Solar Flare Particle Propagation: Comparison of a New Analytical Solution with Spacecraft Measurements

Citation

Lupton, John Edward (1972) Solar Flare Particle Propagation: Comparison of a New Analytical Solution with Spacecraft Measurements. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/RP79-C621. https://resolver.caltech.edu/CaltechTHESIS:05092016-095858999

Abstract

A new analytic solution has been obtained to the complete Fokker-Planck equation for solar flare particle propagation including the effects of convection, energy-change, corotation, and diffusion with ĸ _r = constant and ĸ _Ɵ ∝ r ² . It is assumed that the particles are injected impulsively at a single point in space, and that a boundary exists beyond which the particles are free to escape. Several solar flare particle events have been observed with the Caltech Solar and Galactic Cosmic Ray Experiment aboard OGO-6. Detailed comparisons of the predictions of the new solution with these observations of 1-70 MeV protons show that the model adequately describes both the rise and decay times, indicating that ĸ _r = constant is a better description of conditions inside 1 AU than is ĸ _r ∝ r. With an outer boundary at 2.7 AU, a solar wind velocity of 400 km/sec, and a radial diffusion coefficient ĸ _r ≈ 2-8 x 10 ²⁰ cm ² /sec, the model gives reasonable fits to the time-profile of 1-10 MeV protons from "classical" flare-associated events. It is not necessary to invoke a scatter-free region near the sun in order to reproduce the fast rise times observed for directly-connected events. The new solution also yields a time-evolution for the vector anisotropy which agrees well with previously reported observations.

In addition, the new solution predicts that, during the decay phase, a typical convex spectral feature initially at energy T _o will move to lower energies at an exponential rate given by T _KINK = T _o exp(-t/Ƭ _KINK ). Assuming adiabatic deceleration and a boundary at 2.7 AU, the solution yields Ƭ _KINK ≈ 100h, which is faster than the measured ~200h time constant and slower than the adiabatic rate of ~78h at 1 AU. Two possible explanations are that the boundary is at ~5 AU or that some other energy-change process is operative.

Item Type:

Thesis (Dissertation (Ph.D.))

Subject Keywords:

(Physics)

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California Institute of Technology

Division:

Physics, Mathematics and Astronomy

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Physics

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Public (worldwide access)

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Stone, Edward C.

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Space Radiation Laboratory

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Unknown, Unknown

Defense Date:

2 December 1971

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Funding Agency	Grant Number
NDEA	UNSPECIFIED
NASA	UNSPECIFIED
NASA	NAS5-9312
NASA	NGR-05-002-160
NASA	NGL-05-002-007

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CaltechTHESIS:05092016-095858999

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https://resolver.caltech.edu/CaltechTHESIS:05092016-095858999

