FOOTNOTES:
[1] [Fig. 1] is from a photograph taken by Mr. Evershed at Kodaikanal Observatory, Madras. [Fig. 2] is from the Mount Wilson Observatory, California.
[2] I am indebted to Professor C. T. R. Wilson for Figs. [Fig. 3]-[Fig. 6].
[3] Primarily it is the electric charge and not the high speed of particles which determines their appearance in these photographs. But a high-speed particle leaves behind it a trail of electrically charged particles—the victims of its furious driving—so that it is shown indirectly by its line of victims.
[4] Other substitutions for silver do not as a rule cause greater change, and the differences are likely to be toned down by mixture of many elements. Excluding hydrogen, the most extreme change is from 48 particles for silver to 81 particles for an equal mass of helium. But for hydrogen the change is from 48 to 216, so that hydrogen gives widely different results from other elements.
[5] The mean density of Capella is nearly the same as the density of the air.
[6] Unless otherwise indicated ‘gaseous’ is intended to mean ‘composed of perfect gas’.
[7] For this prediction it is unnecessary to know the chemical composition of the stars, provided that extreme cases (e. g. an excessive proportion of hydrogen) are excluded. For example, consider the hypotheses that Capella is made respectively of (a) iron, (b) gold. According to theory the opacity of a star made of the heavier element would be 2½ times the opacity of a star made of iron. This by itself would make the golden star a magnitude (= 2½ times) fainter. But the temperature is raised by the substitution; and although, as explained on [p. 23], the change is not very great, it increases the outflow of heat approximately 2½ times. The resultant effect on the brightness is practically no change. Whilst this independence of chemical constitution is satisfactory in regard to definiteness of the results, it makes the discrepant factor 10 particularly difficult to explain.
[8] Observation shows that the sun is about 4 magnitudes fainter than the average diffuse star of the same spectral class, and Krueger 60 is 10 magnitudes fainter than diffuse stars of its class. The whole drop was generally assumed to be due to deviation from a perfect gas; but this made no allowance for a possible difference of mass. The comparison with the curve enables the dense star to be compared with a gaseous star of its own mass, and we see that the difference then disappears. So that (if there has been no mistake) the dense star is a gaseous star, and the differences above mentioned were due wholly to differences of mass.
[9] Rougher estimates were made much earlier.
[10] The observed period of Algol is the period of revolution, not of rotation. But the two components are very close together, and there can be no doubt that owing to the large tidal forces they keep the same faces turned towards each other; that is to say, the periods of rotation and of revolution are equal.
[11] It may be of interest to add that although the proper light of Algol B is inappreciable, we can observe a reflection (or re-radiation) of the light of Algol A by it. This reflected light changes like moonlight according as Algol B is ‘new’ or ‘full’.
[11] The mass-luminosity relation was not suspected at the time of which I am speaking.
[13] My references to ‘perfect gas of the density of platinum’ and ‘material 2,000 times denser than platinum’ have often been run together by reporters into ‘perfect gas 2,000 times denser than platinum’. It is scarcely possible to calculate what is the condition of the material in the Companion of Sirius, but I do not expect it to be a perfect gas.
[14] Photographed by Dr. W. H. Wright at the Lick Observatory, California.
[15] Nos. 43, 61, 75 are recent discoveries and may require confirmation. There now remain only two gaps (85 and 87) apart from possible elements beyond uranium.
[16] It does not give both temperature and pressure, but it gives one if the other is known. This is valuable information which may be pieced together with other knowledge of the conditions at the surface of the stars.
[17] Hydrogen (being element No. 1) has only one planet electron.
[18] [Fig. 9] is a photograph of the ‘flash spectrum’ of the sun’s chromosphere taken by Mr. Davidson in Sumatra at the eclipse of 14 January 1926.
[19] The helium line in the Ring Nebula on which we have already commented is not a member of the Pickering Series, but it has had the same history. It was first supposed to be due to hydrogen, later (in 1912) reproduced by Fowler terrestrially in a mixture of helium and hydrogen, and finally discovered by Bohr to belong to helium.
[20] This, of course, is found from the other lines of the spectrum which genuinely belong to the star and shift to and fro as it describes its orbit.
[21] As the word temperature is sometimes used with new-fangled meanings, I may add that 15,000° is the temperature corresponding to the individual speeds of the atoms and electrons—the old-fashioned gas-temperature.
