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504 Progress of Theoretical Physics, Vol. 36, No. 3, September 1966 Central Temperature and Density of Stars in Gravitational Equilibrium Katsuo TAKARADA, Humitaka SATO and Chushiro HAYASHI ·Department of Physics, Kyoto University, Kyoto (Received April 20, 1966) The relation between the central temperature and the central density is investigated for a polytropic star and a polytropic (or isothermal) stellar core, which are in gravitational equilibrium. The ranges of the temperature and density cover the cases of non-degenerate and degenerate, non-relativistic as well as relativistic, electrons. It is shown that, when the mass of the star or of the core is smaller than a certain critical value which depends slightly on the polytropic index N and is equal to Chandrasekhar's limiting mass for N =3, the central temperature takes a maximum value depending on the mass at the onset stage of the electron degeneracy. §I. Introduction and summary Throughout the life of a star, the central temperature and density changes to a considerable extent. The features of the over-all evolution of the star are determined mainly by how far the central temperature rises in its whole life and accordingly how far the synthesis of the chemical elements proceeds 1n the interior. For example, it is well known that a star smaller than about one-tenth of the solar mass is unable to undergo the hydrogen burning phase, since at a certain stage of its pre-main-sequence contraction electron degeneracy sets in and afterwards the central temperature has to decrease. 1), 2),B) On the other hand, if a star is massive enough, it will follow all the phases of nuclear burning, from hydrogen burning to the formation of iron, developing a shell structure composed of zones of different chemical compositions. The evolution of such a massive star has already been studied up to the central carbon-burning phase for a stellar mass of 16 M 0 . 4 ), 5 ) It has been found that, in the stages before the onset of carbon burning, electrons in the central region are relativistically degenerate if the energy loss by neutrino emission is taken into account. In order to investigate the over-all features of stellar evolution and their dependence on the mass, it will be necessary first to know the over-all variations in the central temperature and density which are allowed for stellar configurations in gravitational equilibrium. Strictly speaking, these variations depend on the details c;>f the development of the shell structure. In this paper, however, an attempt is made to find an approximate relation which holds between the central temperature and density of a star or of a stellar core, which satisfies the following simplified conditions. Central Temperature and Density of Stars in Gravitational Equilibrium 505 First, for the case , of a homogeneous star a polytropic relation is assumed to hold between the pressure and density throughout the whole interior. Secondly, for a star having a shell structure our consideration is confined to its innermost homogeneous core. The pressure at the core boundary is assumed to be so small compared to the central pressure that the existence of the outer regions can be neglected (the single star approximation) . This assumption is well satisfied for most of the advanced evolutionary phases which have so far been studied for stars of different masses. 