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10.7907/RP79-C621

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SOLAR FLARE PARTICLE PROPAGATION -- COMPARISON OF A NEW ANALYTIC SOLUTION WITH SPACECRAFT MEASUREMENTS Thesis by John Edward Lupton In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1972 (Submitted December 2, 1971) ii ACKNOWLEDGMENT This thesis is to a great extent the result of the guidance of my faculty sponsor, Professor Edward C. Stone. I am grateful for his sug- gestions and encouragement, and especially for his ability to find time for me within his busy schedule. My thanks go to Dr. Stephen S. Murray, whose thesis was the predecessor of the work presented here, and whose advice concerning many of the particulars of the OG0-6 experiment was indispensable. Steve was responsible for the calibration of the instrument response to protons, and many of his data analysis programs were borrowed directly for this work. I would also like to acknowledge Professor Rochus E. Vogt, who is co-investigator on the OG0-6 experiment. Dr. Vogt's support, both moral and technical, is appreciated. My theoretical efforts were greatly aided by discussions with Caltech Professors J. R. Jokipii, Jon Mathews and Don Cohen, and by talks with Dr. L. A. Fisk of GSFC and Miriam A. Forman of the State University of New York. I also greatly appreciate discussions with Dr. John Fanselow, and with my fellow graduate students Lawrence Evans, Tom Garrard, Alan Cummings, and James Brown. Special thanks go to my undergraduate helpers Paul Sand, who carried out many of the necessary computer calculations, and Don Gunter, whose constant barrage of questions led to the clarification of many problems. iii Caltech's Solar and G:1lactic Cosmic Ray Experiment aboard OG0-6 was the result of the joint efforts of the personnel of the Space Radiation Laboratory, the Central Engineering Services, the Analog Technology Corporation, TRW Systems, Inc., and the OGO Project Office of the National Aeronautics and Space Administration. Among the Caltech staff, William E. Althouse must be mentioned as that indispensable person who really understands the OG0-6 experiment electronics. Thanks also to Ellen Aguilar and Florence Pickett for their help with the data handling and analysis. Valuable support for my graduate work at Caltech was provided by an N.D.E.A. Fellowship and a NASA Traineeship. The research work presented here .was supported in part by the National Aeronautics and Space Administration under Contract No. NAS5-9312 and Grant Nos. NGR-05-002-160 and NGL-05-002-007. I am especially grateful to my wife Kathy, whose nimble fingers at the keyboard of the Selectric transformed my scribblings into the pages which follow. iv ABSTRACT A new analytic solution has been obtained to the complete Fokker-Planck equation for solar flare particle propagation including the effects of convection, energy-change, corotation, and diffusion 2 with Kr = constant and K8 ~ r . It is assumed that the particles are injected impulsively at a sing l e point in space, and that a boundary exists beyond which the particles are free to escape. Several solar flare particle events have been observed with the Caltech Solar and Galactic Cosmic Ray Experiment aboard OG0-6. Detailed comparisons of the predictions of the new solution with these observations of 1-70 MeV protons show that the model adequately describes both the rise and decay times, indicating that K r = constant is a better des- cription of conditions inside 1 AU than is K r boundary at 2.7 AU, a solar wind velocity of radial diffusion coefficient K - 2-8 x 10 20 r ~ r. With an outer 400 km/sec, and a cm 2 /sec, the model gives reasonable fits to the time-profile of 1-10 MeV protons from "classical" flare-associated events. It is not necessary to invoke a scatter-free region near the sun in order to reproduce the fast rise times observed for directly-connected events. The new solution also yields a time-evolution for the vector anisotropy which agrees well with previously reported observations. In addition, the new solution predicts that, during the decay phase, a typical convex spectral feature initially at energy T will 0 move to lower energies at an exponential rate given by TKINK = T exp(-t/TKINK). 0 Assuming adiabatic deceleration and a boundary at v 2.7 AU, the solution yields TKINK ~ lOOh, which is faster than the measured -200h time constant and slower than the adiabatic rate of -78h at 1 AU. Two possible ex~lanations are that the boundary is at -5 AU or that some other energy-change process is operative. vi TABLE OF CONTENTS PART TITLE PAGE I. INTRODUCTION 1 II. INSTRUMENT 4 A. General Description 4 B. The 6E-Range Telescope 6 1. Physical Description 6 2. Detectors 9 3. Anti-coincidence Shield 11 4. Electronics 11 5. Electronics Calibration 15 C. III. Spacecraft 17 1. The Satellite Orbit 17 2. Invariant Latitude and Magnetic Local Time 18 DATA ANALYSIS 22 A. Proton Response of the 6E-Range Telescope 22 1. Accelerator Calibration 22 2. Proton Energy-loss Calculation 23 3. Pulse Height Data Reduction 25 B. Bulk Data Processing 37 1. Tape Merging 38 2. Polar Rate Averages 38 3. Orbit Plots 40 4. Calculation of Proton Flux 45 vii TITLE PART IV. v. PAGE OBSERVATIONS 47 A. Event Identification 48 B. OG0-6 Flare Observations 50 DISCUSSION 65 A. Introduction 65 B. Background 70 1. The Interplanetary Medium 70 2. The Fokker-Planck Equation and Particle 72 Diffusion 3. C. Solar Flare Particle Events Solving the Fokker-Planck Equation for Solar 78 80 Flare Particle Injection 1. Some Boundary Conditions and Simplifying 80 Assumptions D. 2. Separation of Variables 82 3. The Azimuthal Dependence 83 4. The Energy Dependence 85 5. The Radial Dependence 87 The New Solution 89 1. Derivation 89 2. Behavior of the Solution 91 viii TITLF. PART V. PAGE DISCUSSION (cont.) E. F. G. Fits to Data Assuming Pure Adiabatic Deceleration 100 1. Method of Fitting Actual Data 100 2. Forman's Solution 102 3. Fits using the New Solution 104 4. Evaluation 108 Vector Particle Anisotropy 113 1. Definition 113 2. Observations 114 3. Anisotropy Predicted by the New Solution 115 The Energy-Change Effect 120 1. Statement of the Problem 120 2. Observations 121 3. Energy-Change Predicted by the New Solution 124 4. Conclusions Concerning the Energy-Change 136 Process VI. CONCLUSION Appendix A - OG0-6 Monthly Summary Plots for June 1969 141 144 through February 1970 Appendix B - Solution for the Azimuthal Dependence 157 ix TITLE PART Appendix C - Derivation of the New Solution to the PAGE 159 Radial Equation 1. Derivation of the Sol11tion 159 2. The Orthogonality Relation for Regular 162 Coulomb Wave Functions 3. Evaluating the Orthogonality Integral 163 Appendix D - Notes on Calculating the New Solution 165 References 168 x LIST OF FIGURES TITLE NUMBER PAGE II-1 Cross-section of ~E-Range Telescope II-2 ~E-Range II-3 Typical OG0-6 Orbital Trajectories in Geocentric 7 Telescope Electronics Block Diagram 13 19 C.Jordinates II-4 Typical OG0-6 Orbital Trajectories in A-MLT 21 Coordinates III-1 Calculated Average Energy Loss in Various Range 24 Telescope Detectors vs. Incident Energy III-2 Dl vs. D2 Proton Response 27 III-3 D2 vs. D3 Proton Response 29 III-4 Actual Dl vs. D2 Pulse Height Data 31 III-5 Actual D2 vs. D3 Pulse Height Data 33 III-6 Flow Chart of Range Telescope Data Processing 39 III-7 Computer-Generated Rate Plot 42 III-8 . Computer-Generated Orbit Plot 44 III-9 Computer-Generated Plot of Proton Flux vs. 46 Incident Energy IV-la -lb -le -ld Time Histories of Four Selected Solar Flare Particle Events 53 - 56 xi NU¥.BER TITLE PAGE IV-2a Samples of Proton Differential Energy Spectrum -2b During Decay Phase of Four Selected Flare -2c Events 60 - 63 -2d V-1 Diffusion Coefficient K vs. Proton Kinetic Energy 76 V-2 New Radial Solution R(r,t) vs. Radial Distance r 92 for Various Times After Particle Injection V-3 New Radial Solution R(r,t) vs. time t Observed at 93 Different Distances. V-4 Dependence of the Decay Time Constant 'DEC on Kr' 95 V, L, and C. V-5 Radial Distribution of Particles at Late Times vs. 97 r for Various Ratios V/K . r V-6 Particle Current S, and its Components SV and SK 99 vs. r for three values of K • r V-7 Fits to 2 November 1969 Event Using Forman's 103 Solution with L = 2.3 AU. V-8 Fits to 7 June 1969 Event Using New Solution 105 with L = 2. 7 AU V-9 Fits to 2 November 1969 Event Using New Solution 106 with L = 2. 7 AU V-10 Fits to 31 January 1970 Event Using New Solution with L = 2. 7 AU. 107 xii TITLE NUMBER V-11 Best-fit Values of K r PAGE vs. Proton Energy for 109 All Three Events V-12 Diffusive and Convective Components of Aniso- 118 tropy Predicted by New Solution vs. Time V-13 Vector Diagram of Time-Evolution of Anisotropy 119 V-14 Time-Dependence of Intersection Energy During 123 Decay Phase of 7 June 1969 and 31 January 1970 Events V-15 Time-Evolution of Convex Spectrum Generated by 127 Superposition of Power Laws V-16 Dependence of Calculated TKINK on Spectral Shape V-17 Dependence of TKINK on v, L, and K V-18 Dependence of TKINK on TE and L. 133 V-19 Time-Evolution of the Curvature in the Kink 135 V-20 Fits to 7 June 1969 Event with New Solution 137 r . 131 132 Matching Observed Energy-Change Rate V-21 Fits to 31 January 1970 Event with New Solution 138 Matching Observed Energy-Change Rate A-la -lb -le -ld -le OG0-6 Data Summary Plots for June 1969 to February 19 70 148 - 156 x i ii TITLE NUMBER A-lf -lg OG0-6 Data Sununary Plots for June 1969 to PAGE 148 - 156 February 1970 (continued) -lh -li D-1 Computer Routines Used to Calculate the New Solution 166 xiv LIST OF TABLES TITLE NUMBER PAGE II-1 OG0-6 Experiment F-20 Particle Telescopes II-2 ~E-Range Telescope Stack 8 II-3 ~E-Range Telescope Detector Rates 14 II-4 Ground-based Commands for ~E-Range Telescope 16 Telescope Proton Pulse Height Bins 35 III-1 IV-1 ~E-Range Flare-associated Events Observed on OG0-6 5 51 Between June 1969 and February 1970 IV-2 Summary of Four Selected Flare Events 57 V-1 Some Analytic Solutions to Fokker-Planck 69 Equations for Solar Flare Particle Propagation V-2 Power Law Superposition of a Density Spectrum with a Convex Kink 126 1 I. INTRODUCTION The study of solar cosmic rays includes several separate areas of interest: the means by which so l ar particles are accelerated and in- jected into interplanetary space , the transport and storage of these particles in the solar system, and the problem of particle access to the magnetosphere. The subject of this dissertation, the propagation of solar flare protons, is part of the second of these topics. In particular, near-earth observations of solar flare proton fluxes will be used to develop a more complete representation of the physical processes governing energetic particle transport. The basic principles underlying the propagation of cosmic rays in the solar system are at the present time fairly well established(l). The picture of the interplanetary medium with a spiral magnetic field imbedded in an outward-flowing solar wind plasma has won general acceptance ( 2), and the first observational verification of the diffusionapproximation to cosmic ray motion was made by Meyer, Parker, and Simpson in 1956( 3 ~ More recently, Parker added terms for particle con- vection and energy-change in the solar wind to the equation for particle diffusion( 4 ). This Fokker-Planck equation is now widely-used as a description of particle transport in the solar system. In addition, a relationship between the observed fluctuations in the magnetic field and the magnitude of the diffusion tensor has been developed by Jokipii(S, 6 ,?) and others(B, 9 ), and has provided an independent means of estimating the rate of particle diffusion. 2 Many solutions to the Fokker-Planck equation have been developed in an effort to explain the particle fluxes observed subsequent to solar flare injection. The models proposed have become more and more refined, and analytic solutions now exist which include impulsive injection, anisotropic diffusion (due to the presence of the average magnetic field), convection, and energy-change (10 11 12 13 14) ' ' ' ' • Despite these developments, none of these solutions have successfully explained all of the observed features of solar flare events. important questions remain unanswered: Several the exact nature of the diffusion tensor and especially its dependence on radial distance and particle energy; the method of particle injection and the possibility of storage near the sun; the possible existence of a scatter-free region extending outward some distance from the sun; the way in which the particles become distributed in solar longitude; and whether or not an outer boundary to the diffusing region exists beyond which particles are free to escape. The work presented here is a continuation of the process of comparing theoretical solutions with spacecraft observations. Several solar flare particle events have been observed with the Caltech Solar and Galactic Cosmic Ray Experiment aboard NASA's OG0-6 spacecraft. In addition, a new analytic solution has been obtained to the complete Fokker-Planck equation including the effects of convection, energychange, solar rotation, and anisotropic diffusion using a radial diffusion coefficient independent of distance. The predictions of this new solution have been compared with the observed time-profile of 3 1-70 MeV protons, with previous measurements by McCracken et al. (lS) of the anisotropy in the particle flux, and also with OG0-6 observations of the time-evolution of a feature in the proton energy spectrum. These comparisons show that the model is capable of explaining both the rise and decay phases of "classical" solar flare proton events, and allow one to draw definite conclusions concerning the diffusion tensor, the free-escape boundary, the possibility of a near-sun scatter-free region, and the nature of the energy-change effect. 4 II. A. INSTRUMENT General Description Experiment F-20 aboard NASA's OG0-6 spacecraft is a solar and galactic cosmic ray experiment consisting of 3 separate charged-particle telescopes which share a common electronics package. designed and constructed at Caltech. The device was A complete description of the experiment with particular emphasis on the electronics has been published previously(l 6 ). Although this dissertation is concerned only with the data from the 6E-Range Telescope portion of the F~20 experiment, some description of the other parts of the instrument are included here for completeness. The separate charged particle measurements made by the three telescopes are described in Table II-1. By combining measurements of energy loss rate, total energy, range, and velocity, the instrument can separate charges up to Z = 8 and can make accurate measurements of particle incident kinetic energy in the following ranges: electrons: 200 keV to -100 MeV protons and alphas: -1 MeV/nucleon - 1 GeV/nucleon lithium-oxygen nuclei: 350 MeV/nucleon - 1 GeV/nucleon. The incident energy upper limit can be extended by using the geomagnetic field as a particle spectrometer. Since the experiment uses only 72 bits out of the total 1152 bit OG0-6 main commutator data sequence, cosmic ray data from only one of the three telescopes can be accumulated during each sequence. The elec- tronics package includes a logic and priority subsystem that determines Table II-1 OG0-6 Experiment F-20 Particle Telescopes Telescope Physical Characteristics Type of Measurement Ari 2 (cm sr) Electron Sensitivity Threshold Sensitivity to - ---· Nuclei Charge z l Energy* (MeV /nucleon) 7 solid state detectors 5 absorbers active collimation with anti-coincidence cup triple dE/dx, total E, range 0.2 - 1.6 <;200 keV z = 1,2 1 - -300 6E - Cerenkov 2 solid state detectors ql;Jartz window PM Tube Cerenkov detector active collimation with anti-coincidence cup double dE/ dx, velocity - 2.8 ?:3 MeV z = 1- 8 350 - 1000 Flare 2 solid state detectors 2 absorbers passive collimation with copper shield double dE/dx, total E, range - 0 . 02 ?:l MeV z = 1,2 18 - 500 6E-Range v *These are the energy intervals in which a measurement of the differential flux can be made. \.Jl 6 the manner in which telescopes share the analog-processing and telemetry systems. A set of 7 separate ground-based commands can be given to alter this logic and priority structure. These commands allow the experimenter to disable individual detectors in the event of failure, or if necessary, to completely shut down any of the three telescopes. In addition to the individual telescope event data, the telemetry sequence includes samples of 20 different single detector and coincidence cotmting rates, and information on ground command status and telescope temperatures. B. The 6E-Range Telescope Since this dissertation involves only low energy proton data from the ~-Range Telescope, particular emphasis will be placed on this part of the experiment. 1. Physical Description A scale cross-section drawing of the telescope is shown in Figure II-1. The device consists of a stack of 7 totally-depleted silicon solid-state detectors and 5 absorbers, with active collimation provided by a cylindrical plastic scintillator cup viewed by a photomultiplier tube. The entrance aperture is covered by a sheet of 3/4 mil aluminized mylar to provide a light shield for the solid state detector stack. An exit aperture at the bottom of the anti-coincidence scintillator cup permits the measurement of the penetrating particle flux. A list of the relevant stack dimensions and characteristics is provided in Table II-2. Since the investigation discussed here is 7 \ / \ \. \ 31° ~/ \. / \ \ I / Afl (CM 2·SR) Dl-2 1.13 02-3 1.62 D2-4 1.38 02-5 0.82 D2-6 0.48 D2-7 0.26 / I \ \ / \ \ \/ / / I / \ I \ / \ I \ DI D2 D3 D4 A2 A3 RANGE (G·CM-2) A2 AL 0.21 A3 W 2.9 A4 W 27.6 A5 W 31.0 A6 W 38.7 D5 A5 D6 A6 D7 PM RCA 4439 0 2 3 4 5 cm Figure II-1: Scale cross-section drawing of the ~E-Range Telescope. The Afl. values have -6% uncertainty due to the uncertainty in the size of the detector sensitive areas. The absorber thickness values are accurate to ±3%. The values listed here were reported in a previous Ph.D. thesis (17). Table II-2 The ~E-Range Telescope Stack* Sensitive Area cm2 (±6%) Discriminator Threshold MeV (±.005 MeV) Noise MeV (±.002MeV) Proton Incident Energy Threshold MeV (±3%) 2.3 -- -- -- -- Silicon 22.1 2.01 . 398 .038 1.17 D2 Silicon 233 3.80 .147 .020 3.3 A2 Aluminum 206 -- -- -- -- D3 Silicon 227 4.08 .153 .021 17.9 A3 Tungsten 2940 -- -- -- -- DA Silicon 227 3.87 .149 .020 45.2 A4 Tungsten 27570 -- -- -- -- DS Silicon 236 4.01 .142 .022 AS Tungsten 30980 -- -- -- D6 Silicon 227 4.08 .141 .023 A6 Tungsten 38730 -- -- -- D7 Silicon 236 4.08 .148 .022 Detector/ Absorber Material Window Mylar Dl Thickness mg/cm2 (±3%) *The values listed here were reported in a previous Ph.D. thesis (17) • 152 I I I II I -230 -309 I 00 9 concerned with protons below 150 MeV indicent energy, only detectors Dl through D4 will be discussed i n detail. Particle energy-loss can be measured in detectors Dl, D2, and D3 using t h ree separate pulseheight analyzers, while triggers in detectors D4 through D7 are used to indicate particle range. When the experiment is in the normal .--* operating command mode, either a DlD8 initiate pulse-height analysis. or a D2D3D8 trigger will The D8 anti-coincidence shield not only provides collimation for the telescope, but also rejects undesirable the interaction and shower-type events which scatter particles into ~intillator. For protons below 45 MeV incident energy, a triple energy-loss measurement is recorded in Dl, D2, and D3. When any of the "range detectors" D4 through D7 are triggered, this range information replaces the Dl pulse heigh t in the readout sequence. Because of its high -400 keV discriminator threshold and small depletion depth, detector Dl has less than 1% electron detection efficiency for any incident electron energy (18,19) • The problem of separating low energy electrons from nuclei is thus easily solved even for particles which stop in Dl. 2. Detectors The seven solid-state detectors used in the range stack are all totally-depleted silicon surface-barrier type devices manufactured especially for Caltech by ORTEC (Oak Ridge Technical Enterprises Corp.). *The notation DlD8 is used to indicate a Dl trigger in the absence of a D8 trigger. D8 is thus in "anti-coincidence.'-' In the same way, D2D3D8 means a D2-D3 coincidence in combination with a D8 anti-coincidence. 10 With the exception of Dl, they have a nominal lOOOµn depletion depth 2 and 4.0 cm sensitive area. Surface-barrier detectors were used because they have low noise and high reliability in a variety of environments, and because they have high resistance to radiation damage from energetic particles. In the case of Dl and D2, surface barriers were particularly desirable because they can be manufactured with very thin dead regions which allows an accurate measurement of particle total energy. The detectors used in the experiment flight unit were carefully selected on the basis of thickness, sensitive area, bias voltage needed for total depletion, noise at full bias, and performance in a thermalvacuum environment. Since such devices cannot withstand any physical contact from micrometers, fingers, etc., all physical measurements were made in a "remote" fashion using energetic particles. The sensitive area and total thickness were determined by irradiating each detector with a well-collimated monoenergetic electron beam from a magnetic S-ray spectrometer. More exact thickness measurements were made for Dl, D2 and D3 when the completed flight unit was exposed to 1 - 23 MeV protons from Caltech's Tandem Van de Graaff accelerator. Determinations of dead layer thickness and proper operating bias voltage were made by exposing each detector surface to ThB(Pb 212 ) alpha particles. The most important test consisted of a two-week thermal-vacuum exposure for each detector at full bias voltage. During the two weeks the detector noise and leakage current were recorded frequently; an environment of (\ torr and +40 C was maintained. <10- 6 11 Each detector in the operating experiment has a pulse-height discriminator threshold associated with it that has been carefully adjusted to reject detector noise but include all appropriate particle pulses (see Table II-2). This threshold, which is clearly a function of both detector thickness and noise level, was set (for all detectors except Dl) so that 99% of all minimum-ionizing particles cause a trigger. 3. Anti-coincidence Shield The anti-coincidence cup consists of a cylinder of NE 102 plastic scintillator material viewed by a RCA 4439 photomultiplier The PM tube was tested using a Cs tube. 137 source in combination with a N:tI crystal to determine the optimum operating bias voltage. The discriminator was set using ground level muons incident on the assembled D8 scintillator so that at least 99% of these minimum ionizing particles cause a DB trigger. Although this D8 threshold corresponds to only - 400 keV energy loss in the scintillator, the presence of an aluminum housing which surrounds the scintillator raises the D8 incident energy threshold to -9 MeV for protons and 0.6 MeV for electrons. Thus the anti-coincidence cup acts as a mechanical collimator at low energies. 