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The exact theory of linear cyclotron instabilities applied to hydromagnetic emissions in the magnetosphere Jacks, Bruce Raymond 1966

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THE EXACT THEORY OF LINEAR CYCLOTRON INSTABILITIES APPLIED TO HYDROMAGNETIC EMISSIONS IN THE MAGNETOSPHERE  by  BRUCE RAYMOND JACKS B.Sc,  U n i v e r s i t y o f B r i t i s h Columbia, 1964  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of GEOPHYSICS  We accept t h i s t h e s i s as conforming t o the required standard  THE UNIVERSITY OF BRITISH COLUMBIA September, 1966  In p r e s e n t i n g the  this  thesis  Columbia,  I agree that  the Library  a v a i l a b l e f o r r e f e r e n c e and s t u d y . mission  f o r extensive  representatives„  cation  of this  thesis  w i t h o u t my w r i t t e n  It i s understood for financial  Department o f The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a }  i t freely  thesis  per-  for scholarly  by t h e Head o f my D e p a r t m e n t o r by  permission.  S^^jt. XI-  s h a l l make  I f u r t h e r agree that  copying o f t h i s  p u r p o s e s may be g r a n t e d  Date  fulfilment of  r e q u i r e m e n t s f o r an advanced d e g r e e a t t h e U n i v e r s i t y o f  British  his  in partial  \  Columbia  °\LL  that  gain  copying o r p u b l i -  shall  n o t be a l l o w e d  i ABSTRACT The complex d i s p e r s i o n r e l a t i o n which describes transverse plasma waves propagating i n a cold g y r o t r o p i c ambient plasma p a r a l l e l to the back  r  ground magnetic f i e l d as they i n t e r a c t with charged p a r t i c l e streams i s derived by s o l v i n g the l i n e a r i z e d c o l l i s i o n l e s s Boltzmann equation simultaneously with Maxwell's equations using the Fourier-Laplace transform method.  The wave frequency i s allowed to be complex with a p o s i t i v e imaginary  part corresponding to a growing i n s t a b i l i t y .  The r e a l and imaginary parts of  the d i s p e r s i o n r e l a t i o n y i e l d two separate equations.  Under s e v e r a l  assumptions, the equations can be s i m p l i f i e d t o y i e l d an expression f o r the imaginary part of the frequency (the growth r a t e ) and an equation r e l a t i n g the r e a l wave frequency and the wave number. The theory i s then applied to the magnetosphere by choosing a dipole model f o r the earth's magnetic f i e l d and a s u i t a b l e d i s t r i b u t i o n f u n c t i o n f o r the p a r t i c l e s .  The s p e c i f i c case of waves of the ion-resonance  mode i n t e r a c t i n g with mono-energetic,  contra-streaming protons i s considered  i n d e t a i l , and the r e s u l t s of t h i s c a l c u l a t i o n are used i n e x p l a i n i n g hydromagnetic (hm) emissions.  In p a r t i c u l a r , i t i s suggested t h a t the high  frequency c u t o f f i s a r e s u l t of the p i t c h angle d i s t r i b u t i o n of the p a r t i c l e stream. Computer c a l c u l a t i o n s are done i n order to d i s p l a y the general r e s u l t s of the theory.  S p e c i f i c a l l y , when low energy protons (10 - 20 kev),  trapped on a f i e l d l i n e with an L value of 5.6 are considered, i t i s found that the region of i n s t a b i l i t y occurs near the geomagnetic equator, and that the growth r a t e i s a sharply peaked f u n c t i o n of the frequency.  TABLE OF CONTENTS  Chapter I  INTRODUCTION General Discussion Cyclotron Resonance Thesis Outline  II  MATHEMATICAL ANALYSIS  III  APPLICATION OF THE GENERAL RESULTS TO THE MAGNETOSPHERE Discussion The S h i f t e d A n i s o t r o p i c Maxwellian D i s t r i b u t i o n A monoenergetic P i t c h Angle D i s t r i b u t i o n  IV  NUMERICAL CALCULATIONS Normalization o f the Equations Parameter Values Results  V  SUMMARY Discussion Conclusions  Appendix I  Transformation o f Equations and Solution f o r f (v,k,(0)  II  Determination o f the Transformed Magnetic F i e l d s  III  A n a l y t i c Continuation of the Integrals  IV  Wave P o l a r i z a t i o n  V  S i m p l i f y i n g the General Dispersion R e l a t i o n  VI  Coordinate Transformation and C a l c u l a t i o n of CJi  t  Bibliography  iii FIGURES Figure  Page  1.  V a r i a t i o n of sin*iy with y  f o r f i v e values of H .  2.  Dependence o f CJj on the normalized frequency two p a r t i c l e energies. a)X=0°  3.  1  20  for 30  ,b)X-10°  31  c)X=20°  32  Dependence of D  x  on \ f o r two p a r t i c l e  energies.  a) uj'(eq)= 0.3  33  b) 6J (eq) = 0.4  33  c) 6J'(eq)=.0.5  34  d) w'(eq)"0.7  35  l  4.  Integration contours f o r k?0,6Jj;>0.  47  5.  Integration contours f o r k>0,6iJ <0.  47  6.  Integration contours f o r k<0,<*^ >0.  48  7.  Integration contours f o r k<0,CJ <0.  48  r  c  1  iv ACKNOWLEDGMENTS I wish t o thank s i n c e r e l y Dr. T. Watanabe f o r suggesting t h i s problem and f o r h i s assistance i n many h e l p f u l discussions throughout the course o f the research. I a l s o wish t o thank Professor J . A. Jacobs f o r providing the opportunity and the f a c i l i t i e s t o carry out t h i s work and f o r h i s patience while i t was being done.  1 CHAPTER I INTRODUCTION General Discussion Since the earth's upper atmosphere contains a s i g n i f i c a n t number of charged p a r t i c l e s , a p h y s i c a l study of that region involves the concepts o f magnetohydrodynamics and plasma physics.  The theory of plasma waves has been  used t o e x p l a i n such phenomena as atmospheric w h i s t l e r s and geomagnetic micropulsations.  In such studies, the geomagnetic f i e l d i s fundamental.  Atmospheric w h i s t l e r s are electromagnetic waves which occur i n the frequency range 300 - 30,000 cps and propagate i n the e l e c t r o n resonance mode ( f a s t mode) which has an upper frequency l i m i t a t the e l e c t r o n cyclotron frequency - 6J . e  They o r i g i n a t e i n l i g h t n i n g flashes ( H e l l i w e l l and Morgan,  1959) and bounce between the northern and southern hemispheres along paths which approximately f o l l o w the magnetic f i e l d l i n e s ( H e l l i w e l l , 1965). An analogous left-hand c i r c u l a r l y p o l a r i z e d wave e x i s t s i n the i o n resonance mode (slow mode) f o r frequencies below the i o n gyrofrequency OJc . The d i s p e r s i o n r e l a t i o n f o r these two types o f waves propagating p a r a l l e l t o the  background magnetic f i e l d  B  0  i n a c o l d , ambient plasma can be w r i t t e n  (Astrom, 1950)  f o r a plasma with one singly-charged, i o n i c component; frequency, k i s the wave number and c i s the speed of l i g h t . equation, both 6J and k are r e a l q u a n t i t i e s .  A^pe and  Sk^are  and ion plasma frequencies r e s p e c t i v e l y , and are defined by  i s the wave In t h i s the e l e c t r o n  2 firN 1  x  P  pe. (1-2)  where N  p  i s the e l e c t r o n number density o f the plasma, q i s the charge on an  e l e c t r o n or a proton and 3. i s negative or p o s i t i v e r e s p e c t i v e l y , and m  e  and m-  L  are the e l e c t r o n and ion masses r e s p e c t i v e l y .  u n i t s i s used throughout the t h e s i s .  The Gaussian system o f  In equation 1-1, the upper sign i s  used f o r the ion resonance mode and the lower sign f o r the e l e c t r o n resonance mode. I t i s possible that waves o f the ion resonance mode are d i r e c t l y involved with the production of micropulsations i n the pc 1 frequency range 0.2-5  cps.  Tepley and.Wentworth (1962) were the f i r s t t o present the  dynamic spectra (frequency-time p l o t s ) o f such micropulsations.  Those  which showed a d i s t i n c t f i n e structure c o n s i s t i n g o f r e p e t i t i v e r i s i n g tones which often overlapped were c a l l e d hydromagnetic emissions, or b r i e f l y , hm emissions. They have also presented a theory, which accounted f o r t h i s f i n e s t r u c t u r e (Wentworth and Tepley, 1962). Jacobs and Watanabe (1965) have described the h i s t o r y o f the research done on hm emissions and they emphasize the f o l l o w i n g points.  At  hydromagnetic frequencies,, waves o f the ion resonance mode tend t o be  i  I guided by the magnetic f i e l d t o a much greater extent than waves o f the e l e c t r o n resonance mode (Jacobs and Watanabe, 1964).  The dispersion o f 'hm  w h i s t l e r s ' or 'micropulsation w h i s t l e r s ' y i e l d s a t h e o r e t i c a l spectrum which agrees approximately with the observed c h a r a c t e r i s t i c s o f the structured hm emissions. The hm w h i s t l e r s i g n a l s d i f f e r from those o f atmospheric w h i s t l e r s i n that the s i g n a l i n t e n s i t y does not constantly decrease a f t e r the f i r s t bounce but often grows before decaying (Tepley and Wentworth, 1964).  The idea developed i n t h i s t h e s i s i s that the waves gain energy through a c y c l o t r o n i n s t a b i l i t y process i n v o l v i n g low energy protons which are trapped i n the magnetosphere.  