Z。naー Harm。nic Anaーysis 。f C。smic Ray Data fr。m W。rーd-wide

Zonal Harmonic Analysis of Cosmic Ray Data from
World-wide Network Cosmic Ray Stations:
Method and its Application
YAHAGINaohiro
(Received April 1, 1991)
Abstract
The formulation to analyse three−dimensional cosmic ray anisotr6py in interplanetary space has
been. given by Nagashima. ln his paper, an emphasis was pat On formulating the daily variatiop
arising from a 9eneralized anisotropy in space. This paper.gives the details of analytical me中od how
to derive the three−dimensional anisotropies of’cbsrrii6 rays, that is, cosmic ray’northTsbuth
asiymmetry and pole−equator anisotropy in interplanetary space.in reference to the Nagashima’s
formulation. ln addition to it, $ome examples of its application to the practigal ,analysis using cosmic
ray dat,a from the cosmic ray StatiQns of worldwide network through the w6rldLwide data center will
be shown. illustrated and discussed.
’
1.・ lntroduction
Cosmic ray intensities Qbserved at.stations at earth’$ $urface.have usually been
resulted from a combined effect・of・various variations, which have different time and
spatial variations. ln genetal,・it・・is very. difficult.to distinguish .a .single in.tensity
variation from records obtained at a single station at the earth’s surface. lt is necessary
that a close network of cosmic ray Stations is set up’
翌奄р?ly at the earthrs surface to
overcome such a difficulty. Such a close network over the earth’s surface has been set
Up since the lnternational Geophy$ical Year (IGY) , using Sirripson type counter, that
is, IGY−type neutron monitor in the・.period of IGY and thereafter it・has been replaced
by’
Dthe Carmichae17s type neqtron counter ($uper−neutron monitor) ’. ・Many . sl ations .
have been closed for the.per・iod’ td the present since the IGY’, whereas considerable
nqmbers of station have .come into operat.//o,p. ne.wly during the period; Various methods
to distinguish a single componerit of the variatioris from the cosmic ray data observed
’* Department of Physics, College of Humanities and Socidl Sciences, lwate University
Artes Libe.rales
.270
No. 48, 1991
at the earth’s stirface have been proposed (e.g., Yoshida et al., 1971 and 1973) .
.The fgrrpulatio.n t.o analyse. three一・qiMe.pgigpq.1.cggpaic,.ray .ani,s.p. 1ropy. ip ..inter一.
planetary spagg’ has. been ...qeyeloped iri ・a, ne.w sc,heme and a .frqrpg wprk from an
extensive analysis of the.g. bsery. ed. , qq. S4. hqs been p, rovideq b.y Nagashima (Nagashima,
1971, Nagashima et al., 1971 and 1972) . The formulation gives a detailed tnethod・
handling daily variation by solar anisotropy iri which a method how to derive zonal
harmonic components・of cosmic rayS in spac’e is iinplied.
In this paper, we ’give the f6rmulation’ for finding out the zonal harmonic
components and the details of method of analysis for finding out’
狽?e components
which, especially, correspond to the north−south asymmetry and the pole−equator
anisotropy respectively frorp cosrpic ray data obtained at cosrpic ray stations dis−
tributed over the earth’s surface,’in reference. to the.Nagashima’s formulation
(Nagashima, 1971) , together With soMe results of its ptactichl ,analysiS which are
newly ’ 盾b狽≠奄獅?d from re−computation on the basis of somewhat different idea ’from
earlier ones (Tqkahashi e,t aL, 1974, 1975, 1977, 198’1 arid 1983) ..
2. Formulation
The formulation of cosmiC ray daily variation arising from a generalized axis
symmetric anisotropy in space has been d.eveloped by Nagashima (1971) as the basis
of the exterisiye・ analysis which’ utili’zes effectively all the assoeiated harmonic
components of daily variation. Ac60rding to the formulation, the variational intensity
distribution in $pace is defined by the fQllowing equation,
.61(P,.λ:,4ノ/ノ(P,♪=F(X.♪G(P)L.
@・・;,・・一・・・・・・・・・・・・・… 一・・ll・・。・・・… 。・・.・.・・。・・。・鱒・・・… (2一一1)
where P is the ’cosmic ray rigidity, G(P) is the ・diff6rential i igidity spectrum and F(x)
is c’alled the space・ distribution.・ Quantity X defines the’ cosmic ray incident direction
(JO) relative to the refetence. axis (・OR) as shown in Fig. 1..
