Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Oscillations of the earth's outer atmosphere and micropulsations Westphal, Karl Oskar 1961

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1961_A1 W2 O8.pdf [ 6.43MB ]
Metadata
JSON: 831-1.0085477.json
JSON-LD: 831-1.0085477-ld.json
RDF/XML (Pretty): 831-1.0085477-rdf.xml
RDF/JSON: 831-1.0085477-rdf.json
Turtle: 831-1.0085477-turtle.txt
N-Triples: 831-1.0085477-rdf-ntriples.txt
Original Record: 831-1.0085477-source.json
Full Text
831-1.0085477-fulltext.txt
Citation
831-1.0085477.ris

Full Text

OSCILLATIONS OF THE EARTH'S OUTER ATMOSPHERE AND MICROPULSATIONS by K a r l Oskar  Westphal  Diplom i n Physik, U n i v e r s i t a e t Wuerzburg, Germany, 1951 M.A. The U n i v e r s i t y of Toronto, 1959  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY i n the Department of Physics  We accept t h i s  t h e s i s as conforming to the r e q u i r e d  standard  The U n i v e r s i t y of B r i t i s h J u l y , 1961  Columbia  In p r e s e n t i n g the  t h i s thesis i n p a r t i a l fulfilment of  requirements f o r an advanced degree a t t h e U n i v e r s i t y  British  Columbia, I agree t h a t the  a v a i l a b l e f o r reference  and  study.  of  L i b r a r y s h a l l make i t f r e e l y I f u r t h e r agree t h a t  permission  f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may g r a n t e d by  the  Head o f my  It i s understood t h a t f i n a n c i a l gain  be  Department o r by h i s r e p r e s e n t a t i v e s . •  copying or p u b l i c a t i o n of t h i s t h e s i s f o r  s h a l l not  be a l k w e d w i t h o u t my  Department The U n i v e r s i t y o f B r i t i s h Vancouver 8, Canada.  Columbia,  written  permission.  F A C U L T Y OF G R A D U A T E STUDIES  PROGRAMME OF T H E  FINAL ORAL EXAMINATION FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY  of KARL OSKAR WESTPHAL Diplom Physik Wuerzburg, 1951 M.A. Toronto, 1959 Thursday, Aueust 17th, 1961, at 10:00 A . M . in Room 302, Physics Building COMMITTEE IN C H A R G E F. H. SOWARD, Chairman J. A. JACOBS G. M . VOLKOFF R. D. RUSSELL P. R. SMY  R. W. BURLING E. LEIMANIS C. A. SWANSON G. V. PARKINSON  External Examiner: T. WAT A N ABE Tohoku University, Sendai, Japan  OSCILLATIONS O F T H E EARTH'S O U T E R A T M O S P H E R E A N D MICROPULSATIONS ABSTRACT Micropulsations of the Earth's magnetic field are closely related to the properties of the upper atmosphere and probably interstellar space as well. A model which would explain the observed data in a satisfactory manner would also give information on the properties of the outer atmosphere which at the present time are difficult to obtain otherwise. However, in spite of the vast amount of observational data now available, a satisfactory theoretical analysis of the problem has not yet been given. T o a large extent this is due to mathematical and computational difficulties. Geomagnetic micropulsations may be described as magnetohydrodynamic oscillations of which two different modes, namely poloidal and toroidal oscillations, may exist, and which in general are coupled. In this work only the toroidal mode, which may be understood as oscillations of a line of force, is considered. If the phennomenon is studied under the simplifying assumption ^ji^ »o where <t> is the longitude it is possible to obtain the eigenvalues of the oscillating lines of force in simple terms by treating the problem in cylindrical rather than in spherical polar coordinates. For a constant and variable charge density distribution, the eigenfrequencies are obtained as functions of the latitude. For a variable charge density distribution, the agreement between theory and observation is good in middle and low latitudes. However, as the latitude increases towards the poles the model gives periods which tend to infinity. As a result of recent studies it appears that the geomagnetic field does not extend as far into outer space as has been assumed, but that to a first approximation it is confined to a cavity. For this reason the equation of toroidal oscillations is applied to a compressed dipole field. Assuming both a constant and a variable charge density distribution the eigenperiods of the deformed magnetic lines of force are obtained, using an electronic computer.  The results for a variable charge density distribution agree with observational data in the polar regions but not in middle and low latitudes. This may imply that the charge density distribution is in error and it is hoped that further rocket and satellite data will settle this point.  PUBLICATIONS 1. O. Blunck and K . O. Westphal. Zum Energieverlust energiereicher Elektronen in duennen Schichten. Zeitschrift fuer Physik, 130, p. 641-649, 1951. 2. J. A . Jacobs and K . O. Westphal. High frequency  geomagnetic  micropulsations. Physics and Chemistry of the  vol.  Earth,  5,  Pergamon Press.  GRADUATE  STUDIES  Field of Study: Geomagnetic Micropulsations Nuclear Physics Advanced Geophysics Istotope Geology  j . B. Warren j . A . Jacobs R . D . Russell  Related Studies: Applied Electromagnetic Theory Computational Methods Programming of Digital Computers  G . B. Walker C . Froese J . R. H . Dempster  ABSTRACT  Using Maxwell's equations of electrodynamics and the l i n e a r i z e d fundamental equation of hydrodynamics neglecting a l l but the ponderomotive force, the two d i f f e r e n t i a l equations characterizing t o r o i d a l and poloidal modes of o s c i l l a t i o n s are obtained.  Neglecting the coupling  between these modes the t o r o i d a l mode which appears to be connected with the phenomenon of geomagnetic micropulsations i s studied i n d e t a i l . Substituting f o r the constant magnetic f i e l d the undeformed dipole f i e l d of the Earth the eigenperiods of the o s c i l l a t i n g l i n e s of force are computed assuming a :  constant charge density d i s t r i b u t i o n .  Using numerical  methods the eigenperiods are also obtained i n the case of a variable charge density. Since the Earth's dipole f i e l d i s presumably deformed by the solar wind a compressed dipole f i e l d i s introduced into the equation of t o r o i d a l o s c i l l a t i o n s .  The eigen-  periods of the o s c i l l a t i n g l i n e s of force are obtained i n this case, assuming a constant charge density d i s t r i b u t i o n . For the case of a variable charge density a numerical method i s described which could y i e l d the eigenperiods.  - V -  ACKNOWLEDGEMENTS  The w r i t e r wishes t o express h i s s i n c e r e s t thanks t o P r o f e s s o r J . A. Jacobs and P r o f . J . C. Savage f o r suggesti n g the problem and f o r many s t i m u l a t i n g d i s c u s s i o n s the performance of the work.  during  He would a l s o l i k e t o thank  Mr. H. Dempster, Computation Center, U n i v e r s i t y of B r i t i s h Columbia, f o r h i s great  i n t e r e s t and a s s i s t a n c e i n some of  the computational problems.  Further  thanks a r e extended t o  Mr. P. Haas f o r drawing the diagrams and the p r i n t i n g of the e q u a t i o n s and formulae. The f i n a n c i a l a s s i s t a n c e of the N a t i o n a l Research C o u n c i l of Canada and the O f f i c e of Naval Research i s gratefully  acknowledged.  - 1 1 1 -  TABLE  OF  CONTENTS  Abstract Acknowledgements  CHAPTER  I  Introduction  CHAPTER  II  Mathematical  1.  The B a s i c  2.  The E q u a t i o n s  Formulation  Equations of Small  Magneto-Hydrodynamic  Oscillations 3„  4  0  The E q u a t i o n  of toroidal  the  field  The E q u a t i o n the  CHAPTER 1„  2.  3.  case  III  i s a dipole  of toroidal  of a compressed  field  oscillations in dipole  f o r the normal  dipole  constant  charge  Solution  f o r the normal  variable  charge  Solution  f o r the compressed  constant  Solution with  when  field  Solutions  Solution  with 4„  constant  oscillations  with  density dipole  field  with  density  charge  charge  dipole  field  dipole  field  density  f o r the compressed  variable  field  density  -iv-  CHAPTER IV APPENDIX  Discussion of the Results  73  I. The compressed dipole f i e l d II. III. IV.  83  The matrix M  86  Solutions of a cubic equation  88  Tables of matrices and their  lowest  eigenvalues V.  93  A n a l y t i c a l representation of the eigenvalues  VI.  98  Solutions of the l i n e s of force equation, compressed dipole  VII.  Tables of matrices and their eigenvalues  References  103 lowest 113 116  CHAPTER I  INTRODUCTION  During the l a s t ten years pulsations of the geomagnetic f i e l d have received increasing attention.  A great amount of  recorded data from stations a l l over the world has accumulated since such pulsations were f i r s t observed by Balfour Stewart in 1861.  Amplitudes range from a f r a c t i o n of a gamma  (1 gamma (y) =10  gauss) to as much as a few tens of a  gamma, and frequencies from 0.01 to 10 cps. The main reason for studying the phenomenon i s that the generation of such pulsations i s c l o s e l y related to the properties of the upper atmosphere and probably i n t e r s t e l l a r space.  A model which would explain the observed data i n a  s a t i s f a c t o r y manner would also give information on the properties of the outer atmosphere which at the present time are d i f f i c u l t to obtain otherwise.  However, i n spite of the  vast amount of observed data now available, a s a t i s f a c t o r y t h e o r e t i c a l analysis of the problem has not yet been given. To a large extent t h i s i s due to mathematical and computational d i f f i c u l t i e s .  -2-  In  p r i n c i p l e geomagnetic p u l s a t i o n s may be d e s c r i b e d as  magneto-hydrodynamic o s c i l l a t i o n s o f which two d i f f e r e n t modes, namely p o l o i d a l and t o r o i d a l o s c i l l a t i o n s can e x i s t and which i n g e n e r a l are coupled.  In t h i s t h e s i s we d e a l  o n l y with the t o r o i d a l mode which may be understood as o s c i l l a t i o n s of a l i n e of f o r c e . to  C o n s i d e r i n g the phenomenon  take p l a c e i n a meridian plane i t i s p o s s i b l e to o b t a i n  the e i g e n v a l u e s of the o s c i l l a t i n g l i n e s of f o r c e by t r e a t i n g the problem i n c y l i n d r i c a l r a t h e r than i n s p h e r i c a l p o l a r coordinates.  For a g i v e n charge d e n s i t y d i s t r i b u t i o n the  e i g e n f r e q u e n c i e s are o b t a i n e d as f u n c t i o n s of the l a t i t u d e . In  the f i r s t attempt  the E a r t h ' s main f i e l d dipole.  t o t r e a t t h i s hydromagnetic problem  i s taken to be that of a g e o c e n t r i c  T h i s has the u n f o r t u n a t e consequence t h a t the  e i g e n f r e q u e n c i e s tend towards i n f i n i t y  as the p o i n t of i n t e r -  s e c t i o n of the l i n e s of f o r c e with the E a r t h ' s s u r f a c e approach v e r y h i g h l a t i t u d e s - which i s not i n agreement w i t h the o b s e r v a t i o n s . of  finite  presence it  By c o n f i n i n g the d i p o l e f i e l d  to a sphere  r a d i u s which may be the case on account of the s o l a r wind which c a r r i e s the f i e l d  o f the along,  i s p o s s i b l e to remove the d i s c r e p a n c i e s . From the experimental data o b t a i n e d a t d i f f e r e n t  l a t i t u d e s but equal l o n g i t u d e s i t i s p o s s i b l e t o determine those e i g e n p e r i o d s which show a maximum frequency  occurrence  and by comparing t h i s f r e q u e n c y - l a t i t u t d e dependence w i t h  -3-  those c a l c u l a t e d i t appears that the model of an o s c i l l a t i n g magnetic l i n e of f o r c e i s f e a s i b l e . Before d i s c u s s i n g the t h e o r e t i c a l approach to the problem of geomagnetic  m i c r o p u l s a t i o n s which i s based on the  concept of magneto-hydrodynamic waves i t i s i n s t r u c t i v e to o u t l i n e t h i s phenomenon, which was 1942,  d i s c o v e r e d by Alfve'n i n  f o r a r a t h e r simple case. If an e l e c t r i c a l l y c o n d u c t i n g medium moves i n the  presence of a magnetic f i e l d , the  an e l e c t r i c f i e l d  i s induced i n  medium, the c u r r e n t being at r i g h t angles to both the  d i r e c t i o n of motion of the medium and the magnetic However, an e l e c t r i c  c u r r e n t i n a magnetic f i e l d  field.  i s acted  upon by a mechanical f o r c e p e r p e n d i c u l a r to both the c u r r e n t and the magnetic f i e l d , motion or a g a i n s t i t .  i . e . e i t h e r i n the d i r e c t i o n of I t f o l l o w s from simple energy c o n s i d e r a -  t i o n s that the mechanical f o r c e must be d i r e c t e d so as to impede the o r i g i n a l movement.  I f the o r i g i n a l motion was at  r i g h t angles to the magnetic f i e l d  the d i s t u r b a n c e  will  propagate i n the form of a wave along the magnetic l i n e of force. In  o r d e r that such an i n t e r a c t i o n between e l e c t r o m a g n e t i c  and hydrodynamic  phenomena exist i t was shown by Lundquist  (1952) that the i n e q u a l i t y  »l  -4-  raust be s a t i s f i e d , where L i s the l i n e a r dimension of a and p e r m e a b i l i t y ju  l i q u i d conductor, of d e n s i t y presence o f a magnetic f i e l d B. out  this thesis. )  0  i n the  (mks u n i t s are used through-  I t i s e a s i l y seen that under normal  l a b o r a t o r y c o n d i t i o n s t h i s c r i t e r i o n i s not s a t i s f i e d and magneto-hydrodynamic The s i t u a t i o n ,  effects  are not observed.  however, i s c o m p l e t e l y d i f f e r e n t i n  problems of cosmic p h y s i c s .  Because of the enormous dimen-  s i o n s i n v o l v e d i n such cases L u n d q u i s t ' s c r i t e r i o n i s e a s i l y s a t i s f i e d and i n t e r a c t i o n s between e l e c t r o m a g n e t i c and hydrodynamic phenomena may  be c o n s i d e r a b l e .  It i s for this  reason t h a t n e a r l y a l l attempts which have been made i n r e c e n t years t o e x p l a i n the phenomenon of geomagnetic m i c r o p u l s a t i o n s apply the p r i n c i p l e s to  the outer  of  magneto-hydrodynamics  atmosphere.  The s u g g e s t i o n that geomagnetic m i c r o p u l s a t i o n s might be e x p l a i n e d i n terms of magneto-hydrodynamic first the  advanced by Dungey (1954b).  outer atmosphere  oscillations  was  Under the assumption t h a t  i s a v e r y good conductor he d e r i v e d  from Maxwell's e q u a t i o n s and from the fundamental e q u a t i o n of  f l u i d dynamics two c o u p l e d p a r t i a l d i f f e r e n t i a l e q u a t i o n s  g o v e r n i n g the two components  E  w  and V. , v i z . : ?  -5-  and  (2)  From (1) and (2) which are termed the equations o f p o l o i d a l and toroidal  •* V  o s c i l l a t i o n s the f i e l d  { V , Vj, r  V j-  are determined  v e c t o r s E { E , E^,, E ^} and r  i n a unique way.  