UBC Theses and Dissertations

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UBC Theses and Dissertations

Planetary effects on magnetic activity Atkinson, Gerald 1964

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PLANETARY EFFECTS ON MAGNETIC ACTIVITY  by  GERALD ATKINSON B . A . S c , U n i v e r s i t y o f B r i t i s h Columbia,  i960  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n t h e Department of GEOPHYSICS  We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e required standard  THE UNIVERSITY OF BRITISH COLUMBIA August, 1964.  In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree- that the Library s h a l l make i t f r e e l y available for reference and study,  I further agree that per-  mission for extensive copying of t h i s t h e s i s ' f o r scholarly purposes may be granted by the Head of my Department or by his representatives. I t i s understood that,copying or p u b l i cation of t h i s thesis.for f i n a n c i a l gain s h a l l not be allowed without my written permission.?  Department of  Geophyslics  The U n i v e r s i t y of B r i t i s h Vancouver 8, Canada no>.  5  Columbia,  September,, 196*+  i  ABSTRACT  S t a t i s t i c a l evidence i n d i c a t e s t h a t the p o s i t i o n s of the moon, Mercury,  and Venus a f f e c t  magnetic  a c t i v i t y f r e q u e n c y observed a t t h e e a r t h , and the p o s i t i o n of t h e e a r t h a f f e c t s the f r e q u e n c y of b l u e c l e a r i n g s on Mars.  T h i s study shows t h a t these e f f e c t s may be ex-  p l a i n e d as a r e s u l t o f t h e a c t i o n of shock and bow waves formed by these b o d i e s i n t h e s u p e r s o n i c a l l y i n t e r p l a n e t a r y plasma.  streaming  The a t t e n u a t i o n o f l a r g e k i n e t i c  energy v a r i a t i o n s i n t h e s t r e a m i n g plasma b e h i n d  such  b o d i e s i s shown t o be e q u a l t o t h e square o f t h e r a t i o of t h e Mach number upstream  t o t h e Mach number downstream.  F o r t y p i c a l s o l a r i n d u c e d a c t i v i t y , t h i s I m p l i e s an a t t e n u a t i o n c o e f f i c i e n t o f a p p r o x i m a t e l y 1/2 - 1/3.  It is  a l s o shown t h a t an a c t i v i t y i n c r e a s e i s expected i n the bow wave.  The o b s e r v a t i o n a l d a t a f i t s a model w i t h bow  waves o f Mach numbers 2.5 and 15 c o r r e s p o n d i n g t o t h e two bow waves p r e d i c t e d by t h e t h e o r y .  The moon's e f f e c t  v a r i e s from t h a t o f t h e p l a n e t s i n a manner t h a t can be e x p l a i n e d by i t s c l o s e n e s s t o t h e magnetosphere.  vii  ACKNOWLEDGMENT  I am d e e p l y i n d e b t e d t o Dr. T. Watanabe b o t h f o r h i s p a t i e n t s u p e r v i s i o n , and f o r h i s many hours o f d i s c u s s i o n of the ideas expressed i n t h i s t h e s i s . would a l s o l i k e t o thank P r o f e s s o r  J . A. Jacobs f o r h i s  encouragement d u r i n g t h e p r e p a r a t i o n f o r h i s guidance i n i t s f i n a l  I  o f t h i s work, and  formulation.  ii  TABLE OP CONTENTS Page I  INTRODUCTION  1  II  THE BOW WAVE  3  a.  Hydromagnetic shocks  3  b.  Hydromagnetic waves  7  c.  Hydrodynamic  d.  D i f f e r e n c e s between hydromagnetic  b l u n t body problem  11 14  and hydrodynamic bow waves a t l a r g e d i s t a n c e s f r o m t h e source III  IV  V  20  INTERPLANETARY SPACE a.  The s o l a r wind  b.  P a r t i c l e clouds  c.  Use o f magnetohydrodynamic  d.  Supersonic  20  :  flow  PREVIOUS SHOCK WORK  20 theory  21 24 25  a.  K e l l o g g (1962)  25  b.  S p r e i t e r and Jones (1963)  26  c.  Obayashl (1964)  26  d.  Beard (1964)  26  CONDITIONS UNDER WHICH SHOCKS FORM  29  a.  Body w i t h a magnetosphere  29  b„  Immersed c o n d u c t i n g body  29  iii Page VI  MAGNETIC ACTIVITY AT LARGE DISTANCES FROM THE SOURCE  VII  31  a.  A c t i v i t y minimum  31  b.  A c t i v i t y maxima  33  EXPERIMENTAL EVIDENCE AND COMPARISON WITH THE THEORY  38  a.  The p l a n e t s  38  b„  The moon  49  VIII  CONCLUSIONS AND SUMMARY  52  IX  BIBLIOGRAPHY  55  X  APPENDIX  58  iv  TABLES  Page  I  P r o p e r t i e s o f t h e s o l a r wind and p a r t i c l e c l o u d s near t h e e a r t h .  II  V a l u e s o f r e l e v a n t parameters i n i n t e r p l a n e t a r y space.  III  22  23  E x p e r i m e n t a l r e s u l t s i n terms o f Mach a n g l e and Mach number  43  APPENDIX  Symbols and c o n v e n t i o n s used i n this thesis.  58  V FIGURES  Geometry f o r t h e shock  equations  P o l a r v e l o c i t y diagrams f o r magnetohydrodynamic waves (a)  V  A  (b) -Vs  Supersonic  >  V  >  V4  5  f l o w p a s t a sphere  The bow wave (a)  steady s t a t e f l o w  (b)  s i n u s o i d a l Mach number - slowly varying  Mach V and wave p a t h Mach cone and wave f r o n t Geometry f o r K e l l o g g s 1  equation  P l o t of a t t e n u a t i o n c o e f f i c i e n t against  M|  Geometry f o r t a b l e I I I Magnetic a c t i v i t y f r e q u e n c y  - Venus  (a)  g r e a t storms  (b)  o c c a s i o n s when  (c)  f o u r s e t s o f d a t a , each s e t  Kp  g i v e n e q u a l weight  ^30  vi  Magnetic a c t i v i t y f r e q u e n c y - Mercury (a) g r e a t  storms  (b) f o u r s e t s o f d a t a , each  obser-  v a t i o n g i v e n e q u a l weight B l u e c l e a r i n g f r e q u e n c y - Mars Magnetic a c t i v i t y f r e q u e n c y - Moon (a)  112 g r e a t e s t storms )  (b)  s m a l l storms  (c)  o c c a s i o n s when Kp^3o)  (d)  a l l disturbances  (e)  storms  ]  ]  A sudden commencement ) B g r a d u a l commencement]  Earth-magnetosphere-moon system, drawing  scale  1  I.  INTRODUCTION  Bigg  (1963  b) has p r e s e n t e d  statistical  e v i d e n c e f o r l u n a r and p l a n e t a r y i n f l u e n c e s on magnetic a c t i v i t y as o b s e r v e d a t t h e e a r t h ' s s u r f a c e . peaks o f a c t i v i t y f r e q u e n c y d i s t a n c e s from the sun-earth  There a r e  when t h e p l a n e t i s a t l a r g e l i n e , i n d i c a t i n g an e f f e c t  o v e r a r e g i o n o f much g r e a t e r s i z e t h a n t h e p h y s i c a l d i mensions o f t h e body.  This t h e s i s attempts t o e x p l a i n  t h i s phenomenon as t h e r e s u l t o f t h e bow shock wave formed by an o b j e c t i n a s u p e r s o n i c  stream.  The o b j e c t i s a  p l a n e t o r t h e moon, and t h e stream i s t h e s o l a r wind and any p a r t i c l e c l o u d s o r streams which a r e e m i t t e d by t h e sun.  The f l o w i s s u p e r s o n i c  t o magnetohydrodynamic waves  within i t .  The tional,  symbolism used throughout w i l l be conven-  and i s l i s t e d i n t h e Appendix.  I t w i l l n o t be  f u r t h e r e x p l a i n e d w i t h i n t h e t e x t e x c e p t where t h e a u t h o r f e e l s that i t i s unclear or unconventional.  2  3  II.  a.  THE BOW WAVE  Hydromagnetic  shocks  The b a s i c r e l a t i o n s f o r hydromagnetic a r e t h e De H o f f m a n n - T e l l e r e q u a t i o n s .  shocks  An e x c e l l e n t  d e r i v a t i o n o f them i s g i v e n by B e r s h a d e r (1959* PP. 1 8 - 2 0 ) I f a r e c t a n g u l a r c o o r d i n a t e system i s chosen such t h a t the x a x i s i s normal t o t h e p l a n e o f t h e shock f r o n t and the y and z axes l i e w i t h i n t h e p l a n e , t h e e q u a t i o n s have the f o l l o w i n g form: (1)  C o n s e r v a t i o n o f mass  = (2)  Conservation of f i e l d  a;  b.  B  =  O lines  O  B^ -T2.  c.  (3)  B^  - i "2B •x. U  C o n s e r v a t i o n o f momentum  a.  p +- jO U.  B,  =  0  4  -12.  (4)  b.  =  O  c.  =  o  C o n s e r v a t i o n o f energy -I.E.  v  [  = 0  -  1' denotes t h e v a l u e o f t h e e x p r e s s i o n i n t h e b r a c -  k e t s b e h i n d t h e shock minus i t s v a l u e i n f r o n t o f t h e shock.  