,qD-A127 324 ELECTRON LOSSES FROM THE MAGNETOSPHERE4U) AIR FORCE 1/2INST OF TECH WRIGHT-PATTERSON RFB OH SCHOOL OFENGINEERING R S DEWEY MAR 83 RFIT.'GNE/PH/83M-4

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ELECTRON LOSSESFROM THE MAGNETOSP1ITE

THESIS

AFIT/GNE/PU/ 3M-.4 Roger S. DeweyLC)R USN

DEPARTMENT OF THE AIR FORCE

AIR UNIVERSITY (ATC) D IAIR FORCE INSTITUTE OF TECHNOLOG ELECT

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Wright-Patterson Air Force Base, Ohio

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ELECTRON LOSSESFROM TEMAGN4ETOSP13ERE

,TIMIS

- '. AIIT/GNE/PH/83M-4 Rloger S. Dewey

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ELECTRON LOSSESFROM THE MAGNETOSPHERE

'p THESIS

Presented to the Faculty of the School of Engineeringof the Ar Force Institute of Technology

Air Universityin Partial Fulfillment or the

Requirements for the Degree orMaster of Science

Aocession For

by INTIS GRA&I MDTIC TABlWannounoed r3

Roger S. Dewey, B.S. ustifiatioLCDR USN ____________t ____

Graduate Nuclear EffectsMarch 1983 By

Distribution/

Availability CodesAvail and/or

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, piApproved tow public release; distisbutlon unlimited.

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ACKNOWLEDGEMENTS

Dr. Joseph F. Janni of the Air Force Weapons Laboratory sponsored this study. I amgreatly indebted to him for his continuing counsel, for his forbearance, and for making

6*. available such tremendous computer support at AFWL.

Deserving of special thanks are Mr. Harry Murphy and Mr. John Burgio o AFWL andMiss Cherise Jarrett of Computer Sciences Corporation for their outstanding programmingsupport, without which this study would have been impossible.

A strong debt of gratitude is owed to Tsgt. George Radke of AFWL, whose advice wasinvaluable, and who kept me mathematically honest.

Particular thanks is also due to my faculty advisors, Majors Robert Davi" and JamesLange. Their cogent advice and timely direction kept me from going too far atray.

Thanks is extended to Miss Casey Anglada of AFWL and Mrs. Antoinette Aguilar ofTetra Corporation for word-processing the text and ,-quations in final form, and to Mrs.Cindy McDonald for typing a preliminary version of this report in spite of my :iandwriting.

Finally, I wish to thank my wife, Helen, for her continued encouragerrent and forputting up with my extended sojourns in Albuquerque.

Roger S. Dewey

-°i

%............ ,.. .... . . .

CONTENTS

PAGE

Acknowledgementx ..............................

List of Figures. .. ..... ...... ..... ...... ..... ..

List of Tables .. ..... ...... ..... ...... ..... .... v

Abstract. .. ... ...... ..... ...... ..... ........ vi

1. Introduction .. .. ...... ..... ...... ..... ........

O bjective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I

Scope. .. ...... ..... ...... ..... ...... ...

Approach. ... ...... ..... ...... ..... ....... 2

Presentation. .. ...... ..... ..... ...... ....... 2

II. Background .. .... ..... ...... ..... ..... ..... 3

Iff. Theory. .. ...... ..... ..... ...... ...... ... 7

*IV. Analysis of Data .. ... ...... ..... ...... ..... .. 23

V. Results and Discussion. ... ..... ...... ..... ....... 71

Vf. Conclusions and Recommendations .. ... ..... ...... ..... 113Conclusions .. ..... ...... ..... ...... ....... 113

Recommendations. .. .... ..... ...... ..... ..... 113

Bibliography .. .. ...... ..... ...... ..... ........ 15

Appendix A: Demonstration of Zero Diffusion Current at the Magnetic Equator forHalt-Integer Bessel Function Solution .. .... ..... ...... .... 120

Appendix B: Derivation of Kigearunction Derivatives. ... ...... ...... 121

Appendix C: The Specific Solution of the Integral-Energy Omnidirectional FluxEquation for the Half-Integer Bessel Function Solution .. .. ..... ..... 122

Appendix D: The General Solution of the Integral-Energy Omnidirectional. FluxEquation for Bessel Functions of Arbitrary Order an a Function of Sigma . . . . 124

Appendix E: Flow Chart for Program Electrofit .. .. ..... ..... .... 125V ia . . . . . . . . . . . . . . . . . . 2

LIST OF FIGURES

FIGURE PAGE

I Loss of a Trapped Particle by a Random Walk into the Pitch Angle LossCone ........ ............................... 8

2 Postulated Decay or Higher Flux Eigenmodes to the Steady State byPitch-Angle Diffusion ....... ....................... 12

3 Temporal Evolution of 1.9 MeV Electrons from the Russian 3 Burst . . . 12

4(a) Raw Allouette Data Collected Only in Region KU (at burst time) ..... .. 47

4(b) Raw Aliouette Data Collected Only in Region fIt (3 days post-burst) . . . 48

5 Raw Telstar Data Collected in Regions I and M (early time) ....... .49

8 Raw Telstar Data Collected in Region HI (mid-range time) ....... SO

7 Raw Telstar Data Collected in Regions I, II, and III (mid-range timi.). 51

8(a) Raw Telstar Data in Regions I and II (late time) ............... 52

8(b) Raw Telstar Data in Regions 1, II, and III (late time) ... .......... 53

9 Typical Raw Data Coverage for 8 Hours (B-L values not shown) ..... ... 4

10(a)-(g) Raw Explorer 16 Data at Times Corresponding toFigure 3 ..... .... ........................... 56-82

11(aHh) Temporal Progression of Raw Telstar Data from I to 90 DaysPost-Burst ....... ........................... 3-70

12(a)-(i) Temproal Progression of Raw Data and Fitted Flux Curves at L = 2.3for Russian 3 Burst (r. freely varying) .... .............. 72-10

13(a)-(h) Temporal Progression of Raw Data and Fitted Flux Curves at L - 2.4for Russian 3 Burst (r. freely varying) .... .............. 81M-

14(a)-(g) Temporal Progression of Raw Data and Fitted Flux Curves at L 2- 2.3for Russian 3 Burst (r. fixed 1:1/9:1/26) .... ............. 8-96

I5(aHm) Temporal Progression of Raw Data and Fitted Flux Curves at L = 2.4for Russian 3 Burst (r. freely varying with physical constraints) . . 16-108

iv

LIST OF TABLES

ih.+

TABLE PAGEI Iligh Altitude Nuclear Detonations .................... 2r11 Satellites and Their Orbital Parameters ....... .................. 26

III Occurrence Frequencies of Time-Differences Between Raw Data Points,Starfish/Telstar . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

IV Occurrence Frequencies of Time-Differences Between Raw Data Points,Argus 1,2,3/Explorer 4 . . . . . . . . . . . . . . . . . . . . . ... 4)

V Occurrence Frequencies or Time-Differences Between Raw Data Points,Russian I/Allouette ....... .......................... .. 31

VI Occurrence Frequencies of Time-Differences Between Raw Data Points,Russian I/Explorer 15 . . . . . . . . . . . . . . . . . . . . . . . .

VII Occurrence Frequencies of Time-Differences Between Raw Data Points,Russian I/Telstar .. '.3

VIll Occurrence Frequencies of Time-Differences Between Raw Data Points,Russian 2/Allouette ......... .......................... :14

IX Occurrence Frequencies of Time-Differences Between Raw Data Points,Russian 2/Explorer 15 . . . . . . . . . . . . . . . . . . . . . . . .. 3

p4 ' X Occurrence Frequencies of Time-Differences Between Raw Data Points,,R ussian 2 /T elatar . . . . . . . . . . . . . . . . . . . . . . .

X Occurrence Frequencies or Time-Differences Between Raw Data Points,Russian 3/AlIouette ............ ........................ 37

XIT Occurrence Frequencies of Time-Differences Between Raw Data Points,• Russian 3/Explorer 15 . . . . . .. .. . . . . . .. .. . . . .. .

XIII Occurrence Frequencies of Time-Differences Between Raw Data Points,Russian 3/Telstar ......... ........................... 39

XIV Global Magnetic Cutoff Values ........ ...................... 40

XV Initial Estimate of Fundamental Decay Time (days) as a Function or L-S-ielland Energy ...... ... .. ............................. 43

XVI Percent Standard Deviation of Initial Fundamental Decay Times (in Tabe XV) 44

XVII Fitted Linear Amplitude Coefficients and Exponential Decay Times at constant/,Shell and Energy ....... ........................ .

N K ABSTRACT

A semi-empirical method for modelling the loss of electron fluxes in thc .'arth's mag-netosphere was developed. An equation for the integral-energy omnidirectional electron fluxas a futionj. of time and magnetic field strength was derived from pitch-angle diffusionp theory.-

>This flux equation was the basis for a computer data-fitting program written -it theAir Force Weapons Laboratory (AFWL) to fit the AFWL Trapped Electron Da..a Base. Theprogram utilized a least-square. fit and incorporated random variations of the characteristicexponential loss times about their initial values. Au, improved table of initial 1le38 times wascompiled for tise with the program.

7 he derived flux model showed substantial agreement with the empirical data base.Representative pints of computed flux over raw data are shown for L-values of 2.3 andl 2.4.

q:.

io vi

* -,-o.•

I. INTRODUCTION

ObjectIv

Modern technology, both civilian and military, depends greatly on the use of satellitesystems. It is thus important to the Department of Defense to understand and predict theenvironment in which satellites may operate, in order to determine survivability and vul-nerability requirements. One part of that environment is the electron fluxes which may exist

-in the trapping regions of the earth's magnetosphere. Stich fluxes may arise from naturalsources or from injection by high-altitude nuclear bursts. Because such fluxes have greatcapability for inflicting damage to sensitive satellite components, even at great distancesfrom a burst, understanding their behavior is vitally important.

There are a large number of theoretical treatments in the literature which describevarious electron loss mechanisms from the earth's magnetosphere. However, the empiricalcalculations of actual electron losses are often conflicting, and frequently are restricted intheir areas of coverage.

• .- Since the cessation of high altitude nuclear testing in 1962, loss studies have been limitedto electrons injected into the magnetosphere by natural magnetic disturbances such as solarstorms. Thus, much of the theory which has been developed since 1962 has had restrictedopportunity for empirical testing.

.The objective of this study is to produce an improved algorithm to calculate losses..-:. of electrons from the magnetosphere. The algorithm is semi-empirical, because it utilizes

the Air Force Weapons Laboratory (AFWL) Trapped Electron Data Base (Ref. 37) measure-meat of electron fluxes (following the old high altitude nuclear bursts) to determine thecharacteristic loss times of those electrons.

The lees times thus determined and the algorithm for flux 'decay" will then be incor-porated by AFWL into the SPECTER computer code (Ref. 7-9) for improved flux calcula-tions of satellite environments.

Scope

This study is limited to investigation of one los mechanism, pitch-angle diffusion, usinga theoretical formalism developed principally by Shuls (Ref. 47,48). The study has also beennecemsarily'limited to thor regions of space and those energies covered by the satellites fromwhich the data base was complied. The los equations thus developed in this study are onlyimplicitly functions of enery, E, and L-coordinate (explained in Chapter II).

The sheer volume of data and the limited time for this study aim limited the testingof the algorithm developed to a representative subset of the entire data base.

.4.

f' .. . ..o .... . . . .

. .•.. . .. °

0 Appuseek

A specific solution of the pitch-angle diffusion equation in derived using the methodproposed by Shuts (Ref. 47). This solution is then fitted to the electron flux data usinga least-quares fitting program written by AFWL, with the coordination of this author.The solutions are performed for integral fluxes (above threshold energy of the satellitedetector) and for constant L-value. The solutions are thus explicitly functions or magnetic

. field strength, B, but implicitly functions of E and L.

Us of the AFWL/NTCTS computer facilities was essential to this study. Well over 72honrs of actual computer-processing time were used, and over 1000 computer-produced fluxplots were examined.

Presentation

A brief Background section is presented, which covers some physical relations andterminology used in the study of space physics. This section is used to establish uniformsymbol notation, and to provide a common starting point for the reader unfamiliar with thespecialised language of space physics. It may be -skipped by the reader more familiar withthe topic.

Chapter [iD outlines the theory used to develop the flux equations which were fitted tothe datL

Following the theory. chapter is a chapter outlining the method of data analysis.Descriptions are given of the data base and of the AFWL computer codes used in this study.

Representative results of the study are presented in Chapter V, in both tabular andgraphic form. Again, the sheer volume of data makes inclusion of all plots and fitt" valuesprohibitive.

Conclusions and recommendations for improvements for further study are presented inChapter VI.

The Bibliography includes 65 sources and provides a comprehensive summary of existingliterature of relevant topics about radiation trapped in the magnetosphere and or pitch-anglediffusion.

.2

., .. . . ..

IL.BACKGROUNDK The equation of motion of a charged particle in magnetic, electric, and external gravita-

tional fields is: x +q

where

fitm = particle mass,

= gravitational constant,q = charge,B= magnetic field,= total electric field (Ref. 11:21-24.)

In a uniform magnetic field, a charged particle will move in helical fashion along iandaround the field lines with a cyclotron radius, p,, (also called gyroradius or Larmor radius)about its "guiding center". The particle's motion may thus be separated into its rotationaround the field line and the motion of its center of rotation, or guiding center, along Ltefield line (Ref. 11:24-25).

The period of cyclotron rotation, r., is defined by (Ref. 39:5):

=2xm Up 2r,

where m is the relativistic ,"--s:

m = OP Me~ V=

and where vj is component of velocity perpendicular to T.The cyclotron frequency issimply 2w over the period (Ref. 39:5):

2r

The angle between a particle's local velocity, iand the magnetic field, Iis it's Pitch-agle, a, which is defined as:

a =arrens()- aresinQL.)

Tn a uniform static magnetic field, vq is c'onstant and v 1 icnstant, an ec s -sat

The particle will then move with a uniform circulihr mot~ion -irmind the field line and iiniforinrectilinear motion along the lield lint-, resumltinr in a he lical motion abotit thef line, sine

dp3/dt = V Xi< f? mnl,and

Mal q(i x )11 0.

U1 = constant,sad:"a- = q(p x ,).,

"" " = (qu.LB)/m = constant, which is a constant centripetal acceleration(Ref.39:4-7; Ref. 11:23-30).

The concept of the first adiabatic invariant arises naturally from the gi iding center' approximation. The first adiabatic invariant, M, (also called the relativisi- ic magnetic

moment) is defined by:

M A -= constant2moB

where . is the particle's perpendicular momentum in the guiding center ap ,roximation.The assumptions implicit in calling M invariant are that the spatial variation )f n is smallcompared to p, and the time variation of B is small compared to r. (Ref. 39: t 1-23).

If a particle has a constant velocity along a field line in the guiding cerLer frame orreference, then

*sin a(a) sin aisin' - =--- = constant

B(;) Bi7 where s is the arc length along the field line and i is any point on the field line. This assumes

that the particle's kinetic energy remains constant as it follows the field line. If a particleenters the field at point B with pitch angle aj, its velocity along the field line is

VI (8) V cos a(s) = v " -a 9,n-ai.I

It the field is increasing in the direction of vi, then the Omirror point" of the particle, whereits parallel velocity is zero, is

B, B(s)

sina sin2a(S)

and the particle has a local pitch angle of 900 at that mirror point (Ref. 39:34 -42).

When the magnetic field has a geometry like that of a dipole field, incr(asing at thepoles and decreasing in the midpoint, then the particle is in the so-called 'magi;etic bottle",and is trapped between the mirror points. There is some minimum B value between themirror points which is called BO, the equatorial field strength. Using the gI iding centerapproximation and approximating the earth's magnetic field by a dipole fiel 1, a particletrapped in the earth's field has three distinct motions. It circles rapidly about:. field line, itbounces along the line between mirror points, and it drifts slowly in longitud, around theearth. All three motions take place with different speeds, so they are distinct (Ref. 11-:25;Ref. 29:34-65).

From the mirror point definition, and from the velocity equations, we see that

M = l

2mof

and that

VLR vsincti iB3VV

Note that the particle's mirror point depends only on its *injection' point and itsinjection pitch angle, not on its initial energy or velocity. A mirror point is a consequenceof the field alone, and all particles injected at the same point with the same pitch angle willmirror at the same point on the field line. Of course, this is only "WtUe if no external forcesare -cting.

The parallel velocity, il, of the particle will be the maximun at B0, wher I B(s) is theminimum. Thus, a trapped particle spends most of its time near the mirror po nts, and theleast amount of its time transiting the equatorial field regions (Ref. 29:34-44).

The bounce period, 4, of a trapped particle is generally much greater than its cyclotronperiod, r,, and is defined by:

J...,v 2 () v e,, VFB()/B)

where s., and s., are the mirror points on the field line.

As a particle in a dipole field like the earth's bounces along the field line, it also driftsperpendicularly to the field line, due to external forces, field gradients, and fie'd curvature,as well as other effects such as time-dependent field changes. This drift is slcw comparedto the bounce period. As the particle bounces and drifts, it traces out a surface between itsmirror points and around the earth, called a drift shell (Ref. 39:9-19).

The concepts of the second and third adiabatic invariants arise from the abo-ve behavior.If the forces acting on the particle remain almost constant over its bounce periodl, the secondadiabatic invariant, J, is defined by

J -pi d = 4 pl do

where pi is the momentum component parallel to A and do measures arc length from theequator (Ref. 45:12; Ref. 6:3-31). If the forces acting on a particle remain almost constantover its drift period, the third adiabatic invariant, -0, which is the magnetic fiux enclosed bya drift shell, is defined by

where A is the magnetic vector potential and where the integration is performed over acurve, s, which lies in the drift shell. The third invariant, *, is defined and c3mputed forthe drift shell of the guiding center, with constant field, and not for a drift shell which aparticle may physically trace out under short-term conditions (Ref. 45:12; Ref 39:76--79).

The guiding drift shell of a particle may also be referred to as an *invari:Lnt surface,"which is composed of field lines which end at the mirror points. The three adiabaic invariantsuniquely define an invariant surface (Ref. 8:3-31 to 3-32).

The more common set of parameters used to define a particle's position in the mag-netosphere (or to define an invariant surface) is the B-L coordinate system. The R parameteris magnetic field strength, and L is the Mcllwain I, parameter delined by L o/I?. wherr.Ii1p is the earth's radius (o 6371 kin) (Ref. 6:3-33) and (fief. 39:53).

The value of L, therefore, is equal to the distance, in earth radii, or t e equatorialpoint on a field line in a dipole field. If the field is not symmetric (not a pe! feet dipole),the invariant surfrce is not so will defined. However, for most field lines in he trapping

• , - o . , " ° . . I

region of the earth's magnetosphere, L varies by less than one percent along the line, sothe B-L system is adequate. The less-than-exact symmetry or the earth's dipale field andoutaide forces may cause particle drift to deviate from a perfect azimuthal cus.Timeans that its L value will vary in the course of' a drift period. This variatiou of L 'valuesis not significant below L 3, but may be so above that value. The average b-value or allintersecting invariant surfaces is called an L-shell. This &-value defines a set of sitrtaces alongfield lines, which may end at differing mirror points. The &shells are considere . to inters.tthe earth's surface, even though mirror points do not extend through the atmo! phere to thesurface (Ref. 0:3-33 to 3-35; Ref. 39:53).

- .7

. reo The various mechanisms which operate to cause trapped particles to change Lshell orto be lost from the trapping region will necessarily violate one or more or the adiabaticinvariants. For example, pitch-angle diffusion violates M or J, or both; and raial diffusionviolates, (Ref. 45:48). These concepts are discussed further in Chapter Ill.

th

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j TTTf. THEORY

While the motion of particles trapped in a magnetic field is well understood, themechanisms of lomes from magnetic fields such as the earth's magnetosphere are less wellunderstood. It is generally agreed, however, that pitch-angle diffusion into the loss cone isone of the predominant mechanisms for removal of charged particles for mid-range L-values.The loss cone angle is the lower limit of pitch angle for trapped particles. Any particles withsmaller pitch angles will mirror in the sensible atmosphere and will be lost by atmosphericscattering.

Roberts (Ref. 41) has noted that a pitch-angle scattering mechanism must be extantfor pitch-angle diffusion to occur. Such a mechanism would necessarily violate one or moreof the adiabatic invariants.