However,  these two equations a r e f a r too c o m p l i c a t e d t o be o f much use i n s t u d y i n g geomagnetic m i c r o p u l s a t i o n s .  For t h i s  reason  one,assumes a x i a l symmetry, i . e . one supposes that the phenomenon takes p l a c e o n l y i n the plane of a m e r i d i a n - an assumption  which seems t o be confirmed by the l o c a l  dependence. (  <^  Under t h i s assumption  the c o u p l i n g term  = 0) s p l i t t i n g the c o u p l e d system  time vanishes  i n t o two s e p a r a t e  equations, v i z . :  and  The f i r s t attempt  t o compute from e q u a t i o n (4) the  e i g e n p e r i o d s of the t o r o i d a l himself.  T a k i n g the magnetic  o s c i l l a t i o n s was made by Dungey f i e l d o f the E a r t h H  Q  t o be  -6-  that of a d i p o l e he o b t a i n e d under the assumption of a constant charge d e n s i t y jo the  ( = 10  - 1 8  kg/m ) f o r the fundamental p e r i o d 3  approximation  T-  ^  (  where X  0  X > 3 0  °)  i s the l a t i t u d e a t which a p a r t i c u l a r magnetic  of f o r c e i n t e r s e c t s the s u r f a c e of the e a r t h .  ( 5 )  line  Since t h i s i s  a boundary value problem one i n g e n e r a l assumes that the hydro-magnetic wave i s r e f l e c t e d at the p o i n t of i n t e r s e c t i o n . E v a l u a t i n g equation  (5) f o r p a r t i c u l a r l a t i t u d e s g i v e s  the r e s u l t s shown i n t a b l e 1. Table 1 *o  T  l  45°  10 sec  55°  54 sec  65°  11 min  70°  55 min  A d i f f e r e n t method of approach t o the s o l u t i o n of the boundary value problem which i s i n p a r t i c u l a r p e r m i s s i b l e f o r o b t a i n i n g the p e r i o d s of the h i g h e r modes of o s c i l l a t i o n s has been taken by Kato and Watanabe  (1956).  constant charge d e n s i t y d i s t r i b u t i o n , p e r i o d of the n  t n  order  oscillation  Again assuming a  they o b t a i n e d f o r the  -7-  FTT^G  X,=  x  V>- y f ' - a x )  (6)  o/x  0  where y 0  0  the charge d e n s i t y has been taken equal t o i t s  v a l u e a t the most d i s t a n t p o i n t and a = sin A. 2  The  Q  r e s u l t s f o r the fundamental mode as o b t a i n e d from  equation  (6) are summarized i n t a b l e 2. Table 2 A.  0  /> o = 0 . 8 x l 0 ~  1 8  kg/m  1.6xlQ-18 kg/m  3  3  r  2/4xl'Q-  18  kg/m  45°  6 sec  9 sec  11 sec  55°  33 sec  47 sec  57 sec  65°  381 sec  538 sec  659 sec  70°  2071 sec  2929 sec  In equation  3585 sec  (4) the o n l y s p a c i a l d e r i v a t i v e s which occur  are those i n the operator (H ° V 0  Vy,;  3  ) o p e r a t i n g on the f u n c t i o n  t h i s however can be i n t e r p r e t e d as the d e r i v a t i v e of v^ i n  the d i r e c t i o n of H . D  Therefore equation  (4) can be i n t e r -  p r e t e d as a wave.equation where the d i s t u r b a n c e a l o n g a l i n e of f o r c e . equation  Choosing  for H  0  propagates  a dipole f i e l d ,  (4) r e p r e s e n t s the wave equation governing the time-  space r e l a t i o n s h i p of a d i s t u r b a n c e which propagates  along  -8-  the magnetic l i n e s of f o r c e of the d i p o l e .  I t i s then i n t h i s  sense that one speaks of o s c i l l a t i o n s of the l i n e s of f o r c e which, i t must be emphasized,  a p p l i e s i n i t s t r u e s t sense  o n l y to the case of an i n f i n i t e l y c o n d u c t i n g medium. T h i s concept of o s c i l l a t i n g magnetic l i n e s of f o r c e has been u t i l i z e d by Obayashi and Jacobs (1958) f o r computing the charge d e n s i t y d i s t r i b u t i o n of the outer atmosphere use of the observed p e r i o d s of m i c r o p u l s a t i o n s .  making  To c a r r y out  the a n a l y s i s i t was however n e c e s s a r y to make an assumption of the f u n c t i o n a l r e l a t i o n s h i p between the h e i g h t and the charge d e n s i t y . In order to f a c i l i t a t e Siebert  the s o l u t i o n of e q u a t i o n (4)  ( p r i v a t e communication)  chose a charge d e n s i t y  dis-  t r i b u t i o n of a mathematical form which reduces e q u a t i o n (4) to the simple d i f f e r e n t i a l e q u a t i o n of a harmonic  oscillator.  T h i s method of approach however leads to r e s u l t s which are i n disagreement w i t h the observed data i n d i c a t i n g that  this  approach i s not a p p r o p r i a t e . Reviewing the t h e o r e t i c a l work which has been done on the problem so f a r one a r r i v e s at the c o n c l u s i o n that the magneto-hydrodynamical  treatment of the problem leads to  equations which i n t h e i r g e n e r a l form are too complex treated mathematically.  to be  Even under the r e a s o n a b l e assumption  that the phenomenon takes p l a c e i n a m e r i d i o n a l plane only, the r e s u l t i n g equations are q u i t e i n t r a c t a b l e - even, i f i n  -9-  a d d i t i o n , a constant charge d e n s i t y d i s t r i b u t i o n and a simple d i p o l e f i e l d of the E a r t h are assumed. S i n c e c o u p l i n g e f f e c t s , which manifest themselves through the presence of terms i n v o l v i n g of  i n the case  s p h e r i c a l p o l a r c o o r d i n a t e s , are i n i t i a l l y n e g l e c t e d i t  seems f e a s i b l e to i n v e s t i g a t e the problem anew by working i n c y l i n d r i c a l r a t h e r than s p h e r i c a l p o l a r c o o r d i n a t e s and dropping terms c o n t a i n i n g  ?  .  The equations then take  on a s i m p l e r form i n which a mathematical treatment i s p o s s i b l e even f o r the case of a non-uniform charge d e n s i t y d i s t r i b u t i o n and a deformed d i p o l e  field.  -10-  CHAPTER I I  MATHEMATICAL  1.  The B a s i c  In  order  hydrodynamics,  Equations  to relate  hydrodynamics  of  the orders  in  problems  Therefore  where  H  equations  are used.  o f cosmic  Maxwell's  with  first  In a d d i t i o n  H  field  current.  i s simply  strength  and  y  the current  the equation  V- H = o Maxwell's  second  equation  which  intensity  E with  t h e change  that  currents are  the conduction  satisfies  equations  involved  displacement  equation  and  a consideration  of the quantities  physics,  i n comparison  and the b a s i c  I t f o l l o w s from  of magnitude  i s the magnetic  density.  t h e phenomena o f e l e c t r o d y n a m i c s  Maxwell's  of  negligible  FORMULATION  connects  (8) the e l e c t r i c  of the magnetic  V xE =  -I  flux  field  B i s  O)  -11-  where  B  Ohm's  law  =A H  for  a  =  Considering (cr  -> oo)  medium  outer  basic  with  the  velocity  V  is  ( i i )  atmosphere  as  a  perfect  conductor  becomes  E = ~ V  The  moving  £ + V XB)  <T(  the  this  do.  X  equation  £  (12)  of  hydrodynamics  can  be  written  in  the  form  fir where  p  is  external,  the  <">  ' C+j*s density,  non-magnetic  -> V the  forces.  velocity, The  term  and 3. x  > G the B  is  sum  of  a l l  the  -» mechanical element the  force  carrying  mobile  exerted the  operator  by  current  the  magnetic  density  ^.  field d/dt  B  on  a  volume  represents  -12-  It has  been shown by Dungey (1954a,b) t h a t f o r hydro-  magnetic waves i n the outer atmosphere the e f f e c t of v i s c o s i t y which i s u s u a l l y the most important  cause of a t t e n u a t i o n , can  <  be n e g l e c t e d when c o n s i d e r i n g l o n g p e r i o d o s c i l l a t i o n s i n the presence of the e a r t h ' s magnetic f i e l d . Ferraro  A l s o Plumpton  and  (1953 ) have shown that the p e r i o d s of o s c i l l a t i o n  of  a g r a v i t a t i n g l i q u i d s t a r i n the presence of a c e n t r a l magnetic pole and field  are not much reduced i f the  i s neglected.  g r e a t e r than the gas the d i s t u r b a n c e  G  =0  gravitational  Since the magnetic pressure pressure  and  the pressure  i s much  g r a d i e n t due  to  i s s m a l l compared with the magnetic f o r c e ,  and equation  (13) reduces to  These are the b a s i c equations  which w i l l be a p p l i e d to the  problems of geomagnetic p u l s a t i o n s . 2.  The  equations  of s m a l l magneto-hydromagnetic  To o b t a i n the equations necessary  i n general.  i n our case to o b t a i n two equations  governing V.  dependent  Because of the n o n l i n e a r i t y of the  t h i s w i l l be i m p o s s i b l e  and  of s m a l l o s c i l l a t i o n s i t i s  to e l i m i n a t e from the system a l l but one  vector v a r i a b l e .  E  oscillations  one  coupled  equations  However i t proves p o s s i b l e  partial  differential  component of each of the  vectors  -13-  To t h i s end one has t o s u b s t i t u t e equations i n t o equation  where  H  (15) which y i e l d s  i s the sum of a constant f i e l d  disturbance h  (7) and (10)  i n t r o d u c e d through  H  G  and a s m a l l  the motion o f the medium.  Since  "9  -=7  ->  |Hj7> I h|  and  \/,H =0 o  lv  equation  (16) becomes, under the assumption  ^  "  7  that \ ~ \ ^  ^  -"7  fT *y° °'( * ) H  t  7  h  ( l 7 )  T a k i n g the v e c t o r product of t h i s e q u a t i o n with  3^o(H+r)) "y'-'e^o =  and t a k i n g the time d e r i v a t i v e i t  6  follows that  { 1 ( 1 1 , \x\  =  _  u  Wxfir.kWH  (is)  or by v i r t u e of e q u a t i o n (9)  (19)  -14-  C a r r y i n g out the d i f f e r e n t i a t i o n on the l e f t - h a n d s i d e of equation  (19) one o b t a i n s  However s i n c e  J t ~ ~ T t t i -  v  E = - V x B  * T T  W~  1  TF*^  at  2  at  v  Tt^  I n t r o d u c i n g the l a s t e x p r e s s i o n i n t o the expanded l e f t - h a n d s i d e of equation (19)  ^tlat  V" ^VTF dt  i s obtained.  at  +  Since  IE I  v  at  i s proportional  2  )VI  the second  o  term i n the l a s t e x p r e s s i o n i s o f the order V neglected. Hence  B)  -/»#  and can be  -15-  Therefore  equation  (19)  becomes  (20)  i.e.  which  i s a vector  equation H  =  Q  will  wave  (21) i n t o  -^H , r  equation  cylindrical  H^. , oj  for  , the second" term  i n the case  of  z-component  of equation  (21) becomes  *  V * E  from  i n jfche" c y l i n d r i c a l  equation  (r,<^  ), s i n c e  the right-hand  after  Thus  side  the  expanding  coordinates  » d*Ea. _ I ofefr., d*E^  (22)  (12)  E A H^ =  r  equation  on  t h e z-component.  Mzi_ l I l A o - r  Since  Transforming  coordinates  vanish  V  E.  ( 2 2 ) c a n be  v  2  a n d  E^-A^W,  further written  (23 )  -16-  Again, s i n c e  o ( l 4 Mr) .  ij  dU?  ^  drd2-_  and  the r i g h t - h a n d s i d e of equation  ,  (23) becomes  r Li  _ u  ^  "  (24)  d *  Since the constant f i e l d H  Q  obeys  Vx H  = 0  Q  which i n  c y l i n d r i c a l coordinates i s equivalent to  -^Mt =0  L)  equation  (23) by v i r t u e of e x p r e s s i o n  (25)  (24) and equation (25)  f i n a l l y becomes  ?  L  d  '  l  ^  .  d  2  .  !  d )r_ALud 2  n  b\(*Vi  (26  -17-  T h i s i s the d i f f e r e n t i a l equation of p o l o i d a l  oscillations  and one n o t i c e s that on the r i g h t - h a n d s i d e the z-component of  the v e l o c i t y f i e l d appears.  T h e r e f o r e the term on that  s i d e r e p r e s e n t s the c o u p l i n g between the two modes V  z  which i s necessary  E and z  t o d e s c r i b e the magneto-hydrodynamic  problem i n a unique way. To o b t a i n a time-space r e l a t i o n s h i p f o r the z-component of  the v e l o c i t y f i e l d  i t i s necessary  t o e l i m i n a t e the v e c t o r  ->  of  the magnetic d i s t u r b a n c e  by equations equation  (7) t o (15).  h  from the system  represented  To t h i s end one has t o c o n s i d e r  (16) which, n e g l e c t i n g terms of higher order  than  the f i r s t , becomes  ;£!f--rWvxH)  (27)  Using the i d e n t i t y 2  H*(V*HH V H -(H-V)'H and w r i t i n g of  H = H  c  (28)  + h one o b t a i n s f o r the r i g h t - h a n d s i d e  equation (27)  HX(V«H)4V{H^2CH„-IH } - { C H . 4 ) - V } ( H / M !  4vf^va-l04  W?-(H.-V)Hi - (l-V)H -fH -V) 0  0  I  -18-  >  N e g l e c t i n g higher terms i n equation  h  and a p p l y i n g the i d e n t i t y (28)  (27) becomes  +(l-v)k-  -oUn  w ->  S i n c e the c y l i n d r i c a l c o o r d i n a t e s f o r any two v e c t o r s  the z-component of (29) f i n a l l y To e l i m i n a t e equation  h  z  ->  A and B  becomes  i n the l a s t equation one has to s u b s t i t u t e  (12) and (10) i n t o equation  (9) and one o b t a i n s  Expanding the r i g h t - h a n d s i d e y i e l d s ->  jj-  - (H-V)V  which s i n c e N  ->  -  V • H = 0 -»  -»  CV*V)H -  HV-V  +  W'H  becomes ->  ->  >  -ff- = (H-V) V - (V-7)H - UV-V  (32)  -19-  Since H  Q  i s constant i n time, H  z-component of equation  = 0, and  z  ) H | >> 0  |h|  the  (32) i s  H  *  =  6t  (33)  To e l i m i n a t e the l a s t terra on the r i g h t - h a n d s i d e of  (30  the r - and A} - components of Maxwell's equation  are used which a f t e r i n t r o d u c i n g (12) becomes  _  oh.  _  „  M  OF;  I'  _  (34)  and  M u l t i p l y i n g equation  (34) by H  r  and equation  2 and adding y i e l d s , with  H  r  2 + H^.  = H  c  (35) by  2 ,  In order to o b t a i n an equation which c o n t a i n s o n l y E one d i f f e r e n t i a t e s equation  H  z  and  (30) with r e s p e c t to time and  V , z  -20-  s u b s t i t u t e s equation  - JJ . V l k - L  (36), g i v i n g  U  S u b s t i t u t i n g equation  ^  + _L (Hr  (33) f i n a l l y  X  - H  (37)  yields  which can be r e w r i t t e n i n the form  Equation Equations  (38) i s f o r t o r o i d a l  oscillations.  (26) and (38), together with the boundary  c o n d i t i o n s , d e s c r i b e the behaviour  of an i n f i n i t e l y  conduct-  i n g medium of c y l i n d r i c a l symmetry due t o a d i s t u r b a n c e . Apart from the f a c t that these two equations are too complicated to be s o l v e d i n t h e i r general form we are concerned here only with phenomena i n the meridional plane. For t h i s reason the s i m p l i f y i n g assumption  ^/^^ ~ 0  i s made, i . e . the two modes of o s c i l l a t i o n are decoupled, and equations  (26) and (38) become  -21-  and  {^r-W.-^)(HL-V)}^ =0  (40)  I t has been s t a t e d e a r l i e r t h a t a knowledge of V  z  is sufficient  components.  f o r the d e t e r m i n a t i o n of the other  Assuming that the f i e l d  E  and  z  field  q u a n t i t i e s depend on t  through the f a c t o r e x p ( i w t ) , equation  (9) y i e l d s  and  -  i o ) / ^ *{v*E^ = "~J7"  while f o r computing the components of V, equation  ">f>V =  [  r  ~A^{V4}  ?  (  4  2  )  (17) g i v e s  (43)  and  LCOyOV^ = yfc.  