These e q u a t i o n s a r e t o o c o m p l i c a t e d t o use i n t h i s f o r m i n an a n a l y s i s o f t h e bow shock wave i n a t h r e e dimensional s i t u a t i o n . (1963)  K e l l o g g ( 1 9 6 2 ) , S p r e i t e r and Jones  and Obayashi (1964) use t h e Rankine-Hugoniot  t i o n s f o r an o r d i n a r y gas, t a k i n g v a l u e s o f Y —  rela—7^—  from 5 / 3 t o 2 depending on t h e number o f d i m e n s i o n s i n v o l ved i n t h e c o m p r e s s i o n .  The a s s u m p t i o n i n t h i s approach i s t h a t which s i m p l i f i e s f o l l o w i n g form:  = 0,  t h e De H o f f m a n n - T e l l e r r e l a t i o n s t o t h e  5  (5)  C o n s e r v a t i o n o f mass -12. pUyC  (6)  =  0  Conservation of f i e l d  lines -12.  (7)  = 0  B  u«. C o n s e r v a t i o n o f momentum 2-  a  -i2  D 2  = 0  - 0  b.  where Uf  ( u s i n g 5)  O  denotes t h e component of u* t a n g e n t i a l t o t h e  shock p l a n e . (8)  C o n s e r v a t i o n o f energy 12. =  o  As B i s assumed t o be everywhere p e r p e n d i c u l a r t o t h e shock f r o n t , i t e x e r t s o n l y a p r e s s u r e a c r o s s t h e f r o n t , and can t h e r e f o r e be t r e a t e d as a gas ( T = 2) o f p r e s s u r e p^ag (9)  =  -g-p—  .  The e q u a t i o n s t h e n become:  C o n s e r v a t i o n o f mass r - . r  n 2  =  O  6  (10)  Conservation of f i e l d T2. u,  (11)  =  8  lines O  C o n s e r v a t i o n o f momentum -rZ  -1 2.  P'-V  a. 1 2.  b.  =  pu-x?  =0  O  (12) C o n s e r v a t i o n o f energy -t 2.  -I 2. -  2- U where  o  1° P  =  and  If  p«,qej »  p  ;  =- 2. , and i f  p»  I n t h i s form, e q u a t i o n s 9 , 11, 12 a r e i d e n t i c a l w i t h t h e Rankine-Hugoniot e q u a t i o n s .  E q u a t i o n 10 i s an e x t r a one  w h i c h merely s e r v e s t o s p e c i f y t h e magnetic  field.  7  The l a r g e s t e r r o r c r e a t e d by the assumption that  Bjc  = 0 l i e s i n the n e g l e c t of magnetic t e n s i o n  terms t h a t s h o u l d a r i s e f r o m f i e l d l i n e s c r o s s i n g t h e interface.  I f the f i e l d i n the i n c o m i n g plasma i s p e r -  p e n d i c u l a r t o the v e l o c i t y , and i s i n c i d e n t on a s p h e r i c a l shock f r o n t , B i s p a r a l l e l t o the f r o n t o n l y on a g r e a t c i r c l e p e r p e n d i c u l a r t o if.  The a p p r o x i m a t i o n  may  t h u s be q u i t e good on a s t r i p near the g r e a t c i r c l e ,  but  f a r t h e r back towards the a x i s of t h i s c i r c l e magnetic t e n s i o n terms may become i m p o r t a n t .  I t must f u r t h e r be n o t e d i n a p p l y i n g t h i s t o the e a r t h t h a t i n s i d e the shocked r e g i o n (from s a t e l l i t e measurements).  theory  ~2L|0U^~'  Thus t h e hydrodynamic  for-  ces do n o t predominate o v e r t h e f i e l d f o r c e s as i s the case i n t h e i n c i d e n t stream.  b.  Hydromagnetic waves  I n d e a l i n g w i t h the bow wave a t d i s t a n c e s l a r g e compared t o the d i m e n s i o n s of the body, a wave approach i s e a s i e r t h a n an e x t e n s i o n of the d i s c u s s i o n of one s i o n a l shocks.  At f r e q u e n c i e s n e a r and below the i o n c y c l o -  t r o n f r e q u e n c y ( tO^ tromagnetic  dimen-  ), t h e r e a r e t h r e e modes by which e l e c -  waves can p r o p a g a t e .  T h i s t h e s i s i s concerned  m a i n l y w i t h low f r e q u e n c i e s and w i l l h e n c e f o r t h r e f e r t o them by t h e i r hydromagnetic names.  The nomenclature of  8  Denisse  and D e l c r o i x (1963) w i l l be used.  w i l l be l i m i t e d f o r t h e p r e s e n t t o t h e case  The d i s c u s s i o n «0 <£<L u ) ;  ,  and t o dense p l a s m a s — p l a s m a s i n which t h e hydromagnetic v e l o c i t i e s a r e much l e s s than t h e speed o f l i g h t .  Under  such c o n d i t i o n s , a l l t h r e e modes a r e n o n - d i s p e r s i v e , and, to t h e f i r s t o r d e r , e l e c t r i c a l l y n e u t r a l .  The i o n s and  e l e c t r o n s move t o g e t h e r .  The  o b l i q u e A l f v e n mode has a phase v e l o c i t y : 2"  ( 1 3 )  Vp where  ©  ' —  \/ V  a  cos ©  A  i s t h e angle between t h e p l a n e wave p r o p a g a t i o n  vector ( k  ) and t h e steady magnetic f i e l d  p a r t i c l e displacements  ( 6  e  )•  and t h e p e r t u r b i n g magnetic  a r e p e r p e n d i c u l a r t o G» and k e  The field  , so t h a t t h e r e i s no  c o u p l i n g w i t h t h e o t h e r two modes which a r e c o n f i n e d t o the p l a n e d e f i n e d by B g i v e n by ing  V<$ — VA  e  and  k  .  The group v e l o c i t y i s  , and i s p a r a l l e l t o  80  , indicat-  t h a t energy t r a n s f e r c a n take p l a c e o n l y a l o n g t h e  f i e l d l i n e s i n t h e o b l i q u e A l f v e n mode.  The  a c c e l e r a t e d and r e t a r d e d m a g n e t o a c o u s t i c  modes have phase v e l o c i t i e s g i v e n by:  v  P  -  2  the + and - s i g n s i n d i c a t i n g a c c e l e r a t e d and r e t a r d e d modes  9  V5 = y  respectively.  i s t h e speed o f "sound"  i n t h e plasma i . e . t h e speed a t w h i c h a c o m p r e s s i o n wave w i l l propagate t h r o u g h an unmagnetized plasma.  The p a r -  t i c l e v e l o c i t y component r a t i o s a r e g i v e n by:  V*  tan8  V  Vp"  where t h e c o o r d i n a t e axes a r e d e f i n e d such t h a t x i s n o r mal t o Bo and  V£  and k  , and z i s p a r a l l e l t o k  .  I f V\  a r e o f t h e same o r d e r o f magnitude, t h e motions  of t h e a c c e l e r a t e d and r e t a r d e d mode p a r t i c l e s a r e n o t o r t h o g o n a l , and t h e r e i s c o u p l i n g between t h e two modes. P o l a r v e l o c i t y diagrams a r e shown f o r V  s  >  VA  VA >VS  *  i n f i g u r e 2.  For  t h e two extreme c a s e s :  and  , t h e c o u p l i n g becomes s m a l l . the  A N C  a c c e l e r a t e d mode has a phase v e l o c i t y  v  I f \^ »  \4  V,  ,  p  and t h e p a r t i c l e m o t i o n i s p e r p e n d i c u l a r t o B©  .  >  This  mode i s a c o m b i n a t i o n o f an A l f v e n wave a l o n g t h e f i e l d and a c o m p r e s s i o n a l wave a c r o s s t h e f i e l d . has a phase v e l o c i t y  Vp  m o t i o n and energy t r a n s f e r v e l o c i t y being  Vs  - V  s  cos G  The slow mode  , and t h e p a r t i c l e  a r e p a r a l l e l t o Bo , t h e group  . C o n v e r s e l y i f V§ >> VA  f a s t mode has a phase v e l o c i t y m o t i o n i s p a r a l l e l t o l<  Vj=  V4  >  T N E  • The p a r t i c l e  I t i s a combination of a  (b)  V >V  figure  8  2.  A  tolas? v e l o c i t y diagrams f©3? fflagn§t©hydi?©« dynamiG wav§§,  11  sound wave a l o n g t h e f i e l d l i n e and a c o m p r e s s i o n a l wave p e r p e n d i c u l a r t o t h e f i e l d . Vp  a phase v e l o c i t y  -  The slow mode has  VA c o s ©  .  I t I s very  l i t t l e d i f f e r e n t from an o b l i q u e A l f v e n wave, and r e p r e s e n t s energy t r a n s f e r a l o n g t h e f i e l d of  lines.  In either  t h e s e extreme c a s e s f r o m any g i v e n s t a t i o n a r y s o u r c e ,  t h e r e s h o u l d be almost s p h e r i c a l r a d i a t i o n f r o n t s i n the  a c c e l e r a t e d m a g n e t o a c o u s t i c mode, and r a d i a t i o n  a l o n g t h e f i e l d l i n e s i n t h e r e t a r d e d magnetoacoustic and A l f v e n modes.  A p p l y i n g t h i s t o t h e bow wave produced i n s t e a d y s u p e r s o n i c f l o w , t h e a c c e l e r a t e d mode s h o u l d produce an almost c i r c u l a r cone, and t h e slow mode and t h e A l f v e n mode s h o u l d produce t w o - d i m e n s i o n a l V s .  