The process i s exactly analogous to the  i n s t a b i l i t y found by B e l l and Buneman (1964) f o r electrons i n t e r a c t i n g with waves of the w h i s t l e r mode.  I t i s not a s i n g l e p a r t i c l e e f f e c t ( c y c l o t r o n  r a d i a t i o n ) but a plasma i n s t a b i l i t y i n v o l v i n g the t r a n s f e r of some of the transverse k i n e t i c energy of the p a r t i c l e s to electromagnetic  energy i n the  wave ( B r i c e , 1964;.Neufeld and Wright, 1965a). In order to have an i n s t a b i l i t y at a l l , an i n i t i a l wave disturbance must e x i s t so that the wave-particle i n t e r a c t i o n can take place. The a c t u a l source of t h i s i n i t i a l , small 'seed' wave i s not known at present.  The problem has been discussed by Jacobs and Watanabe (1965) and-  Obayashi (1965).  In the present d i s c u s s i o n , the existence of perturbing hm  w h i s t l e r waves i s assumed. Cyclotron Resonance I t i s assumed that the streaming p a r t i c l e s have an i n i t i a l transverse component of v e l o c i t y .  In order to determine whether the wave  grows or i s damped, the v e l o c i t y d i s t r i b u t i o n function f o r the p a r t i c l e s must be s p e c i f i e d . I t has often been noted that growing i n s t a b i l i t i e s require an a n i s o t r o p i c d i s t r i b u t i o n ( S t i x , 1962; Montgomery and Tidman, 1964; Cornwall, 1965). I t can be seen i n t u i t i v e l y that a 'resonance' might occur i f a p a r t i c l e i s gyrating with the same sense of r o t a t i o n as the wave's p o l a r i z a t i o n , and i f the p a r t i c l e sees a wave frequency equal to i t s own cyclotron frequency.  In a laboratory reference frame, the resonant-frequency i s  4  d i f f e r e n t from the cyclotron  frequency because o f the Doppler s h i f t -arising  from the p a r t i c l e ' s l o n g i t u d i n a l v e l o c i t y u. For the case o f protons and a left-hahd polarized'wave, the resonance conditions mentioned above are s a t i s f i e d with a p o s i t i v e  real  frequency OJ given by CL)  - feu =  • In the magnetosphere, D - (0;. < 0 (Booker, 1962).  (1-3) I t can then be seen that  the product ku must be negative, v i z . , the wave and p a r t i c l e s must t r a v e l i n opposite d i r e c t i o n s .  However, i t must be noted that protons can i n t e r a c t  with waves o f the w h i s t l e r mode because o f the anomalous Doppler e f f e c t ( B r i c e , 1964).  When a p a r t i c l e t r a v e l s f a s t e r than the wave and i n the same  d i r e c t i o n , i t sees a r e v e r s a l o f the wave's p o l a r i z a t i o n .  Jacobs and  Watanabe (1965) have discussed the d i f f e r e n t p o s s i b i l i t i e s leading t o cyclotron  instabilities.  Thesis Outline In Chapter I I , a general l i n e a r analysis o f the problem i s c a r r i e d out s t a r t i n g from Maxwell's equations and the c o l l i s i o n l e s s Boltzmann equation.  The Fourier-Laplace transform method i s used, the general  procedure being s i m i l a r t o that outlined by S t i x (1962) f o r l o n g i t u d i n a l plasma o s c i l l a t i o n s .  This method was suggested by Watanabe (1965a), and the  r e s u l t s o f these c a l c u l a t i o n s  agree with those o f Cornwall (1965).  Chapter I I I involves the a p p l i c a t i o n o f the general r e s u l t s t o the magnetosphere.  The proton streams are assumed t o be monoenergetic.  The  p i t c h angle d i s t r i b u t i o n function i s chosen t o s a t i s f y a d i f f e r e n t i a l equation which i s v a l i d f o r p a r t i c l e s trapped i n a strong, steady, magnetic  5  f i e l d i n a tube of f l u x which has a small normal cross-section (Watanabe, 1964). In Chapter IV, the r e s u l t s o f numerical c a l c u l a t i o n s made on a computer are presented.  In order t o carry out the c a l c u l a t i o n s , several  assumptions are made: the earth's f i e l d i s assumed t o be a centered dipole f i e l d having a value o f 0.3 Gauss on the earth's surface at the geomagnetic equator; the Smith model (Smith, 1961) o f the e q u a t o r i a l e l e c t r o n density, which i s v a l i d only f o r distances up t o four earth r a d i i from the earth's surface, i s assumed t o hold i n a l l regions o f the magnetosphere. two assumptions l i m i t the exactness o f the r e s u l t s .  These  A discussion o f the  v a r i a t i o n o f e l e c t r o n density along f i e l d l i n e s has been given by Carpenter and Smith (1964).  Watanabe (1965c) has indicated,how information  about the d i s t r i b u t i o n of electrons at a l t i t u d e s greater than about four earth r a d i i may-be obtained. The f i n a l chapter summarizes several relevant papers which deal with hm emissions and cyclotron i n s t a b i l i t i e s i n the magnetosphere and discusses the l i m i t a t i o n s of the t h e s i s .  6 CHAPTER.II MATHEMATICAL ANALYSIS  To describe the i n t e r a c t i o n between the waves and the p a r t i c l e stream, one must determine the evolution i n time of the p a r t i c l e d i s t r i b u t i o n function.  Knowing the i n i t i a l conditions, the electromagnetic  f i e l d s i n the plasma can then be determined.  I t i s assumed that the  d i s t r i b u t i o n function f ( v , r , t ) s a t i s f i e s the c o l l i s i o n l e s s Boltzmann equation, and i f the plasma i s c o l d , then the f i e l d s produced by the p a r t i c l e density f l u c t u a t i o n s (due t o thermal motions) are n e g l i g i b l e compared to the f i e l d s of the wave and the equation can be w r i t t e n  3t  3£  *m  where t and r are the time and space coordinates r e s p e c t i v e l y , v i s the p a r t i c l e v e l o c i t y , m i s the p a r t i c l e mass and E and B are the e l e c t r i c f i e l d strength and the magnetic f l u x density, r e s p e c t i v e l y , of the wave. 3/8JC represents the s p a t i a l gradient and 9/9y the gradient i n v e l o c i t y space.  The  Maxwell equations used are c u r l B,= * ? j + ^ | f  curl.E=-iH  (2-2)  ( 2  "  3 )  where  COInAp J defines the current density. of the plasma.  The summation i s taken over a l l the components  7  Equation 2-1 i s expanded by assuming that the f i e l d s B and E are f i r s t order q u a n t i t i e s ( B  0  i s zeroth order) and that f can be w r i t t e n  f(v,r,t) = f where f  4  0  (v) + f  ±  (v,r,t)  i s a f i r s t - o r d e r perturbation on f . 0  taken to be i n the p o s i t i v e z d i r e c t i o n .  (2-5)  The background f i e l d B  is  G  Only transverse waves are con-  sidered and the s p a t i a l l y varying q u a n t i t i e s are assumed to depend only on the coordinate z, and not on x and y.  Neglecting terms of second order i n  equation 2-1, the zeroth and f i r s t - o r d e r equations are found to be  and  Introducingthe  c y l i n d r i c a l coordinates (u,w,<|>) i n v e l o c i t y space, the  l a s t term i n equation 2-6 can be w r i t t e n  where ^ c = ^ r  (-) 2  i s the p a r t i c l e gyrofrequency which can be p o s i t i v e or negative.  9  It i s  assumed that  f t = §i° = $ so that equation 2-6 i s s a t i s f i e d .  = °  <- > 2  10  8  Using a F o u r i e r transform i n space and a Laplace transform i n time and using two component equations obtained>from  equations 2-2 and  2-3, the transformed equation 2-7 can be solved f o r ^ ( V j k , ^ ) ) (Appendix I ) . In t h i s way, equation 2-7 i s handled as an i n i t i a l value problem, where the p a r t i c l e d i s t r i b u t i o n f u n c t i o n at time t = 0 must be s p e c i f i e d .  This  method was f i r s t used by Landau (1946) i n d i s c u s s i n g the l o n g i t u d i n a l v i b r a t i o n s of an e l e c t r o n i c plasma.  I f the i n i t i a l d i s t r i b u t i o n can be w r i t t e n (2-11)  then fj.(v,k,6j) i s given by  -±ii»£[w».*)-ie,c<>.vj) w-ku-w i x ^ c ^ * c  +  /aw  9U  B (w,^)+-c& (^» x  y  (2-12)  CO -feu + • 'VnCOt  Using equation 2-4, simple algebra gives (2-13) since f (y_) i s constant with respect t o 0  Using equation 2-12, and the  other two component Maxwell equations a f t e r transformation (Appendix I ) , i t can be shown (Appendix I I ) that B (oo ft)±t& (oJ^) = x  )  y  (2-14)  9  where  x r  . The upper and lower signs correspond respectively  i  3&  t o l e f t and r i g h t polarized waves  (Appendix IV).  In p r i n c i p l e , B(t,z) can now be found by applying the inverse transformations t o equation 2-14. This means that the response o f the plasma system t o an i n i t i a l perturbation o f p a r t i c l e d i s t r i b u t i o n s by a p a r t i c l e beam can be found.  I t i s t h i s r e s u l t that j u s t i f i e s the use o f  the transform method, but i n order t o make the problem f e a s i b l e i t i s not solved i n general.  mathematically,  In the Laplace transformation, the parameter  i s allowed t o be complex, with the r e s t r i c t i o n that i t s imaginary part be positive.  L a t e r , 6J i s i d e n t i f i e d as the wave frequency.  The inverse  transformation must be c a r r i e d out along a path which l i e s i n the upper h a l f OJ-plane above the s i n g u l a r i t i e s of B_((0,k). negative imaginary parts f o r D  should be allowed.  