H {V h}.,  However because of equations  r  x  (9) and (21),  (44)  -22-  -»  and  the r -  and $  - components  of  V  reduce  to  and  If  on  the o t h e r hand  is  known  then  from  the z-component  equation  of the v e l o c i t y  field  (12)  E,= A K v V  (47)  z  and  E,= - A K . V ;  while  from  equation  [  It  follows  mode  from  (33)  U  ^  =  r ^ +  H  equations  of o s c i l l a t i o n  (48)  (41) t o  the f o l l o w i n g  "^Ix (46) that  (49)  f o r the  s e t of quantities  involved (0,  0,  E ) z  ;  (h  r  s  h#,  o)  ;  ( V , V^, r  0)  poloidal are  -23-  while f o r the t o r o i d a l mode of o s c i l l a t i o n s , equations (47) to  (49) show a r e l a t i o n s h i p between (E , r  E.j,  l  0)  (o, o, h )  5  z  Hence from a knowledge of E and V -> -» E and V may be determined. z  3.  (0, 0, V ) z  the two vector f i e l d s  z  The equation of toroidal o s c i l l a t i o n s when the constant f i e l d i s a dipole f i e l d Considering the motion along a l i n e of force given by 2  r = r  where  ^  0  sin^  i s the co-latitude and r  c  (50)  the value at the  furthest point (^ = 90°) the expression (H ° V ) i s given by Q  Since the ^-component of a dipole f i e l d i s given by U  where  M  "  cin f -  ^  i s the magnetic dipole moment, equation (51)  becomes  M-A^rax  >  (52  -24-  and therefore  (53 )  Taking the time dependance i n equation (40) to be of the form exp(itjt) the equation of toroidal  oscillations  i n the  presence of a dipole f i e l d becomes  (54)  It i s frequently convenient to express lengths i n units of the Earth's radius, and one thus writes  r  Q  = » a. 0  Remembering  that the magnetic dipole moment of the Earth i s given by M = H a* 0  where a i s the radius of the Earth and H of the magnetic f i e l d intensity  Q  the maximum value  at the magnetic equator,  equation (54) f i n a l l y becomes 2  ,S  2  0  7" <L2  ft*  (55)  *  This i s an ordinary homogeneous d i f f e r e n t i a l equation of the second order.  Associating the periods of geomagnetic micro-  pulsations with the eigenvalues of equation (55) one has to find  u  such that  V  z  s a t i s f i e s the boundary conditions  -25-  at  the ends of the i n t e r v a l of i n t e g r a t i o n . I t w i l l prove very convenient  i n comparing r e s u l t s t o 2  i n c o r p o r a t e the charge d e n s i t y d i s t r i b u t i o n 2 putting  pu  S/VJ  dJ-  -2 = u .  In t h i s case equation  + A * ^  1  V  2  f> i n u  by  (55) becomes  = 0.  (56)  However t h i s i s only l e g i t i m a t e i f the charge d e n s i t y d i s t r i b u t i o n along the l i n e of f o r c e i s constant and not a f u n c t i o n of the c o - l a t i t u d e 4.  ^ .  The equation of t o r o i d a l o s c i l l a t i o n s i n the case of a compressed d i p o l e f i e l d Parker  (1958) has c o n s i d e r e d i n some d e t a i l the i n t e r -  a c t i o n of the " s o l a r wind" w i t h the E a r t h ' s geomagnetic field.  The s o l a r wind i s the name g i v e t o the outward  streaming o f gas i n a l l d i r e c t i o n s from the sun w i t h v e l o c i t i e s i n the range 500-1500 km/sec.  The s o l a r wind  w i l l compress or sweep away the outer geomagnetic down to a l e v e l where the energy equal t o the k i n e t i c energy  field  d e n s i t y of the f i e l d i s  d e n s i t y of the s o l a r wind.  This  r e s u l t s i n a deformation of the d i p o l e f i e l d which under these circumstances may as a f i r s t  approximation  be con-  s i d e r e d as being c o n f i n e d t o a s p h e r i c a l c a v i t y of r a d i u s P o s t u l a t i n g that the r-component of the magnetic  R.  field  0  -26-  vanishes a t  r = R, Q  i t i s shown i n appendix I that  ^ - ^ r ( - ^ ^ j r ) ^  (57.  and  Using the d i f f e r e n t i a l equation o f the l i n e s of f o r c e , v i z .  (59)  it  i s f u r t h e r shown i n appendix I that the l i n e s of f o r c e of  a compressed d i p o l e f i e l d  are the s o l u t i o n s of the cubic  equation  r  where  a  \  L  h  ±  i s the r a d i u s  c o n f i n i n g c a v i t y and  X  ^  r  , l R  (  of the E a r t h , R  D  the r a d i u s  9  0  )  of the  the c o - l a t i t u d e of the i n t e r s e c t i o n  w i t h the E a r t h of the l i n e of f o r c e which touches the c a v i t y i n the e q u a t o r i a l plane.  Introducing  E a r t h as the u n i t of l e n g t h equation  the r a d i u s o f the (60) becomes  -27-  where  and  =  By l e t t i n g  cx  o< =  A.  tend to l a r g e values equation  cx^  (61) becomes  s/nv  which f i n a l l y i n the l i m i t  c* -» oo  becomes  which i s the equation of the l i n e s of f o r c e f o r the o r d i n a r y d i p o l e f i e l d expressed  i n u n i t s of e a r t h r a d i i .  Using a d i g i t a l computer equation  \) has been e v a l u a t e d from c  (61) as a f u n c t i o n of the c o - l a t i t u d e  values of o<  .  ^  for several  The r e s u l t s are p l o t t e d i n f i g . 2-4.  To c o n s i d e r the e f f e c t of a compressed d i p o l e f i e l d with f i e l d components  c h a r a c t e r i z e d by equations  has to go back to the operator of  (H " V Q  (57) and (58) one  ) which along a l i n e  f o r c e i s given by  Introducing f o r  H^,  the e x p r e s s i o n  (58) one o b t a i n s  field  pattern  of a moqnetic  dir< i t  - 2 9 -  12  Fig field  pattern of ( size  a  2.  compressed  of cavih;  12  magnetic  Earth  radii )  d;pole  -30-  field  pattern ( size  of of  a  compressed  cavity  8  Earth  magnetic radii )  tiipo  Fig. 4.  field  pattern ( size  of of  a compressed cavity  4  magnetic  Earth  renin }  dipole  -32-  CH.-V)and  N  A  V r-  4  R  ' r  hence  (62)  Sin  I n t r o d u c i n g t h i s r e s u l t i n t o equation  (40) y i e l d s  co  (63)  IT Taking  the r a d i u s of the E a r t h as the u n i t of l e n g t h  equation  (63 ) can be w r i t t e n  \>  4  c  letting  ^ <x / 3  c  <LJ  K  2  v -u 2  oC tend to i n f i n i t y and remembering then that equation  (64) reduces to (56)„  (64)  °-33-  CHAPTER I I I  SOLUTIONS  1.  S o l u t i o n f o r the normal d i p o l e f i e l d with constant  charge  density As p o i n t e d  out p r e v i o u s l y the d i f f e r e n t i a l equation of  t o r o i d a l o s c i l l a t i o n s i n the case of a constant  charge  d e n s i t y d i s t r i b u t i o n may be w r i t t e n  dv sin *} 1  (56)  z  d^  Ho  Changing the independent v a r i a b l e by p u t t i n g  (65)  equation  (56) becomes .2  d  V,  i  ,8  -2  f o r which a s o l u t i o n i s e a s i l y Since  = o  found.  one i s p a r t i c u l a r l y i n t e r e s t e d i n f i n d i n g the  (66)  -34-  eigenvalues  of t h i s equation  i t i s necessary  to s p e c i f y the  boundary c o n d i t i o n s which most l i k e l y apply to the p h y s i c a l situation. It has  been shown by Dungey (1954) that hydromagnetic  waves having  long p e r i o d s are almost p e r f e c t l y  when they h i t the ionosphere. ionosphere with  Considering  reflected  the f a c t that  the  i s very c l o s e to the s u r f a c e of the E a r t h compared  the d i s t a n c e s over which the l i n e s of f o r c e extend  can assume without  one  i n t r o d u c i n g much i n a c c u r a c y that the p o i n t  of r e f l e c t i o n i s s i t u a t e d a t the s u r f a c e of the E a r t h . Therefore  at  From equation  3 = <%  (12) i t then f o l l o w s t h a t  at  To f i n d the c o r r e s p o n d i n g  value of  x  equation  (65) must be  integrated, i . e .  X where  =  0  1 8 0 - J ;  hand s i d e i s symmetrical  Since the i n t e g r a n d on the about the p o i n t  rightthis  -35-  becomes  which a f t e r  integration  Ax  cos^(  Equation  (67)  an  gives  Ssin J + 6sintl + 8 sin ^ + 16) 6  a s s o c i a t e s the  i n t e r v a l extending from 3* = ^  tXo -x  Q  x  to  Q  <X  Letting  x = o  to  ^  = 180°-  corresponds to the  to  +x .  Hence  G  3  CosZ(Ssmit+  0  Using the t% <  table  i n t e r v a l from  correspond to  X=  0 <  (67)  2  o  3. The  0  6s'mfy +  s t e p s of  i n  5°  F i g . 5 shows the solution  and  with point interval  i n t e r v a l from  Zsin^+lb)  the  (66)  (68)  been computed f o r  result  p l o t of x  of e q u a t i o n  the  i t f o l l o w s that the  Alwac III E t h i s e x p r e s s i o n has 90  to  0  i s given i n  against  J^, .  i s then  V *Acos  (69)  t  and V. - B"n  (  a  *'*  A  (69b)  -36-  Table 3  X  0.0  -0.4571  50.0  -0.4366  5.0  -0.4571  55.0  -0.4192  10.0  -0.4571  60.0  -0.3926  15.0  -0.4571  65.0  -0.3549  20.0  -0.4571  70.0  -0.3047  25.0  -0.4570  75.0  -0.2422  30.0  -0.4566  80.0  -0.1685  35.0  -0.4554  85.0  -0.0865  40.0  -0.4527  90.0  0.0000  45.0  -0.4470  Because of the boundary c o n d i t i o n i t f o l l o w s from equations (69a) and (69b) that  Kl With  a = 6.371 x 10  meter),  H  Q  fi  (m),  = 0.312 x 1 0 ~  4  Q  -  f  = 1.256 x 10  (voltsec/m )  o  Equation (70) can thus be w r i t t e n  2  (70)  —6  (voltsec/amp.  -38-  w=  where  |X | 0  ^  (71)  i s given by equation  (68) and  I  In equation  (71) the odd v a l u e s of  n  correspond  c o s i n e o s c i l l a t i o n while even v a l u e s of  n  to a  are a s s o c i a t e d  w i t h an s i n e o s c i l l a t i o n . Since  CD = co V~j5~* ( /r)  ^7*~  Zrr  o  n  e  obtains  f o r the  fundamental e i g e n p e r i o d (n = 1)  T, =  3.IS2 x \0 x.V A|X | S  4  O  0  X  The e i g e n p e r i o d s of the o s c i l l a t i n g l i n e s c a l c u l a t e d from equation  [sec]  (72)  of f o r c e have been  (72) and p l o t t e d i n f i g . 6 u s i n g a —19  constant charge d e n s i t y d i s t r i b u t i o n The n e u t r a l p a r t i c l e  of 6,5 x 10  (kg/m°).  d e n s i t y has not been i n c l u d e d i n  because the mean f r e e path i s so long that o n l y particles  "\  p  charged  w i l l c o n t r i b u t e t o the hydromagnetic wave motion.  -39-  Table 4 T(sec )  2.  T(sec )  10°  4.340xl0  5  50°  3.074  15°  1.896xl0  4  55°  1.727  20°  2038  60°  1.036  25°  375.0  65°  0.651  30°  97.61  70°  0.418  35°  31.95  75°  0.267  40°  12.97  80°  0.157  45°  5.972  S o l u t i o n f o r the normal d i p o l e f i e l d with v a r i a b l e charge d e n s i t y In the preceding s e c t i o n the e i g e n p e r i o d s of the  t o r o i d a l o s c i l l a t i o n s of a conducting medium i n the  presence  of a d i p o l e were o b t a i n e d assuming a constant charge d e n s i t y distribution.  Because of t h i s s i m p l i f i c a t i o n  i t was p o s s i b l e  to o b t a i n the e i g e n p e r i o d s without making any  approximations.  However, i n g e n e r a l , the assumption  of a constant  d e n s i t y w i l l not be the case.  Under these  one has to go back to equation  (54) where  of  the d i s t a n c e from the E a r t h ' s s u r f a c e .  circumstances p  i s a function  Since equation  governs the o s c i l l a t i o n s along the l i n e s of f o r c e be a f u n c t i o n of both  r  Q  and  .  charge  p  (54)  will  For the c a l c u l a t i o n s  the  -40-  -41-  charge d e n s i t y d i s t r i b u t i o n which was proposed  by D e s s l e r  (1958) i n h i s c a l c u l a t i o n s of the propagation v e l o c i t y of sudden commencements was used. in fig.  T h i s d i s t r i b u t i o n i s shown  7.  To o b t a i n the e i g e n v a l u e s to r e s o r t to  u  numerical methods.  of equation  (54) one has  The d i f f e r e n t i a l  equation  i s r e p l a c e d by f i n i t e d i f f e r e n c e s and the problem i s then to f i n d the e i g e n v a l u e s of a matrix. To t h i s end one w r i t e s equation  (54) i n the form  + G (X,^) V«. 0  where the independent transformation  (73)  v a r i a b l e has been changed u s i n g the  (68) and where  c( ,^)« X  /  -  y  w  ^  (74,  As i n the previous s e c t i o n one takes v a n i s h i n g v a l u e s of V at the boundaries  x  Q  and  x^.  In order to apply a l g e b r a i c  methods to the problem one d i v i d e s the i n t e r v a l into  n  p a r t s each of l e n g t h  z  ( x , x^) Q  -42-  100  2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 3 0 0 1 0 0 0 I IOC  2.5  3.5  4.5  Charge cf  5.5  6.5  7.5  Densily  height  above  «5  05  Distribution the  P  IQ.5  as  arth's  H.5  "'.5  function surface  1 2 0 0 1300 1 4 0 0 5  I4.S  1500  15.5 x l O k m  -43-  The l a s t form i s p o s s i b l e because of the symmetry about the point  x = 0.  points inside  The second d e r i v a t i v e i n equation (73) at  ( x , x j ) i s r e p l a c e d by a d i f f e r e n c e e x p r e s s i o n 0  r e s u l t i n g from a t r u n c a t e d power s e r i e s , v i z . ,  oix  2  h  *  X  The d i f f e r e n t i a l equation can then be approximated by the f o l l o w i n g l i n e a r and homogeneous r e c u r s i o n  +V (»+M + hG C«,w)\4C«)= 0  V Ca-h) - 2V Ca) ?  relation,  a  2  .  (76)  4  i n which terms of the order  h  formula has t o h o l d at  i n t e r i o r p o i n t s r e s u l t i n g i n n-1  n-1  simultaneous l i n e a r e q u a t i o n s . of the i n t e r v a l ,  This recursion  S t a r t i n g at the l e f t - h a n d end  the f o l l o w i n g system of equations r e s u l t s  \ 4 ( 0 - 2 V , ( x 4 r , ) 4-V (x f z  0  are i g n o r e d .  0  2h)  + h 6 ( x h w) V (x +h) 2  0+  t  }  0  - 0-  V (x -rh)-2V Cx +2h)4-V fX f?h)+h C(x +2h o5 ) Vi(v-2h) - 0 2  2  0  z  0  H  2  <3  With the boundary c o n d i t i o n s at  0  x  Q  )  and  x-^  i t f o l l o w s that  f o r a n o n - t r i v i a l s o l u t i o n of t h i s system of l i n e a r equations  -44-  must have  A  l  1  1  A  0  1  6  0  2  1 A  3  0  0  0  0  1  0  9  a  v  =  0  (77)  n-l  with  Ay = n G CXo+y^ co ) - 2. 2  which, because of the r e l a t i o n  A T h i s may  2  (74), becomes  (78) 1  be w r i t t e n i n a b b r e v i a t e d form  (79)  where because of the dependence f u n c t i o n s of  on  p(d~)  , the  a.  are  t¥ along a l i n e of f o r c e .  I n s e r t i n g the l a s t e x p r e s s i o n i n t o equation  (77) the  problem can be c o n s i d e r e d to be an eigenvalue problem of the f o l l o w i n g k i n d  -45-  2  -2  1  1  0  0  a2W -2  1  0 •  •  1  0  2  0  1  a„<j -2 2  which i s i d e n t i c a l to the matrix problem  where  and  A  and  B  are the m a t r i c e s  -46-  Thus i n matrix n o t a t i o n the problem may  BX =  oo  be w r i t t e n  AX  2  (83)  Since t h i s equation i s not i n the u s u a l form s u i t a b l e f o r d i a g o n a l i z i n g a t r a n s f o r m a t i o n has i n appendix I I I equation  (83) may  My  be w r i t t e n  (84)  the e i g e n v a l u e s are r e a l  r ,  as  from the nature of the problem.  The next s t e p i s to compute the values of equation  As shown  -  Because of the symmetry of M would be expected  to be performed.  (79) which a c c o r d i n g to equation  a^  in  (78) w i l l depend on  i . e . the l i n e of f o r c e , and on the charge d e n s i t y d i s t r i -  Q  b u t i o n which here appears as a f u n c t i o n of  Xj = x  Q  + jh.  However t h i s d i s t r i b u t i o n as a l r e a d y mentioned i s g i v e n as a f u n c t i o n of a l t i t u d e above the s u r f a c e of the E a r t h . furthermore  Since  the a l t i t u d e of a p a r t i c u l a r p o i n t above the  E a r t h ' s s u r f a c e along a c e r t a i n l i n e of f o r c e i s g i v e n by  f-| = r s i n V  it sin  f o l l o w s that f o r a g i v e n 2  d"  has  Xj  to be c a l c u l a t e d .  - a  (85)  the corresponding value of To accomplish  t h i s one  returns  -47-  to  equation  (68) which f o r an a r b i t r a r y  X  where  =  ' 35 (  f  {  S  S  $  3  +  6  "S + 2  S  s  S = 5 m'  about the p o i n t  +  x  , 6  may be w r i t t e n  }  <  8 6  )  Because of the symmetry x=0,  i.e. ^  , equation  =  s o l v e d f o r one h a l f of the i n t e r v a l only. hand h a l f of the i n t e r v a l where  x< 0  (  (86) need be  Choosing  the l e f t -  one o b t a i n s from  e q u a t i o n (86)  | x j ~ ^ / , = ^ L ( | - s ) (£s+6s +8s-r/6)  (87)  2  | 5- j,  where  ^- (  >  n  being an even number of  intervals.  Since f o r each d i f f e r e n t l i n e of f o r c e  w e l l as  are d i f f e r e n t i t i s obvious  to  h  be s o l v e d a g r e a t number of times.  |x |  that equation  as  Q  (87) has  For t h i s reason a pro-  gramme was w r i t t e n f o r the Alwac I I I E u s i n g the NewtonRaphson i t e r a t i o n method f o r s o l v i n g t h i s equation with r e s p e c t to i.e.  s  f o r a r b i t r a r y values of the l e f t - h a n d s i d e  f o r v a l u e s i n the range  0 < |X J 0  h< |x|. 0  s  A s h o r t des-  c r i p t i o n of the method as a p p l i e d to t h i s p a r t i c u l a r problem i s given i n appendix IV. of  equation  Table  ( I l l - a ) g i v e s the s o l u t i o n s  (87) f o r the p a r t i c u l a r value of  needed i n our case.  Since  s = sin  2  3  |x | 0  -jh  , the a l t i t u d e H  above the s u r f a c e of the e a r t h i s r e a d i l y c a l c u l a t e d  (Table  Ill-b).  -48-  With these values the charge d e n s i t y may  be  determined.  p (|x |-^)and  Once the r e l a t i o n between  p (r) has been  0  found f o r each value of the argument, v a l u e s of  a^  may  i  be  r e a d i l y obtained from the equation  1  . where  ju  M  65.028 x 1 0  2  =  -  0  1.256  ( 8 8 )  x 10"  (volt-sec/amp.meter),  (volt -sec .meter )  3 0  2  2  In c a l c u l a t i n g the values of (equation 85) about the For t h i s reason  p (l* l  a^  center w i  0  H  2  one  and  notes  r  =  Q  a/sin ^ 2  the symmetry of H  point a l s o be symmetric about  the same p o i n t , which i m p l i e s that the number of p o i n t s f o r which  and  p(\*ol~j'h)  t h e r e f o r e the e x p r e s s i o n  be c a l c u l a t e d , i s approximately case i t was  halved.  (88) have to  Since i n the  found s u f f i c i e n t to s u b d i v i d e the e n t i r e  i n t o e i g h t p a r t s , one has to compute values of  aj  p o i n t s and the c e n t e r p o i n t of the i n t e r v a l o n l y . r e s u l t i n g values of one may  Aj  are compiled  interval at three  The  i n t a b l e 5 - from them  c a l c u l a t e the elements of the matrix  appendix I I .  present  M  as g i v e n i n  These matrices which i n the present case are of  order seven have been d i a g o n a l i z e d w i t h the h e l p of the Alwac I I I E u s i n g the J a c o b i method.  The m a t r i c e s as w e l l as  t h e i r e i g e n v a l u e s are g i v e n i n appendix IV. the fundamental and  The  periods of  the next higher mode are given i n t a b l e 6.  -49-  Table 5 & =15° A =l4.039xl0 u -2 A =l4.039xl05y2_ A =14.03 9 x 1 0 ^ - 2 A =14.039xl0 w -2  A =50° A = l . 5053x10x-1.2. -w -2 A = l . 1076x10-^ 2_ A = l . 0082x10"^ -2 A =0.97985xl0" u -2  fl> =20° A,=1.6230xl0 u -2 A2=1.6230xl0 w2-2 A =1.6230xl0 w2-2 A =1.6230xl0 w -2  &a =55° A =12.005xl0 w -2 A =5.5187xl0"2w2_ A =4.6 587xl0-2u -2 A4=4.4795x10" w -2  & =25° A7=6\ 0841X10 W 2-2 Ao=5.66l6xl0 w -2 A3=5.4926xlo2w2_ A4=5.4926xl02u)2_  d. =60° A"p9".3571xl0 w -2 A =0.70985x10" Hi -< A =0.34202xl0 u2-2 A =0.30330xl0~ w -2  & =30° A£=6". 5847x10^2-2 A =5.4395x10^ 2-2 A =5.1532xl0 u -2 A4=5.0960x10^ -2  8. =65° A =5 0936u -2 A =5. 7 3 0 3 x 1 0 - ^ - 2 A =2.1648x10"^ -2 A =1.6554xl0" w -2  ^6 =3 5 ° ApF.6500u -2 A =6.0166tj -2 A =5.7000u2-2 A =5.5733u -2  A =70° A =l„ 4209X10 CJ 2-2 A =0.31575X10 CJ -2 A =0. 14472x10^ -2 A =0.113l4xl0 w -2  & =40° A7=T.3394w -2 A2=1.1878w -2 A =1.1373W -2 A4=1.1221u -2  & =75° A ^ T . 2196xl0 w -2 A =0. 8 3 4 4 6 x 1 0 ^ - 2 A =0. 53491xl0 w -2 A =0.44933x10^ -2  5  2  1  2  2  2  3  5  2  4  4  2  4  4  3  4  2  4  2  2  2  2  2  2  1  2  3  2  2  jS  2  2  2  3  1  2  4  _2  2  1  2  2  2  3  2  2  -1  2  2  2  _1  3  1  2  4  2  1  o  2  2  2  3  1  2  4  1  1  2  :L  2  2  2  2  3  3  2  1  4  2  4  2  i  2  2  2  2  2  1  3  2  3  2  2  4  B„ =45  Ai=3.7511x10" \t -2 A =3.1081x10"^ -2 A =2.9473xlO (j -2 A =2. 8 9 3 7 x l 0 w - 2 2  2  2  _1  2  3  _1  4  J  1  2  &  =80° AT=X6061u2_2 Aj=3.8719w -2 A =4„2144W -2 A =4„2442w -2 2  2  3  2  4  -50-  Table 6  <r  H  0  12  15°  1.907xl0  4  20°  2.052xl0  3  1.046xl0  25°  3.804xl0  2  1.965xl0  2  30°  1.174xl0  2  6.184X10  1  35°  3.884X10  1  2.028X10  1  40°  1.735X10  1  45°  8.871  4.673  50°  5.254  2.837  55°  3.759  2.235  60°  6.513  5.370  65°  1.556X10  1  1.265X10  1  70°  2.908X10  1  2.188X10  1  75°  4.104x10*  2.440X10  80°  3.230X10  1.574X10  At t h i s  point  0.9727xl0  sec.  sec.  3  9.049  1  the accuracy  4  1  1  of the approximations  be c o n s i d e r e d - i n other words was  the number of  should  intervals  g r e a t enough to y i e l d the lowest  e i g e n v a l u e even i n the most  unfavorable  accuracy?  case w i t h s u f f i c i e n t  With a f i x e d  -51-  IOOOOQ  Fig. 8.  Fundamental  period, variable charge  normal  dipole  field.  den:, -y  -52-  number of i n t e r v a l s the most u n f a v o r a b l e s i t u a t i o n  occurs  when the e n t i r e i n t e r v a l has i t s g r e a t e s t value s i n c e then has  \h|  i t s g r e a t e s t value too which means that the t r u n c a t i o n  e r r o r i s a maximum.,  To answer t h i s q u e s t i o n one goes back to  the case of constant equation  which p e r m i t t e d a s o l u t i o n of  (54) i n c l o s e d form.  (77) by v i r t u e of equation  -2 1 0  1  determinant  (78) can be w r i t t e n i n the form  0  0  1  0  1  f-2  In t h i s case the  ^2_  1  2  =  0  (89)  where  X  l  2 / i  S  It i s shown i n appendix X d e t e r m i n a n t a l equation  ?  ° P  r  that  y  (J  (90)  which has  to s a t i s f y  the  (89) i s given by  i-z+z^Wi where  M  i s the order of the determinant  S i n c e i n our case  M = 7  ( 9 1 )  and  k = 1,2, •°°<' M. z  the f o l l o w i n g seven values of ^k  -53-  are o b t a i n e d , v i z . ,  = 3.84776  J  = 2.00000  H  ^  = 3.41422  = 1.23464  ^  = 2.76536  = 0.58578  y,  = 0.15224  1  The most u n f a v o r a b l e s i t u a t i o n occurs when this  ^  = 15°. In  case  j  =  A  °  r  p  P  co = 14.03 9 x I0 co 2  (92)  S  M -19  from t a b l e  5 where  p  was taken to be 6.5x10  ^ (kg/m°).  Hence f o r the lowest and next higher eigenvalue one o b t a i n s w  2 x  = 10.84x10  8  (sec ) and  u  2  2 2  On the other hand i n the case of constant  = 41.72x10 ( s e c ~ ) 8  p  , the lowest and  next h i g h e r eigenvalue can be o b t a i n e d from equation p  with  w  2 x  = 6.5xlO"  (71) which,  (kg/m ), y i e l d s  19  3  - 10.98xlO~  Thus the d i f f e r e n c e  2  8  (sec ) - 2  and  u  2 2  = 43.92  between the approximative  the exact s o l u t i o n i s w i t h i n reasonable  (sec ). - 2  s o l u t i o n and  limits.  This  difference  -54-  becomes s m a l l e r as  d'o i n c r e a s e s s i n c e the number of i n t e r v a l s  i s kept constant which i m p l i e s that t h e r e f o r e can be concluded val  h  that s u b d i v i d i n g the e n t i r e  into eight parts yields s u f f i c i e n t Using the values of  becomes s m a l l e r .  ^  It  inter-  accuracy.  given by equation  d i a g o n a l i z i n g the m a t r i c e s i n the case o f constant  (91) and p  one  can check the accuracy of the computer c a l c u l a t i o n s by comparing  the r e s u l t s o b t a i n e d from combining  equations  with those obtained from matrix d i a g o n a l i z a t i o n . p o s s i b l e o n l y f o r the cases £f  0  then  a.  = 15°  i s e s s e n t i a l l y constant.  and ^  (91) and (92) T h i s was  = 20°  I t was found  since  that the  agreement was very good i f the o f f - d i a g o n a l elements d i d not exceed 3.  0.00005.  S o l u t i o n f o r the compressed d i p o l e f i e l d with constant charge d e n s i t y In  the preceding two s e c t i o n s the e i g e n p e r i o d s of  t o r o i d a l o s c i l l a t i o n s were o b t a i n e d under the assumption the magnetic f i e l d was a d i p o l e f i e l d . earlier  t h i s i s not s t r i c t l y  true.  that  As p o i n t e d out  For t h i s reason i n t h i s  s e c t i o n the e i g e n p e r i o d s of t o r o i d a l o s c i l l a t i o n s are s t u d i e d under the assumption  t h a t the d i p o l e f i e l d has been deformed,  the extent of the deformation being governed by the r a d i u s of  the c a v i t y i n which the f i e l d  i s confined.  The d i f f e r e n t i a l equation which governs the motion under  -55-  these circumstances was  +  dd  where V  t  d e r i v e d i n chapter II-4 and i s  S,W  4&  (64)  i s o b t a i n e d from the equation  \i . 3  rL?  A  m  si^o  ,3  (61)  Before computing the e i g e n v a l u e s of equation w i l l o b t a i n an approximate s o l u t i o n .  As may  (64)  we  be seen from  p l o t s of the magnetic l i n e s of f o r c e i n f i g u r e s 2-4  i)  the  c  v e r y soon approaches a constant value f o r l i n e s of f o r c e i n t e r s e c t i n g the E a r t h at very low c o - l a t i t u t d e s  ( s m a l l <^ ).  l i m i t i n g value taken i n t h i s case i s approximately of the c a v i t y oC  .  T h e r e f o r e with  = o(  The  the r a d i u s  equation  (64)  becomes  3 sin& id-  which may  o(4  be w r i t t e n i n the  _J  4. A Y ^ ' v , = 0  equation  (93)  form  _d_  cosec-d' d<& In order t o reduce  d Vg  (94) 5  Ho  (94) to a s i m p l e r form one  change, as has been done before, the independent  may  variable  by  -56-  putting  cos  ecfr  d-d'  = dx  (95)  Then equation (94) becomes  i2  28  u  d X^  d  2  ^  +  ^  2  V/  .^ 2 r  0  2  (96 )  As before one assumes perfect r e f l e c t i o n at the end points as the boundary conditions, v i z . :  V*=0  The r e l a t i o n between J^  0  J , J  at  and  x  Q  i s found by integrating  D  equation (95), i . e .  dx ~ ^  cosect^ dJ'  u  where, as before, ^ becomes  or after integrating  = 180°- i¥ . Because of symmetry this 0  -57-  AX = - 2  Since  &  0  > A X  <  was i n equation i n t e r v a l from x  Q  it  to  x^.  log tan  (97  w i l l be a p o s i t i v e q u a n t i t y as i t  (67). T h e r e f o r e equation -do to  (97) a s s o c i a t e s the  •J' with an i n t e r v a l extending  from  By s i m i l a r r e a s o n i n g to that g i v e n i n s e c t i o n 1  follows that  X = to t a n - ^ 2 -  (98)  3  0  The s o l u t i o n of e q u a t i o n  V-Acos  (  (96) i s  * « * * y A  A  (99a)  and  *  where  GJ) =  CO v^o  V  .  -5  3  H H.  X )  (99b  c  Using the boundary c o n d i t i o n s i t  f o l l o w s from these two equations that  a «  co vu -  3  -X  H  0  0  . I  X0 X •  '  0  7T I  = =  n — n ~  2  (100)  -58-  Since  a  8 r  =  m %.  0.7626X10  3 H, equation  (100) can be w r i t t e n  OJ -  where  a  Y)  I.S2  S2*  10  TT x. <*  4  (101)  |x |  K  0  i s the r a d i u s of the l i m i t i n g c a v i t y expressed i n  u n i t s of E a r t h r a d i i and x  Q  i s g i v e n by equation  ever i n t h i s type of problem one  (98).  How-  i s more i n t e r e s t e d i n the  e i g e n p e r i o d than i n the angular frequency.  From equation  (101) the p e r i o d of the n-th o s c i l l a t i o n i s g i v e n by  7  - 3. OFQx 10 x cx x |X | x \f p 0  j-  5 e c  -j  (102)  However i t should be kept i n mind t h a t t h i s formula a p p l i e s only to those magnetic l i n e s of f o r c e which i n t e r s e c t E a r t h at very low c o - l a t i t u d e s  ( i¥ ^ 0  20°)  the r a d i u s of the c a v i t y should not exceed i n order to guarantee i s approximately equation  (102)  T  and i n a d d i t i o n s i x Earth r a d i i  t h a t the assumption  satisfied.  Taking  V = o<v = c  -J^ = 10°,  for  cx-4  and  T  = 1748 sec  constant  yO - 6.5x10  yields  = 109 sec  the  f o r c x = 8.  K^jyn  -59-  In the m a j o r i t y of cases however i t w i l l not be s i b l e to assume that equation  i s constant.  For these  permis-  cases  (64) has to be s o l v e d r i g o r o u s l y u s i n g l i n e s of  f o r c e g i v e n by equation as i n t h i s s e c t i o n we  (61).  In the f i r s t  s e c t i o n as w e l l  changed the independent  variable  by  putting  In the f i r s t  section,  f C-fJ) =• CoseCi^ .  I t thus appears  l o g i c a l to put  i.e.  Equation  (64) then takes the simple  form  (103) o  The s o l u t i o n of t h i s equation i s  (104a)  -60-  and  a co  where, as b e f o r e ,  co = co Vp  .  (104b  With the same boundary condi-  t i o n s as used b e f o r e , one o b t a i n s from these  equations -9  6.864- x l O x n r , ~'  4.576 x |0 x |X|  [sec ] (105)  IX.I  tf  e  where  ^ —  From equation  =  S£x  2.2  10  (105) the e i g e n p e r i o d s are r e a d i l y  obtained,  viz.  7  In p a r t i c u l a r  ,  9.152 x|  g 0  x  fx./x  fsec]  (106)  the fundamental e i g e n p e r i o d (n = 1) i s given  by  J  ~- 9 . 1 5 - 2 x 1 0 % |X | D  *\fjjT  [sec]  Comparing the equation f o r the e i g e n p e r i o d s of the o s c i l l a t i n g magnetic l i n e s of f o r c e i n the case of an  (107)  -61-  ordinary dipole f i e l d dipole f i e l d  (equ. 72) w i t h that of a compressed  (equ. 107), one sees that they are of the same  form and would be i d e n t i c a l i f the f a c t o r incorporated into  jx |  .  \)*  c o u l d be  For t h i s reason we might  expect  t h a t both e x p r e s s i o n s would y i e l d the same r e s u l t i f the presence of the l i m i t i n g c a v i t y does not i n f l u e n c e the shape of  the magnetic  l i n e s of f o r c e too much.  T h i s would be the  case f o r those l i n e s o f f o r c e i n t e r s e c t i n g the E a r t h a t h i g h co-latitudes. In  order to e v a l u a t e equation (106) i t i s necessary to  f i n d the r e l a t i o n between  and  |X | •  In the present  case the change of independent  v a r i a b l e was accomplished by  D  the t r a n s f o r m a t i o n  dX =  where  V  (108)  has to be determined  c  from the c u b i c equation  (61)  Since i n t e r v a l A3 =  A X = X,- X  0  i s a s s o c i a t e d with the i n t e r v a l  - <& , i n t e g r a t i o n of e q u a t i o n (108) y i e l d s Q  A  X  '  ft 3 * 0 ^ • » /(«<+ 2 ) sm^T  2  f, / (oC+2  » % )sm<7 3  ^  < ' 109  -62-  The l a s t expression i s due to symmetry considerations. Thus  As opposed to the integral which gave x  Q  i n section 1 of this  chapter, i t i s not possible here to evaluate the integral (110) i n terms of known functions.  