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.17M . Nov. 1929. Masses, Luminosities, and Temperatures of Stars. 17 .90. w + Pv — 57. In this formula contains the disturbing mass as factor, and we can therefore substitute in it the elliptic value of v, namely, 9MNRAS. v = w + E _0. 192 w This substitution constitutes the main part of the numerical work, but in asteroid calculations the number of terms to be transformed is usually not very great, and for most of them it is sufficient to put v = w 2e sin (w — 5T). There may be a second approximation necessary for a few terms : this, however, presents no difficulties. The analogy to Hansen’s form mentioned in § 1 is made evident by the transformation. Hansen adds the perturbations to the mean anomaly : substantially the same thing is done in this transformation, but these perturbations, instead of being calculated in terms of the time, are calculated in terms of v. Yale University : 1929 May. The Masses, Luminosities, and Effective Temperatures of the Stars. By E. A. Milne. i. In this paper I investigate the relations between the masses, luminosities, and elective temperatures of the stars from a standpoint which is philosophically different from that adopted by Professor Eddington in his well-known writings. The main conclusions are that it is not possible to infer from the observed masses, luminosities, and temperatures that the interiors of stars are necessarily composed of perfect gas ; and that it is not possible to deduce the value of the absorption coefficient for the stellar interior. Instead, we are led to infer a single definite fact concerning the internal density-distribution. I. Introduction. 2. Contemporary physics makes progress by discarding ideas corre- sponding to quantities which experience shows can never be observed. The “ velocity of a system through the aether ” proves to be an “ un- observable ” ; consequently a dynamics must be constructed which avoids this concept, and the resulting theory of frames of reference is relativity. Einstein’s principle of relativity takes the place of the “ unobservable ” velocity through the aether. The simultaneous pair (position, velocity) when attached to an electron proves to be an un- observable ; quantum mechanics therefore discards this concept, and Heisenberg’s “ uncertainty principle ” takes its place. It may be necessary to introduce into our equations symbols corresponding to unobservables, but the physical content of the assertion which results from a piece of mathematico-physical reasoning must be a relation between observables only. It frequéntly appears, when this mode of 2 © Royal Astronomical Society • Provided by the NASA Astrophysics Data System .17M . 18 Prof. E. A. Milne, The Masses, Luminosities, xc. i, .90. thought is followed, that the same results are deduced as before, but they are now shown to be true independently of the introduction of 9MNRAS. concepts corresponding to unobservables. Theory is enriched by being 192 pruned of unnecessary assumptions. The time seems ripe for applying this method to the structure of stars. The interior of a star can never be directly observed. It is not of course an unobservable in the same way as the velocity through the aether or the simultaneous (position-velocity) pair of an electron, for we do not doubt its “ real existence ” in the sense customary in physics. Yet the same general march of ideas impels us to inquire what are the relations between the “ observables ” of a star which are independent of arbitrary assumptions about the unobservable interior. 3. The observables of a star are its mass M, its luminosity L, and its effective temperature Tj ; L and Tx are equivalent to a determination 2 4 of its radius rx by the relation L = 47rr1 crT1 , where cr is Stefan’s con- stant, for the definition of the effective temperature Tj is that is the energy leaving each square centimetre of the star’s surface. Actually the elective temperature has been determined for very few stars,* but we believe that the colour-temperatures which have been determined for many stars give us a fair knowledge of the efiective temperatures. We know further (i) that the star is in mechanical equilibrium, (2) that its outer layers are gaseous, (3) that its outer layers are in radiative equilibrium. For all these we have observational support : (1) follows because the stars hold together ; (2) follows from their observed spectra ; (3) is in fact only known from direct observation in the case of the sun (from the observed variation of brightness over the disc), but it seems legitimate to assume it for other stars. We shall find that (1), (2), and (3) are sufficient to yield a relation between L, M, Tj and the internal structure ; a specific theory of the interior is not required. We have also a fair working knowledge of the physics of stellar atmospheres. The fact that