A method to determine the directional differential intensity (d.d.i.), expressed in terms of spherical harmonics, from sectored particle data, concurrent interplanetary magnetic field (IMF) and solar wind velocity is presented

Ng, C. K.

English

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/
33O
SH 3.1-11
DETERMINATION OF THE PITCH-ANGLE DISTRIBUTION AND TRANSVERSE
ANISOTRDPY OF INTERPLANETARY PARTICLES
C. K. Ng
Dept of Mathematics, University of Malaya
Kuala Lumpur, Malaysia
I. INTRODUCTION We present a method to determine the directional
differential intensity (d.d.i.), expressed in terms of spherical
harmoni.zs, from sectored particle data, concurrent interplanetary
magnetic field (IMF) and solar-wlnd velocity. In Section 2, we show the
relation between the d.d.i, and the mean sector count rates X,. In
Section 3, we show how to estimate the d.d.i, from the measured X, and
the associated errors due to Polsson statistics. In Sectlon 4, uslng the
above method, we determine the pltch-angle distribution and the
transverse anisotropy of the d.d.i ,oflow energy protons for the Day 118,
197B solar particle event. In Section 5, we discuss an interesting
correlation between the transverse anlsotropy and the IMF direction.
2. RELATION BETWEEN DIRECTIONAL INTENSITY AND SECTOR RATES
We express the particle dlrectlonal differential intensity as
j(p,_,_) = _ P_(cos O) A_(p_ cos m_ + B_(p) sin m_ , (I)
nmO _0
where F_ = the associated Legendre functions, p = particle momentum,
(_= pltch-angle and _ = gyrophase in the standard coordinate system (Ng,
1985). When the telescope points in the direction (y,rp, it measures the
dlfferentlal count rates
C(p,Y,_) = C j(p,8,O) S[ _'(Y,_,8,0) ] sin _ d(_dO, (2)J
where, following Sentman and Baker (unpublished manuscript), we express
the angular response function of the particle telescope as
oo o_
S(_') = _- SkPk(cos g') = Z Sk{Pk(cosY)Pk(cos g) +
k_O kmO
k
+ 2 7[(k-m)'/(k+m) _] P_(cos_)P_(cos _) cos m(_-O)} . (3)
It follows from the orthogonallty of the spherical harmonics that
C(p,_,_) = 41T_Esn/(2n+1)] A,_(p)cos mq + B,,(p)sln mr_ P_(cosY ).(4)
nmO
As the telescope sweeps over sector i, we average (4) to obtain X,(p) the
mean differential count rate over sector i,
oo
x,(p)= E ] { P-. cos )cos >,+
1'_0 mmO
+ B._(p')<P_(cosY_sln m_ >,} , (5)
where < >, = (1/AyJ)_ '+I d_, _ = time spacecraft (s/c) spln-angle
measured from the _ projection of the IMF onto tlmespin plane, and
(_,,_!r,._) defines sector i (see the 2ridcoordinate system in Fig. I).
For multlple-telescope systems (Sanderson & Hynds, 1977), eqn (5) should
/ 330 
SH 3.1-11 
DETERMINATION OF THE 
ANISOTROPY 
PITCH-ANGLE DISTRIBUTION AND 
OF INTERPLANETARY PARTICLES 
C. .<. Ng 
Dept of Mathematics, University of M laya 
Kual a Lur pur, M l aysl a 
TRANSVERSE 
1. INTRODUCTION We present a method to determine the dlrectlonal 
differential Intensity (d.d.l.), expressed In terms of spherical 
harmonics, from sectored particle data, concurrent Interplanetary 
magnetic field (IMF) and solar-wind velocity. In Sect10n 2, we show the 
relatlon between the d.d.l. and the mean sector count rates X,. In 
Sectlon 3, we show how to estlmate the d.d.l. from the measured X, and 
the associated errors due to POlsson statlstlcS. In Section 4, uSing the 
above method, we determlne the pitch-angle dlstrlbutlon and the 
transverse an1sotropy of the d.d.1 of low energy protons for the Day 118, 
1978 solar part1cle event. In Sectlon 5, we diSCUSS an lnterestlng 
correlation between the transverse anisotropy and the IMF dlrectlon. 