4. Electronics The electronics package, which has been described in detail elsewhere(l 6 ), consists of the following separate subsystems: 1) 6E-Range telescope electronics 2) Cerenkov telescope electronics 3) Flare telescope electronics v 12 4) Analog signal processor 5) Coincidence and priority logic 6) Rate accumulators 7) Data storage, formatting, readout, and spacecraft interface 8) Power supply A block diagram of the electronics relevant to the ~E-Range Telescope is shown in Figure II-2. Pulses produced in each detector are passed through a charge-sensitive preamplifier, a shaping amplifier, and finally to a pulse-height discriminator. If the pulse is above the discriminator threshold, a logic pulse is generated which is passed to the coincidence and priority logic subsystem, which in turn decides whether the event should be blessed with analysis by the analog processor. If the decision is yes, then the logic subsystem opens the linear gates and the analog signals are digitized by the three 256channel pulse height analyzers. The logic also passes the appropriate single detector and coincidence rates to the rate accumulator subsystem. The pulse-height data, range detector information, and rates are all read out once during each 143 msec interval during normal spacecraft operation. Fourteen single detector rates and six coincidence rates are accumulated. Since only two rates are sampled during a given readout interval, a commutation sequence is employed to determine how the available telemetry is shared by the rate scalers. to the range telescope are listed in Table II-3. The rates pertinent DI CHARGE~SENSITIVE AMP. H H H DELAY LINE HEIGHTTIME CONVERTER LINEAR GATE I 02 I 03 I ~ CSA ~ csA I ( • .. , DL I 1 cc ... DL I 1 ~I •I HTC I I LG ( C-1 ~HTC LG I I I I C:•I PHA I r (•I •I PHA ~ COINCIDENCE a PRIORITY LOGIC RATE ACCUMULATORS J I DISCR 04 I 05 I .......i CSA I ~CSA I •I DISCR ~1 ~SCR; I 06 CSA 07 CSA 08 I CSA I I INPUTS FROM OTHER TELESCOPES I I I CONTROL TO OTHER TELESCOPES DISCR I .. r;;;;CR Figure II-2: ~E-Range .....I ~ a:: a:: w 0 I- I- DISCR~ DISCR ~ w d5u Telescope Electronics Block Diagram zz 0- u II- LL :::> <( 0 a: oU <( w WU a:: ~ (/) I ....... w Table II-3 ~E-Range Rate Telescope Detector Rates Elapsed time between * readouts Accumulation Period* Contributing Particles 2 An(cm sr) (:~6%) --- ~·- · DlD8 430.9 msec 432 msec protons -1 to -20 MeV 1.1 - - protons ~ 3 MeV D2D8 430.9 msec 432 msec I--' 1.8 electrons ~ 0.2 MeV DlD2D8 430.9 msec 6912 msec D2D3D8 430.9 msec 432 msec protons - 3 to ~ 20 MeV 1.1 protons ~ 18 MeV 1.6 electrons ~ 1.1 MeV *assuming normal spacecraft readout rate of 8000 bits/sec ~ 15 As mentioned previously, a l l three telescopes compete for the use of the analog-processor and telemetry. During normal command-mode operation, the Flare telescope h as highest priority, while the Range v and Cerenkov telescopes compete on an equal basis at a lower priority level. Thus the Flare telescope, which is a miniature version of the Range telescope with passive instead of active collimation, begins to dominate the analysis as the particle flux reaches the saturation v levels for the Range and Cerenkov telescopes. This priority system can be altered easily by means of the seven separate ground commands available. Table II-4 lists the ground based command combinations which are pertinent to the operation of the Range telescope, and shows how these commands affect the logic and priority structure. 5. Electronics Calibration The basic principle behind the use of solid-state detectors is that the charge pulse produced at the detector terminals is proportional to the energy deposited in the active detector volume by the charged particle. In order to convert digital pulse height data into particle energy loss information, one must know the values of the thresholds of all the PHA channels in units of MeV of particle energy loss. The calibration of the pulse-height analyzers was carried out as a two-step process. First the voltage pulses from a Berkeley Tail Pulse Generator were applied across a separate test capacitor at the input of each charge-sensitive preamplifier. By varying the height of this voltage test pulse and observing the pulse height analyzer output, Table II-4 Ground-based Commands for the ~E-Range Telescope ~- ~ ~----·~·..---. - --~.--~-- ----- ·- -- Command Number Command Name Triggering Logic for Analog Processor C7 System Reset DlD8 or D2D3D8 Normal mode C4 Dl dis able D2D3D8 Given if Dl fails cs - - Comment I-' O'\ D3 enable DlD8 or D3D8 Given if D2 fails C6 D2 enable - · DlD8 or D2D8 Given if D3 fails or for electron data accumulation C4C5 -- D3D8 Only D3D8 triggers analysis CSC6 or C4CSC6 -- -- Range telescope shut-down 17 the threshold value of each analyzer channel was determined in units of pulser mV. In a similar way the discriminator threshold in pulser mV was determined for each detector. The second step involved the conversion of pulser mV to energy loss in keV, which is equivalent to determining the value of the individual test capacitors. This was achieved by irradiating each detector with ThB alpha particles (with energies of 6.045, 6.083, and 8. 776 MeV), and comparing these particle-produced pulses with those of the test pulser. The PHA and discriminator thresholds were thus determined 0 to an accuracy sl% for temperatures between -s c and +40°c. Typical analyzer channel widths are -50 keV, yielding a saturation value of -13 MeV for each of the 256-channel analyzers. Tests of the logic and priority structure, command modes, and rate scalers were also made using the ground support equipment. A more detailed description of all the electronics test procedures has been given in a previous Ph.D. thesis(l 7 )_ C. Spacecraft 1. The Satellite Orbit The OG0-6 spacecraft is the last in a series of Orbiting Geophysical Observatories flown by NASA. It was launched on June 5, 1969 into a polar orbit described by the following parameters: perigee 397 km apogee 1098 km inclination 82° period 99.8 minutes 18 Caltech's experiment F-20, which is one of 26 independent experiments aboard the satellite, is mounted on the -Z door so that the telescope entrance apertures always face away from the earth. The satellite orbit can be pictured as nearly fixed in space with the earth rotating beneath. The earth's rotational axis is tilted 8° out of the plane of the spacecraft orbit, so that the satellite never reaches a geog.tta.phi.c latitude greater than 82° N or s. When the satellite crosses the geographic equator from south to north, this is taken conventionally as the beginning of a new revolution. Each of the -14 revolutions per day are numbered consecutively throughout the life of the satellite. Since low energy cosmic ray particles have access to the earth's magnetosphere only in the vicinity of the north and south magnetic poles( 20), the location of the satellite orbit in the polar regions is of particular interest. Figure II-3 shows a series of orbits over the south geographic pole spanning a full day in universal time. 2. Invariant Latitude and Magnetic Local Time Since it is the geomagnetic field that defines the cosmic ray access regions, a coordinate system aligned with the magnetic dipole axis of the earth is commonly used to define the spacecraft location. One approach is to merely set up a spherical polar coordinate system centered on the dipole axis, and to then measure the satellite position in terms of cUpole la.:tl:tude and cUpole long..[tude. Because the earth's field is distorted from a true dipole both by the presence of higher order moments in the source and by the external influence of the solar 19 ±180° SOUTH GEOGRAPHIC -90° + POLE +90° GEOGRAPHIC LONGITUDE Figure II-3: Typical orbital trajectories for OG0-6 across the south pole in geocentric coordinates. The south invariant pole is the point at which A z 900. 20 wind, a non-spherical coordinate system using inva.JUa..n:t fOvtltude_ 11., and magnetic. .tee.a£ time_ MLT, has been found to be more appropriate (21 ' 22) These quantities are defined as follows: MLT = dipole longitude) of spacecraft in ( hours A = dipole latitude adjusted for ) distortions of the geomagnetic ( field from a true dipole dipole longitude ) of earth-sun line + 12 hours ( in .hours The value of the invariant latitude A for a deformed geomagnetic line of force is defined to be the same as the dipole latitude of the equivalent undistorted "dipole" line of force. Note that any distortion of the field lines in the azimuthal direction is neglected by the MLT parameter. When the spacecraft is in the magnetic meridian plane that contains the earth-sun line, it is at MLT = 1200 hours. Figure II-4 shows the orbits of the previous figure plotted in the A - MLT coordinate frame. Since the rotational and magnetic poles 0 differ in latitude by -11.s , the orbit covers most values of A during a 24-hour period. Note that the dUr.e.c.tion. in which the spacecraft crosses the A - MLT plane is roughly constant with time. This cover- age region rotates slowly at the rate of -1.8°/day, and the spacecraft thus covers the entire A - MLT plane once every 100 days. 21 1200 SOUTH INVARIANT 80 1800 0600 POLE < - w -80° 0 ::::::> ...... ~ _J ...... -70° z <[ 0::: g z -60° 0000 2400 MAGNETIC LOCAL TIME :J.igµre -.II-4: Typical orbit.al trajectories for 0~6 - acress the south pole in A-MLT coordinates. The dashed -line indicates where interpolation was necessary to determine the trajectory. 22 III. DATA ANALYSIS This section describes the manner in which the raw data produced by the F-20 Experiment are converted to useful information about the intensity and composit ion of the cosmic ray particles in the vicinity of the OG0-6 spacecraft. A. Proton Response of the ~E-Range Telescope Using the results of the electronics calibration of the analog-processors, one can convert a digital pulse height into a value for the particle energy-loss in the detector depletion region. Given a double or triple energy-loss measurement for a single particle event, the problem remains to determine the particle charge Z and incident kinetic energy E. The following is a discussion of how this problem is solved for Dl through D4 proton events in the Range Telescope. 1. Accelerator Calibration A straightforward way to determine the response of a cosmic ray telescope to low energy protons is to simply expose the device to a monoenergetic proton beam and observe the response directly. Such an experiment has in fact been performed on the assembled F-~O flight unit using Caltech's Tandem Van de Graaff accelerator. Although the primary proton beam of the Caltech accelerator is limited to 12 MeV 10 3 12 energy, 23 MeV protons can be produced by means of the B (He ,p) C reaction. A magnetic spectrometer was used to select the desired beam energy and also to limit the energy spread to ~E/E : 1%. The Ground 23 Support Equipment was used to simulate the experiment-spacecraft interface and to write the digital data on magnetic tape. After some analysis, the incident proton energies needed to penetrate to the top of Dl, D2, A2, D3, and A3 (see Figure II-1) were determined precisely. This information, in combination with the range- energy tables for protons in mylar, silicon, aluminum, and tungsten(2 3 ), yielded an accurate thickness measurement for the Mylar window and for detectors Dl, D2, and D3. Analysis of the pulse-height data from the accelerator runs also determined the most-probable pulse height combinations in Dl, D2, and D3 for various incident proton energies. A detailed description of the Tandem Van de Graaff calibration has been given by S. Murray(l 7). 2. Protort Energy-Loss Calculation A calculation of the instrument response to protons can be made independent of an ~ctual accelerator calibration by using the thickness values for the various layers in the telescope and the rangeenergy tables( 23 ). In particular, the average energy loss (~E) in a given detector can be calculated as a function of incident energy E. These calculated proton and alpha particle energy loss curves for Dl, D2, and D3 are shown in Figure III-1. Note that the Dl - D2 energy- loss combination uniquely determines the particle species and energy from 3 to -40 MeV/nucleon, while the D2 - D3 combination covers the region above -18 MeV/nucleon incident energy. If these calculated curves for the most probable proton energyloss are combined with the electronics calibrations of the pulse-height 24 protons alphas -> 10 CV ~ /\ w <] v '' '' 10-I ..........................~_,__.__,_..........................._~----------..........~--------.........._ 103 10 10 2 INCIDENT ENERGY (MeV/nucleon) Figure III-1: Calculated average energy loss <~E> in various Range Telescope .. 4etectors as a function of incident kinetic energy. The calculation is based on the Janni range-energy .tables. 25 analyzers, then estimates can be made of the expected pulse height channel combinations in Dl, D2, and D3 for a given proton incident energy. These calculated digital pulse height values for Dl vs. D2 are shown in Figure III-2. The locations of the most-probable pulse heights from the accelerator calibration runs are plotted for comparison. The agreement is very good, indicating that the energy-loss tables can be used to extrapolate beyond the 23 MeV limit of the Tandem Van de Graaff runs. in Figure III-3. A similar plot for D2 vs. D3 pulse height is shown The agreement between the energy-loss calculations and the accelerator runs is again good, and the calculated values are shown extended to 150 MeV incident proton energy. 3. Pulse Height Data Reduction The comparison between calculation and experiment just described represents a consistency check between the values for the detector and absorber thicknesses, the energy-loss thresholds of the pulse-height channels, the range-energy tables, and the actual instrument response to protons. This indicates that the telescope proton response is well-understood, and that a given Dl-D2-D3 event can now be easily associated with a specific particle species and incident energy. The dashed linesin Figures III-2 and III-3 are proton regions or bands which have been determined empirically to include essentially all of the proton pulse-height events. Figures III-4 and III-5 show 2-dimensional printouts of actual satellite pulse-height data accumulated during a 10-rninute interval on 3 November 1969. The same proton bands have been superimposed, and they do indeed surround the events 26 Figure III-2 Proton response plotted on the 2-dimensional Dl vs. D2 pulseheight plane. in MeV. The numbers denote incident proton kinetic energy The crosses are the most-probable Dl-D2 pulse-height pairs derived from the results of the Tandem Van de Graaff accelerator calibration. The dots are the results of the average energy-loss calculation based on the Janni range-energy tables and on the best estimates for the Range Telescope detector and absorber thicknesses. The dashed line defines a proton "band" which includes essentially all valid proton events. x Tandem Van de Graaff accelerator calibration 70 • Average energy-loss calculation _J w zz 60 <t I u 50 II <!> w 40 I ~ w 30 (/) _J :::) a_ '.2, 20 ~ ~ 3.6 " ' ~ 4.0 "' " -- --- - - --- --- ......................... 5.0 '-......_ ............. 0 10 6 ·0 30 r-.> -..J -- - 7.0 19.0 -----1 ..::::::-.. --- 00 40 80 120 13.0 15.0 - - - - - - - x_J 200 240 160 02 PULSE HEIGHT CHANNEL 28 Figure III-3 Proton response plotted on the 2-dimensional D2 vs. D3 pulseheight plane. The numbers denote incident proton kinetic energy in MeV. The crosses are the most-probable D2-D3 pulse height pairs derived from the results of the Tandem Van de Graaff accelerator calibration. The dots are the results of the average energy-loss calculation based on the Janni rangeenergy tables and on the best estimates for the Range Telescope detector and absorber thicknesses. The dashed line defines a proton "band" which includes essentially all valid proton events. x Tandem Van de Graeff accelerator calibration 140 • Average energy-loss calculation _J w z z <{ :r: u ------------ -------------==--- -- ---- 120 17.8 18.0 t- :r: 100 (!:) 80 w :r: w 60 Cf) 20.0 20.5 --- / / 40 / 0 ( 20 0 / / 70 100 I~~ - 30,,,,~ / _J - .,,.,.,.., ,,,,,.... / :::> Cl.. C\J 19.0 35 ~5// / / y 40 80 120 160 03 PULSE HEIGHT CHANNEL 200 240 30 Figure III-4 The Dl vs. D2 pulse height array observed during a polar pass on 3 November 1969. The numbers indicate the total number of events which occurred with a particular Dl-D2 pulse height combination. The PHA channel numbers are pseudo-logarithmic (256 actual channels have been compressed into 60). The boundary of the proton band and the calculated locus for a-particles have been superimposed. The exact nature of the pulse-height scales and event number code is explained in reference 17. TW0-0 PLOT FOR: CATA INTERVAL FRCM 59 39900 SEC OF DAY 405CO 307 TO SEC OF DAY 307 VERT. AXIS•Dl HORe AXIS•DZ 2 58 57 ~ 2 55 2 - ·1:--,-- ;- - 2 53 3 522 ~ 51 50 ~9 i- ---,~_ • l • • • • ~---2 1 1 3 1 • • • • ............... _ • 3 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 'l""~······ 3 • • • • l 2 • .......... l • • • • i-----r -- i-- - ~ 47 3 • • • • l 1 46 • • • • • • 4S • • a· • • • • • a ~ 1 _2__________ • 43 2 • • 1 • • • : • : . • : • • • • - .. _ _ _ _ • " ~l 40 7 6 • 1 4 33 l ......... l~l--;-- l "-... __ 1 l • • -----------;-------.• 2 1 1 • • l • 2 • 3 1 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • ••••••••••••••••••••• 2 39 37 35 34 • ~ l r~ • 1 • ~ - 32 31 2 1 • • ---z--1----.------- . . : - - -- - - - · ----·- ··· - --- . • • 1 1 . Ir • • .... . .. . .. .. . . )0 29 ~ 2 8 27 26 25 22 21 20 • I • ..--- • I• 19 18 17 6 15 14 13 E E E 12 11 10 F 9 9 8 7 6 E E F E 5 A • l (j 4 3 0 5 10 ..... .... 15 20 25 35 ~o ~5 50 . 60 w I-" 32 Figure III-5 The D2 vs. D3 pulse height array observed during a polar pass on 3 November 1969. The numbers indicate the total number of events which occurred with a particular D2-D3 pulse height combination. The PHA channel numbers are pseudo-logarithmic (256 actual channels have been compressed into 60). The boundary of the proton band has been superimposed for comparison. The exact nature of the pulse-height scales and event number code is explained in reference 17. 3990D TWO-D PLOT FOR: CATA INTERVAL FROM SEC OF DAY 307 TO 40500 SEC OF DAY 307 VERT. AXIS•02 HORe AlllS•Ol 59 c 511--4 57 E 56-D SS E 5" 0 53 G .......... . . . . . . • ... . . . . . .. . . . . • • • • • • • • • I . . . . . . . . . 52--G 51 D 50 F 49 F 48 F 47 F 4!1 E . • ~-F " 1 43 . . . . . . . . . . . . . . .. . . . . . .. . . . . 1 l• --,, 33 - l • w • l E w . . . . . . . . ... • 1· 8 " 24 IJ 23 B 22 -1 21 9 20 19 111 .. .. .. . . . . 9 c I • • -y 11 u.--11" 15 9 14----i4 n 1 11 10 • • 7 • • 3 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 9 8 .. . . . . . . . . ..... ·~ 1 4 Ci-----s 8 4 ) 8 1 2 9 -1 1 A 1 • • 1 O . u-;-• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • I • • 0 5 10 15 20 25 30 35 40 .. 45 . 50 .. SS . 60 34 of interest. The events which are not within the boundaries of these bands are either electrons, alphas, or in a few cases, protons which have deposited anomalously small amounts of energy in at least one of the detectors. Using the relation between pulse-height response and incident energy displayed in Figures III-2 and III-3, these bands were divided into bins which correspond to different proton incident energy intervals. Specifically, the Dl-D2-D3 pulse height "space" was divided into 37 proton incident energy bins by slicing the proton bands along calculated lines of constant incident energy (see Table III-1). incident energies below ~3 At MeV, protons stop in Dl and no 2-dimensional pulse-height information is available to cleanly separate protons from other charged particles. Since Dl is insensitive to electrons, the only threatening contaminants to the Dl proton analysis are low energy alpha particles. This alpha contamination can easily be estimated and can usually be neglected. Thus, below 3-MeV, the proton analysis consists of a one-to-one map between groups of Dl pulse-height channels and proton incident energy intervals. The physics of the data reduction is contained in the proton pulseheight bins and the incident energy intervals assigned to them. Given such a set of bins, the rest of the analysis consists of deciding whether each pulse-height event lies in the proton band, and if so, in which bin. as follows: The proton differential flux dJ/dE can then be determined 35 TABLE III-1 Proton Pulse-Height Analysis Bins for the ~E-Range Bin No. Telescope Incident Energy Type of Analysis E . nun E max 1 1.17 1.27 2 1.27 1. 37 3 1. 37 1.49 4 1. 49 1.61 5 1.61 1. 73 6 1. 73 1. 87 7 1. 87 2.00 Dl Singles 8 2.00 2.14 Ml 9 2.14 2.28 10 2.28 2.42 11 2.42 2.56 12 2.56 2.70 13 2.70 2.85 14 2. 85 2.99 15 2.99 3.14 16 3.14 3.30 17 3.30 3.60 2 1.13 cm sr + 6% = Dl - D2 An 18 3.60 4.00 = 1.13 cm 2sr + 6% 36 TABLE III-1 (Cont.) Bin No. Incident Energy Type of Analysis E min E 19 4.00 4.50 20 4.50 5.00 21 5.00 6.00 22 6.00 7.00 Dl - D2 cont. 23 7.00 9.00 AQ 24 9.00 15.00 25 15.00 *0.00 26 20.00 30.00 27 30.00 45.00 28 20.00 25.00 29 17.90 20.00 30 20.00 25.00 31 25.00 35.00 max = D2 - D3 An = 1.13 cm2sr + 6% -- no D4 2 1.59 cm sr + 6% 32 35.00 45.00 37 20.00 45.00 33 45.00 70.00 34 70.00 100.00 D2D3 with D4 35 100.00 150.00 AQ 36 45.00 150.00 = 1.2 cm 2sr + 6% 37 = incident energy limits to pulse-height bin i = number of pulse-height events in bin i in given time interval N normalizer between pulse-height data and actual particle rates (DlD8 counting rate in counts/sec) (total no. of DlD8 pulse-height events) A Q. l. geometrical factor in cm sr assigned to bin i = proton differential flux at bin i in (cm2sec sr MeV)-l during time interval dJ. l. dE 2 = = The normalizer N must be used instead of the actual accumulation time because there are frequently many more valid events than can be analyzed and readout. B. Bulk Data Processing The OG0-6 spacecraft has orbited the earth in a fully-opera- tional state for 15 months (from 5 June 1969 to 29 August 1970). During this period, the satellite completed 6500 orbits, and Caltech's experiment F-20 accumulated approximately 2 x 10 data. 10 bits of digital cosmic ray With such a huge data set, it is virtually impossible to completely process all of the data in a reasonable amount of time. Consequently, a quick-processing scheme has been devised which allows 38 one to scan the data and select time periods of particular interest for more detailed analysis. The overall data processing plan for the Range Telescope is described in Figure III-6. 1. Tape Merging The raw spacecraft data are received from Goddard Space Flight Center in the form of Attitude-Orbit Tapes, which describe the satellite location and orientation at 1-minute intervals, and Experimenter Tapes, which contain the decommutated data from Caltech's Experiment F-20. The first step in the data reduction process is to combine the cosmic ray data with the relevant attitude-orbit information and write the result on a single "merged" tape via a program named MERGE (see Figure III-6). These merged tapes are the basic input for all subse- quent steps in the data analysis. 2. Polar Rate Averages Since the work discussed in this thesis is concerned with the intensity of ~n.:teJt..pla.n.eta.Jr..y cosmic rays, the data of interest can be accumulated only in the polar regions (see Section II-C). Although the question of particle access to the magnetosphere is a very complicated and still unsolved problem( 24 ), there does exist a given region in A-MLT space at each pole where the flux is independent of spatial position and identical to the interplanetary flux in the vicinity of the earth( 20). This region is roughly defined by IA! ~ 72° , even though the exact boundary location depends on MLT, particle rigidity, and on the configuration of the geomagnetic and interplanetary fields. P0LRAT Program (Averages polar rates automatically Rate Cards (rotes averaged 1 over polar passes) •I r FLXPLT Program Rate Plot (time-profile plot of individual polar rote overages) I Rate Cards MERGE Program RATES Program _J Orbit Plots (rates plotted continuosly vs. time) w \0 Hand pick polar pass sampling times MTQJTALE and MTW00 Programs (analyze proton PHA data) , I I I Figure 111-6: FLXPLT Program Flux Plot (time-profile of particle flux bins for individual polar posses) ELPL0T0 Program Spectrum Plots (plot of flux vs . incident energy for each polar poss) Flux Cards (proton flux in various energy bins for each polar pass Printed output of 2-dim . PHA arrays Flow chart of the Range Telescope data processing. 40 The program P0LRAT is used to process the merged tapes and sample the cosmic ray data automatically whenever IAJ > 72°. The appropriate range telescope coincidence and single detector counting rates are averaged over these polar "cuts" and the results are punched on cards. A routine called FLXPLT is then used to plot these individual polar rate averages vs. time. Figure III-7 is an example of a typical com- puter-generated "rate plot" which enables one to decide whether the data should be subjected to more intensive (and more expensive) analysis. 