Field expansion effects on omnidirectional proton counting rate dat

De Forest, S. E.

English

NASA Technical Reports Server

Effec ts  of Different Field Expansions 
on 
Omnidirectional Proton Counting Rate Data 
by 
S. E. &Forest 
Department of Physics 
University of California,  San Diego 
La  J o l l a ,  Cal i fornia  
N66 24112 
December, 1965 
https://ntrs.nasa.gov/search.jsp?R=19660014823 2020-03-16T19:39:04+00:00Z
1 
Inves t iga t ions  have been made a t  the Space Sciences Laboratory 
at the  University of California,  San Diego (UCSD) t o  de t e rmhe  t h e  e f f e c t s  
of impro+od expansions of che e a r t h ' s  mzgnetlc f i e l d  on da ta  t h a t  have al- 
ready w e n  reduced using the  previoiis 49-tem expansion. Two new expan- 
s ims have becn used, the Jensen and Cain, (1964) expansion of 64 terms and 
',he more recerit 100 tern- one. 
The dirfErences i n  the magnitude of €3 using t h e  d i f f e r e n t  expsnsions are 
( 2 z i ~ ~  et 51, 1965, 3 e ~ 2 ~ % c k s  2nd Cain, 1966). 
sml l  f o r  t h e  Itost part. Some d i r e c t  comparisons are seen i n  Table 1. 
Table 1 
Effects of D i f f e r e n t  F ie ld  Expansions 
c -30. .58222 .52500 .52614 0.4008 0.1836 
0 -10. -57397 -57077 *57182 0.5598 (I * 3756 
9 t10. .53778 -53548 .53598 0.4303 0.3370 
o +30. ,49308 .49352 .49327 -0.0894 -0.0377 
0 0. - 56139 -55809 55875 0.5900 9.4726 
*By = €3 ca lcu la ted  from N - t o m  expansion. 
2 
A t  some locations,  t he  percent Pference i s  2-3$. However, this 
does not give an indica t ion  of the  e f f e c t  of t he  f i e l d  changes on t h e  
counting rate d i s t r i b u t i o n  i n  B-L space (McIlwain, 1961) f o r  charged 
p a r t i c l e s .  Three methods can be used t o  demonstrate t h e  e f f e c t s :  
1. An e f f e c t i v e  change i n  B can be defined which r e f l e c t s  
t h e  changes i n  counting rates t o  be expected at a given loca t ion  
i n  E-L space. 
2. 
a t  which r a t e s  based on the old expansion are known. 
rates can then  be compared d i r e c t l y .  
?Jew counting r a t e s  can be computed a t  given poin ts  i n  space 
These two 
3. Actual da t a  poin ts  can be used t o  re -ca lcu la te  B and L. 
Tfiese po in t s  can then be arily-zed i n  the  sane mrfier as t h e  
o r i g i n a l  data. 
The first t w o  approaches have t h e  advantage of producing curves 
which show t h e  expected changes as functions of posit ion,  but they  do not 
give any idea of t h e  e f f e c t  on the  s c a t t e r  of da ta  from f i t t e d  curves. 
Therefore, an ana lys i s  of type (3) must a l s o  be conducted s ince  t h i s  last 
r e s u l t  i s  an important d i r e c t  measure of t h e  value of using a p a r t i c u l e r  
f i e l d  expansion. 
Certain constants a r e  needed f o r  use with the  f irst  two techniques. 
To compute these ,  omnidirectional counting rates from Relay I1 w e r e  f i t t e d  
t o  a l e a s c s q u a r e s  curve f o r  s eve ra l  values of t h e  magnetic parameter L. 
From these  curves, t h e  p a r t i a l  der iva t ives  of t h e  logarithm of t h e  counting 
rate with respect t o  both B and 5 were found. 
3 
1. 
using first. the  o ld  expansion and then t h e  newest (100-terms) one. 
Counting rat,es were found f o r  these B, L coordinates from the  l e a s t  
squares cur-Jes, and an e f f e c t i v e  A B w a s  defined as 
For a given set of geographic points,  B and L were ca lcu la ted  
= a Log (Jh9 / [l per second] ) 
hB,ff (&) / ( ' a  B 
and t h i s  w a s  p lo t t ed  aga ins t  longitude f o r  varicus r a t i o s  of B/B, 
t h e  minimu3 value of B on a gi-Jec l i n e  of fo rce ) .  
