Progress of Theoretical Physics, Vol. 23, No.5, May On Stellar Models with Double Energy-Sources* Minoru NISHIDA

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1 896 Progress of Theoretical Physics, Vol. 23, No.5, May 1960 On Stellar Models with Double Energy-Sources* Minoru NISHIDA Department of lvuclear Science, Kyoto University, Kyoto (Received February 6, 1960) To investigate the characteristics of stellar models having double energy-sources, three sample models (M=1.2Me.) consisting of the following regions were constructed using the newest rate of the en-cycle: (1) hydrogen-rich envelope, (2) radiative helium region and (3) convective helium core. A model for which the mass fraction of the helium regions is 0.6 shifts towards the left of the RR Lyrae gap in the HR-diagram from the red giant region, in which the corresponding model of Hoyle and Schwarzschild lies. This result shows that the properties of such models are very sensitive to the rates of both the hydrogen- and helium-burning. 1. Introduction Calculations have been made by Morton l ) on the effect on the main sequence of transfer of control of the CN-cycle from N 14 (p, r) 0 15 to C 12 (p, r) N 13 His results indicate a sideward shift of the main sequence on the mass-luminosity diagram. This shift would be of the order of the observational spread in the width of the main sequence, and he has concluded that the absence of such a shift is an evidence for the absence of a level in the critical region of 0 15 just above 7.30 Mev. Recently, Hebbard and Povh 2 )** have found a new level of 0 15 with an excitation energy of 7.17 Mev, hence it is established that the N 14 (p, r) 0 15 reaction is off-resonant at stellar energies. A few years ago, several authors 3-4 ) investigated the structure of population II stars in the helium-burning phase. Their results showed that the horizontal branch in the HR-diagram of the globular clusters corresponds to the stars in this phase. In their calculation, however, they used the old rate of the CN-cycle which was the most reasonable at that time and which was one hundred times larger than the newest correct value. On the other hand, it has been shown by Hayashi et al. 5 ) that in the case of massive stars the properties of the stellar models with double energy-sources are very sensitive to the rates of both the hydrogen- and helium-burning. One may expect that such a tendency will appear also in the case of population II stars at the helium-burning stage. The purpose of this paper is to reconstruct the models in the helium-burning phase by using the newest rate of the en-cycle. * An outline of this paper was reported at the Symposium on Nucleogenesis and Stellar Evolution, held at the Research Institute for Fundamental Physics, Kyoto University, in October ** The author is indebted to Dr. Povh for the information in advance of publication.

2 On Stellar Models with Double Energy-Sources Definitions and assumptions Closely following the evolutionary scheme as discussed by Hoyle and Schwarzschild,3) we have adopted the following physical parameters and assumptions in our computations. (a) The mass is taken as 1.2 M. (b) The models consist of the following three zones: (i) a hydrogen-rich envelope, (ii) a radiative pure helium zone and (iii) a convective helium core. (c) Radiation pressure is neglected. (d) No mixing occurs in the envelope, or between the envelope and the helium regions, so that inside the convective helium core complete mixing keeps the constant composition of essentially pure helium. (e) The abundance of heavy elements (i. e. elements other than hydrogen and helium) is assumed to be so low that they do not contribute to the opacity. (f) The envelope is divided into two parts, the outer part, in which the opacity is taken as arising from the free-free transitions of hydrogen and helium, and the inner zone, in which the free electron scattering determines the opacity. No intermediate region of mixed opacity has been introduced-the opacity formula is switched abruptly from free-free transitions to electron scattering at an appropriate interface, the opacity being kept continuous. (g) In our models energy generation takes place by the 3a-process in the helium core and by the CN-cycle in a shell just outside the radiative helium region. The rate for the former process is given by p being the density. The numerical values in this expression are adopted from Salpeter's work 6 ) The rate for the CN-cycle is where the newest result of the off-resonant measurements for N 14 (p, r) 0 15 is used,7) and X N is the concentration of nitrogen. (h) The thickness of the shell in which the energy generation by the CNcycle occurs is neglected. Hence the energy flux jumps discontinuously at the shell and is constant outside the shell. (i) The envelope retains the initial composition (1) (2) for which the mean molecular weight in the envelope becomes Pe = This concentration of nitrogen is just the same as adopted by Hoyle and Schwarzschild, and it is a half of the one in the revised table of the cosmical chemical compositions compiled by Cameron. S ) Recently, Greenstein and Keenan 9 ) have found a giant star having a logarithmic deficiency of -:-,1.55.compared with the en abundance in the population I stars. The models, having X1\ one order of magni-