3. Orbit Plots In order to analyze the data in a given time period in more detail, the proton polar access regions must be defined more precisely. A program named RATES is used to generate continuous plots of the key detector counting rates as a function of time for each satellite orbit. Such an "orbit plot" (see Figure III-8) makes the proton polar access regions obvious and allows one to easily hand-pick time "cuts" for further analysis. The dashed lines on Figure III-8 indicate the proton polar cuts chosen for this particular orbit. The orbit plot shown in Figure III-8 is in fact a streamlined version which includes only the most important Range Telescope coincidence rates. The write-up for the RATES program(l 7 ) describes how more complete orbit plots including some of the pulse-height data and all of the 20 experiment counting rates can be generated. 41 Figure III- 7 A computer-generated "rate plot" produced by program FLXPLT . The DlD8 and D2D3D8 rates (labeled DlND8 and D2D3ND8), averaged over indi vidual polar cuts, are plotted vs. time over an 11-day interval. 2100'-I - 2317N 104 µIllliillllllllllllllllfllillliliillilliillilliJllillllllllllllilllililllliillllillililililllil\llililliliilllllillililliiililllililllliillilliillllllllillllllilillilillilllllllillllllililiilliillillllllllllllllllliililliilllllllllllllllllllllllllllllliITTTillD _4 4 NCJV 1969 24 --24 = 24 6 NOV 1969 7 f'lJV 1969 8 NCJV 1969 TJtt: Ct-FS. U. T.l 43 Figure III-8 A computer-generated "orbit plot" produced by the program RATES. The DlD8, D2D8, and D2D3D8 coincidence rates (labeled Dl8, D28, and D238) are plotted continuously vs. time for one full spacecraft orbit. Magnetic Local Time and Invariant Latitude are also included. The proton access regions are clearly evident, and the hand-picked cuts have been marked with dashed lines. Each tickmark along the Y-axis represents one decade in counting rate. Changes in the D2D8 counting rate during the proton polar cuts are due to electrons. ()GO-VI PROCESSED 08/03/71 CIT I I I I I I I I I I I I I I I I I I I I I I I I I I l _j_j__LJ I ~ 018 I 4..J proton polo I cut I . LUl ~----~----~~---+~~~-'----~-t~~Iflp-?---~----~~-: !_--~-----~~~~-r~---~~~---== 028 I~-"""'- . I 11 ' 0238 I \', ~·I.._,. -- I \ I I --- -- -- ....- I I / I I ,c"">t 11 -v- 018 -j 028 11 I ""38 I IU I I ILAT I I l ~EC I .!. I 1 '!!~111!r1,!!~!!!lp~r~111!q!,!~11r!11·1•1~wlf1!,!~11111 j 1 1 _LL 1 1 1 1 I I I 111I111!oL't1 1J 111.!.L• i-' I I I I I I I I l-' I I I I I I I I L' I I I ! I I I I l-' I I I I I I I I I I 'l"'!'!!""' I 11 111 !1111duo!11.','~"'f"!!~m!11,!!~111h'!!~m!t1~~11t I I I I I I I I I I 'I''""'!!#"' I I I '.!. i'' if 1 1 1 11 I 11 '.!.L' 11 / ~ 'I 1 1 '!~' 1_1 I I I I J I I ! ~ I I I I I I I I I !.._' I ! I I I I _1 I I I I I _,.._ __ ____, Hl..T ~J.LJ4 _ _ __JJLAT I fftC ·-' I I ~ ~ 45 4. Calculation of Proton Flux The time cuts chosen from the orbit plots are used as inputs to the programs MT¢TALE and/or MTW¢D which process the pulse-height data on the merged tapes during the time intervals specified. III-6.) (See Figure Printed output of the 2-dimensional pulse-height analyzer arrays can be produced (Figures III-4 and III-5 are examples). The proton differential fluxes in each of the 37 bins are calculated and punched on cards. These cards are in turn used as an input to FLXPLT, which produces plots of the flux time-profiles analogous to the rate profiles of Figure III-7. Alternatively, the flux bins can be plotted together for each polar cut to form an instantaneous proton differential energy spectrum as in Figure III-9. This completes the description of how the instrument and spacecraft function, and in particular how the Range Telescope responds to the low energy cosmic ray protons which are the topic of this dissertation. A brief account has been given of the manner in which the tremendous bulk of satellite data is scanned, sampled, and converted to useful information. The information desired is of course the interplanetary proton flux as a function of energy and time. 46 TO 11:17:30 DAY 307 til -,.... 1-!-i I ~ > w L cr: Cf) + u w 1 Cf) "'L t u .....,, x ::::) 1tt+ _J 10 lJ_ I ..................................~~"'---'--'--'-.L....l....l.Ll....l-'-~-'---1.--1.---'-..J....J....J...J...J lo-·...._~__,~_,__._ 10 ° 10 I 10 2 10 3 ENERGY CMEV ) Figure 111-9: A computer-generated plot of the flux calculated for each 'of the 37 pulse-height bins vs. incident energy for a single polar pass. The plot thus represents an instantaneous sample of the proton dif f erential energy spectrum. 47 IV. OBSERVATIONS Solar cosmic ray phenomena as observed near the earth can be 2 divided into the following four distinct types( S): a) Flare-associated events ~ associated with optical flares, x-ray and microwave emission. Electrons and nucleons observed at the earth generally display a rapid (<2 day) intensity increase followed by a slower decay phase. b) Recurrent events -- particle increases which sometimes appear on the next rotation* after a flare-associated event. These events dis- play roughly the same time history as the original flare and apparently originate from the same active region. c) Energetic particles associated with an active center -- increases which display no velocity dispersion and are not associated with solar flare activity. They are thought to be particles emitted continuously from an active region but confined to a given region of interplanetary space. The time-profile of the event, which persists from 3 - 14 days, is thus produced by the "co-rotation" of this region past the earth. These events display steep proton energy spectra and are often anti-correlated with the MeV electron intensity. d) Energetic storm particle events -- large intensity increases of low energy protons and electrons associated with (and believed to be accelerated by) strong interplanetary shock waves. *The surface of the sun revolves at a rate which depends on solar latitude (the period is -27 days at the equator). 48 This dissertation is concerned only with flare-associated events, and specifically with the means by which these flare particles, which are assumed to be injected impulsively on the sun at the location of the optical flare, propagate through interplanetary space to the earth. The details of the actual flare event on the solar surface are of interest here only when they affect the subsequent particle propagation and the intensity observations at the earth. This section discusses the method by which flare-associated events are identified, and summarizes the OG0-6 proton intensity observations of four such events which are discussed in Section V. A. Event Identification The typical solar flare proton behavior in the vicinity of the earth can be described as a. rapid rise in intensity, followed by a smooth turnover and an approximately exponential decay phase with Tdec "' 5 - 30 h ours (25,26) • However, such "typical" flare time-profiles are rarely observed, and in many cases it is difficult to separate flare events from the other three types of solar phenomena. In order to yield useful clues about the nature of particle propagation in interplanetary space, the following information must be determined about an individual flare event: a) The event identification must be unambiguous -- that is, it must be possible to associate the intensity observations at earth on a one-to-one basis with a -Oingle optical flare event on the sun. particle injection time The (±. a few hours) and the point of injection on the sun (in solar latitude and longitude) can easily be determined once 49 the optical counterpart of the cosmic ray event has been identified. Multiple flare events, which occur frequently, are deemed unsuitable for more detailed analysis. b) It must be determined that no co-rotating features (i.e., recurrent events or active center events) or interplanetary magnetic field boundaries have rotated by the earth during the flare observation period. Such occurrences complicate the situation immensely and make detailed analysis very difficult. c) The average value of the solar wind velocity in the vicinity of the earth must be determined. This is a parameter which affects the particle propagation and thus is needed for complete analysis of the event. The method by which particle events can be accurately identified is provided by the solar geophysical data published by ESSA (now NOAA)(2 7 , 28 ). The emission of Type IV radio bursts and hard x-rays are generally accepted as indicators of solar particle acceleration and injection( 26' 29). Although an intense optical flare is not necessarily a good candidate for a particle event, Type IV radio and/or x-ray emission simultaneous with an optical flare of reasonable size (importance <: 1) preceding the particle increase by only a few hours is taken to be a good event identification. Co-rotating features, magnetic field boundary crossings, and shock wave events can usually be identified and eliminated because they are often associated with disturbances of the geomagnetic field (sudden commencements and sudden impulses) and with sudden large changes in the solar wind plasma velocity. 50 B. OG0-6 Solar Flare Observations For the period from 7 June 1969 to 11 February 1970, the Caltech Solar and Galactic Cosmic Ray Experiment aboard OG0-6 operated normally and recorded 250 days of nearly uninterrupted cosmic ray data. On February 11, 1970, the C4 ground command was given because of a noisy Dl detector( 3o), making subsequent Range Telescope data nearly useless for the purposes of this detailed investigation. These 9 months of data, which were carefully scanned for solar flare events, are summarized in Appendix A. Table IV-1 lists the 12 flare-associated events of significant magnitude that occurred during this period. Of these 12, only 7 ~ould be clearly associated irith an optical flare, and 3 of these were either multiple events or were complicated by magnetic disturbances. Figure IV-1 shows the intensity vs. time profile at several energies for each of the four remaining events. Each intensity point represents an individual polar pass analyzed in the manner described in Section III. The ESSA data for optical flares, x-ray flares, Type-IV radio emission, sudden commencements, sudden impulses, and solar wind velocity have been included. Table IV-2 summarizes the optical identi- fication which has been assumed for each of these events. The 7 June 69 event suffers from some ambiguity as to optical flare identification, although this produces only an uncertainty as to the exact injection time. The precursor to this event, which is quite evident in Figure IV-1-a, may be due to the earlier flare at 0018 0130 UT on the same day. The decay phase of this event has been studied in some detail elsewhere(l 7 ,3l). 51 TABLE IV-1 Flare-Associated Particle Events Observed by OG0-6 Between 7 June 1969 and 11 February 1970 Date Approximate UT of Particle Increase ± 4 hours Optical Flare Identification Comment 7 June 69 1700 yes OK 25 Sept 1100 yes Multiple Flare 27 Sept 2200 yes Magnetic Storm 14 Oct 0000 - 0800 ? Possible Multiple Flare 2 Nov 1000 yes OK - Magnetic Storm beginning on Nov. 7 24 Nov 1200 yes Multiple Flare 18 Dec 1800 no No Identification 19 Dec 2400 no No Identification 30 Dec 2400 yes OK 28 Jan 70 1600 ? 29 Jan 1200 no 31 Jan 2000 yes } Multiple Flares OK 52 Figure IV-1 Time histories of four selected solar flare particle events. The following quantities a r e plotted vs. universal time: Polar averages of the proton differential flux as observed by experiment F-20 aboard OG0-6 for various incident energy bins. Optical Flares of significant magnitude -- the importance is included. X-ray Flares -- observations from Explorers 33 and 35 at 2 - 12X. Type-IV Radio Emission reported as "Solar Radio Spectral Observations" by ESSA. Sudden Commencements and Sudden Impulses -- disturbances of the geomagnetic field often related to fluctuating conditions in interplanetary space. Solar Wind Velocity -- near-earth observations on Vela 3 and 5, and less relevant observations from Pioneer VI and VII. The Pioneer spacecraft are typically separated from the earth by -100° in heliocentric longitude. With the exception of the proton intensity, all of the data 2 were taken from the ESSA Bulletins( l), and further information can be found in the descriptive text provided by ESSA ( 2S). Graph a The 7 June 1969 Event Graph b The 2 November 1969 Event Graph c The 30 December 1969 Event Graph d The 31 January 1970 Event 53 7 JUNE 1969 EVENT • 1.17-1.27 MeV • 2.99- 3.14 MeV -> I ·"""'....,,-..:.__ Q) ~ • 7.0-9.0 MeV J b.N~~. ~•""-• ..... (/) ~; ·~~:A,,~ •• •J'.°" .••"' 10 , .. I x • :::> x I ~ • _J •11- . •"' .... ~ lO\c ~· ¥-~1.. .. ·~· .. ~:,... .. CL .. ! ....... #t . 10-3 Optical Flares X-ray Flares Type - I'il Radio Sudden Comm. Sudden Impulses >- u 600 0 _J w > (.) Q) o~ z .::t::.E ~ _J 0 (/) I ~1N I 28 ta I I I 1:1 1:::1 II o Vela o Pioneer fil " Pioneer ::szn: " " :t. " 0 0 400 38 38 I 500 a:: <l: 28 18 ":!'a I- .• ""' ~.,,.,.. . 10- 1 . . .. .. ~ •.a-•.. ~£11! ... .. 0 I- •• ~'Si~ .... .. ...., ."'-1 ... z ill'""xx ~.. x.. • re • LL ~ xx (L..,f" C{jJ 0 o.o 0 0 0 ooo 0 0 300 00 12 00 12 00 12 00 12 00 12 00 I 2 00 12 00 12 00 7 13 14 8 9 10 11 12 UNIVERSAL TIME-JUNE 1969 Figure IV-1-a 54 " 2 NOV 1969 EVENT • ~ ~~ • • ' ..,,... 0 ~ • 1.17-1.27 MeV ... 7.0 -9.0 MeV • 20-45 MeV • 45-150 MeV • " "'- .. 10 • -~. • • N E ., .1111-., u ..,,.. A.'(. . ~ .....~...~A. •J> ,t.,.-r. • •-r . ·. ... ... l. ......... . ~· •• ~ -· --·-· -- ,,..._i ...:... . . . 4 IQ 1--~3~B--'~---'--~3B"'""'"""'""'..,.__ _.....,..N.,--"3~B,..._.._ _~--~--~--~,=N---1 Optical Flores R a H I I X-ray Flores II Ulll 1:1 Type -nz: Radio 11 Sudden Comm. I I Sudden Impulses 1 - - - - - - - - - - - - - - - - - - - - - - - - . . . , . . . . - 1 1 _ 1=l=--------1 CD<> o Velo >600 1o Pioneer :lZI 0 " Pioneer :szn 0 0 500 g w D > ~ ..,, D ~~400 ~ a:: <! .:¥: " . " D D 300 0 D .. " D A D Q, _J 0 (/) 00 12 00 12 00 12 00 12 00 12 00 12 00 12 00 12 00 12 00 12 00 12 00 2 3 4 5 6 7 8 9 10 11 12 UNIVERSAL TIME- NOVEMBER 1969 Figure IV-1-b , 55 30 DEC 1969 EVENT - •I•"'• I • ,,,, ~ (.) ,. • • 10 ,,,,Q) • 1.17-1.27 MeV • 7.0-9.0 MeV \ N • E (.) • • •I ~ •f,. l 1\. f~••.,_• • . ••-:..it '~ f . : f "'· . 10 4 r-~~~~~~~_._~~__._,.....-~~~~~~---i F 1 Optical Flares X-ray Flores Type-nz:: Radio Sudden Comm. Sudden Impulses >- 1- w 1 II r-~~~~~~~~~~~~~~~~~~---i 600 a Pioneer~ (..) g 16 3N 500 4 4 0 Pioneer JZII 0 4 > ~ z E 00 o~ 400 0 Cb 0 aP 4 0 0 oo 0 04 4 3: ~ a:: 300 <l _J 0 en 00 12 00 12 00 12. 00 12 00 12 00 12 00 12 00 29 30 31 I 2 3 4 UNIVERSAL TIME-DEC 1969 - JAN 1970 Figure IV-1-c 56 4 10 31 JAN 1970 EVENT 3 10 !.-- ... •~ • •• •• ..4' 2 > 10 Q) ~ .... .....t fJ) 10 u . Q) •• C\I E u x :::> 10- 1 _J u... z 0 0 10-2 a:: . -,.... .....,,: ·. ·1 .. .... .,..,·¥.. • •• •¥! t • "Jr.a. ... .A ~ Cl.. "'·· '- • ,._ . .. '• ..,"'"•· .... • • • I .... fJ) I- ._., • 1.17-1.27 MeV • 7.0- 9 .0 MeV • 20.0-45.0 MeV 10- 3 ......• • f ·~ • •••• .-. . • ~ •lfl -.. -: l i # - - .f • •• • • •• • • • 10- 1--~~'---~~'--~---'~~---'~~--'--~._,-----..........~~---1 • 4 28 Optical Flares t.:i X-ray Flares Type-N Rad io Sudden Comm. Sudden Impulses >Iu g II II 1--~~~~~~~~~~~~~~~~~~~~~~---1 o Vela a Pioneer :ID ""Pioneer W 600 0 000 0 500 w 0 000 0 0 00 ~ 400 o~ > z- .:.:.E ~ a:: a 300 <l _J 0 Cf) 00 12 00 12 00 12 00 12 00 12 00 12 00 12 00 12 00 31 I 2 3 4 5 6 7 UNIVERSAL TIME-JANUARY B FEBRUARY 1970 Figure IV-1-d TABLE IV-2 Summary of the Flare Events Selected for Analysis Optical Flare * ------, UT of X-ray TJ'I' of Ty~e-IV Flares* Radio Date Importance Solar Coordinates UT 7 June 69 2N E45-S15 0630-0730 0806-0820 0930-1100 ? -1000 2 Nov 69 3B W90-N15 0939-1130 30 Dec 69 ?t W85-s15t 31 Jan 70 2B W62-S22 Assumed UT of Particle Injection Assumed Ave. Solar Wind Velocitytt km/sec 0800 ± 0200 400 ± 50 0945-1059 1000 ± 0100 325 ± 70 1900-2000t 1903-1933 1930 ± 0030 450 ± 70 1508-1535 1507-1544 1530 ± 0100 400 ± 100 0953-0959 1536-1614 *from the ESSA Bulletins(Z 7) tsince the optical flare data did not exist in the ESSA Bulletins, the optical identification was made directly by the author from solar photographs taken by Harold Zirin of Caltech. ttAlthough the errors listed here are reasonable estimates of the uncertainty in the solar wind velocity, these errors were not used in any of the subsequent analysis. U1 -....J 58 The 2 November 1969 event was one of the largest during Solar Cycle 20( 32 ), and has been described in a special ESSA reportC 33 ). The second particle increase beginning on November 7 is probably not a flare-associated event at all, since it is accompanied by a magnetic storm and by changes in the solar wind velocity. Although the 30 December 1969 event has the symmetric rise and decay that frequently characterize a co-rotating feature, the geomagnetic disturbances and solar wind fluctuations which usually indicate a non-flare event are missing. An x-ray flare occurred at 1903 - 1933 UT, about 2 hours before the particle increase at earth, but there was no flare patrol at this time and thus no optical flare was reported. However, a search through the solar photographs taken by Professor Harold Zirin of Caltech yielded a positive optical event identification. Zirin's solar telescope had recorded a west limb flare event, complete with a prominence, between -1900 and - 2000 UT on December 30, 1969. The 31 January 1970 event is the last in a series of 3 flare events in a 4-day period, but the intensity-time profile is unaffected by the previous events (see Appendix A). The Dl detector noise problem, mentioned previously, was present during this event and affected the data coverage to some extent. The time evolution of the differential proton energy spectrum during the decay phase is shown in Figure IV-2 for each of the four events. Note that, with the exception of the 30 December event, the spectra decay without any significant change in "shape" over a period of at least 48 hours. The change in slope at -3 MeV evident in the 59 Figure IV-2 Samples of the proton differential energy spectrum during the decay phase of four selected solar flare particle events. The observations were made by Caltech's Solar and Galactic Cosmic Ray Experiment aboard OG0-6 and thus represent the near-earth particle flux. In each case, two spectra separated in time are included to demonstrate the time evolution of the proton flux. With the exception of the 30 Dec 1969 event, the spectra decay without significant change in shape over a 2-day period. Graph a The 7 June 1969 Event Graph b The 2 November 1969 Event Graph c -- The 30 December 1969 Event Graph d The 31 January 1970 Event 60 . ... •• •• •REV 79N •REV 108N ...¥, -!- llti > ~ 7 JUNE 1969 EVENT + 10 ++ "- !/) u Q) !/) + ++ N E u + + ~ + + + 16 .__~--~-------------~--~ ............------..........-......--~__, I 10 100 3 INCIDENI ENERGY (MeV) Figure IV-2-a 61 2 NOV 1969 EVENT • REV 2192 S • REV 2222 S -I 10 ~ C/) • I & I u <1) C/) N E (.) x I :::> _J LL z 0 ~ 10 1 0 a:: Q_ 10 INCIDENT ENERGY (MeV) Figure IV-2-b 100 62 10 2 . ••• 30 DEC 1969 EVENT •REV 3031S • REV 3047 S I 10 I -> It Q) ~ft ~ ...... H (f) u Q) (f) C\J E u ' ii 10 1 x :J _J LL !\ ff+ + H!] +++ ,f . t tt+++ z 0 I- 10- 2 0 a::: ++ 0... 10-3 10-4 I 10 + INCIDENT ENERGY (MeV) Figure IV-2-c I 100 63 31 JAN 1970 EVENT • REV 3502N • REV 3530N I .--.. > Q) :?! ~ (/) (.) Q) + (/) N E 10-1 -x (.) ::> _J LL z ~ 10-2 0 0::: 0.. 10 INCIDENT ENERGY (MeV) Figure IV-2-d 100 64 7 June 69 and 31 January 70 spectra are of particular interest because they allow one to determine whether any energy-change processes are occurring as part of the solar particle transport(3 l). In summary, the analysis of about 9 months of OG0-6 cosmic ray data has yielded four solar flare proton events which are uncomplicated and which can be easily associated with an optical flare on the sun. The relevant information about these events consists of the time and position of particle injection at the sun, the interplanetary conditions at the time (such as the solar wind velocity), and the proton intensity vs. time profile for various incident energy bins as observed at earth. This information, which is the result of standard analysis procedures, holds some clues to the physics of the propagation of solar flare particles and to the nature of interplanetary space. These clues are the subject of the next section. 65 V. A. DISCUSSION Introduction In almost every case, attempts to explain the propagation of energetic solar flare particles through interplanetary space have been based on the pioneering work of E. N. Parker. Parker correctly described the general features of interplanetary space< 34 , 3 5)(i.e., a spiral magnetic field imbedded in an outward-flowing solar wind), and also explained the propagation of energetic particles through this medium in terms of the Fokker-Planck equation ( 4 ). The equation can be written: an - J>. • ""} Clt = v a a - 3 ar aT (aTn) (5-1) where n is the particle density, r the radial distance from the sun, -+ V the solar wind velocity, T the particle kinetic energy, a(T) 2 (T + 2m c 2 )/(T + m c ), and 0 0 K = the diffusion tensor describing the random walk of the particles in the interplanetary plasma. This equa- tion, which involves the particle density in a stationary frame of reference, includes the effects of particle diffusion, outward convection by the solar wind, and adiabatic cooling due to the solar wind expansion. The Fokker-Planck equation has won general acceptance, and many solutions to it have been produced both for the steady-state case (to describe galactic particle propagation into the solar system as well 66 as steady-state solar emission), and for the time-dependent case( 3 , 10,ll,12,13,14,36,37) ( transport). to describe solar flare cosmic ray injection and Because the equation is difficult to solve, every success- ful analytic solution has associated with it a set of simplifying assumptions and boundary conditions which make the equation soluble but at the same time critically affect the behavior of the solution. Current discussions about the theory of solar flare particle events are thus more often concerned with the validity of the assumptions used to produce a given solution, than with the physics contained in the differential equation itself. The following is a partial list of unsolved problems concerning the theory of solar flare particle events: 1) The Diffusion Tensor -- Although it is generally assumed that ~can be separated into components K and K.1. parallel and 11 perpendicular to the spiral interplanetary field(!), very little is known about the dependence of ~on distance from the sun and on particle energy, especially at low energies. The most widely-used estimates of the dependence of K on particle energy T and radial distance r have been based on measurements of the power spectrum of the fluctuations in the interplanetary field between 1 and l. 5 AU(l,8,9,38,39). 2) Azimuthal Propagation -- Spacecraft observations of a longitudinal gradient in the solar flare particle density persisting for many days have been reported (15 26 40 41) ' ' ' . . and indicate that, in at least some cases, very little perpendicular particle 67 diffusion occurs in interplanetary space. The idea has thus developed that K .L << K 11 and that flare particles are distributed in solar longitude early in the event by some transport process occurring near the sun, rather than by diffusion "across" the field lines in interplanetary space( 4 Z). However, this idea has not been developed to the point where it can explain the observed dependence of intensity rise time on parent flare position(lO). 3) An Outer Boundary to the Diffusing Region -- In order to explain the often-observed exponential decay of flare events, a perfectly-absorbing outer boundary to the diffusing region at r = L has been introduced in some models ( 3 ' 10 ' 12 ) • . The position and even the existence of this boundary remain points of controversy (l3). 4) Scatter-free Propagation Near the Sun -- The general question of the boundary conditions in existence near the sun is still unanswered. In particular, it has been suggested< 43 ) that the rapid arrival and subsequent slow decay of the particles observed at earth can be explained in terms of a "scatter-free" region extending from the sun outward to some distance r < 1 AU. 5) The Energy-Change Process -- Recently the first experi- mental evidence for the adiabatic cooling effect (predicted by the 3 Fokker-Planck equation) was reported( l), and the notion that a particle acceleration process may be competing with this adiabatic . (44) deceleration was also introduced • This "energy-change problem," which has received a good deal of attention lately< 45 ~ 46 , 47 , 4 s), is still unsolved. 