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On Schwarzschild periodic solutions in the restricted three body problem Olund, Brian Russel 1967

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ON SCHWARZSCHILD PERIODIC SOLUTIONS IN THE RESTRICTED THREE BODY PROBLEM by . BRIAN RUSSEL OLUND . B.Sc. U n i v e r s i t y o f B r i t i s h Columbia  A THESIS SUBMITTED I N PARTIAL FULFILMENT OP THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS  i n t h e Department of Mathematics  We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e required standard  THE UNIVERSITY OF BRITISH .COLUMBIA March, 1 9 6 7  In p r e s e n t i n g f o r an that  this thesis  i n p a r t i a l f u l f i l m e n t o f the  requirements  advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree  che L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e  study„  I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g 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  be  Department or by h i s r e p r e s e n t a t i v e s .  g r a n t e d by the Head o f  w i t h o u t my  written  Department  of  permission.  The U n i v e r s i t y o f B r i t i s h Vancouver 8 Canada  Columbia  S  __^j^4z£^L^_j£  my  I t i s understood that  or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not  Date  and  be  copying allowed  ABSTRACT Consider  the r e s t r i c t e d  h a v e a c e n t r a l body  S  t h r e e body p r o b l e m i n w h i c h we  (the sun), a perturbing planet  J  (Jupit  whose mass i s s m a l l compared t o t h a t o f t h e s u n , and a p l a n e t o i d P  of n e g l i g i b l e  mass.  We c o n s i d e r following  restrictions: 1)  the  The p e r t u r b i n g p l a n e t  The o r b i t o f  as t h e o r b i t o f 3)  J  ;  i  of J  tively  with  by  P  i s a n e l l i p s e i n t h e same  plane  and w i t h t h e s u n a t one f o c u s .  °  of P  J  were  ignored,  w o u l d be r e l a t e d t o t h e mean m o t i o n {  n /n' = p/q ,  where  p and q  are positive rela-  prime i n t e g e r s . The p e r i o d o f t h i s  T  moves i n a c i r c l e  I f the perturbing influence of  t h e mean m o t i o n \ n  1  J  sun as c e n t e r . 2)  n  t h e s p e c i a l c a s e i n w h i c h we have t h e  - 2nrq/n'  .  system i n t h e unperturbed motion i s  We w i s h t o see u n d e r what c o n d i t i o n s a p e r i o d i c  s o l u t i o n c a n be found f o r t h e p e r t u r b e d  motion..  \ Using  t h e method, o f a s m a l l p a r a m e t e r ' S c h w a r z s c h i l d  shqv/n t h a t , u n d e r ' c e r t a i n is  sufficiently  c o n d i t i o n s , i f t h e mass  small, a l l three bodies  will  r e l a t i v e p o s i t i o n as i n i t i a l l y a f t e r a time t h a t t h e e n t i r e system w i l l Tis  of t h e order  of  m'  of J u p i t e r  r e t u r n t o t h e same T = T (i+T)  have r o t a t e d t h r o u g h a s m a l l  m'/m(sun)  has  and v a n i s h e s  with  m  1  except angle.  .  (ill)  The p a p e r i s d i v i d e d i n t o two p a r t s . i s devoted  The f i r s t  t o a method f o r c a l c u l a t i n g t h e p e r i o d and t h e mean  values of the o r b i t a l  elements.  The s e c o n d  p a r t I s devoted t o  a method f o r c a l c u l a t i n g t h e p e r i o d and t h e i n i t i a l orbital  part  elements.  values of t  (iv)  ACKNOWLEDGMENT I w i s h t o e x p r e s s my s i n c e r e t h a n k s t o D r . L e i r n a n i s for  l i i s h e l p i n my r e s e a r c h ; p a r t i c u l a r l y  up t h e t h e o r y o f S c h w a r z s c h i l d P e r i o d i c a method f o r c a l c u l a t i n g t h e i n i t i a l  f o r his help'In  writing  S o l u t i o n s and i n d i c a t i n g  values of the o r b i t a l  ele-  m e n t s w h i c h I have a d a p t e d f o r t h e c o m p u t e r . I would and  also- l i k e t o thank t h e Mathematics  t h e N a t i o n a l Research C o u n c i l f o r t h e i r generous  a s s i s t a n c e d u r i n g my g r a d u a t e  \  \  studies.  Department financial  (v)  TABLE OP CONTENTS Page INTRODUCTION  1  PART I  The mean v a l u e s o f t h e O r b i t a l 4  Elements Section 1.1  The D i f f e r e n t i a l E q u a t i o n s o f M o t i o n 4  of t h e P l a n e t o i d 1.2  The form o f t h e E x p a n s i o n o f t h e 6  Perturbing Function 1.3  Determination of the Secular values 10  o f t h e O r b i t a l Elements 1.4 |  Transformation of the P e r t u r b i n g Components  25  :'  •  i. 1.5  Summary o f t h e Method f o r C a l c u l a t i n g • t h e S e c u l a r mean V a l u e s o f t h e O r b i t a l Elements  •.'1.6  PART I I  2T(  Numerical C a l c u l a t i o n s . The I n i t i a l V a l u e s o f - t h e O r b i t a l v  29  46  \Elements S e c t i o n 2 . 1 ,.• The . . D i f f e r e n t i a l E q u a t i o n s o f M o t i o n  46  of t h e P l a n e t o i d 2.2  P e r i o d i c i t y Conditions i n the Perturbed Motion  ,  4?  (vi)  2.3  The  Perturbation  2.4  Summary o f t h e M e t h o d f o r C a l c u l a t i n g the Period  2.5  and  Equations  the I n i t i a l  50  Values  of the O r b i t a l Elements  5°  Numerical Calculations  53  BIBLIOGRAPHY  •  i.  A • \ "V _  _ \  «  57  (vii)  L I S T OF TABLES Table I  : n' = 2 : 1 ,  n Table I I  Q  : n' = 2 : 1 •, 3i = ir/2  Q  =  Q  ' 5 7  TT  n- = 3 : 2 / i 3 = 0; .  • 38  o  Mean V a l u e s Q  :  =  5  :, 2  ,  5i  : n' = 5 : 3  >  S  /  G3  0  =  T  =•  0.  40  Mean V a l u e s n  Q  0  49  Mean V a l u e s  n • : n' =  5  :  3  \  Table IX  36  o  Mean V a l u e s  n  Table V I I I  '  : n' = 3 : 1 , 0  Q  Table V I I "  1 , «L '= 0  : n' =  n \: Table VI  >  0  Mean V a l u e s n  Table V  32  Mean V a l u e s no  Table I V  = 0  5  Mean V a l u e s  n Table I I I  Pa-?,  Mean V a l u e s  Q  •  I n i t i a l V a l u e s f o r a Symmetric  n  44  = ir  : n' = 3 : 1 > S  7  0  = 0  Solution r-  56  INTRODUCTION Consider  the restricted  we have a c e n t r a l body J  S  ' t h r e e body p r o b l e m i n w h i c h  (the sun), a perturbing  planet  ( J u p i t e r ) whose mass I s s m a l l compared t o t h a t o f t h e s u n , a n  a planetoid  P  o f n e g l i g i b l e mass.  The p e r t u r b i n g p l a n e t  J  moves I n a c i r c l e w i t h t h e s u n a t t h e c e n t e r . At  the i n i t i a l  Instant of time,  t h e p o s i t i o n and  v e l o c i t y - of t h e p l a n e t o i d a r e such t h a t , i f t h e g r a v i t a t i o n a l a t t r a c t i o n o f J u p i t e r were i g n o r e d ,  t h e p l a n e t o i d w o u l d move i n  a n e l l i p s e a b o u t t h e s u n , i n t h e same p l a n e and  v / i t h a mean m o t i o n • n  as J u p i t e r s o r b i t  c o m m e n s u r a b l e v / i t h t h e mean m o t i o n o  n'  of J u p i t e r . The p e r i o d  T  Q  of the unperturbed motion I s given  by T  where prime  n  Q  : n  Q  1  = 2rrq/n  x  = 2irp/n  = p . : q and p a n d q  are positive relatively  integers The mean m o t i o n o f J u p i t e r ""is g i v e n b y  \ n  1  I  = k(-l+m')  I 2  / a'  2  ,  v/here: t h e mass o f t h e s u n i s t a k e n m  !  a',  a s one  i s t h e mass o f J u p i t e r i s the radius of Jupiters  orbit  2 k  i s the constant  of g r a v i t a t i o n  The mean m o t i o n  >f Ikhe p l a n e t o i d  i n the unperturbed motion i s  g i v e n by n v/here in  a  y  = k/a  o  , o *  I s t h e semi m a j o r a x i s  o  of the o r b i t of the planetoid •  the unperturbed motion. By u s i n g P o i n c a r e ' ' s method o f a s m a l l  Schv/arzschild (m!  h a s shown t h a t i f m'  << 1 ) , a p e r i o d i c  by m a k i n g  a suitable  T = T ( 1 + t } , where v/ith  QI ,  i s s u f f i c i e n t l y small  solution of the entire change o f t h e I n i t i a l  elements o f the p l a n e t o i d .  parameter,  s y s t e m c a n be f o u n d  values of the o r b i t a l  That i s t o say a f t e r a t i m e  f=7T(m')  i s of the order of  m'  and v a n i s h e s  a l l t h r e e b o d i e s a r e i n t h e same r e l a t i v e  !  as I n i t i a l l y  b u t the entire  system has been r o t a t e d  position  through a  constant angle J We c o n s i d e r t h e m o t i o n ' o f c o o r d i n a t e system v/ith- S  P and.J  a t the o r i g i n .  