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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.Sc, U n i v e r s i t y of B r i t i s h Columbia, i 9 6 0 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 to the req u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1964. In presenting this thesis i n p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree- that the Library shall make i t freely available for reference and study, I further agree that per-mission for extensive copying of this thesis'for scholarly purposes may be granted by the Head of my Department or by his representatives. I t i s understood that,copying or publi-cation of this thesis.for f i n a n c i a l gain shall not be allowed without my written permission.? Department of Geophyslics The University of B r i t i s h Columbia, Vancouver 8, Canada no>. 5 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 that 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 frequency observed at the earth, and the p o s i -t i o n of the earth a f f e c t s the frequency of blue c l e a r i n g s on Mars. This study shows that 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 of the a c t i o n of shock and bow waves formed by these bodies i n the s u p e r s o n i c a l l y streaming i n t e r p l a n e t a r y plasma. The a t t e n u a t i o n of 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 the streaming plasma behind such bodies i s shown to be equal to the square of the r a t i o of the Mach number upstream to the Mach number downstream. For t y p i c a l s o l a r induced a c t i v i t y , t h i s I mplies an a t -tenu a t i o n c o e f f i c i e n t of approximately 1/2 - 1/3. I t i s a l s o shown th a t an a c t i v i t y increase i s expected i n the bow wave. The o b s e r v a t i o n a l data f i t s a model w i t h bow waves of Mach numbers 2.5 and 15 corresponding to the two bow waves p r e d i c t e d by the theory. The moon's e f f e c t v a r i e s from that of the pl a n e t s i n a manner that can be explained by i t s closeness to the magnetosphere. v i i ACKNOWLEDGMENT I am deeply indebted to Dr. T. Watanabe both 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 of 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 . I would a l s o l i k e to thank Professor J . A. Jacobs f o r h i s encouragement during the pr e p a r a t i o n of t h i s work, and f o r h i s guidance i n i t s f i n a l f o r m u l a t i o n . i i TABLE OP CONTENTS Page I INTRODUCTION 1 I I THE BOW WAVE 3 a. Hydromagnetic shocks 3 b. Hydromagnetic waves 7 c. Hydrodynamic blu n t body problem 11 d. D i f f e r e n c e s between hydromagnetic 14 and hydrodynamic bow waves at la r g e d i s t a n c e s from the source I I I INTERPLANETARY SPACE 20 a. The s o l a r wind : 20 b. P a r t i c l e clouds 20 c. Use of magnetohydrodynamic theory 21 d. Supersonic flow 24 IV PREVIOUS SHOCK WORK 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 V CONDITIONS UNDER WHICH SHOCKS FORM 29 a. Body w i t h a magnetosphere 29 b„ Immersed conducting body 29 i i i Page VI MAGNETIC ACTIVITY AT LARGE DISTANCES FROM THE SOURCE 31 a. A c t i v i t y minimum 31 b. A c t i v i t y maxima 33 V I I EXPERIMENTAL EVIDENCE AND COMPARISON WITH THE THEORY 38 a. The pl a n e t s 38 b„ The moon 49 V I I I CONCLUSIONS AND SUMMARY 52 IX BIBLIOGRAPHY 55 X APPENDIX 58 i v TABLES Page I P r o p e r t i e s of the s o l a r wind and p a r t i c l e clouds near the ear t h . 22 I I Values of re l e v a n t parameters i n i n t e r p l a n e t a r y space. 23 I I I Experimental r e s u l t s i n terms of Mach angle and Mach number 43 APPENDIX Symbols and conventions used i n t h i s t h e s i s . 58 V FIGURES Geometry f o r the shock equations P o l a r v e l o c i t y diagrams f o r magneto-hydrodynamic waves (a) V A > V5 (b) -Vs > V4 Supersonic flow past 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 - s l o w l y v a r y i n g Mach V and wave path Mach cone and wave f r o n t Geometry f o r K e l l o g g 1 s 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 frequency - Venus (a) great storms (b) occasions when Kp ^ 3 0 (c) f o u r sets of data, each set given equal weight v i Magnetic a c t i v i t y frequency - Mercury (a) great storms (b) f o u r s e t s of data, each obser-v a t i o n given equal weight Blue c l e a r i n g frequency - Mars Magnetic a c t i v i t y frequency - Moon (a) 112 greatest storms ) (b) small storms ] (c) occasions when Kp^3o) (d) a l l disturbances ] (e) storms A sudden commencement ) B gradual commencement] Earth-magnetosphere-moon system, scale drawing 1 I . INTRODUCTION Bigg (1963 b) has presented s t a t i s t i c a l evidence f o r lunar 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 observed at the earth's surface. There are peaks of a c t i v i t y frequency when the planet i s at l a r g e d i s t a n c e s from the sun-earth l i n e , i n d i c a t i n g an e f f e c t over a reg i o n of much greater s i z e than the p h y s i c a l d i -mensions of the body. This t h e s i s attempts to e x p l a i n t h i s phenomenon as the r e s u l t of the bow shock wave formed by an object i n a supersonic stream. The object i s a planet or the moon, and the stream i s the s o l a r wind and any p a r t i c l e clouds or streams which are emitted by the sun. The flow i s supersonic to magnetohydrodynamic waves w i t h i n i t . The symbolism used throughout w i l l be conven-t i o n a l , and i s l i s t e d i n the Appendix. I t w i l l not be f u r t h e r explained w i t h i n the t e x t except where the author f e e l s that i t i s unc l e a r or unconventional. 