Effects of core spin-down process on neutrino flux in solar evolution theor

Cameron, A. G. W.

Ezer, D.

English

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I 
I 
, 
S o l a r  Spin-Down and Neutrino Fluxes 
Dilhan Ezer* 
Department of Theore t ica l  Physics 
Middle East  Technical Un ive r s i ty  
Ankara, Turkey 
and 
A.G.W. Cameron 
Be l fe r  Graduate School of Science 
Yeshiva Univers i t y  
New York, N . Y .  
and 
I n s t i t u t e  f o r  Space S tudies  
Goddard Space F l i g h t  Center ,  NASA 
New York, N.Y 
GPO PRICE $ 
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Microfiche (MF) ,-K Abs t r a c t  2 (&&$R $X AD NUMBER) 
ff 653 July 65 
We suggest  t h a t  the f a i l u r e  of Davis t o  d e t e c t  a 
flux of B neu t r inos  f rom the  sun may r e s u l t  from t h e  8 
maintainance of  a high c e n t r a l  hydrogen abundance. 
This could a r i s e  from the c i r c u l a t i o n  c u r r e n t s  assoc i -  
a ted wi th  the spin-down of a f a s t  r o t a t i n g  core.  On 
t h i s  b a s i s  we p r e d i c t  a lower l i m i t  t o  the  f l u x  which 
i s  about t he  same as Davis’ upper limit. 
>kOn leave  from t h e  I n s t i t u t e  f o r  Space S tud ie s ,  
Goddard Space F l i g h t  Center,  NASA, New York, N.Y. 
https://ntrs.nasa.gov/search.jsp?R=19680009667 2020-03-24T00:33:09+00:00Z
2. 
One of the most s i g n i f i c a n t  experiments i n  as t rophys ics  has  been 
the  at tempt  by R. Davis t o  d e t e c t  the f l u x  of B8 neu t r inos  from the sun 
(Davis 1964, 1968). I n  order  f o r  h i s  a n t i c i p a t e d  r e s u l t  t o  be meaningful 
i n  an a s t rophys ica l  con tex t ,  i t  has been important  t o  make p red ic t ions  
of t he  expected f l u x  based on s t e l l a r  evolu t ionary  theory as appl ied 
t o  the  sun. Most of  the  o lde r  ca l cu la t ions  have been rendered u n r e l i a b l e  
by r ecen t  r ev i s ions  i n  the  nuclear  r e a c t i o n  c ross  sec t ions  f o r  the  
r e a c t i o n s  B e  (p,Y)B 
Tombrello 1968; Winkler and Dwarakanath 1967). I t  has  been necessary t o  
r epea t  these  c a l c u l a t i o n s  us ing  the rev ised  nuc lea r  d a t a .  
7 8 3 P a r k e r  1966) and H e  (He3,2p)He4 (Bacher and 
We have repeated our  previous c a l c u l a t i o n s  of s o l a r  evo lu t ion  wi th  
an assumed cons t an t  g r a v i t a t i o n a l  coupl ing c o e f f i c i e n t  (Ezer and Cameron 
1965, 1966). I n  a d d i t i o n  t o  t h e  rev ised  nucJear d a t a ,  a number of  o the r  
improvements were made i n  the code, d e t a i l s  of which w i l l  n o t  be given 
h e r e .  A s  a r e s u l t  of these  changes we now take the  mixing length  i n  the 
convect ion theory t o  be 1 .6  pressure s c a l e  he igh t s .  
The p red ic t ed  B8 neu t r ino  f lux a t  one astronomical  u n i t  throughout 
t he  h i s t o r y  of the  sun i s  shown i n  F igure  1. The va lue  p red ic t ed  f o r  
the  p re sen t  sun i s  1 . 1 7 ~ 1 0  cm. sec .  7 -2  -1 
Meanwhile a much more extensive i n v e s t i g a t i o n  of  evolved s o l a r  
models has  been c a r r i e d  out  by Bahcall and Shaviv (1968). Af te r  consider-  
i n g  the  e f f e c t s  introduced by v a r i a t i o n  of a number of parameters ,  they 
p red ic t ed  a B8 neu t r ino  f l u x  o f  1 .4(1&0.6)~10 cm. sec.  Our 
p r e d i c t i o n  i s  q u i t e  c o n s i s t e n t  with t h i s .  
7 - 2  -1 
3 .  