2. RELATION BETWEEN DIRECTIONAL INTENSITY AND SECTOR RATES 
We express the partlcle dlrectlonal dlfferentlal lntenslty as 
00 ... 
L P~(cos £1) { A",m(P) cos m0 + B",m(P) Sltl m0}, (1) 
m-O 
where F~ = the associated Legendre functlons, p = partlcle momentum, 
a = pitch-angle and 0 = gyrophase In the standard coordinate system (Ng, 
1985). When the telescope pOints In the dlrectlon (Y,~), It measures the 
differential count rates 
C(p,¥,"'j> = f
n
J (p,a,0) S[ a'(y,~,a,0)] sln a da dO, (2) 
wh re, followlng Sentman and Baker (unpubllshed manuscrlpt), we express 
the angular response funct10n of the particle telescope as 
00 oc:> 
S(e') = .L S Pk(COS e') = L Sk { Pk(cos ¥ )Pk(cos e) + 
+ 2 
k 
L [(k-m) '/(k+m)' ] P~(COS II )P~(cos EI) cos m('l.-0)} • (3) 
It follows from the orthogonallty of the spher1cal harmon1cs that 
n 
C(p,'6, rp = 4TI L [S",I (2n+U] L {A",m(P)c,)S m11 + Bnm (P)Sltl mY"(} P~(.:os Y ). (4) 
"-0 m-O 
As the telescope sweeps over sector 1, we average (4) to obtain X,(p) the 
mean d1fferent1al count rate over sector 1, 
00 n 
X, (p) = 4 7T ~ [Snl (2n+l)] ~ { Anm(p)( P~( os '6 )c.;)s mY] >, + 
n-O m-O 
+ Bnm(p'J  P~(cos Y )Sltl mY) >, } , (5) 
whe-re < >, = (1I6P)J)V.+1 d)tr, 'It = the spacecraft (sIc) spin-angle 
measured from the A proJection of the IMF onto th  spin plane, and 
('/f"y.r'+1) dehnes se.:tor 1 (s -e the 2n  c.;).;)rd1nate syste-m Itl Flg. 1). 
For fIlLlltlple--telesc - systems (Satlderson ~ Hytlds, 1977), eqn (5) should 
https://ntrs.nasa.gov/search.jsp?R=19850026502 2020-03-20T17:01:51+00:00Z
331 SH 3.1-11
be repeated for each telescope.
We now illustrate by specialislng (1_ and (5) to 8-sectored data for
a telescope sweeping in the spacecraft's spin plane:
4
j(p,e,@) = _ A,_oP,_(,:ose) + _- A,_P_(cos e) cos 0, (6)
4 m :3
X, = 4_ _-S._AnoQ_o/(2n+l) + 4TF ?-S,_AnIQ_I/(2n+I), ¢7)
t.'1nO _'Iml
where O_o = < _-_ ag sinJe= cosJyi>_, (8)
JnO
^ _-Onll = <(Wil cos _+ W L= sln yr) b_ slnJt_e ,'osJ_>i, (9)
J--O
ag = coefficient of xj in P_(x), bg = coefflclent^of xj in ^P_(x)/(1-x __
g._= angle between IMF and s/c spln-axls, and (Wi_, Wi=, Wire)denotes a
unit vector in the direction of ExB in the 2nd coordinate system
(Fig. 1). Note that the integrations in (8) and (9) may be performed
readily in ,-losedform.
3. DETERMINATION OF"J FROM MEASURED X_ To simplify notation in the
following, let
D_ = A,_o (n=0,4), D,_= A_-4.1 (n=5,7), (I0"3
R_ = Q,_,o(n=O,4), R_ = Q_-4._ (n=5,7). (11)
7
We least-square fit X_ = _-C,_R_, (i=0,7), (12)
to the 8 measured se,ztorrates X_. This yields
7 7
_- H.jC._=_- R_X_, (n=O,7), (13)
J_O i nO
where Hnj = _'-RnRj__ , (n=O,7; j=O,7). (14)
i
Inverting (13), we have
C.,= _H_ _-Rl_Xl, (n=O,7), (15)
k i
and thence Dn = _-Mn_X_, (n=O,7), (16)
where M,,_is ultimately expressed in terms of Q_o, 3
O_ , and S,_. Using (16), we may determine the _
coefficients A,._in (6) by a matrix multiplication _ ,,.]
