SOME ASPECTS OF THREE AND FOUR-BODY DYNAMICS by PETER G.D. BARKHAM 0. B.Sc. ( E n g . ) , Southampton U n i v e r s i t y , 1967 M . A . S c , U n i v e r s i t y of B r i t i s h Columbia, • " 1969 / I A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of E l e c t r i c a l Engineering We accept t h i s t h e s i s as conforming to required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1974 the In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the 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 reference and study. I f u r t h e r agree that permission f o r extensive coyping of 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 granted by the Head of my Department or by his r e p r e s e n t a t i v e s . . It i s understood that p u b l i c a t i o n , i n part or i n whole, or the copying of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. PETER G.D. BARKHAM Department of Electrical Engineering The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada Date W • 117*. ii ABSTRACT Two fundamental problems of c e l e s t i a l mechanics are considered: the s t e l l a r or planetary three-body problem and a r e l a t e d form of the r e s t r i c t e d four-body problem. Although a number of c o n s t r a i n t s are imposed, no assumptions are made which could i n v a l i d a t e the f i n a l s o l u t i o n . A c o n s i s t e n t and r a t i o n a l approach to the a n a l y s i s of four-body systems has not p r e v i o u s l y been developed, and an attempt i s made here to describe problem e v o l u t i o n i n a systematic manner. In the p a r t i c u l a r three-body problem under c o n s i d e r a t i o n two masses, forming a close binary system, o r b i t a comparatively d i s t a n t mass. A new l i t e r a l , p e r i o d i c s o l u t i o n of t h i s problem i s found i n terms of a small parameter e, which i s r e l a t e d to the distance separating the binary system and the remaining mass, using the two v a r i a b l e expansion procedure. O(e^) The s o l u t i o n i s accurate w i t h i n a constant error and uniformly v a l i d as e tends to zero f o r time i n t e r v a l s 0(e- ). 1 4 Two s p e c i f i c examples are chosen to v e r i f y the l i t e r a l s o l u t i o n , one of which r e l a t e s to the sun-earth-moon c o n f i g u r a t i o n of the s o l a r system. The second example a p p l i e s to a problem of s t e l l a r motion where the three masses are i n the r a t i o 20 : 1 : 1. In both cases a comparison of the a n a l y t i c a l s o l u t i o n with an equivalent n u m e r i c a l l y generated o r b i t shows .close agreement, with an e r r o r below 5 percent f o r the sun-earth-moon c o n f i g u r a t i o n and l e s s than 3 percent f o r the s t e l l a r system. iii The four-body problem i s derived from the three-body case by introducing a p a r t i c l e of n e g l i g i b l e mass i n t o the close binary system. Unique uniformly v a l i d s o l u t i o n s are found f o r motion near both e q u i l a t e r a l t r i a n g l e points of the binary system i n terms of the small parameter e, where the primaries move i n accordance with the u n i f o r m l y - v a l i d three-body s o l u t i o n . Accuracy, i n t h i s case, i s Q maintained w i t h i n a constant e r r o r 0(e ), and the s o l u t i o n s are uniforml y v a l i d as e tends to zero f o r time i n t e r v a l s 0 ( e ~ ^ ) . p o s i t i o n errors near and Orbital of the earth-moon system are found to be less than 5 percent when numerically-generated p e r i o d i c s o l u t i o n s are used as a standard of comparison. The approach described here s h o u l d , i n g e n e r a l , be useful i n the a n a l y s i s of non-integrable dynamic systems, p a r t i c u l a r l y when i t i s f e a s i b l e to decompose the problem i n t o a number of s u b s i d i a r y cases. iv TABLE OF CONTENTS Page ABSTRACT ii TABLE OF CONTENTS iv LIST OF TABLES . . LIST OF FIGURES . . . . vii . ACKNOWLEDGEMENTS viii . . . . . . . . . . . . . . . . . . ix Chapter 1. 2. 3. INTRODUCTION . 1 . 1 1.1 Background 1.1 1.2 Methodology 1.7 PERIPHERAL MATERIAL 2.1 2.1 Introduction . . . . . 2.2 L i m i t a t i o n s and R e s t r i c t i o n s 2.3 N-body Dynamics 2.4 nomographic Solutions 2.5 Two-body Motion 2.6 N o n - i n e r t i a l Systems of Reference 2.7 Integrals i n Rotating Systems of Reference . . . . . • • . . . . . . . . . . . . . . . . . . . . . . 2.1 2.1 . . . . . . 2.8 . . . 2.14 . . . . . . .. . . . . . . . . . . . . LIMIT PROCESS CONSIDERATIONS . . . . . . . . . . . 2.16 2.24 2.32 . . . . 3.1 . . . . . . 3.1 3.1 Introduction . . . . . . . . . 3.2 Reduction of the 3-body Problem . . . . . 3.3 Reduction of the 4-body Problem . . . . . . . 3.3 3.10 V Chapter ' 3.4 Small-parameter 3.5 Exact Solutions 3.6 O r b i t a l Perturbations 3.7 4. Expansions 3.13 . 3.15 . . 3.19 3.6.1 Perturbed 2-body motion 3.19 3.6.2 Perturbed 3-body motion . 3.21 The Two-variable Expansion Procedure 3.7.1 Definitions 3.7.2 M u l t i p l e s c a l e methods THE THREE-BODY PROBLEM 3.26 . 3.28 3.34 . 4.1 4.1 Introduction 4.1 4.2 Preliminary A n a l y s i s . . . . . . . . . . . . . . 4.2 4.3 The Uniformly V a l i d S o l u t i o n . . 4.10 4.4 A Uniformly V a l i d S o l u t i o n of the Problem . •. . . 4.5 5. Page Restricted • • 4.4.1 H i l l ' s s i m p l i f i c a t i o n of the problem 4.4.2 H i l l ' s variation orbit 4.4.3 Further terms i n the s o l u t i o n 4.26 restricted . . . . . . 4.28 . . . . . . . . . 4.34 4.38 Discussion . . 4.44 THE FOUR-BODY PROBLEM, 5.1 5.1 Introduction ^.1 5.2 The R e s t r i c t e d Three-body Problem 5.3 Four-body Motion Near L^ and L^ 5.4 Uniformly V a l i d Solutions 5.5 Discussion. . . . . . . . 5.2 . . . . . . . . . . . 5.9 . . . . . . . . . . . . . . . 5.21 • • . • • • 5.42 vi Chapter 6. Page SPECIFIC SOLUTIONS 6.1 Introduction 6.2 Three-body Orbits 6.2.1 6.2.2 7. . . 6.1 . . . . . . . 6.1 . 6.1 P e r i o d i c s o l u t i o n s of y ( t ) = B(t)y + f ( t ) 6.4 I t e r a t i v e method to determine a T-periodic orbit 6.7 6.2.3 P e r i o d i c earth o r b i t 6.9 6.2.4 P e r i o d i c o r b i t s f o r s t e l l a r motion . . . 6.16 6.3 Four-body Orbits Near L 6.4 Discussion 4 and Lg . . . . . . . . . . . . 6.29 CONCLUSION REFERENCES APPENDICES . . . . . . . . . 6.21 7.1 .. .'. . ' : . . . . . . . . . • . R1 vi i LIST OF TABLES Table 3-1 p a 9 e O r b i t a l Motion i n the n-body Problem f o r 2 ^ n < 4 3 - 2 6 viii LIST OF FIGURES Figure Page 2-1 I n i t i a l c o n f i g u r a t i o n f o r the n-body problem . . . . 2.9 2- 2 Primary c o n f i g u r a t i o n and coordinate systems . . . . 2.26 3- 1 Regions of motion f o r p and p^ 3.4 3-2 Lagrange points of the r e s t r i c t e d problem 3- 3 Development of the four-body problem . 3.27 4- 1 Primary c o n f i g u r a t i o n f o r the r e s t r i c t e d problem . . 4.27 4-2 Primary c o n f i g u r a t i o n , with respect to the 5*, H* coordinate system, f o r the r e s t r i c t e d problem 4.30 6-1 P e r i o d i c earth o r b i t about £ = 0 . 0 1 2 1 5 0 , 6.12 6-2 P o s i t i o n e r r o r s f o r the earth o r b i t over one period 6.14 Numerical and a n a l y t i c a l s o l u t i o n s about C =0.012150, n =0.0 6.15 6-3 2 2 6-4 • • • 2 . . . . . n 0-0 • • .' 2 = 2 P e r i o d i c o r b i t f o r y - ^ l O , y = 0 . 5 , r,=50 about. £ = 0 , 5 , n =0.0 . . . . . . . . . . ! . . . . . . 9 2 2 6-5 P o s i t i o n errors f o r the s t e l l a r o r b i t over one period • • • • • 6-6 P e r i o d i c o r b i t s near L^ i n the four-body problem . . 6-7 3.18 . . 6.18 6.25 P e r i o d i c o r b i t s near L^ i n the four-body problem . . . . . . . . . . . . . . 6.26 6-8 P o s i t i o n errors f o r the o r b i t near L^ . . . . . . . 6.27 6-9 P o s i t i o n errors f o r the o r b i t near L5 6.28 • • ix ACKNOWLEDGEMENTS I t i s a pleasure to acknowledge the advice and encouragement of my s u p e r v i s o r s , Dr. V . J . Modi and Dr. A . C . Soudack, during the i n c e p t i o n and presentation of t h i s work. The manuscript was typed by Mrs. M. E l l i s , and my wife Anne spent many hours preparing the mathematics f o r p r i n t i n g . This research was supported by Northern E l e c t r i c , the U n i v e r s i t y of B r i t i s h Columbia and the National Research Council of Canada under grants A-2181 and A-3138. DEDICATION To my parentsj whose generosity made this work and interest possible. 1 T.l 1. 1.1 INTRODUCTION Background The n-body problem of c e l e s t i a l mechanics may be d e f i n e d , quite simply, as f o l l o w s : N p a r t i c l e s of a r b i t r a r y mass a t t r a c t each other according to the Newtonian law of g r a v i t a t i o n and are free to move i n space. I n i t i a l l y they move i n any given manner; determine t h e i r subsequent motion. No general s o l u t i o n of t h i s problem e x i s t s . Only the two-body problem i s considered s o l v e d , because properties of the t o t a l i t y of s o l u t i o n s are known, and even i n t h i s case the o r b i t a l possible coordinates cannot be represented as e x p l i c i t closed-form functions of time. Methods which have been applied to i n v e s t i g a t e n-body motion can broadly be divided into three c a t e g o r i e s : and f o r m a l i s t i c dynamics. qualitative, quantitative Q u a l i t a t i v e methods may be employed, f o r example, to determine conditions f o r s t a b i l i t y , s o l u t i o n existence and bounded motion, but are not u s u a l l y h e l p f u l i n e s t a b l i s h i n g solutions. specific T h i s , i n g e n e r a l , i s the domain of q u a n t i t a t i v e or e x p e r i - mental dynamics. Again, however, we encounter a s i g n i f i c a n t l i m i t a t i o n , as the accumulated e f f e c t of q u a n t i z a t i o n e r r o r over large i n t e r v a l s of time may i n v a l i d a t e any r e s u l t s derived from a numerical process. integration Information about system behaviour along the e n t i r e a x i s cannot therefore be obtained from a purely q u a n t i t a t i v e time approach. 1.2 P a r t i c u l a r s o l u t i o n s of a p e r i o d i c or asymptotic nature have, f o r t h i s reason, assumed a p o s i t i o n of central importance i n c e l e s t i a l mechanics, and they provide f o r m a l i s t i c dynamics with i t s nuclear s t r u c t u r e . The a n a l y s i s presented here i s concerned s p e c i f i c a l l y with p e r i o d i c o r b i t s i n the three and four-body problems. No severe r e s t r i c t i o n s are imposed on the p a r t i c i p a t i n g masses of the three-body problem, but i t i s assumed that two of the bodies form a close system i n o r b i t about the t h i r d . This c o n f i g u r a t i o n w i l l be described as the s t e l l a r three-body problem, although i t may apply e q u a l l y to g a l a c t i c or planetary systems. The four-body case i s derived from the stellar three-body c o n f i g u r a t i o n by i n t r o d u c i n g an a d d i t i o n a l p a r t i c l e o f n e g l i g i b l e mass i n t o the close binary system. There i s no d i f f i c u l t y i n r e l a t i n g these models to our s o l a r system; the s t e l l a r problem, f o r example, i s a g e n e r a l i z a t i o n of the main problem of lunar theory, and the four-body model i s of considerable current i n t e r e s t i n s a t e l l i t e dynamics. The p e r i o d i c o r b i t s i n question emanate from homographic, or c o n f i g u r a t i o n - p r e s e r v i n g , s o l u t i o n s of the two and three-body problems. A l l s o l u t i o n s of the two-body problem a r e , i n f a c t , homo- g r a p h i c , but the p a r t i c u l a r example chosen here corresponds to c i r c u l a r o r b i t s i n an i n e r t i a l coordinate system. A r o t a t i n g system o f axes i s then selected i n which the two bodies appear to be s t a t i o n a r y . The three-body c o n f i g u r a t i o n i s a special case of Lagrange's homographic s o l u t i o n f o r the general three-body problem, where one of the masses i s n e g l i g i b l e and the remaining two move i n c i r c u l a r o r b i t s . The 3 • 1.3 r e s u l t i n g s i t u a t i o n i s known as the r e s t r i c t e d problem. Five homographic t solutions exist where the three bodies move i n coplanar o r b i t s ; the p o s i t i o n s of r e l a t i v e e q u i l i b r i u m f o r an i n f i n i t e s i m a l mass i n the r o t a t i n g coordinate system are c a l l e d Lagrange, l i b r a t i o n or e q u i l i b r i u m p o i n t s , and are u s u a l l y denoted by L-j, L£, •••» L ^ . points (L-,, Three of these and Lg) are located on the l i n e j o i n i n g the two p r i m a r i e s , while the remaining two form e q u i l a t e r a l t r i a n g l e s with the primary bodies at two of the v e r t i c e s . A survey of e x i s t i n g l i t e r a t u r e r e l a t e d to the r e s t r i c t e d problem can be found i n Szebehely's a u t h o r i t a t i v e t e x t •[!]. Major reviews of the general three-body and n-body problems by Lovett [2] and several other authors appeared between 1896 and 1919 [ I ] . Recent 1 work i s mentioned i n a. remarkable c o n t r i b u t i o n to Burrau's problem by Szebehely [3]. Motion i n the v i c i n i t y of L^ and L ^ , at l e a s t f o r an i d e a l i z e d earth-moon system, i s s t a b l e . This s t a b i l i t y i s disturbed by the g r a v i t a t i o n a l influence of a fourth body, but there s t i l l remains the p o s s i b i l i t y of p e r i o d i c or bounded motion i n these regions. By analogy with the three-body case, a four-body problem i n v o l v i n g one i n f i n i t e s imal mass i s known as the r e s t r i c t e d four-body problem, although no s p e c i f i c primary motion i s implied by t h i s d e s c r i p t i o n . Celestial mechanics entered a period of rapid development towards the c l o s e of the nineteenth century, f o l l o w i n g the fundamental researches of H i l l and Poincare, and i n t e r e s t i n the. four-body problem can be traced to These s o l u t i o n s are shown i n Figure 3-1 on page 3.18, q . v . k 1.4 t h i s era; Perhaps the f i r s t s i g n i f i c a n t i n v e s t i g a t i o n was by Moulton [ 4 ] , who considered a r e s t r i c t e d four-body problem i n which the three primaries move according to Lagrange's c i r c u l a r homographic s o l u t i o n s o f . t h e general problem. Twenty-eight homographic c o n f i g u r a t i o n s were found a f t e r formulating an i n t e g r a l of motion f o r the i n f i n i t e s i m a l mass. Moulton's r e s u l t s were extended by Lovett [5] to include p e r i o d i c motion of the p a r t i c l e near these p o s i t i o n s of r e l a t i v e e q u i l i b r i u m . More r e c e n t l y Huang [6] proposed a model that has since been described as the v e r y - r e s t r i c t e d four-body problem, i n which the three primaries move i n c i r c u l a r coplanar o r b i t s . The d e f i c i e n c i e s of Huang's formulation are now widely accepted, but a s u b s t a n t i a l number of numerical and a n a l y t i c a l studies were based on the model. very-restricted The choice of primary motion therefore leads to a dichotomy i n e x i s t i n g methods of a n a l y s i s , and those r e s u l t s which depend on Huang's model are grouped immediately below. DeVries [7] found a n a l y t i c a l s o l u t i o n s f o r a l i n e a r i z e d f o u r body problem near L ^ , and compared the predicted motion with numerical solutions. Some a m p l i f i c a t i o n of these r e s u l t s appeared i n a review of l i t e r a t u r e r e l a t i n g to Lagrange points by Steg and deVries Tapley et al. [9,10,11,12] used numerical methods to [8]. investigate motion i n the v i c i n i t y of L^ f o r a three-dimensional case i n which the earth-moon o r b i t a l plane i s i n c l i n e d at 5.15° to the e c l i p t i c . Cronin et. al. [13,14] considered three-dimensional motion of the f o u r t h p a r t i c l e , and derived-existence conditions f o r p e r i o d i c o r b i t s Lagrange p o i n t s . near In [15] Wolaver examined the e f f e c t of i n i t i a l conditions on motion near L A and L q using l i n e a r i z e d equations. 5 1.5 Bernstein and E l l i s [16] derived l i n e a r i z e d equations of motion and applied Floquet theory to determine a necessary and s u f f i c i e n t c o n d i t i o n f o r bounded motion about L ^ . Kolenkiewicz and Carpenter [17] used numerical perturbation techniques to f i n d p e r i o d i c and a l m o s t - p e r i o d i c o r b i t s near L ^ . A v a r i a t i o n of Huang's formulation has been considered by Matas [18], who extended the r e s u l t s i n [14] to accommodate e l l i p t i c primary o r b i t s and the e f f e c t of s o l a r r a d i a t i o n pressure on s a t e l l i t e motion near earth-moon Lagrange p o i n t s . The a n a l y s i s was f u r t h e r modified [19] to include the e f f e c t of a r e s i s t i n g medium. In [20] the same author investigated the motion of near-lunar s a t e l l i t e s when the primaries move i n e l l i p t i c o r b i t s and the influence of s o l a r r a d i a t i o n pressure i s taken i n t o account. Luk'yanov [21] has considered satellite motion near the e q u i l a t e r a l t r i a n g l e points when the primary bodies form two n o n - i n t e r a c t i n g binary systems, obtaining r e s u l t s s i m i l a r to those given i n [ 8 ] . The p o s i t i o n c o n t r o l of a s a t e l l i t e near a Lagrange point has been investigated by Wolaver et al. [ 2 2 ] , who described the development of an optimal c o n t r o l l e r based on l i n e a r i z e d equations and the very r e s t r i c t e d primary model. This system was tested f o r a r e a l i s t i c model of primary motion, and provided e x c e l l e n t control over long periods of time. Reservations about v a l i d i t y of the v e r y - r e s t r i c t e d primary model were expressed by Szebehely [23] and a l s o Danby [ 2 4 ] , who proposed an a l t e r n a t i v e approach to account f o r secular p e r t u r b a t i o n s . The p r i n c i p a l d i f f i c u l t y i s to f i n d a s u i t a b l e d e s c r i p t i o n of primary motion, and a number of a n a l y t i c a l models have since been employed. Mohn and Kevorkian [25] presented the formulation of four-body equations of motion corresponding to the lunar theories of de Pontecoulant and H i l l , with the i n t e n t i o n of determining asymptotic s e r i e s s o l u t i o n s near Lagrange p o i n t s . p t o t i c expansions primary model. Shi and Eckstein [26] used-.-.^matched -asym- to determine earth-moon t r a j e c t o r i e s f o r a r e a l i s t i c These a n a l y t i c a l s o l u t i o n s were subsequently v e r i f i e d by Kevorkian and Brachet [27]. In [28] Schechter extended an e a r l i e r nonlinear a n a l y s i s by Breakwell and P r i n g l e [29] to i n v e s t i g a t e three- dimensional s t a b i l i t y of the sun-perturbed Lagrange points L^ and Lg. A stable e l l i p t i c o r b i t was found, s i m i l a r i n some respects to a numeri c a l s o l u t i o n determined by Kolenkiewicz and Carpenter [30] and published at the same time. This numerical r e s u l t was l a t e r d u p l i c a t e d a n a l y t i c a l l y by Kamel and Breakwell [31]. employed a d i f f e r e n t primary model, Note however, that Schechter so that exact agreement between the two o r b i t s should not have been expected. The a n a l y t i c a l obtained by Luk'yanov [21] were extended i n [32] solutions to include the i n d i r e c t e f f e c t of s o l a r g r a v i t a t i o n on motion of the earth-moon system. Nicholson [33] has examined behaviour near the col l i n e a r e q u i l i b r i u m points f o r an i n t e r a c t i n g primary model to p r e d i c t station-keeping requirements at the translunar and c i s l u n a r points L ^ a n d l^. and Franca [34] i n t o the Giacaglia have reported some progress i n a s i m i l a r i n v e s t i g a t i o n evolution of e q u i l i b r i u m points of the orbits near earth-moon sun-perturbed system. col l i n e a r Giacaglia [35] a l s o proved an important q u a l i t a t i v e r e s u l t i n the r e s t r i c t e d body problem where three primary masses are located at the has four- vertices of an e q u i l a t e r a l t r i a n g l e according to Lagrange's c i r c u l a r homographic configuration. A r e g u l a r i z i n g transformation i s found f o r t h i s case, and extended to an (n+1) + cf. page 6.22. body problem i n which n equal masses move i n 7 1.7 a regular polygonal c o n f i g u r a t i o n . The numerical a n a l y s i s of f o u r - body motion by Tapley et al. [9,10,11,12] has been extended by Tapley and Schutz [36] to a r e a l i s t i c s i t u a t i o n which includes the i n d i r e c t solar e f f e c t . Bounded motion i n a region about was found to be possible f o r a period exceeding 5000 days. Many of the r e s u l t s i n d i c a t e d here are discussed by Szebehely [l ] 2 i n what i s perhaps the most comprehensive survey of motion near e q u i l i b r i u m points of the r e s t r i c t e d problem. 1.2 Methodology These diverse r e s u l t s do. not, u n f o r t u n a t e l y , a f f o r d any s i g n i f i - cant i n s i g h t i n t o the general" behaviour of four-body systems. No coherent process of development has been adopted, which i s s u r p r i s i n g f o r a problem of such inherent complexity, and as a consequence no detectable patterns have emerged. This a n a l y s i s takes, as i t s p r i n c i p a l o b j e c t i v e , the d e s c r i p t i o n and development of a systematic a n a l y t i c a l approach to the four-body problem. Some of the concepts mentioned i n Section 1.1 are a m p l i f i e d i n Chapter 2. Much of t h i s material i s a v a i l a b l e i n standard t e x t s , but i t was f e l t that an ommission of these fundamental r e s u l t s would detract from the subsequent a n a l y s i s . In Chapter 3 the p a r t i c u l a r three and four-body problems under consideration are decomposed i n t o a number of s u b s i d i a r y cases. The process of r e - c o n s t r u c t i o n leads to a new d e s c r i p t i o n of system s t r u c t u r e , and a r a t i o n a l approach to both the s t e l l a r three-body and r e s t r i c t e d four-body problems now becomes p o s s i b l e . 8 1.8 One p a r t i c u l a r d i f f i c u l t y , known i n c e l e s t i a l mechanics as the problem of small d i v i s o r s , i s frequently encountered when nonlinear systems are studied using q u a s i - l i n e a r methods. The associated resonance phenomena correspond to system d i s i n t e g r a t i o n , but as t h i s r a r e l y happens i n p r a c t i c e the small d i v i s o r problem i n d i c a t e s a l i m i t a t i o n i n the a n a l y t i c a l approach. M u l t i p l e scale perturbation methods have r e c e n t l y been developed to overcome r e s t r i c t i o n s of t h i s nature, and a twov a r i a b l e expansion procedure i s applied i n the present a n a l y s i s to solve the nonlinear equations of three and four-body motion. The d e r i v a t i o n of a unique u n i f o r m l y - v a l i d s o l u t i o n of the s t e l l a r three-body problem i s described i n Chapter 4, and i n Chapter 5 r e l a t e d s o l u t i o n s of the r e s t r i c t e d four-body problem are determined f o r motion i n the v i c i n i t y of and Lg. These l i t e r a l s o l u t i o n s are compared with numerically-generated p e r i o d i c o r b i t s f o r two d i s t i n c t primary models, one corresponding to the sun-earth-moon c o n f i g u r a t i o n of our s o l a r system, the other associated with a problem of s t e l l a r motion. 9 2- 2. 2.1 1 PERIPHERAL MATERIAL Introduction ' Before we embark on a d e t a i l e d study of the three and four-body problems, some general perspective of c e l e s t i a l mechanics i s desirable. Six topics of p a r t i c u l a r importance are c o l l e c t e d here, which serve both to introduce notation and to i n d i c a t e some of the basic s t r u c t u r e the n-body problem. of A number of standard d e r i v a t i o n s are quoted i n condensed form, p r i m a r i l y so that the present work may be s e l f - c o n t a i n e d ; o r i g i n a l sources a r e , i n t h i s case, i n d i c a t e d i n the t e x t . The p h i l o s o p h i c a l basis of Newtonian dynamics i s b r i e f l y considered i n Section Z.Z, witn the i n t e n t i o n O T exposing the assumptions i m p l i c i t i n n o n - r e l a t i v i s t i c mechanics. This i s followed by an i n t r o d u c - t i o n to some general features of the n-body problem, and by a d e r i v a t i o n of equations of motion with respect to n o n - i n e r t i a l axes. The chapter i s concluded with a short discussion of i n t e g r a l s i n r o t a t i n g systems of reference. 2.2 L i m i t a t i o n s and R e s t r i c t i o n s Phenomenon: A thing that appears, or is perceived applied chiefly or observed; to a fact or occurrence, of which is in question. the cause [Oxford] Throughout the subsequent a n a l y s i s we s h a l l be concerned with the motion of a s p e c i f i c number of planetary or s t e l l a r bodies w i t h i n 10 2.2 a p a r t i c u l a r region of the universe. however,' mathematical. The e n t i r e representation As such i t should possess complete i n t e r n a l con- s i s t e n c y , but t h i s i s no guarantee that t h e o r e t i c a l with observable r e a l i t y . is, results w i l l agree The extent of any such agreement i s determined by the mathematical d e s c r i p t i o n of physical phenomena, and a l l s i m p l i f y ing assumptions w i l l tion. lead to imprecision i n the process of representa- We therefore s t a r t t h i s i n v e s t i g a t i o n at the beginning, by examining those fundamental p r i n c i p l e s upon which the development i s to be constructed. Although better models of r e a l i t y now e x i s t , t h i s a n a l y s i s takes as i t s foundation the laws of Newtonian mechanics. Law I. These are [37 ]: 1 Every body continues i n i t s s t a t e of r e s t , or of uniform motion i n a r i g h t l i n e , unless it i s compelled to change that s t a t e by forces impressed upon i t . Law II. The change of motion i s proportional to the motive force impressed; and i s made i n the d i r e c t i o n of the r i g h t l i n e i n which that force i s impressed. Law I I I . To every action there i s always opposed an equal r e a c t i o n : o r , the mutual actions of two bodies upon each other are always e q u a l , and d i r e c t e d to contrary p a r t s . In the d i s c u s s i o n below, these laws w i l l be referred to as the law of i n e r t i a , motion and e q u i l i b r i u m r e s p e c t i v e l y . i f we I t i s necessary, are to avoid a lengthy d i v e r s i o n i n t o philosophy (and, to some e x t e n t , h i s t o r y ) to accept c e r t a i n phenomena as being commonly understood. concepts of r e a l i t y , p o i n t , s t r a i g h t (or r i g h t ) l i n e , plane, space, The 11 2.3 time, distance and force w i l l therefore not be questioned here. . Within a l o c a l region of the universe the laws of Newtonian mechanics generally produce t h e o r e t i c a l results in excellent agreement with o b s e r v a t i o n , but we should be aware at the outset of t h e i r implications. The study of dynamics, or motion i n time and space, i n i t i a l l y requires that the observer possess standards of time and distance measurement which do not a l t e r r e l a t i v e to his system of reference, f o r otherwise observations made under i d e n t i c a l conditions would be i n c o n s i s t e n t . In order to locate a s p e c i f i c event i n time and space, the observer must s e l e c t a s u i t a b l e time o r i g i n and reference coordinate system. I f the system o f reference i s d i f f e r e n t from h i s own, however, there i s no guarantee that an event which he observes appear i d e n t i c a l i n the other system Of reference. will This subtle con- s i d e r a t i o n i s i n t i m a t e l y connected with the process of transforming an event i n one system of reference i n t o a corresponding event i n another system. Such a transformation a f f e c t s not only the p o s i t i o n , but a l s o the time i n s t a n t at which the event occurred, and therefore the influence of both time and space on t h i s operation must be known. The two phenomena may be regarded e i t h e r as being independent or i n t e r dependent; Newton, who had no reason to suspect otherwise, assumed space and time to be independent. The l o g i c a l consequence o f t h i s assumption i s that time a f f e c t s a l l space e q u a l l y , a c o n d i t i o n which Newton defined as "absolute" time [ 3 7 ] . 2 The properties of space are i m p l i c i t i n the law of i n e r t i a . I f t h i s law i s to be v a l i d i n the observer's coordinate system, then that system must have c e r t a i n s p e c i a l p r o p e r t i e s ; i n p a r t i c u l a r the space 12 2.4 defined by the system of reference must be E u c l i d e a n , f o r otherwise r e c t i l i n e a r motion would be a meaningless concept. Suppose we have two systems of reference i n which the standards of time and distance measurement are i d e n t i c a l , one moving with respect to the other through Euclidean space. I f time i s an absolute, unchanging phenomenon i n a l l systems of reference, the mathematical process o f instantaneous t r a n s i t i o n between systems i s e n t i r e l y l e g i t i m a t e . Two points can therefore be transformed instantaneously from one system i n t o another without a f f e c t ing the length which they d e f i n e . Any r e l a t i v e motion of the two systems w i l l have no e f f e c t on the 'transformation, but only because the t r a n s i t i o n is instantaneous. When the r e l a t i v e motion i s known, a sequence of events observed i n one system can, by t h i s process, be transformed i n t o an equivalent sequence with respect to the other system of reference. If the r e l a t i v e motion i s i t s e l f uniform and r e c t i l i n e a r , a uniform v e l o c i t y i n one system w i l l i n the other system. transform i n t o a ( d i f f e r e n t ) uniform v e l o c i t y C o n s e q u e n t l y , . i f the law of i n e r t i a i s true i n the observer's coordinate system, i t w i l l a l s o be true i n a l l systems which move with uniform v e l o c i t y r e l a t i v e to the observer. Now unless the observer's system of reference i s free from a c c e l e r a t i o n , d e v i a t i o n s from the law of i n e r t i a w i l l be observed. This immediately i m p l i e s t h a t , somewhere i n space, there e x i s t s a f i x e d reference coordinate system associated with an unchanging or "absolute" space. A l l systems of reference i n which the law of i n e r t i a i s v a l i d must consequently be moving at uniform v e l o c i t y through t h i s absolute Euclidean space. Newton's law of motion i n t r o d u c e s , i n a d d i t i o n to time and d i s t a n c e , the concepts of force and mass. These two phenomena are 13 2.5 r e l a t e d by an a l t e r n a t i v e statement of the law of motion: a force F a c t i n g on a p a r t i c l e produces an a c c e l e r a t i o n d i r e c t l y proportional to that f o r c e . The constant, of p r o p o r t i o n a l i t y determines the i n e r t i a ! resistance o r "mass" of the p a r t i c l e , which must be an invariant.phenomenon i f the law of motion i s to be v a l i d of reference. w i t h i n the observer's system This d e f i n i t i o n of mass can be applied to determine a f u r t h e r q u a n t i t y , impulse, using the law of motion. We have, i n the d i r e c t i o n of the force F: F and hence = r*v •Lfc where pndv i s the change i n momentum, and the derived quantity J i s defined as impulse. I f two p a r t i c l e s c o l l i d e they w i l l exert impulsive forces on each other which, by Newton's law o f e q u i l i b r i u m , w i l l be equal and i n opposite d i r e c t i o n s . The sum of the two impulses w i l l consequently be z e r o , the t o t a l change i n momentum w i l l also be z e r o , and therefore momentum i s conserved during the c o l l i s i o n . Now mass time and distance must be i n v a r i a n t phenomena i f Newton's law of motion i to be u n i v e r s a l l y v a l i d , for otherwise changes.in momentum would depend on the observer's system of reference. No such r e s t r i c t i o n s required by the law of conservation of momentum, even though i t i s derived from the law of motion. r e l a t i y i s t i c dynamics. Therein l i e s i t s s i g n i f i c a n c e i n • I f the law of i n e r t i a i s v a l i d i n the observer's system of reference i t w i l l be true i n a l l systems moving at uniform v e l o c i t y are Ill 2.6 r e l a t i v e to the observer. This statement applies equally to the law of motion, but only i f mass i s assumed to be an i n v a r i a n t phenomenon. In g e n e r a l , transformations which preserve Newton's laws of i n e r t i a ! motion are of the form |* = fL J + S. t + ^ , (2.1) where ft i s a constant r o t a t i o n m a t r i x , a i s a constant vector d e f i n i n g the uniform r e l a t i v e v e l o c i t y and £ i s a constant displacement vector. A d e r i v a t i o n of t h i s transform, as i t applies to Newton's law of g r a v i t a t i o n , i s given by Wintner [ 3 8 ] . 1 The underlying assumptions of Newtonian mechanics can be summarized as f o l l o w s : 1. Time and space are independent phenomena. 2. Space i s Euclidean and absolute i n the sense that a f i x e d i n e r t i a ! reference must e x i s t . 3. Mass, time and distance are fundamental, i n v a r i a n t phenomena. These a b s t r a c t , but s i g n i f i c a n t , l i m i t a t i o n s influence the analysis. The remaining r e s t r i c t i o n s entire are not s t r i c t l y necessary, but s i m p l i f y the problem structure without departing too f a r from r e a l i t y . (i) The dominant mass of the system i s s u f f i c i e n t l y d i s t a n t that i t s motion i s unaffected by any i n d i v i d u a l motion of the remaining masses. This assumption i s unnecessary the study of 2-body motion. For more than two bodies, except i n c e r t a i n s p e c i a l i n s t a n c e s , little progress in 15 2.7 appears to be possible without i t . on pages 2.25 and 2.26 (ii) (See also the d i s c u s s i o n ). Only g r a v i t a t i o n a l f i e l d s r e s u l t i n g from the masses under consideration a f f e c t the motion of these bodies. The e f f e c t s of r a d i a t i o n pressure, t i d a l i n t e r a c t i o n , gravita- t i o n a l f i e l d s of d i s t a n t masses, l i b r a t i o n a l motion and a l l other complicating features w i l l (iii) be neglected, A l l the masses under consideration are r i g i d , bodies whose concentric l a y e r s are homogeneous. spherical Each may then be treated as a point mass, and Newton's law o f g r a v i t a t i o n can be a p p l i e d d i r e c t l y to determine motion of the point masses. I f the i n d i v i d u a l dimensions of the bodies are s m a l l , compared with the distances them, the inaccuracies negligible. separating introduced by t h i s assumption are This t o p i c i s covered i n d e t a i l by Danby [39 ]. 1 These three i d e a l i z a t i o n s , together with the i m p l i c i t assumptions of Newtonian mechanics, are p o t e n t i a l i r r e g u l a r i t i e s i n the foundation upon which t h i s analysis r e s t s . E i n s t e i n disproved Newton's basic assump- t i o n concerning the independence of time and space with the theory of s p e c i a l r e l a t i v i t y ; general r e l a t i v i t y f u r t h e r disproved the existence of absolute space, and i n d i c a t e d that space-time Euclidean continuum. theoretical predictions. i s not generally a The success of Newtonian mechanics as a v i a b l e basis r e s t s , - however, on the general accuracy of i t s Only at v e l o c i t i e s approaching that of l i g h t , or i n the presence of intense g r a v i t a t i o n a l f i e l d s , i s i t p o s s i b l e to detect 16 2.8 s i g n i f i c a n t c o n t r a d i c t i o n between theory and observation. Neither con- s i d e r a t i o n i s l i k e l y to a f f e c t the present a n a l y s i s . 