Space distribution F(X) in Eq. (2−1) maY be expanded into a ser’ies of Legendre
.function as/,follows : ・ /
YAHAGI: Zonal Harmo.nic Analysis of Cosmic Ray Data/ ・ . 271
F(;り=ΣFn(X)=Σηη、P£ (COS X),.・・
磨h””:’’’’’’””’’”””“”’”’(2−2)
一 一 一
””’”
n=O n=e
whe「e聯ノi・ca11・d中・n−th・pace・di・t・ib・ti…C・effi・1・ntη・i・an・・bit・a・y・・n・ta・t.
and called the magnitude of F.(X), where 1 ny. 1 is called the absolute maghitude Qf F.(X):
Since the n−th space di・t・ib・ti・n姻i・Eq・.(2−2).d・es.・・t・lw・y・.・h・w.th・
maximum in the direction of the axis OR (i.e. X−O) shown in Fig. 1, the OR−axis is
called’ 狽?e “reference axis of anisotropy”・ which is different from the “direction of
anisotropy” hitherto used.
REFERENCE AXIS
rd ’ UF ANisbTRopy
R
DIRECTIQN OF
一・ 1.NclDENcE
J
x
Fig.1. A cootdinate’ system’ illustrating
o
the reference axis・’of anisotr.opy. Quantity X
defines the cosmic ray incident direction (JO)
A
relative to the reference axis (OR)’. (after
Nagashima, 1971)
If we make the trqnsformation of the coordinate system shown in Fig. 1 into the
・quato「ial cOo「dina.te・y・t・m a・i・illu・廿・t・d i・. Fig・2・ it・f・11・w・by the additi・n
theQrem・for spherical harmonics.(Courant and Hilbert, 1943・, or Morsc and Feshbach,
1953) that
P2(C…ノー認。P超.(C・・鱗に・・θ・ノ…{吻7・・ノ}・
where, from spherical trigonometry
一一一・・一一・…・・一一
i2−3)
272
Artes Libe;. ales
No. 48, 1991
N
ロ
α
R
α
θ
DIRECTION OF
J
/
OR
J
INCIDENCE
Fig.2. Equatorial coordintite system.
N : Equatorial north pole
X
.Q : Autumnal equinox.
OR : Reference axis of anisotropy
JO : Direction Qf incident.particle
o
aJ apd eJ.: Right ascensiQn and co−
R
declination of .the direction of incident
particle (OJ),respectively
aR and eR:Right ascension and coL
deglination of’the reference axis (OR),
Q
R,
EQUATORIAL
REFERENCE AXIS
OF ANISOTROPY
respectively (after Nagashima, 1971)
PLANE
cos X=cos OJ cos eR十sin eJ sin OR cos(aJ−aR) ・・・・・・・・・・・・・・… ’・・・・・・・・・・・・・… (2−4)
in which aR and eR are the .right ascension and co−declination of the reference axis
(OR) , respectively and aJ. and OJ,are those of ’ 狽?e incident direction JO, respectivelY as
are shown in Fig.2. The subscripts R and/stand f6r the directions of. reference axis
and of iricident particle, respectively. Since the both functions, P.M (cbs OJ) and P.M (cos
eR) in Eq. (2−3) are the semi−notmalized spherical function by Schmidt (e.g., Chapman
and Bartels, 1940) , if w’e denote them by P.M (cos e) in common, for brevity, P.M (cos e)
may be .giyen as follows :
Pn,m(cos 0)±1勢に。∫砂 when m.=0
三
アコ
一
十
⊥Yノ
砺2
.三
.伽
驚
ノ ︵
and
Pza m(cos 6リ
where the .ordinary “associated Legen.dre function”,
whe江7.勿>0,
YAHAGI : Zonal Harmonic Analysis of Cosrrlic Ray Dqta
273
ph.(xt)=a−it2)g!:’i)SILtPxn,.(X‘) it=’coSe.’
