The Cl37 solar neutrino experiment and the solar helium abundance

ANNALS

OF PHYSICS:

54, 164-203 (1969)

The Cl37 Solar Neutrino The Solar Helium ICKO

IBEN,

Experiment Abundance*

and

JR.

Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

The preliminary upper limit of 3 x 1O-36 counts per second per CP’ atom set by Davis, Harmer, and Hoffman in an experiment to detect solar neutrinos establishes a sensitive, nearly opacity independent, relationship between low energy nuclear cross section factors and an upper limit on the sun’s initial helium abundance. Using currently quoted values for the relevant cross section factors and conventional assumptions regarding the sun’s makeup, consistency between theoretical estimates for the counting rate and the Davis, Harmer, and Hoffman limit can be achieved only if the sun’s initial helium abundance Y (He’ abundance by mass) is less than 0.16-0.20, surprisingly low when compared with values (Y = 0.25-0.30) which have previously been estimated for the sun and for other objects in the galaxy as well. A higher upper limit on the solar helium abundance is possible if one or more of the following is true: (1) One (or more) of the relevant nuclear cross section factors lies beyond commonly quoted limits; (2) One (or more) of the relevant neutrino absorption cross sections has been overestimated; (3) The sun did not condense out of a chemically homogeneous nebula and/or pass through a homogeneous phase; (4) The opacity near the solar center is such that the sun possesses a large convective core. (5) Spin-down or meridional currents mix or have mixed matter over the entire solar interior on a time scale short compared to the age of the sun; (6) The sun is considerably younger than 44 billion years; (7) There are large-scale magnetic fields in the sun with strength up to lo8 gauss near the center; (8) The gravitational constant increases with time. A lower limit on Yin the neighborhood of 0.15-0.18 is set by the choice of 44 billion years for the sun’s age. If, therefore, the eventual upper limit on the counting rate set by the Davis et al. experiment is reduced much below the preliminary limit, a clear discrepency with conventional theory will be established, regardless of external arguments for higher Y.

I. THE ANTINOMY

The Davis, Harmer, and Hoffman (I) experiment to detect solar neutrinos via the CP7(v, e-) A3’ reaction has yielded the preliminary result that C u& < 3 x 10-3s set-l per CP7 nucleus. Here ui is the effective cross section * Supported in part by the National Science Foundation Aeronautics and Space Administration (NsG-496). 164

(GP-8060) and in part by the National

SOLAR

NEUTRINOS

AND

SOLAR

HELIUM

165

for the absorption of neutrinos that impinge on the earth with a flux &, distributed in an energy spectrum of type i. Prior to the circulation of the experimental result, theoretical estimates of C U& were uniformly higher than the experimental limit by factors ranging from 5 to 15 [(2)-(6)]. The early theoretical estimates were based on solar models characterized by an initial helium abundance Y - 0.25. From the work of Sears (.3), it is clear that, all other things being equal, the choice of a lower value for Y leads to a lower theoretical estimate of Cu& . More recent model estimated ((7) (8)) confirm this trend for values of Y lower than those examined by Sears. Extrapolating from the results of these more recent calculations, one finds that, in order to achieve values for 2 ci& at least as small as the Davis, Harmer, and Hoffman limit, it is necessary to accept that, ~46 billion years ago, the fraction Y of the sun’s mass in the form of He* must have been less than 0.16-0.17. Almost every attempt to estimate the helium abundance in galactic objects other than the sun has yielded values for Y in the range 0.20-0.40, the most probable values clustering about 0.25-0.30. In the first comprehensive discussion of solar neutrino fluxes that preceded the C13’ experiment, Sears (3) pointed out that values of Y estimated for Population Z (Disk Population or Metal Rich) objects have “shown a curious uniformity in recent years”. Astronomical evidence for a possibly universal, high value for the Y of Population Z objects has since been amply catalogued (9). Each astronomical determination of Y is, of course, inevitably of low accuracy, but the evidence for the clustering of determined values about a high value near 0.25-0.30 is impressive. A measurement of the relative intensities of radio frequency lines corresponding to high quantum number (n = 109 + 108) transitions in atomic hydrogen and in single ionized helium is in principle capable of permitting fairly unambiguous determinations of Y for matter in HII regions. Several groups ((ZO), (II)) have obtained a lower limit of Y = 0.25 - 0.27 for matter in several HII regions. Members of the most metal deficient stellar component in the Galaxy-the Population II (Halo Population or Metal Poor) stars that are reportedly characterized by heavy element abundances from 10 to 100 times smaller than those in the sun-are, like the sun, too cool at the surface to permit a spectroscopic determination of their surface helium abundance. The possibility that Population II stars might have a He* abundance similar to that of Population I objects was examined quantitatively for the first time by Christy (Z2), through a comparison between theoretical models of stellar envelope pulsation and the observed pulsation properties of RR Lyrae stars, and by Faulkner (Z3), through a study of the stellar model equivalents of the horizontal branch stars in globular clusters. Faulkner’s tentative conclusion that Population II stars are indeed characterized by a high helium abundance has been strengthened in subsequent investigations (14) that give Y 5 0.30, on the assumptions that the total heavy element abundance of

166

IBEN

globular cluster stars is 100 times smaller than that of the sun (taken as 2 per cent by mass) and that little mass loss occurs during hydrogen burning stages. In a parallel development, the existence of a high average galactic helium abundance has been shown to follow “quite naturally” from the assumption of a hot big-bang universe. Reviving the idea of nucleosynthesis in pregalactic matter-an idea eclipsed for over a decade while the general features of nucleosynthesis in stars was being formulated-Hoyle and Taylor (1.5) suggested that, for a wide range of assumed conditions, one might expect the universe to emerge from a fireball stage with Y - l/3. In more detailed calculations for homogeneous, isotropic universes, Peebles (16) and Wagoner, Fowler, and Hoyle (17) have derived values of Y - 0.25-0.28, the precise number depending slightly upon the choices for the present-day densities of matter and radiation in the universe. Higher values of Y would apply if the restrictions of homogeneity and isotropy were relaxed. Fowler and Stephens (18) have provided a comprehensive bibliography of the literature on big-bang nucleosynthesis. In the light of the arguments for an average galactic Y - 0.25-0.30, it comes as a surprise that, for the sun, Y < 0.16-0.17. It is the purpose of this paper to reexamine the dependence of solar neutrino fluxes on several input parameters and on several assumptions regarding the input physics, placing sharply in focus the implications of the Davis et al upper limit for the solar initial helium abundance.

II. THE DEPENDENCE

OF C uicji ON Y

Theoretical estimates of C u& are obtained by weighting the neutrino fluxes from solar models with theoretically calculated neutrino absorption cross sections. Using Bahcall’s (29) calculated cross sections for neutrino absorption, the probability per unit time for the conversion of a terrestial Cls7 nucleus into an A37 nucleus may be written as C a&

= 3 x 1O-36 set-l

(.2 trjOe

+ 1.$(~~o10

+ 3.8W5) x 109 + 1.4&Wx 1010 +

$y;; .

9

)

(1)

where the 4’s are numbers of neutrinos passing in one second through one optimally oriented square centimeter at a distance of 1.5 x 1013cm from the sun’s center. The fluxes #BE), 4(015), and $(N13) are due to beta decays of BE, 015, and Nls; d(Be’) is due to Be7 electron capture; and +(ppe-) is due to the reaction Cp + p) + e- -+ d + v. In expression (l), the quantity in large brackets (call it R) gives the ratio of

SOLAR

NEUTRINOS

AND

SOLAR

HELIUM

167

the counting rate associated with any given solar model to the current preliminary upper limit of Davis, Harmer, and Hoffman. Input physics for stellar models now normally includes extrapolated low energy nuclear cross section factors either identical with or very similar to those compiled by Fowler, Caughlin, and Zimmerman (20). In the fourth column of Table I are the cross section factors given in Ref. (20) and in column five are the limits to these factors as estimated in Ref. (20). The choices of cross section factors used in obtaining results to be described in the present section are given in column 3 of Table 1. With the exception of .S& , these cross section factors are identical with those used by the author in previous stellar model investigations (21). TABLE CENTER

Reaction

He3(He’, Be’(p.

Symbol

..~~.

P+p-d+e++v He3 + He” + Hea + y) Be’ y)Bs

OF MASS

2p

Sp, (KeV barns) 3.5 ;.: ,o-2” 6.5 ,i lo3

S 31 &,

0.6 0.03 1.33 3 10

s lls s 114

Ola(p,

s 118

y) F”

CROSS SECTION

s 11 s 33

P(p, y)Nr3 N’Yp, y) 0’”

1 FACTORS

Szc’(KeV barns) LimitsFCZ(KeVbarns) .--..-.___ 3.36 :< lo-‘” (3.0 -* 3.7) :< 10 22 5 ’ 107 (3 ---f 7) x 109 0.47 0.38 -0.56 0.032 + 0.048 0.04 1.4 1.i --f 1.4 2.5 ---f 3.0 2.75 10.3 7.2 4 13.4

A description of the normal input physics for solar models requires a specification of the approximations to the equation of state and to the opacity. In the present investigation, nonrelativistic electron degeneracy and radiation pressure are included in the equation of state. At temperatures below 5 x lo5 OK, the fact of incomplete ionization of H1 and He4 is taken into account. An approximation to Keller-Meyerott tables (22) has been used for the radiative contribution to the stellar opacity for temperatures above 5 x lo5 “K. Below 5 x lo5 “K, where absorption by H1 and the H- ion dominate, another approximation (23) is used. The contribution of electron conduction is also taken roughly into account (24). A more complete description is given in Ref. (21). As in most model calculations, it is assumed that the sun has been converting hydrogen into helium for approximately 4+ billion years, and that the helium produced at any point in the interior due to fusion processes remains at the site of formation (i.e., no large scale mixing currents are expected). It is implicit here, as in all other calculations, that the sun formed out of a chemically homogeneous nebula and that no differentiation of any consequence preceded the start of the nuclear burning phase.