2.3 N-body Dynamics The n-body ( o r , more a c c u r a t e l y , n-particle) problem i s concerned with the motion of n p a r t i c l e s of a r b i t r a r y mass which a t t r a c t each other according to Newton's law of g r a v i t a t i o n . At some time i n s t a n t their p o s i t i o n and v e l o c i t y vectors are known, and the subsequent motion i s to be determined. This problem has not been s o l v e d . Indeed, the n-body problem i s so r e s i s t a n t to a n a l y s i s that a general s o l u t i o n e x i s t s only f o r the two-body case. The b a r r i e r to progress, which has so f a r proved insurmountable, i s that 6n constants of i n t e g r a t i o n are necessary f o r a cOiiifJicoc 5 u l u c i 0 u a r e , nevertheless, uu t only ceil i i a v e oeeii TOunu. Cci " i u l i i opeolf'iC CuSco amenable to a n a l y s i s ; of these, the homographic s o l u t i o n s considered i n Section 2.4 form an important c l a s s . " c l a s s i c a l " integrals are fundamental to any study of The these special c o n f i g u r a t i o n s , and we therefore give a b r i e f d e r i v a t i o n of the ten known i n t e g r a l s i n t h i s section. Newton's law of g r a v i t a t i o n can be expressed i n the f o l l o w i n g form: There are observed phenomena between two bodies i n space which can be described by presuming that two bodies attract each other with a force d i r e c t l y proportional to the product of t h e i r masses and i n v e r s e l y proportional to the square of the distance separating them. [40 ] Suppose now that n p a r t i c l e s move i n a region of absolute space i n accordance with t h i s law and consider the i n i t i a l c o n f i g u r a t i o n shown 17 Figure 2-1 I n i t i a l c o n f i g u r a t i o n f o r the n-body problem i n Figure 2-1, where we assume there e x i s t s some i n e r t i a l X , Y , Z reference system a r b i t r a r i l y located i n absolute space. p a r t i c l e , then s a t i s f i e s the equation: The motion of p ^ , the i ' t h 18 2.10 where r^. = r^ - r.., i s the mass o f constant of g r a v i t a t i o n . and k denotes the Gaussian An existence c o n d i t i o n f o r s o l u t i o n s of t h i s equation, due to P a i n l e v e , may be found i n P o l l a r d [41. ]; a proof appears 1 i n Wintner [ 3 8 ] . 2 Let the vectors r . , r . be given at some time i n s t a n t t = 0 at i i 3 which a l l the distances r . . are p o s i t i v e ; these w i l l be c a l l e d the i n i t i a l data. If r(t) denotes the smallest o f the distances r . . at time t , then there e x i s t s a unique set of n vector functions r\(t) l a r g e s t i n t e r v a l of time - t and a < t < t-j containing the time i n s t a n t t = 0 2 such t h a t : (i) r\(t) s a t i s f i e s the d i f f e r e n t i a l Equation 2.2 f o r - t (ii) r\(t) and r\(t) / i i i \ \ 4 I t y -J - f 4-l-»V% i • V l l ^ . agree with the i n i t i a l data when t = 0. -Iri+A wi/^ 1 » i t <r<~ I K U . 4* l S,^ then r ( t ) -> 0 as t if t 2 < t < t-j; 2 y 4- ^ S" +• w j *?c n n + • ^ .• ^ +• l-»r» w i wtr>v\»al «i<>- ' I.t v i ^r*-i V . «^ . Also 4- <* - t. m ^ y t-j i f t-j i s f i n i t e and r ( t ) •+ 0 as t -> - t is f i n i t e . The essence of t h i s theorem i s contained i n i t s f i n a l section, which implies that a continuous s o l u t i o n e x i s t s during a time i n t e r v a l determined by the condition r(t) ->- 0. We s h a l l consequently r e q u i r e , f o r the remainder of t h i s a n a l y s i s , that during the time i n t e r v a l of i n t e r e s t r{t) > r * > 0, where r * i s some f i x e d , p o s i t i v e lower bound on r(t). Notice that great care i s taken to avoid s t a t i n g , that a c o l l i s ion occurs when r -* 0. r(t) = The d i f f i c u l t y i s Min { r 1 2 , r^, ••— , that ) n } may tend to zero when none of the i n d i v i d u a l distances tends to z e r o , 2 19 2.11 the r o l e of l e a s t distance being exchanged between them i n f i n i t e l y o f t e n . To prove that the condition r{t) -> 0 corresponds to a c o l l i s i o n at some f i n i t e time i n s t a n t t * i t i s necessary to show that the i n d i v i d u a l distances n > 3. tend to l i m i t s as t -> t * , and t h i s has not been proved f o r A discussion of t h i s enigma can be found i n Wintner [38 ] . 3 The c l a s s i c a l of motion. i n t e g r a l s can r e a d i l y be obtained from the equations Performing the sum over i we o b t a i n : 1=1 The double sum over i and j i s e v i d e n t l y z e r o , as r\ . + r.^ = 0, and therefore fV \ -1 Z * : F, = O . i (,r. I This can at once be integrated to give — ~ \ t i l-1 where a and b are constant v e c t o r s . The time i n t e r v a l - t d e f i n i n g s o l u t i o n existence w i l l n o t , i n g e n e r a l , p o i n t , but should be considered 1 I S. I < t < t-j y be r e - s t a t e d a f t e r t h i s i m p l i c i t i n the a n a l y s i s . defines the centre of mass of the n p a r t i c l e s : 2 Now i f r 20 2.12 From Equations 2.5 and 2.6 we obtain the c o n d i t i o n f o r conservation of l i n e a r momentum: r- \ i* • = a. t + t This r e s u l t , that the centre of mass moves with uniform v e l o c i t y i n the i n e r t i a ! system, i s to be expected i n the absence o f external forces. The i n e r t i a l reference system may now be located with i t s o r i g i n at the centre of mass of the n p a r t i c l e s without a f f e c t i n g Equation 2 . 3 . case a = 6 = 0, corresponding to the c o n d i t i o n r ( t ) = 0, and also - t, \ / In t h i s < t ^ t, . With a and b d e f i n e d , 6 n - 6 constants of i n t e g r a t i o n remain to be determined. We now introduce a force f u n c t i o n F by the equation n. (z.i) If f = f ( r , , r , r ) , the operation - — i s defined by 3r„ 9 5f t U f , V . where a „ , 3,,, Yi/ e a r system. the components of i n a Cartesian coordinate Equation 2.2 may now be w r i t t e n i n the f o l l o w i n g form: 21 2.13 Taking the s c a l a r product with r\ and summing over i «. s 1 and therefore Z . C-I cLt cLt * "* Equation 2.11 can be i n t e g r a t e d , since r. . ? i 1 d — = ^ 7ft ( i * V ' r t 0 g i v e the energy i n t e g r a l where S*, the t o t a l energy, i s an i n v a r i a n t q u a n t i t y . The energy S* reduces the number of undetermined i n t e g r a l s to 6n - 7- constant I f we now take the vector product of r . with Equation 2 . 2 , and sum over i : -I The double sum w i l l be z e r o , since r. x ?..•= -r.. x ? , i and therefore 22 2.14 Consequently IX, where h * i s a constant vector d e f i n i n g the angular momentum. The c o n d i t i o n h . r = 0 defines a plane through the centre of mass c a l l e d the i n v a r i a b l e plane (provided h f 0). the time i n t e r v a l - t 2 Suppose a plane IT contains the n p a r t i c l e s < t < ty during The corresponding s o l u t i o n of Equation i s then known as planar i f the p o s i t i o n of IT i n the i n e r t i a l system does not depend on t , and f l a t i f IT = i r ( t ) . 2.3 coordinate If a solution is p l a n a r , then TT i s the i n v a r i a b l e p l a n e , provided h * f 0; the proof appears in Wintner [ 3 8 " ] . The three i n t e g r a l s associated with h complete the known i n t e g r a l s of the n-body problem, leaving 6n-10 constants undetermined. An exhaustive treatment of t h i s e n t i r e subject may be found i n Wintner [38 ] . 5 l e s s d e t a i l e d a n a l y s i s , see P o l l a r d [41 ] 2 2.4 For a and also Danby [ 3 9 ] . 2 Homographic Solutions A s o l u t i o n of the n-body problem i s c a l l e d homographic i f the c o n f i g u r a t i o n formed by the n p a r t i c l e s at a given time i n s t a n t t moves, with respect to the i n e r t i a l coordinate system, i n such a way that the c o n f i g u r a t i o n i s preserved as t v a r i e s . If, i n i t i a l l y , t = 0, then a homographic s o l u t i o n w i l l be of the form r-(b) = pCb) H C O FCO) , L=l,2,r-, 23 2.15 where p(t) i s a s c a l a r , fi(t) denotes a r o t a t i o n matrix and the i n e r t i a l o r i g i n i s located at the centre of mass. (i) There are two l i m i t i n g cases: i f the c o n f i g u r a t i o n d i l a t e s without r o t a t i o n so that r (fc) L = ^(fc") r ( o ) ( r \2 l , i the s o l u t i o n i s c a l l e d homothetic; (ii) i f , conversely, the c o n f i g u r a t i o n i s . r o t a t i n g without dilation, then which defines a s o l u t i o n of r e l a t i v e e q u i l i b r i u m . Three r e s u l t s are of p a r t i c u l a r importance: (i) (ii) (iii) a homographic s o l u t i o n which i s not f l a t must be homothetic; i f f l a t , a homographic s o l u t i o n must also be p l a n a r ; also a homographic s o l u t i o n i s a s o l u t i o n of r e l a t i v e e q u i l i b r i u m only i f i t i s planar and rotates w i t h a non-zero constant angular v e l o c i t y . A proof of these statements may be found i n Wintner [ 3 8 L 6 S o l u t i o n s of the two body problem are a l l homographic and, unless the angular momentum vector h * = 0, p l a n a r ; Lagrange's e q u i l i b r i u m s o l u t i o n s of the t h r e e body problem are both homographic and p l a n a r . The s i g n i f i c a n c e of homographic s o l u t i o n s i s evident from these two cases, which together form the foundation of t h e o r e t i c a l astronomy. 211 2.16 2.5 Two-body Motion. A s o l u t i o n of the problem of two bodies appears i n most texts on c e l e s t i a l mechanics. Many r e s u l t s from the two-body problem w i l l , however, be needed l a t e r i n the a n a l y s i s ; f o r the sake of therefore, completeness, the elegant and concise development given i n Danby [39 ] 3 is o u t l i n e d here. I f the i n e r t i a l o r i g i n i s located at the centre of mass of the two p a r t i c l e s , then s e t t i n g n = 2 i n Equation r*,F, + = 2.8: o (z. is) We assume that neither mass i s i d e n t i c a l l y equal to zero. Equation 2 . 2 , and using Equation 2.14: r and t h e r e f o r e , if r = ? f where M = 2 Then, from 3 - r^ = - U H F a + m . 2 Taking the vector product of f with Equation (x. .?) 2.17: 25 2.17 and i n t e g r a t i n g : (%'. 18") r x r- where h i s a constant v e c t o r , which we w i l l assume to be non-zero. A l t e r n a t i v e l y , i f Equation 2.13 i s used to determine the angular momentum: The two momenta, although d i f f e r e n t i n magnitude, are e q u i v a l e n t . Expand- ing Equation 2.18 we o b t a i n : 1 1 ' which can be rearranged i n the form 1 +.. *s i-,*r, r * r- 1 <- ! s i n c e , from Equation w , , ^ . The r e l a t i o n s h i p between h 2.14, and h * i s then given by M From Equation 2.18, taking the s c a l a r product with r 26 2.18 so that the motion i s planar. This i s the d i s t i n g u i s h i n g feature of the two-body case; a s o l u t i o n i s possible because h i s normal to the i n v a r i a b l e plane. Now take the vector product of h with Equation using h x r = (r x r) x r from Equation 2.18. 2.17, We obtain = - k = - w^ H d. f r or Integrating t h i s expression t *f = - tcMf where P i s a constant vector. - ? Because P i s i n the i n v a r i a b l e plane i t f o l l o w s that P . h = 0, so one constant of i n t e g r a t i o n s t i l l be found. (a.n ) remains.to A parametric s o l u t i o n can, however, be found by e l i m i n a t i n g t from Equation 2.19. The s c a l a r product with r gives 27 2.19 But r x f = h , and therefore I f the angle between P and r i s denoted by v , t h i s equation can be rearranged i n the form u where p -p— = e ITM ^ and h -p— = p . ITM 2 This i s the equation of a conic with the o r i g i n at one focus; f o r an ellipse p = a(l-e 2 ), and f o r a hyperbola p = a(e 2 - 1). The vector P points along the major axis of the o r b i t toward the p o s i t i o n of c l o s e s t approach between the two p a r t i c l e s ; v i s known as the true anomaly. Equation 2.18'may.be w r i t t e n i n the form ptt and from Equation 2.21 / ( 1 •+ ecoso) . E l i m i n a t i n g r from Equation 2.22 we obtain v 28 2.20 but, although t h i s equation can be i n t e g r a t e d , an a d d i t i o n a l s u b s t i t u t i o n i s necessary to put the r e s u l t i n a useful form. be integrated d i r e c t l y . d i f f i c u l t y ; when Two cases can, however, The c i r c u l a r o r b i t case (e = 0) presents no e = 1 Equation 2.32 can be w r i t t e n r = sec- 0/ , z and the corresponding form of Equation 2.24 becomes It which integrates to give 1* T, the f i n a l constant of i n t e g r a t i o n f o r the p a r a b o l i c case, to the time i n s t a n t when corresponds v = 0. Now consider the energy i n t e g r a l , which can be derived from a force function F = We now w r i t e F and consequently, = hV •, f o l l o w i n g the same procedure as i n Section 2.2: 29 2.21 ± r .r r 2- where S i s an energy i n t e g r a l . from Equation 2.12 The corresponding expression derived is (2.2.?) r where the two energy i n t e g r a l s S and S* are r e l a t e d by n The constant S can.be evaluated from the c o n d i t i o n r*. r (2.2.S) - V After substituting for ^ At. from Equation 2.22, f o r e l l i p t i c o r b i t s we obtain %$ « U M r 0,0 -e*) --.2 + r.. and, since r = 0 when r - a ( l ± e ) , Equation 2.26 may be w r i t t e n r 2. - _l_ r If Equation 2.18 i s squared: (a-*<0 30 C = (F.F)(F. r ) - (r.F) (..*..3.o.) which can be expressed i n the form k M 1. r r t. . O. — f r where r . r i s obtained from Equation 2.29, and h from Equation 2.22. Now we define the e c c e n t r i c anomaly E by r = a- ( I - e cos E ) As the true anomaly n i f f o v o n t i a t i n n v Fnna-Hnn varies from 0 to 2TT, E also varies from 0 to 2u. ? 3?: which can be substituted into Equation 2 . 3 1 ; a f t e r some manipulation we obtain = .1 The e c c e n t r i c anomaly i s chosen so that i t s d e r i v a t i v e i s p o s i t i v e , and therefore (2.33) 31 2.23 Integrating t h i s expression over one complete o r b i t , the period r w i l l ITT be C2.3O PL The mean motion n i s defined by (2 .3<r) P and Equation 2.33 may be w r i t t e n ait = ( i - e<.c*E) crf-E This can be integrated d i r e c t l y to give n.C t - T ) = £ - e,s^ E where T, the time i n s t a n t at which r i s a minimum, i s the f i n a l constant of i n t e g r a t i o n f o r the e l l i p t i c o r b i t ; a s i m i l a r s o l u t i o n may be found f o r hyperbolic o r b i t s . I f we replace M by m^ + i n Equation 2.35, the mean motion i s determined by C23?) n. The preceding a n a l y s i s i s based on the assumption that h f 0. I f , however, the angular momentum h i s z e r o , then from Equation 2.18 motion must be one-dimensional. Although s t i l l homographic, t h i s the one homothetic s o l u t i o n of the two-body problem. An a n a l y s i s is of the 32 2.24 t h i s case, as i t applies to c o l l i s i o n o r b i t s and the process of r e g u l a r i z a t i o n , can be found i n Szebehely [ l 2.6 3 ]. N o n - i n e r t i a l Systems of Reference From t h i s point we s h a l l be concerned with the motion of one p a r t i c l e under the g r a v i t a t i o n a l i n f l u e n c e of the remaining n-1 bodies. The equation of motion f o r t h i s p a r t i c l e w i t h respect to the i n e r t i a l coordinate system may be w r i t t e n i n the f o l l o w i n g dimensionless form n. ~l a.t Z Z . lr-..|3 J ^ where p-j = k m.- ty z 3 normalization. ^pjr = r , \ i s the distance normalization and T the time We w i l l continue to use the convenient notation to denote d i f f e r e n t i a t i o n with respect to normalized time, t. The four body problem under c o n s i d e r a t i o n i n t h i s a n a l y s i s subject to these c o n d i t i o n s : 1. m > m,, > m » m^ ; 2. the mass of p^ i s so small that i t has no e f f e c t on motion 1 3 of the primary bodies p ^ , p 2 and p 3 ; We assume, i n t h i s f o r m u l a t i o n , that the mass m- of the p a r t i c l e i s not i d e n t i c a l l y equal to z e r o , i n which case the d i v i s i o n by m i n Equation 2.2 i s l e g i t i m a t e . i is 33 2.25 3. the i n e r t i a ! o r i g i n i s located at the centre of mass of the three p r i m a r i e s ; 4. i f p* denotes a p a r t i c l e of mass + m^ located at the centre of mass of m^ and mg, then p^ and p* move i n two-body o r b i t s with respect to the i n e r t i a l system at an angular v e l o c i t y 'fi(t); 5. with respect to the i n e r t i a l coordinate system, p^ and p^ move approximately i n two-body o r b i t s about t h e i r centre of mass; 6. p^ moves i n the v i c i n i t y of p 2 and p^. Conditions 2 and 3 are s t r i c t l y v a l i d only as m^ ->- 0; the f o u r t h assumption concerning motion of the centre of mass of p 2 and p-j needs some j u s t i f i - cation. Tf v>* . nncitinn Ar>nr\-t-r\c r - - \iar- + nv . • _- r>-f n r with r o c n o r t tn r>* , ... • , then',- i n the notation of t h i s a n a l y s i s and with respect to the i n e r t i a l system of reference; X* where <?F P_ (co 5 ) = S l\lco S x S -11 i and J cos S = F„ . r * — [42 ]. 1 The f i r s t term of the force f u n c t i o n corresponds to the two-body s i t u a t i o n (see Equation 2 . 2 5 ) ; i f two body motion i s assumed, the r e l a t i v e magnitude p of the f i r s t neglected term i n the force f u n c t i o n i s given by + n) \ x y r* ) The worst case r e s u l t s from m = m , and with the distance normalization 9 q 3k chosen so that ^ 3 = 1 we obtain In p r a c t i c e p f> £ ^ . w i l l be s u f f i c i e n t l y small that deviations from two-body motion can, f o r most purposes, be neglected. 35 2.27 The c o n f i g u r a t i o n of the three primaries i s shown i n Figure 2-2. S,H,Z form an orthogonal coordinate system r o t a t i n g at a normalized angular v e l o c i t y E,H Q with respect to the i n e r t i a l X , Y , Z system, w i t h i n the o r b i t a l plane of the centre of mass of p £,n,C to the 2 and p - coordinate system rotates at an angular v e l o c i t y H,H,Z system, with £,n The 3 u> with i n the o r b i t a l plane of p 2 respect and p^. In the i n e r t i a l system of reference n , the angular v e l o c i t y of the £,n,t; system, i s given by Now suppose three p o s i t i o n vectors same point i n the X , Y , Z , tively. E,H,Z and £,n,C a , 6 and y define the coordinate systems respec- Then: where C i s the p o s i t i o n vector of B (the o r i g i n of the system) i n the r o t a t i n g E,H,Z system. Similarly: £,n,C coordinate 36 2.28 OC (w + .n.) % y + [ C ] + Z " a > < I f r i s the p o s i t i o n vector of p [ ']"HZ C 4 ^ ' ^- * ^ + L < i n the £ , n , £ motion f o r p^ may be obtained from Equation 2.38 r -*- r 2-(l3 +st)x 11- .i + ux(unf) L 3 where Equation 2.41 -f- Z^i-K^^r) a + C * c system, the equation of as: +• -^(xL* f) + ( i r ** J i s used to express r i n the n o n - i n e r t i a l £ , n , S system of reference. Transformations between the three coordinate systems -are determined by the f o l l o w i n g relationships:. A X cos <p — y' o cos y> o o i A y A z. • and o 0-43) A 2 37 2.29 (tos<jfcco£^ + S^H^Sin.^'toS'L) ( - S i r t ( j £ u > S ^ +C o S ^ S i w ^ W u ) - S^^ SiwV (2.4-4-) A A COS V z where fl = H z , n. = t n£ and i denotes the angle between the E , H and The X , E and 5 axes coincide at the time o r i g i n £, n planes( Figure t = 0. 2-2) I f f i s defined by then with respect to the n , C system of reference: (2.45) A l s o , i f C = C E, Z -flay t t then with respect to the E, Sl^St^C.) t AxC =[c.~ H , Z coordinate system: JT- C-] £ a + [ -n- ] I c (2-4-6) 38 2.30 Equations 2.43 and 2.44 impose no r e s t r i c t i o n s on s i n x, C, ft or n . Considerable s i m p l i f i c a t i o n i s , however, p o s s i b l e i f we assume: 1. s i n i = 0; 2. n = 0; 3. the hypothetical body p* i s located at B, the o r i g i n of the £ , n» C system of 4. reference; p* and p-j move i n c i r c u l a r o r b i t s about the i n e r t i a l o r i g i n , i n which case C = C = fi = 0; 5. the three primary bodies move i n coplanar o r b i t s , so that ^(t) = 0 f o r i = 1, 2, 3. Equation 2.42, which defines the motion of p 4 with respect to the £ , n , c system of reference, can now be w r i t t e n = -si c 3 _ N As a consequence of assumption 5 above, one p o s s i b l e s o l u t i o n Equation 2.49 is 39 2.31 £(t) = £ (b) = o (a.jo) which corresponds to motion i n the o r b i t a l plane of the primary bodies p.|, p 2 and p . 3 I t i s convenient, at t h i s p o i n t , to choose the time normalization so that n = 1. I f we denote dimensioned q u a n t i t i e s by an a s t e r i s k , then and consequently x = 1/n* gives the desired r e s u l t . 2.37 i s used to determine the mean distance U tr x PA r 2 3 Now i f Equation *: (2.SZ) j. +• rrt. 3 2-3 where r 2 3 * = Ar 2 3 But we also have and so Equation 2.52 may be w r i t t e n n. = 1 1 3 I f , t h e r e f o r e , the distance normalization A.is chosen so that X = then r 2 3 = 1 and also r ^ ho 2.32 ^ - + p-3 i . a -^O These normalized values are consistent with those i n Szebehely [ i * ] . 2.7 Integrals i n Rotating Systems of Reference I f a dynamical system i s described by the d i f f e r e n t i a l equation and G(x, t ) i s a s c a l a r f u n c t i o n , then G i s an i n t e g r a l of Equation 2.55 if i for a l l x , t f fr(fc.t)l O' s a t i s f y i n g Equation 2.55. body problem (Equation 2.12) d(f , F , t ) = = j \ 2. fr; ll-Sb) The energy i n t e g r a l of the n - i s of t h i s form, where *:r .F. c - ...k N. 1 u, i>c l *jl r In a r o t a t i n g coordinate system the equation of motion f o r a s i n g l e p a r t i c l e p^ of non-zero mass i s : F, + ^iixr. •+ iix ( A x r J + jflx F. - (_VZL' |r..p r 1 i 1 = ° > 1*1 2.33 where ft i s the angular v e l o c i t y of the system with respect to an i n e r t i a l system of reference. L i I f a force f u n c t i o n U i s defined as Ir;.- then Equation 2.57 may be w r i t t e n . >rc . and taking the s c a l a r product of t h i s equation with r\: Now the vectors r . i n Equation 2.58 (which define the p o s i t i o n s of the remaining n-1 p a r t i c l e s ) w i l l i n general be functions of time, so that W- = U. C r • , t ) and Consequently, i f Equation 2.60 i s to s a t i s f y the c r i t e r i o n of Equation 2.56 f o r the existence of an i n t e g r a l , i t i s s u f f i c i e n t to require SI = ^Uat = O , (.2.61 ) 2.34 i n which case: 1. the n o n - i n e r t i a l coordinate system rotates at a constant angular v e l o c i t y ; 2. the n-1 p a r t i c l e s maintain a f i x e d c o n f i g u r a t i o n which i s unaffected by the motion of p^. This l a s t c o n d i t i o n implies that the n-1 vectors r . (j + i ) define a J homographic s o l u t i o n of the (n-1) body problem, and from Section 2.4 t h i s s o l u t i o n of r e l a t i v e e q u i l i b r i u m must be p l a n a r . Subject to these r e s t r i c t i o n s , i f ; . ^ - C" 1 p-j + the s c a l a r f u n c t i o n r y . [ i x * (Cxi* r,.) i s an i n t e g r a l of Equation 2.57, = constant although t h i s expression i s v a l i d only when m.. i s so small that c o n d i t i o n 2 above i s Both of these requirements The corresponding of integral, is where the p o s i t i o n of p^ i s defined by ( x , y , z ) , C i s the Jacobi constant, ft i s normal to the i n v a r i a b l e x , y plane of p^ and p^, normalization i s chosen'so that ft = 1 . cussed i n Szebehely [ 1 fulfilled. are s a t i s f i e d by the assumptions the r e s t r i c t e d problem of three bodies. c a l l e d the Jacobi i n t e g r a l , strictly and the time The Jacobian i n t e g r a l is dis- ] , but note that Szebehely uses a modified force k3 2.35 function 11"= U. + > ^ p . ^ ( defined i n Wintner [38 ] 7 throughout his a n a l y s i s . The i n t e g r a l i s equivalent i n form to Equation 2.63 above. Huang [ 6 ] has used " i n t e g r a l s " of motion to determine curves of zero v e l o c i t y i n a s p e c i f i c four-body problem. s a t i s f y Equation 2.56, consistent The i n t e g r a l s do n o t , and the primary model f o r p.|, p s o l u t i o n of the three-body problem. 2 however, and p^ i s not a Integrals of Equation 2.57 can be determined f o r the four-body case only i f Equation 2.61 valid. is This c o n d i t i o n i s s a t i s f i e d by Lagrange's homographic s o l u t i o n s of r e l a t i v e e q u i l i b r i u m f o r the three primary masses, and corresponding surfaces of zero v e l o c i t y f o r the motion of a fourth p a r t i c l e i n f i n i t e s i m a l mass have been determined by Matas. of The f i r s t paper presents surfaces of zero v e l o c i t y f o r the e q u i l a t e r a l [43] triangle configura- t i o n and c e r t a i n s p e c i f i c values of primary mass; a l a t e r paper [44 extends these r e s u l t s to include the c o l l i n e a r c o n f i g u r a t i o n . ] Note that surfaces of zero v e l o c i t y f o r the r e s t r i c t e d four-body problem were f i r s t determined by MouKon [ 4 ] , given. although s p e c i f i c r e s u l t s were not Matas, i n c i d e n t a l l y , does not mention Moulton's work. This force function i s introduced i n [ 1 ] , Chapter 1, page 18, Equation 49. 3.1 3. 3.1 LIMIT PROCESS CONSIDERATIONS Introduction For the remainder of t h i s a n a l y s i s we s h a l l concentrate on the four-body problem described i n Section 2.5 and defined by Equations 2.47 and 2.48, with c ( t ) = 0. The motion of p.j, p 2 and p ever, be known before attempting to solve these equations. 3 must, howFollowing Huang's i n i t i a l statement of the " v e r y - r e s t r i c t e d four-body problem" [ 6 ] i n 1960, many authors based t h e i r a n a l y s i s on the f o l l o w i n g two assumptions: (i) p-j and p* move i n two-body o r b i t s with respect to the i n e r t i a l frame of reference; (ii) p 2 and p^ move i n two body o r b i t s about t h e i r centre of mass. Although the r e s u l t i n g primary model i s p l e a s i n g l y uncomplicated, i t neglects the g r a v i t a t i o n a l i n t e r a c t i o n of p^ and p s o l u t i o n of the three-body problem. 2 and i s not a v a l i d Discussions of t h i s d i f f i c u l t y can be found i n Danby [24 ] and Szebehely [ 23, l ] . 5 More accurate primary models, derived e i t h e r from numerical a n a l y s i s or lunar theory, have subsequently been employed to overcome t h i s l i m i t a t i o n [25 ] , [30 [31] but unfortunately these precise s o l u t i o n s introduce i n t o the equations of motion a v a r i e t y of parameters which tend to obscure the process of a n a l y s i s and made i t i n c r e a s i n g l y d i f f i c u l t to separate one perturbation from another. ], 1|5 The four-body problem i s u s u a l l y treated as a perturbed version of the r e s t r i c t e d problem i n v o l v i n g p , p 2 3 and p^ so i t i s not unreason- 5 able to expect that Lagrange p o i n t s , which are of such s i g n i f i c a n c e in the r e s t r i c t e d problem, w i l l also be important i n the four-body case. In f a c t , with the exception of velocity surfaces, two papers by Matas [ 43 , 44 ] on zero- Shi and E c k s t e i n ' s paper [ 2 6 ] on Earth-Moon t r a j e c t o r i e s and a r e l a t e d paper by Kevorkian and Brachet [27 ] , a l l recent work on the four-body problem i s concerned with motion near the Lagrange points of p and p . 2 3 A v a l i d s o l u t i o n of the four-body problem can only be obtained i f we f i r s t solve the three-body problem f o r p-j, p a corresponding s o l u t i o n f o r the motion of p^. 2 and p 3 and then derive I n t u i t i v e l y i t seems t h a t some "simplest" form of t h i s four-body problem must e x i s t , i n v o l v i n g o r b i t s of p 2 and p probably that are approximately c i r c u l a r and where 3 p^ moves near the Lagrange points of p 2 and p 3 Three questions have not, however, been r e s o l v e d . 1. What i s the l i m i t process ( i f any) which reduces the threebody problem f o r p-j, p p 2. 2 2 and p to a two-body problem i n v o l v i n g and p ? 3 What l i m i t process reduces the four-body problem to a threebody problem i n v o l v i n g p , p 2 3. 3 3 and p^? What i s e q u i v a l e n t , i n the four body case, to the Lagrange points of p 2 and p ? 3 The f i r s t two queries are answered i n Sections 3.2 and 3.3 below; the l a s t must wait u n t i l Chapter 5 f o r a d e f i n i t i v e answer, but the s o l u t i o n i s i n d i c a t e d i n Section 3.6. 3.3 3.2 Reduction of the 3-body Problem The p a r t i c u l a r three-body c o n f i g u r a t i o n which we s h a l l consider here i s determined by the c o n d i t i o n s : (i) p* and p^ move i n c i r c u l a r o r b i t s about the i n e r t i a l origin; (ii) (iii) the three bodies p ^ , p^ and p^ move i n coplanar o r b i t s ; p 2 and p^ move approximately i n two-body e l l i p t i c o r b i t s about t h e i r centre of mass. The equation of motion f o r p^ w i t h respect to the i n e r t i a l frame of reference i s : where we assume the mass of p 4 i s s u f f i c i e n t l y small to be neglected (see c o n d i t i o n 2 on page 2 . 2 4 ) . Condition ( i ) above implies t h a t fi = C = C = 0, and we may a r b i t r a r i l y s e l e c t <L = 0. s i m p l i f i e s , and Equation 2.42 i n the n o n - i n e r t i a l £ , n , £ coordinate system 3.1 takes the form: which can be w r i t t e n as: then Equation 3.4 where the time normalization has been chosen so that n = 1. The c o n f i g u r a t i o n of p 2 and must now be defined i n a manner consistent with the assumption of (approximately) e l l i p t i c o r b i t s . general p 2 and p^ may be expected to move w i t h i n bounded regions of the £ , n plane, and we f o l l o w Szebehely [ l so that £ 9 In > 0 (Figure 3-1). 6 J i n l o c a t i n g the l a r g e r mass m 2 14-8 3.5 I f the mean distance between p Equation 2.54) u 3 = 1 - i ^ . 2 and p i s chosen to be 1, then (from 3 Now the centre of mass of p 2 and p 3 is permanently located at the o r i g i n of the £ , n axes, and consequently X " x - 0 - / O 2 2 — = — m u m since y 3 . From these two equations: 3 and we may therefore w r i t e 3 1 «I _L_ 3 1 k9 3.6 The equations of motion f o r p S ' t Z 1 + 5 + H + = nc z 2 now take the form + |x, tosC\-.o.)t - g") C3' -) 2 l r , z-3 ( 3 - I 3 ) 1 where, f o r convenience, the s u b s c r i p t on £ 2 ana " n 2 n a s ' 3 e e n o m itted. two equations are s u f f i c i e n t to determine the motion of both p since £ 2 a n are known. d ^3 c a n be found from Equations 3.5 and 3.6 once £ 2 2 These and p^» and n 2 We can, moreover, r e a d i l y demonstrate that as the two quantities and 'Ml tend to z e r o , Equations 3.12 and 3.13 reduce to the equations of two-body motion i n a coordinate system r o t a t i n g with constant angular v e l o c i t y . the two-body case i n v o l v i n g p £,n,£ 2 and p , the motion of p 3 2 In i n the r o t a t i n g coordinate system i s determined by = o , (1.1*") 5P 3.7 where n i s the (constant) angular v e l o c i t y with respect to an i n e r t i a l frame o f ' r e f e r e n c e . This equation may be w r i t t e n (I. ' O jx 4 z 5^- Ix + ( / ix-*tj where i t i s assumed n = I . « O Equations 3.10 and 3.II the present case, and so the motion of p i + 1 I s + - — I t J- 2 are s t i l l i s defined by the valid in equations 1 -1 r which are i d e n t i c a l with Equations 3.12 and 3.13 when the terms i n v o l v i n g 9 'C and u-| are set equal to zero. 2 We should therefore i n v e s t i g a t e the l i m i t process fi C -»- 0 and I f dimensioned q u a n t i t i e s are denoted by an I U asterisk, 3 then and also r* = r, X (3.2.0) 51 3.8 where x and A are the time and distance normalizations defined i n Section 2.6 (pages 2.24 and 2.31). From the c o n d i t i o n that p* and p 1 move i n c i r c u l a r o r b i t s about the i n e r t i a l o r i g i n we have, using Equation 2.37: k V JTV- where p*. ( 3 - zi ) i s the normalized distance between p^ and the hypothetical p a r t i c l e The quantity C, which i s the normalized distance between p* and the i n e r t i a l o r i g i n , i s determined by the c o n d i t i o n ,C r, - C ) = ( wi 3 ) C , so that (3 . ZZ> From Equations 3.21 and 3.22 we can now w r i t e r where the q u a n t i t y of p 2 and p . e i t h e r m-j 3 k "^ / 1 1 3 C3.23) i s completely determined by the c o n f i g u r a t i o n I f Equation 3.23 tends to zero with T and X f i x e d , then 0 or r^ <=°. The f i r s t c o n d i t i o n i s i n c o n s i s t e n t with the 52 3.9 + requirement that m^ > the process r-j -> °°. 2 , and therefore ft C -> 0 must be equivalent to Now and r,. r, X t-, . r - fc r^.. F + t , so that 1 r»\, + •L 0 — (3.24) which i s a form s i m i l a r to Equation 3.23. The same argument can conse- quently be a p p l i e d when Equation 3.24 tends to z e r o , provided the term -1 i s well-behaved as r-j -»• °°. bounded and 2 r < r This c o n d i t i o n i s s a t i s f i e d when r 2 is i » which "> c e r t a i n l y true i n the present s i t u a t i o n . s Question 1 on page 3.2 problem f o r p-j, p 2 can now be answered as f o l l o w s : the and p , defined by Equations 3.12 and 3.13, 3 reduced to a two-body problem i n v o l v i n g p 2 the form e -*• 0, where z -»• 0 as r^ -»• » . ^Condition T on page 2.24 and p three-body can be by a l i m i t process of 3 We s h a l l , i n f a c t , take states that m-j > m > m » 2 3 m^. 53 3.10 e = / 1/ \ 3.3 1 , but t h i s choice w i l l be discussed l a t e r i n Section 3.4. Reduction of the 4-body Problem In view of the s i m i l a r i t y between Equations 2.47 and 2.48 for the four-body case and Equations 3.3 and 3.4 f o r three-body motion, i t might be a n t i c i p a t e d that the same l i m i t process can be a p p l i e d to both sets of equations. To demonstrate the t r u t h of t h i s conjecture, consider the motion of p^ i n the r o t a t i n g £ , n , t, coordinate system under the g r a v i t a t i o n a l influence of p 3.14, 2 and Py 3.15 and 3.16 we may w r i t e : ir**r A- where £> n define the motion of p^. Equations 3.25 and 3.26 when the terms and By d i r e c t analogy with Equations 3 o Equations 2.47 and 2.48 reduce to 5k tend to zero. This condition i s i d e n t i c a l with that derived i n the pre- ceding section f o r the reduction of Equations 3.12 and 3.13, and conse- quently the same l i m i t process can be applied i n the four-body case. This reduction of the three and four-body problems can be i n t e r p r e t e d i n the f o l l o w i n g way: as p-j becomes more d i s t a n t , i t can be assumed with increasing accuracy (i) the motion of p 2 that and p^ i s determined by two-body dynamics, and (ii) the motion of p , p 2 3 and p 4 i s determined by three-body dynamics. This r e s u l t i s independent of the mass of p-j, provided i t is f i n i t e , since terms i n v o l v i n g m^ (through the q u a n t i t i e s ft C and ) i n the equations of motion can always be rendered n e g l i g i b i l y small i f r is s u f f i c i e n t l y large. In a more general a p p l i c a t i o n , t h i s d e r i v a t i o n provides some j u s t i f i c a t i o n f o r neglecting the g r a v i t a t i o n a l e f f e c t of remotely d i s t a n t masses (see r e s t r i c t i o n ( i i ) on page 2 . 7 ) . Suppose, f o r example, that m and m^ correspond to the earth and moon mass 2 r e s p e c t i v e l y , and m^ i s the g a l a c t i c mass. so that Equation 2.52 may be w r i t t e n rvt.. «-3 and, i f n = r ^ = 1, k "^ 7 we obtain =£: i In t h i s case m + 2 * m , 2 55 3.12 From Equations 2.43 and 2.44 the p o s i t i o n of p-j (with respect to the £ , . r i , £ system of reference) i s defined by % x = - r, cos ( l - X O t ; = r, sc*.( t- X L ) t I f these values are s u b s t i t u t e d i n t o Equations 3.12 and 3.13 a c a n c e l l a t i o n occurs (Appendix 1, Equations A1.4 and A 1 . 5 ) , and the leading term i n v o l v i n g m^ takes the form k^r*, 1 - After substituting for .... k" t/ 1 1 we may w r i t e 3 l 17 For the case o u t l i n e d above (very approximately) —- = 0.4 x 10 and 11+ r = 8 x 10 , so that the c o e f f i c i e n t of the l a r g e s t d i s t u r b i n g term m 2 1 i s 0.8 x 1 0 " . 1 9 + From A l l e n [45 ] : 20 Sun's distance from g a l a c t i c centre = 10 kpc G a l a c t i c mass = 1.4 x 1 0 11 m 3.0857 x 10 o 6 We have — * 0.3 x 10 m m m T 2 17 — - 0.4 x 10 . The distance normaliz- 2 ation A = 0.3844 x 10 m (page 4 . 3 7 ) , „so that r 9 m. Q where ni denotes the s o l a r mass. o (from page 4 . 3 7 ) , and therefore = 1 - 8 x 10 . 11 56 3.13 3.4 Small-parameter Expansions We have now derived a reduction c o n d i t i o n , but not i n a useful form. and The d i f f i c u l t y i s associated with the terms i n v o l v i n g sin(l-ft)t i n the equations of motion (2.47, 2.48, 3.12 cos(l-£2)t and 3 . 1 3 ) , but i t can be resolved quite e l e g a n t l y by choosing a new time v a r i a b l e t = (\-Sh) t 4 . ^ , =. k ^ t ^ m , From page 2.18 \ (3.2?) , and from Equation 3.21 S1 L r,» 3 so that i f we set (3-zs) Equation 3.27 may be w r i t t e n t + = [ I - e (n- j O * - ] t 3 We s h a l l immediately drop the notation t , + . . (3.21) and from t h i s point i n the a n a l y s i s t , the independent v a r i a b l e , i s r e l a t e d to dimensioned time t * by f « 0 - . a ) f c * / ' X = C l - J ^ ' t O t * , / "17 = ( a * - J C l * ) t . With respect to the new independent v a r i a b l e , Equations and 3.13 .take the form (3.3a) 3.12 57 3.14 -1 I (3.31) =• cot r + ^Xti-S) • . . ^ J_ -1 (5.32.) where we have used the c o n d i t i o n and £ , n define the motion of p . 2 Equations 2.47 and 2.48 become SL^C = u, / t from Equation 3.23, S i m i l a r l y , f o r the four-body problem, 58 3.15 where, i n t h i s case, £ , n define the motion of p^ (the three and f o u r body cases w i l l be treated s e p a r a t e l y , so there i s no r i s k of c o n f u s i o n ) . The expansions of and - ^, si^b + |*, C 7, - «l) are given i n Appendix 1. Terms i n v o l v i n g (* / % c a n c e l , so that both { 2k expressions i n v o l v e sequences i n e 1 , where k = 3, 4, 5 ••• . We have consequently obtained equations of motion which depend on the small parameter e, where the l i m i t process e -> 0 reduces these equations to a form which, i n c e r t a i n cases, can be solved e x a c t l y . These special cases w i l l now be investigated i n more d e t a i l . 