THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING Department of Aerospace Engineering Technical Report AN ANALYTICAL TREATMENT OF THE OPTIMUM TRANSFER BETWEEN TWO TERMINAL POINTS FOR MINIMUM INITIAL IMPULSE UNDER AN ARBITRARY INITIAL VELOCITY VECTOR Fang Toh Sun ORA Project 06136 supported by: NATIONAL AERONAUTICS AND SPACE ADMINISTRATION GRANT NO. NsG-558 WASHINGTON, D.C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR June 1966

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ACKNOWIEDGMENT The author is grateful to the National Aeronautics and Space Administration for its financial support of this study, and, in particular, to Dr. Raymond H. Wilson, Office of Advanced Research and Technology, NASA, whose continuous interest and support made this study possible. Special thanks are due to Professors Harm Buning and D. T. Greenwood, Department of Aerospace Engineering, University of Michigan for their personal participation in this project. Their many inspiring discussions, their critical review of practically the entire manuscript, and their many helps which enabled the author to carry out the scheduled work at this University are deeply appreciated. Finally, credit is due to Messrs. John Duffendack, Douglas Westerkamp, and Norman F. Harrington for their valuable help in computer programming, calculation, and geometric drawing in the preparation of this report.

TABLE OF CONTENTS Page LIST OF TABLES v LIST OF FIGURES vii NOMENCLATURE ix ABSTRACT xiii CHAPTER 1. INTRODUCTION 1 2. TWO-DIMENSIONAL ANALYSIS OF THE PROBLEM 2 2.1 Formulation of the Problem 2 2.2 The Constraining Hyperbola and the Orthogonality Condition 4 2.3 Criterion of the Nature of the Real Solutions and the Boundary Evolute 9 5. DETERMINATION OF THE OPTIMUM SOLUTION 16 3.1 The Absolute Minimum Solution 16 5.2 Lines of Constant Optimum Trajectory and Lines of Constant Velocity-Increment 22 3.3 The Critical Condition and the Unrealistic Trajectories 22 3.4 Choice of the Realistic Optimum Transfer Trajectory 30 3.5 The Minimum Velocity-Increment of the Optimum Solution 35 3.6 Effects of the Initial Velocity Vector on the Optimum Solution 40 4. HODOGRAPHIC REPRESENTATION OF THE TWO-DIMENSIONAL OPTIMUM TRANSFER 43 4.1 The Orthogonal Net in the Hodograph Plane and the Optimization Chart 43 4.2 The Construction of the Transfer Hodograph 46 4.3 The Hodograph of Optimum Transfer Trajectories in the — Plane 48 iii

TABLE OF CONTENTS (Concluded) CHAPTER Page 5. ANALYSIS OF SOME LIMITING CASES 53 5.1 The Case 4 = 0 53 5.2 The Case Jr = x 57 5.5 The Case n - co 62 5.4 The Case n 0 66 6. TRANSFER FROM A CIRCULAR ORBIT 67 6.1 Analysis 67 6.2 Some Observations 73 7. THE THREE-DIMENSIONAL EFFECTS ON THE OPTIMUM TRANSFER 76 7.1 The Three-Dimensional Analysis 76 7.2 The Three-Dimensional Effects 77 REFERENCES 81 APPENDICES A GLOSSARY OF TERMS FOR TWO-TERMINAL TRAJECTORIES 82 B THE INTERSECTING PROPERTY OF THE NORMALS OF A HYPERBOLA 85 iv

LIST OF TABLES Table Page 1. Principal Formulas in the Nondimensional Form for the Terminalto-Terminal Optimum Transfer 7 2. Principal Geometrical Elements of the Constraining Hyperbola 8 5. Nature of the Real Roots of the Orthogonality Quartic and Number of Real Solutions 11 4. The Real Roots of the v -Equation and the Nature of the Stationary Points on the |Avl-Curve 18 5. Regions in the Hodograph Plane and the Nature of the Optimum Transfer Trajectories 34 6. Optimum Solutions for Vertical Transfer (r = 0) 56 7. Optimum Solutions for 180~ Transfer (* = 5) 59 8. Minimum Velocity-Increment, |Av*|,for the Transfer from a Circular Orbit to a Coplanar Point 70 9. Minimum Velocity-Increment Required and the Critical Range Angles for Interplanetary Flight in the Solar System from the Earth Orbit 75 v

LIST OF FIGURES Figure Page 1. The transfer trajectory and the constraining hyperbola in the hodograph plane. 3 2. Quadrants in the hodograph plane. 9 3. Geometry of the constraining hyperbola and the boundary Lame. 13 4. Variation of |Av| and the corresponding geometry in the hodograph plane. 17 5. Lines of constant optimum transfer trajectory and lines of constant optimum velocity-increment. 23 6. Geometry of the parallel curves. 24 7. Regions in the.._&ograph plane and the nature of the optimum transfer trajectory. 25 8. Choice of the realistic optimum trajectory: Initial velocity vector in the unrealistic region. 31 9. Determination of the boundary point, I|Av| = |Av*j. 35 10. Variation of the minimum velocity-increment with the initial velocity vector of constant direction and varying magnitude. 37 11. Variation of the minimum velocity-increment with the initial velocity vector of constant magnitude and varying direction. 39 12. The optimization chart for minimum initial impulse terminalto-terminal transfer (r = 60~, 1 = 75~). 44 12A. The equi-critical-velocity increment line in the hodograph plane (4 = 60~, c1 = 75 ). 45 13. Construction of the transfer hodograph in the V-plane. 47 14. Geometric representation of the two-terminal constraint in the - and ef-planes. 49 vii

LIST OF FIGURES (Concluded) Figure Page 15. Hodograph of the optimum transfer trajectories in the -- -plane. 50 16. Optimization of vertical transfer (|=0). 54 17. Optimization of 180~ transfer (4 = it). 58 18. The optimum trajectory hodograph for 180~ transfer. 61 19. Optimization of transfer to infinity. 64 20. The optimum trajectory hodograph for transfer to infinity. 65 21. Transfer from a circular orbit to a coplanar point. 68 22. Minimum velocity-increment for the transfer from a circular orbit to a coplanar point. 71 23. The critical configuration of the base triangle for the optimum transfer from a circular orbit. 72 24. Geometry of the three-dimensional transfer. 76 25a. Three-dimensional effect on the optimum transfer from a circular orbit: Low distance ratio (n < n*). 79 25b. Three-dimensional effect on the optimum transfer from a circular orbit: High distance ratio (n > n*). 80 A-1 The two-terminal trajectories. 84 B-l Intersection of two normal lines to a hyperbola. 86 viii

NOMENCLATURE A semi-transversal axis, constraining hyperbola B semi-conjugate axis, constraining hyperbola C center-to-focus distance, constraining hyperbola C1VC2 parameters defined by Eqs. (38) d the perpendicular distance from the field center to the chord of the base triangle e eccentricity, constraining hyperbola h angular momentum per unit mass I an invariant of the orthogonality quartic (see Eq. (18)) J an invariant of the orthogonality quartic (see Eq. (18)) K Godal's compatibility constant = tan d 2 M,N orthogonal projections of a velocity vector on the local radial and chordal axes respectively )7,~ZJk nondimensional form of M and N: M/VS1, N/Vs1 n distance ratio = r2/r1 r radial distance V velocity Vs circular speed = /C- r f2 dimensionless velocity a V h/i x,y displacement coordinates X*,Y* critical coordinates, given by Eqs. (33) A discriminant of the orthogonality quartic AV velocity increment ix

NOMENCLATURE (Continued) Av dimensionless velocity increment = AV/VS1 ~e eccentricity K dimensionless Godal's compatibility constant K/Vs1 strength of the gravity field v dimensionless velocity - V/V1 v value ov v satisfying Eq. (22) p distance of the optimum origin from the radical center, (T) in the hodograph plane p' distance of the optimum origin from the hodograph image of the initial terminal point Q1 a included angle of the local radial and chordal axes (Fig. lc) the path angle with reference to the minimum energy direction the path angle with reference to the local horizontal Cp the interior angle of the base triangle at the terminal point the vertex angle of the base triangle (Fig. la) E! ~ the range angle w the inclination of the initial velocity vector to the plane of the base triangle w a parametric angle, defined by Eq. (B-l), Appendix B Subscripts ~* ortho-point, or orthogonal solution in Chapters 2,3; optimum condition elsewhere ** absolute minimum solution 0 initial condition x

NOMENCLATURE (Concluded) 1 initial terminal, unless otherwise indicated 2 final terminal, unless otherwise indicated d chord perpendicular L lower limit U upper limit opt optimum c,R chordal and radial pair of directions r.~ radial and transversal pair of directions X5 outward directions of the interior and exterior angle bisectors of the base triangle respectively p,n in-plane and out-of-plane components Superscripts * critical xi

ABSTRACT The problem of minimizing the initial impulse required for the transfer between two terminal points in space under an arbitrarily prescribed initial velocity vector is analytically investigated. The chordal and radial components of the in-plane velocity are introduced, and a geometrical approach in the hodograph space is employed. In terms of these velocity coordinates Stark's optimum quartic equation is reformulated and critically examined for the number and nature of its real solutions. Analytical criteria for the unrealistic optimum are derived, and the selection of a realistic transfer trajectory under various conditions of the initial velocity vector is discussed and summarized in some simple rules. Various regions in the hodograph plane concerning the nature of the optimum transfer trajectories are established, and the effects of the initial velocity vector on such a trajectory are analyzed. An optimization chart is developed, and the construction of two versions of the optimum transfer hodograph are introduced. Several limiting cases including the vertical transfer, the 180~ transfer, and the transfer to infinity are investigated, and the particular case of departure from a circular orbit is also reviewed. The analysis is basically two-dimensional with a brief presentation of the three-dimensional effects. xiii

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1. INTRODUCTION The minimization of the fuel expenditure for the transfer between two terminal points by minimizing the initial impulse for a given initial velocity is a problem usually encountered in space flight when the primary objective is to impact a destination planet or to intercept a tar et in space. Such a problem has been previously treated by BattiEl) and Stark,( and numerical solutions for the case of an initial circular orbit have been worked out in the works of both. In particular, Stark's orthogonality consideration for the velocity vectors offers a simple approach to, and yields a general quartic equation for the solution of this problem. However, before such an equation can be broadly applied, several critical questions remain to be answered, regarding the existence of multiple real solutions of the quartic as well as the possibility of the arising of an unrealistic optimum trajectory (a trajectory leading toward the destination terminal point via infinityl). It will be shown here that, while the optimum solution is usually (though not always) unique and realistic when the initial velocity is elliptic and only the short transfers (range angle less than 180~) are considered, the situation may become quite complicated when the initial velocity is hyperbolic, and both short and long transfers are under consideration. The purpose of the present study is thus to investigate analytically Stark's quartic as to these vital questions so as to form a theoretical basis for the selection of a realistic optimum transfer trajectory under broad conditions of the prescribed initial velocity vector. Such an investigation will not only facilitate such a selection, but also reveal clearly the effects of the initial velocity vector on the optimum transfer trajectory. Throughout the following analysis a geometrical approach in the hodograph space will be employed. However, to facilitate the investigation the chordal and radial pair of velocity coordinates will be used instead of the usual transversal and radial pair used by Stark. It will be seen later that such a coordinates pair will reduce Stark's quartic to a simpler form, and also enable the general findings previously found in Ref. (9) for a system of two-terminal trajectories to be readily applied to the present problem. Called "false optimum" in Ref. (6); see also Appendix A. 1

2. TWO-DIMENSIONAL ANALYSIS OF THE PROBLEM 2.1 FORMULATION OF THE PROBLEM Consider a space vehicle, initially at the point Q1 and having an initial velocity VO, to be transferred to a given point Q2 by applying an instantaneous impulse at Ql. The optimum transfer trajectory is defined as the one which requires the minimum impulse, which is equivalent to the minimum velocity increment at the initial terminal Q1. As we know, in an inverse-square central gravity field such transfer trajectories are Keplerian and all lie in the plane of the base triangle OQ1 Q2. Let us assume the initial velocity vector V0 also lies in this plane, then the problem is two-dimensional. Consider an arbitrary transfer trajectory from Q1 to Q2, and let V1 be the departure velocity at Q1 along this trajectory (Fig. la). For convenience we will first restrict the vertex angle to be 0 < l < g so that the base triangle does not degenerate into a line segment. In such a case the departure velocity V1 must satisfy Godal's compatibility condition(5) V VR = tan! (1) C R d 2 where VC and VR are the components of the terminal velocity V1 along the direction of the chord line Q1 Q2 and the local radial direction respectively (Fig. lb). The velocity increment vector is then -4 -^ -4 v = v1- V (2) with its magnitude given by AVI2 = (Vc-Vco) + (VR-VRo) - 2 (Vc-Vc)(VR -VRo)cosp (3) which simplifies to 2 23 c+ vR- 2NoVc- 2 MoV+ v0 - 2 K os cl

Q2 V, POSITIVE BRANCH V C"2 \ (Normal Trajectories) f^\2 ^ \0 L^ VI \ zsv IL1 QINET RANCH^ \ QV\ 0b VEOIYCMOET Cmleetr rjcois (a)TRANSFER TRAJECTORY H Fig. 1ert (Vc V0 VO~~~~~~~~~~~~~~~c~~~n (Vo)RI VI x NEGATIVE BRANC (b) VELOCITY COMPONENTS (Complementary Trajectories) (c) THE CONSTRAINING HYPERBOLA Fig. 1. The transfer trajectory and the constraining hyperbola in the hodograph plane.