It is not the purpose of this study to ascertain the true source mechani m of pitch-angle diffusion. Such a study is beyond the scope of this paper. Indeed, the search forthis mechanism has been going on for decades. Roberts (Ref. 41) has postulated that the

perturbation forces* causing such diffusion may result from turbulent ambienit electric ormagnetic fields or from collisions with other trapped or non-trapped particles. Lyons (Refs.20-28) has formulated extensive theory based on resonant interactions of so-cah d *whistler"VLF waves with the trapped-particles' gyrofrequencies: "cyclotron-resonamce*. Regardlessof the actual physical mechanism behind pitch-angle diffusion, its general treatment ismathematically the same, and the physical results are the same. (Ref. 41:308) This studyfollows the methods of Roberts (Ref. 41), Shuls and Lanterotti (Ref. 45) and Shuls (Ref.47) in esumim that pitch-angle diffusior is an operative process. One of the purposes ofthis study is the formulation of the equations necessary to validate that assumption againstexperimental data, and to perform that experimental validation.

be"It should be emphasised that, while pitch-angle diffusion is assumed in this paper tobe the dominant Ion mechanism for &-values and altitudes considered, it is not the onlyIons mechanism. Several investigators have formulated radial diffusion (cros-L) theories(for example, Walt (Ref. 58) or Tomasian (Ref. 54)), coulombic or eollisio3-scattering-diffusion theories (for example, Wentworth (Ref. 60)) and multiple diffusion theories, such ascombinations of pitch-aogle, energy, and/or radial diffusion (for example, Wait tRef. 68) andLyons (Ref. 26)), all of which show some agreement with experimental data. I particular,atmospheric scattering is obviously a dominant force at very low L values (Ref. 30).

The introduction of multiple los-mechanisms makes explicit solutions of :my diffusionequation extremely difficult. To simplify the problem and render it amenable t, the methodof Shuls (Ref. 47), this study assumes that *as a rule, radial diffusion enables the radiationbelts to become populated from an external source (or rearranges particles injected by aninternal source), while pitch angle diffusion causes particle loss to an atmospheric sink' (Ref.45:48).

Pitch-angle diffusion, while simple in concept, is complicated in detail. Roberts' (Ref.S41:307-337) treatment of the general mechanism is particularly descriptive:

At the magnetic equator, a particle's pitch angle, ao, is determined by

* 1)= os°= - =

VIM* + P , 2

F.7

WIS =

'o.1

r°

; where ljh and Pj, mre the parallel and perpendicluar component. of the particle's

momentum with respect to the field line at the equatorial value of B0. Roberta next defines

( 2 WI° P

1.- 2mo

where me is the rest mass. These are defined since the first adiabatic invariant, M, isproportional to Wj.:

(3) M eBe

and, if W1. 4C WL., the second adiabatic invariant, J, is approximately prcportional toW I :

where.hr r- the particle's bounce period,

relativistic mass M" t mass mo

A particle's path may be defined by plotting its values of WL. and W1 . s in Figure 1.

.4

o 40(

PARTICLE DIFFUSION

o PATH• . 4o a?

,baco

E CONE

... '["iFig. 1. Lous of a trapped particle by a random walk into the pitch-ancle Ibas cone. (Ref. 41:306)

Note that Una of constat equtorial pitc h angle pm through the origin, because

'i "." " tant o W ' •

" "'NLbi' .'.

...,,. ,.- . . . . ..0

F:.,

Any force which violates either M or , the fit orsecond adiabatic i,.aiant, will../ prdue "dift ion" of the particle's pitch angle s shown in Figure I as Wit, md W4ie

change with time. When a particle's equatorial itch angle diffuses to the val e of the loscone angle,

(5) z Cosa, -0

where Bles is the value at 100 km altitude, then it is lost by atmospheric scattering. "Theloss cone serves as a 'sink' for particles' undergoing pitch angle diffusion (Ref. 41:307).

Roberts makes two very important points about pitch-angle diffusion as a oss-mechan-

im:

(1) with no source, the entire radiation-belt would be depleted of particler,

(2) the loss cone approximation is just that, since the atmosphere is not sh:%rply definedat 100 km. However, if dmespewic pitch-angle scattering is not the primary focus,the los come approximation may be useful. "Naturally, when the loss-.:one conceptis used, detailed agreement between theory and experiment cannot be expected inthe region near the edge of the loss cone" (Ref. 41:308).

Both Roberts (Ref. 41) and Shuls and Lanserotti (Ref. 45) develop the pitch angle

diffusion equation from the Fokker-Planck equation using a particle distribution function,

7.

The Fokker-Planck equation is a formalism which arose from the study of Brownian'*..' motion, and which is used frequently in transport theory (Ref. 42:308). The characteristic

Fokker-Planck equation for trapped particles (which ignores radial diffusion) i:;

L7 I- , (-").,7l + .,, +. 2 -T()D + LDat 'YPO [as d J z T(V) OzV a * Y, 91~ OEij

where the first term, subscripted v, represents non-stochastic (mean) ener" loss to theatmosphere, the second term represents pitch-angle diffusion, and the third ter x representsrange-straggling (energy diffusion). The term T (V) is defined below, and the D/ff and D..terms are the characteristic diffusion coefficients; y is relativistic mass ratio (R( r. 45:55-58).

Roberts uses a distribution of particles in a 'tube' of force about a fhtld line, andShuls and Lanserotti use a phe-eape. dealt di.trbution function which :. essentiallyequivalent to Roberts'. Since this study follows the methods of Shuls (Ref. 4?) and Shulsand Lanserotti, a discussion of phase space is necessary (Ref. 45:15-22).

Any moving particle can be described by specifying its three position coo-dinates andits three canonical momentum components. This completes a six-dimension "ihase-spaCein which a particle can travel in time.

If there exist. a system of a large number of particles in phase-space, th,. system canbe described by a six-dimensional distribution function i(A, fi, t) where

(i - 1,2,3) are canonical momentum components,

. (i = 1,2,3) are position coordinates, and is time.

oq

7a71 7117 - . - .. - -. . -. - -I

.- o...,- .. .. - .- -.o--'

Thus, f d3Ad6V is the number of particles instantaneously occupying S-D volumed$Ad34. According to Liouville's theorem, the phase-apace volume containing the systemof particles moves incompressibly through phase-space.

Since A is an awkward quantity to deal with physically, Shulz and Lanzeretti note thatP transforms to the more familiar 0 as

(SI+

where A[ is the electromagnetic vector potential.

Hence, j(A, 4, t) = f(p, F, t) since 4 = F and the P to i transformation has a unitJacobian. The position-momentum distribution function f(p, F, t) defines the particles oc-cupying the S-D volume dOO? at any given t. The pitch angle diffusion equatin is given interms of a phase-space distribution function which is numerically equal to the more easily-definable position-momentum distribution function (Ref. 45:15-22). At constant energy and

*' L,-shell, and under the action of some source 3, the diffusion equation in phase- ;pace can bewritten ' . a T,]+

Ot r T(V) Oz ax

where T(y) is the quarter-bounce integral path length function of v = Vi--- wherez = coas . The quarter bounce integral gives the length of the trajectory of a particle (inunits of Re, the distance to the equatorial crossing) from the equator to the mirror point.

APA The exact delnition of T is T O do' I "'d

wheres is the distance along the field line and a is the local pitch angle (Ref. 10:4029-4030).

D. is the bounce-averaged diffusion coefficient as a function of z (Ref. 41 and Ref.45). The boundary conditions are that 7 = 0 at z = z, (some cutoff value) and J(z - 0) isfinite.

Shut% and Lanserotti (Ref. 45:162) rewrite the above as

()°7 _ 1-[at z z1 D.L Es V 2 TWv ax

and further approximate that the second term is negligible for z C 1 since r2 + 2 1.If T() o T(I), the second term disappears and the equation is a diffusion equation incylindrical coordinates. (These approximations are not used in this work. This ievelopmeutis used to aid in understanding the exact solutions which follow from Shuls ,'Ref. 47)). IfD.. and 3 are then independent of z, and if 7(s, t) - X(z) T(t), the eigenfur ctions of (8)are Bessel functions of order zero. The general solution to (8) would then be

(9) 7(,) - 1.(z) + .Mw Ao 2wshvee7 is the steady-state solution "k il1

• s) .- )-U ', L

", .. 1.

The . are the zeroes of J, (n =1,2,...) and the i,.(t) vary as e-('Iw, where r =z2./(D2 x2). Thus, the pitch angle distribution (and hence the directional or omnidirectionalflux) is shown to be the sum of a steady state and higher order eigenmodes (Ref. 45:160-18).

The steady state can be thought of as the 'normal" or 'quiet-time' value of flux whichexists in equilibrium with the source 3. An 'injection' of particles would then result in aperturbed distribution function with several eigenmodes, each with a charactc ristic decay-time r.. The higher order modes decay faster, and eventually only the fundamental modewould remain, which would decay exponentially to reach the steady state (after 'infinite'time) (Rer. 41 and 45). There is ample evidence that such a process does hadeed occur.Roberts (Ref. 41) cites Explorer XV data for the 28 October 1962 Russian explosion as anexample. Rosen and Sanders (Ref. 43) also note that decay is faster immediately after solarmagnetic storm activity than during quieter periods. The similarities in Figures 2 and 3illustrate the way in which a decay of higher eigenmodes (Figure 2) can approximate thetemporal evaluation of an actual electron distribution (Figure 3).

The primary difficulties with the above discussion arise from the approximations that:(1) T(V) = T(1) = constant(2) Z" ( I (or V a 1)(3) Ds varies little (or not at all) with z.

Roberts (Ref. 41:311) uses the argument that the full spiral path (and hene. T(V)) variesonly by a factor of 1.4 as a varies from 0 to 0.g and by 1.9 as z varies from 41 to 1. Shulsand Lanserotti (Ref. 45:163) use the simplifying assumptions that T(V) - T(l) and D.. isindependent of z. Both use the mumption that x2 < 1.

The present study reqWres that z be allowed to vary from 0 to z, (up to near 1) inorder to adequately examine the AFWL trapped electron data base (Ref. 37). In addition,if z is not small, the approximation that T(V) M T(1) is a poor one, since I is not close to

The function T(y) can be shown, within 0.57% to be approximated as (Rc f. 10:4030)

','' (I ) T (V) Pi T ( ) - [T (0) - T ( )IV3 4

where T(0) = I' [(In(2 + 3))/(2,3)] % 1.3802 and T(1) = (/--)-- rv 0.7405 (Ref.

45:19).

Obviously, if x approaches 1, y approaches 0 and the T(V) T(1) aproximationbecomes invalid.

The three limiting approximations are removed in the treatment of Shuls (Ref. 47) byintroduction of a now 'canonical" variable z such that:

(12) m Z~) = f ' T(y') dz' V' ~ T(y') dt/

and a corresponding diffusion coefficient

(13) D,, = [:T(V)I D,.

it

t.2YAY

o1 2 20

0 0,2 0.4 0.6 0.6 0 0.? 0.4 06 08

Figure 2. Postulated Decay ct Higher Flux Eigenmodes to the Steady Stpte by Pitch-Angle Diffusion (Ref. 41:313).

L=14

.4304

0.10

Figure~~ 3.&mon itto f1 e k~~ollfo b usa us 1.f41:3&9)

JIM-l23

-. . . . _____

Using (11) in (12) and performing the integration:

"*.4

I e " .I

-4'() I T(O)- 1)Iv"-1ld,'~1W f ''[T(o) - [T(O) -7(,,I""j],

[ - T(O) - T()l""dv'

(14) 10(1 - v) 4(T(O) - T(1)[1 - 11/412 i1

where Z(0) - 16/35 and Z(1) - 0 (Ref. 46:5213).

Using (7), (12) and (13), the diffusion equation becnmes:

-t I +

Shals (Ref. 47:6) states that (15) is a "canonical" diffusion equation in that there is noJacobian factor which fails to commute with D,. as was the case in (7).

The assumptions which now must be made are:

(1) D., is some 'suitably simple function of z" (Ref. 47:6) so that ezset ei genfunctions.9 g.(z) may be specified,

(2) .9, the distributed natural environment source, is independent of z.

Assumption (1) is the limiting assumption, since if the function of z is very -omplex, thediffusion equation becomes extremely difficult to solve. Assumption (2) is both simple andreasonable, however. The source of the natural environment must be close tc constant in

* . order to be the 'driver' of quiet-time equilibrium. It must also be distributed "airly closelyto the steady-state quiet time distribution for the same reason.

Shuls (Ref. 47:7) states that even if Ds, is not of a functional form to yield exacteigenfunctions, there may be a D,. which resembles D, 'closely enough" awid for whichexact eigenfunctions j.(z) (resembling g.(z)) are known.

The following derivatior of the exact eigenfunctions and the correspondin T omnidirec-:- tional differential-energy flux arises directly from Shuls (Ref. 47).

The first assumption by Shuls is that

(6) = ) ( D) .

where u is some number, not necessarily an integer, less than 2, and where D.. in the valueof D, at some : < 16/35 where f vanishes. Roherts (160) also uses this asmmption.

* This form of D,, allows a basis set of orthogonal eigenfunctions to be sl own for theinterval of interest: 0 < z < z,. Following the notation above, we seek st me D,, and

13

*--A

eigenfunctions gn(z) which Oclosely resemble' the true functions and which saisfy

.:.(1) D.+ .()=0

for

with the substitution,-." .,,1..)= z,,,, Cz)

where=--- 1 -

2

and

then (17) becomes Bessel's equation. The exact eigenfunctios of (17) are giver. by

(IS) = &..) X .)/ Ar

where x. are the nth zeroes or the Bessel function J. of the order v, where

:'. =2-uY

The p, given by (18) are normalized so that

*- (181 j g,, g-.. d-, .

The eigenvalues of (18) are given by

C n ( )2 Dz,..[-(lg) =,,_ ,. j2 D,

Shult and Boucher (Ref. 48:6) state that, since particles are not lost at z = 0 (butrather at z m= z.), the diffusion current must go to zero at z = 0. A diffusion current atx - 0 would imply diffusion or pitch angles (and hence mirror points) inte the equator,which is not reasonable. The condition or no dilfusion current at z = 0 co'responds to""mlim...e D a ,.(') = 0.

Shulz and Boucher (Ref. 48) use the series expansion of J. from Abramowic ; and Stegun(Ref. 1) to show the limit as z approaches zero. This author shows the limit directly for the

.-N special case eigenrunction (J-1/2) which is used in this report. The derivatio, is shown inAppendix A.

t4

-. . - ' - ' .- ., .- - . . - . -.. .

* In order to arrive at the diffe vetiai-energy onnidirectional Aux, Shuls (Ref. 47) assumesthatf(z, t)-at7(z) as t--oo; that is, the solution distribution function has some steady-statevalue. This also assumes that 9 is constant in time, but not in z or z. The 1. distribution

. is necessary in order to expand 7 as a series of eigenfunctions in the flux equation. To arriveat an expression for 7.. we start by setting 07/t to zero in (15) and integ!'ating twice,which gives:

(20) f(.- I 9.?) J z" V.

Expanding . as a series of orthogonal eigenfunctions gives:

CDn-0(20a) 9(z') == a . 1.(Z')

and using orthogonality (18a) with (20a) gives:

(2Ob) a. = j (') 0.(-") di".

Substituting (20m) into (20) gives

(20c) (Z) = f. Defo a, 1,(z")d? V.

Rearranging the order of summation and integration in (20c) gives:

(20d) 7.(Z) = J,(') d? d.-'.

Now integrating (17a) twice gives:

or since , is a constant,

(20e) IJ(Z)f' l"('~z~.:Be"e J.(2") W" dz.

Substitution of (20e) into (20d) gives:

.,-":L (20f) 7.(z) = -

Substitution of (20b) into (20n) gives:

(21) = !! f(z ) i

where S(z') is positive (hence a source).

Shuls and Lanzerotti (Ref. 45:39-40) point out that the directional flUK Y(E;V) -* ~ ~ P Am ( 1 pjL; F) where f is the saRte distribt~ton function as in this paper. Sh i (Rer. 47)

iT points out that Imust be integrated over all angles in the unit sphere in p-spr' ce to obtain

the emiire, tional flux J:

J(22) - 47p2 j d(cos a).

Sincesin2a sin2ao

B BO

and, .~V ="-- sin2a(-B)

it tollows that

and that,-oe -(B/B) d(V2)(cos,,) 2 - --n/o)

Now sinceCos 2 a =1-A( Z2)= 1-

it cosa = cosa., V:- = v 2; if cosa = 0, v2 = Bo/B and (22) becomes

J - 4'p2f ' -(B/Bo)ld(v)

0s Vr'f / )

which, upon combining terms and reversing the limits (-):

(23) J = 2 ,.(B" ( B,/ D- o(l) 7 d(e2)

Y,. 2 I -2 (B/ BO)

Integrating (23) by parts.a.

fu d u = uv -Jvydu where u=

-- ,= (B/Bo) d(V2 )::.:. /t ylf(iao)'

-:2

gives:

'.fol) (B ) /) 2

i~v(~~i2 1 ~( f

and since

•(.2) - 7(y2 )

f~in

will give (since the definite integral evaluates to zero, because 7is 0 at V, by defintion):

(24)=+ 2 pf 8 8 2 -V()f Zd

B Oz 0(y2 )

But, from (12), dz =-y T(y) dy. So

dz __-y T(y) dy __-v T (V) dg_ .-T(y)d(,r') d(V2) 2ydy2

Hence, (24) ultimately becomes:

(25) J = 12pf *" IV2( ~y) ZdV)

Shuts then expands 7as a weighted series of eigenrunctions:

(26) 7=74)+ A.(E, L; t) ()

This allows i he flux J in (25) to be written:

(27)J 219Ff 1- r.(z) T(V) dy;) 1V ()TvV()~2~

BOB

The following two expressions are derived in Appendix B.

(28)

Mz) 2z. )k S) \ze)

The actual form of the flux equation awaits the choice of a, which als, determinesthe dependence of D. on z. Shulz uses a - 0 for purposes of comparisc a of severalcomputational methoda. This choice is also attractive because it makes D.. independentof z (see 17(b)) which was a major assumption to begin with:

(30) =s z,

This choice is made in the presenit study for the above reasons and also for the sake ofsimplicity in computations.

* 17

With -, 0, the Bessel unctios within cigenfunctions become order -1/1, which canbe simply written as:

(31) .L1 12 (R)= ;r cos(R),

',°R

(32) F2 _ cos (R)_1/2(R ) _- sin(R) ,

V R xR2V6i/7(;rR)

and the zeroes it. occur at (2n + 1)(r/2).

This simplifies (32), since the second term will always be zero if J ) is evaluated

at x. = (2n + l)(r/2).

Equation (32) becomes

2

J-1/2(r-n-- n l)/2 sin ((2n +1)r

.. '(. ) v't( ")-- .2 + 1)1-r/2)2

The integral with respect to z within (28) would thus be evaluated ar:

"" .q2,,+l)(/2J..,/.ocoo (2nc + 1) d'

* .. i -- So" +)(/2 ; sin((2n + 1) ,/) 2z )d

f 2/7Z, cosi((2n + I)(jr/2ze) z')d'" -sin ((2n + 1)(x/2))

-.- -_ sin ((2n + 1)(,/2)(z/zc))((2n + 1)(x/2z ))(-I)n+l

(34) -- 1. sin((2n + 1)(w/2))"" 1(2 n + 1)(,1/2 z))(- I)"+ t"

The S,, in (28) would be evaluated as (rrom (19)):

" .( , 5 1 , (2 n + 1 , 2 _L ) 2 D ...C

The flux equation (27), using (31), (33), (34), and (35), will then simplify to the followingfor U = 0:

. ..,..,...... .2

(36) J r 1 (B/IRI) 3 fY . i ((2n+ 1) (r/2))

-0n +) )(r/2z, )2 D 5

(220

2,z Tcos ((2nt +2)~/)(/e)

cos((2n + 1)(/Zc) 1i,))

(2n + !!L)(712 _ ~ _ y2(Wl T(y) d(y2)l

-0 ~A(E. L; t)Jfy 2,Ln-U/(2fr(fn -I )(w/2)) (1I)n+'

Z )1z / 2 2__ z,~(n t~~"[2zcn+ 1) Cos (n +)(/1.zz ) ' r 2z

+~z. ) 3r(2n + 1)(r/2)(z:,)

cos ((2n + l)(r/2)(z/z.))

x((2n + I)(t/2)(z/Z))2 v'/((2n + )(Z/z0))

Equation (36) reduces to the following upon simplifleation and cancellation of kerms:

(37) 2(Be~/B8) *0 (2/z,)sin ((2n + I)(x~/2)(z/ze)) . _

J=2s9([f, - , (2n + 1)(s/z)(I * !( ) dv]

rI -A (FjfL;

t) f -s~~in 2ll (n+1

(I The first term in Equation (37) represents a steady state value of the flux, az d the second

term represents a sum of eigenmodes. Each A., (Ref. 45:102) has the form am(l)e- ei(?-E).As n increases, the r.'s become shorter; i.e., the higher eignmodes decay fmiter, so thateventually only the fundamental mode is left, which decays exponentially. At t = 00, thefundamental mode has reached zero, and only the flux remaining is the steady state flux,which corresponds to the natural environment background.