The V w i l l n o t  be symmetric about t h e f l o w d i r e c t i o n e x c e p t i n t h e case when t h e i n t e r p l a n e t a r y f i e l d i s p e r p e n d i c u l a r t o t h e streaming v e l o c i t y .  I t w i l l become more symmetric  with  i n c r e a s i n g Mach number.  c.  Hydrodynamic b l u n t body problem  There have been a number o f a t t e m p t s t o c a l c u l a t e the  f l o w near a sphere submerged i n a s u p e r s o n i c stream:  Lin  and Rubinov  (19^8);  D u g u n d j i ( 1 9 4 8 ) ; H i d a ( 1 9 5 3 ) ; Van  Dyke and M i l t o n ( 1 9 5 8 ) ; Van Dyke, M i l t o n and Gordon ( 1 9 5 9 ) ;  12  Inouye,  Mamenu and Lomax  results  by n u m e r i c a l a n a l y s i s ,  tained to  analytical  assume  flow  H i d a assumes et  conditions  assume  are  sphere because perimental that  of  the  i n figure  In three  disturbances  a body  axially  lines are tion the  is  s i m i l a r to  the  rear  to  3.  It  c a n be  that  However,  despite  observed  at  distances  H.A.  size,  Wilson,  For a i r c r a f t  e.g.  Jr., sonic  flow,  Ex-  this  sonic  1962)  effect,  shock  Ap  the  —37-  lines  rarefac-  shocks  for  will  are  the  often  compared  small  at-  indicate.  into  booms f r o m a i r c r a f t  ^  a s Mach  and w a i t  "eating  from a source  If  These  that  and bow waves f r o m  booms,  small  and v i s c o s i t y  rarefaction  shock".  source  the  similar  such a d i s t u r b a n c e  referred  as the  seen  front  This  the  of  t h e s e c a n be t r e a t e d  more r a p i d l y t h a n g e o m e t r y  great  flow.  a pattern  two d i m e n s i o n a l c a s e .  showing  to  whereas  a l o n g Mach c o n e s .  tenuate is  example,  irrotational  that  propagate  c a t c h up w i t h t h e  shock,  and  formed.  symmetric  shown I n f i g u r e tends  Thus f o r  dimensional supersonic  amplitude  shock,  a p p r o x i m a t i o n made.  indicates  3 Is  ob-  generally  i r r o t a t i o n a l flow  of  obtained  three  i n a region i n front  type  observation  first  through the  compressible  v a l i d only  three  The a p p r o a c h i s  behind i t .  incompressible al  The r e s u l t s  to  solutions.  The l a s t  and t h e  Rankine-Hugoniot flow  arbitrary  V a n Dyke  (1962).  (25  to miles,  boats.  (H.A. Wilson,  Jr.,  13  14  1962).  T h i s I m p l i e s t h a t the energy  The  shock i s a s y m p t o t i c  angle a r c s i n —^-—  where M  the i n c i d e n t stream.  (  &z  r" '* -1  t o a Mach cone w i t h i s the Mach number of  I t approaches t h i s cone as i t  d e g e n e r a t e s i n t o an o r d i n a r y s m a l l a m p l i t u d e b a s i c w a v e l e n g t h of the d i s t u r b a n c e w i l l be  wave.  The  somewhat  l a r g e r t h a n the body s i z e .  I f the f l o w i s q u a s i - s t e a d y  ( v a r i a t i o n s of  much g r e a t e r l e n g t h than the body s i z e ) , the p r i n c i p a l d i s t u r b a n c e produced s h o u l d be the bow  wave.  F i g u r e 4b  shows the type of p a t t e r n t h a t would be produced by cons t a n t speed s i n u s o i d a l l y , v a r y i n g Mach number f l o w implies density fluctuations). i n s t r e n g t h and d i r e c t i o n . stream,  a r e c e i v e r at  o f f r e q u e n c i e s and  d.  A  T h i s bow  (which  wave v a r i e s b o t h  I n an i r r e g u l a r l y v a r y i n g would observe a g r e a t v a r i e t y  amplitudes.  D i f f e r e n c e between hydromagnetic and hydrodynamic bow  waves a t l a r g e d i s t a n c e s f r o m the  source.  Perhaps the g r e a t e s t m o d i f i c a t i o n of the hydrodynamic p i c t u r e a r i s e s f r o m the e x i s t e n c e of t h r e e modes of p r o p a g a t i o n  i n a plasma.  The-accelerated  magneto-  a c o u s t i c mode w i l l f o r m a Mach cone, and the o b l i q u e  15  F i g u r e 4.  The bow wave. (a) s t e a d y s t a t e f l o w ; (b) s i n u s o i d a l Mach number - s l o w l y v a r y i n g .  16  A l f v e n and slow m a g n e t o a c o u s t i c modes Mach V's as t h e i r group v e l o c i t i e s a r e p a r a l l e l t o t h e magnetic  field.  The A l f v e n mode has t h e same group v e l o c i t y a s one o f the o t h e r two, and c o n s e q u e n t l y o n l y one cone and one V w i l l be i n e v i d e n c e .  Over a time average o f t h e type  used by B i g g (1963 a, b ) , the f i e l d may be e x p e c t e d t o t a k e up many d i f f e r e n t o r i e n t a t i o n s , and a l l modes o f p r o p a g a t i o n may be expected t o be i n e v i d e n c e .  The Mach a n g l e ( © ) f o r a V w i l l be a f u n c t i o n of t h e a n g l e between t h e i n t e r p l a n e t a r y v e l o c i t y as w e l l a s t h e Mach number.  f i e l d and t h e  F i g u r e 5 shows t h e  geometry of t h e s i t u a t i o n , g i v i n g t h e e q u a t i o n : < ) 16  (V)  -  1  s i n (y sm ©  I n t h e s p e c i a l case 6>c J _ U  ( 1 7 )  M ,  S)  , t h i s reduces t o t h e form:  =  L  S i m i l a r l y , f o r t h e cone, e l l i p s o i d a l wave f r o n t s make t h e relation (18)  _  -  1  sin  0  which i s e x a c t f o r s p h e r i c a l wave f r o n t s , o n l y approximate I f t h e r e i s a l a r g e d i s c r e p a n c y i n t h e v a l u e s o f V£ and  F i g u r e 6.  Mach cone and wave f r o n t .  18  , t h e wave f r o n t s w i l l be a p p r o x i m a t e l y s p h e r i c a l and e q u a t i o n  If  (18) c a n be used.  ^  ~  VA > c o u p l i n g o c c u r s between t h e  two magnetosonic modes, i n d i c a t i n g t h a t a wave o f one type would a c t as a source f o r t h e o t h e r .  I f one i s  much l a r g e r than t h e o t h e r , t h i s e f f e c t i s n e g l i g i b l e . There i s a l s o no c o u p l i n g f o r p r o p a g a t i o n p a r a l l e l t o the f i e l d ,  as one o f t h e modes becomes an A l f v e n wave.  P e r p e n d i c u l a r t o t h e f i e l d , t h e r e i s o n l y one mode o f propagation, exist.  and a s such a d i s t i n c t peak o f energy must  At o t h e r a n g l e s , t h e b e h a v i o r i s v e r y  complicated,  b e i n g a regime i n which t h e f a s t mode i s c h a n g i n g f r o m a combined A l f v e n and p e r p e n d i c u l a r p r e s s u r e wave t o a comb i n e d l o n g i t u d i n a l p s e u d o s o n i c and p e r p e n d i c u l a r wave.  pressure  The energy i s p r o b a b l y spread over a l a r g e r e g i o n  of space, each mode a c t i n g a s a source f o r t h e o t h e r . The  e x i s t e n c e o f u n c o u p l e d modes p a r a l l e l and p e r p e n d i c u -  l a r to the f i e l d ments p r e s e n t e d  s h o u l d r e t a i n t h e v a l i d i t y o f t h e argui n t h i s t h e s i s d e s p i t e t h e c o u p l i n g , and  s h o u l d a c c o r d i n g l y produce magnetic a c t i v i t y peaks a t t h e same p o s i t i o n s a s f o r u n c o u p l e d  As  W  ~  propagation.  u>i i n t h e shock f r o n t i t s e l f , t h e  plasma w i l l be d i s p e r s i v e , p a r t i c u l a r l y f o r t h e r e t a r d e d mode which r e a c h e s a resonance p o i n t a t t h i s  frequency  19  (Denisse  and D e l c r o i x 1 9 6 3 ) .  s m a l l f o r t h e o t h e r modes.  The d i s p e r s i o n w i l l be The e f f e c t w i l l be t o break  the r e t a r d e d mode i n t o a d i s p e r s e d t r a i n o f waves. I n such a t r a i n , each f r e q u e n c y would tend t o f o l l o w i t s own Mach V o r cone.  The geometric a t t e n u a t i o n o f a c c e l e r a t e d mode waves w i l l be s i m i l a r t o t h a t o f sound waves, w i t h an energy drop o f f ©d  a l o n g t h e cone.  g u i d e d modes, t h e r e s h o u l d be no g e o m e t r i c with distance.  