into the measured sector rates. When the IMF pro- _ _O_J
jectlon lles on a sector boundary, the symmetric
matrix H_j becomes singular. So we drop the A._o ,,o(_.._.
term in (6) whenever the IMF projection comes _//__'-_ 2wlthln 2° of a sector bounda y. L
Suppose that |(_counts are registered over the 1 ~
time interval t= in sector i. Assuming Polsson F_g. 1
statistics, we estimate
Xi = K,/t=, var(X_) = K,/t_, var(Dn) = _-H_var(X_). (17)
i
If we define the anlsotropy _,_= D,_/Do, then, providing the counts are
not too low,
var(_..)- (1/D_)_- (Mn_ -_ Mo_)= var(X_). (18)
i
Systematic errors are, of course, much harder to estimate.
31 SH 3.1-11 
f.::.r eac  telesc.::. e. be repeated 
e no  
a telescope 
Ill str t  by s ecialising (1) and (5) t  -sect re  ata for 
s ee i  I  t  acecraft'  s i  l : 
4- 3 
.J(p,9,QI) = ~ A..,oP..,(.:.::.s 9) + L A.., 1 P ~ (c OS 9) cos QI, (6) 
"-0 "-1 4 • 3 
i  T[ ~ SnA..,oQ~ (2n+1)  r L S..,Anl0~11(2n+1), (7) 
..,-0 
.., 
here ~    L (8) 
-,-0 
" 
A .., 
OAI =  ( .l 1 (.::.s '/f + W.l2 Slt1 y.'r) L b'J Sln-'f~e c.::.s-' 'f!>i, ('3) 
-, 0 
I 
'J  .::.efh let1t f -' 1t1 ..,(x), 'J  hClent~of ~ 1t1 ~ ~(x)/(1-~;2)'I 
tie = angle bet een I F and sIc spin-axIs, and ( .L1, 12, W..L3 ) denotes a 
it vector In t e irection of ~x~ In t e 2nd c r i ate system 
(Fig. 1). ote that the Integrations In (8) and (9) ay be performed 
rea il  In closed form. 
3. DETER INATION Dr j rRO  EASURED i To Si plify notation in the 
follOWing, let 
On  no (t1=0, ), 0..,  ..,-4.  ( , ), 
!.  f.o (n ,4), !.  O!.- .1 (n=5, 7). 
 
e least-square fIt i  L CnR~, (1=0,7), 
..,-0 
t  t   eas r  ctor t  ,. iS Yi l  
? ? 
L n-, -,  L ~Xf.' (n= ,7) , 
where (n=O,7; J=O,7). 
, 
Inverting (13), we have 
..,  L H;;i! L R~Xi' (n 0,7), 
Ie , 
 then.:  0..,  LM ,X" (n= ,7) , 
to 
here Mnf, IS ultimately expressed In terms of Q~o, 
$.1, and Sn. USing (16), we may determine the 
coeffiCients Anm In (6) by a matrix multiplication 
Int.::. the measured sector rates. When the IMF pro- B 
.]ectl.::.n lies on a sect.::.r boundary, the symmetrIc 
atrix H..,.J beco es Singular. So we drop the A40 
term In (6) whenever the IMF projection comes 
Wit i     to  ry. 
Suppose that 1(, counts are registered over the 
tim  I terval tc In tor 1. uming Oi on 
statistiCS, we estimate 
i = ilt e , var(X i )  K,/t~, vadO )  LM~ivar(Xi)' 
1 
(12) 
(13) 
(14) 
(15) 
(16) 
3 
2. 
F Jg. 1 
(17) 
If we defIne the amsotr.::.py 1;.., = 0..,/00 , then, pr.:;)vldlng the counts are 
not to.::. low, 
varC.;n) :.;: (lID~) L. (M.." ~" ,)2 vadX t ). (18) 
Syste atic errors are, of course, much harder to estimate. 