2 3 I I . THE BOW WAVE a. Hydromagnetic shocks The ba s i c r e l a t i o n s f o r hydromagnetic shocks are the De Hoffmann-Teller equations. An e x c e l l e n t d e r i v a t i o n of them i s given by Bershader (1959* PP. 18-20) I f a re c t a n g u l a r coordinate system i s chosen such that the x a x i s i s normal to the plane of the shock f r o n t and the y and z axes l i e w i t h i n the plane, the equations have the f o l l o w i n g form: (1) Conservation of mass = O (2) Conservation of f i e l d l i n e s a; B = O b. c. B ^ B ^ -T2. - i "2-B •x. U (3) Conservation of momentum a. p +- jO U. B , = 0 4 b. c. -12. = O = o (4) Conservation of energy v -[ 1' -I.E. = 0 denotes the value of the expression i n the brac-k e t s behind the shock minus i t s value i n f r o n t of the shock. These equations are too complicated to use i n t h i s form i n an a n a l y s i s of the bow shock wave i n a three dimensional s i t u a t i o n . K e l l o g g (1962), S p r e i t e r and Jones (1963) and Obayashi (1964) use the Rankine-Hugoniot r e l a -t i o n s f o r an or d i n a r y gas, t a k i n g values of Y — —7^— from 5/3 to 2 depending on the number of dimensions i n v o l -ved i n the compression. The assumption i n t h i s approach i s that = 0 , which s i m p l i f i e s the De Hoffmann-Teller r e l a t i o n s to the f o l l o w i n g form: 5 (5) Conservation of mass -12. pUyC = 0 (6) Conservation of f i e l d l i n e s -12. u«. B = 0 (7) Conservation of momentum a b. 2- D 2 --i2 = 0 - 0 - O (using 5) where Uf denotes the component of u* t a n g e n t i a l to the shock plane. (8) Conservation of energy 12. = o As B i s assumed to be everywhere perpendicular to the shock f r o n t , i t e x e r t s only a pressure across the 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) of pressure p^ag = -g-p— . The equations then become: (9) Conservation of mass r n 2-- . r = O 6 (10) Conservation of f i e l d l i n e s T2. u , 8 = O (11) Conservation of momentum a. b. -rZ 1 2. = O -1 2. P ' -V pu-x? = 0 (12) Conservation of energy 2- U -t 2. -I 2. where P = 1° - o and I f p«,qej » p ; =- 2. , and i f p » I n t h i s form, equations 9 , 11, 12 are i d e n t i c a l w i t h the Rankine-Hugoniot equations. Equation 10 i s an e x t r a one which merely serves to s p e c i f y the magnetic f i e l d . 7 The l a r g e s t e r r o r created by the assumption that Bjc = 0 l i e s i n the neglect of magnetic t e n s i o n terms that should a r i s e from f i e l d l i n e s c r o s s i n g the i n t e r f a c e . I f the f i e l d i n the incoming plasma i s per-p e n d i c u l a r to 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 to the f r o n t only on a great c i r c l e p erpendicular to if. The approximation may thus be q u i t e good on a s t r i p near the great 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 important. I t must f u r t h e r be noted i n a p p l y i n g t h i s theory to the earth that i n s i d e the shocked re g i o n ~2L|0U^~' (from s a t e l l i t e measurements). Thus the hydrodynamic f o r -ces do not predominate over the f i e l d f o r c e s as i s the case i n the i n c i d e n t stream. b. Hydromagnetic waves In d e a l i n g w i t h the bow wave at d i s t a n c e s l a r g e compared to the dimensions of the body, a wave approach i s e a s i e r than an extension of the d i s c u s s i o n of one dimen-s i o n a l shocks. At frequ e n c i e s near and below the i o n c y c l o -t r o n frequency ( tO^ ), there are three modes by which e l e c -tromagnetic waves can propagate. This t h e s i s i s concerned mainly w i t h low frequencies and w i l l henceforth r e f e r to 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. The d i s c u s s i o n w i l l be l i m i t e d f o r the present to the case «0 <£<L u); , and to dense p l a s m a s — p l a s m a s i n which the hydromagnetic v e l o c i t i e s are much l e s s than the speed of l i g h t . Under such c o n d i t i o n s , a l l three modes are non-dispersive, and, to the f i r s t order, e l e c t r i c a l l y n e u t r a l . The ions and e l e c t r o n s move together. The oblique A l f v e n mode has a phase v e l o c i t y : ( 1 3 ) 2" ' \ / a V p — VA cos © where © i s the angle between the plane wave propagation vector ( k ) and the steady magnetic f i e l d ( 6 e )• The p a r t i c l e displacements and the p e r t u r b i n g magnetic f i e l d are perpendicular to G»e and k , so that there i s no c o u p l i n g w i t h the other two modes which are confined t o the plane defined by B e and k . The group v e l o c i t y i s given by V<$ — VA , and i s p a r a l l e l to 80 , i n d i c a t -i n g t h a t energy t r a n s f e r can take place only along the f i e l d l i n e s i n the oblique A l f v e n mode. The a c c e l e r a t e d and retarded magnetoacoustic modes have phase v e l o c i t i e s given by: v P - 2 the + and - signs i n d i c a t i n g a c c e l e r a t e d and retarded modes 9 r e s p e c t i v e l y . V 5 = y i s the speed of "sound" i n the plasma i . e . the speed at which a compression wave w i l l propagate through an unmagnetized plasma. The par-t i c l e v e l o c i t y component r a t i o s are given by: V* t a n 8 V Vp" where the coordinate axes are defined such that x i s nor-mal to Bo and k , and z i s p a r a l l e l to k . I f V\ and V£ are of the same order of magnitude, the motions of the a c c e l e r a t e d and retarded mode p a r t i c l e s are not orthogonal, and there i s c o u p l i n g between the two modes. Po l a r v e l o c i t y diagrams are shown f o r VA >VS A N C * Vs > VA i n f i g u r e 2. For the two extreme cases: and , the co u p l i n g becomes s m a l l . I f \^ » \4 > the a c c e l e r a t e d mode has a phase v e l o c i t y v p V, , and the p a r t i c l e motion i s perpe n d i c u l a r to B© . This mode i s a combination of an A l f v e n wave along the f i e l d and a compressional wave across the f i e l d . The slow mode has a phase v e l o c i t y Vp - Vs cos G , and the p a r t i c l e motion and energy t r a n s f e r are p a r a l l e l to Bo , the group v e l o c i t y being Vs . Conversely i f V§ >> VA > T N E f a s t mode has a phase v e l o c i t y Vj= V4 • The p a r t i c l e motion i s p a r a l l e l to l< I t i s a combination of a (b) V 8 > V A f i g u r e 2. tolas? v e l o c i t y diagrams f©3? fflagn§t©hydi?©« dynamiG wav§§, 11 sound wave along the f i e l d l i n e and a compressional wave pe r p e n d i c u l a r to the f i e l d . The slow mode has a phase v e l o c i t y Vp - VA cos © . I t I s very l i t t l e d i f f e r e n t from an oblique A l f v e n wave, and repre-sents energy t r a n s f e r along the f i e l d l i n e s . I n e i t h e r of these extreme cases from any given s t a t i o n a r y source, there should 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 magnetoacoustic mode, and r a d i a t i o n along the f i e l d l i n e s i n the retarded magnetoacoustic and A l f v e n modes. Applying t h i s to the bow wave produced i n steady supersonic flow, the a c c e l e r a t e d mode should produce an almost c i r c u l a r cone, and the slow mode and the A l f v e n mode should produce two-dimensional V s . The V w i l l not be symmetric about the flow d i r e c t i o n except i n the case when the i n t e r p l a n e t a r y f i e l d i s pe r p e n d i c u l a r to the streaming v e l o c i t y . I t w i l l become more symmetric w i t h 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 of attempts to c a l c u l a t e the f low near a sphere submerged i n a supersonic stream: L i