4 ),B) Furthermore, the polytropic relation or the isothermal condition is assumed to hold in this core. Under the above approximations, a relation between the central pressure and density is first obtained from the condition of gravitational equilibrium alone, as described in § 2. This relation depends on the star or of the core, but it is rather insensitive to the polytropic index. A general expression for the equation of state of a relativistic electron gas, which include the cases of non-degeneracy and degeneracy, is given in § 3. By means of this expression, a relation between the central temperature Tc and the central density Pc is obtained, in § 4, as a function of the mass. The numerical results are illustrated by Fig. 2. As is well known, Tc increases as p/13 at low densities where the electrons are non-degenerate. When the mass is smaller than a certain critical value, which is equal to Chandrasekhar's limit when the polytropic index is three, Tc attains a maximum value because of the onset of electron degeneracy and afterward Tc decreases with increasing Pc. On the other hand, Tc increases without limit when the mass is greater than the critical value. Using the above results, the critical stellar masses for nuclear burning are estimated in § 5. Finally, a discussion is given of the limitations on the gravitational equilibrium, i. e. of the onset of instability caused by the creation of electron pmrs at high temperatures or inverse beta-decays at high densities. § 2. Relation between the central pressure and density We consider spherically symmetric configurations of stars which are in gravitational equilibrium. If a polytropic relation, Pocpi+l/N (N is a constant), holds between the pressure and the density throughout a star or throughout a stellar core, we have7) M = -4n asP ~2dfJ r . c d~, (1) where Pc and Pc are the central pressure and density, Mr is the mass contained in a sphere of radius r, and fJ (~) is the Emden function of index N which is defined by fJN = p/ Pc and r =a~. If the polytropic relation holds throughout a star of mass M, we have from Eq. (1) a relation between Pc and Pc as a function of M, i.e. 506 K. Takarada, H. Sato and C. Hayashi (2) with (3) where the_ subscript 1 denotes the stellar surface and f is a constant which depends only on the polytropic index N, as shown in Table I. The values in Table I have been calculated using the surface values, ( -ed ()/~) 1 , of the Emden functions tabulated by Chandrasekhar.7) It is to be noticed that the stellar radius is infinite for N = 5, but f takes a finite valve. In realistic stars of homogeneous composition, th~ polytropic index is not generally constant, but its local value lies mostly in the range between 1.5 and 3.0 and thus the value off i~ limited within ·a relatively narrow range, as is seen from Table I. Table I. The value off for a polytropic star of index N. N f 0 0.8060 0.5 0.6379 1.0 0.5419 1.5 0.4780 2.0 0.4310 N f 3.0 0.3640 3.5. 0.3379 4.0 0.3146 4.5 0.2925 5.0 0.2687 2.5 0.3942 For a polytropic index greater than 5, there is no surface boundary of a star. In this case, the U- V curve of an Emden solution starting from the center (U = 3, V = 0) attains a maximum value of V, as shown in ·Fig. 1, and afterwards approaches the point N-3 N+1' U=~- V=2N+1 N-1' (4) by. spiraling around it. For an evolved star in which a shell structure has been developed by nuclear burning, the innermost homogeneous core will, in some cases, be described approximately by a polytrope of index greater than 5. For a polytropic core as mentioned above, the same expressions as Eq s. (2) and (3) are obtained from Eq. (1), if the core mass is taken as M and the core boudary is denoted by the subscript 1. The value off depends generally u Fig. 1. The U- V curves of the Emden functions of polytropic index N which start from the center, U=3 and V=O. The dots represent the points where V take maximum values. Central Temperature and Density of Stars in Gravitational Equilibrium 507 on the position of the core boundary on the U- V curve in Fig. 1, but its change along the U- V curve is very small in so far as the core boundary lies in a region of small U (where the single star approximation as mentioned in § 1 is valid), i.e. in a region near or exterior to the point where V takes a maximum value. The Emden functions and the values of f have been computed on the Kyoto University KDC II for several values of N. For N = ?O, the table of Chandrasekhar and