Chapter V · Stars (continued 2)

We can trace out two possible paths of stellar evolution in the Russell diagram which involve no entry into regions unoccupied by stars—two roads along which the stellar army may march as they transform their substance into radiation. The first is of course the “main-sequence,” which a great number of considerations suggest to be the main line of march of the stellar army. The branch which starts from the red giants in the Russell diagram represents a second possible line of march, a certain number of stars travelling along this branch until they reach the main-sequence as blue or white stars, and then travelling down the lower half of the main-sequence to end as faint red stars passing on to ultimate extinction.

Progress along each of these roads is accompanied by a continuous shrinkage in the size of the star, its diameter steadily decreasing. This is not the same thing as saying that the star’s density continually increases, for the star is continually diminishing in weight, so that even if the star’s density remained the same, its diameter would decrease. Nevertheless a study of Seares’ determinations of mean densities, as shewn in the diagram on p. 282, suggests that there is a continuous increase of density, although this becomes very slight in the middle reaches of the main-sequence.

Practically every theory of stellar evolution which has ever been propounded has imagined the march of the stellar army to be of the same general type as that just described, although perhaps present-day opinion is inclined to treat the main-sequence as the principal line of march, whereas earlier theories supposed the youngest stars to march solely along the red giant branch, only joining the main-sequence with middle age. The first serious theory of all, that of Lockyer, was expressed in terms of branches of ascending and descending temperature, these together forming the last-mentioned line of march in the Russell diagram. A theory which Russell propounded in 1913 again assigned to the stars the lines of march just described. It also attempted a physical explanation, since abandoned, as to why the stars followed these particular paths rather than others. His more recent theory of 1925 only differed from his earlier theory in giving the new explanation, which we have already discussed (p. 293), as to why the stars followed these particular paths.

At present, it is probably fair to say that nearly, and perhaps quite, all astronomers are agreed that the evolutionary paths of the stars are of the general type we have described. Some stars start as red giants, some as blue, some possibly in intermediate conditions. As they age, all move downwards in the Russell diagram, their various paths converging to a point at the fork of the reversed γ shewn in fig. 22, and after passing this point they move down the main-sequence. On the other hand, there is the widest difference of opinion as to the physical interpretation which is to be assigned to these paths. Most astronomers are probably suspending judgment until some definite observational evidence is obtained to decide between conflicting theories.

When the stars first came into being as flecks of fiery spray thrown off by spinning nebulae, they would consist of mixtures of atoms of all kinds, some perhaps being so short-lived as to transform themselves almost at once into radiation, and others having such long lives that they may properly be described as permanent. Except for a small number of radio-active atoms, the earth must consist entirely of atoms of this latter type. Calculation shews that terrestrial atoms must have enormously longer lives than the average stellar atom, otherwise their self-annihilation would make the earth too hot for habitation. The permanent atoms in a star contribute almost nothing to its energy-generating capacity, and so merely add to its weight. The shortest lived atoms of all contribute greatly to the star’s generation of energy while adding but little to its weight. In general the shorter the life of any type of atom, the greater the proportion of its numbers annihilated per year, and so the greater the amount of energy it generates per ton of weight.

A star begins life with a large proportion of short-lived atoms, and so at first generates energy furiously. As it ages, the shortest lived atoms disappear first, and in so doing reduce the average energy-generation of the star per ton so that, as a star’s weight decreases, so also must its rate of generation of energy per ton. Finally all the atoms with much energy-generating capacity have disappeared, and the star is left, a shrunken and diminished mass of atoms which have very little capacity for generating radiation.

To put the same thing in another way, the rate at which a star generates energy per ton is proportional to the death-rate in its population of atoms. To say that Sirius generates 16 times as much energy per ton as the sun is only another way of saying that the average atom in Sirius has only a sixteenth of the expectation of life of the solar atoms; their death-rate is 16 times as high. As those types of atoms which have the highest death-rate gradually die off in any star, the average death-rate of the population decreases, or, in other words, as a star ages its capacity for energy-generation per ton decreases.

This agrees with the findings of observational astronomy. The most massive stars not only generate more energy than less massive stars, as is in any case to be expected; they also generate enormously more energy per ton. This is illustrated by the following list of main-sequence stars:

Star Weight (in terms of sun) Generation of energy (ergs per gramme)

Pearce’s Star A 36·3 15,000

V Puppis A 19·2 1,000

Sirius A 2·45 29

Sun 1·00 1·90

ε Eridani (0·45) 0·26

Kruger 60 B 0·20 0·021

Seares’ diagram of stellar weights (p. 282) shews that this is a general property of the stars. To repeat our former metaphor, the stars squander their substance lavishly in their youth, while they have plenty left to spend, but parsimony comes over them with old age. Theoretical considerations have now given us an explanation of this phenomenon.