168

IBEN

The pertinent results of the solar model calculations-with the “standard” assumptions just described-are displayed in Figure 1. Of greatest interest is the dependence on Y of C U&Q, as given in Figure 1 in units of the current experimental limit. The ordinate of each circle along the curve for C u& singles out that value of Y for which the opacity parameter Z (roughly, the abundance by mass of elements heavier than He? is an integer multiple of 0.01. Rough estimates of the appropriate choice for Z can be obtained from the analysis of spectroscopic data, with results that range from 0.01 to 0.03 for Z. As can be seen from Figure 1, corresponding values for Y lie between 0.22 and 0.30. In this range, the calculated value of C a& exceeds the experimental limit by a factor ranging from 3 to 11. Only if one suspends the literal interpretation of Z as a measure of the abundance of elements heavier than He4 and permits Z to take on negative values, can consistency with the experimental limit be achieved. Extrapolating beyond Z = 0 gives an upper limit on Y of Y,, - 0.16. I

12 It--

l

I /

7

u7797 \

2 = 0.00 I 020 INITIAL

HELIUM

I 0.25 ABUNDANCE,

I 030 Y

FIG. 1. The relationship between Y and c 0~4~ (total counting rate), u& (the contribution of B* neutrinos), u,& (the contribution of Be’ neutrinos), T, (the sun’s central temperature), and pe (the sun’s central density). All U& are given in units of 3 x 1O-SBset-l per Cl*’ atom The solar models are 4t x 10Qyr old, no mixing is permitted, and the canonical cross section factors S& have been employed.

SOLAR

NEUTRINOS

AND

SOLAR

HELIUM

169

It is more reasonable to insist that Z 3 0.0 and to accept that value of Y which corresponds to Z = 0.0 as a lower limit on the sun’s initial helium abundance--for the chosen set of input parameters. In the present instance, we conclude that - 0.174 and that x u~#~ is nearly 1.25 times the experimental limit Y 2 Ymin when Y = Ymin . It is worthwhile to point out that by far the dominant contribution to x a,4, is made by neutrinos from the infrequently occurring reaction B” - Be* -b e- -i- V. This contribution is given in Figure 1 by the curve labeled r~& . Next in importance are neutrinos from the more frequently occurring reaction Be7(em, V) Li5, as shown by the curve labeled o,$, . The curve for T, in Figure 1 demonstrates the well known fact that, under normal circumstances, 1 ai$i is directly proportional to a high power of the temperature at the solar center. Over the range shown in Figure I, one finds 4, cc TC7and +8 oc r,‘” cc $7Ti3. The factor Tp in & represents the temperature dependence of the average Coulomb penetration factor for the Be7(p, y) B* reaction. The factor of $7 in & expresses the fact that the Be’ electron capture rate and the B8 production rate are both proportional to the Be’ abundance. In the sun, Be’ disappears primarily by electron capture, so that the equilibrium abundance of Be7 is essentially unaffected by the rate of Be’ proton capture. That TC and therefore both $7 and +8 increase with increasing Y is a simple consequence of the gas law equation of state and the requirement of a balance between the gravitational force inward and the net pressure force outward on any chunk of matter in the sun. The distribution of the gravitational potential through the sun and the mean solar density are constrained by the specification of solar mass and radius. An increase in Y means a reduction, at all points in the sun, in the number of particles per gram. Therefore, since the mean density is fixed, the mean temperature in the model sun must be increased when a larger initial Y is chosen. It is possible to express quantitatively the dependence of TC on Y. For stars of homogeneous composition, the requirement of a pressure balance may be expressed by the relation kT, - pMH(GM/R), where $kTC is the average kinetic energy of a particle at the stellar center, GM/R is the potential energy per gram at the stellar surface, and PM, is the mass of an average particle in the star. It is remarkable that, even for the inhomogeneous model suns presented here one may write kT, E l.13psMH(GM,/R,), with psMH being the mass of an average particle near the surface, where no fusion of H1 into He4 has taken place. The reason for this simple result is the fact that the luminosity history of the sun is almost independent of initial Y, so that the total amount of H1 processed into He4 in the interior is also almost independent of initial Y. For all model suns, the He4 abundance at the center has increased by about L3Y - l/3, after 4; >: 10" years of evolution, and ps/pcenter - 0.75 + 0.01 (over the range Y = 0.2-0.3).

170

IBEN

III. THE DEPENDENCE

OF c u& ON EXTRAPOLATED

CROSS SECTION

FACTORS

The embarrassment of a theoretical minimum C ai& that exceeds the experimental upper limit can be avoided by altering one or more of the pertinent nuclear cross section factors within quoted limits. In Figure 2, R, the ratio of calculated C U&Q to the experimental limit, is displayed as a function of Y for several different choices of cross section factors. Beside each curve is that cross section factor which differs from the “canonical” set given in column three of Table 1. For the curve labeled Sfj’s, S& = S$ x (4/3.5), S& = S&/2, S& = S&/2, all other Sij = St . All curves in Figure 2 pertain to solar models that have evolved with no mixing for 46 billion years. Circles lying roughly in the same vertical column represent values for the choices Z = 0.01, 0.02 and 0.03. The dependence of C aidi on the Sij occurring in the pp chains is easy to understand. The rate at which H1 is converted into He4 is controlled by the p + p --f d + e+ + v reaction, since this reaction is by far the slowest in the chains. At a given temperature and density, an increase in S,, of necessity means an increase in the energy production rate. However, since the overall energy production rate must remain fixed at the sun’s luminosity, an increase in S,, must

t (Zu+) SOLAR MODEL

(xc+)

DAVIS et 01 LIMIT

I5

015

020

025

0.30

035

FIG. 2. The ratio R of theoretical estimates ofx vi& to the upper limit set by Davis, Harmer, and Hoffman. The ratio is given as a function of the assumed initial solar helium abundance Y for various choices of center of mass cross section factors. The canonical set of cross section factors Sk is given in the third column of Table 1. Each circle designates the ratio for one of three different values of the opacity parameter 2. All results are for an assumed solar age of 44 billion years.

SOLAR

NEUTRINOS

AND

SOLAR

FIG. 3. The relationship between central temperature (10” density (sm cm-3) and Y as a function of the S, . The value differs from the set S,O,is designated beside each curve.

HELIUM

171

OK) and I’ and between central of that single parameter Sli that

be compensated for by a reduction in the temperatures and densities in energy producing regions. A reduction in temperature and density in turns means a decrease in the rate of BE neutrino production. The magnitude of the reduction in central temperature and density is shown in Figure 3. Over most of the energy producing region of all model suns, He3 disappears primarily through the He3 + He3 --f He4 + 2p reaction rather than through the He3 + He4 + Be’ + y reaction. Hence, the rate of Be’ production is essentially proportional to the He3 abundance obtained by formally equating the He3 formation rate to the destruction rate via the He3 + He3 ---f He4 -k 2p reaction. At a given temperature and density, an increase in S,, means a decrease in the equilibrium abundance of He3 and therefore a concomitant decrease in the rate of neutrino production by both the Be’(e-, V) Li’ reaction and the B8 --f Be8 + e-+ + v reaction which follows the Be’@, y) B8 reaction. As is evident from Figure 3, essentially no change in densities and temperatures occurs as a result of a change in S33 . This is to be expected since, when the He3 + He3 --f He4 + 2p reaction dominates He3 destruction, the only effect of altering S,, is to alter the equilibrium abundance of He3, leaving the energy production rate unchanged. Finally, a decrease in S,, means a proportionate decrease (essentially linear) in neutrino production by decays of both Be’ and B8 and a reduction in S,, means a (linearly) proportionate reduction in neutrino production by B8.

172

IBEN

IV.

UPPER

LlMITS

ON

Y AND

EXTRAPOLATED

CROSS

SECTION

FACTORS

For every choice of cross section factors, the specification of an experimental upper limit on C u& establishes a corresponding upper limit on the initial solar Y. The entries in the first column of Table 2 give this upper limit on Y as a function of cross section factors--ifthe Davis et al limit is a hard upper limit, i.e., if R < 1. When the statistics are improved, the Davis et al limit could either decrease (the preliminary result is consistent with R = 0) or increase. Anticipating the latter possibility, upper limits on Y have been entered into successive columns of Table 2 for R = 2, 3, 5. and 10. A theoretical overestimate of the pertinent neutrino absorption cross sections might also make the additional entries of relevance. All italicized entries in Table 2 are extrapolations. Entries distinguished by an asterisk are extrapolations into the range 2 < 0.0. All entries are for t = 44 billion years and no mixing unless otherwise noted.