But p h y s i c a l l y , The procedure followed  i n overcoming t h i s d i f f i c u l t y involves the a n a l y t i c continuation o f a s i n g u l a r i n t e g r a l and has been discussed by S t i x (1962).  Using the  Cauchy P r i n c i p a l Value ( P ) , i t i s found (Appendix V) that the d i s p e r s i o n r e l a t i o n i s given by  where  10  and V  ~  *  Equation 2-16 i s v a l i d f o r both p o s i t i v e and negative imaginary parts o f (J . The presence o f the s i n g u l a r i t y i n the inverse transformation r e s u l t s i n the l a s t term i n equation 2-16 being evaluated under the c o n d i t i o n 60 -  + U) = °  (2-17)  t  This i s how the c y c l o t r o n resonance c o n d i t i o n enters the problem mathematically. Equation 2-16 can be s i m p l i f i e d by s p e c i f y i n g the c o l d , background part of f  0  as f a by w r i t i n g  with b  where N  p  C  u  (2-19)  )  i s the number density o f the background plasma and S  represents  the Dirac d e l t a f u n c t i o n . The d i s t r i b u t i o n f u n c t i o n f ^ i s normalized t o N p and f  s  represents the streaming p a r t i c l e d i s t r i b u t i o n f u n c t i o n . The 3$e/3U  term i n the p r i n c i p a l value i n t e g r a l vanishes under the ,integral over w. The 2§i/dw term can be s i m p l i f i e d using i n t e g r a t i o n by parts so t h a t [ and since  d w  (Pfju,  [G(f ^ B  ^=  = ~ ^ U / p L (<*>TW»  =  - - i k —  (2-20)  0, equation 2-16 becomes  -^Z^jdwlwG^i.^O  (2-21)  11  where  * f l = HirNpl'/'^c.  i s the e l e c t r o n plasma frequency o f the background  p  plasma and i s taken as the t o t a l plasma frequency since N much smaller than N  p  i s assumed t o be  s  . I f CO i s w r i t t e n U) «• CO* +lLJx  ,(2-22)  i t i s a l s o assumed that M « 6 0 R  (2-23)  This condition means that the i n s t a b i l i t y grows or decays v e r y , l i t t l e during a time i n t e r v a l corresponding  t o the period of the wave.  and CJj are f i r s t order q u a n t i t i e s compared t o N  P  Assuming that N  5  and ( J , equation 2-21 R  can be s i m p l i f i e d (Appendix V) and s e t t i n g the r e a l and imaginary  components  separately equal t o zero gives ^ - C ^ - V i t e L  =0  (2-24)  and (2-25)  XL) • + J  ±  ^  where  Equation 2-24 i s the r e a l d i s p e r s i o n equation which r e l a t e s  and k, and  equation 2-25 i s the expression f o r the growth rate o f the i n s t a b i l i t y .  12 CHAPTER I I I APPLICATION OF THE GENERAL RESULTS TO THE MAGNETOSPHERE  Discussion The r e s u l t s of the previous chapter have been derived f o r the case of plane waves i n f i n i t e i n extent propagating p a r a l l e l t o a homogeneous background magnetic f i e l d which extends over a l l space.  In applying these  r e s u l t s to the magnetosphere, i t i s assumed t h a t the region of i n t e r a c t i o n i s small enough that the geomagnetic f i e l d can be considered homogeneous there, but large enough that the hm waves are w e l l approximated by plane waves.  This problem has been mentioned by-Hruska  (1966).  In order to c a l c u l a t e 6J using equation 2-25, an e x p l i c i t t  expression f o r f  s  must be determined i n a meaningful way.  Although much  has been learned experimentally about p a r t i c l e s contained i n the van A l l e n b e l t s , almost nothing i s known about the d i s t r i b u t i o n of low energy protons at higher a l t i t u d e s .  Davis and Williamson (1962) have reported data  obtained from the s a t e l l i t e 'Explorer 12' and Cornwall (1965) suggested these protons might be important i n c y c l o t r o n emissions as w e l l as constituting  a r i n g current. Most of the r e s u l t s concerned protons i n the  energy range 50 kev - 5 mev.  Hoffman and Bracken (1965) have given a more  complete report of the same data.  Some of these r e s u l t s w i l l be quoted  later. Two d i s t r i b u t i o n functions are now considered.  The  shifted,  a n i s o t r o p i c Maxwellian d i s t r i b u t i o n i s used as an example since i t has been used s e v e r a l times before (Sudan, 1963; Guthart, 1964; H u l t q v i s t , Hruska, 1966).  The second  1965;  d i s t r i b u t i o n chosen i s discussed i n d e t a i l below.  The S h i f t e d , A n i s o t r o p i c , Maxwellian D i s t r i b u t i o n This type of d i s t r i b u t i o n represents a p a r t i c l e stream whose spread of random thermal v e l o c i t i e s perpendicular to the background f i e l d i s d i f f e r e n t than the spread p a r a l l e l t o i t , and there i s an organized, uniform v e l o c i t y p a r a l l e l t o the f i e l d .  In t h i s case, the d i s t r i b u t i o n  function i s written i - M i A \ - ^ - < - ^  w  l  (3-1)  where a^T,,  u  f  s  has been normalized to N  s  .  2 )  . In t h i s case,  Setting u = V , and i n t e g r a t i n g R  ( 3  over w and ty, (3-4)  oo and so equation 2-25 gives ^fa), ^ ^ l A - t o ^ A ^ , ill .Ci.'w a* ^ e  and  i V ^ k ^(V*-U.)"\ -rrr&fc" -J t  (3-5)  14  I f only one type o f p a r t i c l e i s streaming, then the summation can be removed and the condition f o r p o s i t i v e  is  2± >  (3-6)  This r e s u l t i s known ( S t i x , 1962). A Monoenergetic P i t c h Angle D i s t r i b u t i o n Using a t h e o r e t i c a l approach, Watanabe (1964) has obtained a d i f f e r e n t i a l equation which governs the d i s t r i b u t i o n function of p a r t i c l e s trapped i n a 'strong' magnetic f i e l d , v i z . , one f o r which the scale o f s p a t i a l v a r i a t i o n s i s much l a r g e r than the gyroradius o f the p a r t i c l e . I f the f i e l d i s steady in,time,  where *+• i s the l o c a l p i t c h angle o f a p a r t i c l e , 1 i s distance measured along a f i e l d l i n e , and B i s the l o c a l magnetic f i e l d strength.  This equation i s  v a l i d only i n the one-dimensional case, when the p a r t i c l e s are confined t o a tube of f l u x f o r which the l i n e a r dimensions o f any normal cross-section are much smaller than the scale length of the trapping region.  A particular  s o l u t i o n i s given by  where e< i s an a r b i t r a r y constant and C i s constant with respect t o 1, *+* , and t and contains the normalization f a c t o r . perturbing wave, s i n ^ ^ / B  In a 'strong' f i e l d , with no  i s a constant of the motion since i t i s  p r o p o r t i o n a l t o the magnetic moment of a p a r t i c l e , the f i r s t adiabatic i n v a r i a n t (Chandrasekhar, 1960; A l f v e n and Falthammar, 1963).  The r e s u l t  15  that f g depends on the adiabatic i n v a r i a n t i s t o be expected (Cornwall, 1965).  The discussion i n the remainder of the t h e s i s concerns only mono-  energetic protons.  The assumption that the p a r t i c l e s are monoenergetic i s  not too r e s t r i c t i v e and helps t o s i m p l i f y the mathematics. electrons have been considered  Monoenergetic  previously (Wentworth and Tepley, 1962).  In the numerical c a l c u l a t i o n s which are done l a t e r , the p a r t i c l e energy i s varied as a parameter.  The d i s t r i b u t i o n function i s w r i t t e n  W S < v - v . ) a a £  (  where V i s the 'pitch angle d i s t r i b u t i o n parameter'.  3-9)  I t i s assumed that  the number density of the streaming p a r t i c l e s i s known at some point i n the magnetosphere, that i s f o r some value o f the main f i e l d B*, f o r instance at the equator. f  s  t o N*  The constant  'A' i s determined by normalizing  at t h i s p o i n t , and i t i s necessary that H > -2 so that the  i n t e g r a l does not diverge.  In t h i s case,  i . ltfB»* * ATTjir V? where p = Jf + 1  +Q p(f|I)  (3-10)  and P represents the gamma function.  The i n t e g r a l over  the p i t c h angle 4* , i s taken from 0 to TT because the p a r t i c l e s are supposed to stream i n the p o s i t i v e and the negative z d i r e c t i o n s , although f o r a given wave at any point, only one-half the p a r t i c l e s can p a r t i c i p a t e i n the cyclotron interaction. Using equations 2-25 and 3-9, GOi i s c a l c u l a t e d (Appendix VI) and i s found t o be non-zero only when y  iN/V  ,  M  \\\< V,,  . In t h i s case,  (Vo --v«)^r / i^. > 3  Y  +  , y ± A » CJT  1  1  ] ( 3 _ 1 5 )  16  The f o l l o w i n g important q u a l i t a t i v e r e s u l t s can be obtained from t h i s  ,  expression. 1.  The growth rate i s d i r e c t l y p r o p o r t i o n a l t o the density of streaming p a r t i c l e s since A^Ng.  2.  The only f a c t o r which can be negative i s  a) The '-2' term represents a constant damping f a c t o r which o r i g i n a t e s i n the expression -2 j ' dw wf . s  b) The quantity + G0!e/GJ* - 1 i s p o s i t i v e f o r both the e l e c t r o n - w h i s t l e r i n t e r a c t i o n and the proton-hm w h i s t l e r interaction.  Therefore, f o r wave growth i n e i t h e r case,  must be at l e a s t p o s i t i v e .  