n and Rubinov (19^8); Dugundji ( 1948); Hida (1953); Van Dyke and M i l t o n (1958); Van Dyke, M i l t o n and Gordon (1959); 12 I n o u y e , Mamenu and Lomax (1962). The l a s t t h r e e o b t a i n e d r e s u l t s by n u m e r i c a l a n a l y s i s , and the f i r s t t h r e e o b -t a i n e d a n a l y t i c a l s o l u t i o n s . The a p p r o a c h i s g e n e r a l l y t o assume R a n k i n e - H u g o n i o t f l o w t h r o u g h the shock, and a r b i t r a r y f l o w c o n d i t i o n s b e h i n d i t . Thus f o r example, H i d a assumes i n c o m p r e s s i b l e i r r o t a t i o n a l f l o w whereas Van Dyke et a l assume c o m p r e s s i b l e i r r o t a t i o n a l f l o w . The r e s u l t s a r e v a l i d o n l y i n a r e g i o n i n f r o n t o f the s p h e r e because o f t h e type o f a p p r o x i m a t i o n made. E x -p e r i m e n t a l o b s e r v a t i o n i n d i c a t e s t h a t a p a t t e r n s i m i l a r t o t h a t i n f i g u r e 3 I s f o r m e d . I n t h r e e d i m e n s i o n a l s u p e r s o n i c f l o w , s m a l l a m p l i t u d e d i s t u r b a n c e s p r o p a g a t e a l o n g Mach c o n e s . I f a body i s a x i a l l y symmetric t h e s e c a n be t r e a t e d as Mach l i n e s s i m i l a r t o t h e two d i m e n s i o n a l c a s e . These l i n e s a r e shown I n f i g u r e 3. I t c a n be s e e n t h a t t h e r a r e f a c -t i o n t e n d s t o c a t c h up w i t h t h e f r o n t shock and w a i t f o r the r e a r shock, showing t h a t such a d i s t u r b a n c e w i l l a t -t e n u a t e more r a p i d l y t h a n geometry and v i s c o s i t y i n d i c a t e . T h i s i s r e f e r r e d t o as t h e r a r e f a c t i o n " e a t i n g i n t o the s h o c k " . However, d e s p i t e t h i s e f f e c t , s h o c k s a r e o f t e n o b s e r v e d a t g r e a t d i s t a n c e s f r o m a s o u r c e compared t o the s o u r c e s i z e , e . g . s o n i c booms f r o m a i r c r a f t (25 m i l e s , H . A . W i l s o n , J r . , 1962) and bow waves f r o m s m a l l b o a t s . F o r a i r c r a f t s o n i c booms, A p ^ — 3 7 - ( H . A . W i l s o n , J r . , 13 14 1 9 6 2 ) . This I m p l i e s that the energy &z r" - 1'* The shock i s asymptotic to a Mach cone wi t h angle arc s i n —^-— where M( i s the Mach number of the i n c i d e n t stream. I t approaches t h i s cone as i t degenerates i n t o an o r d i n a r y small amplitude wave. The b a s i c wavelength of the disturbance w i l l be somewhat l a r g e r than the body s i z e . I f the f l o w i s quasi-steady ( v a r i a t i o n s of much greater l e n g t h than the body s i z e ) , the p r i n c i p a l disturbance produced should be the bow wave. Figu r e 4b shows the type of p a t t e r n that would be produced by con-stant speed s i n u s o i d a l l y , v a r y i n g Mach number flow (which i m p l i e s d e n s i t y f l u c t u a t i o n s ) . This bow wave v a r i e s both i n s t r e n g t h and d i r e c t i o n . I n an i r r e g u l a r l y v a r y i n g stream, a r e c e i v e r at A would observe a great v a r i e t y of frequencies and amplitudes. d. D i f f e r e n c e between hydromagnetic and hydrodynamic  bow waves at l a r g e d i s t a n c e s from the source. Perhaps the greatest m o d i f i c a t i o n of the hydro-dynamic p i c t u r e a r i s e s from the existence of three modes of propagation i n a plasma. The-accelerated magneto-a c o u s t i c mode w i l l form a Mach cone, and the oblique 15 Figure 4. 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 - slow l y v a r y i n g . 16 A l f v e n and slow magnetoacoustic modes Mach V's as t h e i r group v e l o c i t i e s are p a r a l l e l to the magnetic f i e l d . The A l f v e n mode has the same group v e l o c i t y as one of the other two, and consequently only one cone and one V w i l l be i n evidence. Over a time average of the type used by Bigg (1963 a, b ) , the f i e l d may be expected to take 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 of propagation may be expected to be i n evidence. The Mach angle ( © ) f o r a V w i l l be a f u n c t i o n of the angle between the i n t e r p l a n e t a r y f i e l d and the v e l o c i t y as w e l l as the Mach number. Fig u r e 5 shows the geometry of the s i t u a t i o n , g i v i n g the equation: < 1 6) (V) - s i n (y - S) 1 s m © In the s p e c i a l case 6>c J_ U , t h i s reduces to the form: ( 1 7 ) M , = L S i m i l a r l y , f o r the cone, e l l i p s o i d a l wave f r o n t s make the r e l a t i o n (18) _ -1 sin 0 which i s exact f o r s p h e r i c a l wave f r o n t s , only approximate I f there i s a l a r g e discrepancy i n the values of V£ and Figure 6. Mach cone and wave f r o n t . 18 , the wave f r o n t s w i l l be approximately s p h e r i c a l and equation (18) can be used. I f ^ ~ VA > c o u p l i n g occurs between the two magnetosonic modes, i n d i c a t i n g that a wave of one type would act as a source f o r the other. I f one i s much l a r g e r than the other, 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 co u p l i n g f o r propagation p a r a l l e l to the f i e l d , as one of the modes becomes an A l f v e n wave. Perpendicular to the f i e l d , there i s only one mode of propagation, and as such a d i s t i n c t peak of energy must e x i s t . At other angles, the behavior i s very complicated, being a regime i n which the f a s t mode i s changing from a combined A l f v e n and perpendicular pressure wave to a com-bined l o n g i t u d i n a l pseudosonic and perpe n d i c u l a r pressure wave. The energy i s probably spread over a l a r g e region of space, each mode a c t i n g as a source f o r the other. The existence of uncoupled modes p a r a l l e l and perpendicu-l a r to the f i e l d should r e t a i n the v a l i d i t y of the argu-ments presented i n t h i s t h e s i s despite the co u p l i n g , and should a c c o r d i n g l y produce magnetic a c t i v i t y peaks at the same p o s i t i o n s as f o r uncoupled propagation. As W ~ u>i i n the shock f r o n t i t s e l f , the 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 the retarded mode which reaches a resonance p o i n t at t h i s frequency 19 (Denisse and D e l c r o i x 1 9 6 3 ) . The d i s p e r s i o n w i l l be small f o r the other modes. The e f f e c t w i l l be to break the retarded mode i n t o a dispersed t r a i n of waves. I n such a t r a i n , each frequency would tend to f o l l o w i t s own Mach V or cone. The geometric a t t e n u a t i o n of 