where K tan 2 d 2 (4) M- V0 - c VO c0 l osc, N0 - VRO cosl (5) Thus the problem is to minimize IAVI under the constraint Eq. (1). It is to be noted that the parameters MO and No here have the physical significance of being the orthogonal projections of the initial velocity vector tO on the VRl- and VC-axes respectively, as is obvious from the geometry of the velocity vectors shown in Fig. lb and c. 2.2 THE CONSTRAINING HYPERBOLA AND THE ORTHOGONALITY CONDITION It is evident that the constraint Eq. (1) represents a hyperbola in the hodograph plane with the chordal and radial directions at Ql as its two asymptotic directions. Thus in order to insure that the trajectory will pass through the terminal point Q2, the tip of the departure velocity vector V1 has to be constrained on this hyperbola, which in a given Newtonian gravity field is solely determined by the base triangle OQ1Q2. The problem is now reduced to finding the minimum distance from the tip QO of the initial velocity vector VO to the constraining hyperbola, and this requires the vector AV to be normal to the hyperbola (Fig. lc). This is the approach used by Star1 ) in which he employed the velocity coordinates VQ and Vr to obtain an optimum equation by such an orthogonality consideration. In present coordinates this condition may be written AVC* - AVR* cos 1 l dVR = _ c i (6) &VR* - AVC* cos ~ dVC * where AVC* - VC- VCO (7) ^VR VR - VRo and (dVR/dVC)* is to be evaluated along the constraining hyperbola. The sub4

script * here indicates the point on the constraining hyperbola at which the normal line passes through the point Q0. Such a point will be referred to as the ortho-point corresponding to Q0. From Eq. (1) we have, at any point on the hyperbola, dVR VR (8) dVC Vc By substituting Eq. (7) into Eq. (6) and making use of Eq. (8) the orthogonality condition becomes 2 2 VC. - No VcR* -M VR* (9) Further eliminating VR* from Eqs. (9) and (1) yields an equation in the single variable VC*: 4 -N V3 + KM V K2 0 C* 0O C 0 C. (10C) The corresponding equation in VR. is V -M V3 + KN V - = 0 (10R) * O0 R* 0 R* Both Eqs. (10C) and (10R) are of the fourth degree, and in fact they are of the same form. They will be referred to as the orthogonality quartics, and their solutions the orthogonality solutions. Either of them can be solved in closed form by standard method of algebra, or by numerical approximations. With either of the unknown components Vc* or VR* thus determined, the other component and the corresponding velocity increment IAV| can then be easily obtained from Eqs. (1) and (3a), and the principal elements of the transfer trajectory are then obtained from the usual orbital relations. However, it is to be noted that the real solution of either Eqs. (10C) or (10R) is not unique, since a quartic may give 4, 2 or no real solutions. Furthermore, the orthogonality condition expressed by such a quartic is neither sufficient nor necessary for the optimum solution of the problem. It is merely a necessary condition for an interior extremem, and it may yield maxima, minima, or neither. And even if it gives a local minimum, it may not be the absolute one; and even if it is absolute, the resulting trajectory may be unrealistic. Thus instead of going into numerical solutions the following vital questions are now posed: 5

(1) Under what condition will the orthogonality Eqs. (10C) or (10R) have a unique real solution, 2, 4 or no real solutions? (2) If multiple solutions exist, is there any simple rule for the selection of an absolute minimum? (3) Under what condition will the absolute minimum solution yield an unrealistic optimum? And if so, how to choose a realistic optimum trajectory for the problem? These questions will be critically examined one by one in the sections that follow. Before proceeding to answering these questions, the dimensionless velocity parameter defined by v- V /s-V/ (11) will now be introduced and the principal equations developed so far, be non-dimensionalized as summarized in Table 1. Besides, formulas for the principal geometrical elements of the constraining hyperbola are presented in Table 2. Some essential features of the constraining hyperbola worthy of noting are as follows: (1) The conjugate and transversal axes of the hyperbola (vX-,v~axes) are the bisectors of the interior and exterior angles at the initial terminal Q1 of the base triangle respectively. The Vs-axis is in the direction of the minimum energy trajectory through the initial terminal according to Ref. (9) and may be called the minimum energy axis. The pair of directions (x, 0) together with the pair of the asymptotic directions (C,R1) mentioned earlier and their respective normals to be introduced later constitute the most important reference directions of the present problem. (2) The semi-transversal axis (A) of the constraining hyperbola is the minimum velocity satisfying the constraint, and therefore, the departure velocity along the minimum energy transfer trajectory. (3) Of the two branches of the hyperbola, the one on which Vc > 0, and VR > 0 is the constraint for the short transfer or the normal trajectory group,2 and the other one on which Vc < 0 and VR < 0 is the constraint for the long transfer, or the complementary group. 2For the definitions of these terms, see Appendix A. 6

TABLE 1 PRINCIPAL FORMULAS IN THE NONDIMENSIONAL FORM FOR THE TERMINAL-TO-TERMINAL OPTIMUM TRANSFER Compatibility Condition VCVR = K (1') Velocity Increment lav12 = V2+VR -2 ovc-27ovR +Vo2-2K cos (3') The Orthogonality Equation ~in vc T v -2 YoI - - vo2 (9') in VR C* V vR * - R in vC v - 71 + Km2 = (o0-C) VC. o C* o C* in vR V - YoV + K YoR - 2 0= (10'-R) The Constant Product K - tan 2 csccp (4') The Orthogonal Projections 7n (vR ) - (VC) cos c1 of the Initial Velocity Vector ~ ( ( no 0 (vc) - (vR ) cos (P 7

TABLE 2 PRINCIPAL GEOMETRICAL ELEMENTS OF THE CONSTRAINING HYPERBOLA (VCvR = K) Formulas Element SymbolNumbering In terms of K, cpl In terms of J, c1 In terms of r,- n(= r2/rl) Included angle between 2 sin \ the asymptotes a - cp1 tl sn 1 sin-1 _____________k(1 + 1) sec2!- n 2 n Semi-transversal axis A 2 K sin t1 2 tan tan se ( + 1)2 sec2 (1 + ) sec2t (1) 2 2 2 2n n 2 c0 Semi-conjugate axis B 2~ cos ~P1 se( 1 2 Semi-conjugate axis B 22 JK cos 1 |2 tan Y cot /l |sec + se2 |/ + (1 + ) sec2 2 - 2 (14) 2 2 2 2 n n n 2 Center-to-focus distance C 2 ~ K 2 tan 2 csc c 2 sec (1 + )2 sec 2 - (15) q Eoqe~ricit j l csc 21 5 2 1/2 Eccentricity e csc - csc- 2 (16) 2 2 1 - 2 cos2 6 )1- 1) 4 2s,, n n 2

(4) Points on the hyperbola which are symmetrical with respect to its transversal axis correspond to a pair of conjugate trajectories, and will be called the conjugate points; points symmetrical with respect to the origin correspond to a pair of complementary trajectories, and will be called the complementary points. Consequently, points symmetrical with respect to the conjugate axis correspond to a pair of complementaryconjugate trajectories. Such a point pair will be called a complementary conjugate pair. For the convenience of later development the quadrants of the hodograph plane bounded by the symmetrical axes of the constraining hyperbola will be referred to as positive (+) or negative (-) according as it is on the positive or the negative side of the Vx-axis; and high (H) or low (L) according as it is above or below the V —axis. The parts of the constraining hyperbola and all velocity vectors will also be so referred to according to the quadrant in which they lie. Such subdivisions are depicted in Fig. 2. Yg (II) (I) TH- H+ 0 ~ TVX L- H(III) (IV) Fig. 2. Quadrants in the hodograph plane. 2.3 CRITERION OF THE NATURE OF THE REAL SOLUTIONS AND THE BOUNDARY EVOLUTE In order to examine the nature of the solutions of the orthogonality quartic, either the Vc-equation (10C) or the V.R-equation (LOR) may be used since they are identical in form and have essentially the same discriminant. To fix the idea the following discussion will be based on the V~c-equation. 9

The discriminant for such a quartic is given by A - I3 - 27J2 (17) where I =- 4 K (ohO - 4K) (18) 1 2 2 2 16 0 0 By using Burnside's criteria together with Descartes' Rule of Signs we arrive at the conclusions in the first two columns of Table 5, classifying the nature of the real roots. Since multiple roots of the equation give identical solutions, they will be considered as one solution. From such considerations we arrive at the further conclusions in column IV, Table 3. The geometrical implication of such conclusions may be seen as follows. With the expressions (17) and (18) the boundary condition A = 0 may be written 4(>oo - 4.)3 - 27 K - 2)2 = 0 (19) Now introduce the polar coordinates (v,a) for the velocity vector v and express the parameters 1ro and'Yo for the initial velocity vector as MGo = vosin (I + 0o) 1~~~~2 ~(20) )o = vosin (2 - o) where 0 is the path angle referring to the minimum energy axis, and is related to the usual path angle 0 by (D = 0 - (21) 2 By substituting these expressions into the boundary Eq. (19) we obtain 10

TABLE 3 NATURE OF THE REAL ROOTS (vc* OR vR*) OF THE ORTHOGONALITY QUARTIC AND NUMBER OF REAL SOLUTIONS (I) (II) (III) (IV) Real Roots No of Real, Distinct Orthogonality Solutions Number egins Positive Solutions Negative Solutions Regions Discriminant Nature Positive Roots Negative Roots (see Fig. 3) (vC > 0, VR > 0) (VC < 0, VR < 0) TOTAL A < 0 2 real roots, distinct 1 1 S+, S_ 1 1 2 A > 0 4 real roots, all dis- 3 1 N+ 3 1 4 tinct 1 3 N_ 1 3 4 3 1 H+ 2 1 _- (two equal)' branches A = 0 4 real roots, not all 3 1 cusp G 1 1 2 distinct (all equal) 1 3 H 1 2 (two equal) branches L_ 1 3 cusp G' 1 1 2 (all equal)

o2 3 4.n2 2 [v (cos 2o-coscpl) - 8]3 - 54K vO sin sin 2 = o (22) where vo is the magnitude of the initial vector vo, which satisfies the boundary condition. This equation may be transformed into the following standard form in the rectangular coordinates (vX,vt) 2/3 / - 2/3 4/3 (25) (AvG) - (BV) = C where the parameters A, B, and C have been given in Table 3. From Eq. (23) this boundary curve is recognized as one form of the Lame curve, which in the present case is the evolute of the constraining hyperbola. Some essential features of this curve are as follows (see Fig. 5): (1) It is symmetrical with respect to both vx and vS axes. Thus the boundary Lame and the constraining hyperbola are co-axial. (2) It is bounded between the vc- and vd-axes which are normal to the asymptotic directions of the constraining hyperbola (the radial and chordal directions) respectively. (3) It has two portions, one on each side of the vx-axis and each portion consists of two branches with a cusp (G,G') at its vertex given by the coordinates (to G = 2 csc 2l, (c) = 0 (24) G;1' 2 GG' It is well-known that an evolute of a given curve is the envelope of all normals of this curve, or conversely, the given curve is the involute of its evolute. Since to find solutions of the orthogonality quartic according to a pair of given values of nto and 7Q is equivalent to drawing normals to the constraining hyperbola from a given point in the hodograph plane, naturally its evolute should form the boundary separating the regions in which different number of such normals can be drawn. Directly from the concept of an evolute and the geometry of the hyperbola we see that All points of intersections of the different normals to the constraining hyperbola are in the regions beyond the boundary Lame, and no two normals to the hyperbola can intersect in the region between the two portions of the boundary Lame. The latter region will be referred to as simple (S), while the former, non12

y i^, BOUNDARY LAME 03 c v APRl A' \. 2 2...., 0 POONSTRAINING sr1 Q BOUNDARY LAME \. RRE Fig. 5. Geometry of the constraining hyperbola and the boundary Lame. 115

simple (n). For convenience various portions of these regions, together with their boundaries, will be referred to as positive (+) or negative (-), and high (H) or low (L), according to the quadrant in which they are located, just as for the portions of the constraining hyperbola,(Fig. 3). With the foregoing understanding the conclusions previously derived from algebraic considerations may now be stated in geometrical terms as follows: (A) Within the simple region one and only one normal can be drawn from a given point to each branch of the hyperbola. (B) Within the non-simple region four distinct normals can be drawn from a given point, three to the nearer branch, and one to the farther branch. (C) From any point on the boundary three distinct normals can be drawn: two to the nearer branch, and one to the farther branch except at the cusp, where only one normal can be drawn to each branch, both coinciding with the transversal axis. Moreover, further examination of the geometry of a hyperbola shows that, (D) The normals at points of the hyperbola in the same quadrant always intersect in the adjacent quadrant on the opposite side of the transversal axis of the hyperbola. (For example, two normals to the H+ part of the constraining hyperbola can meet only in the L+ portion of the N-region. This property is especially useful in the later treatment of the present problem; an analytical proof is given in Appendix B.) Finally, it is to be noted that for a given vertex angle 4 the distance of either cusp of the boundary Lame from the origin, (Vo)GI G decreases with increasing pl or n, and it has the limiting value o~ G GI sec2 2 > 2 when p-O 0 (n-o) (25) (~O)GjG,G' 2 (2 Thus multiple real solutions can occur in the half-plane (v: > 0 or v: < 0) only when the initial velocity is hyperbolic. Furthermore, owing to the presence of the asymptotic lines of the boundary Lame, such a case cannot occur unless the initial velocity vector is directed above the local horizon but below its conjugate direction, the vd-axis. A necessary and sufficient condition for the occurrence of such multiple solutions in the positive halfplane (v: > 0) may be precisely stated as follows: vo > vO and < < + < (26) 14

where Vo is given by Eq. (22). Similar condition exists for the other halfplane (v < 0) by symmetry. 15