Conseqtiently; ’ 奄煤f ?indS from Eq.. (2−3) that Eq. (2−2) becomes
F(X)= S Fn(X)= S { £ fnM(X)}”o’’’’’”1”’’’””””’””’’’’’’’”‘’’’’’’’’’”(2Ls)
n=o n=o m=o
and
れ
Fn(X)=ΣノhM (X),・…………・・.……∴…・……・・…………・……・・……・・…・・(2−6)
m=0
where
ノ沼6電∫.♪=ηη1)4でヒ。∫θ,)1『hM(cosθR)cos{m(aJ 7αRノ} for 〃z=0,1,2,・。。,n. … (2−7)
The functi’onf.M’ (X) in Eq. (2−7) is called the Projected component・of space distribution.
Especially, components fnM’s assigned by a definite value of n’are called the associate
projected components of F. (X). The n−th sp’ace distribution produces (n+ 1)一terms of
the assoeiated component, which are indePendeht each 6ther. One of the components,
that is, the one assigned by m=O (i.e. f?) does nQt produce any daily variation, owing
to the absence of functional relation to (aJ−a〈R) (cf.. Eq. (2−7)). The north−south
asymmetry of. cosmic ray intensity belongs to this type of projected component arising
from space distributi6n .F.(X).and’ is expresSed by f? . The term fll is similar to fg in
character, corresponding to the pole−equator anisotropy which has been Pointed out by
sorr}e.workers (Takahashi et . al., 1974, 1975 and’1977;EIY, 1977;Duggal and
Pomerantz, .1978) .
Th・fact・岬.@・.θ・ノi・Eq・(2−7)i・call・d th・“qss・・iatib・f・・t・・”whi・h d・fi…
the pautual relation among the associated projected components fnMis when co−de−
cliriation of the reference axis is assigned by eR. ’
Thr f・・t・・珊.(c・・θ・ノi・Eq・.(2−7).m・y b・ ・all・d・・nv・nti・n・lly・th・.“1・tit・d・.
diS・rib・ti・n”gf.・he・r・jec・・d.・・m・・h・n・・nd・…essed b・L鍋ノ.i・.・・d・…dis−
tingu 奄唐? itself from the space distribqtion. But, strictly speaking, q term of “declination
distributi6n” should be used for. this case rathet thap the “latitude ’distribution”,
274 ...・.. .・・…L・b・・al・・...・.■ .…4・,・lg9!
・1・h・・gh…f・t・ili・r i・・h・f・・id・f…mi・・ay mφ中1・・i・n, Thi…becaur・・h・
・・6」ec・・d・・mp・n・nt i・d・fi・・d・fi・・d i・・pace・P・t.1・th…t・ti・g・y・t・m with th・
earth.
S・1・・.・ni・・t・・py・d・fi・・d by Eq・・(2−1)・.(2−5)・nd(2−7)P・・duce・an i・tさ・・ity
(d・ily)・r・i・tig・Dω,・b・e「v・d・t a・t・ti・n on the ea「.th・which may b・gi・・n by・
superPosition of the intensity variations as follows:
Dω一認。鋼・・」……・…・・…・……・∴……・・∴一∵………・・…………・…(2−8)
where’is solar local time in hour.
.Th・p・・jec・・d・6m・・n・n・鯛ih・E・・,(2−6)and(2−7.)・1・・P・・duce・・h・fgll・wi・g
m−th h。,m。nic c。mp。n。nt o岡。f i。t。。,ity(d・ily)va・i・ti・n at・・t・ti・n
伊ω=ΣD鋼.…………・…∴…………・…1・………………・・一……一(2−9)
η=η;
It finds from the combination of Eqs.(2−8)and(2−9)that
D(t)・・ S伊ω=至量D鋼……・・…………一…・…∵……………・(2−10)
m=O m‘On=m
If w・t・k・the ca・e m−O wh・n th・・ph・・ical fun・ti・n・a・e ca11・d・・nal・u・face
harmonics, we have, from consideration of the relation between Eqs.(2−8),.(2−9)and
(2−10),
D9ωとη。P に。、θ。ノ、9・…・……………一・…・一.・・………・∵……………(2−11)
and
・綱∫出品卿…の}.dP・・………」∵…・………・…………(2−12)
where
・一∫ご働・,……一・………….一1.一……・∴・一・……一……・(2 一13)
and furthermore ...
