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145 Progress of Theoretical Physics, Vol. 27, No. 1, January 1962 Evolution of Population II Stars in the Helium-Burning Phase Minoru NISHIDA* and Daiichiro SUGIMOTO Department of Nuclear Science Kyoto University, Kyoto (Received August 31, 1961) Evolution of 1.2 M® Population II stars in the helium-burning phase is investigated, taking into account the effect of conversion of helium into carbon and oxygen both on the mean molecular weight and on the nuclear energy generation. The stellar models consist of the following three regions: (i) a hydrogen-rich envelope, (ii) a radiative pure helium zone and (iii) a convective core in which helium burning occurs. The mass fraction of helium region is fixed respectively at 0.45, 0.50 and 0.54 and is treated as a parameter. Series of models are constructed for various helium contents in the convective core. It is found tl~at the stars stay nearly in the narrow region of the horizontal branch of H- R diagram of the globular cluster M3. But the dependence of stellar radii on the nuclear reaction rates are so strong that a change by one order of magnitude of the rate for CNcycle shifts the stars through the horizontal branch. The calculated lifetime for this phase is 3 X 107 years, which is a little smaller than that estimated by Sandage. § 1. lntJroduction Evolution of Population II stars in the hydrogen-burning phase has been investigated by Hoyle and Schwarzschild1 l and by others, 2> and their results properly interpret the subgiant and red giant regions of the II-R diagram of the globular cluster M3. As the star evolves toward the tip of the red giant branch, the density at the hydrogen-burning shell becomes lower until the energy transfer by radiation cannot be neglected compared with that by degenerate electrons in the outer part of the helium core. Moreover, the evolution becomes so rapid that the rate of gravitational energy release becomes large. As these occur, the central temperature may rise and helium bigins to burn in the core. Then the gas in the central region becomes non-degenerate and the large amount of energy is released, which may make the core expand very rapidly and make it temporarily unstable. The results by Harm and Schwarzschild3l show that the star can proceed to the helium-burning phase without catastrophe. In this paper we lay aside the problems concerning the details of the helium flash and assume that helium begins to burn when the mass fraction of the helium core reaches * Present address: Mount Wilson and Palomar Observatories, California Institute of Technology, Pasadena, California, U. S. A. M. Nishida and D. Sugimoto 146 some definite value, without mixing between the hydrogen-rich envelope and the helium region. Evolution of Population I stars with 15.6 M 0 4>, 5> and 4 M 0 6> in the heliumburning phase has been investigated taking into account the effect of helium depletion on the mean molecular weight and on the rate of helium burning. The results have shown that the stellar radius strongly depends not only on the ratio of the central temperature to that at the hydrogen-burning shell, but also on the mean molecular weight in the convective core. Hence it is very important to take into account the variation of helium content in the convective core; but its effects on the characteristics of models are not considered in the works of Hoyle and Schwarzschild1> and of Obi.?J The present work, in which this effect is properly considered, is the continuation of the previous work by one (M.N.) 8> of the authors. In the case of less massive stars which have a degenerate isothermal helium core at the preceding stage, the mass fraction of the helium core is as large as 40,..._,50 % at the time of th~ helium flash. Therefore it can be expected that the ratio of luminosity by 3a reactions in the core, Lcore, to the total luminosity, L, is larger than in the case of intermediate mass or massive stars. Moreover, due to the smaller mean molecular weight of its envelope