DYNAMICAL THEORY OF THE GLOBULAR CLUSTERS By T. J. J. SEE. (Read April 19, 1912.) (PLATES VIII (bis) AND IX.) I. INTRODUCTORY REMARKS. More than a century and a quarter have elapsed since it was confidently announced by Sir William Herschel that sidereal systems made up of thousands of stars exhibit the effects of a clustering power which is everywhere moulding these systems into symmetrical figures, as if by the continued action of central forces (Phil. Trans., 1785, p. 255, and 1789, pp. 218-226). In support of this view he cited especially the figures of the planetary nebulæ, and the globular clusters, as well as the more expanded and irregular swarms and clouds of stars visible to the naked eye along the course of the Milky Way, which thus appears to traverse the heavens as a clustering stream. And yet notwithstanding the early date of this announcement and the unrivaled eminence of Herschel, it is only very recently that astronomers have begun to consider the origin of sidereal systems of the highest order. The historical difficulty of solving the problem of n-bodies, when n exceeds 2, which dates from the establishment of the law of universal gravitation by Newton in 1687, will sufficiently account for the restriction of the researches of mathematicians to the planetary system, where the central masses always are very predominant, the orbits almost circular and nearly in a common plane, and to other simple systems such as the double and multiple stars: but owing to the general prevalence of the clustering tendency pointed out by Herschel and now found to be at work throughout the sidereal uni verse, it becomes necessary for the modern investigator to consider also the higher orders of sidereal systems, including those made up of thousands and even millions of stars. It is only by such a comprehensive view of nature, which embraces and unites all types of systems under one common principle, that we may hope to establish the most general laws governing the evolution of the sidereal universe. Accordingly, although the strict mathematical treatment of the great historical problem of n-bodies is but little advanced by the recent researches of geometers, yet if we could arrive at the general secular tendency in nature, from the observational study of the phenomena presented by highly complex systems of stars, operating under known laws of attractive and repulsive forces, the former for gathering the matter into large masses, the latter for redistributing it in the form of fine dust, the result of such an investigation would guide us towards a grasp of problems too complex for rigorous treatment by any known method of analysis. Now it happens that in the second volume of the "Researches on the Evolution of the Stellar System," 1910, the writer was able to establish great generality in the processes of cosmogony, and to show that the universal tendency in nature is for the large bodies to drift towards the most powerful centers of attraction, while the only throwing off of masses that ever takes place is that of small particles expelled from the stars under the action of repulsive forces and driven away for the formation of new nebulæ. The repulsive forces thus operate to counteract the clustering tendency noticed by the elder Herschel, and so clearly foreseen by Newton as an inevitable effect of universal gravitation upon the motions of the solar system that he believed the intervention of the Deity eventually would become necessary for the restoration of the order of the world (cf. Newton's "Letters to Bentley," Brewster's "Life of Newton,” Vol. II., and Chapter XVII., and Appendix X). But whilst the argument developed in the second volume of my "Researches" gives unexpected simplicity, uniformity and continuity to the processes of cosmogony, there has not yet been developed, so far as I know, any precise investigation of the attractive forces operating in globular clusters, which might disclose the nature of the clustering power noticed by Herschel to be in progress throughout the sidereal universe. Such an investigation of the central forces governing the motions in clusters is very desirable, because it might be expected to throw light on the mode of evolution of clusters as the highest type of the perfect sidereal system. If it can be shown that a clustering power is really at work, and is of such a nature as to produce these globular masses of stars, it will be less important to consider the details of those systems which have not yet reached a state of symmetry and full maturity; for the governing principle being established for the most perfect types, it must be held to be the same in all. II. GENERAL EXPRESSIONS FOR THE POTENTIAL OF AN ATTRACTING MASS. If we have a mass M' of any figure whatever, in which the law of density is o' = f(x', y', z′), where (x', y', ≈′) are the coördinates of the element dm' of the attracting mass, and this element attracts a unit mass whose coördinates are (x, y, z); then the element of the attracting mass is And the expressions for the forces acting on the unit mass when resolved along the coördinate axes become In spherical coördinates we may take the angle o for the longitude, for the latitude, and r for the radius of the sphere; and then the required expressions become The element of mass dm' defined in (1) has the equivalent form The element of the potential due to this differential element is (4) (5) (6) If we make use of the equations (1), (4), (5) in equation (2) we may obtain the corresponding expressions for the forces resolved along the coördinate axes: These expressions will hold rigorously true for any law of density whatever, so long as it is finite and continuous. In the physical universe these conditions always are fulfilled; and hence if these several integrals can be evaluated, they will give the potentials and forces exerted on a unit mass by an attracting body such as a cluster of stars, or the spherical shell surrounding the nucleus of a cluster. But before considering the attraction of a cluster in detail, we shall first examine the cumulative effect of central forces on the law of density. The problem is intricate and must be treated by methods of great generality, but as it will elucidate the subsequent procedure for determining the attraction of such a mass upon a neighboring point, we shall give the analysis with enough detail to establish clearly the secular effect of close appulses of individual stars upon the figure and internal arrangement of these wonderful masses of stars. III. THE CUMULATIVE EFFECT of the CenTRAL FORCES UPON THE FIGURE AND COMPRESSION OF A GLOBULAR CLUSTER OF STARS. Suppose a globular cluster of stars to be in a moderate state of compression, with density increasing towards the center. Imagine the whole of the mass at the epoch to to be divided into two parts by a spherical surface of radius r, drawn about the center of gravity of the entire system; and let the external boundary of the cluster be R, so chosen that no star, from the motions existing at the initial epoch, will cross the border r=R. The stars in the outer shell, between the surfaces r and R, with coördinates (r', y', '), will give rise to a potential U. Those of the nucleus or series of internal shells, between r=0, and r=r, with coördinates (x, y, z), will give rise to a potential V. Accordingly we have U = o'dx' dy' dz' − SS S √(x − x )2 + (y' − y)2 + (5′ — 2)* odxdydz — √ (x' − x)2 + (y' − y)2 + (≈′ — ≈) And the forces resolved along the coördinate axes are X= дх o'('-x)da'dy'dz' -SSS SSS SSS (9) (10) |