TABLE

2

UPPER AND LQWER LIMITS ON THE INITIAL SOLAR HELIUM ABUNDANCE Sji)S

R=l

S&‘S

R=2

0.166*

0.193

R=3 0.216

R=5

R = 10

YIIli*

0.249

0.294

0.174

&

x 3

.165*

.I91

.212

.243

.288

,174

SE

x 10

.164*

.190

.210

.239

.283

.175

s:, x (4/3.5)

.I85

.217

.241

.271

.317

.177

$1 x 2

.263

.310

.344

-

-

.187

&I2

.189

.226

.252

.280

.316

.172

&I5

.231

.267

.288

.318

-

.171

%I5

.140*

.158*

.I77

.209

.261

.175

s:j’s

.238

.278

.301

.337

-

,177

.165*

.191

.213

.244

.288

.167

t = 6, S,p,‘s

.148*

.I69

.186

.215

.262

.158

t = 3, sfj3

.179*

.224

.254

.285

.326

.191

t = 0, s,q.-s

.276

.315

.342

.382

-

.231

4(1/p)

=

-.Ol

SOLAR NEUTRINOS AND SOLAR HELIUM

(i) PENULTIMATE

173

LIMITS

For every value of maximum R (rg = 49 and no mixing), a “penultimate” maximum to Y may be established by varying all cross section factors to the edge of their stated limits in a direction most favorable for increasing Y. An inspection of Fig. 2 and of Table 2 shows that one wants the largest possible values for S,, and S,, and the smallest possible values for S,, and S,, . Uncertainties in cross section factors for CN cycle reactions shall be temporarily ignored. Despite the fact that it cannot be checked directly by experiment, the theoretically calculated cross section factor for the proton-proton reaction has long been regarded as perhaps the most reliable of all the relevant nuclear cross section factors. A major point of contact between theory and experiment is the axial vector beta decay coupling constant as estimated from decays of, for example, the neutron and O14.Bahcall and May (25) quote a value of (3.78 & 0.15) x 10ezeKeV barns for the leading term in S,, . The bulk of the 12 per cent increase, represented by this result, over the (3.36 & 0.4) x 10ez2KeV barns previously adopted, is due to the choice of a (10.80 f 0.16) minute lifetime for the neutron (26) rather than a (11.7 * 0.3) minute lifetime (27). Accepting the Bahcall and May discussion (and including a three per cent effect occasioned by the omission, in the present calculations, of all but the first term in the standard approximation to the Maxwell-Boltzmann average of ao), an upper limit is S$ective = 4.1 x 1O-22KeV barns. Parker (28) has interpreted his measurementsof the Be’(p, y) Bs reaction cross section to mean that S,, : (0.043 i 0.004) KeV barns. The standard choice for the probability per unit time for Be7to capture electrons from the continuum is given by Bahcall (29). The fact that Be’ is not completely ionized in the solar interior permits the Be7 nucleus to capture electrons from the K shell, as well as from the continuum, leading to an approximately 20 per cent increase in the electron capture probability over that obtained when bound electrons are neglected (30). The net effect is a reduction of about twenty per cent in the total number of Be7(p, y) B8 reactions per second, a result that has been found to be largely independent of the initial composition of the solar model. If only free electron capture is explicitly included in the solar mode1calculations, asis the casefor the calculations reported here, the effect of bound electron capture may be estimated fairly accurately by decreasing the effective Be7(p, y) B6 cross section factor by twenty per cent. Thus, using Parker’s extrapolation for S,, , a lower limit is S$fective > 0.03. From their measurementsof the cross section for the He3(He4,y) Be’ reaction, Parker and Kavanagh (31) extrapolate S,, = (0.47 & 0.05) KeV barns. Adopting

174

IBEN

the more generous f twenty per cent uncertainty estimate of Ref. (20), a lower limit is S, > 0.38 KeV barns. Recent measurements of the cross section for the HeS + He3 -+ He4 + 2p reaction [(32), (33)] coupled with the error estimate of Ref. (20) yield S,, < 7 x lo3 KeV barns. As a cautionary note it should be remembered that (a) the 2 He3 + He4 + 2p reaction is perhaps in principle the most difficult to understand and interpret theoretically and that (b) the recent measurements have led to an extrapolated cross section factor five times larger than that extrapolated from earlier measurements (34). “Penultimate” maxima for Y may now be defined by setting S;ifective = 4.1 x 1O-22 KeV barns, S,, = 0.38 KeV barns,

and

S;iPective = 0.03 KeV barns, S,, = 7 x lo3 KeV barns.

Inspection of Table 2 yields Y(1) s 0.20 as the maximum value of Y permissible under the stated assumption when R < 1. In a similar fashion, Y(2) z 0.24, Y(3) E 0.27, and so forth, for R < 2, R < 3, and so forth. Note that Y(1) N 0.20 is still small compared with the most probable Y estimated for other galactic objects. (ii) ULTIMATE LIMITS Zero energy cross section factors quoted as A & B are not unique interpretations of experimental results. Any quoted value for A is, of necessity, an extrapolation from cross sections obtained at center of mass energies considerably above those relevant in the sun’s interior. A quoted value for B tends to be, in most instances, a statement about the uncertainties in the experimentally determined points rather than a statement of the confidence with which A may be presumed known. Alternate interpretations of the experimental data for the He3(He4, y) Be7 and the Be7(p, y) B* reaction are sketched in Figures 4 and 5 respectively. These figures are reproduced from Refs. (31) and (28), respectively. The solid curves are those which give S,, = 0.47 KeV barns and S,, = 0.043 KeV barns. The dashed curves are based on no theory. They are simply eye-fitted extrapolations from a set of experimentally determined points. In the case of the Be’(p, y) Bs reaction, resonances at 724 KeV and beyond 2000 KeV suggest that the solid curve for S,, might possibly have been normalized differently. The dashed curve labeled as Guess 1 is of the same form as the solid curve but lowered bodily in an effort to abstract the possible effect of the resonances (35). The dashed curve labeled Guess 3 is perhaps somewhat extreme,

SOLAR

NEUTRINOS

,GUESS

-

/

GUESS

SOLAR

175

HELIUM

1

-GUESS

I

AND

‘.GUESS

4

3

G’JiSS

4

2

/ I

FIG. 4. Cross section factor data for the reaction He3 + He4 -Be’ + y, reproduced from Parker and Kavanagh (1963). The solid curve is the fit suggested by Parker and Kavanagh. Dashed curves are alternate choices.

I,,,,,,.,,

0.150

i,,,,,,,,,,

-

0

,

!~‘~~lIlII/,,(~/ 0

500

1000

1500

2000

E, (ke’v’)

FIG. 5. Cross section factor data for the reaction Be7 + p -+ B8 + y, reproduced from Parker (1966a, b). The solid curve is Parker’s choice for a fit to the data. Dashed curves are alternate choices.

176

IBEN

but it is unbiased by any theoretical preconceptions other than that S,, should be a smooth function of energy that fits the experimental points reasonably well. At zero energy, S,, = 0.022 KeV barns for this curve. In the Debye-Hiickel approximation (used in all calculations reported here), the reduction in the Coulomb barrier due to electron screening is expressed (36) by an amplification factor of the form exp(Z,Z&), where Z, and Z, are the charges of the reacting nuclei and S,, is a function of local conditions. The value of S, is fairly constant over energy and neutrino producing regions of solar models and is roughly S,, N 0.05 for all solar models considered in this paper. The maximum uncertainty in C u& , due to an uncertainty in screening factors appears in the rate of the Be7(p, y) BE reaction, for which exp(Z,Z&) - 1.2-1.25. It is surely possible that the simple Debye-Htickel approximation to S,, could be wrong by a factor of two. Thus, the screening uncertainty introduces an additional uncertainty of - 10 to $20 per cent in the effective cross section for the Be7(p, y) B* reaction. “Ultimate” lower limits on S,, and Sa4might now be set at S;SfeCtive = 0.022 x (1 - 0.1) x (1 - 0.2) = 0.015 KeV barns and Ss4 = 0.30 KeV barns. In connection with the “ultimate” lower limit on S,, , reference should be made to earlier measurements by Kavanagh (37); to preliminary measurements by Vaughn et’ al (38); and to a theoretical discussion by Tombrello (39). The limits on S,, stated in Ref. (25) may be somewhat sanguine. There is no quarantee that the neutron half life lies within the limits given by the most recent of two experiments and it is presumptuous to suppose that the most recent calculation of the pertinent matrix element has exhausted all possibilities of error. A range S$fective = (3.1-4.3) x 1O-22KeV barns (including the three per cent effect left out of the calculations) is certainly not out of the question. Adopting as “ultimate” limits (in KeV barns) Sf:fective = 4.3 x 10-22, Ss3 = 7 x 103, S34 = 0.30, and Sz:fective = 0.015, “ultimate” upper limits on Y are, after interpolation in Table 2, y(l) E 0.25, y(2) g 0.29, and y(3) g 0.32, for R < 1, 2, and 3, respectively. (iii) EFFECT OF VARYING CN-CYCLE The dependence of reaction rates has not of ten increase m the values of Y favored by

CROSS SECTIONS

C u& on cross section factors that occur in CN-cycle been emphasized for the simple reason that even a factor controlling reaction rate has a minor effect on C ~~4~ for the Davis et al limit. Ten years ago there was still a question

SOLAR

NEUTRINOS

AND

SOLAR

HELIUM

177

as to whether or not the N14(p, r) 015 reaction might proceed through a resonance at energies relevant to the solar interior. Since that time, it has been concluded on nuclear physics grounds that such a resonance does not occur. The low value of the Davis et al limit provides additional support for this conclusion. In Figure 2, the curve for C u& labeled S,,, x 1000 is equivalent to a curve constructed by assuming that the P(p, r) N13 reaction is the slowest reaction in the CN-cycle. In all cases for which S,,, = S!,, x 1000, the major contribution to 1 u,+( is due to neutrinos from the decays of Or5 and N 13. In all of these cases, C oL4i is an order of magnitude larger than the Davis et al limit. independent of alterations, within “reasonable” limits, of the cross section factors pertaining to the pp chain reactions. Thus, the Davis et al. limit rules out the possibility of a near threshold resonance in the N14(p, y) 015 reaction of sufficient strength to permit the C12(p, y) N13 reaction to control the overall cycling rate of the CN-cycle.

V.