In a relative  Since the motion takes  pia.ce i n a f i x e d p l a n e , f o u r o r b i t a l e l e m e n t s a r e s u f f i c i e n t t o • describe the state  o f .motion o f  and t h e e c c e n t r i c i t y orbit. tude  The semi m a j o r a x i s ' a  d e t e r m i n e t h e s i z e and shape o f t h e  The p o s i t i o n o f t h e o r b i t i s s p e c i f i e d by t h e l o n g i Qj o f t h e p e r i h e l i o n . - The p o s i t i o n o f  s p e c i f i e d by where  e  P .  t  e ,  the' mean l o n g i t u d e a t N  .i s the time of p e r i h e l i o n  i n i t s orbit Is  t = 0 , (e = ft - n t  passage)  Without loss of g e n e r a l i t y ,  P  ,  .  we may c h o o s e t h e i n i t i a l  i n s t a n t o f time t o c o i n c i d e w i t h a p e r i h e l i o n passage o f  e = ty . F u r t h e r we choose t h e p o s i t i o n of t h e l i n e  so t h a t SJ  P  t = 0.  when  t h i s choice,  as t h e i n i t i a l  t h e t r u e anomaly  s i d e o f a l l our a n g l e s . v'  With  o f J u p i t e r I s g i v e n "by  v •= n 1 . 1  W i t h t h e above c h o i c e p f t h e i n i t i a l i n s t a n t o f time and t h e i n i t i a l d i r e c t i o n f o r a n g l e measurement, o n l y t h e eccentricity  e ,  the longitude  of the p e r i h e l i o n  03 , and  the c o m m e n s u r a b i l i t y o f t h e mean motions i n t h e u n p e r t u r b e d motion  n  o  and n  a r e a t our d i s p o s a l , *•  1  P e r i o d i c s o l u t i o n s have t h e f o l l o w i n g 1)  properties:  During a period, the values of the o r b i t a l elements, a r e r e s t r i c t e d between c e r t a i n bound/  | aries. f 2)  !  A f t e r t h e passage o f a p e r i o d , a l l o r b i t a l elements r e t u r n t o t h e i r I n i t i a l  values.  T h e r e f o r e p e r i o d i c s o l u t i o n s may be c h a r a c t e r i z e d by: 1)  The mean v a l u e s o f t h e o r b i t a l i.e.  elements,  by t h e q u a n t i t i e s  \  x  =J 1  x (t)dt-, o ~ \  p J  T  \  \  •  . .  2)  The i n i t i a l v a l u e s o f t h e o r b i t a l  The  f i r s t p a r t o f t h e paper i s devoted t o c a l c u -  l a t i n g the period and  ( i = 1,2,3,4)  elements.  and t h e mean v a l u e s o f t h e o r b i t a l  elements,  t h e second p a r t i s devoted t o c a l c u l a t i n g t h e p e r i o d  initial  values of the o r b i t a l  elements.  and t h e  ii.  PART I The Mean V a l u e s :1.1  o f t h e O r b i t a l E l era e n t s  The D i f f e r e n t i a l E q u a t i o n s  of Motion  We u s e a r e l a t i v e c o o r d i n a t e origin.  P  has c o o r d i n a t e s  system w i t h  equations  of motion of  d x _ 'k x 2  dt"  d%/ d t  2  _ _ k y _ k m' ( y - y ' )  (x ,y ) 1  5  1  y  2  a '  5 A  5  where; r A (1.1)  2  2  = x  + y  = (x-x')  (a') k"  2  fc  ;  + (y-y')  2  2  >  = (x-)- + (y'T y  I s the universal' constant  of gravitation  I f we d e f i n e a p e r t u r b i n g f u n c t i o n ,2", l\ R = K m ' L—--x r  ( x x ' v  A  then  the  equations  of  motion  R  by  + vv') ••^-i-} i  of  P  may  be  written  !  The d i f f e r e n -  _ lc m'y'  2  r  2  !  a'  A  2  o f mass.  k m x  2  r" .  as t r o  P are,.  k m'(x-x')  2  S  ( x , y ) and J h a s c o o r d i n a t e s  The mass o f t h e . s u n i s t a k e n a s t h e u n i t tial  of the Planetoid  i n  the  form  5-  cTx dt  k x._ _ aR 2  ox  2  '  (1-2)  cry , k-  aR  d By u s i n g  t h e method  shown I n t e x t s on c e l e s t i a l the  of variation- of constants  mechanics  time d e r i v a t i v e s of the o r b i t a l  planetoid  have t h e f o l l o w i n g  da  _2 oR d t ~ n a ~$e >  e l e m e n t s - a,e,e,G3  thaof t h  expressions,  =  '£§. _ dt ~  (1-3)  [ 1 , pp. 2 7 3 - 2 8 4 ]  i tI  1  (1-e ) „/2„ na e 2  1 BR ou5  2  (1-e ) 2  2  .1  [l-(l-e ) ] „ 2„ na e 2  2  oR o¥  1 M ( 1 - e ) ' - oR dt \ 2 se na e ° 2  2  =  -  .1 de _ _. _ 1 SR (1-e ) dt ~ na a " 2  2  0  1 [ l ~ ( l - e ] ] aR 2 se ' 2  2  where, n 2 a 3 = k,2 (n-) (a-) 2  = k (l m') ,  3  2  +  2 '-. ,1  R = kSl'  (1.4)  1  A  COS(V'-V)]  -  a'^  ^ 2 2 A" = ( a ' ) + r - 2 a r c o s ( v ' - v ) !  and  v,v  :  are the true longitudes  resoactively.  o f t h e p l a n e t o i d and J u p i '  The d e p e n d e n c e o f t h e p e r t u r b i n g orbital  elements  a,e,e,G3  i s determined  1 + e c o s ( V-QJ)  x  t a n l ^ ] = L ^ f ]  E - e sin(E) v where  1.2  E  :  R  on t h e  by,  3  1 (1.5)  function  ' 2  tan[|]  ,  ~ nt ,  = n't' ,  i s t h e e c c e n t r i c anomaly o f t h e p l a n e t o i d .  The F o r m o f t h e E x p a n s i o n o f t h e P e r t u r b i n g For  Function  the i n v e s t i g a t i o n of p e r i o d i c s o l u t i o n s I t i s  n e c e s s a r y t o e x p r e s s t h e p e r t u r b i n g ; f u n c t i o n i n a new f o r m . restrict In tills  ourselves to the f i r s t  .(1.4)  formulas  may  where t h e R  .  function i s defined  The m a i n p a r t  by t h e t h i r d  of the perturbing  be e x p a n d e d i n a F o u r i e r  1 ,•- =  Hence  p o w e r o f t h e p e r t u r b i n g mass  m  expansion. The p e r t u r b i n g  l/<\ ,  We  of  function ,  series  a  E\ o I=o ..  c' f•  ±  c o s i ( v' - v ) ,  c a n h e e x o r e s s e d .as oower s e r i e s I n - ( r / a ) 1  i s a l s o o f t h e same f o r m , n a m e l y 'P  R = k~m  CO  !  S c. i =o  cos i (  ! v  -v)  •  '  .  The r a d i u s v e c t o r r  c a n he e x p r e s s e d  as  r = a(l+x) . where, x = d  and  +  °  >: d . c o s i(-i-ro) , i=l 1  the c o e f f i c i e n t s  d.  c a n be e x p r e s s e d a s power s e r i e s  The t r u e a n o m a l y so  v  >  in  e .  expressed w i t h the help of the  c a l l e d m i d - p o i n t e q u a t i o n , h a s t h e form,. v = i + y >  wherej y =  For of  R ,  E i--l  e  sini^-ft)  •  1  the determination of the c o e f f i c i e n t s I n the expansion we r e q u i r e a r e p r e s e n t a t i o n ; ' o f t h e p o w e r s o f  From t h e F o u r i e r  x  y  series  = f  a  U  y-  +  = S • a + 1  rt 0  =\  +  for  I  f  x and y  I t follows  cos i(^-.s)  ±  i=l  S S- c o s i('-t-s) 1=1 n  x and y .  that,  ,  ,  1  >; h, s i n i(' *-S) 1=1 -  ,  x  vi he r e  a  Is a positive, integer  Hence t h e c o e f f i c i e n t s are  of the form  and  c.  f . , g. , h.  depend on  I n the expansion of  a  R  c.  =  F.  +  £  The p e r i o d i c series,  F.  . cos  terms i n  s i n c e f o r cos i ( v ' ~ v ) cos  i(v'-v)  A(.l-ii))  R  .  c a n be expanded  we may  i n Infinite  write,  - e o s [ i ( "-I) - i y ] v  .  = c o s ' i ( v' -  cos i y  -K s i n i ( v'  siniy .  Nov; s i n i y = i y - (^i x )  + +  5  2 cos  iy _  .-  i y = G. 1  Hence we  5'.  .  x4  t  H. . s i n j(-t-Oj). >  3=1  cos  5  2  iy = .s  sin  (^, iz)  J  + 0  •£ G j=l  cos j(.i-s) 1 J  have, cos i ( ' - l ) co.s i y v  CO  =  E  k  . c o s U ( v ' - - t ) '+ j ( £ - G 3 ) ]  sin i ( v —t) s i n i y - .S L cos[i( '~^) j=--<»  ,  1  + j(^-a)]  v  X J  ,  and cos  i(v'-v)  =  2 j=-co  M  cos[i(v'.-:<t) + 'j(-t-35)] 1  ^  .  Q  Hence t h e e x p a n s i o n o f t h e p e r t u r b i n g f u n c t i o n  R.  must have t h e f o r m ,  (1.6)  R = k m'  E  2  E  l=o j=-»  N. . c o s D. , , ^ 1 J  where (1.7)  md  D .  = i ( v ' - t - ) + J (-t-Oa) ,  l0  the c o e f f i c i e n t s  • N. .  are functions of  a and e  Nov;, (l.S)  v>  ^ n't ,  and  <£ = -e + n t ,  hence  with a, -h a  1  2  Finally  0  + a = 3  0  we a r r i v e a t t h e f o l l o w i n g e x p a n s i o n s f o r t h e  d e r i v a t i v e s of the perturbing .function w i t h respect t o the o r b i t a l elements.  (  \  . •— = k m oa  1  E E . . .  c o s D. . , 1,1  oa  \ >  p  i „ = k m* 3 e  >R  p  = km'  °°  0 5  7; E i=o J=05  s  "i  v ~  5 e  J  c o s D. . ,  ^  CO  E  JN. . s i n D. .  10  03  9  k'~m'  co  E  Z  i^O  1.3  (i-o)N.  j=-oo  x  1  J  Determination of the Secular Values of the O r b i t a l If  Elements  we c o n s i d e r t h e u n p e r t u r b e d m o t i o n o f t h e e n t i r e  s y s t e m , t h e same c o n s t e l l a t i o n s w i l l complete o r b i t s by J u p i t e r the  s i n D. ..  3  (n  : n' = p : q) .  time f o r J u p i t e r t o complete  period, of the motion w i l l T  be r e p e a t e d a f t e r  q  o r b i t s by  q  I f we d e s i g n a t e T'  then the  be = T  o  .  1  Our a i m now i s t o s e e how t h e s i t u a t i o n i s c h a n g e d when t h e perturbations are taken Into  consideration.  According t o the existence proof of Schwarzschild, t h e r e are. p e r i o d i c  s o l u t i o n s f o r the perturbed motion.  