K CONSTRUCTION (3F STANDARD SOLAR MODEIS

One does nőt need a detailed model.of the internál constitution o f the sün to predict the to tál number of solar neutrinos reaching the earth per cm2 per sec. In fact, this to tá l number is 2 h j(25 MeV • Un

(1 au)2) - 101 1 cm’ 2 sec”1, since on the average the basic fusion reac

tion Up -* a + 2 c+ + 2v releases ~ 25 MeV. However, the proposed methods

of detection recocrd only a portion of the neutrino energy spectrum, which ranges trcm z e ro to more than 10 MeV, and the cross section fór detection

depends strongly on the neutrino energy. Moreover the form o f the neu

trin o energy spectrum is a se n sitive function o f temperature, den sity,

and composition ( e . g . , the .production ra te o f the important high-energy neutrinos from B decay, f i r s t discussed by Fowler (1958) and Cameron8

(l9 p S ), va ries roughly as T

13

) . Thus in the pást ten years a considerable

e f f ő rt has gone in tő the construction o f accurate so la r models and the coraputation o f the neutrino energy spectra they imply.

Table 2 iü u s tr a te s the main process cu rren tly b e lie v e d to conrvert H to He in the sün, the protcn-proton chain. The form o f tha neutrino

energy spectrum depends on the r e la tiv e frequencies o f the three branches

by vhich the chain is cccpieted and a d e ta iled solar model is required to determine the branchlng r a t io s .

Ve s h a ll b r i e f l y desc^ibe the construction o f solar models because the present disagreement betveen calculations and observation may de- pend on one or more assuaptions or parameters o f the models. The stan

dard theory o f main-sequence stars (Schvarsschild I958, Cox fi: G iu li 1963, Chiu 19-53) assumes, at each point in the s ta r, ( i ) hydrostatic equilibrium betvreen g ra v ita tio n a l fo rce and pressure gradient (r a d ia l,

fór a spherically symmetric s t a r), ( i i ) energy transport "by radiation and convection, and ( i i i ) energy production by bydrogen burning. These assumptions may be stated as four first-ord er ordinary d iffe re n tia l equations, together with appropriate boundary conditions and constitu

tive relation s. The latter include ( i ) the equation o f state which connects prescure, density, and temperature; ( i i ) the radiative opacity (which is a function o f chemical composition); and ( i i i ) the nuclear

energy-generation rate, involving the cross sections fó r the several reactions (see Cleyton 19S8 fór a clear and a thorough treatment txf

the nuclear reaction processes). As described by Schwarzschild (1958,

pp. 96-8) the only data required to obtain a solar model are the to tál mass and the distribution of chemical composition throughout the star.

We forego a deocription of the standard numerical techniques (see, e .g «, Sears & Brownlee 19^5» Kippenhahn, Weigert Se Hofmeister 19^7); the re

sults that emerge from the computer include the march of physical variables throughout the star, the to tál radius and luminosity (ergs per second), the distribution of the energy-production processes, and

the central temperature. The calculated neutrino spectrum is

detér-mined by numerical in tegratio n crver masa ^bI^{ís ü b} .

In the case o f the sün the t o t á l mass is accurately known ( I.989 x 10

33

g ) bút the d is trib u tio n o f chemical ccmposition is nőt knovn a

p r io r i. The conventional assumption in most recent vork. has been to

adopt a homogeneoua i n i t i a l composition and then to construct an evolu- tio n a ry sequence o f models (Schvrarzschild 1958, pp. 98-IO O ). In any one evolved model the helium/hydrogen r a t io is g rea te st a t the center

and f a l l s o f f to the primőr d ia i value Beme distance av*ny from the center, v it h the usual asouaption o f no rrbíing. A further dátum requ ired, then,

is the age o f the sün, usually taken as about Ív.7 x 109 years (see

Bahcall & Shaviv I908 fó r r e fe re n c e s ). One constructs an evolution ary

sequence o f a half-dozen or so models, ending v it h "th e" so lar model, vhich is required to have the present-day luminosity a ft e r U.7 x 10 q

y ears. In the case o f the sün a s t r i c t requirement on the computed radius is nőt u seful because the calculated value depends on the uncer**

ta in structure o f the convective envelope; fa rtu n ately th is uncertainty does nőt s ig n ific a n tly a ffe c t the calculated deep in t e r io r structure

(Schvarzochild 19í$> pp. lUlj— ^j Sears 196^ ).

