Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 53
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That equation may be written in the formd2Ȗ PȖ 1 ȡdr2Ȗ2 d Ȗ 1r drPȡ14SG( Ȗ Ȗ 1 Pȡ ) Ȗ 11P[ Ȗ Ȗ 1 c ] Ȗ 1ȡcȖ0,and it is known as the Lane-Emden equation.41 Lane set it up and Emdensolved it by a laborious numerical scheme, and published reams and reams39W. Thomson: “On the convective equilibrium of temperature in the atmosphere.”Proceedings of the Literary and Philosophical Society of Manchester (3) II (1862) pp.125–131.See also: W. Thomson: Philosophical Magazine 22 (1887) p. 287 andW. Thomson: “Mathematical and Physical Papers.” 5 Cambridge (1911) p.
256.40 J. Homer Lane: American Journal of Science and Arts. Series 2 Vol. 4 (1870) p. 57.41 So called by S. Chandrasekhar: “An Introduction to the Study of Stellar Structure.”University of Chicago Press (1939) p. 88. Reprinted by Dover Publications (1957). Pagenumbers refer to the Dover edition.Chandrasekhar also gives much credit to A. Ritter who investigated the condition ofconvective equilibrium in a star independently of Lane’s work. Ritter published 18 papers“Untersuchungen über die Höhe der Atmosphäre und die Constitution gasförmigerWeltkörper.” [Investigations on the height of the atmosphere and the constitution ofgaseous bodies] Wiedemann Annalen (1878–1883).
Those papers says Chandrasekhar…form a classic the value of which has never been adequately recognized…and he istempted to rename the Lane-Emden equation and call it the Lane-Ritter equation.However, Emden gave much credit to Ritter.Convective Equilibrium225of tables for different values of Ȗ.42 Nowadays the solution is very easy withany one of the software systems for doing mathematics, e.g.
Mathematica®.Thus Emden was able to calculate the central values ȡc and Pc of density andpressure for the sun (say). He obtained, cf. Insert 7.7ρc75.6 ¹103kgm3, Pc22.5 ¹1010 bar, hence Tcwhere Ȗ was chosen as 4/3 43 and wherePP0µµ04.03 ¹107 K,is the relative molecular mass ofthe particles of the sun. Such calculations suggested that the central temperatures of stars amount to several 10 million K, and that the central densitiesare many times higher than the density of the densest metal.Solution of the Lane-Emden equation for Ȗ = 4/3With dimensionless dependent and independent variables2 UEU 2EWand\T S)UE2E4the Lane-Emden equation reads for Ȗ = /32dudt22 duz dzu30 with u (0)1,dudz 00.The solution is shown in Fig.
7.5. The radius of the star occurs where u(z) crossesthe abscissa: the table in the figure shows that this happens for z(R) = 6.90. Thetable also shows values of z 2 dduz and, in particular, its surface value z2dudz z ( R )2.015.Fig. 7.5. Solution of the Lane-Emden equation42R.
Emden: “Gaskugeln: Anwendungen der mechanischen Wärmetheorie.” [Gas spheres:Applications of the mechanical theory of heat] Teubner, Leipzig and Berlin (1907).43 Ȗ = 4/ proved later – in Eddington’s work – to be the correct coefficient, although in that3work it is not the ratio of specific heats.226 7 Radiation ThermodynamicsThese values are important for the calculation of Pc and ȡc in terms of the radius Rof the star and of its mass MR.
We havez( R)R SGȡcand z2PcSG 3dudz z ( R )4MRȡc2Pc3,the latter relation from the momentum balance. Hence follows2ȡcz 3 ( R) z 2 dduz z ( R )1 MR4S R 3andPc§·¨¸ G M2z 2 ( R)R.¨¸4udSR16¨ z2¸dz z ( R ) ¹©Insert 7.7Neither Ȗ nor µ were known to the physicists of the time. We recall that for ideal574gases Ȗ can have three values, namely /3, /5 , or /3 depending on whether themolecules have one, two or more atoms. Later, when the significance of the4radiation pressure was recognized, it turned out that Ȗ = /3 is correct, see below,although there are no molecules in the sun.
Indeed, the sun and other stars consistmostly of nuclei and free electrons, because the atoms are largely stripped of theirelectrons at the high temperatures which prevail.44 The particles – nuclei, electronsand a few ions – fly freely through the space that is shielded by the electronic shellsunder normal conditions. It is for that reason that the matter of a star may beconsidered an ideal gas, even when the density in its center is a hundred timesbigger than the density of the heaviest metal.The strong ionisation is also responsible for a rather small value of µ, the meanmass of the particles, because the free electrons contribute a lot to the numberdensity of particles, but only little to the mass density.
