Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 54
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An approximate value for the energy flux J resultswhen we calculate the intejgrals on the left hand side in equilibrium. Thus we obtainpradx jȡk 13 aT 41Jorjc2x jȡk1Jj.c2It was Arthur Stanley Eddington – Sir Arthur after 1930 – who usedthese equations decisively when he presented a consistent and completestandard model of a star.52 He considered the pressure P inside the star asthe sum of the gas pressure pgas = ȡk/µT and the radiation pressureprad = 1/3aT 4. Eddington was lucky, because he found a (pressure, density)relation of the formPțȡ 4 / 3which had been extensively studied already by Lane, Ritter and Emden, seeabove.
It is true though that the exponent 4/3 in Eddington’s work hadnothing to do with the ratio of specific heats of the stellar gas – it is areflection of the T4 law, see Insert 7.8. The coefficient ț, which is equal toPc, turned out to be determined by the mass of the star. So, Eddington4/3ȡcwas able to transfer much of the mathematics from the earlier researches to52A.S. Eddington: “The Internal Constitution of the Stars.” Cambridge, University Press(1926).Arthur Stanley Eddington (1882–1944)229his own. And in a few additional steps he could derive a relation betweenthe luminosity LR of a star – the total power emitted – and its mass MR, cf.Insert 7.8. Using Eddington’s data, one can find a rough analytical fit forthe so-called mass-luminosity relation which readsLRL¤§ MR ·¸¸¨¨© M¤ ¹3.5so that the luminosity of a star grows fairly steeply with its mass.
Thisrelation was confirmed for all stars whose mass was known, and that factprovided strong support for Eddington’s model, e.g. for the ideal-gascharacter of the stars, despite their large mean densities and their enormouscentral densities. After that structure was accepted for stars, the massluminosity relation allowed astronomers to determine the mass of a starfrom its brightness provided, of course, that the distance was known.Mass-luminosity relationThe momentum balance equations for matter and radiation and for radiation alonereaddpraddPMk ρG 2r and 2 ρJ ,drrdrcwhere JdpraddrLris the radiative energy flux density.
Elimination of ȡ gives4 Sr 2k1Lr dP2 4π G M dr and by integration pradcrNLRηMRLR1kNηP,24π c G opacity M Rwhere LR is the luminosity of the star. In Eddington’s standard model the opacity isconsidered homogeneous throughout the star and equal for all stars.If ȕP and (1-ȕ)P are the partial pressures of matter and radiation respectively, wehaveprad(1 ȕ ) P1 aT 43hence Tp gas = ȕ Pȡ kµ T31 ȕ 3k µȕaȡ and PË1 ȕ k 4Ì ȕ ( µ)Í1/33Û4/3 .ȡa ÜÝ230 7 Radiation ThermodynamicsThus P is proportional to ȡ4/3 just like in the Lane-Emden theory for Ȗ = 4/3, wherethe factor of proportionality isPc. Therefore comparison with the results ofȡc4 / 3Insert 7.7 shows that we must set3§k· 3¨ ¸ȕ © µ¹ a1 ȕG3 S163§·MR¨¸¨ 2 du¸¨ z dz z ( R ) ¸©¹2so that ȕ is only a function of MR.On the other hand, the formula for prad provides ȕ as a function of1 ȕLRMR:LR1kη.2MR4π c GLR is reliably measurable 53 for all stars, whose distance is known, and MR ismeasurable for many binaries and, of course, both are known for the sun.
Thereforek Ș can be determined from solar data.The mass-luminosity relation follows in an implicit form by elimination of ȕbetween the last two equations. Eddington solved that equation by numericalmeans, plotted it graphically, and compared the curve with astronomical data formany stars, finding good agreement.Insert 7.8His partisanship for relativity secured Eddington a place in 1919 on theexpedition to Príncipe island in the gulf of Guinea, where the bending oflight rays by the sun – predicted by Einstein’s theory of general relativity –was first observed during a solar eclipse.Eddington was so busy changing photographic plates that he did notactually see the eclipse.54Since we are dealing with radiation in this chapter, the ratio of radiationpressure and gas pressure to the total pressure is of interest. Eddington’scalculations suggest, that that ratio depends only on the mass of the star andthat it grows with the mass, cf.