68 6) Particle Anisotropy -- The anisotropy in the flux of -10 MeV protons has been observed to change magnitude and direct i on d ur i ng a fl are even t (l5,26,40,49,50 ) A completely success- ful solar flare model must explain these anisotropy observations in at least a qualitative fashion. The most important solar flare models that have been developed to date are summarized in Table V-1. The ADB solution of Burlaga(lO) appears to explain the behavior of ~100 MeV protons fairly well. However, no a.n.ai.ytlQ solution exists which can fit both the rise and decay observations for low energy protons (-1-10 MeV) while including the effects of convection and energy-change which have been demonstrated 2 to be significant at low energies(ll,l ). In the discussion which follows, a new analytic solution will be developed which includes a boundary to the diffusing region at r = L, * assumes instantaneous particle injection at one point, uses K = constant, r and includes the effects of convection and energy-change. It will be demonstrated that this solution agrees well with near-earth observations of low energy solar flare protons using reasonable values of K r and L without the assumption of a scatter-free region near the sun. The predictions of the model will also be compared with the existing data concerning energy-change processes. The new solution presented is really concerned with the JW..d.lo..t transport of solar cosmic rays, which can be separated from the problem *Kr is identical to Ku if one assumes a radial magnetic field. TABLE V-1 Some Analytic Solutions to the Fokker-Planck Equation for Solar Flare Particle Propagation Author Diffusion Tensor Outer Boundary? Convection and EnergyChange Effects Included Comments Krimigis(36) 1965 isotropic diffusion K a: rS no no nonexponential decay, inapplicable at low energies Burlaga ( 10) 1967 anisotropic diffusiont K a: r2 K = constant yes no rapid rise * and exponential decay, but inapplicable at low energies Fisk and Axford(ll) 1968 isotropic diffusion no yes slow rise * and nonexponential decay "' e K r = K0 r \() Forman ( lZ) 1971 anisotropic diffusiont Ke a: r 2 Kr = K0 r yes yes exponential decay but slow rise* Ng and (l3 14 ) Gleeson ' 1971 quasi-radial diffusion along spiral field K~ = 0 K,, = K0 r/cos21jJ no yes behavior probably similar to solution of Fisk and Axford The NE~w (3 7) Solution anisotropic diffusiont K a: r2 K = constant yes yes can fit both rise and decay times while including convection and adiabatic dee. e r *As expected, a solution using K = K r * predicts an intensity rise time at 1 AU which is much longer than H8wever, K = constant can result in much faster rise times. r tHere ~ is diagonal in a reference frame aligned with the radial direction, and is characterized by Kr and Ke = K~. If one assumes a radial magnetic field, then Kr = K11 and Ke = K.l.' the 5-10 hrs. frequently observea. 70 of az).mu.tha..t propagation. Evidently this "perpendicular" particle transport cannot be explained with only the simple interplanetary diffusion process used in the discussion which follows. Nevertheless, a brief description of this "longitudinal distribution" problem will be given, even though the real answer must be left to those with access to multiple spacecraft measurements. The predictions of the model will also be compared with observations of the vector particle anisotropy at 1 AU. It will be shown that the new solution is capable of explaining most of the features of these anisotropy measurements. B. Background A short description will now be given of the present state of knowledge concerning the interplanetary medium, the manner in which solar cosmic rays propagate through this medium, and the sorts of particle observations that result from solar-flare type injection. 1. The Interplanetary Medium Unlike the problems of particle propagation, the features of interplanetary space are now fairly well understood, particularly in the vicinity of the earth. Several coherent summaries of the charac- teristics of the interplanetary medium as related to solar cosmic rays . . have been given (1 26 29) ' ' . Interplanetary space consists primarily of a highly conductive but nearly electrically neutral plasma which is an extension of the solar corona and is moving radially outward in all directions. This "solar 71 wind," which "blows" at a velocity of 300 - 500 km/sec at 1 AU, dominates the interplanetary medium out to at least 5 AU and probably to 20 - 100 AU. The plasma, which is composed of roughly equal numbers of protons and electrons, has a temperature of -2 x 10 6 °K at the corona. Near the earth, the plasma has cooled to -10 4 to 10 5 OK and has a density of -3 - 10 protons/cm 3 • Imbedded in the plasma is a magnetic field which originates in the solar photosphere. This B-field is rigidly coupled to the plasma due to the high conductivity, and consequently the field lines are drawn outward by the force of the solar wind. Because the photosphere, where the solar field originates, rotates with a period of -27 days (at the equator), the interplanetary field is rotated into a spiral. Assuming a constant solar wind velocity V, the average field is well approximated by an Archimedian spiral ( 4 ) V(cj> - qi ) r = 0 Q sin (5-2) e where (r,e,qi) are spherical polar coordinates centered at the sun and Q = 2.7 x 10- 6 radians/sec is the angular velocity of solar rotation. The ambient field thus makes an angle ~(r) to the radius vector. For = tan-1 (nr/V) with respect V ~ 400 km/sec and r The field intensity falls off roughly as r -2 = 1 AU, ~ = 48°. due to the solar wind expansion, and has a value of -5 x 10 -5 gauss at 1 AU. Superimposed on this large-scale field are small-scale fluctuations in the field direction and intensity which produce deviations from the average configuration. These irregularities act as scattering 72 centers for the energetic particles and lead to the notion of cosmic ray "diffusion." In addition, the solar wind plasma is permeated with various "transient phenomena" -- discontinuities, shock waves, and hydromagnetic waves -- which contribute to the fluctuations in the medium. Measurements of the average direction of the magnetic field vector over many solar rotations have shown that the ecliptic plane is divided into fairly stable "sectors" of alternating field direction(Sl). These sectors, which require several days each to rotate past the earth, appear to be correlated with geomagnetic storm activity. The sector boundaries may very well act as barriers to energetic particle propagation. The solar wind interacts in a complex way with the quasi-dipole geomagnetic field, creating a bow shock and magnetosphere. The boundary between the geomagnetic and interplanetary fields is a separate branch of geophysics in itself, and is of importance here only as it affects the OG0-6 particle observations (see Section III-B). 2. The Fokker-Planck Equation and Particle Diffusion The details of cosmic ray transport depend on both the large scale and small scale features of the interplanetary field. If the small scale fluctuations were not present, the energetic particles would be confined to gyrate about individual field lines like beads sliding along wires. However, the field irregularities or "kinks" act as resonant scattering centers whenever the scale-length of the irregularity is of the same order as the particle gyro-radius. The scattered 73 particles thus random walk in both pitch angle and position, and the propagation can be approximated by a diffusion tensor ~· In a reference frame with the z-axis along the average field direction, 0 0 0 (5-3) 0 where K 11 characterizes the particle diffusion along the field lines, KL across the field lines, and ~d includes the effects of curvature and gradient drifts(!). The diffusion is anisotropic because of the average magnetic field structure present. The picture is further complicated because the scattering centers are imbedded in the moving solar wind plasma. Since the field irregular- ities are convected outward by the solar wind, there is a net outward particle transport or convection separate from the particle diffusion process. In addition, the solar wind is continually expanding to fill + interplanetary space (V + • V 1 0), and the field irregularities are thus on the average moving away from each other. This expansion produces an adiabatic cooling or deceleration of the cosmic rays at the rate( 4 ) dT dt = 1 . 3 a(T) T (V · V) (5-4) There is some question as to whether this is the only important energychange process at work in interplanetary space< 3i, 44 ). 74 All of these particle propagation effects can be combined into a single differential equation for the time rate of change of the particle density: an at = v <1£, • "Vu) - v . (nV) + ~ V DT (a(T)Tn)] (5-5) The right-hand side of this form for the Fokker-Planck equation includee terms for particle diffusion, convection, and adiabatic decel- eration. This equation gives an adequate description of energetic particle transport as long as the pitch angle scattering can be characterized by a diffusion tensor. The Fokker-Planck equation is not valid whenever a large pitch angle anisotropy is present, (l) and other approaches must be used ( 52 ' 5 3). Although the validity of the Fokker-Planck equation is generally accepted, the exact nature of the diffusion tensor ~is still undefined, largely because of the lack of adequate space-probe observations. The only real agreement seems to be that the antisymmetric terms can in most cases be ignored (Kd ""0) and that K 11 and K.l probably depend on both particle energy T and heliocentric distance r. Jokipii(l) has shown that the diffusion coefficient ~(T) can be related to the power spectrum of the fluctuations in the magnetic field. Using Mariner IV magnetic field observations, Jokipii( l, 54 , 55 ), and Jokipii and Coleman (3S) have obtained the following values for K 11 at r = 1 AU as a function of particle rigidity R (in GV) and velocity 8 =v c 75 (5 x 10 K II [ 21 (1.5 x l0 For rigidities below 1 GV, 2 ) SIR cm /sec 21 K~ 0.1 $ R $ 1 GV (5-6) 2 2 )S R cm /sec R ~ 1 GV is dominated by the random walk of the field lines rather than by actual particle scattering, and Jokipii (1 ' 55) derives R $ 1 GV In a recent paper, Jokipii( 56 (5-7) ) has demonstrated that his pitch-angle scattering approach, once thought to break down for R ~ 0.1 GV, can be safely extended to much lower rigidities. Figure V-1 compares the semi-empirical values for K 11 and K.J. just discussed with values of K derived from measurements during the rise of solar flare events, assuming isotropic diffusion( 26 ). It should be emphasized that the Jokipii and Coleman values for K are based on a smallsample of magnetic field measurements made in 1964 and therefore must be taken only as estimates. The figure makes it obvious that very little is known about K below -10 MeV. The dependence of ~on r is even more of a mystery. Although Mariner IV observations indicate that K 11 is constant between 1.0 and 1. 5 AU, ( 3 8) nothing is known about the important r-dependence of Ko near the sun. Although experimental measurements for KL are largely lacking, some simple theoretical arguments based on the random walk of the 76 } 1-+-l Estimates based on Mariner Ill magnetic f ield measurements Solar particle measurements 1022 (.) Q) ~ C\J E (.) •• 1021 c: ~ 1019.__~.L.-....._........................._~~-"---'--'......................~__.____.__._. . . . . . . . . . . . ..._~..._...._..._._~....... I 10 10 2 10 3 4 10 PROTON KINETIC ENERGY T in MeV Figure V-1: The diffusion coefficient "K is plotted vs. proton kinetic energy T. The lines represent the best estimates for K" and Ki of Jokipii( 1 ,54,55,56) and Jokipii and Coleman(38) based on the power spectrum of fluctuations in the magnetic field. The points are the results of solar particle measurements sununarized by McCracken and Rao(26). 77 field lines, indicate that K..a.. should vary as r 2 . In addition, the assumption of an r 2 dependence for K~ makes the equation easily soluble. As mentioned previously, it is possible that the azimuthal parti- cle transport is dominated by a separate near-sun diffusion region and that perpendicular diffusion in i nterplanetary space is relatively unimportant. Since the Fokker-Planck equation can be solved much more easily if any of the terms can be neglected, it is of interest to calculate the relative magnitudes of the various terms. Us i ng the value of K at 1 MeV from Figure V-1, a solar wind velocity V = 400 km/sec, and assuming that the scale length £ for changes in the density is about 1 AU, the diffusive term beco~es: -+ (~ • 'Vn) For comparison, the convection term is roughly -+ 'V • -+ nV :: Vn £ = n (4 x 10 7 cm/sec) (1.5 x l0 13 cm) This crude calculation indicates that "solar wind" effects are very important when the diffusion coefficient is small. It is thus a violation of the basic physics of the Fokker-Planck equation to ignore the terms for convection and adiabatic deceleration at low proton energies. 78 3. Solar Flare Particle Events McCracken and Rao( 26 ) have classified solar flare particle events as either pnompt, occurring within hours of the parent flare, or delayed, occurring ~ 24 hours after the parent flare or even without an obvious optical flare association. In fact, it is no longer believed that flare particles are always injected impulsively at the 9 sun(Z ) -- proton precursors have been observed, and many flare particle events have been explained in terms of particle storage and/or continual production at the sun. Nevertheless, a large fraction of the so-called "prompt" events appear to be consistent with impulsive injection and can be clearly associated with a parent optical flare. The four event observations presented in Section IV fall into this group, and it is this class of particle events which is the subject of this dissertation. Solar flare particle events have been observed at 1 AU for many years, and as a result, the behavior of the "typical" event has been fairly well defined(ZS,Z 6 , 29 ). The following are a few of the commonly observed characteristics of prompt solar flare events which any solar flare model must attempt to explain: a) The intensity vs. time profile for most prompt events is characterized by a rapid increase to maximum intensity followed by a decay phase which at late times is approximately exponential. The time-to-maximum T m is typically 5 - 35 hours, and the exponen- tial decay time constant b) T ~ 8 - 27 hours. The shape of the profile, especially during the rise, 79 depends strongly on the distance in heliocentric longitude between the flare and the foot of the line of force connected to the observer. (The foot of the line of force passing through the earth is typically at 60°W solar longitude). T m is greater for east- limb events because the longitudinal separation ~ is greater (compare the 7 June 69 event of Figure IV-2 with the events of 2 Nov 69 and 31 Jan 70). roughly as ~ 2 Burlaga(lO) has shown that T varies m 07) has pointed out that particles . • McCracken from west-limb flares are more frequently detected than those from east-limb events, even though the parent optical flares occur in the east and west with equal probability. c) A well-defined velocity dispersion is seen, especially for west-limb events. As expected, the high energy or faster particles arrive first. d) The low energy (-1 MeV) protons last longer (decay more slowly) and exhibit more intensity variation than the higher energy protons. e) The typical vector anisotropy observation is character- ized by a strong anisotropy from -45°W early in the event which later decreases to a smaller equilibrium value directed from -45°E at late times (lS). McCracken (5 7 ) reports that the anisotropy measured using neutron monitor data is strongest for west limb events. This completes the discussion of what is known about solar flare particle propagation through interplanetary space and should set the 80 stage for the following attempt to describe this propagation. C. Solving the Fokker-Planck Equation for Solar .Flare Particle Injecti on 1. Some Boundary Conditions and Simplifying Assumptions The following is a list of assumptions, conventions and boundary conditions which will be applied to the Fokker-Planck equation: a) (r, The particle density n depends only on spatial position e, ¢), time t, and particle kinetic energy T. b) All parameters except for the particle density assumed to be independent of energy T. One can set a(T) n are = 2 for protons below - 100 MeV. c) The solar wind velocity V is radial and independent of (r, e, ¢) and t. d) No attempt is made to describe transient phenomena such as solar wind fluctuations, shock waves, and hydromagnetic waves. e) t = O. The particles are impulsively injected at r = r s at time Specifically, o(r - r ) s n(r, e, qi, t = O) = - - -- - f(e, ¢) 2 (5-8) r n(L, f) It is required that the density n remain finite as r ~ O. g) A perfectly absorbing boundary exists at r = L such that e, qi, t) = o. h) The diffusion tensor will be taken to be diagonal in a frame of reference (r, e, qi) aligned with the radial direction. 81 Specifically, K will be defined by K = K<P = Kl r 8 K = K = constant, both independent of energy T. 2 and r These assumptions are intended to facilitate a solution to the equation without compromising any of the important physics involved. The most serious simplifications are that 1£,iS independent of energy T and aligned with the radial direction. It will be shown that the energy-dependence can be approximated by breaking the energy domain into separate intervals within which K is independent of T, as long as the intervals are large with respect to the -30% energy-change which has been observed to occur during a typical -3 day decay period (31) • Some additional comment should be made about the assumed diagonal form for the diffusion tensor: K = ~ot 0 0 0 0 0 This is r 0 identical to the assumption of a radial magnetic field. However, the simplest and most widely-used interpretation of this assumption is to equate it with the neglect of the spiral field. K r Then = K , K = K = K , and one must make an additional distinction 11 e <P ..!. between actual radial distances in interplanetary space and the radius variable r, which now represents a path length measured along a spiral field line. This interpretation, which sacrifices surprisingly little of the physics involved, will be adopted in this discussion until a treatment of the anisotropy is attempted. 82 The boundary condition at the sun would seem to be more appropriately described by either a perfectly absorbing or perfectly reflecting boundary at r = r s (instead of at the origin). However, a recent paper 5 by Englade( B) indicates that the type of solar boundary condition assumed has little effect on the time profile observed at 1 AU. Evi- dently more appropriate inner boundary conditions serve only to make the solution less manageable. The existence of an outer absorbing boundary is not an accepted point. Several authors have developed solutions using an infinite diffusing region (see Table V-1). 3 In particular, Ng and Gleeson(l ) claim that an "approximately" exponential decay can be produced assum- = K 0 (1 + r 3 ). ing K II However, their K 11 function appears to be just another way of smoothly introducing a region of free escape where K 7 oo and n 7 O. The fact remains that observations of exponential decays are the rule rather than the exception, and a free escape boundary produces such a decay. The r-dependence for K r is a very. important assumption, as will be demonstrated later in the discussion. 2. Separation of Variables We are now in a position to attack the Fokker-Planck equation using the technique of separation of variables. Assuming a radial mag- netic field, a diffusion tensor defined by K (r) and K (r) r e = K1 r 2 both independent of T, and V independent of spatial position, one can rewrite equation (5-5) in spherical polar coordinates as: 83 an at = Kl_ _l_ _a_ r 2 K (r) an+ __ 2 ar r ar sine r a (sine an) ae ae (5-9) Kl + sin 2 2 1 a a n V an _ 2Vn ~ - 3n a-f (a(T)Tn8 ar r aq,2 - Assuming n(r, e , cp , t) = R(r, t) Q( e , cp , t) S(T) the equation separates into the following three equations: (5-10) i a aR at 2-ar r a aT (aTS) (5-11) = 3S(l - C) (5-12) Note that it is not necessary to specify the form of K (r) to carry out r the separation. 3. The Azimuthal Dependence The azimuthal function Q(e, ct>, t), which is the solution to equation (5-10), gives the density distribution in heliocentric latitude and longitude at time t. The solution, which is derived in detail in Appendix B, is identical to the azimuthal function used by Burlaga(lO) and Forman(lZ): Q(y, t) = ~ C.Q. p .Q. ( cosy ) e -.Q. (.Q. + 1) Klt L .Q.=O (5-13) 84 where cosy = case cose (e, ~) (e , ~ 0 0 + sine sine 0 cos(~ - ~ 0 ) (5-14) = point of observation 0 ) = center of azimuthal distribution The Ct are =hosen so that Q(y, t=O) matches f(e, tion profile at the sun. ~), the assumed injec- For o-function injection at y = 0, we have C = 2£ + 1 and £ Q ( y' t) = (5-15) Burlaga(lO) has described the behavior of this Q(y, t) function in :'. Orne detail. Feit(59) has shown that for a source uniform over an emission cone of half-angle y : 0 = (5-16) 1 - cosy Presumably any symmetric f(e, ~) 0 can be generated using the proper coefficients Ct. If we choose our coordinate system coincident with the accepted solar spherical coordinates, then our observation point at earth corresponds to The angle ~' which is 58° for r = 1 AU and V = 400 km/sec, indicates that the earth is typically connected by the spiral field to a point at -58°W solar longitude on the sun. We can thus define a new separation angle y' which is a function of time t and 85 includes both corotation and the spiral angle: = sine 0 cos(~ 0 (t) - ~) cosy' (5-17) = COEA where A ~ 0 n 0 = 2- e cos(~ Qr 0 - ~ + Qt) v solar latitude of optical flare 0 0 (t) = ~o +Qt= the actual time-dependent solar longitude of flare location and ~ 0 = solar longitude of optical flare at t = 0 (~ 0 > 0 for W longitude) It should be mentioned here that the above formulation of the perpendicular transport process is probably far too simple to accurately describe the actual situation. Multiple spacecraft observations of particle fluxes and anisotropies indicate that rapid near-sun diffusion and/or particle injection over a large solar surface area should be . k e d(l5, 26, 40, 41)• invo Despite this evidence, the simple point-injec- tion and 2-dimensional interplanetary diffusion process described by equation 5-15 will be employed in the discussion which follows. The shortcomings of this simple Q(y,t) function, which will be discussed carefully, do not in any way affect the validity of the new solution forn~dla.l 4. particle transport. The Energy Dependence Equation 5-12 for S(T) can be reduced to the following form: a(T)T ~~ = ~ - 3C - a(T) + T ~~J S (5-18) 85 For T proton 100 MeV, a(T) $ = 2 and ..fu! << 1, and the energy equation aT becomes (5-19) with a solution of the form S(T) = S T-y 0 where y = 3C - 1 . is the power law index of the density. 