•* ' 0 and Z, respectively the mass fractions of tydrogen, hélium, and heavier

elments (Z » 1-X-Y); thus orúy tvo free parameters appaar. Once values of the composition parameters axe adcpted, a solar model can be ob

tained. I f the luminosity of the model d iffe rs from the observed lumi- noslty then one uoually changes sligh tly one of the composition párá

mé ters—because the calculated luminosity is sensitive to Z via the opacity (roughly, L ~ ^**^) and is sensitivo to X and Y via the opacity

and the mean atomic veight per free p a r iid é in the perfect-gas

equa-7 5 "

tion of state (roughly, )• Hence one approach (Sears 196U, 1966, Bahcall & Shaviv 1963, Bahcall, Bahcall & Ulrich 1969) has been to adopt the value of Z/X given by spectroscopic observations o f the solar photosphere and to pick successive values of Y, computing an evolutionary sequence fór each, u n til a sequence is obtained in vhich the fin a l model reproduces the observed luminosity. Note' that this approach yields Y, the primordial hélium content of the sün, as a theoretical resu lt vhich can be conpared vith chromospheric

determina-38

tio n s, solar cosmic rays, and the prlm ordial abundance from a v a r ie ty o f cosmologies. Another approach, advocated by Iben (19^8, 1 9 6 ^ ), has been to adopt ab i n i t i o a value o f Y th at one b e lie v e s in on the basis

o f othsr astron caical evidence and to vary Z u n til one obtaina a luml- nosity f i t .

III. 2 RESULTS CiF STANDARD SOLAR MODELS III. 2.1 Review o f Re cent Models

The e a rlie st det&llsd so la r model used t o calculate solar neutrino

fluxes vas described by Bahcall, Fowler, Iben & Sears (1963) . Another early model was used by Pochoda & Reeves ( I96U) to estlmate a neutrino

energy spectrum. The f i r s t systeca tic study o f solar models fó r the purpose o f predictin g neutrino fluxes and th e ir u n certain ties was that

of Sears (190*0 • The baslc parameters o f his p referred Model J vere an i n i t i a l conposition parameter z/X » 0.028; the mass (1.939 x 1 0 ^ g );

and the age o f I+.5 x 10^ years. The equation o f sta te was th at o f an ideál gas, including electron-degeneracy pressure; the opacity was based

on the tables o f K e ller & Meyerott (1955); and the energy-generation rate [including the small g ra v ita tio n a l contraction term (Schwarzschild

» T

1958# Eq. 12.10)] involved the pp-chain and the CN cycle, with the cross section factors In the former from Parker, Bahcall & Fowler ( I96U). Aa

described in the preceding section the model was fitte d to the observed Bolar luminosity, 3»90 x 10 33 erg sec” . The resulting i n it ia l hélium

8 8

content waa Y ■ 0.27 and the resulting B neutrino flux was <p( B)

7 - 2 « l

1*9 x 10 cm sec • The principal p aramé tér 0 of this (19

6

^{h )} model are listed in Table 3• Bahcall (l56Ub)adoptcd an average over several o f Sears's models which gavo Cp(^B) *» 2.5 x 10*^, and th is, plus a small

contrlbution from the Be electron-capture neutrinos, led to a predicted 7

number o f neutrino captures at the earth of Z(<p<r) = 36 SNU.

Bolar models obtained by Ezer & Cameron (1965) and by Weymann &

Sears (1965) contained minor improvements, primarily in the use o f Los Alamos opacities (Cox & Stewart 1966, Cox, Stewart & E ilers 1966) .

Since these are lower than the Keller-Meyerott opacities in the deep

in-; .

terio r the resulting hélium contents and neutrino fluxes were slig h tly reduced compared to those of Sears ( I96I+). Bahcall (1966) combined

fluxes he calculated with the aid of the improved solar models and new

7 8

experimental Information on the maos-37 system and on the B e (p ,r) B

croso eection to fin d a neutrino capture ra te o f Z (c p a ) = 30 SNU.

