The Story of Betelgeuse
This story has not much to do with atoms, and scarcely comes under the title of these lectures; but we have had occasion to allude to Betelgeuse as the famous example of a star of great size and low density, and its history is closely associated with some of the developments that we are studying.
No star has a disk large enough to be seen with our present telescopes. We can calculate that a lens or mirror of about 20 feet aperture would be needed to show traces even of the largest star disk. Imagine for a moment that we have constructed an instrument of this order of size. Which would be the most hopeful star to try it on?
Perhaps Sirius suggests itself first, since it is the brightest star in the sky. But Sirius has a white-hot surface radiating very intensely, so that it is not necessary that it should have a wide expanse. Evidently we should prefer a star which, although bright, has its surface in a feebly glowing condition; then the apparent brightness must be due to large area. We need, then, a star which is both red and bright. Betelgeuse seems best to satisfy this condition. It is the brighter of the two shoulder-stars of Orion—the only conspicuous red star in the constellation. There are one or two rivals, including Antares, which might possibly be preferred; but we cannot go far wrong in turning our new instrument on Betelgeuse in the hope of finding the largest or nearly the largest star disk.
You may notice that I have paid no attention to the distances of these stars. It happens that distance is not relevant. It would be relevant if we were trying to find the star of greatest actual dimensions; but here we are considering the star which presents the largest apparent disk,24 i.e. covers the largest area of the sky. If we were at twice our present distance from the sun, we should receive only one-quarter as much light; but the sun would look half its present size linearly, and its apparent area would be one-quarter. Thus the light per unit area of disk is unaltered by distance. Removing the sun to greater and greater distance its disk will appear smaller but glowing not less intensely, until it is so far away that the disk cannot be discriminated.
By spectroscopic examination we know that Betelgeuse has a surface temperature about 3,000°. A temperature of 3,000° is not unattainable in the laboratory, and we know partly by experiment and partly by theory what is the radiating power of a surface in this state. Thus it is not difficult to compute how large an area of the sky Betelgeuse must cover in order that the area multiplied by the radiating power may give the observed brightness of Betelgeuse. The area turns out to be very small. The apparent size of Betelgeuse is that of a half-penny fifty miles away. Using a more scientific measure, the diameter of Betelgeuse predicted by this calculation is 0·051 of a second of arc.
No existing telescope can show so small a disk. Let us consider briefly how a telescope forms an image—in particular how it reproduces that detail and contrast of light and darkness which betrays that we are looking at a disk or a double star and not a blur emanating from a single point. This optical performance is called resolving power; it is not primarily a matter of magnification but of aperture, and the limit of resolution is determined by the size of aperture of the telescope.
To create a sharply defined image the telescope must not only bring light where there ought to be light, but it must also bring darkness where there ought to be darkness. The latter task is the more difficult. Light-waves tend to spread in all directions, and the telescope cannot prevent individual wavelets from straying on to parts of the picture where they have no business. But it has this one remedy—for every trespassing wavelet it must send a second wavelet by a slightly longer or shorter route so as to arrive in a phase opposite to the first wavelet and cancel its effect. This is where the utility of a wide aperture arises—by affording a wider difference of route of the individual wavelets, so that those from one part of the aperture may be retarded relatively to and interfere with those from another part. A small object-glass can furnish light; it takes a big object-glass to furnish darkness in the picture.
Now we may ask ourselves whether the ordinary circular aperture is necessarily the most efficient for giving the wavelets the required path-differences. Any deviation from a symmetrical shape is likely to spoil the definition of the image—to produce wings and fringes. The image will not so closely resemble the object viewed. But on the other hand we may be able to sharpen up the tell-tale features. It does not matter how widely the image-pattern may differ from the object, provided that we can read the significance of the pattern. If we cannot reproduce a star-disk, let us try whether we can reproduce something distinctive of a star-disk.
A little reflection shows that we ought to improve matters by blocking out the middle of the object-glass, and using only the extreme regions on one side or the other. For these regions the difference of light-path of the waves is greatest, and they are the most efficient in furnishing the dark contrast needed to outline the image properly.
But if the middle of the object-glass is not going to be used, why go to the expense of manufacturing it? We are led to the idea of using two widely separated apertures, each involving a comparatively small lens or mirror. We thus arrive at an instrument after the pattern of a rangefinder.
This instrument will not show us the disk of a star. If we look through it the main impression of the star image is very like what we should have seen with either aperture singly—a ‘spurious disk’ surrounded by diffraction rings. But looking attentively we see that this image is crossed by dark and bright bands which are produced by interference between the light-waves coming from the two apertures. At the centre of the image the waves from the two apertures arrive crest on crest since they have travelled symmetrically along equal paths; accordingly there is a bright band. A very little to one side the asymmetry causes the waves to arrive crest on trough, so that they cancel one another; here there is a dark band. The width of the bands decreases as the separation of the two apertures increases, and for any given separation the actual width is easily calculated.