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 to that of sound waves, wi t h an energy drop o f f ©d along the cone. For the guided modes, there should be no geometric a t t e n u a t i o n with d i s t a n c e . However In a l l cases there should s t i l l be the equivalent of the r a r e f a c t i o n e a t i n g i n t o the 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 to be the expanding s o l a r corona, which Parker (1963) has shown to be un-st a b l e — the sun's g r a v i t a t i o n a l f i e l d being i n s u f f i c i e n t to c o n t a i n the high temperature (10^°K) gas. I t takes about 5 days to p i c k up speed, and then about f o u r more days to reach the earth's o r b i t . As the k i n e t i c energy of the gas i s much greater than the magnetic f i e l d energy, the f i e l d l i n e s from the sun are dragged out wi t h the f l u i d , forming s p i r a l s due to the hosepipe e f f e c t (Parker 1 9 6 3 ) . I t i s , however, a very "gusty" wind, and the f i e l d c o n tains many ki n k s and wiggles. The p r o p e r t i e s of the wind as measured by s a t e l l i t e s ( E x p l o r e r X, Mariner I I , Lunik I and I I ) i n the v i c i n i t y of the ea r t h are given i n t a b l e I . b. P a r t i c l e clouds Another f e a t u r e of i n t e r p l a n e t a r y space i s the p a r t i c l e clouds a s s o c i a t e d w i t h s o l a r f l a r e s . These have higher v e l o c i t y and d e n s i t y than the s o l a r wind, t a k i n g about two days to reach the ear t h , where they cause mag-n e t i c storms. T h e i r p r o p e r t i e s i n the v i c i n i t y of the 21 earth are a l s o summarized i n t a b l e I . Sudden commencement storms are b e l i e v e d to be caused by such a cloud preceded by a magnetohydrodynamic shock wave. c. Magnetohydrodynamic theory Magnetohydrodynamic theory and consequent f l u i d flow methods r e q u i r e s t h a t : ( i ) A l l disturbance dimensions be much greater than the p a r t i c l e g y r a t i o n r a d i u s . ( i i ) A l l frequencies be l e s s than the i o n g y r a t i o n frequency. Three cases are considered. These correspond to c o n d i t i o n s i n the s o l a r wind, the p a r t i c l e cloud, and the re g i o n be-tween the shock f r o n t and the o b s t a c l e . Table I I shows the r e l e v a n t parameters f o r protons i n such regions, u s i n g t y p i c a l values cf the variables.. I t can be seen th a t we must dea l w i t h dimensions much greater than 10G km. and frequences much l e s s than 1 cps. This does not apply to the shock f r o n t i t s e l f which contains the mechanism of t h e r m a l i z a t i o n of the streaming energy. A l l the p l a n e t s f u l f i l l these requirements. The moon i s the smallest body considered having a diameter of about 3500 km. The basic p e r i o d of the disturbance caused by t h i s body w i l l be somewhat l a r g e r than — a f a c t o r of 10 greater than the i o n c y c l o t r o n p e r i o d . 22 Table I P r o p e r t i e s of the s o l a r wind and p a r t i c l e clouds near the e a r t h . S o l a r Wind Clouds streaming v e l o c i t y 300-500 km/sec. 600-1000 km/sec. p a r t i c l e d e n s i t y 1-10 /cc. 5-30 /cc. temperature 5 ~ 2 x 10 °K ~ 7 x 1 0 5 o K contained f i e l d " 5 Y 10-50 r 23 Table I I Values of re 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 Wind Cloud Shock region F i e l d strength ( y ) 5 20 30 Temperature (°K) 2 x 1 0 5 5 7 x i c r Thermal v e l o c i t y = / YkT"_ V (r\: km./sec. 70 100 300 r. _ rr\; V (km.) c^i 6 140 50 100 M* - Qi B ( c y c l e s / s e c . ) .5 2 3 w . B (km./sec.) V* j pop 50 100 I - | -r— (km./sec .) 100 140 24 d. Supersonic flow I t remains to show tha t the flow i s supersonic f o r a l l magnetohydrodynamic modes. Two v e l o c i t i e s must be considered: the A l f v e n v e l o c i t y Vi - > ^  , and the pseudosonic v e l o c i t y which i s given by Denisse and D e l c r o i x (1963) as: Y ^ = HP., m e . Vfr ± n i . N\L Vi  where me-Ve. = Ye.\<Te, and WU =• Yj' k T~i I f rie. =. ni and thermal e q u i l i b r i u m e x i s t s between Ions w „ 2. Y l< T~ and e l e c t r o n s , \A ^= —^—•— . The value of V here i s not c l e a r l y defined, depending on the number of degrees of freedom ( s ) . V =. ^ ^ ^~ For pro-pagation p a r a l l e l to the f i e l d , the magnetoacoustic wave undergoes e s s e n t i a l l y a one dimensional compression, and as such has s = 1, and V = 3 . Perpendicular to the f i e l d , the compression i s two dimensional g i v i n g s = 2, and "V = 2 . The r e l e v a n t values of and Vs are shown i n t a b l e I I . The streaming v e l o c i t i e s are well'above the wave v e l o c i t i e s , i m p l y i n g supersonic flow with Mach numbers i n the range from one to ten. 25 IV. PREVIOUS SHOCK WORK a. K e l l o g g (1962) K e l l o g g makes the assumption that the i n t e r -p l a n e t a r y f i e l d i s normal to the streaming v e l o c i t y , and the plasma can th e r e f o r e be t r e a t e d as a hydro-dynamic gas usi n g Y - 2. . This i s e s s e n t i a l l y the approximation considered i n H a . His f u r t h e r develop-ment uses the r e s u l t s of Hida (1953), i n which incom-p r e s s i b l e flow around the body i s assumed a f t e r the f l u i d has passed through a Rankine-Hugoniot shock. The approxi-mations i n v o l v e d make the s o l u t i o n r e l i a b l e only i n a regio n i n f r o n t of the sphere, as expl 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 flo w w e l l behind the sphere along the c e n t r a l streamlines by c o n s i d e r i n g adia-b a t i c expansion of the gas to the pressure of the incoming f l u i d subsequent to passage through the shock. Combining the Rankine-Hugoniot r e l a t i o n s and the expression f o r a d i a b a t i c expansion: (19) ft he d e r i v e s the expression: (20) = V - \ 2. 26 where J-*2" — ^ ^\ The s u b s c r i p t s i n d i c a t e the corresponding 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 approximations to K e l l o g g . They use the numerical methods-of Van Dyke et a l (1958, 1959)> and allow f o r compressible hydro-dynamic flow behind the shock f r o n t . They a l s o a l l o w f o r the n o n - s p h e r i c i t y of the magnetosphere us i n g the boundary c a l c u l a t e d by Beard ( i 9 6 0 ) r o t a t e d about the sun-earth l i n e . Once again the approximations discussed i n I l a are used. c. Obayashi (1964) Obayashi d i s c u s s e s the s a t e l l i t e , data of 13 space probes i n t e r p r e t i n g t h e i r observations i n terms of shock theory. He a l s o discusses t h e i r f i n d i n g s of the p r o p e r t i e s of the i n t e r p l a n e t a r y plasma. d. Beard (1964) Beard d i s c u s s e s the phenomenon from a p a r t i c l e viewpoint. He dis c u s s e s the case of the 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 Kellogg's equation. 