For the  attenuation  However I n a l l c a s e s t h e r e s h o u l d  still  be t h e e q u i v a l e n t o f t h e r a r e f a c t i o n e a t i n g i n t o t h e shock.  20  I I I . INTERPLANETARY SPACE  a.  The s o l a r wind  The s o l a r wind i s b e l i e v e d t o be t h e expanding s o l a r c o r o n a , which P a r k e r (1963) has shown t o be uns t a b l e — t h e sun's g r a v i t a t i o n a l f i e l d b e i n g i n s u f f i c i e n t t o c o n t a i n t h e h i g h temperature (10^°K) gas.  I t takes  about 5 days t o p i c k up speed, and t h e n about f o u r more days t o r e a c h t h e e a r t h ' s o r b i t . of the  As t h e k i n e t i c  energy  t h e gas i s much g r e a t e r t h a n t h e magnetic f i e l d  energy,  f i e l d l i n e s from t h e sun a r e dragged out w i t h t h e  f l u i d , f o r m i n g s p i r a l s due t o t h e h o s e p i p e e f f e c t 1963).  (Parker  I t i s , however, a v e r y " g u s t y " wind, and t h e f i e l d  c o n t a i n s many k i n k s and w i g g l e s .  The p r o p e r t i e s o f t h e  wind as measured by s a t e l l i t e s ( E x p l o r e r X, M a r i n e r I I , L u n i k I and I I ) i n t h e v i c i n i t y o f t h e e a r t h a r e g i v e n i n table I .  b.  P a r t i c l e clouds  Another f e a t u r e o f i n t e r p l a n e t a r y space i s t h e p a r t i c l e clouds associated with solar f l a r e s .  These have  h i g h e r v e l o c i t y and d e n s i t y t h a n the s o l a r wind, t a k i n g about two days t o r e a c h t h e e a r t h , where t h e y cause magn e t i c storms.  T h e i r p r o p e r t i e s i n t h e v i c i n i t y o f the  21  e a r t h a r e a l s o summarized i n t a b l e I .  Sudden commencement  storms a r e b e l i e v e d t o be caused by such a c l o u d preceded by a magnetohydrodynamic shock wave.  c.  Magnetohydrodynamic  theory  Magnetohydrodynamic t h e o r y and consequent  fluid  f l o w methods r e q u i r e s t h a t : (i)  A l l disturbance  d i m e n s i o n s be much g r e a t e r t h a n t h e  particle gyration radius. (ii)  A l l f r e q u e n c i e s be l e s s t h a n t h e i o n g y r a t i o n f r e q u e n c y .  Three c a s e s a r e c o n s i d e r e d .  These c o r r e s p o n d t o c o n d i t i o n s  i n the s o l a r wind, t h e p a r t i c l e c l o u d , and t h e r e g i o n between t h e shock f r o n t and t h e o b s t a c l e .  T a b l e I I shows  the r e l e v a n t p a r a m e t e r s f o r p r o t o n s i n such r e g i o n s ,  using  t y p i c a l v a l u e s cf t h e v a r i a b l e s . .  I t c a n be seen t h a t we must d e a l w i t h d i m e n s i o n s much g r e a t e r t h a n 10G km. and f r e q u e n c e s much l e s s t h a n 1 cps.  T h i s does n o t a p p l y t o t h e shock f r o n t i t s e l f  which  c o n t a i n s t h e mechanism o f t h e r m a l i z a t i o n o f t h e s t r e a m i n g energy.  A l l the p l a n e t s f u l f i l l these requirements.  moon i s t h e s m a l l e s t body c o n s i d e r e d about 3500 km.  The  having a diameter of  The b a s i c p e r i o d o f t h e d i s t u r b a n c e  by t h i s body w i l l be somewhat l a r g e r t h a n of 10 g r e a t e r t h a n t h e i o n c y c l o t r o n p e r i o d .  —  caused a factor  22  Table I  P r o p e r t i e s o f t h e s o l a r wind and p a r t i c l e c l o u d s near t h e e a r t h .  Clouds  S o l a r Wind  streaming v e l o c i t y  3 0 0 - 5 0 0 km/sec.  600-1000 km/sec.  particle  1-10 / c c .  5-30 / c c .  density  temperature contained  field  5 ~ "  2 x 10 5 Y  °K  ~  7 x 10 10-50  5 o  r  K  23  Table I I V a l u e s of r e l e v a n t parameters i n i n t e r p l a n e t a r y space. Parameter  F i e l d strength ( y ) Temperature  (°K) = / YkT"_ V (r\: km./sec.  Wind  Cloud  5  20  2 x 10  5  Shock region  30  5 7 x  icr  Thermal v e l o c i t y  r  . _ rr\; V c^i  M* w V*  6  Qi B  . j  B pop  I-  | -r—  (km.)  70  100  300  140  50  100 3  (cycles/sec.)  .5  2  (km./sec.)  50  100  100  140  (km./sec .)  24  d.  Supersonic f l o w  I t remains t o show t h a t t h e f l o w i s s u p e r s o n i c for  a l l magnetohydrodynamic modes.  be c o n s i d e r e d : the  Two v e l o c i t i e s must  the A l f v e n v e l o c i t y  Vi - > ^  , and  pseudosonic v e l o c i t y which i s g i v e n by D e n i s s e and  D e l c r o i x (1963) a s :  Y  ^  where If  =  HP., m e .  me-Ve.  rie. =. n i  and e l e c t r o n s , V of  Vfr  ±  = Ye.\<Te,  ni.  N\L Vi  and  WU  =• Yj' k T~i  and t h e r m a l e q u i l i b r i u m e x i s t s between Ions w „ 2. Y l< T~ \A  —^—•—  ^=  .  The v a l u e of  here i s n o t c l e a r l y d e f i n e d , depending on the number degrees of freedom  (s).  V  pagation p a r a l l e l to the f i e l d ,  =. ^ ^ ^~  For pro-  the magnetoacoustic wave  undergoes e s s e n t i a l l y a one d i m e n s i o n a l c o m p r e s s i o n , and as such has  s = 1, and V  =3.  P e r p e n d i c u l a r to the f i e l d ,  the  compression i s two d i m e n s i o n a l g i v i n g  "V  =2.  The r e l e v a n t v a l u e s o f  and  s = 2, and Vs  a r e shown  i n t a b l e I I . The s t r e a m i n g v e l o c i t i e s a r e w e l l ' a b o v e t h e wave v e l o c i t i e s , i m p l y i n g s u p e r s o n i c f l o w w i t h Mach numbers i n t h e range f r o m one t o t e n .  25  IV.  a.  PREVIOUS SHOCK WORK  K e l l o g g (1962)  K e l l o g g makes t h e assumption t h a t t h e i n t e r p l a n e t a r y f i e l d i s normal t o t h e s t r e a m i n g v e l o c i t y , and t h e plasma c a n t h e r e f o r e be t r e a t e d a s a h y d r o dynamic gas u s i n g the  Y - 2.  . This i s essentially  approximation considered i n H a .  H i s f u r t h e r develop-  ment u s e s t h e r e s u l t s o f Hida ( 1 9 5 3 ) , i n which incomp r e s s i b l e f l o w around t h e body i s assumed a f t e r t h e f l u i d has passed t h r o u g h a Rankine-Hugoniot shock.  The a p p r o x i -  m a t i o n s i n v o l v e d make t h e s o l u t i o n r e l i a b l e o n l y i n a r e g i o n i n f r o n t o f t h e sphere, as e x p l a i n e d i n I l a .  In  a d d i t i o n K e l l o g g c a l c u l a t e s the f l o w w e l l behind the sphere a l o n g t h e c e n t r a l s t r e a m l i n e s by c o n s i d e r i n g a d i a b a t i c e x p a n s i o n o f t h e gas t o t h e p r e s s u r e o f t h e i n c o m i n g fluid the  subsequent t o passage t h r o u g h t h e shock.  Combining  Rankine-Hugoniot r e l a t i o n s and t h e e x p r e s s i o n f o r  a d i a b a t i c expansion: (19)  ft  he d e r i v e s t h e e x p r e s s i o n : (20)  =  2.  V- \  26  where  J-*" —  ^  2  ^\  The s u b s c r i p t s i n d i c a t e t h e c o r r e s p o n d i n g  regions i n  f i g u r e 7.  b.  S p r e i t e r and Jones (1963)  S p r e i t e r and Jones use s i m i l a r to K e l l o g g .  approximations  They use t h e n u m e r i c a l methods-of Van Dyke  et a l ( 1 9 5 8 , 1959)> and a l l o w f o r c o m p r e s s i b l e dynamic f l o w b e h i n d t h e shock f r o n t . for  hydro-  They a l s o a l l o w  t h e n o n - s p h e r i c i t y o f t h e magnetosphere u s i n g t h e  boundary c a l c u l a t e d by B e a r d ( i 9 6 0 ) sun-earth l i n e .  r o t a t e d about t h e  Once a g a i n t h e a p p r o x i m a t i o n s  discussed  i n I l a are used.  c.  Obayashi (1964)  Obayashi d i s c u s s e s t h e s a t e l l i t e , d a t a o f 13 space probes i n t e r p r e t i n g t h e i r o b s e r v a t i o n s i n terms of shock t h e o r y .  He a l s o d i s c u s s e s t h e i r f i n d i n g s o f  the p r o p e r t i e s o f t h e i n t e r p l a n e t a r y plasma.  d.  Beard  (1964)  B e a r d d i s c u s s e s t h e phenomenon f r o m a p a r t i c l e viewpoint.  He d i s c u s s e s t h e case o f t h e i n t e r p l a n e t a r y  27  F i g u r e 7.  Geometry f o r K e l l o g g ' s  equation.  