LOWER

LlMITS

ON

Y

A lower limit on the initial solar helium abundance is set primarily by a choice for the age of the sun, rather than by a choice of nuclear cross section factors within the stated limits. For any choice of solar age, the lower limit is determined formally by setting the opacity parameter Z = 0.0. Extrapolated estimates of this lower limit, appropriate to a 4Q billion year old unmixed sun, are entered in the last column of Table 2. If Y is chosen smaller than Ymin , it is necessary to allow the model star to evolve longer than 44 billion years before it shines as brightly as does the present sun. Note that, in several instances, Ymin (column 7) > Ym,(columns 2-6). All entries in columns 2-6 which correspond to Ymin 3 YmaX should, of course, be deleted. Such entries have been marked with an asterisk. The parameters appropriate to each deleted entry are to be viewed as an invalid combination which does not permit the chosen boundary conditions (on R, t, and the extent of mixing) to be satisfied. For example, there exists no Y such that R < I when all Sjj = SFj . It is disturbing that, for all but extreme choices for cross section factors, the lower limit on Y essentially coincides with the upper limit set by the experimental limit on 2 cr& . If the eventual upper limit on the experimental counting rate is reduced much below the current limit, a clear discrepancy with conventional theory will be established, regardless of external arguments for higher Y.

178 . Vl. DEPENDENCE

OF c oiQi ON SOLAR

AGE

In Fig. 6, the ratio R is displayed for models that reach the sun’s present luminosity after 44, 3, 2, 1, and 0 billion years. The cross sections Sioj have been employed and again no chemical mixing has been permitted. If, for example,

(xu#‘kDLAf7

MODEL

( z”+J~~v~s

et 01 LIMIT SEARS

(1964)

15 -

0.15

0.20

0.25

0.30

0.35

Y FIG. 6. The relationship between the ratio R and the helium abundance Y as a function of (1) assumed solar age, (2) opacity, and (3) equation of state. Curves forming the fifteen point mesh are derived from models made with the canonical set of cross section factors Sf, , assuming solar ages of 0, 1, 2, 3, and 4& billion years. The curve labeled d(l/p) = -0.01 is derived from models that differ from those leading to the i = 44 curve only by a slight modification in the equation of state.

the sun were only 3 x log year old and if R < 1 and S,P, for all i and j, then Y < 0.18. The “penultimate” limit on Y, for R ,( 1, would be Y(1) s 0.23 and the “ultimate” limit would be y(l) GZ 0.29. Similarly, if the sun were 6 x lo@year old and R < 1, the corresponding limits would be Y < 0.15, Y(1) zz 0.18, and r(l) E 0.22. The fact that, for a given value of initial Y, C ai& is smaller for a smaller assumed age is simply understood. The lower the age, the less has the conversion of hydrogen into helium proceeded in energy producing regions and therefore the larger is the particle density there. The larger the particle density is, the lower

SOLAR

NEUTRINOS

AND

SOLAR

HELIUM

the temperature and density must be. The effect on central quantities a smaller age is shown explicitly in Figures 7 and 8.

FIG. 7. The relationship between central density and Y and the relationship temperature and Y as a function of assumed age or degree of mixing.

179 of choosing

between central

A convincing lower limit to the sun’s nuclear burning age is provided by age estimates of sediments containing fossil remnants of very elementary life forms. This is becausethe existence and development of life on earth requires that energy be supplied to the earth’s atmosphere at a rate that is neither much larger nor much smaller than that delivered by the sun during its nuclear burning stages. Barghoorn and his associates(40) have obtained morphological evidence for the occurrence, between two and three billion years ago, of an ordered sequenceof rudimentary organisms, the complexity of the organism increasing with proximity to the present. The oldest finds are unicellular organisms that were fossilized an estimated 3.1 billion years ago. How much time elapsed between the formation of the sun and the emergenceof theseunicellular organismsis asyet an unanswered question, but, in any case, the sun as a nuclear-burning star must be older than 3.1 billion years.

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-

0.20

025

030

0.35

Y

FIG. 8. The relationship between central hydrogen abundance and surface Y as a of assumed age or degree of mixing.

An even larger lower limit follows from the conventional assumption (probably fairly safe) that the sun is at least as old as stony meteorites for which ages very close to 4: billion years have been estimated ((4I), (42)). It has been conventional to assume further that the sun (as a hydrogen-burning entity) and the stony meteorites are of approximately the same age. Several dating methods that involve statements about short-lived isotopes suggest no more than a few tenths of a billion years difference in formation times. It should be noted, however, that the assumption of nearly coeval formation is based on arguments that involve models of star and planetary formation and such concepts as the isolation of solar system material out of a presumed parent cloud and “the solidification of planetesimals, etc., out of the solar nebula.” Since theories of both planetary and star formation and, in particular, models of the coupling between star and planetary formation are highly speculative, there would appear to be no overwhelmingly compelling reason for believing that the sun’s nuclear-burning age might not be, say, as much as 6 x log year.

VII. THE EFFECT

OF MIXlNG

BY SPIN-DOWN

OR MERIDIONAL

CURRENTS

The possibility of significant chemical mixing due to currents driven by a spin-down mechanism has been suggested by Ezer and Cameron (43) as a way of accomodating the result of the C13’ experiment without the need of altering either conventional views about the extrapolation of nuclear cross section data or conventional views about the solar composition. As has just been demonstrated,

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181

a sufficient reduction in the assumed age of the sun can lead, for a given Y, to a drastic reduction in the corresponding calculated values of C u& and hence to a significant increase in the upper limit on Y implied by the Davis et al result. Since the larger the assumed age, the further the conversion of hydrogen into helium has proceeded in regions of energy production, there is a strong correlation between assumed age and the degree of large-scale chemical inhomogeneity in unmixed models of the sun. In the extreme limit of t = 0, the abundances of hydrogen and helium are the same throughout the interior (of the conventional model, which assumes that the sun was condensed out of an initially homogeneous cloud). The case t = 0 therefore corresponds to a completely mixed model which could be of any age. Similarly, unmixed models that have been evolved to the sun’s luminosity for assumed ages that vary from 0 to 49 billion years correspond roughly to models which have evolved for 4& billion years but which are characterized by successively decreasing degrees of mixing. The value of initial Y to associate with a solar model that has been completely mixed during its nuclear burning history may be obtained from the model results. One finds that the initial Y increases by d Y E (tJ102) x [l - 5 x (t,/102)] for a fully-mixed model that reaches the present solar luminosity in t,(billion) years. This result is quite insensitive to changes in Z and yields d Y - 0.034 for f = 44 billion years. Thus if the sun is 44 billion years old and has been fully mixed during this time, and if R < 1 and S
182

IBEN

angular momentum by all processses including spin-down must occur on a time scale long compared to the sun’s age. Weber and Davis (48) have constructed a model for the interaction between the solar wind and the solar magnetic field that permits a rough estimate of the rate at which the sun is currently losing angular momentum. This estimate can be interpreted to mean that the sun’s total angular momentum is decreasing by a factor of 2 every 5 x lo9 year. This interpretation assumes that spin-down (or other) currents maintain approximately uniform rotation throughout the sun (i.e., the sun rotates as a rigid body) and that the characteristic decay time for angular momentum is constant. If the Weber-Davis angular momentum loss rate is accepted at face value, an additional argument for a relatively efficient solar spin-down process can be devised. For all of the solar models constructed for this paper, the moment of inertia of the convective envelope is at least 100 times smaller than the moment of inertia of the sun as a whole. If the radiative core were completely uncoupled from the convective envelope, one might expect a loss of angular momentum through the surface to affect only the rotation rate of the envelope. The decay rate for angular momentum could be estimated as dJ/dt = (d/dt)((Z,,/lOO) co) = -Z&T, where I,, is the moment of inertia of the sun as a whole, w is the angular velocity of the envelope, and T is the characteristic decay time estimated by Weber and Davis to be 7 x lo9 year. Thus, if the radiative interior of the sun were to slip past the convective zone without being effectively braked, the rotation rate of the convective envelope would be halved in approximately 0.69 x ~/lo0 or 5 x 10’ year. The implication that, when it first reached the main sequence, the entire sun was rotating at a rate 21°0 more rapidly than at present is clearly untenable and can be avoided only if there is a fairly efficient rotational coupling between the radiative interior and the convective envelope. The transport of matter is quite a “different cup of tea” from the transport of angular momentum. One might suppose that the more effective spin-down currents are in carrying angular momentum, the more effective they are in smoothing out inhomogeneities that arise because of nuclear transformations in the interior. Accepting this reasoning, the arguments of Refs. (44) and (45) would seem to favor appreciable mixing by spin-down currents more than do those of Ref. (46). As an aside, it should be noted that the suggestion that spin-down is inhibited in the solar core, because of a stabilizing gradient of mean molecular weight, is not applicable if spin-down currents are capable of maintaining approximate chemical homogeneity. One might argue conversely that an extremely efficient spin-down process might be very ineffectual in mixing matter on a large scale. That is, the occurrence of appreciable differential rotation, as proposed by Dicke, might be necessary to establish spin-down (or other) currents that are sufficiently strong to cause