3. DETERMINATION OF THE OPTIMUM SOLUTION 3.1 THE ABSOLUTE MINIMUM SOLUTION With the number of real and distinct solutions of the orthogonality quartic determined, the next task is to select the one for absolute minimum. For the time being let us disregard the question of unrealistic trajectory, and consider only the geometrical problem of determining the absolute minimum distance.3 Such questions of maxima and minima can usually be settled by the second derivative test, and the absolute minimum determined by comparing the quantity to be minimized at these stationary points. However, it is simpler here to use a geometrical approach outlined below: A. From the symmetrical nature of the hyperbola, it is evident that the minimum distance solution demands the optimum point on the constraining hyperbola to be in the same quadrant with the tip Qo. of the initial velocity vector. However, in view of the geometrical property of the hyperbola,;given by item (D) of the previous section, there is one and only one such a point on the constraining hyperbola in the same quadrant with the given point (see Figs. 4a, b, c) unless Qo is on either of the symmetrical axes of the constraining hyperbola. This is true whether the point Qo is in the simple or non-simple region. Thus when Qo is off the symmetrical axes, the choice is clear, and the absolute minimum distance solution is unique. Furthermore, directly from this co-quadrant requirement it can be inferred immediately that the trajectory corresponding to such a solution always belongs to the same group (normal or complementary) and the same class (high or low) as the initial velocity vector. B. In case Qo lies on either of the symmetrical axes, then it is on she border of two adjacent quadrants. In such a case the minimum distance requirement is to have the optimum point lie in the half-plane of these two quadrants; and thus two solutions are possibleo (1) If QO lies on the vX-axis, then the optimum point must be on the same side of the vs-axis with Qo. The geometry in the hodograph plane shows that Qo is in the simple region and equidistant from both branches of the hyperbola. Thus there are two and only two normals which can be drawn from Qo, one to each branch, and they are of equal length. Consequently both ortho-points may be admitted, and there are two solutions for absolute minimum distance. The two corresponding trajectories require the same amount of Av, and their departure velocities also have the same magnitude. Obviously 3The absolute minimum distance solution will be indicated by the subscript ** whenever it is to be distinguished from the orthogonality solution. 16

('a) a>O0 no>mo (no>o) v 0 c (d )A<O0 no=-mo>o obs.m min..1 O4'. 1?; "2.RI3 r " 0 - 0* I P4 __ _ _ _ _ _ _ _ __ _ -___X VX Fig. 4. Variation of jAv| and the corresponding geometry in the hodograph plane. I~~~Fi. 4 Vriaionofl:v an te crreponin gemety n te hdorap plne

TABLE 4 THE REAL ROOTS OF THE VC*-EQUATION AND THE NATURE OF THE STATIONARY POINTS ON THE I Av -CURVE (for the case > 0 and )o >?o) A=O Designation A> 0 I2 + J2 O I = O, J = 0 < of the root sign of nature sign of nature sign of nature sign of nature v * VC* o' lAvi VC* of lAvi v* of |Avi vC of |Av c i1 + min + min mi 2 + ma x + min m _ all equal) 3 + min + neither (equal) 4 - min - min - min - min

they constitute a complementary-conjugate pair, one belongs to the normal group, and the other, the complementary group. They will be either both high or both low according as the initial velocity vector is high or low. This situation is depicted in Fig. 4d. (2) If Qo lies on the vs-axis, then the optimum point must lie on the same side of the vX-axis. Now Qo may be either in the simple region (S) or the non-simple region (N). i. Suppose Qo is in the S-region, that is, it lies between two cusps, G and G', of the boundary Lame. Evidently the two and only two ortho-points now coincide with the vertices foD and oP' of the hyperbola, and there is only one on the same side of the vX-axis with Qo. Thus the absolute minimum solution is again unique, and the corresponding trajectory is the minimum energy one. It will belong to the same group as the initial velocity vector. This situation is depicted in Fig. 4e. ii. Suppose Qo is in the N-region, that is, it lies on the parts of the vt-axis which are beyond the cusp points of the boundary Lame in either direction. Then according to property (C) given in Section 2.3, there are three normals on the branch of the hyperbola on the same side of the vXaxis. It is evident from the symmetry of the hyperbola that among the three ortho-points, which are on the branch nearer to the initial point Qo, one coincides with the vertex, while the other two are of a conjugate pair, and equidistant from Qo. The fourth ortho-point coincides with the other vertex. This situation is depicted in Fig. 4f. Evidently, the fourth point should be rejected, and the choice will be between the point Q*2 and either of the points Q*1 and Q*3' It can be shown that it is always the point Q*2 which is at a farther distance. (This can be easily proved by solving the orthogonality quartic with Io = Po, and comparing the distances since in this particular case the quartic admits a simple solution.) Consequently, both points Q*1 and Q*3 may be admitted, and there are two solutions giving the same amount of Av. The two corresponding trajectories are conjugate to each other, requiring the same magnitude of departure velocity, and they are both of the same group as the initial velocity vector. It is interesting to note here that the minimum energy trajectory is no longer the optimum transfer trajectory even though the initial velocity is in that direction; the two optimum directions are now inclined equally on either side of the minimum energy direction instead of lying along it. iii. Finally when the point Qo is at either cusp of the boundary Lame, then both conjugate points coincide with the nearer vertex, and the minimum distance solution is again unique, and the corresponding trajectory is again a minimum energy trajectory. This is the same as case ii. In conclusion, 19

(1) Whenever the point Qo is not on the conjugate axis of the hyperbola nor on the part of its transversal axis beyond the cusps of its evolute, the absolute minimum distance solution is unique. The corresponding trajectory will belong to the same group and same class as the initial velocity vector. (2) Whenever Qo is on the conjugate axis of the constraining hyperbola there are two solutions with the same minimum distance. The corresponding trajectories are a complementary-conjugate pair of the same class as the initial velocity vectort (3) Whenever Qo is on the transversal axis of the constraining hyperbola beyond the cusp points of the boundary Lame, there are again two absolute minimum distance solutions. The corresponding trajectories are a conjugate pair of the same group as the initial velocity vector. Based on such geometrical analysis we may now form the following "rules of thumb": Rules-Geometric (1) Always choose the optimum point which is in the same quadrant with the point Qo whenever no ambiguity arises. (One and only one solution.) (2) If ambiguity does arise such as when the point Qo lies on either of the symmetrical axes of the constraining hyperbola, always choose the optimum point or points in the same half-plane with Qo, and the ones off the minimum energy axis if they are present. As shown above the geometrical rule for the selection of the absolute minimum solution is exceedingly simple. Such a geometrical analysis may in turn guide the selection of the appropriate root from the real solutions of the orthogonality quartic for an absolute minimum without calculating the magnitudes of the corresponding Av's. In view of the symmetry of the constraining hyperbola it is sufficient to consider all the possible cases when Qo is in one certain quadrant, say the second, and center our attention on the variation of |Av| with one variable, say C*, when Qo is in this quadrant. The geometry of such cases are illustrated in Fig. 4, and the corresponding variation of JAvj with vC. and the nature of its stationary points as obtained from usual algebraic analysis are also graphically shown in Fig. 4 for each case, and summarized in Table 4 for reference. It is to be noted that the present restriction of Qo in quadrant II is equivalent to saying YLo >Ylo andlo > 0 in the orthogonality quartic. Keeping this in mind and without going into algebraic details, an examination of the geometry of the hodograph plane shows that: 20

When lo + 7?o (QO off the symmetrical axes), the optimum point in the hodograph plane always corresponds to the highest root vCol of the orthogonality quartic (see Fig. 4a, b, c). When 771n = NO (Q0 on v -axis), the co-half-plane requirement from geometrical considerations indicates that the optimum root vc** must agree in sign with the initial value v0o. Thus, under the present assumption, only the positive roots need be considered. The hodograph shows that there may be either one or three such roots corresponding to the one or three ortho-points on the positive branch of the constraining hyperbola. In the former case the only positive root is necessarily the optimum one. In the latter case the geometry of the hodograph shows that the pair of optimum points correspond to the highest and the lowest roots respectively (see Fig. 4f). Thus both roots may be chosen. It is to be noted that the prerequisite to have vC** agree in sign with v:o hold in general whenever t7o = 720. When )o = - to (Qo on vX-axis), the two optimum points in the hodograph plane, one on each branch, correspond to the two and only two real roots of the quartic, one positive and one negative (see Fig. 4d). Thus again both roots may be chosen. All the foregoing observations were made on the L+ portion of the constraining hyperbola. The symmetry of the hyperbola with its conjugate axis shows that the same is true for the L_ portion if we take the magnitude of the root algebraically. Thus the same conclusions hold in the low-half-plane where "Io > )o0. In the high-half-plane, we have ioZ < _70o. By the symmetrical nature of the hyperbola with its transversal axis, whatever is true for vC in the low-half-plane is equally true for vR in the high-half-plane. Or, in view of the reciprocal relation between vC and vR (Eq. (1)) we may say that whatever is true for the largest vC (algebraic) in the low-half-plane is equally true for the smallest vC (algebraic) in the high]-half-plane. Based on such observations we may form some algebraic rules of thumb as follows: Rules-Algebraic (1) If m i ~ 7?, always choose the root which agrees in sign with the initial value of v:; and if more than one such root is present, choose the largest one if ^f7 and the smallest one if r?0 < 71o (one solution only). (2) If tlo = 7?o, choose both the largest and the smallest roots which agree in sign with v:o (two solutions). (3) If o = -', only two real roots are present, both may be chosen (two solutions). The magnitudes of roots are being considered algebraically. All rules (1) to (5) hold for the VR.-equation (10R) if we interchange the words X7 and t lo 21

352 LINES OF CONSTANT OPTIMUM TRAJECTORY AND LINES OF CONSTANT VELOCITY INCREMENT Before we take up the question of unrealistic trajectories, it is essential to note that when the tip Qo of the initial velocity vector moves along a straight line normal to the constraining hyperbola, the absolute minimim point Q** remains intact, and consequently the corresponding transfer trajectories are the same as long as Qo remains in the same quadrant. Such a trajectory will be the optimum trajectory for the present problem unless it is unrealistic. Thus the part of the normal line intercepted by the symmetrical axes of the constraining hyperbola (e.g., line cd in Fig. 5) may be regarded as a line of constant optimum trajectory. As soon as the normal line crosses either axis. the absolute minimum point will shift to the other side of the axis and move along the constraining hyperbola resulting in a different trajectory for each point on the extended part of this normal line. It is to be noted that along a line of constant optimum trajectory the velocity-increment required varies from point to point depending on the position of Qo on this line, the farther Qo is from the constraining hyperbola, the larger the velocity-increment (absolute value) required. In such a connection we may conceive that, when Qo moves along a curve running parallel to the constraining hyperbola, the amount of velocity increment required will remain the same while the optimum transfer trajectory changes from point to point. Thus such parallel curves may be regarded as lines of constant optimum velocity-increment. As known in geometry, all these parallel curves have the same normal lines and a common evolute. In the present case the boundary Lame is this common evolute, and each of the parallel curves, including the constraining hyperbola is its involute. Thus the lines of constant optimum trajectory and the lines of constant velocity-increment are normal to each other, forming an orthogonal net in the hodograph plane. Such a net will be useful in developing * hodograph charts for the present problem, which will be presented after the question of unrealistic trajectories:has been cleared up. For the time being it is to be noted that such parallel curves though quite similar to the original curve (the constraining hyperbola) when they are close to it, may look radically different from it when they are farther from the hyperbola, especially when they enter the non-simple region. The mathe:. matic equation for the curves parallel to a hyperbola is in general of the eighth degree,(l) A few such typical curves are shown in Fig. 6. 3.3 THE CRITICAL CONDITION AND THE UNREALISTIC TRAJECTORIES From the foregoing consideration of the lines of constant transfer trajectories it is evident that when the tip Qo of the initial velocity vector moves along such a line which passes through a critical point (v = /2) on the constraining hyperbola, the absolute minimum distance solution will call for a parabolic trajectory. Such lines will be called the critical lines. Figure 7 shows the four critical lines, one through each of the four critical points 22

vt \ / \ / /~ BOUNDARY LAME \ LINES OF CONSTANT / Z/ \ \ \WVELOCITY INCREMENT ARALLEL CURVES) Fig. 5. Lines of constant optimum transfer trajectory and lines of constant optimum velocity-increment. 25 r eIrjl C! x ~ 23

V D =Si e Const.Reg Fig. 6. Geometry of the parallel curve /,, / HYPERBOLA BOUNDARY LAME (o) Correspondence Between the Hyperbola and It's (b)Three Parallel Curves Passing Parallel Curve. Through a Point in the NonSimple Region. Fig. 6. Geometry of the parallel curves.