28  f i e l d p a r a l l e l t o the s t r e a m i n g v e l o c i t y , and c o n c l u d e s t h a t except f o r a s m a l l p o c k e t of p a r t i c l e s a t the subs o l a r p o i n t , the p a r t i c l e s w i l l f l o w a d i a b a t i c a l l y the body.  The h o s e p i p e e f f e c t  ( P a r k e r , 1963)  around  implies  t h a t the f i e l d l i n e s are on the average a t 45° t o the streamlines Bi  J) U  ( W a l t e r s , 1 9 6 4 ) , and so the c o n f i g u r a t i o n i s infrequent.  F o r an o b l i q u e  field,  Beard o b t a i n s r e s u l t s s i m i l a r t o those of K e l l o g g .  29  V.  CONDITIONS UNDER WHICH SHOCKS FORM  a.  Body w i t h a magnetosphere  The mechanism o f f o r m a t i o n o f a s t a n d i n g shock and bow wave f o r a body w i t h a magnetosphere has been well described:  K e l l o g g ( 1 9 6 2 ) , S p r e i t e r and Jones  Axford (1962),  (1963),  Obayashi ( 1 9 6 4 ) , B e a r d  (1964).  I t i n v o l v e s t h e i n t e r a c t i o n o f a plasma and a c o n t a i n e d magnetic f i e l d w i t h a magnetic f i e l d .  I t seems l i k e l y  t h a t many o f t h e b o d i e s of t h e s o l a r system have magnetospheres, p a r t i c u l a r l y those w i t h o r b i t s o u t s i d e Venus. These p l a n e t s s h o u l d have a c o r r e s p o n d i n g shock wave.  b.  Immersed c o n d u c t i n g body  I n o r d e r t o cause l i t t l e d i s t u r b a n c e i n t h e f l o w , a body would have t o a l l o w t h e passage o f l i n e s o f f o r c e a t t h e v e l o c i t y o f t h e s t r e a m i n g plasma. p r o c e s s e s a r e much t o o slow.  I t c o u l d be a c c o m p l i s h e d  by a p o l a r i z a t i o n e l e c t r i c f i e l d U  = 400 km./sec. and  volts/m.  Diffusion  E  B = 5r  = a x  B  . For  , E = 2 x 10~  3  F o r a body t h e s i z e o f t h e moon, t h i s r e p r e s e n t s  a v o l t a g e o f 7000 v o l t s a c r o s s i t . Such a v o l t a g e b u i l d u p seems u n l i k e l y f o r a body immersed I n a h i g h l y plasma. the  conducting  I t i s l i k e l y t h a t charge leakage would o c c u r a t  sides.  30  The above p r o c e s s r e q u i r e s t h e " d e f r e e z i n g " o f l i n e s o f f o r c e a t t h e f r o n t o f t h e body, and must r e s u l t In  t h e f o r m a t i o n o f a t r a n s i e n t atmosphere and i o n o s p h e r e  for  such a body.  T h i s has been d i s c u s s e d f o r t h e moon  by H e r r i n g and L i c h t  (1959),  S i n g e r ( 1 9 6 1 ) , Nakada and  M i h a l o v ( 1 9 6 2 ) , W e i l and B a r a s c h ( 1 9 6 3 ) , Gold ( 1 9 5 9 ) , and H i n t o n and Taeusch ( 1 9 6 4 ) .  H i n t o n and Taeusch o b t a i n p a r -  t i c l e d e n s i t i e s o f 10^ / c c . f o r n e u t r a l p a r t i c l e s , and 3  10  /cc. f o r ions.  The gas p r e s s u r e s a r e g r e a t e r t h a n  the s o l a r wind p r e s s u r e s , and hence w i l l a c t as t h e cond u c t i n g body which forms t h e shock.  I t I s l i k e l y that  t h e i o n sheath, o r i o n o s p h e r e , w i l l be d e n s e s t on t h e upstream s i d e o f t h e body. I f a body i s surrounded by an atmosphere and i o n o s p h e r e and no magnetic  field,  the ionosphere w i l l p l a y  the p a r t o f t h e c o n d u c t i n g body, and a shock wave c a n s t i l l be  expected.  A l l t h e b o d i e s o f t h e s o l a r system s h o u l d f a l l i n t o one o f t h e above c l a s s i f i c a t i o n s , and may t h u s be exp e c t e d t o have shock waves u n l e s s s i z e c o n s i d e r a t i o n s exc l u d e them from t h e regime o f hydromagnetic  theory.  31  VI.  MAGNETIC ACTIVITY AT LARGE DISTANCES PROM THE SOURCE.  At t h i s s t a g e , t h e a u t h o r w i s h e s t o suggest t h e following ideas: (i)  There w i l l be a d e c r e a s e i n magnetic a c t i v i t y w i t h i n the r e g i o n b e h i n d a shock wave.  The a c t i v i t y  will  be l e a s t on t h e c e n t r a l stream l i n e . (ii)  There w i l l be an i n c r e a s e of magnetic a c t i v i t y i n a bow wave.  An attempt w i l l be made t o j u s t i f y each  of t h e s e s t a t e m e n t s i n t u r n .  a.  A c t i v i t y minimum  As f l u i d p a s s e s t h r o u g h a shock system of t h e type near t h e e a r t h , energy i s t r a n s f o r m e d f r o m k i n e t i c s t r e a m i n g energy i n t o t h e r m a l energy.  A greater flow  produces a s t r o n g e r shock, r e s u l t i n g i n a g r e a t e r l o s s of s t r e a m i n g energy.  T h i s i m p l i e s t h a t a slow v a r i a t i o n  I n energy f l u x i s a t t e n u a t e d as i t p a s s e s t h r o u g h t h e shock.  A s t r i c t a n a l y t i c e v a l u a t i o n of t h i s i s n o t pos-  s i b l e because o f t h e i n d e t e r m i n a t e n a t u r e of t h e shock equations.  (There i s one more v a r i a b l e t h a n equations.)  I d e a l l y one would l i k e t o e v a l u a t e :  (21)  d  (pi-uu-rpQ  =2-  d  (fft y) u  32  However, i t i s p o s s i b l e t o r e l a t e t h i s t o the work of B i g g (1963 a, b ) , as h i s r e s u l t s are based on the f r e q u e n c y of o c c u r r e n c e of magnetic a c t i v i t y .  He uses  the numbers of magnetic storms and i n t e r v a l s i n which K'p  30  .  As the time i n t e r v a l s on which t h e s e a r e  based a r e l o n g , t h e y a r e a measure of of  ^ ^  .  A 3  r a t h e r than  I f t h e s e f l u c t u a t i o n s are assumed t o be  due t o p r e s s u r e changes a t the magnetosphere we a r e i n d i r e c t l y measuring A (jOU -*-p) X  —  boundary, A (pM***)  ( T h i s assumption s h o u l d be a p p r o x i m a t e l y t r u e u n t i l  ring  c u r r e n t s have had time t o b u i l d up, and even t h e n the magn e t i c d i s t u r b a n c e may g i v e some i n d i c a t i o n of p r e s s u r e f l u c t u a t i o n s a t the boundary.) A (p ^9 u  Thus an a t t e n u a t i o n i n  would i m p l y an a t t e n u a t i o n i n  The a t t e n u a t i o n i n  A  S  i s g i v e n by:  where the s u b s c r i p t s a p p l y t o the r e g i o n s shown i n f i g u r e 7. F o r l a r g e f l u c t u a t i o n s , the minimum terms w i l l be much small e r t h a n the maximum terms.  F o r example c o n s i d e r a f l u c -  t u a t i o n from s o l a r wind t o c l o u d c o n d i t i o n s . of t a b l e I I are used,  (K^P')^Q*  =2== ) ^  I f the v a l u e s .  Under  t h e s e c i r c u m s t a n c e s , i t becomes p o s s i b l e t o i g n o r e the  33  second terms i n e q u a t i o n  (22) l e a d i n g t o an a t t e n u a t i o n  c o e f f i c i e n t (A) f o r l a r g e f l u c t u a t i o n s i n t h e s o l a r wind: (23)  /\  A p l o t of Y M3  A  -  versus  M** wcur  Mj  = 2 using equation  i s shown i n f i g u r e 8 f o r  (20).  must be t a k e n a t maximum  the maximum o f  M  s  .  Higher  The values used f o r M( and M | . M|  T h i s s h o u l d a l s o be  values are u s u a l l y as-  sociated with higher p a r t i c l e f l u x e s .  From f i g u r e 8 i t  can be seen t h a t i f t h i s i s t r u e , t h e l a r g e s t f l u c t u a t i o n s would be a t t e n u a t e d  t h e most.  However, i f t h e models p r e -  sented i n t a b l e s I and I I a r e c o r r e c t , t h e r e i s l i t t l e change i n M.|  b.  with  activity.  A c t i v i t y maxima  The  a c t i v i t y i n c r e a s e I n t h e bow wave c a n be  discussed only q u a l i t a t i v e l y .  The bow wave w i l l  exist  d u r i n g q u i e t c o n d i t i o n s , but i t i s probably of too small an a m p l i t u d e  t o produce a measurable e f f e c t .  the k i n e t i c energy o f t h e s t r e a m i n g f l u i d  During  storms  i n c r e a s e s by one  o r two o r d e r s o f magnitude, and t h e r e s h o u l d be an i n c r e a s e i n bow wave a c t i v i t y — s u f f i c i e n t perhaps t o add n o t i c e a b l e d i s t u r b a n c e t o an e x i s t i n g storm o r a c t i v e p e r i o d .  