In this case one has to  resort to numerical methods in order to obtain X = X (<X d ) 0  0  ;  0  The computation of the i n t e g r a l was performed using Simpson's rule with an i n t e r v a l length of one degree. under these circumstances  Since  the integrand had to be evaluated o  at many i n t e r i o r points of the given i n t e r v a l from  <j£  to  90°, a computer programme for the ALWAC III E was written to l e t the machine take care of this laborious step. contains numerical values of the integrand. of  V  c  Appendix VI  The evaluation  from equation (61) was carried out using the Newton-  Raphson method taking as a f i r s t approximation radius of the l i m i t i n g cavity. values of the i n t e g r a l . function of  -cfc for  for V  c  the  Table 7 gives the numerical  These are plotted i n F i g . 9 as a  cx - 12  <x = 8  and  o< = 4.  As pointed out already i n the short discussion following the derivation of formula (106) for the eigenperiods, the quantities  v/x  |X | 0  i n equation (72) and  equation (106) correspond to one another.  jXj  in  In order to study  -63-  t h i s r e l a t i o n s h i p more c l o s e l y values of the e x p r e s s i o n l*o I * ^  (i?-3o) were computed and the r e s u l t s t a b u l a t e d i n  t a b l e 8 and p l o t t e d  i n F i g . 10.  Table 7 •Jo  x (o<=12)  x (<x =8)  10  57.31xl0  2  I4.53xl0  20  9.827xl0  2  4.194xl0  30  10.07X10  1  7.656X10  2.398X10  40  15.01  13.92  8.317  50  3.641  3.550  2.864  60  1.235  1.222  1.109  70  0.5001  0.4975  0.4730  80°  0.1902  0.1896  0.1836  0  x (cx=4)  0  Q  2  2  1  1.247xl0  2  0.5869xl0  2  1  Table 8 |Xo|<  A5»  V1 0  (<*•'*)  10°  55.293xl0  4  56.873xl0  2  13.410X10  20°  24.411xl0  2  10.458xl0  2  4.5296xl0  30°  116.89  102.23  79.507  26.004  40°  15.534  15.074  14.102  8.8358  50°  3.6814  3.6479  3.5722  2.9830  60°  1.2408  1.2371  1.2278  1.1464  70°  0.50133  0.50075  0.49953  80°  0.19048  0.19040  0.17909  0.48795 0.17864  2  2  0.99584xl0  2  0.60472xl0  2  -66-  We can now e v a l u a t e the e x p r e s s i o n  (106) f o r the e i g e n -  p e r i o d s of t o r o i d a l o s c i l l a t i o n s of a magnetic l i n e of f o r c e belonging to a compressed d i p o l e f i e l d . J_ ^ 3 /  f> =6.S'x 10 ing  3 m  J  Taking the value  f o r the charge d e n s i t y of the conduct-  medium the f o l l o w i n g fundamental p e r i o d s are obtained.  Table 9 <(%  T,  (sec )  T,  ( c< ^  8) ( sec )  T, ( c< -  4)  (sec )  10  4228  1072  92.00  20  725.0  309.4  43.30  30  74.30  56.48  17.69  40  11.07  10.27  6.136  50  2.686  2.619  2.113  60  0.9112  0.9016  0.8182  70  0.3690  0.3670  0.3490  80  0.1403  0.1399  0.1355  The r e s u l t s of t a b l e 9 are p l o t t e d i n F i g . 11 and show c l e a r l y the i n f l u e n c e of the f i n i t e 4.  cavity.  S o l u t i o n f o r the compressed d i p o l e f i e l d w i t h v a r i a b l e charge d e n s i t y In the p r e c e d i n g s e c t i o n the fundamental p e r i o d s o f  t o r o i d a l o s c i l l a t i o n s of a c o n d u c t i n g medium i n the presence of  a compressed d i p o l e f i e l d were obtained assuming a constant  -68-  charge d e n s i t y d i s t r i b u t i o n .  However, i n g e n e r a l , t h i s assump-  t i o n w i l l not be v a l i d . In order to t r e a t the case of a v a r i a b l e charge d e n s i t y i n the presence of a compressed  d i p o l e f i e l d one has t o go  back to e q u a t i o n (103) from s e c t i o n 3, v i z .  where the independent v a r i a b l e x has been r e p l a c e d by y. was shown i n the l a s t s e c t i o n how the i n t e r v a l i s transformed by means of the i n t e g r a l interval  (y ,0). Q  (  ,  It  )  (110) i n t o an  Furthermore the e v a l u a t i o n of t h i s  integral  has shown that the v a r i a b l e y takes a f a r wider range of v a l u e s than the c o r r e s p o n d i n g v a r i a b l e x of the normal d i p o l e field.  For t h i s reason the matrix method cannot be a p p l i e d  to e q u a t i o n (103) immediately u n l e s s one uses a very high order m a t r i x (up to 1,000) which i s p r o h i b i t i v e .  Since i n  g e n e r a l the l e n g t h of the i n t e r v a l i s i r r e l e v a n t f o r the c a l c u l a t i o n of the e i g e n v a l u e i t seems a p p r o p r i a t e to perform a second t r a n s f o r m a t i o n which compresses into  ( x , 0 ) where x o  G  the i n t e r v a l  may be chosen a r b i t r a r i l y .  (y ,0) o  However the  success of such a t r a n s f o r m a t i o n depends l a r g e l y upon the smooth behaviour of the charge d e n s i t y .  F o r the new  interval  ( x , 0 ) the m a t r i x method a l r e a d y d e s c r i b e d can be a p p l i e d o  without  difficulty.  - 6 9 -  In order t o i l l u s t r a t e the method i t w i l l be assumed t h a t a knowledge of the charge is sufficient oscillating  to determine  d e n s i t y at 7 i n t e r m e d i a t e p o i n t s  the fundamental e i g e n p e r i o d of the  l i n e of f o r c e w i t h reasonable accuracy.  Since i t  i 0.\ I  has been shown that w i t h a 7 order matrix and I h I < lowest e i g e n v a l u e i s approximated  with s u f f i c i e n t  h w i l l be taken to be 0.1 and thus x cases  Q  = 0.4.  the  accuracy,  Since i n most  l y l "7 I X 1 i t seems f e a s i b l e t o put 0  0  (in)  where p i s a constant f a c t o r depending upon p  i s determined  where y  i s found from t a b l e 7 ( y o = o ) °  c a l c u l a t i o n t o the case where of  ^ .  from the r e l a t i o n  x  G  C X and  p are obtained :  C o n f i n i n g the  cx= 4 o n l y , the f o l l o w i n g values ;  -70-  T a b l e 10  j%_  /?(CX'4)  10°  3.1170X10  20°  1.4672xl0  30°  5.9948X10  40°  20.792  50°  7.1595  60°  2.7733  70°  1.1826  2  2  1  I n t r o d u c i n g y = (3x i n t o equation  Jl^ /Mf +  (103 ) y i e l d s  V -0  J  where the independent v a r i a b l e x now extends from - x (because  (112)  E  Q  to 0  of symmetry about the o r i g i n ) .  In order t o apply the same numerical methods t o equation (112) as those used i n s e c t i o n 2 one has t o e v a l u a t e the expression z  (113)  Wo at the i n t e r m e d i a t e p o i n t s  \X \ ~ 0  s u b j e c t t o the  same symmetry p r o p e r t i e s as mentioned i n s e c t i o n 2.  In the  -71-  the i n t e r m e d i a t e p o i n t s are X J =0.3; 0.2; 0.1;  present case 0.0.  Since  p> i s given as a f u n c t i o n of a l t i t u d e above the  s u r f a c e of the E a r t h ( f i g . 7) one has t o a s s o c i a t e each X j w i t h y- by means of the r e l a t i o n  (111) and then, by u s i n g the  graph i n f i g . 9, f i n d the corresponding found  the c o - l a t i t u d e ^  co-latitude.  , the corresponding  \)  c  Having  of the  compressed d i p o l e f i e l d can be obtained from t a b l e V l - a of the appendix.  By e v a l u a t i n g  [\~~  6.370 ( V O  C M  the h e i g h t of a p a r t i c u l a r p o i n t corresponding found, and hence the e x p r e s s i o n  <  The  i n t a b l e 11 - from them  one may c a l c u l a t e the elements of the matrix M as g i v e n i n appendix I I . " As before the m a t r i c e s  are d i a g b n a l i z e d with  the h e l p of the A l w a c I I I E u s i n g the J a c o b i method. matrices  as w e l l as t h e i r eigenvalues  appendix V I I  v  The  are given i n  The p e r i o d s of the fundamental modes are  g i v e n i n t a b l e 12 and a r e ' p l o t t e d i n f i g . 11.  )  t o x j can be  (113) can be c a l c u l a t e d .  r e s u l t i n g values of Aj are compiled  114  -72-  T a b l e 11  A =io°  <> =20  o  0  A 8 . 3 9 5 7 0 x 1 0 ^ -2 A =5.44448x10^ - 2 A =4.3 5050xl0lw - 2 A4=3.56181xl0 <j2-2  A-,=34.9494x10^ - 2 A =1.94476xK4u2-2 A =1.35288xl0 w -2 2 A =0.92446 8xl0l<J -  2  2  1=  2  2  2  2  1  3  2  3  1  4  <% =30°  <% =40° A =151.693x10^ -2 A =0. 950914x10^ A = 0 „ 0588661x10^2 A4=0.0301123X10 <J2  A =159.98xl0 u -2 A =0.715206xl0 w -2 A =0.331253x10^ -2 A =0.188212xl0\/ 2-2 1  2  2  1  1  x  2  2  2  2  2  3  3  1  4  ^=50°  <%=6Q°  A =622.804w _2 A =15.5701w -2 Ao=0.134225u -2 A4=0„0566430u -2  A =l04.729w2-2 A =13.2925u -2 A =0.584064u - 2 A =0.0604204u -2  2  1  1  2  2  2  2  2  2  3  2  2  4  <%-70° A =7. 50754u -2 A =19.4097u -2 A =3„66222u2_ A4=2.92977w -2 2  x  2  2  3  2  2  Table 12 T l (sec) 10° 20°  111 124  30° 40° 50° 60° 70°  252 245 159 69 46  -73-  DISCUSSION OF THE  The  aim of t h i s t h e s i s was  to c a l c u l a t e the  of geomagnetic m i c r o p u l s a t i o n s and  observed  eigenperiods  i n p a r t i c u l a r to i n v e s t i -  gate t h e i r dependence on l a t i t u d e . c a r r y i n g out the study was  RESULTS  The main reason f o r  the great d i s c r e p a n c y between the  and c a l c u l a t e d v a l u e s e s p e c i a l l y when the p o i n t s of  o b s e r v a t i o n are i n the p o l a r r e g i o n s . much doubt on the model put forward t h i s reason  This discrepancy casts  by Dungey (1954b).  5  For  the p r i n c i p l e s of magnetohydrody n a m i c s were  a p p l i e d to a somewhat d i f f e r e n t model which, because of the g r e a t e r s i m p l i c i t y of the r e s u l t i n g equations, promised an easier  solution.  From Maxwell's equations and hydrodynamics, two  the b a s i c e q u a t i o n of  p a r t i a l d i f f e r e n t i a l equations were  obtained i n c y l i n d r i c a l coordinates.  The use of t h i s system  of c o o r d i n a t e s seems f e a s i b l e s i n c e the phenomenon of geo-  j  magnetic m i c r o p u l s a t i o n s appears to be c o n f i n e d to m e r i d i o n a l planes.*  Mathematically  t h i s can be expressed  by d i s r e g a r d -  i n g the c o u p l i n g terms c o n t a i n e d i n both e q u a t i o n s . -  Since  t h i s i s e q u i v a l e n t to c o n s i d e r i n g the problem i n a plane i t s h o u l d i n p r i n c i p l e make no d i f f e r e n c e whether c y l i n d r i c a l * The q u e s t i o n of the dependence of m i c r o p u l s a t i o n s on G.M.Ti or L.M.T. i s s t i l l r a t h e r open.  -74-  or s p h e r i c a l c o o r d i n a t e s are used.  On account  of c y l i n d r i c a l  symmetry s o l u t i o n s are obtained more e a s i l y . The e i g e n p e r i o d s of the t o r o i d a l o s c i l l a t i o n s were o b t a i n e d as a f u n c t i o n of c o - l a t i t u d e f i r s t assuming a medium of constant charge d e n s i t y i n a normal d i p o l e f i e l d . be seen from f i g . 6 the e i g e n p e r i o d s tend towards  can  infinity  as z e r o c o - l a t i t u d e i s approached, a r e s u l t which has been observed.  As  not  The r e s u l t i n g high values are e n t i r e l y due  to  the f a c t t h a t the l i n e s of f o r c e i n the case of a normal d i p o l e extend very f a r i n t o outer space.  The  eigenperiods  c a l c u l a t e d by Dungey u s i n g s p h e r i c a l c o o r d i n a t e s agree very w e l l with those obtained i n t h i s t h e s i s which j u s t i f i e s  the  use of c y l i n d r i c a l c o o r d i n a t e s i n d e s c r i b i n g the problem. F i g u r e 6 a l s o shows that the e i g e n p e r i o d s at h i g h c o - l a t i t u d e s tend to z e r o .  Since t o r o i d a l o s c i l l a t i o n s can be  understood  as o s c i l l a t i n g l i n e s of f o r c e i t f o l l o w s that the p e r i o d must tend to z e r o as the l e n g t h of the l i n e of f o r c e decreases it  does  i n t h i s model at higher c o - l a t i t u d e s .  as  T h i s i s the,  p r i n c i p a l d i f f i c u l t y of t h i s model whether s p h e r i c a l or  t  c y l i n d r i c a l c o o r d i n a t e s are used. Since the charge d e n s i t y d i s t r i b u t i o n of the medium above the s u r f a c e of the E a r t h i s probably not constant above model was  the  m o d i f i e d u s i n g a v a r i a b l e charge d e n s i t y as  proposed by D e s s l e r (1958).  A comparison of f i g u r e s 6 and  shows the e f f e c t of a v a r i a b l e charge d e n s i t y on  the  8  -75-  fundamental\ p e r i o d . fig. of  The values on the l e f t  of f i g . 6 and  8 agree e x a c t l y s i n c e the major p a r t of the magnetic  f o r c e run through  lines  r e g i o n s i n which the charge d e n s i t y can  be c o n s i d e r e d to be constant and equal to t h a t used i n the previous c a l c u l a t i o n .  At higher c o - l a t i t u d e s , the magnetic  l i n e s of f o r c e spend longer i n r e g i o n s of lower a l t i t u d e i n which the charge d e n s i t y i s higher than t h a t f u r t h e r out. T h i s r e s u l t s i n a somewhat slower drop of the c a l c u l a t e d e i g e n p e r i o d s with i n c r e a s i n g c o - l a t i t u d e .  Above c o - l a t i t u d e  55°  the p e r i o d s i n c r e a s e , the i n c r e a s e i n charge d e n s i t y completely compensating f o r the s h o r t e n i n g of the l i n e s of f o r c e . So f a r a l l the r e s u l t s were obtained on the assumption t h a t the E a r t h ' s magnetic f i e l d  i s t h a t of a geocentric d i p o l e .  Recent i n v e s t i g a t i o n s (Parker, 1958) t h i s i s not so. field was  One  however have shown t h a t  of the more important  f e a t u r e s of the  i s that i t does not extend as f a r i n t o outer space as,  o r i g i n a l l y thought.  To a f i r s t  approximation  on  the  d a y l i g h t s i d e of the E a r t h i t appears to be c o n f i n e d to a c a v i t y of v a r i a b l e r a d i u s (4-12 E a r t h r a d i i ) .  T h i s may  be  a t t r i b u t e d to the " s o l a r wind" which compresses the f i e l d the d a y l i g h t s i d e of the E a r t h . has an important  Since the form of the  on  field  e f f e c t on the e i g e n p e r i o d s of geomagnetic  m i c r o p u l s a t i o n s a compressed d i p o l e f i e l d was s u b s t i t u t e d i n t o the t o r o i d a l equation and the e i g e n p e r i o d s computed.  |  The r e s u l t s of t h i s c a l c u l a t i o n have been p l o t t e d i n f i g .  12.  -76-  One  of the important  f e a t u r e s i s t h a t the l a r g e values of the  e i g e n p e r i o d s have vanished,  the e i g e n p e r i o d s assuming  v a l u e s as the c o - l a t i t u d e approaches z e r o . l i m i t one Xo to  use equation  i n f i g . 9 as computed from /X |x Q  equation for  cannot  "i)  4  0  |X |  at  0 and  0  0  .  To o b t a i n the  For t h i s purpose however  integral  i n f i g . 10 where  oc  (68) on p u t t i n g  |X |  (98).  finite  (110)  i s n e a r l y equal  i s to be taken  from  Using the l i m i t i n g  the f a c t t h a t  \)  o c  tends  value  to o< , the  r a d i u s of the cavity., the e i g e n p e r i o d of the fundamental mode takes the l i m i t i n g  value  -  which, with The in  o( -  and  4J  8 4 I— 8 3 4 x |0 xcrf x Vyo [>ecj  p ** &.5x/0  g i v e s the value 86 seconds.  s m a l l disagreement between the above value and t a b l e 9 i s due  the i n t e g r a l .  