Estimating the Energy Level of the Van Allen Belts
R. Dodda
The purpose of this experiment is to investigate the factors that cause the energy level of
the Van Allen Radiation Belts to change. Two possible forms of the source functions will
be checked to see if they serve as a good estimation of the energy entering the belts.
INTRODUCTION
The following differential equation
describes the energy in the radiation
belts:
dE
E
 P (t ) 
dt

(EQ1)
where E is the energy of the radiation
belts. P(t) is the source function that
corresponds the energy being pumped
into the belts. A good estimation for the
P(t) based on measurable parameters is a
very useful in Space Physics. Two forms
discussed below have been suggested for
P(t), and in this experiment we will
determine which is a better estimation.
The purpose of this experiment is to
determine the following two quantities:
PROCEDURE
The
procedure
was
fairly
straightforward. We obtained the Geomagnetic data to calculate the following
parameters during the same time period:
dE
dE
E(t) ,
, P1 (t ) , P2 (t ) . (
was
dt
dt
calculated from E(t) ). Then to calculate
 , we assumed at P(t) was zero in
EQ1 and fit a straight line between the
lhs and rhs of the following equation
which is a modification of EQ1 :
dE
E

dt

1. The value of 
2. The form of function P(t). In
fact, two possible candidates for
P(t) have been suggested. The
experiment will only determine
which if the two forms is better.
The two forms are:
P (t )  v | B | if Bz  0
1
where B is the root of the sum of the
squares of the component of B in y and z
directions. The angle is between the y
and z-components of B.
z
(and P1 (t ) = 0 if Bz > 0)
or

P2 (t )  vB 2 sin 4 ( ) L20
2
To deicide the functional form, we
calculate P1 (t) and P2 (t) from the
specified parameters in appearing in
them. We obtained the raw measured
data for the wind velocities and the
Magnetic Flux to calculate these
functions. E(t) was estimated from the
following
Dessler-Parker-Scopke
formula:
E (t )  
2.8  1013 J
Dst
nT
Then, a regression analysis was done to
find the correlation of between the lhs
and the rhs of the following equation
which is a modification of EQ1. The two
forms for P(t) were plugged in it in
turns. Also, since the solar wind takes
time to reach the earth, the correlation
coefficients were also obtained for a
possible delay of 1 and 2 hours.
RESULTS
The value of  obtained was 1.0E+05
with an error of 1.877E-27.
The table of the obtained correlation
coefficients is:
Correlation
for
P1 (t)
0.127733
No
Time
Lag
1
-0.0313
Hour
Delay
2
-0.0783
Hour
Delay
:
Correlation
for
P2 (t)
-0.39366
-0.506206
-0.39383
The error in the correlation coefficient
was, like that of  , was insignificant.
DISCUSSION
From the table it can be concluded that
P2 (t) is a better function for the intended
purpose. There is also an optimum timedelay to obtain maximum correlation,
and we know that it is certainly between
0 and 2 hours.
REFERENCES
NASA’s Coordinated Data Analysis
Web – http://cdaweb.gsfc.nasa.gov
World Data Center for Geomagnetism,
Kyoto – http://swdcb.kugi.kyoto-u.ac.jp