3.5 Exact Solutions When e = 0 Equations 3.31 and 3.32 reduce to Equations 3.17 and 3.18, which define the two-body motion of p 2 problem can be solved (Section 2 . 5 , Now although the two-body Equations 2.32 and 2.36) the s o l u t i o n i s not an e x p l i c i t function of time and, even f o r t h i s comparatively simple case, expressions of the form £ = £ ( t ) , n = n ( t ) are d i f f i c u l t to obtain [ 42 ]. 2 An exact e x p l i c i t r e s u l t can, however, be determined f o r a homographic s o l u t i o n of r e l a t i v e e q u i l i b r i u m . In t h i s s p e c i f i c case a l l d e r i v a t i v e terms are z e r o , and Equations 3.17 and 3.18 become 59 3.16 (3. ) (336) 1 where the expression J_ - 1 has been developed using Equation 3.9. The p a r t i c l e p must therefore be located somewhere on the c i r c l e 2 S % ^ 1 =(1 - ^ J OS?) 1 For convenience we may take n = 0, so that and, from Equations 3.5 and 3.6: (3.3<0 I f the p o s i t i o n s of p and p 2 3 are defined by Equations 3.38 and 3.39, then homographic e q u i l i b r i u m s o l u t i o n s can a d d i t i o n a l l y be determined f o r the three-body case derived from Equations 3.33 and 3.34 (defined by Equations 3.25 and 3.26). i n Equations 3.25 and 3.26: S e t t i n g the d e r i v a t i v e terms equal to zero 60 3.17 = o ",1. '<f3 i = O where 5, n denote the p o s i t i o n of p [l ], 7 we have substituted Equation 3.41 is n = 0, r y 3 = y, 4 and, to f o l l o w Szebehely's notation y 2 = 1 - y. One s o l u t i o n of i n which case - <>- .) s : Equation 3.40 then s i m p l i f i e s to ± ft (3.42) = o which defines the three c o l l i n e a r Lagrange points L-j, L The remaining s o l u t i o n of Equation 3.41 1 - and L 3 [l ]. 8 is = D ti-h> which, when 0 < y < 1, 2 (1.4-3) reduces Equation 3.40 to the form ( 1 '•V 3.4-0 61 3.18 1 -*—> Figure 3-2 Lagrange points of the r e s t r i c t e d problem. The r e s u l t i n g c o n d i t i o n , 43' = C 3.4-5) 1 defines the two e q u i l a t e r a l t r i a n g l e points L^ and L^ (Figure 3-2). An a n a l y s i s of the two and three-body equations of motion (Equations 3.17, 3.18, 3.25 and 3.26) s o l u t i o n s i s now f e a s i b l e , i n the v i c i n i t y of these exact but two a d d i t i o n a l small parameters, a , must f i r s t be introduced. 6 and The i m p l i c a t i o n s of t h i s process are considered i n the f o l l o w i n g s e c t i o n . 62 3.19 3.6 Orbital 3.6.V Perturbations Perturbed 2-body motion The major obstacle preventing an e x p l i c i t s o l u t i o n of the equations determining two and three-body motion i s the presence of nonl i n e a r terms i n v o l v i n g _! Close to an exact s o l u t i o n , however, these terms can be expanded as of a small o r b i t a l parameter. For the two-body motion of p 2 functions t h i s exact s o l u t i o n i s given by Equation 3.38 and, i f we take S x ( t ) = I-p.,. + l e f t ) ; iJ-t} a binomial expansion of the term _J i r form of Equations 3.17 and [ e -2i -3el + 2e] SfCb) leads to a v a r i a t i o n a l x 3 l 3 - o + - 5 3ef (3.46) 3.18: L u\-fS> S [ ? - (3.4*^ J - o (3.48 ) 6 3 3.20 + where 6 « 1. Now suppose we consider the a d d i t i o n a l e f f e c t of p ^ , as determined by Equations 3.31 and 3.32. The motion of p^ w i l l depend both on e and on 6, so that we should make the f o l l o w i n g m o d i f i c a t i o n to Equation 3.46: where the v a r i a t i o n a l terms u ( t ) , v ( t ) correspond to the perturbation caused by p.j and e ( t ) , f(t) determine the o r b i t a l p e r t u r b a t i o n . The equations of motion a r e , i n t h i s case, s i g n i f i c a n t l y more complicated, but as e ->• 0 with 6 f i x e d they reduce to Equations 3.47 and 3.48. When 6 ->- 0 with e f i x e d we obtain 4- To see t h i s , note that from Equation 3.9: and t h e r e f o r e : 0 - f O * *" _ S Now take _ L _ 0- HO +V[e^f ]"? Used-pi) 1 x ... <• and expand (1 + r ) " 3 / z . , J The expressions appearing i n Equations 3.47 and 3.48 f o l l o w d i r e c t l y from t h i s expansion. 6i+ 3.21 U . | 1-26^(1+^,)^ 4-6^1 + /*,)^ - 2 v { l - 6 s ( l Y , ^ £ -3, <«ob + fx, ( g , - g ) x 3uV In., I These equations describe the motion of p and 9 disturbance from r e l a t i v e e q u i l i b r i u m i s caused by p-|. when the only Consequently they define the " s i m p l e s t " v a l i d three-body model f o r p-j, p and p ^ , and 2 the s o l u t i o n of Equations 3.51 and 3.52 forms the basis of the " s i m p l e s t " four-body problem. 3.6.2 Perturbed 3-body motion The e x p l i c i t s o l u t i o n s f o r three-body motion are defined by Equations 3.42 and 3.45. I f the terms i and 3 are expanded about a p a r t i c u l a r Lagrange point _j *3' n » then w i t h L we obtain the f o l l o w i n g v a r i a t i o n a l form of Equations 3.25 and 3.26 (see Section 5.2 and also Appendix I V ) : 65 3.22 Cr[ - + %y where a « 1 0, +/ j +• s r " 0 OC and the c o e f f i c i e n t s v-j, L . , f o r example, o { = - 3/. , ^ = >= O and 3£? (3.5-4) depend on ( 1 - 2 . ^ ) and and n ^ . ° 3 ~ = At • 4- Equations 3.54 and 3.55 define motion about the Lagrange points i n the r e s t r i c t e d problem; a d e t a i l e d a n a l y s i s appears i n Szebehely [ i A more complicated s i t u a t i o n r e s u l t s i f p e l l i p t i c o r b i t s about t h e i r centre o f mass. motion of p ? ]. and p^ move i n For small e c c e n t r i c i t y can be described by Equation 3.46, small parameter 6. 2 2 so that the _i For the expansion of these nonlinear expressions we require v a r i a b l e s , of the form S^/t) =• $^ + -5"e^t:>+ t r s i ( b ) ' f where the v a r i a t i o n a l terms e(t) Y /tr) = Y and f ( t ) a r i s i n g from the e l l i p t i c motion of p 2 u + 3T^t> * .try f t } , correspond to perturbations and p^ (note that e(t) and f ( t ) define v a r i a b l e s d i f f e r e n t from those i n Equations 3.49 and 3 . 5 0 ) . t T This d u p l i c a t i o n i s unfortunate, but these functions take no f u r t h e r part i n the a n a l y s i s and a l l non-ambiguous characters of the Greek and English alphabets have already been (or w i l l be) committed. 66 3.23 We-now have three possible s i t u a t i o n s i f the t r i v i a l case e -> 0, 6.-0 is excluded. 1. When 6 -> 0 with a f i x e d the equations of motion reduce to Equations 3.54 and 3.55 f o r the r e s t r i c t e d problem. 2. When a ->- 0 with 6 f i x e d , Equation 3.56 reduces to . S^(t) - £ +• SeCt) u ; ^ = 7_ + S^Ct^ , O* ) 7 and consequently i t i s not p o s s i b l e to s p e c i f y an a r b i t r a r y i n i t i a l motion of p^ which i s independent of 6. 6 + 0, the functions 6e('t) and 6 f ( t ) A l s o , as must tend to zero uniformly i n t (see Section 3.7.1 f o r the d e f i n i t i o n of l i i l i i.Ci in i t,j'} • iiiiS COnuiClOii c A c l UucS icCti-lui" s o l u t i o n and i s s u f f i c i e n t to determine e(t) uniquely. tciinS 'fi'Ciii tiic and'f(t) A proof of t h i s statement w i l l not be g i v e n , but an analogous r e s u l t f o r four-body motion i s obtained i n Appendix V. The o r b i t s defined by Equation 3.57 f o r the e l l i p t i c r e s t r i c t e d problem correspond to the Lagrange e q u i l i b r i u m s o l u t i o n s of the ( c i r c u l a r ) r e s t r i c t e d problem. 3. When both 6 and a are non-zero the r e s u l t i n g equations describe motion i n the e l l i p t i c r e s t r i c t e d problem where the i n i t i a l motion of p^ can be chosen a r b i t r a r i l y . An- a n a l y s i s of t h i s case, although with d i f f e r e n t forms of the equations of motion, can be found i n Szebehely [ I 9 ] and Nayfeh [ 4 6 ] . 1 I f we next take i n t o consideration the e f f e c t of py, the nonl i n e a r functions _J and _!_ w i l l contain time- 67 3.24 varying terms depending a d d i t i o n a l l y on the small parameter e. When u ( t ) and v ( t ) correspond to the p e r t u r b a t i o n caused by p^ (note again that u(t) and v ( t ) define v a r i a b l e s d i f f e r e n t from those i n Equations 3.49 and 3.50) then, with + (3.5-1?) C3.S"0 e i g h t l i m i t - p r o c e s s combinations are p o s s i b l e . Four of these i n v o l v e the process e •> 0, and consequently lead to s i t u a t i o n s described previously. The remaining combinations are l i s t e d below i n i n c r e a s i n g order of complexity. (i) 6 and a ->• 0 with e f i x e d . p-j, p 2 and p In t h i s case the motion of i s defined by Equations 3.51 3 and 3.52, and we obtain a s i t u a t i o n s i m i l a r to that described by Equation 3.57 where the i n i t i a l motion of p^ cannot be selected a r b i t r a r i l y . Unique o r b i t s , which are the f o u r - body equivalent of Lagrange e q u i l i b r i u m s o l u t i o n s of the r e s t r i c t e d problem, may be determined from the c o n d i t i o n that eu(t) and ev(t) i n Equations 3.58 and 3.59 tend to zero uniformly i n t as e ->- 0. (ii) a 0 with <5 and e f i x e d . case, where p orbits 2 In c o n t r a s t to the preceding and p^ move i n approximately c i r c u l a r ( i . e . e l l i p t i c o r b i t s with zero e c c e n t r i c i t y ) , c f . footnote on page 3. 22. 68 3.25 here the motion -of .p^ and p^ i s .approximately e l l i p t i c and depends both on e and 6. Apart'from t h i s complication the same conditions a p p l y , and corresponding o r b i t s f o r p^ could be determined, (iii) 6 -> 0 with a and e f i x e d . The i n i t i a l motion of p^ can now be chosen a r b i t r a r i l y , and the r e s u l t i n g equations motion around the o r b i t s of case ( i ) (iv) a , 6 and e are a l l non-zero. define above, This f i n a l combination describes the general s i t u a t i o n where p^ and p^ move i n approximately e l l i p t i c o r b i t s , and extends case ( i i i ) to accommodate motion of p^ around the o r b i t s of case ( i i ) . To e l u c i d a t e the process of e v o l u t i o n described i n t h i s • XI-, U l l l ^ j ~ M l / ^ I C ; . j . J I \ > l l -C I I _^ i l l V, I I summarized i n Table 3-1. T ^ I U 4-. — I I, » « U I "~ J.. „.~U4i„ v> t - i j r An a s t e r i s k \s t U k n > l . . .v.4--J ,-n ~ 4-„ I \**J . • n . ^ ^ . v / i . section, A ri ~i in the e, 6 or a column denotes a non-zero value of that parameter, and the comment " v a r i a t i o n a l o r b i t " in the f i n a l column denotes motion about the preceding unique o r b i t f o r an a r b i t r a r y i n i t i a l p o s i t i o n and v e l o c i t y of p^. The morphosis of t h i s 4-body problem i s i l l u s t r a t e d i n Figure 3-3. Interactions between the various p a r t i c l e s are represented by f i g u r e s which correspond dimensionally to the number of bodies i n v o l v e d , with the exception of the enclosing four-dimensional hypersphere which i s depicted by a conventional sphere. Nodes i n Figure 3-3 are equivalent to the cases of Table 3-1, and the small parameter responsible f o r a t r a n s i t i o n from one node to another i s shown beside the corresponding branch. The least-complicated s i t u a t i o n (node 1) i s associated with 69 3.26 Case Motion of p Motion of p^ and e 6 1 0 0 Circular 2 0 * El 1 i p t i c 3 0 Perturbed c i r c u l a r 4 * * * Perturbed e l l i p t i c 5 0 0 0 Circular Lagrange point 6 -0 0 * Circular Variational 7 0 0 Elliptic 3-body unique o r b i t 8 0 * * * Elliptic Variational 9 * * * * 0 0 Perturbed c i r c u l a r 4-body unique o r b i t 0 * Perturbed c i r c u l a r Variational * * 0 Perturbed e l l i p t i c 4-body unique o r b i t * Perturbed el 1 i p t i c Variational 10 11 12 Table 3-1 0 2 O r b i t a l motion i n the n-bodv Droblem f o r orbit orbit orbit orbit 2*n^4. e q u i l i b r i u m s o l u t i o n s of the r e s t r i c t e d problem (node 5) and with the simplest three-body model f o r p.|, p 2 and p^ (node 3). form of the four-body problem, represented The " s i m p l e s t " by node 9, i s the l o g i c a l consequence of these three fundamental cases. A l l t h a t we now require i s a method by which the n o n l i n e a r , non-homogeneous equations of case 3 and case 9 can be s o l v e d . 3.7 The Two-variable Expansion Procedure Several perturbation methods have r e c e n t l y been developed to determine approximate s o l u t i o n s of equations which, f o r a v a r i e t y of reasons, cannot be solved by conventional a n a l y s i s . A review of these 71 3.28 perturbation techniques i s given by Nayfeh [46 ] , and also Cole [47 Certain concepts which r e l a t e to approximation methods i n general ]. will repeatedly be applied i n the present a n a l y s i s , arid f o r the sake of completeness these are defined here. Nayfeh [46 3.7.1 Further d e t a i l s can be found i n ]. Definitions Gauge f u n c t i o n s : I f f(e) denotes a function which depends on the small e, then, provided the l i m i t as e ->• 0 of f ( e ) parameter e x i s t s , when e •*• 0 we could have (i) f(e) (ii) +0; f(e) ->- A where (iii) f(e) -> °° A is f i n i t e ; (or f(e) -»-•-«>) . The rate at which f(e) -*• 0 f(e) or f ( e ) -> °° with a known gauge function g ( e ) , most useful can be expressed by comparing and of these the simplest and are e e The comparison of f(e) , I e , , e j • •• with g(e) employs the Landau symbols, 0 and 0 . Large 0 I f there e x i s t s a p o s i t i v e number A independent of e and some e„ > 0 such that 0 72 3.29 I iU) I £= fl I gCe>l V |e| ^ fe o » then we w r i t e He) = 0[ gCe)] as e -> O The c o n d i t i o n f o r Equation 3.60 to be v a l i d can also be stated i n the form e -5>e> U) 5 I f f i s a f u n c t i o n of another v a r i a b l e x i n a d d i t i o n to e, and g ( x , e) i s a corresponding gauge f u n c t i o n , then as € —> O i f there e x i s t s a p o s i t i v e number A independent of e and an e (3 .62.) Q > 0 such that When A and e Q are independent of x , then the c o n d i t i o n defined by Equation 3.62 i s s a i d to hold u n i f o r m l y . 73 3.30 Small o: I f , f o r every p o s i t i v e number 6 independent of e, there e x i s t s an e Q such that I He) I £ l (e)| 3 for U l as £ ± 6 then we w r i t e He) o [ g(e)J = —> O ( 3 . 6 3 ) The d e f i n i n g c o n d i t i o n can be replaced by £0) = (3. 64) o which i s a form corresponding to Equation 3.61 Asymptotic Expansions: A sequence o f functions <5 (e) i s c a l l e d an asymptotic sequence n if n L r»--i J as e ->• 0 I f 5 (e) i s an asymptotic sequence and a m sum \ a HI - O f i f and. only i f S (e) m (3.65) i s independent of e , the i n f i n i t e i s the asymptotic expansion o f a function 71+ 3.31 & = 2^ ^S^Cfe} as e + 0 ° L" +" . (3.66) We then w r i t e Uniform v a l i d i t y : I f x i s a s c a l a r or vector v a r i a b l e independent of e, then the asymptotic expansion of f ( x ; e) i n terms of the asymptotic sequence nr A/ : V a ( O T ^ e ^ as e + 0 where the functions a (x) are independent of e. m This expansion i s uniformly v a l i d i f / . — i -o where uniformly f o r a l l x of i n t e r e s t . satisfied, a m ( ) x 5 m ( ) E m u s t b e For these uniformity c o n d i t i o n s to be small i n comparison to the preceding term 75 3. a _ ( x ) 6 _ ( e ) f o r each m. m 1 m Now from Equation 3.65 6 (e) = o[6 1 ^ e ' ) ] so t h a t , f o r the expansion to be uniform, a (x) must be no more s i n g u l a r m than _-|(>0- a m Each term i n the expansion must consequently be a small c o r r e c t i o n to i t s predecessor, independent of the value of x . This concept forms the foundation of the method of m u l t i p l e scales [ 4 6 ] , of 2 which the two-variable expansion procedure i s a s p e c i f i c case. Elementary operations on asymptotic expansions: Consider the two asymptotic expansions as e -> 0 C3.7i) as e + 0 , C3.72) and where <f> (e) i s an asymptotic sequence. The operation o f l i n e a r combin- m a t i o n i s j u s t i f i e d i n g e n e r a l , and we have 00 •«K*jO + Cx;0 ^ 1 ^ s e - ^ * P" » ^tS^ as e 0. (3.73 ) I f , however, the two expansions are f o r m a l l y m u l t i p l i e d , the product 4>,-(e)<J>.(e) (where i , j = 0 , 1, 2 , •••• °°) cannot i n general be arranged in an asymptotic sequence. The operation o f m u l t i p l i c a t i o n i s therefore j u s t i f i e d only when the product 6. (e)<j>-(e) forms an asymptotic sequence or possesses an asymptotic expansion. This c o n d i t i o n i s s a t i s f i e d f o r the important case <j> (e) = e , and we obtain m m 76 3. 6 (x^a) 3 C;x ; e) A / \ ( O e " e -> 0 as (3.74) where (3.7S) K_^> 2_ I f f ( x ; e) and a (x) m i n Equation 3.71 are i n t e g r a b l e functions of x , then »c as e ->• 0 so that term by term i n t e g r a t i o n of an asymptotic expansion i s j u s t i f i e d . When f ( x ; e) and <i> (£). are integrable functions of e a corresponding m r e s u l t holds f o r i n t e g r a t i o n with respect to e. The process of d i f f e r e n t i a t i o n with respect to x o r e cannot, however, be j u s t i f i e d i n the general case. We cannot, consequently, assume without r e s e r v a t i o n that CO as e -> 0 (3.?0 S i m i l a r l y the process of exponentiation i s not j u s t i f i e d i n g e n e r a l , so that 2^ as s •+ 0 (3.7f> . 77 3.34 may not be v a l i d . In both cases, however, when these operations are not j u s t i f i e d they introduce non-uniformities i n t o the r e s u l t i n g expansions. The basic reference f o r a l l the r e s u l t s o u t l i n e d above i s Van der Corput's 1956 paper [ 4 8 ] . A d d i t i o n a l discussions appear i n Nayfeh [46 ] 3 and E r d e l y i [49 ] . 3.7.2 M u l t i p l e scale methods The two-variable expansion procedure introduced by Cole and Kevorkian [50 ] , and developed by Kevorkian [51 ] , i s one of a c l a s s of methods i n v o l v i n g transformations of both dependent and independent variables. A d e t a i l e d d e s c r i p t i o n of these m u l t i p l e scale asymptotic methods can be found i n Nayfeh [46 ] , and only those aspects which apply to the present a n a l y s i s are discussed here. We are interested i n d e r i v i n g r e s u l t s which are v a l i d over long periods of time, and t h i s c o n d i t i o n w i l l be s a t i s f i e d i f the s o l u t i o n s are uniformly v a l i d i n t over some " l a r g e " time i n t e r v a l . Consider, f o r example, Equations 3.51 body motion of p and p^. 2 and 3.52 which describe the three- The v a r i a b l e s u ( t ) and v ( t ) are assumed to possess uniformly v a l i d asymptotic expansions i n terms of two t i m e - l i k e v a r i a b l e s x and r , so that oo u. (t^) ro \ L—j or L—i u( m t v) e m as e 0 (3.?s) 78 3.35 where R ( x , T; e) = 0(e ) uniformly i n T and r as c + 0. n n The formal operation of d i f f e r e n t i a t i o n (as defined i n Equation 3.76) i s assumed to be l e g i t i m a t e , and from Equation 3.78 we obtain oo A/ m V 6. as e 0 . C3.8o) m -o Corresponding expansions apply f o r v ( t ) and ^ f o r u ( T , r ) - i n Equations 3.78 to when v ( x , r ) i s substituted m 3.80. The t i m e - l i k e v a r i a b l e s T and r are selected so that one, r instance, i s a n e a r - i d e n t i t y transformation of t and the o t h e r , x, f i e s the c o n d i t i o n x = 0(et) as 0. z for satis- For the three and four-body problems considered i n the present a n a l y s i s the equations of motion contain t e x p l i c i t l y , and i t i s therefore to set the f a s t v a r i a b l e r equal to t . both convenient and expedient The slow v a r i a b l e i s chosen so that x = e t , which i s the simplest case s a t i s f y i n g i n x then requires that R ( ^ , T; z) = 0(e ) n n that t = 0(e~^). x = 0(et). Uniformity f o r x f i x e d as z -> 0, so This c o n d i t i o n consequently determines, f o r small values of e, the required large time i n t e r v a l during which the s o l u t i o n i s uniformly v a l i d . The g e n e r a l i z a t i o n of t h i s procedure to the many v a r i a b l e case and a time i n t e r v a l 0(e~ ) (where n" = 2 , 3 , ••••) i s n described i n Nayfeh [ 4 6 ] . tf When the asymptotic expansions f o r u(t) and v ( t ) are s u b s t i t u t e d i n the equations of motion secular terms appear i n the r e s u l t i n g general s o l u t i o n , and i t i s the e l i m i n a t i o n of these non-uniform terms which defines the uniformly v a l i d s o l u t i o n . With t h i s section the foundation of the development i s completed, and we can now proceed to the a n a l y s i s of three and four-body 79 3. motion. A uniformly v a l i d s o l u t i o n of the s t e l l a r three-body problem i s derived i n Chapter 4, which may be reduced to a corresponding s o l u t i o n of the r e s t r i c t e d problem by applying the l i m i t process m^ + 0. In t h i s form a comparison with G.W. H i l l ' s v a r i a t i o n o r b i t i s f e a s i b l e , and we obtain d i r e c t v e r i f i c a t i o n of the uniformly v a l i d s o l u t i o n from w e l l established r e s u l t s . The general three-body o r b i t i s then applied to the four-body problem, and i n Chapter 5 a uniformly v a l i d s o l u t i o n i s derived f o r motion near the e q u i l a t e r a l t r i a n g l e points of m 9 and m-. 80 4.1 4. 4.1 THE THREE-BODY PROBLEM Introduction Equations 3.51 and 3.52, together with Equations 3.5 and 3.6, describe the least-complicated form of three-body motion f o r p-|, p and Pg, and the s o l u t i o n of these equations w i l l chapter. 2 be o u t l i n e d i n t h i s The i n i t i a l a n a l y s i s presented here i s s u f f i c i e n t to reveal the basic s o l u t i o n s t r u c t u r e , but i n the i n t e r e s t of r e a d a b i l i t y much of the d e t a i l e d development i s relegated to Appendix I I I . It will, perhaps, be helpful at t h i s point to summarize .the sequence .of r e s t r i c t i o n s which generated Equations 3.51 and 3.52. 1. m-| > m^ > m^. 2. p.| and the hypothetical p a r t i c l e p* move i n two-body o r b i t s about the i n e r t i a l o r i g i n . 3. p 2 and p^ move approximately i n two-body o r b i t s about t h e i r centre of mass. From page 2.30: 4. p* i s located at the o r i g i n of the.?;, n , C coordinate system. 5. p-| and p* move i n c i r c u l a r o r b i t s about the i n e r t i a l o r i g i n . 6. The motion of p.j-, p F i n a l l y , from Section 2 3.6.1: and p 3 i s coplanar. 81 4.2 7. As the small parameter P e tends to z e r o , the l o c a t i o n of i s defined by the s o l u t i o n of r e l a t i v e e q u i l i b r i u m 2 C (t) = 1 - y 2 2 ; n (t) = 0 . 2 Note that conditions 5, 6 and 7 combine to produce the simplest form of three-body motion. assumption. The only approximation i s associated w i t h the second This source of inconsistency was discussed i n Section (Equation 2 . 3 9 ) , but as e -> 0 The a n a l y s i s r e s t s , t h e r e f o r e , 4.2 2.6 the r e s u l t i n g e r r o r reduces to zero. on a comparatively f i r m foundation. Pre!iminary A n a l y s i s From Equations 3.49 and 3.50 the p o s i t i o n of i s determined by where the o r b i t a l parameter 6 has been set equal to zero. We now express u(t) and v ( t ) as asymptotic expansions i n the two time v a r i a b l e s x and t , with $JtO x = e t , so that ^ •«-/*,.•+ -T L—i 1 u^r,t)e~ 4- OC<L^) C4 3) 82 4.3 where these expressions are to be uniformly v a l i d i n T and t . formal operation of d i f f e r e n t i a t i o n i s assumed to be v a l i d Section 3 . 7 . 1 ) , then f o r the d e r i v a t i v e s of eu(t) If the (see and e v ( t ) we o b t a i n : C4.?> N 6 /v/ Jib 4- & 3y„, 5" Pb^Tr Pb"" ^b^r 1 Equations 3.51 and 3.52 can now be expressed as a sequence of non- homogeneous p a r t i a l d i f f e r e n t i a l equations i n ascending powers of e, For the i ' t h power of e we have <)t 2- h» 3^- ^ c k £ * C u,v, b, v ) v In the absence of any information about i t s v a l i d i t y , we have no a l t e r n a t i v e but to proceed i n the hope that t h i s operation i s justified. (4.1) 83 4.4 . ¥ ( u,v b , v ) K ; ^t - , 0*--'°) at 1 where the functions E* and F* involve ( i n general) the terms u^ to u^._^ and v-| to v^_^ but not u^ and v ^ . Now the ordinary d i f f e r e n t i a l equations £ -?.</- 3u_ = o (>•") have the f o l l o w i n g s o l u t i o n : where A , B,C and D are a r b i t r a r y constants. The homogeneous s o l u t i o n s f o r u.j ( t , x) and v ^ ( t , x) i n Equations 4.9 and 4.10 therefore assume an i d e n t i c a l form of x . except t h a t , i n t h i s case, A , B, C and D are functions Equations 4.13 and 4.14 can be rearranged so that i^lt/e) = ZolL-v) - izCv)c°$[t +• 0 0 c ) ] which, i n p r a c t i c e , i s a more convenient form. (4.1?) The functions a ( x ) , 3 ( T ) , Y ( ) and 9(x) are to be determined from the c o n d i t i o n s f o r uniform t v a l i d i t y i n t and x . Note, i n c i d e n t a l l y , that Equation 4.16 already 4.5 contains a secular term. In the absence of f o r c i n g terms i n v o l v i n g Uy, Equations and 3.52 possess the exact s o l u t i o n discussed i n Section 3.5. ~ ^ ~ ^2' I t i s therefore n 2 ^ = ^' 3.51 a s reasonable to assume that any disturbance from r e l a t i v e e q u i l i b r i u m caused by the f o r c i n g terms w i l l be of a comparable order i n e, so we can take £ <^ ) f c = J " ^ ( t , T ) r Z > + - OU ) n . (4.1?) r>.-l . I, (4. 18) »~ =. k where k > 1. To determine the value of k we must now examine the non- hnmnopnpnu?; f u n r t . i n n f ; and | r „ ! 3 in Equations 3.51 and 3.52, which can be expanded using Equations A1.6 and it Al.7 A1.7 i n Appendix I. As the non-homogeneous functions are O'(e^), i s not u n r e a l i s t i c to assume that k ^ 4. we then obtain From Equations A1.6 and 85 4.6 + 6 3. cob 3b JS_ S (4-.2°) since £ 2 = 1 - y 2 + °( e 4 ) From Equations 4.15 and 4.16 a n d ^2 = 0 ( e u ^ ( t , x) and 4 when ) k = 4. ( t , x) are p e r i o d i c i n 8 8 t = 2TT, S O that terms of the form e acost and e gsint i n Equations 4.19 and 4.20 w i l l generate secular terms i n the non-homogeneous s o l u t i o n f o r Ug(t, x) and V g ( t , x ) . I f the complete s o l u t i o n i s to be uniformly v a l i d i n t these secular terms must be e l i m i n a t e d , which can only be.accomplished i f the q u a n t i t i e s and 8 i n Equations 3.51 and 3.52 are 0 ( e ) . This c o n d i t i o n i s s a t i s f i e d when k = 5. The same r e s u l t can be o b t a i n e d , i n a l e s s h e u r i s t i c manner, 86 4.7 using the complete expansions of Equations 4.3 and 4 . 4 , although at the expense of s u b s t a n t i a l l y increased complexity i n the a n a l y s i s . A more complete expansion of Equations 4.19 and 4.20 i s given in Appendix I (Equations A1.8 and A'1.9) f o r L j m > 5 At the s t a r t of the sequence of non-homogeneous p a r t i a l d i f f e r e n t i a l equations corresponding to Equations 4.9 and 4.10 we have and t h e r e f o r e , from Equations 4.15 and 4.16: ^(t,f) = 2yScO sc*.[ t + 9CT)] +XCV) - 3-bocCv) .. Higher order s o l u t i o n s w i l l , i n g e n e r a l , c o n s i s t of both homogeneous and non-homogeneous terms, so that we can w r i t e Z.—x m a t (*-*0 87 4.8 where the subscripts H and NH denote homogeneous and non-homogeneous parts of the s o l u t i o n . solutions into I t i s convenient to accumulate a l l the homogeneous u (t, T ) 5 h and v ( t , T ) , but i n t h i s case we must 5 h express the functions a(-r), B ( T ) , y("0 and 6(x) as asymptotic expansions of the form I. = 2^ i ^ ^ ^ ^ Equations 4.27 and 4.28 can then be w r i t t e n C + -3I) 88 4.9 feu. Z . — i yt-l Z — - i m- 6 Now i t i s always p o s s i b l e , by r e - l a b e l l i n g x as e t , to express the nonhomogeneous s o l u t i o n s u . and (where i > 6) as functions of t , and therefore no d e r i v a t i v e s of these terms with respect to x need appear i n the sequence of p a r t i a l d i f f e r e n t i a l equations. With t h i s s i m p l i f i c a t i o n we obtain -a dO-5 it* + E it (4-.»0 - 5. J Oil) -3u. f = -Z — o(0 - 2 — J Of.) 89 4.10 >Jt< - it r- - his — oCfe) (4.31) + ^ +- 23u it J E, O(0 g it 2 0 + - p,) ". 1 [5 where the subscripts 0(1), 0(e) order i n e and the functions (Equations A1.8 and A 1 . 9 ) . and 0(e ) denote terms of corresponding and F. are defined i n Appendix I The extension of t h i s sequence to higher orders of e i s given i n Appendix I I I , 4.3 Equations A3.20 to A3.29. The Uniformly V a l i d S o l u t i o n We can now solve the sequence of p a r t i a l d i f f e r e n t i a l equations under the condition that no secular or unbounded terms remain i n the solution. A development as f a r as s o l u t i o n s f o r Ug and Vg i s to reveal the e s s e n t i a l sufficient features of the a n a l y s i s , and d e t a i l s of the f u r t h e r development can be found i n Appendix I I I . A l l non-homogeneous terms i n the sequence of p a r t i a l d i f f e r e n t i a l equations may be w r i t t e n i n the form &*Ctp) ~ a„ 4- P*(fc,V) = y + o a. b x 4- a ^ j U v b + b^<-b 4- A ^ C s o t by l o t 4- •*• c «i^ wfc + c . t o o k <^ AV<0b -4- cL C o « o b D o { (4.4l) (4.4*) 90 4.11 (the functions E* and F* are defined i n Equations 4.9 and 4.10).Corresponding non-homogeneous solutions are given i n Appendix I I I (Equations A3.10 to A3.13), and the conditions under which secular terms are eliminated w i l l determine the uniformly v a l i d s o l u t i o n . When the asymptotic expansions defined by Equations 4.29 to 4.32 are s u b s t i t u t e d i n Equations 4.25 and 4.26 we obtain "" 6./», + fc /3*.*--- J COS £ b + 0 f £0, + e?9^ir -• ] l o The functions c o s ( t + 0 ) and s i n ( t + 9 ) are expanded as Cn C t + P) - <^b coP - <- b «u. ©„ 0 - «E> I s—t c<o (4.45) ( t + 6) * sCtb c«o9 4- cr> b sJ-v. P„ + £ P, I dr»t CoPo p — ^1 b P ] e +_of O so t h a t , f o r the s o l u t i o n of Equations 4.23 and 4.24, we may w r i t e fr 91 4.12 (a. 4 y ) The secular term in Equation 4.48 could be eliminated by s e t t i n g CXQ(X)=0, but t h i s assumes, without any a d d i t i o n a l proof, that «Q(T) = constant. I t i s not, i n g e n e r a l , necessary to make any assumptions of t h i s nature, so we w i l l proceed as i f CCQ(X) ^ 0. A f t e r taking the p a r t i a l d e r i v a t i v e of Equations 4.47 and 4.48 with respect to x and s u b s t i t u t i n g f o r Eg and Fg from Appendix I (Equations A1.8 and A 1 . 9 ) , we obtain f o r Equations 4.35 and 4.36: 2 <-<-< - ^ 2s.«-b - x *• % SC«-b + 4- <ht Z« 2v Oil) + 6b 2c<rz> b 5 — 2. dot ^/3„ C«Pp - p> ^^ 0 0 p_Pe I f terms i n these equations are i d e n t i f i e d with q u a n t i t i e s i n Equations 4.41 and 4.42, a corresponding non-homogeneous s o l u t i o n can then be derived from Equations A3.10 to A3.13 i n Appendix I I I . example, We have, f o r 0 92 4.13 2>V « ' To eliminate terms i n v o l v i n g t }x 2 i . [ Wo*****] ; from the non-homogeneous = o 0 solution (A- * > so we are now j u s t i f i e d i n taking * - 0 ° to remove the secular quantity i n Equation 4.48. C^.rz) I f terms containing t s i n t and t c o s t are to be e l i m i n a t e d : P I ^ f f I =. © , (A-.SO 1 so t h a t , i f 3 Q ^ 0 , both BQ and 6Q must be constant. 4.53 and 4.54 are s a t i s f i e d we have When Equations 4 . 5 1 , 93 4.14 6 u pit '?*-} = ^6 1 [ "<« X c e r > i : - £ ,zt * 0 If u 6 c b ] 0t? Ur i s to be free of secular terms, then *u.t - [ 2-t - 4jiK.t + £ t <LXo must be constant. its Now contains a term -e[3ta^] which, i f a-j i s constant, could be used to eliminate the secular terms i n Equation 4.56. the r e s u l t that = o (Equation 4 . 6 4 ) . Suppose we a n t i c i p a t e Then, i f the l a r g e s t secular terms i n the complete s o l u t i o n are 0(e ) when t i s f i x e d as e -»- 0. A l s o , i f we take then To j u s t i f y t h i s choice of a-j, suppose We wil1 then have O I O 9k 4.15 .where TQ and are constant, and Equation 4.57 may be w r i t t e n as 17Exactly the same c o n d i t i o n i s obtained as i n Equation 4.58, but f o r + r.j instead of a-p Y Q ( T ) "= F-| is therefore redundant, and we are l e f t with r , which i s equivalent to Equation 4.59. Q The non-homogeneous s o l u t i o n s Ug(t) and Vg(t) now reduce to ii ? $ul fc. - + £b 4- s o l u t i o n by the c o r r e c t choice of For Equations 4.37 and 4.38 we have, from Equations A 1 . 8 and A l . 9 i n Appendix I : o With the s i m p l i f i c a t i o n s r e s u l t i n g from Equations 4 . 5 1 , 4 . 5 3 , 4 . 5 4 and 4.59 we obtain f o r the d e r i v a t i v e terms: 3 u. x s Otl) 4- sc~b V, iv 95 4.16 I at- \ L \ ivJ air ar ' 5i a_P c av L a*r _ «*p„ ) z^s, aj> 40 I - ^ v -t5 . aT^tr a\ Ptr av ' ar' Z f^, s ar ar P/S Z c « f f , J / J , £P, *• I +- /5,a_P 0 I W ? i v ) av J L at" A ^ aira r - coj k a"v v <aW J 1 (. a r a-r ' at\) _ Secular terms i n v o l v i n g t , t s i n t and t c o s t w i l l be e l i m i n a t e d from the s o l u t i o n s f o r Uy and Vy i f : 1«, = o 96 4.17 (4.6$) at? Tv j-c* V a w ' ar a-r s and these conditions are a l l s a t i s f i e d when o and 6^ are , constant. The remaining secular term i n Vg i s removed i f atr aV + 3^9 av at: n air where 1 a-t" 1 I i s the only term which i s undetermined. airy From the same argument as before f o r YQ i t follows that A l l the q u a n t i t i e s in Equations 4.62 and 4.63 are consequently e l i m i n a t e d , so that we obtain H (fc) = V (t) ? ? = O . (4.61) The conditions f o r uniform v a l i d i t y of the complete s o l u t i o n have, at t h i s point i n the a n a l y s i s , produced e x p l i c i t values only f o r ctg and , and to determine BQ and 6Q we must consider the s o l u t i o n s 97 4.18 for Ug and v . From Appendix I , Equations A1.8 and A1.9: g 1 tosl + ]S COS 3fc « 8 3_s^b + 8 If 5 ^ 3b 8 and the non-homogeneous terms i n Equations 4.39 and 4.40 can be w r i t t e n i n the form - J U co^St ix hx */ 60s ]£. j * , ( l - ^ . ' ) t ix iX e + jc*.b 2 C 0 O ^ 5 t - Zs^B a ^K 0 0 • + 2-0+-/O A> X ^ " ^ o (4.?3) 98 4.19 To remove terms i n v o l v i n g t 2 from the non-homogeneous s o l u t i o n ^ 2 . - 0 C4.?0 o and f o r the e l i m i n a t i o n of the q u a n t i t y -e [ 3 t a , J from Equation 4.44 we must take = 0. (4-.7O Secular terms containing t s i n t and t c o s t are eliminated i f I J 2V lb and bx . 16 Now, unless s i n 6Q = 0, Equation 4.77 v i o l a t e s uniform v a l i d i t y i n x , and therefore From Equation 4.76 we have 99 4.20 and since 3 n and 9 u 0 constant, a r e it i s also constant. I f , however, i s also constant, then i t f o l l o w s that 2 fi N u = is 0 cos 9 C*-*0 V Although t h i s d e r i v a t i o n f o r gg and 9 cannot be j u s t i f i e d at 2 the present stage of the a n a l y s i s , i t w i l l be shown i n Appendix III that + 6^ v i o l a t e s uniform v a l i d i t y i n T unless 9 a p p l i e s , i n f a c t , to Qy i n Appendix I I I ) , 2 = 0- t n e same c o n d i t i o n 9 ^ , and 9 ^ (which i s the l i m i t of the a n a l y s i s and i t appears l i k e l y that 9^ = 0 f o r a l l i > 0. A f t e r some m a n i p u l a t i o n , the one remaining secular term i n Vg can be obtained i n the form « when Y 2 iv i s constant (by the argument of page 4.14), and i f is also constant, then the term -e [3ta.j] i n Equation 4.44 i s e l i m i n a t e d by oi 3 = Z j^O-fO^ . At present we have only assumed that 0 (4.22) 2 and are constant. Both these assumptions a r e , however, v e r i f i e d i n Appendix I I I , and f o r Ug and Vg we then have Page A3.24, Equation A3.83. 100 4.21 - 2_ + 4 8? c o s t 64 + « tos H (4-33) »4 31 s « t + rs s c ^ 3 b 3732.' 2-1 b where the secular term i n v ( t ) i s eliminated by the c o r r e c t choice of g a. This process o f a n a l y s i s , which i s continued i n Appendix 111 : leads to the f o l l o w i n g r e s u l t s : (*•«*) 3r ^5 * 31 -t- &4 Z%£ 26 .(4.*t) f*,) (4.S?) 64 0 + ^ , ) ^ S?6 II. 64 (4.88) (4.89) 141 + 135 3z( p-, l + (4.1. .p >J l (4.12.) O u^(t) = f*,('+^,) (i-j*0 i ) -9- +• M t o j t £> - i_5 c o s 2-t 6 (4-«3> 101 4.22 ?> 17. (4.14) 2. 2 +• 22-5" p. i 1 v ft) 10 ^,(i- / O | -T = 4 + -63 - i 7 Go2t ^ /"*• 7 '*" ~ •( Jt si^i+b 3 ^ t - s ^ . k ) - t •f 15' I 64- + 6 + 7 + t^i x % s ? t y.xCj+w,) > 7 A-t Co5 S ^ Six. 2.b 1j <eS/*,(l-jO ^ ( l - c a t " } (A-.9fc) 16 C.\.+.i*, ) > (. - f O where the development i s terminated at O ( e ^ ) . Note that the secular terms i n Equations 4.94 and 4.96 are eliminated i n the complete s o l u t i o n f o r ev(t) when and a,- s a t i s f y Equations 4.85 and 4.86 r e s p e c t i v e l y . 3Y No s p e c i f i c conditions apply to y » except that ^± = 0, so ax that y appears to be an a r b i t r a r y constant q u a n t i t y . no reason to exclude the c o n d i t i o n are v a l i d f o r 0 < ^ < 1. ->- 1 , since the equations of motion In t h i s case Equations 4 . 8 6 , 4 . 8 8 , 4.95 and 4.96 contain unbounded terms unless Y q and Y-J are both z e r o , and i t appears l i k e l y t h a t , i n g e n e r a l , Y ( T ) = 0. When Y s t a t e from Equation A 3 . 