28 f i e l d p a r a l l e l to the streaming v e l o c i t y , and concludes that except f o r a small pocket of p a r t i c l e s at the sub-s o l a r p o i n t , the p a r t i c l e s w i l l f low a d i a b a t i c a l l y around the body. The hosepipe e f f e c t (Parker, 1963) i m p l i e s t h a t the f i e l d l i n e s are on the average at 45° to the streamlines (Walters, 1964), and so the c o n f i g u r a t i o n Bi J) U i s i n f r e q u e n t . For an oblique f i e l d , Beard obtains r e s u l t s s i m i l a r to those of K e l l o g g . 29 V. CONDITIONS UNDER WHICH SHOCKS FORM a. Body wi t h a magnetosphere The mechanism of formation of a standing shock and bow wave f o r a body w i t h a magnetosphere has been w e l l described: K e l l o g g (1962), S p r e i t e r and Jones (1963), Axford (1962), Obayashi (1964), Beard ( 1 9 6 4 ) . I t i n v o l v e s the i n t e r a c t i o n of a plasma and a contained 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 that many of the bodies of the s o l a r system have magneto-spheres, p a r t i c u l a r l y those w i t h o r b i t s o utside Venus. These p l a n e t s should have a corresponding shock wave. b. Immersed conducting body In order to cause l i t t l e disturbance i n the flow, a body would have to al l o w the passage of l i n e s of fo r c e at the v e l o c i t y of the streaming plasma. D i f f u s i o n processes are much too slow. I t could be accomplished 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 E = a x B . For U = 400 km./sec. and B = 5 r , E = 2 x 1 0 ~ 3 volts/m. For a body the s i z e of the moon, t h i s represents a voltage of 7000 v o l t s across i t . Such a voltage buildup seems u n l i k e l y f o r a body immersed In a h i g h l y conducting plasma. I t i s l i k e l y t hat charge leakage would occur at the s i d e s . 30 The above process r e q u i r e s the " d e f r e e z i n g " of l i n e s of f o r c e at the f r o n t of the body, and must r e s u l t In the formation of a t r a n s i e n t atmosphere and ionosphere f o r such a body. This has been discussed f o r the moon by H e r r i n g and L i c h t (1959), Singer (1961), Nakada and Mihalov (1962), Weil and Barasch (1963), Gold (1959), and Hinton and Taeusch (1964). Hinton and Taeusch o b t a i n par-t i c l e d e n s i t i e s of 10^ /cc. 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 i o n s . The gas pressures are greater than the s o l a r wind pressures, and hence w i l l act as the con-d u c t i n g body which forms the shock. I t I s l i k e l y that the i o n sheath, or ionosphere, w i l l be densest on the up-stream side of the body. I f a body i s surrounded by an atmosphere and ionosphere and no magnetic f i e l d , the ionosphere w i l l p l a y the p a r t of the conducting body, and a shock wave can s t i l l be expected. A l l the bodies of the s o l a r system should f a l l i n t o one of the above c l a s s i f i c a t i o n s , and may thus be ex-pected to have shock waves unless s i z e c o n s i d e r a t i o n s ex-clude them from the regime of hydromagnetic theory. 31 V I . MAGNETIC ACTIVITY AT LARGE DISTANCES PROM THE SOURCE. At t h i s stage, the author wishes to suggest the f o l l o w i n g ideas: ( i ) There w i l l be a decrease i n magnetic a c t i v i t y w i t h i n the region behind a shock wave. The a c t i v i t y w i l l be l e a s t on the c e n t r a l stream l i n e . ( i i ) There w i l l be an increase of magnetic a c t i v i t y i n a bow wave. An attempt w i l l be made to j u s t i f y each of these statements i n t u r n . a. A c t i v i t y minimum As f l u i d passes through a shock system of the type near the earth, energy i s transformed from k i n e t i c streaming energy i n t o thermal energy. A greater f l o w produces a stronger shock, r e s u l t i n g i n a greater l o s s of streaming energy. This i m p l i e s that a slow v a r i a t i o n In energy f l u x i s attenuated as i t passes through the 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 not pos-s i b l e because of the indeterminate nature of the shock equations. (There i s one more v a r i a b l e than equations.) I d e a l l y one would l i k e to evaluate: (21) d ( p i - u u - r p Q =2- d ( f f t u y ) 32 However, i t i s p o s s i b l e to r e l a t e t h i s to the work of Bigg (1963 a, b ) , as h i s r e s u l t s are based on the frequency of occurrence 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 3 0 . As the time i n t e r v a l s on which these are based are long, they are a measure of A 3 r a t h e r than of ^ ^ . I f these f l u c t u a t i o n s are assumed to be due to pressure changes at the magnetosphere boundary, we are i n d i r e c t l y measuring A (jOU X-*-p) — A (pM***) (This assumption should be approximately true u n t i l r i n g c u r r e n t s have had time to b u i l d up, and even then the mag-n e t i c disturbance may give some i n d i c a t i o n of pressure f l u c t u a t i o n s at the boundary.) Thus an a t t e n u a t i o n i n A ( p u ^ 9 would imply an a t t e n u a t i o n i n A S The a t t e n u a t i o n i n i s given by: where the s u b s c r i p t s apply to the regions shown i n f i g u r e 7. For 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 smal-l e r than the maximum terms. For example consider a f l u c -t u a t i o n from s o l a r wind to cloud c o n d i t i o n s . I f the values of t a b l e I I are used, (K^P')^Q* =2== ) ^  . Under these circumstances, i t becomes p o s s i b l e to ignore the 33 second terms i n equation (22) l e a d i n g to 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 the s o l a r wind: (23) /\ - M** wcur A p l o t of A versus Mj i s shown i n f i g u r e 8 f o r Y = 2 u s i n g equation (20). The values used f o r M( and M 3 must be taken at maximum M | . This should a l s o be the maximum of M s . Higher M| values are u s u a l l y as-so c i a t e d w i t h 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 that i f t h i s i s t r u e , the l a r g e s t f l u c t u a t i o n s would be attenuated the most. However, i f the models pre-sented i n t a b l e s I and I I are c o r r e c t , there i s l i t t l e change i n M.