the observed spectra of the stars fall into a linear sequence confirms both qualitatively and quantitatively Saha’s theory of high-temperature ionization. The quantitative relations— the observed temperatures of maxima—amount in principle to a fixation of the photospheric opacity f of the star, though the values of the opacity so far determined may be liable to considerable uncertainty. In addition to the foregoing, when we consider the stars as a whole we have the two sets of facts which are summed up in the observed mass- luminosity law and the observed luminosity-colour law. The former of these was discovered by Eddington, but it must be remembered that it has never received theoretical explanation ; the formula derived by Eddington allows a star of given M to have an arbitrary luminosity by appropriate choice of its efiective temperature T-l, and it has never been explained theoretically why the efiective temperatures of the stars have their observed order of magnitude. How exact is the correlation between mass and luminosity for the majority of the stars remains for * For the sun, by direct observation, and for a few other stars by measurement of their angular diameters. f See Bakerian Lecture (1929), Phil. Trans. Roy. Soc., A. 228, 421. © Royal Astronomical Society • Provided by the NASA Astrophysics Data System .17M . Nov. 1929. and Effective Temperatures of the Stars. 19 .90. future investigation, but the existence of a fair degree of correlation is a fact we must take into account. The luminosity-colour relation, 9MNRAS. summed up in the Hertzsprung-E-ussell diagram and exhibited more 192 particularly (for main sequence stars) by Hertzsprung’s recent studies * of the Pleiades, is very well established. The general problems are {a) whether any of these relations can be explained or predicted in the light of present-day physics ; (b) what information they yield as to the state of affairs in the interior of a star. 4. It is customary to speak of “ luminosity-formulæ,” and we begin our inquiry by asking what is meant by “ calculating ” the luminosity of a star. There are in principle two ways of calculating the luminosity of a star. I. We may reconstruct as best we can the physical conditions— temperature, density, state of ionization, etc.—^holding in the interior and then attempt to calculate what energy will be liberated under these conditions. It is known from various lines of evidence—astronomical, geological, and biological—that gravitational contraction is insufficient, und we deduce that the energy liberated must come from subatomic ¡sources. If we could calculate the rate of liberation of subatomic energy per gram of matter under stellar conditions, simple addition through the whole mass of the star would give its luminosity in the steady state. Unfortunately, in the present state of physics, we know too little of the laws governing the rate of liberation of subatomic energy to be able to perform the calculation. We are therefore compelled to use another method. II. We can build up a star in mechanical equilibrium of the same mass as the given star. We then let it cool freely for a short period dt. The different parts of the star interchange energy, and the star as a whole radiates to space. In the process some parts lose energy and cool down ; others may gain energy and warm up. We now introduce a distribution of sources (and if necessary sinks) of energy through the star just sufficient to restore the star to its state at the beginning of the interval dt. The algebraic sum of the sources and sinks is equal to the amount radiated by the star as a whole to space. We then assume that the actual star is provided with just this distribution of sources and sinks. Its luminosity is then determined. The result of the calculation will be a formula connecting L with the mass M and any other quantities we find it necessary to introduce to build up the star in equilibrium and calculate its cooling. The important point to notice is that the problem is essentially a cooling problem. In any cooling problem boundary conditions are paramount, and they prove to be so in stellar problems. We must not treat the problem as a cooling problem for the innermost TT) of the radius, and then assume the energy flowing past this point can flow through the outer ^ of the radius without affecting the conditions inside.| That would be to change horses in mid-stream. We must carry our cooling problem to the bitter end. * M.N., 89, 674, 1929. f For an illustration taken from physics, see Appendix. © Royal Astronomical Society • Provided by the NASA Astrophysics Data System .17M . 20 Prof. E. A. Milne, The Masses, Luminosities, xc. ir .90. 5. Method II. is in fact the method always followed in calculations of stellar luminosity, though it is not always analysed in this way.

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