I t can  be shown t h a t under i d e a l i z e d c o n d i t i o n s , s u f f i c i e n t energy i s a v a i l a b l e t o cause measurable a c t i v i t y .  35  The energy f l u x i n c i d e n t on t h e f r o n t o f a body (a c i r c l e o f r a d i u s 15 e a r t h  t h e s i z e o f t h e magnetosphere  r a d i i ) , d u r i n g storm c o n d i t i o n s (n = 20, u = 1000 km/sec.)  22 i s o f t h e o r d e r o f 10  ergs/sec.  I f energy o f t h i s amount  i s assumed t o spread o u t i n a Mach 3 cone, i t i s spread 2 ir s t a n O ^= 2s a t a  over a c i r c l e o f c i r c u m f e r e n c e distance  s  behind the source.  For typical  interplanet-  8 a r y d i s t a n c e , t h i s i s o f t h e o r d e r o f 10 km. A r e c e i v e r the s i z e o f t h e magnetosphere would i n t e r s e c t a l i n e s e g -  5 ment 10 km. l o n g , and might be e x p e c t e d t h e r e f o r e t o lO 19 receive ~ ^ t t t o t a l energy o r 10 ergs/sec. A x f o r d (1963) e s t i m a t e s t h e energy d i s s i p a t i o n d u r i n g a . 2  =  1  0  o  f  t  n  e  18 magnetic storm t o be  ~  10  e r g s / s e c . Even a f a c t o r  of 10 l o w e r would be c a p a b l e o f p r o d u c i n g measurable turbance.  dis-  The above c a l c u l a t i o n o f a t t e n u a t i o n t o o k i n t o  account o n l y g e o m e t r i c e f f e c t s .  I f a dropoff  o<. p  i s used, a s i n t h e case o f t h e s o n i c boom (H. A. W i l s o n , J r . .1962), t h e a v a i l a b l e energy i s reduced by a f u r t h e r f a c t o r of 10. A second independent e s t i m a t e o f t h e s i z e o f t h e bow wave i s d e v e l o p e d h e r e .  From t h e second Hugoniot shock  e q u a t i o n (11),  relation  (24)  p  x  the f o l l o w i n g  - p , -  A p - (^u^) -(pu^) l  i  ~((° *)i u  36  h o l d s a c r o s s the s t a n d i n g and out t o t h e s i d e s . Ap  decrease i n supersonic cone.  shock I n f r o n t o f t h e e a r t h ,  The assumption i s made t h a t t h e  w i t h d i s t a n c e i s t h e same as f o r a  a i r c r a f t , d i s t a n c e s b e i n g measured a l o n g the  Prom H. A. W i l s o n ,  J r . (1962):  (25)  Now i f e q u a t i o n  (24) i s t r u e a t p o i n t  i n f i g u r e 14,  (25) i s assumed t o h o l d f r o m t h e r e onwards  and e q u a t i o n  out a l o n g t h e Mach cone, A p distances.  A  c a n be c a l c u l a t e d a t l a r g e  F o r a Mach 3 cone,  A  i s a t about 27 e a r t h  r a d i i f r o m t h e e a r t h ( f i g u r e 14) and about 3 x 27 = 8 l e a r t h r a d i i f r o m the v e r t e x of t h e cone. i n t e r p l a n e t a r y d i s t a n c e 5 x 10 that  A p e  -03 A p *  •  km., e q u a t i o n  f5U>t  1 t h i s represents  few gammas a t a r e c e i v e r .  (25) i m p l i e s  By comparison w i t h sudden storm  commencements, which must a l s o r e p r e s e n t ~  For a t y p i c a l  a pressure  jump  magnetic f l u c t u a t i o n s of a  T h i s i s perhaps a l i t t l e low  t o produce much d i f f e r e n c e i n magnetic a c t i v i t y , but t h e u n c e r t a i n t i e s i n t h e assumptions c o u l d e a s i l y make a d i f ference  of a f a c t o r of 10.  F o r a Mach V, t h e r e i s no g e o m e t r i c a t t e n u a t i o n , and t h e r e f o r e no energy problem.  I f \^  > V5  , the  A l f v e n mode V w i l l be c o i n c i d e n t w i t h t h e f a s t mode, i n g e x t r a energy i n t h e cone, and e x t r a magnetic  supply-  activity.  37  The t h e o r y t h e n p r e d i c t s t h a t t h e r e w i l l be an a c t i v i t y minimum b e h i n d such a body, and two maxima on each s i d e c o r r e s p o n d i n g t o t h e t h r e e modes o f propagation.  The remainder o f t h i s t h e s i s w i l l be concerned  with  the comparison of e x p e r i m e n t a l e v i d e n c e w i t h t h e t h e o r y .  38  VII.  a.  EXPERIMENTAL EVIDENCE AND COMPARISON WITH THE THEORY  The P l a n e t s  I t has been suggested by B i g g and De V a u c o u l e u r s ( B i g g , 1963 b) t h a t t h e b l u e c l e a r i n g s of Mars may be r e l a t e d t o magnetic a c t i v i t y f r e q u e n c y . of  F o r t h e purpose  t h i s t h e s i s i t w i l l be assumed t h a t t h e f r e q u e n c y of  b l u e c l e a r i n g s i s a maximum when a minimum o f magnetic a c t i v i t y e x i s t s a t Mars and/or i n t h e l i n e of s i g h t between the e a r t h and Mars.  F i g u r e s 10 - 13,  a l l from B i g g (19^3  a and b ) ,  show t h e dependence o f magnetic d i s t u r b a n c e f r e q u e n c y on the p l a n e t a r y p o s i t i o n s o f Mercury, Venus, and t h e moon, and t h e b l u e c l e a r i n g f r e q u e n c y f o r Mars.  T a b l e I I I shows  the Mach a n g l e s t h a t c o r r e s p o n d t o t h e peaks i n d i c a t e d by the arrows i n f i g u r e s 10 and 11, 12.  and t h e minima i n f i g u r e  The geometry on which t h e c a l c u l a t i o n s a r e based i s  shown i n f i g u r e 9-  The BB maxima a r e l a t e r a s s o c i a t e d w i t h  a guided slow mode Mach V, and t h e use o f e q u a t i o n r a t h e r t h a n (16)  (18)  t o c a l c u l a t e Mach numbers f o r t h e s e peaks  must i n t r o d u c e some e r r o r .  I f d u r i n g storms t h e i n t e r p l a -  n e t a r y f i e l d i s n o t o r i e n t e d , on t h e average, i n a d i r e c t i o n w i t h i n about 45° o f t h e s t r e a m i n g v e l o c i t y , t h e e r r o r s s h o u l d not be l a r g e .  I t must be p o i n t e d o u t t h a t t h e a n g l e  oL  in  39  F i g u r e 9.  Geometry f o r t a b l e  III.  40  A  40  30  20  Degrees East  j  10,. Inferior  (A)  \|/A  10  20  30  40  SO  Degrees West  F i g u r e 10. Magnetic a c t i v i t y f r e q u e n c y - Venus. (A) g r e a t storms; (B) o c c a s i o n s when Kp£* 3 0 > (C) f o u r s e t s o f d a t a , each s e t g i v e n e q u a l w e i g h t .  41  I  F i g u r e 11.  A  Magnetic a c t i v i t y f r e q u e n c y - Mercury. (A) g r e a t storms; (B) f o u r s e t s of d a t a , each o b s e r v a t i o n g i v e n e q u a l w e i g h t .  42  30  F i g u r e 12.  20 Before Opposition  10  0 ANGLE E.M.S.  10  A f t e r  20 Opposition  B l u e c l e a r i n g f r e q u e n c y - Mars.  30  TABLE I I I .  Fig.  E x p e r i m e n t a l r e s u l t s I n terms of Mach a n g l e and Mach number.  Angle cLe*. b e t ween peaks, degrees  b  f*|, = S\fY'6  a  Mach a n g l e B degrees  Mach number  M  AA  BB  AA  10a  32  ?  .382  1.38  22  2.6  10b  27  .324  1.38  19  3.1  10c  34  ?  .405  1.38  24  2.5  11a  13  none  .292  2.58  17  3.4  lib  18  2.7  .405  .061  2.58  24  3.5  2.5  16  12  31  5.0  .408  .067  1.52  "24  3.8  2.5  15  AA  AA  1  BB  AA  AA  1  BB  1  AA  AA  10b  34  ?  .405  1.38  24  2.5  11a  16  ?  .360  2.58  21  2.8  BB  1  44  G i  I80*W  90*W  0*  90*E  180'E  90*E  ISO*E  Lunar position  c  90*W  o* Lunar position  F i g u r e 13.  Magnetic a c t i v i t y f r e q u e n c y aT 112 g r e a t e s t storms; b) s m a l l storms.  Moon,  45  Lunar position  F i g u r e 13.  Magnetic a c t i v i t y f r e q u e n c y - Moon, fcT o c c a s i o n s when Hp ^ 3 0 (d) a l l d i s t u r b a n c e s  F i g u r e 13.  Magnetic a c t i v i t y f r e q u e n c y - Moon. (e) storms A sudden commencement B g r a d u a l commencement  47 f i g u r e s 10 t o 12 i s the a c t u a l angle  sun-receiver-source,  and not the angle from i n f e r i o r c o n j u n c t i o n .  Venus, a t  i n f e r i o r c o n j u n c t i o n , i s a t a n g l e s f r o m o° t o 8° f r o m the sun-earth  l i n e , r e s u l t i n g i n a s c a r c i t y of p o i n t s i n the  r e g i o n 6° e a s t t o 6° west.  