The same diagram shews that two stars of the same weight do not usually have the same luminosity. In general, giant stars on the spur branch leading out to the red giants have substantially higher luminosities than the main-sequence stars of equal weight. We have already noticed how a red giant may emit as much as 10 or 20 times the radiation of an equally massive main-sequence star. The same story is repeated when we pass from the main-sequence stars to the white dwarfs. Main-sequence stars emit enormously more radiation—anything up to 500 times more—than white dwarfs of equal weight. This is illustrated by the three following white dwarfs, which may be compared with the last three stars of the preceding table:

Star Weight (in terms of sun) Generation of energy (ergs per gramme)

Sirius B 0·85 0·0027

ο₂ Eridani B 0·44 0·002

van Maanen’s star (0·20) (0·00055)

We have hitherto supposed generation of energy to be spontaneous and so unaffected by changes of physical conditions. Yet the facts just mentioned seem to suggest that this can hardly be the whole truth of the matter. To state the objection in terms of a concrete instance, Sirius A and its white dwarf companion Sirius B must in all probability have been born at the same time out of the same nebula (p. 284), yet the former generates 4000 times as much energy per ton as the latter. It seems improbable that so great a difference can be attributed to different types of atoms; the common origin of the two stars almost precludes this. We know that the atoms are in different physical conditions in the two stars; in Sirius A they have retained their K-rings intact, while in Sirius B, the white dwarf, they are completely broken up into bare nuclei and free electrons. If the two components of Sirius consist of essentially the same types of atoms, as their common origin would lead us to expect, then the enormous difference in the rates at which these atoms generate energy would seem to depend on the different physical conditions of their atoms.

The considerations brought forward in Chapter III make it highly probable that a star’s rate of generation of energy depends on the physical condition of its atoms. We there supposed stellar energy to be generated through electrons coalescing with protons; protons exist only in atomic nuclei, and purely physical considerations led to the conjecture that the only electrons which can coalesce with a particular proton are those which are momentarily describing orbits around the nucleus in which the proton resides. A study of stellar structure supports this hypothesis, for if energy could be generated by free electrons falling into nuclei, it can be shewn that the whole star would be unstable and would explode in a flash of radiation. On this hypothesis, a star in which only a few atoms have any electrons left in orbital motion can of course generate but little energy. This at once explains the feeble energy-generating powers of the white dwarfs, and also gives an inkling as to why the red giants, in which L- and M-rings of electrons survive, generate more energy than main-sequence stars of equal weights.

As a star ages and its weight decreases, it continually has to pick out new configurations such as make its emission of energy equal to its internal rate of generation of energy. The same star may be a red giant, a main-sequence star and a white dwarf in turn. Stripped of technicalities this means that a star continually adjusts its diameter to suit its varying rate of generation of energy.

On the hypothesis just considered, a star alters both its emission and its generation of energy on changing its diameter. At every instant it has to select a diameter for which the two exactly balance. The star has so large a range of rates of generation, according as it has few or many electrons left in orbital motion, that it is likely always to be able to find a configuration of equilibrium. At any rate all the stars in the sky appear to have done so, with the exception of the long period variables which are continually expanding and contracting as though they could not hit upon a diameter at which their income and expenditure of energy would just balance.

This same hypothesis immediately makes it possible for all the great variety of stars in the galactic system to be of approximately the same age, and so to have been all born out of the same nebula. The most luminous galactic stars can hardly have been generating energy at their present rate for more than about 100,000 million years—any longer age would require an impossibly high weight to start with. Yet the motions of the stars indicate that even these highly luminous stars must have been in existence for at least 50 times this period. The apparent contradiction disappears if we admit that the extreme luminosity of the very brightest stars may be a recent development, and that for perhaps 98 per cent. of its life such a star was losing but little energy because most of its atoms were stripped bare of electrons and so were immune from annihilation. The requisite proportion of 98 per cent. may seem suspiciously large, but we have stated an extreme case; we need only demand so large a proportion in the case of a very rare type of star: probably not more than one star in ten million is of such a type.

In their earlier dormant state, these stars, which are now so luminous, would in effect have been white dwarfs of enormous weight. Observational astronomy provides no evidence that any such stars exist, but there is certainly no evidence that they do not exist. Very massive stars are known to be very rare objects, so that in all probability we should have to travel a long distance from the sun before finding one, and then it might be so distant as to be invisible from the earth. In any case, a very distant star of feeble luminosity would be exceedingly likely to escape detection. We could not infer that no such stars existed from the fact of none having yet been found.