3 898 M. lvishida tude smaller than the one used by Hoyle and Schwarzschild, will also be discussed in 4. The mean molecular weight in the helium regions is taken to be 4/3. (j) The boundary condition that the temperature and density to zero at surface of the star is used. (k) for the outer convective zone, our models have tema perature than 4700oK, hence this zone is not expected to playas role as In stars. (l) will be used with the following meaning: e for III the envelope, l for quantities in the radiative helium region, c for values, 1 for values at the interface between the envelope and the radiative helium zone, d for values at the interface between the radiative helium region and the convective core, and s for the values at the interface where the opacity formula switches. 3. Basic equations and construction of models In terms of the dimensionless variables/oj P=p GM 2 T=tp.H GM M(r) =qm, r=xr, 47rR4 ' k R' the basic equations for the conditions of mechanical and thermal equilibria take the following forms: where O<x<1 ; dp _ pq -----, dq = px 2 dx tx 2 dx t O<X<Xd; ~ dp =2.5, P dt xs<x<1 ; dt =-CI1_L dx 0 t 4 x 2 ' (4) (5) (6) (7) (8) (9) (10) (11) (12) and

4 On Stellar lt10dels with Double. Energy-Sources 899 The homology variables are (13) V=~ xt, (14) 4 n+1=--~ ~- eel p q t n+l=-- 4 ~ ei~ p conditions at q=o lip =0 dx constants x" as a U.L"!-,U''-'u.,U"~ and (7) ment (7) specified X rl IS made with the for 1 >x>x s, for Xs> x> Xl, for Xl> X> X d center are The value of ej;"!t is given by the continuity of Provided that values of CRT and Xs integrated inward under the boundary at =0. (15) (16) (17) (18) i. e. tc and Pc- Regarding O<X<Xl from (5) by the requirefrom Eq. (6) to Eq. conditions. (19) Eqs. (5), (8) and (9) can be p=o, q=l, t=o at x=l. (20) The fitting conditions at Xs are such that dimensionless variables and their derivatives are continuous. The value of eel is determined by the continuity of dt/ dx, that IS, C -c ps Ez- KT--5' t 4. 8 (21) If we regard ql as given, then Pc, tc, X a, e Kn Xs, and R, which is required for transforming the dimensionless variables to physical ones, remain to be determined. The continuity of U /p. and V //1 at Xl gives two conditions. The third one is that the envelope solution has a specified ql at Xl' Three other conditions are supplied by the following physical requirements. (i) The energy generation in the core due to the helium-burning is given

5 900 M. lvishida LSa=7.12X 10_4Y3(~)3!2(_3_)3/2 PC3/ 2 T 3/2 ( Tc )29.0 (22) GPcH (' 1.35 X 10 8 and must equal the outward flux Lcorr. as determined by Eq. (12). (ii) The energy production in the hydrogen-burning shell, L RheIl, must equal where L-L core, and L is determined by (11). (iii) The opacity from electron scattering must be equal to the opacity from free-free transitions at x= Xs' 4. Results and discussions M vis -4~-~1-T'--~1---r-'-'--~1r--r-1-'--'1--' xx I OX~~ m~ n (23) We have constructed a model for ql = 0.60, which is designated as Model I. In Table I, the mathematical and physical characteristics of this model are summarized and are compared with those of the corresponding model of Hoyle and Schwarzschild, which is referred to as H-S. The results of these two models are plotted in a color-magnitude diagram in Fig. 1, where the shaded area is that occupied by the stars of the globular clusters M3 12 ) and M92.13) Color index has been derived on the same system as used by Hoyle and Schwarzschild. As seen in Fig. 1, Model I shifts towards the left of the RR Lyrae gap in the Hertzsprung-Russell diagram from the red giant region, in which the H-S model lies. Since the re- +4- I J Col. Fig. 1. Hertzsprung-Russell diagram for the star with M=1.2Mi.!) at the helium-burning stage The shaded area indicates the observed sequences in globular clusters.

