be expected to have escaped the common fate." Jeans gives a careful calculation from which it is possible to derive some idea of the probability of any given degree of approach of the sun and some other star. Of course all such calculations must be based on certain assumptions. The assumptions made by Jeans are such as to make the probability of close approaches as great as possible. For example, he allows only 560 million years for the entire evolution of the sun, whereas some astronomers and geologists would put the figure ten or more times as high. Nevertheless, Jeans' assumptions at least show the order of magnitude which we may expect on the basis of reasonable astronomical conclusions.
According to the planetary hypothesis of sunspots, the difference in the effect of Jupiter when it is nearest and farthest from the sun is the main factor in starting the sunspot cycle and hence the corresponding terrestrial cycle. The climatic difference between sunspot maxima and minima, as measured by temperature, apparently amounts to at least a twentieth and perhaps a tenth of the difference between the climate of the last glacial epoch and the present. We may suppose, then, that a body which introduced a gravitative or electrical factor twenty times as great as the difference in Jupiter's effect at its maximum and minimum distances from the sun would cause a glacial epoch if the effect lasted long enough. Of course the other planets combine their effects with that of Jupiter, but for the sake of simplicity we will leave the others out of account. The difference between Jupiter's maximum and minimum tidal effect on the sun amounts to 29 per cent of the planet's average effect. The corresponding difference, according to the electrical hypothesis, is about 19 per cent, for electrostatic action varies as the square of the distance instead of as the cube.
Let us assume that a body exerting four times Jupiter's present tidal effect and placed at the average distance of Jupiter from the sun would disturb the sun's atmosphere twenty times as much as the present difference between sunspot maxima and minima, and thus, perhaps, cause a glacial period on the earth.
On the basis of this assumption our first problem is to estimate the frequency with which a star, visible or dark, is likely to approach near enough to the sun to produce a tidal effect four times that of Jupiter. The number of visible stars is known or at least well estimated. As to dark stars, which have grown cool, Arrhenius believed that they are a hundred times as numerous as bright stars; few astronomers believe that there are less than three or four times as many. Dr. Shapley of the Harvard Observatory states that a new investigation of the matter suggests that eight or ten is probably a maximum figure. Let us assume that nine is correct. The average visible star, so far as measured, has a mass about twice that of the sun, or about 2100 times that of Jupiter. The distances of the stars have been measured in hundreds of cases and thus we can estimate how many stars, both visible and invisible, are on an average contained in a given volume of space. On this basis Jeans estimates that there is only one chance in thirty billion years that a visible star will approach within 2.8 times the distance of Neptune from the sun, that is, within about eight billion miles. If we include the invisible stars the chances become one in three billion years. In order to produce four times the tidal effect of Jupiter, however, the average star would have to approach within about four billion miles of the sun, and the chances of that are only one in twelve billion years. The disturbing star
would be only 40 per cent farther from the sun than Neptune, and would almost pass within the solar system.
Even though Jeans holds that the frequency of the mutual approach of the sun and a star was probably much greater in the distant past than at present, the figures just given lend little support to the tidal hypothesis. In fact, they apparently throw it out of court. It will be remembered that Jeans has made assumptions which give as high a frequency of stellar encounters as is consistent with the astronomical facts. We have assumed nine dark stars for every bright one, which may be a liberal estimate. Also, although we have assumed that a disturbance of the sun's atmosphere sufficient to cause a glacial period would arise from a tidal effect only twenty times as great as the difference in Jupiter's effect when nearest the sun and farthest away, in our computations this has actually been reduced to thirteen. With all these favorable assumptions the chances of a stellar approach of the sort here described are now only one in twelve billion years. Yet within a hundred million years, according to many estimates of geological time, and almost certainly within a billion, there have been at least half a dozen glaciations.
Our use of Jeans' data interposes another and equally insuperable difficulty to any tidal hypothesis. Four billion miles is a very short distance in the eyes of an astronomer. At that distance a star twice the size of the sun would attract the outer planets more strongly than the sun itself, and might capture them. If a star should come within four billion miles of the sun, its effect in distorting the orbits of all the planets would be great. If this had happened often enough to cause all the glaciations known to geologists, the planetary orbits would be strongly elliptical instead of almost circular. The considerations
here advanced militate so strongly against the tidal hypothesis of solar disturbances that it seems scarcely worth while to consider it further.
Let us turn now to the electrical hypothesis. Here the conditions are fundamentally different from those of the tidal hypothesis. In the first place the electrostatic effect of a body has nothing to do with its mass, but depends on the area of its surface; that is, it varies as the square of the radius. Second, the emission of electrons varies exponentially. If hot glowing stars follow the same law as black bodies at lower temperatures, the emission of electrons, like the emission of other kinds of energy, varies as the fourth power of the absolute temperature. In other words, suppose there are two black bodies, otherwise alike, but one with a temperature of 27° C. or 300° on the absolute scale, and the other with 600° on the absolute scale. The temperature of one is twice as high as that of the other, but the electrostatic effect will be sixteen times as great.[118] Third, the number of electrons
that reach a given body varies inversely as the square of the distance, instead of as the cube which is the case with tide-making forces.