material, the total luminosity of Population II star is smaller by a factor 2.4,-.,.,5.2 than that of Population I star of the same mass, hence Lcore/ L increases until Lcore becomes comparable to or larger than the luminosity by hydrogen burning, Lshezz. The nuclear energy release from unit mass in helium burning is a tenth of that in CN-cycle. Therefore, in the course of helium burning, the change of the mass fraction (q1 ) of the helium region is as small as 1 %. In our computation, q 1 is fixed to be 0.45, 0.50 and 0.54, and is treated as a parameter. Recently, Cox and Salpeter9> published a work on "Helium Stars with Hydrogen-Rich Envelopes". In their work, the mass fraction of the helium core is larger than 0.80. These values of q1 are too large, if the peaceful evolution of the star at the helium flash is assumed, as in the case of this paper. In the stars with double energy sources, the following two factors play the important role in determining the stellar radii : (i) The larger Lcore and smaller 20 reaction rates of 3a->C12 and C12 +a-7>016 (016 +a-7>Ne being neglected from the 20 experimental factl 0),* that the spin and parity of the 4.97 Mev level of Ne is 2-) make the central temperature higher, and the lower temperature at the hydrogen-burning shell results from the smaller Lshezz and larger rate of CNcycle. These make the stellar radius larger. (ii) The larger jump of mean molecular weight at the interface between the convective core and intermediate helium zone makes it smaJler. These effects are thoroughly considered in our computation. * The authors are indebted to Dr. H. E. Gave for this information in advance of publication. Evolution of Population II Stars in the 1-Ielium-Burning Phase 147 In § 2, the details of definitions and assumptions adopted are described. The basic equations and construction of models are discussed in § 3. Finally, m § 4 the results are compared with the horizontal branch of the 1-I-R diagram of the globular cluster Jlv13. § 2. Definitions and assumptions In constructing the models the following physical parameters and simplifying assumptions are adopted. 1. The mass is 1.2 M ®· 2. The envelope retains the initial chemical composition: (1) The characteristics of models, particularly the stellar radii, strongly depend on the value of XN or the nuclear reaction rates. This dependence will be discussed in § 4. 3. The models consist of the following three zones: (i) a hydrogen-rich envelope, (ii) a radiative pure helium zone and (iii) a convective core composed of helium, carbon and oxygen in various concentrations corresponding to the time elapsed since the helium flash. The thickness of the hydrogen-burning ·shell surrounding the radiative helium zone is neglected, hence the energy flux is taken to jump discontinuously at the shell. In the core, 3a-reactions and subsequent C12 (a, r) 0 16 reactions take place. 4. The nuclear energy generation rates are T 3X107 EoN= 4.20 X 105 nX XN ( --------1 e )16.3 ergs/ g ·sec, (2) for CN cycle, and (3) for helium burning where T is in units of 108 °K and f is a correction factor due to the occurrence of the C 12 +a reaction and given by f = __Q_ila ~~-j- Q_(J_t:~!:__q_j-_et:_ = 1 + Q3aP3a Q 0 +_a__ ICY- 2 X 012 ' (4) Q3a where P's denote the reaction rates, and K 1s the function of temperature and density only. The numerical value of E3a 148 M. Nishida and D. Sugimoto is due to the work by Hayakawa et al. 11 ),* which is by a factor of 3 smaller than Salpeter's value. 12 J 5. The value of the mean molecular weight in the convective core is very important in determining the stellar radius. This importance will be discussed in § 3. The formula (6) :s adopted. 6. The opacity is caused by free-free transitions of hydrogen and helium in the outer part of the envelope, and by electron scattering in the inner part of the envelope and in the radiative helium zone. The opacity formula is switched abruptly from free-free transitions to electron scattering at an appropriate interface, keeping the opacity continuous. 