The next major development vas a substantial increase (a fa c to r o f f i v c ) in the reported lov-energy experimental cross-Eection fa c to r fó r the ^líe (^Ke, 2p) ^He rea ctio n in the period 1965-67 (Dwarakanath

& Winklcr 1971* Bacher fis Tcmbrello 19^3; see Tombrello 19&7, Earnes 1971, and Kavanagh 1972 fó r r e v ie v s ). This enhanced the branching

through termination I o f the pp-chain (see Table 2) and lavered the predicted neutrino flu x ; the f i r s t modelo incorporating the név ex

perimental reBults vere reported by Shaviv, Bahcall & F o vle r (1967) . General uncertainty in the lov-energy cross sections a t th at t i me led

Bahcall, Bahcall, F ovler & Shayiv (1968) to construct tvo models based on assumed extreme values fó r four inportant cross-section fa cto rs in the pp-chain. A p referred model, incorporating the be6t parameters then a v a ila b le , vas constructed by Bahcall & Shaviv (1968) , vho found

£(cpo) = 22 SNU. They alsó discussed the e ffe c t s o f Beveral parameter v a r ia tio n s .

Shortly a fterva rd s, in e o rly 1958, tvo név developments stimu- lated furthcr model b u ild in g. A redetermination o f the h a lf l i f e o f

the free neutron by Cbristensen et a l (1967, 1971) and a refinement in the calculation of various nuclear parameters led to an increase in the cross-section factor of the proton-proton reaction (Bahcall & May 1968, 1969)* A redetermination of the solar photospheric chemical camposition,

by Lombért (1967a ,b , 1968) , Laxnbert & Warner (1968a ,b , c ), and Warner (1968) , gave a new lower value of

z/x

« 0.019, thus suggesting a de

crease in the solar interior opacity. Both of these developments inplied

a further reduction in the predicted neutrino flux and new models were obtained by Bahcall, Bahcall & Shaviv (1968) (see alsó Torres-Peimbert, Ulrich & Simpson 1969) • At the same time the f i r s t results froui the Brookhaven experiment became available: Davis, Harmer & Hoffman (1968)

found an observational upper lim it to the solar neutrino capture rate of 2(</?cr) = 3 SNU. This upper lim it was less than that

predicted by the most probable nodel of Bahcall, Bahcall & Shaviv (1968) : 2(<?<r) = 7. 5 SNU.

The model work of the latter authorB was refined by Bahcall, Bahcall

rr 0

& Ulrich (1969) , who incorporatsd reviscd values fór the B e(p ,r) B cross section and fór the ^Be(e",v) \ i . capture rate, both of which tended to

. lover the predicted value. Their (19&9) model (see Table 3) gave a capture rate of Z(^cr) = 6 SNU, only a factor of two above the observa- tional lim it. Their paper gave an extensive discussion -of parameter variations, including effects of pcrturbations on the oapcity and on the equation of state.

Another diocucsion of the f i r s t results of _Daviset al vaa _givenby Iben (₁₉o₈, ₁₉^₉a) vho constructed a large number of models to illufl- trate the parameter dependence of neutrino flu xes. He utillzed _Y,the primőr d ia i heliua content, as the compositicn parameter rather than Z/X, the observed heavy-elenent-to-hydrogén ra tio derived from the pho-

tosphere. Iben cmphasized that consistency vith the experimental re

sults could almost be achieved vith values o f Y considerably belov those determined in other objects in the galaxy. This conclusion is s t i l l possible at the present time although la tér developaents have modified

the argusients in Iben (1969a ) . Kis paper is particu lariy useful be

cause of the extensive illu stration s and physical descriptions of the effects of parameter changeo.

The next tvo dcvelopnenta vere a sűbstantial increace in the Fe

4 2

-abundance in the photo6phere from revised f-valu e determinations (Garz, Holweger, Kock & Richter 1 9 ^ 91 Martiné js-Garcia, Whaling, Mickey &

Lawrence 1971, Wolnik, Berthel & Wares 1971) c f . Ross 1970, Cowley 1970, 1 9 7 1) and the in clu sion o f e le ctro n -co rre la tio n e ffe c t s in tbs opacity

(Diesendorf 1970). Watson (19^9, 1970) pointed out that an increaae by a fa cto r o f 10 in the iron number abundance would lead to a

sübstan-t i a l (~ 3056) increaae in the opacity o f the eolar core and thua an in -crease in the predicted neutrino flu x . The e le ctro n -co rre la tio n e f fe c t s , as Diesendorf noted, tended to decrease the op acity. A név

model incorporating these e ffe c t s vas canputed by Bahcall & U lrich (1971);

see Tahié 3* At the same time név resultB frcm the Brookhaven experiment

became a va ila b le (Davis, Rogers & Radeka 1971): a p o s itiv e d etection o f 37A, 72(<pcr) = 1. 5 ± 1. 0 SNU was suggested. The m odel p re

-d ic tio n by Bahcall & Uilrich (1971), 2(<pcr) = 9. 0 SNU, was g re a te r by a fa cto r o f s ix ; sim ilar resu lts vere reported hy Abrahara & Iben (1971) and by E z e r & C am eron (1971).