(H Hyperbolic Tronsfer (Realistic) / Constraining Hyperbolica or bit ECVI Line;. U Unrealistic re:ion: no d egion of Unrealistic CRITICAL CIRCLE Transfer! lEji^ / I | Equi- critical velocity-increhentrline'Elliptic o,i.... R ealistic) r n _ _ a l Line ______hproiorbitu_______________requiresellipticopt Fig. 7. Regions of the hodograph plane and. the nature of the optimum transfer trajectory. 25 ~. i""':-~iIntalelitc ritrqurs~ hyeblc piu ~_-! Inital elliptc orbit rqursneaitiopmu

on the constraining hyperbola, forming a critical circuit a-b-a'-b'-a. These four critical points are given by the intersections of the hyperbola with the critical circle centered at the hodograph origin and having the radius /2. When Qo moves along such a circuit, the trajectory corresponding to the absolute minimum distance solution will first be a parabola of the high class and normal group when Qo remains on ab, and as soon as it passes the point b, the trajectory will shift to its conjugate, and so forth. As seen from the hodograph geometry, as long as Qo is inside the rhomboidshaped region bounded by the four critical lines, the corresponding absolute minimum point on the constraining hyperbola will remain inside the critical circle, consequently the transfer trajectory will be elliptic. This region will therefore be called the elliptic region. When Qo is on the boundary of this rhomboid and beyond, the corresponding absolute minimum distance trajectory will first become parabolic and then hyperbolic. Thus the regions beyond the critical boundaries are hyperbolic regions. As shown in Ref. (9), a transfer trajectory between two fixed terminal points will be unrealistic only when it is parabolic or hyperbolic, and of the high class. Consequently the hyperbolic region on the high side is the region for unrealistic optimum transfer, and will be called the unrealistic region, while that on the low side, and the elliptic region in between are regions for realistic optimum transfer, and will be called the realistic region. Thus the boundary b-a'-b' separates the region for hyperbolic transfer from that for elliptic transfer, all realistic; while the boundary b'-a-b separates the elliptic realistic region from that of unrealistic transfer. Hence the two critical lines on the high. side will hereafter be referred to as the realistic barrier. Beyond the vertices b and b' of the rhomboid aba'b' the realistic and the unrealistic regions are further separated by the vP-axis, which itself belongs to the realistic region. In short, the broken line b'-a-b and the part of vt-axis beyond either b or b' form the entire realistic barrier which divides the whole hodograph plane into two main regions, the realistic region and the unrealistic region for the optimum transfer. With such a partition established in the hodograph plane we may say that the absolute minimum distance solutions obtained in the preceding analysis is actually the optimum solution of the problem whenever the tip Qo of the initial velocity vector lies in the realistic region. It ceases to be the optimum only when Qo is beyond the realistic barrier, or on the boundary b'-a-b, excluding the two end points b and b'. The various regions in the hodograph plane are shown in Fig. 7, and further divisions of the unrealistic region will be presented in the next section. It is interesting to note that the type of the optimum transfer trajectory, whether elliptic, parabolic, or hyperbolic does not necessarily agree with that of the initial velocity. The shaded region beyond the critical lines on the low side but within the critical circle is the region where the. initial velocity is elliptic, but the optimum solution calls for a hyperbolic transfer. Similarly, the shaded region beyond the critical circle but within the rhomboid is the region where a hyperbolic initial velocity calls for an optimum elliptic transfer. 26

It is also evident from the hodograph that even an elliptic initial velocity, if at sufficiently high path angle, may introduce an unrealistic optimum. The geometrical criterion for an unrealistic optimum transfer obtained so far will be analytically formulated as follows: First, we note that there always exist such critical points, where v =$2, on the constraining hyperbola, because the minimum velocity along this hyperbola is always elliptic according to Eq. (13), Table 2, Vmin = A = tan tan < 2 (27) since cp1 < - - 1r. The condition to be satisfied by the initial velocity vector in order that its tip lies on the critical line through a critical point (vj, vR) is then, according to Eq. (9'), * * *2 *2'o0 VC - Qo VR =VC- -R (28) Proceeding from the oblique coordinates (vC,vR) to the rectangular coordinates (vX,v(),Eq. (28) may be transformed into vv sin2 1 + vvXcoS2 1 = vv (29) XY to 2 - 2VxV which finally reduces to the polar form Vo( VX cos sin2 - + v: sin 0 cos2 = v* (5~) 2 2 with vXo= vo sin bo (51) Vto= Vo cos Co The coordinates of the four critical points as given by the intersection of the critical circle and the constraining hyperbola are found as follows: 27

V - Vx 1* + X* + fY* 2* - ^ + IY* (32) 3* -.X* - JY* 4* + IX* - Y where X* = 2 sec cos (P cos + c1) (55) 2 2 2 Y* = 2 sec sin sin + The four points are numbered according to the quadrant they are in (see Fig. 7). Let vo be the initial velocity vector satisfying Eq, (30) then by inserting Eqs. (32,33) into Eq. (30) we may express the critical condition along the boundary a-b-a'-b'-a as summarized below (where the usual subscripts H+ etc. are used to indicate the quadrant where the tip of Qo lies): along a-b (0 S 00 < (2) (o) (C co08 0 + C2 sin *o). 1 (54-1) along b-a' (- i < o S< ) (o)* (C1 CO8 0o - C2 sin o0) - 1 (34-2) along a'-b' (- < so <_ - ) (V)L- (C1 CO. 00 + c2 sin 0) - (-3) 28 HBn CB, |O^C C, ~/ —-s —-^^^^ -— T —-: qh

Recalling that the realistic barrier is along b'-a-b, a criterion for unrealistic optimum transfer may now be stated as follows: o < o vo > (o)H+ (36) < o < t: Vo > (Vo*)HSimilarly, recalling that the realistic critical boundary is along b-a'-b', and that realistic hyperbolic transfer exists along the vt-axis beyond b and b', a criterion for parabolic and hyperbolic optimum transfer is <- o <0: vo > (vo) 2- - L+ - 2o > > —' Vo > (vo)L- (37) DO = 0,': Vo > Vb,b' By setting vXo= 0 in Eq. (29) and using formulas (32) and (33) we find the distance from the origin to either corner point, b or b', Vbb = b = 12 sec I csc- sin ( + 1) (38) It can be shown by comparing Eq. (38) with Eq. (24) that, Vbb' > VGG' That is, the corners of the elliptic region always extend into the non-simple regions. This should be expected since either point b or b' is an intersection of two normals to the constraining hyperbola. This situation implies that two realistic optimum solutions exist in the elliptic region when the initial velocity is in the minimum energy direction, and has the magnitude VGG' < vo < Vb,b' As discussed before, the optimum solution in such a case does not give the minimum energy trajectory, but instead it gives a conjugate pair of two trajectories. And, within the present range of vo they are both elliptic of course. The same situation exists when vO > Wb,b' except that the optimum trajectory is now hyperbolic, and the realistic optimum solution is unique since its conjugate becomes unrealistic. 29

Finally, as the hodograph shows, there is a minimum initial speed (Vo)L below which neither a critical nor an unrealistic optimum can occur, for whatever the path angle may be. This is given by.the length of the perpendicular drawn from the origin to any of the critical lines, e.g., the line segment oe in Fig. 7. From the trigonometry of the triangle oab, we find sec J sin(*+-pl) (V)L = Oe 2 ( l (39) (1*)L ==3 Cl 1(*+Cpl)+ COS3 (P sin 1 Jsin 2 cos (4+cp9)+ cos sin(+) 2 2 2 2 For example, if r = 60~, c1 = 75~ (corresponding to the transfer to a target point at the distance ratio n = 1.366) we have (vo)L = 1.22. Besides, it is evident that unrealistic optimum cannot occur when the initial velocity vector is in the low half-plane (o_< 0). 3.4 CHOICE OF THE REALISTIC OPTIMUM TRANSFER TRAJECTORY From the preceding analysis the absolute minimum solution of the orthogonality quartic is the optimum solution of the problem whenever the tip Qo of the initial velocity vector is in the realistic region. However, whenever Qo is outside this region, the absolute minimum solution is an unrealistic optimum, from the physical point of view, and it remains to select a realistic optimum trajectory for the problem. Such a selection will depend on whether the point Qo is in the simple or non-simple region of the.hodograph plane. A. Suppose Qo is in the simple region and off the vX-axis. Then the absolute minimum distance solution is unique. In such a case it is evident that the best choice will be the point on the constraining hyperbola sufficiently close to the critical point in the same quadrant with the initial point Qo but still within the elliptic region. Thus, strictly speaking, there is no definite optimum solution for the problem in this case. The transfer trajectory so chosen will necessarily be highly eccentric, of the same class (high) and same group as the initial velocity vector. If Qo is on the v -axis, then the two critical points on the realistic barrier, one on each side of the vX-axis may be the reference points, and points close to either critical point may be chosen. B. Suppose Qo is in the non-simple region. We recall that in such a region three normals can be drawn from the point Qo to the nearer branch of the constraining hyperbola. For definiteness let us assume Qo is the H+ portion of the region N (see Fig. 8). Then the three ortho-points on the constraining hyperbola will be distributed as follows: 50

ir CONSTRAINING i (i ) Qo HYPERBOLA.. JA VI tC..... Unrealistic U;o ='L.~ P01"1.\ ii\ V^ 0optimu\m ~~Optimum Point I'rCV Fig. 8 Choice of the realistic optimum trajectory: initial velocity vector in the unrealistic region. velocity vector in the unrealistic region.

Branch of the Constraining Ortho-Point Hyperbola Nature of the Solution *1 H+ Min., absolute, unrealistic *2 L+ Max. *3 L+ Min., local, realistic Thus, besides the unrealistic minimum there is a second minimum for consideration, which is realistic. Let IAv13 and |Av|* be the velocity-increments required at the point 3 and the critical point under consideration (e.g., point 1* in Fig. 8) respectively. Then the choise will depend on the magnitudes of these two quantities. (1) If |Av13 < |Avl*, then the optimum trajectory is definite and unique, as given by the point *3. (2) If |Av13 > IAvI*, then some point close to the critical point but within the elliptic region should be chosen. This case is the same as case A. C. Suppose Qo is on the boundary Lame, then the points *2 and *3 coincide, giving neither minimum nor maximum, leaving the unrealistic point *1 to be the only minimum solution. This case is again the same as case A. In making the foregoing comparison, the concept of constant velocity-increment introduced in Section 3.2 is helpful. It is to be noted that while such lines are curves parallel.to the constraining hyperbola in the realistic region, they are concentric circles centered at the reference critical point in the unrealistic region, since in this latter region the velocity increment at the critical point is the standard for comparison. The point in the unrealistic region at which I|v13 = lavl* is then given by the intersection of such a circle with one of the parallel curves of the same constant JAvJ as illustrated in Fig. 9. Of course only these intersections within the non-simple region are of interest at present. The locus of. all such points of intersections in the unrealistic region will be called the line of equi-critical-velocity-increment (E-C-V-I line for short), and there is one such line on either side of the vx-axis. As shown in Fig. 7 these two lines further divide the unrealistic region into the following subregions: the one (U2) bounded by each E-C-V-I line and the VC-axis is the one in which we have jAv31 < jAvj* and therefore the realistic optimum solution is definite and unique; and the one (U2) bounded between these two lines is the subregion in which either |Av31 > |Av|* or Av3 does not exist, therefore the realistic optimum solution of the problem is definite. On the boundary 1AV3J = IAVJ* the realistic optimum solution is 32

<.,, /., —- t-Line of Equi-Critical-Velocity-C' ^U Increment r'CONBOUNDARY HYLAMPE b CRITICAL \ CIRCLE 3 2*^^^^~~~~~~ ^^ ^Unreolistic 2* - 1^^ ^ ^ ^ ^ ^ I Optim um 00 CONSTRAINING HYPERBOLA Fig. 9. Determination of the boundary point, |Av13 = jAvl* 533

TABLE 5 REGIONS IN THE HODOGRAPH PLANE AND THE NATURE OF THE OPTIMUM TRANSFER TRAJECTORIES Number of Region Realistic Nature of the Realistic Optimum Optimum Transfer Trajectory Name Designation Description Subregion SOptim Solutions Off vrve -axes 1 Elliptic On v -axis: between 2 Elliptic, complementarya and a' conjugate pair Elliptic Within the rhomboid Region | aba'b' On vs-axis: between 1 Elliptic, minimum ener and at' and On Vs-axis: between 2 Elliptic, conjugate pair e-and b or a' and b' Critical b-abIncluding end 1 Parabolic b-a'-b' 1 Parabolic A-n Line points b and b' On the low side of Off y,V -axes 1 Hyperbolic Hyperbolic | the line b-a'-b' and Region the.O.n a. Hyperbolic, complementarythe v-axis On vc-axis: beyond a' 2 conjugate pair v~-axis Beyond b or b' 1 Hyperbolic Realistic ealistiEcluding end Elliptic, highly eccentric, Barrier Excluding end Line b'-a-b oint nd Indefinite close to the unrealistic points b and b' parabolic trajectory Between the two Elliptic, highly eccentric, lines of equi-critilines of eui-criti- Indefinite close to the unrealistic cal-velocity-incre- parabolic trajectory Unrealistic ^ On the high side ment (U1) Region and the vs-axis Between the vs-axis and each line of equi- 1 Hyperbolic critical-velocityincrement (U2+,U2 )

also definite and unique. The subregion U2 falls entirely within the nonsimple region of course. Following the previous analysis we see that in this subregion the realistic optimum solution is to be found by following the normal line through the initial point Qo. to the constraining hyperbola in the low half plane. This practically extends the applicability of the normal lines originated from the low-half plane to the high half-plane until the E-C-V-I-line. Evidently the same is true for the parallel curves. Furthermore, the geometry of the hodograph shows that, whenever a definite realistic optimum solution exists while the tip Qo of the initial velocity vector is in the unrealistic region the optimum transfer trajectory will always be hyperbolic of the low class, since the two E-C-V-I lines terminate at the corners b and b' of the elliptic region. The foregoing analysis completes the discussion on the selection of the realistic optimum transfer trajectory for the problem. All the previous conclusions on such selections are summarized in Table 5. 5.5 THE MINIMUM VELOCITY-INCREMENT OF THE OPTIMUM SOLUTION Following the previous analysis the realistic optimum solution of the problem is indefinite whenever Qo is in the region U1, and is definite everywhere else. The definite optimum solution is provided by the orthogonality Eq. (10'-C or 10'-R), and the corresponding minimum velocity-increment is given by Eq. (3'), which may be written alternately, lpvy.2 2. 2 (3'-C) Av* = 2vC* - 3tLov - + v - 2 tan- cot C1 (3C) VC* o 2 2 2VR - 3 R - + v 2 tan 2 cot c (3'-R) =2R- 3oVR. - CVR* o by using Eqs. (10'-CR). The indefinite optimum solution may be written approximately, v = v (40) lAv* = IAV*l where * is the critical velocity vector co-quadrant with vo, and |Av*i is given by IAvj- = -o = 2 f cos(io —*) + 2 (41) 55