T h i s makes i t I m p o s s i b l e  to  determine i f BB type maxima appear i n f i g u r e s 10 a, b, The  few p o i n t s a v a i l a b l e and the approaches t o the  i n d i c a t e that there probably  I n f i g u r e 10b,  region  are maxima w i t h i n t h i s r e g i o n .  i t i s not o b v i o u s which peak  s h o u l d be used on the e a s t s i d e of the AA maxima. i s not a c l e a r l y d e f i n e d peak r i s i n g w e l l above the S i m i l a r l y i n f i g u r e 11a  c.  There others.  t h e r e are f o u r competing maxima.  I f i n s t e a d of t a k i n g the i n s i d e maxima, the c e n t r e  A  of  the r e g i o n of u n c e r t a i n t y (as shown by the shaded r e c t a n g l e ) i s used, the r e s u l t s f r o m these two f i g u r e s are more c o n s i s t e n t w i t h the o t h e r s .  A l s o the e a s t e r l y maximum i s now  at  a s l i g h t l y l a r g e r angle t h a n the w e s t e r l y maximum, as i t i s f o r a l l c a s e s w i t h c l e a r l y d e f i n e d peaks.  The  r e s u l t s f i t a model w i t h Mach numbers 2.5  15 f o r the two t y p e s of bow  wave.  and  These we a s s o c i a t e w i t h  the a c c e l e r a t e d and r e t a r d e d m a g n e t o a c o u s t i c modes i n t u r n w i t h the o b l i q u e A l f v e n mode a d d i n g t o one i n g on the r e l a t i v e s i z e s of  VA  and  Vs  of these depend-  48  The l a c k of BB type maxima i n f i g u r e 11a  is  i n t e r e s t i n g and s u g g e s t s a number of p o s s i b l e p r o p e r t i e s of  g r e a t storms as an e x p l a n a t i o n :  (i)  The i n t e r p l a n e t a r y f i e l d may  be a l i g n e d n o r t h - s o u t h  d u r i n g a g r e a t storm, so t h a t no g u i d e d mode waves would r e a c h the r e c e i v e r . (ii)  The i n t e r p l a n e t a r y f i e l d may be broken up t u r b u l e n t f l o w ) d u r i n g g r e a t storms,  (highly  restricting  p r o p a g a t i o n by g u i d e d modes t o s h o r t d i s t a n c e s . ( i i i ) The A l f v e n and sound v e l o c i t i e s may  be e q u a l i m p l y -  . i n g e q u i p a r t i t i o n between t h e r m a l and magnetic (iv)  The f i e l d l i n e s . m a y be  energies.  radial.  At p r e s e n t the a u t h o r sees no r e a s o n t o e x p r e s s p r e f e r e n c e for  any one of t h e s e p o s s i b i l i t i e s .  The t h e o r y p r e d i c t s t h a t , because of g e o m e t r i c a l a t t e n u a t i o n , guided,modes w i l l have more energy a t  suffi-  c i e n t l y l a r g e d i s t a n c e s than modes which produce c o n i c a l bow waves.  T h i s i s c e r t a i n l y i n d i c a t e d by f i g u r e l i b ,  and  i t i s u n f o r t u n a t e t h a t i t i s not p o s s i b l e t o compare the r e l a t i v e magnitudes of the two t y p e s of peaks w i t h those i n f i g u r e 10c .  However, I f b l u e c l e a r i n g f r e q u e n c y does  decrease m o n o t o n i c a l l y w i t h magnetic  a c t i v i t y increase,  f i g u r e s 12 and 11 b show t h a t t h e a t t e n u a t i o n i s i n agreement w i t h the t h e o r y .  49  b.  The moon  Any  e f f e c t of the moon on magnetic a c t i v i t y a t  the e a r t h may  be e x p e c t e d t o be s m a l l as a r e s u l t of  l a r g e s i z e of the magnetosphere (see F i g u r e 14). s t a n c e , a t new bow  moon the a t t e n u a t e d  the  For i n -  r e g i o n i n s i d e the  inner  wave r e p r e s e n t s a spot of r a d i u s 3 e a r t h r a d i i on  the  f r o n t of the magnetosphere, and cannot be e x p e c t e d t o have a l a r g e e f f e c t a t the s u r f a c e of the e a r t h . v a r i a t i o n i n magnetic a c t i v i t y f r e q u e n c y  Any  observed  with lunar  angle  must be a f u n c t i o n of the a n g l e of i n c i d e n c e of the  bow  wave on the magnetosphere, the s e n s i t i v i t y of the magnetosphere t o p r e s s u r e  f l u c t u a t i o n s a t the p o i n t o f i n c i d e n c e ,  and the e f f i c i e n c y of t r a n s f e r of energy f r o m t h i s p o i n t t o the s u r f a c e of the e a r t h .  (1963  Bigg f i g u r e s 13a  to  e.  a, b) o b t a i n s the v a r i a t i o n s shown i n There i s , however, some q u e s t i o n as t o  the s t a t i s t i c a l s i g n i f i c a n c e o f these r e s u l t s . s u l t s are a c c e p t e d ,  I f the r e -  the maxima i n a c t i v i t y f o r 45°<L oi < 8 o °  i n d i c a t e t h a t the energy t r a n s f e r i s most e f f e c t i v e when the o u t e r bow sphere.  The  wave i s i n c i d e n t on the s i d e of the magnetominimum a t new  moon c o u l d i n d i c a t e t h a t  energy i s r e c e i v e d f r o m the o u t e r bow The  wave a t t h i s  no  time.  maximum n e a r f u l l moon can o n l y be a s s o c i a t e d w i t h  the  F i g u r e 14.  Earth-magnetosphere-moon system, s c a l e drawing  51  passage o f t h e moon t h r o u g h one o f t h e f o l l o w i n g : (i)  The t a i l o f t h e magnetosphere.  (ii)  F i e l d l i n e s shared by t h e magnetosphere  and t h e  i n t e r p l a n e t a r y plasma. (iii)  A subsonic r e g i o n i n the l e e of the e a r t h .  W i t h t h e moon i n such r e g i o n s , t h e " t a i l wagging" o f t h e magnetosphere  due t o v a r i a t i o n s i n t h e i n c i d e n t magnetized  plasma, and any o t h e r f l u c t u a t i o n s , would be e x p e c t e d t o produce a v a r i e t y o f hydromagnetic waves w i t h i n t h e magn e t o s p h e r e which c o u l d cause a d i s t u r b a n c e i n c r e a s e .  The t o p l i n e o f f i g u r e 13 e i s f o r sudden commencement storms.  T h i s i n v o l v e s the i n t e r a c t i o n of a  hydromagnetic shock wave w i t h t h e two bow shock waves — a s i t u a t i o n very poorly understood.  No attempt w i l l be made  here t o e x p l a i n t h e r e s u l t i n g a c t i v i t y f r e q u e n c y p l o t .  52  VIII.  CONCLUSIONS AND SUMMARY  The i n t e r p l a n e t a r y plasma and magnetic can be t r e a t e d as a f l u i d w i t h s u p e r s o n i c  field  velocity.  As  such, i t forms shock and bow waves a t any p l a n e t w i t h i n i t s flow.  The bow wave t a k e s t h e f o r m o f a Mach cone  for  t h e a c c e l e r a t e d m a g n e t o a c o u s t i c mode, and Mach V s  for  t h e guided modes ( t h e r e t a r d e d m a g n e t o a c o u s t i c and  the A l f v e n ) .  The A l f v e n mode c o i n c i d e s w i t h e i t h e r t h e  cone o r t h e o t h e r Mach V, r e s u l t i n g I n o n l y two bow waves.  There i s an a t t e n u a t i o n o f magnetic frequency  behind  such an o b s t a c l e , t h e c o e f f i c i e n t of  energy a t t e n u a t i o n b e i n g g i v e n by e q u a t i o n i s a l s o an I n c r e a s e wave.  activity  (23).  There  of magnetic a c t i v i t y a l o n g a bow  Thus, t h e e x p e r i m e n t a l  r e s u l t s should e x h i b i t a  minimum o f a c t i v i t y a t i n f e r i o r c o n j u n c t i o n , and two maxima on each s i d e . numbers o f 2.5 Vf\  and Vs  i s greater. "^p"  The o b s e r v e d maxima c o r r e s p o n d  t o Mach  and 15 i m p l y i n g a v e l o c i t y r a t i o between  of 6:1.  The t h e o r y does n o t i n d i c a t e which  There s h o u l d be g e o m e t r i c energy a t t e n u a t i o n  a l o n g a cone, and none a l o n g a V.  by t h e e x p e r i m e n t a l g i v e n by e q u a t i o n  results. (16).  This i s implied  The Mach angle f o r a V i s  53  The  a c t i v i t y f r e q u e n c y diagrams f o r the moon are  q u i t e d i f f e r e n t f r o m those f o r the p l a n e t s .  T h i s i s due  t o the s i z e of the magnetosphere b e i n g of the same o r d e r as the earth-moon d i s t a n c e .  The  r e s u l t i s t h a t the mag-  n e t o s p h e r e i n t e r s e c t s l a r g e s e c t i o n s of the bow  wave p a t t e r n ,  and t h e r e f o r e l i t t l e magnetic e f f e c t should be f e l t a t the s u r f a c e of the e a r t h .  