[22] Photograph taken by E. T. Cottingham and the author in Principe at the total eclipse of 29 May 1919.
[23] We refer to calcium as it occurs in the chromosphere, i. e. with one electron missing.
[24] There is an awkwardness in applying the term ‘apparent’ to something too small to be seen; but, remembering that we have armed ourselves with an imaginary telescope capable of showing the disk, the meaning will be clear.
[25] Densities below that of air have been found for some of the Algol variables by an entirely different kind of investigation, and also for some of the Cepheid variables by still another method. There are also many other examples of stars of bulk comparable with that of Betelgeuse.
[26] From a photograph taken at the Royal Observatory, Cape of Good Hope.
[27] For comparison, the nearest fixed star is distant 4 light years. Apart from clusters we rarely deal with distances above 2,000 light years.
[28] One cannot always be sure that what is true of the cluster stars will be true of stars in general; and our knowledge of the nearer stars, though lagging behind that of the stars in clusters, does not entirely agree with this association of colour and brightness.
[29] The term nebula covers a variety of objects, and it is only the nebulae classed as spirals that are likely to be outside our stellar system.
[30] This can be checked because uranium lead has a different atomic weight from lead not so derived. Ordinary lead is a mixture of several kinds of atoms (isotopes).
[31] You may wonder why, having said that the sun contains 2,000 quadrillion tons of energy at the most, I now assume that it contains just this amount. It is really only a verbal point depending on the scientific definition of energy. All mass is mass of something, and we now call that something ‘energy’ whether it is one of the familiar forms of energy or not. You will see in the next sentence that we do not assume that the energy is convertible into known forms, so that it is a terminology which commits us to nothing.
[32] Aston in his latest researches has been able to detect that the oxygen atom is just appreciably lighter than the four helium atoms.
[33] A measurement of the heat observed to flow from a continuous fountain of heat is a measurement of the output of the fountain, unless there is a storing of energy between the output and the outflow. The breakdown of the Kelvin time-scale indicates that the storing in the stars (positive or negative) and consequent expansion or contraction is negligible compared to the output or outflow.
[34] The stars all put together cover an area of the sky much less than the apparent disk of the sun, so that unless their surface-layers are generating this radiation very much more abundantly than the sun does, they cannot be responsible for it.
[35] The term ‘dwarf stars’ is not meant to include white dwarfs.
[36] We can scarcely suppose that all stars after reaching the main series pass through precisely the same stages. For example, Algol, when it has become reduced to the mass of the Sun, may have slightly different density and temperature. But the observational evidence indicates that these individual differences are small. The main series is nearly a linear sequence; it must have some ‘breadth’ as well as ‘length’, but at present the scatter of the individual stars away from the central line of the sequence seems to be due chiefly to the probable errors of the observational data and the true breadth has not been determined.
[37] Exhaustion of supply without change of mass would cause the star to contract to higher density; it would thus have a combination of density and mass which (according to observation) is not found in any actual stars.
[38] This increase was assumed in our detailed description of the automatic adjustment of the star, and it will be seen that it was essential to assume it.
APPENDIX
[Further Remarks on the Companion of Sirius]
I HAVE preferred not to complicate the Story of the Companion of Sirius with details of a technical kind; some further information may, therefore, be welcome to those readers who are curious to learn as much as possible about this remarkable star. I am also able to add a further instalment of the ‘detective story’ which has just come to hand, the sleuth this time being Mr. R. H. Fowler.
The star is between the eighth and ninth magnitude, so that it is not an excessively faint object. The difficulty in detecting it arises entirely from the overpowering light of its neighbour. At favourable epochs it has been seen easily with an 8-inch telescope. The period of revolution is 49 years.
The Companion is separated from Sirius by a distance nearly equal to the distance of Uranus from the Sun—or twenty times the earth’s distance from the sun. It has been suggested that the light might be reflected light from Sirius. This would account for its whiteness, but would not directly account for its spectrum, which differs appreciably from that of Sirius. To reflect ¹⁄₁₀₀₀₀th of the light of Sirius (its actual brightness) the Companion would have to be 74 million miles in diameter. The apparent diameter of its disk would be 0"·3, which, one would think, could scarcely escape notice in spite of unfavourable conditions of observation. But the strongest objection to this hypothesis of reflected light is that it applies only to this one star. The other two recognized white dwarfs have no brilliant star in their neighbourhood, so that they cannot be shining by reflected light. It is scarcely worth while to invent an elaborate explanation for one of these strange objects which does not cover the other two.