Each point of the star’s disk is giving rise to a diffraction image with a system of bands of this kind, but so long as the disk is small compared with the finest detail of the diffraction image there is no appreciable blurring. If we continually increase the separation of the two apertures and so make the bands narrower, there comes a time when the bright bands for one part of the disk are falling on the dark bands for another part of the disk. The band system then becomes indistinct. It is a matter of mathematical calculation to determine the resultant effect of summing the band systems for each point of the disk. It can be shown that for a certain separation of the apertures the bands will disappear altogether; and beyond this separation the system should reappear though not attaining its original sharpness. The complete disappearance occurs when the diameter of the star-disk is equal to 1⅕ times the width of the bands (from the centre of one bright band to the next). As already stated, the bandwidth can be calculated from the known separation of the apertures.
The observation consists in sliding apart the two apertures until the bands disappear. The diameter of the disk is inferred at once from their separation when the disappearance occurred. Although we measure the size of the disk in this way we never see the disk.
We can summarize the principle of the method in the following way. The image of a point of light seen through a telescope is not a point but a small diffraction pattern. Hence, if we look at an extended object, say Mars, the diffraction pattern will blur the fine detail of the marking on the planet. If, however, we are looking at a star which is almost a point, it is simpler to invert the idea; the object, not being an ideal point, will slightly blur the detail of the diffraction pattern. We shall only perceive the blurring if the diffraction pattern contains detail fine enough to suffer from it. Betelgeuse on account of its finite size must theoretically blur a diffraction pattern; but the ordinary diffraction disk and rings produced with the largest telescope are too coarse to show this. We create a diffraction image with finer detail by using two apertures. Theoretically we can make the detail as fine as we please by increasing the separation of the two apertures. The method accordingly consists in widening the separation until the pattern becomes fine enough to be perceptibly blurred by Betelgeuse. For a smaller star-disk the same effect of blurring would not be apparent until the detail had been made still finer by further separation of the apertures.
This method was devised long ago by Professor Michelson, but it was only in 1920 that he tried it on a large scale with a great 20-foot beam across the 100-inch reflector at Mount Wilson Observatory. After many attempts Pease and Anderson were able to show that the bright and dark bands for Betelgeuse disappeared when the apertures were separated 10 feet. The deduced diameter is 0·045 a second of arc in good enough agreement with the predicted value (p. 78). Only five or six stars have disks large enough to be measured with this instrument. It is understood that the construction of a 50-foot interferometer is contemplated; but even this will be insufficient for the great majority of the stars. We are fairly confident that the method of calculation first described gives the correct diameters of the stars, but confirmation by Michelson’s more direct method of measurement is always desirable.
To infer the actual size of the star from its apparent diameter, we must know the distance. Betelgeuse is rather a remote star and its distance cannot be measured very accurately, but the uncertainty will not change the general order of magnitude of the results. The diameter is about 300 million miles. Betelgeuse is large enough to contain the whole orbit of the earth inside it, perhaps even the orbit of Mars. Its volume is about fifty million times the volume of the sun.
There is no direct way of learning the mass of Betelgeuse because it has no companion near it whose motion it might influence. We can, however, deduce a mass from the mass-brightness relation in Fig. 7. This gives the mass equal to 35 x sun. If the result is right, Betelgeuse is one of the most massive stars—but, of course, not massive in proportion to its bulk. The mean density is about one-millionth of the density of water, or not much more than one-thousandth of the density of air.25
There is one way in which we might have inferred that Betelgeuse is less dense than the sun, even if we had had no grounds of theory or analogy for estimating its mass. According to the modern theory of gravitation, a globe of the size of Betelgeuse and of the same mean density as the sun would have some remarkable properties:
Firstly, owing to the great intensity of its gravitation, light would be unable to escape; and any rays shot out would fall back again to the star by their own weight.
Secondly, the Einstein shift (used to test the density of the Companion of Sirius) would be so great that the spectrum would be shifted out of existence.
Thirdly, mass produces a curvature of space, and in this case the curvature would be so great that space would close up round the star, leaving us outside—that is to say, nowhere.
Except for the last consideration, it seems rather a pity that the density of Betelgeuse is so low.
It is now well realized that the stars are a very important adjunct to the physical laboratory—a sort of high-temperature annex where the behaviour of matter can be studied under greatly extended conditions. Being an astronomer, I naturally put the connexion somewhat differently and regard the physical laboratory as a low-temperature station attached to the stars. It is the laboratory conditions which should be counted abnormal. Apart from the interstellar cloud which is at the moderate temperature of about 15,000°, I suppose that nine-tenths of the matter of the universe is above 1,000,000°. Under ordinary conditions—you will understand my use of the word—matter has rather simple properties. But there are in the universe exceptional regions with temperature not far removed from the absolute zero, where the physical properties of matter acquire great complexity; the ions surround themselves with complete electron systems and become the atoms of terrestrial experience. Our earth is one of these chilly places and here the strangest complications can arise. Perhaps strangest of all, some of these complications can meet together and speculate on the significance of the whole scheme.