We- m a y - r e s t r i c t o u r s e l v e s , w i t h o u t l o s s o f g e n e r a l i t y , to  the following i n i t i a l 1)  At time v  Q  = 8  situation.  t = 0 ,  the planetoid  has t r u e l o n g i t u d e  j, I . e . a t t i m e , t = 0--, ...the p l a n e t o i d  passes the p e r i h e l i o n of i t s o r b i t as i n the unperturbed motion. is  2)  theri\also ^  At time  t = 0  thence a t time is  v  ' = n't  e  o  The mean l o n g i t u d e a t  t = 0  = 03 •o  J u p i t e r has t r u e l o n g i t u d e t  v' = 0  the true longitude of Jupiter  V  I I  Since the  independent v a r i a b l e  oeriod  P  t  of motion  of  m'  .  P  are nonlinear  and  does n o t appear e x p l i c i t l y , t h e  of t h e motion i s of t h e form  of t h e order of  the equations  T = T (l~:-T)  The c o r r e s p o n d i n g  .  where  mean a n g u l a r  T  i s  velocity  i s g i v e n by n = 27rp/T. We b e g i n  planet  J  has t h e l o n g i t u d e  planetoid. the  a t a moment o f t i m e I n w h i c h t h e p e r t u r b i n g s  of the p e r i h e l i o n of the  Without the e f f e c t s of the p e r t u r b i n g planet  system would r e t u r n t o t h i s i n i t i a l  passage of t h e p e r i o d  T  Q  .  situation after the  The p e r t u r b i n g p l a n e t  h e l i o n o f t h e p l a n e t o i d would a g a i n  J  and t h e p e r i - ,  have t h e l o n g i t u d e  ©  .  As  a. r e s u l t o f t h e p e r t u r b a t i o n s , h o w e v e r , t h e p e r i h e l i o n i s s h i f t e d . The  longitudes  T  but after a slightly .  o  longitude  ~ OJ  )q  o f J a n d p e r i h e l i o n do n o t c o i n c i d e a f t e r t h e p e r i o d d i f f e r e n t time  but I n a s l i g h t l y  ~ w  +  HJ-^T  o  T •  and n o t I n t h e  different longitude  .  J  I f we a r e t o have t h e same s i t u a t i o n a t t i m e as a t t i m e  t = 0 x  As of  ,  we must a l s o have  a t time  t  T .  a r e s u l t o f t h e c o m m e n s u r a b i l i t y o f t h e mean m o t i o n s  P and J ,  t h e p l a n e t o i d must have a p p r o x i m a t e l y  mean a n o m a l y a t t i m e  t = T  Schwarzschild the  e = s  t = T  semi-major a x i s  a.  as a t time  t h e same '  t• = 0 .  h a s shown t h a t one c a n s u i t a b l y change by a q u a n t i t y o f t h e o r d e r  of  m' ,  so t h a t , a t t h e same t i m e .as t h e p e r t u r b i n g p l a n e t longitude  ft  on t h e I n i t i a l  T ,  attains the  o f t h e p e r i h e l i o n , t h e mean a n o m a l y o f t h e p l a n e t o i d  returns to i t s i n i t i a l  semi-major  J  value  v a l u e ; and t h a t w i t h c e r t a i n r e s t r i c t i o n s  o f t h e mean a n o m a l y t h e p e r t u r b a t i o n s o f t h e  a x i s and t h e e c c e n t r i c i t y , a v e r a g e d  over the period  vanish. A f t e r passage  of.the period, therefore, a l lthree  bodies  a r e I n t h e same r e l a t i v e p o s i t i o n b u t t h e e n t i r e s y s t e m has b e e n rotated  through the angle  u-T  ( t h emotion of the p e r i h e l i o n ) .  I f we i n t r o d u c e a new c o o r d i n a t e stant angular velocity remains a t r e s t .  ^  system w h i c h r o t a t e s w i t h a con-  about t h e sun then t h e p e r i h e l i o n .  I n t h i s ' c o o r d i n a t e s y s t e m we have p e r i o d i c  s o l u t i o n s i n a t r u e sense o f t h e word. i  I f the perturbations of the period  T ,  a r e t o v a n i s h , a and e +  o  >"] A. ^ •j n t j cos  where  a  e , A .\, E .  Since the I n t e r v a l  ,  E. jnt , j cos >. s  i  n  0  are constants.  t h e p e r i h e l i o n i s s h i f t e d b y a n amount  t = 0  to  t = T , ft must have t h e f o r m , CO  ft = 03 + ^ t + o  over  must b e o f t h e f o r m  V  .  E  Integrated  ,  0  CO  e = e + o  —  sin  ro  a = a  a and e ,  S 3 -**  an n Q.  C  Q  S  —  jnt.  :J.J~T  i n  13  Similarly  e  must have t h e f o r m ,  tude o f the epoch  d  0=1  J  are c o n s t a n t s and  it  l o n g i t u d e o f .the e p o c h  e• j n t, j .cos d  7,  u ) ,fi..and e.  v/here  .—  Q-T S IP  0 3  e = e + wi.t + o 4  t = 0 f o r 0 _< t < T •  t = T f o r T <_'t < 2T  The  mean v a l u e s o f t h e o r b i t a l  J. -  1  —  Tm  T J  1  I s t h e mean t h e mean l o n g i -  and so o n . elements  a and e  are  ,  o -  T  e = ~ I J  a(t)dt  e  o  e(t)dt  I n b o t h Integrals., t h e r e a r e terms o f t h e form  c o s  (jnt)dt  which vanish since  «j  . • ( j = 1,2/...) ,  takes only i n t e g e r values.  Hence  a and e  a r e g i v e n by a = a  o  3  ,  e = e  o  ana we have • a = a +\  e = e +  i : A. (jnt) J cos ' b X U  w  2 ''E .  }3  ( jnt) • . j cos^ s  i  n  J  y  I f we a p p l y t h e same r e a s o n i n g md  e - m.,t  a s a b o v e t o ffi - y.^t  we s e e t h a t t h e mean v a l u e s o f . t h e s e Q u a n t i t i e s a r e  J.  exactly  e„  s  and we have  + , _ t -i- 2  = 0  u  o  >  j,„l  Q_ ^ g ( n j t ) J  ;  (1.10) .1 =1  J  I n a c o o r d i n a t e system r o t a t i n g w i t h c o n s t a n t about S ,  velocity tills  c o o r d i n a t e s y s t e m we must r e p l a c e t h e a n g u l a r ( n ) ~ OJ-, w h e r e  by  t h e p e r i h e l i o n remains a t r e s t .  value of  n  (n)  must b e c o n n e c t e d  angular But I n  velocity  n  with the previous  by t h e e q u a t i o n  (H)  -  = n  = 27T /T  ,  P  I n o r d e r t h a t 'the m o t i o n s h o u l d  have t h e p e r i o d  T  .  Hence  (n)  I s g i v e n by (n) = ^ [ 2 r r p + -u, T ]  The  quantity  that I T  (n)  2rrp + oi^-T  i s t h e mean v a l u e o f t h e mean d a i l y m o t i o n I n  i s the angle  of t h e p l a n e t o i d .  The q u a n t i t y  n  P  i n t h e course  i s t h e mean a n g u l a r  have t h e same f o r m a s I n  Apart  I n r a d i a n s w h i c h t h e mean a n o m a l y  moves t h r o u g h  i n the rotating  everywhere bv  .  of the period  velocity of  c o o r d i n a t e system t h e o r b i t a l ( 1 - 9 ) & o\ (1.10) n  with  n  P . elements  replaced  ( n ) - w-. .  from constant  terms,  t h e elements  p e r i o d i c f u n c t i o n s o f t h e time w i t h p e r i o d  T .  a and e  are  The e l e m e n t s  15.  05 and e  both contain  a time p r o p o r t i o n a l  (secular)  term.  Now we want t o d e t e r m i n e how t h e d i f f e r e n t i a l e q u a t i o n ; of motion o f  P  c a n be s a t i s f i e d (1.3)  We p u t e q u a t i o n s (1.11)  = x  . dt  ,  i  by t h e o r b i t a l  ( i=  elements.  i n t h e form  1 , 2 , ,  where x  = a, x  1  = e,  2  = ob x ^ = e  and 1 • n a Se  3  I (1-e  X  na  2  )  i 3 R  2  ,  1 [ l - ( l - e  2v2  e  na  2  )  2  j B R  e  (1.12) (1-e )  X  2  "  5  n  a  2  *R  2  e  S  X '  —  I f  we  X  ,  P  4  X  '  s  n a Ba  +  ^ 2  ^"V  integrate  these  ( t ) = x (0) + | X x  x  J  [ l - ( l - e ) ] S R 2  2  ^  equations,  (t)dt,  we  obtain  ( i = 1,2,3,4) ,  o  where t h e x^(0)  are,the values of the o r b i t a l  t = 0 .  At time  t = T ,  x (T)  = x (0) + J  ±  i  elements a t time  we have X.(t)dt  ,  ( i = 1,2,3^1-). •  Since period  T  we  a and e  are periodic functions  of time  with  have x (T) = x (0)  = x (0) ,  x (T)  1  1  2  ,  2  and T  \ = i I (" ) = > i = ^ ) • X  fc dt  0  1  x  2  i  Since the s o l u t i o n s a r e p e r i o d i c , developed I n F o u r i e r X  sine  series with  and Xg  may he  constant c o e f f i c i e n t s  n  =  E. A ? . s i n D? . + t e r m s i n ( m )  x.  =  2  1  2  o / \2 E. . s i n D. . -f t e r m s i n (m') 0  v  •2  ±  j  j  i j  i j  where D  i j  =  ( i ' a  n  +  a  2 o^ n  +  a  2 o( £  +  3®  a  3  and cu •+ a  1  2+ 0  a.,  p  In general,  =  0 .  the perturbing  a t i m e dependent p a r t and a p e r i o d i c  function  R  consists of  part.  The p e r i o d i c t e r m s b r i n g i n t o t h e e x p r e s s i o n s orbital  elements terms o f t h e form T  o o A; . s i n D". d t o J . J  (1.13)  N  X  if would  f o r the  give  X  the orbital  e l e m e n t s were c o n s t a n t t h e above i n t e g r a l  .1  F-A?i ,.1  .. .  a n  T  o  D°  -A. . c o s ct^n'  +  a  n 2  :os D: . o i j _ 1  n  1  + a_n 2  o-  I f c a n be shown t h a t b e c a u s e p e r t u r b i n g mass, (1.13)  the latter  very closely. cos  expression  vanish. X  1  X  '  i.V  I.V  = ct =  2)  a^n' +  t  appears  consider  10-  explicitely.  