We must first explain that in order to obtain a clear view of the construction of the visible universe, it would be necessary to know the relative distances of a large number of stars; but as the distances of only a few stars from the earth have yet been determined by actual measurement, and the results hitherto obtained are open to much uncertainty, we must have recourse to some other method of estimating the distances. While travelling in a railway carriage, if we fix our attention on trees, buildings, and other objects we pass on our journey, it will be noticed that all objects apparently move past us in the opposite direction to that in which we are travelling, and that the nearer the object is the faster it seems to move with reference to distant objects near the horizon. So it is with the stars. As we showed in Chapter III., the sun is moving through space, carrying along with the earth all the planets, satellites, and comets, forming the solar system. The effect of this motion is to cause an apparent small motion of the stars in the opposite direction, and the nearer the star is to the earth, the greater will this apparent motion seem to be as in the case of the railway train. In addition to this apparent motion, the stars are themselves—like the sun—moving through space, and this real motion is also visible. If this real motion takes place in the opposite direction to that in which the sun and earth are moving, it will add to the apparent motion, and will increase the star’s “proper motion,” as it is termed. If, on the other hand, the real motion is in the same direction as the earth’s motion, the proper motion will be diminished. In either case, the nearer the star is to the earth, the greater will be its apparent annual displacement on the background of the heavens. The amount of the “proper motion” is, therefore, considered by astronomers to form a reliable criterion of the star’s distance from the earth, and the actual measures of distance which have been made show that this assumption is approximately true. Of fourteen stars which have proper motion of over three seconds of arc per annum, eleven have yielded a measurable parallax, or displacement, due to the earth’s annual motion round the sun; that is to say, eleven out of fourteen fast-moving stars are within a measurable distance of the earth, and are, therefore, near us, when compared with the great majority of stars which are not within measurable distance, or, at least, are beyond the reach of our present methods of measurement.

In the case of small groups of stars, we may assume that the real motions of the individual stars take place indifferently in all directions, and that consequently, taking an average of all the motions of the stars composing the group, the effects due to the real motions will destroy each other, and there will remain, as the most reliable criterion, the effect due to the sun’s motion in space. If, however, we compare the proper motions of groups situated in different parts of the sky, there is a consideration which, to a great extent, vitiates this conclusion. For, near the point of the heavens, towards which the sun and earth are moving, known as the “apex of the solar way,” and probably situated not far from the bright star Vega, as indicated by recent researches, and near the point away from which the sun is moving known as the ant-apex, about 15° south of Sirius, there will be no apparent displacement due to the solar motion through space, as this motion takes place in the line of sight with reference to these points of the sky. The observed proper motion at these points will, therefore, be solely due to the real motions of the stars themselves in those regions. In other parts of the heavens, however, the total proper motion will be a combination of the apparent and real motions of the stars, and for stars in different parts of the sky, it will not follow that stars having equal proper motions are necessarily at the same distance from the earth. To make this point clearer, let us suppose that there are two stars at absolutely the same distance from the earth, one situated at or near the solar “apex,” and the other at a point 90° from the apex, and let us suppose that both stars are moving through space with exactly the same velocity and in the same direction, say at right angles to the direction of the solar motion. Then in the case of the star near the apex, the observed “proper motion” will be solely due to the star’s real motion, and in the star 90° distant from the apex, the proper motion will be solely due to the solar motion, as the star’s real motion, being in the line of sight, will not be visible. Now, unless the stellar motion and the solar motion happen to be equal, the observed “proper motions” will not be equal, although both stars are at the same distance from the earth. If both the stars are really at rest, the star at the apex will have no proper motion, while the star 90° distant will have an apparent proper motion due to the sun’s motion. To overcome this source of error in estimating the distance of a star from its proper motion, Professor Kapteyn made use of another measure, which is independent of the solar motion. This is the component of the proper motion measured at right angles to a great circle of the sphere passing through a star and the solar apex. The amount of motion in this direction will evidently not be affected by the sun’s motion, and from a discussion of the stars, contained in the Draper “Catalogue of Stellar Spectra,” which were observed by Bradley (and of which the proper motions are now known with accuracy), Professor Kapteyn finds that this motion is “nearly inversely proportional to the distance,” that is, the greater the motion, the less the distance of the stars, and the smaller the motion, the greater the distance. Excluding stars with proper motions greater than half a second of arc per annum, Professor Kapteyn found that for stars at various distances from the Milky Way this component of the “proper motion” forms a good measure of distance.

As the result of his investigations on the subject, Professor Kapteyn arrives at the following conclusions. Neglecting stars with small or imperceptible proper motions, we have a group of stars which no longer show any condensation in a plane. Stars with very small or no proper motions show a condensation towards the plane of the Milky Way. This applies to stars of the second or solar type, as well as to those of the first or Sirian type of spectrum, and evidently indicates that the stars composing the Milky Way lie at a great distance from the earth. The extreme faintness of the majority of the stars composing the Galaxy seems in favour of this conclusion. The condensation of stars of the first type is more marked than those of the second, and this agrees with the fact which has been noticed by Professor Pickering, that the majority of the brighter stars of the Milky Way have spectra of the Sirian type.