to the use of the approximation  As one would expect  T,  f o r c*=4  r a t h e r than  the r a d i u s of the c a v i t y  i n f l u e n c e s the s i z e of the e i g e n p e r i o d s c o n s i d e r a b l y , i n p a r t i c u l a r those which are a s s o c i a t e d with l i n e s of f o r c e i n t e r s e c t i n g the E a r t h ' s s u r f a c e at low c o - l a t i t u d e s .  As  h i g h e r c o - l a t i t u d e s are approached the e i g e n p e r i o d s c o r r e s ponding to d i f f e r e n t values of  CX  almost  c o i n c i d e with  those o b t a i n e d i n the case of a normal d i p o l e f i e l d . P h y s i c a l l y t h i s can be understood The  l i n e s of f o r c e which i n t e r s e c t  from f i g u r e s 2,  3 and  i 4.  the E a r t h ' s s u r f a c e at  -77-  high c o - l a t i t u d e s  (  > + 5 " ° - 50°) h a r d l y s u f f e r any deforma-  t i o n even f o r o< = 4- and hence have the same shape as those of a normal d i p o l e with the r e s u l t  that the e i g e n p e r i o d s are  equal to those of a normal d i p o l e .  As a consequence the  e i g e n p e r i o d s tend to z e r o as the equator f a c t which i s i n disagreement  i s approached, a  with the observed  data.  F i n a l l y i t seemed promising to c a l c u l a t e the e i g e n p e r i o d s for  a compressed f i e l d  tribution.  assuming a v a r i a b l e charge d e n s i t y d i s -  In that way one c o u l d expect t o i n c o r p o r a t e both  the advantages o f model 2 and model 3 ( s e c t i o n s 2 and 3 of chapter I I I ) i n t o one model which would remove the t r e n d t o i n f i n i t y a t low c o - l a t i t u d e s and a t the same time r a i s e the s m a l l values of the eigenvalues at high c o - l a t i t u d e s . c a l c u l a t i o n was c a r r i e d out f o r the case result  indicated i n f i g .  too s a t i s f a c t o r y .  11.  cx = 4  The  and the  The r e s u l t as i t stands i s not  One would have expected from simple  con-  s i d e r a t i o n s t h a t at low c o - l a t i t u d e s the curve would a p p r o x i mately  f o l l o w that o b t a i n e d f o r the compressed d i p o l e f i e l d  w i t h constant charge d e n s i t y w h i l e a t high c o - l a t i t u d e s the behaviour  of the curve would be s i m i l a r t o that of a normal  d i p o l e w i t h v a r i a b l e charge d e n s i t y d i s t r i b u t i o n .  Instead  the curve r i s e s s l i g h t l y with i n c r e a s i n g c o - l a t i t u d e about 30°) and there decreases c o n t i n u o u s l y .  (up t o  Before drawing  any c o n c l u s i o n s from t h i s r e s u l t one must c o n s i d e r the following point.  As a l r e a d y mentioned the i n t e r v a l  over  -78-  which the d i f f e r e n t i a l equation had to be i n t e g r a t e d was q u i t e l a r g e a d i f f i c u l t y which was overcome by i n t r o d u c i n g a second variable.  S u b d i v i d i n g the i n t e r v a l i n t o 8 p a r t s and a s s o c i a t -  i n g the e q u a l l y spaced p o i n t s with p o i n t s along the l i n e s of f o r c e , those p o i n t s near the ends of the l i n e s of f o r c e are favored.  Since these p o i n t s are s i t u a t e d i n that p a r t of the  medium having  the higher charge d e n s i t y an i n c o r r e c t p i c t u r e  may e a s i l y be obtained.  In order t o a v o i d t h i s the i n t e r v a l  s h o u l d be s u b d i v i d e d i n t o more p a r t s i n order  to o b t a i n a  b e t t e r r e p r e s e n t a t i o n with r e s p e c t to the charge d e n s i t y t r i b u t i o n along a l i n e of f o r c e . l e n g t h 0.1 a seven  order matrix  with s u f f i c i e n t accuracy  dis-  Since f o r an i n t e r v a l of g i v e s the lowest  i n the case of constant  eigenvalue charge  d e n s i t y one can be sure t h a t any change i n the e i g e n p e r i o d with  i n c r e a s e i n the number of i n t e r v a l s i s e n t i r e l y due t o  the i n c r e a s e d i n f o r m a t i o n on the charge d e n s i t y along the l i n e s of force.  (  The number of p o i n t s which would be s u f f i c i e n t  c o u l d only be found by r e f l e c t i n g the c a l c u l a t i o n u s i n g 16 i n t e r v a l s i n s t e a d of 8 as i n the present performing  case.  Before a c t u a l l y  the c a l c u l a t i o n i t would be a d v i s a b l e to supplement  t a b l e 7 by c a l c u l a t i n g the values of x That would allow higher accuracy f o r determining  the value o f  Q  f o r <t- 15° 2 5°  75°  i n the i n v e r s e i n t e r p o l a t i o n  d' which belongs to a c e r t a i n  value of x. In view of the d i f f i c u l t i e s i n v o l v e d i n c a r r y i n g out the  -79-  improvement d e s c r i b e d above i t seems f e a s i b l e to c o n s i d e r a d i f f e r e n t method of approach.  Assuming that the  disturbance  t r a v e l s along a l i n e of f o r c e the time r e q u i r e d to t r a v e l from one  end p o i n t to the other i s g i v e n  where V i s the A l f v e n v e l o c i t y g i v e n  by  by  - ° fa \U H  v  m  T h i s i s the f o r m u l a t i o n which has  been used by Jacobs  and  Obayashi (1958) to f i n d the charge d e n s i t y from a knowledge of the p e r i o d of m i c r o p u l s a t i o n s .  The  i n t e g r a t i o n has  to be  c a r r i e d out along the l i n e of f o r c e u s i n g the compressed dipole f i e l d .  The  d i f f i c u l t y l i e s i n the e v a l u a t i o n of the  i n t e g r a l and a r i s e s mainly  from the f a c t that the  equation  of a l i n e of f o r c e i s a c u b i c equation which makes the  line  element ds more complicated.  Using modern computing devices  however the problem should be  tractable.  So f a r the d i s c u s s i o n has been concerned o n l y with r e s u l t s of the c a l c u l a t i o n s .  Since i t was  our aim  the  to  determine a model which would e x p l a i n the phenomenon of m i c r o p u l s a t i o n s i t i s necessary  to c o n s i d e r the  data which have been obtained.  The  experimental  u s u a l procedure to o b t a i n  -80/ l  4 Va: ' i a b l e chaj -ge d e n s i t y  Exper im< ;ntal valu< !S  © t  >  ©®  \.  \  oc - 4-  Cons';ant  charge i d e n s i t y •  Referent :es Kato & £ l a i t a Q95< 1) Campbell . (1959) Maple (].909 ) Ber thole I (1960) oCc n1Mo i i-3-c & v e x u K a m j> v.J.yoo / Duffus 8 c Shand (1£ >58) 1  i-k  0.1-  Fig. Comparison  12  of experimental and t h e o r e t i c a l  values,  -81-  the dependence of the e i g e n p e r i o d s on the c o - l a t i t u d e i s t o take those e i g e n p e r i o d s which e x h i b i t a maximum of  occurrence a t a c e r t a i n l a t i t u d e .  frequency  Only data which have  been observed s i m u l t a n e o u s l y a t s t a t i o n s l y i n g on the same m e r i d i a n s h o u l d be used.  In s p i t e of the g r e a t l y i n c r e a s e d  e f f o r t s which have been put i n t o r e s e a r c h on m i c r o p u l s a t i o n s , data s a t i s f y i n g these requirements obtain.  are very d i f f i c u l t t o  For t h i s reason i t i s necessary t o r e s o r t t o data  o b t a i n e d under l e s s r i g o r o u s c o n d i t i o n s .  In f i g . 12  e x p e r i m e n t a l v a l u e s of p e r i o d s a r e p l o t t e d a g a i n s t geomagnetic co-latitude.  The tendency  of the e i g e n p e r i o d s t o decrease  i n c r e a s i n g c o - l a t i t u d e i s c l e a r l y apparent.  with  For comparison  some of the c a l c u l a t e d e i g e n p e r i o d s a r e p l o t t e d i n the same diagram.  The agreement between the observed  and c a l c u l a t e d  v a l u e s i s not too s a t i s f a c t o r y e i t h e r f o r the compressed d i p o l e f i e l d w i t h c o n s t a n t charge  d e n s i t y , or f o r the  compressed f i e l d with v a r i a b l e charge  density.  In the f i r s t  case the magnitude as w e l l as the s l o p e do not agree; the disagreement magnitude.  i n the second  case i s o n l y w i t h r e s p e c t t o  Apart from the f a c t t h a t the model may not be  a p p r o p r i a t e the d i s c r e p a n c y might be due t o an i n c o r r e c t charge  d e n s i t y d i s t r i b u t i o n or t o the wrong c h o i c e of oc.  S i n c e the value of CX i s not c r i t i c a l  i n the range o f c o -  l a t i t u d e s under d i s c u s s i o n (see f i g . 10) the major cause f o r the d i s c r e p a n c y has to be sought  i n the charge d e n s i t y  -82-  distribution.  From s i n g l e measurements i n low c o - l a t i t u d e s  which y i e l d e i g e n p e r i o d s of the order of 100 seconds, i t appears almost c e r t a i n from our model that d u r i n g d i s t u r b e d conditions  must be around 4 i n agreement with  other  calculations.  The t h e o r e t i c a l e x p l a n a t i o n of geomagnetic  m i c r o p u l s a t i o n s i n terms of t o r o i d a l o s c i l l a t i o n s of a compressed d i p o l e f i e l d  i s promising  and  mathematically  t-ractabie i f c y l i n d r i c a l c o o r d i n a t e s are used.  Further  s t u d i e s on the s u b j e c t u s i n g the methods d e s c r i b e d i n t h i s t h e s i s are s t r o n g l y recommended.  -83-  -.X-r  APPENDIX I  The f i e l d from  "'"  i n t e n s i t y of a magnetic d i p o l e can be d e r i v e d  the p o t e n t i a l  V = - — — — 2 ; — :  where dipole.  coiJ  (i-i)  i s the c o - l a t i t u d e and M the magnetic moment of the In order to allow f o r the i n f l u e n c e of the s o l a r  wind, one m o d i f i e s e q u a t i o n superimposing  (1-1) a c c o r d i n g to Obayashi  a p o t e n t i a l a s s o c i a t e d with a constant  (I960)  field.  Hence one w r i t e s  (1-2)  where the constant A i s chosen so t h a t the r-component o f the magnetic f i e l d vanishes a t r = R  G  (compressed  field).  Therefore, since  one takes (1-3)  -84-  With t h i s value of A, the r - and  -components of the com-  pressed d i p o l e f i e l d are  and  The l i n e s of f o r c e are o b t a i n e d by i n t e g r a t i n g the d i f f e r e n t i a l  o  equation  dr  _  rdd'  which upon i n t r o d u c i n g the equations  (1-4) and (1-5) becomes  l .e  (1-6)  I n t e g r a t i n g t h i s equation y i e l d s  = 2Ln sir>& f C  -85-  l.e.  <-7R To determine the constant  = C sin d-  C i n equation  (1-7) we  (1-7)  postulate  that the l i n e of f o r c e i n t e r s e c t s the s u r f a c e of the E a r t h (r = a) at  &-  J^, .  r  L  Thus  and equation  (1-7)  becomes  CL  -86-  APPENDIX I I  Letting X = A equation  -1/2  -1/2 Y, where A i s defined l a t e r ,  (83) becomes  BA"  Multiplying  1 / 2  Y =  2 U  AA"  1 / 2  t h i s from the l e f t - h a n d s i d e one o b t a i n s  A  -1/2  -1/2 - / - / 2 Y = u*A AA Y = w Y 1  BA  2  1  2  or  MY = u Y  where one has a b b r e v i a t e d  M = A  -1/2  With  /foi"  0  0 - '/^ -A  A =  Y  0  0  0  0  BA  -1/2  2  -87-  the r e a l symmetric matrix M becomes  -88-  APPENDIX I I I  In order to apply the Newton-Raphson method, equation (87) has t o be w r i t t e n i n the form ij  f CS j = - L . ( | - s ) ( 5 4 - 6 s + S 1  3  2  s  s  +16) -(\Xo\-jh)=  (III-D  0  Taking the d e r i v a t i v e of f ( s ) w i t h r e s p e c t t o s y i e l d s  ^ - f o i I f t h e r e f o r e an approximative  (III  s o l u t i o n s^ of equation  i s known an improved s o l u t i o n s - ^ i s o b t a i n e d i +  av  As may be seen from equation i n the i n t e r v a l  0 < S < I  (III-l)  from  '« ffS  s  (iii  ( I I I - 2 ) f ' ( s ) w i l l never  "  used q u i t e s a f e l y to o b t a i n a s o l u t i o n of any d e s i r e d accuracy.  the "flow c h a r t " looks as f o l l o w s  3)  vanish  and hence the method may be  Leaving out a l l d e t a i l s concerned  -2)  with s c a l i n g , e t c .  -89-  ~l  (/  Coma  fe5gi-S/r/K5s ^)s,-»g3Si.-n^} t  CCT.p.  5Sj+6  I  Accuracy  ~r....  (5S,+<5)S,-  (5S/+0  +S  Ccmp. L ^  (*V-1 <*"»  [(5S -46)S/+8]Sj+l6 (  i/  Ccmp.  ^  / ci-s){r(ss,+^s,-+as,-+i6} 35  A  J  Test  -90-  The necessary s t a r t i n g values s^ were obtained from the graph  in fig.  5.  For speeding up the i n p u t both  l* l 0  " ^  n  and S i were punched out on tape. The  f o l l o w i n g t a b l e c o n t a i n s the s o l u t i o n s of equation  (87) which were r e q u i r e d f o r the computations elements.  The p a i r s of v a l u e s are arranged  of the matrix  i n groups each  c o r r e s p o n d i n g t o a c e r t a i n c o - l a t i t u d e of i n t e r s e c t i o n .  -91-  Table  Ill-a  <% =50  c  =0.342855 =0.228570 =0.114285  s=0.83872 =0.94129 =0.98658  ='0.342839  s=0.83874 =0.94130 =0.98659  0.327425 0.218283 0.109142  s=0.85836 =0.94706 =0.98779  =0.314405 =0.209603 =0.104802  8=0.87306 =0.95163 =0.98877  =0.294476 =0.196317 =0.098158  s=0.89284 =0.95811 =0.99017  =0.266161 =0.177441 =0.088720  =0.91639 =0.96633 =0.99200  =0.228556 =0.152371 =0.076185  s=0.94130 =0.97563 =0.99413  =0.181626 =0.121084 =0.060542  s=0.96460 =0.98489 =0.99631  =0.126380 =0.084254 =0.042127  s=0.98349 =0.99280 =0.99822  X =20  c  ki-*i.  0.220559 0.114280  ki- 3  =25  =60°  c  =0.342754 =0. 228503 =0.114251  s=0.83886 =0.94133 =0.98659  <% =30  c  ki  =65°  <%  0.342447 0.228298 0.114149  s=0.83927 ; =0.94145; =0.98662 ;  |X |0  ^=35°  =70° 0.341575 0.227717 0.113858  J  0  Jl  s=0.84o45 ; =0.94178 ; =0.98669 ;  A,  =40° =0.339506 =0.226337 =0.113169  s=0.84321 ; =0.94258 ; =0.98685 ;  =75°  kl  =80° IXj-jji,  : =0.335244  =0.223496 =0.111748  s=0.84873 J =0.94419 ; =0.98719 ;  lx | 0  -92-  Table  A  =15< s=0.83872 =0.94129 =0.98658 =1.0000  <^=20 s=0.83874 =0.94130 =0.98659 =1.0000  <£=50 H=82259km =92319 =96761 =98077 &  c  <% =25 s=0.83886 =0.94133 =0.98659 =1.0000  Ill-b  H =40736km =46496 =49040 =49793 &  c  H=24486km =28255 =29921 =30414  c  s=0.85836 =0.94706 =0.98779 =1.0000  H=3240km =4233 =4689 =4826  =55° s=0. 87306 =0. 95163 =0. 98877 =1.0000  H=2178km =2948 =3311 =3421  =60° s=0. 89284 =0. 95811 =0. 99017 =1. 0000  H=l451km =2023 =2304 =2390  ^ =65°  <% =30  c  0  s=0.83927 =0.94145 =0.98662 =1.0000  H=15686km =18371 =19558 =19910  s=0. 91639 =0. 96633 =0. 99200 =1. 0000  H=959km =1359 =1564 =1628  -^-35° s=0.84045 =0.94178 =0.98669 =1.0000  H=104l4km =12437 =13334 =13600  <% =70° s=0. 94130 =0. 97563 =0. 99413 =1.0000  H=633km =889 =1026 =1070  H=7039km =8618 =9322 =9531  s=0.,96460 =0,,98489 =0,,99631 =1,,0000  H=4 22km =565 =646 =672  H=4782km =6037 =6602 =6770  <% =80° s=0. 98349 =0. 99280 =0. 99822 =1. 0000  H=292km =355 =392 =404  ^=40° s=0.84321 =0.94258 =0.98685 =1.0000 t%=45 s=0.84873 =0.94419 =0.98719 =1.0000 c  -93-  APPENDIX IV  As can be seen from t a b l e 5 a common f a c t o r i n the a j which i s of course d i f f e r e n t f o r the d i f f e r e n t l i n e s of f o r c e appears q u i t e f r e q u e n t l y .  For t h i s reason i t seems f e a s i b l e o  to i n c o r p o r a t e t h i s f a c t o r i n t o the value of u .  One then has  the f o l l o w i n g m a t r i c e s and t h e i r e i g e n v a l u e s : J;=i5° 1.42460 -0.71230 0 0 0 0 0  -0.71230 1.42460 -0.71230 0 0 0 0  0 -0.71230 1.42460 -0.71230 0 0 0  0 0 -0.71230 1.42460 -0.71230 0 0  Scaling: Eigenvalues <£=20  0 0 0 -0.71230 1.42460 -0.71230 0  10 w =X 6  2  0 0 0 0 -0.71230 1.42460 -0.71230  0 0 0 0 0 -0.71230 1.42460  2  A.