YAHAGI.: Zonal Harmonic Analysis’ of Cosmic Ray Data・
275
.乙晃 {θ.R(P)}〒.㌍.{cosθOR(IFり} ・… 。。・・一。。… 。・・・・・・… 。・・∴・… 。・。… 一一。.一・・。・・・… (2−14)
In the above equations, Pd, Y(P) and 1 are the cosmic ray cut−off rigidity, the response
function, and the average iptensity ot cosmic ’rays at the station respe6tively, and
LhO { OoR(P)} is the declination (or latitude) distribution defined by NagaShima (1971) ,
where’ @denotes ’coTdeclination of eosmic ray asymptotic orbit at a .station.
D.O’
@(t)’s in Eq. (2−11) are the soTcalled n−th zonal harmonic components, which do not
produce any daily variation. Therefore, hereafter D.O (t)’s ’ [email protected]
DS.
ユ8 m・an・i・・t・・pic c・血P6・・nt・f…mi・・ay i・t…ity・hd P? which is relat・d t・
f? as mentioned before is observed as the north−south asymmetry of cosmic ray
intensity and interpreted as the cosmic ray flow in the direction of the earth’s rotation
axis ・(Nagashima et al.) 1968) . Higher order terms D.O ’
it)’s which have not obseryed
yet,.have complicated declination distributions ’respectively, as iS known from Fig. 3.
As is seen in Fig.3, P92 (cos’OJ), i.e. LS (OJ) is a north一$outh symmetric distribution,
showirig an’ anomaly at the equator, and is called the equatorial anoma16us distribution
→P8
SPHERICAL. FUNCTION
1.0
P2 P?
O.5
Fig.3..Zonal hqrmonic functipns
..P男(cosの, for n==0」1,2」respectively.
o
ecorresponds to the latitude; e=90e
corresponds to the equator.
Pg eorresponds to ・the isotropic component
of cosmic rays in space.
一〇.5
一
一
P曾(60sθ)
一1.0
o
30
60 90 120
150
Pg corresponds to the north−south
asymmetry of cosmic rays in space.
,
P. e2 corresponds to the pole−equator
F
anisotropy of cosmic rays in space.
180
e(o)
(north)
(equator)
(south)
276 .’
D ’” Artes Liber.ales. . ・ .・ ’ No. 48, 1991
.in contrast with the north−south asymmetry. Therefore we see that P8 corresponds to
the pole equator anisotropy.
The quantity.c.O is. the coupling coefficient・(or. 6Qnstant)・ be.tw.een the zonal
harmonic corpbonent D.O・ and the projected component pf the・ space distribution f.O .
A・is seen i・Eq・(2−9).・th・二一t坤・㎜9・ic c・皿P・P・nt.Dmωi・.・・mp6S・d・f a
series of D.M(t), which originates fro’m the.projegted component f.M in Eq. (2−7).
Similarly to the definition of the components, D.M(t)’s.assigned by’ a definite value of n
are・called the associated harmohic components’ of.intensity (daily) variation, which
cah be produced by the n−th space distribution F.(x).
3. Method of Analysis
In Eq. (2−11) , for simplicity if we take the harMonic components up to the second
order, that is, n=O, 1, 2, an intensity variation of cosmic rays at i−th station, lf・XP, which
is expected theoretically, may be giVen as fOllows :
△1野ρ=Σ{ηη」pg (cosθRノ}6㌦=α808.ご十a?c?,i十α26呈,i,……….………・・…・(3−1)
π=0
where
iOi,1・ ・ii・lli’iliiSl elii,/ ・・・・…i・・・・・・・・・・・・・・・・・・・・・・・・・…’・…一‘・・・・・・・・・・・・・・・…L・・…1・・・…(3−2)
If the intensity’ @decrement of cosmic rays observed actually’ at the i−th station is
denoted by AI9・bS (i一一L 2, 3, ......, 1’) the Values of a8, a9 and a20 in Eq. (3−2) are
determined at every day (as function of UT) . frotn the data of cosmic ray intensities
at stations, i一一 1, 2, 3, ’....., ]’
C by the least square fittirig method (see Appendix) , which
is to minimize’ the following sum of square variations weighted by w’s, ’
YAHAGI : Zonal Harmonic Analysis.of Cosmic Ray Data
,己
7
﹂ Σ.=
=
S
277
w,(AI9・bS−AI9・XP)2 ’・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・… 一・・・・・・・・・・・・・・・・… (3−3)
where. api. may be tak.en from the mean cbunting raVe$ at’ the i−th station resPectively.
