THE OPACITY AND THE INTERNAL STRUCTURE OF THE SUN
by SHU-MU KUNG
Purple Mountain Observatory, Academia Sinica
I. Introduction
The purpose of this paper is to obtain the distributions of the
temperature, density and pressure from the surface to the center of the sun
on the one hand and to get the composition, i. e. the content of
hydrogen, helium and heavy elements, of the sun and the relative
importance of the proton-proton reaction and the carbon cycle to the
contribution of the solar energy on the other hand.
There are many papers, such as those of Eddington [1], Strömgren
[2], Cowling [3], Chandrasekhar [4], Ledoux [5], Canadian [6] and Soviet
[7] workers, to investigate the stellar structure in general. Only a few
articles, however, have been published to deal with the structure of the
sun by means of numerical calculation. Since 1939 Bethe [8] and
Weizsäcker [9] suggested independently that the carbon cycle provided
the stellar energy, Schwarzschild [10] first computed the solar model
with the carbon cycle as the source of the solar energy and obtained the
composition of hydrogen, helium and heavy elements for the sun. He
employed a simple approximate formula for the opacity law inside the sun.
Later in 1949, Harrison [11] investigated the problem again along
the line of Schwarzschild's, but used a better opacity formula which
followed generally the values obtained according to the temperature and
the density distributions of her solar model. In 1950 Epstein [12]
pointed out the importance* of the proton-proton reaction to the
contribution of the solar energy.
* Epstein misused Mrs. Harrison's results and reached a
conclusion which exaggerated the importance of the proton-proton
reaction in his article, but his general conclusion is still true.
Until 1953 the opacity inside the sun was considered to be wholly due
to heavy elements. Later, it was realized that the contribution of the free-free
absorptions of hydrogen and helium to the opacity could not be neglected,
mainly due to the extremely low content of heavy elements. Then Naur [13],
later Epstein and Motz [14] have taken the free-free absorptions of hydrogen
and helium into consideration and computed the solar models.
Table 1 lists the essential results of the solar models of current authors.
All the models except one of Naur's in Table 1 are the so-called Cowling model
which has a convective core and a radiative envelope. Each model has a
common assumption that the mean molecular weight is the same inside the
sun. The validity of this assumption depends wholly upon the thoroughly
mixing of the material between the core and the envelope, otherwise the
mean molecular weight in the core will be higher than that in the
envelope due to the more effective transformation of hydrogen into
helium in the core.
Table 1 Results of Current Solar Models
* In this article the values of (n + 1) discontinue at some point. At
the interface of the core and envelope the value of (n + 1) is equal to
2.75 instead of the correct value 2.50 (Ap. J. 117, 311, 1953)
Whether is there thoroughly mixing of the material inside the sun is
not settled. However, Greenstein and others [15] pointed out that the
appearance of beryllium and lithium in the solar atmosphere seemed in
favor of non-mixing.
We have investigated this problem in general along the lines of the
previous works except that we assumed the mean molecular weight in the core
different from that in the envelope. Since 1955 we have computed a series of
solar models with both the proton-proton reaction and carbon cycle as the
source of the solar energy. The latest data of nuclear reactions given
by Fowler [16, 20] are employed. In the computation of opacities accordingly
to the temperature and the density distributions of our assumed models, we
found out that not only the free-free absorptions, but the bound-free
absorptions of hydrogen and helium should be taken into account as well
in the outer layers of the sun, where the temperature is lower than
2 x 10^6 K. This is partly due to the extremely low content of heavy
elements, 0.1 per cent in our model and partly due to the fact that at
low temperatures the peak of energy distribution of the Planckian
function shifts to the longer wavelength and thus the photo-ionization
of K and L electrons of hydrogen and helium plays a leading role in
contributing to opacity, as pointed out by Keller and Meyerott [17].
The values of opacity are computed according to the density
distribution of assumed models at 14 points with assigned temperatures.
