It is shown that the results of maximum likelihood treatment of Monte Carlo simulation with constant production rate of 7.6 SNU and 1.Epsilon SNU are consistent with the constant production rate when the tests of hypotheses (e.g., t-test, sigma squared-test, Wilcoxon-Mann-Whitney test, run test, etc.) are applied to the two groups of data formed from sunspot minimum range and sunspot maximum range, whereas the real data pulsates with the solar activity cycle. It is shown that SN flux-change is in opposition phase to the solar activity cycle and lags behind the latter by about one year. A correlation between SN flux and the cosmic rays is suggested

Raychaudhuri, P.

English

NASA Technical Reports Server

48S
SH 8-_-7
sOLAR _UTRIN0:TL_GOSMIC BAYS A_D THE 11YEAR SOLAR CYCLE
_ohbas Raychaudhuri
Department of Applied Mathematics
Calcutta University, Calcutta-700009, INDIA
It is shown that the results of maximum likelihood treatment of Monte
Carlo simulation with c_stant production rate of 7.6 SNU and 1-8 SNU
are consistent wi%h the constant production rate when the tests of hypo-.
theses(e._g, %-testp3tm-test_ilco_Dn-Mamn-_itaey test_run test etc.)has
been applied to the two groups of data formed from sunspot _Lnimu_ range
and sunspot maximum range whereas the real data pulsates with the solar
activity cycletIt is shown that SN flux-change is in opposite phase to
the solar activi_ cycle and lags behind the latter by about one year.A
correlation between SN flux and the cosmic rays has also been suggested.
Ra_chaudhuri(1,2,3,4,5,6_d Gavrin et al(7)sho,ed that the solar neu-
tri_e(SN)flux data is _ with the solar activity cycle.Fmychaudhuri
(3,_5,6)showed that the fluxes will be higher in the sunspot minimmm
range than in the s_%tmaximum range. _kis is supported by the data
presented by Rowley et al(8)at the solar neutrino and neutrino astronu%y
conference held at Lead, South Dakota.Apart from the abo_ variation Ray-
chaudhuri(5,6)showed that there is also a quasiblemliel _ariation from
19_3-1975 and 1979-1981.This type of variation is already observed by
Gne_Ishe_9)and Filisetti(lO)in the cosmic ray intensity(proton flare),
In thispaper an estimste of the moving a_erage(5 successive run numbe$
data is considered to find the statistical significant variation of SN
flux with the solar activity cycle of 11 years by t-testj_L_-test,Wilcoxon
=Manm_k_itney test and run test-To search variation in a convincing man-
her the above tests of _potheses have been applied te the data that had
beem generated by Monte Carlo simulation and background parameters are
typical of those in the actual experiment-
SOIAR ACTIVITY C__ AND(a)MOVING A_2_AC_ DATA_blMONTE C_Q SIMULATED
DATA, _ITH CONSTA_ pRODUCT!ON RA_(i)%6 _ A_n _ _._.SNUND(ii)!.8 8_J..iHere we apply the _ proceedure as in Raychaudhuri(5,6)
l)8tudent t-re st:-a(i)Movin_, average data
We form the two groups,the first group o_mprises sunspot m/_mtm range
(about 4"6 years)and the second group i.e.,the rest c_mprises sunspot
maximum range.Let us now collect the data from the run number 36 to 58
from August 1974 to February 1979(about _.6 years)for the first group.If
we now set up the null hypothesis H : m, = m_against the alternati_s m,>
mA o_he statistics t is given by(here the difference of variance _s not
+ _ •
where4_ = Nt_N_-_In the above case N,--_3,N_=_7_9=48 which gives t-_.5_
Thus we can conclude that 99.9% of the data pulsates with the solar acti
vity cycle._e standard error of the difference of means is _ 51(.Hence
the data are __nt with the ass_m_ that the two means are
(_ Y___ average of the movingaverage data:-Here one group c_mprises
the yearly average data from 1975 to 1978 an_ the other gr_up c_mprises
•rest of the moving a%_rage data. _ get for 9 d.f. t--2.76.The stamdard
error of the difference of the means is 3.1g which suggests that the
https://ntrs.nasa.gov/search.jsp?R=19850026799 2020-03-20T16:50:01+00:00Z
43@
SH 8e 1-7
• '(_------------__o _,_.1.3_- _oductionrates"
i Ipbserved in the Chlorine SNL I , _ . I -_*-o_ experiment 1970-1982 a d the