The’valueS of cO,,’ i for 41mo$t 411 stations distrl.buted over Yhe ea’rth’s surface haye .
been g.alcg.lated from Eq: (2−12・ and 2/ 13) by Yasue et al., (1982) , using the fo116wing
two types of the differential.rigidity speetrum as G(P) in the equation,
for P〈 PL
・の一
for PL$P〈=PH ・・・… 一・一・一,… 一一一…一…・・一…(3−4)
for P> PH
which is normalized at 10 GV .(fot brevity, this tyPe of spectrum is referred to as
“poWer type spectrum”) and
G(P)一
i扮卿一£ノ・…………・1…・∵一・……・……・…一…・…・(・.一5)
which is als’ [email protected], giving a peak intensity in the above
spectrum for the positive 7 (for brevity, this type of Spectrum is .referred to’ as “power
−exponential spectrum”).PL ’ [email protected]. (3−4) are lower and higher cut−off
rigidity in the power type spectrum, respectively and Po in Eq.・ (3−5) characteri4es’the
decaying form in.the higher rigidity side f6r the power−eXp6nential type spectrum.
The valuep of S in Eq.. (3L3)’can’be thus calculated by .usip.g the va.1ties of bb.,i f.or
the gases vihen the assumed spectrum.form is either power.type or power−exp6nential ’
type, or the both types.’
4. Discussion of the method and examples 6f the analysis
As ’is knowp from Eq, (2−11) , we can expect a great pumber of the Zonal harmonic
components correspphding’to n=.O, 1, ... , oo,・ theoretically. .
The c’ase n=O (i.e: D8) corresponds to’ the isotropic component of cbsmic rays in
sPace.’The ease n=1 (i.e. D?) corresponds to the cosmic ray north−south.asymmetry..
・The case n’=2 (i.e. D;) cbrresponds to the pole−equator anisotropy. The higher terms
278
D£’s’
.Artes Liberales
No. 48, 1991
@corresponding to. n ) 3 have rtot been Qbserved yet and moreover, any mechanism
or phenomenon・is’ @expeqted neither theoretically nor experimentally. Even such a
phenomenon corre’sponding to n=2 iS difficult to detect by practical ’observation, as is
shown by such a fact that we.hayc,only a few ’reports (E!y,.1977;Duggal and
Pomerantz, 1978) till the Present since the beginning Qf the cosmic ray observation. ln
other words, it may.be said thqt, in general, such a pheriomenon is unfamiliar with us,
because it may be masked by the north−south asymmetry: lf the change of cosmic ray
intensity at polar region is taken place, that is, when the north−sQuth asymmetry of the
intensity is taken place due to such a change, the pole−equator anisotropy is regarded
as being taken place, even in case with no intensity change at the equator region. To
be detected, such a pole−equator anisotropy have to’ b? distinguished clearly from the
north−south asymmetry.
The method (Eq. (3−3)) given here may be useful to distinguish the pole−equator
anisotropy from the. north−south asytnmetry and to find out the pole−equator
anisotropy, independently on the north−south asymMetry.
Some examples of the analysis using Eq. (3−3) are shown below (Figs. 4 and 5) .
In these analyses we used cg,b c?,i and cg,i for each of which the following spectra are
used fQr their evaluations from Eq. (2−12) , respectively.
(i) power type spe6trum with 7=0, 一〇.2, 一〇.5, and 一1.0. For simplicity, each
spectrurrt with such a 7−value designates their spectrum numbers as Nos. 1, 2, 3 and
4. ln other words, each of the numbers 1, 2, 3 and 4 corresponds to the power type
spectrum with 7=0, 一〇.2, 一〇.5 and 一1.0, respectively.
(ii) power−exponential type spectrum with the pairs (7=2.0, P=20 GV), (7=2.0, P=
30 GV), (7=2.0, P=50 GV), and (7=2.0, P= 100 GV). Similarly to the case of the
power type spectrum, each spectrum designates as Nos. 5, 6, 7, 8’and 9, respectively.
Stating in more detail, for example, the power exponential spectrum with the pair (7
=2.0, P=20 GV) is called .the “No. 5 spectrum” and similarly the others are called
like that.