where (o and 0 are the path angles of vo and v respectively, both referring to the minimum energy direction at the initial terminal. For a given base triangle the effects of the initial velocity vector v on the magnitude of AIv may be easily seen from the hodograph geometry. Let us first consider the case when vo has a constant direction but varying magnitude, that is, when Qo moves along a directed straight line through the hodograph origin at an arbitrary angle 0o such as the t -lines in Fig. lOa. When QO moves from the origin outward, the hodograph shows that IAvI| first increases and then decreases. In the realistic region, it will have its least value when Qo is closest to the constraining hyperbola. The point of closest approach, Qc, will be at the intersection of the a -line and the hyperbola if they do intersect. This will be the case when the e-line falls in neither of the inner and outer forbidden regions for the direction of departure4 (like;l and i2 in Fig. 10a). In such a case, Av* = 0, and the initial velocity is the correct departure velocity along the given direction for the two terminal transfer. If the c-line falls within the inner forbidden region, no such an intersection is possible; however, Qc may still exist on the S-line (see -t^ in Fig. lO), but the corresponding Av* will not be zero. If the i -line falls within the outer forbidden region, then it lies partly in an unrealistic region, and |IAv* will be least when Qo is closest to the critical point in the same quadrant with the Liline. The point of closest approach, Qc, will then be given by the foot of the perpendicular drawn from this critical point to the o - line if the foot lies also in the unrealistic region. A rectangular plotting of IAv*. versus v is shown in Fig. lOb. To avoid confusion the constant 0o lines have been separated into two groups: |Iol = 0 to O, and |Iol = 0 to t/2, where O* is the critical angle indicated in Fig. lOa. It is worth noting that, in the elliptic region, due to the symmetry of the hodograph geometry with respect to the vs-axis, |Av*| remains the same when 0o changes to -0 at the same vo. Thus, in this region the IAv I versus vo curves are identical for +~0o However, after vo reaches its value on the critical boundary a-b-a'-b'-a, such a symmetry no longer exists due to the presence of the unrealistic region, and IAv*I in the high half-plane is higher than its conjugate part in the low half-plane. Consequently the |Av*| vs Vo curves splits into two branches, one for the +%o and one for the -0o as shown in Fig. lOb. It is also to be noted that, on the positive branch, the optimum solution is indefinite, and the values used in the plotting are in fact those of IAv*l which is the lower limit of the indefinite|Av.|. Finally it should be mentioned that, as the hodograph geometry is symmetrical with respect to the vX-axis in both the realistic and the unrealistic regions, each constant 0o curve also holds for its supplementary angle of the same sign (e.g. the curves for 0o = 70~ and 110~ are indentical, and so are those for -70~ and -110~). 4For terminology see Appendix A and Ref. 9, pp. 10-12. 56

2.0 / - 1.8 " I I 1<>oI; i: ~o goo 1.6 "5~~. u1.4. 1.2 ( 2 ) I /.+ 9(00 - = 60~' >= *5~ Critical Bou.nndary'"-'.-' 0. 0. v 2.0 _________ ConstrainingElliptic Transfer \0.8~~~ ~ ~70 (+i 110) 0., L)\ Q 1.8 -.- Elliptic Transfer, Nearly Critical |/ -q ~ * ~ "~,i.6 Parabolic Transfer ~/ ~~~HRearlistic Barrier 0.4p Ta Constraining.1.2 departure direction 1.0 oInner forbidden region fostraining (a) Hodograph Geometry 0.6,, 0.4 0 ~ /. 0.2.% \k0 0 0.5 1.0 1.5 2.0 2.5 3.0 Z0 (b) IAZI vs /,, at constant,o ( =: 600 ~9 -750) Fig. 10. Variation of the minimum velocity increment with the initia velocity vector at constant direction and varying magnitude.

Now let us consider the case when vo has a constant magnitude but varying direction, that is, when Qo moves along a circle centered at 0 with the radius v0. There are several sub-cases to be distinguished. a. When vo < (vo)L, the Vo-circle is entirely within the boundary aba'b' and the transfers are all elliptic. (a-l) vo < A First suppose vo < A, then as the hodograph shows (Fig. lla-l), the vo-circle intersects the constraining hyperbola at no point, and Qo is closest to the hyperbola when it is at the point D or itscomplementary point D' (not shown in Fig. 11). Consequently I Av*l is least at to = 0 or + n, and is given by AV lmin =A - v (42) Thus the best direction for the initial velocity vector is the local minimum energy direction. The same is true when vo = A except that the vo-circle now touches the constraining hyperbola at its vertices e and ^'. Thus the points D and D' coincide withf and A respectively, and Av* = 0. (a-2) vo > A The v -circle now intersects the constraining hyperbola at four distinct points, one in each quadrant (two of them are shown in Fig. lla-2), where Av* = 0. Consequently the best directions for v shifts from the minimum energy direction to either of the four directions determined by these four points. They are the correct directions for the 2-terminal transfer at the departure speed vo. There are one pair of such directions, a conjugate pair, for each of the trajectory groups, the normaland the complementary. b. When vo > (vo)L, a part of the vo-circle is outside the boundary aba'b' and the transfers are not all elliptic. (b-l) v0 < f2 Four points of intersection of the vo-circle with the constraining hyperbolic exist in the elliptic region like in case (a-2). (see Fig. 11 b-l) (b-2) vo > f2 The vo-circle will intersect the constraining hyperbola in the realistic region at two points only, both in the low half-plane (one of them is shown in Fig. llb-2 as QC2)' Thus there are two optimum directions for the low transfer with zero velocity-increment. In the high half-plane the Vo-circle extneds partly into the unrealistic region, and IAv*| will be least when QO is closest to either of the critical points 1l and 4*. As the hodograph shows, it is given by the point where the vcircle intersects the radial line through each of these critical points. Thus the optimum directions for to for high transfer are given by to = * and n-c*. The corresponding minimum |Av * will be nonzero unless vo =12. Figure 11 also shows the rectangular plottings of | Av* versus Do for severa constant values of vo. Note that the portions of the lAv* -curve beyond the critical points in the high and low half-planes are not symnetrical. This asymmetry is negligible when vo is close to (vo)L (Fig. llb-l), but becomes increasingly obvious as vo grows (Fig. llb-2). 38

(a-i) (a-2) (b-I) (b-2) ___ _____ 0 o<(VO*C)L PO SA ( __ O ) )o L O>A (I) < vo > /' L P P r I1 uw hIb b b b CD 0 0 a C P~ i YR 2'. 1' "n'c'IC' YC I*' K 0 a a' K 0 a K 0 J K a' k -o-4 —O 4 L —Vo1L I — l.........'o —---.... o oO -, —Hyperbolic Transferb..-..-Parabolic Transfer 2Critical Point Hyperboli c Transfer C C =(3~~~~~~~~~~~~~~~~~ All Elliptic Transfer A __Elliptic Transfer - - ~ 1.6 1.6 1.6 - Elliptic Transfer, Nearly Critical 3.2.-Elliptic Transfer, Nearly i All Elliptic Transfer All Elliptic Transfer ~ Critical Point. — -Hyperbolic Transfer 1.2- 1.2 1.2 ( Curves beyond their zero-I A v I points 124 Critical Line / *..- /' \ are exaggerated in the vertical scale.) / / ^. \ V.=O / \ //\\ /'/" -/. -8,A- / * 1.6w //A\ 04 0.4 04 \-8 -\?'. 0 o(,) 1.22 -.. - -90 -60 -30 0 30 60 90 -90 -60 -30 0 30 60 90 -90 -60 -30 o 30 -o 90 -90 -60 -30 0 30 60,o -90 -120 -150 180 150 120 90 -90 -120 -150 180 150 120 90 -90 -120 -150 180 150 120 90 -90 -120 -150 180 150 120 90 t(o (degrees) (DO (degrees) (o (degrees) Do (degrees) Fig. 11. Variation of the minimum velocity-increment with the initial velocity vector of constant magnitude and varying direction.

Finally it is interesting to note that I Av I is a local maximum when Qo is at the point J or K (0o = + x/2) in all the previous cases. In the case (a-l) it is also the absolute maximum since no other local maximum is present. Consequently the worst direction for 0 in this case is along the vX-axis, which is the bisector of the base angle cpl. This is also true in the case (a-2) even here a second local maximum for JAv*l exists at D or D', corresponding to the local minimum energy direction. This second local maximum is also present in case b. As shown in Fig. llb-1,2, it grows as vo increases, and it may eventually becomes the absolute maximum. Its location will shift to the high side of the vs-axis instead of lying on it when vo > vb. Thus the minimum energy direction and its neighbourhood in the high half-plane may become the worst direction for vo at high initial speed. 3.6 EFFECTS OF THE INITIAL VELOCITY VECTOR ON THE OPTIMUM SOLUTION: SUMMARY OF FINDINGS As seen from the preceding analyses the optimum solution for the problem is determined by the geometry of the base triangle and the initial velocity vector. Based on the previous findings the effects of the initial velocity vector on the optimum solution fotr a given base triangle may be summarized as follows: (1) Corresponding to every initial velocity vector Vo there exists at least one definite realistic optimum trajectory for the problem provided by the orthogonality quartic unless the tip of Vo exceeds the realistic barrier in the hodograph plane. Such a barrier is analytically defined by Eqs. (34-1) and (34-4). (2) If such a limit is not exceeded, the initial velocity vector is said to be in the realistic region, then the realistic optimum solution is unique whenever Vo is not directed along the bisector of either the interior or the exterior base angle at the initial terminal. If this is the case, then the optimum trajectory will be of the same group and the same class with the initial velocity vector Vo. However, the type of the trajectory, whether elliptic, parabolic, or hyperbolic, does not necessarily agree with that of Vo, but is determined by the particular region in the hodograph plane in which its tip Qo lies (see Table 5 and Fig. 7). (3) In a realistic region, if'o is directed along the interior base angle bisector, then there are two optimum solutions for the problem, corresponding to a complementary-conjugate pair of trajectories of the same class with the initial velocity vector, and of the same type which is determined by the region in which the tip Qo lies. 40

(4) The minimum energy direction of departure is along the exterior base angle bisector. If to is directed along this direction, then the optimum solution may be unique or not, depending on the magnitude of Vo or the location of its tip, Qo. Consider to in the positive half of the hodograph plane (see Fig. 7): (a) When Qo moves from the origin up to the cusp G of the boundary Lame along the minimum energy direction such that 0 < Vo < VG (where VG is given by Eq. (24)), the optimum solution is unique, the trajectory is elliptic, and of minimum energy, and the velocity-increment vector is to be directed along the minimum energy axis. (b) When Qo moves between the cusp G and the point b, where the boundary of the elliptic region meets the minimum energy axis such that VG < VO < Vb (where Vb is given by Eq. (38)), then there are two optimum solutions for the problem corresponding to a conjugate pair of trajectories of the same group with the initial velocity vector. They are both elliptic, but no longer of minimum energy, and the optimum directions for the velocity-increment vector deviate from the minimum energy direction with equal inclinations on either side of it even though the initial velocity vector is along that direction. (c) When Qo moves further along the minimum energy direction such that Vo > Vb the realistic optimum is again unique. Like case (b) the optimum AV is no longer in the minimum energy direction, and the trajectory is no longer the minimum energy one. It is parabolic when Vo = Vb, and hyperbolic when Vo > Vb. Situations similar to the foregoing three cases (a) to (c) exist when Vo is in the other half plane. (5) Different initial velocity vectors may call for the same optimum transfer trajectory. This statement is necessarily true when these velocity vectors all lie on the same normal line in the same quadrant in the realistic region. (6) Similarly, different initial velocity vectors may call for the same amount of velocity increment. This statement is necessarily true when these velocity vectors all lie in the realistic region and on the curve parallel to the constraining hyperbola with the same common distance on either side of it. (7) No unrealistic optimum will arise when the initial velocity is directed below the minimum energy direction regardless of its magnitude, or when its magnitude is below the lower critical limit (Vo)L (given by Eq. 39)) regardless of its direction. 41

(8) When the tip Qo of the initial velocity vector exceeds the realistic barrier, it is said to be in the unrealistic region. In such a region a definite realistic optimum solution can be found only when Qo is inside or on the boundary of the strip bounded by the vt-axis and the line of equi-criticalvelocity-increment (see Fig. 7). In such a subregion the realistic optimum trajectory is hyperbolic of the same group with the initial velocity vector, but of the low class. Outside this subregion no definite optimum solution can be found. The possible choice will be an elliptic one, of high eccentricity, close to the unrealistic parabolic trajectory given by the critical point or points nearer to Qo. (9) For a given direction of the initial velocity, there is a best magnitude for which the optimum velocity-increment is an overall minimum. This is given by the point of closest approach on the direction line.to the constraining hyperbola. in the realistic region, or to the critical point co-quadrant with the direction line in an unrealistic region. This best magnitude will be the correct departure speed in the given direction for the 2-terminal transfer if it falls in neither of the inner and outer forbidden regions for the direction of departure. The velocity-increment required is thus zero. (10) For a given magnitude vo the best direction for the initial velocity from the initial impulse standpoint is in the local minimum energy direction only if V0 is not greater than A, which is the departure speed along the minimum energy trajectory. When vo exceeds A, the best directions are those for a realistic transfer (long or short) with vo as the departure speed. The corresponding velocity-increment is again zero From the same standpoint, the worst direction for the initial velocity is that along the bisector of the base angle at the initial terminal, either inward or outward, when vo is less than A. At higher initial speed a. second worse direction exists in the minimum energy direction; it and its neighbourhood in the high side may eventually become the worst when vo grows. 42

4. HODOGRAPHIC REPRESENTATION OF THE TWO-DIMENSIONAL OPTIMUM TRANSFER 4.1 THE ORTHOGONAL NET IN THE HODOGRAPH PLANE AND THE OPTIMIZATION CHART As seen from the previous analysis the normal lines to the constraining hyperbola and its parallel curves form an orthogonal net in the hodograph plane. Such a net may be looked upon as the curvilinear coordinates of the initial velocity vector, and it forms naturally the basis for the development of the optimization chart for the present problem. A typical example of such a chart is shown in Fig. 12,5which is constructed for the case of 1 = 60~ and cp = 75~ corresponding to a transfer distance ratio of n = 1.366. As soon as the tip Qo of the initial velocity vector is located on the chart, the optimum velocity increment vector and the optimum departure velocity vector can be readily determined by noting the normal line and the parallel curve passing through this initial point Qo. In case unrealistic optimum arises it can be seen at once from the chart, and in such a case a realistic optimum solution may also be obtained directly from the chart by noting the subregion (U1 or U2 in Figs. 12 and 12A) in which the point Qo is located, and the rules given in Section 3.4. The type of the optimum transfer trajectory, elliptic, parabolic, or hyperbolic, will be indicated by the region in which the selected optimum point lies. To illustrate the use of this chart an example is given below: Consider a transfer from an initial point to a target point at a distance of r2 = 1.366 rl, an angle of separation 600, and an initial velocity given by 0 = o.80, = -250 By locating the initial point Q% according to (VoPO) in Fig. 12, we find the optimum solution approximately as follows: Velocity increment: magnitude IAv*. = 0.672 direction (,v* = 54.5 Departure velocity: magnitude Vl. = 1.14 direction 1* = 11~ The transfer trajectory is elliptic. 50nly one half of the hodograph plane is shown owing to symmetry; the normal lines are arbitrarily numbered for convenience. 43

REALISTIC BARRIER,. _,\ -... g' ECVI LINE k U^ - y ^v X X / Transfer y O |I C THE BASE TRIANGLE AND REFERENCE DIRECTIONS / Region lliptic / -a/,;, CRITICAL CIRCLE Transfer -3.0 -2.5 -2.0 -1.5 -10e -0.5 0 +0.5 +1.0 +1.5 +2.0 +25 +3.0 Fig. 12. The optimization chart for mini initial impulse terminal-to-terminal 2 +25 transfer (12 = 600 pt o = 75~) transfer (* = 600, pl = 750).