2 In principle, any initial spectrum S(T) can be described by this solution by an appropriate super-position of power laws: S(T) L = (5-20) Ai T-Xi i 2 The separation constant C is the Compton-Getting factor(l ) c = 1 - 3~ aaT ~ (T) T~ (5-21) For low energy protons, assuming that adiabatic deceleration is the c = 2x3+ 1 (5-22) only energy-change process operative, this becomes The solution can be generalized to describe any combination of energychange processes by introducing a variable • energy-change time constant (17) • 'E(r) = , 0 r, which is the The generalized Compton-Getting factor then becomes: c = 1 - (y - 1) 2VT (5-23) 0 which will account for any energy-change process which varies as l/r. Note that pure adiabatic deceleration corresponds to 3 'o = 4 V , and a value of C = 1 corresponds to no energy-change effect at all. 87 5. The Radial Dependence Most of the features of a solar flare event observed at 1 AU are due to the "radial" transport involving convection, energy-change, and diffusion parallel to the field lines. For this reason, much effort has been devoted to solving the radial equation (equation 5-11). if the perpendicular diffusion is neglected (K 8 Even = 0), the equation describing the radial propagation remains unchanged. Burlaga 0.0), has solved the radial equation by neglecting the terms for convection and energy-change, including a perfectly absorbing boundary at r = L, and assuming K (r) = K. r His differential equation becomes a2R = .!. .QR + ~lR r ar K (lt -2 ar (5-24) He derives the following eigenvalue expansion for the radial dependence: 00 R(r,t) = A rr L s I n=l nn -n TI t sin(nz rs) sin(L r) exp ( 2 L 22 K ) (5-25) where the constant A is an arbitrary normalization determined by the number of particles injected. His overall solution which includes the azimuthal distribution, is n(r, e, qi, t) = R (r,t) Q(e, qi, t) (5-26) Burlaga's solution produces a rapid rise and an exponential decay and thus adequately describes the time profile of high energy (T ~ 100 MeV) protons observed at earth. However, this solution is inapplicable to 88 observations of -1 MeV protons because it does not include the convection and energy-change terms which become important for small values of K • r Forman(lZ) has solved the radial equation in the form ~) 1R - 2CV R = 3r 2 a2 R + ( l 2 r K r or 0 ~ K 0 (5-27) r including convection, energy-changes, and a boundary, but using K r (r) = Ko r. Her radial solution is R(r,t) = A 2 r L s ./r /L) J s n (j n n ./r /L) exp (-t/T ) n ~1+1<Jn,nJ where J ( x ) is the Bessel function o f or d er n, an d j n zero of J (x). n n ,n 1• (5-28) s the nth Also n + 2CV = (5-29) K 0 and •n = 41 (5-30) As will be demonstrated later, this solution adequately describes the decay phase of flare events, but because of the K r = K o r dependence 89 chosen, predicts a rise time which is slower than the 5-10 hours observed. D. The New Solution 1. Derivation It is apparent that a solution to the radial equation (equation 5-11) similar to Forman' s but using K (r) = constant r yield desirable results. might The equation becomes _ 2CV R= !_~ (~ - :) ~ K ct er Kr (5-31) A solution has in fact been found using the boundary conditions and assumptions listed in Section V-C-1. The details of the derivation are given in Appendix C. The new solution, which was recently reportedC 37 ), can be written as the following eigenvalue expansion: R (r, t) r r s (5-32) 2: Fo(s/2/an,ran r s ) Fo Cs/2/an ,;an r) ~~~~~~~~-N~~~~~~~~~ n=l e -t/, n n where F (n,p) is the regular Coulomb wave function( 60). 0 The an are the eigenvalues defined by the outer boundary condition F (S/2/a ,/a o n n L) =0 (5-33) 90 The other parameters are defined as follows: 8 = V(2C - l)/K c = Compton-Getting Factor = 1 + (y - l)/2VT0 (5-35) 2 4K/ (4K Cl + V2) n (5-36) Tn N n (5-34) z J L~o (f!!zr.n,r.nxJdx 0 = ~ t::or [here A 8 - 4u n Fo (n,p) 20 2 cm /sec. (5-37) = Fo (8/2/an ,/ax~ n = arbitrary normalization This expansion converges rapidly for K ~ 10 (::o1 f:~o11.L t ~ 5 hours, C ~ 4, and The details of calculating the radial solution are discussed in Appendix D. In the limit as V + O, we have 8 + 0, F (8/2/a ,/ax)+ sin(nTrX/L), o n n 2 2 2 Tn + L /n 1T K, L Nn + 2 and the solution reduces smoothly to Burlaga's solution (equation 5-25) as expected. The complete solution including the energy-dependence can now be written as: 00 n(r,6,cj>,t,T) Q(6,cj>,t) I Ai T-Yi Ri (r,t) (5-38) i=l which consists of a sum over different radial functions R.(r,t) each l. 91 corresponding to a different Compton-Getting factor C. i = 1 + (y.1 - l)/2VT0 • Equation 5-38 is thus the response to impul- sive injection of the form n ( r, e, <P, t=O, T) with S(T) = f(8,<P) I A T-Yi i i 2. o(r - r ) s S(T) 2 r = (5-39) (5-40) Behavior of the Solution The radial dependence R(r,t), defined by equation 5-32, defines most of the features of the particle time profile. Figure V-2 shows the distribution of particles as a function of radial distance r at various times using reasonable values of the parameters. The peak in the particle distribution, which is initially near the sun due to the a-function injection, moves outward until the effect of the absorbing boundary is felt and eventually assumes a stable "shape" at late times. The time-profiles R(r,t) vs. t observed at different radial distances r are shown in Figure V-3. As expected, the time-to-maximum T depends on the radial position of the observer because the maximum m intensity occurs roughly when the peak in the radial distribution (Figure V-2) moves past the observation point. observed at 1 AU is < 20 hours. Note that the T m At late times, the intensity decays exponentially with a time constant TDEC roughly independent of radial position. A glance at equation 5-32 makes it clear that only the first eigenvalue (n=l) survives and that the decay phase is described by the 92 100 20 2 cm /sec = 5 x 10 v = 400 km/sec c = 1.0 L = 2.3 AU rs = .0047 AU Kr t = 5 hours 10 20 - Q) 0 u (/) 40 >~ --.. 0 ~ . .0 ~ d 10- 1 ~ ~ 10- 3 ....__~~--~~~~~~~~~--~......__._~....._......_.._~---..J 0 0.4 0.8 1.2 1.6 2.0 2.4 r 1n AU Figure V-2: The new radial solution R(r,t), defined by equation 5-32, is plotted vs. radial distance r for various times after particle injection. Typical values of the parameters were chosen and are indicated on the graph. Note that the distribution, which is initially peaked near the sun due to the a-function injection, reaches a stable configuration at late times. 93 cm2/sec V =400 km/sec Kr= 5x10 20 c = 1.0 L=2.3AU r5 = .0047 AU 10 r = 0.4 AU Q) 0 u (/) >- ~ 0 ~ +- ...c ~ 0 -;:- - r =1.0 AU -I ...: 10 16 2 Tl ME tin hours Figure V-3: The intensity R(r,t) predicted by the new solution is plotted vs. time for various observation distances r. Note that although the time-to-maximum depends on radial distance, the time-profile eventually reduces to an exponential decay independent of observer position. The values of the parameters, which are the same as in Figure V-2, are listed on the figure. 94 following equation: , exp (Vr /2 K ) F (s/2 !;., , la.. r) R(r, t) a: 0 ~~~~~~~~~~i~~i-e-t/Tl (5-41) r Thus if one is concerned only with the exponential decay phase of an event, the predictions of the new model can be summarized by the single quantity 'DEC = T1 , which is independent of position and time and depends only on the boundary location L, the solar wind speed V, the diffusion coefficient Kr' and the energy-change parameter C. Figure V-4 shows how varies with K , L, V, and C. 'DEC r For small values of Kr' 'DEC becomes essentially independent of Kr because the particle transport outward to the boundary is dominated by convection and energy-change processes. For large values of K , the solar r wind effects become unimportant, and the decay time constant approaches Burlaga's (K) r -1 dependence. Increasing the solar wind speed hastens the decay, as expected (Figure V-4a), because the contribution of both convection and energychange to the decay rate is increased. In the limit V + O, 'DEC becomes identical to Burlaga's expression (see equation 5-25). Changing the boundary position L has a marked effect on the decay rate, as shown in Figure V-4b. The decay in the particle intensity is largely produced by the outward transport of the particles to the boundary, where they "escape." A more distant boundary means that the processes of diffusion and convection cannot d~iver particles to the boundary at the same rate, and 'DEC must increase accordingly. A close look at the behavior of the complete solution shows that at short 100~ L=2.3AU <fl '- r~ 1100~ I I~ ~ I V=400krn/sec L=2.3AU II t=__~ :::J 0 ..c ~w 10[0 1--- 400 ~ 600 "' - - 1i 1020 I p I I I •• I I 1021 I I(~ I) ,,I 1022 2 Kr in cm /sec 10[ 2 .7 2.3 ~ ___, 'i ~~~ 10 ~ -;· ~_/// ~ I 2 .0 V=400 km/sec C= 1.0 11 1020 I I I I I 11 I 1021 2 I Kr in cm /sec I I I (I~)"' 1022 (c) 1! 1020 I I I I I I 11 I 1021 I I I I I 1022 2 Kr in cm /sec Figure V-4: The exponential decay time constant 'DEC predicted by the new solution is plotted vs. Kr for a) various values of the solar wind velocity V, b) various values of the boundary position L, and c) different values of the energy-change parameter C. ~ V1 96 times, when the particles have not yet reached the boundary, the profile is independent of L. The Compton-Getting factor C characterizes the extent to which the random walk of the particles in energy-space affects the decay. While a value of C = 1 corresponds to ignoring the energy-change effects completely, a typical value of C = 3 means that at L = 1 AU the intensity is decaying, due to the energy-change process, at a rate that can be estimated from equation 5-5: 'tENERGYCHANGE = -n s (an/at) (2 ~v) •17 hours (5-42) This is of the same order as the actual decay time constant and indicates that the contribution of C to the decay rate is significant -a conclusion which is substantiated by Figure V-4c. The relative distribution of particles f(r) in radial distance r at late times is given by (5-43) and depends only on L, C, and the ratio V/K. This function is shown in Figure V-5 for various typical values of this ratio. condition f(L) = 0 is quite evident. reduces to Burlaga's sin(n~r/L)/r Again, for The boundary V/K ~ O, f(r) which is peaked at the origin. As V/K is increased, the peak in the distribution becomes more pronounced and moves outward due to the increased convective force. The depen- dence of the position of this peak on V/K has been discussed by Forman <4a). 97 L=2.3AU c = 1.0 V = 400 km/sec c 0 Kr= 10 21 cm 2/sec 0 N 0 10 E ...... 0 c ~ ...... 0 ...... . .0 ~ ...... 0 ~ 6 VJ+ e 'l/ 11 t 4 (J rv\~Qj -...:- ~ ~ (J~ -? '\ '1 2 0.4 0.8 "" " 1.2 1.6 "" 2.0 2.4 r in AU Figure V-5: The radial distribution of particles at late times is plotted vs. r for various values of the ratio V/K • r 98 The transport process occurring at any position r is best described by the particle current S, which is simply the vector parti- cle flux. s = CVn + ~ + (5-44) • 1111 For the radial B-field assumption used so far in this discussion, S is radial and can be treate:d as a scalar composed of two terms (5-45) and s K = -Kr (5-46) 'd R/ar which are the convective and diffusive components of the total particle current. Note the azimuthal part of the solution Q(y,t) has been neglected, and that SV is proportional to the radial solution R(r,t), while s K is proportional to the slope aR/ar. Figure V-6, which shows the decay-phase values of S, SV' and SK as a function of r, makes it clear which transport process is most important at any given radial distance. At the outer boundary, the net particle flux is outward and is due entirely to diffusion. typical particle distribution at late-times is peaked at r Because the ~ 1 AU (see Figure V-5), the gradient over much of the distance is positive and the diffusive current S K in this region is directed inward. This inward diffusion, which impedes the process of particle escape, exactly balances with outward convective current at the origin to produce S(r=O) = O. 99 L=2.3AU V = 400 km/sec = 1.0 c +6 4 (l) 8 (/) Kr=IO 2 20 2/ cm/sec ...... 0 ...... ...... ...... a 9 t +6 ... 4 (j) f- z 2 SK w ""-1 I I 1\ 20 Kr=2.5xlo cm 2/sec \ / 0:: 0:: :J u w _J u ~ +6 <! Q_ Kr= 10 4 21 21 cm 1sec --- Sv-- 2 0 0.4 0.8 1.2 1.6 2.0 r in AU 2.4 2.8 Figure V-6: The particle current S at late times as well as the convective and diffusive components, Sv and SK, are plotted vs. r for three values of Kr. The shaded area emphasizes the region where the diffusive current SK is inward. 100 The decay time constant 'DEC = n/ ( 'rh/ 0t) is a measure of the total "stored" particle population relative to the rate at which particles reach the boundary and escape. Figure V-4 shows that a broad maximum in 'DEC vs. Kr occurs typically for This maximum in 'DEC' which has been discussed by Forman (48) , can be understood qualitatively as the certain combination of parameters which produces the lowest loss rate per unit stored particle. E. Fits to Data Assuming Pure Adiabatic Deceleration 1. Method of Fitting Actual Data Attempts were made to fit the flare event time-profiles des- cribed in Section IV using the solution of Forman and the new solution. In each case, the following assumptions and parameter values were employed: a) It was assumed that adiabatic deceleration due to the solar wind expansion was the only energy-change process operative. The Compton-Getting factor C was thus defined in terms of the power law index y of the particle density as C = (2y + 1)/3 (see Section V-C-4). b) Observations of the intensity within different energy bins were treated separately. The density was assumed to be a power law within each incident energy bin, and a separate value of C was thus assigned to each energy interval by evaluating y(T) at the center of the interval. = d(ln n)/d(lnT) The fact that the spectra of Figure IV-2 are stable with time indicates that the "mixing" 101 between various energy bins is not very pronounced, and that the description of each bin by means of an independent solution to the differential equation is reasonable. In the same way, the energy change rate reported by Murray et al. (Jl) indicates that the "typical" particle does not lose enough energy to move from one bin to another during the typical observation period. c) The spiral character of the interplanetary field was par- tially taken into account by assuming that the observations at earth correspond to r = 1.15 AU, which is the path length along the spiral field for V = 400 km/sec. The spiral field and the corotation effect were also included by using the azimuthal function Q(y' ,t) defined by equations 5-15 and 5-17. The depen- dence of the longitude of the directly-connected field line on the solar wind velocity V was included as well. velocity was defined as Q = 2.69 x 10- 6 The solar angular radians/sec, the value at the solar equator. d) in r, The particles were assumed to be injected as a a-function e, and cp. Thus the cone half-angle was y 0 = 0 in all cases. The coordinates .(A 0 ,~ 0 ) and time of injection for each event were defined precisely in terms of the parent flare identification summarized in Table IV-2. The radial position of injection was assumed to be the surface of the sun e) r s = 0.0047 AU. The solar wind velocity was set to the best estimate based on the ESSA data (see Section IV). 102 Thus the only nJz.e.e. parameters for each fit were the f) diffusion coefficients Kr and Ke; the boundary position L, and the arbitrary normalization A. An initial choice of L = 2.3 AU was made based on the fits made by Burlaga(10). Because of the complicated nature of the flare profiles, a leastsquares fitting technique could not be easily used. Instead, a simple but tedious optimization with respect to K ; K and A was made by eye r e for each profil~. 2. Forman's Solution An attempted fit to the 2 November 69 event profile using Forman's solution (equation 5-28) with K (r) r V-7. K r is shown in Figure 0 With L = 2.3 AU, her solution duplicates the decay profile using reasonable values of K and K (evaluated at 1 AU), but cannot match the e r rapid rise observed. Forman has attributed the long rise time predicted by her solution to the simple boundary condition assumed at the origin and to the K r « 2 r dependence used(l ). As will be demonstrated next, the r~dependence assumed for K inside 1 AU has a drastic effect on the r time-to-maximum, for an obvious reason: early in the event the particles are peaked near the sun, the propagation is due almost entirely to diffusion, and the rate of particle transport is thus very sensitive to the near-sun value of K • r Concerning the boundary condition at the sun, Englade has recently used Burlaga's ADB model to study the effect of different boundary conditions at r = rs <58 ). He reports that the type of boundary condition assumed has surprisingly little effect on the profile observed at 1 AU. 103 2 NOV 1969 EVENT FORMAN'S SOLUTION Kr (r) =K0 r ... L=2.3 AU V= 325 km/sec • .~+. , Q) :f - .~: 0 u (/) ~, 1.17-1.27 MeV =1.8 c ••. :- >. ....... 0 ....... -+- ....... 0 cm2/sec Ke (IAU)=4x10 20 •I • ....0 Kr(IAU)=3XI0 20 • • x • ~ _J LL z 0 f0 0::: • CL <...') 0 7.0-9.0 MeV c = 1.97 Kr=3Xl0 2 0 Ke= 6X10 20 _J 2 3 4 5 6 7 8 9 10 DATE - NOVEMBER 1969 Figure V-7: Fits to the 2 November 1969 event using Forman's solution with L = 2. 3 AU. 104 3. Fits Using the New Solution The new solution developed in Section V-D can be described as identical to Forman's except t hat it assumes Kr(r) = constant. A preliminary attempt to fit all four flare events discussed in Section IV using L = 2.3 AU produced the following results: a) The 30 Dec 69 event appears to belong to a completely different class than the other three events: its decay rate is twice as fast as the usual and is inversely correlated with energy (which is very atypical). Although the time-profile of this event can be fitted using the new solution, it will be omitted from the analysis here on the basis that it is not a flare-associated feature or at least not an example of the "classical" prompt flare event which this discussion is attempting to explain. b) Some difficulty was encountered in fitting both the rise and decay rates of the 2 Nov and 31 Jan events. It was found that this small problem could be remedied by moving the boundary out to L = 2.7 AU. Figures V-8 through V-10 show fits to the remaining three "classical" flare events using the new solution with L = 2.7 AU. With the value of L fixed, using only Kr' Ke' and A as free parameters, the predictions of the new solution can be made to agree very well with the observed profiles of all three events at energies from -1 MeV to -70 MeV. 105 7 JUNE 1969 EVENT NEW SOLUTION Kr (r) = K L=2 .7 AU V = 400 km/sec 1.17-1.27 MeV C= 2.0 20 2 Kr =2.5x 10 cm /sec Ke ( 1AU)=6.5 x 10 20 Q) 0 (.) (f) • 3.0-3.2 MeV x :J _J LL z c = 2.65 ,. Kr= 2.5 X 10 20 •• Ke= 5X 10 20 • • • 0 0 f0:: • Q_ - C) 0 _J 7.0-9.0 MeV c =3.33 • Kr=3xlo 20 20 1• Ke=4.5x 10 • ,,-• 7 8 9 10 II 12 13 14 15 DATE - JUNE 1969 Figure V-8: L == 2. 7 AU. Fits to the 7 June 1969 event using the new solution with 106 2 NOV 1969 EVENT NEW SOLUTION Kr(r) = K L = 2 .7 AU V = 325 km/sec • , , .....-.. •• • Q) 0 (.) 1.17 - 1.27 MeV c = 1.8 20 2 Kr= 2.3 x 10 cm /sec K (IAU) = 4 x 10 20 (./) •• •I >. ~ 0 ~ • '• x => _J e LL • 7.0 - 9.0 MeV z 1. c = 1.97 20 0 ~ ~ b 0:::: ' •. Q_ K = 5 X 10 r 21 Ke= 1.0 X 10 (.'.) 0 15 - 20 MeV ~ c = 2.43 • r'It Kr= 7 X 10 20 ·~t' Ke= 1.8 X 10 21 _J 45-70 MeV 0 c =3.0 Kr= 8X10 Ke= 2X10 0 2 3 4 5 6 7 8 20 21 9 10 DATE - NOVEMBER 1969 Figure V-9: Fits to the 2 November 1969 event using the new solution with L = 2. 7 AU. 107 31 JAN 1970 EVENT NEW SOLUTION Kr (r) =K L=2.7 AU V= 400 km/sec • Q) - • 0 u 1.17-1.27 MeV =2.05 2 20 Kr=3x 10 cm /sec Ke(IAU)=1.2x10 21 • • c • C.f) ....... >. ~ 0 ~ •• -+- t\, ! •. ~ ..0 ~ 0 f x 7.0-9.0 MeV = 2.50 20 Kr=7Xl0 Ke-- I.BX 10 21 =::) c _J LL z 0 • I- 0 0::: Q_ (_9 0 _J 15-20 MeV =2.3B Kr=BXl0 2 0 ...!If· ., c • • ~ f \ Ke= I.BX 10 21 • 1 31 2 3 4 5 6 7 B DATE - JAN 8 FEB 1970 Figure V-10: Fits to the 31 January 1970 event using the new solution with L = 2. 7 AU. 108 4. Evaluation Figure V-11 summarizes the best-fit values of K r for all three flare events determined using the new solution with L = 2.7 AU. It is immediately apparent that the K r values agree well with each other and are also at least consistent with the low energy estimate for K 11 given by Jokipii and Coleman( 3 S) (see Section V-B-2). The consistency between these three events is encouraging, especially since one of them (the 7 June event) was separated by -100° in solar longitude from the foot of the near-earth field line, while the other two (2 Nov and 31 Jan) were "west-limb" or "directly-connected" events. It should be emphasized that the absolute values of K needed to r achieve reasonable fits depend on the boundary position L chosen. Thus the agreement within a factor of 3 with Jokipii's numbers is really better than could be expected. Note that the dependence of these K r values on energy T is not very strong below -10 MeV, indicating that the built-in assumption that K r is independent of T is a reasonable approximation. The value L = 2.7 AU should by no means be taken as a location for an actual physical shock transition in the solar wind. The repeated observation of exponential decay profiles seems to indicate that some kind of free escape region exists, but a sharp boundary is not necessary. Specifically, while the success of the new solution at explain- ing both the rise and decay times indicates that K (r) r better estimate of the r-dependence of K r K r, o = constant is a inside 1 AU than is K (r) the results reported here yield no evidence that K r r is constant = 109 FITS TO DATA L = 2.7 AU • 7 JUNE 69 EVENT • 2 NOV 69 EVENT o 31 JAN 70 EVENT (.) Q) (/) ~ E 1021 (.) c Estimate of Jokipi i and Colemon 1020..._~~__.,~~.....__.___.___._.__._.._._~~~_.._~_._~.......__.__.__._..l.....I.~ I 10 100 PROTON KINETIC ENERGY T in MeV Figure V-11: The best-fit values of Kr are plotted vs. proton energy T for all three flare events. The values were determined using the new solution with L = 2.7 AU. The estimate of Jokipii and Coleman based on Mariner IV field measurements is included for comparison (see Section V-B-2 ). 110 beyond 1 AU. on whether Thus the physical significance of L = 2.7 AU depends K r = constant is a good estimate of actual conditions between 1 AU and the boundary. It is interesting to note that the best-fit values of K (1 AU) r obtained using Forman's solution are nearly identical to those obtained using K (r) =constant (compare Figures V-7 and V-9). r One would expect that the decay rate at late times would depend on the average value of K beyond 1 AU, since the majority of the particles r are stored in this region. However, Forman's solution yields identi- cal decay times with a K (r) that is everywhere beyond 1 AU greater r than the K r of the new solution. sensitive to the r-dependence of K The decay rate is thus not very r probably because the particle , transport at low energies is dominated by convection at late times. In contrast to the radial particle propagation described by the new solution, the azimuthal diffusion does not appear to be explained very well by the simple model used. Although the actual data have been fitted quite well (even for the 7 June event for which perpendicular diffusion is important), the best-fit values of K e are not con- sistent from flare to flare and disagree with the estimates based on Mariner IV observations. Worst of all, the K case larger than the corresponding K r , e value is in almost every which is in disagreement with the observations of McCracken et al. (l 5) as well as with the calculations of Jokipii( l). McCracken et al. (l 5) have presented the picture that the particles spread rapidly in longitude near the sun, and at the same time begin to 111 move outward along the field lines. Later in the event, when the near- sun diffusion region has been depleted, the particles assume an azimuthal distribution in interplanetary space which has a shape defined by the specifics of the near-sun propagation, but which is smoothed out very slowly because Ki is small in interplanetary space. This explanation, which seems to be a reasonable one, is completely different than the simple point-injection and 2-dimensional diffusion used here. It was found, however, that the solution for the directly- connected events was relatively insensitive to the value of K e , simply because the azimuthal part of the solution doesn't play a very important role in determining the profile. The large values of K e deter- mined for the events of 2 Nov and 31 Jan can thus be taken as indications that the azimuthal part of the solution can (and should) really be ignored complet~ly whenever the parent flare is close to the line-offorce passing through the earth. However, for the 7 June 1969 event, the longitudinal solution cannot be neglected. Nor can the rise profile for this event be explained away as the eifect of a rigid azimuthal distribution corotating by the earth: the gradient in longitude needed to explain the rise is much steeper than that observed by McCracken et al. (lS) In fact, the effect of corotation on the event profile observed at earth seems to be small, contrary to the claim of McCracken et al. that the decay rate observed should depend critically on whether the earth is corotating toward or away from the peak of the longitudinal distribution. Based on McCracken's observations of the longitudinal 112 gradient dn/d6 ~ n/40° (l 5 ), one finds that corotation will super- impose a ±5 hour effect on a typical decay constant of 'DEC= 20 hours. Also, in our admittedly small sample of flare data, no correlation of decay time with corotation direction is evident -- in fact the measured decay times are remarkably similar for east and west limb events (compare 7 June with 2 Nov and 31 Jan). It seems that the slow rise of the June event must be due to the finite evolution time of the near-sun longitudinal distribution. It is then not such a surprise that our simple Q(6,~,t) function produces a profile which, when combined with the radial dependence R(r,t), matches the actual data quite well: although our azimuthal solution was intended to describe a 2-dimensional diffusion process in interplanetary space, it can just as well describe a similar random walk within a narrow region near the sun. The values of Ke determined by fitting the 7 June event can thus be taken as an indication of the rate of nearsun propagation, rather than as a measure of K~ near the earth. Finally, the results of the fitting procedure are consistent with the assumption that adiabatic deceleration is the only energy-change process operative. However, this question can only be answered by comparing the predictions of the new model directly with experimental evidence for the energy-change effect, such as that reported by Murray et al. (Jl) section. This energy-change problem will be discussed in a later 11 3 F. Vector Particle Anisotropy Measurements of the so l ar flare proton flux in interplanetary space have shown that this flux i s not completely isotropic, and that t he anisotropy is most pronounced early in a flare event 0.5 26) ' . It is of i nterest to compare these observations with the predictions of t he new solution. 