There is a gratifying accordance in general order of magnitude between the opacity inside the star, determined from astronomical observation of leakage, and the opacity of terrestrial substances to X-rays of more or less the same wave-length. This gives us some assurance that our theory is on the right track. But a careful comparison shows us that there is some important difference between the stellar and terrestrial opacity.
In the laboratory we find that the opacity increases very rapidly with the wave-length of the X-rays that are used. We do not find anything like the same difference in the stars although the X-rays in the cooler stars must be of considerably greater wave-length than those in the hotter stars. Also, taking care to make the comparison at the same wave-length for both, we find that the stellar opacity is less than the terrestrial opacity. We must follow up this divergence.
There is more than one way in which an atom can obstruct ether-waves, but there seems to be no doubt that for X-rays both in the stars and in the laboratory the main part of the opacity depends on the process of ionization. The ether-wave falls on an atom and its energy is sucked up by one of the planet electrons which uses it to escape from the atom and travel away at high speed. The point to notice is that in the very act of absorption the absorbing mechanism is broken, and it cannot be used again until it has been repaired. To repair it the atom must capture one of the free electrons wandering about, inducing it to take the place of the lost electron.
In the laboratory we can only produce thin streams of X-rays so that each wave-trap is only called upon to act occasionally. There is plenty of time to repair it before the next time it has a chance of catching anything; and there is practically no loss of efficiency through the traps being out of order. But in the stars the stream of X-rays is exceedingly intense. It is like an army of mice marching through your larder springing the mouse-traps as fast as you can set them. Here it is the time wasted in resetting the traps—by capturing electrons—which counts, and the amount of the catch depends almost entirely on this.
We have seen that the stellar atoms have lost most of their electrons; that means that at any moment a large proportion of the absorption traps are awaiting repair. For this reason we find a smaller opacity in the stars than in terrestrial material. The lowered opacity is simply the result of overworking the absorbing mechanisms—they have too much radiation to deal with. We can also see why the laws of stellar and terrestrial opacity are somewhat different. The rate of repair, which is the main consideration in stellar opacity, is increased by compressing the material, because then the atom will not have to wait so long to meet and capture a free electron. Consequently the stellar opacity will increase with the density. In terrestrial conditions there is no advantage in accelerating the repair which will in any case be completed in sufficient time; thus terrestrial opacity is independent of the density.
The theory of stellar opacity thus reduces mainly to the theory of the capture of electrons by ionized atoms; not that this process is attended by absorption of X-rays—it is actually attended by emission—but it is the necessary preliminary to absorption. The physical theory of electron-capture is not yet fully definitive; but it is sufficiently advanced for us to make use of it provisionally in our calculations of the hindering factor in the leakage of radiation from the stars.
[The Relation of Brightness to Mass]
We do not want to tackle too difficult a problem at first, and so we shall deal with stars composed of perfect gas. If you do not like the technical phrase ‘perfect gas’ you can call it simply ‘gas’, because all the terrestrial gases that you are likely to think of are without sensible imperfection. It is only under high compression that terrestrial gases become imperfect. I should mention that there are plenty of examples of gaseous[6] stars. In many stars the material is so inflated that it is more tenuous than the air around us; for example, if you were inside Capella you would not notice the material of Capella any more than you notice the air in this room.
For gaseous stars, then, the investigation will give formulae by which, given the mass of the star, we can calculate how much energy of heat and light will leak out of it—in short, how bright it will be. In [Fig. 7] a curve is drawn giving this theoretical relation between the brightness and mass of a star. Strictly speaking, there is another factor besides the mass which affects the calculated brightness; you can have two stars of the same mass, the one dense and the other puffed out, and they will not have quite the same brightness. But it turns out (rather unexpectedly) that this other factor, density, makes very little difference to the brightness, always provided that the material is not too dense to be a perfect gas. I shall therefore say no more about density in this brief summary.
Here are a few details about the scale of the diagram. The brightness is measured in magnitudes, a rather technical unit. You have to remember that stellar magnitude is like a golfer’s handicap—the bigger the number, the worse the performance. The diagram includes practically the whole range of stellar brightness; at the top -4 represents almost the brightest stars known, and at the bottom 12 is nearly the faintest limit. The difference from top to bottom is about the same as the difference between an arc light and a glow worm. The sun is near magnitude 5. These magnitudes refer, of course, to the true brightness, not to the apparent brightness affected by distance; also, what is represented here is the ‘heat brightness’ or heat intensity, which is sometimes a little different from the light intensity. Astronomical instruments have been made which measure directly the heat instead of the light received from a star. These are quite successful; but there are troublesome corrections on account of the large absorption of heat in the earth’s atmosphere, and it is in most cases easier and more accurate to infer the heat brightness from the light brightness, making allowance for the colour of the star.