The Einstein effect, which is appealed to for confirmation of the high density, is a lengthening of the wave-length and corresponding decrease of the frequency of the light due to the intense gravitational field through which the rays have to pass. Consequently the dark lines in the spectrum appear at longer wave-lengths, i.e. displaced towards the red as compared with the corresponding terrestrial lines. The effect can be deduced either from the relativity theory of gravitation or from the quantum theory; for those who have some acquaintance with the quantum theory the following reasoning is probably the simplest. The stellar atom emits the same quantum of energy hν as a terrestrial atom, but this quantum has to use up some of its energy in order to escape from the attraction of the star; the energy of escape is equal to the mass hν/c2 multiplied by the gravitational potential Φ at the surface of the star. Accordingly the reduced energy after escape is hν(1 - Φ/c2); and since this must still form a quantum hν', the frequency has to change to a value ν' = ν(1 - Φ/c2). Thus the displacement ν' - ν is proportional to Φ, i.e. to the mass divided by the radius of the star.
The effect on the spectrum resembles the Doppler effect of a velocity of recession, and can therefore only be discriminated if we know already the line-of-sight velocity. In the case of a double star the velocity is known from observation of the other component of the system, so that the part of the displacement attributable to Doppler effect is known. Owing to orbital motion there is a difference of velocity between Sirius and its Companion amounting at present to 43 km. per sec. and this has been duly taken into account; the observed difference in position of the spectral lines of Sirius and its Companion corresponds to a velocity of 23 km. per sec. of which 4 km. per sec. is attributable to orbital motion, and the remaining 19 km. per sec. must be interpreted as Einstein effect. The result rests mainly on measurements of one spectral line Hβ. The other favourable lines are in the bluer part of the spectrum, and since atmospheric scattering increases with blueness, the scattered light of Sirius interferes. However, they afford some useful confirmatory evidence.
Of the other white dwarfs ο2 Eridani is a double star, its companion being a red dwarf fainter than itself. The red shift of the spectrum will be smaller than in the Companion of Sirius and it will not be so easy to separate it from various possible sources of error. Nevertheless the prospect is not hopeless. The other recognized white dwarf is an unnamed star discovered by Van Maanen; it is a solitary star, and consequently there is no means of distinguishing between Einstein shift and Doppler shift. Various other stars have been suspected of being in this condition, including the Companions of Procyon, 85 Pegasi, and Mira Ceti.
If the Companion of Sirius were a perfect gas its central temperature would be about 1,000,000,000°, and the central part of the star would be a million times as dense as water. It is, however, unlikely that the condition of a perfect gas continues to hold. It should be understood that in any case the density will fall off towards the outside of the star, and the regions which we observe are entirely normal. The dense material is tucked away under high pressure in the interior.
Perhaps the most puzzling feature that remains is the extraordinary difference of development between Sirius and its Companion, which must both have originated at the same time. Owing to the radiation of mass the age of Sirius must be less than a billion years; an initial mass, however large, would radiate itself down to less than the present mass of Sirius within a billion years. But such a period is insignificant in the evolution of a small star which radiates more slowly, and it is difficult to see why the Companion should have already left the main series and gone on to this (presumably) later stage. This is akin to other difficulties in the problem of stellar evolution, and I feel convinced that there is something of fundamental importance that remains undiscovered.
Until recently I have felt that there was a serious (or, if you like, a comic) difficulty about the ultimate fate of the white dwarfs. Their high density is only possible because of the smashing of the atoms, which in turn depends on the high temperature. It does not seem permissible to suppose that the matter can remain in this compressed state if the temperature falls. We may look forward to a time when the supply of subatomic energy fails and there is nothing to maintain the high temperature; then on cooling down, the material will return to the normal density of terrestrial solids. The star must, therefore, expand, and in order to regain a density a thousandfold less the radius must expand tenfold. Energy will be required in order to force out the material against gravity. Where is this energy to come from? An ordinary star has not enough heat energy inside it to be able to expand against gravitation to this extent; and the white dwarf can scarcely be supposed to have had sufficient foresight to make special provision for this remote demand. Thus the star may be in an awkward predicament—it will be losing heat continually but will not have enough energy to cool down.