In f a c t , f  must s a t i s f y  I f (O i s taken as the e q u a t o r i a l value, then t h i s c  condition allows wave growth at any point on t h a t f i e l d l i n e provided  v^" >  .  c) Suppose a p a r t i c l e stream trapped i n the magnetosphere can be described by a s p e c i f i c value of If , I t f o l l o w s that there i s an upper frequency l i m i t f o r waves that w i l l be a m p l i f i e d . This maximum frequency  TTt  i s given by  <3  y  Any waves with frequencies higher than t h i s w i l l be damped, and using the value of U  c  at the e q u a t o r i a l  plane w i l l i n d i c a t e approximately the maximum frequency  17)  17  of any a m p l i f i e d waves.  I f the mechanism f o r wave  a m p l i f i c a t i o n suggested here i s c o r r e c t , then the existence of a maximum frequency gives a method of determining Y , provided the guiding magnetic f i e l d l i n e can be determined.  This problem i s discussed  qualitatively later. 3.  At a given point i n space and f o r a given p a r t i c l e energy, as waves of lower frequencies are considered,  V -* R  v„ , and the  waves are not a m p l i f i e d . This f a c t may be used t o e x p l a i n the observed minimum frequency of hm emissions. 4.  For a given p a r t i c l e energy and wave frequency, only i f ^ . v a r i e s .  a.  can vary  o  approaches v  0  as k ) gets l a r g e r and c  t h i s occurs as the region under consideration moves down the f i e l d l i n e away from the e q u a t o r i a l region. point, V  R  w i l l always be greater than v  e  Past a c e r t a i n and i n s t a b i l i t y can  no longer take place so i n t u i t i v e l y i t seems that the unstable region tends to be s i t u a t e d near the e q u a t o r i a l plane.  The  numerical c a l c u l a t i o n s show that t h i s i n t e r p r e t a t i o n i s v a l i d .  The existence of maximum and minimum frequencies as discussed above can be thought of as roughly d e f i n i n g a band width f o r the emissions The suggestion that the i n s t a b i l i t y tends to occur i n the e q u a t o r i a l region i s due to Watanabe (1965b) and has been mentioned by Jacobs and Watanabe (1965).  The same idea has been put f o r t h by Tepley  and Wentworth (1964) f o r d i f f e r e n t reasons.  They suggest that streaming  protons i n the magnetosphere can sometimes be superluminous  with respect  18  to hm waves and that since the p a r t i c l e s move f a s t e r near the e q u a t o r i a l plane, i t i s i n these regions that the proton c y c l o t r o n r a d i a t i o n i s subject to the anomalous Doppler s h i f t and i s l i k e l y to be most intense. They suggest that on each pass through the e q u a t o r i a l region, the same process occurs.  In order that the wave be reinforced each time, they  suggest that the bounce periods o f the wave packet and p a r t i c l e s be approximately equal so that the p a r t i c l e s pass through the wave packet at the equator each time.  I t i s not obvious that by the time the p a r t i c l e s  return to the e q u a t o r i a l plane, they w i l l s t i l l be i n phase with the wave which they i n i t i a l l y generated.  Tepley and Wentworth also had to assume  that the p a r t i c l e stream was coherent to begin with i n order to obtain a s i g n i f i c a n t amount of r a d i a t i o n i n the f i r s t emission. Obayashi (1965) has discussed t h i s point.  Besides t h i s , there i s no 'a p r i o r i ' reason  f o r the two t r a v e l time periods to be the same.  But the most important  f a u l t i n the theory i s the suggestion that such a superluminous p a r t i c l e stream can i n t e r a c t i n a c o l l e c t i v e manner with the hm waves.  Since the  p a r t i c l e s see anomalously Doppler s h i f t e d waves, the sense of the waves' p o l a r i z a t i o n i s opposite t o that of the gyration of the p a r t i c l e s ( B r i c e , 1964) and cyclotron resonance cannot occur.  Other d i f f e r e n t attempts have  been made to determine l i k e l y regions of wave growth i n the magnetosphere and several of these are discussed i n Chapter V. I t was suggested above that X must exceed a minimum value before wave growth can occur.  The energy which i s gained by the wave i n  the a m p l i f i c a t i o n process comes from p a r t i c l e k i n e t i c energy.  It is  possible that although some energy i s t r a n s f e r r e d from the p a r t i c l e s , i t i s not enough to balance the constant damping which i s present and the wave decays.  I t i s seen then, that the transverse component of v e l o c i t y  19  of the p a r t i c l e s cannot be a r b i t r a r i l y small (Neufeld and Wright, 1965b); Obayashi, 1965).  ,  Since the p i t c h angle d i s t r i b u t i o n i s given by s i n <y , when V i s p o s i t i v e , more p a r t i c l e s have l a r g e p i t c h angles than small ( f i g . 1 ) . Increasing  from zero e f f e c t i v e l y increases the average transverse  k i n e t i c energy of the p a r t i c l e s while decreasing the average l o n g i t u d i n a l k i n e t i c energy.  When the p a r t i c l e s lose transverse energy t o the wave i n  a growing i n s t a b i l i t y , there i s a general reduction of p i t c h angles and some p a r t i c l e s may be l o s t because they have p i t c h angles which are inside the 'loss-cone' (Cornwall, 1965; B r i c e , 1964).  20  3-  0  15  30  45 M-  1  1  60  75  (degrees)  Family of curves of sm^H* . Values of the parameter Jf are w r i t t e n beside the corresponding curve i n the diagram.  90  21 CHAPTER IV NUMERICAL CALCULATIONS  Normalization o f the Equations I t i s assumed that only protons are contained i n the p a r t i c l e stream.  The background plasma contains t h e r e f o r e , more e l e c t r o n s than  protons by a small amount i n order t o preserve o v e r a l l charge n e u t r a l i t y . Since f g i s non-zero only f o r protons, the summation over components m the numerator of the expression f o r the growth rate o f the i n s t a b i l i t y i s not required. I t i s often convenient t o w r i t e the important equations obtained i n a study i n normalized form i n v o l v i n g dimensionless v a r i a b l e s so that the general r e s u l t s can be seen without employing numerical values which are v a l i d f o r a s p e c i f i c case only. In equation 3-15, the term 2 6l) i n the denominator o r i g i n a t e s r  i n the displacement current term i n Maxwell's equations and at hm frequencies i t can be neglected (Jacobs and Watanabe, 1965).  Eliminating  t h i s term allows the equation t o be put i n t o dimensionless form with the help of the f o l l o w i n g r e l a t i o n s h i p s .  <A.i*Ui  where  CJ =-6J t t  e  ( 4  _  1 }  22  I t i s then convenient to write  U  =  ^-"W^  V~  ( 4  "  3 )  by d e f i n i n g Ul  ?  (4-4)  I f the 2CJt term i n equation 3-15 i s neglected, then by d i v i d i n g both sides of the equation by 60i , i t i s found that  _  _  I —  t  (4.5)  i  In a s i m i l a r manner, the r e a l d i s p e r s i o n r e l a t i o n (equation 2-24) can be put i n t o normal form by neglecting the  term f o r the same  reasons as above, and then i t i s w r i t t e n ^  ^  ±1 - UJ*  M  ^  (4-6)  ±h +60'  Parameter Values The f i r s t requirement of any theory of hm emissions i s that the emitted frequency be i n the Pc 1 range from 0.2 to 5 cps.  I f a proton's  v e l o c i t y and p i t c h angle are known, then the resonant frequency f o r that p a r t i c l e can be determined from the resonance condition  (OR +  = (Oi  <- _ 4  I f the d i s t r i b u t i o n function f o r the p a r t i c l e s i s a n i s o t r o p i c such that there are more p a r t i c l e s with p a r a l l e l components of v e l o c i t y s l i g h t l y l e s s than |ul than there are p a r t i c l e s with components s l i g h t l y greater, then energy w i l l be t r a n s f e r r e d to the wave.  Transverse Landau damping  7)  23  of the wave occurs i f the p a r t i c l e d i s t r i b u t i o n i s i s o t r o p i c (Scarf, 1962; S t i x , 1962).  I f 6J i s s p e c i f i e d , and the emission i s t o be of a t  c e r t a i n frequency, then the value f o r \u| can be c a l c u l a t e d from equation 4-7, and a lower bound f o r the energies of the p a r t i c l e s involved can be c a l c u l a t e d . In order t o s p e c i f y OJi , the earth's main magnetic f i e l d i s assumed to be a centered d i p o l e f i e l d with a value B«J = 0-3 on the surface of the earth at the geomagnetic equator.  I f X i s the  geomagnetic l a t i t u d e and L i s the Mcllwain coordinate (Mcllwain, 1961) i n t h i s case applied to a d i p o l e f i e l d , then the t o t a l f i e l d strength at a point with coordinates ( L , X ) i s given by  \*  U o  cos * 4  L represents the distance, measured i n u n i t s of earth r a d i i , that a given f i e l d l i n e i n the e q u a t o r i a l plane l i e s from the centre of the earth. For a dipole f i e l d L  where ^  e  =  I^T.  (4-9)  i s the geomagnetic l a t i t u d e at the point where the relevant l i n e  of force i n t e r s e c t s the earth's surface.  