| w i t h a c t i v i t y . b. A c t i v i t y maxima The a c t i v i t y increase I n the bow wave can be discussed only q u a l i t a t i v e l y . The bow wave w i l l e x i s t 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 amplitude to produce a measurable e f f e c t . During storms the k i n e t i c energy of the streaming f l u i d i n c r e a s e s by one or two orders of magnitude, and there should be an increase i n bow wave a c t i v i t y — s u f f i c i e n t perhaps to add n o t i c e a b l e disturbance to an e x i s t i n g storm or a c t i v e p e r i o d . I t can be shown that 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 to 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 the f r o n t of a body the s i z e of the magnetosphere (a c i r c l e of r a d i u s 15 e a r t h 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 of the order of 10 ergs/sec. I f energy of t h i s amount i s assumed to spread out i n a Mach 3 cone, i t i s spread over a c i r c l e of circumference 2 ir s t a n O =^ 2s at a dis t a n c e s behind the source. For t y p i c a l i n t e r p l a n e t -8 ary d i s t a n c e , t h i s i s of the order of 10 km. A r e c e i v e r the s i z e of the magnetosphere would i n t e r s e c t a l i n e seg-5 ment 10 km. long, and might be expected t h e r e f o r e to l O 2 19 r e c e i v e ~ ^ t t = 1 0 o f t n e t o t a l energy or 10 ergs/sec. Axford (1963) estimates the energy d i s s i p a t i o n d u r i n g a . 18 magnetic storm to be ~ 10 ergs/sec. Even a f a c t o r of 10 lower would be capable of producing measurable d i s -turbance. The above c a l c u l a t i o n of a t t e n u a t i o n took i n t o account only geometric e f f e c t s . I f a dropoff o<. p i s used, as i n the case of the sonic boom (H. A. Wilson, J r . .1962), the 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 estimate of the s i z e of the bow wave i s developed here. From the second Hugoniot shock equation (11), the f o l l o w i n g r e l a t i o n (24) p x - p , - A p - ( ^ u ^ ) l - ( p u ^ ) i ~ ( ( ° u * ) i 36 holds across the standing shock In f r o n t of the ear t h , and out to the s i d e s . The assumption i s made that the decrease i n A p w i t h d i s t a n c e i s the same as f o r a supersonic a i r c r a f t , d i s t a n c e s being measured along the cone. Prom H. A. Wilson, J r . ( 1 9 6 2 ) : (25) Now i f equation (24) i s true at p o i n t A i n f i g u r e 14, and equation (25) i s assumed to ho l d from there onwards out along the Mach cone, A p can be c a l c u l a t e d at l a r g e d i s t a n c e s . For a Mach 3 cone, A i s at about 27 e a r t h r a d i i from the earth ( f i g u r e 14) and about 3 x 27 = 8 l earth r a d i i from the vertex of the cone. For a t y p i c a l i n t e r p l a n e t a r y distance 5 x 10 km., equation (25) i m p l i e s that A p e -03 A p * • By comparison wi t h sudden storm commencements, which must a l s o represent a pressure jump ~ f5U>t 1 t h i s represents magnetic f l u c t u a t i o n s of a few gammas at a r e c e i v e r . This i s perhaps a l i t t l e low to produce much d i f f e r e n c e i n magnetic a c t i v i t y , but the u n c e r t a i n t i e s i n the assumptions could e a s i l y make a d i f -ference of a f a c t o r of 10. For a Mach V, there i s no geometric a t t e n u a t i o n , and th 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 with the f a s t mode, supply-i n g e x t r a energy i n the cone, and e x t r a magnetic a c t i v i t y . 37 The theory then p r e d i c t s that there w i l l be an a c t i v i t y minimum behind such a body, and two maxima on each side corresponding to the three modes of propaga-t i o n . The remainder of t h i s t h e s i s w i l l be concerned with the comparison of experimental evidence w i t h the theory. 38 V I I . EXPERIMENTAL EVIDENCE AND COMPARISON WITH THE THEORY a. The P l a n e t s I t has been suggested by B i g g and De Vaucouleurs (Bigg, 1963 b) tha t the blue 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 frequency. For the purpose of t h i s t h e s i s i t w i l l be assumed that the frequency of blue c l e a r i n g s i s a maximum when a minimum of magnetic a c t i v i t y e x i s t s at Mars and/or i n the l i n e of s i g h t be-tween the earth and Mars. F i g u r e s 10 - 13, a l l from Bigg (19^3 a and b ) , show the dependence of magnetic disturbance frequency on the p l a n e t a r y p o s i t i o n s of Mercury, Venus, and the moon, and the blue c l e a r i n g frequency f o r Mars. Table I I I shows the Mach angles t h a t correspond to the 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, and the minima i n f i g u r e 12. The geometry on which the c a l c u l a t i o n s are based i s shown i n f i g u r e 9- The BB maxima are 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 the use of equation (18) r a t h e r than (16) to c a l c u l a t e Mach numbers f o r these peaks must introduce some e r r o r . I f duri n g storms the i n t e r p l a -n etary f i e l d i s not o r i e n t e d , on the average, i n a d i r e c t i o n w i t h i n about 45° of the streaming v e l o c i t y , the e r r o r s should not be l a r g e . I t must be pointed out that the angle oL i n 39 Figure 9. Geometry f o r t a b l e I I I . 40 A j \|/A (A) 40 30 20 10,. Inferior 10 20 30 40 SO Degrees East Degrees West F i g u r e 10. Magnetic a c t i v i t y frequency - Venus. (A) great storms; (B) occasions when Kp£* 3 0 > (C) fo u r s e t s of data, each set given equal weight. 41 I A F i g u r e 11. Magnetic a c t i v i t y frequency - Mercury. (A) great storms; (B) f o u r sets of data, each observation given equal weight. 42 30 20 10 0 10 20 30 Before Opposition ANGLE E.M.S. A f t e r Opposition Figure 12. Blue c l e a r i n g frequency - Mars. TABLE I I I . Experimental r e s u l t s In terms of Mach angle and Mach number. F i g . Angle cLe*. bet-ween peaks, degrees f*|, = S\fY'6 b a Mach angle B degrees Mach M number AA BB AA BB AA BB AA BB 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 l i b 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 1 AA 1 AA 1 AA 1 10b 34 ? .405 1.38 24 2.5 11a 16 ? .360 2.58 21 2.8 44 G i I80*W 90*W 0* Lunar position 90*E 180'E c 90*W o* Lunar position 90*E ISO*E F i g u r e 13. Magnetic a c t i v i t y frequency aT 112 greatest storms; b) small storms. Moon, 45 Lunar position F i g u r e 13. Magnetic a c t i v i t y frequency - Moon, fcT occasions when Hp ^ 3 0 (d) a l l disturbances Figure 13. Magnetic a c t i v i t y frequency - Moon. (e) storms A sudden commencement B gradual commencement 47 f i g u r e s 10 to 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 conjunction. Venus, at i n f e r i o r c o njunction, i s at angles