III. 2.2 Dependence o f Predicted Neutrino Fluxes on Parameter Values

As noted ahove, the e ffe c t s o f parameter changes have been explored in many papers. We s h a ll nőt give a d eta iled discussion, p a r tly to

avoid repetitlon and partig* because of the posBibility that a ^qualita- tiv ely név development may changé the present situation.

The dcpsndence of tte predicted counting rate in the 37o l experiment on most of the relevant parameters is succinctly illu strated by the re- lation

l&<?)» 1.35 x 10"36 sec’ 1 per 3^C1 atom

The denomlnators ere the paremeter values used by Bahcall, Bahcall &

Ulrich (1969) (see Table 3 ); X^^(O) is the ^B3 electron-capture rate from Bahcall £•. Moeller (l9o9) • Equation(l) vas derived numerically by

Bahcall, Eshcall & Ulrich (19o9) frca a comparison of many models.

Setting the ratios equal to unity gives their standard value, 6.0 SNU (Table 3) ia 19^9• Thü slgns of the exponents give the directions of the dependences vhich are nőt obvioua in a l l cnses (usually because of

the strong influence of the boundary condition that the calculated present lum inosity equal the observed lum inosity; see Section III. 1). M ore detailed discussions of individual dependences are given in a number o f papers (Sears 1964, Bahcall, Cooper & Demarque 1967, Shaviv, Bah

call & F o w ler 1967, Bahcall, Bahcall, Fow ler & Shaviv 1968, Bahcall &

Shaviv 1968, Bahcall, Bahcall & Shaviv 1968, Iben 1968, 1969a, Bahcall, Bahcall & U lrich 1969, Bahcall & U lrich 1971). M ost of these papers have graphs illustracing the dependences.

The various factors in Equation (1) enable one to find the in di--vidual e ffe c t of each of the input param eters on the predicted counting rate. F ó r exam ple, the effect of the fiv e -fo ld in crease in the

3 3 4

H e( He, 2p) He c ro s s -s e c tio n fa ctor, was to reduce the 1964 counting rate by a fa ctor o f 0. 57. Other effects w ere, in order o f decreasing im portance, the d ecrease in Z/X (0.68), the in crease in S^ (0.75), the d ecrease in JL^q (0. 84) and the in crease in Be electron -captu re 7 rate due to bound electron -captu re (Iben, Kalata & Schwartz 1967, Bah

call & M o e lle r 1969) (0.87). The in crease in assumed age led to an increase in predicted counting rate by 1. 06; the in crease in S t by 1.23. M ultiplying all the factors together gives a totál reduction of 0. 3 which is the ratio of Z(<p<r) (1971) to Z(<p<r) (1964) in Table 3. The calculation from Equation (1) does nőt include refinem ents in the equa

tion o f state, which are m inor, or changes in opacity. The gross de

pendence of opacity on Z is o f course included, bút the sp ec ific

in-46

crea se in opacity (hence in (p{ B )) due to the in creased iron abundance (Watson 1970) is nőt; evidently this is nearly balanced by the decrease in in terio r opacities between the K e lle r - M e y e r o tt and Cox-W atson tables

and by the electr on -cor r elation effects. In cid en tally, we note that Bah

call & U lrich (1971) have shown that so la r-m o d e l calculations by d if

feren t w ork ers with the same param eters give the same results to an accuracy of ~ _+ 10 percent.

III. 2. 2 M ost Recent Results

The m ost recent calculations by Bahcall, Huebner, M agee, M erts, and U lrich (1972) have used re v is e d Los Alam os op acities.

The results (nőt yet in final fo rm ) with all other p arem ters having their standard values [Bahcall and U lrich 1971] are: E(<pcr) ^ 6 SNU, with

8 7

4. 2 SNU1 s coming from B neutrinos, 0. 83 fro m Be neutrinos, 0.26

13 15

fro m pép, and 0. 26 from N and O. It appears that the e a r lie r M IT opacity corrections w ere invalid. When m eson exchange effects on the p-p reaction are included [M. G ari and A. H. Huffman (1972); D. O.

Riska (1972)], (meson exchange) ö (1. 09)S (B ah call-M ay 1969)),

we find 2(<p<r) = 4. 35 with 3.15 coming from B neutrinos, 0.74 from 7Be neutrinos, 0. 26 from pép, and 0. 2 fro m 13N and 150 . These latter calculations take intő account the rev ised Los Alam os opacities and estim ated m eson exchange effects fó r the p-p reaction.