^ =0.10844 ; X =0.41725 5 \ =0.87944 2  2  2  2  3  c  1.232286 -0.616143 0 0 0 0 0 0 0 •0.616143 1232286 -0.616143 0 0 0 -0.616143 1232286 -0.616143 0 0 0 -0.616143 1232286 -0.616143 0 0 0 0 0 0 0 -0.616143 1232286 -0.616143 0 -0.616143 1232286 -0.616143 0 0 0 0 0 -0.616143 1232286 0 0 0 0 Scaling: Eigenvalues  2  X! =0.093802  10 u =X 2 ; X =0.360924 4  2  2  2  j X  2 3  =0.760715  -94-  <X=25° 0.657451 -0.340771 0 0 0 0 -0.340771 0.706514 -0.358650 0 0 0 0 0 -0.358650 0.728252 -0.364128 0 0 0 0 0 0 -0.364128 0.728252 -0.364128 0 0 0 0 0 -0.364128 0.728252 -0.358650 0 0 0 0 -0.358650 0.706514 -0.340771 0 0 0 0 0 -0.340771 0.657451 Scaling; Eigenvalues  A.! =0.054579 2  2xl0 w =X 2  2  2  ; A =0.204527 ; A =0.426525 2  2  2  3  <#=30° 0 ,607469 -0.334182 0 O 0 0 0 ,334182 0.735362 0.377756 0 O 0 0 -o -0.377756 0.776217 -0.390280 O O O O 0 0 0 0 -0 390280 0.784929 -0.390280 0 0 0 -0.377756 O -0.390280 0.776217 O 0 0 ,334182 0 -0.377756 0,735362 -0 O 0 O -0.334182 O ,607469 0 O Scaling: Eigenvalues:  2xl0 u =A. 1  2  2  A-x^O. 057250 j A. ^=0. 206443 J A. *=0.422851 2  3  A =35° • 0.601504 -0.316186 O O 0 0 0 -0.316186 0.664827 -0.341521 0 0 0 0 0 -0.341521 0.701754 -0.354844 0 0 O O -0.354844 0.717708 -0.354844 O O O -0.354844 0.701754 -0.341521 0 0 0 0 0 -0.341521 0.664827 -0.316186 0 0 0 0 0 -0.316186 0.601504 0 O O S c a l i n g : 2w =A. 2  Eigenvalues  2  1^ =0.052352 j A =0.191944 » 1 =0.401217 2  2  2  2  3  -95-  <%=40°  1.493206 -0.792820 0 0 0 0 0 -0.792820 1.683785 -0.860385 0 0 0 0 0 -0.860385 1.758551 -0.885214 0 0 0 0 0 -0.885214 1.782372 -0.885214 0 0 0 0 0 -0.885214 1.758551 -0.860385 0 0 0 0 0 -0.860385 1.683785 -0.792820 0 0 0 0 0 -0.792820 1.493206 Scaling: Eigenvalues:  w =A 2  2  A^^O. 131089 j A. =0.482142 ; A. "=l. 00078 2  2  3  <%=45° 0.533177 -0.292869 0 0 0 0 0 -0.292869 0.643480 -0.330400 0 0 0 0 0 -0.330400 0.678587 -0.342421 0 0 O 0 0 -0.342421 0.691157 -0.342421 O 0 0 0 0 -0.342421 0.678587 -0.330400 0 0 0 0 0 -0.330400 0.643480 -0.292869 0 0 0 0 0 -0.292869 0.533177 Scaling: Eigenvalues:  A.^0.050164  lO" ^ ^ 1  2  2  ; A. =0.180788 j 2  2  X "=0.371086 3  £,=50' 1.3281639 -0.774455 0 0 0 0 0 -0.774455 1.805706 -0.946316 0 O 0 0 O -0.946316 1.983733 -1.006113 0 O 0 0 0 -1.006113 2.041129 -1.006113 0 0 0 O 0 -1.006113 1.983733 -0.946316 0 0 0 O -0.946316 1.805706 -0.774455 0 0 O -0.774455 1.328639 O 0 Scaling: Eigenvalues:  X^O.  lO w =X _ 1  2  2  143013 I \ =0.490445 .} A. =0.990559 2  2  2  -96-  X =55  c  Oo333194 -0.245714 0 0 0 0 0 -0.245714 0.724808 -0.394438 0 0 0 0 0 -0.394438 0.858609 -0.437807 0 0 0 0 0 -0.437807 0.892957 -0.437807 0 0 O 0 0 -0.437807 0.858609 -0.394438 O 0 0 0 0 -0.394438 0.724808 -0.245714 0 0 0 0 0 -0.245714 0.333194 Scaling; Eigenvalues:  ^=60  l  =0-055870  X  2xlO~ w =A. 2  2  2  A =0.158036 5 X  =0.303705  2  2  3  c  0.427483 -0.776024 0 0 0 0 0 -0.776024 5.63499 -4.059018 0 0 0 0 0 -4.059018 11.69522 -6 209676 0 0 0 0 0 -6.209676 13 18826 -6 209676 0 0 0 O O -6 209676 11 69522 -4.059018 0 -4 059018 5.63499 -0.776024 0 O 0 0 0 0 0 O -0.776024 0.427483 0 Scaling: Eigenvalues:  t£=65  A.^ =0.18612 2  2xlO w =A -1  2  ; Xg =0.27378 2  2  § A. =0.153292 2  3  c  0.0392650 -0.0585326 O 0 O 0 0 -0.0585326 0.349022 -0.283924 0 0 O 0 O -0.283924 0.923873 -0.528251 0 0 0 0 0 -0.528251 1 208170 -0.528251 O 0 -0.528251 0.923873 -0.283924 6 O O O O -0.283924 0.349022 -0.0585326 0 0 0 -0.0585326 0.0392650 0 O Scaling: Eigenvalues:  X =0.016305 2  x  10~\* =A. 2  2  ; A. =0.024666 2  2  ;  X =0.118095 2  3  -97-  ^=70° 1,40756 -1,49296 0 0 0 0 0  ,49296 ,33412 ,67806 0 0 0 0  0 -4,67806 13,8198 -7,81500 0 0 0  Scaling: Eigenvalues:  0 0 0 -7.81500 13.8198 -4.67806 0  0 0 -7.81500 17.6772 -7.81500 0 0  10 y =A. 1  2  0 0 0 0 -4.67806 6.33412 -1.49296  0 0 0 0 0 •1.49296 1.40756  2  A. =0.4669 ; A. =0.8250 I A. =2.3658 2  2  1  2  2  3  <&=75° 1.63988 -0.991257 0 -0,991257 2.39676 -1.49678 0 -1.49678 3.73895 0 0 -2.03975 0 0 0 0 0 0 0 0 0  O  -2 ,03975 4 ,45107 -2 ,03975 0 0  Scaling: Eigenvalues:  ^=80  0 O 0 -2,,03975 3 ,73895 -1.49678 0  o  t  0 0 0 0 -1.49678 2.39676 -0.991257  O O 0 0 0 -0.991257 1.63988  12 2 10 y =X  A-i =0. 234426 ; A. =0.662810 J X = l . 45850 2  2  2  2  (  0.767430 -0.314805 0 0 O 0 O 0 -0.314805 0 516542 -0.247554 0 0 0 0.474563 -0.236447 0 0 O -O, 247554 0 O O 0 -0 236447 0.471231 -0.236447 0 0 O O -0.236447 0.474563 -0.247554 O O 0 0 -0.247554 0.516542 -0.314805 -0.314805 0.767430 0 0 0 0 0 Scaling: Eigenvalues :  A.^0.037834  w =X 2  2  j \ *=0« 2  159243 j A. =0.352488 2  3  -98-  APPENDIX V  Putting  Y  - Z= X the determinant i n (89) becomes  x 1  1 x  0 1  0 0 = O  O  l  x  l  T h i s determinant i s of the n *  n  order and can be expanded to  o b t a i n a r e c u r s i o n formula, viz„  D  where  D_ = n  0,  D  Q  n  = 1,  =  x D  n-l  = x  -  0  D  n-2  (V-l)  In order t o f i n d a  s o l u t i o n of e q u a t i o n ( V - l ) , one w r i t e s  (V-2)  S u b s t i t u t i o n then y i e l d s , a f t e r c a n c e l l i n g the terms on both s i d e s ,  -99-  n-i  n U  - X U.  n-2  -  (V-3)  U  Therefore f o r n = 2  u  2  = xu - 1  or  x z.  D  X  = u + 1/u  = u (l+u ) 1  From t h i s i t f o l l o w s from equation  D  (V-4)  ( V - l ) , that  = xD -l = u~ (l+u +u ) 2  2  2  2  4  x  T h e r e f o r e by i n d u c t i o n  D  n  = u  -n ,, 2 4 2n (l+u^+u*+ •••• u )  (V-5)  As r e q u i r e d t h i s e x p r e s s i o n w i l l v a n i s h i f the numerator i s  -100-  zero without  the denominator being zero at the same time.  Hence the s o l u t i o n i s  NX.  Substituting  or  I 2; — M  =e  (V-7) i n t o e q u a t i o n  (V-4) one  , (v-7)  obtains  since  X =-  1  As. an i l l u s t r a t i v e  -2  J  2.  jrK  (V-8)  example l e t us take the simple case g i v e n  by  0 i  r  2  Expanding y i e l d s the e q u a t i o n  fjr-0  = 0  -101-  l  which has the s o l u t i o n s hand from equation determinant one  1  f - 3 y  and  =  ' •  ^  n  other  (V-8) w i t h M = 2 f o r the order o f the  obtains  z  ft =  2 + 2 coo "/3 z  I  -102-  vAPPENDIX VI  To e v a l u a t e it  i s necessary  the i n t e g r a l  (110) u s i n g Simpson's r u l e  t o compute values o f the i n t e g r a n d a t  e q u a l l y spaced p o i n t s depending on the l e n g t h of the interval  A &  .  Since the i n t e g r a n d c o n t a i n s powers of  being the s o l u t i o n s of the c u b i c equation e s s e n t i a l to incorporate a subroutine  (61) i t was  i n t o the program  which by means of the Newton-Raphson method s o l v e d equation  (61) whenever necessary.  The program was  w r i t t e n i n such a way that a f t e r i n p u t t i n g C< the values of  ~S) and those C  computed and t h e i r numerical  of the i n t e g r a n d  values p r i n t e d out.  advancing from The  <J^ , A  ( I ) were The  was done a u t o m a t i c a l l y .  r e s u l t s of the computation f o r d i f f e r e n t l i n e s of  f o r c e are given i n t a b l e V l - a .  and  -103-  Table V l - a  0  10 n 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50  r  I 1.0000 1. 2069 1. 4320 1. 6746 1. 9339 2. 2090 2.4983 2. 8004 3. 1133 3. 4347 3. 7620 4. 0925 4. 4233 4. 7515 5, 0744 5.3896 5. 6951 5. 9890 6. 2702 6. 5379 6. 7917 7. 0314 7, 2571 7. 4693 7. 6684 7. 8549 8. 0296 8.1931 8. 3460 8. 4891 8. 6230 8. 7484 8. 8657 8. 9756 9. 0786 9. 1752 9. 2658 9. 3508 9. 4307 9. 5057 9. 5763  5. 7521 11. 0967 20. 1544 34. 7677 57. 3412 90. 8584 138. 8311 205. 1456 293. 7638 408. 3312 551. 6578 725. 2407 928. 8699 1160. 4488 1416. 0477 1690. 3194 1977. 0020 2269. 5168 2561. 5905 2847. 6279 3123. 0187 3384. 2850 3628. 9160 3855. 4458 4063. 1582 4252.0059 4422. 4043 4575. 082 4711. 0649 4831.4375 4937. 3962 5030. 1391 5110.8491 5180. 6476 5240. 5982 5291. 7328 5334. 8917 5371. 0117 5400.8423 5425. 1042 5444. 3859  cx = V  1. 0000 1. 2056 1. 4282 1. 6663 1. 9179 2. 1808 2.4519 2. 7278 3. 0049 3. 2794 3. 5479 3. 8074 4. 0555 4. 2905 4. 5113 4. 7177 4. 9096 5. 0874 5. 2517 5. 4034 5. 5434 5. 6723 5. 7912 5. 9009 6. 0022 6. 0957 6. 1822 6. 2623 6. 3365 6. 4053 6. 4692 6. 5287 6. 5840 6. 6356 6. 6837 6. 7287 6. 7707 6. 8100 6. 8468 6. 8813 6. 9137  5. 7364 10. 9978 19. 7855 33. 6590 54.4304 83. 9858 123. 9768 175. 4592 238. 5493 312. 2418 394. 4669 482. 3869 572.8415 662. 7687 749. 5470 831. 1975 906. 3749 974. 3327 1034. 8203 1087. 9201 1133. 9743 1173. 4612 1206. 9340 1234. 9973 1258.2190 1277. 1676 1292.3617 1304. 2547 1313. 2942 1319. 8595 1324. 2966 1326. 8861 1327. 9041 1327. 5862 1326. 1190 1323. 6923 1320. 4662 1316. 5588 1312. 0941 1307. 1784 1301. 9001  I 1. 0000 1. 1940 1.3946 1. 5964 1. 7939 1. 9822 2. 1578 2.3188 2. 4646 2. 5954 2. 7123 2. 8164 2. 9091 2. 9917 3. 0653 3. 1311 3. 1901 3. 2431 3. 2908 3. 3339 3. 3729 3. 4083 3. 4405 3.4699 3. 4968 3. 5214 3. 5440 3. 5648 3. 5840 3. 6017 3. 6181 3. 6333 3. 6474 3. 6605 3. 6728 3. 6841 3. 6947 3. 7046 3. 7139 3. 7226 3. 7307  5. 5843 10. 1121 16. 7721 25. 6174 36. 2638 47. 9687 59. 8586 71. 1593 81. 3427 90. 1328 97. 4616 103. 3907 108. 0535 111. 6149 114. 2377 116. 0797 117. 2750 117. 9420 118. 1819 118. 0798 117. 7023 117. 1082 116. 3423 115. 4458 114. 4475 113. 3745 112. 2483 111. 0844 109. 8966 108. 6974 107.4964 106. 2999 105. 1135 103. 9431 102. 7932 101. 6644 100. 5600 99. 4820 98. 4321 97. 4111 96. 4195  -104It  51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90  9.6426 9.7051 9.7639 9.8192 9.8714 9.9205 9.9668 10.0105 10.0517 10.0905 10.1270 10.1615 10.1940 10.2246 10.2534 10.2805 10.3059 10.3299 10.3523 10.3734 10.3931 10.4115 10.4286 10.4445 10.4593 10.4730 10.4856 10.4971 10.5076 10.5171 10.5256 10. 5332 10.5398 10.5456 10.5504 10.5543 10.5574 10.5596 10.5609 10.5613  I  I  5459. 3161 5470.4115 5478. 1292 5482. 8620 5485. 0732 5485. 0166 5483. 0422 5479. 4430 5474. 4204 5468. 2313 5461. 0382 5453. 0566 5444. 4419 5435.3191 5425. 8129 5416.0598 5406. 1360 5396. 1925 5386. 2273 5376. 3632 5366. 6736 5357. 1850 5347. 9847 5339.0997 5330. 5594 5322. 4454 5314. 7502 5307. 5201 5300. 8007 5294. 5858 5288. 8664 5283. 7448 5279. 1951 5275. 2122 5271. 8284 5269. 0563 5266. 9094 5265. 3508 5264.4147 5264. 1159  6. 9440 6. 9726 6. 9994 7. 0246 7. 0483 7. 0706 7. 0916 7. 1114 7. 1300 7. 1476 7. 1641 7. 1797 7. 1944 7. 2081 7. 2211 7. 2333 7. 2448 7. 2556 7. 2657 7. 2751 7. 2840 7. 2922 7. 2999 7. 3071 7. 3137 7. 3199 7. 3255 7. 3307 7. 3354 7. 3396 7. 3435 7. 3469 7. 3499 7. 3524 7. 3546 7. 3564 7.3577 7. 3587 7.3593 7. 3595  1296. 3296 1290. 5468 1284. 6012 1278. 5458 1272. 4262 1266. 2790 1260.1454 1254. 0516 1248. 0180 1242. 0700 1236. 2260 1230. 5043 1224. 9273 1219. 4890 1214. 2064 1209.0962 1204. 1642 1199.4082 1194. 8457 1190.4759 1186. 3048 1182. 3376 1178.5742 1175. 0145 1171. 6661 1168.5232 1165. 5986 1162. 8823 1160. 3860 1158. 1025 1156. 0380 1154. 1899 1152. 5622 1151. 1458 1149. 9548 1148. 9728 1148. 2073 1147. 6542 1147. 3316 1147. 2296  \ 3 .7383 3 .7454 3 .7522 3 .7585 3 .7644 3 .7700 3 .7752 3 .7801 3 .7848 3 .7891 3 .7932 3 .7971 3 .8008 3 .8042 3 . 8074 3 .8104 3 .8133 3 .8160 3 .8185 3 . 8208 3 .8230 3 .8251 3 .8270 3 .8288 3 .8305 3 .8320 3 .8334 3 .8347 3 .8358 3 .8369 3 .8378 3 .8387 3 .8394 3 .8400 3 .8406 3 .8410 3 .8414 3 .8416 3 .8417 3 .8418  T  95. 4581 94. 5266 93. 6262 92. 7557 91. 9152 91. 1055 90.3244 89. 5726 88. 8499 88.1546 87. 4896 86. 8504 86. 2390 85. 6543 85. 0962 84. 5626 84. 0552 83. 5731 83. 1147 82.6811 82. 2706 81. 8831 81. 5194 81. 1787 80. 8594 80. 5616 80. 2861 80.0317 79. 7984 79. 5864 79. 3956 79. 2243 79. 0732 78. 9437 78. 8346 78. 7452 78. 6769 78. 6268 78. 5960 78. 5874  -105-  cx--IZ  20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62  1. 0000 1. 0977 1. 1991 1.3042 1.4128 1. 5246 1. 6396 1. 7574 1. 8780 2. 0011 2. 1265 2. 2539 2.3831 2. 5139 2. 6460 2. 7791 2. 9129 3. 0472 3. 1817 3. 3161 3. 4501 3. 5835 3. 7160 3. 8473 3. 9772 4. 1055 4. 2319 4, 3562 4.4783 4. 5979 4. 7150 4. 8293 4. 9407 5.0492 5. 1547 5. 2570 5.3561 5.4521 5. 5448 5.6342 5. 7203 5. 8031 5. 8827  2.9204 4.0450 5.5086 7.3859 9.7617 12.7318 16.4003 20.8807 26.2951 32.7722 40.4463 49.4534 59.9296 72.0095 85.8184 101.4746 119.0812 138.7241 160.4720 184.3681 210.4301 238.6510 268.9950 301.3944 335.7636 371.9892 409.9184 449.4057 490.2703 532.3201 575.3559 619.1740 663.5670 708.3264 753.2507 798.1510 842.8403 887.1447 930.9038 973.9618 1016.2009 1057.4907 1097.7198  1. 0000 1. 0972 1. 1980 1.3020 1.4093 1. 5193 1. 6320 1. 7470 1. 8640 1. 9826 2. 1025 2. 2233 2.3447 2. 4663 2. 5877 2. 7086 2. 8285 2. 9471 3. 0642 3. 1793 3. 2923 3. 4029 3. 5109 3. 6160 3. 7182 3. 8174 3. 9133 4. 0061 4. 0956 4. 1819 4. 2648 4. 3445 4. 4211 4. 4944 4. 5647 4. 6319 4. 6962 4. 7575 4. 8161 4. 8719 4. 9251 4. 9757 5. 0238  2. 9124 4. 0232 5.4610 7. 2929 9. 5926 12.4385 15. 9127 20.0986 25. 0771 30. 9248 37. 7099 45. 4891 54.3014 64. 1723 75. 1025 87. 0751 100.0473 113. 9578 128. 7237 144. 2493 160. 4232 177. 1278 194. 2317 211.6105 229. 1409 246. 7028 264. 1847 281. 4834 298.5103 315.1851 331. 4349 347. 2100 362. 4641 377. 1615 391. 2730 404. 7890 417. 6928 429. 9821 441. 6657 452. 7456 463. 2306 473. 1370 482.4840  1.0000 1.0926 1. 1868 1. 2822 1.3780 1. 4735 1. 5683 1.6616 1. 7530 1. 8419 1. 9280 2.0109 2.0905 2. 1667 2. 2392 2.3081 2.3735 2.4354 2. 4939 2. 5492 2. 6013 2. 6504 2. 6968 2. 7404 2. 7815 2. 8203 2. 8567 2. 8911 2. 9235 2. 9541 2. 9828 3. 0100 3. 0356 3. 0598 3. 0826 3. 1041 3. 1244 3. 1435 3. 1616 3. 1787 3. 1948 3. 2100 3. 2243  2.8352 3.8205 5.0333 6.4891 8.1941 10.1416 12.3148 14.6852 17.2151 19.8616 22.5773 25.3164 28.0383 30.7033 33.2804 35.7456 38.0811 40.2750 42.3198 44.2124 45.9560 47.5514 49.0067 50.3273 51.5210 52.5973 53.5628 54.4275 55.1980 55.8837 56.4899 57.0263 57.4972 57.9098 58.2690 58.5812 58.8502 59.0801 59.2761 59.4415 59.5786 59.6919 59.7828  -106cx.=lj  C*=IZ  0  /&  ?c  63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90  5. 9590 6. 0322 6. 1021 6. 1688 6. 2323 6. 2928 6. 3503 6. 4047 6. 4561 6. 5046 6. 5502 6. 5929 6. 6329 6. 6700 6. 7045 6. 7362 6. 7652 6. 7917 6. 8155 6. 8367 6, 8554 6. 8716 6, 8852 6. 8963 6. 9050 6. 9111 6..9148 6. 9160  I 1136. 8051 1174. 6727 1211. 2482 1246.4633 1280.2898 1312.6881 1343. 6226 1373. 0877 1401. 0521 1427. 5197 1452.4922 1475. 9490 1497. 9242 1518. 3933 1537. 3851 1554. 9046 1570. 9725 1585. 5905 1598. 7578 1610.5143 1620.8639 1629. 8008 1637. 3445 1643. 5011 1648. 2942 1651. 7061 1653. 7504 1654. 4251  Vc  I  ^  I  5.0696 5. 1130 5. 1541 5. 1931 5. 2299 5. 2648 5. 2976 5.3285 5.3576 5.3848 5.4103 5.4341 5.4562 5.4767 5.4957 5. 5131 5. 5289 5. 5434 5. 5563 5. 5678 5. 5779 5. 5867 5. 5940 5.6000 5. 6046 5.6080 5.6100 5.6106  491. 2790 499. 5536 507. 3152 514. 5930 521.4001 527. 7643 533. 6952 539. 2252 544. 3609 549. 1263 553. 5369 557. 6081 561. 3558 564. 8000 567. 9465 570. 8075 573. 4011 575. 7346 577. 8140 579. 6540 581. 2599 582. 6381 583. 7936 584. 7314 585. 4532 585. 9724 586. 2844 586. 3856  3 .2378 3 . 