Insert 7.8. For the relatively small sun theradiation pressure amounts to only 5% of the total, but it runs up to 80% fora massive star of 60 times the solar mass. Since there are very few moremassive stars than that, Eddington assumes that a high radiation pressure is53Eddington remarks that…it is said that the apparatus on Mount Wilson [in California] isable to register the heat radiation of a candle on the bank of the Mississippi river. Thatwas in 1926; I wonder what astronomers can do now.54 According to I.
Asimov: “Biographies …” loc. cit. p.603.Arthur Stanley Eddington (1882–1944)231dangerous for the stability of a star 55 … although one cannot, a priori, seea good reason why the radiation pressure acts more explosively than thegas pressure.56Eddington was an infant prodigy of the besttype, – the type that grows into an adultprodigy.
He was one of the first personsto appreciate Einstein’s theory of relativity, andadvertised it to British scientists.At that time it was generally said that onlythree persons in the world understand the theory ofrelativity. When Eddington was asked aboutthat by a journalist he answered: Oh? And whois the third? 57Fig. 7.6. Arthur Stanley EddingtonThere is a group of fairly massive stars – between 5 and 50 solar masses–which exhibit a possible sign of instability by a regularly oscillating luminosity.
These are the Cepheids, named after Delta Cephei for which thatbehaviour was first observed. Naturally Eddington’s attention was drawn tothe phenomenon, and he investigated it without, however, clearly relating itto the predominance of the radiation pressure. I suspect that now stellarphysics can answer that question decisively; if so, I would not have heardabout it.The Cepheids play an important role in astronomy, because theastronomer Henrietta Swan Leavitt (1868–1921) has detected – in 1912 – aclear relation between the mean luminosity of those stars and the period oftheir oscillation: The more luminous stars oscillate more slowly.
No reasonwas at first known, but nevertheless the observation led to the Cepheidyardstick for measuring the distance of galaxies. Since the brightness ofequally luminous Cepheids depends on their distance, while the period ofoscillation does not, of course, the relative distance of two Cepheids fromthe observer could be determined. Eddington’s mass-luminosity relationprovides a plausible explanation for Leavitt’s observation: Indeed, moremassive stars are more luminous and presumably more sluggish in theiroscillations.55A.S. Eddington: “The internal Constitution of the Stars.” loc.cit p.
145.Ibidem, p. 21.57 Nowadays meetings on Relativity Theory are visited by up to 2000 participants. One mustassume that, perhaps, all of them understand what the theory is about.56232 7 Radiation ThermodynamicsEddington’s book “The Internal Constitution of Stars” – written in 1924and 1925 – is crystal clear in style and argument, and when assumptionsoccur, as they invariably must, they are made plausible either by referenceto observations, or by convincing theoretical arguments.
Some things hecould only guess at, most notably the origin of the stellar energy. But heguessed well, albeit without being specific:… after exhausting all other possibilities we find the conclusion forced uponus that the energy of a star can only result from subatomic sources.58Eddington did not identify the subatomic sources. However, his insightinto the enormous temperatures of stellar interiors made it feasible thatnuclear fusion occurs which – basically – forms helium from hydrogen, atleast to begin with.
Hans Albrecht Bethe (1906–2005) is usually creditedwith having worked out the details of this nuclear reaction in 1938,although there were forerunners, most notably Jean Baptiste Perrin (1870–1924).Strangely enough Eddington sticks to the obsolete ether waves when hespeaks of radiation:Just as the pressure in a star must be considered partly as the pressure ofether waves and partly as pressure of material molecules, the heat contentis also composed of ethereal and material components.
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