The remaining substitutions in (37) are :, z, T(y), and ye:

L ( -(3/L)

(38) z = ( I= , 2)T(O) 4IT(0)- T(1)1(1 .Y (11/4)

(as shown in Equation (14)) T(y) is given by Equation (1I).

The variable of integration in (37) is taken as V rather than 2 for !zimplicity ofintegration.

ie preceeding derivation has explicitly followed Shuls (1981). However, Shulz didnot sm ow how the differential-energy flux related to the integral-energy flu . Shuls andLanserotti (Ref. 45:163) state that the integral flux 'will then scale as' the diff'rential flux,without derivation. Since the AFWL trapped electron data base (Ref. 37) exists primarily as

X1. % omnidirectional fluxes integrated over broad energy bandpass, the relation must be. derived.

It will be shown in the following and Janserotti (Ref. 45) are indeed ..orrect: Thedifferential-energy flux equation (37) will scale as the integral flux, and the energ3 dependencewill be imbedded in the constants for the source and the eigenmodes.

From Roederer (Ref. 39:86), the integral energy omnidirectional flux can lie written:

'"(39) J /=fd"- J> --- J dE

where J is the differential-energy flux, as in the previous equations, and J>S ir the integralenergy flux above threshold energy E.

A principal assumption will be that energy and pitch angle are both independentvariables of the flux. Thus, (26) would really look like

fln:;!i 139a) R'z, E)= 7,,() 7. ,(z.) A ,q.(C,, L; t) (

and hence in (25) O/0z would look like

(3gb) ---- 7.,(E) Y]",(Z) + A.,(E,; ',; t)

Using (27), (39.t) and (30b), Equation (39) becomes, at a given L-shell:

* .

,_:. ,. .~ ~ -... .- ; - ', ., . . i, . " ' , -. i . ; , . . .

,°-i

(40) C Z

4g= 2f p27,,(E) dEf 7',.,(z) dx - Zwr p2 dE A,(V) f Z z),rf, -- (,

where J7.(z) and 9'.(z) represent all of the angular-dependent runctions of z, ,, or z shownin (25) through (27).

Using the previously stated assumption (Ref. 45:162-1603) that A.L(E) are of the forma.(E)e- '/I('r)0, 40) becomes:

J> p = -2wf p 2 7 ,(E) dEL:' 7.,,(z) dz

(41) - 2w f p2G,(E)eh/v()) dE P.(z) dz

at a given L-shell.

To "remove* the energy dependence or the decay-time terms, the expwetation valueof e- '1 1'- ) is computed, assuming that the second term in (41) represents a listribution-function of the energies.

f; P24a4(EK)t dE(42) fp2a.(E) dE

Then (41) can be written, using (42):

J.- = 2,w.f p27 1 (E) dE]f I.(z)d

(43) -2W 4 [ 2 jP2 (E) dF, (e-' '), f '.(z) dz.

Now since it is assumed that the energy and angular shape functions are separable,the 7.,,(E) must include only those parts of (28) which have energy dependence. The onlyterms in (28) which could have any energy dependence are 9 and .

Shuls (Ref. 47:23) notes that, while 7.,(z) resembles go(z) in functional rorm, 7,.(z)will coincide exactly with go(z) only if 9 is directly proportional to go(z). Fcr simplicity,the assumption is made in this study that .9 is constant over the interval 0 < z < z.. Inphysical terms, this really corresponds to a steady-state isotropic source. It is known thatthe natural environment is not isotropic over either space or Lime, but the viriations aregenerally small compared to the variations from an injection by a nuclear eveit.

Now if S and i. are the only terms in (28) which may have energy depe idence, thenfe (9) in (43) would correspond to

* and since the only portion of the n which may have any energy dependence I'rom (10)) isDS, then . ,(I,) should be proportional to

4 21-!-. - - -....- - - -- -- - -- - - -( v--,--- - -

where D,,(E) may be a constant in E. There may, of course, be some other functionalenergy-dependence included in 7,,,(E) which is yet-undetermined.

Equation (43) is generally of the same functional form as equation (37) if the terms insquare brackets in (43) correspond to the combined constant terms in (37) as d&monstratedabove and as shown in Appendix D. Equation (43) also bears out Shuls and Lanz,rotti's (Ref.45:163) prediction that the integral flux scales with the differential flux; i.e., the 'pitch-angleshape' dependence is invariant for differential- or integral-energy fluxes.

Since it is impractical to evaluate an infinite number of terms in (37), only the termsup to n = 3 will be shown. Shuls and Lanzerotti (Ref. 45:163) state that the I igher modesvanish for n > 2. This in fact means that the higher modes decay so rapidly that modeshigher that n = 3 should not be seen except at very early times. The final rorm of Equation(43), taken to n = 3, is shown in Appendix C. This is the form of th. flux eqi:.tion used in

". the data analysis, except that fewer terms were used than are shown in Apperdix C.

The general solution for the differential-energy flux (which, from (43), is dir.,ctly propor-tional to the integral-energy flux), using Bessel rmnctions of arbitrary orde.r and with ff-dependence explicitly shown, is given in Appendix D.

r.

a'

'*

TV. ANALYSIS OF DATAThe AFWL Trapped Electron Data Base consists of a set of computer tapes of satellite

counting data which have been assembled from historical archives (such as I he NationalSpace Sciences Data Center, NSSDC) (Ref. 37). The data rover injections of elctrons frompnine high-altitude nuclear detonations shown in Table I (Ret. 6:6-2).

The Data Base ha been organised by Pfitzer into a coherent set of tapes whichpresent, by satellite, net electron omnidirectional integral-enerKy (above threshold) fluxes asa function of time, 8, and ,.shell. The computed error% for each data point, in units of fluix,are alo given. The background which was subtracted to give net flux is als(, given (Ref.

.* 37:11, 188-189).

S-•Additional data have been collected by AFWL from plots published in eay literaturp.These plots have been photographically enlarged, and the data points digiti l and placedin computer files with formats similar to the Pfitzer data (Ref. 12:670--671, iRet. 31:646--40,,Ref. 38:637-638).

The satellites which provided the data, and their orbital parameters, are sh.)wn in Table! (Ref. 44:41-46).

- The analysis of the data base required development of three computer ;)rograms bype rsonal at AFWL, in coordination with this author:

(1) OProgpam DTABASE* to read raw data, organize it for processing, and output it*into uniformly formatted files (Ref. 15).

(2) A program to plot the raw data and to plot flux curves generated by the fittingprogram.

(3) 'Program Electrofit', a fitting program to take data points from DTBASE at agiven L-shell and energy and to fit them with the theoretical model teveloped inthis study. It also generates flux curves using the fitted functions at predeterminedtimes to compare with the raw data (Ref. 34).

The first two programs were written with limited input from the author. T'ie third waswritten under close and extensive coordination with this author, an discussed in this chapter.

* 23

The DTABASE program, written by Min Cherise Jarrett of Computer SciencesCorporation, reads the raw data files, in various formats created by Pitter (R,,. 37), satel-lite by satellite, and stores them in a three-dimensional array. The points are then ordered(within each energy group) by increasing time since burst. To generate a file of points at aspecified L-value, each data point L-value is compared to the desired L and is recorded inthe output fle If a match occurs. Additional points are 'created" by linear intorpolation oftwo successive data points if they fall on both sides of the desired L and if the data pointsdiffer in time by no more than a specified value. The interpolation is perforred in L, I,time, flux, and error. The output file can be restricted to a specified time *window* and toa specified range of B values, if desired; however, the data were analyzed over the frll rangeof B and over all times which existed in the Data Base. The program has L e additionalcapability to *B-average" data points at the same L if their individual B values are withina tuer specified limit of each other and they are within the required time difference. Thisprocedure has the effect of eliminating "double' points, and was used for the majority ofthis study with a limit of one percent. Another capability of the program is to interpolateto a specified B-value in a manner analogous to the L-interpolation describedl above; thiscapability was unused in this study (Ref. 15:1-2).

24

...........

'3 .~~'(j00 0 V

I 0 CID 9-

-C

C4 C4 I - I- - -

CL

-w C5 b- M

-vi

In C-1 en .o .0 '0 .0-1 0. I I

0 C4M

01 a1 %A~ IA %ft.0

25

q TABLE 11

Satellites and Their Orbital Parameters

Name Launch Failure Decay Period Perigee Apo-ee Inel.

Date Date Date (min.) (ml.) (mi.) (deg.)

Alouette 1 g-28-62 uak. - 105.4 620 631 80.5

AlphaUpsilon I q-1-62 - 10-26-64 94.4 189 411 82.8

Explorer 15 10-27-62 2-0-6 - 312.0 194 107.30 18.0

l-2un 1 62-61 3-6-63 - 103.8 534 631 67.0

Tram 11-15-61 7-62 - 105.6 562 720 32.4

Telstar 7-10-62 2-21-63 - 157.8 593 35(13 44.8

Explorer 4 7-26-58 10-6-58 10-2359 110.1 163 1372 50.1(Ref. 44:41-45)

S.2

,

. .. .

-In erder to determine the optimum time difference to use for interpolation across the

dsired-,value, the DTABASE program wans modified under direction of this author. Theprogram now presents a summary table of the raw data points for each satellite, showing thefrequency of occuranee of time differences between successive data points from 0.1 minuteto 6.0 minutes, by tenths of a minutes. The cumulative percentage of points with a specifiedtime difference or less is also shown. These summaries are presented for th. bursts andsatellites examined as Tables III through XlU (Ref. 15:67-70). Perusal of these t:ibles showedthat the optimum time difference for interpolation was 2.2 minutes, primarily to include themaximum number of points in the Starfish Telstar file. This time difference waus used for alldata examined in this study.

A final capability of DTABASE is to add data points to a specified L-value file byrounding points in L to the desired value. This procedure assumes that poins which arecloser to the desired L than some specified limit may be considered to occur at the desiredL. Within the errors in satellite measurements, this procedure is reasonable, and in factappears (by data comparison) to have been used by Roberts (Ref. 41:310), although he doesnot so explicitly state. This capability exists in the program, but was not used :n this study(Ref. 15).

The plotting program was developed by Mr. John Burgio of AFWL to generate all theplot@ of raw data and fitted curves shown in this report. The program plots raw data pointsbeginning at a specified start time since burst (in days) and covering the time period in aspecified time *window" (shown in Figure headings as *TW"). The L-value, equatorial B-value, and B-cutoff values ae also shown. I the B-range of points is restricted, the minimumand maximim values of B are shown. The equatorial B value is computed from L by the

494 relation

AO The B-cutoff value was initially computed by using the dipole formula (Ref. 39i:55)

"= 1( B...

Roweer, it was later determined that this approximation might be less than optimum; soa table of L-values versus B.3 , was used and was linearly interpolated across the tabulatedL.values to fnd the B., value. This table (Table XIV) was prepared by AFNVL from the48-term Jensen-Cain model of the magnetosphere (Ref. 16). It was found that use of TableXIV produced B.., (or ji.., or z..,) values more indicative of the global cutoff values. Thistable was used thereafter in both the plotting program and the fitting program (discussedbelow). This procedure is reasonable in view of the use or the same Jensen-Cain model toproduce the B-L coordinates for the Pfitter Data Base (Ref. 37).

The plotting program plots flux as a function of z = /l - (oH/B). It also has the

capability to plot flux data points versus time over all values of L and B, or flux versusenergy at specified B or L. Next to the satellite name on each plot which has fitted curves

*. is a number in square brackets which corresponds to the serial number of the run of thefitting program which prodeuced the curves.

This author's input to the plotting program was limited to supplyivog header information* and specifying layout and format.

The third program used in data analysis is 'Program Rlectrofit," written by Mr. MarryMurphy of AFWL with the close coordination of this author (Ref. 34). Electroit is a least.squares fitting program which uses the functions developed by this author (Appendix C)

27

to it data pointswhich awe output from DTABASE in order to compute the characte'risticdecay time (r.) and linear amplitude constants (a.). These decay times and constants aredetermined at the specific energy and L5-value which are input with the data.

47

24

Table mI

COUNTS OF TIME DIFFERENCES BETWEEN NONZERO RAW DATA POINTSFOR ALL CHANNJELS OF SATELLITE(S): TELSTARFOR BURST: STARFISH

TIME DIN NO. DELTA TIMES CUMULATIVE(MIN) IN DIN PERCENTAGE

0.0 0.1 607 1. 40.1 0.2 3 1. 40.2 0.3 7 1. 50.3 0.4 33 1.50.4 0.5 0 1.50.5 0.6 22 1.60.6 0.7 9 1.60.7 0.9 58 1.70.5 0.9 2 1. 70,9 1.0 219 2.31.0 1. 1 29 2.31.1 1.2 4 2.31.2 1.3 56 2.51.3 1.4 12 2.51.4 1.5 15 2.51.5 1.6 0 251.6 1.7 35 261.7 1.9 it 26198 1.9 19 271.9 2.0 33421 91.42.0 2.1 179 91.92.1 2.2 26 81.92.2 2.3 8 91.92.3 2.4 5 91.92.4 2.5 1 81.92.5 2.6 5 81.92.6 2.7 0 81.92.7 2.8 2 91.92.9 2.9 91.92.9 3.0 50 8203.0 3.1 6 9213.1 3.2 0 9213.2 3.3 3 932.13.3 3.4 1 92.13.4 3.5 1 9213.5 3.6 0 92. 13.6 3.7 2 9213.7 3.8 1 92.13.9 3.9 5 92.13.9 4.0 1451 95 54.0 4. 1 5 95. 54.1 4.2 3 9554.2 4.3 0 85 54.3 4.4 0 9554.4 4.5 0 95.54.5 4.6 2 85 54 6 4.7 6 95 54.7 498 2 855

*498 4 9 2 8554.9 5.0 23 956

5.0O<m 6115 100 0

"2qai.

r - - . . .- : - + - . ' .+ •

" -

UI.UT-%ble TV

COUNTS OF TIME DIFFERENCES BETWEEN NONZERO RAW DATA POINTSFOR ALL CHANNIELS OF SATELLITE(S). EXPLORER4FOR BURST: ARGUS 1 .2 .3PTIME DIN No. DELTA TIMES CUMULATIVE

(MIN) IN DIN PERCENTAGE0.0 0.1 442 63701 02 133 9290.2 0.3 46 9.950.3 0.4 10 90 90.4 0.5 8 92.10.5 0.6 0 92 10.6 0.7 1 9220.7 0.9 1 924098 0.9 0 92 40.9 1.0 0 9241.0 1. 1 0 924.o1 1.2 0 92.41.2 1.3 0 924.41.3 1.4 0 9241.4 1.5 0 92.41.5 1.6 0 92.41.6 1.7 0 92.41.7 1.3 0 92.41.9 1.9 0 92.41.9 2.0 0 92.42.0 2.1 0 92.42.1 2.2 0 92.42 2 2.3 0 92.42.3 2.4 1 9252.4 2.5 0 9252.5 2.6 0 92.52.6 2.7 0 92.52.7 2.8 1 92,729. 2.9 1 92.92.9 3.0 0 92.83.0 3.1 1 92.93. 1 3.2 1 93.13.2 3.3 0 93.13.3 3.4 0 93.13.4 35. 0 93 13.5 3.6 0 93. 13.6 3.7 0 93.13.7 3.8 0 9313.8 3.9 0 93.13.9 4.0 0 93.14.0 4.1 0 93.14.1 4.2 0 93.14.2 4.3 0 93.14.3 4.4 0 93.14.4 4.5 0 93 14.5 46 0 93.14.6 4.7 0 93.14.7 4.9 0 93.14.9 4.9 0 93.14.9 5.0 0 93.1

5.- O< 49 1000.

" '.4.6 .7. .3

Table'V

COUJNTS OF TIM'E DIFFERENCES BETWEEN NONZERO RAW DATA POINTSFOR ALL CHANNIELS OF SATELLITE(S): ALLOUETTE1FOR BURST. RUSSIAN I

TIME DIN NO. DELTA TIMES CUM.ULATIV(MIN) IN DIN PERCENTAGE

0.0 0.1 0 0.00.1 0.2 3413 7270.2 0.3 0 7270.3 0.4 572 84890.4 0.5 1S0 88.70.5 0.6 15 9.00.6 0.7 105 91.20.7 0.9 0 91.20.9 0.9 46 92.20.9 1.0 46 9321.0 1.1 1 9321.1 1.2 5 9331.2 1.3 0 9331.3 1.4 15 9361.4 1.5 4 9371.5 1.6 3 93981.6 1.7 5 9391.7 1.9 0 9391.9 1.9 12 94.11.9 2.0 1 94.22.0 2.1 1 9422.1 2.2 2 9422.2 2.3 0 9422.3 2.4 2 94.32.4 2.5 1 94.32.5 2.6 0 9432.6 2.7 3 9442.7 2.8 0 94.42.8 2.9 0 94.42.9 3.0 0 94.43.0 3.1 1 94.43.1 3.2 0 94 43.2 3.3 0 94.43.3 3.4 3 94.43.4 3.5 0 94.43.5 3.6 0 9443.6 3.7 0 94.43.7 398 0 94 4398 3.9 1 9453.9 4.0 1 9454.0 4.1 0 94 54.1 4.2 3 94 54.2 4.3 0 94 54.3 4.4 1 94 64.4 4.5 0 9464.5 4.6 0 94 64.6 4.7 0 94 64.7 4.9 0 946&44.9 4.9 0 9464.9 5.0 0 94 6

5. 0<- 255 100 0

-.

Table V1

COUNTS OF TIME DIFFERENCES BETWEEN NONZERO RAW DATA POINTSFOR ALL CHANNELS OF SATELLITE(S): EXPLORER15FOR BURST. RUSSIAN 1

TIME BIN NO. DELTA TIMES CUMULATIVE(MIN) IN BIN PERCENTAGE

0.0 0.1 0 0.00.1 02 0 0.00.2 0.3 0 0.00.3 0.4 2 1.70.4 0.5 12 12 00.5 0.6 0 12 00.6 0.7 5 16 20.7 0.8 6 21.40.8 09 1 22.20.9 1.0 0 22 21.0 1.1 13 33.31.1 1.2 a 40.21.2 1.3 0 40.21.3 1.4 1 41.01.4 1.5 0 41.01.5 1.6 0 41.01.6 1.7 i1 50.41.7 1.8 3 53.01.9 1.9 0 53.01.9 2.0 0 53.02.0 2. 1 0 53.02.1 2.2 3 55.6

22 2.3 4 59.02.3 2.4 0 59.02.4 2.5 0 59.02.5 2.6 0 59.02.6 2.7 0 59.02.7 2.8 0 5902.8 2.9 1 5982.9 3.0 0 59.83.0 31 2 61.53.1 3 2 0 61.53.2 3.3 0 61.53.3 3.4 3 64. 13.4 3.5 7 70, 13.5 3.6 0 70 13 6 3.7 3 72 63.7 38 0 72 63,9 39 0 72 6

3.9 4 0 1 73 54.0 4 1 0 73 54.1 4.2 0 7354.2 4.3 0 7354.3 44 0 7354.4 4.5 0 73,5

4.6 4.7 0 7444.7 4.9 0 74 448 4.9 0 74 4

4.9 5.0 0 74 4.o<= 30 100 0

Table VTl

COUNTS OF TIME DIFFERENCES BETWEEN NONZERO RAW DATA POINTSFOR ALL CHANNELS OF SATELLITE(S): TELSTARFOR BURST: RUSSIAN 1

TIME BIN NO. DELTA TIMES CUMULATIVE(MIN) IN BIN PERCENTAGE

0.0 01 0 0.00.1 02 0 0.00.2 0.3 0 0.00.3 0.4 0 0.00.4 0.5 1 0.00 5 06 0 0,00.6 0.7 0 0.00 7 0.8 0 0.00.8 0.9 0 0.00.9 1.0 7 0.21.0 1.1 1 0.21.1 1.2 0 0.21.2 1.3 0 0.21.3 1.4 0 0.21.4 1.5 0 0.21.5 1.6 0 0.21.6 1.7 0 0.21.7 1.9 1 0.21.B 1.9 2 0.31.9 2.0 3637 86.62.0 2.1 3 86.621 2.2 3 86 72.2 2.3 0 86. 72.3 2 4 0 86.72.4 2.5 0 86.72.5 2.6 0 86.72.6 2.7 0 86.72.7 2.9 0 86.72.9 2.9 0 86.72.9. 3.0 6 86.93.0 3.1 0 89.83.1 3.2 0 86.83.2 3.3 0 86.93.3 3.4 0 8683.4 3.5 0 96 93.5 3.6 0 1693.6 3.7 0 86 93.7 3.8 0 S693.8 39 1 8693.9 4.0 134 90 04.0 41 0 9004.1 4.2 0 90.04.2 4.3 0 90.04.3 4.4 2 90. 14.4 4.5 0 90,14.5 4.6 0 9014.6 4.7 0 90 14.7 4.8 0 90. 14.8 4.9 0 90.14.9 5.0 5 90 2

5. 0<- 413 100. 0

-------...-...--..- '-....p.. . ......