The solar model with a hydrogen content of 82.5 per cent gives the value
of energy generation very close to the observed one, apart from that the
values of density distribution of the resultant model is much higher
than that of assumed mode at the region near the surface. This will
produce the same order of discrepancy between the values of opacity
of the resultant and assumed models, and, so we have to start
new models. From successive computations with different density
distributions we obtain an empirical relation between the density
distributions of the assumed and the resultant models. It can be stated
generally as follows: the higher is the values of the density
distribution of the assumed model, the lower is the values of the
resultant model. With this empirical relation in mind, it is easier for
us to assume a density distribution more close to the one of the
resultant model.
II. The Opacity of the Sun
The opacity kappa is defined by
(1)
where B_nu is the Planckian function, k_nu is the absorption coefficient
per gram at frequency nu and the other quantities have their usual meanings.
According to Strömgren's computation [18], kappa can be written as
(2)
Gamma, Gamma_R are, the composition, respectively, for the adopted
composition and for the Russell mixture,
(3)
c_z, the percentage in the composition of certain element whose atomic
number and weight are, respectively, Z and A. tau in formula (2) is the
guillotine factor and calculated by the following formulae:
(4)
(5)
(6)
(7)
On applying the above formulae to obtain the guillotine factors, we
compute them according to the following two principles: (1) in the
computation of the absorption coefficients we, taking into account the
effect of the excluded volume, have included only those terms having
quantum number n < = 4 ; (2) with regard to the depression of the
continuum we have excluded the terms having quantum number, n > = 2, for
the bound-free absorption of hydrogen and helium when the density is
high. This latter limitation is not important for the solar model
because the bound-free absorptions of hydrogen and helium become
negligible due to the increasingly high temperature when the density is
high.
The composition of heavy elements we have adopted is 40 per cent
Russell mixture and 60 per cent oxygen. As the values of composite
factor for heavy elements, hydrogen and helium are, respectively, 4,8, 1
and 1, we have the resultant D_n, at nu_n as,
(8)
where
The values of guillotine factor so computed at 14 assigned
temperatures and at proper values of psi/kT for including the
temperature-density points of our model are given in Table 2. The
resultant opacities consisting of the atomic opacity and electron
scattering are obtained by means of the tabulated values given by Morse [19].
Table 2 Guillotine Factors Log_10 tau and Densities Log_10 rho
for the Equilibrium model of the Sun X = 0.820, Y = 0.179, Z = 0.001
(60% oxygen and 40% Russell mixture)
The values of the resultant opacity are
represented by the following four equations:
(9)
In the current works there are three papers in which the free-free
absorptions of hydrogen and helium have been taken into consideration. The
correctness of their formulae to combine the opacity of heavy elements and of
hydrogen and helium is worthy to be discussed. Naur's opacity formula [13] is
(10)
in which the resultant opacity is equal to the sum of the opacity of heavy
elements, that of free-free absorptions of hydrogen and helium and 1.5 times
the scattering of the electrons. Next, the Abell's opacity formula [20] is
(11)
in which the resultant opacity is equal to the sum of the opacity of the
heavy elements, 1.5 times that of the free-free absorption of hydrogen
and 1.5 times the scattering of electrons. Actually, the resultant
opacity, as given by equation (1), is equal to the weighted harmonic
mean of absorption coefficients of various constituents and electron
scattering, and naturally, will not be equal to those given by their
formulae. Therefore, both Naur and Abell did not employ their formulae
just given, in actual computation of opacity, but used the more general
ones as indicated in our Table 1. Furthermore, Strömgren's empirical
formula (2) for the resultant opacity (i. e. the larger value plus 1.5
times the smaller one) consisting both of the absorptions of heavy
elements and of electron scattering is good approximately for the case
of early B-stars. It is questionable whether it can be applied as well for
the G-type star such as the sun and even more doubtful when it is applied
to the case for the free-free absorptions of hydrogen and helium.
Epstein and Motz [14] combine the free-free absorption of hydrogen with
that of heavy elements to get the guillotine factor t. Their opacity formula
is given by
(12)
As to check our computation we have computed the opacity according to
their composition (X = 0.931, Y = 0.067 and Z = 0.002) and the temperature
and density distributions of their model. With equations (2) and (3)
of the present paper, we obtain
(13)
Since (1 - X - Y) = 0.002, the relation between t and tau is
(14)
The values of log_10 tau obtained from [14] and those of us are tabulated in
table 3 in which for the purpose of comparison both values, besides consisting
of the absorptions of the heavy elements and free-free absorptions of
hydrogen and helium, the one with (with the bracket (b)) and the other
without the bound-free absorptions of hydrogen and helium, are included.