HQwever, for cg,i in case of n= 0, in Eqs. (2−11) and (3−3),only the power type
spectrurp (spectrum numbers 1,・ 2, 3 and 4) is used, because such a type of spectrum .
is assumed to be plausible for the isotropic component. Thus, combination of these
・pect・a紬・・e・ゆ・・is rapgi・g l t・9・m・叩tS t・4・9×9−324 ca・e・. S−Vql…a・e
calculated on the basis of 27−day period whi/ch corresponds to each solar rotation
YAHAGI : Zonal Harmonic Analysis of Cosmic Ray Dq.ta
1
5
10
15
20
279
25
[o/o]
D.R. g[
・㌧忙
[
1
N−S
o
E
l
a? o
−1
1
P−E o
−1
1
aig o
E
E
−1
p8 2[.R.poooooooeQooooOooaDo6...
pg iE
・窪.;l
l竃三身dl
1 5 10 15 20’ 25
Day in a Solar Rotation lnterval
Fig. 4. lllustrating the results of analysis ’ rigidity spectrum corresponding to the O−th,
for the period, Solar Rotation No. 1873 (28 ’・”一lst and 2nd zonal harmonic components,
June−24 July, 197Q). ’ ’・ respectively. Figures’for POo, Pg and pS indicate
Horizontal: days (from 1 to 27) during a the spectrum number ranging from 1 to 9.
solar rotation period (Rot. No. 1873) .・ N−S indicates the observed cosmic ray’
Vertical:D. R. shows cosmic ray intensity, intensity・ difference between cosmic ray
in O/o at the cosmic ray station, Deep river station at North region (Thule) and South
(Height 145 m;Geograph.’Lat.’46.10e N, ’ region〈MgMurdo).
Long. 77.50“ W;Cut−off rigidity 1.41 GV, ’P−E indicates the cosmic ray intensity’dif−
Count. Rate 19.35× 105/hr).ag, ag and a8 ’・ ference betweeri polar regions:((Thule−
correspond to O−th, lst and 2nd zonal har一 一McMurdo)/2) and equatorial region
monic components, respectively. (Huancayo) stations.
pg, Pg and PO2 show the best−fit differential
.Artes Liberales
280
No. 48, 1991
一t 10..1? 20 25
lo
.一 一 一
0 一 01 0 1 1 0 1 1 0 1
1
.N
一
P E
一
﹁l﹂﹁一﹂IlL﹁1卜1し.﹁ートー﹂
コ
院
D R
1
1 0
一 1
∂
02
P・E
ps 4,[’aoSooooenb.A.oAb.Rb..coauDD
pg ”
pg・
@i[
$ec−tor Polarity ’
鱒dE
L.m−L一.wwL一一
1 5 10 15 20 25
Day in a Solar Rotation lnterval
Fig. 5. The result of analysis of the’
@cosmic
ray data during the soJar rQtation period
(Rot. No.・ 1841). Vertical and Horizontql
axes are the.same as those of Fig. 4.
YAHAGI: Zonal Harmonic Analysis of Cosmic Ray Data
281
interval for these cases by Eq. (3−3) , using the daily mean of the neutroh intensities
from the world−wide network cosmic ray stations through the world data center, VVDC
.一
b2, ltabashi, Tokyg. .The cosmic ray . intensity decrement, AI9・bS, is taken as the
difference between the mean interisity fot the daily mean intensity for each day at each
cosmic ray station during each solar. rotation period and the cotresponding 27−day
interval to be analysed on 27−day basis, as shown in Figs. 4 and 5. The number of the
・cosmic ray stations whose data are used ’in the analysis is about 44 in’Fig. 4 and about
45 in Fig.5 in the maximuln, respectively. Such a low limit of the number of the
statidns in the analysis is mainly caused by lack of data at some stations, although the
upper lipait of the number of the stations was 65 ・at that time, as mentioned before.
It may be clear from Figs. 4 and 5 that the distinction among the isotropic
component, the northLsouth asymmetry and the pole−eqtiator anisotropy is relievabl’
?.