2 4 1 There i s , however, that = Q 0 we can also 102 4.23 The values given i n Equations 4.85 to 4.97 are e n t i r e l y l e g i t i m a t e , i n that no assumptions whatever are necessary. If, however, we make the p l a u s i b l e assumption that ctg, 6g and 6^ are constant, then we may also wri te (from Equation A3.250 with YQ = Y-j = 0) ; 5*c Sec e, 0„ ( A . too In a d d i t i o n to these concrete r e s u l t s , the d e r i v a t i v e s of a , 3, y with respect to T are at l e a s t O(e^), ||- = O(e^) and tP = 0(e°) From Equations 4.85 to 4.100 constant), solutions for ^ ( t ) a constant e r r o r O(e^). and r ^ t ) can be accumulated that have Secular terms i n the s o l u t i o n s f o r 2 and v ^ ( t ) ( and assuming 8g and 0^ are could involve t , t u^(t) 3 and t expressions (see Appendix I I I , Equations A3.8 and A 3 . 9 ) , but i f secular terms are eliminated from the ) 103 4.24 lower-order s o l u t i o n s no t J expressions w i l l be generated. I t a l s o appears 2 highly l i k e l y that t terms are e l i m i n a t e d i f a ( r ) = constant. With a reasonable degree of confidence we can therefore state that the s o l u t i o n s for ^ ( t ) a n c ' a r e u n i ^ o r n i ^ y v a l i d f o r t = 0 ( e ~ ^ ) , but even the most p e s s i m i s t i c estimate gives a time i n t e r v a l of uniform v a l i d i t y 0(e" ). 7 A f t e r some m a n i p u l a t i o n , these s o l u t i o n s can be derived as: J. + Cos - cos - ^ It 3 fc &4- (4. e ^ / l f ^ ' i M i - ^ ) - £ ID l o S d - ^ ) 3 c^t + 1 1 'oO cos 2.1 ] co^ 4-t + IS ^ Ci-p,_") sc*.b ( * + .b £ t l f o • Ifed-t-fJ,) J - tS" sin. 32- J 3b (4.IOZ ) - »2- 101}. 4.25 Zb +• fe3 ScV4b 10x4 Ci+^i 7) 3 + ocv) , where i t i s assumed, i n Equation 4.102, that y{x) = 0(e ). Note that any ambiguity concerning the choice of 8Q i s resolved i n t h i s f i n a l form of the s o l u t i o n (see Equation 4 . 7 8 ) . . Corresponding r e s u l t s f o r £ ( t ) and n ( t ) 3 from Equations 4.101 3 can r e a d i l y be obtained and 4.102 using Equations 3.5 and 3.6. We then have: > A<ifc.£'-/0 s /6 (1+ -r 6 U ' z? + 3 2 - >35 pi, est do 6 2b ) ~* - 2£ ? «jt 3^1+^1) J io5(l-p^") + cost cos 3 fc ^4- + co^2.b n (4. * 103) cos4b 32. OCfe") ^ 0 11 Si*>.t — _i£ SCn. 3 b 32. -+L 640+/X,") SH 12. Si«,?-b J ] sc^zb (4. 1 0 4 ) 105 4.26 4.4 A Uniformly V a l i d S o l u t i o n of the R e s t r i c t e d Problem Quite a r b i t r a r i l y we have chosen m^ > m^, so that Now suppose U £ •+ 1 o r , conversely, 0. > In t h i s case Equations 4.103 and 4.104 reduce to a s o l u t i o n of the r e s t r i c t e d problem, but i n a coordinate system where the p o s i t i o n of m-j changes with time. When U £ -> 1, Equations 4.103 and 4.104 take the f o l l o w i n g form: r L 6 O f t ' ) (4.«oS) 1 / \! i± Si*, i t 12. 9 (A-.lofc") where these s o l u t i o n s are uniformly v a l i d as e -> 0 f o r t = 0(e" Note that Ur*. $ (t) = o t ^ Ct) - o , ). and consequently m i s located at the o r i g i n of the £, n. coordinate system. ? 106 4.27 Figure 4-1 Primary c o n f i g u r a t i o n f o r the r e s t r i c t e d problem I f we s e l e c t the H , H coordinate system shown i n Figure 4 - 1 , so that both and m are located on the E a x i s , then the transformation 2 equations are (4.|o?) H = ^ d.ib -<• cos t (A--log) With respect to the ~, H axes, Equations 4.105 and 4.106 can now be w r i t t e n as: C fc) = . cos b 4-g it *4- 2 107 4.28 - - Sin.t 4? A 3-4 (4.l«o) 6 8 if. In t h i s form the s o l u t i o n f o r the motion of can be compared with H i l l ' s v a r i a t i o n o r b i t , which we consider i n the f o l l o w i n g s e c t i o n . 4.4.1 H i l l ' s s i m p l i f i c a t i o n of the r e s t r i c t e d problem G.W. H i l l based his a n a l y s i s of lunar motion on a three-body model which corresponds to a s p e c i f i c s i m p l i f i c a t i o n of the r e s t r i c t e d problem with the sun and earth as primaries [52 * ] . A discussion of H i l l ' s a n a l y s i s can be found i n Brown [ 5 3 ] Hagihara [ 5 4 ] ; detailed and f o r a concise summary see Brouwer and Clemence [ 4 2 ] . 3 The p r i n c i p a l assumptions of H i l l ' s approach a r e : the s o l a r p a r a l l a x i s zero .t 1. More s p e c i f i c a l l y the a n a l y s i s requires that a / r ' be n e g l i g i b l e , where a and r ' denote the mean earth - s a t e l l i t e and earth - sun distances r e s p e c t i v e l y . I f the e a r t h ' s equatorial radius i s denoted by r * , then the lunar and s o l a r p a r a l l a x TT and T T are defined by: o = and, since r a_ r' L arc s i n « a « a arc s i n _r r' r' 7L- C I f the s o l a r p a r a l l a x i s z e r o , a / r ' i s therefore zero. 108 4.29 2. the s o l a r e c c e n t r i c i t y i s -zero; 3. the lunar i n c l i n a t i o n i s z e r o ; 4. m / i s s u f f i c i e n t l y small "to be neglected. x • / lA | Note that although H i l l ' s a n a l y s i s does not e x p l i c i t l y require m^ to be n e g l i g i b l y s m a l l , t h i s condition i s implied by s e t t i n g the e c c e n t r i c i t y equal to zero. solar The sun's o r b i t about the e a r t h ' s mass centre i s consequently c i r c u l a r , so that m-j and f i g u r a t i o n independent of the motion of m^. maintain a f i x e d con- Note also that the Jacobi i n t e g r a l , which H i l l uses to e l i m i n a t e nonlinear terms from the equation of motion, i s i n v a l i d when m^ d i f f e r s s i g n i f i c a n t l y from z e r o , as indicated i n Section 2.7. Kevorkian [55 ] has demonstrated, using an elegant and d i r e c t nieuiGu, Midi, uic r t i u i c u e u |j I UU t ein can ue ICUULCU l i m i t process to give H i l l ' s equations of motion. also appears i n Szebehely [ l 1 0 ], uy an as,ym[j lui, i C A s i m i l a r approach but i n the summary given here we f o l l o w Kevorkian i n l o c a t i n g the l a r g e r mass (m-j) on the negative H a x i s (Figure 4-2). The c o n f i g u r a t i o n of the planar r e s t r i c t e d problem i s shown i n Figure 4-2, where the larger mass i s associated with the sun and JL - ^^/^ • The time and distance normalizations are s e l e c t e d so that the H * H* coordinate system rotates at u n i t angular v e l o c i t y w i t h respect to i n e r t i a l axes, and the earth-sun distance is unity. We next locate m^ at the o r i g i n of a new system by s e t t i n g * = <j = - H ' 0 - / 0 . . . (4.ni> (.4.WZ.) 109 4.30 ri Figure 4-2 Primary c o n f i g u r a t i o n , with respect to the H * , H* coordinate system, f o r the r e s t r i c t e d problem. i n which case m^ i s located at x = - 1 . The equations of s a t e l l i t e motion can then be w r i t t e n as: & - 2.. at + + l - r.3 4- i - 0-/O where An asymptotic expansion of the form p. X 110 4.31 C4-.ll?) (4. i s not v a l i d near the smaller primary, as the g r a v i t a t i o n a l J i and us) terms appear as 0(y) whereas in p r a c t i c e , for a r b i t r a r y u, there e x i s t s a region i n the neighbourhood of m w i t h i n which 2 these terms dominate the equations of motion. defined by Equations 4.117 and 4.118 Near the o r i g i n a s o l u t i o n i s consequently not uniformly v a l i d i n the space v a r i a b l e s x and y as u -»• 0. This non-uniformity can be avoided i f we take t and, i n terms of these new v a r i a b l e s , + « t U.II<O Equations 4.113 and 4.114 can be wri tten o(.-3-/3 i - 2. ^ s (4.U0) i - Ci->) dLk a dt where, f o r convenience of n o t a t i o n , t q u a n t i t i e s r^ and now take the form r 3 - ^ r (4.13.1) 3 has been replaced by t . The 111 4.32 so t h a t , provided a > 0: y.-> o At t h i s point i t i s only necessary to consider one component, and i f we s u b s t i t u t e JLt for i n Equation 4.120 using Equation 4.123: 1 (4.12 4-) '4 The values of a and 3 can now be selected from the c o n d i t i o n that s p e c i f i c terms i n Equation 4.124 If, are of a p a r t i c u l a r order i n y . f o r example, s a t e l l i t e motion i s predominantly determined by planetary g r a v i t a t i o n , then: I - 2-OL = O 112 4.33 so that a = 1/2, 6 = 1/4. This case, where s a t e l l i t e motion takes place in the . v i c i n i t y of m^, i s considered i n d e t a i l by Kevorkian [55 ] . Con- v e r s e l y , when the planetary g r a v i t a t i o n , C o r i o l i s and c e n t r i f u g a l accelerations are a l l of the same order we have: oi - %fi and = dL-p therefore a = 1/3, 4.120 and 4.121 d-t I f terms 0(\i%) 3=0. With these values f o r a and 3, Equations can be w r i t t e n as: r , ,-,3/, are neglected and we set (see Equation 4.123 with a = 1/3), Equations 4.125 and 4.126 i d e n t i c a l to H i l l ' s equations of motion [ 5 2 ] . 2 are In the terminology of asymptotic expansion theory, H i l l ' s equations are therefore obtained from Equations 4.113 and 4.114 the v a r i a b l e s f o r the planar r e s t r i c t e d problem by holding 113 4.34 f i x e d as u -> 0. The p a r t i c u l a r symmetric p e r i o d i c s o l u t i o n of Equations 4.125 and 4.126 corresponding to the i n i t i a l 5 / 0 ) > o ; ^Co) - defines H i l l ' s v a r i a t i o n o r b i t [ 5 2 ] . 3 conditions o j 5^°^ = 0 This s o l u t i o n i s compared with the o r b i t defined by Equations 4.109 and 4.110 i n Section 4.4.2 below. » 4.4.2 H i l l ' s variation orbit From the general s e r i e s s o l u t i o n , H i l l computed a p e r i o d i c lunar o r b i t assuming the f o l l o w i n g sidereal periods T L = 27.321 661 T $ = 365.256 371 . [l 1 1 ]: Here T^ and T<. denote the lunar and s o l a r periods r e s p e c t i v e l y , and these numerical values agree with the observed periods i n ephemeris days f o r the earth-moon and earth-sun systems. + S l i g h t l y d i f f e r e n t values are i n current use: Sidereal mean motion of moon (1900) i n r a d i a n s / s e c . E.T. [ 5 6 ] 2.6616 99489 x 10" K Sidereal mean period of moon (1900) .in ephemeris days 27.321 66140 Sidereal year 1900 i n ephemeris days [45 ] 365.256 36556 D 114 4.35 The corresponding v a r i a t i o n o r b i t , which appears i n Brouwer and Clemence [42 *] i n a form compatible with Equations 4.109 and 4.110, 1 i s given below. X Y = =- o. <Ho4o s?-&og cost • 4 o. ooooo SS?$8 COS. St •*• o. ooooo ooioo cos ? t + o. ooooo 00007. i, C O 7 7 8 iso-il •» o. oo 151 4i + o. oo o o o 56 7 •*• o . 4 0* CDS 'J C*4. I Z 7 ) fc sc^t 13 sc«. 3b 09 S"fc ooooo 06300 s^7b ooooo 00007. s;«. 1 1 (4.iae) I t i s assumed, f o r t h i s s o l u t i o n , that new moon occurs at t = 2nTr (|n| = 0, 1, 2, ••••) and that the l a r g e r mass m^ i s located on the positive X axis. Equations 4.109 and 4.110 were derived f o r the same i n i t i a l c o n f i g u r a t i o n but with m^ on the negative H a x i s (see 4-1). H = -Y Figure The two o r b i t s should therefore be equivalent i f we set E = - X , so that Equations 4.109 and 4.110 take the form: 115 4.36 X(b) = cost 1 - 65 46 + cos 3fc 3 .+ 7 6*p,(i+n,Y 16 « *4 4-8 9 6 1 Now from Eauation 3.21 where fi = T^/T^. Consequently, f o r the s i d e r e a l periods used by Hill: 9. = 7.4801 32633 x 10 -2 The earth-sun distance can be obtained by expressing the a . u . i n dimension-free form. 1 a.u. = From A s h , Shapiro and Smith [57 ] : 4.9900 4785 x 1 0 and from Clemence [56 ] : 2 L i g h t seconds, 116 4.37 Mean earth-moon distance i n metres q A = 0.384400 x 10 V e l o c i t y of l i g h t i n metres/second q c = 0.2997925 x.10 . We then have the f o l l o w i n g numerical values f o r r-j and u-|: r 1 = 3.8917 24558 x 10 2 y ] = 3.2979 46189 x 10 5 , and from Equation 3.28 f o r e: e = 5.0690 77750 x 1 0 - 2 . Equations 4.129 and 4.130 can now be evaluated, and we obtain the r e s u l t s given below. 1 - 16 7.5768 6*38*1 - 3 l.6llS 5862? % \0~* (4.13O S L-.P-VW 1 ^ l o " + 5". 7 1 1 7 *..lo" 8ZS6? y. l o " 3 •+• 3 + 3.66ZI l. M i S 373&S S"€6S8 * l o * 10"* - 3 (V(33) 117 4.38 xct) = o. <?<) O ? V i^gl co t 5 (4. + OCV') •+.O.. o o . l A - l S3i 7e _s;~-.3t v + 0 ( t " ) (4.(35) A comparison of H i l l ' s s o l u t i o n (Equations 4.127 and 4.128) with Equations 4.134 and 4.135 shows reasonable agreement. vergence of i n d i v i d u a l c o e f f i c i e n t s 3 Vz quantity e u-j ; i n the present case Note that the con- i s determined approximately by the e 3 'A = 0.07480, and consequently higher-order terms w i l l have a r e l a t i v e l y s i g n i f i c a n t e f f e c t . 4.4.3 Further terms in the s o l u t i o n I t i s evident from the general behaviour of Equations and 4.106 that the next c o n t r i b u t i o n s w i l l be 0(e 4.105 ), and although a 12 complete determination of 0(e ) terms i s not f e a s i b l e , u-j (t) and v - j ( t ) can be evaluated i f we assume 2 2 case Equations A3.206 and A3.207 can be w r i t t e n : the functions = o • In t h i s 118 4.39 766 ^ f l - j O 4£ 16 (Vl3<0 4 67J- p ^ O - j O cos 4fc ccs 3 b 175 17-8 - s.>-2.b 32. 6 4. 115 / * | C " ' - f O 32- 12.9 where we have taken YQ = Y-J = scw$b 12-8 Y 2. 0 . = From Equations A 3 . 8 to A 3 . 1 1 , the corresponding s o l u t i o n s f o r u^(t) and v ^ ( t ) are: 23-6? 768 + (OS «>5b 48 - 13-7- -t-. ?£_1 COJS t 4e<r?2-b + j-5 c-n? 4 b 2-^6 - 137 (U. OS) cosZb I% -4/ ... 3 x + MS cos b + JZ4S c c j 3 t i<*-4 This s i m p l i f i c a t i o n i s discussed on page 4.22. < + 44! <^s S " t 119 4.40 5 «3L? t - 5is s^b 4- .1 £>4 ^4- 2-^6 4-S -t- ^ sivv^-t ? x 3072. %S(>o ( 4 . 13*1) where the secular terms i n Equation 4.139 are eliminated when Equation satisfies 4.88. The c o n t r i b u t i o n s to Equations 4.103 and 4.104 f o r £ ( t ) and 3 n-jCt) therefore take the form: - \$5\ e. L + 105 4S ?-3°4 ZfJ cost coot - 1 M5 11-8 -t 4-cco2-t + cos4t ;z$& 137 &o 2. b 18 Ceot + 3-4 S ie>2-^- cos 3 t •+ 441 cos *t 3t>7l. (4.1*0) 120 4.41 '1 fx 6 |_o w b r> -t- i 3 £ s^Z-t - _ 2 £ s^ij-b 4-g 1 S j 8 5in.b 1 + e .2-5 7?- sc-vb + 77o sc.v3b -t- 3'*) °i (4.I4l) Sivv-5t Note, however, that these expressions are incomplete because the terms fxf>^ Co and $t from and - 1 fxf>^. n a v e D e e n omitted r e s p e c t i v e l y (see Equations 4.25 and 4 . 2 6 ) . The e f f e c t of t h i s omission i s purely a matter f o r s p e c u l a t i o n , but from previous s o l u t i o n behaviour i t seems probable that constant and cost terms should be removed from A , and s i n t terms eliminated r from A t I f , t h e r e f o r e , these terms are neglected, then applying the l i m i t process (1 - l ^ ) -.1 V b ) .+ 0 to and we o b t a i n : CA.14-2.) b Co5 2 b ^5 2S6 e' 7 - ft, " 1 cos 4h. Compare Equations 4.93 and 4.94 with 0(e ) terms i n Equations 4.101 and 4.102. Note also that the remaining terms i n cost and s i n t involve the f u n c t i o n (1 -^V» and as ( l - j ^ ) these w i l l a l s o be e l i m i n a t e d . 0 121 4.42 Cfc) = - s>v^t (4.143. ) 2.56 Equations 4.109 and 4.110, d e f i n i n g motion with respect to the E , H axes, can now be revised as shown below. z: ct) = -cash 48 *4- cc-s 3 1 - SIS L £ "-p, " cos 5"t S [_ 144- 4-S I t J o a r ) 1 J (4. 2-St H3Ct) = - S<«.t l*S /6 £ 5 2-^ G,l^y <S Jin. I Ur St (. 4-S 144 • ?t 3 <?4 (4.14?) 122 4.43 I f we evaluate the a d d i t i o n a l , c o e f f i c i e n t s values f o r e and y-j given on page | 114-77 L _ I-*? p, L 42 ^£ Z <?6 ~l = 2.4/708 3 2 - 0 IS *. .3.0572 7?fc?7 * lO (.4.146) \0~** ~> e ^^," 1 4 . 3 7 , then 3 16 144 using the numerical = 1 ( _ f c . . ...(**. I 4 « ) t i n which case Equations 4.134 and 4.135 f o r X ( t ) and Y ( t ) can be revised as: X(fc) = t O . <no4<3 . 441S7 o . O o ! 54 ^ 3 4-1P cos3t 3e>S72 tc. St + o. ooooo cost s C4I49) +oa' ) 5 Yet) =• I . oo765 -+ o. 001S4 + 0 . ooooo 04305 si-t S34TO s^3t 3e>5 72. 5i«-?t.. 123 4. This s o l u t i o n shows a c l o s e r correspondence with H i l l ' s v a r i a t i o n o r b i t that i t s predecessor (Equations 4.134 and 4.135). must, however, be emphasized that Equations 4.149 and 4.150 It are based on an incomplete a n a l y s i s and that the conjecture applied i n d e r i v i n g Equations 4.142 and 4.143 cannot be s u b s t a n t i a t e d . The r e s u l t s of t h i s section should, i n consequence, informed be regarded as estimates. 4.5 Discussion In contrast to H i l l ' s approach, which a p p l i e s only to the r e s t r i c t e d problem, the a n a l y s i s of three-body motion presented here i s v a l i d f o r any system s a t i s f y i n g the conditions o u t l i n e d i n Section At l e a s t i n p r i n c i p l e , using t h i s method we should be able to compute lunar o r b i t s of high accuracy, p a r t i c u l a r l y when the mass r a t i o m^/m^ i s s i g n i f i c a n t l y greater than zero. As mentioned on page 4.29 H i l l ' s s o l u t i o n i s no longer v a l i d i n t h i s case because the Jacobi i n t e g r a l does not e x i s t . For the earth-moon system m^/ny i s substan- t i a l l y greater than z e r o , so that some inaccuracy might well be a n t i c i p a t e d when Equations 4.127 and 4.128 are used to describe lunar motion. We can, i n f a c t , estimate the e r r o r introduced, by comparing Equations 4.109 and 4.110 f o r the r e s t r i c t e d problem w i t h a c o r r e s ponding s o l u t i o n derived from Equations 4.101 to 4.104, as shown below. The p o s i t i o n "of nu r e l a t i v e to m i s determined by: 9 4.1 121+ 4.45 - 1. ,- e' tost + e p, 6 .J. .+ c&si-b 6 Sirvb * cn-p,y*. J (4. I f 5 , H axes are chosen so that the o r i g i n i s located at on the negative E axis (as i n Figure 4 - 1 ) , IS *) - and m-j l i e s then from Equations 4.107 and 4.108 we o b t a i n : .3. CO = TTT7" - L 7T- c^xt £5 it 3. e - n,.**s 3 t + 6 o( *) £ J 1..32CI.4- p,)s- L (4-1*3) J ^ '*;<,(•-/O L $2- C H - H . ) i J ( 4 . 154) 1 25 4.46 where these equations are compatible with Equations 4.109 and 4.110. The earth-moon mass r a t i o deduced from radar observation i s given by Ash, Shapiro and Smith [57 ] as: m /m 2 = 3 81.3024 + and t h e r e f o r e : (1 - u ) = 2 1.2150 31396 x 1 0 " 2 . We can now evaluate those terms which do not appear i n Equations 4.109 and 4.110, and using the previous values of e and y-j (from page 4 . 3 7 ) : e? = 3.Z840 ^6S"<?6 x io~ 6 , (4-.i56) Neglecting the mass of m^ w i l l therefore introduce errors of approximately 4.3788 x 1 0 " and 1.0947 x 1 0 " 6 components r e s p e c t i v e l y . 6 i n t o the 5 and H S i m i l a r errors can be a n t i c i p a t e d i n Equations 4.127 and 4.128, which define H i l l ' s lunar v a r i a t i o n o r b i t , + A s h , Shapiro and Smith derived values f o r a number of astronomical cons t a n t s , using both general r e l a t i v i t y and Newtonian theory to determine a model of planetary motion. The value of the mass r a t i o quoted here corresponds to Newtonian theory. •1 26 . 4.47 although i t should be noted t h a t a comparison of H i l l ' s o r b i t w i t h r e s u l t s derived from the present uniformly v a l i d s o l u t i o n may be misleading. H i l l ' s i n i t i a l conjecture concerning symmetry of the v a r i a t i o n o r b i t [52 ] i s , f o r example, neither necessary i n t h i s a n a l y s i s n o r , tt i n f a c t , evident i n Equation 4.153. A d e t a i l e d r e v i s i o n of H i l l ' s approach can be found i n Eckert and Eckert [58 ] , where the v a r i a t i o n o r b i t i s determined assuming: 1. the mean s o l a r distance 2. the s o l a r e c c e n t r i c i t y 3. the lunar i n c l i n a t i o n i s zero; 4. the mass r a t i o + M ( + M . ^ r-| -* °°; i s zero; + 0. " 3 + j rvv Their a n a l y s i s consequently takes i n t o consideration the combined mass of the earth and moon. J a c o b i ' s i n t e g r a l i s s t i l l , however, a p p l i e d to reduce the equations of motion, so that i n e f f e c t the lunar mass must be n e g l i g i b l e f o r the r e s u l t s to be v a l i d . The a n a l y s i s of Section 4.4 i n d i c a t e s that the convergence Equations 4.101 e^u-| , where of to 4.104 depends, to a s i g n i f i c a n t e x t e n t , on the q u a n t i t y j-i, and £ = y •/ We should therefore expect p a r t i c u l a r l y convergent s o l u t i o n s f o r r e l a t i v e l y small values of u-j i n combination with large values of r-j. This s i t u a t i o n would a p p l y , f o r example, when two s t a r s forming a c l o s e binary o r b i t a comparatively d i s t a n t s t a r , the masses being of s i m i l a r magnitude. from page 2.26, that provided Note, TX- « 1 we are j u s t i f i e d i n assuming ^ l two body o r b i t s f o r m-| and the mass-centre of m and m This s p e c i f i c r 9 v 127 4,48 case w i l l be given more d e t a i l e d consideration i n Chapter 6. Next, how- ever, we apply these general three-body r e s u l t s to the four-body problem. 128 5.1 5. 5.1 THE FOUR-BODY PROBLEM Introduction Now that a uniformly v a l i d s o l u t i o n f o r the motion of p-j, p and p 3 2 has been d e r i v e d , we can proceed to i n v e s t i g a t e the motion of p^ as o u t l i n e d i n Section 3 . 6 . 2 . This three-body s o l u t i o n i s based upon the c i r c u l a r ( r e l a t i v e e q u i l i b r i u m ) s o l u t i o n of the two body problem. S i m i l a r l y , the a n a l y s i s of four-body motion takes as i t s foundation e q u i l i b r i u m s o l u t i o n s of the r e s t r i c t e d problem. the r e s t r i c t e d problem can be found i n Szebehely's [1 A d e t a i l e d treatment of authoritative text ] , and only those aspects which are necessary f o r the subsequent development are described here. Although the method of s o l u t i o n i s b a s i c a l l y i d e n t i c a l to that adopted f o r the three-body case, the a n a l y s i s of four-body motion i s s i g n i f i c a n t l y more complicated. If, f o r example, the uniformly v a l i d s o l u t i o n f o r a four-body o r b i t i s to have an e r r o r ©(e ), then i n 1 p r i n c i p l e the motion of p-j, p 0(e ). 2 and p^ should be known to an accuracy In the present case t h i s four-body s o l u t i o n i s consequently o l i m i t e d to an e r r o r 0(e ) by the r e s u l t s of Chapter 4. The a n a l y s i s should therefore be regarded, not as exhaustive, but as p r o v i d i n g the basis f o r f u r t h e r development. + This appears to be a feature of the expansion process not found i n the one degree of freedom case [51 ] . Note that i n the previous chapter i t was necessary to continue the a n a l y s i s as f a r as e r r o r terms 0(e '*) to define a uniformly v a l i d s o l u t i o n w i t h i n an e r r o r 0 ( e " ). A s i m i l a r l i m i t a t i o n a f f e c t s the present d e r i v a t i o n . •129 5.2 5.2 The R e s t r i c t e d Three-bo.dy-Problem In Section 3.3 we considered the process by which the f o u r - body problem could be reduced to a r e s t r i c t e d problem i n v o l v i n g p^, Pg and p^. The motion of p^ i n the r e s t r i c t e d problem i s described by the f o l l o w i n g equations: (Equations 3.25 and 3.26). Equations 5.1 and 5.2 possess f i v e homo- graphic s o l u t i o n s of r e l a t i v e e q u i l i b r i u m , and i t i s therefore feasible to i n v e s t i g a t e motion i n the v i c i n i t y of these points of e q u i l i b r i u m . I f we set then Equations 5.1 and 5.2 can be w r i t t e n as 1 Mr. - ~ hit: - o 130 5.3 1 since = n.^ = 0 = o C5.G) (from Equations 3.38 and 3 . 3 9 ) . To cast Equations 5.5 and 5.6 into a t r a c t a b l e some method of expanding the nonlinear functions in x and y . form we require _J as s e r i e s I t i s , however, important that the f i r s t few terms of these expansions should represent the nonlinear functions accurately w i t h i n a region containing the complete s o l u t i o n . For the case of perturbed two-body motion considered i n Section 3.6.1 t h i s region of motion i s s m a l l , and binomial expansions are adequate. S i m i l a r l y , f o r motion near Lagrange points of the r e s t r i c t e d problem, a binomial expansion of the »-» r\ e> I ~i -> v» the numerical work of Kolenkiewicz and Carpenter [ 3 0 ] indicates motion i s not, i n g e n e r a l , confined to such a small r e g i o n . that We consequently require an expansion which p r o v i d e s , with r e l a t i v e l y few terms, an accurate representation inside a s i g n i f i c a n t l y larger region. The nonlinear functions can be rearranged I as: + 4- where Now so that l -jl r L denotes the distance between p^ and a Lagrange p o i n t . 131 5.4 Cs.l) Equation 5.7 can therefore be w r i t t e n i n the f o l l o w i n g form: (5.|o) I f we set _L r i + ^c^ ) b *> then a general s e r i e s expansion of the f u n c t i o n [1 + p ( x , y ) ] interval -a 6 p ^ g •3/4 on an can be w r i t t e n as Cs". i i ) In p r a c t i c e the summation i s terminated at some f i n i t e value of i , l e a v i n g a r e s i d u a l e r r o r which i s , h o p e f u l l y , s m a l l . The constant coefficients k . w i l l then, i n general, depend both on the type of expansion and the terminating value of i . We s h a l l return to t h i s subject l a t e r , but at present the general form given i n Equation 5.12 is sufficient. The foregoing d i s c u s s i o n a p p l i e s to motion which takes place i n s i d e a region of f i n i t e s i z e . As i n d i c a t e d on page 5 . 1 , however, s o l u t i o n s of the l i n e a r i z e d r e s t r i c t e d problem are c e n t r a l to the a n a l y s i s of four-body motion. Consider, t h e r e f o r e , the e f f e c t of reducing t h i s region to an i n f i n i t e s i m a l s i z e , so that nonlinear terms i n v o l v i n g 2 x , xy, y 2 and higher powers of x and y can be neglected. We then have, 132 5.5 f o r Equation i^ 5.10: I, Equation 5.12 can be s i m p l i f i e d as: and consequently, from Equation 5.13: A l i n e a r i z e d version of Equations 5.5 and 5.6 can now be d e r i v e d , using the expression given i n Equation 5.15 to s i m p l i f y the functions. A f t e r some rearrangement nonlinear the l i n e a r i z e d equations of motion can be w r i t t e n — X- = o 133 5.6 where and Hg have been retained to preserve symmetry. Now the Lagrange ^ - points s a t i s f y — o >-3 = O 0.1%) (see Section 3 . 5 , page 3.17 ), and i f we set (5-22) 134 5.7 Equations 5.16 and 5.17 take the f o l l o w i n g form: = CK-O r I ^ 3 • t 3 1 Motion w i t h i n an i n f i n i t e s i m a l l y small region about a Lagrange e q u i l i b r i u m point i s now defined by the s o l u t i o n of these equations. Note that kg = 1 f o r a binomial expansion, which then reduces Equations 5.23 and 5.24 to the conventional v a r i a t i o n a l form [l 1 3 ] (/. The homogeneous s o l u t i o n of Equations 5.23 and 5.24 at) i s derived i n Appendix IV (Section A 4 . 1 , pages A4.1 to A4.7), but the character of t h i s s o l u t i o n depends on the p a r t i c u l a r Lagrange point to which i t r e l a t e s . There a r e , i n f a c t , four d i s t i n c t classes of s o l u t i o n , each of which must be treated (i) separately. At the col l i n e a r Lagrange points the homogeneous s o l u t i o n of the v a r i a t i o n a l equations can be expressed as :<-b) A; e B: e Cs.zf) Mb C5.28) 135 5.8 where, i n g e n e r a l , there w i l l always be one unbounded term [1 (ii) 1 k ]. Close to the e q u i l a t e r a l t r i a n g l e points the homogeneous s o l u t i o n i s bounded when r < 2- -J- - 4-(k +k,) 3k* (see Appendix IV, Section (iii) 0 A4.1) When V-3 _\_ — -X - . 4-Ckp + 3 k. O C5-.30) 1 the v a r i a t i o n a l s o l u t i o n about and contains terms of the form t s i n t , t c o s t which, i n g e n e r a l , the s o l u t i o n unbounded [ l (iv) 1 5 secular render ]. If hi (S-31) 17- 3k the s o l u t i o n near the e q u i l a t e r a l t r i a n g l e points and Lj. contains terms of the form e ^ cos(cot + 8 ) so t h a t , i n g e n e r a l , ± t the s o l u t i o n i s again unbounded [ I 1 6 ]. The appearance of unbounded terms i n the l i n e a r i z e d s o l u t i o n s i n d i c a t e s that motion cannot, i n these s p e c i f i c cases, be confined to an i n f i n i t e s i m a l region about the points of e q u i l i b r i u m . may however, s t i l l Bounded motion be possible i n c e r t a i n instances when nonlinear 136 5.9 e f f e c t s are taken into consideration [I 1 7 ]. In the f o l l o w i n g a n a l y s i s of four-body o r b i t s we consider only those values of y 3 f o r which Equation 5.29 i s s a t i s f i e d . As e -»• 0 the equations d e s c r i b i n g four-body motion reduce to corresponding equations f o r the r e s t r i c t e d problem near to use and L ^ . We s h a l l consequently continue and Lg as points of reference, although i t should be noted that i n the four-body case no p o s i t i o n s of e q u i l i b r i u m a c t u a l l y e x i s t . 5.3 Four-body Motion Near t and For the p a r t i c l e p^ l e t where e = ( -p-yj and the v a r i a t i o n a l terms u and v denote perturbations from the reference point n^) caused by p-j alone. to case 9 i n Table 3-1 ( i n Section 3 . 6 . 2 , pages 3.21 This corresponds to 3.26). Equations d e f i n i n g the motion of p^ were derived i n Section 3.4 (Equations and 3 . 3 4 ) , and f o r convenience these are 3.33 r e - s t a t e d below. P o s i t i o n s of e q u i l i b r i u m at s p e c i f i c time i n s t a n t s can be defined by equating d e r i v a t i v e terms i n the equations of motion to z e r o ; a d i s c u s s i o n of t h i s topic and i t s i m p l i c a t i o n s i s given i n Appendix I I , q.v. 137 5.10 S ! ^ — * r - l 3 t- *" I r I * When Equations 5.32 and 5.33 are s u b s t i t u t e d i n t o these expressions we obtain: 3 138 5.11 .where S(K,V, b) = f-i cost •+ f., t ^ . - O (s--3s) Note t h a t , because ( ? » n ) i s a f i x e d p o i n t , no d e r i v a t i v e s of ^ L appear i n Equations 5.36 and 5.37. S(u,v,t) and „ H -i— ~ 2 . u n u ~ We can evaluate the expressions and T ( u , v , t ) from Equations A1.6 and A1.7 i n Appendix I pages A1.2 - A 1 . 4 ) . J — ~J - [-/^ » Next, however, the nonlinear functions must be expanded i n terms of e. 3 y J i v c n l i t and L r - , L ^ u u . -u ^ i> i UII / i j T • i r \ i » O also depend on the small parameter e. i x. _ / i -r • i n / i i WT _J The p o s i t i o n s of u...«. 9 (see x i UMUJC . — v> l , , x . v i u u This feature complicates • O i i o the process of expansion c o n s i d e r a b l y , and d e t a i l s of the a n a l y s i s are therefore relegated to Appendix V( Section A 5 . 1 ) . the f o l l o w i n g | k u..+ 0 Eventually we obtain results: k, j \ it. + V/vfJ i - 2 . j O J ^ C5-.40) 139 5.12 - £ O - CV) e Note here that eu and ev are both assumed to be 0(e ); the j u s t i f i c a t i o n f o r these values i s given on page 5.12 . ^ 11—- 2 V - «<- [ * - ( k 0 + k, > We now have 114.0 5.13 £ ^ V •+ ZU, ILW^^CI- 2/0 + - and, <f ' l - by s u b s t i t u t i n g f o r £ L O - + and n L vy £ I + 3k, ) j O^fc ") ^ , 1 (5.43) i n Equations 5.20 to 5.22, a t and these equations can be w r i t t e n a s : e £ 11 - zv...+ 0,u-+ e £ v •+ zi. Ar o^u. O^v^ + o^v J = ^ (i_ k ^ 0 - k = In t h i s form the l i m i t process e 0 c V + ^ 0-44) OCVO o a O .. (5.45) 0 appears to be i n v a l i d , but the constant terms i n v o l v i n g kg a r e , i n f a c t , dependent on e, as demonstrated below. When kg f 1 there w i l l be some e r r o r i n representing the functions _L_ and at L - and U (where eu = ev = 0 ) , 1 which i s caused by the process of t r u n c a t i o n described on page 5.4.. It i s not unreasonable to assume that t h i s e r r o r i s of the same order as the i n t e r v a l w i t h i n which the expansion i s v a l i d , and therefore i f the i n t e r v a l i s 0 ( e ) i t follows that 1 ( k - 1) = O f e ) . Q 1 A proof of t h i s conjecture f o r Chebyshev polynomial approximations i s given i n Appendix VI ( i n Section A 6 . 4 ) . For the present problem the region of v a l i d i t y i s chosen to contain the f i n a l o r b i t (see page 5 . 3 ) , so that the order c l a s s of. the truncation e r r o r w i l l depend on the leading term i n the 141 5.14 s o l u t i o n s f o r eu(t) and e v ( t ) . These s o l u t i o n s depend, i n t u r n , on q u a n t i t i e s i n Equations 5.40 and 5.41 that are independent of u and v and of the lowest order i n c, namely e | 2 k ^ . ^ e^j and e J 2 k , p ^ J f j • s s i I f e^ and f g generate no secular terms i n the s o l u t i o n s f o r eu(t) and e v ( t ) , we are therefore j u s t i f i e d i n assuming ( ) from Equations 5.44 and 5.45, = O(e^) so t h a t , as e -* 0 the o r b i t described by eu(t) and ev(t) w i l l contract to one of the e q u i l a t e r a l t r i a n g l e points or Note p a r t i c u l a r l y that the homogeneous s o l u t i o n s of Equations 5.44 and 5.45 contain only p e r i o d i c terms of the form cos(to t + 0) and s cos(co t + <j)). From Equations A4.19, A 4 . 2 1 , A4.28 and A4.29 we have: + L o ^ ^ I + k *• (s.*o + W," - 3 k % ( i - / 0 ] V <*.*?0 + k, -+ [ -vdc^w,) + k * - k e +k ( - [ 4(k and consequently, since k Q 0+ 0 and k-. are 2 ^^c\-fA )] Z ( constant: j.. Appendix IV, Section A 4 . 1 . iij.2 5.15 For a binomial expansion of the nonlinear f u n c t i o n [1 + p] 3 A k A Q - 1 3 and k-j. = y » i n which case ^ - 1 V ^ ° (S.5o) . (5.50 where the e q u a l i t y condition applies when u -»- 1. The change i n these 2 values of k Q and k-j v/hen a d i f f e r e n t expansion method i s employed w i l l depend on the accuracy of t h i s f i n i t e expansion, but i n p r a c t i c e changes are s u f f i c i e n t l y small that Equations 5.50 and 5.51 used (see Appendix V I , Section A 6 . 5 ) . generated by e ( t ) 5 the can s t i l l I f no secular terms are to be and f ( t ) , we must therefore require that y 5 2 f 1•' When t h i s c o n d i t i o n i s s a t i s f i e d , the functions eu(t) and ev(t) can be expressed as asymptotic expansions i n the two t i m e - l i k e v a r i a b l e s t and T w i t h , as before, T = e t . From Equations 5.32 and 5.33 we then have *— i i~—» •n. « 5 . + . . . For the general two body s o l u t i o n i t i s assumed that neither mass i s i d e n t i c a l l y zero (see page 2.11 and the footnote to Equation 2.38 on page 2.18). The r e s t r i c t i o n u f 1 i s consequently i m p l i c i t i n t h i s entire analysis. 2 be 11+3 5.16 where these expansions are to be uniformly v a l i d i n t and T as e -»• 0. The formal operation of d i f f e r e n t i a t i o n i s again assumed to be j u s t i f i e d , and the d e r i v a t i v e s of Equations 5.52 and 5.53 are,, i n f a c t , i d e n t i c a l to the expressions defined i n Equations 4.5 to 4 . 8 . We can now expand Equations 5.44 and 5.45 as a sequence of nonhomogeneous p a r t i a l d i f f e r e n t i a l equations where, f o r the i ' t h power of e : hi?- 1 J Here the functions S u.j and v ^ . it*- and T involve u to u . , and v to v . , , but not o l-l 5 i-l c c The f i r s t p a i r i n t h i s sequence are *t *" l\ + . + o u. + o v = e" /y• Ci-iO ^ft) ts.S7) dl k It. and i f the homogeneous s o l u t i o n s of Equations 5.56 and 5.57 are denoted A 7 5 3 5 s t d by u ( t , x ) 5 and v ( t , x ) H 5 ' i<-*F')fr = * - u 5 H r e s p e c t i v e l y , then from Equations A4.30 and A4.31: o-Ct?) o-s [ fc> t + 6<lv)^ s o s a c e ) cc.5 [ <o fc + 5 9*ttO] + h(t) + ffl cos u [ iO t + u e/>Cr)J (5.58) U r ) c« ]_ m fc + < £ * ( - & ) ] u (5.51) where, from Equations A4.34 and A4.35: &* Co) - 9Cv) + 2-3 y (s.6o) 11*4 5.17 Homogeneous s o l u t i o n s of higher order can be accumulated i n t o and V g ( t , r ) H i f the a r b i t r a r y functions a ( x ) , Ug(t,x)^ b ( x ) , 9(x) and <J>(x) are expressed as asymptotic expansions of the form L—. z> Z i Z—- (see page 4.6 f o r a comparable process i n the three-body c a s e ) . Also, by r e - l a b e l l i n g x as e t , the non-homogeneous s o l u t i o n s f o r u^ and v.. (where i > 5) can always be considered functions of t , and consequently no d e r i v a t i v e s of these q u a n t i t i e s with respect to x need appear i n the sequence of p a r t i a l d i f f e r e n t i a l Within an e r r o r O ( e ^ ) , equations. Equations 5.38 and 5.39 can be evaluated d i r e c t l y from Equations A1.6 and A1.7 by s u b s t i t u t i n g £^ and ri|_ i n place of £ and n . and i f we express these equations using the compact notation SCV.v, t ) = \ S.CO & Is TtV,v,fc) + l OtVO b = \ T. C t ) £ + OCV') : the sequence of p a r t i a l d i f f e r e n t i a l equations can be w r i t t e n i n form given below. d\ s + 2_5u cJt - 5b 5t 5b 1 s + 0 u. z 5 tOjV s e~ rf(i-k } «. 0 %\Y^"I • e Ct) s L Ct) 5 l - S - - 2- <3tatr T CO 6 - 5v __5 OC.) Z k f ^ e / t } - 75t5V •it* 5b 3V 5.19 x C O T 5"? f - a i c , ^ ^ t ? f t ) ? • aV - 2- 5 L atar - •4- J, a + oy ? l 9 at at 1 - 2 - 2>J aV- at +o a^ + 0 v g x 3 g 'oC.) - + - ^ \ J z>a CO _ $ TV + - av - T It - CO at J 1 oCt) - Z W , ^e s ( 0 at - 2- i- OCe ) a, -v p V5 -t- ciu.5 ^Ct+jO' 1 OCi) 06O v ^_«-<} ...-.2 a_Vq + 0 ^ + O^V^ a if s/t) - ^ . ^ A ^ C O fc -I r ^ atr . aTai av . v - ay at h\ at 1 4 aaa, + 0 ^ a, + = + za u x t atar T^CO - ^ at e ] J ( </0 (???) u 2- ^"Vj - au. 5 0( O j 1 ae- + ^.5 7 -»- aj* -f at J o 6 ^ e L t at 1 at a%/ " it; 5 1 £ 1 - av a* s s + 2a\ s ik^V ^w.^ ? avj 114-7 5.20 c>fc - 2- a\i 1 - 5 ar 1 bv- at 1 c , ^ L it*- 0 bt L bt^ u e 5 5 (3k, + k^) a> ^tatr bt *\ — • J OCe~)> bv at*- J > , oCl 1- V ^ Ck,+3k,.) 5 s o(0 at - 7- at* oCe*) aw- ^ oCO J o oCe ) 3 1i+8 5.21 J As before (page 4.8) the s u b s c r i p t 0(e ) 0CO denotes terms of order e 1 in 1 the associated asymptotic expansions. - 5.4 Uniformly V a l i d Solutions The method of s o l u t i o n adopted here i s e s s e n t i a l l y i d e n t i c a l to that employed i n Section 4.3 f o r the three-body problem, where a uniformly v a l i d s o l u t i o n i s determined by e l i m i n a t i n g a l l secular or unbounded terms. Because the a n a l y s i s becomes both p r o l i x and r e p e t i t i o u s , only Equations 5.68 to 5.71 are considered i n d e t a i l , although the e n t i r e development i s given i n Appendix V, Section A 5 . 