1. Definition We first define the following quantities: j(8, ¢)dS1 = particle flux directed from a ) ( given solid angle increment ds-2 j (8' <jl ) ds-2 w n ( 8, <P) ds-2 contribution to the density from\ ( a so l id angle increment ds-2 } w = particle velocity When the particle flux j(8,¢) is nearly isotropic, it can be written as j (8, <ji ) I (1 + i;:cos8) 0 (5-47) where i;; is the particle anisotropy. The omnidirectional flux is then simply (5-48) and the net flux through a surface is identical to the particle current S discussed in Section V-D-2: 114 Jan s = j(e, cp) case= Jan 1 0 (1 +~case) case (5-49) 47T .,,. = 3 I:" .Lo" The aver~ge density n, which is the quantity predicted by the solution, can be written as n Finally, the = J vect.o~ anisotropy 47Tl ' 0 = dn n(e,cp) -+ ~ can (5-50) w be defined in terms of the densi- ty n and the previously discussed particle current S: -+ ~ -+ = 3 -+ 1 (CV - - K w n = 3S nw • (5-51) 'Vn) The magnitude of the anisotropy can easily be related to the maximum and minimum values of j(e,cp) measured with a cosmic ray telescope: jmax - jmin i!i = jmax + jmin 2. (5-52) Observations No anisotropy measurements can be made from the OG0-6 space- craft, because its orbit lies within the magnetosphere. We must therefore rely on measurements such as the Pioneer observations reported by McCracken et a 1 • (15,26) . For 7 - 20 MeV protons at -1 AU, they observe a large 25-50% anisotropy early in a solar event which is directed roughly along the field (from -45°W). This strong initial anisotropy then changes in magnitude and direction so that at late times it has a strength of about 5-10% directed approximately 115 perpendicular to the field (from ..45°E). The details of their aniso- tropy observations vary greatly from event-to-event, but the general features just described are nearly always present. 3. Anisotropy Predicted by the New Solution At this point, the fo l lowing conclusions have been drawn con- cerning the new solution and its ability to predict the features of solar flare events: a) By taking the magnetic field as radial, and thus equating Kr = K and Ke = K~ = K , reasonable fits to the 1-20 MeV proton data have been achieved using L ~ 2.3 - 2.7 AU, C = 1.5 - 3.5, and K r b) 2 x 10 20 to 1 x 10 21 2 cm /sec (all reasonable values). The azimuthal diffusion is probably only significant in a small region near the sun, and as suggested by the observations of McCracken et al. (15) , Ke << Kr in interplanetary space. Except when the near-sun longitudinal diffusion process affects the event rise profile (as in the case of the 7 June 69 event), the azimuthal part of the solution is best ignored completely (Ke= K~ = O). Ng and Gleeson(l 3 ,l 4 ) have used another interpretation of the assumption that the diffusion tensor is diagonal in a frame aligned with the radial direction. Instead of neglecting the spiral nature of the field, they have completely neglected all perpendicular diffusion (Ki= 0) and azimuthal gradients and have taken K (r) = K r r o -1 2 K.. ( r)cos ij; where ij; = tan 0.r/V is the spiral angle. Thus their rather .. special form of K11 (r) produce K (r) = K r. r o = K0 r/cos 2ij; is simply the r-dependence needed to 116 The nature of this assumpt i on by Ng and Gleeson can be seen more clearly by writing the diffusion tensor in a reference frame aligned with the B-field, K II 0 0 0 0 (5-53) 0 0 K .1.. and then transforming it into a frame aligned with the radial directi on K C " ' Kij 2 + Kl. S 2 (K ii - K..l.)cs Kii s (K,, - Kl.) CS + K.1. C 0 0 where c = cos~ and s 2 = sin~. 2 0 (5-54) 0 K.L Even if one sets K~ = 0, the tensor is still non-diagonal due to diffusion in ~ caused by K • " Ng and Gleeson have neglected all the elements of the tensor except K = Kr rr = K11 cos 2 ,,,'I' and have thus used purely radial diffusion to describe diffusion along the spiral field. This interpretation, which is only an approximation of the actual situation, can easily be applied to the solution developed in this dissertation. The r-dependence for K11 becomes K11 (r) gradient Vn = cos~(an/ar). = K/cos 2 ~'and the Then we can write separate expressions for the diffusive and convective parts of the vector anisotropy: -+ E;;K 3 K = - -wR- cos~ aR - ~ ar B A 3 w (5-55) 117 = l c v; w (5-56) r where R(r,t) is the radial part of the solution, eB is the unit vector directed outward along the spiral field, and e vector. is the radial unit r The vectors er and eB meet at an angle the resultant ~ ~ 48 0 at 1 AU, and + s has a direction which depends on the relative magni- The calculated magnitudes of sK and sv as a function of timeare shown in Figure V-12 for several values of K • Note that r independent of time, while s K sv is decreases in magnitude and eventually reaches an equilibrium value which is negative for typical values of K • r The vector combination of these separate components is shown in Figure V-13, which depicts the evolution of t in one-day steps. A typical observation borrowed from McCracken et al. (l 5 ) is shown for comparison. The parameter values chosen are typical for -10 MeV protons, and the agreement with the observations of McCracken et al. is quite good: + s decreases in magnitude and changes direction from the west to the east. The strength of the anisotropy predicted at late times is -3-8% (depending on the values of V, K , r and C chosen), a value which is in good agreement with the McCracken observations. Figures V-5 and V-6 make it obvious why the anisotropy observed at 1 AU is directed from the east at late times. The typical equili- brium particle distribution has a peak situated beyond 1 AU. The diffusive anisotropy at 1 AU is thus directed inward along the field, L = 2.3 AU v = 400 km/sec 6 c - 2.0 >CL 4 o~ n::: 0 ..__ c o:.: 2 Cf)z- :::) <! <! I II :J 'LL LL 'u...P - 0 f.v r ~1 = 1.0 AU 10 MeV Tproton = \ \~ ""'--- - - r K r = 1021 cm 2/sec _L 0 wO > _. Cf) \_ \ -2 ' \',, ' .....__~,--..... 20 -~~Q __ _ ------ -------- --------~ -----= 1020 cm2/sec K -4 r -6 --} =---= ~K 0 100 200 300 400 TIME t in hours Fi gure V-12: The magnitudes of the diffusive a~d convectiye components of the anisotropy ;K and ;V predicted by the new solution are plotted vs. time for various values of Kr· Note that ;V is independent of time and of K • r t-' t-' o:i to sun " -er 30 MARCH "es B-field at IAU I 20°/o I ANISOTROPY SCALE DAY 12 DAY 12 Kr = 10 20 cm2/sec (a) Kr= 5 X 1020 (b) (c) Figure V-13: A vector diagram of the time-evolution of the vector anisotropy at 1-day intervals. a) and b) are the predictions of the new solution using 1 = 2.3 AU, V = 400 km/sec, C = 2.0, r = 1.0 AU, and T2 roton = 10 MeV. c) is a typical Pioneer 8 observation as reported by McCracken et a1.(15) for 7.5-21.5 MeV protons during 30 March to 9 April, 1969. f--" f--" l.O 1 20 and produces an eastward anisotropy when combined with the outward radial convective current. At early times, the large density grad- ient produces a strong outward diffusive anisotropy directed along the field. This is exactly the description used by McCracken et al. . t h e f eatures o f t h eir . anisotropy . . (lS). to exp 1 ain ob servations. G. The Energy-Change Effect 1. Statement of the Problem Recently, Murray et al. reported a direct observation of the 3 energy-change effect during the 7 June 1969 solar flare event( l). Their observation consisted of mapping the time-evolution of a "kink" or "knee" in the proton differential energy spectrum. They found that this kink moved to lower energies at a rate characterized by an exponential time constant 'KINK = 210 ~ 10 hours, a result which appears to be at variance with the energy-change rate at 1 AU predicted assuming only adiabatic deceleration. They interpreted this as an indication that an acceleration process may be competing with the adiabatic deeeleration and thus producing a slower energy-change rate. In a companion paper, Jokipii suggested that solar flare particles could be energized via second-order Fermi acceleration by hydromagnetic waves travelling in the interplanetary medium< 44 ). He presents an estimate of the strength of the effect, and concludes that this Fermi acceleration may well be fast enough to offset a sizable fraction of the adiabatic deceleration. 121 Subsequent discussions of t his problem have pointed out that the decay rate in energy of a spectral feature depends not only on whether or not a competing acceleration process is present, but also on the spatial distribution of particles with respect to the observer< 45 , 46 , 47 , 4 s, 6 1). Based on an earlier solution by FormaJ 62 ), which ignored diffusion and radial gradients, Murray et al . compared the adiabatic time constant a.:t. 1 AU given by •E(l AU) = 3r/4V = 78 hours (for V = 400 km/ sec), with the measured 'KINK;:; 210 hours. This approach is not generally correct when solar particles are dif fusing over a large region of interplanetary space. 'E;(r) « The time constant r characterizes the instantaneous energy-loss rate experienced by a particle at position r in a frame moving with the solar wind. Thus, especially during the decay phase, a typical solar flare particle undergoes a random walk in position and thereby samples many different rates of energy-loss. One might then expect the energy-change rate to be described by the 'E(r) at the position where the "average" particle spends its time, rather than by the 'E(l AU) at the position of the observer. This simple characterization will be investigated more rigorously. 2. Observations Thus far two "well-behaved" flare events have been observed aboard OG0-6 which have a clearly-defined kink in their decay-phase spectra at -3 MeV: Figure IV-2). the events of 7 June 1969 and 31 Jan 1970 (see The following procedure, similar to that employed by Murray et al. ( 3 l), has been used to investigate the energy vs. time 122 characteristics of these spectral features: a) Separate power laws were fitted above and below the "kink" to each differential density spectrum over a -3 day period. Cn 1 Both the slopes (Y and Y ), and the intensities 1 2 and n ) were used as free parameters. 2 b) The average power law slopes above and below the kink (respectively ~ 2) and y vs. time. 2 (r1J were determined fro_m plots of y1 For both events these slopes were found to have negligible time-dependence. c) 1 The spectra were t h en re-fitted using the average values 2 ( y ~and~y ~with only n 1 and n 2 as free parameters. d) A value of the intersection point EINT(t) of the power laws n T- Y1 1 and n T- Y2 2 was determined for each spectrum fitted. Figure V-14 shows a plot of EINT(t) vs. t for both events. The open circles represent points which were deleted from the 31 Jan 1970 decay because of an energy-correlated particle increase which occurred between -0700 UT on 3 Feb 1970 and -1400 UT on 4 Feb 1970. Because of this puzzling feature, which is evident in Figure IV-1-d, the evidence for an energy-change process is not as unambiguous for the 31 Jan event as for the 7 June event. The large error bars for the 31 Jan data reflect the goodness-of-fit achieved and are the result of poor counting statistics. In each case there is definite evidence for an energy-loss effect, and a least-squares exponential fit yields TKINK(7 June) 182 + 6 hours and TKINK(31 Jan) = 160 + 20 hours. = The disagreement with the TKINK(7 June) of Murray et al. is a systematic effect due to the 6.-~~~~~~~~~~~~----. 6..-~~~~~~~~~~~~~~ 7 JUNE 1969 EVENT 31 JAN 1970 EVENT 5c_ 4 ~\~~11 I i I ~l W - 5 - 4 > 3 (1) ~ .~ -..... I- ....z TKINK =182±6 2 w 2 hours 0 0 oo 0 t-' N 0 w 0 (Y1) =2.77 <Yi> =2.69 (Y2) =4.26 (Y2) = 3.32 I 10 I 11 I I 12 DATE - JUNE 1969 I 0 0 I 13 2 3 4 5 DATE-FEBRUARY 1970 Figure V-14: The time-dependence of the intersection energy EINT(t) is shown for a period during the decay phase of the 7 June 1969 and 31 Jan 1970 flare events. This intersection energy was defined by fitting separate power laws with fixed slopes <y 2> and <y 1 > above and below a kink at -3 MeV. The cpen circles represent points which were deleted beaause of an unexplained energy-dependent particle increase which occurred during the decay of the 31 Jan event. The dashed lines are least-squares exponential fits. 124 specific choice made for (r ~and 1 (y 2). and the difference of - 28 + 12 hours can be taken as an indication of this systematic uncertainty. The fact that the power law asymptotes y 1 and y 2 as well as the "kink" are stable with time is an important observation in itself: it indicates that the energy "smearing" caused by the particle random walk in position is not so great as to erase a spectral feature within the -3 day period needed to define the energy decay rate. 3. Energy-Change Predicted by the New Solution Although considerable discussion has been waged concerning the energy-change process for solar f lare particles, no prediction has yet been made using any model which c an be compared directly to the available energy-decay observations of a convex-shaped spectral feature. Palmer( 46 ) and Urch and GleesonC 47 ) have carried out calculations concerning the way in which an energy "spike" propagates. Gleeson and Ng( 6 l), on the other hand, have applied their recent flare propagation solution(l 3 ) directly to the observations of Murray et al. They make the simple geometrical calculation that a eoneave power law kink (steeper at lower energies) defined by (5-57) will move to lower energies at a rate defined by '1'2(Y2 - yl) (Tl - T2) where 'l is the decay constant associated with slope y , etc. 1 (5-58) 125 Unfortunately, energy spikes and concave spectral kinks are rarely observed, and it is not immediately obvious how the evolution of such features can be related to the existing observations of a convex kink. For this reason, an attempt has been undertaken to predict the evolution of a convex kink using the new solution. The method is straight- forward (see equation 5-38) but difficult to implement. At late times, the new solution predicts that a pure power law spectrum defined by slope y will decay exponentially at TDEC' Any initial spectrum S(T) that can be written as a superposition of power laws z S(T) = l A T -Y·l i (5-59) can then be propagated in time by n(T,t) = (5-60) where the one-to-one correspondence between each y. and T. involves l l the calculation of the first Coulomb wave function eigenvalue and depends on TE, L, V, and Kr. With some difficulty, a set of Ai's and y, 'shave been found which, when superimposed, produce a convex l density spectrum S(T) with power law asymptotes of y above and below 3 MeV. 1 2 and r 2 =4 These coefficients as well as a representative set of T, values are listed in Table V-2. l Figure V-15 shows the S(T) spectrum thus generated and also the way it evolves over a 200 hour period using typical values of the parameters. Several results are evident: a) The S(T) generated is a good approximation to the observed 126 TABLE V-2 Power Law Superposition of a Density Spectrum with a Convex Kink assuming L = 2.3 AU V = 400 km/sec K = 3 x 1020 cm2/sec r 'E (1 AU) = 78 hours i Yi A.l. c.l. 2y.+l l. 'i 3 hrs -3 cm 1 1. 75 3.8133 1.50 31.88 2 2.00 -29.128 1.67 29.43 3 2.25 0.3227 1.83 27.38 4 2.50 561.42 2.00 25.62 5 2.75 -1767.9 2.17 24.10 6 3.00 1495.4 2.33 22. 77 7 3.25 909.92 2.50 21.59 8 3.50 -954. 86 2.67 20.54 9 3.75 2.83 19.60 10 4.00 -872. 78 3.00 18.75 11 4.25 711. 74 3.17 17.98 12 4.50 177. 88 3.33 17.28 13 4.75 -115.51 3.50 16.63 14 5.00 -2.9039 3.67 16.03 15 5.25 -58.594 3.83 15.48 16 5.50 26.527 4.00 14.97 -0.4838 127 Figure V-15 An initial spectrum with a convex kink at -3 MeV and power law asymptotes of r S(T) 1 = 2 and r 2 = 4 is shown. This spectrum = ~Air-Yi was generated by superimposing 16 separate power laws. The time-evolution of this input spectrum was calculated by assigning (via the new solution) a separate decay time constant Ti to each power law element of S(T). For the parameter values chosen, the spectral shape is reasonably stable over an -8 day period, and the "kink" moves exponentially to lower energies at a rate characterized by TKINK ~ 106 hours. The power law slope y(T) of the spectra are shown to emphasize the kink movement and to illustrate the slight rounding effect which occurs. 128 4 10 L=2.3AU V = 400 km/sec 2 20 Kr= 3 x 10 cm /sec 10 3 10 rE (IAU)=78 hours 2 Asymptotes: ~ =2 ~ = 4 I ' 2 >- ICf) 10 z w 0 _J <! I- z w w 10- 1 - 10-2 0::: LL LL 0 z 0 I0 10- 3 0::: o__ 10- 4 10- 5 u ...--.. 4 ~ I- I~ l- o::: w 3 <! o__ <..9 0 0 _J _J Cf) 2 10- 2 10- 1 I 10 PROTON KINETIC ENERGY T (MeV) 100 129 spectra for 7 June 69 and 31 Jan 70. b) The spectrum decays both in density and energy but retains retains its shape over a long (-8 day) period. c) There is some "rounding" of the kink -- a result expected due to the particle random walk. d) The kink decays at TKINK = 106 + 4 hours for this choice of parameters -- a value still in disagreement with the observations. The assumption that K is independent of energy T (built into the r new solution) must be discussed carefully when making predictions concerning the energy-change process. A strong dependence of Kr on T would affect the time-evolution of the spectrum and make the model inapplicable. The K energy dependence is in fact relatively weak: r the best-fit values of K for a given solar flare vary by only about a r factor of 2 in the interval from 1 to 10 MeV (see Figure V-11). As will be shown in Figure V-17, the dependence of TKINK on Kr is also weak, another indication that an energy-independent Kr is a reasonable approximation. The application of the formula of Gleeson and Ng using the asymptotes y 1 = 2 and y 2 = 4 yields a similar result: = (29.43h)(l8.75h)(4 - 2) = 103 hours (29.43h - 18.75h) Thus, even though this convex spectrum cannot be written as the superposition of two power laws (as in equation 5-57), the rate of kink 130 movement can be predicted by considering the independent evolution of the power law asymptotes using equation 5-58. A glance at Figure V-5 shows that for V = 400 km/sec and Kr ~ 3 x lo 20 cm2 /sec, the particle distribution is peaked at r ~ 1.4 AU and thus the "average" particle should lose energy at TE (1. 4 AU) "' 109 hours, in agreement with the more exact analysis just completed. Further investigation has shown that the value of TKINK is relatively insensitive to the choice of the asymptotes y 1 and y . 2 The complete power law superposition technique was used to generate a different convex spectrum with asymptotes of 2.6 and 4.2, a more accurate approximation to the 7 June 1969 spectrum. This kink was found to propagate at the same rate as the kink defined by r 1 = 2 and Y2 = 4. In addition, the simple formula (equation 5-58) was used to investigate the dependence of TKINK on r 1 and Y = r + ~y in more detail. 1 2 Figure V-16 shows that TKINK varies by only -10% over the entire range of reasonable values for the power law slope. One would then conclude that any spectrum, as long as it contains no very sharp features, will decay in energy at rate close to (TKINK) -1 independent of shape. The dependence of TKINK on V, L, Kr and TE is explored in Figures V-17 and V-18 using both the simple formula and the power law superposition. The agreement between these two methods of calculating TKINK is almost exact. Increasing the distance to the boundary L decreases the energy-loss rate because the particles on the average spend their time further from the sun and thus experience a larger TE(r). Increas- ing the solar wind velocity also increases the distance to the peak in 110 (/) ...... 6 105 ..c c ~ z ~100 ..... ~ w ~ L = 2.3 AU V =400 km/sec 20 Kr= 3x10 cm 2/sec 95 'E (IAU) =78 hrs 2 3 4 '1 Figure V-16: The dependence of TKINK as calculated by the simple formula of Gleeson and Ng is shown as a function of the power law asymptotes y1 and y 2 = y 1 + ~y. The conclu- sion is that the value of TKINK is relatively insensitive to the spectral shape used. 132 L = 2.7 AU = 78 hours TE (IAU) 4 00 100 . - -__ 600 (f) \.... :J 0 _c c O L-~~~_.l_~~-L-~L-_L_--'---'-----'---'-L:-~~~_,_~~-'--~~~~~~~ 10 20 I 0 21 !022 300 ~~~~~~~~~~~~~--,~~~~~~~~~~~~~---, V = 400 km / sec TE (I AU ) = 78 hours IOO r----_ • L = 2.7 power low superpos ition method 2.3 2.0 OL-~~~_j_~~.J..___1L.___J_----'------'--_L_-'---''---::-:-~~~-'-~~L:-----1~-'--'--'--'-.J..._J 1020 Kr Figure V-17: I 02 1 2 I 022 in cm /sec The dependence of 'KINK on V, Land Kr is shown using both the simple formula and the complete power law superposition method. 133 V =400 km/sec Kr= 10 21 cm 2/sec 300 observations 200 100 'E for adiabatic deceleration . . . . ~~_._~~-'-~----J o~~__.~~---~~-"."'"'.~~~~~ 0 20 40 60 80 100 120 140 160 TE (I AU) in hours Figure V-18: 'KINK is plotted vs. 'E(l AU) for various values of the boundary position L. The cross-hatched areas indicate the T (1 AU) for pure adiabatic deceleration and the 'KINK observations E for the 7 June 69 and 31 Jan 70 flare events. The uncertainty in 'E reflects the observed fluctuations in solar wind velocity. 134 the particle distribution, but this effect is overshadowed by the direct dependence of TE(r) on V -- an increase in V thus also increases the energy-loss rate because the solar wind expansion is everywhere occurring more rapidly. The dependence on K r is surprisingly weak, and is no doubt due to the variation of the particle distribution with the ratio V/K (see Figure V-5). Figure V-18 shows the linear dependence of TKINK on TE(l AU) for various boundary positions L. The observed TKINK values for 7 June 69 and 31 Jan 70 are included for reference. This figure makes it clear that in order to duplicate the measured TKINK of 160 - 190 hours one must either a) move the boundary out to L = 4-5 AU while retaining pure adiabatic deceleration TE(l AU) ~ 78 hours, or b) keep the boundary at the usual 2.3-2.7 AU but assume a weaker energy-change process characterized by TE(l AU) ~ 110 - 140 hours. The observations could also be matched with the new solution by using V ~ 250 km/sec, but this is in direct conflict with the -400 km/sec value measured in interplanetary space at the time. In addition to the kink movement, it is evident from Figure V-15 that the kink in the power law superposition spectrum becomes slightly more rounded as it evolves. A quantitative expression of this "kink rounding" is given in Figure V-19, which plots the curvature dy(T)/dlnT evaluated at y = 3.0 as a function of time. The smearing effect, which is an expected result of the particle diffusion process, is unambiguous but small: the curvature changes by less than 10% over a 200 hour interval for this particular spectrum. Presumably a sharper 135 L=2.3AU V = 400 km/sec .58 Kr= 3 x 10 0• f'() II >--.. rE .57 20 cm 2/sec (IAU) = 78 hrs ~ 0 I- .56 >-.... c '"O ~ -0 w .55 0:: :J tr> .54 0:: :J u .53 0 50 100 150 200 Tl ME in hours 250 Figure V-19: The quantity dy(T)/dlnT evaluated at y = 3.0 via the power law superposition method is plotted vs. time. This quantity is a measure of the curvature or sharpness of the kink in the spectrum, and it is apparent that a finite but small (-10%) smearing effect occurs over the 200 hour time interval shown, 136 spectral feature would be initially smeared more rapidly. Unfortunate- ly, this rounding is an order of magnitude too small to be measured by the OG0-6 experiment during a typical 3-4 day observation period. The fact that the analytic solution predicts a small finite rounding for this particular spectrum suggests that experimental verification could be made by comparing more accurately determined solar flare proton spectra with the evolution of better power law approximations. 