SOLAR

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AND

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HELIUM

183

significant mixing. Evidently a clear-cut and unambiguous solution is not yet at hand and further carefully modeled investigations are called for. A high rotation rate promotes meridional circulation currents that are capable of mixing core and surface material. On the assumption that departures from relatively uniform rotation are negligible, the time scale for large scale mixing by meridional circulation has been estimated (49) as T - 8 x 1012 year (m/CS2R3). Here R is the radius and M is the mass (both in solar units) of the region over which mixing occurs; 6 is the average angular velocity in the interior (in units of w,, , where woR, = 2 x lo5 cmjsec). This estimate for mixing time may be rewritten as 7 N 2.4 ;: IO* year/(v2/Rg), where (r2/Rg) is the average ratio of centripetal acceleration to gravitational acceleration. In order that T - 1OSyear, it is necessary that (02/Rg) - l/4. Such a large ratio for matter in the interior of the present sun would imply an oblateness many orders of magnitude larger than that observed. It is therefore highly improbable that meridional currents are now effectively mixing matter throughout the sun on a time scale comparable to the sun’s age. This does not, however, exclude the possibility that meridional currents may have produced appreciable mixing earlier when the sun was rotating more rapidly. The sun’s surface temperature has been increasing and its convective envelope has been shrinking for the past 43 billion years. The strength of both the solar wind and the solar magnetic field may have also been declining during this time. A rotation rate for the early sun given by the WeberDavis time scale, assuming uniform rotation, could therefore be a severe underestimate. Other forms of evidence suggest that this may be the case. The mean angular velocity at the surface of lower main sequence stars with deep convective envelopes is considerably smaller (by two orders of magnitude) than the mean angular velocity at the surface of upper main sequence stars. One possible interpretation of this fact is that the rotation rate of matter at the surface of the sun, which possesses a deep convective envelope, has also decreased by several orders of magnitude during its main-sequence lifetime. Effective mixing by meridional circulation may therefore have occurred in the early sun on a time scale close to 10” yr - 8 x 1012/1002. How long mixing of this sort might have played an important role can be inferred from a statistical study by Kraft (50) of rotation rates of Hyades and Pleiades stars. Kraft estimates a half-life of -4 x lOa year for the angular momentum of young main sequence stars with convective envelopes. This suggests that, although meridional circulation was of possible importance as a mixing process in the early sun, it remained an important process only during the first l/2 billion years or so of the sun’s life. In conclusion, the conventional assumption that significant mixing by meridional circulation has not occurred during most of the sun’s life is probably justified.

184

IBEN

However, it is possible that appreciable mixing by such currents may have occurred during the first half-billion years of the sun’s life with the consequence that the present sun is best described by a model of a slightly younger star (younger by, say, A to - l/2) in which no mixing has been permitted. Whether or not mixing by spin-down currents has had a similar or an even more important effect is still an open question. VIII. OPACITY,

EQUATION

OF STATE

AND c u<#~ VS Y RELATIONSHIPS

In Figure 6 are shown the values of C ai& resulting from models obtained by Sears (3). The Sears’ results which are connected by a continuous curve pertain to models that were constructed with almost the same equation of state and with exactly the same opacity as were the models described in this paper. Sears used the cross section factors S& x (0.47/0.60), S,, = S&,/S, and Sfj for all other Sii . The results of the present study are in essential agreement with those of Sears. The solitary point labeled “Sears (1964) Model D” is derived from a model for 2 = 0.035, differing from other Sears models in that Cox-Stewart (52) opacities were used. It is highly significant that, although the relationship between Y and the opacity parameter 2 is quite different for the model made with the Cox-Stewart opacities, the associated value of C ai+i lies very close to the curve for C ai& derived from the Sears models constructed with an approximation to KellerMeyerott opacities. This indicates that, for a given set of cross section factors, the relationship between C ai& and Y is, to first order, independent of the choice of opacity. To put it slightly differently, the R-Y relationship is nearly independent of the details of the relationship between Y and the opacity parameter 2. This near independence is very fortunate since the relationship between spectroscopic estimates of heavy element abundances at the solar surface and the appropriate value for an opacity parameter 2 is far from being firmly established. Estimates of the appropriate mean interior 2, which is approximated roughly by the total abundance of elements heavier than He4, are normally obtained on the basis of estimates of spectroscopic abundances near the solar surface. Surface abundances, even if they were known exactly, are not necessarily identical with abundances in the deep interior. To exhibit but a few examples, Cl2 has been converted almost completely into N14 over the inner one-fourth of the sun’s mass (21), while spallation reactions and selective diffusion have possibly affected surface abundances relative to interior abundances. Estimates (51’) of diffusion rates suggest that, at the solar surface, the abundances of all elements heavier than helium may now be smaller by a factor of about 2 than the original surface abundances. Thus, if estimates of surface abundances suggest Z(surface) - 0.015, a value of 2 closer to 0.03 might be appropriate to the interior.

SOLAR

NEUTRINOS

AND

SOLAR

HELIUM

185

Finally, even if the interior heavy element abundances were known exactly, there remain many known sources of large errors in the opacity. A Z estimated from spectroscopic data is therefore at best a crude approximation to the appropriate choice for the opacity parameter Z. Recent investigations by Watson (52) dramatize the fact that, for any given choice of element abundances, many uncertainties remain in the specification of the opacity. Watson has shown that, over most of the solar interior, the inclusion of photon induced transitions between bound and autoionizing atomic levels increases the opacity by 5-20 per cent relative to that obtained without the inclusion of autoionizing lines. He has also shown that the use of the most recent line breadths for bound-bound transitions increases the opacity in many parts of the solar interior by an additional 5-10 per cent relative to that calculated with the CoxStewart choice of line breadth. Finally, he has pointed out that since, in many regions of the solar interior, the wave Iength of a typical photon is large compared to the Debye length, the contribution of electron scattering to the opacity may be quite different from that customarily employed. Carson, Hollingsworth, Mayers, and Stibbs (52’) have shown that, due to the uncertainty in determining the position of absorption edges, the contribution of bound-free absorption may be uncertain by as much as fifty percent over much of the sun’s interior. Because of the demonstrated incompleteness and uncertainty in the opacity, it is to be cautioned that the near independence of the R-Y relationship to the approximation chosen for the opacity will not persist if the sun possesses a convective core. it is conceivable that, near the sun’s center, the correct solar opacity varies sufficiently slowly with temperature and density that energy is transferred predominantly by convection over a finite region about the center. Shaviv and Salpeter (53) have shown that mixing in a central adiabatic core on a time scale large compared to the lifetime of a Be7 nucleus (4-6 months) will lead to an increase in C ui&, all other things being equal. Iben (54) has shown that if the mixing time in the adiabatic core is small compared to the Be7 lifetime, the net effect is a reduction in C ai& . A careful study of the possibility of a convective core and of convective mixing time scales is called for. The sensitivity of results to slight differences in the choice of the equation of state is displayed in Figure 6, where the curve marked d(l/p) = -0.01 results when: t, = 4+, no mixing is allowed, and Sij = Sio, for all i and j; but when the quantity p-l in Pg,, = p(M;;%T) p-l is reduced by 0.01. A reduction in p-l acts qualitatively in the same way as (a) including screening, (b) reducing the exclusion-principle contribution to the electron pressure, and/or (c) slightly increasing the mean molecular weight for a given composition. An increase in p-l acts in the same way as the inclusion of large-scale magnetic fields. Under solar interior conditions, the reduction in pressure due to electron screening is much less than the reduction corresponding to O(p-l) N &O.Ol.

186

IBEN

The neglect of degeneracy and differences among authors in the dependence of the mean molecular weight on composition are, however, comparable to a change on the order of O(,&) - 10.01. Any positive contribution to the pressure over and above that due to the thermal motions of hydrogen and helium nuclei and associated electrons acts in the same way as an increase in p-l and thus has the effect of reducing C u& for any given initial Y. Another way of stating this is: for a given value of C a& , additional sources of pressure must be compensated for by a reduction in the number of particles per gram, i.e., by an increase in Y. A large-scale magnetic field could contribute significantly to the pressure at the solar center if H center - lo9 gauss. Then $(Hzenter/8n) - & Pcenter . Assuming that magnetic field strength and density are related by H = const x p2j3, then a surface magnetic field strength of -200 gauss would occur in photospheric layers at a density of lO-s gm cm-3. Since density and temperature in the sun are related roughly by p - const T3, the ratio of magnetic pressure to gas pressure would be nearly constant throughout the sun if indeed H cc p213.From Figure 6 it may be inferred that, for Sii = Sz’j , ts = 4+, and no mixing, a large-scale magnetic field of the strength chosen for illustration might increase the upper limit on Y, for R < 1, from -0.17 to-O.21 and increase other upper limits by comparable amounts. Whether or not such large fields could occur in the sun has not been investigated. IX.