2 . The non-homogeneous functions i n Equations 5.68 to 5.79 can, i n g e n e r a l , be w r i t t e n i n the f o l l o w i n g form: • —I r ( 5 . 8c») '4 / 1 J 11+9 5.22 -where E and F are defined i n Appendix I V , Equations A4.J36 and A4..37. Corresponding non-homogeneous s o l u t i o n s are given i n Section A 4 . 2 , and these f u r n i s h the required conditions f o r the e l i m i n a t i o n of secular terms. For Equations 5.68 and 5.69, and f ( t ) from Equations 4.101 K and a f t e r s u b s t i t u t i n g f o r e,-(t) and 4.102, we have: A" ( ' F* Cb,X) = fe" ^ C\- k ) D + i u C i HlJ III oi i* »v- V-^MU ^ « W I I ^» ***.v- + |A, ) i ^ K l f i f t d - ^ 8 A I . _ _ . < v • (sVs'i) O + | 0 . . . . > "I "' w * " ' •*' equations A4.40 and A4.41, the corresponding non-homogeneous s o l u t i o n s u (t) 5 N H and v ( t ) 5 N H can be obtained from Equations A4.44 and A4.45 as: 150 5.23 -S *^.5 Si.«vt 4- O ^ Q o b L (S-.84) ' 5 { O - O { )co t S o u C u> * - t J ^ ) C o J - 5 I ) In these s o l u t i o n s f o r U g ( t ) ^ and v ^ ( t ) NH terms associated with can be obtained d i r e c t l y from the corresponding short period (co ) terms s 2 if u s 2 i s replaced by (note, however, that (u) s 2 + co^) becomes 2 - ( u > - w ) a f t e r the interchange). s L A s i m i l a r instance of t h i s type of symmetry i s mentioned i n Appendix IV, page A4.13. We next consider non-homogeneous s o l u t i o n s of Equations 5.70 and 5.71 f o r U g ( t , r ) and v ( t , T ) . g The functions S ( t ) g and T ( t ) g can 151 5.24 be evaluated from Equations A1.6 and A 1 . 7 , and from Equations 4.101 and 4.102: e Cb) 6 *s (b) b * p., C i - j O £ n. s» z t I t therefore f o l l o w s that (5.S8> hilt- Sin.Zb - S:~2.b { + 1^1 toZb The expansion of d e r i v a t i v e terms using Equations 5.62 to 5.65 i s a straightforward but lengthy o p e r a t i o n , and f o r t h i s reason i s relegated to Appendix V (Section A 5 . 2 ) . When the d e r i v a t i v e expressions i n Equations 5.70 and 5.71 are accumulated using Equations A5.50 to A5.70 we o b t a i n : 152 5.25 5tr L at ire ' av J t ar 3 av (s\ qo) t I J ar ar 2- -.2. J 5a 0 Qo C w b -f s av ae> 0 at av (see page A5.15). and at; ar 153 5.26 (5.1') Secular terms i n the non-homogeneous s o l u t i o n s f o r Ug and Vg w i l l be generated by a l l the expressions i n Equations 5.90 and 5.91 involve p a r t i a l d e r i v a t i v e s with respect to x. If, that f o r the moment, we neglect the remaining q u a n t i t i e s , then Equations 5.90 and 5.91 can be written: 2 co. + COS (J t L +• COS O j t 151+ 5.27 •+• c o s c O t u (S.<?3) The symmetry between short and long-period terms, already noted on pages 5.23 and A4.13, i s p a r t i c u l a r l y apparent here. I t i s only necessary, t h e r e f o r e , to consider s h o r t - p e r i o d terms i n the f o l l o w i n g a n a l y s i s , as corresponding r e s u l t s f o r the l o n g - p e r i o d terms can be obtained by i n s p e c t i o n . I f we now take av av I s 6 - zo 5, = 5 o> J atr 5 -S av av L atr p L av 0 o yc J L s^0 * + A 39„ <*>8*? - 2.5" ^o. o o L av 0 av j t aatv J av c ^ 0 / - a a £ , ^ e / ? •+ 2-5" J ^ s ^ B , + f | ^ L <feo .a^ 9 7 c> J ar then, from Equations A4.50 and A 4 . 5 1 , secular terms i n v o l v i n g u> 'are s eliminated from u and v when: g g 155 5.28 A f t e r s u b s t i t u t i n g f o r YQ» Y-J » <$ and 6-j, Equations 5.98 to 0 101 can be expressed as: +- a, = o = o• = ftptoC ^ Z.^^^" -0-4^0^ 1 + « tt^$ ^ 9 0 2(« - o Now from Equation A4.34: (if. io«.) o 5 l -0 - ^ s ' j (£. las') 156 5.29 0 6 T 2.60 arctcxiv- f so that we have = si* 0 Cos B = sc^, 9 cos 9 COS 9 $ + to* 4 ZcJ 5 4 ^ ) ^ - S in.fii 2c0 ' A f t e r s u b s t i t u t i n g f o r the corresponding values of sinGg Equations 5.102 (5. s and cos6g to 5.104 can be w r i t t e n i n the f o l l o w i n g form: 2 - O 1 : 2 C 3 i «-o CL 0 0 o 2_ where c-|, C2> c^ and c^ are constant q u a n t i t i e s i n v o l v i n g Q , to , s and v-,. s , V2 'o?) 157 5.30 The only possible s o l u t i o n of these equations i s JL ( Q-o ") = O and consequently we obtain A l s o , from the symmetry of Equations 5.92 and 5.93: A l l the d e r i v a t i v e terms i n Equations 5.90 and 5.91 f o r E*( t,x) and F * ( t , x ) are therefore e l i m i n a t e d , so that we have (5.117-") 158 5.31 F*£lO - + Cos Zb The non-homogeneous s o l u t i o n s u ( t ) g and v ( t ) can now be derived g from Equations A4.40 to A4.45 ( i n Appendix IV, Section A4.2.1) a - lb) V> j * , cos cJ t s r S - 1°*-^ KM«-JO+J. ^ as: 159 5.32 jK Cos e«?i.b + t + ' 3_ 2. V Cfc) 6 4 - p x = - ^ si-_ to b s cos o s k ( [«oJ-+ 4 - - o ] 3 + I" 1 ] 160 5.33 f*, Si*.Zt 4 . ( ^ - 4 X ^ - 0 - p j cos Z t The f o l l o w i n g r e s u l t s are condensed from the continuation of t h i s a n a l y s i s , which i s given i n Appendix V as f a r as s o l u t i o n s f o r Equations 5.78 and 5.79. air at = ^ 3 av air 4k = av a_p 3 = constant O 161 5.34 = constant (5.127) = constant CS.\T3) = constant . (s.af) I t i s not u n r e a l i s t i c to set k Q = 1 (see Appendix V I , Section A6.5), and i n t h i s case the s o l u t i o n s f o r Ug(t) and v,-(t) reduce to the form given below: (5.134) 162 5.35 The s i g n i f i c a n t reduction i n U g ( t ) ^ and v ^ t ) ^ (Equations 5.84 and 5.85) r e s u l t s from s p e c i f i c values of and which cancel short and long-period terms i n the non-homogeneous s o l u t i o n s . For a^ and bg we have: C4o9 t^s" The values of a . and b. depend on r e s p e c t i v e l y (see page 4.20 1 0 -CiO, Sl«.0 a n c ' ^2" 7 (ffSS) - O ^J&Ui at? and ua. at f o r an equivalent d i f f i c u l t y i n the three- body c a s e ) , which prevent the rigorous determination of a-j, a^, 6-| > 02' *h D b-j, b^, However, from the general trend d i s p l a y e d by |^ and jj^ i n equations 5.115, 5.117, 5.124 and 5.125, we might well be j u s t i f i e d i n assuming that constant terms i n Equations 5.126 to 5.129 are z e r o . a, When t h i s c o n d i t i o n i s s a t i s f i e d : I P 5 c^e o - iO A s s 9 e ^ (^13?) 163 5.36 ( * . « 5 l ) 8, - 9, L - ] ( ,4 t0 5 A s sCvv © t 1 142.) 143) where: rs 161+ 5.37 + 6 2, (S.lSo) *3 (s.isz) and ( i = 1,2, ••• 4). corresponds to Equations 5.149 to 5.152. when w i s replaced by s in 165 5.38 Considering only the s h o r t - p e r i o d terms i n u g and v ^ , from Appendix V, Section A5.2 we. have: H 5 5 which can be expanded as: CoS (5.1 S 3 ) H When 3 30j av and d j ^ "are z e r o , from Equations A5.245 and A5.246 1 h-e (Appendix V, page A5.69) both - a a 9? j and ja.Q, +a ej_J o 166 5.39 are z e r o , so that terms 0(e ) can be eliminated from Equations 5.153 and 5.154. • A f t e r s u b s t i t u t i n g f o r a^ and a 8 ^ from Equations 5.135, 5.137 Q and 5.141, u ( t ) 5 H + and v ( t ) 5 can be w r i t t e n i n the form: H oC*fc*) .... (s.i s o When the homogeneous and non-homogeneous s o l u t i o n s are combined, the terms aQCOs(w t + 6 ) , s Q Q aQCOs(w t + 9^*) and corresponding long s s period terms are eliminated from u^(t) and v ^ ( t ) (see Equations 5.133, 5.134 and Appendix V , pages A5.42 to A5.45). A similar cancellation might be a n t i c i p a t e d between components of the s o l u t i o n s f o r u ( t ) , g Vg(t) and corresponding" 0(e) terms i n Equations 5.155 and 5.156. The functions u ( t ) and v ( t ) are defined i n Equations 5.120 and 5.121, g g but i n more compact n o t a t i o n : 16? 5.40 52 - * * u 0 u . •2 v Ct) 6 *J C « U s l - co^) J ,i ^ «i. . / i . N _ 168 5.41 Although a proof i s not given here, the c a n c e l l a t i o n a f f e c t s a l l short and long-period terms i n Equations 5.157 and 5.158. Solutions of Equations 5.34 and 5.35, which we are now i n a p o s i t i o n to accumulate, are given below. (sr. 151) l * ' 7 ./*</ ^'-/0 k / / t - € I 3«Vs:».fc cosb 169 5.42 - £ /<-, co$ Z t & 5.5 Discussion Equations 5.159 and 5.160 describe the motion of i n the 8 v i c i n i t y of and Lg w i t h i n a constant e r r o r 0(e ), where the s o l u t i o n i s uniformly v a l i d f o r t = 0(e~^). As the small parameter e tends to zero these o r b i t s contract to the Lagrange points L^ or L g , so that we may consider each e q u i l a t e r a l t r i a n g l e point as the l i m i t i n g member of a f a m i l y of four-body o r b i t s . I t should be emphasized that these unique s o l u t i o n s do not account f o r any perturbations other than those r e s u l t i n g , from the i n f l u e n c e of p-j. The question of s t a b i l i t y cannot, f o r example, be resolved without an a n a l y s i s of the more complicated s i t u a t i o n described as Case 10 i n Table 3 - 1 . ^ Note, however, that Equations 5.159 and 5.160 both define the l i m i t i n g members of a f a m i l y of o r b i t s depending on the parameter 6 and provide the generating o r b i t s f o r Case 10. Section 3 . 6 . 2 , q . v . We may deduce from the general 170 5.4 character of Equations 5.159 and 5.160 that p e r i o d i c terms i n v o l v i n g s i n n t , cos nt are associated with the c o e f f i c i e n t ! (o.'-rvSC^-n. ) 1 For n > 2 t h i s expression can be approximated by -^r , and consequently higher-frequency terms w i l l exert comparatively l i t t l e influence on s o l u t i o n behaviour. 171 6.1 6. 6.1 SPECIFIC SOLUTIONS Introduction The s o l u t i o n s derived i n Chapters 4 and 5 f o r three and f o u r - body motion permit, w i t h i n the l i m i t a t i o n s of the primary model, a comp l e t e l y general choice of values f o r between p^ and the mass-centre of , y,, and a n d r^, the distance Three p a r t i c u l a r cases are considered here, two of which r e l a t e to three and four-body motion f o r the sun-earth-moon c o n f i g u r a t i o n of our s o l a r system. The remaining three-body problem was selected to i l l u s t r a t e the accuracy with which unusual o r b i t s can be p r e d i c t e d , and a p p l i e s more to s t e l l a r than to planetary motion, in each case the o r b i t i s p e r i o d i c , and a standard of comparison may be obtained from a numerically-generated o r b i t having the same p e r i o d . 6.2 Three-body Orbits The s o l u t i o n s defined by Equations 4.101 to 4.104 are ordered i n terms of e, but when computing a s p e c i f i c o r b i t i t i s more convenient to rearrange these expressions £ (b) = Ii- y.^ i n the f o l l o w i n g form: - fe1, p / i - f O - pJ^Vx.) 1, 2. Cos t + . 10S e^pt, ( I- j / O l ? S ( I + p, ) + l*\ ~l 172 6.2 -cc.s2t • JL "VI ('-/-O cos k'e 6 £4- (o.l) € ( H-J* r- Sin. 2-fc S 12. L 33- 63 3^ e' 0 -^z.") I0Z40+/O J J r:*u4t 256 .(.t.O If ( t ) 3 : p cos b = - Pa. ' ^ V ^ i + £ 6 */z ( |+ M i ) . 0 - f O L 5 z 3zd\+fO ^ 173 6.3 + cos 2 k 64- 32. (6.3) 8 12- L 3z 10x4.(14^ , ) J J 2.5-6 32. We need only consider the o r b i t of one p a r t i c l e , ' and i n t h i s section a t t e n t i o n i s therefore confined to the motion of p 2 Equations 6.1 and 6.2 d e s c r i b e , with an accuracy O(e^), an o r b i t which i s p e r i o d i c i n t = 2TT. With t h i s as an i n i t i a l 'cf. Equations 3.5 and 3.6. estimate, 6.4 .we .should be .able to f i n d p e r i o d i c s o l u t i o n s of the o r i g i n a l equations motion (Equations 3.31 and 3.32) using a numerical approach. It of is important, however, that the computed o r b i t be p e r i o d i c i n t = 2TT and not t = 2TT -i- 6, where 6 i s some small q u a n t i t y . One method s a t i s f y i n g t h i s c r i t e r i o n has 'been o u t l i n e d by Bennett and Palmore [59 ] , f o r which the t h e o r e t i c a l 6.2.1 basis i s given below. P e r i o d i c solutions of y ( t ) = B(t)y + f(t) I f X(t) i s a fundamental matrix s o l u t i o n of the equation x = A(t)X A X '(t) = - X~'(t) (6.5) fl(t) (6.6) Now suppose where Y(t) i s a fundamental matrix s o l u t i o n of U.8) 175 6.5 We then have, from Equation j u. L 6.6: -* tit' and t h e r e f o r e , i f t = 0: Q j Y~ Cb)y(b)= ] 1~'LX)$CZ)AX + constant. (6.io) o I f Y(t) i s the p r i n c i p a l matrix s o l u t i o n , Y(0) = I (the i d e n t i t y m a t r i x ) , so that +| lb) c^Co) We now assume that the l i n e a r system y = B(t)y has s o l u t i o n of period T except the t r i v i a l s o l u t i o n y = 0. i f f(t) is T-periodic,^ Equation 6.7 no p e r i o d i c In t h i s case there e x i s t s a unique T - p e r i o d i c s o l u t i o n of [60 ]. 2 I f Equation 6.11 i s to be T - p e r i o d i c , then: T ycT) (o') 4 -f f y c T ) T h i s contraction period t = T. + y~ c-v) icx) AX l = ^ to') c&.>o i s used to denote that a f u n c t i o n i s p e r i o d i c with a , 176 6.6 which defines y(0) Co) = as: [ y-'Cr) -i - l ] r y.-'CuKctOAV . . c& . ) l3 The T - p e r i o d i c s o l u t i o n can then be w r i t t e n i n the form t-t-T Now suppose that p(t) i s the s o l u t i o n of Equation 6.7 s a t i s f y i n g i n i t i a l c o n d i t i o n p(0) = 0. In t h i s case [61 the ]: t p(t) - j" YCt) W r and t h e r e f o r e , from Equation tjCt) I f Equation 6.16 = YCtOyCo) ) K r ) ditr <6.»0 6.11: + j=(t) i s to be T - p e r i o d i c : . ' (6.ifc) 177 6.7 YCT) 3 CO) + pCT) = yCo*) Cfe-i?) so that 3co) =• - [ v c r ) - x ] pCT) . a.is) We now have s u f f i c i e n t information to generate a T - p e r i o d i c . solution i t e r a t i v e l y . The algorithm given i n the next s e c t i o n is s i m i l a r to that described by Bennett and Palmore, although they propose a d i f f e r e n t updating c o n d i t i o n . The proof of convergence appearing i n [59 ] does, however, apply to both algorithms. 6.2.2 I t e r a t i v e method to determine a T - p e r i o d i c o r b i t The system equations are w r i t t e n i n the general form iCt) = ^ [ xCt") , t] U.n) and we expand the function g [ x ( t ) , t ] about a T - p e r i o d i c estimate to o b t a i n : o-vc x (t) k 178 6.8 Setting x = x ^ , + 1 Equation 6.19 where the terms 0(|| * w + , - * can be expressed as II*) k are neglected. ing a s s o c i a t i o n s between Equations 6.21 H (t) x = ^9 and VJe now make the f o l l o w - 6.7: Ct) • U.ix.) [* . (6.as) =c .' k k+| ax fCt) s .-3[=c .tJ " ^ [ * k k J t ] * 0 * 0 k 2x and require that the s o l u t i o n x ^ ^ ( t ) of Equation 6.21 + be T - p e r i o d i c . The algorithm to determine x ^ ^ ( t ) i s summarized below. + 1. Integrate Equation 6.21 over the i n t e r v a l 0 ^ t ^ T f o r the s o l u t i o n estimate x^(t) with zero i n i t i a l c o n d i t i o n s , and denote the r e s u l t by p ( t ) . 2. Integrate the matrix vct) « . ^ [ x equation k l t ] yet) C6.zs) ax over the i n t e r v a l 0 s= t ^ T with Y(0) = I (the matrix). identity This generates the p r i n c i p a l matrix s o l u t i o n of 179 6.9 equation 6.21 when the terms <j [ x , k t] - ^ t 3 C k . ] t a k a r e neglected. 3. Determine x -|(0) from Equation 6.18. 4. Determine x^ -|(t) from Equation 6.16. 5. Set x^(t) k+ + = x ^ - j C t ) and i t e r a t e from step 1 u n t i l satisfactory convergence i s obtained. The matrix [Y(T) - I] can be almost s i n g u l a r , and i t i s p o s s i b l e , because of accumulated e r r o r , f o r the i n v e r s i o n process to destroy convergence. In t h i s case the t h i r d step of the algorithm may need some m o d i f i c a t i o n to avoid inaccurate elements i n the c a l c u l a t e d vector x k + 1 (0). 6.2.3 + P e r i o d i c earth o r b i t For the sun-earth-moon c o n f i g u r a t i o n of our s o l a r system the constant ft i s determined by the r a t i o of the s i d e r e a l mean motions of the sun and moon as: ft = 7.4801 32855 x 10~ 2 . Note the s l i g h t d i f f e r e n c e between t h i s r e s u l t and that quoted f o r ft on page 4.36 which a r i s e s from the use of current values f o r the mean This d i f f i c u l t y was encountered i n d e r i v i n g the. three-body o r b i t s of Sections 6.2.3 and 6 . 2 . 4 . The two elements adversely a f f e c t e d by the matrix i n v e r s i o n were, i n i t i a l l y , both z e r o , and convergence was restored by r e t a i n i n g these values i n subsequent i t e r a t i o n s . 180 6.10 motions (given i n the footnote to page 4.34). We then have, from Equation 3.21: y 1 = 3.2979 46384 x 10 5 No a l t e r a t i o n i s necessary to the numerical values of r-j and ( l - y ) 2 given on pages 4.37 and 4.46 r e s p e c t i v e l y , so that r 1 = 3.8917 y 2 = 0.98784 l-y 2 = 1.2150 24558 x 10 2 96860 31396 x 1 0 - 2 The corresponding o r b i t f o r the earth can now be obtained from Equations 6.1 and 6.2 a s : 5 C t "i = 1.7-132 +• %.X iiO £ <?2334 63^34- *• lo"" " 1 x lo~'° Ccr^3t 181 6.11 + i. • - * oi~?2-6 v (©"^ s;~zfc 1 . f^2- f&joo sin s xo ° X 3fc 83o<?4 mo'^siiv 4 t 1-630? . This s o l u t i o n i s shown i n Figure 6-1 i n terms of the displacements A£ and An from the point £ = l - ^ , n = 0. Equations 6.26 and 6.27 were evaluated at the time instants t = 2nir/100 (n = 0 , 1 , 2 , * • • ,100), and the data points of Figure 6-1 therefore define the p o s i t i o n of p^ a ^ successive i n s t a n t s of time separated by At = 2IT/100. To v e r i f y t h i s s o l u t i o n the o r i g i n a l equations of three-body motion (Equations 3.31 and 3.32) were solved f o r a 2ir-periodic o r b i t using the algorithm described i n Section 6.2.2 with Equations 6.26 t and 6.27 as the i n i t i a l estimate. When Equations 6.21 and 6.25 were evaluated at 201 equally-spaced time i n s t a n t s 0 ^ t < 2TT the quantity |x k+1 ( 0 ) - x (0)|| k over the i n t e r v a l reduced below 1 0 " f i v e i t e r a t i o n s , i n d i c a t i n g rapid convergence. [62 ] 1 13 in A Fourier i n t e r p o l a t i o n of the r e s u l t i n g p e r i o d i c s o l u t i o n generated the f o l l o w i n g expressions f o r and r^Ct): A l l the numerical r e s u l t s presented here were obtained i n doublep r e c i s i o n on an IBM 370 Model 168 computer, using a Runge-Kutta i n t e g r a t i o n routine with v a r i a b l e step s i z e and e r r o r c o n t r o l . • • • 0.8 - • O.t* 1 • - 1 I - o.t*. -0.2 0.4-0-4 - • • • - 0.8 - • • • • Figure'6-1 • P e r i o d i c earth o r b i t about £ 9 = 0.012150, n 9 = 0.0 183 6.13 - 8 . 7 l t o 362-31 + 'I . 12-04- (fc) = q. io$\ * I O " " ^ cos Z 2 0Z0 XI42,7 co$ 3 t x 10 .* i o " . q - 8 _ •V 1 . ^ 3 ^ 5 bb<\o"> -X.H10S 2-42-7^ A. * io * <o"\ -8 n^Zt 5C^3t + 6 . & o 3 7 Soifcl * io a*. 4 t I.H6£> ? j 5 ' 4 3 y; i o S i * . £fc - c ,.. . (6.3.1) -9 where terms l e s s than 10 i n magnitude have been neglected. We now assume that Equations 6.28 and 6.29 represent an exact p e r i o d i c s o l u t i o n of the three-body equations of motion. The d i f f e r e n c e s 6 ? and 6n between t h i s o r b i t and the a n a l y t i c a l s o l u t i o n defined by Equations 6.26 and 6.27 are shown i n Figure 6-2 f o r the i n t e r v a l 0 ^ t < 2TT. The dominant e r r o r s are associated with terms i n v o l v i n g cos 2t and s i n 2 t ; by comparing c o e f f i c i e n t s i n Equations 6.26 to 6.29 the percentage e r r o r of these terms i s found t o be 3.5256% and 4.4130% r e s p e c t i v e l y . Both o r b i t s are p l o t t e d i n Figure 181+ IO 6.14 0<5 -I -I 2.TT 0-3 -J S/>^ x 0-S io S -, o-2S A .TT O-XS -i -OS -I Figure 6-2 ZTT P o s i t i o n e r r o r s f o r the earth o r b i t over one period t Arjxio ^ 4 — • • \ \ . . \ > / \ /. /. 0.8 N •\ - \. • / / \ • \ /'• \ /'• 0.4 •\ - / • 10' -0.4 0.8 Figure 6-3 0 0.4 Numerical and a n a l y t i c a l solutions about ^ 0.8 = 0.012150, - 0.0 186 6.16 6-3 f o r the f i r s t quarter p e r i o d , 0 ^ t ^ TT/2, to provide some v i s u a l comparison between the numerical and a n a l y t i c a l s o l u t i o n s . As i n Figure 6-1, d i s c r e t e data points along the i n d i v i d u a l o r b i t s are separated by a time i n t e r v a l At = 2TT/100. 6.2.4 P e r i o d i c o r b i t s f o r s t e l l a r motion In Section 4 . 5 a s t e l l a r model was proposed i n which two s t a r s forming a close binary o r b i t a r e l a t i v e l y d i s t a n t s t a r of comparable mass. Configurations of t h i s type may be described by Equations and 3.32 provided the c o n d i t i o n l / 2 r « 3.31 1 i s s a t i s f i e d , and i n t h i s s e c t i o n we i n v e s t i g a t e the p a r t i c u l a r case defined ....by y^ = 10, ^ 0-5 = and r-j = 50. Thi? Choice Of n a r a m p t p r s Hp>$pr\/ps ^nmp rnirmpnt.. Foliations 3 4.101 >/T. to 4.104 w i l l e x h i b i t rapid convergence i f the quantity e t 5 6 i s s m a l l ; a l s o , i f terms of order e and e i n the equations are to be of comparable magnitude, ( l - y g ) - • i t f o l l o w s that e * 0.5/y^ " , and therefore 75 When y^ = ( l - ^ ) e y-|' ~ 3 /2 = - 0.125/y^. 0-5, If y-j = 10 t h i s l a s t c r i t e r i o n i s s a t i s f i e d when r^ = 50, which completes the set of parameters. We can now derive the corresponding p e r i o d i c o r b i t f o r p^ from Equations 6.1 and 6.2 S^(fcO - - c f . page 4.32. as: 6.66 9 0 4.. I S S o mo? * io~ * lo~ S fc c&t 187 6.17 -7- />f f t . ) .. - 8-4"5oo I , 2?S 446i5 "A I c " ^ + t.«43?5 x v. i o ' sin. b S S i r i 3 t lo~'° 4-fc . (6.31) The e r r o r incurred by neglecting terms O ( e ^ ) by the quantity e u-j , i s l i k e l y to be dominated which i n t h i s case w i l l be approximately 2 x 10" . 8 The looped o r b i t described by Equations 6.30 and 6.31 shown i n Figure 6-4, where A£ and An denote the respective is displacements from % - 0.5 and r\ = 0; the d i s c r e t e data points define the l o c a t i o n of p 2 at successive time i n s t a n t s t = 2nTr/100, where n = 0 , 1 , 2 « * * ,100. A 2ir-periodic o r b i t , with Equations 6.30 and 6.31 as the i n i t i a l estimate, was determined using the algorithm of Section 6 . 2 . 2 . Only two i t e r a t i o n s were needed to reduce the q u a n t i t y below 1 0 " ^ , 11x^^(0) - x^(0) j| and the f o l l o w i n g p e r i o d i c s o l u t i o n was obtained: - 4 . i < ? 2 - 5 <»I4?5 •,* to' cost s 188 6.18 0.8 H 0.4 H - i •0.8 •0.4 0.4 0.4 H -0.8 Figure 6-4 P e r i o d i c o r b i t f o r .u, = 10, u C = 0.5, n = 0.0 2 2 9 = 0.5, r , = 50 about A ^ x l O 4 189 6.19 - S . $"34? 4o$zo g.44«?- = + S.6g<J3 where terms l e s s than 1 0 ~ IU |0 co5 4 b (6.32) 6$||5 * io~. Si«.fc 5 IS3S5 bb$$o .+.1.3316 * * | o " % io""'' ? Swy.Zfc s ^ 4 b , (6.33) i n magnitude have been omitted. Differences between t h i s r e s u l t and the a n a l y t i c a l s o l u t i o n (which would not be detected on the s c a l e of Figure 6-4) i n Figure 6-5. are shown The dominant c o e f f i c i e n t e r r o r s i n Equations 6.30 and 6.31, 2.9968% and 0.2154%, are associated with s i n 2t and s i n t respectively. Note, i n Equations 6.1 and 6 . 2 , that terms belonging to the sequence appear i n the c o e f f i c i e n t s of cos 2t and s i n 2 t , which are therefore the most s u s c e p t i b l e to e r r o r . 190 6.2C 0-4 H t -I IT o-4 H -0-8 H 0-2 0-1 H i TT o-i t 2ir H 0-2 Figure 6-5 P o s i t i o n errors f o r the s t e l l a r o r b i t over one period 191 6.3 Four-body Orbits Near and L Equations 5.159 and 5.160 define two 2iT-periodic o r b i t s , i n the v i c i n i t y of and Lg r e s p e c t i v e l y , corresponding to s o l u t i o n s f o r the motion of p and p which s a t i s f y Equations 4.101 to 4.104. 2 3 With, the s o l a r system constants given on page 6.10 we have, from Section A 4 . 1 : %^ = i. £674 - -l.2t?4 at l _ ?o6£4- 4 at U 5 =. - 4 . S 7 2 4 - 96€6o K. /o~' S • 6&OZ. £404° * 10 54-04-0 * 1 * . - 8 . 6662 co~'- at L c b Note that v 2 and change i n sign but not magnitude at L g . corresponding o r b i t near The of the earth-moon system can now be determined from Equations 5.159 and 5.160 as: 192 6.22 gCt) 1 Ct) r 4.8731 474^1 * 10 - i "5.?<iZl 4 * 4 i . o68D ^ o z l * io" + S". 6?o8 84-639. -«- 4- £547 4ST63S - = g. - 65"2| * 78033 IO v. <x>s t 6 io" *io" 3 3 sc^Zb * U - » 0 co 2 b 5 io -1> 2 . 1361 *?4o58 x. io I . 8560 7411 3 2-2281 I«'j44 + 5". 6 8 3 0 sc«. t sc*-b fe v. l o ~ Cost x io" x 7oS4S 3 $i*,Zt lo"-..cos2t : An equivalent o r b i t near Lg f o l l o w s d i r e c t l y from t h i s s o l u t i o n when terms i n d i c a t e d by an a s t e r i s k are changed i n s i g n . The o r i g i n a l equations of four-body motion (Equations 3.33 and 3.34) were solved f o r a 2ir-periodic o r b i t using the algorithm outl i n e d i n Section 6 . 2 . 2 , with Equations 6.34 and 6.35 d e f i n i n g the i n i t i a l estimate. Three i t e r a t i o n s were s u f f i c i e n t to reduce the q u a n t i t y _ i p ll k -|(°) x + _ x |<(0)ll below 10" , and a f t e r a numerical F o u r i e r i n t e r - p o l a t i o n the f o l l o w i n g r e s u l t s were obtained. • 193 6.23 4~. 8 7 3 6 - 63i?6 K ID' 9 . 9 3 3 4 - . 6<i74o * to s c ~ t . + 3.35VS A-BbZ^ x I O -4 cos t - + S- 63S2. 3oxo5 -3 * io s^Zfc 4 4.Z493 734(2- _3 * io c 4 2o647 x io s;«.3t - 2.. 7o4-3 "37909 K I D — 1 . 4 8 23 O63ol -<> * |o cos 4 b 6.o64o + z.fc465 o i i o ? -6 i o s;*_-4--t x oS 2-t co5"3t (6.36) 8. 87532. *. io" -2.4-366 9I Z 3 3 J< i o ->.././.v>j J 6 6 T | y . i O 4 3.6460 67Z74 v io -1.364-0 .3SIST- io x Su^. t Sii*. £ siOb . fc sc^4t 4 7.4-9X3 6 4 4 2 - 4 * 'O cost + i . .B*0'Yt> 1C l O Co5 4 4 . 4-436 80674 * »o ~ 2 . 6666 SS304 x i o « ct>s3t -6 c©s4t C6-3?; 4 . 2 7 36 63| 76 n io" - s a "> ' s ^ t S. 8 3 3 4 6SII4 5. b?*Z -3 -bozoS t. \o ' n^.7Jt 6..064O 2.142-i- x. 6 4 6 4 <??pio + - fc> * io K i o " $j^3t • 6 s;«-4-fc - 3. 3 5 < S 4-87S8 ^lo^cej.t x io~ co 2.k +4.2-49 3 734.11 - 8864-2-* Z . 704.3 .1.48X3 Z5"9?-9 3 \o~ x l o " S cbs 31 6 cos4fc (6.38) 19k 6.24 —i 1 * 7 - 4 3 6£> - - 2 . ZZ-S*? 1 M 3 I 3S67I * ?t I O " * c^b 5 i o " + 3. 64-6C? -6 £ > 6 o Z £ •< to - ?71 7 I I . -3640 3 * 10' s ^ i t si*. 4 b - 7 . A H 2-3 6 7 ? 7 2 ^ lo" -4.4.436 ^0ZZ« -< < o " c o S 3 f c + 2.6^66 M7§Z. - 5. cosh S bb\ 8 6 * lo" 6 cos4fc O3<0 Terms l e s s than 10" have been omitted from these expressions. The symmetry evident i n Equations 6.34 and 6.35 f o r o r b i t s near L . and L 4 i s maintained here, although the agreement between c o e f f i c i e n t s always complete. c 0 i s not These s l i g h t i r r e g u l a r i t i e s a r e , however, more l i k e l y to r e s u l t from the e f f e c t s o f q u a n t i z a t i o n noise and accumulated e r r o r i n the i n t e g r a t i o n process than from any non-uniform system behaviour. The a n a l y t i c a l and n u m e r i c a l l y - d e r i v e d o r b i t s near and L g are shown i n Figures 6.6 and 6 . 7 , where A£ and An denote displacements from the e q u i l a t e r a l t r i a n g l e point ( i . e . ? - £^ = A£, n - = An) and d i s c r e t e data points define the l o c a t i o n o f p^ at successive instants 2TT of time separated by At = J^Q . Differences between the a n a l y t i c a l and numerical s o l u t i o n s over one period are p l o t t e d i n Figures 6.8 and 6 . 9 , which show that the p r i n c i p a l source o f , e r r o r i s p e r i o d i c i n t = 2TT. Note, at l e a s t i n the present case, that the p e r i o d i c o r b i t defined by Equations 5.159 and 5.160 i s dominated by perturbations p e r i o d i c i n t = TT and d i r e c t l y associated with p-,. Terms p e r i o d i c i n 195 6.25 Am <- x 10 Analytical t = o,TT, xtr z Figure 6-6 P e r i o d i c o r b i t s near L, i n the four-body problem solution 196 6.26 t =3J 0.8 - - 0 . 6 -J . | t = 0,Tr,2TT Analytical Figure 6-7 P e r i o d i c o r b i t s near U i n the four-body problem solution 197 x 0-4- ** to" 1..- TT ATT TT .aTT o-z 0-4 J >7 O-l J 0-4 J Figure 6-8 P o s i t i o n errors f o r the o r b i t near 198 5£ O-br 6.28 x Io" 1 o.i . 0-4 H rr ZTT Tf —i ZTT J 5*7 x io* 0-4 -i 0-3. •o-Z H J Figure 6-9 P o s i t i o n errors f o r the o r b i t near L ( . 199 6.29 t = 2TT r e s u l t from the i n d i r e c t influence of p-j on the motion of p and pg, and are r e l a t i v e l y small i n magnitude. Perturbations 2 associated with p-j and p e r i o d i c i n t = 2TT w i l l , however, be encountered when higherorder terms are included i n the s o l u t i o n . I t would be misleading to assess accuracy of the o r b i t s described by Equations 6.34 and 6.35 by comparing c o e f f i c i e n t s with the numerical solutions. If, i n s t e a d , we express the p o s i t i o n e r r o r s <5£ and 6n i n terms of. the maximum displacements |6C| * 0.048913 |A5| i <5nI ^ 0.043673 An 1 6.4 a x 1 and [An| max » then: max max Discussion Although the numerical s o l u t i o n s described here are i n r e l a t i v e l y close agreement with predicted r e s u l t s , we cannot be c e r t a i n that the numerical and a n a l y t i c a l o r b i t s are e q u i v a l e n t . This uncertainty i s a consequence of step 5 i n the algorithm of Section 6 . 2 . 2 , and can be explained i n the f o l l o w i n g way. When the i n i t i a l conditions are modified there i s no guarantee that the eventual o r b i t w i l l be unique i . e . for the three-body case we may obtain o r b i t s corresponding to case 4 i n Table 3 - 1 , rather than case 3. Section 3 . 6 . 2 , q . v . S i m i l a r l y o r b i t s i n the four-body 200 6.30 problem may correspond to case 10 instead of case 9. No adequate r e s o l u t i o n of t h i s d i f f i c u l t y i s p o s s i b l e without a d e t a i l e d a n a l y s i s of motion i n v o l v i n g more general perturbations. Comparatively few e x p l i c i t s o l u t i o n s of the three and four-body problems have appeared i n a form compatible with the r e s u l t s of t h i s chapter, which makes any f u r t h e r comparison d i f f i c u l t . H i l l ' s solution f o r lunar motion i n the r e s t r i c t e d problem i s p o s s i b l y the c l o s e s t equivalent to the o r b i t defined by Equations 6.26 and 6.27, case has already been discussed i n Section 4 . 4 . but t h i s An a n a l y s i s of p e r i o d i c motion i n the general three-body problem by Moulton [63 ] a p p l i e s to o r b i t s described as case 3 i n Table 3-1; more r e c e n t l y A r e n s t o r f [64] derived corresponding r e s u l t s f o r case 4 o r b i t s . has Unfortunately neither a n a l y s i s can r e a d i l y be compared with the present approach. The four-body s i t u a t i o n i s e q u a l l y problematic, with only one o r b i t showing s i m i l a r behaviour to the s o l u t i o n described by Equations 6.34 and 6.35. motion near Kolenkiewicz and Carpenter [ 1 7 ] investigated of the earth-moon system f o r Huang's very r e s t r i c t e d model, using a numerical perturbation scheme to determine an o r b i t with a period of one synodic month. Although s i g n i f i c a n t l y l a r g e r i n s i z e , the r e s u l t i n g o r b i t e x h i b i t s some of the features shown i n Figure 6-6. Equations 5.159 and 5.160 i n d i c a t e that the motion of p 2 and pg exerts comparatively l i t t l e influence on the four-body o r b i t (as discussed on page 6.29 )which may account f o r the success of the very r e s t r i c t e d model i n t h i s instance. near Three r e l a t e d s o l u t i o n s should also be mentioned, although s t r u c t u r a l d i f f e r e n c e s in the methods of a n a l y s i s prevent any comparison with the present approach. 201 6.31 In a subsequent paper [ 3 0 ] Kolenkiewicz and Carpenter examined motion about when the primary bodies move i n coplanar o r b i t s and s a t i s f y the equations of three-body dynamics. Two nearly i d e n t i c a l one- month o r b i t s were found, with a phase d i f f e r e n c e of 180°; the d i s p l a c e ment from i s comparable to the earth-moon d i s t a n c e , and i t appears l i k e l y that these o r b i t s correspond to case 12 of Table 3-1. Similar r e s u l t s have been derived a n a l y t i c a l l y by Kamel and Breakwell [31 using the Von Zeipel technique. Schechter [28] has considered ], the s t a b i l i t y of p e r i o d i c one-month o r b i t s f o r the three-dimensional case i n which the earth-moon o r b i t a l plane i s i n c l i n e d to the e c l i p t i c . stable p e r i o d i c o r b i t was found, smaller i n s i z e than the numerical s o l u t i o n described i n [30 ] and which corresponds to case 10 of Table 3-1. A 202 7 7. ' CONCLUSION Two fundamental problems of c e l e s t i a l mechanics have been considered i n t h i s a n a l y s i s : r e s t r i c t e d four-body problem. the general three-body problem and a Although a number of c o n s t r a i n t s are imposed, no assumptions are made which could i n v a l i d a t e the f i n a l solution. A consistent and r a t i o n a l approach to the a n a l y s i s of four body systems has not p r e v i o u s l y been developed, and an attempt made here to remedy t h i s d e f i c i e n c y . is The basic s t r u c t u r e of the four body problem described i n Chapter 3 should f a c i l i t a t e extension of the present work to include the e f f e c t of more general perturbations. In the p a r t i c u l a r three-body problem under i n v e s t i g a t i o n two masses, forming a close binary system, o r b i t a comparatively d i s t a n t mass. A new a n a l y t i c a l s o l u t i o n of t h i s problem i s found i n terms of a small parameter e, which i s r e l a t e d to the distance the binary system and the remaining mass. separating The asymptotic s o l u t i o n , which i s determined using the two-variable expansion procedure, i s _14 uniformly v a l i d as e 0 fortune i n t e r v a l s 0(e a constant e r r o r O ( e ^ ) . ) and accurate w i t h i n Although many terms are l o s t i n the process, i t i s p o s s i b l e to reduce the s e r i e s s o l u t i o n to a form which can be compared with G.W. H i l l ' s r e s u l t f o r the lunar v a r i a t i o n o r b i t . This c o n s t i t u t e s a severe t e s t , but comparatively close agreement i s found between the two . o r b i t s . The predicted s o l u t i o n i s p e r i o d i c , a feature that allows f u r t h e r comparison with numerically-generated orbits. Two examples are considered, the f i r s t of which r e l a t e s to 203 7.2 the sun-earth-moon c o n f i g u r a t i o n of the s o l a r system. The second example a p p l i e s to a problem o f s t e l l a r motion where the three primary masses are i n the r a t i o 2 0 : 1 : 1 . In both cases the numerical and a n a l y t i c a l s o l u t i o n s show close agreement, with an e r r o r below 5% f o r the sun-earth-moon c o n f i g u r a t i o n and l e s s than 3% f o r the stellar system. The four-body problem is. derived from the three-body case by introducing a p a r t i c l e of n e g l i g i b l e mass i n t o the close binary system. Unique u n i f o r m l y - v a l i d s o l u t i o n s are found f o r motion near both e q u i l a t e r a l t r i a n g l e points of the binary system i n terms of the small parameter e, where the primaries move i n accordance with the u n i f o r m l y v a l i d three-body s o l u t i o n . Accuracy, i n t h i s case, i s maintained Q w i t h i n a constant e r r o r 0(e ), and the s o l u t i o n s are uniformly v a l i d as E + 0 f o r time i n t e r v a l s 0 ( e ~ ^ ) . The predicted p e r i o d i c o r b i t s are compared with corresponding numerical s o l u t i o n s f o r motion near and Lg of the earth-moon system. O r b i t a l p o s i t i o n e r r o r s are found to be l e s s than 5%, and i t appears l i k e l y that an extension of these r e s u l t s to the next order i n e would produce a s u b s t a n t i a l improvement i n accuracy. Further work on t h i s t o p i c could take a number of p o s s i b l e directions. The s o l u t i o n s presented here may f a i r l y r e a d i l y be continued to higher orders i n e, but an extension to the three-dimensional case would probably be of greater i n t e r e s t . e i t h e r development. No d i f f i c u l t y i s forseen i n B.eyond t h i s p o i n t , however, the process of a n a l y s i s i s l i k e l y to become considerably more i n v o l v e d . The s i t u a t i o n s which r e s u l t from more general perturbations have been i n d i c a t e d 201; 7 i n Section 3 . 6 , and i t would be important to proceed from one degree of complexity to the next. In t h i s context Table 3-1 should prove helpful i n d e s c r i b i n g system s t r u c t u r e . On a much more general l e v e l the e n t i r e approach may be used i n the a n a l y s i s of non-integrable dynamic systems, p a r t i c u l a r l y when i t i s f e a s i b l e to decompose the problem into a number of s u b s i d i a r y cases. 205 Rl REFERENCES 1. Szebehely, V . , Theory of O r b i t s . 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A . , and Levinson, N . , Theory of Ordinary D i f f e r e n t i a l Equations. McGraw H i l l , New York, 1955, pp. 74-75. 62. Lanczos, C. Applied A n a l y s i s , P r e n t i c e - H a l l , New York, 1956. *pp. 229-235, p p . 438-507, p p . 455-457, " p p . 229-248, p . 454, p . 516. 2 5 6 3 210 R6 63. Moulton, F . R . , A c l a s s of p e r i o d i c s o l u t i o n s of the problem of three bodies with appl4cation to the lunar theory. American Mathematical Society Transactions, V o l . 7 (1906), p. 537. 