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A model of the pitch angle distribution of particles in the natural radiation belts
Radiation Measurements,Vol. 26, No. 3, pp. 391-393, 1996 Pergmon P H : S1350-4487(96)00013-3
Copyright © 1996 ElsevierScienceLtd Printed in Great Britain. All rights reserved 135o-4487/96 $15.0o+ 0.00
A MODEL OF THE PITCH ANGLE DISTRIBUTION OF PARTICLES IN THE NATURAL RADIATION BELTS I. V. GETSELEV* and E. A. ULANOVt *Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow 119899, Russia and tThe Central Research Institute of Machine Building, Kaliningrad, Moscow Region 141070, Russia Abstract--An algorithm for calculation of the equatorial pitch angle distribution of the natural radiation belt (NRB) particles is presented. The suggested model permits only the particle flux values to be kept in data files. Utilization of this model for the calculation of radiation effects gives more accuracy than an omnidirectional flux model. Copyright © 1996 Elsevier Science Ltd
1. INTRODUCTION A model which presents the particle flux as an omnidirectional flux J in L,B coordinates has some disadvantages: • the necessity of keeping both necessary values of B and J values in data files; • the absence of information about the angular distribution of the particle flux which is desirable for the calculations of radiation effects in the elements of space vehicle (SV) equipment; • incorrect application of the model for L > 5, owing to appreciable influence of the interplanetary magnetic field on spatial distribution and on B values in near-earth space.
and after that the particle flux on the external surface of an SV may be calculated taking into account its angle distribution. Thus the calculation of the radiation influence of NRB will be more accurate than with the model J in L,B coordinates. Using EPAD, more accurate calculations of the NRB influence on high orbital SV could be made if the integral invariant I is known. The following equation connects the equatorial pitch angle distribution J0 and omnidirectional flux J (Hess, 1968) in the dipole magnetic field:
J(L,B) = 4riB •l "~*{n°/a~':2 jo(L,%) Bo ~
Our model of the equatorial pitch angle distribution (EPAD) of particle flux in NRB will help to eliminate the above-mentioned drawbacks• 2. THEORY The EPAD in this case is a tabular presentation of the functionj0(L,~o, > E) (sm -2 s-L st-m), where ~0 is the equatorial pitch angle of the particle's velocity vector. The necessity of specifying j0 at the equator is caused by the laws of motion of particles trapped by the geomagnetic field. Because of these laws, the distribution o f j in any other point of the magnetic field line does not carry the complete information about NRB particles• The scale of ~ in distributionA(L,%, > E) may be chosen as arbitrary and identical for all L. It permits only a series of j0 values, to be kept in the data files• The functionj(a0 at any other point of space may be found from the following relations: sin2a~ Bi
sin2% Bo
jo(L,ao, > E) =jt(L,a,, > E)
(1)
sinao'cos~0.dao
~/1 - (B/Bo)'sin2ao
. (2)
From this formula we see that by knowing j0 it is easy to find J• The inverse transformation also does not cause any difficulties. The algorithm for the J0 calculation from the distribution of J is given below. For simplification of the formulae the index 0 on j0 and ~ will be neglected. Let N values J and B be known for a certain L, where Js = 0 at an appropriate equatorial pitch angle as = as (as is an angle of loss cone), j~ is obviously equal to 0. F o r every point of space in which the measurement of J, was made, the equatorial pitch angle % may be calculated from: a,=arcsin~.
(3)
From equation (3) in particular, ~L = n/2 follows when B0 = B~. The particles with equatorial pitch angles greater than a~ do not reach the point of space in which the measurement of £ was made• Using equation (2) and equation (3) the expression for J~ 391
I. V. GETSELEV and E. A. ULANOV
392 J,L = 1.5
7•
10s
J
l0 s
10 7
~
107
"~
106
~
1° 6
7
105
~
lOS
104
&
104
~
Jo,L = 4.2
"2
103
t
I
I
t
I
0
0.2
0.4
0.6
0.8
I
I
I
0
0.2
0.4
103 1.0
/
\11 0.6
0.8
1.0
Parameter x
Parameter x
jo(x)
Fig. 1. Pitch angle distributions and appropriate omnidirectional fluxes of electrons for E0.04 MeV (sunspot minimum), x = cos ~0 = (1 -
J(x)
Bo/B)~:~.