It has the re fo re  come as a s u r p r i s e  t h a t  the  pre l iminary  r e s u l t s  from 
6 - 2  -1 Davis '  experiment give an upper l i m i t  t o  t he  f l u x  of 2x10 cm. sec .  e 
This  r e s u l t  poses a s t r o n g  chal lenge t o  s t e l l a r  evolu t ionary  theory ,  
Of course the  p red ic t ed  f l u x  i s  s u b j e c t  t o  var ious  poss ib l e  e r r o r s .  
These inc lude ,  i n  p a r t i c u l a r ,  element abundances i n  the sun, and a l l  of 
t h e  va r ious  nuc lear  r e a c t i o n  da ta .  However, the  p red ic t ed  f l u x  i s  not  
changed by the  necessary f a c t o r  i f  one accepts  the  e r r o r s  claimed f o r  
t h e  abundance de te rmina t ions  and the  nuc lea r  d a t a .  It i s  the purpose of  
t h f s  note  t o  p o i n t  o u t  t h a t  i t  may be p o s s i b l e  t o  r e t a i n  the  accepted 
va lues  f o r  abundances and nuclear  d a t a  by p o s t u l a t i n g  t h a t  c i r c u l a t i o n  
c u r r e n t s  e x i s t  i n  the s o l a r  core .  
It  may be seen  i n  Figure 1 t h a t  the p red ic t ed  B8 neu t r ino  f l u x  
rises by a f a c t o r  30 throughout s o l a r  h i s t o r y .  This  i s  p r imar i ly  due 
t o  the  inc rease  i n  c e n t r a l  temperature as the  c e n t r a l  hydrogen i s  
dep le t ed .  It  i s  ev iden t  t h a t  the f l u x  would be s t rong ly  suppressed i f  
t h e  c e n t r a l  hydrogen remained near  i t s  i n i t i a l  abundance. This  would 
r e q u i r e  t h a t  an ex tens ive  mixing process  has been ope ra t ive  throughout 
so  1 ar  h i s  t o r y  
Such a m i x i n g  process  may e x i s t  i n  connection with t he  s o l a r  spin-  
down process &ich has r e c e n t l y  been d iscussed  i n  connection wi th  a 
p o s s i b l e  s o l a r  ob la teness  (Howard, Moore, and Spiege l  1967; Dicke 1967; 
McDonald and Dicke 1967). Any reasonable model f o r  t he  formation of 
t h e  sun p r e d i c t s  t h a t  i t s  i n i t i a l  s p i n  i s  much g r e a t e r  than  the p r e s e n t l y  
observed va lue .  The o u t e r  s o l a r  convect ion zone w i l l  be r a p i d l y  slowed 
4 .  
down through outward t r a n s p o r t  of angular  momentum by the s o l a r  wind. 
The ques t ion  which then  a r i s e s  is whether i n s t a b i l i t i e s  w i l l  occur  which 
w i l l  l ead  t o  mixing between the  convection zone and t h e  r a d i a t i v e  co re  
w i t h  outward t r a n s p o r t  of angular  momentum, l ead ing  t o  a spin-down of 
t h e  core .  Howard e t  a1 have claimed t h a t  t h e  core  w i l l  spin-down too  
r a p i d l y  t o  a l low a mass quadrupole moment s u f f i c i e n t  t o  account f o r  t h e  
s o l a r  ob la teness  measured by Dicke and Goldenberg (1967) e Dicke (1967) 
and McDonald and Dicke (1967) claim t h a t  s o l a r  s t r a t i f i c a t i o n  w i l l  
s u f f i c i e n t l y  impede the  spin-down process  t h a t  a s i g n i f i c a n t  quadrupole 
moment i s  poss ib l e .  
For t h e  purposes of ou r  p re sen t  a n a l y s i s  we assume t h a t  t h e  s o l a r  
spin-down process  keeps the  composition of t h e  sun f a i r l y  homogeneous 
throughout a l a r g e  p a r t  of i t s  i n t e r i o r .  Since the re  i s  much d i s p u t e  
over  t h e  f l u i d  dynamics of t h e  spin-down p rocess ,  i t  i s  n o t  c l e a r  whether  
ou r  p re sen t  assumption favors  one s i d e  o r  t h e  o t h e r  o f  t he  above 
d i scuss ion .  However, i t  i s  ev ident  from t h e  r e l a t i v e l y  h igh  abundance 
of be ry l l i um i n  t h e  s o l a r  photosphere t h a t  t h e  o u t e r  convect ion zone 
has  n o t  been d i l u t e d  by a l a r g e  f a c t o r .  This  need not  r u l e  o u t  ex tens ive  
. mixing i n  the  core  i f ,  as Spiege l  (1967) has  suggested,  t he  convect ion 
zone acts as a ' r r i g i d r r  boundary giving r ise t o  an Ekman l a y e r  i n  t h e  
o u t e r  l a y e r  of t he  core .  Meridional c i r c u l a t i o n  may a l s o  ex is t  i n  t h e  core .  