64. A r e n s t o r f , R . F . , New p e r i o d i c s o l u t i o n s of the plane three-body problem corresponding to e l l i p t i c motion i n the lunar theory. Journal of D i f f e r e n t i a l Equations, V o l . 4 (1968), p. 202. 65. Henrici, P., 1964. 66. F l e t c h e r , R . , A new approach to v a r i a b l e metric algorithms. Computer J o u r n a l , V o l . 13 (1970), p. 317. Elements of Numerical A n a l y s i s . W i l e y , New York, The 211 SOME ASPECTS OF THREE AND FOUR-BODY DYNAMICS PART I I : APPENDICES by PETER G.D. BARKHAM FACULTY OF APPLIED SCIENCE THE UNIVERSITY OF BRITISH COLUMBIA September 1974 r 212 A ii TABLE OF CONTENTS Appendix I. II. Page EXPANSIONS IN TERMS OF e Al.l THE REFERENCE LOCUS CONCEPT . A2.1 Equations of Motion A2.1 A2.2 Numerical Explorations A2.4 A2.2.1 Intersections with the £ a x i s at t = 0 A2.2.2 A2.5 P o s i t i o n s of e q u i l i b r i u m when tfO A2.3 III. A2.14 Limitations A2.16 SOLUTION OF THE THREE-BODY PROBLEM A3.1 A3.1 The Homogeneous S o l u t i o n A3.1 A3.2 The Non-homogeneous S o l u t i o n A3.2 A3.3 Expansions of s i n ( t + 0 ) and cos(t+6) A3.4 A3.5 Continuation of the Uniformly V a l i d Solution IV. A2.1 A3.6 VARIATIONAL ORBITS IN THE RESTRICTED PROBLEM . . . A4.1 . A4.1 A4.1 The Homogeneous S o l u t i o n A4.2 Non-homogeneous S o l u t i o n s A4.7 A4.2.1 A4.2.2 Non-resonant case . . . . . . . . . Resonances i n v o l v i n g s h o r t period terms A4.8 A4.2.3 Resonances i n v o l v i n g l o n g period terms A4.10 A4.12 213 A iii Appendix V. Page SOLUTION OF THE FOUR-BODY PROBLEM A5.1 A5.1 Expansion of the Nonlinear Terms A5.1 A5.2 Asymptotic Expansion of the D e r i v a t i v e Terms A5.9 A5.3 Continuation of the Uniformly V a l i d Solutions VI. A5.16 CHEBYSHEV POLYNOMIALS A6.1 A6.1 Introduction A6.1 A6.2 Function Approximation Using S h i f t e d Chebyshev Polynomials A6.2 A6.3 Coefficients of T * ( y ) A6.4 A6.4 Approximate Expansions A6.4 A6.5 Expansion of the Function | r . | k di A6.9 A iv LIST OF TABLES Table Page A2-1 Astronomical Constants A2.9 A2-2 Parameter Values A2.10 A6-1 Chebyshev and Binomial C o e f f i c i e n t s f o r the expansion of f(x) = [ l + x ] " 3 / 2 A6.13 215 A v LIST OF FIGURES Figure A2-1 ,A2-2 A2-3 A2-4 A2-5 A2-6 Page Primary c o n f i g u r a t i o n at t = 0 . . A2.5 Locus near L-j A2.16 A2-2-1 Polar representation . . A2.16 A2-2-2 £ component • . A2.17 A2-2-3 n component Locus near L A2.17 A2.18 2 A2-3-1 Polar representation A2.18 A2-3-2 £ component A2.19 A2-3-2 n component . . . . A2.19 Locus near L A2.20 3 A2-4-1 Polar representation A2-4-2 £ component A2.21 A2-4-3 n component A2.21 Locus near L . . . A2.20 A2.22 4 A2-5-1 Polar representation . A2.22 A2-5-2 £ component . . A2.23 A2-5-3 n component A2.23 Locus'near.Lg • A2.24 A2-6-1 Polar representation A2.24 A2-6-2 5 component A2.25 A2-6-3 n component . . . . . . . A6-1 f(x) = [ l + x ] " A6-2 Region of v a l i d i t y f o r the expansion of f ( x ) =• [l+x]-3/2 near L and L A2.25 A6.10 3 / 2 4 g A6.ll 216 Al.l APPENDIX I EXPANSIONS IN TERMS OF e Consider the two expressions £ tot + j*, L%<-%) CAM) and where £,n define the p o s i t i o n of p ^ . From Equations 2.43 and 2.44 the p o s i t i o n of p^ i n the £,n>£ system of reference i s given by t = - r cos t where t i s the independent v a r i a b l e defined by Equation 3.30. therefore express the quantity l>c,l since r as b -/^s^b) + i We can 5 + 1 Cfil.3) 217 A1.2 2 A binomial expansion of Equation A1.3 w i l l be v a l i d when 2 E, +n 2 « r-j , and consequently the two q u a n t i t i e s defined by Equations A l . l and A1.2 can be developed i n terms of the small parameter - L . The f i r s t few terms of t h i s expansion a r e : J*l *oSt + = £ , cost - ^ t o s t l " J_ - 3. ( j c © s t - ^ $ l n . t ) CAi.r) which demonstrates the c a n c e l l a t i o n of terms i n v o l v i n g and 3.15). With e = — A x (see pages 3 . 1 2 (from Equation 3 . 2 8 ) , a f t e r some t e d i o u s , but s t r a i g h t f o r w a r d , algebraic manipulation we obtain the f o l l o w i n g expansions f o r Equations A l . l and A 1 . 2 . In, IV 1 8 1 8 X x • 1 5 (A(.£>) 218 Al .3 { 1 /». 6 +£ it> L &oZfc 15 ^4-t 4- - - - s«vb - i o £ «~-3b 3Z- Cot <^3b - 1>S - tfc iz8 l r u l c^Sfc 5 + 3|5 yi^5b ? 3?J 32- 64- si~Sfc ? ^ 5 4 4 ^4- / 6 15-8 - 6 V« £ i J V - <•{ [" I - 3<*,2.b][ ^ 3 »1* to <L zb - tos It <.4t • - 219 A1.4 35 (^,3fc -5-2- «C3fc to^t •315 1-2- - 'JJhS <C5fc 64 Gob + 64- Cj33tr IfS — 3*5 <3-S Qrp 5"b «*« 1 1- oCe'O When £>n define the p o s i t i o n of p and 2 Equations A1.6 and A1.7 take the form J L 6 io / L* 8 s.3 ^/'-/O It 7£- 4- cosb <6 + Ijf5 cos 3b + 3T? cos 5k ( A I . * ) 220 A1.5 it — • 5 fc'^O-fO 2 a 5 9 c o s t +• if c e ^ t 1 j - - V -* 3 si^fc + ] S St«.3t $ L- oCO J 13 + /i' 6 Oct) 3^zt) - 3v< siM.2-tr 3_ si«.b Xb 8 it iS_ s;«.fc •+- w £ VS_ «;^3fc « ^ S + s 0 - 3 « o 2 f c ) - 3 u . S^aZb^ s 2- s^'St +• 3iS 64- oft) (A l . l ) Si«v £ fc x-xe it * Hi X. e t' 3 11,6-fO cxTfe) £ 1 5."-5 [ 3. «.~b L * + is • * $^3b . -v/ s 3_ Q r o b - _1S < t e 3 t d ©CO 221 Al 15 h 6 Ji 1 1- 3coZt~) - 3M. where the subscripts 0(1), 0(e) and 0(e ) denote terms of t h i s order i n the asymptotic expansions of u^ and (i = 5 , 6 , 7 , « « « ) . 222 2.1 A APPENDIX II A2.1 THE REFERENCE LOCUS CONCEPT " 1 Equations of Motion The a n a l y s i s of three and four-body motion given i n Appendices I I I and V was preceded by a l e s s - f r u i t f u l i n v e s t i g a t i o n , but the r e s u l t s of t h i s approach are summarized here to i n d i c a t e the l i m i t a t i o n s which were encountered. These two paths of i n q u i r y s t a r t to diverge at Equations 2.47 and 2.48, i n the ri plane. which describe the motion of p 4 near p 2 and p 3 We have, f o r the planar motion of p^ : I— 1 Lagrange configurations i n the r e s t r i c t e d problem s a t i s f y the c o n d i t i o n s f o r r e l a t i v e e q u i l i b r i u m a t a s p e c i f i c i n s t a n t of time. When the same conditions are a p p l i e d to the four-body s i t u a t i o n then, corresponding to a sequence of time i n s t a n t s , there w i l l be a sequence of p o s i t i o n s of r e l a t i v e e q u i l i b r i u m . The locus i n the £> n plane described by these points of e q u i l i b r i u m w i l l be c a l l e d a reference l o c u s . + T h i s material was presented at the 13th I n t e r n a t i o n a l Congress of Theoretical and Applied Mechanics, Moscow, August 21-26, 1972, and at the 23rd International A s t r o n a u t i c a l Congress, Vienna, October 8-15, 1972. 223 A2.2 I f the point s^, n L i s located on such a locus then, when the v e l o c i t y and a c c e l e r a t i o n of p^ are both zero, we obtain LEI * This device was mentioned, although not i n v e s t i g a t e d , by Steg and deVries [ 8 ] , and appears also i n the work of Tapley and Lewallen [ 9 In contrast to the r e s t r i c t e d problem the terms £ . and ]. in Equations A2.1 to A2.4 are functions of t , so the s o l u t i o n of Equations A2.3 and A2.4 w i l l also depend on t . The method by which these s o l u t i o n s can be found i s described i n the f o l l o w i n g s e c t i o n , but f o r the present we s h a l l assume that K^(t) and n^(t) are known functions of t . Now l e t 5 = O- ( +• x ) = 5^ +• roc where a i s a normalizing constant, p 4 so that x and y define the motion of with respect to the .reference l o c u s . then be w r i t t e n (A»-0 Equations A2.1 and A2.2 can 2214. A2.3 - <v a. +• \ -5, 7 4— 1 IT- | 3 - <. = ( a Z—i i_ i - lit (A I f the term LI 3 ' 3 x.S) i s expressed as J in-1 , .,-J r and we expand Equation A2.9 u s i n g , f o r example, Chebyshev polynomials, then f o r Equations A2.7 and A2.8 we obtain II) where the functions r , A-j and contain t e x p l i c i t l y and i n v o l v e terms i k of the form x y J (j,k i s s u f f i c i e n t l y small = 0,1, 2, ••••). When the o r b i t a l parameter a these equations may be w r i t t e n 225 A2.4 (A*.13) where a , b and c contain t e x p l i c i t l y . I f X Q and denote s o l u t i o n s of Equations A2.12 and A2.13 when the terms 0(a) are neglected, the approximate s o l u t i o n s of Equations A2.1 and A2.2 are given by S = 1 * %u °-*o + (A2.I4) (R2..IS ) "lc The character of the reference l o c i w i l l determine whether or not t h i s approach i s f e a s i b l e , so we should next attempt to solve Equations A2.3 and A 2 . 4 . A2.2 Numerical Explorations At a p a r t i c u l a r i n s t a n t of time the p o s i t i o n s of r e l a t i v e e q u i l i b r i u m are defined by the c o n d i t i o n s 3 3 ft fe Ir.;!' These equations correspond to Equations 3.40 and 3.41 f o r the r e s t r i c t e d problem but, i n contrast to the equivalent r e s u l t f o r the 226 A2.5 r e s t r i c t e d case, i n general i t i s not p o s s i b l e to p r e d i c t i n advance the t o t a l number of reference l o c i . D e f i n i t e r e s u l t s can, however, be obtained i n one s p e c i f i c i n s t a n c e , and the approach i s described below. A2.2.1 Intersections with the E, a x i s at t = 0 When t = 0 the three primary bodies are c o l l i n e a r and l i e on the E, a x i s , as shown i n Figure A 2 - 1 . Figure A2-1 Primary c o n f i g u r a t i o n at t = 0. Note that t h i s i n i t i a l c o n f i g u r a t i o n i s chosen so that both p-j and p are located on the negative t h e r v f (0) = 0, and we have - X E, a x i s . 3 One s o l u t i o n of Equation A2.17 i s 227 A2.6 This equation can be w r i t t e n (s,-O f^-O" x <*,-0* from which a seventh-order polynomial i n E,^ can be d e r i v e d . i e n t s of t h i s polynomial depend on the signum functions The c o e f f i c - sgn(£.j-£;^), and i n f a c t there are four d i s t i n c t cases corresponding to the conditions When Equation A2.19 i s expanded we obtain — 228 A2.7 + 5 t -1, + sC-c 6C7-) + ^ ^ s ( 5 , - O +• fx-S^^C^f r O = o (fli.^o) where &(4> = - 2. 229 A2.8 We must apply numerical methods to obtain s o l u t i o n s of Equation A2.20, but t h i s w i l l require s p e c i f i c values f o r n , fi, C , u. and 5. (where i = 1, 2, 3). As an approximation to the motion of p-j, p p , suppose we take the model described i n Chapter 2 (Section 3 on page 2.30 ), with the a d d i t i o n a l r e s t r i c t i o n that p i n c i r c u l a r o r b i t s about t h e i r centre of mass. 2 2 and 2.6, and p^ move The l i m i t a t i o n s of this approach have been discussed i n the i n t r o d u c t i o n to Chapter 3, but any e x p l i c i t s o l u t i o n f o r the motion of the three primary bodies could i n general be used. An i d e a l i z e d model f o r the sun, earth and moon, c o n s i s t e n t with these r e s t r i c t i o n s , can be derived from the f o l l o w i n g expressions: X = r Z3 n* = X 3 L S r, = At X C hx = r, - ( jry R 3 H 230 A2.9 Symbol Parameter Value Astronomical Unit A 0.4990 04785 x TO l i g h t sec. V e l o c i t y of Light c 0.29979.25 x TO m. s e c . " Ephemeris Seconds Per Tropical Year (1900) s 0.3155 6925 9747 x 10 Earth-Moon Mass Ratio . Moan Fsrth-Mnnn llictaiiro 9 m /m 2 3 3 0.813024 x 10 * 8 sec. 2 q r Z6 n 3«4-inn y i o m PI Geocentric Constant •: IT Gravitational 3.1415 9265 3589 79324 Km 2 0.398603 x 1 0 Table A2-1 Astronomical Constants 1 5 m sec? 3 1 2 231 Parameter C r l r 2 r 3 Value 0.3891 7127 26 X 10 3 0.3891 7245 58 X 10 3 0.1215 0313 96 X I D " 0.9878 4968 60 1.0 n 0.7470 2544 92 X l O " y l y 2 y 3 1 0.3289 2414 85 X 10 1 6 0.9878 4968 60 0.1215 0313 96 X l O " Table A2-2 Normalized Constants 1 232 A2. where the q u a n t i t i e s A , c, s and Km^ are defined i n Table A 2 - 1 . numerical values f o r the astronomical The u n i t and earth-moon mass r a t i o are from Ash, Shapiro and Smith [57 ] ; the remaining constants are taken from Clemence [56 ] . Corresponding numerical values f o r n , £2, C, r.j and y . (i = 1, 2, 3) are given i n Table A2-2. Now that s p e c i f i c values f o r the parameters of Equation A2.20 have been obtained, we must s e l e c t a numerical method f o r s o l v i n g n ' t h order polynomials with real c o e f f i c i e n t s . difference algorithm [ 6 5 ] Rutishauser s quotient- 1 provides a p a r t i c u l a r l y elegant method of s o l u t i o n , and has the d i s t i n c t v i r t u e that convergent approximations all to roots are derived simultaneously. There a r e , i n f a c t , four d i f f e r e n t polynomials to s o l v e , corresponding to the cases defined on page A 2 . 6 . Roots o f Equation A2.20 f o r these four separate cases were found using a version of quotient-difference the a l g o r i t h m , and the s o l u t i o n accuracy was tested by evaluating the polynomial with these s p e c i f i e d numerical v a l u e s . The r e s u l t s are given below. Case 1 £ L > 0.12150 31396 x 10' •1 REAL COMPLEX O. 39 O 2. S"8 3 4 -*. io 0.39 02. S834 x io" O .1363 66 3 2 O. 7 - 9 4 9 l9 9o * io" 0 . 1 8 6 3 6632. O. 7 - 8 4 9 19 9 0 * O . 490 0 2 8 ? € 0 . 8S57 Il o I O .4-9 o o O . 8557 If o I O . IOO 3 287-8 l 39 I 3 x Io' 0.2.905 I 841 - 0.7-<\OS O.o x 18 4 2 i o 1 K IO Z 1 io"' 233 A2.12 Case 2: -0.98784 96860 < £. < 0.12150 31396 x 10" REAL COMPLEX - o. 3102. S"834- - o. 3°02- S"8 3 4 ~ o. 1 D 62. 5 ? o 3 - o. | 0 6 2 -o. - o. X io3 O - 1 1 062. 5"?o3 x S?oS - 0 10 ' .o . 0 . 1 — 0 .0' 0 8372, -0.38971 24558 x 10° < K < L < to" - o. 3 l O i 5834- * vo - o . lozz T-i 1 I - o . <\07-Z. O. 5o3S A-5SI O. Ilb3 <?4-8o ? L 31 3 2 I4S3 3l 3 8 I ft S 3 31 3 8 I4-S3 31 3 g 1 . O -0.98784 96860 3 -o.ZloS" - o - 7 184-2. O . I 3>06 * 10 K 10 " 0. X°ioS 5 O . S03S 4-581 Case 4 : (o - COMPLEX o . 3<jo2 - I 4 ? 3 . REAL - l S 4 2 s - 0 • Z1 o5 10 < - 0 . 8360 Case 3: O . 2-10 57o3 1 0 6 2 1 . 1306 x IO*" 7<?78 71 7 8 o . ? 6 4-8 o . 8 6 04- 76 4- 8 O - O 1 < -6.38971 24558 x 10 3 COMPLEX REAL - O . 35€« o 2 1 I * 10 - o. 4 l ? l Zo3fc x io - o. 4-44-4- 73S"S x 10 0.0 3 o.o 3 0.0 1 O. 3627 14-74 x io~' o . 36. 8? I 4 74 - O . <?8 6<? oo$l - o. 1861 oo$i x io"' - - O . 472k 3©4I o . 4-72-6 3o4-l O. 5 Z 8 3 o 16 2 K O. S Z O l6 Z xi 1 o 8 3 10' 23k A2.13 In a l l cases the residual e r r o r was below 10 -15 i n magnitude. Note that there i s no ambiguity i n these r e s u l t s , as the complex part of each root i s e i t h e r zero or a w e l l - d e f i n e d quantity s i g n i f i c a n t l y d i f f e r e n t from zero. Now only those real roots w i t h i n the regions of v a l i d i t y f o r £|_ a c t u a l l y define points on reference l o c i , so t h a t we can s t a t e the f o l l o w ing result:When t = 0, f o r the p a r t i c u l a r numerical model chosen here p r e c i s e l y four reference l o c i i n t e r s e c t the £ axis. The points of i n t e r s e c t i o n are € = 0.1001 3913 x 10 K = -0.8360 8372 K = -0.1153 9480 x 10 € 1 1 = -0.4173 2036 x 1 0 . 3 I t might be a n t i c i p a t e d that three of these i n t e r s e c t i o n s be located near c o l l i n e a r Lagrange points of the corresponding problem. From Szebehely [ l 1 8 ] , when points are located (approximately) £ = -1.15568 K = C = = 0.01215 restricted the c o l l i n e a r at (1^) -0.83692 (L ) 1.00506 (L ) 2 3 , so that the conjecture i s j u s t i f i e d . This r e s u l t c a n , however, only be a p p l i e d when Equations A2.16 and A 2 . 1 7 reduce to the form of Equation A 2 . 1 8 , which i s true only when (n - ft)t = 2kir, where k = 0, ± 1 , ± 2 , •••• . will 235 A2.14 A2.2.2 P o s i t i o n s of e q u i l i b r i u m when t f 0 The functions f ^ ( t ) and f ^ U ) i n Equations A2.16 and A2.17 define two nonlinear a l g e b r a i c expressions, but the formulation i s not particularly helpful. then F ( £ , n , t ) > 0 If, however, we take and a s o l u t i o n f o r £ and n which s a t i s f i e s the condition at a s p e c i f i e d instant of time w i l l a l s o s a t i s f y Equations A2.16 and A2.17 at t h i s time i n s t a n t . Fletcher [ 6 6 ] has developed an e f f i c i e n t algorithm to determine the l o c a l minimum of a nonlinear algebraic f u n c t i o n of n v a r i a b l e s i n the v i c i n i t y of an i n i t i a l estimate, which can be a p p l i e d to solve Equation A2.22 f o r s p e c i f i c values of t . The extension of t h i s procedure to generate a complete locus i s summarized below. 1. At t = 0 s e l e c t an i n i t i a l estimate of the p o i n t on a particular locus. 2. From t h i s estimate, determine the exact l o c a t i o n of the point using F l e t c h e r ' s method to solve Equation A2.22. 3. Increment t by A t . 4. With the preceding point as an i n i t i a l estimate, the process by r e t u r n i n g to step 2. iterate 236 A2.15 In t h i s approach we must, however, assume that some point on the p a r t i c u l a r reference locus s a t i s f i e s t. Equation A2.22 f o r every value of There i s some evidence to suggest that F ( £ , n , t ) = 0 cannot be s a t i s f i e d f o r a l l values of t along two l o c i , but a d i s c u s s i o n of t h i s d i f f i c u l t y w i l l be postponed u n t i l Section A 2 . 3 . Six l o c i were found f o r the sun-earth-moon system by t h i s method, using the approximate primary model of the preceding s u b - s e c t i o n . For each i n d i v i d u a l point the f u n c t i o n F ( £ , n , t ) was scaled so that F = 1 with the i n i t i a l estimate of the minimum. F l e t c h e r ' s algorithm was -13 terminated when the p o s i t i o n c o r r e c t i o n was l e s s than 10 -20 which i n a l l cases reduced F below 10 . i n magnitude, Four of these l o c i emanate from the points of i n t e r s e c t i o n given on page A2.13,and the remaining two l o c i correspond to e q u i l a t e r a l t r i a n g l e points of the problem. The locus containing the point forms a c i r c l e i n the £ , n. restricted £ = -0.4173 2036 x 1 0 ° , n = 0 plane so that the point of r e l a t i v e e q u i l i b r i u m l i e s on a l i n e j o i n i n g the sun and the earth-moon mass centre. Results obtained f o r the f i v e l o c i near the earth-moon system are shown i n Figures A2-2 to A 2 . 6 . 1_2 are s i m i l a r i n character, and Lg. Note that the two l o c i near L-| and and so a l s o are those associated with The locus near L ^ , shown i n Figure A 2 - 5 , i s b a s i c a l l y equivalent to that found by Tapley and Lewallen [ 9 ] (Figure 9 i n [ 9 although t h e i r primary model includes the i n c l i n a t i o n of the earth-moon o r b i t a l plane at an angle of 5.15° to the e c l i p t i c and takes i n t o cons i d e r a t i o n the e f f e c t of s o l a r r a d i a t i o n pressure. The u n i t of time i n Figures A2-2 to A2-6 i s chosen so that new moon corresponds to the instants t = n , where n = 0, ± 1 , ± 2 , •••• . As one might a n t i c i p a t e ]), 237 A2.16 1. l 1 -\-\SS -1-154 h -0-2 0-4- Figure A 2 - 2 A2-2-1 Locus near L^ Polar representation 238 I A2.17 -1-154- -r •1 -tSS 1-156 1 t OS Figure A2-2-2 0-? E, component -1 o -is -4 1 o-S o-s -J Figure A2-2-3 n Figure A2-2 component Locus near L^ 1 -o t. 239 A2.18 r 0 . 2 V h -0.837 o.i -0.836 -0.1 *• - 0 . 2 Figure A2-3 A2-3-1 Locus near L, Polar representation 1 0 22+0 A2.19 5 -0.836 i -0.837 H L -0.838 r — 0 0.25 Figure A2-3-2 £ > 0.5 • 0.75 . t 1.0 component 0.2 0.1 10' 0.25 0.5 -0.1 -0.2 J Figure A2-3-3 n Figure A2-3 component Locus near L 0.75 t 1.0 214-1 A2.20 1.0 0.75 H 0.5 0.25 -i 0.5 0.75 -0.25 -0.5 -0.75 -1.0 Figure A2-4 A2-4-1 Locus near L^ Polar representation 1.0 1 1.25 2i+2 A2.21 1.0 0.8 0.6 0.4 0.25 Figure A2-4-2 1.0 ' -T- £ 0.5 0.75 t 1.0 component 1 0.5 ^ 0.25 -0.5 -1.0 0.5" 0.75 H J Figure A2-4-<3 n Figure A2-4 component Locus near L 3 1.0 t 0.8 0.7 0.7 •0.5 -0.3 Figure A2-5 A2-5-1 -0.1 Locus near L^ Polar representation 0.1 A2.23 0.1 1 0.1 i 0.3 1 0.5 1 0.7 1 1 0 0.25 Figure A2-5-2 1.0 1 0.5 1 0.75 1 t 1.0 E, component n 0.9 0.8 0.7 -\ 0.25 Figure A2-5-3 n. Figure A2-5 0.5 component Locus near L^ 0.75 1.0 t 1L •0-7 -o-3 o-S o-i -o-l -0-7 H ro -O-S l-O Figure A2-6 A2-6-1 Locus near L^ Polar representation ro -p. 2i;6 A2.25 - 0'\ 0-3 i -OS - 0... 7 —i .0-2.5 Figure A2-6-2 £ 1 r- 1 0-5 0-75 i-o t component -0.7 -0-8 -o.<\ J - i-o H t i o-S O'XS Figure A2-6-3 n Figure A2-6 component Locus near L, •o 2k7 A2.26 f o r coplanar primary o r b i t s , a l l f i v e l o c i near the earth-moon system are p e r i o d i c functions of time. A2.3 Limitations One p o s i t i v e r e s u l t of t h i s a n a l y s i s i s the extension of Tapley and Lewallen's i n v e s t i g a t i o n [ 9 ] to include a l l p o s i t i o n s of r e l a t i v e e q u i l i b r i u m f o r the sun-earth-moon system at a p a r t i c u l a r time A number of f e a t u r e s , instant. however, render the reference locus approach u n a t t r a c t i v e i n the a n a l y s i s of four-body motion. (i) The reference l o c i only apply to one s p e c i f i c primary model, and i f any parameters are a l t e r e d the entire a n a l y s i s must be repeated, (ii) When the o r b i t a l parameter a = 0, the functions £^(t) and r i ^ ( t ) are not s o l u t i o n s of the equations of motion (because the q u a n t i t i e s E,* and and A2.8 are 0 ( a ) ) . - 1 i n Equations A2.7 Although a may be s m a l l , the l i m i t process a -> 0 cannot, t h e r e f o r e , j u s t i f y neglecting terms 0(a) be used to i n Equations A2.12 and A2.13. (iii) An expansion of the term _j (where i = 1, 2, 3) leads to a form of Equations A2.12 and A2.13 which cannot e a s i l y be s o l v e d , p a r t i c u l a r l y i f s o l u t i o n s are to be accurate over large i n t e r v a l s of time, (iv) Not only the l o c u s , but also i t s f i r s t and second d e r i v a t i v e s , must be evaluated. The process of differentiation 214-8 A2. i s i t s e l f s t r a i g h t f o r w a r d , although the computation time increases, but rapid changes i n C[_(t) and Hj_(t) present severe obstacles to any accurate a n a l y s i s . i s evident in the two l o c i near and A 2 - 6 ) . This behaviour and Lg (Figures A2-5 An attempt to continue the l o c i through these abrupt t r a n s i t i o n s d i d not succeed because no s o l u t i o n f o r Equation A2.22 could be found. The only inference which can s t r i c t l y be drawn from t h i s r e s u l t i s that algorithm f a i l e d to f i n d a l o c a l minimum, but i t Fletcher's appears l i k e l y that no s o l u t i o n of Equation A2.22 e x i s t s in these regions of the n plane f o r c e r t a i n small i n t e r v a l s of time. T+ i.fac Aar-A A r\A K n r a n r n r\-f + h "i c arr-nmul ati'rin rt-F r l i f f i n i l f i abandon the a n a l y s i s i n v o l v i n g reference l o c i and concentrate instead on a l o g i c a l development from fundamental p r i n c i p l e s . 2k9 A3.1 • APPENDIX III — SOLUTION OF THE THREE-BODY PROBLEM With the exception of three d e r i v a t i o n s (which are given below), the a n a l y s i s of Chapter 4 up to Equation 4.83 requires l i t t l e f u r t h e r amplification. We now continue the development beyond t h i s point to / -14\ obtain s o l u t i o n s uniformly v a l i d f o r t = 0(e ). A3.1 The Homogeneous Solution I f we take x-j = e, x 2 = e, x 3 = f , x^ = f , then Equations 4.11 and 4.12 can be w r i t t e n in the form o I 3 o o o o -X o o o o o X (A3.f) I o which i s equivalent to ^ = A where x i s the vector (x-|, x , x , x^) . 2 transform of x ( t ) , 5E.Cs:> 3 I f x(s) denotes the Laplace then from Equation A3.2 we obtain = ( s i - A ) -i 51 C o ) . CM. 3-") Now det(sI-A) = s (s+1), and a f t e r some s t r a i g h t f o r w a r d manipulation: 250 A3.2 4 01-/0" - 2±_ S--r\ z s **•*•• - 6 6 2s - 2L s+r l *Vl S " 3- _ J_ . A- - 3 4-S - 3_ S - 2. 6$ o o s 0*3.4) Taking the inverse transform of Equations A3.3 and A 3 . 4 , we can w r i t e xCt) = <f>tl) x C o ) , where 0 . - . 2.<<ofc 4 - 3<<r>t 3 *.vfc Cos b (A3.S) X 4*J^b - 3 1 Equations 4.13 and 4.14 now f o l l o w d i r e c t l y from Equation A 3 . 5 . A3.2 The Non-homogeneous S o l u t i o n For the non-homogeneous case we have ii - Z v ' - 3UL = £ (t) (A3.0 2^1 A3.3 Cfi-5.7) where the general form of E* and F* i s given i n Equations 4.41 and 4.42. If B denotes the vector (0, E * , 0, F * ) , Equations A3.6 and A3.7 can be T w r i t t e n i n the form 5. = and P5c •+ B therefore s£s) = _(sJ-A)" x('o) V + (si -A)' 1 where B(s) i s the Laplace transform of B ( t ) . BC*s"> When B(s) i s known, the non- homogeneous part of the s o l u t i o n corresponding to the term (sI-A)~^ B(s) can be derived from Equation A3.4. I t i s convenient to p a r t i t i o n Equations 4.41 and 4.42 so that periodic terms i n v o l v i n g wt are considered separately. If e ^ ( t ) and f^l^(t) denote the non-homogeneous s o l u t i o n s of Equations A3.6 and A 3 . 7 , then f o r I F * Cfc) +• b,b . t . b ^ t . + >j«ot. we obtain 4- ( Z b , - 2 b ^ - a ) ^*b 0 3b s;».b (fl 3.8) 252 A3.4 - [ + _ L 3 J . J l> * - fe, J t 3 x + ( ft + 2 - 0 b *~b + t ( Cft^O - * - 0 t *=,b S i m i l a r l y , when the corresponding non-homogeneous s o l u t i o n s a r e : [ X3L + o t { 0 b - I su.«jb 3 f to %3- t - (3+w 1 5'' cost (A3, io") x 0 b (B3.il) ZkJC ItOC. , cJ b Q Co -0 L - :, - %u3d~ + 0 + onb 41. ifa>C 0 + (.3> + «J cos <~>h 253 A3.5 A3.3 Expansions of sin(t+8) and cos(t+8) From Equation 4.32 the expansion f o r 8 i s of the form so that we can write ««. ( b + P ) = suvb or>9 + c«fc <*> CsQ, + e" e ^ . - - ) + c Q ^ C e P , v t ^ P ^ ... + <i,fcJVs^P )J 1 <> tr> t> and, s i m i l a r l y , - COS b - 5.K o o C f e P , ^ ^ P , . * ••• > - s : ~ P s i ^ C t P , 0 ••• *) j b £ * ~ P co C eD, * G" 0 _+. • ) -t- Cr,t904^C 69, + t i e v + - •) ^ 1 0 i 0*3.13") 2 2 The functions sinCeS-j+e e * * * ) and cos(e8.|+e 8 «««) can be expanded as 2 + 2 •51 .(AVI4) 1 - e^f* z! CA3-15-) 25k A3.6 We therefore obtain f o r sine and cos9: 8 - I P x ^ 9 X 0 o - e," ,^P 1 0 + e ( x C qo 0 0 X and Equations A3.12 and A3.13 can now be evaluated without d i f f i c u l t y . A3.4 Continuation of the Uniformly V a l i d S o l u t i o n The sequence of p a r t i a l d i f f e r e n t i a l equations defined i n Chapter 4 (see Equations 4.35 to 4.40) can be continued as f o l l o w s : • -7-2^ - 3a,, = - z it 1 (fl3.l«) 255 A3.7 = if- - X '5 i\ it?- it OCe 1 I it - Z^,„ - 3 a l e = - 2v - Z it [ 1 it""- J >) - OCfc ") 1 "f 2^ 5 7 J OCt OCe*) xo-j-o . oCc + z ^ Vs - s itZt; L (A3-19) 5 ^.5 + OCt'") 1 if*- °t itz-c i ri oo) (A3, io) Ofl) - 2 d*V$ J -+• 0)0.5 J <?Cfe*) "(3.XV-) + p. <'-/**) J O C t ' ° ) oCt) (fl*.ai> 256 A3.8 - it • i — — + 5" }\*? X iC + bj* ~l s i t J ^ J ^ ) +5 1 - 7 ^ at 1 - Joco _ it* • - 2 +- . L it* at — b <j s itir ^ J o C O I Jb* 5V Joco J OCi) 257 A3.9 it* it Z G - ^ O L J - z oCe'*\) — i^J-f OCi ) 6 1 Xt*" Xt ) - I o C 6 ^> 1 bb it* bt • d^Socrf it L it Sou) 0-»o Li ^,5 dfc»- -2-^ ( > it o t f e 3 ) lit*- it J oC.) J — . «3b^r I V L*t* J J ° ^ " 0 z '.•I. bV Ttir atjoc£<0 J (_ i t ^ i b OCeO atar oft . 1 CA325) jfc - 3u 258 A3.10 _I i it" it joceO at ^j^) L. * ^ ^ dt L ^ ^ J o^) it JoCe^ L at* at J oco ©CO £y - + l3 at* F it - I J> LTb* 2. it )t J • 5 r v o f i t 0 o i at* a O at La t 1 ^a-i? a r .at J a ) & £. T L <>t ht v ; ' l ^ J o ( 6 ' ) •' The q u a n t i t i e s Al.8 E g L a t - •• aba*} o o o «fc J t- " dtl and F. are defined i n Appendix I , oCO J ..^Joco Equations and A l . 9 , and f o r . E g and Fg i n Equations A3.18 and A3.19 we have = Fg = 0 . and A3.19. Now consider the remaining q u a n t i t i e s i n Equations A3.18 A f t e r d i f f e r e n t i a t i n g Equations 4.25 and 4.26 we o b t a i n : 259 0>\t - X A3.11 — s ibiv bx bx 2t S*- - Z (b+£>3/>_ . + ^>JLg 5 obtr bv + a-cr 2t? bv which can be rearranged i n the form: 2 -.1 - 2v = s at 2 s.«. 2-5^ z b v/ •+. s ^a • t Zcos t - 61 bfi_ 5 ^ f cose ^5 Si^9 -2-cost ^ (b COS 8 - ph9 SLVLB <r 1 b"C At the present point i n the a n a l y s i s we have shown that | | = 0 ( e ) , f j = 0(e ) and | f = 0 ( e ) . 3x 9T 3 ~bt +6* 3 59^ i 3/>, -f> ^bK 0 I ar With these s i m p l i f i c a t i o n s : 2 QoP - p>z® bV -/j.e.af* *«. 6>. -/s p,5p 0 ^ = 0(e), 3 < C«0 Po 2. b 260 A3.12 3& c*>9 t D + e• +6* l }v ytf zx 2. 3 Equations A3.28 and A3.29 can now be evaluated, and hence 2 5^, - 2- + 2 cob" - 6t ^ [I CA3.32) to CA3-33) + 2s;*.fc [ t 5" <?/s" -/S.P,5fix 0 o + ] 261 A3.13 From Equations 4.25 and 4.26: - 5^t 4- 5 •Zy~!r + Ze^fc 8 - - (G3-34-) e 2-^ l 1 Six- do S'". i (.AS- 3 5 ) I I 3 t £fk and we tnen obtain A F ] /*. F I t has already been shown that ||ii s constant quently both these q u a n t i t i e s are zero. (on page 4.20) and conse- When Equations 4.25 and 4.26 are d i f f e r e n t i a t e d with respect to t : — 3 v" (A3.36) 5 - bt ^ v - C t +p) (ft3.3?) 262 3.14 and from these expressions: 2fu. - Pv $ s (A-3. 3S) ib> it oCt) (A3. The terms 1 2 a V -4- }u. ~) L at-av atav AAt? V J . and I s a* J at it? s are both 31) 0(e), and therefore do not c o n t r i b u t e to the non-homogeneous s o l u t i o n of Equations A3.18 and A3.19. For Ug(t) and v ( t ) we have (from Equations g 4.60 and 4 . 6 1 ) : «?V 6 - Xr it* it if - it 1 iVo Zb b 2c<o b - _£ CA3-40) (A3-4-0 and the non-homogeneous terms can now be accumulated. The r e s u l t i n g expressions are: _* = %H (\+H>VC\-^')\ y £ c»2.b -£ \ + 6(i +/ 0 \ « * , + zip, - 6b£^ i-v I ar L ai? J iv 3 263 A3.15 + Ceo t J I air + *0 - ?^,Ci+fOVi-jO iX! + 5;~b f 2PA, '• *- dX Jx Ceo b CA3.43) When a-| s a t i s f i e s Equation 4.58: so that Equation A3.42 s i m p l i f i e s s l i g h t l y . The symmetry of terms i n E* and F* i n c i d e n t a l l y provides a useful t e s t f o r e r r o r i n these f u n c t i o n s . 26*4. A3.16 I f terms i n v o l v i n g t 2 are to be eliminated from the non-homogen- eous s o l u t i o n , from Equation A 3 . 9 : o which j u s t i f i e s the assumption that We have already shown that s i n 0 Q i s constant (see Equation 4 . 8 2 ) . = 0, and with t h i s condition on 6 containing t s i n t and t c o s t are eliminated from u g and v g Q terms if iv 0+fO'>> - «»9o\ L From Equation p W* 0 iV d~C (A3-4-0 4.76: (A 3-4?) it and therefore Equation A3.45 takes the form 16 Xv (A3-4S) The only value of 9^ f o r which 3g does not v i o l a t e the uniform v a l i d i t y condition i s 9, - consequently o CP 3 . 4 0 i n which case air (A3 s-o) r 265 A3.17 Equation A3.46 can be rearranged as and i f we again use Equation 4.76: • since cosBg f 0. The q u a n t i t i e s and BQ»8-J 6Q are a l l constant, and therefore <^3 = constant . At present we have no way of d e f i n i n g 3 N > 3 N , these q u a n t i t i e s are known to be constant. that both 9 2 and 6 3 and , although I f , however, i t i s assumed are constant, then from Equations 4.81 and A3.49 we obtain where s i n 6 Q the q u a n t i t y = 0. The one remaining secular term i n v ( t ) must e l i m i n a t e g -e (3ta ) 4 4 i f v ( t ) i s to be uniformly v a l i d to 0 ( e t ) . 5 Consequently, from Equations A3.9 and A3.11: 5 266 A3.18 and, i f we s u b s t i t u t e f o r the quantity t o s 9 1 / ^ f ^ + /*, ^ P r i from Equation A3.46, t h i s can be w r i t t e n as From the argument of page 4.15: I t i s , i n f a c t , l e g i t i m a t e to assume that and 6 ^ are constant (see Equations A3.83 and A3.170), so t h a t , from Equations A3.53 and A3.54: assuming is constant- E x p l i c i t expressions f o r Ug(t) and v ( t ) g are not needed at t h i s stage of the a n a l y s i s , and we can postpone evaluating these functions u n t i l the s o l u t i o n s f o r u-j and v-j are considered. 2 2 Now we proceed with the s o l u t i o n of Equations A3.20 and A3.21 for u 1 Q and v 1 Q . This p o r t i o n of the a n a l y s i s i s p a r t i c u l a r l y i n t e r e s t i n g because here nonlinear terms f i r s t enter the sequence of p a r t i a l d i f f e r e n t i a l equations. From Appendix I , Equations A l . 8 a n d ' A I . 9 : 267 A3.19 ] (.A3. SrO (03.58) L g |6 and from Equations A3.28 to A3.31 we have, with 0^ = s i n 0 = 0: Q COS I + 2-eosk - C<JD 9, ^/>x^2- 1 >f A > ^ a ar £ /j as<, +/\a^2 1 av ' PT- ' j . 1 - 2 cost General expressions f o r and A3.35. and 4 or 2 2o o*<i and d V * are given i n Equations A3.34 s A t present, however, = o(e ), (ft 0 a t; _ so that z ]>jf$ at* ||=0(e ), 4 and }j*s bx |f=0(e ), are both x i t follows that: * 0Ct») oU ) 1 0(e ) 2 o 4 0(e ). 4 Consequently 268 A3.20 From Equations 4.47 and 4.48: -t- o a ) (ft3.6l) 6 (A3.62) since 0-j = 0 and a Q = 0 (see Equation 4 . 5 2 ) . The nonlinear terms i n Equations A3.20 and A3.21 can now be evaluated, and we obtain • ( 3 / - - feu.*) a - 9 //** o cos 2.b ( A 3.63.) I (l- /*>) (see a l s o pages A3.26 to A3.39 f o r a d e t a i l e d d e r i v a t i o n ) . When sin0 = 0 Q and 0.| = 0, the equations f o r sin0 and cose s i m p l i f y t o : OcV) «~ 9 Cos » L 2. . 0 (A3.65) 269 A3.21 (see Equations A3.16 and A3.17 on page A3.6 ), so that s i n ( t + 9 ) and c o s ( t + 9 ) can be w r i t t e n : s.v. Ct + B) 5 ~ t | I - £*fC ^ + cosh | ocv; CO ft, (A3.iS) We then o b t a i n , from Equations A3.36 and A3.37: r ^b* at — (i e^,9 east e 0 +- £ |^ 3<*, - C«0 t - *~b ~* <<o b J 270 A3.22 3b* L at 4- 0 ( t f ) , and therefore cIVj - a>/ ^ 3oi s - x &o at - <*>9, 033-92) From Equations 4.25 and 4.26: z. dldX - 7- S~^s + a t a r a_u. $ aj/ = 5 -at a»t atr a-r a_t< % 5 f/ 3 (A3.7S) ar ^ t + O - 3 / i s i ^ C t + a") ' a-r ar - A- < u <ar 2 ^ ; dx~ L a-u" - 3x 1 - 3ta ar* L ar ~ixr 2to$Cb-*-e) a-cj 2s^b+e) (A3.?6) 271 A3.23 At the present stage of the a n a l y s i s , i n a d d i t i o n to the order r e l a t i o n ship given on page A3.19, we have and ^lif = 0(e ) . dr - ||> = 0(e ), = 0(e ), 4 4 9x 0(e ) 4 None of the expressions defined i n Equations A3.74 2 to A3.76 therefore makes any c o n t r i b u t i o n to Equation A3.20 and A3.21. From Equation 4.69 u ( t ) = Vy(t) = 0, and a l s o 7 (see Equations A3.40 and A 3 . 4 1 ) , so that we can now accumulate the nonhomogeneous terms i n Equations A3.20 and A3.21. The corresponding f u n c t i o n s , E* and F * , are given below. 6b L ' + o o b" C«>9/ + . QoZb :* - ar L * r atr ' a t d - ^ O J J ' 1 f*,(>-fO 4- L iX It, 2. 9^-^ IT 1^ i TTT^ ( A3.?« ) - 35 Y C<X S£~4-b 272 A3.24 To eliminate secular terms i n v o l v i n g t for v from the non-homogeneous s o l u t i o n : 1 0 - which j u s t i f i e s ° (A3.rO the assumption i n Equation A3.56 that is Terms containing t s i n t and t c o s t are eliminated from u ^ ( t ) constant. and v^(t) if: - f>„)K f>o^->-\ L ar + ar C^— y + br J fc\ ^ I >r ' (AS.so) =• ° ? J We can again s u b s t i t u t e f o r the quantity - ° (Ai.sO • iP> - ( l + ^ ^ ^ J from Equation A 3 . 4 7 , and Equation A 3 . 8 0 then takes the f o l l o w i n g form: The exact values of 3 Q and 8 2 have been i n doubt since Equation 4 . 8 0 . Now, however, t h i s uncertainty can be resolved because, unless Q z = O (A3.S3) the f u n c t i o n 8^(x) v i o l a t e s the c o n d i t i o n f o r uniform v a l i d i t y i n x. When e 2 = 0 we have 273 A3.25 i/ ^3 = O • 0*3.84) Equation A3.81 can, i n t h i s case, be w r i t t e n : and, since $Q,3-|,$2 and H.* = £f>3 are a l l constant constant: . (fB.sO I f we now assume that 8 and 6 ^ are constant;, 3 then from Equation A3.85: (see Equations A3.170 and A3.211). The remaining secular term i n v^g(t) i s generated by the expressions i n Equation A3.77 and 16 i n Equation A3.78. I f we assume t h a t a_ i s constant (see Equation b A3.167), t h i s secular term must e l i m i n a t e the quantity - e ( 3 t a g ) i n 5 27k A3.26 the expansion of V c ( t ) . By the usual argument O From Equations A3.9 and A3.11 we then have: (page 4 . 1 5 ) : , (A3.S8) and, a f t e r s u b s t i t u t i n g f o r B from Equation 4 . 8 1 , Equation A3.87 takes Q the form 31 + 2ZS" S12-C1 + Note that Equation 4.81 f o r 3 Q 0 i s now known to be v a l i d (since = 0). and consequently the s p e c i f i e d value o f 3 can l e g i t i m a t e l y be used Q i n the a n a l y s i s . Before proceeding with the s o l u t i o n o f Equations A3.22 and A3.23 f o r u ^ and v ^ , i t w i l l be h e l p f u l to evaluate the nonlinear functions and e' C3u.v^ 2_ in detail. 2-Cl-fO (which i s rather tedious) i s only continued to 0 ( e ) . 1 4 This process 275 A3.27 We have, from Equations 4.27 and 4.28: - e*^ + fe'^ 4- e*v f + 6*v .... 8 . («»,iO Some e f f o r t can be avoided i f the c a n c e l l a t i o n of secular terms i n v.-(t) o i s taken i n t o c o n s i d e r a t i o n . In t h i s case the quantity - e ^ t o u ) is omitted from the expansion f o r V g ( t ) , and corresponding secular terms i n v^ .j(t) + are ignored. The p r a c t i c a l consequence of t h i s s i m p l i f i c a t i o n i s t h a t , instead of taking v 5 = 2 ^ 5 ^ + ^ + y - itcL, when expanding the nonlinear terms, we use and modify the terms Vg, v ^ , Vg, e t c . to exclude the secular c o n t r i b u t i o n . Equation A3.92 can only s t r i c t l y be j u s t i f i e d up to a c e r t a i n order, and to be precise we should w r i t e , f o r example: L « J 1 '2. 3* J 276 A3.28 From Equation 4.69 u (t) = v (t) 7 7 = 0, so that we have: Each term in these equations must now be evaluated. Equation For u ^ ( t ) , from 4.25: and we obtain Now, from Equations 4.52 and 4.75: and therefore The expansion f o r 6 does not s i m p l i f y (because Equation A3.86 cannot, at present, be j u s t i f i e d ) , and so 277 A3.29 A l s o , from Equations A3.98 and A3.100: d.f> = £<*xf>o * + 6 * [ ,4 i/J,u '' 3/*o] + + 8T 0 .