Wares 8 ) has been used. The values of f, P/Pe, U and V at the points of maximum V are tabulated in Table II. Again the value off is insensitive to the polytropic index. Table II. The boundary values of a polytropic core of index N. N 5.0 6.0 8.0 20.0 f 0.2777 1.182X 10-3 u1 0.2687 0 0 V1 6 0.2809 5.9126 X 10-3 0.5845 3.744 0.2863 1.851X 10-2 0.8623 2.907 Pt!Pe 0.3563 4.509 I I 00 0.2904 3.113X10-2 1.000 2.5176 i I Finally, we consider a partially-degenerate isothermal core in which relativistic effects are taken into account. By using the equation of state for an electron gas which will be described in § 3, the solutions for the jsothermal core, which are specified by the two parameters a=kT/mc 2 and ¢e (the degeneracy at the center), have been tabulated by Hayashi and Cameron9) for the case Table III. a .10 c/Je The boundary values of a partially-degenerate isothermal core. 0.00 I 20 l 100 log (Pel Jl.E) f 0.4594 5.5x10-3 2.1x10-2 -0.5 0.3450 7.6765 Pt!Pe PtfPe ¢1' U1 V1 a c/Je log (Pel Jl.E) f Pt!Pe PtfPe ¢1 ul V1 0.4690 0.4772 7.7X 10-4 2.7x 10-6 5.8x 10-3 1.1 X 10-4 -0.8 -2.5 0.1104 0.0021 14.0714 71.1787 I 10 I 10 6.305 0.4264 5.4x10-3 1.9X 10-2 -0.3 0.3657 6.9980 I 20 6.883 0.4125 6.6X 10-4 4.2x w-3 -0.5 0.1204 11.6236 20 I 100 I 10 I I 100 I 100 0.10 0.08 I 20 8.073 6.531 7.357 5.475 5.978 5.986 0.3839 0.4253 0.4031 0.4345 0.4448 0.4429 5.1x10-3 7.8x10-4 1.8x 10-6 5.2x10-3 7.4>< J0-4 1.2X 10-6 I1.9X 10-2 5.4x10- 3 5.7x 10-5 1.9X 10-2 4.9x1o-3 3.6x10-5 -2.0 -0.5 -0.6 -2.1 -0.4 -0.5 0.0021 0.1245 0.3526 0.3393 0.1212 0.0023 I 7.1877 12.2379 45.2222 7.4120 13.0217 52.6272 0.06 I 0.04 0.02 )1 10 I 20 8.524 6.542 7.148 0.3759 0.4214 0.4034 1.1X 10-6 6.1xl0-3 6.4X 10-4 2.9x w-5 2.0x10-2 3.9X lQ-3 -1.8 -0.4 -0.1 . 0.0021 0.3971 0.1242 41.1895 11.1298 6.8668 I 100 10 I 20 7.362 8.857 6.734 0.3965 0.3718 0.4161 8.8x10-7 6.3x10-3 6.4X 10-4 2.4X lQ-5 2.1x10- 2 3.8x w- 3 -0.3 -1.7 0.0 0.4095 0.1283 0.0019 38.6477 10.7278 6.6984 100 I- - - - - - - I I 9.121 0.3694 7.8xl0- 7 2.1X10- 5 -1.6 0.0019 36.9007 508 K. Takarada, H. Sato and C. Hayashi of negligible ion pressure. In terms of these solutions, the expressiOn for which replaces Eq. (3) is given by f= (4rr)lf3 G(cfJe, a)/cp1 2;3{F(¢e, a)} 4; 3 , where cp IS (5) a variable defined by Eq. ( 4C. 26) in reference 3), Mr = J1E 2 I. e. (2m kT) 3f4 (2m G) 3; 2 cp ' h3f2 4n f I-J2 the subscript 1 denotes the core boundary as before, and G and F are functions defined by Eqs. (11) and (12) in § 3. The core boundary is chosen as a point of maximum V on the U- V curve, as mentioned previously. The results, which are given in Table III, show that the value off lies intermediately between the cases, N = 1.5 and 3.0, of a polytropic star. § 3. Pressure and density of an electron gas When relativistic effects are taken into account, the number density n and the pressure P of a gas composed of free electrons are given by7) (6) (7) with (8) where JlE and cpkT are the mean molecular weight and the thermodynamic potential of an electron, respectively. The above expressions can be re-written as n=A F(cp, a), P=AkTG(cjJ, a), 4n (2mkT) 3 A=~ ha ; 2 ' a = _kT_ = _T_____:_(°K_) mc 2 5.93 X 109 (9) (10) ' where (11) (12) Central Tenzperature and Denszty of Stars zn Gravztatwnal Equtlzbrzum 509 are the generalized Fermi-Duac functiOns which reduce to F1; 2 ( cjJ) and (2 /3) X F 3; 2 (¢), respectively, In the non-relativistic limit, a =0 We have a general relatiOn F(¢, a) = aG (¢, a) ja¢ (13) In the case of non-degeneracy (¢~ -1), the functiOns F(¢, a) and G (¢, a) can be expressed In power senes 00 F= (2a) -1!2~ ( -1) n+1n-1en<t+1fa) K2 (n/ a) ' (14) n=l 00 G = (2a) -1!2~ ( -1) n+1n-2en<t+1fa) K2 (n/ a) , (15) n=1 where K 2 IS the modified Bessel functiOn In the case of extreme degeneracy (¢~ 1), the asymptotic expressiOns for F and G are giVen by (16) where x = (2a¢ + a2 ¢2) 1f2 ' (18) r/ f(x) =3ln{x+ (1 +x2 2} +x(2x2-3) (1 +x2) 112 - 8x5/5- 4x7/7 + x 9/3 + 0 (x11 ) 2x4 -2x 2+3 ln 2x-7 /4+ 0 cx- 2) 1 for x~1 (19) for x~1 For the Intermediate case of Incomplete degeneracy, the functiOns F and G have been tabulated by Hayashi and Cameron 9) by direct IntegratiOn with an IBM 7090 In the ranges and Intervals, cjJ = -4 0 (0 1) 20 0 and a= 0 00 (0 02) 0 38 § 4. Relation between the central temperature and density Now, a relatiOn between the central temperature and density will be obtamed from Eqs (2) and (9) To take mto account the arb1trarmess of llE and the small vanatwn off, the mass In Eq (2) IS measured In units of _ ( -3) Mon 1 2 1 ( 1 \ ~- /lbH) 2 ( -he-- ) 8Gf 3 2 1 _ 1 26 M0 11E2 f3!2 (20) It IS to be noticed that for N = 3, M 0 IS exactly equal to Chandrasekhar's limitIng mass The radiatiOn pressure, which IS Important for a mass greater than M 0 , IS mcluded, but the IOn pressure, which Is not Important for a gas of heavy elements, Is neglected for simplicity Then, using 510 K. Takarada, H. Sato and C. Hayashi P=AkTG(cjJ, a) +a T 4/3, a=8n 5 k4/15c 3 h 3 Eq. (2) IS (21) , expressed in the form M Mo = 2 1114 ~c 314 {G(cjJ, ac) + (n /~5v~)ac~ }~~ 3 112 {F(cfJc, ac)} 2 4 2 (22) 8 I I I I 20 I I I I 7 -0.5 3 4 5 -02 -0.1 -0.05 6 7 8 9 10 .Qog(fc/JJE) ( g/cm3) Fig. 2. The curves of constant M/Mo in the central temperature-density diag~am. The dashed curves represent the states of given cPc· The stars become unstable when they enter the shaded region where r is smaller than 4/3 because of the creation of electron-positron pairs and the decomposition. of. iron nuclei. Central Temperature and Density of Stars in Gravitatzonal Equilibriwn 511 Using the asymptotic expressions and the table for the functions F and G, the values of ac and </Jc (and hence the values of Tc and Pc) which satisfy Eq. (22) have been calculated for a variety of values of M/M0 • The upper limit of M/ Mo has been taken to be 10 so that the presence of electron-positron pairs can be neglected. The results are shown by the curves of constant M/Mo in the Tc- Pc diagram in Fig. 2. If M 0 is a constant, each of these curves represents the variation 'of Tc with Pc in a star or a stellar core of given mass. In this case, Tc increases as p/13 as long as the electrons are non-degenerate. The behavior of Tc in the degenerate region is defferent depending upon whether the mass is greater or smaller than Mo For a ~ass greater than M 0 , Tc increases continuously with Pc though the gradient is relatively small at the stage of incipient degeneracy. This effect of the degeneracy is conspicuous when the mass is close to M 0 , as shown by the curve of log (M/ Mo) = 0.01 in Fig. 2. On the other hand, for a mass smaller than Mo, Tc attains a maximum value which depends on the mass, as shown m Table IV, and decreases with further increase of PcTable IV. The values of Tc, • The maximum values of Pc and ¢c at Pc ar~ added the tops of the for comparison. Tc-Pc curves for constant JI.;JjM0 • II log(M/M0 ) I! ------- logTc The stage of maximum ¢c ---- ---------- max1mum density I; 9.096 6805 4.87 log(pc/.Up;) temperature --------- -o.o5 -0.10 -0.20 -0.30 -0.50 -1.00 8.968 6.388 3.90 8.768 5.882 3.17 8.601 5.525 2.85 8.291 4.974 2.58 7.593 3.873 2.30 --~------ -- - - - - - - - - 7.562 log (Pel .UE) - -------- 7.029 6.442 5.459 6.050 4.254 When the mass is sufficiently smaller than M 0 , we can use the non-relativistic approximation and at the same time the radiation pressure can be neglected. In this case, we have from Eqs. (9) and (22) a relation between Tc and Pc of the form Tc = fJ (</Jc) ( M)4;s o K, Mo (23) where the functions fJ C</Jc) and rJ C</Jc) are tabulated in Table V. On each of the curves of constant M) M 0 in the Tc- Pc diagram, Tc attains a maximum value at </Jc = 2.276. Relativistic effects become important for M/ M 0 >0.3 as shown in Fig. 3. Table V. -4.0 I log 0 log cr 7.7896 1.8619 1 1 -2o • I I The values of {) (¢) and cr (¢) in Eq. (23). o.o -------- --------- 8.3406 1 3.5398 .1 8.7661 4.9503 2.o __ I __!