From table 3 one can see, there are quite differences between the two
sets of the values of Log_10 tau.
Table 3. Comparison of the two sets of values of Log_10 tau,
for X = 0.931, Y = 0.067, Z = 0.002
* The values with bracket (b) contain also the contribution of the bound-free
absorptions of hydrogen and helium.
Furthermore, some values of kappa (i. e. kappa in Epstein and Motz's paper)
obtained by equation (6) of their paper by employing values of t in their
Table 2 corresponding to the temperature and density distributions of their
model have appreciable differences with those obtained by equation (5)
of their paper. (The second formula of equation (5) has a printing error. The
constant should be +0.588, instead of -0.588). These two sets of values should
have little difference. They are shown in Table 4.
Table 4. Values of Log_10 kappa
According to equations (5) and (6) of this paper, it is evident that
the values of Log_10 tau at the same temperature should decrease as the
values of psi increase. This, is clearly illustrated in Harrison's
Tables (11). It is rather perplexing to read some values of Log_10 t
in Table 2 of Epstein and Motz's paper that at T = 0.625, 0.833 and
1x10^6 degrees, the values of Log_10 t at psi/kT = 5, 6, and oo increase
with the values of psi/kT, instead of decreasing.
III. The Structure of the Sun
The differential equations used in this investigation are based upon
the following assumptions:
(1) The sun may have a convective core or an isothermal core or no core
at all. If a convective core exists, the material inside the core, like
an ideal gas, has a value of gamma = c_p/c_v = 5/3. If an isothermal core
exists, the material inside the core has exhausted its hydrogen and so
generates no energy. The material outside the core is in the state of
radiative equilibrium.
(2) The mean molecular weight of the core must be larger than, or at
least equal to that of the envelope.
(3) Radiation pressure can be neglected.
(4) The thermal nuclear reactions operate both in the convective core
and in the radiative envelope.
The assumption (4) is introduced because the proton-proton reaction
generates energy rather effectively under lower temperature range
(10^7 deg - 1.3*10^7 deg). Since the rate of energy generation depends on the
content of hydrogen and nitrogen, we must assume before-hand the content
of hydrogen and heavy elements so as to carry out the numerical integration
in the envelope. Thus the problem is solved by successive steps point by point.
The four differential equations are:
(15)
(16)
(17)
radiative equilibrium (18a)
convective equilibrium (18b)
Where Epsilon_p, Epsilon_c are, the energy generated per gram per second,
respectively, by the proton-proton reaction and carbon cycle. The other
quantities have their usual meaning (4). As usual we employ the non-dimensional
quantities as follows:
(19)
The energy generation equations [16] [20] can then be put in the following
forms:
(20a)
(20b)
Where chi_CN is the portion of carbon plus nitrogen content in the heavy
elements, we take chi_CN = 0.2.
In the process of numerical integration when log y = - infty, Delta log t = 0,
it means that the energy generated outside the sphere of, the radius in
question attains the total energy of the sun. If the solution of the isothermal
core does not exist, it infers that our assumed model is wrong. Quite a few
models we assumed are such case of over production of energy.
When the value of d (log p) /d (log t) decreases and is equal to 2.5 in the
integration, it speaks that the boundary of the convective core is reached.
With the values of t, p and q at the interface we can calculate the ratio of
the values of non-dimensional quantities U and V by usual means. Then the
Emden variable ksi_f corresponding to the radius at the interface by finding
in Emden table the ratio of U and V equal to that just found. The relations
between the Emden variables theta, ksi and our previous ones are:
(21)
where subindex f indicates the value at interface and omega is the ratio of
the mean molecular weight in the core to that in the envelope. The total energy
generated inside the core is
(22)
where X_c X_e are, respectively, the hydrogen content in the core and in the
envelope.
After a series of trials, finally we obtain a satisfactory solar model
in which the contents of hydrogen, helium and heavy elements are, respectively,
by weight, 0.820, 0.179 and 0.001 in the radiative envelope and in the
convective core are, respectively, 0.779, 0.220 and 0.001.