Bigg  (1963  b) shows an  activity  dependence on l u n a r p o s i t i o n t h a t can be e x p l a i n e d by presented  theory.  the  However, t h e r e i s doubt t h a t h i s o b s e r -  v a t i o n s are s t a t i s t i c a l l y  significant.  Other t h e o r i e s of p l a n e t a r y e f f e c t s on magnetic a c t i v i t y a t the e a r t h are few. p l a n e t and the c l o u d may i s d i f f i c u l t t o see how  Bigg suggests that  the  have e l e c t r o s t a t i c c h a r g e s .  It  t h i s c o u l d produce two maxima on  the magnetic a c t i v i t y f r e q u e n c y p l o t .  Indeed, the  cloud  has t o be n e a r l y n e u t r a l i n o r d e r not t o f l y a p a r t .  Houtgast and van S l u i t e r s (1962) suggested t h a t Venus a c h i e v e s  i t s e f f e c t by h a v i n g a v e r y l a r g e magnetic  field.  T h i s has been d i s p r o v e d by the r e s u l t s of  Mariner  2 space probe.  the  There i s a n o t h e r p o s s i b i l i t y t h a t i s e s p e c i a l l y a p p l i c a b l e t o the case of  o i . J_ L|  .  This i s that  p a r t i c l e s are squeezed out a l o n g the t u b e s of f o r c e as  the  54  field  i s compressed.  They would move from a f i e l d ^= 50 V  i n the shock t o a r e g i o n where ^  = constant.  o  implies that  V/y  3=2* | O v , conserving  T h i s constancy of magnetic moment  (distant)  t£=  V_L  (shock)  U  t  •  I t would give the appearance o f a Mach V w i t h Mach number 2-  .  T h i s produces o n l y one a c t i v i t y peak, but c o u l d  combine w i t h one hydromagnetic cone o r V t o produce the complete p a t t e r n . of  As the v e l o c i t y of these p a r t i c l e s i s  the same o r d e r as the waves, t r a v e l l i n g wave tube ampli-  f i c a t i o n of one of the guided modes i s p o s s i b l e .  55  IX.  BIBLIOGRAPHY  Axford, W. I . , I n t e r a c t i o n between s o l a r wind and the e a r t h ' s magnetosphere. J . Geophys. Res. 6 7 , 3791*  1962.-  Axford, W. I . , V i s c o u s i n t e r a c t i o n between the s o l a r wind and the e a r t h ' s magnetosphere. C.R.S.R. 153, C o r n e l l U n i v e r s i t y P u b l i c a t i o n , 1963. Beard, D.B., The i n t e r a c t i o n of the t e r r e s t r i a l magnetic f i e l d with the s o l a r c o r p u s c u l a r r a d i a t i o n . J . Geophys. Res. 6 5 , 3559,  I960.  Beard, D.B., The e f f e c t of an i n t e r p l a n e t a r y magnetic f i e l d on the s o l a r wind. J . Geophys. Res. 6 9 , 1159, 1964. Bershader, D., The Magnetodynamics of Conducting F l u i d s . S t a n f o r d U n i v e r s i t y Press, 1 9 5 9 . B i g g , E . K., The i n f l u e n c e of the moon on geomagnetic d i s turbances. J . Geophys. Res. 6 8 , 1409, 1963 a. Bigg, E. K., Lunar and p l a n e t a r y i n f l u e n c e s on geomagnetic disturbances. J . Geophys. Res, 6 8 , 4099, 1963 b. Courant, R., K. 0 . F r i e d r l c h s , Supersonic Flow and Shock Waves. I n t e r s c i e n c e P u b l i s h e r s , New York, 1 9 4 8 . De Hoffman, F., E. T e l l e r , Magnetohydrodynamic Phys. Rev. 8 0 , 6 9 2 , 1 9 5 0 .  shocks.  Denisse, J . F., J . L. D e l c r o l x , Plasma Waves. P u b l i s h e r s , New York, 1 9 6 3 .  Interscience  Dugundji, J . , J . Aero. S c i . 15, 6 9 9 ,  1948.  Gold, T., i n d i s c u s s i o n f o l l o w i n g R. Jastrow, Outer atmospheres of the e a r t h and p l a n e t s . J , Geophys. Res. 64, 1798,  1959.  H e r r i n g , J-. R., A. L. L i c h t , E f f e c t of the s o l a r wind on the l u n a r atmosphere. Science 1 3 0 , 266, 1 9 5 9 . Hida, K. An approximate study on the detached shock wave i n f r o n t of a c i r c u l a r c y l i n d e r and a sphere. J . Phys. Soc. Japan 8 , 7 4 0 , 1 9 5 3 .  56  Hinton, P. L., D. R. Taeusch, V a r i a t i o n of the l u n a r a t mosphere w i t h the s t r e n g t h of the s o l a r wind. J . Geophys. Res. 6 9 , 1 3 4 1 , 1 9 6 4 . Houtgast, J . , I n d i c a t i o n of a magnetic f i e l d Venus. Nature 175, 6 7 8 , 1 9 5 5 . Houtgast, J . , A. van S l u i t e r s , gth o f t h e magnetic f i e l d 196,  462, 1962.  of the p l a n e t  A new e s t i m a t e of the s t r e n of the p l a n e t Venus. Nature  Inouye, Mamanu , Lomas, Comparison of e x p e r i m e n t a l and n u m e r i c a l r e s u l t s - b l u n t body. NASA TN P, 1 4 2 6 , 1 9 6 2 . K e l l o g g , P. J . , Flow of plasma around the e a r t h . phys. Res. 6 7 , 3805, 1 9 6 2 . Lin,  C. C ,  S. I . Rublnov,  J . Geo-  J . Math. Phys. 27, 105,  1948.  Montgomery, D., Development of hydromagnetic shocks from l a r g e amplitude A l f v e n waves. Phys. Rev. L e t t e r s , ' 2, 36,  1959.  Nakada, M. P., J . D. Mihalov, A c c r e t i o n of the s o l a r wind to form a l u n a r atmosphere. J . Geophys. Res. 6 7 , 1 6 7 0 , 1962.  Neugebauer, M., C. W. Snyder, P r e l i m i n a r y r e s u l t s from Mariner I I s o l a r plasma experiment. S c i e n c e 1 3 8 , 1 0 9 5 , 1962.  .Obayashi, T., I n t e r a c t i o n of s o l a r plasma streams w i t h the o u t e r geomagnetic f i e l d . J . Geophys. Res. 6 9 , 8 6 l , 1 9 6 4 . Parker, E . N. I n t e r p l a n e t a r y Dynamical P r o c e s s e s . s c i e n c e P u b l i s h e r s Inc., New York, 1 9 b 3 . P i d d i n g t o n , J . H., The CIS-Lunar magnetic f i e l d . Space S c i . 9 , 3 0 5 , 1 9 6 2 . Singer, S. F., Atmosphere near the moon. 135,  1961.  InterPlan.  A s t r o n a u t . Acta 7,  S p r e i t e r , J . R., W. P. Jones, On the e f f e c t of a weak i n t e r p l a n e t a r y magnetic f i e l d on the i n t e r a c t i o n between the s o l a r wind and the geomagnetic f i e l d . J . Geophys. Res. 68,  3555,  1963.  Van Dyke, D. M i l t o n , The supersonic blunt-body problem review and e x t e n s i o n . J . Aeron. S c i . 25, 4 8 5 , 1 9 5 8 .  57  Van Dyke, D. M i l t o n , H. D. Gordon, a f a m i l y of b l u n t axisymmetric  Supersonic flow past b o d i e s . NASA TR R - l ,  1959.  Walters, G. K., E f f e c t of an o b l i q u e i n t e r p l a n e t a r y magnetic f i e l d on the shape and behaviour of the magnetosphere. J . Geophys. Res. 6 9 , 1?69', 1 9 6 4 . W e i l , H., M. L. Barasch, A t h e o r e t i c a l l u n a r ionosphere. I c a r u s I, 3^6, 1 9 6 3 . Wilson, H. A., J r . , Sonic Boom.  S c i . Amer., Jan., 3 6 , 1 9 6 2 .  58  APPENDIX - Symbols and Conventions  Used.  A  attenuation  B  magnetic  Bo  unperturbed  Bi  i n t e r p l a n e t a r y magnetic  Cp  s p e c i f i c ')heat a t constant p r e s s u r e  Cv  s p e c i f i c heat a t constant volume  t=  electric  e  as s u b s c r i p t denotes "of e l e c t r o n "  L  as s u b s c r i p t denotes "of i o n " (except  k  Boltzmann's constant  coefficient  field magnetic  field field  field  Bi  )  wave p r o p a g a t i o n v e c t o r m  mass o f p a r t i c l e  n  number d e n s i t y  P = pmag + p P  gas p r e s s u r e magnetic p r e s s u r e charge  a  ion gyration radius  r  distance number of degrees of freedom  T  temperature  t  as s u b s c r i p t denotes the t a n g e n t i a l component  U  streaming v e l o c i t y of a f l u i d  59  4u //  p e r p e n d i c u l a r and p a r a l l e l components of q  V  particle  \^  Alfven velocity  \/^  v e l o c i t y of sound  u  group  velocity  velocity  Vp  phase v e l o c i t y  V  volume  x>\j,B c o o r d i n a t e axes. components  As s u b s c r i p t , they  oC  angle sun - r e c e i v e r - source  ©  angle between  ©  Mach angle  ^  angle between  Y*  r a t i o of s p e c i f i c  Y  d e f i n e d i n the t e x t p. 3  |Jo  p e r m e a b i l i t y of f r e e  p  density  u) •  If  and  S  P  B l and TJ* heats  space  angular frequency i o n c y c l o t r o n frequency  denote  

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