L = 1.6 arpears I n  Figure 1. The e f f e c t i v e  A B w a s ,  on t h e  average, 
(Bo i s  
The results f o r  
2-4 times g rea t e r  than t h e  ac tua l  change i n  B. 
2. 
as 
F x  t h e  next compariso_rl, a new counting rate w a s  defined 
where J, = counting rates "or 3-Term expansion 
This repyesents t h e  predicted counting rate f o r  t h e  sane pos i t ion  using 
t h e  r,ew expansion. Then t h e  quantity 
w a s  p lo t t ed  aga ins t  longitrude f o r  s eve ra l  \-aides of B/B 
The r e s u l t s  a r e  shown i n  Ilgure 2 f o r  t h e  cases where N i s  64 and 100. 
For t h e  r a t i o s  of B/Bo much g rea t e r  than  2, t he  $ A J increases  rap id ly ,  bu t  
%he actual values of t h e  counting rates a l s o  decreases rap id ly ,  and t h e  least 
squares f i t  upon which t h e  analysis i s  based, becomes less accurate.  
and L = 1.6. 
0 
4 
3. A s  can be seen from Figure 2, the  f r a c t i o n a l  changes are approxi- 
mately f (1-2)$ f o r  both cases. 
i n  t he  old 49 term f i e l d  then a s ign i f i can t  reduction i n  the s c a t t e r  of 
d a t a  should r e s u l t  from t h e  use of t h e  newer f i e l d  representations.  
g rea t  d e a l  of computer t i m e  would be required t o  reca lcu la te  B and L f o r  
If these changes are i n  f a c t  due t o  e r r o r s  - 
A 
till poiiits ir? ii gL-Jzr; reglor;  f r a  t h e  s s t e l l i t e .  I=stead, t h e  p i E t s  that 
have been interpolated t o  standard values of L are used as these  form a much 
smaller  s e t .  The new values of B a r e  kept with the  in te rpola ted  values of 
L and a new counting rate defined as 
[" a YgL'"' I exp Jn = "49 
This type of correct ion f o r  t h e  changes i n  L should be s u f f i c i e n t  f o r  the 
small changes expected from Figure 2. These correct ions were made on 
Relay I1 omnidirectional da ta  f o r  both the  64 and 100-term expansions. 
The corrected poin ts  were t h e n  f i t t e d  t o  least square curves as before. 
The R.M.S. deviat ion from these  curves a re  given i n  Table 2. 
of L and two ranges of B w e r e  analyzed. 
Figure 2, t h e  difference between the  r e s u l t s  due t o  the  two improved ex- 
pansions is  much less than  the difference between e i t h e r  one and the un- 
corrected data .  It i s  a l s o  seen that the  cor rec t ion  lowers t h e  s c a t t e r  
considerably as long as t h e  r a t i o  of B/B 
than about 3.5. For the  case where t h i s  w a s  v io la ted  
(L = 2.0, B/Bo = 3.6C - 5.141, the s c a t t e r  a c t u a l l y  increased. 
probably due t o  two causes. 
Two values 
A s  might have been predicted from 
considered is  not much g r e a t e r  
0 
This i s  
The p a r t i a l  der iva t ive  with respect t o  L i s  
. .  5 
not h o w n  accurately i n  t h i s  region. And the  f r ac t iona l  change due t o  
the  change i n  the f k l d  i s  p-abably t c o  la rge  t o  j u s t i f y  the  simple 
correct ion t h a t  was vsecl. In t ha t  region it may be necessary t o  re- 
ca lcu la te  B-L from t& oril;inal data before in te rpola t lon .  
Table 2 
Analysis of Corrected Relay I1 Data 
Expan- L sir: BIJn BmN/Bo &/Bo NO. of R.M.S. Aver. 
points  E r ro r  E r ro r  s i a  
49 1.6 .076 .15 1.0 1.97 184 4.07 3.09 
64 1.6 .076 .15 1.0 1.97 186 3.01 2.25 
100 1.6 .076 .15 1.0 1.97 186 2-97 2.22 
49 2.0 .039 -15 1.0 3.85 262 4.55 3.44 
64 2.0 .c39 .15 1.0 3.85 263 3.94 2.95 
106 2.0 ,039 -15 1.0 3.85 263 3.93 2.94 
49 1.6 .14 .20 1.84 2.52 29 3.71 2.80 
64 1.6 .14 .20 1.84 2.52 28 2.71 2.10 
1GO 1.6 .14 -20 1.84 2.52 28 2.68 2.09 
43 2.3 .I4 .20 3.60 5.14 aa 5.45 4.46 