We conclude that refinem ents in the input data fór standard solar models are nőt lik ely to elim inate the present la rg e discrepancy be-

tween calculated and observed neutrino fluxes. M ost notable in the

i m p r o v e m e n t s o f the pás t d e c a d e has b een the r e m e a s u r e m e r i t o f the v a r i o u s p p - c h a i n c r o s s s e c t i o n s - - b y now e ac h c r u c i a l r e a c t i o n has been r e m e a s u r e d ( o r r e c a l c u l a t e d ) at l e a s t t w i c e by d i f f e r e n t gr oups w i t h good a g r e e m e n t throughout [ s e e the e x c e l l e n t r e v i e w s by T o m b r e l l o (1967) and K a v a n a g h (1972)].

I I I . 3 N o n - S t a n d a r d S o l a r M o d e l s T h e p r e c e d i n g d i s c u s s i o n has b e e n c o n c e r n e d w i t h standard s o l a r m o d e l s . S i n c e the f i r s t e x p e r i m e n t a l r e s u l t s o f Davi s et al. (1968) r e v e a l e d a d i s c r e p a n c y w i t h s ta nda rd t h e o r y , m a n y s ug g e s t i o ns have been o f f e r e d and d ebat ed. W e l i s t s o m é o f these s u gg es ti on s b e l o w .

T h e p o s s i b i l i t y o f m i x i n g , w hi ch could m a i n t a i n a high amount o f h y d r o g e n in the c e n t e r and thus l o w e r the Central t e m p e r a t u r e c o m - p a r e d to the s t an da r d m o d e l s , was f i r s t s u g g e s t e d by E z e r & C a m e r o n

(1968), who found that c o m p l e t e m i x i n g o v e r the e n t i r e l i f e t i m e o f the

sün could r e d u c e (p ( B ) to o n e - f o u r t h o f the standard r e s u l t ( s e e a l s ó S h a v i v & B e a u d e t 1968, Iben 1968). V a r i o u s a r g u m e n t s ha ve be en g i v e n a ga i n s t the l i k e l i h o o d o f m a i n t a i n i n g such e x t r e m e m i x i n g [ s e e B a h c a l l , B a h c a l l & U l r i c h (1968) and S h a v i v & S a l p e t e r (1968, 1971)]. A l s ó U l r i c h (1969) s h o w e d that a r a p i d l y r o t a t i n g s o l a r c o r e had l i t t l e e f f e c t on the c a l c u l a t e d n e u t r i n o f l u x e s . Iben (1969b, c) i n v e s t i g a t e d the p o s s i b i l i t y that the sün had a c o n v e c t i v e c o r e and c onc lude d this wa s u n l i k e l y to

significantly change the calculated fluxes.

The effect of a large-scale interior solar magnetic fie ld has been explored by Bahcall & UIxich (1971) [c f . Iben (1968^ Abraham and Iben

(1971)],

who find (taking the primordial

z/x

from observation) that the predicted neutrino flux increases.

Diffusion of heavier elements toward the solar center (Allec &

Chapman 1960) would tend to increase the predicted neutrino flu x.

Schatzman^(1969,1970) has suggested turbulent diffusion of ^He, bút the sign of his efxect has been questioned by Shaviv & Salpeter (1971) (who showed i t was small) and by Bahcall & Ulrich (1971).

Kocharov & Starbuncv (1970, a .b ) , pointed out that i f the sün presently contains enough 3Ke so that 3He(3He, 2p) \fe is the major energy source,.the central temperature would be too low to produce

solar neutrinos detectable in the 37C1 experiment. Abraham & Iben (1970) have constructed models to check this suggestion and find that the in i-t i a l mass abundance required is X = Ö.09, which they note is

sub-He

atantially lsrger than the values found in II I I regiona, in meteorites,

and in c o s m o l o g i c a l m o d e l s ( s e e a l s ó U l r i c h 1971).