A-.-/ IS<~ /'=4., 7 ^/_2.0o ~ ^/ ^/ \^ /1t) 3 \1.5 Fig. 12A. The equi-critical-velocity-increment line in the hodograph plane. (* = 60~, cp1 = 75~)~ 45

While such a chart yields immediately the optimum solution corresponding to a specified initial velocity vector, it does not give directly the principal elements of the transfer trajectory except its type. For such information the hodograph circle for the transfer trajectory should be constructed, and it will be presented in the next section. Finally it is to be noted that although such a chart is constructed on the basis of a hyperbolic constraint, it may well be applied when the departure velocity is constrained not on this hyperbola, but on any one of its parallel curves, since all of them have common normals and the same Lame as their involute. The only change necessary is to shift the datum curve, on which Av = 0, from the hyperbola to the new constraining curve and to make corresponding adjustment on the constant value of Av on each of the parallel curves. Graphical techniques on the extensive use of such optimization charts, however, will not be elaborated here. 4.2 THE CONSTRUCTION OF THE TRANSFER HODOGRAPH With the optimum departure velocity vector determined analytically or graphically, the hodograph for the transfer trajectory may be constructed by using the terminal relations given in Ref. 9, -4 -4 V -v V =V Cl1 C2' R2(43) from which we see that once the hodograph image of the initial terminal Q1 is determined, so is that of the final termina Q2. In fact the point Q2 in the hodograph plane is also constrained on a hyperbola defined by VR = - K (44) which is Godal's compatibility condition applied at the second terminal. The negative sign here signifies the fact that the vector VR2 is directed in the negative direction of the local vertical at Q2. However, the construction of this second constraint is not necessary since following Eq. (43), the point Q2 may be easily located in the hodograph plane by completing the two velocity parallelograms with the common side V; and the other sides of equal length VR lying along the directions of rl and r2 respectively as shown in Fig. 15. With the two terminals on the transfer hodograph thus determined, the next step is to locate the center of the hodograph circle. According to the general correlation established in Ref. 7 this center must lie on the local horizontal line at each terminal. Thus by drawing the lines perpendicular to the local radial directions at Q1 and Q2 respectively we find their intersection at C, and by using C as center the hodograph circle can be drawn to pass through 46

Vc Q2 / \I o r\ /"WOrtnsfer \ 1 Vy CONSTRAiNING Q, HYPERBOLA ^ Q, (\nitial Termino') cONSTRAIN-NG 8YPERBOLA VRI 5g in the V-pla-ne h~.aot-.____-h.ao-vph construction of i

the points Q1 and Q2.6 This completes the construction, and the circular arc between the points Q1 and Q2 subtending a central angle * represents the transfer trajectory. The principal geometric as well as kinematic elements of the trajectory can then be determined from the hodograph according to the correlation given in Ref. 7. 4.3 THE HODOGRAPH OF OPTIMUM TRANSFER TRAJECTORIES IN THE v-PLANE So far the analysis has been made exclusively in the v-plane. Such a hodograph, though nondimensionalized, is essentially different from the dimensionless hodograph in the V-plane defined by - h v (45) where h is the angular momentum per unit orbiting mass and 4 is the Newtonian gravitational constant, as introduced in Ref. 7. To distinguish the two we will call them the v-hodograph and the V-hodograph respectively according to their planes. In a v-plane the velocity is nondimensionalized by dividing through by the circular speed at a fixed point, which is a constant in the problem. Thus the v-hodograph is in fact the same as the hodograph in the usual V-plane, except for the scale of plotting. However, in the U-plane the velocity is being divided through by the parameter p/h which varies from one trajectory to another. Such a nondimensionalization has the advantage of reducing the hodograph of all Keplerian orbits into a unit circle. Having made the analysis and representation of the present problem in the v-plane, it is appropriate to introduce here the hodographic representation of the same optimum solution of the problem in the U-plane. The locus of the hodograph origins in the J-plane, as shown in Ref. 9, is a straight line for all two-terminal trajectories of the same group. Thus, the two straight lines parallel to the chord of the base triangle in the 2plane are comparable to the two branches of the constraining hyperbola in the v-plane, one for each group (see Fig..14). Thus while the tip of the departure velocity vector is constrained on the two branches of the hyperbola in the vplane, the origin of the transfer hodograph is confined on these two straight lines in the Y-plane. Detailed discussion on the lines of origins are found in Ref. 9. Consider a normal group. Let 0 be an arbitrary point on the straight line locus, and p its distance from the radical center T as shown in Fig. 15. Then by definition 6+ + 6Note here the vector V2 - V1 is in the direction of the bisector of the vertex angle 4 in the physical plane, see Ref. 5 or 7, pp. 897-898. 48

K'P2~ ~Line of Origins v I2 \ (Complementary Group) Constraining Hyperbola 4 \ (Normal Group) 2 Hod -1 rvri ^ 1Hodograph Circle Base 4' Vc Triangle Q (Complementary Group) (Normal Group) - H" I*H unrealistic constraint - _ 1 T " / ~~~~/ ~ Qz~~f, \ 2~ / Q2' Constraining Hyperbola Line of Origins (Complementary Group) (Normal Group) vZ /:: unrealistic constraint f. -.f. (a) The v- plane (b) The Z I'-plane Fig. 14. Geometric representation of the two-terminal constraint in the v - and %-planes.

TRANSFER TRAJECTORY T \ QI nn-O _ HODOGRAPH CIRCLE.. R n OPTIMUM TRANSFER /:TRAJECTORIES vo 0 1 Q1 0 (a) Physical Plane LINE OF OPTIMUM ORIGINS (b) The'I2- Plane Fig. 15. Hodograph of optimum transfer trajectories in the r-plane (Jr = Const.).

p = VC (46) Comparing this with the definition of v given by Eq.. (11), and noting here, h = Vc d = Vc r1 sin cp (47) we find the relation p = v2 sin cp c 1 (48) Thus corresponding to each optimum value of vC* for a given base triangle and a given initial velocity vector there is a unique value of p*, from which the origin 0* of the optimum transfer hodograph is determined. Such an origin will be called the optimum origin for the present problem, and the locus of such origins in the V-plane, the locus of optimum origins, or simply the 0*locus. A typical example of such a locus for a constant vertex angle 4 is shown in Fig. 15b. By substituting Eq. (48) into the orthogonality Eq. (10C) we find the p*-equation (p2 - tan2 )2 = p sin 1l (roP* -o~tan )2 (49) 2 2 where nTO = Vo sin So (50) L.O = vo sin (P1 — o) according to Eqs. (20) and (21). For constant * such an equation may be looked upon as the polar equation of the p*-vector with the angle pl as the polar angle, and the directed tangent line T Q1, its polar axis. It represents the 0*-locus whenever the orthogonality Eq. (10) yields a realistic optimum solution. Some essential features of a p*-curve are to be noted as follows: (1) It is bounded between the two tangent lines at Q1 and Q2 on the hodograph circle since for a given vertex angle 4' the angle 1- can only vary between 0 and it-4. (2) For a constant 4 each value of the angle cpl corresponds to a 51

unique value of the distance ratio n. Thus the radial lines drawn from the radical center T are also the lines of constant n. (3) The point Q1 lies on the p*-curve since the p*-equation (49) is satisfied by ql = 0 and p* = tan V/2 there. (4I The origin of the initial orbit as given by the initial velocity vector Vo in the v-plane lies on the p -curve, as its coordinates also satisfy the p*-equation (49). The corresponding values of c1 and n at this point give the configuration of the base triangle such that the initial orbit passes through the final terminal Q2, and thus itself may be regarded as the optimum transfer trajectory. (5) The point where the p*-curve intersects the hodograph circle is the critical point, and the portion of the curve beyond it is hyperbolic. (6) From the critical point beyond, the p*-curve will be unrealistic (corresponding to unrealistic optimum trajectories) if it is on the high side, otherwise it is realistic. (7) When the p*-curve is unrealistic, it ceases to represent the 0*-locus, and should be modified according to.its corresponding realistic optimum value of vC*. (8) The point where the 0*-locus meets the bounding line T Q2 gives the optimum transfer trajectory from Q1 to infinity. Such a transfer will be further discussed in Section 5.3. Finally it is to be noted that the orthogonality principle does not directly apply in a v-plane since the initial velocity and the velocity along the transfer trajectory to be optimized are not represented by the same scale there. However, it has the advantage over the v- or V-hodograph in that it shows the totality of the optimum transfer trajectories for all possible configurations of the base triangle (given by the variable cp1 or n) under a given vertex angle 4r and a prescribed initial velocity vector (see Fig. 15a). Furthermore, unlike in the v-plane where a hodograph circle is to be drawn for each transfer trajectory, the v-hodograph enables one to use the same arc of the unit circle for all transfer trajectories between the fixed terminal points Ql and Q2, and from which all the principal geometrical as well as the kinematic elements of the transfer trajectory associated with a particular optimum origin can be readily determined according to the correlations given in Ref. 7. All the foregoing features are also true for the p*-curve or the 0O-locus of the complementary group. Such a hodographic representation can be easily obtained by turning the corresponding hodograph for the normal group through 180~. 52

5. ANALYSIS OF SOME LIMITING CASES So far the analysis has been restricted to 0 < 4 < A, and 0 < n < o. An examination of each of these limiting cases is now in order. 5.1 THE CASE 4 = 0 Physically this case corresponds to a vertical descent if rl > r2 and a vertical ascent if rl < r2. In either case the base triangle OQ1Q2 degenerates into a line segment with Q1 and Q2 on the same side of O. The geometry in the physical plane and that in the hodograph plane for each case are shown in Fig. 16. The constraining hyperbola also degenerates in each case, and its principal elements are as follows: r1 > r2 r1 < r2 (n < 1) (n > 1) C00 1 1 0 A 0 j2(l - ) (51) B 2( 1) 0 e ~~ 1 (a) Vertical Descent: rl > r2 (n < 1) The degenerate constraining hyperbola is a straight line parallel to the line OQ1Q2 in the physical plane. Consequently all normal lines are parallel to the local horizontal at Q1 and Q2, the orthogonal net becomes rectangular, and the transfer trajectory is a vertical straight line. The entire hodograph plane is divided into three main regions as usual: the hyperbolic region on the low side, the unrealistic region on the high side, and the elliptic region between them. However, it is to be noted that the usual closed elliptic region is now open since its sides are parallel. Furthermore, as a straight line trajectory is identical to its conjugate, as well as its complementary-conjugate, the optimum solution is unique everywhere in the realistic region even on the v -axis which now coincides with the vrl-axis. The optimum solution of the problem is very simple in this particular case. As seen from the hodograph (Fig. 16a) Qo is in the realistic region whenever vo sin o < (52-la) 55

0z rgO l Q__ 2_ r r2 Radial Direction at Q Radial Direction at QI Chordal Direction Chordal Direction:* fff' \ff,, rV ~O;^iraf. Q ~ =5le.~_iiii RAI_,CONSTRAINING LINE....(ao) r,.>r (n.~l)-~ (.Degenera.te Hyperbola) ( 2 (b) r < r2 (n>l) Fig. 6. Optimization of vertical transfer ( = 0). _/-~-_ ]/.'JJ-~ l~ v r l:' -r~.............. iiiii- d~' —......... Fig. 16. Optimization of vertical transfer 0).