S o l a r  evo lu t ion  wi th  an unmixed core  l eads  t o  an inc rease  i n  t h e  
9 luminos i ty  by a f a c t o r  1.35 during t h e  evo lu t iona ry  age of 4 . 5 ~ 1 0  
yea r s .  This  r e s u l t s  pr i .mari ly  from the  conversion of hydrogen i n t o  
5. 
helium, which reduces the  number of e l e c t r o n s  i n  the  i n t e r i o r  and hence 
decreases  the  opac i ty .  It i s  l i k e l y  t h a t  s o l a r  evo lu t ion  w i t h  a mixed 
co re  would have much the  same luminosi ty  h i s t o r y ,  s ince  the o v e r a l l  
r educ t ion  i n  opac i ty  would be about the  same wi th  the hydrogen-burning 
products  r e d i s t r i b u t e d  throughout the  i n t e r i o r .  Under these  circumstances 
t h e  mean mass f r a c t i o n  of hydrogen i n  t h e  sun would be reduced from 
0 , 7 4  (assumed i n  our  i n i t i a l  models) t o  0 ,70 ,  and the mean helium mass 
f r a c t i o n  would r ise  from 0 , 2 4  t o  0.28. For s i m p l i c i t y  we assume a 
uniform evolved composition wi th  these  a l t e r e d  abundances. W e  assume the  
c e n t r a l  d e n s i t y  would r i s e  by three p e r  cent  t o  compensate f o r  the  
sma l l e r  number of p a r t i c l e s  pe r  gram of ma t t e r  wi th  a consequent r educ t ion  
of p re s su re  
8 Tn t h i s  extreme model the B neu t r ino  f l u x  i s  no t  t h a t  i nd ica t ed  i n  
the  lower l e f t  hand corner  of Figure 1, The proton-proton cha in  has  an 
energy genera t ion  ra te  which va r i e s  a s  T 
where T i s  the  temperature (see fo r  example Reeves 1965),  Hence the  
temperature  r i s e s  by a f a c t o r  (L035) The B8 f l u x  i s  
cor  re s p ond ing  1 y inc  re as  ed 
under c e n t r a l  s o l a r  cond i t ions ,  4.06 
= 1,0765. 
The abundance of B8 i n  the  s o l a r  i n t e r i o r  w i l l  i nc rease  over  t h a t  
i n  our  i n i r i a l  main sequence models by a f a c t o r  depending on the  product  
of the  temperature s e n s i t i v i t i e s  of the  H e  (cY,y)Be and Be (p,y)B 
r e a c t i o n s ,  s i n c e  the He (He ,2p)He r e a c t i o n  i s  the  dominant energy- 
genera t ing  r e a c t i o n  a t  the cen te r  of t he  sun. From Reeves (1965) w e  
can w r i t e  approximately f o r  the number d e n s i t y  of B e  
3 7 7 8 
3 3  4 
7 
6.  
. 
. 
3 7 
7e where Cov> is a measure of  the  r a t e  of t he  H e  (a,y)Be r e a c t i o n ,  34 
i s  the  r a t e  of e l e c t r o n  capture  of Be7, and N 
d e n s i t i e s  of He3 and He  . The abundance of N 
and N a r e  the  number 3 4 
3’ 
4 i n  t u r n ,  i s  approximately 
where <ov> measures the  r a t e  of the p-p r e a c t i o n ,  <ov> t h a t  of t he  
H e  (He ,2p)He r e a c t i o n ,  and N i s  the  abundance of hydrogen. A t  
1 . 4 ~ 1 0 ~  OK, the  q u a n t i t i e s  <ov>ll, <av> 
temperature  r a i s e d  t o  the  powers 4.06, 16.30, 17.08, and -0.5, The 
Be (p,Y)B r e a c t i o n  v a r i e s  a s  T 
11 33 
3 3  4 
1 
<ov>34, and p vary  a s  t he  
33’ 7e 
7 8 13.50 
P u t t i n g  toge ther  the  abundance, d e n s i t y ,  and temperature  v a r i a t i o n s ,  
8 we thus  e s t ima te  t h a t  t he  B 
a f a c t o r  1.14x(1.0767) 24*96 = 6.7 over t h a t  i n  the  i n i t i a l  model. 