T h e p o s s i b i l i t y o f a s e c u l a r ( t h e r m a l ) i n s t a b i l i t y , in w h i c h the o b s e r v e d l u m i n o s i t y m i g h t nőt e q u al the e n e r g y p r o d u c t i o n r a t e on a K e l v i n - c o n t r a c t i o n o r p h o t o n - d i f f u s i o n t i m e s c a l e , has nőt been t h o r o u g h l y e x p l o r e d . An i n v e s t i g a t i o n b y A i z e n m a n & P e r d a n g (1971) o f a s l i g h t l y m o r e m a s s i v e e v o l v e d m o d e l did nőt r e v e a l any g r o w i n g m o d e s o f t h e r m a l i n s t a b i l i t y . M o s t r e c e n t l y H a r m and S c h w a r z s c h i l d

(1972) h a v e shown that t h e r e a r e no t h e r m a l l y unst abl e r a d i a l m o d e s w i t h p e r i o d s g r e a t e r than 10 y e a r s and L i t t l e t o n et a l . (1972) have shown that l o n g i tu d i na l w a v e s a r e u n i m p o r t a n t f ó r e n e r g y t r a n s p o r t .

T h e p o s s i b i l i t y that the g r a v i t a t i o n a l const ant G m i g h t d e c r e a s e on a c o s m i c t i m e s c a l e has b ee n i n c o r p o r a t e d intő s o l a r m o d e l s by s e v e r a l w o r k e r s ; E z e r & C a m e r o n (1966) found that a v a r y i n g - G m o d e l ( B r a n s & D i c k e 1961) w o u l d h av e f o u r t i m e s the cp( 13) o f a standard m o d e l ( s e e a l s ó S h a v i v & B a h c a l l 1969). A m o d e l by R o u s e (1969) i m - p l i e d that the C N c y c l e d o m i n a t e s in the sün; this was a l r e a d y r u l e d out on the b a s i s o f the r e s u l t o f D a v i s et al. (1968) by B a h c a l l , B a h c a l l

& S h a v i v (1968) ( p r o v i d e d the n e u t r i n o s r e a c h the e ar t h ) . It has been s u g g e s t e d that q u a r k c a t a l y s t s m i g h t be i m p o r t a n t in the sün ( L i b b y &

T h o m a s 1969; cf. S a l p e t e r 1970).

N ő n e o f the a b o v e s u g g e s t i o n s has be en w i d e l y a c c e p t e d .

I V . I M P L I C A T I O N S O F T H E O B S E R V A T I O N S

T h e i m p l i c a t i o n s o f the 37C l e x p e r i m e n t a r e b e st a p p r e c i a t e d

w h en one d i s t i n g u i s h e s the v a r i o u s counting l e v e l s at w h i c h d i f f e r e n t things a r e b e i n g t e s t e d . I f the e n t i r e e n e r g y s o u r c e f ó r the sün w e r e the C N c y c l e , then the counting r a t e in the 37C l e x p e r i m e n t w o u l d be

35 SNU i ndepe nde nt o f m o d e l p a r a m e t e r s ( B a h c a l l 1966). Thus at the l e v e l o f tens o f S N U ' s one is d i s t i n g u i s h i n g b e t w e e n the p r i n c i p a l e n e r g y

s o u r c e m e c h a n i s m s ( p p - c h a i n o r C N O c y c l e ) . M o d e r n s o l a r m o d e l c a l - cul at ions a l l s u g g e s t that the p r i n c i p a l e n e r g y s o u r c e f ó r the sün is the p p - c h a i n and that, w i t h p r e s e n t l y a c c e p t e d p a r a m e t e r s , the e x p e c t e d counting r a t e should be ~ 4 - 9 SNU ( B a h c a l l & U l r i c h 19^1, A b r a h a m &

Iben 1971, B a h c a l l , H é u b n e r , M a g e e , M e r t s , & U l r i c h 1972 ). M o s t o f the c a l c u l a t e d counting r a t e w i t h p r e s e n t p a r a m e t e r s i s due to the r a r e

B r e a c t i o n s ( s e e T a b l e 2) that hav e no e f f e c t on the s t r u c t u r e o f the sün.

On the o t h e r hand, the b a s i c t e r m i n a t i o n s , pp I and pp II ( s e e T a b l e 2), c o n t ri b ut e ( al ong w i t h a s m a l l amount ~ 0. 3 SNU f r o m the C N n e u t r i n o s ) a counting r a t e ~ 1 to 1. 5 SNU w h i c h i s i n s e n s i t i v e to m o s t p a r a m e t e r v a r i a t i o n s ( B a h c a l l , B a h c a l l & U l r i c h 1969). Thus a counting r a t e b e l o w 1 SNU woul d be in c o n f l i c t w i t h the b a s i c i d e as o f s t e l l a r e v o l u t i o n as d e s c r i b e d in the s t andar d b ooks of, f ó r e x a m p l e , S c h w a r z s c h i l d (1958) and C l a y t o n (1968). The b a s i c i d e a that n u c l e a r f usi on among light e l e m e n t s is the e n e r g y s o u r c e f ó r m a i n s e q u e n c e s t a r s gua r an t ee s ( a s s u m -

ing v flux c o n s e r v a t i o n ) a counting r a t e o f 0. 3 SNU ( B a h c a l l , B a h c a l l