The assumed values of density distribution agree very well with the
resultant ones; the differences between the logarithms of the
corresponding ones are all within 0.035. They are shown in Table 8. The
characteristics of the present solar model are given in Tables 5 and 6.
Table 5. The characteristics of the present solar model
Table 6. The various quantities at the interface
From Table 5, we know that the energy generated inside the core amounts
30.6 per cent of the total energy of the sun, while the energy generated in the
envelope reaches 64.7 per cent. The sum of them is 95.3 per cent, about 5 per
cent less than the observed value. The model with a hydrogen content of 82.5
per cent results overproduction of energy. Unless there is convective core
or isothermal occurs, there would not have any appreciable change for
the present solar model if the above 5 per cent of energy discrepancy is
eliminated. This is clear when one read the run of the values in Table
7, of the characteristics of the three models closest to the present
adopted one.
Table 7. Characteristics of 3 models closest to the adopted solar model
It is well known generally that the error in the energy generation formulae
due to the extrapolation of experimental data is much higher than 5 per cent.
And the same might be true for the current method to compute the opacity.
It is, we think, quite justifiable to regard our result as an equilibrium
model.
Since the radiative envelope produces about 65 per cent of the total solar
energy, it is not surprise to find the ratio of mean molecular weights in the
core to that in the envelope close to 1, namely mu_c/mu_e = 1.029.
Table 8.
NUMERICAL INTEGRATION OF THE PRESENT SOLAR MODEL
A = log_10(1/x-1)
B = log_10 t
C = d log t/d log(1/x-1)
D = log_10 p
E = d log p/d log(1/x-1)
F = log_10 q
G = d log q/d log(1/x-1)
H = log_10 y
I = d log y/d log(1/x-1)
J = d log p/d log t
A B C D E F G H I J
-1.000 -1.70357 1.00005 -5.22274 5.05324 - -0.00002 - - 5.05209
-0.950 -1.65357 1.00002 -4.97008 5.05324 - -0.00004 - - 5.05314
-0.900 -1.60357 1.00002 -4.71742 5.05324 - -0.00006 - - 5.05314
-0.850 -1.55357 1.00000 -4.46476 5.05324 - -0.00010 - - 5.05324
-0.800 -1.50357 1.00000 -4.21210 5.05324 - -0.00017 - - 5.05324
-0.750 -1.45357 0.99998 -3.95944 5.05312 -0.00001 -0.00029 - - 5.05322
-0.700 -1.40357 0.99995 -3.70679 5.05289 -0.00003 -0.00048 - - 5.05311
-0.650 -1.35357 0.99988 -3.45415 5.05254 -0.00006 -0.00079 - - 5.05315
-0.600 -1.30348 1.01498 -3.20154 5.05091 -0.00011 -0.00130 - - 4.97636
-0.550 -1.25176 1.04971 -2.94942 5.02991 -0.00020 -0.00210 - - 4.79171
-0.500 -1.19882 1.06522 -2.69878 4.99448 -0.00033 -0.00333 - - 4.68868
-0.450 -1.14541 7.06965 -2.45008 4.95302 -0.00054 -0.00523 - - 4.63051
-0.400 -1.09175 7.08418 -2.20352 4.90772 -0.00087 -0.00807 - - 4.52667
-0.350 -1.03694 1.10448 -1.95957 4.84808 -0.00137 -0.01232 - - 4.38947
-0.300 -0.98155 1.10910 -1.71885 4.78002 -0.00212 -0.01816 - - 4.30982
-0.250 -0.92617 1.10494 -1.48162 4.70923 -0.00322 -0.02646 - - 4.26198
-0.200 -0.87112 1.09618 -1.24794 4.63767 -0.00482 -0.03786 - - 4.23076
-0.150 -0.81658 1.08515 -1.01785 4.56573 -0.00707 -0.05320 - - 4.20746
-0.100 -0.76402 1.01967 -0.79116 4.50609 -0.01022 -0.07370 - - 4.41917
-0.050 -0.71408 0.98154 -0.56699 4.46190 -0.01456 -0.10108 - -0.00001 4.54582
0 -0.66561 0.95885 -0.34499 4.41713 -0.02047 -0.13708 - -0.00002 4.60670