s h o u l d be i n d e p e n d e n t o f t i m e  2  0  ,  CL n^ = 2 o  0  0  c o n d i t i o n demands t h a t : n' - = - a , o  a l l  explicitely  does n o t c o n t a i n t h e t i m e  D?.  (1.12)  those terms o f  c o n d i t i o n i s independent of t h e o r b i t a l  n  since  i n t w o ways';  a  first  second  D? . i j  1)  1  two e q u a t i o n s o f  o f t h i s , we n e e d o n l y  condition that  c a n be f u l f i l l e d  The  -  i n which  The  form vanish  ;  i n which the time  p c  Because  and X „ 2  approximates the i n t e g r a l  D?.(T) = cos D ° . ( 0 )  and X  1  of the smallness of the  Hence t h e t e r m s o f t h i s  and u p o n i n t e g r a t i o n o f t h e f i r s t terms o f  o  . 1  : a  0  elements.  The  = p : q ,  2  w h i c h i s t h e same c o n d i t i o n t h a t was n e e d e d i n t h e u n p e r t u r b e d motion f o r  periodicity.  I n o r d e r t h a t t h e t i m e i n d e p e n d e n t t e r m s i n X.^ and X  2  18.  should  v a n i s h , we must have f o r t h e s e  terms,  s i n D? . = 0  w h i c h may.be s a t i s f i e d e  =3  o  o  • Besides f o l l o w i n g type. un 1  '  these  or  e  o  s e c o n d . c o n d i t i o n demands t h a t  have t h e • -o 1- , 2- q  a^,'a_  = u (j • = ir  o  s o l u t i o n s , there are s o l u t i o n s of the  + c' n = 0 , 2 o '  1  - cu = 1 a^,  = 0 ,  The  These e q u a t i o n s  where  by  with  cu + a + a - = 0 . 1 2 . j? 0  solution '  a  a- = — J q  o v  (p-q) . , •  must a l l be I n t e g e r s .  I n order t h a t the s e c u l a r terms I n v a n i s h , we must  X^ and X  have  2 o  y  v  2  ;r o w  = 0., i r . o r air . Putting  a  = q  e  Hence we  Q  have  we  = S  Q  2p  obtain  = o£ ,  a = 1,2, . . . , 2 p - l  groups of p e r i o d i c s o l u t i o n s  p  should  19.  lie now This w i l l  "be shown I f we  corresponding  t o ft =  = (a+2)Tr/p  3  tions  show t h a t t h e r e a r e o n l y two can  distinct  groups.  show t h a t t h e g r o u p o f s o l u t i o n s  air/p i s e q u i v a l e n t t o t h e g r o u p o f s o l u with a different  choice  of the  initial  I n s t a n t of time. . Suppose a t t i m e helion at longitude (where  k  t = 0  $ = onr/p  I s some i n t e g e r )  the angle  2irk  and  time, J u p i t e r w i l l  take  .  =  a g a i n be  time  t = 2irk/n  the p l a n e t o i d w i l l  27Tkq/p  the  the p l a n e t o i d i s a t i t s p e r i -  Then a t time  have t r a v e l l e d  27rkn'/n  Hence i f we  will  9  have g o n e ' t h r o u g h  at i t s p e r i h e l i o n . through  the  as  our  During  this  angle  .  t =  2/rk/n  initial  i n s t a n t of  o time tive  and  the p o s i t i o n of the l i n e  x - axis, 05  o  at  = ^  the I n i t i a l To  the p e r i h e l i o n w i l l  k  group  can be  some I n t e g e r  are Integers  as the p o s i -  have l o n g i t u d e  [a+2k(p-q) ]£  '  3  show t h a t t h e g r o u p o f s o l u t i o n s 0  Q  c h o s e n so  [a+2k(p-q)]| =  for  .=  at t h i s time  Instant.  e q u i v a l e n t to the Integer  + 2w\,(l-f)  SJ  = (a+2)ir/p  ,  we  ® = air/p  is  must show t h a t  the  that  (a+2)|+  2J7T  =  [a+2(l+jp) ] | ,  j  .  T h i s i s e q u i v a l e n t t o showing t h a t  , j and  k  such t h a t  there  20.  I + jo = k(p-q) > or (k-o)p - kq = 1 . But  since  p and  integers  q  are  ( k - j ) and  relatively  (~k)  prime i n t e g e r s there  such that  (k-j)p +  (-k)q  Hence t h e r e a r e o n l y t h e f o l l o w i n g two  exist  = 1 .  independent  groups of s o l u t i o n s 1)  S = cnr/p  ,  a = 0,2,4,...,2p~2 ,  2)  a = cnr/p  ,  a  We  c o n s i d e r now,  = 1,3, 5, . . . , 2 p - l . the t h i r d  and  f o u r t h of e q u a t i o n s  (1.11),  dx. dt Por  the elements  x_  and  x  -  i  X  1  o  amount  ,T  quantity  x  j  fulfilled.  d u r i n g the n  (T)  (i=3,4)  1  s o l u t i o n s I f we (ih  condition  X (t)dt = 0 ,  can i n g e n e r a l n o t be periodic  the  •  T  = ± T 1  ,  u  2  permit course  In  s p i t e of t h i s , t h e r e  t h e p e r i h e l i o n t o s h i f t by o f a p e r i o d and  xtfith a s l i g h t l y d i f f e r e n t v a l u e .  = x,(0)..+  w  >  T = x^(0)  ^  \  ^  + £•! ..  J  O  replace We  are an  the  have  X_(t)dt ,  whence 1 '= rn  T  I  ^(t)dt o ^  Since  X  = X\, . '  the period  T  • v  I s unknov;n, we  want t o  express  I n terms of the known period T  1 u»  3  I  X (t)dt  - ?p  7  T  r  J  T . We have o  o X ,dt +  " ) "  0  ' T  T o  X_dt > .  Now T /T = T /[T + (T-T ) ] o' o o o' J  v  nence  1  1  T  T  1  Q  1  T  + T  T  o  Substituting the expression f o r 1/T to-  T o  into the expression fo:  we obtain T  1  T  ° x (i- ^ 3  o  0  o  v  -  T  )dt + i1  o  T T  X dt v  o  or, r e s t r i c t i n g ourselves to the f i r s t power of the perturbing mass,  (1.14)  = i  X^dt  o "o In a similar way, we allow the mean longitude the epoch  t = 0 t o be changed by an amount x  &;,T  e of  i n the course  of a period and, restricting'ourselves to the f i r s t power of the perturbing mass, obtain" T ( 1 .15)  U);, -= m  i  Xudt  .  o °o  Now we want to calculate the period of the perturbed motion.  As a result of the mean motion  of the perihelion,  the  perturbed  rotating  motion i s only p e r i o d i c i n a coordinate  w i t h constant  angular  as- a b o u t S .  velocity  d a i l y motion of J u p i t e r I n t h i s  system  coordinate  system I s  The mean n  1  - sy . v  Hence n'T  = 2-n-q  Q  (n'-^T  3  = 2irq ,  f r o m w h i c h we g e t  T/T Restricting mass, we  = nV(n'-u) )  Q  .  3  ourselves again to the f i r s t  power o f t h e p e r t u r b i n g  obtain T = T (l+^/n')  .  o  •  N e x t we w i s h t o f i n d  o u t what q u a n t i t y  n o  J  r e p l a c e d by I n t h e p e r t u r b e d  motion.  must be  F o r t h i s p u r p o s e we  consider  the p e r t u r b i n g f u n c t i o n I n the form (l.l6)  R = k m'  where- t h e by  v  2  N. .  formula  N. . c o s D. . ,  s  are f u n c t i o n s of  a and e  and t h e  D. .. a r e g i v e n  (1.7) ^  D  ij  =  i  ( '~^ v  j(-f~ ) •  +  a  \  x  Expressing  v'.and  the formulas D  ij  1  (1.10) =  -  by t h e f o r m u l a s we  (1.8)"  and  obtain  i(n'-n)t - i e - i u ^ t+ j n t + j e o  + jiuijt - J S  Q  - juo-1  + a p e r i o d i c f u n c t i o n of. t'- ,  Q  e and &  by  23-  or  D  (1.16)  =  e  Q  - js  o  +[in'-(i-j)n -  + a periodic function of The perturbing function the time  t  with the period  R  - (i-j) ,Jt ( ; J  t .  i s a periodic function of  T . The c o e f f i c i e n t s N. . are  periodic' functions of time since  a and e  are periodic functions  of tl'.ne. Hence  (1.170  D..  must have the form.  D. . = constant + a periodic function of t + a(n -uu^)t . ,  ?o obtain this form of  D. .•' we put ij'  nt = (n n-a)t + a periodic function of t where n o = n'p/q The quantity tions  ( l . l o ) and (L.17)  a  may be determined by' comparing equa-  . Equating c o e f f i c i e n t s of  a(n'-iu ) = (n.'-(l~j)n 5  o  -  _(i-j)  (iJi  t  we get  ,  -(i-j)a , OJ  a(n'-  ti)5  ) = [i-£(i-j) ]n' -  - ( i - j ) ^ - (i-j)a ,  2k  f r o m w h i c h we o b t a i n a  a  (1.18)  n = n  Q  -  Finally  - ^  we c a l c u l a t e t h e s e m i - m a j o r 2  perturbed motion from K e p l e r s t h i r d given by  (1.18)  .  L  n  "5  a  i n the  2  law n a^ = k  where  n  Hence  2  a=  axis  2  "  J  L  n  J  °  •  —p * (1+TI)-3  where  r\ --  i r /-p-qy o  i  ^ ••  'Thus we have  [§ I [ l - l n + . . . ] 5  o  or r e s t r i c t i n g  ourselves t o the f i r s t  mass (1-19) where (1.20)  power'-of  ' a = a  o +  ^-t(£zfl)  u ) 5  + ^ ] ,  the disturbing  25-  1.4  Transformation To  o f t h e Components o f t h e P e r t u r b i n g  calculate  a n d r.jj,  ^  Function  we n e e d . t o e v a l u a t e t h e  integrals T , o X_dt o °o ^  T - °  l  ,  a n d -=n  o  X^dt  o  Introducing the perturbing function  R ,  t h e problem I s reduced  to evaluating the i n t e g r a l s T  T  , O  ->r>  (fa) o  o °o To  evaluate  d  >  t  these  i  i o  a n d  ~O o  3R\  d t  ^ 'o e  i n t e g r a l s , we e x p r e s s t h e d e r i v a t i v e s  of t h e p e r t u r b i n g f u n c t i o n i n terms o f the p o l a r r .and v • radial  1'T ™  put  e  ~  = S ,  and  = rT ,  component o f t h e p e r t u r b i n g f o r c e aiid  component.  