"_ I | ,| , I $ _i:i_o',d._ a_era6e data(dashed
constant production rate of
! i_ • _ ....
_o means of the two samples are
!not equal.
"_ : :_S; , . _, ...,,, b)Fonte G_rlo simulated data
_0 _verage data. In the case of
- 7.6 SNU the difference of
'. variance is s_nificant. W_
• j . , | '_, ,o have to use the formula for"
_or d.f. 47,t=1.19 The stand.
error of the difference
two means is 1.1 E .H_nce
• i_he tw_ means are equal i.e,,
__ Ithe stead_ flux is accepta-
'hlrm,,. _ '.Im,,_-_,_'._-_,li,..., , : _ 'b_e. In the case of 1.8 SNUi- :the difference of _Aanee
__o_--._:::-:_-:....--_.__._.__.--,,._ . ............../ 1- ' ..................... ]"°'° l!S not _cant. Hence %= "
!_ / / , / t:_.Sk,d-_. $_._e _'._.ia___r,
I_ " i / / , I " _iHence the data are consist-
i_ / _ I I /_/!t111_ /d $,/ /,o _e._xt we_e the _'_kY
i_ ' . .: [ I a_rage of the Monte (_z'lo
O in t-test.Nere in both cases' ..t ' 3_ f_ thetwo group, as1
S_,t_A9 d-f ._9.In,case
i_: :%_i-l_l_._ ............... .. !il.8 l_14U,t=l.01 d-f. 9.H_e
the standard e_or is _'o_I to be (i)0-_3 e _(dl) i e _ospoct_::1_.F_n-
ce the data are consistent with the assumption that the stea_ fl_
acce] t _ble_ " : .._
II)_-_st:- k)We form the group as in the t-test_Jb_ movAng average
data 6"_=0.02 _ for d.f..49,_S--.73.19.'Zbus the _esis of s%_2
flux is not acceptable-For the case of 7-6 S_, 6" _0,291. _=47.84 d.f.4?,
again for 1.8 SNU_ _m=O, 10, _6.83 d.f. 4_.Heacethe d_fere'_ce "oft%D_"
samples are not significant for the case of 7.6 _NU and 1.8 SN_
440
SH 8.I-7
(ii)if we take the yearly averageof the moving average data and F_nte
Carlo simulated data.Ne form the groups as in t-test.We get G_01_ _=
18s55,d.f.10formo data.£,thiscasesteady fAux£s aot
acceptable_Again (i) G _09,o_'=9;39,d.f • 9 for 7-6 SNU and (ii)6._05,
_=10%_S,d.f- 9 for 1-8 S_Hence the steady SN flux is acceptable iI
Monte Carlo simulated data.
(III)MLlcoxon-_um-_bitney test:-_re we take the yearly average data of
moving average data and Honte Carlo simulated data 7.6 S_U and 1.8 SNU
and form the two groups as in t-test and _-test. If we set up the null__'-
pothesis H o :the distribution of x and y are identicalolf a two sided te_%
at 5% is desired for m=_pn=7 we use 3_U_25 as the accepted region,For
this test _ =P,o (V d 3 er U _2.5)=.0/-,2..,
e)_oving average data..-we write the two sets of data as y:.3552,-4740#
-3546,-_07,.3089,.2469,-4707 and x:.5305p.$¢37_.7_82,-5_21.Putting these
numbers in order and keep track of which numbers came from x and which
from y:yy/yjxy_x_x:_here are _2 inversions of the y's and the rank of ySs
is I_.3_$.5_ 7_8 _30 =2._ Since the number of inversion of the y's is
not within 3 and _ _we reje_ct'thenull _pothesispi-e. the constant SN
flux is not acceptable.
b)(i)7.6 S_U :-y:I._08,0.993,I-Z_qO,1.286,1.277,I"928 and x:1600,1.396,
It320_1.098.Puttingthese numbers in order and keep track of which n_m-
bets came from x and which cams from y: y_yyx_yy_.There are 16 inversi-
ons of the y's a_d the rank of y's is 1+3+4+7+8+I0+11=44,
(ii)I_8 SNUI-y:1,545_*_001_.165_.805_.537,*_07_.378 and xl.3755_.4900,K_
._18#%581.A_nereare 8 inversion of the y's and the rank of y's is 36.
Since the number of inversion of the y is within 3 and 25_we accept the
null hypothesis i.e.#steady SN flux is acceptable in both the Monte Carlo
(IV)Run test:-In the case of run up and down the total n_ber of runs is
approximately normally distributed u_der the assure.prioriof.ra_,omness _ .
with mean and variance is given by E(r) =V3(2n-1) and V(r)=_/(90)(C_-_9).
Fgr move, average data:- n=50,_E_33,V(r)=8-57"In this case there are
2i runs either p0sitive or negat_%e.lfwe take the critical region to be
=0%20 would yield a K of 33-0'84_7 =30,_3,Thus we cannot accept
the randumness since 21 _30,5_3.Thus we can suggest that the _ns_ SN
flux is not acceptable.For case 7.6 Sl_Up_$9_E(r)=32*33and V(r)=8.389._
this case there are 31 runs either positive or negative.For the case 1-8
SNU#n--47_E(r)=O*31and V(r)=8.O3.,Inthis caee there are 29 runs is either