Of course, µ depends on thecomposition of the star: The heavier the atoms are that compose a star, the more1closely µ/µo is equal to 2, because a heavy atom contributes approximately /2 µ/µoelectrons. If, on the other hand, the sun consists mostly of hydrogen – as we nowthink it does – µ/µo is equal to ½, because a hydrogen atom provides two particles,a proton and an electron. Thus the above calculation suggest an interior solartemperature of 20.000.000K.4544Of course, neither Lane nor Emden – nor anybody else at the time – knew anything aboutthe atomic structure, or that an atom contains largely empty space, and that there arenuclei and electrons. This knowledge came with Rutherford in 1913, I believe, and it wasprimarily Eddington in the 1920’s who made use of the new knowledge, see below.45 By good luck, perhaps, we have thus calculated an interior temperature of the sun that iscurrently considered to be right: According to the “Fischer Lexikon zur Astronomie” thetemperature lies between 17 and 21 million K.Arthur Stanley Eddington (1882–1944)227Arthur Stanley Eddington (1882–1944)Although the pressure of radiation or, more generally, of electro-magneticfields had been known since Maxwell’s equations were known, cf.
Chap. 2,its role in stellar physics was not recognized at first. The reason must havebeen that the radiation pressure prad exerts a minimal force in everyday life,so small that it cannot be felt when we hold out our hand to absorb sunlight(say). Note that, by Table 7.1, the momentum density Pj is related to theenergy flux Jj by Pj c12 J j , so that the momentum density is a lot smallerthan the energy flux. But then, prad equals 1/3 aT 4 according to Table 7.2. Itgrows with the fourth power of T and as it became clear, or at leastprobable, that the interior temperature of stars reaches millions of degrees,researchers decided that it might be worthwhile to look at the momentumbalance equation of radiation rather than only at the energy balance.It seems that Karl Schwarzschild (1873–1916) was first to take theradiation pressure into account in his investigations of the solaratmosphere.46 Another influential astrophysicist was S. Rosseland.47Between them they worked out a plausible expression for the source densityS in the photon transport equation.
In obvious analogy to Einstein’s ansatz 48for absorption and emission – spontaneous and stimulated – of photonsRosseland setsSρ ( cn (a bf ) cm cf ) ,49where cn and cm are the concentrations of atoms in the energetic statewith İm, and İn = hȞ + İm.50 ȡ is the mass density. Following ideas ofSchwarzschild 51 about thermodynamic equilibrium Rosseland gives thisequation a suggestive form: In equilibrium the right hand side must vanishand, since we know fequ, we obtain a/c = h3, b/c = 1, while ccn must be equalm46K.
Schwarzschild: “Über das Gleichgewicht der Sonnenatmosphäre.” [On the equilibriumof the solar atmosphere] Göttinger Nachrichten 1906 p. 41.K. Schwarzschild: “Über Diffusion und Absorption in der Sonnenatmosphäre.” [Ondiffusion and absorption in the solar atmosphere] Berliner Sitzungsberichte 1914 p. 1183.47 S. Rosseland: “Note on the absorption of radiation within a star.” Monthly Notices Vol.
84(1924) p. 525.S. Rosseland: “The theory of the stellar absorption coefficient.” Astrophysical JournalVol. 61 (1925) p. 424.48 A. Einstein: “Quantentheorie der Strahlung.” (1917) loc.cit.49 Note that this expression goes a little beyond Einstein’s ansatz in that the emissionstimulating ray produces a photon with its own Ȟ and in its own direction n.
This is thephenomenon exploited in lasers.50 E.g. see: S. Rosseland: “Astrophysik auf atomtheoretischer Grundlage.” [Astrophysicsbased on atomic theory] Springer, Berlin (1931).51 K. Schwarzschild: Göttinger Nachrichten (1906).228 7 Radiation Thermodynamicstohν )exp( kTAssuming that the coefficients a, b, and c obey the samerelations in non-equilibrium, or near-equilibrium, Rosseland could cast thesource term into the plausible formSȡ ccm (1 exp[ kThν ]) ( f equ f ) .kThus very plausibly the source is proportional to the difference betweenthe photon density function and its equilibrium value with k – the Ȟ- and Tdependent coefficient of absorption – as factor of proportionality.According to Tables 7.1 and 7.2 we may therefore write the momentumbalance of radiation in the formÔhνn j fdp hν n n fdpÔ jkctxkρ k Ôhνn j fdp ,cwhere the Ȟ-dependence of k has been neglected. The integral on the righthand side represents the momentum density or c12 energy flux.
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