We t a k e  the magnitude o f  S  T  coordinates then  S  I s the  I s the transverse  a s p o s i t i v e when i t t e n d s t o i n c r e a s e  r and T  a s p o s i t i v e when i t t e n d s t o i n c r e a s e  v • To system  of  (% r\) 3  e  tive  calculate  P  with origin at  direction. a.re  (^'n)  (C ,-n') ,  In this  S and T  ;  coordinate  S  and t h e l i n e  I n s u c h a, c o o r d i n a t e  = (r 0)  ,  we i n t r o d u c e , a r o t a t i n g  = (a'  s y s t e m we  have  3  as t h e p o s i -  system t h e c o o r d i n a t e s  and t h e c o o r d i n a t e s  cos(v'-v)  SP  coordinate  of  J- a r e  a' s i n ( v ' - v ) ) •  ^  r  do.  A Hence  2  = (--r)  S and T  Wf  +  2  a r e g i v e n by , 2  , -»-r  |»  r  A The f i r s t  i  a'^  3  d e r i v a t i v e s of  R  with respect to  a and e  a r e g i v e n by _ 5R or oa ~ 3 r Sa  >,R  5R 3  t  _  v  V  sa  Sa Q R  S Sa  oSr  ,  3e ~ °se ^  r  fjiciy ,:  oe  Using formula's turbed motion, Sr Sa  we  ^  P  where  Hence we  S» Sa v  3  3v /2-i-e c o s i . -a c o s x , - ~ = ( —p ) sin f , ' Se 1-e"  se  i  f o r the s o l u t i o n of the unper-  obtain  r a  or  (1.5)  = v -  have  a  aa 3R  ae  = -Sa c o s f + rT. s i n f (  and  2+e  cos f  1  sin f ] ,  [- S cos f + T-  nae (1.21)  1 2r  S + [l-(l-e ) ].X^ . 2  O  2  na  1.5  Summary o f t h e M e t h o d f o r C a l c u l a t i n g t h e S e c u l a r Mean Values of the O r b i t a l Elements. The s e c u l a r  mean v a l u e s o f t h e o r b i t a l  p l a n e t o i d .are c a l c u l a t e d 1)  elements  according t o the following  T he u n p e r t u r b e d  of the  scheme.  m o t i on.  a)  The e c c e n t r i c i t y  b)  a' =•. 5 . 2 0 2 8 0 3  e = e  ,  i sarbitrary.  Q  ( t h e semi m a j o r a x i s o f t h e  e a r t h s o r b i t i s t a k e n as t h e u n i t of l e n g t h ) . c)  k = .017202099  rad./mean s o l a r d a y . 2  d)  n'  e)  n  f) '  a  g)  The p e r i o d  Q  o  i s calculated  "5  2  from  n' a ' ^ = k  from  n o/  -  i s g i v e n by  T  = pn'/q . i s calculated T  2  0 0  k  2  = 2irq/'n' .  23.  2)  The p e r t u r b e d m o t i o n . a)  The e c c e n t r i c i t y  b) ' The q u a n t i t i e s equations c)  e = e  remains  Q  r and v  arbitrary.  a r e f o u n d by  solving  (1.5) •  v' = n ' t , K ' = a ' c o s ( v'-vK.o - ' = a' s i n ( v ' - v ) , n  , ^2 , , , 2 = (V-*r + (V)  2  N  a  d)  S = k m'  -  3 A'  i5  fl  T = k m'[^ - -^]r,' A" a'^ 2  ] .  .  X (a ,e. ,ffi ,e )  e)  0  5  0  P  N  2  =  0  \2  o  r T •• ' (2±e^cos_f) sin f ., 2, ] ,  [_s c o s f + ^ -  )  T  1 V  V  V  where  V  f = o f o o  ^  v  = - — 2 n o~ ao  £l-(l-e ) ]X ,  +  2  o  - S  T  f)  = |  ^  J  X.(a ^  e  5 o  ,e )clt  ,  o  T  U!  4  =  o TT~ .,J V V o o rt  e  o ' V  6  o  )  d  t  2  3  n = n  g)  o  a = a  -  i f ) - - coti  i~—-  a  -!-  4  o  2a, q ^3  ^4  1  T = T Ll-t—?] . o n !  1.6  Numerical  J  Calculations  N u m e r i c a l c a l c u l a t i o n s were done f o r t h e c o m m e n s u r a b i l i ties  n  Q  : n' = 2 : 1,  3 : 1,_  3 : 2 ,  Since the q u a n t i t i e s  5 : 3 -  X_ and X,,  are calculated  only  I n terms of t h e unperturbed motion, w h i c h i s symmetric, I t I s s u f f i c i e n t to' a v e r a g e them o v e r h a l f t h e p e r i o d . n  Q  : n' = 2 : 1,  S  = TT/2 ,  0  averaged over the I n t e r v a l  the quantities [- ~ o / 4  j  o/4]  Por t h e case  X_. and X^ .  were  Por a l l other  cases  T X_, and X^  .Por  (N  quantities  b e i n g chosen t o give  X-, and X  , t_,...,t , i T  O  .  o/2]  t h e i n t e g r a t i o n , t h e I n t e r v a l was d i v i d e d  equal parts  t  [0 3  were averaged over t h e i n t e r v a l  T  where  iL  were e v a l u a t e d a t t h e t  O  i n t e r v a l of i n t e g r a t i o n . e v a l u a t e d by u s i n g  a suitable  and t  T T  Into  step size) .  N + 1  - r e p r e s e n t t h e end. o o i n t s  the trapezoidal  cu- and cui,  rule. I.e.  The  points of the  IM  The q u a n t i t i e s  N  were t h e n  An i n t e r v a l o f small  3°  i n t h e anomaly o f J u p i t e r was  enough t o g i v e a n e r r o r o f l e s s t h a n  a- and ajj  provided the planetoid ( i . e . & 2. ° -  Jupiter  2  .001  sec/day i n  d i d not pass t o o c l o s e t o  everywhere i n the i n t e r v a l o f I n t e g r a t i o n )  The r e s u l t s o f t h e c a l c u l a t i o n s f o r t h e u n p e r t u r b e d motion are as f o l l o w s  a' = 5-20280 n' = 299.12838 s e c / d a y . Case 1:  n  n' = 2 : 1  o  = 598.2568 sec/mean s o l a r d a y .  n a  T C a s e 2:  n  = 3.276519 o Q  o  = .'+332.'588 d a y s : n' = 3 : 1  n o = 897.3351 O  T C a s e 3:  n n  D  o o o  2.500451 = 4332.588 : n  1  = 3 : 2 ^ •  = 448.6926 y  3.96220  T o = 8665.176 C a s e 4:  : n' = 5 : 3  n  n  Q  = 498.547:  a, = 3.699987 o  o  = 12997.763  3>l  The  results of the calculations  motion are given I n the following t a b l e s . UJ-V  oj|, and n  i n mean s o l a r  f o r the perturbed The q u a n t i t i e s  a r e g i v e n I n s e c o n d s / mean s o l a r day. days.  1  32.  TABLE I \  MEAN VALUES  n  : n' = 2 : 1  © '= 0 o  o e  ^4  n  a  T  .01.  -41.321  -.229  639.808  3.12481  3 7 0 8  .02  -19.959  - .210  . 618.425  3.20288  4043•51  • Op  -12.857.  - .191  611.305  3.22888  4146.37  .04  -.9.519  -.174  607.750  3.24186  4197.61  .05  - 7.207  -.157 •  605.621  3.24963  4228.20  .06  - 5-307  - .142  604.205 •  5.25480  4248.48  .07  - 4.813  -.127  603.197  3.25848  4262.87  .08  - 4.074  -.113  602.443  3.26123  4273.59"  .09  ' - 3.503 .  -.099  601.859  .3.26337  4281.85  .10  - 3.050  -.087  601.394  3.26507  4283.41  .11  - 2.683  -.075  601.015  3.26645  4293.72  .12  - 2.380  '-.064  600.701  3.27660  4298.ll  .13 ^  - 2.127  -.053  600.436  3.26856  4301.78  .14  - 1.912 '  -.043  600.211  3.26938  4304.90  .15  -  1.727  -.033  600.017 •  3.27009  4307.57 .  .16  -  1.568  -.023  599-848  3.27071  599-700  3.27125  4311.90  599•569  3.27173  4313.67  •  4309.88  • IT  -  - 1.428  ''''-.015  .18  ;  - 1.306  -.006  -  - i - . 002  599- +53  3.27215  4315.23  .010  599.349  .3.27253  4316.63""  . 043  598.963  3.2739^  4321.73  .19  1  1.198  •  ]  .20  ~ 1.102  .25  -  .749  • 30  -  . 530  .071  593.716  3.27484  4324.91  • 35  -  .383  . 094  598.5^-5  3.27546 -  4327.04  .  33  TABLE 1 (cont'd') 2 : 1  0)^  £  o  - 0  n  1%  a  T  3.27591  4328.54  .40  -  .280  . 113  598.424  M  -  . 204  .129  598.332  3.27625  4329.63  .50  -  .148  .144  598.260  3.27651  4330.45  • 55  -  .104  .157  • 598.203  3.27671  4331.09  .60  -  .068  .169  598.156  3.27689  4331.60  -65  -  .039  .180'  598.II6  3.27703  4332.02  .70  -  .015  .191  598.O8O  3.27716  4332.38  .007  .202  598.047  3.27728  4332.69  .028  .213  598.016  3.27740  4332.99  . 048  .225  .3.27752  4333.28  • 75 j-  .80 . G O  .  '  • 597-984  ,  .90  JL.  .O69  .238 '  597.950  3.27764  4333.59  • 95  +  .095  . 249  597.912  3.2777  4333.97  TABLE I I MEAN VALUES . n e  o  : n' =  2:1.  ffi  UU-. -  .01  o  =  TT/2  n  a  T  44.378  -  .274  554.152  3.43755  4975.36  22.989  -  .298  575.565  3.35937  4665.56  .03  15.883  -  .323  582.696  3.33333  4562.64  .04  12.3^9  -  .350  586.257  3.32033  4511.45  .05  10.245  - ' .378  588.390  3.31254  4480.97  .06  8.856  -  .409  . 589.809  3.30736  4460.86  .07  7.877  -  .442  590.821  3.30367  4446.68  .08  7.155  -  .477  591.578  3.30090  4436.22  .09  6.6O5  -  .514  592.166  3.29876  4428.25  .10  6.175  -  .55^  592.635  3.29704  4422.05  5.834  -  .597  593.019  3.29564  4417.09  .12  5.561  - ' .644  593.3 !-0  3.29447  4413.13  .13  5.339  -  .693  593.611  3.29348  4409.91  5.158  -  .747  593.845 . • 3.29263  4407.30  .02  ,  .11  .14  •  -  l  • 15  5.012  -  .805  594.049  3.29188  4405.18  .16  4.893  -  .867  59 !--231  3.29122  4403.46  .17  4.793  -  .935  59^.393  3.29063  4402.09  .18  4.. 724  -  1.008  59^.5^1  3.29009  4401.00  .19  4.667  - I.087  594.677  3.28959  4400.18  .20 •"  4.625  -  1.173  594.805  3.28912  4399.58  .25  4.600  -  1.730  595.387  3.28700  4399.21  .30  4.801  - 2.599  596.054  3.28456  4402.13  .35  5.152  - 3.983  597.088  3.28079  4407.21  i  35-  TABLE I I ( c o n t ' d )  2 : I  e  u>  3  T  si = ir/2  n  a  T  .40  5.556  " 6.193  598.894  3.27419  4413.06  • 45  5.929  - 9.663  601.990  3.26289  4418.47  • 50  6.667  -15.382  606.971  3.24470  4429.15  • 55  11.752  -3I.352  617.857  3.20496  • . 