& S h a vi v 1968). T h e v a r i o u s l e v e l s o f m e a n i n g o f the Cl e x p e r i m e n t

T h e c a l c u l a t e d p r i m o r d i a l h é l i u m abundance i s , h o w e v e r , u n c o m f o r t a b l y s m a l l ( Y = 0.1). N e v e r t h e l e s s i t is i m p o r t a n t to i n v e s t i g a t e f u r t h e r w h e t h e r o r nőt i t can be e x c l u d e d that in the e a r l y s t a g e s o f the f o r - m a t i o n o f the sün the h e a v y e l e m e n t s w e r e l a r g e l y e x p e l l e d f r o m the i n t e r i o r (by r a d i a t i o n p r e s s u r e o r b y c o l l e c t i o n in s o m é dúst cloud).

V . DO S O L A R N E U T R I N O S R E A C H T H E E A R T H ?

In o r d e r to r e a c h the e a r t h , s o l a r ne ut r in os m u s t t r a v e l ~ 1013 c m (= 1 A . U. = e a r t h - s u n d i s t a n c e ) and t r a v e r s e — 10 3 6 e l e c t r o n s ( or

. - 2 (" A u n . (-.36 - 2

p r o t o n s ) p e r c m I n dí ~ 1 0 c m It is at l e a s t c o n c e i v a b l e - o

that s o l a r n e u tr i no s do nőt r e a c h the e a rt h . I f this w e r e known to be t r u e , the upper l i m i t s o f the Da vi s e x p e r i m e n t woul d ha ve to be i n t e r - p r e t e d as c o n s t r a i n t s on the p h y s i c s o f the e l e c t r o n ' s n eu tr in o, nőt the

s o l a r i n t e r i o r .

T h e r e a r e t h r e e p r o c e s s e s by w h i ch s o l a r neutr inos m i g h t be p r e v e n t e d f r o m r e a c h i n g the e ar t h: a b s o r p t i o n , s c a t t e r i n g , or d e c a y . A s l ong as w e r e s t r i c t o u r s e l v e s to p a r t i c l e s that h a ve a l r e a d y been d i s c o v e r e d , nőne o f these p r o c e s s e s a r e p o s s i b l e w it h the r e q u i r e d

- 36 2

c r o s s s e c t i o n s (~10~ c m at 1 M e V ) o r h a l f l i f e (< 500 s e c . ) A n a m u s i n g p o s s i b i l i t y w h i ch I hav e c o n s i d e r e d is s t r o n g l y f o r w a r d - p e a k e d e l a s t i c s c a t t e r i n g , so f o r w a r d - p e a k e d that m o s t o f the i n t e r a c t i o n s a r e nőt d e t e c t e d by e x p e r i m e n t a l i s t s who w o r k at a p p r e c i a b l e m o m e n t u m t r a n s f e r s . E v e n this is nőt n e a r l y enough g i v e n the e x p e r i m e n t a l

con-54

s t r a i n t s [Cundy et al. (1970); R e i n e s et a l . (1960)], as one can e a s i l y show using, f ó r e x a m p l e , an a s s u m e d d i f f r a c t i o n s c a t t e r i n g l a w f ( q ) oc

, 2

exp - bq .

W e h a v e r e c e n t l y s u g g e s t e d [ B a h c a l l , C a b i bb o , & Y a h i l 1972]

that e l e c t r o n n eu tr i no s o f 10 M e V e n e r g y m a y b e u nst abl e w i t h a h a l f l i f e

< c (the d i s t a n c e b e t w e e n the e a r t h and the sün) ~ 500 s e c o n d s . T he v a r i o u s p o s s i b i l i t i e s w e r e d i s c u s s e d i n our o r i g i n a l p a p e r , bút a g e n e r á l r e q u i r e m e n t o f a l l the p r o p o s a l s i s that t he y r e q u i r e a p r e v i o u s l y un- known i n t e r a c t i o n o r p a r t i d é . One m a y a l s ó c o n s i d e r the p o s s i b i l i t y ( R e g g e 1971) that t h e r e a r e p a r t i c l e s p r e s e n t in the sün (bút nőt y e t o b s e r v e d on e a r th ) w h i c h i n t e r a c t s t r o n g l y w i t h n e u t ri n o s bút o nl y v e r y w e a k l y w i t h o r d i n a r y m a t t e r . T h e a s t r o p h y s i c a l c o n s e q u e n c e s o f this

l a t t e r s u g g e s t i o n a r e s o m e w h a t m o d e l dependent so I w i l l r e s t r i c t m y

-f

s e l f to a d i s c u s s i o n o f n e ut r i no d e c a y .