Fig. 2. Pitch angle distributions .~(x) and appropriate omnidirectional fluxes o f protons for E0.1 MeV (sunspot minimum), x = cos =o ffi (1 -
J(x)
Bo/B)|/~.
may be written as:
1 •
~ j(u). sin____~.cos~.d.____~+
{;:":
sin2~ ~
1
f
='+ |
j(u). sin__.__~.c._.osu.d.____~a
~=,+|
/1 - sin2~ q sin2ai
t
sin2=~ + 1
sina.cosa.d~ + 4n:](=,)- /1
'£. J(~) N/1
sin2(xi
sin~ sin2~
(5)
(4) From equation (4) follows:
The greater the number of J values known for any L, the higher the accuracy of equation (5). Using equation (5) for omnidirectional flux the following system of equations may be written:
1
J, = 4~" sin2-----~
Js=O
JN- I = JN-~ = 4 n ' - -
|
47[d(~,V- t)"
X/
sin2%
1 -- sin2~N - I
'~|j(o~N-k+t)"SlnO~N-k+~'COSntN-t+((O~N-k+i-0[N-k+t+l)
sin20~u- k . .
I sin2=#-*+ |
+
1
sin2=N-*+| --
sin2~N- k
sin2~N - k
(6)
where 2 < k < N - 1. The system of equation (6) may be solved for j :
~=0
=JN_|.I4n.;l
sin2=N ~ -~
- k+l)"sino~N-k+dCOSCg~-k+ "(~N-k+~--O~N--*+l+')'(4~'~l--~)sin2~N-k+|~- ! Js-k ----JN-k -- 41r" sin2~N-kl .a~.lk-~(~N I --
sin2~N-k+| (7)
where 2 < k ~ N - 1 also.
PITCH ANGLE DISTRIBUTION 3. RESULTS AND CONCLUSION The above-stated algorithm of j0 calculation was made into a computer program. The data (The Model of Space, 1983) was used for the computer calculations. The most typical results of calculations are presented in Fig. 1 and Fig. 2. Four curves are given on every figure. In these figures x is equal to (1 Bo/B) ~/2or cos ~, since cos ~o = (1 - Bo/B) j/2. All curves j0(cos ~ ) have a maximum at cos ~o = 0 and they decrease with increasing cos ~. The form of curves appears higher at small L (L < 2), and for L > 4 they resemble a step function. The form obtained for the curves and their
393
changing dependence on L conforms with available ideas about the movement of NRB particles. The model of the pitch angle distribution of particles in NRB presented here would be used for working out algorithms and computer programs for calculation of the NRB influence on SV.
-
RM 26/3--E
REFERENCES Hess W. N. (1968) The Radiation Belt and Magnetosphere. Blaisdell, Waltham. The Model of Space (1983) Vol. 3, pp, 335--481, Skobeltsyn Institute of Nuclear Physics, Moscow.
Copyright © 1996 ElsevierScienceLtd Printed in Great Britain. All rights reserved 135o-4487/96 $15.0o+ 0.00
A MODEL OF THE PITCH ANGLE DISTRIBUTION OF PARTICLES IN THE NATURAL RADIATION BELTS I. V. GETSELEV* and E. A. ULANOVt *Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow 119899, Russia and tThe Central Research Institute of Machine Building, Kaliningrad, Moscow Region 141070, Russia Abstract--An algorithm for calculation of the equatorial pitch angle distribution of the natural radiation belt (NRB) particles is presented. The suggested model permits only the particle flux values to be kept in data files. Utilization of this model for the calculation of radiation effects gives more accuracy than an omnidirectional flux model. Copyright © 1996 Elsevier Science Ltd
1. INTRODUCTION A model which presents the particle flux as an omnidirectional flux J in L,B coordinates has some disadvantages: • the necessity of keeping both necessary values of B and J values in data files; • the absence of information about the angular distribution of the particle flux which is desirable for the calculations of radiation effects in the elements of space vehicle (SV) equipment; • incorrect application of the model for L > 5, owing to appreciable influence of the interplanetary magnetic field on spatial distribution and on B values in near-earth space.
and after that the particle flux on the external surface of an SV may be calculated taking into account its angle distribution. Thus the calculation of the radiation influence of NRB will be more accurate than with the model J in L,B coordinates. Using EPAD, more accurate calculations of the NRB influence on high orbital SV could be made if the integral invariant I is known. The following equation connects the equatorial pitch angle distribution J0 and omnidirectional flux J (Hess, 1968) in the dipole magnetic field:
J(L,B) = 4riB •l "~*{n°/a~':2 jo(L,%) Bo ~
Our model of the equatorial pitch angle distribution (EPAD) of particle flux in NRB will help to eliminate the above-mentioned drawbacks• 2. THEORY The EPAD in this case is a tabular presentation of the functionj0(L,~o, > E) (sm -2 s-L st-m), where ~0 is the equatorial pitch angle of the particle's velocity vector. The necessity of specifying j0 at the equator is caused by the laws of motion of particles trapped by the geomagnetic field. Because of these laws, the distribution o f j in any other point of the magnetic field line does not carry the complete information about NRB particles• The scale of ~ in distributionA(L,%, > E) may be chosen as arbitrary and identical for all L. It permits only a series of j0 values, to be kept in the data files• The functionj(a0 at any other point of space may be found from the following relations: sin2a~ Bi
sin2% Bo
jo(L,ao, > E) =jt(L,a,, > E)
(1)
sinao'cos~0.dao
~/1 - (B/Bo)'sin2ao
. (2)
From this formula we see that by knowing j0 it is easy to find J• The inverse transformation also does not cause any difficulties. The algorithm for the J0 calculation from the distribution of J is given below. For simplification of the formulae the index 0 on j0 and ~ will be neglected. Let N values J and B be known for a certain L, where Js = 0 at an appropriate equatorial pitch angle as = as (as is an angle of loss cone), j~ is obviously equal to 0. F o r every point of space in which the measurement of J, was made, the equatorial pitch angle % may be calculated from: a,=arcsin~.