( *^ & • CA3.ID2) When G = 0, Equations A3.67 and A3.68 f o r s i n ( t + 9 ) and c o s ( t + 6 ) reduce 2 to i^Ct-h9y = cas9 si^b cos cos & cost - e 0 0 9^ + z 6? CPS0 c o s t + O 3 to$9 0 s^fc O f f c O tXfc*-") (/u.ioj) , From Equation A3.104 we therefore have which may be w r i t t e n c^Cfc + e) > _L ( i+• coZb") - e ^ s ^ z b -t- oft } 4 (AS.IOO 2 since cos 6 = 1 (see Equation 4 . 7 8 ) . Q We are now i n a p o s i t i o n to expand Equation A3.97, and the r e s u l t i n g expression i s given below. 278 A3.30 + fc" ^ 4- et, " + 1 + 1 + Oft*) / 3 ^ ^ ( l f o o Z t ) - 4- * f i t n 9 t t 0 cosh . J CflrioO From Equation 4.60 f o r u (.t) we have: g 2 and /*|Cl-^i - /3<to(b + e) ^ j" 2 Cot - Cco^b - 1 j , (ft 3.107) using Equations A3.98, A3.100 and A3.104 t h i s can be w r i t t e n 2 n.j l i j = 2 . ^ , 0 - / 0 ^ / 3 «r>9 £ - 1 + 1 c o b - coj2fc + ± cos 3 b 0 5 0 £ 2 * , | 2oob - Qo2tr - 1 ^ + / S , & o £ ^_ 0 " ^ ] + & The remaining terms i n Equation A3.94 can r e a d i l y be evaluated, and from Equations 4.60, 4 . 8 3 , A3.100 and A3.104 we o b t a i n : 279 A3.31 ltvr>4t (fl-3.|0<j) Crfo^t +• Z £ c o 4-b 64- (A3.\io) u.. 2- 1^ 87 64 - 7 coot +7. " 5 - 4 - With the s i m p l i f i e d expression f o r v ^ s ^ t + p ) + 4-^^s-^t+e; + ^ c (see Equation A3.92) , and from Equation A3.104: because cos 8 ^ = 1 . The expansion f o r y i s of the form (A*.!! *) - so t h a t , from Equations A3.100 and A3.103, we can w r i t e : +9) + ^ ^ [ ^>^ = + ^(Zo^o /V> + cos 9 / V ^ > j cos-B s^t 0 (03.114-) 280 A3.32 Equation A 3 . I l l can now be expanded as V^ 1 - 2^0-co zt) 5 + t+fiofa 4 / 5 ^ , 0 -CP52t) + 6 4. .cos 9 D s^t ^|/3a^, + + / } , ^ ^ CPsPoff^t + 4 £' When the secular term i n Equation 4.61 i s neglected (see the d i s c u s s i o n on page A3.27 ) we have: $ - and consequently = 2 t Zb - 4-s;~b (A 3. H o ) A f t e r s u b s t i t u t i n g f o r 3 , sin(.t+6) and y from Equations A3.100, A3.103 and A3.113, the expansion of 2 v v 5 g can be w r i t t e n : 1L s ^ l b - 4 j ; ^ b ^ - 4 t o s 9 | 4 - - J l / +e 0 o c o b -4-oo"2.fc +_Mco.?3 S 5] 281 A3.33 4- £ (A l. u?) When evaluating the q u a n t i t i e s Vg and 2VgVg, the reduced form of and Vg('t) must be used (see page A3.27). Vg(t) Apart from t h i s c o n s i d e r a t i o n , the remaining terms i n Equation A3.94 are r e a d i l y evaluated, and from Equations 4 . 6 1 , A3.100, A3.103 and 4.84 we o b t a i n : "4S lV V $ g - u_ c o t - ScsaZfc •/*,r«-/*0 + 11 c o 3 t Cs>l - iii c» 4 t CA3-H8) (A3.\n) For the product u^v^: U. \/ s S = £ 2.^-^SecoCt+p) J j 2^}<cCk+P) + which can be developed as From Equations A3.102 and A3.103: (A3.no ) 282 A3.34 and also 2±y e*,tfo + ^ ° < ^ ° ]J + e\*,^r, + (*3-*3) 3 (see Equations A3.98 and A3.113). From Equations A3.103 and A3.104 f o r sin(t+9) and cos(t+9) we obtain s u t ( t + 8>c©s(t + e ) and t h e r e f o r e , - e j. s*ib = e 9.c^Zt 3 4- + (/ij a t ) OfeO using Equation A3.101: * ^/s,*" + 2 /*<»/V ^ b - e3 £ z s-zfc s + 2^ X 0 0 3 <*>zbJ + oCfeO The f i n a l term i n Equation A3.121 can be w r i t t e n - e 5 where 3 , cos(t+9) and y are given i n Equations A3.100, A3.104 and A3.113. When the reduced forms of v , v^ and v 5 ing terms i n Equation A3.95 take the form: g are used, the remain- 283 A3.35 n> c<nQo | II 5 « t - 2s:«.2b + i i sc«.3t^ i ifc It J (A 3.ix?) D 1 Jkrab - GoJU: - 1 ' *1>4 4 £7 s ~ b - 4 2 sCv2.fc + W s 8 L [ 32. 3 : I *.W3b - J i ;~4-b j bk bit 32. 32. J s sc~2.b + j £ $^.A-b M- I T ' e -+- O ^ ) J M- (A*.1*1) (A*.r&o) ( A 3 . 13-1) ] 28^ A3.36 We are now in a p o s i t i o n to assemble the various terms i n 2 2 Equations A3.92, A3.93 and A3.94. The r e s u l t i n g expansions f o r e u , 2 2 2 e v and e uv are given below. 4of,' - 1 - Zp,(l-fO^ £ I - 3^ cob + toJ-t-.1 c^>3b ^ ( -t — otto i, t- •*- t » <"-U — Cao > i 13 ZfS' ~ l t- e ^to5 |l-3 0 + c^Zt- - ± to 3b ^ + 4 p , ( l - fO<*-, f ^b-<*>2.fc - 285 A3.37 3 Zp/l-fO + p c ,9 >0 er c> j 2^,0.-^0 ^ 4 - ii. c*>b - 4 J i si«.ib felt 1 -t- JL c o 3 b ^ (A3.»33) -4s«.b? Ca b - S c < o 2 . t -t- j | c o 3 f c - I?-] Co 4 b 7 2. 13 siwZb — 4 s:»vt - 2/*,(l-/0 f e*>9 x 0 3 £ A- -JL <vot - 4 <*>ab +. - 3_ 2- Xb - \± ^3b-^ is o o 4 b ^ J6 J -4- . 0 ^ * ) 286 A3.38 (i 0 5 .+ p y ;^t o o coPoCeot +6 +fi^-fO - 1 « . ~ Z b -t- i i s ^ n ^ ^ 11 ««-Zb-gsiwt ^ - ^ 5 , c o P | Jl 0 (h4. •+ p C\-p _') x l 7 L 5 7JJ- s;«.b - 43 s « 2 b +• xj $ w3b - Ji ; ~ 4 t " £ L s & s xtJ t b -2«;_x.b 4- Ji s^-ib ^ + /3^<ic9 ^ s ^ b - Z s ^ Z b + 5Uv3b 0 L i'*4) I- 3 Z b^t- ^ s - ^ 5 Zcob- c o l t - i j i - 4- *>*- 64- J 287 A3.39 The nonlinear functions and can now be evaluated f o r a s p e c i f i c order i n e when they are r e q u i r e d . A f t e r t h i s necessary, but lengthy, d i g r e s s i o n , we return to the a n a l y s i s of Equations A3.22 and A3.23 f o r u ^ and v ^ . From Appendix I. Equations A l . 8 and A l . 9 : CA3.I3S) (fl3.'30 ^ OCi> Now s i m i l a r terms also appear i n the expressions f o r E-| » F 2 F 1 3 (see pages A1.4 to A 1 . 6 ) , 1 S U.,- = ' E 13 a n d o i t w i l l be expedient to e v a l u a t e , i n a d d i t i o n , the c o n t r i b u t i o n s 0(e) and 0 ( e ) A3.136. 1 2 from Equations A3.135 and We have, from Equations 4 . 2 5 , A3.98 and A3.104: £. 2erf, - CA3-I3?") (A3.I3S) Note, i n c i d e n t a l l y , t h a t ' t h e complete form of v ( t ) g i s used here, i n contrast to the reduced version given by Equation A3.92. elementary manipulation: A f t e r some 288 A3.40 r -€ 1 $ p <^>9 ^ i 0 ] 3^,«<o^ (l -3coZk) ~ I px^e* + 6 V tot, c o z t ^ot,- + = ^ 4oP 0 ] ^ 0 ^ 5j^b-3 s ; ^3b"^ C v-3c«Xb') + + 6 s;«.Xb = - 2- tio^o | s^b+ «c*-3k ^ X I ] ] 289 A3.41. From these equations we obtain d i r e c t l y : 2- The nonlinear functions can be evaluated from Equations A3.132 to A3.134. A f t e r some rearrangement, the nonlinear terms may be expressed i n the f o l l o w i n g form: It 290 A3.42 General expressions f o r the q u a n t i t i e s -2 - -2h>j£s . + a_u LdbSV at J a r e 5 £jf-s - ^fs Ibtbv and bV J given i n Equations A3.28 and A3.29, but at t h i s stage of the development Equations A3.30 and A3.31 are no longer adequate. now know that | | = 0(e ), § £ = 0(e ), 5 6 = 6 Q + 0(e ). | * + 0(e ), 5 5 | | = 0(e ) We and 3 With t h i s value of 8, equations A3.14 and A3.15 can be 3 revised so that a^'C e*9y Acos C + -*• ••• ) • ) = e = 1 % 0 3 ^ +• OC «) . +'e* + £ fe 0 s oC**> s CA3.IV?) (A?.UtS) Consequently Equations A3.16 and A3.17 take the f o l l o w i n g form: since Hv sine Q = 0 (see Equation 4 . 7 8 ) , and we can w r i t e ' ar av t j r ^ J (fl3.i5i ) - (i.h*2* - AcQ+Zjj -fi, * 3£J \ 9 (. " ^ r ' ar t- atr *6*j/3„<^*.+ L a av • ' ar / av 5 -• • . + air J aV [_ u . aV ^12? \to 6 + a*c at 0 Oft') av j ' ' , <>vj .(fl3.'J2"> 291 A3.43 where Equations A3.151 A3.30 and A3.31. and A3.152 are the revised form of Equations From Equations A3.28 and A3.29, and using the above expressions: d U5 -% — is J - 6b ^ O i y -Z .b L ab«?r {' ix iv i r 1 - i v S t*>® df> 2 0 s ar Cost 2V In a d d i t i o n to the o r d e r r e l a t i o n s h i p s given on page A3.42. we have 3V= 0(e ), = 0(e ), 5 tly' 5 at" and £r = 0(e ) 5 at t l y (from Equations A3.34 and A3.35) 1 fu* at* and and [*aV = 0(e ). 5 at* 3V at S Consequen- are both 0(e ), 2 so that From the expansions of db L iv-_ given i n Equations A3.69 and A3.70, and with 8 = 0 (see Equation A 3 . 8 3 ) , i t f o l l o w s that 2 1 <^ dVj ^b 1 <h x ^b J / , V > L 0 +- c)u 5b 5 - era ' 292 A3.44 With the order r e l a t i o n s h i p s given immediately above, together with those on page A3.42, none of the expressions defined by Equations A3.74 to A3.76 has any c o n t r i b u t i o n at 0(e) and A3.23. or 0(e ) i n Equations A3.22 There i s , s i m i l a r l y , no c o n t r i b u t i o n from terms i n v o l v i n g the d e r i v a t i v e s of Ug, Vg, Uy and v^ (see Equations 4.55, 4.56 and 4.69), A f t e r d i f f e r e n t i a t i n g Equations 4.83 and 4.84 f o r u ( t ) g and V g ( t ) , we o b t a i n if- Zi + _5_ ts^fc £ 64- bb fO^ if- j" ~3_ s;^k + I3S Co 3b 64- + WHS s»\v3.b bb A l s o , from Equations 4.25 and 4.26 f o r Ug and v ^ : = p> c^> ( b + 6 0 (A3. 16 o) <)b^ .1- a_v - 2/b J<«-(b + P ") 5 and t h e r e f o r e , using the expansions of sine and cose i n Equations A3.149 and A3.150: cos b b_U$ (A.3.lfrl) bt x aS/ s ^ - 2 c o 0 J * /s d 0 +• e * A I x s;^.b + o (A3. 143^ 293 A3.45 The non-homogeneous terms i n Equations A3.22 and A3.23, which we can now accumulate, are given below. 4- L a^ o - / o 8 at? a^ 3 5 2 L Cft3.ifc'5). at I av a IT av j 3s. 29k . A3.46 <*3 p fb c^9 x 0 - 0 us /i 0 + l When ctg s a t i s f i e s Equation 4.82, - . i l / t t C l + w,)^ ( i - / x . ^ ) 4 2- / t | . W - / O the two terms X 5C«. 3 t 6( 1 + j O *~°i- 3 and i n Equation A3.164 cancel (see also page A3.15 f o r a s i m i l a r r e s u l t concerning a-j), which s i m p l i f i e s the expression f o r E* s l i g h t l y . 2 I f no term i n v o l v i n g t i s to appear i n v ^ ( t ) , then from Equation A 3 . 9 : Now (from Equation 4.58) and consequently Equation A3.166 s i m p l i f i e s so that av which j u s t i f i e s the assumption made i n the d e r i v a t i o n of Equation A3.89. To e l i m i n a t e t s i n t and t c o s t terms from u-j~|(t) and v . ^ ( t ) : . 295 A3.47 32- From Equation A3.168, 3g(x) w i l l v i o l a t e the condition f o r uniform v a l i d i t y i n x unless 9 ^ i s zero, so we have 6> - 3 Chz.no) o - o. (A3.17-0 When 6 ^ i s constant, Equation A3.53 i s v a l i d and we can l e g i t i m a t e l y use the s p e c i f i e d value of 3 , . The term 1 i n Equation A3.169 i s now oX z e r o , and a f t e r some manipulation we can w r i t e tO* £ p Note that cos 9 N >4j (i+p.r*- = 1, so that u 4.81). of +• \->s a ' ? cosB = H a 1 6 = u, / c '~ 0+ o (A3. . \z (see Equation A l l the terms i n Equation A3.172, with the exception at present sL©5 , are constant (see Equations A3.50 and A 3 . 8 6 ) , and consequently d©5 - constant (A3.t?3) 296 A3.48 I f we also assume that and 6^ are constant: sec 9, (ft 3.174 (see Equations A3.211 and A3.244). From Equation A 3 . 9 , one secula r term i n v ^ ( t ) i s generated by the expression I (>-f J) 1 6 -J ^- 3 We have, from Equation A3.169: -t 13-oi, and when p coiB ) 0 - 2^,0-^) i t f o l l o w s that (A3.(?S) (see Equation A3.53), so that the s e c u l a r c o n t r i b u t i o n . c a n be w r i t t e n ) 297 A3.49 The remaining secular terms i n -]](t) r e s u l t from q u a n t i t i e s i n Equation v A3.165 which involve s i n 2 t and s i n 3 t (see Equation A3.11). Note, however, that by Equation A3.175 the c o e f f i c i e n t of s i n 2 t i n Equation A3.165 i s zero. Consequently, i f we assume that ag i s constant (see equation A3.208) then to cancel the term -e (3tag) 6 in v (t): 5 (A 3 . i ? 0 As u s u a l , we take -which leads to the f o l l o w i n g value o f SfjL,( | _ fO^" [ l < ? + ^ 3 ^ , ] - CA3.IT8) The functions Ug(t) and Vg(t) can now be evaluated, since the values o f 0-j, 6 2 and 8 3 are known. c o n s i d e r a b l y , and we then have Equations A3.42 and A3.43 s i m p l i f y 298 A3.50 Non-homogeneous s o l u t i o n s corresponding to these functions can r e a d i l y be derived from Equations A3.10 and A 3 . 1 1 , and when the secular term i s included ( c f . page A3.17) u ( t ) g and v ( t ) g + take the f o l l o w i n g form: vsb 6± - 6 \± Cos, (A3.(61 > 2-k 6 £ 2 s* 3 Xb + XI b I (A3.ISO Next we consider the s o l u t i o n s of Equations A3.24 and A3.25 f o r u.j2 and v - ^ . From Equations A1.8 and A1.9 i n Appendix I : '7-S (to3tr 6A- ' + i»9 3|5 ^Sfc- (A3-lS3^ 1-2.8 i- JS si^t 64- + IOS ; ^ 3 b is-S S -+ ?rS <c Sb ra-S CA3.I24-) H Solutions f o r Ug(t) and Vg(t) are given on page 4.15, and a f t e r some manipulation we o b t a i n : H,A 6 ' 3<*>Zb) - 1 CA3-I85) 3V. <^.xt • /6 i6 4- 299 C 6 •A I - "3 Ccr> 2 - 0 «» - A3.51 3 u . , S<«-2b 8 (A3.I86) 4- /6. From Equations A3.139 to A3.142, the c o n t r i b u t i o n s i n v o l v i n g u^ and v^ can be w r i t t e n : 2b^ - 3v, -2.1 } (f\3.lg?) Oft) £ 6 +£<^ab - il c ^ t +3<o>3t + 2- ^ , Cl- 3 + -M-.CI- /O 2. IS_ tS i - 2-b J \ c n i t l (A3.I88) ? Equations A3.183 and A3.184 can now be evaluated, and we have: 96 7£ 4 cob + ]75 cr>3t 64- i5 64/ J6 p.^i Cfl3.i81) + ioS iaS 3Z + 3i£ <v»Sb l>8 t3.8 t i t 2 3-2. s ^ b + 3jj> b (A3.l<*o) U S rV2Tr Cl- 3c«2.t) 2. ' Note the c a n c e l l a t i o n of secular terms i n v o l v i n g t s i n 2 t and t c o s 2 t , which appear i n Equations A3.185 to A3.188. A f t e r some a l g e b r a i c manipulation the nonlinear functions can be 300 A3.52 arranged i n the f o l l o w i n g form: II 5x (A3. Ill") [ Zfc sew Z t (see Equations A3.132 to A3.134). The q u a n t i t i e s -zf^jfs - "1 |_atav J and - 2 T i*y> +^5*1 Lata* ar J oCfc ) 6 oCe ) i n Equations A3.24 and A3.25 can be determined from Equations A3.28 and A3.29, using Equations A3.151 and A3.152. We then have: -Z oCe ) 6 + Zc*o© ( I ' br dv J 6 301 A3.53 [ S A , a P +/>,iJ£f + ^ ^ 7 ^c L >v ' dx bxr S - x a v/g + }tdx S£ t Tx (ft 1 .ISO dx because, from Equation A3.170, 6 ^ = 0. At present the d e r i v a t i v e s o f a , B and y with respect to T are all 0(e ), |^-=0(e ), 6 4 ^9 = 0(e ) and 6 = 9 + 0 ( e ) . 6 A3.34 and A3.35 i t f o l l o w s that $j±_s and ^y_s From Equations 4 n are 0 ( e ) , and 6 consequently =. o dx* _ OCeO With 2 9 = 9 3 = °' E < l u a t i° n s A3.69 and A3.70 s i m p l i f y c o n s i d e r a b l y , and we o b t a i n : L at 1 at J (p^*) COS OCe*) None of the q u a n t i t i e s defined i n Equations A3.74 to A3.76 2 3 c o n t r i b u t e to Equations A3.24 and A3.25 at 0 ( e ) or 0 ( e ) . The remaining non-zero terms i n Equations A3.24 and A3.25 are given below. From Equations A3.181 and A3.182 f o r u ( t ) and v ( t ) : g g 302 A3.54 (A3.119) • at" 1 '<Jb . L 6 (A3. 3 I l l ) From Equations A3.162 and A3.163: cofi(, c o t •2/3, e o P 0 (A3. *-.t 2O0) (A3. From Equations 4.60 and 4.61 f o r Ug(t) and Vg(t) = piC'-/oF 4t<olt C » - 4 - s u v t -2<uot - ] (As. J i ^ Z b J XOTL) (A3.-3.03) and t h e r e f o r e , using Equation A3.53 f o r B^COSOQ: cn (A3- ze>4-) It aV at* (AS.ZoS) We are now i n a p o s i t i o n to assemble the non-homogeneous terms i n Equations A3.24 and A3.25. E* and F* then take the form: 768 *>r 1 ar J ('-^O J 303 A3.55 + cot £' a*" ' ar ' a v J 6^- - OS 3• -j**.") fco3fc 13.8 2- Sot t •+ sovcb r ** <?v a*p" J £4- C -ro aV (j 17-S - r o 4- 3Z Secular terms i n v o l v i n g t are e l i m i n a t e d from v - ^ t ) if (A1.208) so that equation A3.178 f o r a and t c o s t from u ^ and v - ^ : g is valid. To remove terms c o n t a i n i n g t s i n t 3014. A3.56 *- ' air From Equation A3.209, i f 9 d~c 4 J i s not zero 3g(x) v i o l a t e s the c o n d i t i o n f o r uniform v a l i d i t y i n x, and therefore ^4 - Now when 6 (A-i.ait) ° 4 = 0, we obtain from Equation A3.85: i n which case Equation A3.210 can be w r i t t e n 'A-. Iff. / « , 0 - / O -+ '£L K . ^ f O ^ - j O =o V 64 From Equations A3.84 and A3.173 both 3. and 4 3_f ar 5 are constant, so that 305 A3.57 SSfc at? = constant . (A3.2IS } I f we now assume that e 5 and e 6M b g are constant, then from Equation A3.214: + 105 ( i - - f O ~ —. sec 9 (A3.lib) r s i« ( n- JJ. , ) The quantity - 3 l' av ' J 64- 3 - z ?6g a-rr which generates a secular term i n v - j ( t ) , 2 can be s i m p l i f i e d using Equation A3.214, so that the secular c o n t r i b u t i o n takes the form [ fa. 3S43S4- 32- .0.-/0 ~ A d d i t i o n a l secular terms in v-^Ct) r e s u l t from those expressions i n Equation A3.207 which contain s i n 2 t , s i n 3 t , s i n 4 t and s i n 5 t Equation A3.11). (see The combination of these s e c u l a r q u a n t i t i e s must e l i m i n a t e the term -e^(3tay) i n V g ( t ) , and t h e r e f o r e : 4 We t a k e , as before, v* (A-s.ai? ) 306 A3.58 and if i s assumed to be constant (see Equation A3.240), from Equation A3.217 i t f o l l o w s that i X f 0+|<) + ££ Yi + t ix- 64- 3£ (A3.2l<0 Now that the values of ^ to evaluate u ^ ( t ) and v ^ ( t ) . a n d 6^ are known, we are i n a p o s i t i o n Equations A3.77 and A3.78 can be w r i t t e n i n the reduced form: <Vo2-t +• 3f p,(i~pj) Cr>A-b ( A 3 . 2 X O ) 16 0 -pO ^ and TT^o 8 from Equations A3.8 to A3.11 u ' 1 Q 6 ( t ) and v-|g(t) can therefore be derived as: fl (l-ftj.^ t L 8 s-ia-Ci+jJ.) J 37 + 6 7 S ^t, 3a cob 307 A3.59 - i fr3 s^A-b L - 3 ( s^.b) b - - 64- + X-zS fi, SixCl-t- u,) ^ 4-S If. 0 J (fl5 ( V c o b ) .«-3) 0-/**) Note that secular terms i n v o l v i n g t are eliminated i n the complete s o l u t i o n f o r ev(t) when s a t i s f i e s Equation A3.89. We c o n s i d e r , f i n a l l y , the s o l u t i o n of Equations A3.26 and A3.27 f o r u-j2 and v-jg. The functions Uy(t) and v^(t) are both z e r o , so that from Equations A1.8 and A1.9 we have - u,(l - fO 'IS | f I ' 1 <»k + l l c « 3 b I L « e J - V 5 J L s^2-b? Now U r - p e 9 Cob s >0 tr> 0 N/ = 2y3„ Co0 s~b & o •+ O f t ) + y 0 + oCe^> .', 3_ s . v b +• *5 s ; * 4- 5C^.3b ( A 3 , aa-S) 308 A3.60 and consequently: 1 tot +• .3 5 4 3_ ;~b + to?t s 4 « t t if sc^3b 4- 0 Csp >x 0 if 8 + i f 5J«.3b 4 6 co 3-t - if <<o4k « (A3.1Z6 ) — - [ -y 0 & £,•», c _ 9 1 tot - l£ < o 3 t 4 4- I? [" 2. c ^ b - if c*>3b [_ A4 Because 3 and V 2 - if (A3.Z2>) j;^4k 8 = 0 (see'Equation A3.213) Equations A3.139 to A3.142 s i m p l i f y , for E^'and = si^Zfc e>Ci> we therefore obtain p / ' - f O - j /3«,<«'Po| l l + feCeoXb - If Ceo 4bJ •*• jTo ^ 1. Si-b + is j . - ^ t j 2L = - y , ( l - ^ D c ^ 9 0 2 7 si^ab - i f s ^ 4 b | + y e £ 3 e ^ k - IS «to Sbj ^ CA». 2.3-1) From Equations A3.132 to A3.134, a f t e r some manipulation the nonlinear c o n t r i b u t i o n to Equations A3.26 and A3.27 can be w r i t t e n as: 3x ' ^ 309 A3.61 . 4- CA3.230) [i lit CA3 . 23\^) 64 8 For the expressions • -2 | d "-s |_ d t a r ^ s l and - 2 J }L?S + ^-s , we have, using Equations A3.151 and A3.152: it7r -2. + L ;>k3 tatr X t)/3 ix ar J QCtT) ? ceoPp s ^ t + >x 2&9 f 0 A bB? 0 ( A 3 . 2.32.) 1-/3,^6 "I ^st 2 a*,. atr ( A - i .133) CoP„ cost , 2 •tr 310 A3.62 since $ 2 = 9 4 = 0 The d e r i v a t i v e s o f a , 8 and y with respect to T a r e , at t h i s stage of the a n a l y s i s , a l l 0 ( e ) , | ^ = 0 ( e ) , 7 5 $9 = 0 ( e ) and 6 = e_ + 0( 7 (see Equations A3.208, A3.211, A3.212, A3.215 and A3.218). the q u a n t i t i e s v z 3_!f5 and a_v> It*- bX*- Consequently 5 do not contribute a t 0(e ) to Equations A3.26 and A3.27. I t i s necessary to extend equations A3.69 and A3.70 to include terms 0 ( e ) , but as 9^ = 5 = 9g = = 0 t h i s presents no d i f f i c u l t y , -and we o b t a i n : I a t *' at j a_y + h± c Lafat J s 5 There i s no c o n t r i b u t i o n , e i t h e r at 0 ( e ) o r 0 ( e ) , from the expressions 4 3 defined i n Equations A3.74 to A3.76, and f o r the remaining non-zero terms i n Equations A3.26 and A3.27 we have: dj* to at' - a*, at /-",(<-r 3- V-3.Z5 p , ? tost + f + 2 i l cos4t 64- + 3^ [ X - .e^-b 1 -f- -45 j< , 6 - j O - p., coj? 2.t S T3 + 6 ? £ ^ _ L 64- 2o s ^ t 7 S-ia-Ci^p.) J (A3.a34) 311 A3.63 - + ^,0-/0* - 3 ^ s u l : f ^. 4- za-g'/x, - 4-5 ^ . Q - f Q y ?-5^b -'/i!> e to 1 CA3.23?-) 5 + 16 The non-homogeneous terms i n Equations A3.26 and A3.27, which we can now assemble, take the f o l l o w i n g form: - 6 t ^ , . ' H . a v }Xr _ 3 _ f / 4 . / i 3 + ^ 3 * " ^ ] f'l-^,^ - p/'-fO/* + 64- J 1 3 8. 2 •+ «o b •'+11 (*,yx^« + 75 su-3fc •+ cos4b r *5 j « . & - f O ' / v * 0 . - + i i i p / ' + f O ' ^ ' - r O 64- 31 Z 3 * 3o£> -t-S^b (.' a r <ar J 3 CR3.2.38) 312 A3.64 - <vob - , - s;».zb * 64 L 4- g 2 I f no term i n v o l v i n g t i s to appear i n v 1 3 (t), then from Equation A 3 . 9 : ^ ? = O Cfl3.Z4o^ , ar and therefore Equation A3.219 is v a l i d . Terms containing t s i n t and * c o s t are eliminated from u-j^ and v - ^ when: a* it t ar ar Although i t i s not required i n the present a n a l y s i s , i n p r a c t i c e we must take Y Q = 0. The reason f o r t h i s choice i s discussed i n Chapter 4 (on page 4.22 ). With t h i s value f o r y n we then have: 313 A3.65 Unless 0g i s z e r o , B^CT) w i l l v i o l a t e the requirement of uniform v a l i d i t y i n x , and consequently Zfe ~ O ( A » . * « ) . Now that 6g i s known to be constant, Equation A3.174 i s v a l i d , and the s p e c i f i e d value of 8 3 can l e g i t i m a t e l y be used i n the a n a l y s i s . Equation A3.215 - From i s constant, and therefore using Equation A3.242: constant . fjnW) at A l s o , i f we assume that both 6 g and Qj are constant, sec 9 Q then . -(A3.24-?-) o The secular term i n v ( t ) must e l i m i n a t e the q u a n t i t y -e 1 3 (3ta ) g i n v ( t ) , and from Equations A3.9 and A3.11 we then have: 5 + W } - By the usual argument: C jr.^*^.^] .- ^ia> v 3 * 8 ( A * . * * * ) 3114A3.66 and, assuming ctg i s constant, from Equations 4.81 and A3.174 f o r 3Q and $ 2 we o b t a i n : 0+ j*,)*" ( ~ / O ifj l pi + -| 2-175 4 - 2.0 25 p., ( (A3. 2-So ) where Y Q i s included to f a c i l i t a t e Equations A3.89, A 3 . 1 7 8 This completes solutions for ^'(t) Chapter 4 . and comparison with e a r l i e r r e s u l t s (see A 3 . 2 1 9 ) . the general a n a l y s i s of 3-body motion, although and r ^ U ) derived from these r e s u l t s are given i n 315 A4.1 APPENDIX IV — VARIATIONAL ORBITS IN THE RESTRICTED PROBLEM We are concerned here with the homogeneous and non-homogeneous s o l u t i o n s of Equations 5.23 and 5.24, which define v a r i a t i o n a l motion about the f i v e e q u i l i b r i u m points of the r e s t r i c t e d problem. For the non- homogeneous s o l u t i o n s i t i s necessary to consider motion near the c o l l i n e a r points separately from that near the e q u i l a t e r a l t r i a n g l e p o i n t s . The present a n a l y s i s , however, i s l i m i t e d to the i n v e s t i g a t i o n of motion near the e q u i l a t e r a l t r i a n g l e p o i n t s , so non-homogeneous s o l u t i o n s f o r the c o l l i n e a r case are omitted. A4.1 The Homogeneous S o l u t i o n I f we take x =x-j, x = x , y = x 2 3 and y = x^, then (neglecting the non-homogeneous terms) equations 5.23 and 5.24 can be w r i t t e n i n the form 3c = A (A 4. 0 Here x = (x-j, x , x^, x ^ ) ^ , and the matrix A i s given by 2 A o I "°. o •o. o o -z o o 2. 0 i O ( A4. % ) 316 A4.2 Taking the Laplace transform o f Equation A4.1 we obtain ( A 4 . 3) where I denotes the u n i t m a t r i x . From Equation A4.2: and i f we now define then Equation A4.4 can be w r i t t e n deb ( s i - A ) = r - + 2-X, r 2 + x (A4.6) z where (A4.7) (A4.g) From Equation A4.6 we therefore - have deb C s l - f l ) = j s V X , + A, - ( A * - \ ) ^ A f t e r some manipulation the f o l l o w i n g r e s u l t can be obtained: (A4.<0 317 A4.3 (si-A) —i C 3K ( A 4 . lo^) <Lefc ( s I - A ) where C denotes the matrix: 2s-u, 20,-o^s s -*- o. 1 Note that the q u a n t i t i e s £^ and and depend on the p a r t i c u l a r values of ( c f . E q u a t i o n s 5.20, 5.21 and 5.22 on page 5.6 ). « • n c - - • - HUH ICOOI ... u u i — J. J u i . -I o i,ci i o i u i ... i u O .!. I. - - ~ . . J o n e c i { u i T - J. _ T IUICIUI _ ^ _ ... _ T Ci _ i c i n y i c ~> — — ,~ J - — i m , o and 1_ (see Figure 3-2 on page 3 . 1 8 ) , f o r which 5 (_A4. i z ) 2, yr/ - 1 L2> L7- Note, a l s o , that ^3 = 1 + 5.21 and 5.22: o, - — ' - K - k, z 2 at L Y at L C (04.14-) In t h i s case we have, from Equations 5.20, 318 A4.4 at L : 4 3L and at L^: 2. From Equations A4.7 and A4.8 i t f o l l o w s that \ = V and 1' + k 0 + K (A4.n) = (l-^o)'' - * k / l - k ) 0 + 3 1 ^ , 0 - / 0 (A4.ZO) consequently \ x - \ x = A-C^+O + k, " - 3 k ^ J i 7 r , _ ) . Note that Equations A4.19 and A4.20 are v a l i d both at (A4.20 and L g . The character of the homogeneous s o l u t i o n i s determined by Equation A4.20, and bounded motion only o c c u r s , i n g e n e r a l , when 4(k -rk,) 0 + k,- - S k ^ j ^ f i ' / O 2 > r«4.ix) O. I t i s possible f o r bounded motion to take place when Equation A4.21 not s a t i s f i e d , but only f o r s p e c i f i c i n i t i a l c o n d i t i o n s . is We now r e s t r i c t the s o l u t i o n f u r t h e r to those cases f o r which Equation A4.21 i s valid, 319 A4.5 which implies that > - _L j - 4- C i< + k, > i e "5 or, k alternatively "7 J - < j_ - -J 2- since u 11- 3 k ^ VJ3 requires that u ^ 0 - 5 - 2 2 I f the constants kg and k-j corres- pond to a conventional binomial expansion, then kg = 1 and k-j = - 3 / 2 . Equation A4.23 can then be w r i t t e n * it 1 - (r,)' 1 which i s i n agreement with the value given i n Szebehely [1 19 ]. Taking the inverse Laplace transform of Equation A4.3 we o b t a i n , using Equation A 4 . l l . : • Cb^ - C05 co = s fc si*. u>< b Zo^Tcto^ + (oi^-O^ £Co"} + 20^ LJCO } + 0 ^ <j<o) - 320 A4.6 + Cos i(fc) k - - s^i^ b s 2 0 , 3t£o"> - o^ito) 4- 2 0 ^ 3 ( 0 " ) -(- ( o , - ^ 7 + (o,-^ - - ^ y(o) to. + COS o><t + Sit tJ t - cos ojt u 2 0 , =c(o") - 0^ i C o " ^ + ZOa. LJCO") 0 ) 3(0) (A4.1?) where (ft4.ZS) I t i s more convenient, however, to express x ( t ) and y ( t ) i n the f o l l o w i n g form: CA4.3o) (04.31) where a , b , 9 and <>j are a r b i t r a r y constants. Q ,Q - , 6* and <j>* are defined below: The remaining q u a n t i t i e s 321 A4.7 Q = u [ 4-oJ- -»- 0? ] " (A4.33) ^ (A4.34) Note the symmetry between expressions associated with the s h o r t - p e r i o d (u> ) and long-period (a^) terms which i s revealed by t h i s f o r m u l a t i o n . s A4.2 Non-homogeneous Solutions The non-homogeneous form of the v a r i a t i o n a l equations can be written £ - 2 ^ +o,x + -O^tj = E * ( t ) y + Z i + t o ^ » F*Ct) 3 (ft4.36 ) > . (44.3?) I f we s e l e c t the vector B so that B = (0, E * , 0, F * ) , then Equations T A4.36 and A4.37 can be expressed as S = f\5c + 5 , CA4-36) 322 A4.8 . where the matrix A i s defined i n Equation A 4 . 2 . Taking the Laplace transform of Equation A4.38: 5L( ^ = S (sI-A) _ , S C o ) + ( s l - f l ) " ' ZCs) ( f t * . 3*0 and the non-homogeneous s o l u t i o n can consequently be determined from Equations A4.10 and A 4 . l l i f B(s) i s known. When E*(t) and F * ( t ) contain terms of the form s i n u t and cosoot, three d i s t i n c t cases are p o s s i b l e : f co f 1. w 2. w = oo 3. oo = co^ . s s u L ; ; These are considered separately i n the a n a l y s i s that f o l l o w s . A4.2.1 Non-resonant case When co f w f co , E*(t) and F * ( t ) can be w r i t t e n i n the general s L form £ * ( b ) = F * = Cf) C<„ /S 0 and we therefore 2» +. Ct, Swvtofc- -t /S, sf^tJb + + /J,. cx>5 t J t have sHiO 1 - '. o L , . cos o b S ^ + O *- , . (fl+**o) . ( A « r . <*i) 323 A4.9 B(s) i s now known (from Equations A4.42 and A4.43), and the nonhomogeneous s o l u t i o n may be.derived from Equations A4.10 and A 4 . l l as the inverse transform of ( s ! - A ) B ( s ) . _1 A f t e r some manipulation we o b t a i n : A 2 UJJ- ( toot, »2|Q + + •oCft.Q.,.-*^) 2/3, + cos o b t ^ — cx>s tob — SiVv u><- b ( CO*- (A*.44-) 32k A4.10 — Cos t->jt- — — 2-ot, StK-0>fc - to$ cob I ( co - O L -+ c i O x (w *- W" ) 1 s A4.2.2 -V 2-coot, x Resonances i n v o l v i n g s h o r t - p e r i o d terms For the second case, when co = to , E*(t) and F*(t) can be s expressed i n the f o l l o w i n g general form: F* Ct) = '3".**.o,b . + S", w s c j j b t (A4.A-?) 325 A4.ll. so that ( A4. The corresponding non-homogeneous s o l u t i o n of Equation A4.38 i s given below. MH — Siw-lO.fc to. - cos to. b t Si«- to-b Zto ( s - t cos (O b s to^-cojO 4- 2^,05*1 (A4, 5-o) 326 A4.12 NH COS t CoS«0, t tcc-5 ^ A4.2.3 t (fi4.Sl) Resonances i n v o l v i n g long-period terms When to = to^ we have: (A4.Sz) (A4.53) 327 A4.13 and therefore The non-homogeneous s o l u t i o n s can, i n f a c t , be obtained d i r e c t l y from Equations A4.50 and A4.51 by interchanging w and o ^ . s ? 2 that i n t h i s case (to (a£ - tto£) transforms to - ( t o s change. We then o b t a i n ; v - cos t ^ k 0^ e VZJ, 2 s Note, however, 2 - w ) a f t e r the L inter 328 A4.14 Cb) = " sew. cjj b- ^ L S^C^-O +*-y«>l - 2jr, V J u - cos <o s 11 C JJ^-«Q t + t cos to. b + °j-}f\ V° °'T 329 A5.1 APPENDIX V — SOLUTION OF THE FOUR-BODY PROBLEM Two r e l a t i v e l y lengthy d e r i v a t i o n s , associated with the expansion of nonlinear and d e r i v a t i v e terms r e s p e c t i v e l y , are given below i n Sections A5.1 and A 5 . 2 . In the f i n a l section we continue the a n a l y s i s of Chapter 5 to derive s o l u t i o n s f o r eu(t) and e v ( t ) that are uniformly v a l i d when t = 0 ( e ~ ^ ) and which define the p o s i t i o n of p^ to w i t h i n a 8 constant e r r o r 0(e ). A5.1 Expansion of the Nonlinear Terms For the function \r d | we have, using Equations 5.32 and 5.33: Uniformly v a l i d expressions f o r the motion o f p and p are given i n 2 3 Equations 4.101 to 4.104, so that 1. r 330 A5.2 I f Equations A5.2 and A5.3 are rewritten i n the f o l l o w i n g form: then we obtain Now because of the l o c a t i o n of and (the e q u i l a t e r a l triangle points): - C> ~ - • + = Cfx+Sj' + I?'' = In t h i s case Equations A5.6 and A5.7 can be w r i t t e n o r , more s i m p l y : • .. (AS.*) 331 A5.3 43 where . From equations A 5 . l l and A5.12 we then have the f o l l o w i n g general expansions of ' I r r 3 4 l | anrl and I series 3 P k o + k, ^ -t- Lr g t v + . •• (A*.iO ( c f . page 5.4), where the constant c o e f f i c i e n t s of expansion i n iuse. ( f o r i = 2,3) 0 . - * i * = - £) and (n. - n) can b e ' w r i t t e n as: " ^ 1 * 0 s ^ - 5 The expressions depend on the type - A O . \'.(AMI) (AS.H) (AS.**) 332 A5.4 and consequently the nonlinear terms i n Equations 5.36 and 5.37 take the form: + k. + k. (ftS.Zl) where we have made use of the c o n d i t i o n . (A5.0.3 ) i - which i s v a l i d at and L ^ . 333 A5.5 Equations A5.21 and A5.22 cannot, u n f o r t u n a t e l y , be used i n t h i s compact form, and a development i n terms of e i s now necessary. expressions f o r The , T^, A-| and A are more r e a d i l y handled i f we take 2 IO /•—J «--S A » « eu- + where e^(t) C 1 e ^ O ^ and f ^ ( t ) + OCe") (As.**) correspond to terms i n v o l v i n g e 1 i n Equations 4.101 and 4.102. From Equations A5.13, A5.14 and A5.23: Now i f i t i s assumed that eu = O(e^) and ev = O(e^) on pages 5.13 P, 1 - cV-- (see the d i s c u s s i o n and 5.14),-we obtain from Equations A5.24 to A5.29: 2-fcVe s + fe'°c * s + OCV') (A5-3o) 33k A5.6 e V + L I - -tL Ode") (A«.3S). I- 0 - / O 0 - p * ^ (A5.35> A l s o , i f q u a n t i t i e s ( H e ) are neglected, the f o l l o w i n g approximations 11 are j u s t i f i e d : CA5.-56) (AS.Ifi') - A. + , —3(AS-Ao) (AS.A-0 From Equations A5.38 to A5.41 we then have: (AS. A-*) 335 A5.7 and from Equations A5.36 and A5.37: = x C V* + - A-^ C P,P,_-A,^) u + ( P . V A , ^ (AS.44) The nonlinear expressions defined by Equations A5.21 and A5.22 can now be evaluated, using Equations A5.24 to A5.27 and A5.30 to A5.35. Note t h a t , when q u a n t i t i e s 0(e 2 11 2 2 ) are neglected, the terms f r ^ , g , 2 f and g A make no c o n t r i b u t i o n to Equations A5.21 and A5.22. 2 some rearrangement, After the nonlinear functions can be w r i t t e n i n the form given below. + <^ 2^*v ( k , + O + P - g ^ ^ l ^ O + v ^ k , ^ ^ ) ^ CAS.4t) i-^O L *• • • 336 A5.8 u.k , * f 6 - £ / 0 + fe V | k t 3k, | u CA5.47) 0 + 6* 0- . Note t h a t , although the value of ri|_ depends on the choice of Lagrange 2 p o i n t , n£ = 3/4 at both L^ and L g . This value has been s u b s t i t u t e d 2 f o r n£ i n Equations A5.46 and A5.47. terms i n v o l v i n g e Equations 4.101 ^ s «• ' 1 0 The d i v i s o r ( l - y ) associated with 2 c o u l d , p o t e n t i a l l y , create d i f f i c u l t i e s , but from and 4.102: l? 8 jO* S-K-b , (*!+ and therefore as y , 1 the terms i n question tend to z e r o . (AS. 4--0 337 A5.9 A5.2 Asymptotic Expansion of the D e r i v a t i v e Terms From Equations 5.58 and 5.59, u ^ ( t , T ) and V g ( t , x ) the homogeneous s o l u t i o n s f o r are defined by: and t h e r e f o r e : av ^ U-s = Titx — u> y bo. s^»v s I bx - u> / u I iP'VLf ^tr = COS S £W s:-vEto k + ^ ] u bX + bj> cos E^ut+-</>] Z bX (A5.S3) J "5' (BS -5-4) - Corresponding r e s u l t s f o r the d e r i v a t i v e s of V g ( t , x ) inspection from Equations A5.52 to A5.54. can be obtained by 338 A5.10 Expansions f o r sine and cose are given i n Equations A3.16 and A3.17 (see Appendix I I I , page A3.6 ) , and using <O t cos 9 + Co5u>jt « > 5 # - s Cos(^ b+ 5 9) = coso b $ S^«O t s si^9 (As , ss) S<^-0 (AS. ? 6 ) - we obtain -s;^o k |Vv.9 5 + feD.cosB* + t ^ c o s 0 1 e x o ©, *H&OJ + • - J (As.*8) 1 Equivalent expressions f o r sin[w^t + <j>] and cos[o^t + <j>] f o l l o w from these r e s u l t s by symmetry. From Equations 5.62 to 5.65 the functions can be w r i t t e n as: ax dv ' $v , , a-^- and 339 A5.ll a.b9_ [ bxr av av oC O at- (AS. fe 61) bbj> bV ar bX? d~c 0Ce ) 3 and using Equations A5.57 and A5.58 we now have 3a co* ( W j k 0 ) = co5«J b / s cos 0 O - si^o t- sc*-0 7 5 o at I +el (p t + cos Lai? s •/ r ^ 1 L ^ z- J } 9) a at? arj cos C to b +- D ) 5 0 } [ + CA^.65) 0 ( V ) di> cos(to fc+^>) dY u L a*c - at D « ° S cosO t cos 6 1 ' L 1 dv - sc* o b si*. d> 1 J u ° J 31+0 A5.12 { bh, - cos C u> t + C^ ) -J3bV 0 u Co. t + c O + o(e0 e)a si>v( u t + G ) dX s = 0 ba. S si*.u> b c*z>0 bV I 0 J Sa, SC«- (u> 1 + 6 ) +• G, 5 I L bx i- 0 s o + co5 0 t s ^ _ E ) 7 J 0 5 cosCcO t + 0 ) O s "I bxr j 9, ^ce, + G^cte.,, } cos(u> b+ 8 ) s 0 (AS. 6 5 ) } 1 + O l V ) ar Tx cos ( ojc + <j> ) 0 x (A5.^) A5.13 cosC u> t +9) o. 5 bX - a c>Qo I cos u> bcos & - $i^u> t s<.v. 9 bX L J 0 a^bQo -f- a ^ 6 | 0 s Cos( u) l + 6>) s I — ' o o h + I- - Q s \ £ <O fc + 0 ) - a 0, 3P 0 s 0 i f J L rrtC ( l-> r> 0 O d v 3- \ r£0 ( oX cos ((j tr -f </>) u + 6. 1 - 0 J l h j> ~c\i> s ^ ( o _ t + ^ > ) at? o x Ir i 0 <?r ar3 }xr d9 5 ^ ( i o t + © ) 5 ^ a 0 d 9 I s ^ v . ^ b cos 9 0 s & 0 dX ^ J + tosu> \:si^Q "I 5 0 3k2 A5.14 +• e a 0, C o t 4-9 ) s ar ar ar- 0, d0 *>r b 3c£ Sin, ( <O t 4- •i- ar 0 b ar 0 £ to 1 + £> ) 0 c ^ , | s ^ ( J b COS ^ L I ar + ( k L <ar e ^ l<h> a r dr + 0 co5 (to t+ { 9) 5 4- CoScJ^t ar L r b. ScL 4- •!>_ Sc6.~l sCw ( u). b 4- (h.") 5 d r + ©, a-, a_9 +• a d9 av ar o = u COS o 0 0 s^(j> \ b 0 4- k. ^ j r * c ( ,S. \- ^ A> } "£ a r 3 £ [" b , ^ L a r 0 + b o ^ . l] a rJ j c o 5 ( ' o b + ^. ) u Ode ) 3 From Equations 5.62 and 5.64: a r (AS.?I) 7 ar : ar' 3if3 A5.15 ( ft • ar r / +• e \ ar a„ Cud_!f> 4- 0 oC °> a_a 6 L aa tr ar- ar bl9 a r a-c ar" fe 4- f a, b^9 i ar a r 0 a ar" 4- a^£. d ar - O^*^ (AC? 4) 1 # and Equation A5.54 can now be expanded as: aV = / aV 5 av 1 L 0 ar 2 - i " 2a_ aj 1 a r ©1 t av -1 I X 3],,, ^ I ar ar + k e I I ar-- a r 4Sa ae i[h,^o t> dt? 1 V a r / o ar • 0(e ) l ar 1 e ar 4 9, 4 0 O ) 2 S ~ ( o t 4 cf> a-c -) u 1 a r + a.,d2iV 4 a a 2 p , h S ~. ( fto *\, a r ^ j -ike. te)'I3 c 3¥i A5.16 As mentioned on page A 5 . 9 , the d e r i v a t i v e s of v,-(t,T) can be deduced from the corresponding r e s u l t s f o r Ug(t,-r). Note, however, that from Equations A4.34 and A4.35 (see Appendix IV, page A 4 . 7 ) : dr ar dj>* = 14> and (A?.??) c>-c 5v therefore only the terms s i n 8 , c o s e , sincf> and cos(j) need be a l t e r e d Q Q 0 0 whenever 6 and <>j appear. A5.3 Continuation of the Uniformly V a l i d Solutions Tr-» +• It ^ i- o /-N /-> +• -i +• U, e\ i n • >1 \ t r* the s o l u t i o n of Equations 5.78 and 5.79, r\ •€ A <~* P r\ 4-r\ which w i l l CT -t« then lead to results that are uniformly v a l i d when t = 0 ( e ~ ^ ) . For Equations 5.72 and 5.73 we have .1. O- - ar L afc-af - Z- a v/ 5 atar as e ( t ) 7 = f (t) ? a o.^ Jr - — (AS. + a_v/«; ar- = 0 (from Equations 4.101 Q using Equation A5.75, the terms and 4.102), and S ( t ) Q Q (see Equations and <J> are constant, fsVsl = ? In Section 5.4 to 5.117) i t was shown that a , b , e (AS . ?<)) a-c*- ea) = 0 from Equations A1.6 and A 1 . 7 . 71?) 1 Q and [^l/sT and T (t) y 5.114 therefore, can also be 3k$ A5.17 eliminated from E* and F*. The functions [ be assembled from Equations A5.52 and A5.53 5 d_Js atar atar ++ <5<A^ c a n a-c a-c J j" - j and in the form given below. - <Hs a>ar = u> $ a_a. L <?v - s ^ I jc^Co b+9"> +- a£i9 cos 6os b+• s a*-? av 9) 7 5 S ( AS. • aV* ix^ + d** av = fi u f 3 t cpsfu^b +<£*) - b^>*s<~fo b u I a r Ja. cos (^1 ar + 9) a r - - -<CjoJ a«. s-*w(^t+-9*) L at- r \ a 3 9 5i«- ( o j t < - P ) ar +- cLaD^cosdo^b+e*) av (AS. a t cesCu> u 4-(jJ>) — t ar t>a^> s-^Cuijb-tcf>^) dxr L ar a r As before i n Section 5.4 ( o n pages 5.27 j to 5 . 2 9 ) , any r e s u l t s i n v o l v i n g long-period terms can r e a d i l y be obtained by inspection from corresponding s o l u t i o n s f o r the s h o r t - p e r i o d terms. If, therefore, s-i) 8o) 34-6 A5.18 we neglect expressions containing long-period q u a n t i t i e s , Equations A5.78 and A5.79 can be w r i t t e n - £*(. = k,"0 Sin,od t & ' Z«>< , c*,$ 'O ^-o —- n ae, <a 0 av bx 3V (AS.ffZ) 2Q 5i ?i^9 40 <i c5£i c o ^ S c ' tf + cos <-> t s where Equations A5.80 and A5.81 have been evaluated using the expansions given i n Section A 5 . 2 . Note, i n c i d e n t a l l y , the correspondence between these expressions and equivalent terms i n Equations 5.92 and 5.93. To pursue t h i s comparison f u r t h e r , we can redefine the q u a n t i t i e s Y g , Y-|» 6 n and 6, i n Equations 5.94 to 5.97 by s u b s t i t u t i n g place of i2o and a ^° 0 5.101 and 0L d9, o In t h i s case Equations 5.98 in to again define conditions f o r the e l i m i n a t i o n of secular terms, but from u ( t ) 7 0Q are respectively. ^3' and v^(t) instead of u^(t) and v ( t ) . g constant: 2*., s^P 0 4- a. 29, 0 O c A l s o , because a Q and 3if7 A5.19 So., fco & 0 av - fl. 0 • 0 2. r ^ . ^ & o - a r a r aa, c ^ P / - a a_9 o ( ^ - S^-^o j L J i f * , * ar ar av av a_£. Si^9 + a ^ c o D , (AS.tffc) L ar L J and we now obtain conditions analogous to those defined by Equations 5.102 a. to 5.105 of the form: s-^f.