4502.80  • 56  15.659  -40.954  623.552  3.18416  4559 .40  .61  -21.442  +37.967  581.733  3.33685  4022.01  .62  -14.447  24.657  588.047  3.31380  4123-34  .65  - 6.941  11.282  593.915  3.29237  4232.06  .70  - 3.298  5.471  596.083  3.28446  4284.82  . 7 5  -  1.914  3.465.  596.706  3.28218  4304.87  .80  - I.205  2.489  596.973  3.28121  4315.13  -  .786  1.925  597.118  3.28068  4321.21  .90  -  .511 •  I.562  597.205  3.28036  4325.19  • 95  -  .312  1.311  597.258  3.28017  4328.07  .  <"> r0 5  3b.  TABL E I I I MEAN VALUES n e  Q  : n' = w  3  .: 1  o  ffl  3  =  0  n  a  T  .01  -.610  -  .128  896.294  2.50248  4341.42  .05  .611  -  .132  896.295  2.50248  ii^.in  .10  .615  -  .146  896.3OI  2.50246  4341.49  .15  .621  -  .169.  896.313  2.50244  4341.58  .20  .629  -  . 204  896.33O  2.5.0241  4341.70  ..25  . 640  -  .251  ' 896.357  2.50236  4341.85  . 3 0 '•  .652  .315  896.396  2.50229  4342.03  • 35  .665  -  .398  396.454  2.50218  4342.21  .40  . 677  -  .506  896.537  2.50203  4342.39  .45  .687  -  .645  896.656  2.50181  4342.54  .50  .692  -  '.823  896.823  2.50150  4542.62  .55  .688  -  1.046  897.055  2.50107  4342.56  .60  .670  - .1.323  897.368  2.50048  4542.29  .65  • •633  -  1:658  897.778  2.49972  4341.75  .70  .572  -  2.047  8 9 8 . 2 8 8 '•  2.49877  • ! J  .490  -  2.482  898.888  2.49766  4339.68  .80  .393  -  2.956  899.556  2.49642  4538.28  ,85.  .293  -  j?.4op  900.282  2.49507  .90  .200  -  4.135  901.120•  2.49351  4555.49  .95  .102  -  5.117  902.298  2.49132  4 5 5 4 ."Oo  .066  -  6.448  903.965  2.48823  4531.63  • 99  J  -  .  i'ii  4540.87  t  '  4356.84  37.  TABLE IV MEAN VALUES n  0  e  : n' = 3 : T  i;i  ©  3  =  7T  n  a  T  .01  -  -375  , -  .128  898.263  2.49882  4327.15  .05  -  '.372  -  .125  898.253  2.49884  4327.21  .10  -  .360  -  .116  898.221  2.49890  4327.38  .15  -  . 3^1  -  .103  898.170  2.49899  4327.65  .20  -  .317  -  .086  989.IO5  2.49911  4328.00  .25  -  .289  .066  . 898.030  2.49926  4328.40  -50  -  .259 •  -  .044  897.947  2.49941  4328.84  • 35  -  .228  -  .022  897.862  2.49956  4329.29  .40 .  -  .197  +  . 001  897.777  2.49973  4329.74  -  .167  .023  897.695  2.49988  4330.17  .50  -  .138  .045  897.616  2.50002  4330.59  • 55 •  -  .111  .065  897.5^-2  2.50016  4330.98  .60  -  .086  '• .084 •  897.473  . 2.50029  4331.34  .65  •-- -  .062  .102  897.408  2.50041  4331.68  .70  -  , 040  .119  897.347  2.50052^  4332.00  .75  -  .020  -.134  897.291  2.5OO63  4332.30  .80  .000  .147  897.238  2.50073  4352.59  .85  .020  .159  897.187  2.50082  4332.87  .90  .040  .166  897.140  2.50091  4533.16  .95  .061  .166  897•097  2.50099  4333.47  /  J  TABLE V MEAN ViiLUES n o : n' = 3 :  a  2  e  Q  = 0  • n  3.  T  .01  --62.O35 J \  -  .444  •480.155  3.78367  6868.13  .02  -2Q.147  -  .395  463.661  3.88094  7820.85  .03  -18.278  . 351  458.183  3.91325  8135.68  .04  -12.908  -  .311  455.^57  3.92933  8291.26  .05  -  9.732  -  . 274  453.833  3.93891  8583.27  .06  -  7.649  -  . 241  452.758  3.94524  8443.61  .07  -  6.188  -  .211  451.997  3.94973  3485.93  .08  -  5.H3  -  ,133  451.432  3.95306  8517.06  .09  -  4.294  -  .158  450.997  3.95563  8540.78  .10  -  3.653  -  .134  450.653  3.95766  .11  - 3-141  -  .113  450.375  3.959 30  8574.19  .12'  -'2.723  -  .093  450.147.  3.96064  8586.29  .13  -  2.378  -1  .074  449.956  3.96177  • l '- .  -  2.O89  -  .057  449.794  3.96272  8604.65  • 15  -  1.845  -  .041  449.656  3.96354  8611.73  .16  - 1.636  -  .027  449.537  3.96424  8617.77  .17  -  1.457  -  .013  445.434  3.96485  8622.97  .18  -  1.301  .000  449.343  3.96538  3627.49  .19  -  I.165  .012  449.263  3.96586  8631.43  .20  -  1.045  .023  449.192  3.96628  8634.89  .25  -  .621  .070  448.933  3.96780  8647.20  .30  -  .368  .106  448.770  3.96876  8654.53  .35  -  .204  .135  448.659  3.96942  8659.27  .40  -  .O89  .159  1  +  448.578  x  '  3.96990  '  '  8559.34  8596.28  8662.59  39TABLE V (cont'd) 3  e  2  U)!j,  j  %  -0 11  a  T  .003  .181  448.513  3.97028  8665.O8  • 50  .067  .201  448.458  3.97060  8667.ll  .55  .129  .220  448.408  3.97090  8668.94  .60  .193  .238  448.357  3.97120  8670.77  .65  ,264  .256  448.304  3.97151  8672.83  .70  .356  .270  448.244  3.97187  8675.48  .75  .491  .269  448.178  3.97226  8679.4i  .80  .736'  - .208  448.116  3.97261  8686.50  .83  1.031  .O65  448.112  3-97264  8695.04  .45  -  :  -  01.  1.192 •  -  . 032  448.129  3.97255  8699.70  Q r-  1.417  -  .181  448.165  3.97233  8706.23  ,86  1.762  -  .428  448.239  3.97189  8716.21  .87  2.364  -  .882  448.393  3.97098  8733.65  op  3.716  - 1.945  448.780  3.96871  3.511  3.836  446.612  3.98149  8563.47  .92  - 1.955  2.510  447.160. '•  3.97826  8608.53  .95  -  1.403  447.647 .  3.97539  8644.48  •  0 0  .91 ....  ...  -  .714  '  8772.81  TABLE V I MEAN VALUES n o e  : n' = 3 : 2  GD =  i'0-7  TT  n  a  T  .01  70.506  -  . 560  413.999  4.17382  10707.59  .02.  37.657  -  .623  430.492  4.07656  9756.03  .03  26.856  -  .704  435.968  4.04426  9443.15  . 04  21.581  -  •.789  438.691  4^.02820  9290.35  .05  18.530 .  -  .886  440.314  4.01863  9201.96  .06  16.603  -  .996  441.387  4.01231  9146.15  .07  15.332  -  1.121  442.148  400781  9109.31  .08  14.482  - 1.264  442.715  4.00447  9084.70  .09  13.928  - 1.429  443.157  4.00186  9068.64  .10  13.594  443.514  3.99976  9053.96  .11  13.436  - 1.839  443.815  3-99799  9054.35  .12  13.423  -  2.097  444.078  3-99643  9054.OO  13.540  -  2.399  444.322  3.99500  9057.43  -  2.756  444.559  3.99360  9064.34  • - 3.181  444.807  3.99214  9074.62  .13  •  •  •  -  1.619  '  .14  13.780  .15  14.134  .16  14.605  -  3.689  .445.079  3.99053  9088.24  .17  15.191  -  4.300  445.397  3.98866  9105.24  .18  15.898  -  5.040  445.783  3.98638  9125.71  .19  16.728  .-  5-9^0  446.268  3.93352  9149.74  .20  17.682  -  7.041  446.892  3.o7984  9177.40  .22  19.968  -10.060  448.769  3.96377  9243.60  .24  22.768  -14.667  451.975  3.94986  9324.74  •"  4 1 .  TABLE V I 3  :  2  (cont'd)  CL = tr u  e  n  a  T  -21.868  457.294  3.91849  9433.74  <%  .26  26 • 531  .28 ' .  3^  .819  -34.921  466.204  3.86595  9673.80  .29  4o .462  -48.484  • 473.945  3.82029  10011.09  • 33  ' -50 .033  29.241  444.468  3.99414  7215.81  ~^4  -31 .850  17.400  447.217  2-97792  7742.54  -17  .368  • 8.832  448.544  3.97009  8162.06  ~  (  .905^  3.9^4  448.701  3.96917  8436.18  .45  -  3 .980  2.159  448.524  3.97021  8549.86  .50  -  2 .325  1.449'  448.406  3.97091  8597.84  .55  -  1 .'473  1.088  448.341  3.97129  8622.50  .60  -  .982  .876  448.307  3.97149  8656.74  . 65  -  .675  .739  448.291  3.97159  8645.63  .70  -  .472  .646  448.282  3.97164  • 8651.52  .75  -  .330  -579  448.278  3.97166  8655•61  .80 • -  -  .227  .530  448.276  3.97168  8658.59  .85  -  .149  .494  448.273  3.97170  3660.86  • 90  -  .085  .468  448.266  3.97173  .6882.72  • 95  ' -  .027  .451  448.255  3.97180  8664.41  • 36 .40  '  .  • '  .  4 2 ,  TABLE V I I MEAN VALUES n  Q  : • n»  = 5 :  3  S5  e  0  =  0  n  a  T  .01  2.704  -  .376  497.120  3.70705  I3II5.27  .02  2.711  -  .380  497.120  3.70705  13115.53  .03  2.724  -  .388  497.120  - 3.70705  .04  2.741  -  .400  497.120  3.70705  13116.86  .05  2.763  -  .415  497.120  3". 70705  13117-84  .06  2.791  -  .^33  497.120  3.70705  13119.04  2.824  -  .456  497.120  3.70705  13120.48  .08  2.863-  •-  .483  497.121  3.70704  13122.17  .09  .2.908 .  -  . 51^-  497.123  3.70703  13124.12  .10  2.959  -  • 551  497.126  3.70702  13126.34  .ii  3-017  -  .594  497.130  3.70700  13128.86  .12  3.082 .  -  .643  497.136  3.70697  13131.68  3.154  - ' .700  497.144  3.70693  13134.83  3.235  -  .764  497.155 .  3.70688  13138.33  .15  3-324  -  .838  497.169  3.70681  13142.21  .16  3.423 •  -  .923  497.139  3.70671  13146.49  .17  3.531  -  1.021  497.214  3.70659  13151.20  .18'  3.650  - 1.132.  .497.246  3.70642  13156.37  .20  3.923  - 1.409  497.341  3.70596  13168.21  .25  4.830  - 2.578  497.905  3.70316  13207.65  .30  6.048  - 5.O60  !-99.575  3.69490  13260.57  • 35  7.972  •-IO.578  503.810  3.67395  13344.18  .07  .  1^ .14 •  -  2  i3il6.ll  43.  TABLE V I I (cont'd) '5 = 3 o = S  e  .  0  n  Ml  a.  T  . 