(3)
From equation (3) in particular, ~L = n/2 follows when B0 = B~. The particles with equatorial pitch angles greater than a~ do not reach the point of space in which the measurement of £ was made• Using equation (2) and equation (3) the expression for J~ 391
I. V. GETSELEV and E. A. ULANOV
392 J,L = 1.5
7•
10s
J
l0 s
10 7
~
107
"~
106
~
1° 6
7
105
~
lOS
104
&
104
~
Jo,L = 4.2
"2
103
t
I
I
t
I
0
0.2
0.4
0.6
0.8
I
I
I
0
0.2
0.4
103 1.0
/
\11 0.6
0.8
1.0
Parameter x
Parameter x
jo(x)
Fig. 1. Pitch angle distributions and appropriate omnidirectional fluxes of electrons for E0.04 MeV (sunspot minimum), x = cos ~0 = (1 -
J(x)
Bo/B)~:~.
Fig. 2. Pitch angle distributions .~(x) and appropriate omnidirectional fluxes o f protons for E0.1 MeV (sunspot minimum), x = cos =o ffi (1 -
J(x)
Bo/B)|/~.
may be written as:
1 •
~ j(u). sin____~.cos~.d.____~+
{;:":
sin2~ ~
1
f
='+ |
j(u). sin__.__~.c._.osu.d.____~a
~=,+|
/1 - sin2~ q sin2ai
t
sin2=~ + 1
sina.cosa.d~ + 4n:](=,)- /1
'£. J(~) N/1
sin2(xi
sin~ sin2~
(5)
(4) From equation (4) follows:
The greater the number of J values known for any L, the higher the accuracy of equation (5). Using equation (5) for omnidirectional flux the following system of equations may be written:
1
J, = 4~" sin2-----~
Js=O
JN- I = JN-~ = 4 n ' - -
|
47[d(~,V- t)"
X/
sin2%
1 -- sin2~N - I
'~|j(o~N-k+t)"SlnO~N-k+~'COSntN-t+((O~N-k+i-0[N-k+t+l)
sin20~u- k . .
I sin2=#-*+ |
+
1
sin2=N-*+| --
sin2~N- k
sin2~N - k
(6)
where 2 < k < N - 1. The system of equation (6) may be solved for j :
~=0
=JN_|.I4n.;l
sin2=N ~ -~
- k+l)"sino~N-k+dCOSCg~-k+ "(~N-k+~--O~N--*+l+')'(4~'~l--~)sin2~N-k+|~- ! Js-k ----JN-k -- 41r" sin2~N-kl .a~.lk-~(~N I --
sin2~N-k+| (7)
where 2 < k ~ N - 1 also.
PITCH ANGLE DISTRIBUTION 3. RESULTS AND CONCLUSION The above-stated algorithm of j0 calculation was made into a computer program. The data (The Model of Space, 1983) was used for the computer calculations. The most typical results of calculations are presented in Fig. 1 and Fig. 2. Four curves are given on every figure. In these figures x is equal to (1 Bo/B) ~/2or cos ~, since cos ~o = (1 - Bo/B) j/2. All curves j0(cos ~ ) have a maximum at cos ~o = 0 and they decrease with increasing cos ~. The form of curves appears higher at small L (L < 2), and for L > 4 they resemble a step function. The form obtained for the curves and their
393
changing dependence on L conforms with available ideas about the movement of NRB particles. The model of the pitch angle distribution of particles in NRB presented here would be used for working out algorithms and computer programs for calculation of the NRB influence on SV.
-
RM 26/3--E
REFERENCES Hess W. N. (1968) The Radiation Belt and Magnetosphere. Blaisdell, Waltham. The Model of Space (1983) Vol. 3, pp, 335--481, Skobeltsyn Institute of Nuclear Physics, Moscow.