^o^o^-OjV^oJ +- £>,*[z^C^-O^-4<%^~\ ar" . s 5 a. o (ft s.ss) O (fr*.**?) 3^8 A5.20 1 ar (A*.«»0' With the exception of and 6^ a l l q u a n t i t i e s i n Equations A5.88 to A5.91 are constant, so that we may again use the compact notation I [ •4. a? L 3_ " (AS.«S) C 4*. ~ C J % 8 , ( c f . Section 5.4, page 5.29), where c-j, c , c 2 3 and c 4 are constant. however, that the s p e c i f i c values of these constant terms i n Section Note, 5.4 31+9 A5.21 d i f f e r from those f o r the present case. We s h a l l , in f a c t , f i n d that the general form of Equations A5.92 to A5.95 i s preserved when higher-order s o l u t i o n s are considered pages A5.33, A5.52 and A5.67). (see A f t e r s u b s t i t u t i n g f o r Q from Equation s A4.32, the constant terms can be w r i t t e n : c, = ZcJ s s^9 D |O ---o ) s 3 + -4- a-J*- 7 (K.U) From Equations A5.92 and A5.95, and using the symmetry property to derive corresponding r e s u l t s f o r b-j and <j>-j, we obtain 3a., d_0, 3b, at- av •=. O •= O O (A*, IOO) (AS.IOI) (AS.iox) 350 A5.22 The non-homogeneous terms i n Equations A5.80 and A5.81 are therefore both z e r o , and consequently U.^. Cb) = O (As.ioO ^ 4. \J Ct) = O ' . f . t>s. ie>S) We now proceed with the s o l u t i o n of Equations 5.74 and 5.75 Ug and Vg. for The a n a l y s i s , beyond t h i s p o i n t , i s rather more i n t e r e s t i n g because here the functions u^(t) and v^(t) re-enter the system of p a r t i a l d i f f e r e n t i a l equations. From Equations 4.101, 4.102, A l . 6 and A1.7 we have: fxY*- ^-fO^" *" 1 t, cos t Note that [ + IS cos 3 b ^ u.(t) 2 = u-(t). 1 v.(t) = I V i t I l i + _!l£ii!_ 16 Si-v3b^ CH- p,> + 7 sc«.t - j 5 J 32- 3 - lu N H f o r a l l i > 5 (see the d i s c u s s i o n on pages 5.16 and J tf cos 3 b ^ l \\W (t) CAS s£«-3b 5.17 ) ."ot.) 351 A5.23 tosb - Z$ (AS. cosZb 64- At present (from Equations 5.114 8 a 3T = 0(e ), f£=0(e ), 2 neither 2 or dVs dx x | | = 0(e ) 2 to 5.117 and A5.100 to A5.103) and aVj- contribute at 0(e) at* (see Equation A5.54). -2 f + ^ | L btbr jOLeJ r x The functions , |^=0(e ). 2 Consequently to Equations 5.74 and 5.75 -2|^s - }Js \_dbdv | and bv J oce ) z which can be evaluated from Equations A5.80 and _. and A5.81, take the f o l l o w i n g form: (AS.log) -2 Son. (O t s ar I m; I atr 3 ar 3 aV J i a-r av l <*>S> - Q., aj> a-r air at z 0 t <-av - to*** j lo?) 352 A5.24 atar ar J e 1 ar j + co5 «0 t s ar a r a r I <?r + Cos C0 t u z ^ ^ J 5 ^ ^ * at t ( at* + cos^ *2 o 1*. ar and and - z f ab,. J a r a SO. a*c 0 a r a r L 3 IT J b,^ a r Note that i f replaced by L L a v «*>f o ~ b a r o ^ a r J i n Equations A5.82 and A5.83 are a Z9i. r e s p e c t i v e l y , the r e s u l t i n g expressions <>T 0 are i d e n t i c a l to the corresponding s h o r t - p e r i o d components of Equations A5.108 and A5.109 above. We s h a l l soon have occasion to e x p l o i t t h i s symmetry (see pages A5.33 and A5.34). The two remaining terms i n Equations 5.74 and 5 . 7 5 , a*g - av at- bt c and s 0(1) 20+- p , ) 1 a Vg + au.^ include at d t J o(i) 7 both homogeneous and non-homogeneous s o l u t i o n s f o r u ( t , x ) and v ( t , x ) 5 5 so that UjCb.r) « a-tt&Cus t + + • b Si*-Cu>Lk+0") -+- U 5 ( t ) N r t (AS.Uo) 353 A5.25 I t i s convenient to consider homogeneous and non-homogeneous components of Equations A5.110 and A5.111 separately. For the non- homogeneous terms, we then obtain (from Equations 5.84 and 5 . 8 5 ) : (AS. ^ 1+ p, 112.) )i C^ 5 -^X^^ l : (A5."3) L «O .5 u (^-u^xv-o 20^s^cO b + <->,_0,.c<ocO,_b ? u + / 2 o sc^.b + O Cob 5 x 351+ AS. " ^ r 26 ~i 3 H« (AS. U4) (^-•vx^-o -5 f . j . N 4- " r [ J . A, r ZOj.tj^b -^3+o )sCv k ( "\ 1 . . .s V 1 (AS."5 > S (1+ p , ) ^ E C s ^ - ^ X « V - 0 These equations l e a d , a f t e r some rearrangement, to the f o l l o w i n g expressions: 355 A5...27 { Cos u> s 4- ^ . f -/>. . . . . . . 1 Jt 7 Jb . y 2.1 ~~? - £ 7 b 356 A5.28 4- I) Note that the usual symmetry between short and long-period i s preserved here ( c f . terms page 5.23). For the homogeneous components of Ug(t,x) ana V g ( t , x ) we have: Ti St*- ««? a . Cso(jo> b + 9) v s + *b s bcoCio b+-c^>) u - <C w - <Q ^ bs^*- C t o b + <£>*) S U s a . s c * _ ( c o b •+• U 5 u 0*0 ( AS.il<?) Xt x + UJ D **•«.(* «o b +j>} U u + ffl L «*C b c ^ ^ b -t-d>*) (AS. n«j> 357 A5.29 In contrast to the non-homogeneous terms given i n Equations A5.116 and A5.117, these homogeneous functions enter i n t o the a n a l y s i s f o r higher orders of e (see Equations 5.76 to 5.79 i n Chapter 5 ) . It w i l l con- sequently be expedient to evaluate equations A5.118 and A5.118 to w i t h i n 3 an e r r o r 0(e ) , which i s the accuracy required by Equations 5.75 and 5.79. The process o f expansion i s s t r a i g h t f o r w a r d , and from Equations 5.62, 5.63, A5.57 and A5.58 we obtain the expressions given below. < r - "J'-bo j — ^O tc^j> u o b, £ t w ^ t c o ^ (AS.ixo) <^.S - C u , b vL^ ^. 0 - sc~«Jfc s ^ ^ c . ^ u +• ^ u ° u b ^ s^«j b 0 + u +• «*> o b t *c^ * ^ ^ J " b <^> ^ w o ^ b s o ^ +• co^^b 0 ( 0 * ^ 0 ^ 358 A5.30 z + Q s s |ftx- - o-cf^j t ^j -[ --- " ^ J a a $^Co t+• u ^JQ.e.-t-a e J j ^ c ^ u e j c + Q s tosja,©, " " ^ ^ T 3 a>sCtJ t 0 c o s ^ t + e,,) + +^ A 9 i ^ C ^ k + e*) t> ^j s^(o^b4.^) + 0 (f>f. L ^ at J H HI) . fcsc-9: - o 5 a , ^vi^fcc«P +- < i o e S b C P 0 ^ - ^ s a 0 © , ^ 359 A5.31 Note again the symmetry between short and l o n g - p e r i o d terms i n Equations A5.120 and A5.121. This symmetry i s e x h i b i t e d by a l l non- homogeneous expressions i n Equations 5.74 and 5.75 that involve u) and s 0^, and we may therefore consider only s h o r t - p e r i o d terms when accumulating E*(t,x) and F * ( t , x ) from Equations A5.108, A5.109, A5.117, A5.120 and A5.121. Contributions to E * ( t , x ) and F * ( t , x ) from Equations A5.106 and A5.107 may a l s o be ignored, as the functions { S c 8 -0 - 2k,^^^ ct)j g and I T a) ? no secular terms i n the s o l u t i o n s f o r U g ( t ) s i m p l i f i c a t i o n s we can w r i t e : NH Zk-,^e Ct)J generate g and ^ ( t ^ H ' W ^ t h these 360 A5.32 I ar + + Ceo 2e' s at? j (_av av Cn-jOHi-O l <)v - - -V av .5 (^/-"VO cat? i atr ~~~ (fl5. 127.) f 1 - ^ ^ ! ^ + -2.6 + ^,)"«-1 cOg 6^* - co a. fe P ? s 0 0 361 A5.33 + C05tO t $ ] ox •15 bV J> k,^ / * , ^ , 0 - / * / { ^ [ 4 4 - o , - o ^ f - 2 o 7 s 3 \ bx ar J -ICU^HQ,^ I f secular terms are to be eliminated from Ug(t) and Vg(t) Equations 5 . 9 8 to 5.101 must be s a t i s f i e d , where Y Q , Y-| » <5Q and 6-j denote expressions i n Equations A 5 . 1 2 2 and A 5 . 1 2 3 when we take A f t e r some rearrangement, Equations 5 . 9 8 to 5.101 lead to the following conditions: - ie^O+^.^Ci- k ) | ^ ( ^ - O e + 0 ^ 362 A5.34 (ft*. ^j/^Ci-^o" -if W,/^ av 1 fc» 0^«toOo 5 - ( c J j - * . 0,) 1 s^P 0 1 V [ 363 A5.35 where c-j, and are defined on pages A5.20 and A5.21. d e r i v i n g these expressions, In some reduction has resulted, from the equalities (AS.1*1) which correspond to Equations A4.7 and A4.8 when a ) ' and i o s a t i s f y L Equations A4.28 and A4.29. A f t e r s u b s t i t u t i n g f o r the constants c^(i = 1,2,•••4) from Equations A5.96 to A5.99, the conditions defined by Equations A5.124 to A5.127 above can be w r i t t e n as: (A*.i3cO 2 u 5 ( <v - ar 4. 0 *i«-8 0 39* ? (ftS.l*l) CAS. I 3 2 - ) 36ii A5.36 av j where the constant expressions d . ( i = 1,2,•••4) can be arranged i n the form given below: «o. /L. *- ' ' ( A S . 134) (AS.J340 365 A5.37 - In d e r i v i n g these values from Equations substantial reduction has resulted from using Equations and 5.107 f o r Q , sin0Q* and cos9g* s Now because Equations u e ttir'ni i l i e u to A5.124 quan oi uie b , o^x. auu should be redundant. A4.32, a 5.106 respectively. A5.130 bX A5.127, " ^ t to A5.133 , involve only two un~ uwu u i uiesc iutti c<-|uu o i ZX We can rewrite Equations i ., A5.132 • A5.133 as: i = .f. and 3L. (flS.I3S) Ctn>9 6 (AS. 131) 366 A5.38 and therefore: (AS.Kl) The expressions C ^ i - 2 o i ) and (2«J» i + v ^ ) 3 s v -4e~ 6+j0*(l.-0 5 s 3 can be arranged i n the form: C^- A l s o , from Equations A5.128 and A5.129: 0,) fAS.lA-3) 367 A5.39 and we therefore - have: CO, By comparing these expressions with Equations A5.134 and A5.135, i t i s evident that Equations A5.132 and A5.133 are equivalent to Equations A5.130 and A5.131. If we now w r i t e Equations A5.130 and A5.131 as: 368 A5.40 then from Equations'A5.147 and A5.148 i t follows that -i5k 6 S. With the exception (at present) of ^ and a 3 P , a l l terms i n a*r at these expressions are constant, so that 0 Z ISO 369 A5.41 ^•2. constant - and a l s o , i f a ( x ) 2 daz. = (fl?. i f i ) , i s to s a t i s f y the c o n d i t i o n of uniform v a l i d i t y i n T: (As. i s i ) O. S i m i l a r l y , using the property of symmetry, we can w r i t e = constant OKi?3> bx oxr Eauation A5.152 Drovides. toaether with Eauation A5.149, the necessary condition to determine 8Q. In p a r t i c u l a r , i f ( l - k g ) = 0, we obtain the f o l l o w i n g r e s u l t : and .by symmetry: Corresponding values f o r 9Q and <J> can obviously be obtained when k 0 Q } 1, but not of such an elegant form. At t h i s point i t i s tempting to assume that 2[£ = 0 (see t dx A5.248), and from Equation A5.150 we then have Equation 370 A5.42 The s h o r t - p e r i o d components of u ( t , x ) 5 and v ( t , - r ) H 5 H a r e , from Equations 5.58 and 5.59: CAS. is*) where 8„ can take two d i s t i n c t values i n the ranqe -TT <• 6 n A f t e r expanding cos(w t + 6 ) and Q c o s ( w t + 0 * ) , s Q s s Q < IT. from Equations A4.32, 5.106 and 5.107 we can w r i t e . (ASM 6°) <C ftoCco b $ s Co o? k 4- ^J^CeoPo - Z O j S ^ P o ^ r 3 - S^. t J k s («S -0 )L l 3 Z ( 0^O % 4- cO 0 ) x X = 9 0 9 0 4 - 4 - S j J " & 0 (As, 371 A5.43 Now f o r e i t h e r value o f 9g: P r = 'if (AS.lt*) 0* COS i n which case cosCw t + 6 > 0 ) { < ° A 5 -•^s C c o 5 9 o - 2 0 sin.e j. o , - ^ * ] ceocojfc - = t^O^toOjt ^ Z o e 3 2-O sc^to t $ ^ t - (AS. IAS) (A*, u t ) . 5 t 5 I f we s e t kg = 1, then from Equations 5.84, 5 . 8 5 , A5.157, A5.165 and A5.166 the complete s o l u t i o n s f o r u^(t) and v ^ ( t ) can be reduced to the f o l l o w i n g form: U. Cb)= $ -l$lf,^ u 5 20 I v $ Ct) = ;L0 Kv^t + Oncost £ ( ^ - 1 X 0 ^ - 1V -) lL? »- lSk 3 Slwtr * (-Z + Oj) ^,|* (l-/A».) x +• X (AS.It,^) sO+fO"*\ cost 7 ^,^ Cl-/jQ a x + OU) (AMfcg) 372 A5.44 where the symmetry property has been invoked to e l i m i n a t e long-period 5 terms. Note t h a t , although they do not appear i n the 0 ( e ) s o l u t i o n , a ^ , b 0Q and <J)Q w i l l be present i n the s o l u t i o n s f o r higher orders o f e. The present a n a l y s i s i s s u f f i c i e n t to determine e u ( t ) and e v ( t ) o in Equations 5.32 and 5.33 to w i t h i n a constant e r r o r 0(e ) , although the s o l u t i o n s are uniformly v a l i d f o r t = 0 ( e " ^ ) as e ->- 0. quently nothing to be gained by evaluating u ( t ) ^ i H There i s conse- and v . ( t ) N H .for i > 7. Before we continue with the s o l u t i o n of Equations 5.76 and 5.77 f o r Ug and Vg, the expansions o f d e r i v a t i v e terms must be extended to include higher orders i n e. = da. £ 3 ^ +- 3 b~F A t t h i s stage of the a n a l y s i s : e^}a. 4- H OCe ) (hS. Ifcq) S bV b& + bX~ ?9 bx 1 and therefore 3a. coiC u bi-9) s X 0 Zx*- from Section 5 . 2 : bv (AS. 173) ««.(«,t+9 ) 0 Ib-x 4- B 3a 3 Gos(eJjt+ 9 )~l +OU ) s 0 373 A5.45 a. d&i 0 c«o £ <->, b -t- So') av iX + L I ? £<„ Ceo (o } atr ar j a*c dv av ax j L dv av j (AS. 174) av L + .e- ar I av a^j a-r ar ^r) 6 at- at- 3 CAS.17S) ar a 0.5 a^• , a^ - e ar- x. a\-j c o ( i o k + © ) - d ajj&j si«.r"u>,b+ © ) > J ax 1 0 0 e at"1 a a2fs 0 *M.(*«J,fc*0, at*CAS.I7?) 37k A5.46 Note t h a t , i n the i n t e r e s t of b r e v i t y , l o n g - p e r i o d terms have been omitted from Equations A5.172 to A5.177, as complete expansions i n c l u d i n g the long-period components can r e a d i l y be obtained by i n s p e c t i o n from these r e s u l t s (see Section A 5 . 2 ) . The functions 3_j* av s and \js to consequently do not c o n t r i b u t e 1 z o to Equations 5.76 and 5.77 at 0(e ), and f o r the non-homogeneous terms we now o b t a i n : CAS For the remainder of t h i s a n a l y s i s s p e c i f i c non-homogeneous s o l u t i o n s are not required (see the d i s c u s s i o n on page A5.44). If secular terms are to be eliminated from Ug and Vg, we may therefore r e s t r i c t our a t t e n t i o n to q u a n t i t i e s i n E* and F* that i n v o l v e s h o r t period terms alone, since corresponding l o n g - p e r i o d r e s u l t s are r e a d i l y obtained from the symmetry property. With these s i m p l i f i c a t i o n s 5 ( t ) ^ and V g ( t ) can be w r i t t e n i n t h e f o l l o w i n g form ( c f . NH N H 5.33). u pages 5.31 to H 375 A5.47 lL,(t) = - JA x Sin- 6*) t s - 0. (AS.180) r o ( o / - - u j-y s - | t , <vo 4 j s b J O f i ^ - ^ k . ^ C - z O r 11-2.(^-0^1 - ^ [ i f ^ - o , ] and i f Equations A5.180 and A5.181 are r e - s t a t e d using the compact notation («V--<ojO 376 A5.48 then we can w r i t e : (RS.IS4) if- 5V *b +3a t » S^0 bC^-> ') b s 5 5 H- c « O b s - X, ) . (fls.iss) From Equations A5.52 and A5.53, together with Equations A5.172 to A5.175, and again considering only s h o r t - p e r i o d terms: CAS. 1 8 b ) L 5.^0 b s 1o s <to&. 15 377 A5.49 - Z. (A?. - S^0o av i f >*cr \ 4 C<o tOjk z<& 5 - z. o 1 5 r J «te e r We are now i n a p o s i t i o n to evaluate Equations A5.178 and A5.179, as the remaining q u a n t i t i e s 2-| fV 5 - J ^vj y, — c and have already been determined (see Equations A5.120 and A5.121). After some rearrangement E * ( t , x ) and F * ( t , x ) can be w r i t t e n i n the form given below. 187) 378 A5.50 ar L> / zci+^y*- } i<i L <D + <*>P 0 <j E, e i + *<ps L I 2a, ar L 1 >\ i r r 1 ar. - * 0 M A a-cj a r L ^r J j J 379 A5.51 Sin. u>,b + 1 + z^Otn.y 1 s +- COS u> t s QQ-i — a. bf £ x^ - x *> 3 ar ar -«r>9. (_ I bV dV ar ar J J 380 A5.52 I f these equations are w r i t t e n as ( A ? . I<»l > then we can again use the conditions given i n Equations 5.98 to 5.101 e l i m i n a t e secular terms from Ug and Vg. A f t e r some rearrangement f o l l o w i n g r e s u l t s are obtained: ^ 3 av - ^ R ^ iv JV ^ 3 -*o°*>j>z IX bv CO - c, at av . iV the to 381 A5.53 ,r + 4 ^ 3 - 0>* 1 a D, 0 ar av ) a P. ar i c 3 Q| a&x 4- ar - 382 A5.54 -2U (I + JI,)* S where the constants c^ to c^ are defined on pages A5.20 and A5.21 . These four equations correspond to Equations A5.124 to A5.127, and by d i r e c t analogy with Equations A5.130 to A5.133 we can therefore w r i t e l at- 5 I atr 3] ( ^, -|*x,3£ I L a_p - t av at- t 7 +ft a^ ] [ o r ?^p 4- z ^ e Z«a,<.P e - s 0 t 0 3 av j 0 ar 1 0 coP ] S£^.9 + X ^ Ceo 0 L 0 (ft 5.1*7) ar S o 1 (flS'.m). 383 A5.55 where the q u a n t i t i e s ( i = 1,2,•••4) denote constant terms i n Equations A5.192 to. A5.195. I t should again be possible to reduce Equations A5.198 and A5.199 to the form of Equations A5.196 and A5.197. Using the previous r e s u l t s of Equations A5.145 and A5.146, to demonstrate t h i s reduction we must therefore show that (4*4*+ V ) The evaluation of Equations A5.200 and A5.201 presents l i t t l e d i f f i c u l t y and, although the d e t a i l e d c a l c u l a t i o n i s omitted here, both of these conditions a r e , i n f a c t , s a t i s f i e d . Before s o l v i n g Equations A5.196 and A5.197 f o r a 1 and 6 - | , it would be helpful to reduce the functions H and H to a more compact form. 1 2 A f t e r s u b s t i t u t i n g f o r Q , s i n 8 * and cos 9 * (see Equations A4.32, s and 5.107 Q Q 5.106 r e s p e c t i v e l y ) . t h e c o e f f i c i e n t s of a 8^ and a^ i n Equation A5.192 Q can be expressed as: (RS.20Z) 381; A5.56 ( A S . 203) and f o r Equations A5.196 and A5.197 we now have: L air 6Xr j L a r j 1 4J S C u J ^ - cjj") (AS.zot*-) 1 av a* j t av av j ] (ftS.ZoS) Equations A5.128 and A5.129 are p a r t i c u l a r l y important during t h i s of r e d u c t i o n . process 385 A5.57 The expressions i n v o l v i n g ( i = 1,2,•••4) may be re-grouped, and Equations A5.204 and A5.205 then lead to the f o l l o w i n g c o n d i t i o n s : ^3 bX dX ar ue^x.E^-Oj-z] (05.26?) Now from Equation A5.151 and 3 9 ar 3 must be constant. L i w 5 c to^ -<JJ"!) 1 ^_S>z = constant, so that both da. 3 at In t h i s case we can w r i t e : ^ r" L (fl?.20S) 386 A5.58 and, i f a~(-r) i s to s a t i s f y the c o n d i t i o n of uniform v a l i d i t y i n T: Sen. <9 e £ ^ + X ^Oj 1 [ W s l - 0 -Z 3 1 ( A S : io<0 Without d e f i n i t e values f o r 3_e a and ot ao at 3 Equations A5.208 and A5.209 cannot be solved f o r a-j and B - j , but i f we assume that both 02 and 6^ are constant: ' t f JivPo^ 5 [ ^ - O j - Z ] + ..... CflS.Zio) ^ 0 ,<VoPo^ X, [ ^~O i - 2.] - • ^ 387 A5.59 where a Q i s defined by Equation A5.157. Note t h a t , corresponding to the two possible values of 0Q w i t h i n the i n t e r v a l -IT < 6Q ^ IT, there are two values of 6^. The functions i n v o l v i n g X . ( i = 1,2,•••4) do not reduce s i g n i f i c a n t l y , and i t i s easier to r e t a i n the present form of Equations A5.208 and A5.209. S p e c i f i c values of X^ can be obtained from Equations A5.180 to A5.183, but note that these equations r e l a t e only to s h o r t - p e r i o d terms. D i f f e r e n t values of X^ consequently apply i n the l o n g - p e r i o d case, and t h i s must be taken into consideration when e v a l u a t i n g A5.209 with b , <f> and Equations A5.208 and i n place of a , e and OJ . F i n a l l y we consider the s o l u t i o n of Equations 5.78 and 5.79 f o r U-|Q and V.|Q. Only those q u a n t i t i e s which generate secular terms i n the s o l u t i o n need be considered, . A <t> D - 2k,|^$ l o so that the functions a> and T l D Cb) - Z k . ^ c . / t ) may both be ignored (see Equations A l . 6 , A 1 . 7 , 4.101 and 4.102). Equations A5.100 and A5.151 both a, and d j i are constant. from Equations A5.197 and A5.198, 3f-3 and a_9 3 1 This s i m p l i f i c a t i o n i s discussed on page A5.46. From Consequently, are constant. case Equations A5.176 and A5.177 reduce to the f o l l o w i n g form: av - ' In t h i s 388 A5.60 and therefore neither of these terms contribute at 0(e ) to Equations and 5.79." At t h i s stage of the a n a l y s i s a ~ and | | are o(e ) 2 and 0(e ) r e s p e c t i v e l y (see Equations A5.151 and A5.152), and from page Uy(t) and Vy(t) are both zero. 5.78 A5.22 Consequently the d e r i v a t i v e s of u^ and v , together with the expressions 7 c o n t r i b u t i o n to the non-homogeneous terms of Equations 5.78 and 5.79 (see Equations A5.52 and A5.53)1 I f we include only short period terms,^ A5.52, A5.53.and A5.172 to A5.175: . bxr t c f . page A5.46. A then from Equations 389 A5.61 I 2o< ] I • z.7 ajv s I at-ar + a^. 7 CflS.2.1S") a r 3 2<3?, £0 5 { ^ Co9 dr -s^D *[ 0 a L i .a3 ar LP* "1 ar \ ar + a r + «. f aj^. - & * a & A e v ^ d i r > ; +^3*3! ^ J 390 A5.62 Note the equivalence i n s t r u c t u r e between these two expressions and the corresponding r e s u l t s f o r lower orders of e (see Equations A 5 . 8 2 , A5.83, A5.108, A5.109, A5.186 and A5.187). . We next require expansions o f : t and at t h i s point the advantages of assuming k n u siderable. = 1 and 0 are con- bX With t h i s s i m p l i f i c a t i o n , from Equations 4.101, 4.102, A5.167 and A5.168 i t f o l l o w s that none of these q u a n t i t i e s involves short (or long) period terms, and we may therefore s i m p l i f y the non-homogeneous expressions i n Equations 5.78 and 5.79 e x t e n s i v e l y , as shown below. d <X - z S av i at x at S oaO (PL*.211,) Note that u 5 and v u v cf. 5 5 page . A5.24. g here denote complete s o l u t i o n s , so that = u (t)j + u (t) 5 = v 5 ( t ) | H 5 + v 5 ( t ) N H NH ' 391 A5.63 b\ + s du. fO-f a\/ -t- Ji* ^ L s }tr atar 5 We') These functions can be evaluated d i r e c t l y , using Equations A5.120, A5.121, A5.214 and A5.215. For E * ( t , x ) and F * ( t , x ) we then o b t a i n : - z |_ a r yc L *v - 3- a r / ar ar a rJ J 3 j it Z arc V ar ar (AS". 218) av ar 392 A5.64 r av + z - <?ir avj fex h 3 ]1 $ * } } ] A f t e r s u b s t i t u t i n g f o r Q , s i n 6Q* and cos 8 * from Equations G A4.32, 5.106 and 5.107: Q 393 A5.65 © 5 u> CJO 5 +5^9,, = u>^0 cos9 - s;^9 x b 0 (u) ' 5 l + o ^ , 3 so that Equations A5.218 and A5.219 can be w r i t t e n as: 1 CAf.^2-3) 39i+ A5.66 F * ( t , r ) 4 co^jb! I < -P P + <te0 >C V+ 3) ? < e 0 t o (AS. 12-5) where H L (fl*.2Zt) i L y a^ - e, ar 1< ^r ^ 3 ] arj * (ft&zz?) « e,i?3 ar 0 L M - C*e>. +<*O ae* ] 0 I f we now w r i t e E * ( t , x ) and F * ( t , x ) i n the form t ar J 395 A5.67 F * ( t , t ) = o~ b Sir*. £0j t -v J ( COS CO t $ C AS-.zs.i') , then to e l i m i n a t e secular terms from U-JQ and V-JQ Equations 5.98 to 5.101 must be s a t i s f i e d . A f t e r some s t r a i g h t f o r w a r d manipulation the f o l l o w i n g t conditions are obtained: CflS.2.30) o - o (AS".i32.> t^ 5 C* tOf t^t?} - (AS.233 ) By comparing these r e s u l t s with Equations A5.130 to A5.133 we can state d i r e c t l y that Equations A5.232 and A5.233 are equivalent to Equations A5.230 and A5.231 r e s p e c t i v e l y . Note a l s o , from Equations A5.124 to A5.127 and A5.130 to A5.133, that Equations A5.230 to A5.233 can be written as: c, U - c,, 8, x c z + t , H, = o = O To obtain expressions of t h i s form i t i s necessary to use Equations A5.128 and A5.129. 396 A5.68 c H s i . ~ c 4 H , = ° ( AS.2.36) which i s consistent with previous r e s u l t s (see page A5.20). From Equations A5.230 and A5.231 we now have: H , C O H x £ r ) = O = O CAS-.zSg) , (AS.i3<0 so that ..... <?r . . . a_tr a r . _ _ w j e 2- a r ^ 3. J [ . a r a r L (AS.Z4.0) J (AS.2M) At the present stage of the a n a l y s i s : ar £§3 ar = constant (from page A 5 . 5 7 ) , and from Equation A5.151 Z9 Z = constant 397 A5.69 From Equations A5.240 and A5.241 i t therefore f o l l o w s that ^f-q. = bO^ constant = constant (AS.a-4-i-) . CftS*4-3) To maintain uniform v a l i d i t y i n x we must require - O (AS. 2.4-4) i n which case Equations A5.240 and A5.241 can be w r i t t e n as: a r av bv We can s u b s t i t u t e f o r a i n Equation A5.246 from Equation A5.208 Q to o b t a i n : CAS. 24?) Ceo - «,s~.P<> ^ \ [ t O ^ - O j -2-] + . - ^ 398 A5.70 ^ "I and therefore, since the function j^Ci + p,) *• -.|P». ^ is constant and i non-zero: 0 , - constant («i**<0 (see Equation A5.209). I t i s convenient a t t h i s p o i n t to set and from Equation A5.247 we now have: P^ - /*,P, ( T c ^ P 0 . .• a . « V 5 tf^s^Q ... . (ftr.^i) -COJ)" I f i t i s assumed that both Equation A5.245: ^ 6 3 and 19^ are z e r o , then from at at 2. where, i n t h i s case, 9-j i s defined by Equation A5.211. Corresponding r e s u l t s f o r b 2 and (J> can r e a d i l y be obtained 2 using the symmetry property, but note that the q u a n t i t i e s r , A and A., ( i = 1,2,•••4) i n Equations A5.249 and A5.250 must be re-evaluated i n terms of the long-period v a r i a b l e s o ^ , b and-<j> (see the d i s c u s s i o n on page A5.59). 399 A5.71 This completes the d e t a i l e d a n a l y s i s of four-body motion near and L g , although the r e s u l t s derived here are condensed i n Chapter 5. A6..1. APPENDIX VI A6.1 CHEBYSHEV POLYNOMIALS Introduction Suppose we wish to approximate a f u n c t i o n f ( x ) over some i n t e r v a l , f o r example interpolation. -l^x^l, to w i t h i n an e r r o r ±e using a polynomial A number of i n t e r p o l a t i o n schemes may be used [ 6 2 ] , 2 but to obtain the prescribed accuracy a' Targe number of terms may be necessary, a feature which i s p a r t i c u l a r l y s i g n i f i c a n t when .accuracy to be maintained over the e n t i r e range of x . In t h i s case an i n t e r - p o l a t i o n procedure based on Chebyshev polynomials leads to the most "economical" approximation, in the sense t h a t , f o r a c e r t a i n accuracy The Chebyshev polynomials T ( x ) are simple trigonometric k functions cosk6 s IX- = but expressed i n terms of the v a r i a b l e cosB , where - I T ^ 6 ^ IT and, consequently, -1 ^ x ^ - 1 . I t i s frequently * more convenient to work with the s h i f t e d Chebyshev polynomials T ^ ( x ) , f o r which T * Cx) = cos k9 k with cos 9 =- 1 is 2+01 A6. As 6 v a r i e s from 0 to i r , we now have 0 ^ x ^ l . mation f o r an a r b i t r a r y f u n c t i o n f ( x ) The method of a p p r o x i - i s o u t l i n e d i n the f o l l o w i n g section. A6.2 Function Approximation using S h i f t e d Chebyshev Polynomials Consider the problem of approximating some f u n c t i o n f ( x ) by a polynomial in x over the i n t e r v a l -a 4 x ^ 3. I f s h i f t e d Chebyshev polynomials are used, then a new v a r i a b l e y must be defined such that the i n t e r v a l -a £ x ^ 3 corresponds to 0 ^ y 4 1. Of the many t r a n s - formations s a t i s f y i n g t h i s c o n d i t i o n , the most obvious i s and we now take [62 ] 3 Z In t h i s case f ( x ) i s transformed i n t o an even f u n c t i o n of 6 which can be expanded as CO rK*0 = + j. ... % lest cos\< & . (06.3) -- However, since |<(,y) = coske, t h i s expression i s equivalent to T eo The constants a^ are obtained from a numerical i n t e r p o l a t i o n ii02 A6.3 procedure [62 *]. and i n p r a c t i c e the summation may be terminated a f t e r 1 n terms. I f |a | denotes the magnitude of the l a r g e s t c o e f f i c i e n t , the m summation could be truncated when, f o r example, |a -|l < 10 n+ 1 0 |a l • m We then have Z — i k si where e i s s u f f i c i e n t l y small that i t s e f f e c t may be neglected. To derive the c o e f f i c i e n t s a^, f ( y ) i s evaluated at the points tj^ = 1 + cos C S " T T / V ) S = O f \ 1 > j - •• , n. . which y i e l d a l i n e a r d i s t r i b u t i o n o f data points 6 over the i n t e r v a l 6 0 is- TT. Equation A6.5 can be rearranged i n t o the power s e r i e s iAx} = N n. 21—» . . .. . ' t\ . ... + e. , and the corresponding s e r i e s i n x can now be w r i t t e n ^ or, alternatively: J (A4-6) k03 A6.4 where denotes the binomial c o e f f i c i e n t m For computation, however, the f o l l o w i n g form of Equation A6.8 i s more convenient: J A6 3 C o e f f i c i e n t s of (y) I f the s h i f t e d Chebyshev polynomial of order k i s w r i t t e n as k 2-. i — O then, from Lanczos [ 6 2 ] , the c o e f f i c i e n t c^ i s defined by: ( A * - II ) W.-C - C o e f f i c i e n t s of the f i r s t twelve s h i f t e d Chebyshev polynomials are given i n Lanczos [ 6 2 * ] . In the present case, however, Equation A 6 . l l provides the information needed to evaluate Equations A6.6 and A6.9. A6.4 Approximate Expansions Suppose the expansion defined by Equation A6.9 i s s u f f i c i e n t l y accurate t h a t . w i t h some f i n i t e value of n , f o r a l l p r a c t i c a l purposes k-ok the e r r o r e may be taken as zero. A6.5 In t h i s instance Equation A6.9 can be rearranged as the i d e n t i t y which, from Equation A 6 . 6 , we can a l s o w r i t e as £6c"> - t k <j k . ( f i t . 13 > k=.o Now from Equation A6.10: k so that kH k * where |T^(y)| ^ 1. I f we s u b s t i t u t e f o r y which can now be w r i t t e n i n the form n i n Equation A6.13: 1+05 A6.6 CA6.I7 ) where the e r r o r 6 s a t i s f i e s 1 5 1 • - . CA6-I8) This process of contraction can be continued so that f ( x ) i s approximated by p r o g r e s s i v e l y fewer terms, but with an i n c r e a s i n g e r r o r [ 6 2 ] 7 We next i n v e s t i g a t e the behaviour of |S| when n-1 terms are r e t a i n e d but the i n t e r v a l - a ^ x ^ 3 i s decreased. I n t u i t i v e l y one would expect the e r r o r to d i m i n i s h , and i n f a c t the reduction i s associated with b . n Let DC = =c 0 (M,. •+ CT£ |q ) where a t l , the i n t e r v a l of i n t e r e s t i s defined by -oo. ^ and f ( x ) 5 i s bounded on t h i s i n t e r v a l . value f ( X g ) . As a.-»• 0, f ( x ) w i l l tend to some We can therefore w r i t e 4Gt> = SCO + v$*C$) , Cfl6.ao) it • where f (£) i s bounded and f ( x ) - f ( x ) n = 0(a) as a -*• 0. The i n t e r v a l A6.7 -a ^ C - B corresponds to and s u b s t i t u t i n g f o r x i n Equation A6.1 we obtain ^ - % + oc The function cj>(6) i s ) now defined by which we can w r i t e as Equation A6.13 i s equivalent to rt. (see Equation A 6 . 5 ) , where the F o u r i e r c o e f f i c i e n t s numerical i n t e r p o l a t i o n process. a k are derived from a Note t h a t , because n i s the exponent of y i n Equation A6.13, b n using i n t e r p o l a t i o n , the c o e f f i c i e n t s i s equal to a n < largest If, instead of a^ were obtained from the i n t e g r a l IT • « k = i. (f>C0) co lt& d9 | S o , (fl6.a-5; . 4 0 • 7 A6.8 the two values of a^ would be i d e n t i c a l only when f ( x ) contained no harmonics higher than cosnG, i n which case Equation A6.20 would be exact [62 ]- Although t h i s c o n d i t i o n may not be s a t i s f i e d , we assume here 8 that Equation A6.25 can be used to define a n and b n > Consequently we can w r i t e If (9) cosr\9 <19 . , . and t h e r e f o r e , from Equation A6.22 II H Cos T? J ir For b we then obtain n ir Cos m. 9 <&9 so that b n = 0(a) as a 0 . (fl&-2s) When n » 1, the q u a n t i t y c n i n Equation A6.18 w i l l be a large i n t e g e r , ^ and i t f o l l o w s that O ( r ) = \l\ as 0- _> o The e r r o r 6 associated with a Chebyshev i n t e r p o l a t i o n of f ( x ) on the i n t e r v a l X Q - oa ^ x From Equation A 6 . l l : c n c X Q + a3 therefore diminishes as a -> 0. 408 A6. Note t h a t , f o r t h i s s p e c i f i c i n t e r p o l a t i o n process, if g(x) denotes the approximating polynomial: I f ( x ) - g(x) so that f ( x ) | ^ |5| f o r a l l x such that x Q - aa ± x ^ X Q + a3, i s approximated uniformly over the e n t i r e i n t e r v a l and, f o r a given order of g ( x ) , with increasing accuracy as 0 tends to zero. A6.5 Expansion of the Function | r . . |" From Chapter 5, Equation 5.7, the expansion of J i s given by: J_ r1 + and i f we take ... -x.. . = then ..[ For L . and Lj- we have | r.. 1 + | *= 1 (Equation 3 . 4 5 ) , and i f f ( x ) = _L about k-09 A6.10 = [ I + (A6.3l) J 2 where x = | r^. \ - 1. If f ( x ) i s approximated i n the i n t e r v a l - a ^ x ^ 3, as shown in Figure A 6 - 1 , then: Figure A6-1 f ( x ) = [1 + x ] •3/z The approximation w i l l consequently be v a l i d w i t h i n a region about p . , bounded by inner and outer r a d i i (l-a) ^ 1 and (1-3) fx respectively. For the a n a l y s i s of Appendix V, Section A 5 . 1 , i t i s necessary to evaluate the functions and over i d e n t i c a l i n the sense that a and 3 must be equal f o r both expansions. intervals I f t h i s were 410 A6.n not so, the c o e f f i c i e n t s and A5.16 would d i f f e r . near p , 0 ( i = 0, 1, 2, 3, •••• ) i n Equations A5.15 The corresponding region of the and p ~ , w i t h i n which the expansions of ,3 and (l+B)' _j r\ plane and are both v a l i d , i s shown i n Figure A6-2 f o r ( l - a ) ' * = 7 /a = 1.35. 0.65 14-11 A6.12 Note that i f a and 3 are chosen so that l - ( l - a ) ' " = (1+3) /a the expansions are v a l i d i n s i d e a c i r c l e of radius p about For the function f(x') = [1+x] /z -1 = p , and L g . , and with these p a r t i c u l a r values of a and 3, we have: -0.5775 ^ x ^ 0.8225 , 3.6413 ^ f ( x ) ^ 0.4064 . The e r r o r e i n Equations A'6.5 to A"6.9 i s reduced below T 0 ~ ^ f o r n > 25; -8 the corresponding Chebyshev i n t e r p o l a t i o n i s accurate w i t h i n 10" the i n t e r v a l -0.5775 < x 4 0.5985. over A s l i g h t error becomes apparent f o r 0.5985 ^ x ^ 0.8225, reaching a maximum of 4 x 1 0 " when x = 0.8225. 7 The c o e f f i c i e n t s of x fl y tvi S I E 4- A n ^ +- U m i n the expansion of f ( x ) are tabulated below f o r ,.,^4-U o^wv*^/.^««rl4^r. U4« -I 1 * . 1 . . ~ <- A6.13 Chebyshev - Derived Coefficients Binomial Coefficients 0 1.0000 000 1.0 1 -1.5000 000 -1.5 2 1.8750 000 1.875 3 -2.1874 999 -2.1875 4 2.4609 366 2.4609 375 5 -2.7070 411 -2.7070 313 6 2.9326 656 2.9326 172 7 -3.1417 974 -3.1420 898 8 3.3371 447 3.3384 705 9 -3.5284 891 -3.5239 410 10 3.7215 732 3.7001 381 11 -3.8294 740 -3.8683 262 12 3.8122 951 4.0295 064 13 -4.3456 147 -4.1844 875 14 5.7458 168 4.3339 334 15 -4.4987 347 -4.4783 979 Table A6-1 C o e f f i c i e n t s of x i n the Series Expansion of [1+x] f o r -0.5775 $ x ^ 0.8225 m PUBLICATIONS Barkham ; P.P.D. and Soudack, A.C., An e x t e n s i o n t o . t h e method o f K r y l o f f and B o g o l i u b o f f . Intv Journ. of C o n t r o l , V o l . 10 (1969), pp. 377-392. Barkhazn, P.G.D. and Soudack, A.C., A p p r o x i m a t e s o l u t i o n s of n o n - l i n e a r , non-autonomous s e c o n d - o r d e r d i f f e r e n t i a l equations. I n t . Journal of C o n t r o l , V o l . 11 (1970), pp. 101-114. Soudack, A.C.', and Barkham, P.G.D. , F u r t h e r r e s u l t s on "Approximate s o l u t i o n s o f n o n - l i n e a r ^ non-autonomous s e c o n d - o r d e r d i f f e r e n t i a l . equations." I n t . Journal of Control, V o l . 12 (1970), pp. 763-767. Soudack, A . C , and Barkham, P.G.D., On t h e t r a n s i e n t s o l u t i o n of t h e u n f o r c e d D u f f i n g e q u a t i o n w i t h l a r g e damping. I n t . J o u r n a l o f C o n t r o l , V o l . 13 (1971), pp. 767-769. Barkham, P.G.D. and Soudack, A . C , A p p r o x i m a t e d i f f e r e n t i a l equations w i t h s a t u r a t i n g .nonlinearities. I n t . Journal of Control, V o l . 19 (1974) pp. 941-946. Barkham, P.G.D., Modi, V . J . and Soudack, A . C , The concept of reference L o c i a p p l i e d t o f o u r - b o d y dynamics. P r e s e n t e d a t t h e 1 3 t h I n t e r n a t i o n a l Congress o f T h e o r e t i c a l and A p p l i e d M a t h e m a t i c s , Moscow, August 21-26, 1972. Barkham, P.G.D., M o d i , V . J . and Soudack, A . C , The concept o f r e f e r e n c e L o c i a p p l i e d t o f o u r - b o d y dynamics. P r e s e n t e d a t t h e 23rd I n t e r n a t i o n a l A e r o n a u t i c a l C o n g r e s s , V i e n n a , October 8-15, 1972.
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Some aspects of three and four-body dynamics Barkham, Peter George Douglas 1974
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Title | Some aspects of three and four-body dynamics |
Creator |
Barkham, Peter George Douglas |
Date Issued | 1974 |
Description | Two fundamental problems of celestial mechanics are considered: the stellar or planetary three-body problem and a related form of the restricted four-body problem. Although a number of constraints are imposed, no assumptions are made which could invalidate the final solution. A consistent and rational approach to the analysis of four-body systems has not previously been developed, and an attempt is made here to describe problem evolution in a systematic manner. In the particular three-body problem under consideration two masses, forming a close binary system, orbit a comparatively distant mass. A new literal, periodic solution of this problem is found in terms of a small parameter e, which is related to the distance separating the binary system and the remaining mass, using the two variable expansion procedure. The solution is accurate within a constant error O(e¹¹) and uniformly valid as e tends to zero for time intervals 0(e¹⁴). Two specific examples are chosen to verify the literal solution, one of which relates to the sun-earth-moon configuration of the solar system. The second example applies to a problem of stellar motion where the three masses are in the ratio 20 : 1 : 1. In both cases a comparison of the analytical solution with an equivalent numerically-generated orbit shows .close agreement, with an error below 5 percent for the sun-earth-moon configuration and less than 3 percent for the stellar system. The four-body problem is derived from the three-body case by introducing a particle of negligible mass into the close binary system. Unique uniformly valid solutions are found for motion near both equilateral triangle points of the binary system in terms of the small parameter e, where the primaries move in accordance with the uniformly-valid three-body solution. Accuracy, in this case, is Q maintained within a constant error 0(e⁸), and the solutions are uniformly valid as e tends to zero for time intervals 0(e¹¹). Orbital position errors near L₄ and L₅ of the earth-moon system are found to be less than 5 percent when numerically-generated periodic solutions are used as a standard of comparison. The approach described here should, in general, be useful in the analysis of non-integrable dynamic systems, particularly when it is feasible to decompose the problem into a number of subsidiary cases. |
Subject |
Two-body problem Three-body problem |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-01-22 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0064964 |
URI | http://hdl.handle.net/2429/18937 |
Degree |
Doctor of Philosophy - PhD |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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