I t i s recognized that a dipole  representation of the earth's main f i e l d i s not perfect because of the compression on the daytime side but i t i s a good approximation and very easy to describe mathematically. Under the dipole model, iO- i s i n v e r s e l y p r o p o r t i o n a l to L^. L  At  very low frequencies, the wave's phase v e l o c i t y i s very nearly the A l f v e n v e l o c i t y . V , and using equation 4-7, the frequency of emission can be A  approximated by  24  6J  B  =  ^-7—  (4-10)  Using the density model of Smith (1961), the l o c a l e l e c t r o n number density i s l i n e a r l y p r o p o r t i o n a l t o the gyrofrequency, or N  P  ~ B ~ - j i  (4-n)  I t i s assumed that t h i s model holds not only i n the e q u a t o r i a l plane below L = 5, but t h a t i t i s v a l i d along f i e l d l i n e s away from the e q u a t o r i a l plane and at a l t i t u d e s which correspond to L values greater than about 5 ( B r i c e , 1964; Carpenter and Smith, 1964).  Using equation  4-10, i t can be seen that 6J  R  ~  -r;  1  ;—5TT  (4-12)  and i t can be seen t h a t f o r a given emitted frequency, the p o s i t i o n i n the magnetosphere at which the i n t e r a c t i o n takes place strongly determines the energy range of the p a r t i c l e s involved. Cornwall (1965) has suggested that since the data from the Explorer 12 s a t e l l i t e ,  f i r s t reported by Davis and Williamson, (1963) and  l a t e r i n more d e t a i l by Hoffman and Bracken (1965), i n d i c a t e a large f l u x of protons with energies of the order of hundreds o f kev at L — 3 . 5 , these protons may be very important i n emission processes.  The energy range i s  the r i g h t order f o r resonance i n the Pc 1 range i n a dipole f i e l d . Cornwall (1965) also mentions that 10 - 20 kev protons at L—  5.6 have been suggested as the energy source f o r the emissions.  Hoffman and Bracken (1965) have reported the presence of protons i n the region of the magnetosphere between these two extremes, with the f l u x o f low energy p a r t i c l e s increasing with increasing a l t i t u d e .  I f these  25  energies are i n the correct range, then resonance could occur on any l i n e of force having an L value between about 3.5 and 5.6.  Qbayashi (1965)  i n d i c a t e s that a l l hm emissions should occur i n the region between L = 4.0 and L = 5.6.  However, out of nine examples, Watanabe (1965c) found  no L values f o r the guiding l i n e of force below 4.98 f o r a dipole f i e l d . For the d i s t o r t e d dipole f i e l d which he used, t h i s value becomes 4.75. Taking i n t o account the o u t l i n e of hm emissions given i n Chapter I , the r e p e t i t i o n of r i s i n g tones separated by a constant time i n t e r v a l i s interpreted as an hm wave packet bouncing between ionospheric r e f l e c t i o n s i n the northern and southern hemispheres, being guided by the geomagnetic f i e l d l i n e s .  I t i s suggested that the wave i s strengthened  by the cyclotron i n t e r a c t i o n with the proton stream each time i t traverses the f i e l d l i n e .  The bounce period of hm waves has been c a l c u l a t e d  t h e o r e t i c a l l y by Jacobs and Watanabe (1965) as a function of the frequency and the L value, and i t involves c a l c u l a t i n g an i n t e g r a l numerically. Using t h e i r t a b l e , the bounce periods f o r a wave with a frequency of 1.3  cps  for L = 5.6 and L = 3.5 are found to be approximately 280 sec. and 60 sec. respectively.  Tepley and Wentworth (1964) mention that the r e p e t i t i o n  period of the r i s i n g tones i n hm emissions can vary from one to f i v e minutes so t h a t these values are not outstanding.  This r e s u l t i s  p h y s i c a l l y reasonable since the d i s p e r s i o n r e l a t i o n i n d i c a t e s that the phase v e l o c i t i e s of ion resonance mode waves tend toward zero as 6J« approaches 6o , and so, f o r a given frequency range, the wave goes slower c  at higher a l t i t u d e s since the cyclotron frequency decreases.  At the same  time, the path which the wave follows i s longer at higher a l t i t u d e s . Such d i f f e r e n c e s of r e p e t i t i o n period between d i f f e r e n t emissions i s noticeable even by making very rough measurements on d i f f e r e n t  26  dynamic spectra.  In the example presented by Cornwall (1965), the  period measured over the i n t e r v a l between 7 min. and 14 min. i s approximately 84 sec. with a mean d e v i a t i o n o f about 10 sec.  In the  sonogram given by Jacobs and Watanabe (1965), the period measured between 13^ hr. and 14 hr. i s never l e s s than 130 sec. these two spectra i s measureable.  The d i f f e r e n c e between  This type of measurement has been  made by Watanabe (1965c). The frequencies o f the emission i n the example above i n which the bounce period i s 130 sec. are low, around 0.3 cps.  I f t h i s event i s  to have occurred on the l i n e o f force given by L = 3.5, then at the l e a s t , the protons would have had t o have energies o f about 30 mev. At t h i s energy, the protons are r e l a t i v i s t i c and such p a r t i c l e s are not mentioned i n Cornwall's presentation (Cornwall, 1965). Besides the p a r t i c l e energy, the streaming p a r t i c l e density and the p i t c h angle parameter must be chosen.  Hoffman and Bracken (1965)  f i t t e d t h e i r data t o a p i t c h angle d i s t r i b u t i o n and found that the best f i t was made when they considered two ranges o f p i t c h angles separately, 0° to 30° and 30° t o 90°. The parameter values which gave a good f i t ranged from 1 t o 4 over the region L = 2 t o L = 7. Each c a l c u l a t i o n here i s done f o r 3" = 2. The r a t i o Ns/N  p  i s taken to-be 1 at the equator so  that r e l a t i v e sizes can be seen from the r e s u l t s , but the calculated rates are too large. 10~^.  growth  Jacobs and'Watanabe (1965) assumed the r a t i o t o be  The correct r a t i o probably v a r i e s from case t o case.  f a c t o r i n the choice of N  5  One l i m i t i n g  i s that the growth rate must be very much  smaller than the r e a l frequency. I t was mentioned above that Watanabe (1965c).found no emissions taking place on a l i n e of force with an L value l e s s than 4.98 i n nine  27  examples considered.  Cornwall (1965) suggests that the emissions would  not take place at lower a l t i t u d e s i f the r a t i o N /N S  i s too small and  P  t h i s might occur because the background plasma density i s large. A l l the c a l c u l a t i o n s have been done using an L value of 5.6. For each CJ , the quantity v* - V* X  i s set equal t o zero.  i s c a l c u l a t e d and i f v^ 4 v£ , 6 J  X  The wave frequencies have been varied from  6Ji(eq)/20 to 19 6J (eq)/20 and at each frequency, the growth rate i s £  c a l c u l a t e d f o r twenty-nine values of the geomagnetic l a t i t u d e from 0° to 29° i n half-degree steps. The Smith model of e l e c t r o n density (Smith, 1961) can be w r i t t e n N= P  \%,00O ^S*  (4-13)  This model has been used by B r i c e (1964) i n the form Xl = K  (3.T)^. i.OOO C J \  (4-14)  e  In the c a l c u l a t i o n s , equation 4-13 i s w r i t t e n N = P  v r . 5>5!>0 B c 7 v f  (4-is)  3  e  and using t h i s value f o r the density, the A l f v e n v e l o c i t y can be c a l c u l a t e d f o r the magnetosphere.  Values corresponding to p a r t i c l e energies i n the  range approximately 10 - 20 kev can then be assigned to Uj. 7 the value of V  A  at the e q u a t o r i a l plane i s 4.95 x 10  energy of 10 kev corresponds t o a v e l o c i t y of 1.38 x 10 case, U (eq) = 2.79. 5  For L = 5.6, -I  cm-sec.  A proton  cm-sec so i n t h i s  Since the v e l o c i t y increases as the square root of  the energy, f o r 20 kev, U (eq) = J~2 (2.79) = 3.95. 1,  s  In the programme, v  0  i s given the f o l l o w i n g seven values; 2.8V (eq), 3.0V (eq), ... , 4.0V (eq). ft  A  A  28  I t should be noted that equation 4-13 describes Np e m p i r i c a l l y and i s an average value.  Since Np can vary by as much as a f a c t o r of 2 at d i f f e r e n t  times, the c a l c u l a t e d r e s u l t s are not exact and d i f f e r e n c e s e x i s t between one s p e c i f i c example o f hm emissions and another.  The r e s u l t s  w i l l serve as an i n d i c a t i o n o f general e f f e c t s which r e s u l t from the mechanism which has been considered. The c a l c u l a t i o n s have been done at the U n i v e r s i t y of B r i t i s h Columbia Computing Centre on an I.B.M. 7040 computer using Fortran IV language. Results The l a r g e s t growth rate at any point was found t o occur at the e q u a t o r i a l plane ( X = 0) at a frequency o f 0.65 cps.  Figure 2-a i n d i c a t e s  how the growth rate v a r i e s with frequency and p a r t i c l e energy at the e q u a t o r i a l plane. i s narrow.  The lower c u t o f f i s very sharp and the peak, i t s e l f  Figures 2-b and 2-c show the same type o f p l o t f o r A = 10°  and X = 20° r e s p e c t i v e l y . I t can be seen that as X increases, the frequency band o f a m p l i f i c a t i o n moves toward higher frequencies f o r a given v . 