a82 SH 3.1-11
Fig. 2: 15-m_nute _tenslty and anzsotrop=es of 1.4-2.5 MeV protons
4. _°PLICATION As an example, we show in Fig. 2 the 15-rnln averages
of Aoo, E,1o, _=o, _=o, and -_11 for 1.4 - 2.5 MeV protons, determined
g 1.5 
<C 
~I.O 
i...U'-
12 ) 
\ 
\ 
o 
o 
r<> 
.8 
o 
W 0 
4 
co 
01 -80 
(IJ 
\ 
~-40 J ~-
'&,rfJ 
o 
.2 t 
t~ I ~ 
-
)JJ' o 
I 
-.2 
22 o 
ADO -
~20 
• I 
... + + 
4 8 
332 
v- L 
( 
+ 
t 
H f--; H 
12 16 
DAY 11Q, 
H 
.20 
1978 
o 
SH 3.1-11 
4 
Flg. 2: 15-11JlntJ.te lntenslty and anlsotroples of 1.4-2.5 MeV protons 
4. APPLICATION As an example, we sh,:::ow ltl FIg. 2 the 15-mltl averages 
,:::of Aoo, E,10, E,:zo, ;30, and -~11 for 1.4 - 2.5 MeV protons, deterrrnned 
SH 3.1-11
333
with the above method using 8-sectored particle data (P.I.: R.E. Vogt,
CalTech), concurrent IMF (P.I.: N.F. Ness, GSFCI, and hourly solar wind
speed (P.I.: H. Bridge, MIT), measured aboard IMP-8 and accessed through
NSSDC. Some typical standard errors due to Polsson statistics only are
indicated by vertical bars.
Estimating the spectral slopes by using the corresponding results
for 4-13 MeV protons, we have found the Compton-Gettlng corrections (Ng,
1985) for transformation to the ExB drift frame to be small, 1.0(]2Aoo,
<.01, <.01, <.02 for Aoo, _o, K2o, and _o respectively. (For transfor-
mation to the solar wind frame, the corrections are of the order of
0.04 Aoo, 0.1, 0.3, and 0.5 respectively). Thus Aoo, _o, _2o, _=o essen-
tially characterlse the pltch-angle distribution in the E_xBdrift frame.
5. DISCUSSION The Compton-Gettlng correction, _Wi(3 - pA_o/Aoo +
pAio/5Aoo_ to the transverse anlsotropy -_ is indicated by the dots in
Fig. 2. The ,observed anlsotrc,py varies in phase with this correction but
is much larger It_magnitude. The same feature, even more marked, is seen
for 4-13 MeV protons. What is the cause ,0fthis large discrepancy 9
For Fig. 2, IMP-8's GSE coordinates in R_- varies from (21.6, 21.5,
5.5) to (5.7, 29.5, 18. I). The times when the IMF is connected t,0 the
nominal bow shock (BS) are indicated by horizontal bars in Fig. 2 (Ng &
Roelof, 1977). At _1537 UT Day 119, some solar particles with guiding
centres below the IMF through IMP-8 are probably shadowed by the nose of
the BS, resulting in the observed peak value of -_ = 0.53. However BS
connection does not account for the general variation ,of-_ in Fig. 2.
Close inspection reveals that the sector plot of X_ lags behind the
IMF in directional changes. Hen,=e a tentative explanation is that sc,me
observed 15-mln averages contain a substantial fraction of non-gyrotropic
distributions which reside a short distance (~I gyroradlus) beyc,nd a
'kink' in the IMF. An alternative explanation is as follows. When _s
swings rapidly in an averaging interval such that its average value is
close to one end of the range of values, then a fleld-aligned anisoptropy
_o would "induce" a nonzero value of -_, i.e., the apparent value
of -_ is not real. Further studies with smaller averaging intervals
would clarify this matter.
6. CONCLUSION We have shown how to obtain the directional differen-
tial intensity referred to the standard coordinate system (Ng, 1985) from
sectored particle data and concurrent IMF and solar wind data. The
corrections for tran_formatlon to the ExB drift frame are explicitly
calculated and found to be small for these ~1.5 MeV protons. However,
the new correction formulae would be important for < 500 KeV protons. It
is tentatively suggested that the 'observed' transverse anisotropy may in
large part be induced by a rapidly changing IMF in the presence of a
fleld-allgned anlsotropy.
Acknowledgement Prof E.C. Stone's hospitality and the advice of Drs.
R.A. Mewaldt and T.G. Garrad on CalTech EIS experiment are gratefully
acknowledged. I thank B.L. Ng for helping to prepare the manuscript.
References
Ng, C.K. (1985) Paper SH 3.1-10, this conference.