Especially, it may be found that even thc pole−equator anisotropy which is supposed to
be difficult to detect・ in the first−hand data from the observing stati6ns may be
recognized ・clearly in contrast to the other components: From the other poirit of view
it follows that the analysis method is usefu1 and besides powerful to distinguish the pole
−equator anisotropy from.the north−south asymmetry and also from the isotropic
component. As is convenient for a comparison with the corresponding quantities, that
iSl the n.eutron inten$ity at Deep River in contrast to the isotropic componerit a8, N−S
component from the corresponding cosmic ray stations in contrast to a9, and P:E.
component from the.corresponding cosmic ray stations (given as only a ・rough.
.measure) in cbntrast, to a8, are shown ip.the figures (Figs. 4 and 5) .’ lt might be no’
狽奄モ??
that there exists a good correlation between ’the Deep River neutron intensity and ag
through the both figures (Figs. 4 and 5) . Such a tendency has beeh ’seen in the previous
results (Takahashi et al., 1974, 1975 and otherS) . An examination of the significance
of the features (inclusive of a?, a20) will be left for the future.
5. Conclusion
From these, it may be concluded .’that this formu’lation and, accordingly, the
m・th・d.・f・n・ly・1・a・e plq・・ib1・・At th・・a皿・tim・, it fi・d・th・t th・p・le−eq・・t・・
anisotropy phenomena as well as the north一$outh asymmetry ones have been fre一
282
Artes Liberales
No. 48, 1991
quently happened beyond our imagination. The critical examination and also a search
of the mechanism of the ’results・obtained will be a future problem.
Acknowledgements
Data used for the present analysis are complied by the wotld data center, WDC−
C2, ltabashi, Tokyo・in the form of data books. The author expresses.his sincere
appreciation to the director, Dr. Wada, at that time and Mrs. lnoue and the other
members at the. data center. Particular thanks are due to Profs. K. Nagashima, Cosmic
Ray Research Laboratory, Nagoya University and H. Takahashi, Emeritus Professor,
Iwate University for their kind, heartfelt assistance and encouragement throughout
this・ .work. ・The author also would .like to ’ ≠垂垂窒?ciate to Profs. J. Hiura and T.
Takatsuka, Department of Physics, .lwate University for their encouragement and
support through the course of preparing this paper./
This work was done by support of the scientific research fund from the Ministry
of Education in Japan. Computations were made at the Computer Center, lwate
University, Morioka, Japan.
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Nagashima, K., S. P. Duggal and M. A. Pomerantz, Planet. Sci., Lt, 29, 1968.
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Nagashima, K., H. Ueno, K. Fuj’imoto, Z. Fujii and 1. Kbndo, ibid, 26, 31, 1972.
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Taka獅?aShi, H・,.N:th1!fFh.a.gnirand K・ NagasPima, ?roc.’14Fh lnt. Cosmic Ray.Conf.T rvlanchen, West一
’Germany, 4, 1236, 1975.
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YAHAGI: Z6nal HarmQnic Analysis of Cosmic Ray Data
283
1979.’
Thkahashi, H., N. Yahagi and K. Nagashimai Proc. 17th lnt. Cosmic Ray Conf., Paris, Fran6e, 10,
197, 1981.
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Appendix
Details of the least square fitting method used in this analysis・
It is given in the section 3 in the body of this paper that the zonal harmonic
components of cosmic rays in space are obtainable by minimizing the following sum
(Eq. 3−3)
=
7巴
.
﹂ユ
Σ=
S
勧仏準ε一△穿吻2
by the 1’ ?ast square fitting method. This is the purpose to give the details of the least
square fitting method used in the practical analyses.
In the following, the notations a.and ’w are. the same ones as in the body of the
paper,. whereas fS may be corresponded to AleXP, di may be corresponded to cO.’, and x
may be.correSponded to i. ln other words,’if the station number i is designated as a
variable, ‘function’ which is given as a numerical value.may be corresponded to・ the
coupling coefficient c.
A function f(x) of vatiable x may be .expanded .as follows :
f(X)=abdio(x)十aidii(x)十a2ip2(x)’ ・・・・・・・・・…i‘・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・…(A一] ),
where x is an independent variable, di is a functiori of x (e.g. trigonometric series ot・
spherical functions),and a is a constant to be determined by the method ・of’least
squares to’be mentioned below.
If an observed value is designated by y and also its theoretiCal value from the
.284
@. .. .’
D . Artes Libergles.’