SUMMARY

AND

FURTHER

COMMENT

If one accepts the Davis, Harmer, and Hoffman limit on solar neutrino fluxes and adopts conventional assumptions concerning the sun-initial homogeneity, no subsequent mixing, 44 billion year age, current estimates of cross section factors, etc.-then the sun’s initial helium abundance lies in the range of 0.16-0.17. Since a lower limit on the sun’s helium abundance is in the range 0.15-0.18, even a relatively modest reduction in the upper limit on solar neutrino fluxes would require modification of one or more of the conventional assumptions. If one believes that the sun’s initial helium abundance is comparable with values estimated for other galactic objects (Y - 0.25-0.30), then even the present upper limit on neutrino fluxes demands that one or more of the conventional assumptions be modified. The most prosaic solution of the dilemma is to suppose that standard extrapolation procedures fail to give the correct result for one or more of the relevant nuclear cross section factors. That is, the difficulty may rest with the assumption that cross section factors are essentially constant over the entire range from typical interaction energies in the solar interior to energies one hundred times larger, where ,experimental data exist. Since, for almost every calculated model, neutrinos from the B* decay provide the major contribution to C u& , the most

SOLAR

NEUTRINOS

AND

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HELIUM

187

likely candidates for modification are those cross section factors that determine the formation rates of Be7 from He3 and He4 and of B* from Be7 and H1. On the supposition that one surprising change is more likely than two, one might guess that a considerable reduction in S,, may be the eventual solution of the puzzle. The calculated cross section for the absorption of BE neutrinos by C13’ could also be in error. Other possibilities include any modifications of the conventional assumptions that lead to a more chemically homogeneous present sun. Several variants are: (a) The sun is considerably younger than 44 billion years. (b) Spin-down or meridional currents mix matter from regions where the conversion of H1 into HeA is actively occurring into regions which are thermonuclearly inactive. (c) The nebula out of which the sun was presumably born was characterized by pronounced concentration gradients, He” being more concentrated in peripheral areas and H1 being concentrated more toward the center of the forming protosun. Most of these possibilities seem rather unlikely: (a) If anything. the sun is probably older than 49 billion years. (b) A rotation rate high enough to cause a substantial mixing by meridional currents on a time scale comparable with the sun’s age would betray itself through an oblateness far in excess of observational estimates. (c) Fractionation and diffusion might be expected to build up a higher concentration of He4 in central regions of the protosun, provided time were available and conditions were right for these processes to occur. On the other hand, we have minimal knowledge of the prior history of the matter which formed the initial solar nebula. Matter in clouds forming the nebula could exhibit large fluctuations in composition that could have persisted throughout the pre-nuclear burning phase of the sun, especially if the contraction to the first opaque stage occurred on a dynamic time scale and if the completely convective Hayashi phase were by-passed (55). The efficacy of spin-down currents in mixing core and surface material is as yet unknown. Spin-down mixing can therefore not be summarily dismissed. The curves in Figure 9 summarize what might be accomplished by altering simultaneously several center of mass cross section factors and by assuming varying degrees of mixing. All curves in Figure 9 pertain to S;, :~:- Stj and again, an unmixed model of age t,,&less than 49 billion years is roughly equivalent to a partially mixed model with t = 49 . The degree of mixing is greater, the larger the departure of t,,( from 4: The only uncertainty in the equation of state that appears capable of producing a significant reduction in C oiCi for a given Y is that associated with the presence or absence of large-scale internal magnetic fields. An average field strength which drops off (according to the two-thirds power of the density) from IO9 gauss at the center to 200 gauss near the surface would lead to an increase of about 0.04 in all upper limits on Y.

188

IBEN

The consequences of assuming a time-varying gravitational constant have been discussed from time to time in the literature. It has been shown (5) that, if the gravitational constant increases with time, then C ai& is larger than if the gravitational constant is maintained fixed at its present value. Thus, the assumption of a decreasing G means a reduction in upper limits on Y. Since, with conventional choices for cross section factors, etc., the Davis ef al. limit leads to an upper limit on Y that is already dangerously close to the lower limit, the assumption of an appreciable decrease in G over the past 44 billion years would be difficult to support. Then, again, a G that increases with time is in principle no less acceptable than a G that decreases with time. Those not averse to time changes in fundamental constants might therefore find evidence here favoring the idea of a gravitational constant that increases with time. 4

,,,,

,I,,

I,,,

(~d)SOLAR

MODEL

(x0+)DA”,S

et 01 LIMIT

,,,.I

,,I[

Y FIG. 9. The relationship between R and Y as a function of assumed solar age when &, = Sf, . The value of R for any given Y and t is appropriate also for an older star of age t,, that has been partially mixed but has the same current surface value of Y. The degree of homogeneity increases with increasing values of (tm - t)/tm .

It is, of course, entirely possible that the sun’s initial average He4 abundance really is in the neighborhood of 0.16-0.17. It is interesting that, during periods of a quiet sun, measurements of the He4 to H1 ratio in the solar wind yield values of Y that are not too different from 0.16-0.17. Neugebauer and Snyder (56), Hundhausen et al. (57), and Ogilvie, Burlaga, and Wilkerson (58) quote helium

SOLAR

NEUTRINOS

AND

SOLAR

HELIUM

189

to hydrogen number ratios of 0.046, 0.042, and 0.02 to 0.05, respectively. These are equivalent to Y - 0.155, 0.144, and 0.074 to 0.167 for matter in the solar wind during a quiet sun. On the other hand, during magnetic storms, the Y for matter in the solar wind has risen as high as 0.3 to 0.4. The helium to hydrogen ratio found by Ogilvie et al. (58) during one storm yields Y = 0.37 f 0.03. Is it conceivable that, during quiet times, the low He4 to H1 ratio in the solar wind is the result either of (a) a steady state abundance of He4 in the corona that is lower than the subsurface abundance or of(b) an accelerating mechanism in the corona that evaporates hydrogen more readily than helium; whereas, during magnetic storms, the high ratio of He4 to H1 in the solar wind is the result of an accelerating process that throws matter in bulk out from subsurface layers without discriminating as to atomic weight or as to charge-to-mass ratios? The high He4 to H1 ratio could then be interpreted as a true measure of subsurface composition. There would then be no question but that, in order to achieve consistency with the Davis et al. limit, at least one of the conventional assumptions regarding the sun and solar input physics must be significantly modified.

X. SOME FURTHER

DETAILS OF COMMENTS

INTERIOR ON THE

STRUCTURE AND MIXING QUESTION

The primary reason for studying the limitation that a neutrino counting rate places on solar input parameters is to obtain greater insight into the internal structure of the sun. There are many general features of interior structure that are quite insensitive to variations in input parameters and input physics over even much wider ranges than have been considered here and elsewhere. These general features are worth reiterating in the context of a model which produces neutrino fluxes that are at least within shooting distance of the Davis, Harmer, and Hoffman limit. Figures 10 and 11 describe conditions in a 49 x 10” year old unmixed sun for which Y = 0.25 and Sij = S,! . The calculated value of C ~(4~ for this model is only 25 per cent larger than the current experimental limit. With two exceptions, all variables in Figures 10 and 11 are scaled so that the maximum value of each variable in the sun corresponds to unity. One exception is the He4 abundance variable in Figure 11 that approaches 0.57 at the center and approaches Y = 0.25 at the surface. The other exception is the Cl3 abundance variable that is on the same scale as the Cl2 abundance variable. The horizontal variable in both figures is the mass contained between the center of the sun and the point in question. One remarkable feature of solar structure that has been known for many years is illustrated in Figure 10. To a good approximation, density p and temperature T

190

IBEN

are related over most of the sun’s interior by p/T3 = constant. The particular value of the constant varies with the choice of opacity and with other input parameters, but, in a given model sun, density varies rather accurately as the cube of the temperature. The approximate constancy of p/T3 is a general characteristic of the radiative portions of all main sequence stars. It is interesting that p/T3 = pT/T4 expresses the ratio of two energy densities-the kinetic energy density of material particles relative to the energy density of the radiation field in equilibrium with this matter. It is tempting to suppose that the empirical theorem might be a derivable general property of those portions of stars that communicate by radiative diffusion, just as the constancy of p/T3j2 in regions that communicate by convective motions is a consequence of the fact that convection tends to smooth out differences in the entropy per unit mass.

0.0

01

0.2

03

0.4 M/MO

0.5

0.6

07

FIG. 10. Reaction rates as a function of mass fraction in an unmixed solar model of age 4+ x lo9 yr. Cross section factors ,Sij have been used and the initial He4 abundance has been chosen as Y = 0.25. The temperature and the cube root of the density are also shown. Each variable is scaled in such a way that the maximum of that variable is unity. In conventional units, maximum values are as follows: central energy generation rate via pp reactions = 14.3 erg gm-1 se+ (compared to an energy generation rate by CN cycle reactions = 1.1 erg gm-l se&); central p--p reaction rate = 5.93 X lo6 gm-l set-‘; central HeS (HeP, r) Be’ reaction rate = 9.19 x lo* gm-1 set-I; central Be’ (p, y)B8 reaction rate = 18.9 gm-l se+. For completeness, central CN cycle rate = 2.75 x 10’ gm-l set-l. Central temperature = 15.23 x 10’ OK, central density = 148.4 gm cm-3.

SOLAR

NEUTRINOS

AND

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HELIUM

191

From Figure 10, one sees that energy generation by the pp chains (E,,) is appreciable over the inner one-half of the sun’s mass. For the particular set of input parameters chosen, the final reaction in the formation of He” is predominantly the He3(He3, 2~) He4 reaction, so that the curve for epp represents, to a good approximation, the rate of thepp reaction. Additional curves indicate that neutrino production by both Be7 electron capture and by the reaction B*(#~+v) B*. is much more concentrated toward the center than is energy production.

FIG. 11. The abundances by mass of He3, He4, Be7, P, CL3 and N” as a function of mass fraction in the unmixed solar model partially described in the caption to Fig. 10. The He’ abundance varies from the initial value of Y = 0.25 near the surface to 0.57 at the center. All other variables but C’” are scaled so that the maximum of that variable is unity. Maximum values are: He3 abundance 1 3.06 x 10-3; Be’ abundance = 9.41 x 10-l*; P abundance = 3.14 :, lo-“; and Nl’ abundance : 5.10 x 1OM3.Cl3 is on the same scale as Cl”.