Table VII

COUNTS OF TIMIE DIFFERENCES BETWEEN NONZERO RAW' DATA POINTSFOR ALL CHANNJELS OF SATELLITE(S): AL.LOUETTE1FOR BURST: RUSSIAN 2

TIME DIN NO. DELTA TIMES CUMIULATIVE(MIN) IN BIN PERCENTAGE

0.0 01 45 1. 10.1 0.2 3107 79.9

0.3 0.4 409 80.2 0.3 01 9.

0.7 0.9 0 92980.8 0.9 29 9380.9 1.0 25 9421.0 1. 1 3 9431. 1 1.2 13 9461.2 1.3 0 94 61.3 1.4 7 94.91.4 1.5 1 94.91.5 1.6 0 94.91.6 1.7 6 94.91.7 1.9 0 94.91.9 1.9 4 9501.9 2.0 1 95. 12.0 2.1 0 95,12.1 2.2 4 95.22.2 2.3 0 95.22.3 2.4 1 95 22.4 2.5 3 95 32.5 2.6 0 95 32.6 2.7 0 9532.7 2.8 0 95 3298 2.9 0o 95.32.9 3.0 0 95 33.0 3. 1 0 95.33.1 3,2 0 9533.2 3.3 0 95. 33.3 3.4 0 9533.4 3.5 0 9533.5 3.6 0 9533.6 3.7 0 9533.7 398 0 9533.9 3.9 0 95.33.9 4.0 0 9534.0 4. 1 0 95 34. 1 4.2 0 9534.2 4.3 0 95 34.3 4.4 0 95 34.4 4 5 0 95 34.5 4,6 0 95. 34.6 4.7 0 9534.7 49 0 95344.1 4,9 0 95 34.9 5.0 0 95 350o- 199 1000

• N

~qI Table IX

COUNTS OF TIME DIFFERENCES BETWEEN NONZERO RAW DATA POINTSpFOR ALL CHANNELS OF SATELLITE(S): EXPLORER15FOR BURST. RUSSIAN 2

TIME "IN NO. DELTA TIMES CUMULATIVE(MIN) IN BIN PERCENTAGE

0.0 0.1 0 0.00.1 0.2 0 0.00.2 0.3 0 0,00.3 0.4 71 2.90. 4 0.5 264 13. 80.5 0.6 344 28 0

0.6 0.7 142 33.90.7 0.8 77 37.00.9 0.9 31 3830.9 1.0 83 41. 71.0 1. 1 65 44.41. 1 1.2 68 47.21. 2 1.3 64 49.91.3 1.4 63 52.51.4 1.5 115 57,21.5 1.6 75 6031.6 1.7 125 65.417 198 36 66 91.8 1.9 11 67.41.9 20 33 6872.0 2.1 12 69 22. 1 2.2 29 70.42.2 2.3 32 71.92.3 2.4 17 72. 52.4 2.5 42 74.22.5 2.6 14 74. 82.6 2.7 26 75.82.7 2.8 11 76.32.8 2.9 16 76.92.9 3.0 10 77.43.0 3.1 10 77.83.1 32 10 78.23.2 3.3 7 7853.3 3.4 7883.4 3.5 22 79.73.5 3.6 23 90.73.6 3.7 45 2 537 3.9 14 83 13.8 3.9 19 83 93.9 4.0 27 94.94.0 4. 1 15 95 64.1 4.2 to 1. 04.2 4.3 6 86 24.3 4.4 10 9& 64.4 4.5 4 86.94.5 4.6 15 87.44.6 4.7 10 879.4.7 4.9 4 88904.9 4.9 14 98 64.9 5.0 9 98,9

5.0<w 269 100 0

4 .13;

Table X

COUNTS OF TIME DIFFERENCES BETWEEN NONZERO RAW DATA POINTSFOR ALL CHANNELS OF SATELLITE(S): TELSTARFOR BURST: RUSSIAN 2

TIME BIN NO. DELTA TIMES CUMULATIVE(MIN) IN BIN PERCENTAGE

0.0 0.1 0 0.00.1 0.2 0 0.00.2 0.3 0 0 00.3 0.4 0 0 00.4 0.5 0 0,00.5 0.6 0 0 006. 0 7 0 0.00.7 0.8 0 0.00.8 0.9 0 0.00.9 1.0 2 01 11.0 1.1 0 0.11.1 1.2 0 0.112 1.3 0 0.11.3 1.4 0 0.11.4 1.5 0 0.11.5 1.6 0 0.11.6 1.7 0 0. 11.7 1.9 1 0. 11.9 1.9 0 0.11.9 2.0 2244 95.62.0 2.1 49 97.52.1 2.2 1 97.52.2 2.3 0 97.52.3 2.4 0 87. 52.4 2.5 0 87.52.5 2.6 0 87.52.6 2.7 0 97. 52.7 2.9 !0 87.52.8 2.9 0 97.52.9 3.0 3 87.630. 3.1 0 97.63.1 3.2 0 97.63.2 3.3 0 87.63.3 3.4 0 87.63.4 3.5 0 7.63.5 3.6 0 87.63.6 3.7 0 97.63.7 3.9 0 87.63.8 3.9 2 87. 73.9 4.0 100 91.54.0 4.1 2 91.64.1 4.2 0 91.64.2 4.3 0 91.64.3 4.4 0 91.64.4 4.5 0 91.64.5 4.6 0 91.64.6 4.7 .0 91.6

4 4.7 4.9 0 91.64.9 4.9 0 91.64.9 5.0 0 91.6

5. 0<0 221 100. 0

.1-

.. 4:

Table " I

COUNTS OF TIME DIFFERENCES BETWEEN NONZERO RAW DATA POINTSFOR ALL CHA*IELS OF SATELLITE(S): ALLOUETTEIFOR BURST: RUSSIAN 3

TIME DIN NO. DELTA TIMES CUMULATIVE(MIN) IN BIN PERCENTAGE

0.0 0.1 123 0980.1 0.2 10901 74.00.2 0.3 45 74.30.3 0,4 1595 85.00.4 0.5 491 9.3

: 5 0.6 55 9. 70.6 0.7 289 9060.7 0.8 11 90.709 0.9 129 91. 50.9 1.0 74 92.01.0 1.1 a 92.11. 1 1.2 55 92 51.2 1.3 3 92.51.3 1.4 62 92.91.4 1.5 28 93. 11.5 1.6 9 93.11.6 1.7 19 93.31.7 1.9 0 93.31.8 1.9 19 93.41.9 2.0 15 93. 52.0 2.1 2 93.52.1 2.2 10 93.62.2 2.3 0 93.62.3 2.4 13 93.72.4 2.5 7 93.72.5 2.6 1 93.72.6 2.7 7 93.82.7 2.8 0 93.82. B 2.9 "4 93982.9 3.0 2 93.83.0 3.1 2 9383.1 3.2 5 93.93.2 3.3 0 93 933 3.4 2 93.93.4 3.5 0 93.93.5 3.6 1 93.93.6 3.7 1 93.93.7 3.9 0 9393.9 3.9 1 93.93.9 4.0 0 93.94.0 4.1 0 93.94.1 4.2 0 93.94.2 4.3 0 93.94.3 4.4 0 93.94.4 4.5 0 93.94.5 4.6 1 93 94.6 4.7 0 93.94.7 4.6 0 93.94.9 4.9 0 93 94.9 5.0 0 93.9

5. 0<- 911 100.0

Z-

Table XII

COUNTS OF TIME DIFFERENCES BETWEEN NONZERO RAW DATA POINTSFOR ALL CHANNELS OF SATELLITE(S): EXPLORER15FOR BURST: RUSSIAN 3

TIME DIN NO. DELTA TIMES CUMIULATIVE(MIN) IN BIN PERCENTAGE

0.0 0. 1 46 0.30.1 0.2 0 0.30.2 0.3 77 0.80.3 0.4 588 4.80.4 0.5 1654 16.00.5 0.6 2144 30.50.6 0.7 1629 41. 50.7 0.8 970 4800.8 0.9 274 49.90.9 1. O 194 51.21.0 1. 1 226 52.71. 1 1.2 210 54. 11.2 1.3 293 56. 11.3 1.4 273 9.01.4 1.5 474 61. 21.5 1.6 442 64. 11.6 1.7 618 68.31.7 1.9 313 70. 41.9 1.9 194 71.71.9 2.0 131 72.62.0 2.1 82 73.22.1 2.2 92 73.82.2 2.3 74 74.32.3 2.4 54 74.72.4 2.5 45 75.02.5 2.6 35 75.22.6 2.7 53 75.62.7 2.9 34 75.82.9 2.9 37 76.02.9 3.0 40 76.33.0 3.1 29 76. 53. 1 3.2 47 76.83.2 3.3 52 77.23.3 3.4 66 77 63.4 3.5 53 78.03.5 3.6 79 78.53.6 3.7 155 79.63.7 3.8 91 80.23.9 3.9 127 81.03.9 4.0 111 81.84.0 4.1 93 82.34.1 4.2 117 93. 14.2 4.3 87 93.74.3 4.4 110 84.54.4 4. 5 73 85.04.5 4.6 100 85,64.6 4.7 76 8.24.7 4.8 63 96.64.8 4.9 90 87. 14.9 5.0 59 97.5

5. 0<- 1848 100.0

. . .

T.ble Xm

COUNTS OF TIME DIFFERENCES BETWEEN NONZERO RAW DATA POINTSFOR ALL CHANNIELS OF SATELLITE(S): TELSTARFOR BURST: RUSSIAN 3

TIME DIN NO.' DELTA TIMES CUMULATIVE(MIN) IN BIN PERCENTAGE

0.0 0.1 5 000.1 0.2 0 0.00.2 0.3 0 0.00.3 0.4 0 0.00.4 0.5 0 0.00.5 0.6 0 0.00.6 0.7 0 0.00.7 0.9 0 0.00.9 0.9 1 0.00:9 1.0 64 0.31.0 1.1 4 0.31.1 1.2 2 031.6 1.3 0 0.3,""1.3 1.4 0 0 3

''.1.4 1.5 1 0.31- .5, 1.-- 3 0.3

-,1.6 1.7 3 0.31.7 1.9 6 0.41.9 1.9 15 0.41.9 2.0 20107 62.92.0 2.1 121 93.42.1 2.2 7 63,52.2 2.3 6 8352 3 2.4 1 83.52.4 2.5 0 63 525 2.6 0 93 52.6 2.7 0 83.527 2.9 0 83 52.9 2.9 0 83 52.9 3.0 25 93 63.0 3.1 1 83 63.1 3.2 3 93 63.2 3.3 1 8363.3 3.4 0 93 63.4 3.5 0 83 63.5 3.6 0 93 6

" 3.6 3.7 3 83 63.7 3.8 2 83 63.9 3.9 9 83 73.9 4.0 1027 87.94.0 4.1 5 97 94.1 4.2 1 8794.2 4.3 0 9794.3 4.4 3 97.94.4 4.5 1 87 94.5 4.6 0 97 94.6 4.7 0 87 94.7 4.6 0 87 94.8 4.9 1 97.94.9 5.0 12 8 0

K 5. 0<0 2929 100. 0

Table XIVGlobal Magnetic Cutoff Values

L-Value B.,,..11

1.0949 0.239011.1152 .229981.1356 .224821.1559 .222881.1762 .22353t.2628 .24O71.4817 .267701.8331 .275732.3168 .297622.9330 .333113.6815 .370014.5626 .402745.5750 .431786.7218 .455028.0000 .46951

Note: The Table may be interpolated for other L-vaues (Ref. 16).

4

.'40

The overall flow for the progran is as follows (see Appendix E):

(1) Use an initial guess for the fundamental decay time and the theoretical ratiobof the higher-mode dek'y times to compute the starting decay tines. Tile firstguess at fundamental decay time is shown in Table XV as a function of L andenergy. This table is interpolated in L and E for starting value.

(2) Numerically compute the integrals of the functions shown in Appendix C at eachdata point using the initial guessed decay times. The integration is performedby a 12-point Causs/legendre Quadrature at each data point for e:-ch function.Only the fundamental decay mode and two higher modes were c( mputed.

(3) Compute the linear amplitude constants using a least-squares matrix-solvingroutine. Compute the standard error of the fit. The individu-,l points art-weighted by the reciprocal of their fractional errors, which have been normal-ized to an average value of one.

(4) Optimize the initially computed ye.t value by using a "Golden ,;ection Min-imim" function. This section attempts to let the data determin: the "best"cutoff value. If the data are not distributed down toward the loss cone, theGolden Minimum function will provide an erroneous y/.t value. Hence, thefunction specifies that the computed y/,, must be within ± 10 prcent of theinitial input value. If, in the iteration process, the function tries to exceed theselimits, the initial guess of Vett is specified as the correct value.

S(5) Repeat the steps (2) and (3) with the optimum v..t value.

(6) To determine the best combination of decay times and amplitude constants,the decay times are changed by a "random-walk" method, eitfier increasedor decreased. The amount by which the decay times may randomly vary isdetermined by using a cumulative normal probability distribution about theinitial value.

The standard deviation used was estimated according to this author's best gnms orthe accuracy of the tables and curves shown by Stassinopoulos (Ref. 53:31-32, 40-44) andWest (Ref. 63:50-54). The Stassinopoulos tables do not always correlate with each otheror with the Stassinopoulos curves, and the West curves do not always correl Ite with theStaasinopoulos in the 1-regions of overlap. The initial fundamental decay times in Table XVwere computed by this author by interpolation of the tables and curves in Reference 53 and

* '. by reading and extrapolation of the curves in Reference 63. The author's estimates of theirstandard deviations are shown in Table XVI. Of particular note is the very lo'w confidencein decay times at high L values (above 2.2) and at high energies (above 2.0 MeV). Therehave been few computations of decay times in these regions, and the results vary widely(Refs. 53, 03).

The effect of using the cumulative normal probability distribution and the above stan-dard deviations is to restrict the "random-walk" of the decay times away from their initialvalues. The farther away from the initial "mean value" that the random val te g,,neratorplaces the new decay time, the greater the "push" that is created back toward he mean forthe next walk. The distance away, however, is deftned in terms of the standa.d devition,so that r values in which there is low confidence may vary by larger ,amounts th:in rmn, rvalues in which there is greater confidence. The r's ar, :aiso restricted by th, o istrint thatro > r I > r.

With each chage or decay timps, steps (2) and (3) are repeated up to 96 times, in anattempt to get a 'better* fit by reducing the standard error of the fit.

,.A4

42

-W 4c C4 wq V) In~ -4 r. P, r r N r, - a. &. 0. Nc. N 0 0. e. col 0 0.0

C., V: C: 0!o O o 000!0000

in N NNAD bDy .0.O oIn O.O0

N 01N:e 9 -;1, c C c 7

C) o0 mo0N nc mv w, c rm eqv IN 0.-r.Ca JP'oll4w fG w.o

* ~ - N 4UCI ' I-O. C'. l.N 0l

* c cc x v -W cc -r N ~ a CC -, N, Nl r, 4c -r - .0, -C, cc -Go 0. N

A. C4

A N 'r ,C0z c. C A00 L.r .c

*; -CLr*40 ~ C N V;N ~ Wt O O O -z0 ; c : r 4r

Z :L 0; 0 00; V O0L:1;00 C; 1: 04 0; -Z1 ;C

C. AN AV. v51 WC 0 O ~ W C c ~ -- N e C

A0A nWmr . Ar 4 -A - - - -

N --- 0 - l - 70i

.. . .a. ....

0- An VN V) t- . - -0 -0 -0 -0 Ll OD r'. .0 0c in N 0 q .0-VA NC4MN. fl)r. mM " F t. " t. " " n M n r N 4 C C

L4o b ;A ;4 ;d ;;0 ;4 ;LZL Z ;b ;4CD An0. a. ,Nf ,r 0 40V '- WWP . C.0C% m0 r"A n"i oP "r -o nMP .i nr . r

on. W* V;W ;1 ;4 94 ;I ;C ZV W 'C '4 ;- 0 0' 0 V) if- 10 7 V V) qe -r Fe M " n CA C N C~d CAC ~ o*-

w 400 -040ev O 00 e ~ wI W . L0WMr 'O I' bD V ""I rnt nC ACA A C I

- g eq CA eq I -0 eq Nq Ce N rqe, 10 V5 eq Vq 1" 420 0q ODe qe q qe q

e! p 4q n v- w~p 0q eeqq0Ow w 4v - r " C4 eq g0 w r 0 V'- A C4 CA C4 q' ANA IO CA CAN N C4 C CA r4 C

4

i o~qep 0 a 0 a 0 0 0 - - - - - CA eqeq CAqe e rd 'IANC C N4

(AW 'II0 ~

N 00 OO @0 00 OO~I O0 O~~b43~i

ri

Table XVI

Percent Standard Deviation of InitialFundamental Decay Tims (in Table XV)

LiRange .02- 1.21- 1.41- 1.81- >1.20 1.40 1.80 2.20 2.20

Energy RangeMeV

E < 0.0 50 30 20 35 200

0.0 < F < 2.0 30 20 20 75 too

E > 2.0 50 20 50 100 100

(estimated from Ref. 53:31-32, 40-44 and Re.f. 6-3:50-54)

44

A44

.. -- - - - . . -

(7) If a better fit is found in less than 951 random walks, and if less than 32 "better"values have been found, steps (2), (3) and (0) are repeated up to 3. times, using

L.:- the latest "best" decay times.

(8) Up to five repeats or steps (4) and (5) may be carried out, but only "32 "bettr"fits are found. Otherwise, only the first yet optimization is p.rformed.

(9) The program will always end at the point where 96 random-walk have failedto reduce the error of the fit from the previous "best-fit'. The lecay times,yeut, and amplitude constants are printed for each 'better" fit, v ith the ir:ilvalues being those of the "best" fit.

(10) After the best fit is found, the program uses the fitted constants and decaytimes to compute fluxes, at specified times, from the equator to the best (.utoifvalue. These specified times correspond to default times plus 1/2 .ime windowin the program which plots raw data. Thus, the computed fluxe: are used togenerate curves which are plotted over raw data points at approimately thesame times.

Variations in the program were tried in order to study the effects of the fiting process.The principal modifications were in the random decay-time variations.

The first modification was to fix the ratios of the r's to their theoretical values of1:1/9:1/25 (Ref. 48:16-19). The effect of this modification is to significantly reduce thedegrees of freedom available to fit the data.

The second modification was a compromise between the above and complete freedom torandom walk. This variation allowed r0 and r, to vary freely (subject to the usial contraintthat re > r1 ), but r, and r2 were fixed at their theoretical ratio of 25:9. The principal reasonfor this attempt was the very short time-span covered by some satellite/burst combinations.For example, the Russian 2 Burst covered only 4 1/2 days prior to the Russian 3 injection.If the data cover a time-span which is small compared to the fundamental decay constant,then the fitted value of that decay time may be suspect. However, if the time-s -)n is not sosmall compared to the decay times of the higher modes, one should have mote confidencein the fitted values of those decay time. Stated another way, the fit may not b. sensitive tothe fundamental, but may still be sensitive to the higher modes.

