Download Evolution of Population II Stars in the Helium

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.