from o° to 8° from 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° east to 6° west. T h i s makes i t Impossible to determine i f BB type maxima appear i n f i g u r e s 10 a, b, c. The few p o i n t s a v a i l a b l e and the approaches to the region i n d i c a t e that there probably are maxima w i t h i n t h i s region. In f i g u r e 10b, i t i s not obvious which peak should be used on the east side of the AA maxima. There 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 others. S i m i l a r l y i n f i g u r e 11a there 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 centre A of the region of u n c e r t a i n t y (as shown by the shaded rectangle) i s used, the r e s u l t s from these two f i g u r e s are more c o n s i s -tent with the others. Also 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 than the w e s t e r l y maximum, as i t i s f o r a l l cases w i t h c l e a r l y defined peaks. The r e s u l t s f i t a model wi t h Mach numbers 2.5 and 15 f o r the two types of bow wave. 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 retarded magnetoacoustic modes i n t u r n w i t h the oblique A l f v e n mode adding to one of these depend-i n g on the r e l a t i v e s i z e s of VA and Vs 48 The l a c k of BB type maxima i n f i g u r e 11a i s i n t e r e s t i n g and suggests a number of p o s s i b l e p r o p e r t i e s of great 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 north-south dur i n g a great storm, so that no guided mode waves would reach the r e c e i v e r . ( i i ) The i n t e r p l a n e t a r y f i e l d may be broken up ( h i g h l y t u r b u l e n t flow) d u r i n g great storms, r e s t r i c t i n g propagation by guided modes to short 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 equal imply-. i n g e q u i p a r t i t i o n between thermal and magnetic energies. ( i v ) The f i e l d lines.may be r a d i a l . At present the author sees no reason to express preference f o r any one of these p o s s i b i l i t i e s . The theory p r e d i c t s t h a t , because of geometrical a t t e n u a t i o n , guided,modes w i l l have more energy at s u f f i -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. This 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 unfortunate t h a t i t i s not p o s s i b l e to compare the r e l a t i v e magnitudes of the two types of peaks w i t h those i n f i g u r e 10c . However, I f blue c l e a r i n g frequency does decrease monotonically w i t h magnetic a c t i v i t y i n c rease, f i g u r e s 12 and 11 b show that the a t t e n u a t i o n i s i n agree-ment wi t h the theory. 49 b. The moon Any e f f e c t of the moon on magnetic a c t i v i t y at the earth may be expected to be small as a r e s u l t of the l a r g e s i z e of the magnetosphere (see F i g u r e 14). For i n -stance, at new moon the attenuated r e g i o n i n s i d e the inner bow wave represents 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 expected to have a l a r g e e f f e c t at the surface of the e a r t h . Any observed v a r i a t i o n i n magnetic a c t i v i t y frequency with lunar angle must be a f u n c t i o n of the angle of incidence of the bow wave on the magnetosphere, the s e n s i t i v i t y of the magneto-sphere to pressure f l u c t u a t i o n s at the p o i n t of 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 from t h i s p o i n t to the surface of the e a r t h . Bigg (1963 a, b) obtains the v a r i a t i o n s shown i n f i g u r e s 13a to e. There i s , however, some question as to the s t a t i s t i c a l s i g n i f i c a n c e of these r e s u l t s . I f the r e -s u l t s are accepted, the maxima i n a c t i v i t y f o r 45°<L oi <8o° i n d i c a t e that the energy t r a n s f e r i s most e f f e c t i v e when the outer bow wave i s i n c i d e n t on the side of the magneto-sphere. The minimum at new moon could i n d i c a t e that no energy i s received from the outer bow wave at t h i s time. The maximum near f u l l moon can only 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, scale drawing 51 passage of the moon through one of the f o l l o w i n g : ( i ) The t a i l of the magnetosphere. ( i i ) F i e l d l i n e s shared by the magnetosphere and the i n t e r p l a n e t a r y plasma. ( i i i ) A subsonic re g i o n i n the l e e of the ea r t h . With the moon i n such regions, the " t a i l wagging" of the magnetosphere due to v a r i a t i o n s i n the i n c i d e n t magnetized plasma, and any other f l u c t u a t i o n s , would be expected to produce a v a r i e t y of hydromagnetic waves w i t h i n the mag-netosphere which could cause a disturbance i n c r e a s e . The top l i n e of f i g u r e 13 e i s f o r sudden com-mencement storms. This 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 wit h the two bow shock waves — a s i t u a t i o n very p o o r l y understood. No attempt w i l l be made here to e x p l a i n the r e s u l t i n g a c t i v i t y frequency p l o t . 52 V I I I . CONCLUSIONS AND SUMMARY The i n t e r p l a n e t a r y plasma and magnetic f i e l d can be t r e a t e d as a f l u i d w i t h supersonic v e l o c i t y . As such, i t forms shock and bow waves at any planet w i t h i n i t s f l o w . The bow wave takes the form of a Mach cone f o r the a c c e l e r a t e d magnetoacoustic mode, and Mach V s f o r the guided modes (the retarded magnetoacoustic 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 the cone or the other Mach V, r e s u l t i n g In only two bow waves. There i s an a t t e n u a t i o n of magnetic a c t i v i t y frequency behind such an o b s t a c l e , the c o e f f i c i e n t of energy a t t e n u a t i o n being given by equation (23). There i s a l s o an Increase of magnetic a c t i v i t y along a bow wave. Thus, the experimental r e s u l t s should e x h i b i t a minimum of a c t i v i t y at i n f e r i o r c o njunction, and two maxi-ma on each s i d e . The observed maxima correspond to Mach numbers of 2.5 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 Vf\ and Vs of 6:1. The theory does not i n d i c a t e which i s g r e a t e r . There should be geometric energy a t t e n u a t i o n "^ p" along a cone, and none along a V. This i s i m p l i e d by the experimental r e s u l t s . The Mach angle f o r a V i s given by equation (16). 