and the geometry of the hodograph gives readily the solution summarized in Table 6, column a. As seen from the hodograph the optimum velocity increment vector in this region is everywhere in the local horizontal direction. It is simply to nullify the horizontal component of the initial velocity if any-a fact which is evident from physical considerations. However, whenever v0 sin o_ > 2 (52-2a) Qo is in the unrealistic region, the point on the vrl-axis close to the critical point 1* but inside the elliptic region has to be chosen as discussed in Section 3.4. No second choice is possible at present since no evolute exists for a straight line and the hodograph plane is simple everywhere. Consequently, the optimum velocity increment vector is no longer in the normal direction, and in addition to nullifying the horizontal component of the initial velocity, it has a vertical component opposed to that of the initial velocity so as to keep the resultant velocity below that for escape. The optimum solution in this case is indefinite, and, as seen from the geometry in the hodograph plane, it may be written approximately as summarized in Table 6. (b) Vertical Ascent: rl < r2 (n > 1) This case looks similar to the previous one, but there are some radical differences: 1) the constraining hyperbola degenerates into two semi-infinite lines along the radial axis instead of a single line as in case (a); and between the vertices and p' of these two branches of the velocity constraint, there is a gap of length 2A where no normal lines to the constraint line can be drawn, and consequently the orthogonality principle cannot apply there; 2) trajectories of the complementary group are out of the question since in such a transfer all physically realistic trajectories must go in one direction only, that is, from Q1 to Q2 not through 0. Thus the negative portion of the degenerate constraining hyperbola is meaningless. Consequently the straight line normal to the positive branch of the constraining line at its vertext forms a realistic barrier instead of the usual critical line.7 The geometry of the hodograph plane and various regions are shown in Fig. 16b. As seen from the hodograph, whenever v sin 0 > A (52-lb) QO is in the realistic region, the solution is definite and unique, and formulas are identical to those for case (a) in Table 6. Whenever vO sin o < A (52-2b) 7Note here in the region between the horizontal lines through oP and e' there exists no optimum solution, realistic or unrealistic, and in the region to the left of the horizontal line through ~D' (not shown in Fig. 16b) the unrealistic solution consists of elliptic trajectories in addition to the hyperbolic ones as encountered in the case of +# O, owing to the consideration 2). 55

TABLE 6 OPTIMUM SOLUTIONS FOR VERTICAL TRANSFER (^ = 0) (a) (b) r1 > r2 (n < 1) rl < r2 (n > 1) Vertical Descent Vertical Ascent Condition on vo V0 sin 4, < %2 (52-la) vo sin. > 2(1 - -) (52-lb) ^ Velocity-Increment 0 H r-component (AV )opt. 0 9 -component (Av;)opt -v cos 5(3 H ^ Departure Velocity CYN r-component (Vr)opt. sin (,. ~~~~~~~~~~~~~~~~~(54-1) 9-component (V) opt. 0 Condition on v V0 sin 0 > \/2 (52-2a) v sin 0 < 2( - ) (52-2b) g Velocity-Increment (Av)opt. = (Av)p H r-component (Avr)opt /2 -V sin = 2(l-) -v sin o 0.(55-2a) n (53-2b) H H 9-component (AvQp)pt. V- cos O- Vg cos J& I Departure Velocity r-component (vr)opt 2 2(1 - 9cmp n(54-2a) ()(54-2b) 9-component (vQ)opt. 0 0

Qo is in the unrealistic region, the orthogonality principle no longer applies. In such a. case the vertex-P should be chosen as the optimum point, giving Vopt = A.= 2(1 (54-2b) which is the minimum departure velocity for such a transfer (see item (2) on the "Constraining Hyperbola," Section 2.2). Formulas for this optimum solution are summarized in Table 6, column b. 5.2 THE CASE = = The case is of practical importance. Like the previous one the base triangle degenerates again into a. line segment but with the two terminal points on the opposite sides of 0. The elements of the constraining hyperbola have the following limiting values according to Table 2: A -f (A - Jr ( 55) B - 00 e - 0 Thus the constraining hyperbola degenerates into two straight lines parallel to the vrl-axis at the distances ~ A. Consequently, all normal lines are again in the horizontal direction everywhere in the hodograph plane, the orthogonal net is again rectangular, and the plane is divided into the three regions, elliptic, hyperbolic, and unrealistic, by the two critical lines just as in the case \r = 0, and rl > r2. With the absence of the boundary Lame the entire hodograph is again simple, and a definite and unique optimum solution exists everywhere in the realistic region except on the vrl-axis along which a complementary-conjugate pair of optimum solutions exist. The geometry in the physical plane and that in the hodograph plane are shown in Fig. 17. As seen from the hodograph, Qo is in the realistic region whenever v sin < 2 (56-1) 0 0o n+l and in this region the optimum direction of Av is horizontal everywhere. The optimum solution can be readily obtained from the geometry of the hodograph, and is summarized in Table 7, column 1. Whenever v sin > (56-2) 57

VELOCITY CONSTRAINT (Degenerate Hyperbola)i;.../~, -U,1;== —!( i\~ ~ 1: (H) E)?: a TRANSFER (S I":~"';:' ~=:;'J:ji TRAJECTORY! >- ^^e\,'*+ / ^^A'^/^^^r'+0 /^,~~80.1 Z l^ / ^ > -2 RE ——'-r — REAlISTIC r^ \SB _'- BARPIER ~ 3* ~!i~:"'~;....~~ Radial Direction at Q! 22 ~ 1.i [ ~ Chord al Direction * (a) Physical Plane (b) Hodograph Plane Fig. 17. Optimization of 180~ transfer (* = n).

TABLE 7 OPTIMUM SOLUTIONS FOR 180~ TRANSFER (~ = i) (1) (2) Realistic Region Unrealistic Region Condition on vo vo sinoo < -(56-1) Vo sin0 > (56-2) 0n+l (nb-1) Vo sin+o 2 Velocity-Increment (Aov)ot v* r-component (AVr)opt. 0 - v sinko (57-2) Ag opt. +- n+l - v~(7-)+_ 9-component (AV.)opt i + (I - vo cos0o (27-1) | v0 cosn0 n+1 + 2+1 vo cosnO Departure Velocity (v)opt v* (v* = J2) r-component (Vr)opt. vo sino0 | V n+l 9-component (v)opt + (58-1) + (58-2) I n+l n+l NOTE: For the double sign: take the upper sign when (ve)o > 0- (1|ol < () (1 solution) take the lower sign when (vqe) < 0 (1Iol > 2) (1 solution) 2

Q is in the unrealistic region. Since no non-simple region exists, the only choice for the optimum is then the one close to the nearer critical point, 1* or 4*, and remains in the elliptic region. The optimum solution is again indefinite, and may be written approximately as summarized in Table 7, column 2. With vl* thus determined the hodograph for the transfer trajectory can be constructed in the v-plane by noting that for a 180~ trajectory we have vl1: Ve2 = n: 1 (59) Thus once the image point Q1 is determined in the hodograph plane, so is the image point Q2. Since the center of the hodograph circle is necessarily halfway between Q1 and Q2, the hodograph of the transfer trajectory is now completely determined, a.s shown in Fig. 18a. It is interesting to note that,' in a v-plane the center of the hodograph circle for such transfer trajectories is constrained on a. line also parallel to the vrl-axis and at a. distance (l-n)/2n(n+l) from it, which follows directly from Eqs. (55) and (59). In the )-plane the radical center T recedes to infinity, and all constantn-lines become parallel. Consequently, p* also tends to infinity and the p*equation is no longer suitable for the description of the 0*-locus. In such a case the use of an alternate coordinate system is necessary. A convenient choice is a rectangular system with its axes coinciding with the directed lines TQj and Q21 in the if-plane (which are in the local horizontal and vertical directions at Ql respectively). Let p4 be the radius vector from the point Q1 to the optimum origin 0*, then evidently, (see Fig. 18b). = - I (60) with their rectangular coordinates related by * +r = - r* (60a) (P.) = - "e Temporarily let us consider only the trajectories of the normal group, that is we restrict'uV to be non-negative (-Xr/2 < 0o < rt/2), then directly from their definitions the components of Zra.nd v are related by o = e = (61) 0r = VrVe 60

High Branch, Low Branch ai a n —n \ 4 TRAJECTORY Y' TRAJECTORY....,-. /*"- Vy *1+A I-n V0 + n=,._. _ ~ g Line of / t \, /7/^ VVRI CY Line of Centersl' \ — of the Hodo- %' 7 graph Circles / s/ Unrealistic LINE OF OPTIMUM ORIGINS (a) The V-Hodograph (b) The r-Hodograph Fig. 18. The optimum trajectory hodograph for 1800 transfer.

It follows that r =~ r (62) Do But according to Eqs. (58-1) and (50) we have for realistic optimum, vr* = V0 sin o0 x o (65) Substituting Eq. (63) into Eq. (62) gives 2 2 = Mn ) (64) r* o 9* In terms of pr and p~, this becomes,p)2 =:-m 2(p) (65) ~r o * e Thus the p. -equation is a parabola tangent to the v 1-axis at Q and having the te8l-axis as its axis of symmetry (see Fig. 18b). Note that the line of optimum origins must pass through the initial point 00 determined by the initial velocity vector J4o, according to Section 4.3. Thus the positive branch of this parabola corresponds to initial velocity vectors at negative path angles (Do < 0) and will be designated as the low branch; while the negative branch corresponds to those at positive path angles (0o > 0) and will be designated as the high branch. The low branch therefore always gives a realistic optimum, and its portion beyond the hodograph circle is the hyperbolic portion. The high branch corresponds to a realistic optimum only up to the critical point, and beyond that the optimum origin will move closely around the circumference of the hodograph circle, but remain inside it. It is to be noted that, when the initial velocity is directed in the local horizontal direction (Xo = 0), the p —parabola degenerates into the line Q1Q2, and all optimum transfer trajectories are realistic and elliptic. As is evident from both the V'-hodograph and the v-hodograph, such an optimum transfer trajectory is always the Hohmann transfer ellipse independent of the magnitude of the initial velocity vector. When the path angle of the initial velocity vector exceeds the limit ~ I/2, the optimum solution calls for a trajectory of the complementary group. The corresponding transfer hodograph can be obtained by rotating the present one for the normal group through 180~ as usual. 5.3 THE CASE n +oo (r2 + o) When r2 increases indefinitely while the angle 4 is fixed, the final terminal point Q2 recedes to infinity along a given direction, and the problem becomes an escape from a given point Q1 along a given asymptotic direction 62

specified by r. The base triangle is now open with 1 -,X -I,'2 0 (66) cq-~x-g J ~-+O (66) and the principal elements of the constraining hyperbola. have the following limiting values according to Table 2: A - v2 (67) B -e J2 tan i 2 e - see Besides, the boundary Lamb has its cusps G and G' given by VGG _ 2 sec2 (68) G,G 2 2 The geometry in the physical plane and that in the hodograph plane are shown in Fig. 19 a,b. The minimum velocity along the constraining hyperbola, as given by A, is the escape speed; thus all possible transfer trajectories are hyperbolic, or at least parabolic, a. fact which is self evident. In the hodograph plane the critical circle now touches the constraining hyperbola at its vertices fad p, and', and the entire hodograph plane is divided into the realistic (all hyperbolic) and the unrealistic regions by the vt-axis. It is to be noted that, although nonsimple regions exist in the hodograph plane for the present case, no realistic conjugate optimum solutions exist along the vs-axis since no elliptic region exists, and the high half-plane is all unrealistic. Furthermore, a parabolic trajectory should not be admitted as a solution since it has no definite asymptotic direction as required by the problem. Thus whenever the tip Qo of the initial velocity vector lies between the points G and G' on the vs-axis, a point on the constraining hyperbola in the realistic region and close to the nearer critical point, or t' is to be chosen as the optimum point. For points on the vs-axis beyond either G or G', of course hyperbolic realistic optimum solutions always exist. A simple criterion for realistic optimum transfer is then 2 (t+*) < < 2 (t-*) (^ < 0 < 0) any YO (69) 0 \ (~ ) (,o ~o ): v >J2 sec2 Note that the E-C-V-I lines here coincide with the vt-axis and the subregion U2 does not exist. Consequently, the optimum solution is indefinite whenever Qo is in the unrealistic region, and such a solution is given by Eqs. (40,41) with $ = O. The optimum solution when Qo is in the realistic region 65

TRAJECTORY Q2 (at infinity) 4d/ X V~oW~v~ 0v1 v C O r, Q (a) Physical Plane BOUNDARY LAME CONSTRAINING/ f V. HY(b) HodoraphPER BOLAPlane / CRITICA RI (b) Hodograph Plane Fig. 19. Optimization of transfer to infinity (r2 + ao). 64

TRAJECTORY A \ 12iVC V TRAJECTORY RZ2. ci \ 0 ZV / CONSTRAINING Q C 0 v, ~~HYPERBOLA C 7/2y "RI HODOGRAPH CIRCLE HODOGRAPH CIRCLE LINE OF OPTIMUM ORIGINS yx (a)The I/ - Hodograph (b) The2 - Hodograph Fig. 20. The optimum trajectory hodograph for transfer to infinity.

cannot be readily written from the geometry in the v-plane as was done in the previous particular cases. However, it is given by the point where the line of the optimum origins meets the line TQ in the D-plane as shown in Fig. 20b, and essential information concerning the transfer trajectory can be obta.ined from the IU5-hodograph. For example, the eccentricity of the optimum trajectory is given by 0*C, its apsidal axis by the line normal to C, and the residual velocity, the vector O*Q2^ The v-hodograph can be constructed as usual. In this case the point Q2 in the hodograph plane can be easily located by drawing a straight line passing through Q1 and parallel to the bisector of the angle *. The point where this line meets the vrl-axis gives the point Q2 required (see Fig. 20a), The hodograph circle will of course be tangent to the vrl-axis. 5.4 THE CASE n + 0 (r2 + 0) In this case the final terminal Q2 is approaching the field center 0, and the constraining hyperbola in the hodograph plane is approaching the vrl-axis. In the limit the situation reduces to that of a vertical descent analyzed in Section 5.1a. with r2 = 0. The hodograph geometry is the same, the transfer trajectory is again a vertical line segment, and all formulas of Section 5.1a apply to the present case. 8See footnote 6. 66

