against an infinite universe of stars of the same average density as the portion we see.
Telescopic Evidence as to the Limits of the Star System
Throughout the earlier portion of the nineteenth century every increase of power and of light-giving qualities of telescopes added so greatly to the number of the stars which became visible, that it was generally assumed that this increase would go on indefinitely, and that the stars were really infinite in number and could not be exhausted. But of late years it has been found that the increase in the number of stars visible in the larger telescopes was not so great as might be expected, while in many parts of the heavens a longer exposure of the photographic plate adds comparatively little to the number of stars obtained by a shorter exposure with the same instrument.
Mr. J.E. Gore's testimony on this point is very clear. He says:—'Those who do not give the subject sufficient consideration, seem to think that the number of the stars is practically infinite, or at least, that the number is so great that it cannot be estimated. But this idea is totally incorrect, and due to complete ignorance of telescopic revelations. It is certainly true that, to a certain extent, the larger the telescope used in the examination of the heavens, the more the number of the stars seems to increase; but we now know that there is a limit to this increase of telescopic vision. And the evidence clearly shows that we are rapidly approaching this limit. Although the number of stars visible in the Pleiades rapidly increases at first with increase in the size of the telescope used, and although photography has still further increased the number of stars in this remarkable cluster, it has recently been found that an increased length of exposure—beyond three hours—adds very few stars to the number visible on the photograph taken at the Paris Observatory in 1885, on which over two thousand stars can be counted. Even with this great number on so small an area of the heavens, comparatively large vacant places are visible between the stars, and a glance at the original photograph is sufficient to show that there would be ample room for many times the number actually visible. I find that if the whole heavens were as rich in stars as the Pleiades, there would be only thirty-three millions in both hemispheres.'
Again, referring to the fact that Celoria, with a telescope showing stars down to the eleventh magnitude, could see almost exactly the same number of stars near the north pole of the Galaxy as Sir William Herschel found with his much larger and more powerful telescope, he remarks: 'Their absence, therefore, seems certain proof that very faint stars do not exist in that direction, and that here, at least, the sidereal universe is limited in extent.'
Sir John Herschel notes the same phenomena, stating that even in the Milky Way there are found 'spaces absolutely dark and completely void of any star, even of the smallest telescopic magnitude'; while in other parts 'extremely minute stars, though never altogether wanting, occur in numbers so moderate as to lead us irresistibly to the conclusion that in these regions we see fairly through the starry stratum, since it is impossible otherwise (supposing their light not intercepted) that the numbers of the smaller magnitudes should not go on continually increasing ad infinitum. In such cases, moreover, the ground of the heavens, as seen between the stars, is for the most part perfectly dark, which again would not be the case if innumerable multitudes of stars, too minute to be individually discernible, existed beyond.' And again he sums up as follows:—'Throughout by far the larger portion of the extent of the Milky Way in both hemispheres, the general blackness of the ground of the heavens on which its stars are projected, and the absence of that innumerable multitude and excessive crowding of the smallest visible magnitudes, and of glare produced by the aggregate light of multitudes too small to affect the eye singly, which the contrary supposition would appear to necessitate, must, we think, be considered unequivocal indications that its dimensions in directions where these conditions obtain, are not only not infinite, but that the space-penetrating power of our telescopes suffices fairly to pierce through and beyond it.'2
This expression of opinion by the astronomer who, probably beyond any now living, was the most competent authority on this question, to which he devoted a long life of observation and study extending over the whole heavens, cannot be lightly set aside by the opinions or conjectures of those who seem to assume that we must believe in an infinity of stars if the contrary cannot be absolutely proved. But as not a particle of evidence can be adduced to prove infinity, and as all the facts and indications point, as here shown, in a directly opposite direction, we must, if we are to trust to evidence at all in this matter, arrive at the conclusion that the universe of stars is limited in extent.
Dr. Isaac Roberts gives similar evidence as regards the use of photographic plates. He writes:—'Eleven years ago photographs of the Great Nebula in Andromeda were taken with the 20-inch reflector, and exposures of the plates during intervals up to four hours; and upon some of them were depicted stars to the faintness of 17th to 18th magnitude, and nebulosity to an equal degree of faintness. The films of the plates obtainable in those days were less sensitive than those which have been available during the past five years, and during this period photographs of the nebula with exposures up to four hours have been taken with the 20-inch reflector. No extensions of the nebulosity, however, nor increase in the number of the stars can be seen on the later rapid plates than were depicted upon the earlier slower ones, though the star-images and the nebulosity have greater density on the later plates.'
Exactly similar facts are recorded in the cases of the Great Nebula in Orion, and the group of the Pleiades. In the case of the Milky Way in Cygnus photographs have been taken with the same instrument, but with exposures varying from one hour to two hours and a half, but no fainter stars could be found on one than on the other; and this fact has been confirmed by similar photographs of other areas in the sky.
The Law of Diminishing Numbers of Stars
We will now consider another kind of evidence equally weighty with the two already adduced. This is what may be termed the law of diminishing numbers beyond a certain magnitude, as observed by larger and larger telescopes.
For some years past star-magnitudes have been determined very accurately by means of careful photometric comparisons. Down to the sixth magnitude stars are visible to the naked eye, and are hence termed lucid stars. All fainter stars are telescopic, and continuing the magnitudes in a series in which the difference in luminosity between each successive magnitude is equal, the seventeenth magnitude is reached and indicates the range of visibility in the largest telescopes now in existence. By the scale now used a star of any magnitude gives nearly two and a half times as much light as one of the next lower magnitude, and for accurate comparison the apparent brightness of each star is given to the tenth of a magnitude which can easily be observed. Of course, owing to differences in the colour of stars, these determinations cannot be made with perfect accuracy, but no important error is due to this cause. According to this scale a sixth magnitude star gives about one-hundredth part of the light of an average first magnitude star. Sirius is so exceptionally bright that it gives nine times as much light as a standard or average first magnitude star.
Now it is found that from the first to the sixth magnitude the stars increase in number at the rate of about three and a half times those of the preceding magnitudes. The total number of stars down to the sixth magnitude is given by Professor Newcomb as 7647. For higher magnitudes the numbers are so great that precision and uniformity are more difficult of attainment; yet there is a wonderful continuance of the same law of increase down to the tenth magnitude, which is estimated to include 2,311,000 stars, thus conforming very nearly with the ratio of 3.5 as determined by the lucid stars.
But when we pass beyond the tenth magnitude to those vast numbers of faint stars only to be seen in the best or the largest telescopes, there appears to be a sudden change in the ratio of increased numbers per magnitude. The numbers of these stars are so great that it is impossible to count the whole as with the higher magnitude stars, but numerous counts have been made by many astronomers in small measured areas in different parts of the heavens, so that a fair average has been obtained, and it is possible to make a near approximation to the total number visible down to the seventeenth magnitude. The estimate of these by astronomers who have made a special study of this subject is, that the total number of visible stars does not exceed one hundred millions.3
But if we take the number of stars down to the ninth magnitude, which are known with considerable accuracy, and find the numbers in each succeeding magnitude down to the seventeenth, according to the same ratio of increase which has been found to correspond very nearly in the case of the higher magnitudes, Mr. J.E. Gore finds that the total number should be about 1400 millions. Of course neither of these estimates makes any pretence to exact accuracy, but they are founded on all the facts at present available, and are generally accepted by astronomers as being the nearest approach that can be made to the true numbers. The discrepancy is, however, so enormous that probably no careful observer of the heavens with very large telescopes doubts that there is a very real and very rapid diminution in the numbers of the fainter as compared with the brighter stars.