-~~~-- J -~~_]__2?___ 8.9152 . 8.9168 5.7411 5.8032 I I 8.8771 6.0467 I I 10.0 15.0 1 8.74781 6.1942 - - - 8.6283 6.2411 8.4726 6.2692 512 K. Takarada, H. Sato and C. Hayashi llog·fc(''K) 75 3~------~------~5--~---L~s---L--~7 flog UUJ.lE) (g/cm3 ) Fig. 3. The curves of constant M/M0 in the Tc-Pc diagram. The solid and dashed curves represent the relativistic and non-relativistic cases, respectively. § 5. Application and discussion For a star to experience a certain kind of nuclear burning in its life, its mass has to be greater than a certain critical value. The results in § 4 will be used in the following to estimate the minimum masses for burning of hydrogen, helium, carbon and so on. The effect of the ion pressure, which has so far been neglected f<?r simplicity, is important for the accurate determination of the critical masses, especially for burning of hydrogen and helium. This effect can be included by replacing G (</Jc, ac) in Eq. (22) by G C</Jc, ac) + (!1E/ !lr) FC</Jc, ac)' where /lr is the mean molecular weight of the ions. The result is to lower the curves of constant M/Mo in Fig. 2 and to shift their maximum points to slightly higher densities. For the case of hydrogen burning, we abopt !1E = 1.24 and flr= 1.40, corresponding to the chemical composition X=0.61 and Y=0.37. Further, we take f = 0.478 which corresponds to a wholly convective structure (N = 1.5) in the pre-main-sequence stage. For helium burning, we take /1E = 2, /lr = 4 and f = 0.453 which corresponds to a point-source model of constant opacity and negligible radiation pressure. 10 ) The temperatures of hydrogen burning and- helium burning are simply taken to be 5 X 10 6 °K and 1.0 X 10 8 °K, respectively. The final results for the corresponding minimum masses and the central densities are given in Table VI. These values are 1n good agreement with the results of other authors. 1 ). 2 ).s).u) For carbon burning, we adopt flg=2, !1r=12 and f=0.453 for the above Central Temperature and Density of Stars in Gra·vitational Equilibrium 513 point-source model. The burning temperature Tc has been determined as follows. The minimum mass and the central density are calculated as a function of Tc for the case of non-relativistic degeneracy. Then, the luminosity is obtained using the mass-luminosity relation of the above stellar model, which is given by __L =21.2/14 ( M) Le !Co (24) s' Me where the opacity /Co is taken to be 0.2 crn 2/ g and the effect of degeneracy Is adequately included by taking 1/,u=1/,uEA+l/J1r with A=0.7. On the other hand, the nuclear luminosity is obtained using Eq. (4D. 4) in reference 3) and the nuclear energy production rate given by Reeves 12 ) where the fractional abundance by mass of carbon is taken to be 0.5. The above two luminosities agree with each other for Tc = 5.5 x'10 8 °K. The minimum mass for this temperature and the corresponding central density are given in Table VI. It is to be noticed that in the case of relativistic degeneracy the critical mass will be somewhat larger than the tabulated value since the polytropic index in the convective core is slightly larger than 1.5 and correspondingly the value of f is smaller than the adopted value. Table VI. The minimum masses for nuclear burning. Relativistic degeneracy Helium Carbon 1Xl08 5.5Xl08 0.30 0.72 3.66 3.88 Non-relativistic degeneracy 5.004 0.32 4.16 3.38 6.265 0.89 2.66 2.902 4.939 5.925 Recently, Deinzer and Salpetey1 3 l have obtained 0.697 Me for the carbonburning critical mass in the case of non-relativistic degeneracy and no neutrino loss. This low mass value is probably due to their incorrect normalization of the physical quantities in their stellar model. For example, the values of Tc and IIc in Table I of their paper gives for Pc only a half of the tabulated value. If the universal Fermi interaction between electrons and neutrinos is assumed, the energy loss by neutrino production becomes important at temperatures above 3 