One suggestion for avoiding this dilemma is like the device of a novelist who brings his characters into such a mess that the only solution is to kill them off. We might assume that subatomic energy will never cease to be liberated until it has removed the whole mass—or at least conducted the star out of the white dwarf condition. But this scarcely meets the difficulty; the theory ought in some way to guard automatically against an impossible predicament, and not to rely on disconnected properties of matter to protect the actual stars from trouble.
The whole difficulty seems, however, to have been removed in a recent investigation by R. H. Fowler. He concludes unexpectedly that the dense matter of the Companion of Sirius has an ample store of energy to provide for the expansion. The interesting point is that his solution invokes some of the most recent developments of the quantum theory—the ‘new statistics’ of Einstein and Bose and the wave-theory of Schrödinger. It is a curious coincidence that about the time that this matter of transcendently high density was engaging the attention of astronomers, the physicists were developing a new theory of matter which specially concerns high density. According to this theory matter has certain wave properties which barely come into play at terrestrial densities; but they are of serious importance at densities such as that of the Companion of Sirius. It was in considering these properties that Fowler came upon the store of energy that solves our difficulty; the classical theory of matter gives no indication of it. The white dwarf appears to be a happy hunting ground for the most revolutionary developments of theoretical physics.
To gain some idea of the new theory of dense matter we can begin by referring to the photograph of the Balmer Series in [Fig. 9]. This shows the light radiated by a large number of hydrogen atoms in all possible states up to No. 30 in the proportions in which they occur naturally in the sun’s chromosphere. The old-style electromagnetic theory predicted that electrons moving in curved paths would radiate continuous light; and the old-style statistical theory predicted the relative abundance of orbits of different sizes, so that the distribution of light along this continuous spectrum could be calculated. These predictions are wrong and do not give the distribution of light shown in the photograph; but they become less glaringly wrong as we draw near to the head of the series. The later lines of the series crowd together and presently become so close as to be practically indistinguishable from continuous light. Thus the classical prediction of continuous spectrum is becoming approximately true; simultaneously the classical prediction of its intensity approaches the truth. There is a famous Correspondence Principle enunciated by Bohr which asserts that for states of very high number the new quantum laws merge into the old classical laws. If we never have to consider states of low number it is indifferent whether we calculate the radiation or statistics according to the old laws or the new.
In high-numbered states the electron is for most of the time far distant from the nucleus. Continuous proximity to the nucleus indicates a low-numbered state. Must we not expect, then, that in extremely dense matter the continuous proximity of the particles will give rise to phenomena characteristic of low-numbered states? There is no real discontinuity between the organization of the atom and the organization of the star; the ties which bind the particles in the atom, bind also more extended groups of particles and eventually the whole star. So long as these ties are of high quantum number, the alternative conception is sufficiently nearly valid which represents the interactions by forces after the classical fashion and takes no cognizance of ‘states’. For very high density there is no alternative conception, and we must think not in terms of force, velocity, and distribution of independent particles, but in terms of states.
The effect of this breakdown of the classical conception can best be seen by passing at once to the final limit when the star becomes a single system or molecule in state No. 1. Like an excited atom collapsing with discontinuous jumps such as those which give the Balmer Series, the star with a few last gasps of radiation will reach the limiting state which has no state beyond. This does not mean that further contraction is barred by the ultimate particles jamming in contact, any more than collapse of the hydrogen atom is barred by the electron jamming against the proton; progress is stopped because the star has got back to the first of an integral series of possible conditions of a material system. A hydrogen atom in state No. 1 cannot radiate; nevertheless its electron is moving with high kinetic energy. Similarly a star when it has reached state No. 1 no longer radiates; nevertheless its particles are moving with extremely great energy. What is its temperature? If you measure temperature by radiating power its temperature is absolute zero, since the radiation is nil; if you measure temperature by the average speed of molecules its temperature is the highest attainable by matter. The final fate of the white dwarf is to become at the same time the hottest and the coldest matter in the universe. Our difficulty is doubly solved. Because the star is intensely hot it has enough energy to cool down if it wants to; because it is so intensely cold it has stopped radiating and no longer wants to grow any colder.
We have described what is believed to be the final state of the white dwarf and perhaps therefore of every star. The Companion of Sirius has not yet reached this state, but it is so far on the way that the classical treatment is already inadmissible. If any stars have reached state No. 1 they are invisible; like atoms in the normal (lowest) state they give no light. The binding of the atom which defies the classical conception of forces has extended to cover the star. I little imagined when this survey of Stars and Atoms was begun that it would end with a glimpse of a Star-Atom.
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