37  10.453  -15.500  507.093  3.65771  13451.09  • 38  13.473  • -20.282-  509.844  3.64410  13583.40  .44 .45  -  -10.544  8.823  496.754  3.70886  12539.59  - 7.913  6.430  497.393  3.70570  12653.92  • 50  -  3-088  2.477  498.129  3.70206'  12863.58  • 55  -  1.642  1.463  498.179  3>70l8l  12926.40  .60  -  .931  I.O36  498.165  3.70188  12955.15  • 65  -  .616  .813  498.145  3.70198  12970.99  .70  -  .391  .682  498.126  3.70207  I298O.73  .75  -  .238  -  .597  498.109  3.70216  12987.40  .80  -  .126  .A  .5^0  . 498.091  3.70224  12992.28  .85  -  . 034  .496  498.074  3.70233  12996.28  498.059  3.70240  13000.07  498.080  3.70230  13004.77  .90  .053  .95  .161  .  •  .452 .360  .  TABLE V I I I MEAN VALUSS  n  o  : n' = 5 ° 3  a  e  o  = ir  n  a  T  .01  - 1 .713 .  • 373  500.066  3.69247  i2923.ll  .02  - 1 .706  • 370  500.055  . 3.69253  12923.63  .03  - 1 .686  .365  5OO.O36  3.69262  12924.49  .04  -  i .659  .358  500.011  3.69274.  12925.66  .05  - 1 .625  .3^9  499.98O  3.6929O  12927.13  .06  - 1 .586  • 333  499.9^3  3.693O8  12928.87  . 07  - 1 . 540  .327  499.901  3.69329  12930.84  .08  - 1 .490  • 313  499.854  3.69352  12933.01  .09  - 1 .436  .299  499.804  3.69377  12935.35  .10  _ i .3S0  .284  499.751  3.69403  12937.82  .ii  _ i_ . 3 2 0  . 268  499.696  3.6943O  12940.39  .12  -  i . 260  .252  499.639  3.69459  12943.02  .13  -  i .198  \ .235  499.581  3.69487  12945.69  ... - 1 .136  .219  499.523  3.69516  12948.38  3.69544  12951.06  .14  '  -  .15  -  i .075  .202  499.466  .16 "  - 1 .014  , -i->-0  fie  499.403  3.69573 .  12953.71  .17  -  .954 '  .169  499•352  3.696OI  I2956.3I  .18 •  -  .895  .152  499.296  3.69628  12958.86  .20  -  .783  .121  499.190  3.6968I  12963.75  • 25  -  .535  .049  498.953  3.69798  12974.51  • 30  -  • J s s  .010  498.761 '  3.69893  12983.19  • 35  -  .176  .058  498.606  3.6997O  12990.13  .40  -  . 044  . 097  498.479  3.70032  12995.85  1  ~  •  TABLE  VIII  5 e  5  1%  (cont'd) SL = TT o n  a,  T  . 071  .129  498.371  3.7OO86  13000.83  • 50  .180  .153  498.274  3.70134  13005.59  • 55  .298  .158  493.181  3.70180  15010.70  .60  .444  .165  498.086  • 3.70227  13017.05  -65  .661  .117 '  497.989  3.70275  15026.49  .70  1.079  -  .081.  497.908  3.70315  13044.67  .72  1.406  -  .288  497.898  3.70320  15058.86  .74  2.014  -  .729  497.934  3.70302  13085.26  .76  3.646  - 2.007  498.124  3.70208  15156.19  2.213  497.567  3.70484  12917.47  A  5  .82  +  •- 1.848  .  .oo  -  .916  1.423  497.735  -3.70401  12957.95  .90  -  .410  .977  497.343  3.70347  12979.96  • 95  -  .190  .783  497.890  3.70324  12989.55  46.  PART I I The I n i t i a l V a l u e s o f t h e O r b i t a l 2.1  Elements  T h e ' D i f f e r e n t i a l Equations of Motion of the Planetoid L e t t h e mass o f t h e Sun be  1-u ,  that of Jupiter  t h e d i s t a n c e f r o m t h e Sun t o J u p i t e r be o n e , and t h e p e r i o d 2TT .  a b o u t t h e s u n be  a , of Jupiter  W i t h t h e above c h o i c e o f u n i t s , t h e mean  2 a n g u l a r m o t i o n o f J u p i t e r and t h e G a u s s i a n c o n s t a n t  k  are also  unity. We c o n s i d e r t h e m o t i o n o f t h e p l a n e t o i d  in. a coordinate  s y s t e m w h i c h r o t a t e s * , w i t h c o n s t a n t a n g u l a r v e l o c i t y one a b o u t t h e Sun a s c e n t e r . system..  J u p i t e r remains at r e s t r e l a t i v e t o t h i s  We c h o o s e t h e l i n e The  SJ  as t h e  x - axis .  e q u a t i o n s of motion of the p l a n e t o i d i n t h i s co-  o r d i n a t e system a r e o  d x • dv , , , /,• \X —2 dt - H " (l-n)-vdt" =  2  0  +  a  (x-1) i- — % >  r  x  1  u  A  (2.1) a y  — P  dt^  where  odx  =  2-rp - y aXj  fi  \Y  - ii-jjj—^  V  ^  p o  The  = x  2  uy  - —5 , A  (x,y) are the coordinates of r  coordinate  P ,  and  p  + y~ ,  = (x-1)  p  p  P  + y ~ = 1 + r - - 2x .  equations  (2.1)  have t h e J a c o b ! i n t e g r a l  k  (2.2)  + y  2  where  2•2  quadruple  + y  2  + 2(i^)+ ^  2  Conditions i n the Perturbed Motion .  state of motion of  P  ( x ( t ) ,,y(t) ; x ( t ) , y ( t ) )  (x(0) Y{0);x(0) y(0)) >  J u p i t e r a t time  t  I s c h a r a c t e r i z e d by t h e  a t time  at time . t = 0 .  }  We  - C ,  C ' I s t h e Jaco'bi c o n s t a n t .  Periodicity The  = (x-u)  2  I s c h a r a c t e r i z e d by  t  and b y  The s t a t e o f m o t i o n o f (1.0;0,0)  designate the values of the o r b i t a l  perturbed motion a t time  t  . elements  In the  by  x(t) = x ( t ) + *(t) ,  y ( t ) = y ( t ) + n(t)  x(t) = x ( t ) + |(t) ,  y('t) = j j t )  Q  Q  Q  -!- -q(t) ,  where 2 d x  „2  at  dy ° ~ 2 d- t^  +' x.o  r  -j o  (2.3)  o  and o  o  (x y ) \ o o' motion.  = x  o  o  2 + y -o  are the coordinates of the planetoid  i n the unperturbed  Vie ? and J For  c o n s i d e r t h e c a s e I n w h i c h t h e mes.n m o t i o n s o f  I n the unperturbed motion are i n the r a t i o  t h i s case, the period  : n' = 3  n  of the unperturbed motion I s  T  = 2ir o  According t o the r e s u l t s of Part I ,  periodic  p e r t u r b e d m o t i o n e x i s t when t h e i n i t i a l h e i i o n ox  ?  is  3 = 0 , — , uj  ' p a r t i c u l a r case time  t =0  o  a  = 0  longitude of the peri--  2~, y=; 4—, 5— .  y  J  y  y  y  s o l u t i o n s of the  we c o n s i a e r t h e  y  f o r which the s t a t e of motion of  P at  I n t h e unperturbed motion i s c h a r a c t e r i z e d by  ( x ( 0 ) , 0 ; 0 , y ( 0 ) ) •. Q  Q  Since the Jacobi-constant C Is  i n the I n t e g r a l  a r b i t r a r y , we may v a r y t h e q u a n t i t y  quantities motion of  x ( 0 ) , y ( 0 ) , x(0) P o  °-|(0)  The q u a n t i  w i t h t h e above I n i t i a l  T = T (1+9)  where  9  conditions  I s o f t h e orde  u . In  The  = 0 .  I s a t o u r d i s p o s a l and we w i s h t o choose I t so t h a t t h e  i s periodic with a period  line  Consequently the state of  P(0) = ^ ( 0 ) = \{0)  or  s o l u t i o n o f e q u a t i o n s ,(2.1)  of  while keeping the  i n t h e p e r t u r b e d m o t i o n may be t a k e n a s  (x (0),0:0,y (0)-!-^(0)) o  fixed.  y(0)  (2.2)  SJ  t h e r o t a t i n g c o o r d i n a t e s y s t e m o f S e c , 2.1 ,  I s a symmetry a x i s o f t h e p e r i o d i c p l a n e t a r y  v e l o c i t y o f t h e p l a n e t o i d a,t t i m e  the, l i n e  t = 0  the  orbits.  Is perpendicular t  SJ . In  the unperturbed motion, the planetoid w i l l  again  cross the  x - a x i s a t r i g h t a n g l e s a f t e r h a l f t h e p e r i o d , and  hence w i l l  be s y m m e t r i c w i t h r e s p e c t t o t h e  x-axis.  I n the perturbed general If  motion however, t h e p l a n e t o i d w i l l I n  not cut the x-axis a t r i g h t angles a f t e r half the period.  I t c u t s t h e x - a x i s a t an a c u t e a n g l e t h e n we must c a l c u l a t e  a new o r b i t i n w h i c h we s t a r t t h e p l a n e t o i d f r o m t h e same p o s i t i o n , again perpendicular initial  velocity.  initial  to the x-axis but with a d i f f e r e n t  A f t e r a f e w a t t e m p t s , we f i n d  o r b i t v/hich i s s y m m e t r i c v / i t h r e s p e c t  the desired  t o - t h e x - a x i s and t h e r e f o r e  periodic. The • • c o n d i t i o n t h a t t h e o r b i t x-axis a t right angles i s equivalent x  r e a c h e s i t s maximum v a l u e ,  y  should  Intersect the  t o t h e c o n d i t i o n t h a t when  should  be z e r o .  t h e f o l l o w i n g I n t e r p o l a t i o n method t o e s t i m a t e v/hich should  be a p p l i e d t o t h e I n i t i a l  a symmetric  orbit.  Suppose t h a t f o r two t r i a l s y, (0) and y ( 0 ) o  the values  y (x  are. a t t a i n e d a f t e r a p p r o x i m a t e l y mate o f t h e c o r r e c t i n i t i a l  If  yi  y (0)  correct I n i t i a l  ?  y(0)  with I n i t i a l 0  m  ;)  half the period.  to obtain  velocities respectively Then a n  esti-  v e l o c i t y f o r a symmetric s o l u t i o n i s  y(o).= (o) - (x ^ : 3  the correction  velocity  •) and y ( x  :  Hence we use  yi  "  yo(o) 2 ' K  y  ax  2<  I s n o t an a c c u r a t e  x  - (o)  1  "' yi I '  max^ -  enough e s t i m a t e  of the  v e l o c i t y f o r a s y m m e t r i c s o l u t i o n , t h e n we  i n t e r p o l a t e again,, t h i s t i m e u s i n g  (y.(0), y (x  ))  ,  q u a d r a t i c I n t e r p o l a t i o n on t h e  I = l,2  j 5  50.  