4. Conclusions Concerning the Energy-Change Process One would like to be able to decide either for or against the existence of a competing acceleration process, but the investigation thus far shows that the lower rate of kink movement observed can be produced equally well by a ~arger boundary distance or by an energy- change process roughly half as strong as the adiabatic process. A final attempt was made to settle the issue by re-fitting the event intensity vs. time profiles using parameters which produce a 'KINK of 160 - 190 hours. Figures V-20 and V-21 compare fits to the 7 June and 31 Jan events at 1.17 - 1.27 MeV using a) a larger boundary and b) a weaker energychange process. For the directly-connected 31 Jan 70 event only the radial part of the solution was used (per the discussion in Section V-E-4). Unfortunately the fits are equally good for the two cases, and no conclusion can be drawn. It is clear that moving the boundary out to L = 4-5 AU merely changes the peak in the particle distribution out to r ~ 2.3 AU, so that 137 7 JUNE 1969 EVENT 1.17-1.27 MeV NEW SOLUTION Kr(r) = K V = 400 km/sec a) L = 5.0 AU TE ( IAU) = 78 hrs c = 2.0 Q) 0 u (/) >- Kr= 4.5 x ~ 0 -~ K ...0 8 21 2 10 cm /sec (IAU) = 5 x 10 20 ~ 0 ...__ x :J _J LL z 0 f0 CL • .'t' • ,. b) L = 2.7 AU TE (IAU) = 140 hrs c = 1.56 •• 1.0 X 10 21 20 K 8 (IAU) -- 5 x 10 Q_ Kr= (_'.) 0 _J •te• ,. • 7 8 9 10 11 12 13 14 15 DATE - JUNE 1969 Figure V-20: Fits to the 7 June 1969 event at 1.17 - 1.27 MeV using the new solution. The observed 'KINK. z 182 hours was matched by two different methods: by a) moving tne-ooundary out to L = 5.0 AU and b) decreasing the strength of the energy change process to TE(l AU) = 140 hours. 138 31 JAN 1970 EVENT 1.17 - 1.27 MeV -... <l> 0 - u (j) NEW SOLUTION Kr (r) = K V = 400 km/sec ." • >- 1.- 0 • 1.- +- a) L = 4.5 AU TE (IAU) = 78 hrs c = 2.03 20 2 Kr = 5 x 10 cm /sec ..0 1.- 0 x •• . ::::::::> __J LL z r0 0 b) L = 2.7 AU TE (IAU) = 110 hrs c = 1.73 • 0::: Q_ (.'.) 0 Kr= __J 31 2 3 4 5 6 X 10 20 6 7 8 DATE - JAN 8 FEB 1970 Figure V-21: Fits to the 31 Jan 1970 event at 1.17-1.27 MeV using the radial part of the new solution. The observed TKINK ~ 160 hours was matched by two different methods: a) moving the boundary out to L = 4.5 AU and b) by decreasing the strength of the energy-change process to TE(l AU) = 110 hours. 139 TE(2.3 AU) matches the observed 'KINK ~ 180 hours. Since any means of peaking the distribution at 2.3 AU will produce essentially the same energy-change effect, a third method for reproducing the observed energy-change rate can be evoked by employing a K (r) which depends on r distance r (beyond 1 AU) in such a way that the particles are peaked much closer to the boundary. Although this approach is not within the assumptions of the new solution, it can be understood qualitatively in terms of the discussion present earlier in Section V-D-2. It can easily be seen that a K (r) which decreases beyond 1 AU will cause the r particle distribution to be peaked at a larger radial distance. This effect can be explained by noticing that a smaller K increases the r ratio V/K on the average and thus pushes the peak outward (see Figure V-5). It is interesting that this third method requires a form for K (r) which has the opposite radial dependence to the frequentlyr used K (r) which increases with distance. r It is apparent that this energy-change question cannot easily be answered with observations at 1 AU. One would like either to follow the energy vs. time history of a single particle or alternatively to at least determine the spatial distribution of particles so that the energy loss rate •E(r)-l of the "average" particle is known. Gleeson and Palmer( 4 S) have suggested (but not carried out) a rather difficult experiment which would afford a direct measurement of the energy-loss rate: they propose the simultaneous observation from two spacecraft situated on the same heliocentric radial line of a "pulse" of nearly 140 rnonoenergetic particles such as might be convected outward during a flare event. 141 VI. CONCLUSION The new solution presented here consists of resolving the differential equation for the radial part of the particle propagation, using = constant, and including the effects of convection and energy- ~(r) change known to be important at low energies. This new radial solution, when combined with the azimuthal solution used by Burlaga (lO) and Forman (lZ), is capable of accurately reproducing both the rise and decay of the "classical" prompt solar flare proton event observed at 1 AU using reasonable values of the parameters. The principal limitations of the new solution are that it assumes a diffusion tensor which is independent of energy and diagonal in a reference frame aligned with the radial direction (rather than with the B-field). Within these limitations, the comparison of the predic- tions of the new solution with actual data leads to the following conclusions concerning solar flare particle propagation in the interplanetary medium: a) The Diffusion Tensor -- The success of the new solution in fitting both the rise and decay of flare events indicates that K r = constant (r) is a better approximation to the actual condi- tions inside the 1 AU than is K r (r) ~ r. It is not necessary to invoke a scatter-free region near the sun in order to reproduce the fast rise time observed for directly-connected flare events. The K r values derived from fits to actual data have a weak energy 142 dependence and agree reasonably well with the estimates for K based on Mariner IV magnetic field measurements. b) " The Azimuthal Propagation -- The simple point-injection and 2-dimensional longitudinal diffusion adopted here does not seem to be an accurate description of the actual processes at work. Although it appears that the azimuthal solution can be neglected for directly-connected events, some kind of longitudinal propagation must be invoked to explain the slow rise times of east limb events. Even though this propagation may occur near the sun, the 2-dimensional interplanetary diffusion adopted in this discussion is capable of explaining at least qualitatively the slow rise profiles of these east limb flares. c) Free-Escape Boundary -- The success of the new solution in fittjng the decay phases of flare events indicates that i:he assumption of a free-escape boundary at some 2 - 5 AU is at least a reasonable approximation to the actual interplanetary conditions. For example a sharp boundary may not exist, but there might instead be a finite region of rapidly increasing diffusion coefficient. d) Anisotropy -- Despite the limitations of the solution with respect to the actual spiral nature of the interplanetary field, it is possible to understand essentially all of the features of the observed vector anisotropy in terms of the solution. In fact, even the absolute value of the residual anisotropy predicted agrees well with the 5 - 10% observed. 143 e) The Energy-Change Process -- By superimposing power law spectra, it was possible to reproduce a spectrum with a convex kink and to verify, via the new solution, that such a feature will move to lower energies as has been observed. The existing energy- change observations can be explained in the light of the new solution in at least three different ways: 1) by means of a boundary at -2.7 AU and an energy-change process weaker than pure adiabatic deceleration, 2) by pure adiabatic deceleration and a boundary at 4-5 AU, or 3) by a somewhat different solution with a boundary at -2.7 AU and a f orm for K (r) which decreases beyond r 1 AU. The resolution of this question must await more sophisti- cated spacecraft measurements. 144 Appendix A OG0-6 Monthly Summary Plots fer June 1969 through February 1970 Figure A-1 consists of computer-generated plots which summarize the OG0-6 range telescope polar counting rates (see Section II) as well as certain geophysical data from the ESSA Bulletins~ 7 28 nine months of June 1969 through February 1970 are covered. , ). The Full credit Evans~ 3 ), for developing the for these summary plots must go to L. C. necessary computer coding, and to T. L. Garrard(6 4 ) for generating the OG0-6 plots shown here. For each month the following information, starting at the top, is plotted vs. time: 1) The average polar DlD8 counting rate in cts/sec (labelled Dl) is plotted logarithmically. This rate is nearly insensitive to electrons but responds to nuclei from -1.2 to -20 MeV/nucleon. 2) The average polar D2D8 counting rate in cts/sec (labelled D2) is plotted logarithmically. ~ 3) This rate responds to electrons 200 keV and to nuclei ~ - 3 MeV/nucleon. The average polar DlD2D8 counting rate in cts/sec (labelled DlD2) is plotted logarithmically. This rate responds to nuclei from - 3 to -20 MeV/nucleon. 4) The average polar D2D3D8 counting rate in cts/sec (labelled D2D3) is plotted logarithmically. ~ This rate responds to electrons 1 MeV and nuclei ~ 19 MeV/nucleon. 145 5) The >10 MeV solar proton fluxes measured by the Solar Proton Monitoring Experiment aboard Explorer 41. This cosmic ray tele- scope which is described briefly in the ESSA descriptive text(ZS), also has some electron sensitivity. The large rate excursions repeated at -4.3 day intervals are due to the periodic passage of the satellite through the earth's radiation belts. These excur- sions have been largely suppressed by the plotting program. 6) Normalized hourly average counting rates for 2 neutron moni- tors: 7) Alert (upper line) and Deep River. The standardized K-index of geomagnetic activity from twelve observations are averaged to obtain K • p The quasi-logarithmic K p scale ranges from O(quiet) to 9 (very disturbed). The legend for the plots is identical to that adopted by ESSA. 8) Geomagnetic storm sudden commencements (labelled SC) are indicated by solid triangles if confirmed and by open triangles if unconfirmed. 9) Magnetogram sudden impulses (labelled SI) are indicated by solid diamonds if confirmed and open diamonds if unconfirmed. 10) Optical solar flares (labelled SOLAR FLARE) of importance greater than 2F observed by the world-wide system of solar observatories are indicated by a small vertical line plotted at the beginning time of the flare. included. The importance (2N, 3B, etc.) is Periods of no flare patrol are indicated by horizontal lines of appropriate length. 11) 0 2 - 12A solar x-ray flares (labelled X RAY) with a peak flux 146 at least 4 times the ambient value are indicated by a verticAl line. These data were collected by experiments aboard Explorers 33 and 35. 12) Occurrences of type-IV radio emission (labelled TP IV RADIO) are indicated by vertical lines. This radiation is nonnally asso- ciated with the acceleration of solar flare electrons. The gaps in the OG0-6 data after November 1969 are in most cases due to incomplete processing rather than actual data coverage problems. 147 Figure A-1 OG0-6 Data Stm1I11.ary Plots A description of these plots is given in the text of Appendix A: Graph a: June 1969 Graph b: July 1969 Graph c: August 1969 Graph d: September 1969 Graph e: October 1969 Graph f: November 1969 Graph g: December 1969 Graph h: January 1970 Graph i: February 1970 148 ·•. ' .. ; . .. I~~: ;;_. ·.. I ..,. "•:•.:: ;..: .. ·.. f I ~ .l i.------------~-·~_ _ _ _"_,_...._"_~_........ __:_:-~·:~~~·~'~,"""-.:_.._--"'~M~··~·~~·-'~'~··~'~';'~~·~·-·~~~·~··~•...,..;..,_,,~,.~·~··~·..._'-"'--'--"'"-"'~....._--'t r::: i'o I' ---- ----·· -- .I I 10 r ~ 110' f--r--ir-.-----,--,--,-'"-,r'l,."""'~=~~;,;;;i="""'q'"'"'"":r::::;::i=::;i:::::i=;:;:;;=~"='l--=f=...............-"T"""°l"=-=-j 100 ~ i~r--~·----------------------------------------~90 rs • SC SJ 2 N ,,...,,, 11 2 "'"' F X AR T ...,,. TP IV 2 2 B N 11 111 I 31 2 2 2 lfJ NB B II 1111 Ill 11 I I .. ·2 B 2 N 11 1-c-,~2~~3~~~-.~~~~~ 9~~10~~11~1~2~1=3~14c--r~1so-r-1=sCT"C1~1-rc1=s~19~=20~2~1~2=2-r-:2=3~24C"T"2=sC"T"C2=s~2=1-r-:2a~= 29~3=0~ JUNE 19G9 Figure A-1-a 149 ,-1o'k-c ~ 11o'r--E 1 ~10~- \\ 110~ fs -2 Io "' SI • 0 I "'-"' "~ 2 N IPIY RRJIO I 2 3 JULY 1969 5 6 7 8 9 10 II I2 I3 I4 IS I6 17 18 Figure A-1-b 19 20 2I 22 23 24 25 25 27 28 29 30'3I 150 .. ... .. .._.... .•.·,.·......:. ...... . ......;.-. '·· . . ... . .. ·.· l1 ··.. · ,.~_- . ..... . ; . ... . ·:. \ ... , _ ·.;. . 1 .. · ,..._. ·'.. . r:::. . . _ _ _ _ - - , j i - 100--~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~--d . rs >< l0 SC 0 " 0 0 "'-"' """" XMfl 111 I 11 ...,,. T,.l't 1 RUG 2 3 1969 5 6 7 8 9 10 11 12 13 lij 15 16 17 18 Figure A-1-c 19 20 21 22 23 ~ 25 26 27 28 29 ~ 31 151 1-:: _ _ _ _ _ _ _ _ " ----.J -100-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~--~~~..,._~~~~ . ~......,~ .. ,, '\. ..... ' • \ • I ~F'' °V".~M .#;N•d*f",,,....,.;... ~~~, ~\,~~~~.,,.p ·~~"*'' I-·~· " 0 51 A 00 I I ·c 11 F I 2 8 8 I 2 2! N .. "'o• l!. @l o 0 3 ....... '""" ....,,. l!. 4t -: Ill 3 N I 3 8 I I 11 1 I I 111 Tl" IV f-,-1~2~~3~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ SEPT 1969 s 6 7 8 IO II 12 13 14 IS 16 17 Figure A-1-d 18 19 20 21 22 23 24 25 26 27 28 29 30 152 (i . ' f ~ ."'_ ~ - -,----.----:- ~~---~~---,.,.~.:.~ -,.-: ~~-~~-~,-~~-~--~-~--~-~--~-~--.---..-----.-:-r ~.-.. ···, ;_ ,,.~.·'L. . --... -: ........·=VJ; ·, ..•:.. """'r-e ·:<....,..<:.i> ...':·>.,_·.~r, ... -·: .v_"';: ,:_...N:-... Po .., ..... ~- .,.. .... ..~ .:· -.. g-·.._.:.: -_. . . - ~ . - - - - - - - - - - - - -- - -- - - - - - - - - - - - - - - -- - - - · - -· ' ~ 't: I.. I' E c I " ·: 10 ~- -- 110 90 rs ~ Io &o t; SC t; 0 • 0 " I I 2 2 N N I 2 "'"" N fLi:!fli , HM ! I 11 11 II 11111 :;.,:~: I I r 2- '"1 - '""ii"TS"'G '?"" 8 1 ~(! I 1 9 II II I II I II 11 I I I I 10 I)) I 12 1 13 1 14115I16 1 17 1 18'19T20 I 21 I 22 1 23 r ·24125 '26 '27 1- 28 1- ? 9 f .111 1 I 9G9 Figure A-1-e 153 ..,......_...... rI , .,.,,,... ,•#> r .. ,, ..... ~. .?\... .. · " ·"'"!#,. ~- ~ g 1 o r-~~~~~~~~~~~-Y, .-:-:·'--~··"--,---~,---~~~~~~c'-..~~~~~,.....::c_.:___.::c__•~~:.,,,..,~_:_~~~_:___] l 1 F-'""=-:-.-.----,,---,---,~-.--r~'TJ_.·_:,~_'_·..,~_~-;-~-;;_;r~A-~-'~r._•~,r·~:-->,_"_;,~·~;-~T:·_..,_'·T·;:__:_· ,-. ,/-~~·,·~·,~-~~~;..,::__~,---,··~·~~·~·:_~-~~·~-·~··::..J' .-,,....~~:· . •:\·.... ': ..... . . .. - ..... .., , ~, ..... ......... . : :, ~ \ 1-:: ~ 10 •• SC " ....... A 0 0 00 2 I I 2 2 2 88 8 "'"" ...,,. I Tl" IY 3 I 2 NOV 19 6~ I 5 8 II I 22 I 2 NB I I I c 2 8 8 2 2 NN N 11 11 I 111111111 II 111 I I 111 11 II 11111111111 '"'" I I II 6 7 8 9 10 11 12 13 14 15 '16 17 Figure A-1-f 18 II I 19 20 21 22 23 24 25 26 27 I 28 29 30 154 r--,--.--.--,-.-.--.--.-.-.-.--.-.-.-.--,---,,..._,__,---,-----.-----.-----.-----.-~~~~~~ " F ·.• ~ .,_ :. ; '\, r I r- ,---,---,----,----,-.,.--.,.--,-r--,--.---.----,------;--.--r---r'"~·_:,--,-.""-,.--...-..~ ··.---7--.--r--.-----.-r-->---J o .1 i-------- - - - - - - - - - - - - - - -;_ _ _ __ __ _ _ ___ __ _.:..;,___j ,...--:: -. .....,. ,,... I"\ C.1 · u r --- :4·- · ~ -· t .. . • "' ·.·. . ..._ ~ .. . . ':", l I :,. .... . .. 1. ·.':"..> .... ~.I 0---------------------------'.~.'-------·' ~·8'''°""'"-"'_,,,___ __ : .-.v _ __ _:...:_.~_J ~ ~100 l90 rs ~ l0 SC " 00 !J. 0 0 I - 2 N "''""""" 111 I I 11 I I 2- -2 . B I I I 111 0 N 111 ....,,. I,. IV 1· T2 "3 -·" "ij I s DEC 1%9 1 6 '7 'B 'g I 10 ' 11 1 12' 13' 14l 1s' 1G ' 11' 1B' 19l20 1 21 1 22 '23' 24l2s ' 2G'21'28'29T]or31 Figure A-1-g 155 ,. •'. fI a .I ~10 l I .. .._. .. .·. . . . . :·."'..._:~ ~- I l Ei ·I ~ ~. •. .· . ': . ' .·,··~ ..·-.. : -:~·. ·:· .. '"".......... . ........\' ,. ':.,·.,·- .. °':"'.' I l~ . I ,. .·•· ... r: ~ 10 ·' :"''-\ r5 ~ ... l0 • SC " " •" ""-"' 2 N """ I I """' 2 8 I 2 B I II . 2 22 N 8N I Il l I I 11 II I ,,..,,. Tr'IY 3 I 2 JAN 1970 5 6 8 10 11 12 13 14 15 16 17 Figure A-1-h 18 19 20 21 22 23 24 25 26 27 28 29 30 31 156 .•••• :.... '·' , -·· ,.: ....... ·: /°\ · . ·~ . .. ~· p. ·~- ,:------.-- . '"''. :· ~ . ··:... -- ---- - -----:----~----------------------- ~~--------------------------------~ 0 • ll .. ~-1 ' ....... ~ . --:.... '"· ....:~ .. . ......._... . . ~. ~ r: ~-----------~-~-----.l I ~ -IQ ~-':-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~- '.... '"'.......... .. ... . ... ._..,......,,,.;;.-. ......... . . ~~-, ·.~ _.,,....:,..~ ~--~~~(;~~ ~ ·~~ .lf--~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~---l f-~~~~~~~~~~~~~~~~~--~~~~~~~~~~~~~~~~~~~~ 110 ~ ~100 l 90 rs ~ l0 2 B II 2 ........ • A SC " I 2 B B 5'>."' 'L""' II r ''" 2 B I 2 B I -2 N 111111 1 11 II I '""" I 2 3 FEB 1970 4 5 5 09123171 B 9 10 11 12 13 14 15 15 Figure A-1-i 17 IB 19 20 I I I 21 11 II I Ill fl• IV 22 23 24 25 25 27 26 157 Appendix B Solution for the Azimuthal Dependence Assuming Ke(r) = K 1 r 2 , the equation for Q(e,4>,t) in spherical polar coordinates centered at the sun is (B-1) The separation of variables can be carried further by writing Q(e,4>,t) (B-2) Then only a differential equation for 0 remains 1 sine a . ae as sine as - with the well-known solutions 2 m 0 . 2e sin -!l(!l + 1)0 P~ (case), Q~ (case). (B-3) The solution has the general form (B-4) Assuming a-function injection at Ceo, 4>o), we must require = (B-5) The coefficients A!lm and B!lm can easily be determined by using the 65 appropriate orthogonality integrals ( ), and the specific solution 158 becomes: R, CO · Q(8,¢,t) I I = £=0 m=-£ (B-6) e where µ = cos 8 and µ 0 = cos8 0 • ±im(¢-cf> 0 ) e -£(£+l)K 1 t The addition theorem for spherical . ( 66) can now .be app l'ie d h armonics Q, PQ,(cosy) 47T -2-£-+-1 ~ m~£ (B-7) y is the angle between (8,cf>) and (8 0 ,cf> 0 ) given by cosy cos8 cos8 0 + sin8 sin8 0 cos (cf> - cf> ) o (B-8) Our solution for 2-dimensional diffusion thus reduces to the 1-dimensional solution co Q(y,t) k 2£ + 1 47T P Q, ( cosy ) e -£(£+l)K t 1 (B-9) Evidently the 2-dimensional random walk in 8 and cf> is completely equivalent to a 1-dimensional diffusion problem because the gradient -+ VQ is directed radially outward from (8 , cf> ). 0 0 This will be the case as long as the injection profile f(8,cf>) is symmetric about the point 159 Appendix C Derivation of the New Solution to the Radial Equation 1. Derivation of the Solution A solution R(r,t) will now be derived to the equation 2 a R + (:!:.. _ E ar2 r K 3r Y) _ 2CV R = ..!. 1.R Kr (C-1) K 3t using the assumptions and boundary conditions listed in Section V-C-1. It can easily be shown that if we write R(r,t) (C-2) then the function y(r) is the solution to iL2 + (a-~)y = 0 (C-3) dr v2 2 4K where ~ 1 KT -- > 0 (C-4) and 8 y_ (2C - 1) > 0 (C-5) K A simple change of variable, p = lar reduces equation C-3 to a form of the Coulomb wave equation = 0 (C-6) with solutions which are 1he regular and irregular Coulomb wave functions, F (8/2/a, lar) and 0 6 G (8/2/a, lar). ( 0) 0 160 The general solution to equation C-1 can then be written R(r, t) = e Vr/2K r (C-7) F (f3/2/;. ,/;. r) + B G (S/2/;. ,/;. r)l e-t/Tn o n n n o n n j O, then B = 0 and only If we require that R(r,t) remain finite as r ~ the regular Coulomb functions are involved. The eigenvalues a n = 0. defined by the outer boundary condition R(L,t) equation for a n n are The eigenvalue is thus F o cs12ra, ran L) = o n (C-8) which must be solved by an iterative technique. The coefficients A are determined by the requirement of impulsive n o-function injection at r = rs: o(r - r ) s R(r, t=O) r (C-9) 2 We can thus write o(r - r ) s R(r,O) r 2 where y (r) = F (f3/2/;., /;. r). n o n n f 0 L e Vr/2K r 00 I n=l A n y (r) n (C-10) If we left-multiply by V yrn(r) r exp(- 2K r)dr, we have o (r - r ) (- _y_ r\ 2K j d r y (r) ~~~-"-S- e m r ! n=l y (r) y (r) dr n m (C-11) 161 In Section 2 of this Appendix, the following orthogonality relationship for the regular Coulomb wave functions will be derived: y (x) y (x) dx n m where 8 nm = Nm 8nm (C-12) is the K.nonecker delta and y (O) = yn (L) n 0. Using this relation, we can write - -v r 2K S y (r ) m s e r 2An Nm 8nm = Nm Am n = s (C-13) Thus v - -2-K rs ..::e_ _ _ F A N r n n o s (pj2/a , la r ) n n s (C-14) and the radial solution for the specific boundary conditions assumed is exp [vcr - rs)/2KJ R(r, t) a: --=------r r s fr F (f>/2/a ,/a r ) F (B/2/a ,/a l_. o n n s o n n with B = V(2C-l)/K T n = 2 4K/ (4K ct n n Jo ~ n (C-15) (C-16) + V2 ) L N -t/T r) e (C-17) 2 2 0 CS/ /;n, ran x~ dx (C-18) 162 2. The Orthogonality Relation for Regular Coulomb Wave Functions J L We wish to demonstrate that 0 y (x) y (x)dx = 0 for n ~ m n m given that y (x) is the solution to n a - ~) y x n ( n with boundary conditions y (L) n (C-19) 0 = y n (x) = 0. We merely write down separate equations for two different eigenvalues a y m and y n n and a , multiply by m respectively, and subtract: y y" m n y y" m n + = 0 y" y m n J L If we left-multiply by dx, we have 0 (C-20) = But since y (O) = y (L) = O, the left side vanishes, and n n for n I: m (C-21) Q.E.D. 163 3. Evaluating the Orthogonality !ntegral Equation C-21 can be generalized as L f 0 y (x) y (x) dx n m N o n mn = (C-22) The problem remains to evaluate the normalization integral N . Using n equation C-21, one can write this as the following limit: [ymy~ - y~yn] ~ . limit N n a m -+ a. a. n m - a. (C-23) n applying l'Hopitals Rule, we have L =t~ aa. Nn '1ca. ,x) y' (a. ,x) - y' (a. ,x) y(a. ,x)l } [ m n m n Ja. =a. m n m 0 (C-24) = [ y a. (a. n ,x) y' (a. n ,x) - y'a. (a. n ,x) where However, y(a. ,x) = y (x) and n n y(a ,O) n y(a. ,L) n a y (a. ,x) = ~ y(a. ,x) a. n y (a. ,O) a. n 0 and N n = y (a. ,L) y' (a. ,L) a. n n In terms of the Coulomb wave functions, this becomes (C-25) 164 [~ N n (3 where F (n,p) = F o . (F') 2 ~ , / ; x) and o 2a n n (C-26) 0 oF F' = _o_ o ap Equation C-26 is not a simple evaluation of N , but it greatly facilin tates the calculation because time-consuming numerical integration can be avoided. 165 Appendix D Notes on Calculating the New Solution = R(r,t) Q(6,~,t), as expressed by equa- The solution n(r,6,~,t) tions 5-13, 5-17, 5-23, and 5-32 through 5-37, has been implemented on the computer in FORTRAN IV via a main control program and 8 subprograms. These routines, which are shown with their linkages in Figure D-1, can be described as follows: FO -- Calculates the regular Coulomb wave function F (n,p). 0 For 0 ..::_ n ..::_ 10 and 0 ..::_ p ..::_ 30, the standard power series is adequate, but for P > 30, the "asymptotic" expansion is neeessary. A useful discussion of the methods for calculating this function at various locations in the n - p plane has been given by Fr8bergC 67 been published by Tubis( FOPR -- Calculates F' 0 ). 68 The most complete tables have ). = cF 0 (n,p)/ap. Again, both the power series and the asymptotic formula are used depending on the value of p. Calculates the nth zero of F (n,p) by making an initial FOZER 0 estimate of the zero and then closing in with Newton's method. This subprogram obviously uses both FO and FOPR. DFODET -- Calculates cF (n,P)/an using the power series derived in 0 the NBS Tables( 6 g). This method breaks down for p > 30 and aF /an is then calculated directly by varying F (n,p). 0 0 Input Parameters: M(i1DPLT V, r, r , L, O, y , s A0 ' 0 Plots n ( r , e , <P , t ) MAIN I' K, Kl, c 0 i-----------i~ Printed } Output_ LUPT¢N L(i1NGTD t--' ~ Figure D-1: °' °' DFODET FO FOZER FOPR Computer routines used to calculate the new solution. The input and output parameters are defined in Section V. 167 LUPTO'N -- Calculates the new radial solution R(r,t) using the subprograms FO, FOPR, FOZER and DFODET. Using the input parameters V, r, rs, L, K, t and NMAX, LUPT¢N carries out the following operations: a) Defines B = V(2C-l)/K b) Calculates the eigenvalues a(n) for n = 1, NMAX using FOZER. c) Calculates the value of the normalization integral N for n n=l, NMAX (see Appendix C) without any numerical integration using FOPR and DFODET. d) P -- Sums the eigenvalue series (Equation 5-32) and finds R(r,t). Evaluates the Legendre polynomial Pt(y) using a recursion relation. 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