Pertinent composition variables are shown in Figure 11. CE has been converted almost completely into N14over the central portions of the sun’s mass.The second rise in the N14abundance parameter toward the very center indicates that some 016 has been converted into N14 near the center in 46 x log years. Between the sun’s center and the maximum for the He3 abundance, He3 is locally in equilibrium between creating and destroying reactions. Beyond the maximum, He3 has not yet

192

IBEN

reached equilibrium values in 49 x lo9 years. Similarly, the peak in the Cl3 abundance distribution marks the point below which Cl3 is in equilibrium and beyond which it has not yet reached equilibrium. The general features of the distributions of composition variables represented in Figure 11 are common to all unmixed models of the sun. In approximately the inner 20 per cent of the unmixed model sun, all CN cycle elements have attained their equilibrium abundances. In particular, the number ratio of Nr4 to Cl2 varies from about 200 at the center to about 400 at the edge of the equilibrium region and the number ratio of Cl2 to Cl3 is approximately 4 over the entire equilibrium region. The lifetime of a Cl2 nucleus against conversion into a Cl3 nucleus and thence into an Nr4 nucleus varies from about IO6 years at the center to about 7 x lo8 years at a mass fraction of about 0.2. If large scale mixing currents were present in the sun, Cl2 would continually be fed into the central equilibrium region where it would be converted almost completely into N14. Additional quantities of Cl3 would be formed in the C12-C13 transition region centered at a mass fraction of about 0.28. At the same time, Cl3 and N14 would be swept out of the equilibrium region and out of the C12-C13 transition region toward the surface, so that the ratio of Cl2 to N14 and the ratio of Cl2 to Cl3 at the surface would continually decrease with time. In the unmixed model isolated for reference, the total number of Cl2 nuclei (averaged over the entire model) is approximately 40 times the total number of Cl3 nuclei. If matter were thoroughly mixed over the sun on a time scale comparable to the sun’s age, one might expect the ratio of Cl2 to Cl3 at the surface of the present sun to be no larger than 40. If mixing were to occur on a time scale smaller than lo’-lOa year, one might expect a surface ratio approaching the limiting value of about 4. Spectroscopic estimates of the Cl2 to Cl3 ratio at the solar surface give lower limits for the ratio of 36(59) and 40(60). One might conclude that overall mixing, if it has occurred at a steady rate, has proceeded on a time scale no shorter than 5 x lo9 year. Further information on mixing may be derived if one knows the surface ratio of Cl2 to N14 in the sun. Whatever the primordial ratio of Cl2 to N14 in the sun, nuclear transformations have converted at least 20 per cent of the initial Cl2 almost completely into N14 in the first one billion years of the sun’s life. If at least one complete “turnover” has occurred in 4& billion years, the present surface ratio of Cl2 to N1* must therefore be less than 4. Five complete turnovers, spread over 46 x IO9 years, would lead to a surface ratio less than -(0.8)5/{0.20[1

+ 0.80 + (0.80)2 + (0.80)3 + (0.80)4] - 0.5.

Since the observed ratio is near 4(61) or 5.5(62), one may conclude again that

SOLAR

NEUTRINOS

AND

SOLAR

HELIUM

193

194

IBEN

large scale mixing currents have not succeeded in mixing matter thoroughly throughout the sun more than once since the sun reached the main sequence. At the same time, however, one may speculate from the observed surface ratio that some mixing between core and surface may have occurred, perhaps during the first $ billion years of the sun’s life when it may have been spinning much more rapidly than at present. Studies of more advanced stages of stellar evolution have shown that most of the N14 built up in stars during main sequence phase is converted into Ne22 by the reactions N14(~, r) Fls(/3+v) Ol*(,, r) Ne22. This occurs just before the dominant phase of core helium burning during which Cl2 and 016 are synthesized in roughly equal numbers. If, then, the initial solar system abundances of the heavy elements were the result of processing through the interiors of a previous generation of stars, one might expect the initial Cl2 to N14 ratio to be much larger than that observed at the solar surface. In order to explain the observed ratio, one could invoke either a mild degree of continuous mixing, or a very rapid mixing that lasted for perhaps 4 x log year. Further details of the Y = 0.25 unmixed model are given in Tables 3-5. Results for two additional models are also given for comparison. The “flux” $(33) in TABLE

4

NEUTR~NOFLUXESATTHEE~~DUETOREACTIONSINSOLARMODELSWHEN~~ Y

tiPI

“9(33)”

MW

0.20

6.38(10)

3.12(10)

1.31(9)

7.04(5)

KW

0.25

6.25(10)

3.02(10)

2.08(9)

1.81(6)

0.30

5.96(10)

2.82(10)

3.21(9)

4.62(6)

+ Units

of 4 are cm-% set-I.

INDIVIDUAL

CONTRIBUTIONS

TABLE TOCU& MODELS

= 4&S,,

= Sij.O

d(W

w15)

W”)

6.90(7)

6.44(5)

3.99(S)

4.64(7) 3.24(S)

1.37(9)

1.24(9)

1.88(7)

4.77(6)

5

RESULTING FROM ~-DECAYS OF Be’, OF THE SUN WITH & = .S$

Be, N13, AND 016 IN

0.20

0.125

0.320

0.00493

0.0122

0.25

0.198

0.823

0.0285

0.0853

0.562 1.23

0.30

0.306

2.10

0.098

0.326

2.93

Table 4 is a measure of the relative frequency of He3(He3, 2~) He4 reactions and &P’) gives the flux of neutrinos due to the p-decay of Fl’. The increase

SOLAR

NEUTRINOS

AND

SOLAR

HELIUM

195

in the central He4 abundance is shown in Table 3 to be approximately

independent of the initial He4 abundance, a fact which implies that the prior luminosity history of the sun is independent of the choice of initial Y. Finally, as indicated by the fluxes +(N13) and +(015), the extent to which Cl2 and N1” have reached equilibrium abundances in energy producing regions is greater, the greater Y. The total fraction of initial Cl2 converted into N14 over 4$- :A’ 10” year of unmixed evolution is 21, 24, and 27 per cent for Y = 0.20, 0.25, and 0.30, respectively. One complete turnover of solar matter would therefore lead to a surface ratio of Cl* to N1” less than 3.8, 3. I, or 2.7 for the three cases.

XI. AN ANALYTIC APPROXIMATION RELATING STRUCTURE PARAMETERS TO NEUTRINO FLUXES

All results in this paper have been obtained after rather involved computations to achieve solutions of detailed structure equations. It is understandable that the general reader might experience dissatisfaction with exclusive reliance on detailed computational results. It is therefore worthwhile to demonstrate that the main features of the relationship between interior structure parameters and neutrino fluxes can be revealed by means of a simple model. Let us suppose that all of the intermediate elements in the ~JI chains and in the CN cycle are locally in equilibrium at all points in the sun. Let us call rrj = RijXjX, the rate (per gram per second) at which a reaction between nuclear species i and j occur. Here X, is the abundance by mass of the ith nuclear species and i = I(Hl), 3(He3), 4(He4), 7(Be7), 14(N14), e (free electrons per nucleon), etc. The assumption of local equilibrium means that rI1 = 2r,, + rztl , r31 m=r17 -+ re7 , rllz = rll, , etc. Defining 01= rQl/rll, /I = rI14/yll , y = r17/r,7. 6 == a(0.9573 I 0.4574y)( 1 7 ~1. I. we can express the nuclear energy output of the sun as L, = En9 s ‘+ r,,(l 0

+ 6) dM + 6C.J i’ltf -’ rl14 d.W, ‘0

(2)

where the integrals are over the solar mass distribution. Here epl, = 13.10 MeV is the energy liberated per p-p reaction if the pp chain is terminated exclusively by the He3 + He3 -+ He4 + 2p reaction and EcN = 25.03 MeV = 1.91~,,~ is the energy liberated in one complete CN-cycle. Defining an average of any quantity Q over rll dM by (Q> = \ Qr,, dM + I rll dM, we can rewrite

(3)

Eq. (2) as

Lo= %w srll

dM[l

+ (S) + 1.91@?].

(4)

196

IBEN

The flux of neutrinos at the earth due to pp reactions in the sun can then be written as $(PP) = (4vD2)-l

1 ~II dM

= 6.44 x lOlo cm-2 set-l [l + (6) + 1.91(p)]-1,

(5)

where D = 1.5 x 1013 is the mean distance between the earth and the sun. Neutrino fluxes due to other reactions in the sun can now be written as +(Be’) 4W w15)

= (41

+ 9)

= (~(1

+(PP),

+ Y)-9

NPP),

(6)

= dW31 Gz 4(PP).

Inserting these expression into Eq. (1) of Section II gives 1 u&

= [6.13(01(1 + r)-I)

+ 29300(&l

+ y)-l)

+ 21.%/DlU + (6) + 1.91<8)1-1,

(7)

where +(ppe-) has been neglected and C u& is in units of 3 x 1O-36set-l. From all of the models, one has y Q 1, so that, to a very good approximation, #Be’) = (4 &PP), 0’) = CY~ $(PP), and (6) = 0.9572(a). Accepting the current experimental limit on x u& , we conclude from Eq. (7) that ~(Be7)14(w) Ws)l+(~~) $W5)l$(~~)

= () < l/5.17 = (
< l/29300

C-9

= () < V9.6

These are general results that have required no assumptions other than that of local equilibrium. Further progress requires an awareness of the dependence of 01,p, and y on local conditions in the sun. When all Sij = Sij , one has /I = (X1,/X,) exp(49.005 - 118.491T;1/3 + $S,), y = (2X,/l

+ X1) T;1/6 exp(36.749 -

102.639T;1/3 + So), (9)

01= h{[l + (X/2)“]‘/” - h/2}, h = (2X,/X,)

exp(18.080 - 49.994T;1/3 + $S,),

where So g 0.752(~/T,2)l/~

(Q + Xl/2)lj2

(10)

SOLAR

NJZJTRINOS

AND

SOLAR

197

HELIUM

is an approximation to the Salpeter (36) screening factor. When (A/2)2 is less than one, as it is in all models that satisfy the experimental limit on C u&, it is permissible to approximate 01by a g A(1 - +A + &X2 - a**)

(11)

For our purposes, it is useful to convert the expressions in Eq. (9) into power law approximations. We have p Eg 4.50(X1,/X,)( T/To)15.94 y E 7.34 x 10-3(2X,/l + Xl)(T/TO)13.63 h GS 0.267(X,/X,)(T/To)6.71,

(12)

where T, = 15.23 x IO6 “K and S, has, for simplicity, been standardized at 0.20. It is apparent that all quantities to be averaged are, to a good approximation, independent of density but very sensitive to temperature. The major contribution to the averages (Q) will therefore come from near the solar center and we might expect to find approximations of the sort
approximations

to the AB based on the fact

that the quantity p/T3 is nearly constant throughout all solar models. It is quicker to make use of model results already available. Results shown in Table 6 permit us to approximate (a) - 0.21cY.C)

and

- 0.11ac,

- 0.43yc (a) - o.o9oy,cX, .

