194Oap J 92 . . 32IM the ASTROPHYSICAL JOURNAL AN

32IM . 92 J 194OAp THE ASTROPHYSICAL JOURNAL AN INTERNATIONAL REVIEW OF SPECTROSCOPY AND ASTRONOMICAL PHYSICS volume 92 NOVEMBER 1940 number 3 THE INTERNAL TEMPERATURE OF WHITE DWARF STARS* R. E. MARSHAK ABSTRACT The state of matter in the interior of a white dwarf star is investigated in detail. The main part of this paper is concerned with the calculation of the temperature distribution in the interibr. For this purpose accurate expressions for the radiative and conductive opacity are derived, both for the nondegenerate envelope and the degenerate core of the stars. It is shown that the energy transport is chiefly by radiation in the envelope and by electronic conduction in the core. Furthermore, the free electron density is investigated, and it is found that in the transition region between degeneracy and nondegeneracy the atoms are only partially ionized, whereas both in the envelope and in the core the ionization is almost complete (in the envelope, temperature ionization; in the core, pressure ionization). Quantitative calculations are performed for the two well-known white dwarfs, Sirius B and 40 Eridani B; of the two the observational material is better for Sirius B, so that most attention is paid to this star. The extension of the nondegenerate envelope and the internal temperature are calculated for several widely different assumptions regarding the chemical composition of the star as well as possible errors in the observed luminosity L and the radius R. The depth h of the envelope of Sirius B is about 6 • 107 cm, when the observed values of R, L and mass M are used and the star is assumed to consist exclusively of “Russell mixture.” For a pure helium star h is even smaller, and, if the theoretical value for the radius is assumed, h is only 8 • 106 cm. Thus, the envelope constitutes only a very small fraction of the total radius of the star (~ 109 cm). o The central temperature Tc of Sirius B is ^ 15,000,000 for pure Russell mixture and is almost independent of the assumed radius; moreover, the degenerate core is practical- ly isothermal. At this temperature and at the high densities of 105 to 3 • 107 gm/cm3 which prevail in the core, nuclear reactions between protons and other nuclei would go at a very rapid rate. Since the observed luminosity is very small, we must conclude that no appreciable number of protons is present inside of white dwarfs. If Sirius B contains carbon and nitrogen in the same abundance as do main-sequence stars, the hydrogen * The major part of this work was done while the author was President White Fel- low at Cornell University; preliminary results were reported with H. A. Bethe to the American Physical Society (cf. Phys. Rev., 55, 681, 1939, and 57, 69, 1940). 321 © American Astronomical Society • Provided by the NASA Astrophysics Data System 32IM . 92 322 R. E. MARSHAK J concentration X cannot be more than 2 • io-8; and even if C and N are completely H _s absent, XH must be less than 2 • io , because the protons will still combine with each other. A lower reaction rate is obtained for a lower central temperature; for this pur- 194OAp 6 pose a pure helium star was investigated for which rc âî 7 • io ° was found; but even -3 at this temperature the proton-proton reaction is so fast that XH must be less than 10 . For 40 Eridani B it turns out, with the assumption of Russell mixture and observed M, L, R, that the envelope has a depth of 3 • 107 cm, the central temperature a value of 30 • io6°, the hydrogen content an upper bound of 8 • io~s. We must, therefore, conclude that no hydrogen is present in white dwarfs. Under these circumstances the radius of the star is uniquely determined by its mass in ac- cordance with Chandrasekhar’s theory of degenerate configurations. The theoretical radius of 40 Eridani B agrees well with the observed radius, but the radius of Sirius B is only 5.7 • 108 cm, as compared with an observed radius of 13.6 • 108 cm. This large discrepancy is very serious, since the observational value is considered good and there does not seem any way out from the standpoint of theory. Thus, no reasonable modifications of the theory (which, however, still keep the hydrogen content small) will ap- preciably increase the extent of the nondegenerate envelope and therefore the radius of the star. The only faint possibility would be to assume that Sirius B does not contain any light nuclei other than helium and hydrogen and that, moreover, the combination of two protons is much less probable than follows from well-established nuclear theory. A further result of these investigations is the conclusion that the energy production of the white dwarfs is not due to thermonuclear reactions at all, but to gravitational con- traction. With this source of energy it will take at least 108 years before the white dwarfs will become dark