6 On Stellar Models with Double Energy-Sources 901 Table 1. Mathematical and physical characteristics of inhomogeneous models with helium-burning in convective core (c to d), radiative zone in helium (d to 1), hydrogen-burning shell (at 1), and radiative envelope (1 to surface) H-S Model I Model II Model III {}d ~d log Xd log qd log ta log Pa (n+1hi U le VIe (n+1he log Xl log qi log t log PI log em log Xs log Ps log ts log RjR@ log LjL@ log Tc log Pc log TI log PIe Lshell j Leore I log Teff B.C Mvis C.l markable differences between Model I and H-S result from using the newest rate of the CN-cycle, it must be emphasized that the characters of the model having a double energy-source strongly depend on the ratio of the hydrogen-burning rate to the helium-burning one. In our computations we did not take account of the effect of the conversion of helium to carbon in the core. A rough estimate indicates that the degree of depletion of the helium content in the core amounts to 0.2. A sequence of models having the helium-depleted cores will be studied in a forthcoming paper. Recently, Eccles and Bodanskyl4) have attempted to determine the probability that the 7.65 Mev state of C I2 decays to its lower states and found that it is less than 0.1 'J~. This limit is an order of magnitude smaller than the best previous experimental value l5 ) from which Salpeter estimated the rate of the helium-burning. Hence it may be of interest to construct a model by using the value of one tenth of Salpeter's reaction rate. In this case, Model I should be replaced by Model II in Table I and Fig. 1. The following feature can be pointed out by comparison of these two models. The reduced radius of the hydrogen-burning shell, XI. now

7 90Z Iv!. Nishida become smaller by a factor 0.78 for Model II than Model 1. This gives rise to an increase of radius R by a factor 1.29 and consequently to a decrease of the effective temperature. Finally, we consider the case where a smaller value of nitrogen concentration is adopted according to the argument given in 2. The results are given in Table I as Model III. In this case, it should be noted that Xl becomes larger than that of Model I while L 8hell / L core decreases, due mainly to the smaller X N The author wishes to thank Prof. C. Hayashi, Dr. J, Jugaku and Mr. S. Sakashita for their valuable discussions. References 1) D. C. Morton, Ap. J. 129 (1959), 20. 2) D. F. Hebbard and B. Povh, Nuclear Physics 13 (1959), ) F. Hoyle and M. Schwarzschild, Ap. J. SuppL 2, No. 13 (1955). 4) S. Obi, Publ. Astro. Soc. Japan 9 (1957), 26. 5) C. Hayashi, J. Jugaku and M. Nishida, Prog. Theor. Phys. 22 (1959), ) E. E. Salpeter, Phys. Rev. 107 (1957), ) E. M. Burbidge, G. R. Burbidge, W. A. Fowler and F. Hoyle, Rev. Mod. Phys. 28 (1956),547. 8) A. G. VV. Cameron, J. 129 (1959), ) J. L. Greenstein and P. C. Keenan, Ap. J. 127 (1958), ) M. Schwarzschild, Structure and Evolution of the Stars, (Princeton, Princeton University Press, 11) S. Hayakawa, C. Hayashi, M. Imoto and K. Kikuchi, Prog. Theor. Phys. 16 (1956), ) H. C. Arp, W. A. Baum and A. R. Sandage, A. J. 58 (1953), 4. H. C. Arp, Handbuch der Physik, Vol. LI (1958), ) A. R. Sandage, A. J. 58 (1953), 61. H. L. Johnson and A. R. Sandage, Ap. J. 122 (1955), ) S. F. Eccles and D. Bodansky, Phys. Rev. 113 (1959), ) c. W. Cook, W. A. Fowler, C. C. Lauritsen and T. Lauritsen, Phys. Rev. 107 (1957), 508.