Professor Kapteyn finds that this condensation of stars with small proper motions is very perceptible even for stars visible to the naked eye, and is as well marked in those stars which have spectra of the second type as for all the stars of the ninth magnitude; but for stars of the first type the condensation is still more marked. He considers that this condensation is either partly real, or that there is a real thinning out of stars near the pole of the Milky Way. As already mentioned (in the beginning of this chapter), Celoria’s observations with a small telescope, compared with Sir William Herschel’s observations with a large telescope, indicate clearly that there is a real thinning out of stars near the poles of the Galaxy.

Professor Kapteyn concludes that the arrangement of the stars suggested by Struve—a modification of the “disc theory”—has no real existence.^[153] He attributes the fallacy in Struve’s hypothesis to the fact that the mean distance of stars of a given magnitude in the Milky Way, and outside it, is not the same.

Professor Kapteyn finds that the vicinity of the sun is almost exclusively occupied by stars of the second or solar type, a conclusion which evidently tends to strengthen Dr. Gould’s theory of a “solar cluster.” He finds that the number of Sirian type stars increases gradually with the distance, and that beyond a distance corresponding to a proper motion of about ¹⁄₁₄th of a second of arc per annum, the Sirian stars largely predominate. In the group of stars known as the Hyades, however, the components of which have a common proper motion both in amount and direction, stars of the first and second types appear to be mixed, and Professor Kapteyn assumes that the two types represent different phases of evolution, and that as the brightest stars of the group are chiefly of the solar type, these stars must be the largest of the group. From this fact he concludes the solar type stars are in a less advanced stage of evolution than those of the Sirian type. This does not agree with the generally accepted view. Professor Vogel considers the Sirian stars to represent an earlier stage of stellar evolution. Mr. Proctor held the same opinion, and in Professor Lockyer’s hypothesis of increasing and decreasing temperatures in stars of various types, he places the Sirian stars at the summit of the evolution curve, and the sun and solar stars just below them on the descending branch of the curve.^[154] These hypotheses are in conformity also with the current opinion that the sun is a cooling body. The discrepancy may perhaps be explained by supposing that the brighter stars of the Hyades form a connected group, and that some, at least, of the fainter stars do not belong to the group, but lie at a great distance behind it. In the case of the Pleiades, which form a more evident cluster, I find from the Draper “Catalogue of Stellar Spectra” that the great majority of the brighter stars have spectra of the Sirian type. Most of the stars in the Pleiades have a very similar proper motion, both in amount and in direction, and there can be no doubt that most of the brighter stars, at least, form a connected system. As already stated, it seems highly probable that the fainter stars in the Pleiades lie far beyond the brighter components, and have merely an optical connexion with them, and the same may be the case in the Hyades. The superior brilliancy of the stars composing the Hyades would suggest that they are nearer to the earth than the Pleiades group, and they may possibly form members of Gould’s “solar cluster.”

Assuming that the distances are inversely proportional to the proper motions, Professor Kapteyn computes the relative volumes of the spherical shells which contain the stars with different proper motions (from one-tenth of a second to one second of arc and more). Comparing these volumes with the corresponding number of stars, we arrive at an estimate of the density of star distribution at various distances. The result of this calculation shows that the distribution of stars of the Sirian type approaches uniformity when a large number of the faint stars (ninth magnitude) are considered. With reference to the stars of the second type, however, the larger the proper motion the greater the number of the stars; or, in other words, the second type, or solar stars, are crowded together in the sun’s vicinity. Evidence in favour of this conclusion is afforded by the fact that, of eight stars having the largest measured parallax (and whose spectrum has been determined), I find that seven have spectra of the solar type. The exception is Sirius, which is evidently an exceptional star with reference to its brightness and comparative proximity to the earth, no other star of the first magnitude having nearly so large a parallax. Indeed, the average distance of all the first magnitude stars is about forty times the distance of Sirius.

Professor Kapteyn finds that the centre of greatest condensation of the solar type stars lies near a point situated about ten degrees to the west of the great nebula in Andromeda, and that this centre nearly coincides with the point which, according to Struve and Herschel, represents the apparent centre of the Milky Way considered as a ring. This would indicate that the sun and solar system lie a little to the north of the Milky Way, and towards a point situated in the northern portion of the constellation of the Centaur. The fact is worth noting, that the nearest fixed star to the earth, Alpha Centauri, lies not very far from this point. Possibly there may be other stars in this direction having a measured parallax, as the southern portion of the heavens has not yet been thoroughly explored.

Professor Kapteyn finds that for stars of equal brightness, those of the Sirian type are, on an average, about two and three-quarter times farther from the earth than those of the solar type. Now, as light varies inversely as the square of the distance, this would imply that the Sirian stars are intrinsically brighter than those of the solar type. This conclusion is confirmed by the great brilliancy of Sirius and other stars of the same type in proportion to their mass. I have shown in Chapter IV. that Sirius is about ten times brighter than the sun would be if placed at the same distance, although its mass is only twice the sun’s mass, as computed from the orbit of its satellite.

The general conclusions to be derived from the above results seems to be that the sun is a member of a cluster of stars, possibly distributed in the form of a ring, and that outside this ring, at a much greater distance from us than the stars of the solar cluster, lies a considerably richer ring-shaped cluster, the light of which, reduced to nebulosity by immensity of distance, produces the Milky Way gleam of our midnight skies.