But our mathematical astronomers can find no indications of such stability of the stellar universe as a whole, if subject to the law of gravitation alone. In reply to some questions on this point, my friend Professor George Darwin writes as follows:—'A symmetrical annular system of bodies might revolve in a circle with or without a central body. Such a system would be unstable. If the bodies are of unequal masses and not symmetrically disposed, the break-up of the system would probably be more rapid than in the ideal case of symmetry.'
This would imply that the great annular system of the Milky Way is unstable. But if so, its existence at all is a greater mystery than ever. Although in detail its structure is very irregular, as a whole it is wonderfully symmetrical; and it seems quite impossible that its generally circular ring-like form can be the result of the chance aggregation of matter from any pre-existing different form. Star-clusters are equally unstable, or, rather, nothing is known or can be predicated about their stability or instability, according to Professors Newcomb and Darwin.
Mr. E.T. Whittaker (Secretary to the Royal Astronomical Society), to whom Professor G. Darwin sent my questions, writes:—'I doubt whether the principal phenomena of the stellar universe are consequences of the law of gravitation at all. I have been working myself at spiral nebulæ, and have got a first approximation to an explanation—but it is electro-dynamical and not gravitational. In fact, it may be questioned whether, for bodies of such tremendous extent as the Milky Way or nebulæ, the effect which we call gravitation is given by Newton's law; just as the ordinary formulæ of electrostatic attraction break down when we consider charges moving with very great velocities.'
Accepting these statements and opinions of two mathematicians who have given special attention to similar problems, we need not limit ourselves to the laws of gravitation as having determined the present form of the stellar universe; and this is the more important because we may thus escape from a conclusion which many astronomers seem to think inevitable, viz. that the observed proper motions of the stars cannot be explained by the gravitative forces of the system itself. In chapter VIII. of this work I have quoted Professor Newcomb's calculation as to the effect of gravitation in a universe of 100 million stars, each five times the mass of our sun, and spread over a sphere which it would take light 30,000 years to cross; then, a body falling from its outer limits to the centre could at the utmost acquire a velocity of twenty-five miles a second; and therefore, any body in any part of such a universe having a greater velocity would pass away into infinite space. Now, as several stars have, it is believed, much more than this velocity, it follows not only that they will inevitably escape from our universe, but that they do not belong to it, as their great velocity must have been acquired elsewhere. This seems to have been the idea of the astronomer who stated that, even at the very moderate speed of our sun, we should in five million years be deep in the actual stream of the Milky Way. To this I have already sufficiently replied; but I now wish to bring before my readers an excellent illustration of the importance of the late Professor Huxley's remark, that the results you got out of the 'mathematical mill' depend entirely on what you put into it.
In the Philosophical Magazine (January 1902) is a remarkable article by Lord Kelvin, in which he discusses the very same problem as that which Professor Newcomb had discussed at a much earlier date, but, starting from different assumptions, equally based on ascertained facts and probabilities deduced from them, brings out a very different result.
Lord Kelvin postulates a sphere of such a radius that a star at its confines would have a parallax of one-thousandth part of a second (0".001), equivalent to 3215 light-years. Uniformly distributed through this sphere there is matter equal in mass to 1000 million suns like ours. If this matter becomes subject to gravitation, it all begins to move at first with almost infinite slowness, especially near its centre; but nevertheless, in twenty-five million years many of these suns would have acquired velocities of from twelve to twenty miles a second, while some would have less and some probably more than seventy miles a second. Now such velocities as these agree generally with the measured velocities of the stars, hence Lord Kelvin thinks there may be as much matter as 1000 million suns within the above-named distance. He then states that if we suppose there to be 10,000 million suns within the same sphere, velocities would be produced very much greater than the known star-velocities; hence it is probable that there is very much less matter than 10,000 million times the sun's mass. He also states that if the matter were not uniformly distributed within the sphere, then, whatever was the irregularity, the acquired motions would be greater; again indicating that the 1000 million suns would be ample to produce the observed effects of stellar motion. He then calculates the average distance apart of each of the 1000 million stars, which he finds to be about 300 millions of millions of miles. Now the nearest star to our sun is about twenty-six million million of miles distant, and, as the evidence shows, is situated in the denser part of the solar cluster. This gives ample allowance for the comparative emptiness of the space between our cluster and the Milky Way, as well as of the whole region towards the poles of the Milky Way (as shown by the diagrams in chapter IV.), while the comparative density of extensive portions of the Galaxy itself may serve to make up the average.
Now, previous writers have come to a different conclusion from the same general line of argument, because they have started with different assumptions. Professor Newcomb, whose statement made some years back is usually followed, assumed 100 million stars each five times as large as our sun, equal to 500 million suns in all, and he distributed them equally throughout a sphere 30,000 light-years in diameter. Thus he has half the amount of matter assumed by Lord Kelvin, but nearly five times the extent, the result being that gravity could only produce a maximum speed of twenty-five miles a second; whereas on Lord Kelvin's assumption a maximum speed of seventy miles a second would be produced, or even more. By this latter calculation we find no insuperable difficulty in the speed of any of the stars being beyond the power of gravitation to produce, because the rates here given are the direct results of gravitation acting on bodies almost uniformly distributed through space. Irregular distribution, such as we see everywhere in the universe, might lead to both greater and less velocities; and if we further take account of collisions and near approaches of large masses resulting in explosive disruptions, we might have almost any amount of motion as the result, but as this motion would be produced by gravitation within the system, it could equally well be controlled by gravitation.
DIAGRAM OF STELLAR UNIVERSE (Plan).
1. Central part of Solar Cluster. 3. Outer limit of Solar Cluster.
2. Sun's Orbit (Black spot). 4. Milky Way.