positive or negative.If ew take the oritical re_re_i_onto h_ r_K_a choice
of _ =0.20 would yield a K of (i)32.33-O.85_8.389 =29.9|(ii)31-O.85_803 "
=28.62._Thuswe accept the randu_ness in both the case since(i)31>29,9
and (ii)_28_62.So we can conclude that the F_nte Carlo simulated data
d@ not pulsate with the solar activity cycle.
_unsDot cycle and '_N flux:- We have taken the yearly average smooth suns-
pot from solar geophysical data#yearly averq_e of the moving average data
from fight.and yearly average of the cosmic ray data I with E =O.I-_8
Gev(11)._Ne ha_ displayed the data in Table I.
The correlation coefficient betwee_ sunspot cycle and3_ Ar production
data are r(S_Q_ )= -O._61_when_T=-I_r(S_Q_)---0.49when_T=O_r(S_ _ )=_-
whe,. ,,. the
rate anticorrelates with the sunspot cycle.The correlation coefficl nt
between the galactic cosmic r_y intensity a_ Y_Ar production data r(lu_
QM )=O.r_8when_T=O_r(l s ,QM )=O.31 when_ T=-I_r(_Q_)=0.39 when/
441
. SH 8. i-7
Table 1-Year3y averages of smooth sunspot n_jber(S), Ar prcduction data
(Q_),yearly average of commic r_y data iowith E=O. I-5.8 Gev.
1971 1972 1973 1974 1975 1976 19_7 1978 1979 1980 1981
Yearly average 66.6 68.9 38*5 3_4 15.5 12.6 27.5 92.7 155.3 155 140Sunspot n_ber
(s)
Moving &verage *36 "44 */_) *,/I_,
of the yearly .35 .53 ,75 ,5/+ .31 .25 "47
Ar production
data(QM)
 (1 m'2s"st-'0 2.01 2.S02.7 3.,,11 1.891.241.13Falactic cosmic ..........
ray intensity
between the SN flttxin Davis, data and galactic cosmic rays. Ibis shows
that the galactic cosmic rays and SN flux undergo the essential changes
in the solar acivity cycle from 1970-1982.
From the SN flux data we see that SN flux is higher during the ascend.
ing phase(about P_3 years)and descending phase(about 2-3 years)of the
solar cycle avoiding the sunspot maximtm time.We call the time 2-3 years
before the sunspot maximum as the second minim_ time. Thus SN flux is _
higher in both second miuim_ time and second maxim_ time and SN flux is
lower in both first minim_nand first sunspot maximtm stage. Zu addition to
two prominent maximum in the year 1957 and 1960 they(9_lO)h_ve found thi-
rd maximum of smaller amplitude in the com_i_ r_ iutemsi_ but veru clear
in 1963 .From fig-1 we see that run nt_ber 37 is connected with third max-
imum.IT_is m_y have some connection with the quasibien_iel _riation of SN
flux found from 1970-1975 and 1979-198_,Frcm the moving average data in
fig. I it appears that SN flux varies with a period of about 2.4 to 2.5
years from 19_-1975-
In conclusion it may be rsmarked that the present SN data exhibits a
high level of statistically significant variation of SN flux with the
solar activity cycle.The above results strong3/f suggests that the s@lar
activity cycle i@ due to the pulsating character of the nuclear ener_
generation inside the core of the s_utWe suggest that the chlorine SN
experiment should be continued to get the data like the sunspot data etc.
and this data could be of great imp_Mmace _o our understamdim_ o_ the
" sun and cosmic r_st
• Acknowledgement:-The author would like to thank Professor R%Davis, Jr. for
s_d_ug the SN data prior to publication.
" References :-
1)PtR_chaudhuri (1976)JINR communications E_- 10019,USS_,_)_ Raychaudhuri
_9)176 Talk gl'v_ at the international School of Cosmic ph_sics_Erice_Ita3y,
3)P.Raychaudhuri( 198.t)SpectScizTech* ,_, 271. _t)P-R_ycha',xlhuriQ 1983)Pr00.
18th.ICRC, Bangalore _ 108.1)P,R%ychaudhuri(1984)Sol_r Phi/s,._397.6)Pb
R_ychaudhuri( 198$)Proc-3rd-Asian Pacific regional meeting of IAU held at
F_oto(to be published)A_Science._)V. • G_vrin et al. (1982)JETP left
608,8)J.K.Rowley et al.(1984)American institute of Physics .c_:nference
proceedings(to be published)held at South Dskota, USA.,_)M.N 0nevyshev
(1967)Solar Ph_._67., 10)@ Fillisetti et al. (1974)LetttN_ovo Cimento I_/,
_9,_)C_A- Bazilevskaya et al. (1983 )Proc._Sth. i_RC,Bangalore/._2_8.


Solar neutrino: Flux, cosmic rays and the 11 year solar cycle

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