0  This i s t o be expected and r e s u l t s from the requirement  which means that as 6J; increases with X ,  that  must increase i n order t o  a, r e s t r i c t the s i z e of V^. This e f f e c t can be seen i n another way by observing how CJj changes with X f o r s e v e r a l frequencies.  I t i s found that the growth  rate i s p r a c t i c a l l y zero f o r a l l frequencies l e s s than or equal t o  (  29  0.2 (^i (eq). Remembering that the l a r g e s t growth rate occurs near = 0.25 CJi (eq), f i g u r e  3-a gives the r e s u l t s f o r £*JR = 0.3  (eq).  At the higher energies, i t i s noticeable that the growth rate maximum occurs  near X — 10°.  In f i g u r e s 3-b and 3-C, t h i s e f f e c t i s much more  noticeable and as the wave frequency i s increased  s t i l l further(figures  3-d and 3 - e ) , the wave i s damped s l i g h t l y near the equator and amplified i n the region around 25° geomagnetic l a t i t u d e .  This damping e f f e c t occurs  near the equator because at that point on the f i e l d l i n e , the r a t i o i s smallest and i f X does not happen t o be large enough, the expression tf(k->t/6JR - 1) - 2 can e a s i l y be negative. For energies i n the range 10 - 20 kev, the sharp low frequency c u t o f f occurs at 0.2 (Ji(eq) and i t i s important t o note that the maximum proton energy determines t h i s c u t o f f point.  The proton energy also i s  very important i n determining the s i z e o f the growth rate.  30  U  s  = 4.0  X = o°  o to  OS o o  0.2  0.4  0T6  0.8  Normalized Frequency F i g . 2-a  Growth rate as a function of frequency at X = 0° f o r two p a r t i c l e energies. The frequency i s normalized to the e q u a t o r i a l cyclotron frequency, 16.4 s e c ! . -  1.0  31  F i g . 2-b  Growth rate as a function of frequency at A = 10° f o r two p a r t i c l e energies. The frequency i s normalized to the e q u a t o r i a l cyclotron frequency, 16.4 sec -'-. -  32  40 ~  X=  30  20  c  o TO 20 . U  U  s  = 4.0  o u o 10 -  0.2  0.4 ,  0.6  0.8  Normalized Frequency F i g . 2-c  Growth rate as a function of frequency f o r two p a r t i c l e energies f o r X = 20°. The frequency i s normalized to the e q u a t o r i a l cyclotron frequency, 16.4 s e c . - x  33  40V  6L) = 0.3 CJ: (eq) = 4.9 s e c " r  30h  U  20L  s  = 4.0  10  5 F i g . 3-a  10  15 20 Geomagnetic L a t i t u d e (degrees)  25  V a r i a t i o n of GO with X f o r CO^= 4.9 sec-1. x  40 L\ = 0.4 6 J i ( e q ) = 6.54  sec  - 2  30  20  U  s  = 4.0  10  10  15  20  25  Geomagnetic L a t i t u d e (degrees) F i g . 3-b  V a r i a t i o n of 6 J w i t h X f o r CO = 6.54 x  R  sec . - 1  1  1  5  10  15  20  25  Geomagnetic L a t i t u d e (degrees) F i g . 3-d  V a r i a t i o n of k\ with X f o r 60 = 11.4 s e c . - 1  K  35 CHAPTER V SUMMARY Discussion I t i s important to remember that the a n a l y s i s i s v a l i d only to f i r s t order.  The growth of waves can be i n d i c a t e d but soon a f t e r i t begins,  the l i n e a r theory becomes i n v a l i d and nothing f u r t h e r can be said about the behaviour of the system.  I t might happen that when the p a r t i c l e s have  l o s t a s u f f i c i e n t amount of t h e i r energy, many are dumped i n t o the ionosphere so that the wave growth becomes n e g l i g i b l e compared to i t s attenuation.  In t h i s case, the thermal background plasma may be an  important damping agent, but i t was assumed to have zero temperature i n the above a n a l y s i s . Kennel and Petschek (1966) have considered the s t a b i l i t y of trapped p a r t i c l e s i n d e t a i l and Cornwall (1966) and Watanabe (1966) have discussed some non-linear aspects of the problem. conceivable that  N  s  It is  might sometimes be large enough that the r a t i o  N /N s  p  i s of the order of unity and t h i s would i n v a l i d a t e the l i n e a r theory. I t i s also known that the earth's f i e l d i s not accurately represented by a dipole but t h i s representation makes the a n a l y s i s much simpler and general r e s u l t s can s t i l l be obtained. Considering only monoenergetic p a r t i c l e streams i s an overs i m p l i f i c a t i o n , although the v a r i a t i o n of t o with energy has been r  c a l c u l a t e d numerically.  Hoffman and Bracken (1965) i n d i c a t e that a  doubly sloped exponential energy spectrum describes w e l l the d i s t r i b u t i o n of proton f l u x e s which they observed.  I f t h e i r d e t a i l e d observations were  taken at lower energies, then the i n t r o d u c t i o n of an energy d i s t r i b u t i o n of the form e~^/^° where E r e s u l t s more q u a n t i t a t i v e .  e  i s e m p i r i c a l l y determined would make the  36  In order to determine more exactly the a c t u a l growth of a wave of some f i x e d frequency, the wave amplitude must be integrated over the region i n which  i s non-zero.  Jacobs and Watanabe (1965) have  used Sturrock's a n a l y s i s of growing waves (Sturrock, 1961) and shown that the mechanism which has been considered here gives r i s e to a non-convective i n s t a b i l i t y (the point where the i n s t a b i l i t y occurs i n i t i a l l y remains f i x e d i n space, although the disturbance can spread out around i t ) and they have discussed b r i e f l y the problem of how such a disturbance might come t o be observed on the earth's surface. In Chapter I I I , the theory presented by Tepley and'Wentworth (1964) has been discussed and references to the papers by Cornwall (1965) and Obayashi (1965) have been made i n several places. Gendrin (1965) and Hruska (1966) have considered the problem of c y c l o t r o n emissions i n the magnetosphere.  Both authors f i n a l l y consider  only the (L, e) and (R, p) i n t e r a c t i o n s where 'R' and 'L' r e f e r to l e f t and right-hand p o l a r i z e d waves and 'e' and 'p' r e f e r to e l e c t r o n s and protons respectively. Gendrin comments b r i e f l y on i n s t a b i l i t i e s which a r i s e when the transverse v e l o c i t y components of the p a r t i c l e s are important but suggests i m p l i c i t l y t h a t t h i s would never occur except near the mirror points. This assumption does not seem reasonable.  He suggests t h a t hm emissions  occur when super-luminal protons i n t e r a c t with R waves and describes the process of repeated emissions as Tepley and Wentworth (1964) do.  This idea  has been c r i t i c i z e d above. Hruska (1966) considers a plasma i n s t a b i l i t y by considering the net t r a n s f e r of energy between waves and p a r t i c l e s .  He chooses a s h i f t e d  37  Maxwellian d i s t r i b u t i o n f o r the streaming p a r t i c l e s but gives no reason f o r t h i s choice.  Since he takes the temperature d i s t r i b u t i o n t o be  i s o t r o p i c , the contra-streaming (L,p) and (R,e) i n t e r a c t i o n s do not give r i s e t o a growing i n s t a b i l i t y and he does not discuss them any further. Conclusions I t i s suggested that hm emissions r e s u l t from hm wave packets propagating along the earth's magnetic f i e l d l i n e s guided between ionospheric r e f l e c t i o n s i n the northern and southern hemispheres.  The  ion resonance mode of wave i s considered because i t i s guided by the earth's f i e l d at hydromagnetic frequencies much more than the e l e c t r o n resonance mode of wave and because the dispersion c h a r a c t e r i s t i c s o f the ion resonance mode are the same as the observed spectra o f structured hm emissions. Since the s i g n a l s sometimes increase i n i n t e n s i t y i n time before dying out, i t i s suggested that the wave packets gam energy v i a a c y c l o t r o n i n s t a b i l i t y process as they i n t e r a c t with low-energy protons which are trapped i n the magnetosphere.  An expression f o r the growth o f  the waves was developed s t a r t i n g from Maxwell's equations and the c o l l i s i o n l e s s Boltzmann equation. Choosing a pitch-angle d i s t r i b u t i o n function containing the f a c t o r sin*M'/B ' r e s u l t s i n an upper c u t o f f y/a  frequency o f k - V ( l +  at any given point.  I t i s not known whether t h i s  e f f e c t i s more important than the damping which r e s u l t s from the thermal background plasma. Computer c a l c u l a t i o n s ( f o r Y = 2 ) i n d i c a t e three important features o f the theory. F i r s t , the growth rate i s a sharply peaked  38  function of the frequency.  Second, changes i n the proton energy greatly  influence the magnitude of the growth rate as w e l l as the frequency of the steep lower c u t o f f .  This suggestion was also made by Qbayashi (1965).  F i n a l l y , the l a r g e s t growth rates f o r the i n s t a b i l i t y are found tb occur near the e q u a t o r i a l plane, although at some frequencies, there are two regions o f l a r g e s t growth, each s l i g h t l y removed from the equator by ten or twenty degrees. The observed l a t i t u d e dependence of hm emissions may be explained by the f a c t that none of the L waves i n the hm packet can have a frequency above the i o n cyclotron frequency at the geomagnetic equator and that i t decreases as the l a t i t u d e of the point where the l i n e of force i n t e r s e c t s the earth's surface increases.  