and  so o n u n t i l  2.3'  the c o r r e c t i n i t i a l c o n d i t i o n s are found.  The P e r t u r b a t i o n E q u a t i o n s (2.3)  Subtracting equations  f r o m ( 2 . 1 ) we o b t a i n t h e  per tu r b a t I o n e quatIons  d s  .pdjQ  2  —  -  z  ,  ^  -r  -  5  ,  x  O x  x  ,  x  * - l - - - 5 ) -.-  ,x-lx  - u(-^f) »  5  o  (2.4)  -2 dt where  x  o  , 2  d t  and y -o  v  '  n  r  p  ^ r  5  of  the previous section.  Summary o f t h e Method f o r C a l c u l a t i n g the' P e r i o d and t h e I n i t i a l Values of the O r b i t a l Elements it  axis  'P  a r e o b t a i n e d -from t h e s o l u t i o n o f t h e two  body p r o b l e m ( e q u a t i o n s (1.5))  2.4  y  e, o  I s more c o n v e n i e n t t o s t a r t w i t h t h e semi  and t h e e c c e n t r i c i t y  e  o  major  as o r b i t a l elements, since 3  these a r e t h e elements used I n the s o l u t i o n o f e q u a t i o n s for  (1.5)  t h e two body p r o b l e m . The p e r i o d and. t h e I n i t i a l  values are calculated  a c c o r d i n g t o t h e following scheme. 1.  After the eccentricity  e  i s c h o s e n , we c a l c u l a t e t h e  o 'following f o r the unperturbed motion a)  The mean angular m o t i o n of n  o  = 3n' = 3 ' '  P  b)  The s e r a i - m a j o r a x i s O n  n a"" - k o o  p  The p e r i h e l i o n  d)  The I n i t i a l  distance  * -  The e x p r e s s i o n f o r  v  ~  = [a(l-e )] 2  of  P  S)  0  ) =  y (0)  P  In a fixed  as  follows.  ,  y = r sin(v-t)  ,  v  y = r sin(v-t) + r ( v - l ) cos(v-t) . t = 0  we  have  = x (0) = a ( 1 - e J .  = 0 ,  v  coordinate  and' f r o m t h e e x p r e s s i o n s  x = r-cos(v-t). - r ( v - l ) s i n ( - t ) ,  Q  0  ,  2  1 l r '/-, 2\ T 2 rv = -La(l-e )J >  r  (^-~ ) e  0  I s obtained  n  At time  8 j  o(°)  i n the r o t a t i n g coordinate  x = r cos(v-t) have  x  i s t h e t r u e anomaly of  system w i t h o r i g i n a t  we  (  second l a w 2  coordinates  o  velocity  0  (where  x  '~l+e  y (o)  r  from  = 1  c)  From K e p l e r s  a,  - t = 0 ,  x (0) = 0 Q  f o r the  s y s t e m o f S e c . 2.1  r  \) - r o •o o o  U^l-e^)]  o 2  - a (l-e )  (1-eJ  1+e  2.  • ~i  °  2  o  The p e r t u r b a t i o n e q u a t i o n s  i c a l l y u s i n g any s u i t a b l e method. are 'obtained X  where  =  O  r ^ and  a r e i n t e g r a t e d numei  The u n p e r t u r b e d  T  O  COs(v-t) \  '  •o r  J  are obtained  v  coordinates. "  sin( -  i  v  from the s o l u t i o n of equations  S  -5  y = yo  x  n  2  , 2 + y  c o n d i t i o n s ' used I n t h e I n t e g r a t i o n a r e  -= ,(0) = 5 ( 0 ) ; = Tl(0) = 0 . r  p.  I f the o r b i t  corresponding  to  y„(0)  I s not symmetric,  o t h e n a second t r i a l y-j-(O)  a  n  ( l  c o o r d i n a t e s a r e g i v e n by  x = .x o The I n i t i a l  (2.4)  from  The p e r t u r b e d  g('o)  °  . • We  I s made w i t h , a d i f f e r e n t I n i t i a l  calculate l+e  . from 1-  1' .  :om  ,(0) = a (l-ej , r  velocity  Jj-  a,nd  from  - \1~1  - • 1  The p e r t u r b a t i o n e q u a t i o n s ditions and  v  with  are integrated with i n i t i a l  | ( 0 ) = n(0) = p(0) = -n(O) = 0 are obtained  .  The v a l u e s  from the s o l u t i o n of equations  con-  of  r  Q  (1.5)  e = e^, a = a ^ , n = n ^ „  4.  Further  corrections to  y(0)  are then  estimated  u s i n g i n t e r p o l a t i o n , u n t i l a symmetric (hence p e r i o d i c ) s o l u t i o n , has b e e n f o u n d . 5.  Once a p e r i o d i c o r b i t has b e e n f o u n d , t h e p e r i o d i s  c a l c u l a t e d as f o l l o w s .  Suppose t h e o r b i t I n t e r s e c t s t h e  x - a x l s a t r i g h t a n g l e s when t h e a n o m a l y o f J u p i t e r has t h e value  v' = Tf + a ,  the period  i s given  v/here  a  i s of the order  of  u .  Then  by  T = °v ( 1 + ^ ) o ~ TT v  2.5  J  Numerical Calculations The p e r t u r b a t i o n e q u a t i o n s  as a system o f f o u r f i r s t  order  (2.4)  differential  were  expressed  equations.  54  dt - " = y. ,  Ori  /  y  y  oN ,  1These  integrated  we.72  10  s t e p was l e s s t h a n grated  A  enough so t h a t  z(t*h)  ) , [ 2 , pp. 20-24].  = z(t)-•+ h [ i •+ |y+ Z  h  I s theInterval size,  o p e r a t o r and  t h e e r r o r a t each  They were a l s o  Inte-  t h e backward  differ-  [ 2 , p. 2 9 ] .  z(t-i-h) = ( t ) + h [ i -  where  pro-  -  "by a p r e d i c t o r - c o r r e c t o r method u s i n g  ence f o r m u l a e  first  r-  "by t h e s t a n d a r d R i m g e - K u t t a  cess ( w i t h i n t e r v a l s i z e small -6  y  y_  z  ^v  + Iy  ^  -  2  V  5  + 251^- _ +  - yllv  :  ^  ...]z>(t+h) ,  I s t h e backward  r e p r e s e n t s any o f t h e v a r i a b l e s  ] z T ( t )  difference  ^^-o^c, r\  The  }  f o r m u l a I s t h e p r e d i c t o r and t h e s e c o n d t h e c o r r e c t o r .  s i z e f o r t h e p r e d i c t o r - c o r r e c t o r ' method was c h o o s e n s m a l l  The  enough  -6 so  that  t h e c o r r e c t o r would.converge t o w i t h i n  10.  i n less  than f i v e I t e r a t i o n s . The two n u m e r i c a l methods w e r e ' c o m p a r e d f o r t h e c a s e s e  Q  = . 0 5 , . 2 0 , .40 a n d w e r e f o u n d t o a g r e e v e r y w e l l .  predictor-corrector and  was t h e r e f o r e  The  method was f a s t e r t h a n t h e R a n g e - K u t t a method  used I n t h e c a l c u l a t i o n s .  The f o l l o w i n g i n t e r v a l s i z e s w e r e u s e d I n t h e c a l c u l a t i o n s .  2  I n t h e anomaly o f J u o i t e r f o r e < . 4 0 , ° " o —  1°  for  .40  0.5°  for  .55 < e  0.25°  f o r .70 < e  After f o r the cases  e  themselves.  Q  < .70 ,  Q  <_ .80 .  = .20, .40. .60 '  t o see I f t h e o r b i t s  A c h e c k was a l s o  (2.2)  and I t was f o u n d t h a t  they  did.  results of the calculations are given i n the  table.  compare w i t h p a r t  The p e r i o d s a r e g i v e n i n mean s o l a r d a y s t o I .  The u n i t s  of the other q u a n t i t i e s a r e  those s p e c i f i e d a t the beginning of part The  closed  cases t o see I f t h e o r b i t s s a t i s f i e d t h e  Jacob! i n t e g r a l  following  ,5'5 ,  I n a l l three cases they d i d .  made I n s e v e r a l  The  Q  t h e s y m m e t r i c o r b i t s w e r e f o u n d a c h e c k vras mado  on  <e <  II .  c a l c u l a t i o n s f o r the unperturbed motion  a o = .480597 and To = 4 3 3 2 . 5 8 8 - .  yield  TABLE.IX Initial  n  : n' = 3  e  ^o  V a l u e s f o r a Symmetric  :  1  = 0  a  a.  Solution  -x (0)  T  o  y(0)  y (°) 0  .05  . 0499  .48055  .10  .0999  .43054  .15  .1498  .20  4343 "  .456567  1.05993  1.05986  .432537.  1.16218  1.16211  .48048  4346  .408507  i . 269 33  1.26918  .1996  .48036  4346  .384478  1.38219  1.38190  •25  .2492  .48009  .360448  1.50178  1.50119  •30  .2989  .47984  4343  .336418  1.62935  1.62852  .47964  4343  J  • 35  •  .312388  1.76645  •  1.76544  .40  .3985  .47940  4 34 3  .288358  1.91506  1.91388  .45  '.4483  .47912  4342  .264328  2.07781  2.07643  .50  .4981'  .47878  4343  .240298  2.25815  2.25657  • 55  .5480  .47824  4343  .216269  2.46086  2.45912  . 6o  .5978  .47797  4342  .192239  2.69272  2.69074  .64?6  .47732  4342  .168209  2.96263  2.96148  .70  .69736  .47647  4341  .144179  3.28816  3.28693  .75  .74711  .47527  4340  .120149  3.69474  3.69317  .80  .79683  4340  .096119  4.23138  4.22781  .05  .  •  .47489  57.  BIBLIOGRAPHY 1)  B r o u w e r , D. ' and  C l e m e n c e , G,M„  M e c h a n i c s - New 2)  Fox,  L.  - Numerical  Differential  S c h v m r z s c h i l d , K. des  4)  Addison  Wesley,  Partial  Pergamon P r e s s  -  1962,  '  - Uoer e i n e C l a s s e p e r i o d ! s c h e r  Dreikorperproblems  S e h w a r z s c h i l d K.  - Oxford;  Celestial 1961.  Academic P r e s s ,  S o l u t i o n o f O r d i n a r y and  Equations  R e a d i n g , Mass: 5)  York:  - Methods of  ft  Losimgen 17-24.  - A s t r , Nachr. 1 4 7 ( 1 8 9 8 ) ,  - Uber w e i t e r e C l a s s e n p e r i o d i s c h e r  Losungen 'es Dreikbrperproblems  - A s t r . Nachr.  148(1898),  289-298.  5)  S c h w a r z s c h i l d , K.  - Uber d i e p e r i o d i s c h e n Bahnen  Hecubatypus - A s t r . Nachr. 1 6 0 ( 1 9 0 3 ) ,  285-400.  vom  

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