7. Effective temperatures are so high, a jJosteriori, that the outer convective layers can be neglected. 8. Radiation pressure is neglected throughout the star. 9. Subscripts are used with the following meanings: e for the quantities in the envelope, s for those at the opacity switching point, l for those at the hydrogen-burning shell, d for those at the interface between the convective core and the intermediate helium zone and c for the central values. § 3. Basic equations and construction of models In terms of the dimensionless variables, Gl'v1 2 P=P-4:-;](4 , T=t pll GM k-- R , M(r) =qM and r=xR, the basic equations for conditions of mechanical and thermal equilibria take the following forms : dpdx pq tx 2 O<x<l: -- O<x<xa: -------.'.J.5-~' ·------~--, djY _') dt p t dq =_px~ dx t (7) (Sa) * In reference 11) Eq. (3 · 5), the numerical value 1.12 must be replaced by 2.40 because of calculational error. Since in our work the former value is used, our results correspond to the case when TE 2 =3.27XlO-IO Mev with the numerical value 2.40 instead of 7X10-10 Mev. If the factor 2.40 and r E 2 = 7 X 10-10 Mev are used the stellar radii become smaller, for example, by a factor of 1.05 for the model II -a. However, a recent experiment by Alburger 16l shows a 3.23 Mev r-ray branch of (3.3±0.9) Xl0- 4 per decay of the 7.66 Mev level. If the reasonable r a is 0.5 ev, Tp; 2 becomes nearly equal to 2 X 10-4 ev. The authors are indebted to Dr. D. E. Alburger for this information. E'volution of Population II Stars in the I-Ieliunz-Burning Phase dt _ dx ---.------ C* Et - p- (Sb) --, t 4 ::c2 dt - -CEl ___ p--------, X1 <x<xs: ----2 4 dx t x where c1z = -4~;- . --t~:~ 2 CEl= 4 ~~ · --~~~~ 2 CKr= (Sc) dt -., --- j} ------Cx -dx r t8.5 x2 ' Xs<x<1: • • 149 (Sd) (--,j~~~~0c r.-f;;~e ' (I+Xe) · (-Pe~!Gr · ja-, 4~c . __?·_?~?5-~~=;~+ Xe). (--fl:~G r·v. -;~o~o-. For O<x<xa the solution of the polytrope n=1.5 is used, for xa<x<xb (7) and (Sb) are numerically integrated from point d outward as a one-parameter family for definite Pc, and for x 1<x < 1 available solutions are tabulated as a two-parameter family. 13) These solutions are then fitted. In terms of homology variables, d logM(r) u -_ ----------- --- -d log r ' · d]ogP v-_ ------------------- ' n+ 1=--dlogP --- ---- (9) d logT' dlogr the fitting conditions which demand the continuity of physical quantities are expressed by ( n + 1) d e.v = 2. 5, and vl in --v1 ---- ex (10) fliie The condition of the continuity of opacities at point s eEl= CK __ jJ_s_ " r t4.5 s IS given by (11) • Another condition demanding the physical consistency of the model IS (12) where Lcore as IS approximated expressing (3) in the form of E = E0 fY 3 2 ,0 (T /To) S, (13) 150 M. Nishida and D. Sugimoto and Lshell is giVen by Lshell=4.20 x 105 XeXNpl~ ( 3rio7-) 16.3 Vle[2+ 14.~J (1:+ i~-eJ=3, (14) under the approximation that the thickness of the hydrogen-burning shell i~ infinitesimally small. These conditions uniquely determine a model for given Ye or fle· It is found as a result of the above numerical integrations that for the same values of ul and Te/Tl is approximately proportional to Pel f1Ho the jump of mean molecular weight at the interface between the the convective core and the intermediate helium zone, and from the fitting of the solutions, that the stellar radii are proportional to Ten/T1 m where n is nearly 2 and m is nearly 4,..,_,5 o.s..--------.---------, due to the both effects of the solutions of the intermediate helium zone and of the corresponding variation of opacity switching point of the fitting envelope solution. Hence the stellar radii strongly depend on the value of f!-e • In our 0.2 computation Eq. (6) is adopted for fle. f and fle depend on the ratio of the rates of the 3a and C 12 +a nuclear reactions and the concentrations of the reacFig. 1. Correction factor for the 3a tion products, C 12 and 0 16 • IC in Eq. (5), nuclear energy generation rate due to calculated from the reaction rates given the occurrence of the C12+a reactions. Vertical bars indecate the range of by Hayakawa et al., is larger by factor uncertainty of log f, and horizontal 13 than that from Burbidge et al.'s reacbars indicate the uncertainty of Ye for tion rates. 