Temperature in the Interior
By mathematical methods it is possible to work out how fast the pressure increases as we go down into the sun, and how fast the temperature must increase to withstand the pressure. The architect can work out the stresses inside the piers of his building; he does not need to bore holes in them. Likewise the astronomer can work out the stress or pressure at points inside the sun without boring a hole. Perhaps it is more surprising that the temperature can be found by pure calculation. It is natural that you should feel rather sceptical about our claim that we know how hot it is in the very middle of a star—and you may be still more sceptical when I divulge the actual figures! Therefore I had better describe the method as far as I can. I shall not attempt to go into detail, but I hope to show you that there is a clue which might be followed up by appropriate mathematical methods.
I must premise that the heat of a gas is chiefly the energy of motion of its particles hastening in all directions and tending to scatter apart. It is this which gives a gas its elasticity or expansive force; the elasticity of a gas is well known to every one through its practical application in a pneumatic tyre. Now imagine yourself at some point deep down in the star where you can look upwards towards the surface or downwards towards the centre. Wherever you are, a certain condition of balance must be reached; on the one hand there is the weight of all the layers above you pressing downwards and trying to squeeze closer the gas beneath; on the other hand there is the elasticity of the gas below you trying to expand and force the superincumbent layers outwards. Since neither one thing nor the other happens and the star remains practically unchanged for hundreds of years, we must infer that the two tendencies just balance. At each point the elasticity of the gas must be just enough to balance the weight of the layers above; and since it is the heat which furnishes the elasticity, this requirement settles how much heat the gas must have. And so we find the degree of heat or temperature at each point.
The same thing can be expressed a little differently. As before, fix attention on a certain point in a star and consider how the matter above it is supported. If it were not supported it would fall to the centre under the attractive force of gravitation. The support is given by a succession of minute blows delivered by the particles underneath; we have seen that their heat energy causes them to move in all directions, and they keep on striking the matter above. Each blow gives a slight boost upwards, and the whole succession of blows supports the upper material in shuttlecock fashion. (This process is not confined to the stars; for instance, it is in this way that a motor car is supported by its tyres.) An increase of temperature would mean an increase of activity of the particles, and therefore an increase in the rapidity and strength of the blows. Evidently we have to assign a temperature such that the sum total of the blows is neither too great nor too small to keep the upper material steadily supported. That in principle is our method of calculating the temperature.
One obvious difficulty arises, The whole supporting force will depend not only on the activity of the particles (temperature) but also on the number of them (density). Initially we do not know the density of the matter at an arbitrary point deep within the sun. It is in this connexion that the ingenuity of the mathematician is required. He has a definite amount of matter to play with, viz. the known mass of the sun; so the more he uses in one part of the globe the less he will have to spare for other parts. He might say to himself, ‘I do not want to exaggerate the temperature, so I will see if I can manage without going beyond 10,000,000°.’ That sets a limit to the activity to be ascribed to each particle; therefore when the mathematician reaches a great depth in the sun and accordingly has a heavy weight of upper material to sustain, his only resource is to use large numbers of particles to give the required total impulse. He will then find that he has used up all his particles too fast, and has nothing left to fill up the centre. Of course his structure, supported on nothing, would come tumbling down into the hollow. In that way we can prove that it is impossible to build up a permanent star of the dimensions of the sun without introducing an activity or temperature exceeding 10,000,000°. The mathematician can go a step beyond this; instead of merely finding a lower limit, he can ascertain what must be nearly the true temperature distribution by taking into account the fact that the temperature must not be ‘patchy’. Heat flows from one place to another, and any patchiness would soon be evened out in an actual star. I will leave the mathematician to deal more thoroughly with these considerations, which belong to the following up of the clue; I am content if I have shown you that there is an opening for an attack on the problem.
This kind of investigation was started more than fifty years ago. It has been gradually developed and corrected, until now we believe that the results must be nearly right—that we really know how hot it is inside a star.
I mentioned just now a temperature of 6,000°; that was the temperature near the surface—the region which we actually see. There is no serious difficulty in determining this surface temperature by observation; in fact the same method is often used commercially for finding the temperature of a furnace from the outside. It is for the deep regions out of sight that the highly theoretical method of calculation is required. This 6,000° is only the marginal heat of the great solar furnace giving no idea of the terrific intensity within. Going down into the interior the temperature rises rapidly to above a million degrees, and goes on increasing until at the sun’s centre it is about 40,000,000°.
Do not imagine that 40,000,000° is a degree of heat so extreme that temperature has become meaningless. These stellar temperatures are to be taken quite literally. Heat is the energy of motion of the atoms or molecules of a substance, and temperature which indicates the degree of heat is a way of stating how fast these atoms or molecules are moving. For example, at the temperature of this room the molecules of air are rushing about with an average speed of 500 yards a second; if we heated it up to 40,000,000° the speed would be just over 100 miles a second. That is nothing to be alarmed about; the astronomer is quite accustomed to speeds like that. The velocities of the stars, or of the meteors entering the earth’s atmosphere, are usually between 10 and 100 miles a second. The velocity of the earth travelling round the sun is 20 miles a second. So that for an astronomer this is the most ordinary degree of speed that could be suggested, and he naturally considers 40,000,000° a very comfortable sort of condition to deal with. And if the astronomer is not frightened by a speed of 100 miles a second, the experimental physicist is quite contemptuous of it; for he is used to handling atoms shot off from radium and similar substances with speeds of 10,000 miles a second. Accustomed as he is to watching these express atoms and testing what they are capable of doing, the physicist considers the jog-trot atoms of the stars very commonplace.
Besides the atoms rushing to and fro in all directions we have in the interior of a star great quantities of ether waves also rushing in all directions. Ether waves are called by different names according to their wave-length. The longest are the Hertzian waves used in broadcasting; then come the infra-red heat waves; next come waves of ordinary visible light; then ultra-violet photographic or chemical rays; then X-rays; then Gamma rays emitted by radio-active substances. Probably the shortest of all are the rays constituting the very penetrating radiation found in our atmosphere, which according to the investigations of Kohlhörster and Millikan are believed to reach us from interstellar space. These are all fundamentally the same but correspond to different octaves. The eye is attuned to only one octave, so that most of them are invisible; but essentially they are of the same nature as visible light.
The ether waves inside a star belong to the division called X-rays. They are the same as the X-rays produced artificially in an X-ray tube. On the average they are ‘softer’ (i.e. longer) than the X-rays used in hospitals, but not softer than some of those used in laboratory experiments. Thus we have in the interior of a star something familiar and extensively studied in the laboratory.
Besides the atoms and ether waves there is a third population to join in the dance. There are multitudes of free electrons. The electron is the lightest thing known, weighing no more than ¹⁄₁₈₄₀ of the lightest atom. It is simply a charge of negative electricity wandering about alone. An atom consists of a heavy nucleus which is usually surrounded by a girdle of electrons. It is often compared to a miniature solar system, and the comparison gives a proper idea of the emptiness of an atom. The nucleus is compared to the sun, and the electrons to the planets. Each kind of atom—each chemical element—has a different quorum of planet electrons. Our own solar system with eight planets might be compared especially with the atom of oxygen which has eight circulating electrons. In terrestrial physics we usually regard the girdle or crinoline of electrons as an essential part of the atom because we rarely meet with atoms incompletely dressed; when we do meet with an atom which has lost one or two electrons from its system, we call it an ‘ion’. But in the interior of a star, owing to the great commotion going on, it would be absurd to exact such a meticulous standard of attire. All our atoms have lost a considerable proportion of their planet electrons and are therefore ions according to the strict nomenclature.