T h e r e a r e t h r e e d i r e c t a s t r o p h y s i c a l c o n s e q u e n c e s o f the hy-p o t h e s i s that the i n s t a b i l i t y o f v is r e s p o n s i b l e f ó r the u n e x p e c t e d l y l o w counting r a t e in the 37C l e x p e r i m e n t . T h e s e c o n s e q u e n c e s a r e :

(1) the 37C l e x p e r i m e n t should y i e l d a r e s u l t c o n s i s t e n t w i t h 2(<pcr) = 0;

(2) a t t em pt s to d e t e c t the a s t r o p h y s i c a l l y - g u a r a n t e e d pép ne ut ri no s [ e. g. , by the p r o p o s e d 7L i e x p e r i m e n t - - s e e B a h c a l l (1969)] should f a i l ; and

(3) a l l a s t r o n o m y w it h w i l l be i m p o s s i b l e ( i . e . , e l e c t r o n neutri nos f r o m b l a c k h o le f o r m a t i o n o r s u p e r n o v a c o l l a p s e w i l l a l s ó nőt r e a c h us) .

T h e b a s i s f ó r t hese p r e d i c t i o n s is the r e m a r k that i f n e u t r i n o - i n s t a b i l i t y i s the e x pl a na t io n o f D a v i s ' r e s u l t then the t é r r e s t r i a l l y o b s e r v e d flux o f s o l a r n e u tr i no s w i t h e n e r g y E , <p(E _ , s a t i s f i e s

* e a r t h ( E > ’ * Su „ (E) Í « P -- ( E ) .

w h e r e a (E ) = [ ( e a r t h - s u n d i s t a n c e ) / c T j(10 M e V ) ] (— '^) and a^lO M e V ) >

2 ^

2 b e c a u s e o f the a b s e n c e o f B n e u t r i n o s . T h e e n e r g y dependence o f ar(E) is due to t i m e - d i l a t i o n ; l o w - e n e r g y p a r t i c l e s d e c a y m o r e r a p i d l y

than t h e i r h i g h - e n e r g y c o u n t e r p a r t s . I f the h i g h e r e n e r g y B ne ut ri no s d e c a y , then the l o w e r e n e r g y ^ B e , ^N, O, and pép ne ut ri no s m u s t

37 .

c e r t a i n l y do so. T h e C l e x p e r i m e n t o f Da vi s p r o b a b l y has an u l ti -m a t e s e n s i t i v i t y ~ 0. 5 SN U c o -m p a r e d to a p r e d i c t e d counting r a t e ~

5-10 S N U. It is e x t r e m e l y u n l i k e l y that the p a r a m e t e r s f ó r v d e ca y a r e j ust such that m o s t o f the n e u t r i n o s d e c a y in 1 A . U. bút énough a r e l e f t o v e r to p e r m i t a d e t e c t a b l e counting r a t e in the 37C l e x p e r i m e n t .

O f c o u r s e i f n e u t ri n o s f r o m the sün do nőt r e a c h us, then v 's f r o m e

m u c h m o r e d is tant a s t r o n o m i c a l s o u r c e s w i l l a l s ó d e c a y al ong the w a y (the a n t i c i p a t e d e n e r g i e s a r e nőt v e r y d i f f e r e n t f r o m those o f s o l a r neu

t r i n o s bút the d i s t a n c e s a r e m o r e than a m i l l i ó n t i m e s l a r g e r ) . In the a b s e n c e o f e x p e r i m e n t a l i n f o r m a t i o n , I ’woul d l i k e to s u m m a r i z e the a r g u m e n t s , p r o and con, as to w h e t h e r or nőt s o l a r n e u t r i no s r e a c h the e a r th . T h e r e a r e two " a r g u m e n t s " s up po rt ing the

In document NEUTRINO 72 (Pldal 43-101)