Another important consideration i s the  e f f e c t of the ionosphere on the wave as i t t r a v e l s from the lower regions of the magnetosphere to the observation point on the earth's surface. Ionospheric wave guiding may r e s t r i c t the wave packet frequencies because of a l a t i t u d e v a r i a t i o n o f the duct c h a r a c t e r i s t i c s . I f measurements on wave propagation above the ionosphere could be made to determine p o l a r i z a t i o n s and i f d e t a i l e d records of low-energy proton f l u x e s (1 - 100 kev) could be obtained, many u n c e r t a i n t i e s i n the theories of hm emissions would be eliminated.  39 APPENDIX I  Transformation o f Equations and S o l u t i o n f o r f (y_,k,6j). t  Let G(t,z) represent any of the q u a n t i t i e s which are t o be transformed.  I f G(t,z) i s well-behaved, then the Fourier-Laplace transform  of G e x i s t s and i s defined by G(OJ,M= ( d t J j i i G t t . i ) t " o -eo  t (  **"  (Ai-i)  C o t )  For the F o u r i e r transform t o e x i s t i t i s s u f f i c i e n t that G(t,z) be of bounded v a r i a t i o n and absolutely integrable, i . e . , +00  (|&(t,^a*  < co  ( -2) A1  and i t i s implied that G(t,z)-^0 as z-* + oo.  In order t o assure existence  of the Laplace transform, i t i s convenient t o assume ( S t i x , 1962; S o k o l n i k o f f and Redheffer, 1958) that f o r some choice of the constants M and  , |Gct»u n^  (AI-3)  1  and ImGj>ja. Applying the transformations defined above t o the f o l l o w i n g component Maxwell equations  2* ^  c a t c  9t  ( A 1 _ 6 )  ( A 1  -  7 )  40  and using i n t e g r a t i o n by parts and the conditions o u t l i n e d above, i t i s found that (AI-8)  -c-fe & CoJ,&)= T T f c t o . f e ) -£[«.uE (u>,fe) + E x ( o , ^ y  t*B  l c  x  Cu)>)=My Cw>)-i[t^E ( y  y  t*E (co,IO=.  - i [ t ( 0 ^ ( w »  L^E^iuiM^  {[iufyCuM  y  W j  W+E Co M] Y  (Ai-9)  J  + & <©A)]  (Ai-io)  X  + B (o^)] y  (  A  1  _  n  )  The f i r s t - o r d e r Boltzmann equation which was derived i n Chapter I I (equation 2-7) can be w r i t t e n  +  c M -  V  y  W  l  =  °  (Al-12)  Applying the combined transformation defined by equation A l - 1 t o t h i s equation and e l i m i n a t i n g E (k>,k) and Ey(60,k) from the r e s u l t i n g x  equation  using equations Al-10 and Al-11, i t i s found that  i s the d i f f e r e n t i a l equation governing the transformed d i s t r i b u t i o n f u n c t i o n f (y,k,(0). x  41  I t i s now convenient t o transform coordinates from ( v , v ) t o x  y  (w,4>) using the f o l l o w i n g r e l a t i o n s 3V  9w'^v  X  ^.2-  3*'?^  K  (Al-14)  +  _  ( A 1  and i t was assumed above that 9f /94> vanishes. 0  Since w = v£ + v 2  2 5  1 5 )  i t is  easy t o show that _  -  w  - cos* -  ^  ( A 1  2 * = * = si* • - < ^ J L £ 1 _  _  ( A 1  1 6 )  .  1 7 )  By w r i t i n g ^ ( Y j k j O ) as a F o u r i e r s e r i e s in<p, i^.t.oi^Zr'e^*  <Ai-i8)  equation Al-13 can f i n a l l y be w r i t t e n  " i ^ M f - u)|fc • *2i°] .-«[  6 x C ( u  ,i)  + 1  This equation i s l i n e a r , non-homogeneous, f i r s t - o r d e r with constant c o e f f i c i e n t s and can be solved using standard methods.  Coddington (1961)  42  uses the notation y  1  + ay = b(x)  (Al-20)  where a i s a constant and b i s a continuous f u n c t i o n on an i n t e r v a l I. A l l s o l u t i o n s must have the form  = §CX) where x  0  =  e"  •ax  X re J  * b C t ) d t + c e -  a  i s i n I and c i s any constant.  ax  X  ( M  _  Note that i f the a n t i - d e r i v a t i v e  the integrand i s evaluated at x„, then t h i s 'constant 'c' i n m u l t i p l y i n g e~  4  1  2 1 )  of  can be grouped with  and the i n t e g r a l evaluated at the upper l i m i t of  'x' i s j u s t the i n d e f i n i t e i n t e g r a l . zero, and the r e s u l t i s the same.  The new constant i s then chosen to be  In t h i s way, the term i n v o l v i n g the i n d e f i n i t e  i n t e g r a l represents the p a r t i c u l a r s o l u t i o n of the inhomogeneous equation and the term containing the redefined constant i s the general s o l u t i o n of the homogeneous equation.  The redefined constant i s chosen to be zero  because £j_(v, z,t) must be zero when the wave and p a r t i c l e stream perturbations are removed, so the s o l u t i o n to the homogeneous equation must be dropped from the general s o l u t i o n of the inhomogeneous equation. s o l u t i o n f o r ^ ( v , k,(o) i s given by  T  Z_j  60--feu•'w^t  The  resulting  43 APPENDIX I I Determination of the Transformed Magnetic F i e l d s Applying the Fourier-Laplace transform defined i n Appendix I t o equation 2-13 gives  ^A)±c> (ojA)=Z S.UveT *w5 (yA w) t  y  1  and using the expression f o r f ^ ( ^ ,  (A2-1)  >  k,lo) found i n the f i r s t appendix, equation  A2-1 gives the current density (transformed) i n terms of the transformed magnetic f i e l d s .  An expression f o r the current density can a l s o be found  from the transformed Maxwell equations.  By e l i m i n a t i n g E (w,k) and E (o;,k) K  y  from the four transformed equations i n appendix I , i t i s found that i(CO*-C  6 {tOj (?) = -HTVC-  a  X  - CJ &„( » 0  + c& EyCoA)  (A2-2)  and  c(^-^\t)Z CLoM= y  H i r c f e y ^ w . ^ - w a y C o ^ )  - c ^ E x ( o ^ )  ( A 2  and hence  +cfe[e (o fe)±tE co^ x  y  y  ( A 2  _  Noting that when the s o l u t i o n f o r f (y,k,6j) i s s u b s t i t u t e d i n t o 1  equation A2-1, the i n t e g r a l i n v o l v i n g f ^ e m = + 1, i t i s found that  i l 1 n  ^  i s non-zero only when  4 )  _  3 )  44  ">*p  J  f  60 - -feu ^ 60  L  c  r  C : F l )  F i n a l l y , using equations A2-4 and. A2-5,  i t i s found that  ^ - . M -  (A2-6)  where  1  " J -  CO-fea^COc  (A2-7)  45 APPENDIX I I I  A n a l y t i c Continuation of the I n t e g r a l s The o u t l i n e of the procedure as given by S t i x (1962) w i l l not be copied,in d e t a i l .  There, the case of l o n g i t u d i n a l plasma o s c i l l a t i o n s i s  considered as an example.  The d i f f e r e n c e s f o r the case of transverse waves  i n t e r a c t i n g with a p a r t i c l e stream are noted.  The problem i s s i m p l i f i e d by  c a l c u l a t i n g the asymptotic value of B(t,k) as t-9°o . The expression f o r the magnetic f i e l d s given i n equation 2-14 i s v a l i d f o r Imk) >jX , where /* was defined i n Chapter I i n connection with the d e f i n i t i o n of the Laplace transform. equation 2-14 must be  The a n a l y t i c continuation of  determined.  I t i s assumed t h a t B ^ ( 0 , k ) , B ( 0 , k ) , Ex.(0,k), E ( o , k ) , y  r\  v  9f„/aw,  (-1)  2 f o / 9 u , and f j .  are a l l e n t i r e functions o f u.  One must then consider  i n t e g r a l s o f the form + 00  -oo  with \, 6J +• 60c V= ^ where the i n t e g r a l i s t o be taken along the r e a l a x i s . be an e n t i r e f u n c t i o n of u.  (A3-2) F(u) i s assumed t o  I f one i s considering u as a complex v a r i a b l e ,  then the path of i n t e g r a t i o n i n equation A3-1 can be changed i n accordance with complex v a r i a b l e theory. There are four cases, depending on the signs of k and  46  Case 1.  k>Q, Ux> 0. In t h i s case, Im(V)>0.  The path o f i n t e g r a t i o n can be raised  ( f i g . 10) from the r e a l axis above the s i n g u l a r i t y using the residue theorem. I(v)(p*tki) =• K V K P * ^ ) -  F(V)  (A3-3)  Case 2. k>Q,0Ji< Q. In t h i s case, Im(V)<0 and the a n a l y t i c continuation i s obtained by deforming the path of i n t e g r a t i o n down from the r e a l axis ( f i g . 11) so that KvKpaUi)  = Icv)Cp tka.)-^iF(v) a  ( A 3  _  4 )  and path 2 can e a s i l y be chosen t o be the r e a l a x i s . Case 3. k<0,  0->JL>  0.  In t h i s case, Im(V)<0.  The path i s taken along the r e a l axis and  i t i s t r i v i a l t o write ( f i g . 12) I ( V H p * t k l ) =• K V K p . t k i ) Case 4.  k<0,  (A3-5)  0.  Im(V)>0 and the continuation by contour deformation i s analogous to case 2.  From f i g u r e 13, i t can be seen that I C v K p o c t k l ) = K v H p a t K 2.)  (A3-6)  Using the residue theorem I ( V ) ( W a.cs) - K v X p a t K 2 ) « - ^  F^V)  ( A 3  _  7 )  47  path 2  X W  = V ->Re( )  path 1  Fig. 4  Contours f o r i n t e g r a t i o n when  Va  k> O  and 6 J "> ° X  48  path 2  V  4  = V  path 1  Fig. 7  Contours f o r i n t e g r a t i o n when k < O  and OJj< O  49  and so  I(VHp*^1)  = I ( V ) < W  «xi ) - i ^ F ( V ) S  (A3-8)  since k<0 i n t h i s case. I t can be seen from the above d i s c u s s i o n that the i n t e g r a l s over 'u' f o r k J j X ) are taken s t r i c t l y as they appear, along the r e a l u,', a x i s (cases 1 and 3).  F o r ^ i < 0, (cases 2 and 4 ) , again the i n t e g r a l s are taken  along the r e a l a x i s but there i s an a d d i t i o n a l residue term i n each o f the two cases. Following S t i x (1962), i t i s seen that both the numerator and denominator of B (W,k) + iBy(lO,k) are a n a l y t i c functions o f u i n the whole x  plane ( e n t i r e f u n c t i o n s ) and so the poles of B(k>,k) come only from the zeroes of the denominator.  