In this analysis, not all L-values covered by the satellite in Table i were examined.Over 1000 plots of flux versus z were examined at various L-values for the s:.tellite/burstcombinations of Tables I and 11. In many cases, the satellite orbits were (,f such highinclination (toward polar) that data were not available close to equatorial B-values. Becauseof this, and because of the very large volume of data, those satellite/burst/L combinationswhich provided the most consistent coverage from the equator to the loss cone were chosenfor study.

Generally, the data fell into three regions of z-space (with some overlap). legion I wasfrom the equator to the point where flux dropped off sharply into the loss cone (z = 0 toz f 0.6 or 0.8). Region II was the area where the flux "turned the corner" friom a rather"horizontal" curve to a rather "vertical" curve into the loss cone (z f 0.7 to 0.9). Region ifwas the loss cone region, where the flux levels dropped sharply to the cutoff v- lue (z

* for a typical L-value).

45

Representative plots of raw data in the three regions are shown in Figures 4 through 9.These plots are typical in that data seldom cover all three regions in a short timo-span. Capsin the data appear betcause or satellite orbital coverage and because of lack or on-satelliterecorders. When a satellite lacked a recorder, dlata were' colleceted on!,- -.s the sal.ellite pwimedabove the radio horizon of a ground tracking station. Figre 9J is -in eight-hour 1pot of typiealsatellite coverage over all B-L space.

4r

BURST : RUSSIAN 2 SAT :ALLOUETTE1TIME 0 (dy) 0 (hr) 0.00 (min,) 0.000(days)L 2.40 Beq : .022 8CUT : 0.511 TW (days) : 0.5

Bmin (Gauss) :0.001 Bmax (Gauss) :1.000

10 I LEGENDo= 3.90 MEV

0 = 3.90 MEVA= 3.90 MEV

10 10'

z 1 0"10

z0

Li

z

0 0.5 1/X =(1-Beq/B)1/

Figure 4(a). Raw Allonette Data Colletted Only in Region III (at burs', time).

,17

BURST RUSSIAN 2 SAT : ALLOUETTE1TIME : 3 (dy) 0 (hr) 0.00 (min) 3.000(days)L 2.50 Beq: .020 BCUT : 0.515 TW (days) : 1.0

Bmin (Gauss) : 0.001 Bmax (Gauss): 1.000

10' 10

S107 - 10'

0

x id'. id'

z 10 10

0

_J

z

;T -- LEGEND]]T o0 = 3.90 uEV/

i. 0 = 3.90 MEV I

::~ ~ ~~d 10 A = 3.90 MEVL..' -- - '10

X =(1-Beq/B) V

• ~Figure 4(b). Raw Alloette Dt Collected Only in Region 111 (3 days potburst).

4110

BURST STARF.SH SAT 71 TS-ARTIME 1 (dy) 0 (hr) 0.00 (mi) 1.OOC(days)

L .180 Beq 3 BCUT 0.470 TW (days): 1.0Bmin (Gauss) 0.001 Bmax (Gcuss) :1.000

-- li . 10 I 10

><. 4o o

z 10 - -10010

.- ,Li

L-

'. Fi1r . iowTltrDtaClef di Rgos[adll ery&m)

k,,-j

z0

C°w-i 10,Z LEGEND

o = .220 MEV0 = .320 MEV

a= .420 MEVx =.60 MEV 1 10,

Figure 5. Raw Teistatr vData collected in Regions I and III (arly timne).

49)

BURST STARFISH SAT TELSTARTIME 15 (dy) 0 (hr) 0.00 (mn) 15.000(days)L : 1.70 Beq : .063 BCUT . 0.460 TW (days) : 5.0

Bmin (Gauss) : 0.001 Bmax (Gauss) :1.000

: .10 LEGEND 10'0 .220 MEV0 .320 MEV

.420 MEVx .660 MEV

"' 108t 10Wid1' 10

0

Lai

x 1c 10'A" 3

z

103 - ..--- - , . . 10')0 0.5 1

X- (I-Beq/B)" 2

Figure 5. Raw Telstar Data Collected in Region III (mid-range time).

e;

* 5i

,, . . : . : - ...: . . .-. . . , ... . .. . .. .- _ , . . .. .. . . , , :

.l I- . . / --

BURST : STARFISH SAT : TELSTARTIME 80 (dy) 0 (hr) 0.00 (min) 80.000(dcys)

L :1.70 Beq : .063 BCUT 0.460 TW (days) : 5.0

0 Bmin (Gauss) : 0.001 Bmax (Gauss) : 1.000

1I' '"LEGEND 10.0 = .22MEV

-l 0 = .320 MEV4+_4 A = .420 MEV+~~ + .l ~ X660MEV

x 10'-S

zo0

-J -

S 10z

0 0.51

:!IL X = (1-Beq/B) 1/ 2

.'" Figure 7. Raw Telatar Data Collected in Regions ,I, U, ci L (mid-ra nge IqmeJ,

,,t o

li!..:.:. .. .. ./. . -. -. -, ..-s .. . . ..

* . . .. , r " '

I , , - r - - " " - ." . . " •. . " - " .

BURST : STARFISH SAT : TELSTARTIME :100 (dy) 0 (hr) 0.00 (min) 100.000(days)L :1.80 Beq : .053 BCUT : 0.470 TW (days) : 5.0

10' B in (Gauss) : 0.001 Bmax (Gauss) :1.000m, o-• - . - - ---- . doi... !LEGEND

0 = .220 MEVo = .320 MEV4 4 i = .420 MEV

11 x = .660 MEVII I?

$ ,oo10' -0 7

o 107

× 106 I ' 0

U- tz

10 ,10'0

::I 0l' 10'

0 0.5X = (1-Beq/B)1/ 2

Figure $(a). Raw Telatar Data in R'gions I and I (late time).

52

*. .. ", . 7 . . . . .t - .- . - - -:. . .•

- - -74.

BURST STARFISH SAT TELSTARTIME :100 (dy) 0 (hr) 0.00 (min) 100.000(days)L : 2.00 Beq : .038 BCUT: 0.487 TW (days) : 5.0

Bmin (Gauss) 0.001 Bmax (Gauss) :1.000

10 : * LEGENDo = .220 MEVo = .320 MEV

A = .420 MEV

IL 10x = .660 MEV

C(J.o•- -'I

Z- iL. 4-. TXI id o' T 4oLi 6 I t~ 10z

'z

o4( 10' 10

0 0.5X =(1-Becq/B)/"

Figue 3(b). Raw Telstar Data in Regons 1, HI, and Ml (late time).

.53

AFWL TRAPPED ELECTRON DATA BASEBURST : STARFISH SATELLITE TELSTAR

DAY:237 HOUR: 1 MIN: 0 #: 98CHANNEL NUMBER: 4 TIME WINDOW: 8.00

10 LEGEND ,= .620 MEV

z

0'

old

-J ILi

J04

id Ai0 .WL° A4L± L

0 1 2 3 4 5 6 7 8

TIME (Hours)Fige g. Typical Raw Data Cov.rage for S Hour (,-L values ot shown).

54r- - - -

Figures 10 (a) through (g) shown raw data from Explorer XV over the time period of theRumian 2 and 3 bursts, at times corresponding to those shown in Figure 3, for comparison.It is apparent that fewer points are plotted from the AFWL data base in Fgures 10 (a)through (g) than were plotted by Roberts in Figure 3 for the 1.9 Mc- electrons, even thoughthe satellite and time periods of coverage were the same. These discrepancies are not resolvedin this study, although the previously alluded-to L-rounding procedure may account for thedifferences.

The temporal progression of flux after one burst from one satellite (Telstar, at a typicalL-value of 1.9, is shown in Figures 11 (a) through (h). Again, typical gaps in coverage ofRegions 1, 11, and III are readily apparent over the 90 days shown.

As previously noted, only fundamental and two higher eignemodes (fo, fl, f2) are lioedto fit the satellite data (Appendix C). The third eigenmode (f.t) is not used hecai se Shilz andLanserotti (Ref. 45:163) suggest that the modes greater than two rapidly vanish. Initially, thesteady-state function (fee) was used along with the fundamental, but this solution was rotidto be 'competing" with the fit for the fundamental, in the sense that the lincar constantstended to be roughly equal in magnitude and tended to alternate in sign. Sholz (Ited. 47:2223) indicates that the f. and fe functions should resemble each other in shape. Also, l'litzer(Ref.37) has subtracted the background from all flux data, and the background shouldapproximate the steady-state, fe, solution. For these reasons, the steady-state solution wasnot used in the data fit.

2o.

,I°

.o

---- nr- .r-.. C .r- 'C ..- . - -

BURST RUSSIAN .2 SAT EXPLORER 15/EXPLO RER 15TIME: 0 (dy) 4 (hr) 58.07 (min) 0.207(days)L :1.90 Beq : .045 BCUT 0.479 TW (days):- 1.0

B min (Gauss) 0.001 Bmax (Gauss) :1.000

9 9

01

LA-

.: BRTt:RSIN2 ST:EXLRR5EPOE

" M1:0 d7 (r)5 .07(an .0(as

id') 10 T 10'

0 1"

-° -, .,o

-.J

2.90 + 10d0

0.5

. . 10'10'

X =1-Beq/B)1/

Figre 10(). Raw Explorer 15 Ota at rmes Correspondfing to Fig'ure

-J

BURST :RUSSIAN 2 SAT EXPLORER15/EXPLORER15TIME : 2 (dy) 19 (hr) 22.09 (min) 2.807(days)L :1.90 Beq : .045 BCUT 0.479 TW (days) : 2.0

Bmin (Gauss) :0.001 Bmax (Gauss):1.00010' 10

V)10' d10

0, 7

L4.

z .-

LEGEND- TM =2 .500 MEV1

z 0 = 1.90 MEV

A= 2.90 MEVx .500 MEV

=1.90 MEVFigure Raw 0. * V 2.90 MEV o

X (1-Beq/B)"

Fiue10(b). RwExplorer 15 Data at Times Corresponding to Figure 3.

.57

4 . _

BURST RUSSIAN 2 SAT EXPLORER15/EXPLORER15TIME : 9 (dy) 19 (hr) 22.09 (min) 9.807(days)

L :1.90 Beq : .045 BCUT 0.479 TW (days) : 2.0Bmin (Gauss) : 0.001 Bmax (Gauss) :1.000

id' id-=1'

10 10

x i0' 10

kid 4t le

• I-JUj

%10,

4 1o

.LEGEND

"'"0 = .500 MEV

::'" -- 1.90 MEV

id' A- 2.90 MEV 10,

0i 0.5 1~

v .',;: X = (-Beq/B) 1/2

i'- Figure 10(c). Raw Explorer 15 D-ta -it Timps Corresponding to Figure 3.-JJ

:'.' , i -

l1 ' 0

!'-.". -- - "..- ., .- -.... ' -. LEGEND .

BURST RUSSIAN 2 SAT EXPLORER15/EXPLORER15TIME :20 (dy) 19 (hr) 22.09 (min) 20.807(days)L :1.90 Beq : .045 BCUT :0.479 TW (days) : 2.0

Bmin (Gauss) :0.001 Bmax (Gauss) :1.00010 LEGEND 1

0 .500 MEV0 = 1.90 MEV

A= 2.90 MEV

Lj100

0

0

W 0 10,

z

id 10,

0 0.5X =(1-Beq/Bj')

Figure 10(d). Raw Explorer 15 Dita at Times Corresponding to Figuire 3.

BURST :RUSSIAN 2 SAT EXPLORER15/E)(PLORER15TIME: 29 (dy) 19 (hr) 22.09 (min) 29.807(days)L :1.90 Beq : .045 BCUT 0.479 TW (days) : 4.0

Bmin (Gauss) : 0.001 Bmax (Gauss) :1.00010. LEGEND

' = .500 MFV0 = 1.90 MEV

= 2.90 MEV

,.,,18 106

U-I

z 10 7

I- It1i

110X id_J

z0Z- -oLoi' I(fLj-j

Iz

10' - ,, _________ '- , - ---- ko- 10 0.5 ,

X (1-Beq/B) / 2

Figure 10(e). Raw Explorer 15 Data at Times Corresponding to Figure 3.I

*

BURST RUSSIAN 2 SAT : EXPLORER15/EXPLORER15TIME : 47 (dy) 19 (hr) 22.06 (min) 47.807(days)L : 1.90 Beq : .045 BCUT 0.479 TW (days) : 2.0

Bmin (Gauss) 0.001 Bmax (Gauss) :1.000

LEGEND 100 - .500 MEV

()

I I~ e 10

(I.) ,

z 10' 1010,0

I-

L.:-j

x 16 10'

D

LA.

z0

1-

i' ---- 10,

Lii

L-J

(0L- 10) 10'z

0 0.5 1X =(1-Beq/B)I/

Figure 10(of). Raw Explorer 15 Data at Times Correqponding to Figure 3.

6 1

BURST RUSSIAN 2 SAT EXPLORER15/EXPLORER15TIME : 63 (dy) 19 (hr) 22.06 (min) 63.807(doys)

1 7 L : 1.90 Beq : .045 BCUT 0.479 TW (days) : 10.0

Bmin (Gauss): 0.001 Bmax (Gauss) :1.000i' LEGEND I10 .51 0EV

0X ", 10 10'

* U)

A-

0

L_

._J

Wid-J

*Z -

z

io1" "'' ' 10"0 0.5 1

X = (1-Beq/B)1/ 2

Figure 10(g). Raw Explorer 15 Data -it Times Corresponding to Figure 3.

BURST STARF'SH SAT TELSTARTIME 1 (dy) 0 (hr) 0.00 (min) 1.000(days)

L :1.90 Beq : .045 BCUT 0.479 TW (days) : 1.0Bmin (Gauss) :0.001 Bmax (Gauss) :1.000

10

0le~

0O .220 MEV0o .320 MEV

=.420 MEVx .660 MEV _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 1

0 05 ,12X =(1-Beq/B)1 '

Figure 11(a). Temporal Progression for Raw Teistar D-A rrom I to 90 Days Post-Burst.

BURST STARFISH SAT : TELSTARTIME 2 (dy) 0 (hr) 0.00 (min) 2.000(days)

L :1.90 Beq : .045 BCUT 0.479 TW (days) : 1.0Bmin (Gauss) 0.001 Bmax (Gauss) :1.000

10oLEGEND0 .220 MEVo0= .320 MEV

..420 MEV

" Z10' -10

0

Lka

x 0 lo'

z0

-LaJ-AJ

1 10'z

10'I * 10'I

0 05 1/2X =(-Beq/B)

Figure 1l(b). Temporal Progression for Raw Telstar Data from I to 90 DaY.T Post-Burst.

4

...

BURST STARFISH SAT : TELSTARTIME : 4 (dy) 0 (hr) 0.00 (min) 4.000(days)L :1.90 Beq : .045 BCUT 0.479 TW (days) : 1.0

Bmin (Gauss) : 0.001 Bmax (Gauss) : 1.000L.10- I 'J 10'

Li dI

C~4

0• 0

I-

L-

z0

-AJ

d,z10

LEGEND13 = .220 MEV

io 3 .420MEL ~i0 05 ,i

X=(1-Beq/B)"

Figure 11(c). Temporal Progre1ion for Raw Telstar Data from I to 90 Day, Post-Burst.

WW

- :'-5

BURST STARFISH SAT TELSTARTIME: 6 (dy) 0 (hr) 00(m) 6.000(days)

L :1.90 Beq : .045 BCUT :0.479 TW (days) : 2.0*Bmin (Gauss) :0.001 Bmox (Gauss) :1.000

10' ~ 1 ~LEGEND 10'T~ 'j 30 .220 MEV

* 40 .420 MEV'I' =.660 MEV

C'4

0

-

z

0 .

* fi6

..............................

BURST: STARFISH SAT: TELSIARTIME 10 (dy) 0 (hr) 0.00 (min) 10.O00(days)L :1.90 Beq : .045 BCUT :0.479 TW (days) : 5.0

8min (Gauss) :0.001 Bmax (Gauss) :1.000

id 1

01

zT 1(10

I-

**~ 10e 108

lo' idlo

:!0'

z10' LEGEND 10'

o = .220 MEVo = .320 MEVA = .420 MEV

Sx =.6o MEV 3103 , , , , 100O.5 / 1

X = (1-Beq/B)/ 2

Figure 11(e). Temporal Progression for Raw Telstar Data from I to 90 Day Post-Burst.

137,>. .

-- . - -o •. .- - . --

',o

BURST STARFISH SAT TELSTARTIME: 15 (dy) 0 (hr) 0.00 (min) 15.000(days)

L :1.90 Beq : .045 BCUT 0.479 TW (days) : 5.0Bmin (Gauss) 0.001 Bmax (Gauss) :1.000

10' 10 LEGEND 10a .220 MEV

+ o0 .320 MEV& = .420 MEVx .660 MEV

) 1r id

0

x lod'

z 10'tYt10' 1

,0d,

7. 0 0.5 1

X = (1-Beq/B)

Figure 11(f). Temporal Progreion for Raw Telstar Data from I to 0 Dayi Post-Burst.

:I

,-. "..'M

BURST : STARFISH SAT TELSTARTIME 60 (dy) 0 (hr) 0.00 (min) 60.000(days)

L :1.90 Beq : .045 BCUT 0.479 TW (days) : 5.0K 10Bmin (Gauss) : 0.001 Bmax (Gauss) :1.000

z id7

0LaLq

x i' id'

0 4

iiid

Z LEGENDi00 = .220 MEVo = .320 MEVA = .420 MEV

xc = 660 IEV *1

0 0.5 , 1X = (I-Beq/B) / 2

Figure 11(g). Temporal Progresion for Raw Telutar Data from I to 90 Day, Post.-Burst.

4n

• ,I - " ",. " " . ; " -, : -- . - " -. . - . . . ... . -,-. .. . . . . . ... ... .... . . . . . . . . .. .. ... ... ...: "

BURST STARFISH SAT TELSTAR

TIME: 90 (dy) 0 (hr) 0.00 (min) 90.000(days)

L :1.90 Beq : .045 BCUT : 0.479 TW (days) : 5.0

Bmin (Gauss) : 0.001 Bmax (Gauss) :1.000

i' . . 10'-.. , , ---- -- --T - ' - EGENDo = .220 MEVo = .320 MEV

A=.420 MEV

x 10.6 d1

z

L.-z0

Sl,,10,

• '0 0.5

1112X (1-Beq/B)/ 2

Figure 11(h). Tempord Progresion for Row Telstar D a From I to 90 DaY, Po4L-P1irst.

I°

-. 71'7

V. RESULTS AND DISCUSSIONFigures 12 and 13 show the typical temporal progression of raw data at two representativeL values, L = 2.3 and L = 2.4. Included on those figures are the flux curves cal'ulated fromthe amplitude coefficients and decay times fitted to that data by Electrolit. As previouslynoted, the curves shown on any given figure may vary slightly in time from the individualdata points, since the curves represent only the one time-instant correspondiny to the starttime plus one-half the time window of the plot. Figures 12 and 13 represelit maximumfreedom of the fit, since each r is free to vary independently of the other r's.

Figure 14 shows the same temporal data progression with different fitted curv,:. The curvesin this figure were calculated under the restriction that the ratios of the r's were fixed atthe theoretical values of 1:1/9:1/25; hence, only rT was really varying in a ran,;om sence.

Figure 15 shows similar data progression and fitted curves where the r's were allowed tofreely vary, subject to the constraint that ro > r, > r2. Also, this was an initial run with 500tries for convergence, vice 90 tries. In addition, the computed flux was used in the iterationprocess to eliminate non-physical (negative) solutions. The solution was restricted to beingpositive at four different times and at five different equatorial pitch angles. Alt tough this isan initial run with only a small amount of data used, it indicates that the recoinemndationsin Chapter VI should be pursued.

Table XVII shows the fitted linear amplitude coefficients and exponential decay times (indays) at constant Energy and L-shell for a representative subset of the data examined inthis study. The sheer volume of resultant fits and calculated curves made it impossible to

41 include all results in this report. The values shown in Table XVII were comp'.ed with ther's allowed to vary freely.

Figures 12 and 13 are considered representative of the results of the fitting model. It isapparent from examination of Figures 12 and 13 that the pitch-angle diffusion theoryand flux computation method developed in this report are consistent with ,xperimentalmeasurements. Higher eigenmodes are readily discernible at the lowest energy in the Figuresor the first five days, and are not visible at later times. These higher eigenmades are notdiscernible in the plots at the higher energies; Table XVII shows that thrs modes arepresent, but at much lower levels, in some cases. The calculated curves show "1 reasonablefit to the data even at late times, consistent with the assumption of exponenti d decay.