objects. I. INTRODUCTION Recent investigations1’2’3’3*1 into thermonuclear processes as the source of energy of main-sequence stars have proved quite conclu- sively that the carbon-cycle is responsible for the energy production in main-sequence stars brighter than the sun. For main-sequence stars fainter than the sun the evidence is less conclusive; calculations on the proton-proton reaction, using the Fermi ß-decay theory, the Gamow-Teller selection rules, and the average of a set of empirically determined decay constants, seem to indicate that’ this reaction can account for the energy evolution of the fainter stars. For the sun itself theory predicts that the carbon-cycle and the proton-proton reaction give about equal contributions to the luminosity.4 On the other hand, it is well known that white dwarf stars differ from main-sequence stars of the same mass in two fundamental 1 H. A. Bethe, Phys. Rev., 55, 434, 1939. 2 H. A. Bethe and C. L. Critchfield, Phys. Rev., 54, 248, 1938. 3 H. A. Bethe and R. E. Marshak, Reports on Progress in Physics, 6, 1, London: Physical Society of London, 1939. 3aH. A. Bethe, Ap. /., 92, 118, 1940. 4 R. E. Marshak and H. A. Bethe, Phys. Rev., 56, 210, 1939. © American Astronomical Society • Provided by the NASA Astrophysics Data System 32IM . 92 WHITE DWARF STARS 323 J aspects : they have extremely low luminosities and exceedingly high densities (as a result of their very small radii), compared with the 194OAp main-sequence stars. The high densities in the white dwarfs have been explained in terms of the theory of degenerate matter, and it has seemed likely that their low luminosities are due to an energy- production process other than that occurring in main-sequence stars. In an attempt to settle the matter of the energy production and, moreover, to determine the modifications introduced into Chandra- sekhar’s theory of degenerate configurations5 by the presence of finite temperatures, we have investigated in detail the temperature distribution in the interior of the white dwarfs. To make any head- way at all it was necessary to obtain fairly accurate expressions for the radiative and conductive opacities not only under conditions of extreme degeneracy but also at the onset of degeneracy. Such expressions have been derived by the author,6 and the results are sum- marized in section II. Similarly, the free-electron density, the pressure, and the Fermi energy are calculated for the transition region (between the radiative envelope and the degenerate core), and the results given in section III. In section IV the star equations are integrated through the sur- face region of the two best-known white dwarfs, Sirius B and 40 Eridani B; both the temperature and the extent of the radiative envelope are found under different assumptions as to chemical composition, luminosity, and radius. In section V the integration of the equations of stellar equilibrium is carried through to the center, quite independent of any explicit assumptions concerning the energy-production process. The resulting temperature-density distribution is used to determine upper bounds on the hydrogen con- tents and radii of Sirius B and 40 Eridani B. In the final section (sec. VI) the theory is compared with the observational material, and some conclusions are drawn regarding the sources of energy, the radii, and the evolutionary behavior of white dwarf stars. 5 S. Chandrasekhar, Introduction to the Theory of Stellar Structure, esp. chaps, x and xi. 6R. E. Marshak, Annals of New York Academy of Sciences, 1940. © American Astronomical Society • Provided by the NASA Astrophysics Data System 32IM . 92 324 R. E. MARSHAK J II. THE RADIATIVE AND CONDUCTIVE OPACITIES Under conditions of nondegeneracy the radiative opacity Kr is 194OAp given by Kramers5 formula corrected by the insertion of a “guillo- tine55 factor.7 This formula may be written 6.0 • io^Xr ç-^p/kT * Vf2 (1) The quantity T is the temperature, k is the Boltzmann constant, Xr is the concentration of the Russell mixture of heavy elements, r is the guillotine factor, and \[/ is the negative of the Fermi energy (\p/kT ^>> o implies nondegeneracy, \f//kT <K o gives strong degeneracy, and \f//kT ^ o holds for the intermediate region of in- cipient degeneracy). Equation (1) is valid for \¡//kT o; it is found6 that the following expression is a good approximation for Kr in the regions \(//kT « o and \f//kT o: 6.0 - ioI7X* T i + e~*/kT ] Kr = l0 tT2 ^ + e-*/kT-7j * (2) In the limit \[//kT equation (2) goes over into the usual formula for the degenerate radiative opacity.8 In the region « o, r ^>> i, so that the contribution from the bound-free transitions is cut down considerably; ior \f//kT ^ — 5, r attains its maximum value of 196.5, and for larger negative values of \¡//kT only the free- free transitions contribute.

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