64 2.0 .14 .20 3.60 5.14 86 6.48 5.10 
100 2.0 .14 .20 3.60 5.14 86 6.30 5.01 
*I3 i n  gauss 
The decrease i n  s c a t t e r  i s  a c t u a l l y  more than  it seems. If aT i s  
t h e  t o t a l  s c a t t e r ,  and 0 is the s c a t t e r  due t o  e r r o r s  i n  t h e  f i e l d ,  and 
0 i s  t h e  s c a t t e r  due t o  a l l  other e f f e c t s ,  then 
F 
0 
2 
0 
o2 r (52 + (5 
T F 
if it i s  assumed t'mt tne XG-term expansion completely i=aiaves G theii F 
f o r  t h e  first case (L = 1.6, Low B) 
2 
0 
0 R 5 9  
and 
+ g 16. 2 0 T DF 
This gives 
0 x 2.2% F 
as t h e  t o t a l  e r r o r  t h a t  has b e e n  removed. 
It can be seen from Figure 2 t h a t  t h i s  i s  ,ndeed t h e  approximate 
decrease i n  s c a t t e r  t h a t  would be expected i f  the  d i f fe rences  i n  the fields 
were i n  f a c t  primarily due t o  e r ro r s  i n  the  49 term f i e l d .  The s c a t t e r  
Jsing t h e  64 and 100 term f ie lds  is not incons is ten t  with that expected 
from causes o ther  than  magnetic f i e l d  e r ro r s .  It i s  the re fo re  not poss ib le  
t o  estimate t h e  s c a t t e r  which can be attributed t o  t h e  e r r o r s  i n  the 64 and 
100 term fields. 
Figure 3 shows how t h e  counting rate varies with magnetic f i e l d  f o r  
t h e  first two ranges considered i n  Table 2. 
cbiiig ~i ie  +y- L ~ L I U  ~ A ~ U ~ L U U .  
These curves were p lo t t ed  
1 7  1.n I 2 - -  
7 
Figure 4 shows a n  enlarged view of t h e  segment of t h e  curve f o r  
L = 1.6 that I s  enclosed i n  t h e  rectangle. 
as obtained using t h e  49-term expansion and t h e  100-term expansion. 
A l s o  shown are t h e  poin ts  
I n  conclusion, it can be seen t h a t  a s i g n i f i c a n t  reduction i n  
- 
t h e  s c a t t e r  of trapped p a r t i c l e  data can be obtained by t h e  use of t h e  
newer f ie ld  representations.  
Acknow1edgmer;ts 
I thank Professor C. E. McIlwain f o r  he lpfu l  discussion and 
&Trice during t h e  course of t h i s  work. The work w a s  supported i n  p a r t  
bj. t h e  National Aeronautics and Space Administration Contract NASr-116. 
8 
Figure Captions 
Figure 1. 
F i g ~ l r e  2. 
Effect ive change i n  magnetic f i e l d  versus longitude. 
Predicted change i n  cclmting r a t e  v e r s w  longitude. 
Figure 3. Counting r a t e  lrersus ragnet ic  f i e l d .  
Figure 4. Counting r a t e  -7ersus nagnetic f i e l d  with s c a t t e r  of 
data shown. 
9 
References 
Cain, J. C., W. E. Daniels, S h i r l e y  J. Hendricks, and Duane C. Jensen, 
An evalua t ion  of t h e  main geomagnetic f i e l d ,  1940-1962, - J. Geophys. 
- -  Res. ,  70, 3647-3674, 1965. 
Hendricks, SI J., and J. C. Cain, Magnetic f i e l d  data f o r  t rapped-par t ic le  
evaluation, - J. Geophys. - -  Res. ,  77, 346-347, 1966. 
McIlwain, C. E., Coordinates f o r  mapping t h e  d i s t r i b u t i o n  of magnetically 
trapped p a r t i c l e s ,  - J. Geophys. - -  R e s . ,  66, 3681-3691, 1961. 

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c I 
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p o o  - 
9 
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c 
11 
c 
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m 
r 
. -  
49-TERM EXPANSION 
I m  I 
IUI in  Gauss 
. 
900 
800 
700 
cu 
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0 
0 
900 
800 
a 
700 
I I I 
X 
Uncorrected Data - 49-Term Expansion 
X 
Corrected Data - 100 - Term Expansion 
0.120 0.122 0. I24 0.126 
(0) IN GAUSS 
0.128 


Effects of different field expansions on omnidirectional proton counting rate data

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