- -0.66561 0.95885 -0.34499 4.41713 -0.02047 -0.13708 - -0.00002 4.60670
+0.050 -0.61808 0.94319 -0.12546 4.36154 -0.02844 -0.18368 - -0.00005 4.62424
+0.100 -0.57126 0.92974 +0.09086 4.28766 -0.03904 -0.24297 - -0.00015 4.61168
+0.150 -0.52512 0.91578 +0.30290 4.18938 -0.05297 -0.31702 -0.00001 -0.00044 4.57466
+0.200 -0.47973 0.89937 +0.50931 4.06172 -0.07102 -0.40765 -0.00004 -0.00120 4.51618
+0.25O -0.43525 0.87929 +0.70853 3.90131 -0.09404 -0.51608 -0.00014 -0.00314 4.43689
+0.300 -0.39188 O.85456 +0.89887 3.70629 -0.12294 -0.64261 -0.00040 -0.00773 4.33707
+0.350 -0.34987 0.82473 +1.07861 3.47752 -0.15860 -0.78628 -0.00101 -0.01781 4.21656
+0.400 -0.30919 0.78962 +1.24613 3.21862 -0.20182 -0.94467 -0.00236 -0.03812 1.07616
+0.450 -0.27100 0.74912 +1.40007 2.93596 -0.25325 -1.11378 -0.00511 -0.07537 3.91921
+0.500 -0.23467 0.70349 +1.53947 2.63858 -0.31330 -1.28861 -0.01031 -0.13731 3.75070
+0.550 -0.20074 0.65322 +1.66385 2.33684 -0.38211 -1.46342 -0.01937 -0.23055 3.57742
+0.600 -0.16942 0.59918 +1.77325 2.04113 -0.45955 -1.63279 -0.03394 -0.35805 3.40654
+0.650 -0.14087 0.54270 +1.86822 1.76056 -0.54522 -1.79213 -0.05571 -0.51755 3.24408
+0.700 -0.11517 0.48531 +1.94968 1.50186 -0.63854 -1.93798 -0.08611 -0.70170 3.09464
+0.750 -0.09232 0.42872 +2.01885 1.26926 -0.73877 -2.06828 -0.12611 -0.89993 2.96058
+0.800 -0.07225 0.37456 +2.07707 1.06454 -0.84509 -2.18192 -0.17614 -1.10055 2.84211
+0.850 -0.05480 0.32415 +2.12576 0.88740 -0.95668 -2.27882 -0.23603 -1.29306 2.73762
+0.900 -0.03976 0.27832 +2.16625 0.73631 -1.07270 -2.35896 -0.30517 -1.46876 2.64555
+0.950 -0.02689 0.23757 +2.19978 0.60895 -1.19231 -2.42237 -0.38252 -1.62091 2.56324
+1.000 -0.01592 0.20201 +2.22749 0.50263 -1.31467 -2.46894 -0.46680 -1.74518 2.48814
This means that our solar model has little difference with the model which
assumes a single value of mean molecular weight throughout the core and the
envelope. From the comparison of our result shown in table 5 with those
of Naur, and Epstein and Motz, tabulated in table 1, one finds that the
hydrogen content of our model is intermediate between theirs. The
central temperatures and central density both are a little higher than
theirs.
The model of Epstein and Motz who have taken into account of the free-free
absorption of hydrogen and computed opacity according to the density and
temperature distributions of the solar model should be close to the true state
inside the sun and should be comparable to our model. The hydrogen content of
their model is much higher, more than 10 per cent, than ours. This may be due
partly to (1) the different energy generation laws that we employ a new one
for proton-proton reaction which generates energy about 1.4 times slower than
theirs, partly to (2) the inconsistence of their opacity laws mentioned above,
and partly to (3) the including both of the bound-free and free-free absorptions
of hydrogen and helium in our model while only the free-free absorption of
hydrogen is considered in theirs.
Table 8 gives the values of numerical integration of the present solar
model.
The writer is indebted to Mr Ching Lee for his assistance in computation
in the first part of the work and especially to Miss Hsieh-chen Chen who
did the most of the computation in this investigation.
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