There are two equations, one f o r each of  p o s i t i v e and n e g a t i v e ^ .  60  f o r lx) > 0, and x  A  +  6J - f c u * CUc  ™ If*  W - T O  w  - ^ ^ ^  +  i ^ U •VJ = °  ( A 3  -  1 0 )  f o r GJi< 0. Using the Cauchy p r i n c i p a l value ((P) n o t a t i o n , these two equations can be combined i n t o one equation which i s v a l i d f o r both p o s i t i v e and negative values o f the imaginary part o f 6J . (A3-11)  50  where the f u n c t i o n a l form of G i s given by frttiJ-jLJCl  7 ^ ) ^  + "7J  (A3-12)  51 APPENDIX IV  Wave P o l a r i z a t i o n Consider the case o f pure wave propagation when no p a r t i c l e stream perturbation i s present.  In t h i s case, D = 0, since f  A  = 0* "  Consider waves of the form  with only x and y components present.  L e f t hand p o l a r i z a t i o n i s defined by  writing tVt,"*) - K ^ y t t , ^ = O  ( A 4  _  2 )  I t then f o l l o w s from Maxwell's equations that E * ( t , % ) + £ E  y  C t ^ ) - 0  (A4-3)  The Fourier-Laplace transformation o f these two equations i s s t r a i g h t forward so that at t = 0, the condition B * ( 0 , k ) ± l B ( 0 > ) =r O  (A4-4)  E C0,fc) t  (A4-5)  y  implies that  x  (o,fe) = O  The t o t a l f i e l d i s w r i t t e n as the sum o f the separate f i e l d s of two waves of opposite p o l a r i z a t i o n .  6(to, fc)- B % , k ) +a cu) «,> a)  i  ( A 4  _  6 )  (A4-7)  52  where the superscript ( 1 ) i n d i c a t e s the left-hand mode and ( 2 ) the right-hand mode.  Using equations A4-6 and A 4 - 7 , i t i s found that £>(0,  l O = &\oM  + &\o,to  (A4-8)  = E ( o » +• E ( o »  (A4-9)  and  EioM  0 )  a )  S u b s t i t u t i o n of the above values o f B^(6J,k), By(6J,k), B ( 0 , k ) , B ^ O j k ) , x  Ex(Oj^) & an  Ey(0,k) i n t o equation 2-14 and choosing, say, the upper sign  everywhere, y i e l d s an equation i n v o l v i n g only the superscript ( 1 ) because those q u a n t i t i e s i n v o l v i n g the superscript ( 2 ) a l l vanish because of the d e f i n i t i o n of the r i g h t and l e f t modes o f p o l a r i z a t i o n . When the lower sign i s chosen, the r e s u l t i n g equation contains only the superscript ( 2 ) . Therefore, i n equation 2-14, the upper sign corresponds t o left-hand p o l a r i z a t i o n and the lower sign corresponds t o right-hand p o l a r i z a t i o n .  53 APPENDIX V  S i m p l i f y i n g the General Dispersion R e l a t i o n Equation 2-21 i s  (A5-1) The f i r s t and t h i r d terms can be approximated.using the f a c t that 6 0 * ^ 6J« *a<J*<Jx  (A5-2)  and CO  ^  t\J«  ±CJc  (A5-3)  The fourth term i n equation A5-1 contains the product o f G ( f ) and UJ s  and the product CO£H i s second order. At the- same time, two of the s  three terms i n G ( f ) contain a m u l t i p l y i n g f a c t o r 1/co, and when t h i s s  f a c t o r i s w r i t t e n i n the normal form o f a complex number with a r e a l denominator, 6J  tJfi + i W i  £o£+-6j£  UK  £0*  (A5-4)  Exactly the same procedure can be c a r r i e d out f o r the denominator of the integrand of the fourth term, with the r e s u l t that V|Gc(5Q  ^  W £(^s)  In the l a s t term, the m u l t i p l y i n g 60 can be replaced by6J*. s i m p l i f i c a t i o n can be made i n the l a s t term.  (A5-5) One f u r t h e r  Expanding the numerator of  54  the integrand i n a Taylor s e r i e s about u = V , i t i s found that R  «[w6(W] « u  (A5W5)  V ) t  since V - V ^ ^ a n d G ( f ) contains the f i r s t order quantity N . s  s  Using a l l the above approximations, equation A5-1 can be w r i t t e n  +  i  H ^  +  S  f  f e ^ i - ^  £  i  >  ^  4  -°  («-)  S e t t i n g the imaginary part o f t h i s equation equal t o zero gives equation 2-25.  A second equation r e s u l t s when the r e a l part i s set equal t o zero.  The fourth  term can be neglected since i t i s o f f i r s t order while the  t h i r d term i s zeroth order. r e l a t i o n given i n Chapter I.  This approximation y i e l d s the d i s p e r s i o n  55 APPENDIX VI  Coordinate Transformation and C a l c u l a t i o n of 6Jx In c y l i n d r i c a l coordinates (u,w)  Equation 3-9 gives f  i n s p h e r i c a l coordinates (V,HO as  s  i,*AS<v-v.) Siltlg.  (A6-2)  I t i s convenient t o transform G ( f ) using the chain r u l e s s  9w  ~ 0V  3W  0 u  9 V  * u  0q> SW  9  (A6-3a) (A6-3b)  u  and the transformation  V = +• (Wl4-  ) ^  (A6-4a)  »+> = tav>-' M!  (A6-4b)  Equations Al-3a and Al-3b then become  and u and w can be eliminated using the inverse transformation. However, the problem can be s i m p l i f i e d because the i n t e g r a l i n the numerator o f equation 2-25 must be evaluated at u = V . R  I t w i l l be  56  written (A6-6) In equations Al-5a and Al-5b, u and w are eliminated by p u t t i n g u = V , and R  w = + ( v - V^)3. 2  This r e s u l t f o r w implies that W<Jw » V<W  (A6-7)  and s  W  l  * . * . J v ^  ( A 6  .  8 )  Using these r e s u l t s , i t then f o l l o w s by algebra that  R  6_>  V**  L  R  V**  1  * w ;  i  C  V  V  '  )  V- ]  1  1  v ^ ^ J  ( -9) A6  with 60*  W  *  y has been r e s t r i c t e d t o values greater than -2. I f IVr\ > then the i n t e g r a l i s zero.  v  a  I f I v J = v , the i n t e g r a l may be d i f f e r e n t from 0  zero because of the presence of the d e l t a functions.  In t h i s case, )S must  57  be f u r t h e r r e s t r i c t e d i n order t o avoid d i v i s i o n by zero i n f a c t o r s containing the expression v^ - V*. i t can be seen that  Y^  From the t h i r d term i n the integrand,  -2, and t h i s i s weaker than the o r i g i n a l r e s t r i c t i o n .  The second term requires that ^ ^ 0.  In the f i r s t term, the presence of the  d e l t a - f u n c t i o n d e r i v a t i v e means that the function which m u l t i p l i e s i t i n the integrand must be f i r s t d i f f e r e n t i a t e d and then evaluated at v = v . 0  This  c a l c u l a t i o n leads t o an expression containing terms with the f a c t o r s (v*  - V * ) k and (v** - v j ^ * 1  1  .  I t i s seen, then, that f o r \\«\ = v., V  must be at l e a s t zero, and i n t h i s case, whether V i s zero or p o s i t i v e , the i n t e g r a l i s zero. This r e s u l t i s not unexpected since | v l R  = v  0  represents the  resonance condition f o r the case of protons i n t e r a c t i n g with contrastreaming left-hand p o l a r i z e d waves when the proton beam i s ' l i n e a r ' , that i s , when the beam has no transverse k i n e t i c energy.  Neufeld and  Wright (1963) have shown that f o r t h i s case, a contra-streaming  instability  does not e x i s t . I t i s found then, that f o r W*l < v„ and V> -2,  S u b s t i t u t i n g t h i s r e s u l t i n equation 2-25, i t i s immediately found that  ( A 6  -  U )  58  where N  s  i s defined by  Ns -  N s ( f ^  (A6-12  and represents the streaming p a r t i c l e density at a point where the magnetic f i e l d has the value B.  59 BIBLIOGRAPHY  A l f v e n , H., and C.-G. Falthammar, (1963), Cosmical Electrodynamics, Oxford U n i v e r s i t y Press. Astrom, E., (1950), On waves i n an ionized gas, Ark. Fys., 2, 443-457. B e l l , T.F., and 0. Buneman, (1964), Plasma i n s t a b i l i t y i n the w h i s t l e r mode caused by a gyrating e l e c t r o n stream, Phys. Rev., 133A, 13001302. Booker, H.G., (1962), Guidance of r a d i o and hydromagnetic waves i n the magnetosphere, J . Geophys. Res., 67, 4135-4162. B r i c e , N., (1963), An explanation of triggered VLF emissions, J . Geophys. Res. 68., 4626-4628. B r i c e , N., (1964), Fundamentals of very low frequency emission generation mechanisms, J . Geophys. Res., 69, 4515-4522. B r i c e , N., (1965), Generation of very low frequency and emissions, Nature, 206, 283-284.  hydromagnetic  Carpenter, D.L., and R.L. Smith, (1964), W h i s t l e r measurements of e l e c t r o n density i n the magnetosphere, Rev. Geophys., 2, 415-441. 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Drummond, McGraw-Hill Book Co. Sudan, R.N., (1962), Plasma electromagnetic i n s t a b i l i t i e s , Phys. F l u i d s , 6, 57-61. Tepley, L.R., and R.C. Wentworth, (1962), Hydromagnetic emissions, X-ray bursts and e l e c t r o n bunches, part 1: experimental r e s u l t s , J. Geophys. Res., 67, 3317-3333. Tepley, L.R., and R.C. Wentworth, (1964), Cyclotron e x c i t a t i o n of hydromagnetic emissions, Rep. Contr. NAS5-3656, Lockheed M i s s i l e s and Space Co. Watanabe, T., (1964), D i s t r i b u t i o n of charged p a r t i c l e s trapped i n a varying strong magnetic f i e l d (one-dimensional case).with a p p l i c a t i o n s to trapped r a d i a t i o n , Can. J . Phys., 42, 1185-1194. Watanabe, T., (1965a), Private communication  (May).  Watanabe, T., (1965b), Private communication  (October).  Watanabe, T., (1965c), Determination of the e l e c t r o n d i s t r i b u t i o n i n the magnetosphere using hydromagnetic w h i s t l e r s , J . Geophys. 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