14> However, because of rather the corresponding llc. Both come from large values of K even in the latter case, the uncertainty of the ratio of reaction vb rates of 3a and C12+a. Table I. The run of the correction factor for the 3a nuclear energy generation rate due to the occurrence of the Cl 2 +a reactions, log}; and the mean molecular weight of the convective core material, log /le (cf Eq. (6)) for the corresponding value of helium concentration, Ye, in the course of evolution. Y is the helium concentration which is derived by usual formula for the mean molecular weight instead of Eq. (6). Model ---~--~·----- Yc -··---~~-~ logf ------~---------- log IJ.e --~--------· f -~-------·~------~-------- a 1.00 0.00 0.125 1.00 b 0.85 0.16 0.140 0.90 c 0.58 0.30 0.171 0.70 d e 0.31 0.41 0.204 0.50 0.09 0.54 0.235 0.33 Evolution of Population II Stars in the Helium-Burning Phase 151 carbon is almost completely converted to oxygen, so the effect is small. The run of the values, logf and log flc with Yc adopted to our models are seen in Table I and Fig. 1. In the latter the vertical bars show the width of f and the horizontal bars show the ranges of Yc for specified flc, due to the ambiguity of the reaction rates. The helium concentration, which is derived from flc - I = (3/ 4) Y + (1- Y) 12 instead of (6), is listed also as Y in Table I. It should be noted that the difference between Y and Y is large. § 4•. Results und discussions Physical characteristics of the constructed models are summarized in Tables II, III and IV. Model series of I, II and III correspond to fixed q1 's, 0.45, 0.50 and 0.54, respectively. The symbols a, b, c, and so on are designated for various Yc (cf. Table I). Since the change of q1 in the course of evolution amounts only to 1 ~ 0 , each series is approximately an evolutional sequence. The slight retreat of the convective core and the resulting slope of ~ean molecular weight are neglected. The results are plotted in the log Teff-log L/ L® diagram, Fig. 2, and they fall in the range of the RR Lyrae gap which is in color, B- V =0.17 ,..._,0.39, and in magnitude, Mvis= -0.15,...._, +0.15 for M3, as indicated by the hatching in Fig. 2. However, this fact must be interpreted with some care. The characteristics of the models strongly depend on the concentration of nitrogen XN in the envelope or the rate of CN-cycle. This dependence is shown in Fig. 3 for the model I-c. The larger XN or EcN is, the lower the effective temperature Table II. Physical characteristics of the models of 1.2 M@, Population II star in the phase of helium burning. Mass fraction of the helium core, q1 is 0.45. -- - Model -------·-----------~- log R!R® log LfL® log Teff I- I-a i I-b I-c ·---- ----·-- ---------------- -------- I-------- I-d ---·-···- ------------- -- I I I-e ------------------------ 0.73 0.73 0.78 2.03 0.75 2.02 2.01 2.01 2.03 3.88 3.89 3.90 3.90 3.87 0.77 log Tc 8.11 8.12 8.14 8.16 8.21 log Pc 4.17 4.16 4.17 4.19 4.32 Lcoref LtotaZ 0.58 0.62 0.71 0.81 0.85 qd logTa 0.14 0.14 0.14 0.13 0.13 7.93 7.94 7.95 7.96 8.00 log T 1 7.48 log P1e log Ule log vle 1.94 7.48 1.94 7.47 1.93 7.46 1.90 7.46 1.88 1.07 1.08 1.10 1.11 1.07 0.50 0.50 0.50 0.49 0.50 log r 1 /R@ 2.50 2.50 2.51 2.52 2.52 t (10 7 yr) 0.0 0.5 1.3 2.1 2.7 I 152 M. Nishida and D. Sugimoto Table III. The same as Table II, in the case of q1 =0.50. Model II-a II-b II-c li-d II-e log RJR® 0.75 0.72 0.71 0.72 0.82 log LJ L0 2.12 2.11 2.11 2.11 2.14 log Teff 3.92 3.93 3.93 3.93 3.88 log Tc 8.12 8.13 8.14 8.17 8.22 log Pc 4.10 4.09 4.10 4.13 4.24 Lcore/ Ltotal. 