64

We may now write: 1 ui$i s [1.2901, + 264Oyc(uc + 2.26/I,] (13)

f [I + 0.2001, + 0.21&] TABLE AVERAGES

Y



RELATIVE

- UC

TO CENTRAL

6

VALUES

- Ye

Q,

FOR MODELS

(Yd Ye<4 ~~~.-.__-.___

WITH

Srr = S$ .

Bc ___----

0.20

0.218

0.0932

0.428

0.122

0.25

0.210 0.201

0.0910

0.433

0.112

0.0831

0.413

0.0916

0.30

-

198

IBEN

Our final task is to obtain an approximation to composition variables at the solar center. It has already been demonstrated that, for models of 49 x lo9 yr. old, unmixed suns, 1, s Y + Q at the center. It has also been shown that, in the inner one-fifth of the sun, Cl2 has been converted almost completely into N14, whereas very little conversion of 016 into N14 has occurred. For the particular choice of heavy element abundances used here, this means that, at the center, x14 - (0.272). Then, ignoring the effect of Z in the equation of state, so that Xl + x4 - 1 at the center, we may write Ac g 0.267(Y + $)(8 po z 1.21Z(Q -

Y)-l (Tc/To)6.’ S34(SllS33)-1/2

Q-l (Tc/Tp~”

ycX, = 3.92 x 10-3(Y + &)(j -

(14)

&4/S,,)

Y)-’ (TJT,)2o.3 S34S1,(S11S33)-1/2S;

where the Sij are in units of the S$ . Our final requirement is an approximation to the relationship between Tc and Y. From the models with Sej = Stj , one finds that Tc g 0.84~,(GM,M,/kR,)

(15)

z 19.4~~ x lo6 “K when pc , the molecular weight at the center, is approximated p,l g 1.583 -

1.25Y

by (16)

For fixed Y and S,,, , the greatest change in central temperature occurs when S,, is altered. Empirically one finds T,‘S,, - const. This result may be simply understood. When the CN cycle provides a negligible contribution to the total energy output, one has L, a Jr,, dM(1 + (6)). To a first approximation, when Y is fixed, s

rll dM -

const pcTc4S,, .

A general consequence of hydrostatic equilibrium when Y is fixed is that const. Hence, for low enough Y that (6) may be neglected, T,‘S,, - const. Thus, we may set

pc/Tc3 -

T, g (19.8@;,~‘)

x lo6 “K,

(18)

where S,, is in units of Si, . When the approximations represented by Eq. (14), (16), and (18) are inserted, Eq. (13) reproduces the detailed model results to roughly 30 percent. More

SOLAR

NEUTRINOS

AND

SOLAR

HELIUM

199

importantly, the analytic approximation gives quick insight into the relationship between internal structure parameters and neutrino fluxes. For example, it is clear that, for a fixed set of cross section factors, C o,c$; increases: (a) with Y, when age is fixed; (b) with age, when Y is fixed; and (c) with a decreasing degree of mixing, for fixed Y and fixed age. In each of these cases. central temperatures are increased because of an increase in the mean molecular weight near the center. The relative insensitivity of C ai& to the opacity law, for a fixed Y, is also clarified by the analytic approximation, which contains no explicit reference to the parameter 2 as it applies to the opacity. One basic reason for the lack of a strong dependence on opacity is the fact that, in all cases, pressure/ (density)4i3 E constant over the central regions that contribute most strongly to the pertinent averages (p’). Perhaps the major reason is that mean interior temperatures are set primarily by the choice of molecular weight (which is very little affected by changes in the parameter Z); whatever the choice of opacity law. the parameter Z must perforce be chosen to achieve the distribution in temperature that leads to the correct luminosity after 44 billion years. without altering the mean temperatures set by the choice of initial Y. The sensitivity of 2 ~~4~ to changes in the equation of state, as, for example, the change due to the inclusion of internal magnetic fields, is abundantly clear from the high powers of T, that appear in the approximation to x CJ~+> Qualitatively, one may write (T, + T,,,) CCpC where T,,, is a “magnetic temperature”. For fixed pLc, the larger T,, , the smaller T, and the smaller C ffi+i . Similarly, for fixed Y, errors in the equation of state are reflected by errors in the effective value of pC , errors that are then propagated to a high power in x G& . XII.

WHAT

OF

THE

PAST

AND

OF

THE

FUTURE

No story of the sun would be complete without a description of the sun’s probable past and future, long though these may be relative to man’s probable past and future. Ideally, one would like to know how sensitive solar history is to variations in all input parameters for the present sun. Attention will be confined here to the effect of varying Y. Figure 12 contains evolutionary tracks in the Hertzsprung-Russell diagram for three models when S/j = Sij and Y = 0.20, 0.25, and 0.30. All models have been normalized to pass through the sun’s present location after 4+ x IO9 year. This has been accomplished by adjusting the opacity parameter Z (to obtain the right luminosity) and adjusting the convection parameter Z/H (to obtain the correct radius). The parameter I/H is the ratio of mixing length to density scale height employed in the standard treatment of envelope convection. lt has been given the values 0.065, 0.36, and 0.45 for Y = 0.20, 0.25, and 0.30, respectively. Several other characteristics of the three model suns have been given in Tables 3-5.

200

t-

‘\

86-F

‘3, -

-

NOW

MASSACHUSETTS

IN THE

“WINTER”

‘. \

(8.5)

(123)

,Combrdqe

6000-K

‘\.

IllLIIl,, 3.80

IN

3.75

50000K

‘\

‘,\ J I,,, \ \.( 3.70

3.65

40000K

I,,,

11,

L

360

FIG. 12. Evolutionary tracks in the Hertzsprung-Russell diagram for models which mimic the sun’s size and power output after 4~3 x lO@years of evolution. The sun’s surface temperature T. is in degrees Kelvin and luminosity is in units of the sun’s present luminosity. Estimates of terrestial temperatures assume that T&s is proportional to the sun’s luminosity. Numbers in parentheses give the CP neutrino absorption rate relative to the current experimental limit for the Y = 0.25 case. Lines of constant radius are inserted for reference. Numbers beside circles along each track give ages in billions of years.

SOLAR

NEUTRINOS

AND

SOLAR

HELIUM

201

It is apparent from Figure 12 that the previous history of the sun, insofar as it has affected the evolution of terrestrial life, is quite independent of the value of Y that we eventually assign to the sun. From additional model results one can show that the sun’s prior history is also essentially invariant to the choices for the Sij (within “acceptable” limits). The past luminosity history is also invariant to whether or not mixing has taken place. We can therefore conclude that, over the past 42 ;i log yr, the sun’s surface temperature has increased by -225 “K, and the sun’s luminosity has increased by a factor of -1.4. The wavelength at which the peak in the sun’s surface emission occurs has increased by -200 a. If the early earth had not been internally heated by Potassium 40 decay and if it had possessed an atmosphere somewhat similar to that which covers it now, the mean temperature at the latitude of Cambridge, New England. would have been -22” F, some 44 x lo9 yr. ago. Minor divergences occur among models of the future sun. Hydrogen is exhausted sooner at the center of the model with a smaller hydrogen abundance to begin with. so that the main sequence lifetime of the sun is smaller, the larger the initial Y. It is interesting that, when the sun’s luminosity has increased sufficiently so that the mean temperature in Cambridge is roughly the present mean temperature in Pasadena, California (and average Pasadena temperatures approach present day maximum temperatures in Needles), the neutrino absorption rate by Cl37 will have increased by a factor of about seven over that which presumably occurs now. One might wish that man’s appearance had been delayed by several billion years. The maximum luminosity and maximum radius of the sun during its major nuclear burning lifetime will occur when the 3 Hea --, Cl” reactions are ignited at the red giant tip. One can estimate the maximum luminosity at log(L/L,)

-

2.8-3.2.

One can further estimate, for the Y = 0.30 case, that the sun’s surface temperature and radius will be given by (7’JRGT - (3200-3350) “K, and RRGT - (75-130) R 1 at the red giant tip. For Y = 0.20, one may estimate (7’JRor Y (3040-3180) ‘K and RRGT - (83-144) R, . Since the distance from the sun’s center to the earth is 216 R,, the earth will not be engulfed by the sun at its maximum extent. But, sincethe earth’s surface, if it absorbsand radiates on the average like a black body, will have reached a temperature of (1500-1890) “K, mountain ranges on the earth will be transitory waves and bubbles in a seaof molten lava. RECEIVED:

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SOLAR

NEUTRINOS

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HELIUM

203

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H. PINSON,

JR., C. C. SCHNETZLER,

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Geochemica and Cosmochemica Acta, Pergamon

42.

43. 44. 45. 46. 47. 48. 49. 50. 51.

51’. 52. 52’.

53. 54. 55. 56. 57. 58.

5Y.

60. 61. 62.