Examination of Figures 14 and 15 shows that some inconsistencies remain to h, resolved inthe use of this fitting method. The caluclated flux curves for the low-energy data points alsoappear to be slightly lower than optimum, while the high-energy data appear, to be fittedbetter by the model. This may be partially explained by the existence of many lower datapoints in the vicinity of the loss cone which tend to "drag down" the overall lit. Any changein the cutoff value significantly affects the fit by shifting the 'vertical" portion of the curvetoward or away from those points. However, this cannot account for the entire differenceof the curves from the data. The energy dependence may be treated less th:tn optimallyin the choice of functional solution made in this study. If the chosen function d solution isnot the correct one, it may he nevertheless close enough to correct to give a r :asonable fitat high energy, but be significantly in error at lower energy. A different functional solutionmay fit the lower-onergy groups better and yet still lit the lhigh-,nergy elect ons as w,,ll.Recommendition for the ,ise or other functions is mad, in Chapter VI.

71

BURST :RUSSIAN 3 SAT EXPLORER15 [0451TIME : 0 (dy) 0 (hr) 0.00 (min) 0.000(days)

L :2.30 Beq : .025 BCUT 0.506 TW (days) : 0.5

id* Bmin (Gauss): 0.001 Bmax (Gaus Is) :1.00016

0

LiLi

-

-,- 10 = .500 MEV 1

z ri.500 MEVA= 1.90 MEVX= 1.90 MEV0= 2.90 MEY

4 10' V==2.90 MEV ~0 05,

X =(1-Beq/B)"

Figure 12(a). Temporal Progression or Raw Data and Fitte Flux Curves at , 2.3 forK Russian 3 Burst (r. rreely varying).

BURST RUSSIAN 3 SAT EXPLORER15 [045]TIME : 0 (dy) 12 (hr) 0.00 (min) 0.500(days)

L :2.30 Beq : .025 SCUJT 0.506 1W (days) : 0.58min (Gauss) 0.001 Bmax (Gauss) :1.000

U 10~~LGEND. 10 = .500 MEV0= .900 MEVA= 1.90 MEV

9 9 .0 E10 = 2.90 MEV -10

= 2.90 MEV

0

id' 108

X 1/2(1-Beq10

Figue 1(b) Teporl Pogresio ofRawDat an Fited luxCures t 1 2. fo

Russan 3Bur-t (,, feelyvaryng0

id :BURST:RUSSIAN 3 SAT: EXPLORER15 [045] 6

0 (hr 0.000(min

0 50MEV

~ 40 1.90 MEVV2.90MEV

CN

0

F-U

A'-

D

z0

00.

X (1-Beq/B) 1/2

Fiue12(c). Temporal Progresuion of Raw Data and Fitted Flux Curyr' at h, 2.3 forRussian 3 Burst (r., freely vurying).

* 74

BURST :RUSSIAN 3 SAT :EXPLORER15 [0451TIME :2 (dy) 0 (h r) 0.00 (min) 2.000(days)

L :2.30 Beq : .025 BCUT :0.506 TW (days) :1.0Bmin (Gauss) 0.001 Bmax (Gauss) :1.0000

10 LEGEND 1O= .500 MEVO= .500 MEVA= 1.90 MEV

U(x = 1.90 MEVL c =29 MV1

V) = =2.90 MEV

0

x id 10d-

z0

id 1060 .5 1/

-J1-eqB

igue1() eprlPrdmqo fRwDt n itdFu uvsaf . o

-J sa os r.fel nyn)

...

BURST RUSSIAN 3 SAT EXPLORER15 [045]

TIME : 3 (dy) 0 (hr) 0.00 (min) 3.000(days)

L : 2.30 Beq : .025 BCUT : 0.506 TW (days) : 1.0

Bmin (Gauss) 0.001 Bmax (Gauss) :1.000

00

CN

L-

7u 7

0 l106--

10id.1- 1 10p

o LEGENDid' o0 [] .5GA MEY id'

Z o0- .500 MEV

,." -"1.90 MEV

x "1,90 MEV" 0 = 2.90 MEV

_,4-- .. ..v, ._ .. . . . . _ _ = 2.90 MEv n0 0.5

X =(1-Beq/B)1/

Figure 12(e). Temporal Progresion or Raw Dta and Fit.ted Flux Curves at h - .3 rorRussian 3 Burst (r,, rreely varying).

1

BURST RUSSIAN 3 SAT :EXPLORER15 1045]TIME :4 (dy) 0 (hr) 0.00 (min) 4.000(days)

L : 2.30 Beq : .025 BOUT :0.506 TW (days) : 1.0id EGN mi (Gauss) 0.001 Bmax (Gauss) :1.000 10

0 .500 MEVo = .500 MEV

A= 1.90 MEVC) 'K= 1.90 MEVLAJ 1d' = 2.90 MEVid

"*5.. ~ 2.90OMEV

C)zLid

Li

x 10L-

z0

C-

ILI

- 0 0.5 ,

X =(1-Beq/8)1 '*.Figure 12(o). Temporal Progression of Raw Data and Fitted Flux Curyes at L, 2.3 ror

Rusupian 3 Burst (r., freely vnrying).

* 77

Alti

BURST: RUSSIAN 3 SAT EXPLORER15 [045)TIME: 5 (dy) 0 (hr) 0.00 (min) 5.000(days)

L :2.30 Beq : .025 BCUT :0.506 TW (days) : 1.08min (Gauss) - 0.001 Bmax (Gauss) 1.000O

10 -~ ILEGEND

o=.500 MEVo=.500 MEV

=1.90 MEVx =1.90MEV

(, 10 = 2.90 MEV 10=2.90 MEV

C."

0

U

Li

L-

x0 1

10-jLiJ

id'

10'005 / 10'X (1-Beq/B)"'

Figure 12(g). Temporal Progressior of Raw Data and Fitted Flux Curves -it L , 2.3 forRutssian 3 Butrst (r,, freely vnryinq).

* 79

iaBURST:RUSSIAN 3 SAT: EXPLORER15U[0451)L 2.0 Bq : 025BCUT :0.506 TW (days) : 5.0

10-(as) .0 mx(Gus .0 10i d II-- - - L G N

zz

0

x 1Q 106

1010

X (1-Beq/B)~/

Figure 12(h). Temporal Progresuion of Raw Data and Fitted Flux Curvm at L 2.3 oarRuissian .3 Hurst (r,, freely varying).

79)

. °"

BURST: RUSSIAN 3 SAT EXPLORER15 [0451

TIME : 60 (dy) 0 (hr) 0.00 (min) 60.000(days)L : 2.30 Beq : .025 BCUT : 0.506 TW (days) : 5.0

SidI° Bm im (Gauss) : 0.001 Bmax (Gaus ) :1.000 1:• ~~LEGEND [. .. , ,,

i!-i 3 = .500 MEV|.: .. o = .•500 MEVI

0

LU-j

._J

z0

•i

M

-

* 0 0.5SX = -Beq/8

Figure 12(i). Temporal Progression of Raw DAta and Fitted Flux Cuntm a't 1, = 2.- forRussian 3 Burst (r,, freely varying).

4i

., .. -.. .. .. .- - , , • . . - ' - - - - -7

-. M. 7917

BURST RUSSIAN 3 SAT EXPLORER15 [045]TIME: 0 (dy) 0 (hr) 0.00 (min) 0.000(days)L 2.40 Beq : .022 BCUT : 0.511 TW (days) : 0.5

Bmin (Gauss) 0.001 Bmax (Gauss) :1.000

i i0' i0* (N

U

0

..--

UAr

07

6U10 10

.-- O

Li

-J

10' 0 =2.500MEV id,_______________ 0

0 5

a = 1.90eq/B)"

x =o 1.o'ME

10, 0 0.

Figure 13(a). Temporal Progression of Raw Data and Fitted Flux Curves at 1, 2.4 frRussian 3 Burst (r. freely varying).

- - .'' i

BURST RUSSIAN 3 SAT EXPLORER15 [045]TIME: 0 (dy) 12 (hr) 0.00 (min) 0.500(days)

. L : 2.40 Beq : .022 BCUT : 0.511 TW (days) : 0.50 Bmin (Gauss): 0.001 Bmax (Gauss) :1.000

10 LEGEND ,r, 10o = .500 MEV

o = .500 MEV, 1.90 MEV

C)1 X 09 1.90 Mrv

1' 10' 2.90 MEVv 2.90 MEV

U

-

x<10' 10'

z010

-j

1d' 0'

0 0.5 1X = (1-Beq/B)I 2

Figure 13(b). Temporal Progression of Raw Data and Fitted Flux Curves at L = 2.4 rorRussian 3 Burst (rn rreely varying).

-.... . .. .....

iU

BURST : RUSSIAN 3 SAT EXPLORER15 [045]TIME : 1 (dy) 0 (hr) 0.00 (min) 1.000(days)

L : 2.40 Beq : .022 BCUT : 0.511 TW (days) : 1.0Bmin (Gauss) 0.001 Bmax (Gauss) :1.000_________ ___10__________________________10 0

LEGEND I 1- -- .500 MEV

0 .500 MEV

.1.90 MEV

-1.90 MEVw 109 =2.90MEV

v =290 MEV

o01z id" lO0:': I--

x 10d id7

z01

L j

1z id( '-.- J

10"0 10

2-"2

4 l"" 0 0. / 10

X" (1-Beq/B) 1/2Figure 13(c). Temporal Progremion of Raw Data and ittd Flux Cnrvm -it 1, 2.4 forRu-tjian 3 Burst (r,, freely varying).

*°i"

.-- . -. , • . - .

BURST RUSSIAN 3 SAT EXPLORER15 [0451TIME : 2 (dy) 0 (hr) 0.00 (min) 2.000(days)

L 2.40 Beq : .022 BCUT : 0.511 TW (days) : 1.0Bmin (Gauss) 0.001 Bmax (Gauss) 1.000 0

LEGENDr --. 500 MEV

0 = .500 MEVA = 1.90 MEV

0x 1.90 MEV1 ;i---- 2.90 MEV 10

7 = 2.90 MEV

,i-0z id' i

Li

1 6-

Li 5

bi- i .. iO'

0 0.5X = (1-Beq/B)

1/ 2

Figure 13.d). Temporal Progression of Raw Data and Fitted Flux Curv ". at , 2.4 forRussian 3 Burst (r. freely varying).

* R4

BURST :RUSSIAN 3 SAT EXPLORER15 [0453TIME :3 (dy) 0 (hr) 0.00 (min) 3.000(days)L 2.40 Beq : .022 BCUT :0.511 1W (days): 1.0

id'Bmir, (Gauss) :0.001 Bmax (Gaus .s) 1.000

10 10p

1c10

Lld' 1 = .50 M6

0 = 2.90 MEV0.110

X (I-Beq/B)"12

Figure 13(e). Temnpotul Progr'.sion. of Raw Data and Fitted Flux Curvre,, at r, -2.~4 for

Russian 3 Burst (r. freely varying).

BURST RUSSIAN 3 SAT EXPLORER15 [045]TIME :4 (dy) 0 (h r) 0.00 (min) 4.000(days)L :2.40 Beq : .022 BCUT : 0.511 TW (days) : 1.0

Bmrnr (Gauss) 0.001 Bmnax (Gauss) :1.00010, LEGEND -10

01 = .500 MEV0 = .500 MEV

A= 1.90 MEVx = 1.90 MEV

id 10 > = 2.90 MEVid7 =2.90 MEV

(-4

z id'100

I-

L-

w>(0

01id'

Fiur 106 eprlPoreso fRwDt adFte lxCrvs't1 . o

Id

Irv

-4

BURST : RUSSIAN 3 SAT EXPLORER15 [045]TIME: 25 (dy) 0 (hr) 0.00 (min) 25.000(days)L :2.40 Beq : .022 BCUT : 0.511 TW (days) : 5.0

10l Bmin (Gauss) 0.001 Bmax (Gauss) :1.000

10 10

,,1 i, id,,--J 10 10

Z9

(/

C)

Z 10

xL -0

Ofldi

U -J

× d' lO'

x10 50 E 104

C1/co 106 106

-J

X(1-Beq/B)1~

; x =

Figure 13(g). Temporal Progression of Raw DA3 and Fitted Flux Curvesat L 2.4 '.OrRuss41n 3 Btirst (r, rrf'iy vvyiri,).

BURST RUSSIAN 3 SAT :EXPLORER15 F045]TIME . 60 (dy) 0 (hr) 0.00 (min) 60.000(days)L :2.40 Beq :.022 BCUJT : 0.511 TW (days) :5.0

id' B~~mi (Gauss) :0.001 Bmax (Gaus Is) :1.000 1 d10 LEGEND 1

.!5 10

0io 90 .00JE

C)

LU -

L I - 1

z0

10

idU id'

*0 0.5 11X =(1-Beq/B)1/

Figure 13(h). Temporal Progression of Raw Data and Fitted Flua Curve-, 'it , =2.4 forRussian 3 Burst (r,, freely wirying).

AD-Ri27 324 ELECTRON LOSSES FROM THE MAGNETOSPHERE(U) AIR FORCE 212INST OF TECH NRIGHT-PATTERSON RFB OH SCHOOL OFENGINEERING R S DEWJEY MAR 83 AFIT/GNE/PH/83M-4

UNCLASSIFIED F/G /4 N

EhhhhhhhhhhiEsmhhhhEmhhhhh

• - • • . • • . . . . - - . . . . ."- " - . - . o

1111 1.0 0

1125 1.j 4

lIE--

MICROCOPY RESOLUTION TEST CHART

NATIONAL BUREAU OF STANDARDS-1963-A

......... *I~I ii - . -i~ , -__ - * - --..... .....

..I-.

2,7"

BURST RUSSIAN 3 SAT EXPLORER15 [052]TIME : 0 (dy) 0 (hr) 0.00 (min) 0.000(days)L :2.30 Beq: .025 BCUT 0.506 TW (days): 0.5

Bmin (Gauss) : 0.001 Bmax (Gauss) :1.000ld 10

z" ld' o

00'C4

0

" LEG END

Jicf =ic00fE

0 = .500 MEV1 .90 MEV

I.-%

_x = 1.00 10'

~ 0 102906E

0 0.5

X = (1-Beq/B)l 2

Figure 14(a). Temporal Propasion of Raw Data and Fitted Flux Curves it, L, 2.3 forRumiam 3 Burst (r. fixed 1:l/g:1/2$).I

i-9" A. :

A.

. . . . . . . . . .

- - ''-

BURST - RUSSIAN 3 SAT: EXPLORER15 [052]TIME : 0 (dy) 12 (hr) 0.00 (min) 0.500(days)

L : 2.30 Beq : .025 BCUT : 0.506 TW (days): 0.5min (Gauss) : 0.001 Bmox (Gauss): 1.000

GEND 1= .500 MCV

0 .500 MCVA 1.90 MEV

S1.90 MEV

"1 ) = 2.90 MEV 10'S V= 2.90 MEV

-. C-.

0

0

0 0.51

• .X = -Beq/"::: rr 14(b). Temporal P"oleggion of Pa* Date and Fitted Flux Curm at L 2 .3 for

R*'" 3-"i

"' ~..-"

"' ' " .: .2 .-.. - . ' : , , .

Ii_ - _ . _"-" "_ -"" -- - ' - " " " " """"Z" "

BURST: RUSSIAN 3 SAT: EXPLORER15 [052]TIME : 1 (dy) 0 (hr) 0.00 (min) 1.000(days)

L : 2.30 Beq : .025 BCUT : 0.506 TW (days) : 1.0Bmin (Gauss) : 0.001 Bmax (Gauss) :1.000

id1 LEGEND 10*o3 = .500 MEV

o=.500 UCY1 .90 MEV

X=1.90 MEYL" 1 0 = 2 90OMEV

N. = 2.90 MEV

0

xiO 10

z13 0

lict

W-W

0 0.5 1/2

a. X =(1-Beq/8)'

7ipml4(c). eupwn PeUolon fRw Dto ad Fittd Fux Curm at 1, .3 forRuuuis 3 80rs (T. fixed 1:1/9:1/26).

4,f

BURST :RUSSIAN 3 SAT :EXPLORER 15 [052]TIME : 2 (dy) 0 (hr) 0.00 (min) 2.000(days)

L :2.30 Beq : .025 SCUT : 0.506 TW (days): 1.0Bmin (Gauss) :0.001 Bmax (Gauss) :1.000 1

id0 .r ....-. '-LGEN 100 = .500 14EV0 =.500 IWEV

4..A =1.90OMEV

Ox = 1.90 MEVLi1d' =2.0E 0

=2.90 MEV l

0

Li.

4. zo

idc

z

id- ------ _

0 0.51X = (1-Seq/B)'

Fire 14(d). Tempora Progression orftRa Data and Fitted Flux Cw-m. at L 2.3 for

Russian 3 Burst (r, fixed 1:1/10:1/25).

BURST RUSSIAN 3 SAT EXPLORER 15 [0521TIME :3 (dy) 0 (hr) 0.00 (min) 3.000(days)

L : 2.30 Beq : .025 8CUT : 0.506 TW (days) : 1.0

Bmn(Gauss) 0.001 Bmax (Gauss) .000

LiJ

0K 12E

olcr_10

~ 10~ .500 MEVio__ o = .500 MWEY

A = 1.90 MEV

x = 1.90 MEY* = 2.90 MEV

io '=2.90 MEV LJ . - 00 0.5 1/1

X =(1-Beq/B)"'

Figuis 14(o). TempoiaI Propeeion of Raw Data and Fitted Flux Cnrves at L , 2.3 forRuaan 3 Bunt (r, fixed 1:1/9:1/25).

BURST :RUSSIAN 3 SAT :EXPLORER 15[052]TIME : 4 (dy) 0 (hr) 0.00 (min) 4.000(days)

L : 2.30 Beq : .025 SCUT : 0.506 TW (days) : 1.0

1010 LEGNO mm (Gauss) :0.001 Omax (Gauss): 1.000 101

00 MEV0=500 MEV

A1.90OMEV

UX = 1.90 MEV1d0 O=2.0QMEV

= 2.90 MEV

0

( 0 0.5

.44

-; BURST RUSSIAN 3 SAT rEXPLORER15 [092TIME : 50 (dy) 0 (hr) 0.00 (min) 50.000(days)

' L :2.30 Beq: .025 BCUT 0.506 TW (days): 5.0to rin (Gauss): 0.001 Bmax (Gauss) :1.000

10° 10

X 0

i! 1cr lo'

Llz

lee L . E:.-.

;: 10? 10'

: LEGEND I

0 0.5 .50 M

X = (1-Beq/B)" 2

Figure 14(g). Tempor Progreuion of Raw Dsta and Fitted Flux Curmv at L 2.3 for

Ruuia 3 Burst (r, fixed 1:1/9:1/265).

.i":..........." "

, BURST : RUSSIAN 3 SAT : EXPLORER15 1568]TIME : 0 (dy) 0 (hr) 0.00 (min) 0.000(days)L : 2.40 Beq : .022 Bcuf 0.302 TW (doys) : 0.5

Bmin (Gauss): 0.001 Bmax (Gauss) :1.000

0

,.'.

I.LJ --

0

LEEN4 =# 1.90ME

10 " 10

.4'

"± i =s uv

X =(10Beq/B) 2

Figure IS(%). Temporal Progreoion of Raw Data and Pitted Flux Ctuvs at f, 2.1 forRussian 3 Burst (r. frey varying with physical contraints).:', v= 2.9 MEV! lfl

4' /

~ ~BURST RUSSIAN 3 SAT :EXPLORER15 1568]TIME : 0 (dy) 12 (hr) 0.00 (min) 0.500(days)

KKL 2.40 Beq - .022 Bcut : 0.302 TW (days) : 0.5Bmin (Gauss) :0,001 Bmax (Gauss) :1.000 11

Jd '1 s 1 10

V) Lidid 'io7 7

0

01

L-

Lid'

10' (3=2.500MEV~ 10'

0 0.5 ,1/ 1X (1-Beq/B)"

Figut. 16(b). Temporal Prwpmion of Raw Data and Fitted Flia Cuvwi at! , 24 forRuusino 3 Borst (r., freely vnryinq with pbysicaiI contraints).