0.65 0.16 0.70 0.80 0.89 0.92 qd 0.16 0.15 0.15 0.15 log Td 7.94 7.94 7.95 7.97 8.01 logT1 7.49 7.48 7.47 7.46 7.46 log Pie log U1 e log vle log r1JR® t (107 yr) 1.87 1.86 1.84 1.81 1.06 0.50 1.07 1.09 1.09 1.77 1.03 0.50 0.50 0.50 0.51 2.58 2.59 2.60 2.61 2.60 0.0 0.4 1.1 1.8 2.3 Table IV. Model III-a The same as Table II, in the case of q 1 =0.54. III-b --- III-c -------,------------ I III-d I III-e ---~-~---------------~ log RJR 0 0.76 0.74 0.76 0.79 0.91 log LJL® 2.19 2.19 2.19 2.19 2.23 log Teff 3.93 3.94 3.93 3.91 3.86 log Tc 8.13 8.13 8.15 8.17 8.22 log Pc 4.05 4.04 4.05 4.08 4.20 Lcoref Ltotca 0.68 0.75 0.83 0.91 0.93 0.16 q<l 0.17 0.17 0.17 0.16 logTd 7.95 7.95 7.95 7.98 8.02 log T 1 7.48 7.48 7.47 7.46 7.46 log P1e log ~e log vle 1.87 1.85 1.83 1.79 1.75 1.10 1.11 1.11 1.10 1.04 0.51 0.51 0.52 2.64 2.64 0.51 2.65 0.52 logr1 fR® 2.66 2.66 0.0 0.3 1.0 1.5 2.0 becomes. By a change by one order of magnitude of the former on either side, the position of the star in the H-R diagram is dispersed over the horizontal branch. Evolutional tracks calculated here are confined to considerably narrow regions in the I-I-R diagram. This comes from the fact that the effect of the change of flc/ flue nearly compensates that of the change of fY 3. The lifetime is calculated from the energy release per unit mass by 3a and Evolution of Population II Stars zn the 1lelium-Burning Phase 153 2.3 r - - - - - - - - - - - - - - - - - , ~ql=0.51 2.2 '-1 -- 0.7 j - ~ql=0.50 2.1r.· O.fi ~ ~- ~q1=0.4;) ~ ,§ 4.0 2.01-- 1.9 "" R.R.T• • 1.8 .4.0 ;].9 t--.' I 3.9 bJ:J ..Q /, 3.8 , ~3.8 log TrJJ Fig. 2. Evolutional tracks of 1.2 M@, Population II star, with fixed mass fraction of the helium core, q1 . The hatching indicates the RR Lyrae region of the globular cluster M3. ~3.7 2.1 0' >--..1' 2.0 bJ:J _g C+a reactions with q1 fixed, i.e. assuming the energy generation per unit 1.8 mass ·by the CN-cycle is quite large compared with that by 3a and C+a 1.0 reactions, and it is tabulated in Tables ~ II, III and IV. The total lifetimes are ~· 7 bJJ computed to be 3.0, 2.5 and 2.1 X 10 _g years for q1 = 0.45, 0.50 and 0.54, respectively. The above calculation is made 0.5 -1 0 +1 using the reaction products from the log (X' NIXN) or log (E' aN/EoN) reaction rates of Hayakawa et al. BurFig. 3. Dependence of radius, luminosity, bidge et al.'s reaction rates give values effective temperature and Lcorel L on the smaller by a factor of 1.2. Sandage15l concentration of CN group elements or estimated the lifetime of the horizontal nuclear reaction rate of CN-cycle; in the case of model I-c (q1 =0.45 andY= branch of M 3 to be 2.3 X 108 years on 0.58). the basis of star counts, Salpeter's luminosity function, and the evaporation ratio computed by van den Bergh. He estimated also the time spent in the RR Lyrae phase to be 8 X 107 years. Our result seems a little too small to explain the relatively high density on the horizontal branch. A more accurate observational determination of the lifetime of the phase on the horizontal branch seems to be desirable. The authors would like to express their sincere thanks to Prof. C. Hayashi for many valuable discussions. .;;. '• 154 M. Nishida and D. Sugimoto References 1) 2) 3) 4) F. Hoyle and M. Schwarzschild, Ap. J. Suppl. 2 (1955), No. 13. C. Haselgrove and F. Hoyle, M. N. 116 (1956), 515, 527. L. Biermann, R. Kippenhahn, R. Lust and St. Temesvary, Z. Astrophys. 48 (1959), 172. R. Harm and M. Schwarzschild, Astron J. 66 (1961), 45. C. Hayashi, J. Jugaku and M. Nishida, Prog. Theor. Phys. 22 (1959), 531; Ap. J. 131 (1960)' 241. 11) 12) 13) 14) C. Hayashi and R. C. Cameron, to be published in Ap. J. C. Hayashi, M. Nishida and D. Sugimoto, Prog. Theor. Phys. 25 (1961), 1053. S. Obi, Publ. Astro. Soc. Japan 9 (1957), 26. M. Nishida, Prog. Theor. Phys. 23 (1960), 896. J. P. Cox and E. E. Salpeter, Ap. J. 133 (1961) 764. A. E. Litherland, J. A. Kuehner, H. E. Gove, M. A. Clark and E. Almqvist, Phys. Rev. Letters 7 (1961), 98. S. Hayakawa, C. Hayashi, M. Imoto and K. Kikuchi, Prog. Theor. Phys. 16 (1956), 507. E. E. Sal peter, Phys. Rev. 107 (1957), 516. R. Harm and M. Schwarzschild, Ap. J. Suppl. 1 (1955), No. 10. E. M. Burbidge, G. R. Burbidge, W. A. Fowler and F. Hoyle, Rev. Mod. Phys. 29 15) 16) A. Sandage, Ap. J. 126 (1957), 326. D. E. Alburger. Phys. Rev. 124 (1961), 193. 5) 6) 7) 8) 9) 10) (1957)' 547.