Sanderson, T.R. & Hynds, R.J. (1977) Planet. Space Sc1., 25, 799.
Ng, C.K. & Roelof, E.C. (1977) EOS Trans. A.G.U., 58, 1205.
 . - 1 
3 
it  t  ove ethod Si g - ct red article ata ( .I.: . . ogt, 
aITech), concurrent I F ( .I.: .F. ess, SFC1, and hourly s lar ind 
s  ( .I.: . ridge, IT), easured oard I P-   ccessed t r h 
SS C. So e tYPical standard err rs due t  POisson st tistics c'nly are 
I dicated  ertical ars. 
Estimating the ~pectral slopes by uSing the corresponding results 
for 4-13 MeV protons, we have found the Compton-Getting corrections eNg, 
)  t r ation  t   rift a    all, (. 02 oo, 
<'01, <'01, <.02 for Aoo, C,10, c,:zo, and t.so respectively. ('For transfor-
ation to the s lar Wind fra e, the corrections are of the order of 
0.04 oo, 0.1, 0.3, and 0.5 respectively). Thus oo, t.,10, z,:zo, c'so essen-
tially characterise the pitch-angle d1strlbutlon In the ~~~ drift frame. 
. IS SSI N he ompt'Jtl-Gettlng c'Jrrectl'Jn, E WJ. (3 - p [)t)/Aoc:>  
p '0/5 oo) t  the transverse an1sotropy -~11 1S Indicated by the dots 1n 
F1g. 2. he observed anisotropy aries 1n phase With th1s correction but 
IS c  l r r 1~ agn1tude. he s e f t r ,  ore ~  IS s  
f r -  e  r t ns. hat 1S t e ca se of t iS l r  lscre a cy? 
For ig. 2, I P-8's SS  coordinates Itl F<:E anes from (21.E., 21.5, 
. ) toj ( . , '3. , .1>. he ti es he  t  I  IS tltlected to t  
inal     I i at   ri ntal ars I  i .  N   
~oelof, 1977). t ~1537 UT ay 119, so e solar particles 1th gUiding 
ntres el  t  I F t r  I P-  r  r ably s a ed  t  se f 
t  , r s lti  I  t  ser e  ea  al e f ~11  . . o ever S 
connection does not account for the general variation of -~11 In Fig. 2. 
lose Inspect10n reveals that the sector plot of Xi lags behind the 
I  I  ir ti al anges. e c   t t ti e planati n IS t t some 
observed 15- ln a erages c ntai  a s stantial fracti  f n-gyrotroplc 
i tri ti ns hic  r Si  a s rt ist  (Nl r r dl s) e o  a 
'~Ink' In the I F. An alter ati e e~planatlon IS as follo s. hen 0& 
sWings rapidly In an averaging Interval such that ItS average value IS 
l  t    f t  r  f l s, t   fl l - llg  ls tr y 
~10 ould "Induce" a nonzero value of -~11' 1.e., the apparent value 
of -~11 lS not real. Further studies With s aller averaging Intervals 
ould clar1f  t 1s atter. 
. CLUSI N e     t i   al l -
tlal Intensity referred to the standard coordlnate syste  (Ng, 1985) from 
sectored part1cle data and concurrent IMF and solar Wind data. The 
r ti ns  tra s ation  t  !;~f! n ft ram   ~;pll l t  
l l t   fou  t   all for t se-l.  e  r t s. o ever, 
the new correction for ulae would be l portant for ~ 500 KeV protons. It 
IS tentat1vely suggested that the 'observed' transverse anisotropy may In 
large part be lnduced by a rapidly changing IMF In the presence of a 
f1eld-allgned an1sotropy. 
ckno ledge ent rof . . t e's spltal1t  and the adVice of rs. 
. . ewaldt a  . . arra   al ec  I  e erl ent r  r t f ll  
acknowledged. I thank B.L. Ng for helping to prepare the anuscript. 
ferences 
Ng, C.K. (1985) Paper SH 3.1-10, thiS conference. 
Sanderson, T. .  ynds, .J. (1977) lanet. Space SCl., 25, 799. 
Ng, C.K.  Roelof, E.C. (1977) EOS Trans. A.G.U., 58, 1205. 


Determination of the pitch-angle distribution and transverse anisotropy of interplanetary particles

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