BURST: RUSSIAN 3 SAT EXPLORER15 [568]TIME : 1 (dy) 0 (hr) 0.00 (min) 1.000(days)

L 2.40 Beq : .022 Bcut: 0.302 TW (days) : 1.0Bmin (Gauss) 0.001 Bmax (Gauss) : 1.000

,o,10 10'

Sz 108 I { --.- :10'

0id

I-

LJ

.- x 1o0 7 O

.

.-. j

0

-LJ

~LEGEND.LJ ld" 0 500 MEv

z 0 500 UEV

A = 1.90 MEV10"'x = 1.90 MEV

;.'. 0 = 2.90 MEV4" v = 2.90 MEV

i::, 10 ' ' '' .. . 10i!0 0. o/ 2

X = (1-Beq/B) 2

- IFiguro 16(c). 1reporal Prclression of Raw Data and Pitted Flux Curves at I, = 2.4 foraRusian 3 Burst (r. freely varying with physira! contraintsV

e2R

.'.. . .

BURST :RUSSIAN 3 SAT EXPLORER15 15681TIME: 2 (dy) 0 (hr) 0.00 (rin) 2.000(doys)

K. L 2.40 Beq: .022 Bcut :.302 TW (days); 1.01010 Bmin (Gauss) : 0.001 Bmax (Gauss) :1.00010T -r -

10'

0

C,)

Z

x 0s : 10sI-

0L.

-J

L&.

zid

Z o = 3.00 MEV ',

,"-1.90 MEV"'-×Y 1.90 MEV:-.-. o0 2.90 MEV -

10 -- 2.90 MEV 4 O

01

0 0.5X = (.-1eq/B) 1/ 2

Figure 15(d). Temporal Progression of Raw Data and Fitted Flux Curves at I, = 2.4 'orRussian 3 Burst (r. freely varying with physical contr.ints).

... 2-.

U lBURST RUSSIAN 3 SAT EXPLORER15 15681TIME :3 (dy) 0 (hr) 0.00 (min) 3.000(days)L 2.40 Beq :.022 Bcut :0.302 TW (days) :1.0

1oBmnin (Gauss) 0.001 Bmax (Gauss):1.000

0

x 0 10~

Li

zi

-JJ

LEGEN

0 =0.50ME

*~~ ~ =iur 500 TemprlPorsino o aaadFitdFu uV tI !4t,

Rus = 1.9 BusMrfelvarigwt hsclcnrit

x = 1.90 ME

-L -~.-.. .- -e/B

wo.,- ~nV--

BURST :RUSSIAN 3 SAT :EXPLORER15 [5681TIME :4 (dy) 0 (hr) 0.00 (min) 4.000(days)L 2.40 Beq : .022 Bcut :0.302 TW (days) :1.0

8min (Gauss) 0.001 Bmax (Gauss) :1.00010

0

wv 108id

0

w I

x i

z0

-LJ

LEGENDF-- 10' = 500 MEVZ = .500 MEV

= 1.90 MEV

= 1.90 MEV0 = 2.9 0 ME

10' ~' =2.90 MLE--.--S-.J-

0 0.5X -1Beq/B)1/2

Figare 15(f). Temporal Progression o-f Raw NOt and Pitted Flux Cure at L , 2.4 forRlussian 3 Bursit (r,, freely varvinm 00Ah physicalI contraints).

a 101

"" ' BURST RUSSIAN 3 SAT EXPLORER15 [5681

TIME : 5 (dy) 0 (hr) 0.00 (min) 5.000(days)L 2.40 Beq : .022 Bcut : 0.302 TW (days) 1.0

Bmin (Gauss) : 0.001 Bmax (Gauss) :1.000i__-- ------- - - --

' 10' 1oP

0

x 1d 10O -

LA

0.10

") . LEGENDLa 0 = .500 MEV

z = .500 MEV= 1.90 MEV

X = 1.90 MEV* = 2.90 MEV

4 v = 2.90 MEV10

': 0 0 ' 0.5 '. .. .110

X (1-Beq/B)112

Figure 1.5(g). Tempor-l Progres'ion of Raw D-ita and Fittel Flux Curvs nt , 2.4 forRussian 3 Burst (r. rTely varying with physical contrints).

1(12

.• . .

BURST RUSSIAN 3 SAT EXPLORER15 [568]TIME: 6 (dy) 0 (hr) 0.00 (min) 6.000(days)

L 2.40 Beq: .022 Bcut 0.302 TW (days): 2.0

Bmin (Gauss): 0.001 Bmax (Gauss) 1.000

106 10

0

Li

i

0ij:I '10' 0

:..1

i:':(..9LEGEND-010* .0 E 0

LaJ

::,:,

I.LJ = 0.500 MEV

= 2.90 MEV 100 0.5 ,/

X =(-Beq/B)"2

'igre 15(h). Tempid Progression of Raw Data and Fitted Flux Curves at . 2 '4 forR,mian 3 Burst (r. rreely varying wiLh physical contrmnts).

M3"

BURST: RUSSIAN 3 SAT EXPLORER15 f568]TIME: 8 (dy) 0 (hr) 0.00 (mtn) 8.000(days)L : 2.40 Beq : .022 Bcuf 0.302 TW (days) : 2.0

1010Bmin (Gauss) 0.001 Bmax (Gauss) :1.000

d1' d10'

0

N.

0 Z li ' = 108

0

io

- r

" 0

- r-J

i~i , lO & = .9oo MEV10i± o~ ---00MEV 1

- = 1.90 MEV= .90 MEV

1' V =2.90 MEV10IU, I I

0 0.5 1

X = (1-Beq/B)" 2

Figure 15k3). Temporal Progresion of Raw Data nad Fitted Flux Curves at I, 2 2.4 forRussian 3 Tlur- (r. rreeIy varying with physial rontrnints).

1n14

1WBURST :RUSSIAN 3 SAT EXPLORER15 f 568]TIME : 10 (dy) 0 (hr) 0.00 (min) 10.000(days)L:- 2.40 Beq: .022 Bcut :0.302 TW (days) : 5.0

1& mi (Gau?,'s): 0.001 Bmax (Gauss) :1.000

0

x id, t

ld0' 10p

LEEN

II.- %)

TIME 1 = .50 0

xri =Gus 1.9 0.0 Bo Eo ): .0

00 . .= .90 MEV0 005 10

":,~ (A-i

" Z 10 .500MEV 10

' ': " =1.90 MEV _" x =1.90 ME'/

o= 2.90 MEVl' .. "2.90 MEV ' ... " 10

0 0.5X (1-Bq/B) 1/ 2

?. IFiglure 15(J). ILmporal Progvf'uioa of Paw Data and F~itted Flux Curve ait L = 2.4 for' Ruaian 3 Burt.t (r,. freey varying with phyeiwal contraints).

al

'10-s

i

BURST RUSSIAN 3 SAT EXPLORER15 [568]TIME : 35 (dy) 0 (hr) 0.00 (min) 35.000(days)

L 2.40 Beq : .022 Bcut 0.302 TW (days): 5.0Bmin (Gauss) 0.001 Bmax (Gauss) :1.000

z 10p 10'

106n ,. 10"

0

0

-j

loe .o" 10'

' X =(1-Beq/B) '/

":Fjgee 15(k). "Tfmporal Pyogrino of Raw Data and Fitted Flux Curm, at 1. 2.4 for',!i Runi.n 3 Brt (rreely varying with physic.al contrmints).

I Or

,"

5

%" 1qz

LEEN. .... ... . 50 0 M.. . . . .V

|% ; .. . . . t . . . - - . . -. o , ,-. • • - . - - - "-77 7:7 77** - 7 .1. -

BURST RUSSIAN 3 SAT EXPLORL 15 [5681'"TIME.: 50 (dy) 0 (hr) 0.00 (min) 50.00O(doys)

L 2.40 Beq : .022 Bcut 0.302 TW (days) : 5.0Bmin (Gauss) 0.001 Bmax (Gauss) :1.000

1°'°

.-. 110

N0uo- log

!.I-oojo

0

x id, " d0

-

z0

10-"b 10'Sz

" LEGEND- - .500 MEV

10' .500 v0 0.5 1/

X (1-Beq/B)1/ 2

Figure 15I). Temporal Progression of Raw Data and Fitted Flux Curves at I, 2 2.4 forRiusian 3 Burst (r,. freely varying with physical contraints).

I:1010

- 0 %-.

- BURST RUSSIAN 3 SAT EXPLORER15 [568]

TIME : 60 (dy) 0 (hr) 0.00 (min) 60.000(doys)L : 2.40 Beq : .022 Bcut: 0.302 TW (days) : 5.0

Bmin (Gauss) 0.001 Bmax (Gauss) :1.00010 , 10

7Z:~10' -10'

00I-

z0

x 10' 10'

_ic 10 0

LEGEND0= .500 MEV

10' 0 = .500 MEV 0

X -eq/B) 1

rigure 15(m). Tremporal Progrluion of Raw Data and Fitted Flax Cur .,,t -A b 2.4 forRussian 3 Burst (r. freely varying with physical contraints).

,-n-

0n in- 0- N A O n r , r, i no inq -- in . r .-.- t-4) -0 0.0 r-i-. w@ qi ..a ~ 0 0in oo

-.- wn we nn nn -.- nn -.- uq nn

r ~.0 w# qtW0 4r- V# Vn M n Mn Mo m on. M62 C, 0 0 . 6-

00 00 00 00 00 00 00 00 00 00 .we on tw0o) d. wI r, 4w -V ; - W (0 00 M0Ot 0I N( Q4' r-..w)0 of We tolV#0 4' CD- AdVn

m o -- 6n .rd rn o r 6 q o w o- N -co

-0--0 r 'i

Mu r 04 in mm Cn) r)o -a #r , o m n- 'c-d 1), Co4t i

4w W 0-. 1) tr' o- 6# o- on .r on C1)-4.

6 4 ) od~ -...n 4i oi Cu .0 dia. 0-- wm i

coo

0+ +4 +4 4

WW .2 ;ww W wW ww wW w ww

-Inl 09 29 OR? Wi0 .0b MWo %n is It 9 onE- rtn. cr-. 0 " nIn rn in -orn0 . C - W r% c

= 1) 4 i- and* ed -' +4 Nr m4 'C-Lnd

8200 8 00 00 00 00 00 00 00 8 00 00 1

WW WW LgWW ww -. wwwww

0, 0 0 0D 0d 0I4 0 0 in 0 04 0- -40 0- N0 vn .4 ( 0 - 0- % 0) .C - 0 & .

M4 0 -O 0 M CD rd 0 nin tw nr o 00 C rI.i

00~ 00 0 000 00 000 00 00

om~~~~~ 0 0i n0VNrdi Pr- CD 4 .4 M rd 0d 0 l NCdN0 ri n

M V C i 0004

40 t I V 4 W W o o M r ni

o04*4* 0~~~~ 0 9. 0 N 0 0

p.7

+oi +o +W 44 1f 4 +-! O0 +l)~ W +-~- n+ nn fW WJLLW W W W W WW W W IAJ WlW Id W W l w l

CA rl d 0d 00 iO V 0r 0 ) j'0 ' i n. 1"1,0 Od 4t 00 -W O b

V) mlO n- .4 Ia 0fl 0"0- ,00 -04 o

00 00 00 00 00 oo 0 6 o o 00 00 00 00 00

0

I44W 00 00 01!- 0 V1 0 d 1 r0'0 V. ., 4 0- p a) N NW 0 f"--0 D 0 00 oo oo In t-a r- in o ~. ;r n In r) * I cr-N inri li d In I

C' Ol 0 0 N0 , Coj O ...ri 0 Od .0,-d d 0 Org 0 0 001

0 rd 0, ri i f g n i cen -a0 ) n M M a l(0," d 0 00 O 00 0 1 Of-0 Of ow) Ao0 d t - -. 0-, 4e

rif fto 010 rdf .tn -ni -. n m0. .0. In l o0 m t

0 0O' 00 00 0W 0D 0W in 0r o - 0 O0 Oo O0 rm o0 0- 00 00 Ow 04 - 04 - o- O 00 00 040 v u 0- .. 0 I

,0 - CJi NI~ CDd CDWl iF l i 0ri t. , ,o im dr 4I

-in0 --. 0 000 nt -0 q inn , n In: -r, n r, or -a n -"in -e inS O01 W0 O 0 O0 O0 T 0 ? 0 O 0 O0 0 0 0 0 OT= .0 4 0 0 - I 0 "I+ 4- + W) 4 + +, + +Ct- 4 W) 44) 4 WtuJ W *W W w ww ww L W W W *W uw~ til II W wW ll Il IAJ W UW -

a o oo o -00 r- .ud 0r " -" r'N , 0i -, -,F if,' -) Oi f N" U

ga- n' % N 4 -0 4 1u-- iJ'CI, 0 '-00- 00 00 00 010 M i (VM n m r ' rd.- r r. 1 -t 41 00- aio 0 in

-v o c ut n ..- a- -. an (i ' fe f0 'l 0

6 c nin i I-M- 00 00 00 N Nt 4Cv f- MCm

.0 0 0 0 0 0 00 0 00 00 00 00 00+ ~ ~ ~ ~ ~ ~ ~~~4 +4 +4 4+++++++++ 4c- W W LW W uJ bI w w WW g. l WW W W ww w 1*w w W w wW WC% d 0- 00 nt. rl 8 -. f Q 10M N O cj 0l~ M 0 n 0,-W@Z ) -in WD VN- N -;:. 1 0 Ill .0 IAoCD- n .I in W, " d Cd- q0o -N V t- -n (14 q fD- .t- 1i ~. .10 O -i 4-C v( m -W 0'a ..

0 ".0 I 0 t 0 0q wrd nflr 0 4W NN NN r 0 -a OD

. . .

.0 d0 4d 4-C n o. r4 M M -6- rn 0 CD iON N M N100 00 00 00 00 00 00 00 00 00 00 00 00 00

1 IIn

MO t- in -. rd 00 0- 0 -0 -0-. inn m fi n n. I n CO- ( rd-CID-in - in emo 0 d -o . -i in fIn 1 V, C -0- CD NW ON in tq40c 0-0 -Go, NN r -- 0- 00 NO t4'0 -In N- rd0 0 in ,- -CCD .4 M. N10 0- -V0 !4 -0 N,- N010 -G o0. mn on co t0- in -

>. 0 0 0 0 0 0 0 0 0 0 0 0 00 Id rd 0 - Cd 0 0 (d 4J Ii iD, 01 q4 0- 0- o. in 0- qt 01 O4 0- 0- 0A1 0 0 ml0 C 0 - d 01 0 0i 0 m

0 0 0 0 0 0 0 'D 0 0 0 0 0A0 0 01 01 (v 01 C4 9 Cd V qt qt4

.Pt N1 ft ft 01 (V 01 0 d 01 I'd 01 Cd rd

I.;

. ... ". . .. .. .- -.. .

01 I. 00 0

00. anS P 0 1" 00 0!SI,.w 40 0 & 4

- UP.: ri..

664 44 45e

IV CP ON' 4 nIV*-W )in ... 0 0. In (14. r 6 6 d r i

fi~i in6o00000 00 OC'

~i v1 OCS 0W a) -~

I ~ ~ n 6. 0 0IVr

AMIN,

41 lP -Win q*1 41) 8 1 0 Oi

.4 +.

W IA U~ 6I tU aow WUS '#U0, v9. to9 0-4 a ,;

1 .04W 44 ro I

1,10Id 0.01 Who -t onn CA

It! 1(d In I. I 'lr, U 144 1U.

*- 41i **-t

PrS- - to : ;f IN,inr14~ .. r (al

wo, to I',

gj 0 in in inW

t. 5.604 N t CI 1 rI 4w

It, -* .m i 04'( l le4 .14, fallI to 9 I 9D P . I" I d a I, C .

0 0 0

0 4u CV o

00 0 0 0

ri 19 9 (4 C

it

The short time-spans covered by most of the data sets iu the Trapped Electron Data Base

have not allowed adequate determination of ro. The values of ro presented it, Table XVIIae suspect for this reason. However, more confidence is held in the values f r, and r2computed by this study. In most data sets studied, the time period of the ,l:tta was longenough compared to the values or r, and r, to compute reasonable values. More consisti-neybetween data sets was found for r, and r2 titan was found for ro. or course, th,.n ro may be.simply computed by multiplying r, by the theoretical ratio corresponding to tle Solution.

.1

;'-,4

r-J

:11

U ~ VI. CONCLUSIONS AND RECOMENDATIONS

Conelulouslm

The pitch-angle diffusion theory and flux calculation method developed in this studyare reasonably consistent with experimental data in most cases. The method slows promiseas a way of calculating improved decay times over a broad region of the manetosphere,particularly for higher energy electrons. Additionally, a comprehensive summary table ofthe best available literature values for ro has been prepared (Table XV). A complete tablecovering such a broad range or energy and L has not heretofore been compiled.

Time and computer limitations have precluded examination of the entire data base.However, the overall efficiency of the model developed in this study has been demonstrated. Itis expected that improved predictions of fluxes and improved inputs to the AFW1, SPECTERCodes (Ref. 7-9) will be available as a result or this work. This will enhanice tje capabilityfor calculation of satellite operational environments, and will improve the abili .y to predictsatellite survivability and vulnerability.

Roeommendatlons

The following recommendations for further study are proposed to improve the modeldeveloped in this study, and to resolve the problem or non-physical solutions f,,r -ome datasets. At the direction of this author, several of these recommendations are presently beingimplemented at the Air Force Weapons Laboratory.

(1) Incorporate an estimation of the error in the magnetic field value, B, ofindividual data points. This corresponds to a "horizontal" error bar on theplots such as Figure 12. It is known that errors exist in the computation of theB value associated with each data point, and this error would prove significantin the region of the loss cone. Improved weighting of the data points with thecombined errors should improve the fit.

(2) Utilise the b-rounding procedure of Chapter IV to add data points to each 1,value considered. This should aid in filling gaps in the data, whic't necessarilywill improve the fit.

(3) Discard the highest eigenmode if the fitted value of its decay tim,, is less that0.25 day, and fit the data with only the fundamental and one higher mode.A value of 0.25 day is not reasonably fitted within the errors in the TrappedElectron Data Base. This mode would be essentially lost within thi first day inany case.

(4) Perform an "energy-scaling" of the data to combine data from different satel-F. lites. This involves assuming some form of energy spectrum, such asincluding an estimate of the error in the assumed value of Eo. The new fluxwould be simply the old flux multiplied by a factor e-(RR---.4IE. If theerror in the old flux were .50 percent (a value consistent with the Data Base)and the error in Eo were 25 percent, and if Re f 1.1 (a fission speftlrum), then

Ui the resultant error in the shifted flux would be less than I percent differentfrom that of the old flux, when shifting from 660 KeV to 500 KeV. Thus, itshould be possible to combine data sets and fill the gaps in data to achieve a

S". "better fit. However, the limitation of the assumed spectrum still ecists.

113

(5) Utilise only late-time data where it is available, and fit only the rundamentalmode to it, under the assumption that higher modes will have decayed away.Then, with the fundamental mode constrained, go back to early #time data andattempt to fit higher modes, if any. This recommendation is or limited valuebecause only two data sets cover a significant period or time: Starfish/Telstarand Russian 3/Explorer 5. However, it may provide a better value of r0 than ispresently available at some L values.

- (5) Revise the fitting functions to other Bessel runction solutions th:mn the J-t/2r:-. solution and compare results. This was performed only on one otli..r functional

solution, J0 , with only one small data set, and the results slowod no improve-

ment over the present study. However, to be certain that the au.hor has notsimply chosen the wrong functional form, more functional forms ust be in-vestigated.

(7) In the fitting program, constrain the directional flux to physical reality at eachiteration (i.e., compute flux at each iteration and reject non-physi'al solutionsat each iteration, rather than after converging on a final solution). This shouldconverge to a physically real solution if it converges at all; however, significantincreases in computer processing time may accrue.

(8) Increase the number of tries for a solution from 96 to 200 and reduce the numberof outer loops from 32 to 10 or 15. This may improve the convergence of the

least squares error, if non-physical solutions are also rejected at each iteration.

(9) Compare the background subtracted from the flux with the latest availablenatural environment, to ascertain the correctness of the backgroind subtr.c-tion. If significant differences are found, the background should bc re-added tothe flux and the natural environment subtracted.