Moreover, it is far from absolutely certain that such stars have not been found. Very massive white dwarfs ought to have higher surface-temperatures than either massive main-sequence stars or than the known white dwarfs, all of which are of small weight. A whole group of stars is known—the O-type stars—whose spectra indicate very high temperatures indeed. These are usually interpreted as stars of enormous luminosity at enormous distances, but it is possible that some at least of them may be stars of feeble luminosity at moderate distances. In particular the central stars of the planetary nebulae are of types O and B. But whereas the normal main-sequence star of these spectral types is generally about a thousand times as luminous as the sun, the central stars of the planetary nebulae are found to be considerably less luminous than the sun. Dr Gerasimovič has recently shewn that 52 of these stars have an average luminosity only about two-thirds of that of the sun. Other observers have found even lower luminosities. As the surface-temperatures of these stars must be of the order of 30,000 degrees, they must be minute in size; indeed a simple calculation shews that they can hardly have more than about a tenth of the sun’s radius.

Not only are they small in size, but there are indications that they are of great weight. Spectroscopic evidence shews that in many cases the surrounding nebulae are in rotation, and, just as with the galaxy (p. 69) and the huge extra-galactic nebulae (p. 71), their weights can be calculated from their observed speeds of rotation. These seem to indicate average weights of something like 40 or 50 times the weight of the sun. Campbell and Moore, of Lick Observatory, have calculated that the planetary nebula N.G.C. 7009 must have 162 times the weight of the sun, but many factors combine to make this estimate rather uncertain.

All this goes to suggest that the central stars of the planetary nebulae must, in all probability, be regarded as “white dwarfs” of enormous weight. It seems possible that they are exactly the type of star needed by our theory. The 52 planetary nebulae already mentioned have the same average distance from the sun, and the same distribution with respect to the galaxy, as the O-type stars, so that it is possible that O-type stars and planetary nebulae are forms assumed either successively or alternatively in the course of evolution of similar bodies. It should be mentioned that the planetary nebulae exhibit spectral displacements which, if interpreted in the ordinary way, would indicate that they are moving with very high speeds, in some cases reaching hundreds of miles a second. Such speeds would be entirely inappropriate to stars of large weight. Yet this objection rather defeats itself, for if the spectral shifts really arose from high speeds of motion, the highest of the observed speeds would suffice to carry the planetary nebulae possessing them clear of the galactic system altogether; they would be random travellers through our system of stars, whereas their orderly arrangement and concentration near the galactic plane suggests very forcibly that they are permanent members of it. Thus the observed large shifts of the spectral lines can hardly indicate large speeds in space; some other interpretation must be found. Professor Perrine considers that they arise in the main from internal motions, contraction, expansion and rotation of the nebulae. If so, the motions in space cannot be determined, but at least they no longer present any difficulty. If these stars are, as our hypothesis requires, white dwarfs of very great weight and small radius, their spectra ought to exhibit a relativity displacement to the red of the same type as that actually observed in Sirius B. It is possible that this may be found to present a more serious difficulty, although it is also possible that it again may prove to be merged in the large observed displacements.

In view of all this, it seems quite possible that both the planetary nebulae, and also other O- or B-type stars of feeble luminosity, may be very massive stars in the dormant condition contemplated by the hypothesis we have just been discussing. In brief, we imagine that a massive star may have its weight conserved through existing as a planetary nebula or a dwarf O-type star for millions of millions of years, and then burst out as a highly luminous star with all the appearance of extreme youth. But there is at present insufficient observational evidence either for or against such a hypothesis. Some pieces of the puzzle are missing, and we can only wait until they turn up.

WHITE DWARFS. Apart from these hypothetical massive white dwarfs, astronomers generally regard ordinary white dwarfs as the final stage in stellar evolution. There is general agreement that they are stars with central temperatures so high that their atoms are stripped bare of electrons, but there is no general consensus of opinion as to why stars shrink to this condition.

On the liquid-star hypothesis, the unoccupied regions in the Russell diagram represent unstable configurations. Usually a slight loss of weight by a star merely moves it to a new position in the diagram contiguous to the old one. Sometimes, however, this slight move may happen to carry the star into an unstable region of the diagram, in which case it will hurriedly traverse this region, until finally it ends up in some entirely different stable configuration.

The liquid-star hypothesis explains the white dwarf state quite simply as the final state to which a star shrinks cataclysmically when its generation of energy is no longer sufficient to entitle it to a place in the main-sequence. In this state the star radiates so little energy that annihilation and decay are almost entirely checked. We have seen that if the sun went on radiating at its present rate for 15 million million years, its whole weight would be transformed into radiation. By contrast, van Maanen’s star can, and probably will, go on radiating at its present rate for 15 million million years without losing more than about a thousandth part of its present weight. We may think of the white dwarf state as a final state from which change and decay have so nearly disappeared that a star which shrinks to this state acquires a new lease of life for a period of thousands of millions of millions of years—we can only wonder to what purpose.