@. ... No. 48, 1991.
calcu [email protected].bY f, the reg.idual, v, may be reprgsenlgd bY
v=y−f ”一一…‘;:・・・・・・…一’一’・一・・…一一一・…一・一・・:一・’一・一一’:一・一・・一一一…(A−2)
Determination of a is ’
≠狽狽窒奄b浮狽?d t6 .find the most probable value of a whlch satisfie$ the
’following relation
S= [v2] =minimblm “””・・’・・’・・’’’’’’”””””’・’’””’”:””””’’’””’’’’”(A−3)
by means of the least square method, where
[v2] i [v.v] i2. (vi’vi)
i
Th・・yゆ・1[]mean・ ・umi・q・i・n・紬・quan・i・i・・and・Will・b・u・ed・h・・ugh・ut thi,
section in place of 2L
Expressing the.above relations in ・more detail,
v(x, y;の≡y−f(x) ノ;¢1,2.
s(x, y∫の≡[v2].=[砂一ノ)2]
The necessary condition for which S takes its minimum value is
i3.:1一.=o・ ・・・…1・1・・・・・….・・一….・・・…1・・・・・・・・・・….・・・・・・・・・・・・・・・・・・・・・・・・・・・・・……・・…(A−4)
Namely
謡{Σ@1}=2Σ剛力}=o.
Consequeptly
[ip,・・f] = [di,・’y] ]’=O・ 1・ 2・ ’・・…r・・・・・・・・・・・・・・・・・・・・・・・・・・・・・…;・・…(A−5)
Expressing Eq. A−5 in more detail
YAHAGI ’: Zonal H4rmonic Analysis of Cosrr}ic Ray Data 2ss
ope’〉・・ ・1・・ r ao> ・H l r qo.
epi 1,・ i (epp・/opb.op2),・ 1.ai 1 },1 = H opi ・1 ・.y
op2/・ ,1, .. , .. ・k az,7 1・il i l’K op・2・
and furthermgre...
opoipo qoipi ・ qoop2 〉・’,r ao .〉 ・11 1’r epo y
・opi opo epiepr . oprq2 .1 ・1.ai ’1,1 =・・’j 1・ qi y
ψ・q・.卿1.ψ・ψ・・ a・ . q2.ツ
Therefore
[・…][卿・1[・…] 砺..[ψみ]
[ipiipo] [qiopi] [qiq2] 1 ・ 1 ai [=1 [q, y]
[ψ2ψo][ψ2ψ1][q2q2] α2 [ψ2y] .………・・……・…轡…………(A−6)
Slnce the observed data .are not’always equi−weighted ones,. the “weight” ’for v2 should
be taken i [email protected]. Then Eq. A−4 may be replaced by the following
51.X. {2(w・v2)} =o・・・・・・・・・…’・・・・・・・・・…一・・・・・・・・・・・・・…”・・・・・・・・・・・・・・・・・・・・・・…(A−7)
and furthermore ’
[w・φゴ・ノ] = [ω・φブ・y] , ブ=o,1,2
namely
[卿ψoψo][WOPgOP1][痴・の・] [Zゆ・夕]
[刎9)1ψo][ωψ、q、ゴ[卿ψ1ψ2] ; [Z〃ψ、 y]
[Wop2epO] [Wop20p1 n[聯].[WOP2 Y]’.’’’”1●●●’”の”’曾曾●.”””.●”’”●..〈A−8)
The bractical analyses by this・method were done oh the basis of 27−day interva,1 which
corresponds to a solar rotatiop periQd. y is takeri. as a deviatio’n in tt of the 60Srpic ray .
286
Artes Liberales’
intensity in daily rr!ean at each station at ea’
No. 48, 1991
モ? day for .the 27−day interval,from the
mean value of cosmic ray intensities at each station dtiring the ’cortesponding solar
rotation interval. w is deterrhined from the counting rate at each cosmic ray station
which is divided by a proper number (e.g., 103) ’under such considerat−ion’ that the data
do not become less’precise. Sipce’the coupling coefficient c is dimensionless, a is・
naturally evaluated in unit of O/o. The ’ [email protected]65, which
is dependent of the date when the cosmic ray.data.are obtqined at the station.s. At some
date, some stations may ptovide their data; but some lother stations may lack their data
at the same date. For such a reasOn, the data from about 45 stations were, on an
average, available for each day during each solar rotation period throughout the
course of the practical analyses in this study.