53 The a c t i v i t y frequency diagrams f o r the moon are q u i t e d i f f e r e n t from those f o r the p l a n e t s . This i s due to the s i z e of the magnetosphere being of the same order as the earth-moon d i s t a n c e . The r e s u l t i s that the mag-netosphere 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 at the surface of the e a r t h . Bigg (1963 b) shows an a c t i v i t y dependence on lunar p o s i t i o n that can be explained by the presented theory. However, there i s doubt that h i s obser-v a t i o n s are s t a t i s t i c a l l y s i g n i f i c a n t . 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 at the earth are few. Bigg suggests that the planet and the cloud may have e l e c t r o s t a t i c charges. I t i s d i f f i c u l t to see how t h i s c ould produce two maxima on the magnetic a c t i v i t y frequency p l o t . Indeed, the cloud has to be n e a r l y n e u t r a l i n order not to f l y apart. Houtgast and van S l u i t e r s (1962) suggested that Venus achieves i t s e f f e c t by having a very l a r g e magnetic f i e l d . This has been disproved by the r e s u l t s of the Mariner 2 space probe. There i s another p o s s i b i l i t y that i s e s p e c i a l l y a p p l i c a b l e to the case of o i . J_ L| . T h i s i s that p a r t i c l e s are squeezed out along the tubes of f o r c e as the 54 f i e l d i s compressed. They would move from a f i e l d =^ 50 V i n the shock to a region where 3=2* | O v , conserving ^ = constant. This constancy of magnetic moment o implies that V/y (distant) t£= V_L (shock) U t • It would give the appearance of a Mach V with Mach number 2- . This produces only one a c t i v i t y peak, but could combine with one hydromagnetic cone or V to produce the complete pattern. As the v e l o c i t y of these p a r t i c l e s i s of the same order 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 possible. 55 IX. BIBLIOGRAPHY Axford, W. I., Interaction between solar wind and the earth's magnetosphere. J. Geophys. Res. 67, 3791* 1962.-Axford, W. I., Viscous i n t e r a c t i o n between the solar wind and the earth's magnetosphere. C.R.S.R. 153, Cornell University Publication, 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 solar corpuscular rad i a t i o n . J. Geo-phys. Res. 65, 3559, I 9 6 0 . Beard, D.B., The e f f e c t of an interplanetary magnetic f i e l d on the solar wind. J. Geophys. Res. 69, 1159, 1964. Bershader, D., The Magnetodynamics of Conducting F l u i d s . Stanford University Press, 1959. Bigg, E. K., The influence of the moon on geomagnetic d i s -turbances. J. Geophys. Res. 68, 1409, 1963 a. Bigg, E. K., Lunar and planetary influences on geomagnetic disturbances. J. Geophys. Res, 68, 4099, 1963 b. Courant, R., K. 0 . F r i e d r l c h s , Supersonic Flow and Shock Waves. Interscience Publishers, New York, 1948. De Hoffman, F., E. T e l l e r , Magnetohydrodynamic shocks. Phys. Rev. 8 0 , 692, 1950. Denisse, J. F., J . L. Delcrolx, Plasma Waves. Interscience Publishers, New York, 1963. Dugundji, J., J . Aero. S c i . 15, 699, 1948. Gold, T., i n discussion following R. Jastrow, Outer atmos-pheres of the earth and planets. J, Geophys. Res. 64, 1798, 1959. Herring, J-. R., A. L. Lic h t , E f f e c t of the solar wind on the lunar atmosphere. Science 130, 266, 1959. Hida, K. An approximate study on the detached shock wave i n front of a c i r c u l a r cylinder and a sphere. J. Phys. Soc. Japan 8, 740, 1953. 56 Hinton, P. L., D. R. Taeusch, V a r i a t i o n of the lunar at-mosphere with the strength of the solar wind. J. Geophys. Res. 69, 1341, 1964. Houtgast, J., Indication of a magnetic f i e l d of the planet Venus. Nature 175, 678, 1955. Houtgast, J., A. van S l u i t e r s , A new estimate of the stren-gth o f t h e magnetic f i e l d of the planet Venus. Nature 196, 462, 1962. Inouye, Mamanu , Lomas, Comparison of experimental and numerical r e s u l t s - blunt body. NASA TN P, 1426, 1962. Kellogg, P. J., Flow of plasma around the earth. J. Geo-phys. Res. 67, 3805, 1962. L i n , C. C , S. I. Rublnov, J. Math. Phys. 27, 105, 1948. Montgomery, D., Development of hydromagnetic shocks from large amplitude Alfven waves. Phys. Rev. Letters,' 2, 36, 1959. Nakada, M. P., J . D. Mihalov, Accretion of the solar wind to form a lunar atmosphere. J. Geophys. Res. 67, 1670, 1962. Neugebauer, M., C. W. Snyder, Preliminary r e s u l t s from Mariner II solar plasma experiment. Science 138, 1095, 1962. .Obayashi, T., Interaction of solar plasma streams with the outer geomagnetic f i e l d . J . Geophys. Res. 69, 8 6 l , 1964. Parker, E. N. Interplanetary Dynamical Processes. Inter-science Publishers Inc., New York, 19b3. Piddington, J. H., The CIS-Lunar magnetic f i e l d . Plan. Space S c i . 9 , 305, 1962. Singer, S. F., Atmosphere near the moon. Astronaut. Acta 7, 135, 1961. Spreiter, J. R., W. P. Jones, On the eff e c t of a weak int e r -planetary magnetic f i e l d on the in t e r a c t i o n between the solar wind and the geomagnetic f i e l d . J. Geophys. Res. 68, 3555, 1963. Van Dyke, D. Milton, The supersonic blunt-body problem -review and extension. J . Aeron. S c i . 25, 485, 1958. 57 Van Dyke, D. Milton, H. D. Gordon, Supersonic flow past a family of blunt axisymmetric bodies. NASA TR R-l, 1959. Walters, G. K., E f f e c t of an oblique interplanetary magnetic f i e l d on the shape and behaviour of the magnetosphere. J. Geophys. Res. 69, 1?69', 1964. Weil, H., M. L. Barasch, A t h e o r e t i c a l lunar ionosphere. Icarus I, 3^6, 1963. Wilson, H. A., J r . , Sonic Boom. S c i . Amer., Jan., 36, 1962. 58 APPENDIX - Symbols and Conventions Used. A attenuation c o e f f i c i e n t B magnetic f i e l d Bo unperturbed magnetic f i e l d B i interplanetary magnetic f i e l d Cp s p e c i f i c ')heat at constant pressure Cv s p e c i f i c heat at constant volume t= e l e c t r i c f i e l d e as subscript denotes "of electron" L as subscript denotes "of ion" (except Bi ) k Boltzmann's constant wave propagation vector m mass of p a r t i c l e n number density P = pmag + p P gas pressure magnetic pressure charge a ion gyration radius r distance number of degrees of freedom T temperature t as subscript denotes the tangential component U streaming v e l o c i t y of a f l u i d 59 4uu// perpendicular and p a r a l l e l components of q V p a r t i c l e v e l o c i t y \^ Alfven v e l o c i t y \/^  v e l o c i t y of sound group v e l o c i t y Vp phase v e l o c i t y V volume x>\j,B coordinate axes. As subscript, they denote components oC angle sun - receiver - source © angle between If and S P © Mach angle ^ angle between Bl and TJ* Y* r a t i o of s p e c i f i c heats Y defined i n the text p. 3 |Jo permeability of free space p density u) angular frequency • ion cyclotron frequency 

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