2505 3 .2626 3 .2739 3 .2845 3 .2945 3 .3039 3 .3127 3 .3209 3.3286 3 .3358 3 .3425 3 .3486 3 .3544 3 .3596 3 .3645 3 .3689 3 .3729 3 .3765 3 .3796 3 .3824 3 .3848 3 .3868 3 .3885 3 .3898 3 .3907 3 .3912 3 .3914  59. 8553 59. 9107 59. 9517 59. 9798 59. 9968 60.0036 60.0037 59. 9958 59. 9830 59. 9651 59. 9446 59. 9202 59. 8946 59. 8675 59. 8401 59. 8121 59. 7853 59. 7598 59. 7353 59. 7113 59. 6904 59. 6713 59. 6554 59. 6418 59. 6305 59. 6231 59. 6192 59. 6165  o  30 31 32 33 34 35 36 37 38 39  1. 0000 1. 0609 1. 1230 1. 1861 1. 2501 1.3150 1.3807 1.4470 1. 5140 1. 5815  I  \  I  1. 9977 2.4565 2. 9964 3. 6267 4. 3575 5. 1995 6. 1632 7. 2599 8. 4998 9. 8942  1. 0000 1.0607 1. 1224 1. 1850 1. 2485 1.3127 1. 3776 1. 4431 1. 5089 1. 5752  1. 9922 2.4460 2. 9780 3. 5970 4. 3119 5. 1317 6. 0655 7. 1219 8. 3091 9. 6350  I  1 .0000 1 .0580 1 .1163 1 .1748 1 .2334 1 .2918 1 .3499 1 .4076 1 .4646 1 .5209  1. 9394 2. 3455 2. 8081 3.3289 3. 9091 4. 5487 5. 2463 6. 0000 6. 8058 7. 6592  -107-  40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90  1.6494 1.7176 1.7861 1. 8547 1 . 9234 1.9920 2.0605 2.1288 2. 1968 2. 2644 2.3314 2.3979 2.4638 2. 5289 2. 5931 2.6565 2.7188 2.7801 2. 8402 2. 8992 2. 9568 3.0131 3. 0680 3. 1215 3. 1734 3. 2237 3. 2724 3.3195 3.3648 3.4084 3.4501 3.4901 3. 5281 3. 5643 3. 5985 3.6308 3.6611 3.6893 3. 7156 3. 7398 3. 7620 3. 7821 3. 8001 3. 8160 3. 8299 3. 8416 3. 8512 3. 8586 3. 8640 3. 8672 3. 8682  I  v.  L  11.4539 13.1889 15.1088 17.2228 19.5396 22.0658 24.8082 27.7708 30.9580 34.3719 38.0116 41.8763 45.9640 50.2692 54.7844 59.5015 64.4101 69.4985 74.7536 80.1584 85.6966 91.3521 97.1022 102.9280 108.8072 114.7186 120.6401 126.5468 132.4143 138.2215 143.9427 149.5540 155.0343 160.3621 165.5119 170.4646 175.1987 179.6961 183.9359 187.9045 191.5831 194.9566 198.0139 200.7384 203.1251 205.1584 206.8364 208.1467 209.0845 209.6508 209.8398  1.6416 1.7082 1.7749 1.8414 1.9077 1.9738 2.0395 2.1047 2. 1693 2. 2331 2. 2963 2.3585 2.4198 2.4801 2. 5393 2. 5973 2.6540 2.7095 2.7635 2. 8162 2. 8674 2.9172 2.9654 3.0120 3.0570 3. 1004 3. 1421 3. 1822 3. 2206 3. 2573 3. 2923 3.3256 3.3572 3.3871 3.4152 3.4416 3.4663 3.4893 3. 5105 3. 5301 3. 5479 3. 5640 3. 5784 3. 5911 3.6021 3.6115 3.6191 3.6250 3.6292 3.6317 3.6326  11. 1069 12. 7312 14. 5129 16.4567 18.5646 20. 8393 23. 2798 25. 8859 28. 6539 31. 5783 34. 6548 37. 8746 41. 2297 44. 7085 48. 3001 51. 9930 55. 7729 59. 6266 63. 5394 67. 4952 71. 4807 75.4805 79.4786 83. 4606 87. 4116 91. 3192 95. 1683 98. 9453 102.6409 106. 2417 109. 7360 113. 1146 116. 3692 119. 4890 122.4669 125. 2953 127. 9699 130. 4802 132. 8249 134. 9987 136. 9951 138. 8117 140.4472 141. 8940 143. 1531 144. 2227 145. 1003 145. 7821 146. 2713 146. 5650 146. 6644  c  • r. 1.5762 1.6305 1.6837 1 . 7356 1.7862 1. 8354 1 . 8832 1.9295 1.9743 2.0175 2.0592 2.0994 2. 1380 2.1751 2. 2106 2. 2448 2. 2774 2.3086 2.3385 2.3670 2.3941 2.4200 2.4447 2.4681 2.4904 2. 5115 2. 5316 2. 5505 2. 5684 2. 5854 2.6013 2.6163 2.6303 2.6435 2.6558 2.6672 2.6778 2.6876 2.6966 2.7048 2.7123 2.7190 2.7250 2.7302 2.7347 2.7385 2.7416 2.7440 2.7458 2.7468 2.7471  8.5557 9.4883 10.4512 11.4371 12.4394 13.4511 14.4660 15.4779 16.4808 17,4700 18.4403 19.3887 20.3110 21.2056 22.0693 22.9011 23.6992 24.4638 25.1939 25.8894 26.5508 27.1784 27.7729 28.3345 28.8644 29.3637 29.8335 30.2741 30.6875 31.0744 31.4352 31.7716 32.0849 32.3749 32.6440 32.8917 33.1198 33.3289 33.5191 33.6921 33.8481 33.9872 34.1102 34.2177 34.3099 34.3876 34.4509 34.4997 34.5345 34.5556 34.5617  -108-  0  40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82  1. 0000 1.0416 1.0835 1. 1255 1. 1675 1. 2096 1. 2517 1. 2937 1. 3356 1. 3773 1.4188 1.4600 1. 5008 1. 5413 1. 5814 1. 6210 1. 6600 1, 6985 1. 7364 1. 7736 1. 8100 1. 8457 1. 8807 1. 9147 1. 9479 1. 9802 2.0115 2. 0419 2.0712 2. 0994 2. 1265 2. 1525 2. 1774 2. 2010 2. 2234 2. 2447 2. 2646 2. 2833 2.3007 2. 3167 2. 3314 2. 3448 2. 3568  1.5539 1.7920 2.0565 2.3486 2.6697 3.0214 3.4044 3.8203 4.2696 4.7532 5.2716 5.8250 6.4137 7.0369 7.6951 8.3869 9.1117 9.8677 10.6543 11.4688 12.3094 13.1741 14.0601 14.9641 15.8832 16.8149 17.7544 18.6987 19.6443 20.5861 21.5210 22.4448 23.3534 24.2414 25.1060 25.9430 26.7471 27.5157 28.2440 28.9290 29.5678 30.1551 30.6898  I  I  I  Sf  1. 0000 1. 0415 1. 0831 1. 1248 1. 1666 1. 2083 1. 2500 1. 2916 1. 3331 1. 3743 1.4152 1.4558 1. 4960 1. 5358 1. 5751 1. 6139 1. 6521 1. 6896 1. 7265 1. 7627 1. 7981 1. 8327 1. 8665 1. 8994 1. 9314 1. 9625 1. 9926 2. 0216 2.0497 2. 0766 2. 1025 2. 1272 2. 1508 2. 1732 2. 1945 2. 2146 2. 2334 2. 2510 2. 2673 2. 2824 2. 2963 2.3088 2. 3200  1. 5497 1. 7853 2.0463 2.3340 2.6496 2. 9941 3. 3687 3. 7739 4. 2103 4. 6784 5.1787 5. 7105 6. 2741 6. 8685 7. 4934 8. 1475 8. 8295 9. 5376 10. 2708 11. 0267 11. 8027 12. 5966 13. 4058 14. 2277 15. 0590 15. 8967 16. 7375 17. 5782 18.4156 19. 2453 20. 0650 20. 8709 21. 6594 22.4268 23. 1704 23. 8873 24. 5732 25.2262 25. 8423 26. 4199 26. 9567 27.4491 27. 8960  1. 0000 1. 0397 1. 0792 1. 1186 1. 1577 1. 1964 1. 2348 1. 2727 1. 3101 1.3470 1.3832 1.4187 1. 4535 1. 4875 1. 5208 1. 5532 1. 5848 1.6154 1. 6452 1.6741 1. 7020 1. 7289 1. 7549 1. 7800 1. 8040 1. 8271 1. 8492 1. 8703 1. 8905 1. 9097 1. 9280 1. 9453 1. 9616 1. 9770 1. 9915 2.0051 2. 0177 2. 0294 2.0403 2.0502 2. 0592 2. 0674 2. 0747  1. 5086 1. 7205 1. 9507 2. 1993 2.4660 2. 7506 3. 0524 3. 3706 3. 7042 4. 0520 4. 4130 4. 7854 5. 1679 5. 5592 5. 9571 6. 3600 6. 7668 7. 1751 7. 5838 7. 9912 8. 3956 8. 7956 9. 1902 9. 5777 9. 9573 10.3278 10.6881 11. 0374 11. 3751 11. 7007 12. 0131 12.3123 12. 5973 12. 8682 13. 1245 13. 3659 13. 5922 13. 8031 13. 9992 14. 1791 14.3441 14.4933 14. 6267  5" 83 84 85 86 87 88 89 90  50 51 52 53 54 55 56 57 58 59 60 61 62 63 , 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79  -109-  a = i2  0  I 2.3674 2. 3766 2. 3844 2. 3908 2.3958 2.3994 2.4015 2. 4022  31.1678 31.5875 31.9463 32.2429 32.4751 32.6419 32.7420 32.7748  V,  r  1 .0000 1 .0291 1 .0581 1 .0867 1 .1151 1 .1431 1 .1708 1 .1981 1 .2250 1 .2514 1 .2773 1 .3027 1 .3275 1 .3517 1 .3754 1 .3983 1 .4206 1 .4423 1 .4632 1 .4833 1 .5027 1 .5212 1 .5390 1 .5559 1 .5720 1 .5872 1 .6015 1 .6149 1 .6273 1 .6388  1.3039 1.4416 1. 5882 1. 7437 1. 9080 2. 0810 2. 2624 2.4520 2.6495 2. 8542 3. 0658 3. 2839 3. 5075 3. 7361 3. 9690 4. 2052 4. 4438 4. 6842 4. 9250 5. 1655 5.4046 5.6409 5. 8738 6. 1019 6. 3242 6. 5395 6. 7470 6. 9453 7. 1336 7. 3106  0^=4  i 2.3300 2.3386 2.3459 2.3519 2.3566 2.3599 2.3619 2.3625  I  28. 2947 28. 6439 28. 9419 29. 1873 29. 3799 29. 5176 29. 6006 29. 6277  2. 0811 2. 0866 2.0913 2.0951 2. 0981 2. 1002 2. 1015 2. 1019  o\=4  I 1 .0000 1 .0290 1 .0578 1 .0863 1 .1145 1 .1424 1 .1699 1 .1969 1 . 2236 1 .2497 1 .2754 1 .3005 1 .3251 1 .3490 1 .3723 1 .3950 1 .4170 1 .4383 1 .4589 1 .4787 1 .4978 1 .5161 1 .5335 1 .5501 1 .5659 1 . 5808 1 .5948 1 .6079 1 .6201 1 .6313  1.3003 1.4366 1. 5815 1. 7349 1. 8968 2. 0670 2. 2452 2.4310 2. 6242 2. 8242 3. 0306 3. 2427 3. 4599 3. 6816 3. 9068 4. 1350 4. 3651 4. 5962 4. 8275 5. 0578 5. 2865 5. 5122 5. 7340 5.9508 6. 1619 6. 3661 6. 5621 6. 7495 6. 9272 7. 0942  14.7443 14.8462 14.9326 15.0032 15.0582 15.0974 15.1208 15.1287  1.0000 1.0278 1.0552 1.0823 1.1089 1.1351 1.1607 1.1859 1.2105 1.2345 1.2580 1.2808 1.3029 1.3244 1.3453 1.3654 1.3848 1.4035 1.4214 1.4386 1.4550 1.4707 1.4856 1.4997 1.5130 1.5256 1.5373 1.5482 1.5583 1.5677  I 1. 2659 1.3888 1. 5177 1. 6524 1. 7926 1. 9377 2.0875 2. 2413 2. 3988 2. 5593 2. 7223 2. 8870 3. 0530 3. 2196 3. 3862 3. 5522 3. 7169 3. 8797 4. 0401 4. 1974 4. 3511 4. 5007 4. 6455 4. 7853 4. 9194 5.0474 5. 1691 5. 2838 5.3915 5.4916  -110-  <*=IZ  0  I  80 81 82 83 84 85 86 87 88 89 90  1 .6494 1 .6590 1 .6676 1 .6752 1 .6818 1 .6874 1 .6920 1 .6956 1 .6982 1 .6997 1 .7002  7. 4760 7. 6282 7. 7668 7. 8912 8.0000 8. 0933 8. 1705 8. 2309 8. 2740 8.3001 8. 3086  1.6416 1.6510 1.6594 1.6669 1.6733 1.6788 1.6833 1.6868 1.6893 1.6908 1.6913  arty  I  v .  7. 2495 7. 3926 7. 5227 7.6393 7. 7413 7. 8285 7. 9008 7. 9573 7. 9979 8.0222 8. 0301  1 .5762 1 .5839 1 .5908 1 .5969 1 .6022 1 .6066 1 .6103 1 .6131 1 .6152 1 .6164 1 .6168  f  5.5841 5.6684 5.7445 5.8120 5.8710 5.9212 5.9625 5.9945 6.0175 6.0313 6.0361  I 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86  1. 0000 1.0199 1. 0394 1.0584 1.0770 1.0950 1. 1125 1. 1295 1. 1459 1. 1617 1. 1769 1. 1915 1. 2055 1. 2188 1. 2314 1. 2434 1. 2546 1. 2651 1. 2749 1. 2840 1. 2923 1. 2998 1.3066 1.3126 1.3178 1.3222 1.3258  1. 1534 1. 2356 1.3201 1.4065 1.4946 1. 5839 1. 6742 1. 7651 1. 8564 1. 9475 2. 0380 2. 1276 2. 2160 2.3026 2.3869 2.4687 2. 5474 2.6228 2.6944 2. 7617 2. 8247 2. 8826 2. 9352 2. 9825 3. 0242 3. 0596 3. 0889  1.0000 1.0198 1.0392 1. 0582 1.0766 1.0945 1. 1119 1. 1288 1. 1451 1. 1608 1. 1759 1. 1904 1. 2043 1. 2174 1. 2300 1. 2418 1. 2529 1. 2634 1. 2730 1. 2820 1. 2902 1. 2977 1.3044 1.3103 1.3155 1.3198 1.3234  1.1502 1.2316 1.3151 1.4005 1.4874 1.5754 1.6644 1.7540 1.8437 1.9332 2.0220 2.1099 2.1964 2.2810 2.3636 2.4435 2.5204 2.5939 2.6636 2.7294 2.7904 2.8468 2.8981 2.9441 2.9845 3.0189 3.0475  1.0000 1.0190 1. 0375 1. 0556 1.0731 1.0901 1. 1065 1. 1223 1. 1376 1. 1523 1. c-664 1. 1799 1. 1927 1. 2049 1. 2164 1. 2273 1. 2375 1. 2470 1. 2558 1. 2640 1. 2715 1. 2782 1. 2843 1. 2897 1, 2943 1. 2982 1.3015  1.1197 1.1933 1.2681 1.3440 1.4203 1.4972 1.5742 1.6508 1.7270 1.8023 1.8766 1.9494 2.0204 2.0895 2.1561 2.2203 2.2816 2.3398 2.3947 2.4460 2.4935 2.5372 2.5768 2.6120 2.6428 2.6690 2.6906  -111OUIZ  0  CX--4  <x = g  jd_  }±  E  v±  I  v±  1  87 88 89 90  1.3286 1.3306 1.3319 1.3323  3 .1119 3 .1284 3 .1384 3 .1417  1.3262 1.3282 1.3294 1.3298  3 .0697 3 .0857 3 .0953 3 .0985  1.3040 1.3058 1.3069 1.3072  2.7076 2.7198 2.7270 2.7295  & 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90  CA=IZ  1.0000 , 1.0124 1.0243 1.0356 1.0463 1.0565 1.0661 1.0750 1.0833 1.0910 1.0981 1.1045 1.1103 1.1154 1.1198 1.1236 1.1267 1.1291 1.1308 1.1318 1.1322  I 1 .0629 1 .1098 1 .1559 1 .2012 1 .2452 1 .2881 1 .3292 1 .3686 1 .4061 1 .4414 1 .4742 1 .5046 1 . 5322 1 .5569 1 .5786 1 .5973 1 .6126 1 .6246 1 .6332 1 .6385 1 .6402  r 1.0000 1.0124 1.0242 1.0354 1.0461 1.0562 1.0658 1.0747 1.0829 1.0906 1.0976 1.1040 1.1097 1.1148 1.1192 1.1230 1.1260 1.1284 1.1301 1.1311 1.1315  1 .0600 1 .1065 1 .1521 1 .1968 1 .2404 1 .2827 1 .3234 1 .3622 1 .3991 1 .4339 1 .4664 1 .4963 1 .5234 1 .5478 1 .5691 1 .5875 1 .6026 1 .6144 1 .6228 1 .6279 1 .6297  L 1.0000 1.0119 1.0232 1.0339 1.0441 1.0538 1.0628 1.0713 1.0791 1.0864 1.0930 1.0990 1.1044 1.1092 1.1133 1.1169 1.1197 1.1220 1.1236 1.1245 1.1249  1.0319 1.0740 1.1150 1.1551 1.1940 1.2314 1.2674 1.3016 1.3339 1.3642 1.3924 1.4182 1.4417 1.4626 1.4810 1.4967 1.5096 1.5198 1.5270 1.5314 1.5328  -112-  a 80 81 82 83 84 85 86 87 88 89 90  T  I  1 .0000 1 .0058 1 .0111 1 .0157 1 .0198 1 .0232 1 .0260 1 .0282 1 .0298 1 .0307 1 .0310  1.0143 1.0351 1.0541 1.0712 1.0861 1.0989 1.1095 1.1178 1.1238 1.1273 1.1285  1 .0000 1 .0058 1 .0111 1 .0157 1 .0197 1 .0231 1 .0259 1 .0281 1 .0296 1 .0306 1 .0309  1 .0115 1 .0322 1 .0510 1 .0679 1 .0826 1 .0953 1 .1057 1 .1140 1 .1199 1 .1234 1 .1245  I  1 .0000 1 .0056 1 .0106 1 .0150 1 .0189 1 .0222 1 .0248 1 .0269 1 .0284 1 .0293 1 .0296  .9847 1 .0034 1 .0204 1 .0357 1 .0490 1 .0604 1 .0698 1 .0771 1 .0824 1 .0856 1 .0866  -113-  APPENDIX V I I  h  =10°  0 0 0 0.238217 -0.147909 0 0 0 0 0 -0.147909 0.367344 -0.205472 0 0 O O 0 -0.205472 0.459717 -0.254037 0 O 0 O -0.254037 0.561512 -0, 254037 0 0 0 0 -0.254037 0.459717 -0.205472 -0.205472 0.367344 -0 ,147909 0 0 0 0 0 -0.147909 0 ,238217 0 0 0 0 Scaling. Eigenvalues;  \  lo\i =X 2  2  A.^0.031931 ) A. =0.096597 2  2  ; \g =0.201148 2  = o° 2  O O 0.057226 -0.121296 0 0 0 0 O 0 0 -0.121296 1.028404 -0.616504 0 0 0 -0.616504 1.478328 -0.894182 O 0 0 -0.894182 2.163406 -0.894182 O 0 O O 0 -6.894182 1.478328 -0.616504 O 0 0 0 -0.616504 1.028404 -0 121296 0 -0.121296 0 057226 0 0 0 0 O Scaling. Eigenvalues;  lo\i ^\ 2  2  A-i =0.025495 I X =0.037008 § Xo =0.225847 2  2  2  2  -114-  ^  =30  c  0.01250 -0.09349 0 0 0 0 0  -0.09349 2.79640 -2.05449 0 O 0 0  0 -2.05449 6.03768 -4.00495 0 0 0  0 0 -4.00495 10.62632 -4.00495 0 0  Scaling: Eigenvalues:  1^=0.00620  0 0 0 -4.00495 6.03768 -2.05449 0  l0 u =A. 1  2  0 0 0 0 -2.05449 2.79640 -0.09349  0 0 0 0 0 -0.09349 0.01250  2  2=, J A. ^=0.00830 ; A, *=0.74523 3  =40° 0.01318 -0.08326 0 0 0 0 0  -0.08326 2.10324 -4.22665 O 0 0 0  0 0 0 0 -4.22665 0 O O 33.97541 -23.75178 0 0 -23.75178 66.41804 -23.75178 O 0 -23.75178 33.97541 -4.22665 0 0 -4.22665 2.10324 0 O O -0.08326 Scaling:  Eigenvalues:  &  A.^0.00655  lohi **\ 2  0 0 0 O 0 -0.08326 0.01318  2  ; X =0.00877 ; X =0.97673 2  2  2  3  =50°  0.00321 -0.01016 0 0 0 0 O  -0.01016 0.12845 -0.69173 O 0 0 0  0 0 0 0 -0.69173 0 0 0 14.90035 -11.46859 0 0 -11.46859 35.30886 -11.46859 O 0 -11.46859 14.90035 -0.69173 0 0 -0.69173 0.12845 0 O 0 -0.01016 Scaling:  Eigenvalues:  0 0 0 O 0 -0.01016 0.00321  w =A. z  X =0.00157 j 1 =0.00210 j A. =0.06522 2  1  2  2  2  3  -115-  =60° 0.01909 -0.02680 0 0 0 0 0  -0.02680 0.15046 -0.35889 0 0 0 0  0 -0.35889 3.42428 -5.32325 0 0 0  0 0 -5.32325 33.10140 5.32325 0 0  Scaling: Eigenvalues:  £  0  X  =0.00829  ±  0 0 0 -5.32325 3.42428 -0.35889 0  u =A. 2  ; \  0 0 0 0 -0.35889 0.15046 -0.02680  0 0 0 0 0 -0.02680 0.01909  2  =0.01197 ; A. ^=0.08253  2  3  =70°  O 0 O 0.266399 -0.082840 O O 0 0 O -0.082840 0.103041 -0.118609 0 O 0 0 -0.118609 0.546117 -0.305288 0 0 0 -0.305288 0.682647 -0.305288 0 0 O -0.305288 0.546117 -0.118609 0 0 0 O -0.118609 0.103041 -0.082840 0 O O -0.082840 0.266399 Scaling: Eigenvalues:  <J =\ 2  2  X =0.018424 ; \^=0.044137 2  x  \  A =0.211214 2  3  (  -116-  REFERENCES  D e s s l e r , A . J . (1958) The p r o p a g a t i o n v e l o c i t y of world-wide sudden commencements of magnetic storms. J . Geophys. Res. 63, 405-408. Dungey, J.W. (1954a) The p r o p a g a t i o n of A l f v e n waves through the ionosphere. Penn. S t a t e Univ. Ionos. Res. Lab. S c i . Rep. No. 57. Dungey, J.W. (1954b) E l e c t r o d y n a m i c s of the outer atmosphere. Penn. S t a t e Univ. Ionos. Res. Lab. S c i . Rep. No. 69. Kato, Y. and Watanabe, T. of g i a n t p u l s a t i o n s . Geophys. 8, 19-23.  (1956) F u r t h e r study on the cause S c i . Rep. Tohoku Univ. Ser. 5,  Lundquist, S. (1952) S t u d i e s i n magneto-hydrodynamics, A r k i v f o r F y s i k , Bd. 5, nr 15, p.297. Obayashi, T. and Hakura, Y. (I960) Enhanced i o n i s a t i o n i n the p o l a r ionosphere caused by s o l a r c o r p u s c u l a r e m i s s i o n . Rep. of Ionos. and Space Res. i n Japan, V o l . XIV, No. 1. Obayashi, T. and Jacobs, J.A. (1958) Geomagn. p u l s a t i o n s and the E a r t h ' s outer atmosphere. Geophys. J o u r n a l of the Royal A s t r o n . Soc. V o l . 1, No. 1 (1958). Parker, E.N. (1958) I n t e r a c t i o n of the s o l a r wind w i t h the geomagn. f i e l d , P h y s i c s of F l u i d s 1, 171-187. Plumpton, C. and F e r r a r o , V.C.A. (1953) On the magn. o s c i l l a t i o n s of a g r a v i t a t i n g l i q u i d s t a r . Mon. Roy. A s t r . Soc. 113, 647-652.  Not.  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0085477/manifest

Comment

Related Items