6 - 2  -1 gives a lower l i m i t  t o  t he  pred ic ted  f l u x  of 2 . 5 ~ 1 0  cm. sec .  . This  
neut r ino  f l u x  w i l l  i nc rease  by a t  l e a s t  
This  
i s  about the same as Davis’ upper l i m i t .  
It i s  ev iden t  that element abundances and nuc lear  d a t a  can r e a d i l y  
be adjusted. .saithin the  claimed e r r o r s  t o  reduce our lower l i m i t  below 
Davis’ upper l i m i t .  Hence w e  be l i eve  t h a t  the  s o l a r  spin-down c i r c u l a -  
t i o n  problem deserves  much more c a r e f u l  examination, s i n c e  i t  i s  p o s s i b l e  
t h a t  i t  may have a major e f f e c t  on s o l a r  evolu t ion .  
7. 
We wish t o  po in t  ou t  t h a t  the evo lu t ion  of o t h e r  s ta rs  may be 
s i m i l a r l y  a f f ec t ed  by a core  spin-,down process .  Since the  age of our  
galaxy i s  based on the s t e l l a r  evolu t ion  ages of s t a r  c l u s t e r s ,  t h i s  age 
must now be regarded as un re l f ab ly  determined. 
We are g r a t e f u l  t o  R. Davis f o r  communicating h i s  r e s u l t s  t o  us i n  
advance of publ icac ion .  This research has  been supported i n  p a r t  by 
g r a n t s  from the  Nat ional  Science Foundation and t h e  Nat ional  Aeronautics 
and Space Admins t r a t i o n .  
F igure  Caption 
8 Figure  1. The B neu t r ino  f l u x  a t  one astronomical  u n i t  dur ing  the  
evo lu t ion  of the  sun. Also shown are Davis '  experimental  upper l i m i t  
and the lower l i m i t  c a l cu la t ed  i n  t h i s  paper f o r  a con t inau l ly  mixed 
s o l a r  core ,  
I 8. 
Re f e r e  nce s 
. 
. 
Bather, A.D., and Tombrello, T .A . ,  1968, t o  be publ ished.  
Bahca l l ,  J . N . ,  and Shaviv,  G . ,  1968, t o  be publ ished.  
Davis, R. ,  Jr., 1964, Phys. Rev. Letters, l2, 302. 
Davis, R., Jr., 1968, P r i v a t e  communication. 
Dicke, R.H., 1967, Astrophys. J.,JS, L121. 
Dicke, R.H., and Goldenberg, H.M.,  1967, Phys. Rev. Let ters ,  l8, 313. 
.- Ezer ,  D . ,  and Cameron, A.G.W., 1965, Can. J. Phys., 43, 1497. 
Ezer ,  D . ,  and Cameron, A.G.W., 1966, Can. J. Phys., 44, 573. 
Howard, L.N., Moore, D.W., and Sp iege l ,  E.A., 1967, Nature ,  2, 1297. 
McDonald, B.E. ,  and Dicke, R.H.,  1967, Science,  g, 1562. 
Pa rke r ,  P.D., 1966, Phys. Rev., z, 851. 
Reeves, H. , 1965, i n  "Stars  and S t e l l a r  Systems," Vol. VI11 ( A l l e r  
and McLaughlin, S t e l l a r  S t ruc tu re )  Un ive r s i ty  of Chicago P r e s s ,  
Chicago. 
S p i e g e l ,  E.A., 1967, Colloquium, Be l f e r  Graduate School of Science.  
Winkler ,  H.C., and Dwarakanath, M.R., 1967, Bull. Am. Phys. SOC., 2, 16. 
\ 
8 
10’ 
IO6 
I I I I 
L O W E R  
L I M I T  
t 
PREDICTED FOR 
UNMIXED SOLAR CORE - 
- 
- 
D A V I S  
UPPER 
i iMlT 
I 2 3 4 
AGE (BILLIONS OF Y E A R S )  
Figure 1 
5 


Solar spin down and neutrino fluxes

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