(10) A suggestion proposed by Professor D.C. Shankland, Air Forre !ustitute ofT6-;6uulogy, involves a significant modification to the fitting wethod. Thismethod involves an interpolating function, or measure of 'roughness" or thefit, which m~y be made arbitrarily smooth. The advantage of this method isthat no limiting assumptions need be made about the data. The chief disad-vantage is that the method is a strictly mathematical method, with no physicalconstraints. Although the method would involve extensive reprogramming, itpromises worthwhile information about the fix curves, and shoulI be tried.

%

BIBLIOGRAPHY1. Abramowics, M. and I.A. Stegun (editors). Handbook of Mathematic3l Functions,

U.S. Department of Commerce, Applied Mathematics Series 55, Washington, D.C.,1988.

2. Baker, D.N., et. al., "'High-EnerTy Magnetospheric Protons and Their Dependenceon Geomagnetic and Interplanetary Conditions," Journal of Geophysi-al Research

Ss4:7838 (1979).

3. Chan, K.W., et. al., "Modelling of Electron Time Variations in tie R.aliationBelts," Quantitative Modeling of Magnetospheric Processes, Geophh, %ical Mono-graph 21:121-149, W.P. Olson, ed., American Geophysical Union, Washington,D.C., 1979.

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6. Cladis, J.B., et. al., (editors), The Trapped Radiation Handbook, DNA 2524 HRevision 2, Headquarters, Defense Nuclear Agency, Washington, D.C., November1973. (AD 738841)

7. Cladis, J.B., et. al., Improvement of Spector II Code, AFWL-TR-75-286, Air ForceWeapons Laboratory, Kirtland AFB, New Mexico, September 1970.

S. Cladis, J.B., et. al., Improvement of Specter H Code, Injection and Evolution ofAn Artificial Radiation Belt, AFWL-TR-78-236, Air Force Weapons Laboratory,Kirtland AFB, New Mexico, August 1979.

9. Cladis, J.B., et. al., Improvement of Specter Ht Codes, Models of GeomagneticField, Fission Products Decay, Debris Tube Dynamics, and Evolution of TrappedElectron Distribution, AFWL-TR-80-117, Vol. 1, Air Force Weapons Laboratory,Kirtland AFB, New Mexico, July 1981.

10. Davidson, G.T., 'An Improved Empirical Description of the Bounce Motion orTrapped Particles," Journal of Geophysical Research 81:4029-4030 (1,170).

11. Hess, W.N., The Radiation Belt and Magnetosphere, Blaisdell Publishing Co.,Waltham, Massachusetts, 1968.

12. lies. W.N., 'The Artificial Radiation Belt Made on July 9, 1962," Journal of7 Geophysical Research 88:6617-883 (1963).

13. Higbie, P.R., et. al., "High Resolution Energetic Particle Measurement at 0.11 Rp,"Journal of Geophysical lesenrch 83:4851 (1978).

14. Imhof, W.L., et. aL., "Long-Terum Study o F, lectrons Trapped on Lo>w L Shells,'. .•Journal of Geophysical Resenrch 72:2371. (10617).

i " 115

15. Jarrett, C., Programmer, Computer Sciences Corporation, 'Program DTABASE,'unpublished FORTRAN computer program, Air Force Weapons Laboratory, Kirt-land AFB, New Mexico, November 1982.

16. Jensen, D.C. and J.C. Cain, 'An Interim Magnetic Field," Journal of GeophysicalResearch 67:3568-3569 (1082).

17. Kuck, G.A., Pitch-Angle Diffusion of Relativistic Electrons in the Pasmasphere,AFWI-TR-73-115, Air Force Weapons Laboratory, Kirtland AFIB, r4ew Mexico,June 1973. (AD 762-952)

18. Kivelson, M.G., et. al., 'Satellite Studies of Magnetospheric Sabstorms on August15, 1908,* Journal of Geophysical Research 78:3079 (1973).

19. Lanserotti, L.J. and A. Wolfe, "Particle Diffusion in the Geomagnetosphere: Com-parison of Estimates from Measurements of Magnetic and Electric Field Fluctua-tious," Journal of Geophysical Research 85:2345 (1980).

20. Lyons, L.R., "A Theory for Energetir Electron Lifetimes within the Plsmasphere,"in Folkestad, Kristen, Magnetosphere-lonosphere Interactions, 57-81, Universitets-forlaget, Oslo, 1971.

21. Lyons, L.R., R.M. Thorne and C.F. Kennel, 'Electron Pitch-Angle Diffusion DrivenBy Ob!ique W istklr-Mode Turbulence,* Journal of Plasma Physics 6:589-006(1971).

22. Lyons, L.R., R.M. Thorne and C.F. Kennel, 'Pitch-Angle Diffusion of Rakdiation BeltElectrons within the Plasmasphere,' Journal of Geophysical Research 77:3455-3474(1972).

23. Lyons, L.R., and R.M. Thorne, 'Parasitic Pitch Angle Diffusion of Rrdiation BeltParticles by Ion Cyclotron Waves," Journal of Geophysical Research 77:5808-5616(1972).

24. Lyons, L.R., and R.M. Thorne, 'EqAilibriun Structure of Radiation Bel" Electrons,"Journal of Geophysical Research 78:2142-2149 (1973).

25. Lyons, L.R., 'Comments on Pitch Angle Diffusion in the Radiation Belt&," Journalof Geophysical Research 78:0793-5797 (1073).

26. Lyons, L.R., "General Relations for Resonant Particle Diffusion in Pitch Angle andEnergy," Journal of Plasma Physics 12:45-49 (1974).

27. Lyons, L.R., 'Electron Diffusion Driven by Magnetospheric Electrost:%tic Waves,"Journal of Geophysical Research 79:575-580 (1974).

* -28. Lyons, L.R., "Pitch Angle and Energy Diffusion Coefficients from Resonant Internc-tions with ion-Cyclotron and Whistler Waves," Journal of Plasma Ph) sics 12:417-

• .. 432 (1974).

29. Mihalov, J.D., 'Energetic Trapped Electron Observations During the Fall of 1964,'Journal of Geophysical Research 72:1081-1094 (1907).

11

30. MacDonald, W.M. and M. Walt, *Distribution Function of Magnetically Confined

Electrons in a Scattering Atmosphere,* Annals of Physics 15:44-62 (1981).

31. Moser, F.S.. et. al., 'Preliminary Analysis of the Fluxes and Spectrum; of TrappedParticles after the Nuclear Test of July 9, 1962," Journal of Geophysical Research68:641-849 (1963).

32. McCormac, B.M. (Editor), Radiation Trapped in the Earth's Magnetic Field, ReidelPublishing Co., Dordrecht, lolland, 1966.

33. McLachlan, N.W., Bessel Functions for Engineers, Oxford University l'res.s, London,1955.

34. Murphy, Harry, "Program Electroflt,' unpublished FORTRAN computer program.Air Force Weapons Laboratory, Kirtland AFB, New Mexico, December 1982.

35. Palmer, I.D. and P.R. Higbie, 'Magnetosheath Distortion of Pitch Angle Distribu-tions of Solar Protons,' Journal of Geophysical Research 83:30 (1978).

36. Pflitser, K.A., An Experimental Study of Electron Fluxes from 50 KeV to 4 MeV inthe Inner Radiation Belt, TR-CR-123, School of Physics and Astronomy, Universityof Minnesota, Minneapolis, Minnesota, August 1968.

37. Pftser, K.A., Summary of Trapped Electron Data, AFWI,-TR-81-223, Air ForceWeapons Laboratory, Kirtland AFB, New Mexico, October 1982.

.38. Pieper, G.F. and D.J. Williams, "Trnaa Observations of the Artificial RadiationBelt from the July 9, 1962 Nuclear Detonation," Journal of Geophysical Research63:635-640 (1963).

39. Roederer, J.G., Dynamics of Geomagnetically Trapped Radiation, Springer-Verlag,Berlin, 1970.

40. Roberts, C.S., "On the Relationship Between the Unidirectional and Omnidirection-al Flux of Trapped Particles on a Magnetic Line of Force,' Journal of GeophysicalResearch 70:2617-2527 (1965).

41. Roberts, C.S., "Pitch-Angle Diffusion of Electrons in the Magnetosphere,' Reviewsof Geophysics 7:305-337 (1969).

42. Rossi, B. and S. Olbert, Introduction to the Physics of Space, McGraw-Hill, NewYork, 1970.

43. Rosen, A. and N.L. Sanders, "Loss and Replenishment of Electrons in the InnerRadiation Zone During 1955-1967,' Journal of Geophysical Research 76:1 10-121(1971).

44. Scott, Edie (Editor), TRW Space Log, Redondo Beach, California, T(PW SystemsGroup, TRW Inc., One Space Park, Redondo Beach, California 00278 (1970).

45. Shuls, M. and L.J. Lauserotti, Particle Diffusion in the Radiation Bells, Springer-Verlag, New York-Heidelberg-Berlin, 1974.

.117

117

!~~~~...... .... ....-............. •-.-............. . .-.. ,.1 ._..1-....-,. . .. :.,....

46. Shuls, M., "Relation Between Boune-Averaged Collisional Transport Co.fllcientafor Geomagnetically Trapped Electrons," Journal of Geophysical Resea-eb 61:5212 -13(1976).

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48. Shulz, M. and D.J. Boucher, 'Pitch-Angle Diffusion in Canonical Coordinates: SomeNumerical Results," The Aerospace Corporationn, El Segundo, California, October1979, presented at 1979 Fall Meeting of American Ceophysical Union, S:n Francisco,California, 3 December 1979.

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50. Stasuinopoulos, E.G. and P. Verzariu, "General Formula for Decay Lifetimes orStarfish Electrons,* Journal of Geophysical Research 76:1841 -1844 (I171).

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53. Teague, M.J. and E.G. Staauinopoulas, A Model of the Starfish Flux in theS .' -. Inner Radiation Zone,* X-601-72-487, National Space Science Data Conter, NASA

Goddard Space Flight Center, Greenbelt, Maryland, December 1972.

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55. Tranter, C.J., Besel Functions with Some Physical Applications, Ilart PublishingCompany, New York, 1968.

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~IS

I' '''" " ' " " ' '" " "' ''' " "''

61. West, H.!., R.M. Burk and J.R. Walton, 'Electron Pitch Angle DistributionsThroughout the Magnetosphere as Observed on Ogo 5," Journal of GekophysicalResearch 78:1064 (1973).

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Ito

APPENDIX A

Demonstration of Zero Diffusion Currentat the Magnetic Equator for Ilair-Integer

Bessel Function Solution

To demonstrate ero diffusion current, one must show thatlim,_.oD,z ljz) 1

for the a = 0 case. We Know that from (31):

J-i/.2/lx) ca o(z)

and from (16):

and from (18):

= cos ((2n + ( 1 rJ.11 Z- (2 + l 1)( 1,2)( 1z/ + ) ze)

[)c os Gn+ l s')

Hence, the derivative of the eigenfunction:

J~(z) = + '))][(29t + I)j!L] sin ((2nP + 1

Finally, the quantity which must go to zero is:

DxV = [D3 5 . V4/j22 ++) (n+1 r(z)n; )""="- " 2 //1 +'xl1l(n ls/(.1 sin (2n + 1)-r

The term in square brackets is a finite constant for any n, and clearly the sine goes to zeroas z goes to zero. Hence, the limit is zero.

* 1°o*f°

,.

APPENDIX B

Derivation of Eigenfunction Derivatives 7oo and 1

To find the derivative of the steady state function, J.e, one starts with (21):

-.= --- j (') t,,(z') d(')m-O '0

If the X. are ausumed (from (19)) independent of z, then the series can simply bedifferentiated tarm-by-term, since the definite integral is simply a constant for each valueof n.. x.]

To nd the derivative of the eigenfunction , one starts with (18):

Using the product rule for differentiation:

ds

44

121

:. .- .(,.(.)

APPENDIX C

The Specific Solution of the Integral-EnergyOmnidirectional Flux Equation for the Half-Integer

Bessel Function Solution

The Ibtepnrlenerly omnidirectional flux Equation (43), which is the energy-integratedform of Equation (37), is represented as the steady state term, f.., plus the sun, of exponen-tially decaying eigenmodes f. through f3:

>(, ON 1B)=,.. , , 2h f2 f3

In this appendix, each term is shown, a derived from Equations (37) and (43). Notethat the eigenmode sum should be infinite, but only the fundamental mode and three highermodes are shown.

The constants in each term are defined as follows (Ref. 39:55):

A~. To - 1 + ((n42 + V531)/(2vi)}re 1.3801730

T, - (wV2)iS P 0.7404805

K - (4/1 I)(To - T,! P 0.232 155

Z. = ToO - V.2) - K(i .- )

The ting eonstants are defined as in Equation (43) or in Appendix D:

=- 2 f ... () dE

a f p2a.(S) dl

[ ,v~i (4 Z.w) [vIZ) r - (B/uN)][To - (To -Ta)V1I4jV]

{sift [((/27,)( [(1 -y' 2)rnj/2 - K(l _

(sin [(3r/27.)(((I - 02 )To)/2 - K(I - y'

(sin [(5r/2Za)(((l - V)To)/2 - K(l -

25

1 2-p , . . . . - . -. ... - . . . . 2 . . .. • ;. . . .

m .4- .. . ... : ,r, -- d "" .. . -" ' ' - '" -

VP (sin [(7w/27,,)(((1 -2)To)/2 - K(I - 14V

fe -2wagee0/"o V V(B~(F/e1 1 - v27I/flo)f To (To

v- ( Z))(( -2)To)/: - K (I - V 1/ Dd

f , + 2w . s11 / s df v - / R . Ha[ / -Z . 3 - Z ) 1- _v2 (I)B . )I T o - (T o _ T d v '1 4] u ]

(sin [(3i/(2Ze))(((l - y")To)/2 - K(l - V/4)@d

f2= -2ae 1 "' [V2/1(5w/Z.4Vl- - -y2(B/Bo) ][. - (r. - ,3/

(sin [(Sir/(2z,.))(((1 - y2 )To)/2 - K(I - Y 1 /4))1)d

12 =~~[V' _.11 C") [ii( c[ V - (B/Bo)1 [To - (To _-,Y34

(sin [(7w/(2Z.))(((1 - vtYTo)12 - K(I - y 1 t/4 )))

Theme fitting constants are really tuctions of energy' and &rshell.

The ratio@ ofro, ?i, r2~, and r3 should be (as computed from the eigen value~ ratio of theseroes of the function):

rqf 25rj or alternatively r2 (t/25)raro ow49r3 f = (1/ 4 9 )ro

123

APPENDIX D

The General Solution of the Integral-Energy

Omnidirectional Flux Equation for Bessel Functionsor Arbitrary Order, ; a Function of Sigma

Substitution of (18), (28), and (2g) into (27) completes the general expre;sion ror thedifferential energy omnidirectional flux, J, with Bessel functions or order

2-

where iv is some number less than two:

J ~ 69 = 9 1Z5 v/-~z'='-dt-L.._0,- J"'(". ) Z.--[: )=

• -L+:-L;( - a) , I - f Z -- i-- -

+ :'-" '. -z v(R/.I) T(V) )(y)))Th. ,g Lb th abo -qut io n ti y(

-o dr 1) 2 y ,( -,, )

!. ( -° e t -( ' + ',= p 2 '-A-L(-):::f: L(t1 ,, t.+T(V) dy )

Th aencea)ssloy te en rithe oe qatio d tEm o (437) iw f

.' . , _.,- . _'_'_ _

S... .J

... 1 . _ ,,/0 -,)___-_ )}

Pre2( = l , ,,,.,,,.

"?-" Hene, (43) is simply the nergy-intpg.ted form of (37).

APPENDLX E

Flow Chart for Program Eletrorlt

Initial gues ro (and ,, ,t).

Numerically integrate functions at,' , each data point.

easts

I. .,Least s~pares fit for the constants:

'%',-.' Optimizr e

fit tuing least squares residual. better_

UlKte "best" r's and a'q.

no better

')" :"'[ 1r~v m e nC h ec k for 00 tries with no ira-

: ~Yes l

l ] Check for 32 better fits ]yes

""no no

Rantdom walk r-s L

;. Print results.',° ( ,,n mlp lle. iesX ..)2 I'hot co'mputed flILIeS With raw

' data.

VITARoger Scott Dewey was born on Aulgnst 15, 1947 in Akron, Ohio. fie graduated from

Tallmadge High School, Tallmadge, Ohio, in 19fib and attended Brown Uni,.ersity, from

which he received the Degree of Bachelor of Science in Chemistry in January 1970. Upongraduation, he received a commission in the USN through the NROTC progr:im. lie com-pleted flight training and received his wings in April 1971. lie has served as :in SII-3 pilotand flight instructor in Helicopter Antisubmarine Squadrons At Naval Air Stations Quon. tPoint, Rhode Island and Jacksonville, Florida. lie served as Assistant Air Oper:,tions Officer

an 1-IN pilot aboard the IISS Guadalcanal, LPIJ-7, out of Norfolk, Virginia.utletring the School of Engineering, Air Force Institute of Technology in August 19.

Permanent Address:

co Robert A. DeweyRD IBemus Point, New York 14712

. ..

UNCLASSIFIEDSECURITY CLASSFICATION OF THI O4 _ 'W, Ao r)Are 'refred)

REPORT DOCUMENTATION PAGE RFA IN INSTRUCTIONSI.C E ( ,NI'ILFTIN I' F RM

I, REPORT NUMDER 1 GOVT ACCEFSSICN NO. 3 PF(I'ri,'

T- r AT A G NUMFr R

AFIT/GNE/Plt/83M-4 At_ ID93aI~~q4. TITLE (and Subtitle) 5 .'L J; NLP(,&T & PERIOJ COVE+ -

ELECTRON LOSSES FROM THE MAGNETO3PHERE MsU Thesis

"" PF , I , Ir-rf f PO T NUMf

7. AUTHOR(s) S. CONIPACT O0 GRANT NuMItFR .

Roger S. DeweyLCDR USN

9. PERFORMING ORGANIZATION NAME AND AODRESS 10. FPOAj;M I FMF pT. -rJ- T T A

Air Force Institute of Technology(AFIT-EN) APrA W*., UNIT NUMf ,,

Wright-Patterson AFB, Ohio 45433 Y091901

I. CONTROLLING OFFICE NAME AND ADDRESS 12. -FOORT IDA I-

Air Force Weapons Laboratory(AFWL/NTCTS) iAarch, 1983Kirtland AFB, N.M. 57117 13. JJMSEP Or PAGES

__-'_134I4. MONITORING AGENCY NAME & ADDRESS(I1 different from Controlling Office) 15. SE(CURITY CL. ASS. (.f this .,,p,01

Unclassified

Ia. OfCLASSIFtCATION DOWN GRtI;i N CSCH.DULr

16. DISTRIBUTION STATEMENT (of this Report)

Approved for Public Release; distribution unlimited

17. DISTRIBUTION STATEMENT (of the ebtrect entered in Block 20, If differnt fro. Pp.rf

Approved for public release; IAW AFR 190-1

* - 1S. SUPPLEMENTARY NOTES 1AV~ezh a V2 mIWns.

Je~ld w Aseatch amd ProPelenhmiel Devel.p NO

Aft Ifse h--Utut, of Teobuslogy (ATC)

I•. KEY WORDS (Continue an reverse ede if necessary and identify by block number)

electron flux electron decay timemagnetosphere

* pitch-angle diffusiontrapped radiationelectron loss time

20. ABSTRACT (Continue an rearse aide If necessary and Identify by block number)

A semi-empirical method for modelling loss of electron fluxesin the earth's magnetosphere was developed. An equation for

.! integral-energy omnidirectional electron flux as a function oftime and magnetic field strength was derived from pitch-anglediffusion theory. The flux equation was the basis for a com-puter data-fitting program written at the Air Force WeaponsLaboratory(AFWL) to fit the AFWL Trapped Electron Lata Base.The program utilized a least-sguares fit and incorporated random

* DD , P2'°NP 1473 licT.o. OF INOV 65, OBSOLETE UNCLASIFIE-S;F.CURITY CLASS1 ' , ! II ,N 1"t ''.*U PAt,r .. ,, P ' , ,

UNCLASSIFIE DSE'URIT CLASSIFICATION OF THIS PAGE(Whan Date Entefed)

variations of the characteristic exponential loss times abouttheir initial values. An improved table of initial loss times was

* q compiled for use with the program. The derived flux modelshowed substantial agreement with the empirical data base.

Representative plots of computed fluxover raw data are shown for L-values of 2.3 and 2.4.

'*1

UNiCLASSIFIE D

I 141T r o tF 1A 4