In order to use these three principles in calculating the effect of the stars we must know the diameters, distances, temperature, and number of the stars. The distances and number may safely be taken as given by Jeans in the calculations already cited. As to the diameters, the measurements of the stars thus far made indicate that the average mass is about twice that of the sun. The average density, as deduced by Shapley[119] from the movements of double stars, is about one-eighth the solar density. This would give an average diameter about two and a half times that of the sun. For the dark stars, we shall assume for convenience that they are ten times as numerous as the bright ones. We shall also assume that their diameter is half that of the sun, for being cool they must be relatively dense, and that their temperature is the same as that which we shall assume for Jupiter.
As to Jupiter we shall continue our former assumption that a body with four times the effectiveness of that planet, which here means with twice as great a radius, would disturb the sun enough to cause glaciation. It would produce about twenty times the electrostatic effect
which now appears to be associated with the difference in Jupiter's effect at maximum and minimum. The temperature of Jupiter must also be taken into account. The planet is supposed to be hot because its density is low, being only about 1.25 that of water. Nevertheless, it is probably not luminous, for as Moulton[120] puts it, shadows upon it are black and its moons show no sign of illumination except from the sun. Hence a temperature of about 600°C., or approximately 900° on the absolute scale, seems to be the highest that can reasonably be assigned to the cold outer layer whence electrons are emitted. As to the temperature of the sun, we shall adopt the common estimate of about 6300°C. on the absolute scale. The other stars will be taken as averaging the same, although of course they vary greatly.
When Jeans' method of calculating the probability of a mutual approach of the sun and a star is applied to the assumptions given above, the results are as shown in Table 5. On that basis the dark stars seem to be of negligible importance so far as the electrical hypothesis is concerned. Even though they may be ten times as numerous as the bright ones there appears to be only one chance in 130 billion years that one of them will approach the sun closely enough to cause the assumed disturbance of the solar atmosphere. On the other hand, if all the visible stars were the size of the sun, and as hot as that body, their electrical effect would be fourfold that of our assumed dark star because of their size, and 2401 times as great because of their temperature, or approximately 10,000 times as great. Under such conditions the theoretical chance of an approach that would cause glaciation is one in 130 million years. If the average visible star is somewhat cooler than the sun and has a
radius about two and one-half times as great, as appears to be the fact, the chances rise to one in thirty-eight million years. A slight and wholly reasonable change in our assumptions would reduce this last figure to only five or ten million. For instance, the earth's mean temperature during the glacial period has been assumed as 10°C. lower than now, but the difference may have been only 6°. Again, the temperature of the outer atmosphere of Jupiter where the electrons are shot out may be only 500° or 700° absolute, instead of 900°. Or the diameter of the average star may be five or ten times that of the sun, instead of only two and one-half times as great. All this, however, may for the present be disregarded. The essential point is that even when the assumptions err on the side of conservatism, the results are of an order of magnitude which puts the electrical hypothesis within the bounds of possibility, whereas similar assumptions put the tidal hypothesis, with its single approach in twelve billion years, far beyond those limits.
The figures for Betelgeuse in Table 5 are interesting. At a meeting of the American Association for the Advancement of Science in December, 1920, Michelson reported that by measurements of the interference of light coming from the two sides of that bright star in Orion, the observers at Mount Wilson had confirmed the recent estimates of three other authorities that the star's diameter is about 218 million miles, or 250 times that of the sun. If other stars so much surpass the estimates of only a decade or two ago, the average diameter of all the visible stars must be many times that of the sun. The low figure for Betelgeuse in section D of the table means that if all the stars were as large as Betelgeuse, several might often be near enough to cause profound disturbances of the solar atmosphere. Nevertheless, because of the low
temperature of the giant red stars of the Betelgeuse type, the distance at which one of them would produce a given electrical effect is only about five times the distance at which our assumed average star would produce the same effect. This, to be sure, is on the assumption that the
radiation of energy from incandescent bodies varies according to temperature in the same ratio as the radiation from black bodies. Even if this assumption departs somewhat from the truth, it still seems almost certain that the lower temperature of the red compared with the high temperature of the white stars must to a considerable degree reduce the difference in electrical effect which would otherwise arise from their size.
| [TABLE 5] | ||||
|---|---|---|---|---|
| THEORETICAL PROBABILITY OF STELLAR APPROACHES | ||||
| 1 Dark Stars | 2 Sun | 3 Average Star | 4 Betelgeuse | |
| A. Approximate radius in miles. | 430,000 | 860,000 | 2,150,000 | 218,000,000 |
| B. Assumed temperature above absolute zero. | 900° C. | 6300° C. | 5400° C. | 3150° C. |
| C. Approximate theoretical distanceat which star would cause solar disturbance great enoughto cause glaciation (billions[121] of miles). | 1.2 | 120 | 220 | 3200 |
| D. Average interval between approaches close enough to cause glaciation if all stars were of given type. Years. | 130,000,000,000[122] | 130,000,000 | 38,000,000 | 700,000 |
Thus far in our attempt to estimate the distance at which a star might disturb the sun enough to cause glaciation on the earth, we have considered only the star's size and temperature. No account has been taken of the degree to which its atmosphere is disturbed. Yet in the case of the sun this seems to be one of the most important factors. The magnetic field of sunspots is sometimes 50 or 100 times as strong as that of the sun in general. The strength of the magnetic field appears to depend on the strength of the electrical currents in the solar atmosphere. But the intensity of the sunspots and, by inference, of the electrical currents, may depend on the electrical action of Jupiter and the other planets. If we apply a similar line of reasoning to the stars, we are at once led to question whether the electrical activity of double stars may not be enormously greater than that of isolated stars like the sun.