X 108 °K. In the the phase of gravitational contraction, the existence of this energy sink has the effect of lowering the temperature gradient in the central region. Therefore, the polytropic index in this region increases, at least, when the electrons are non-degenerate. Thus, the value off is lowered correspondingly and the value of M 0 increases according to Eq. (20). This means that the curves of constant mass in the Tc- Pc diagram are lowered in the high ternperature region where neutrino loss is effective. Thus, the minimum masses for the onset of carbon burning and burning of the heavier elements are increased. Now, we consider the phase of carbon burning with neutrino loss. Since 514 K. Takarada, H. Sato and C. Hayashi the energy liberated by carbon burning is mostly carried away by neutrinos, more energy is required to maintain a stellar luminosity. Therefore, the central temperature becomes slightly higher than in the case of no neutrino loss. In the ranges of temperature and density relevant to carbon burning, the photoneutrino process is predominant. The minimum mass and the corresponding central temperature are estimated by the same method as in the case of no neutrino loss, except that the neutrino luminosity is taken into account. We adopt the same value off as in the case of no neutrino loss, since f is determined essentially by the structure or the polytropic in,dex of a relatively inner region of a star. 14 ) In this manner, we obtain Tc = 7.0 X 108°K, and Mmm = 1.06 M 0 (nonrelativistic case) or 0.82 M 0 (relativistic case). The minimum mass given by Deinzer and Salpeter13 ) is not in agreement with the above value 1.06 M 0 , probably for the same reason as stated previously. So far, our consideration has been confined to the stable configuration of gravitational equilibrium. It is known that instability sets in when the adiabatic exponent, r = (d ln P 1d ln p) ad, of a gas becomes smaller than 4/3 or when the effects of general relativity become important. At high temperatures, r becomes smaller than 4/3 owing to the creation of electron pairs and the decomposition of Fe56 into He 4 and neutrons, as shown by the shaded region in Fig. 2, which is taken from the results of Sato. 15 ) Another region of instability is known to exist at high densities and at relatively low temperatures. At densities above 1010 g/cm3 , the electron Fermi energy is higher than about 10 lVIeV and most of the stable nuclei are subject to inverse beta-decays. Further, Chandrasekhar and Tooper 16 ).l 7 ) have recently shown that an instability due to general relativistic effects occurs at a density of about 1011 g/cm3 • References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) S. S. Kumar, Astrophys. J. 137 (1963), 1121. C. Hayashi and T. Nakano, Prog. Theor. Phys. 30 (1963), 460. Y. Osaki, Publ. Astron. Soc. Japan 15 (1963), 336. C. Hayashi, R. Hoshi and D. Sugimoto, Prog. Theor. Phys. Suppl. No. 22 (1962). C. Hayashi, Stellar Evolution (edited by R. F. Stein and A. G. W. Cameron, Plenum Press, New York, 1965), p. 253. C. Hayashi, R. Hoshi and D. Sugimoto, Prog. Theor. Phys. 34 (1965), 885. S. Chandrasekhar, An Introduction to the Study of Stellar Structure (The University of Chicago Press, Chicago, 1939 and Dover Publications, New York, 1957). S. Chandrasekhar and G. W. Wares, Astrophys. J. 109 (1949), 551. C. Hayashi and R. C. Cameron, unpublished. J. P. Cox and R. T. Giuli, Astrophys. ]. 133 (1961), 755. ]. P. Cox and E. E. Salpeter, Astrophys. J. 140 (1964), 485. H. Reeves, Stellar Structure (Vol. 8 of Stars and Stellar Systems, edited py L. H. Aller and D. B. McLaughlin, The University of Chicago Press, Chicago, 1965), p. 113. W. Deinzer and E. E. Salpeter, Astrophys. J. 142 (1965), 813. D. Sugimoto, Prog. Theor. Phys. 32 (1964), 703. H. Sa to, Prog. Theor. Phys. 35 (1966), 241. S. Chandrasekhar and R. F. Tooper, Astrophys. J. 139 (1964), 1396. S. Chandrasekhar, Astrophys. J. 140 (1964), 417.