6. TRANSFER FROM A CIRCULAR ORBIT Since the transfer from an initial circular orbit is of frequent occurrence in space flight problems, it is worth a brief treatment in the light of the present analysis. For the time being the two-dimensional case will be considered, that is, the final terminal will be restricted to the plane of the initial orbit. 6.1 ANALYSIS The initial condition for the transfer from a circular orbit to a coplanar point is as follows (Fig. 21a): Short transfer Vo = 1, (o = 0 (70a) Long transfer v = 1 = (70b) As is evident from the optimization chart (Fig. 12), an unrealistic optimum is possible only for a long transfer. Thus a definite realistic optimum solution exists for a short transfer, and also for a long transfer before the realistic barrier is reached. Such a solution is provided by the orthogonality Eq. (lO'-R), which takes the simple form 2 2 va. + VR. tan -K2= 0 (71) under the conditions (70a,b). The upper sign in Eq. (71) pertains to the short transfer, and the equation has one positive real root (according to Table 3) giving the optimum solution. Similarly, the lower sign pertains to the long transfer, and the equation has one negative real root for the optimum solution. Evidently, these two roots differ in sign only and the two solutions are a complementary pair. The corresponding 0*-locus in the v-plane is given by Eq. (49), which reduces to 2 2 3 ( 3 (p* - tan i) = p* sincp (72) for the present case. The locus passes the center of the hodograph circle as shown in Fig. 21b. Whenever an unrealistic optimum arises from Eq. (71) in the case of a long transfer, the realistic optimum solution becomes indefinite, and is given by Eqs. (40,41). The corresponding 0*-locus is then to follow the arc of the hodograph circle but remain inside it as discussed in Section 4.3, item (7). Formulas for the minimum velocity-increment for both the defi67

T Qi n=O //initial Orbit 2 Q 2 Transfer I locus Short f Trjetoy Trajectory c Transfer (Normal),, h\'//al Plan H o O \H / lodograph i. / T e f r /A Circle "v \ y Hyperbolic CrlJe Transfer Critical Point \,.-, \,, Critical Point Initial Orbit Q Long \ Hodograph Tr ansfer I /Xl V Circle C c^ 1 Unrealistic 4. -— 0- locus / Transfer n=O 01 T Transfer Trajectory v 1V0 (Complementary) Physical Plane Hodograph Plane Fig. 21. Transfer from a circular orbit to a coplanar point.

nite and indefinite optimums as specialized to the circular case are summarized in Table 8, and the variation of the minimum velocity-increment under various configurations of the base triangle is shown in Fig. 22. In view of the foregoing analysis it is of importance to safeguard the occurrence of an unrealistic optimum in the case of a long transfer. An analytical criterion for the occurrence of a critical optimum (including both the realistic and unrealistic cases) has been derived by Battin (4) in an approximate form. An exact form of such a criterion can be obtained here by applying the circular condition (70a) or (70b) to the general critical condition (28), which, after some trigonometric simplifications, reduces to 3 co oscos2 4(-)2 cos - + (- + ) 0 (75) 2 2 n* 2 n* n* where' is the vertex angle of the base triangle, and is related to the range angle E by for short transfer u = 4 (76) L2~t- for long transfer and n*, the critical distance ratio, is the value of n which satisfies the critical criterion (75) for a given 4. For a fixed initial terminal Q1, Eq. (75) defines for the final terminal Q2 in the physical plane a boundary on which the optimum transfer trajectory given by Eq. (10'-R) would be parabolic. Such a boundary is shown in Fig. 23. It is the critical boundary for a short transfer, but an unrealistic barrier for a long transfer. For convenience the configuration of the base triangle will be called sub-critical, critical, or super-critical according as Q2 is below, on, or above this boundary. As Fig. 23 shows, along this boundary n* extends to infinity at 0 = O,r, and it has a minimum value of approximately 3.845 at 7 - 71~. Such a boundary line has also been depicted in Fig. 22. The lAv*. -curve at a constant I, as Fig. 22 shows, holds for both the short transfer and the long transfer before it reaches the critical line, that is, the curve for a range angle 4 also holds for the range angle 2rT-4. However, this breaks down after it crosses the critical line, and the lAv*l-curve splits into two branches, one for the short transfer, and one for the long transfer with the latter branch above the former one. Thus the region enclosed by the critical boundary is the region of definite hyperbolic optimum for short transfers, but of indefinite elliptic optimum for long transfers. 69

TABLE 8 MINIMUM VELOCITY INCREMENT I Av I FOR THE TRANSFER FROM A CIRCULAR ORBIT TO A COPLANAR POINT Definite Optimum Indefinite Optimum For short transfer 0 < n < oo or long transfer with n < n* For long transfer with n > n* 1 0 < 3 < X L [2v*- tan (1 n) sec2 J + (73a) | - 2e 2 cos (D* - 2 (75b) VR-j 2 n - 1 @ < D 0 (n = ) (73a-l) $jf = 0 15 - 2/n Cn > l) 2n 3 = ^ 1n< -1 (73a-2) | n = 0 1 (73a-3) 1 1 n- v -o i-2 1 tan - sec2 + ec 2 (75a-4) - 2 2 sin 2 (73b-4) P1 T ^4 2iir~i 22 l Hcot = an tan t (+cp1) Auxiliary Equations VR* t VR tan e = sec2 j - (71') (74) -24 n 2 n (O < ( * <') Note: 1) For indefinite optimum, the formula listed in each case is its lower limit, IAv*I. 2) For the double sign in Eq. (71'), the upper sign is for the short transfer, while the lower one, the long transfer. 3) n* is defined by Eq. (75).

2.C 2I0 0T - Elliptic Transfer 2~^^^><~~~~ -~~_-...Elliptic Transfer, Nearly Critical,Q ______ ______ ______,~ ~ R2/^M^\~ ~~-....- Parabolic Transfer 1.8 Q --— Hyperbolic Transfer 1.6 Inner Transfer' —o-Outer Transfer;; IA 1.4 _- 0 300~ <3!!. =l0~ / I -' _ _ _ o^"' 1. -4 E< ",i\ ^ / 270~ 0.6'o,- --- ---- _o ~ —0 o4 240~ E, E 0.2 380 -'" — "'-'- 345~ 0 0.2 0.4 0.6 0.8 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 DISTANCE RATIO n = 4 / ti nl: l"= unit) Fig. 22. Minimum velocity-increment for the transfer from a. circular orbit to a coplanar point.

\~ X X \ \ ~c> \ \ |1 H Elliptic Long Transfer r X Ho x 7^ / ^Hyperbolic Short Transfer ~~Fig. 235. The critical cong Tfiguration of the base triangle for the optimum transfer from a circular orbit. optimum transfer from a circular orbit.

6.2 SOME OBSERVATIONS Based on the foregoing analysis and the graphs of Figs. 22 and 23, a number of observations may now be made as summarized below: (1) There exist definite configurations of the base triangle for which the optimum trajectory is parabolic, which will be realistic for the short transfer, but unrealistic for the long transfer. Such critical configurations are defined by Eq. (75). (2) For a base triangle of sub-critical configuration, the optimum trajectory is elliptic, and the minimum velocity-increment is the same whether the transfer is short or long. (see Fig. 22) (5) For a base triangle of critical or super-critical configuration the realistic optimum will be definite, parabolic or hyperbolic for the short transfer, and it will be indefinite, elliptic but nearly parabolic for the long transfer. (see Fig. 22) The minimum velocity-increment is higher in the latter case. (4) For each vertex angle 4 between 0 and t, there is a minimum distance ratio, n*, below which no critical optimum, realistic or unrealistic, may occur (see Fig. 23). An overall minimum n = 3.845 exists, below which no such a critical optimum may occur for whatever the vertex angle r. In the solar system this distance ratio corresponds to a transfer from the earth orbit to somewhere between the orbits of Mars and Jupiter. (5) No critical optimum, realistic or unrealistic, may arise for either a vertical transfer or an 180~ transfer, through any finite distance ratio since n*-oo in both cases. (6) For a given distance ratio n > n*, there are two critical values of \ir beyond which no critical optimum, realistic or unrealistic, may occur (see Figs. 22, 25). Thus the two values of t define a range of Ir for the definite hyperbolic optimums for the short transfers or the indefinite elliptic optimums for the long transfers, both will be referred to as the critical range for brevity. Definite parabolic optimum exists at the end points in the case of the short transfers of course. In the solar system such a critical range exists in the interplanetary transfer from the earth orbit to that of Jupiter and beyond. Values of these critical angles together with some numerical data pertaining to the solar system as obtained from the present analysis are shown in Table 9. These angles related to Jupiter, Saturn, and Neptune confirm the previous results of Battin (4). (7) At a constant distance ratio n, the closer the range angle to 180~ the smaller the minimum velocity-increment required (see Fig. 22). Thus, from the viewpoint of fuel economy, transfer close to 180~ range is desirable. In the limiting case of 180~ transfer, the optimum trajectory will be an Hohmann ellipse. 75

(8) At a constant range angle, the closer the values of rl and r2 to each other, the smaller the minimum velocity-increment required. The overall minimum IAv*. is zero at l1 = r2 for all values of the range angle, since in this case the initial orbit passes through the final terminal point. (Note in Fig. 22, the lAv* l-curve for * = 0 is discontinuous at n = 1 with an isolated point at n = 1, and jAv*| = 0 in accordance with Eqs. (73a-1).) (9) As the distance ratio n increases indefinitely at a constant range angle, the minimum velocity-increment increases and approaches a finite limit de-. pending on the range angle according to Eqs. (73a,b-4). Similarly, when n decreases indefinitely, the minimum velocity-increment also increases; however, it approaches the value of unity as its limit regardless of the range angle. (see Eq. 73a-3 and Fig. 22) (10) There exists an overall upper limit for the minimum velocity-increment for all possible configurations of the base triangle. It is given by lAv*1 _IPP %T5 or IAVjI =%f5 V,1 (77) y^.1 - 3 o r IV* upper limit 3 sl ( according to Eq. (73b-4). Thus, in principle, any propulsion device capable of producing a velocity-increment of 29.8 43 I 51.6 Km/sec will be enough for the transfer from the earth orbit (orbital speed = 29.8 Km/sec) to any terminal point in the solar system. All the foregoing observations are made on the assumption of the twodimensional transfer from an initial circular orbit. The three-dimensional effects will be presented in the chapter that follow. 74

TABLE 9 MINIMUM VELOCITY INCREMENT REQUIRED AND THE CRITICAL RANGE ANGLES FOR INTERPLANETARY FLIGHT IN THE SOLAR SYSTEM FROM THE EARTH ORBIT Mean Minimum Velocity Increment Required! Critical Range Angle Distance i 8I = 180~ Short Transfer Long Transfer Destination Ratio (Orbital Speed of Earth = 29.8 s-T) IL _ _ Pec a VU YL LU Planet - r2 IAVI*, n = I Av- = — I AV-1. rl Vs1 Mercury 0.39 0.250 7.45 Venus 0.72 0.085 2.553 _ Mars 1.52 0.098 2.92 Jupiter 5.20 0.295 8.79 52.0~ 100.2~ 259.8~ 308.0~ Saturn 9.54 0.346 10.3 40.2~ 124.8~ 2355.2 319.8~ Uranus 19.19 0.379 11.3 30.3~ 142.1~ 217.9~ 329.7~ Neptune 50.07 0.392 11.7 25.1~ 150.00 210.0~ 554.9~ Pluto 39.46 0.397 11.8 22.3~ 153.9~ 206.10~ 57.7

7. THE THREE-DIMENSIONAL EFFECTS ON THE OPTIMUM TRANSFER 7.1 THE THREE-DIMENSIONAL ANALYSIS When the initial velocity vector is not coplanar with the base triangle, the problem is three-dimensional. In such a case the in-plane and out-ofplane components of the velocities are to be considered. Thus Eq. (1) may be written. AV = V - ( + Ion) (78) (Vop n where Vop = Vo cos 6 (79) Vn = V sin X and U0 is the inclination angle of the initial velocity Vo with the plane of the base triangle. The geometry of the transfer is shown in Fig. 24. "i>(VO (VO)N Q/ Q,Q Transfer Plane 0 Fig. 24 Geometry of the three-dimensional transfer. 76

In view of the fact that the departure velocity V1 along the transfer trajectory must be in the plane of the base triangle, Eq. (78) may be written AV = p -on (78a) where AVp is the in-plane velocity-increment, defined by AVp = V1- Vp (80) The magnitude of the total velocity-increment is then given by IAVI2 = AV2 + V2 (81) For a given.initial velocity vector and a given base triangle, Von is constant. Thus the optimization of |AVI amountsto the optimization of IAVp|, and the problem becomes two-dimensional. Consequently we have v, l2 l Vp. 2 2 1AV12 = |AVp2 + Von (82) where IAVp*. is given by the two-dimensional optimum solution corresponding to the initial velocity vector VOp. Thus by replacing Vo by VO cos w we obtain AV * from the previous two-dimensional analysis, and the three-dimensional solution follows from Eqs. (78a and 82). Such a reduction of the three-dimensional case to the two-dimensional case has been pointed out by Stark and some numerical solutions are found in (6). Thus no elaborate analysis is necessary here. However, in the light of the present analysis a few remarks on the three-dimensional effects will be given below. 7.2 THE THREE-DIMENSIONAL EFFECTS First, the effect of tilting the initial velocity vector from the plane of the base triangle may be investigated by using Eq. (82). Let IAV*l3D and IAV*I2D be the minimum velocity-increments for the three-dimensional and twodimensional problems respectively, both referring to the same base triangle and the same initial speed and path angle (Vo, 0o) except 0 = 0 in the latter case. When CD is small, we have V Vo, |AVp*| |AV.*2D and Eq. (82) may be written |AVwl n ^av+ V^2 v (85) V* 77V2D+ Von 77

from which we see that the presence of the out-of-plane component Von is of importance when the term IAV*12D is comparatively small. Thus we may say that, the smaller the two-dimensional solution, the more significant the three-dimensional effect. In the case of the transfer from an initial circular orbit, such is the situation in the neighborhood of n = 1. That is, the closer the distances r1 and r2 are to each other, the more significant is the effect of the inclination between the orbital plane and the plane of the base triangle. As Fig. 25a shows, the maximum deviation of JAV*.3D from IAV*12D occurs at n = 1, and is the same regardless of the range angle. It is in fact equal to the magnitude of the out-of-plane component of the initial velocity. The same reasoning accounts for the fact that at a constant distance ratio n the deviation of IAV*I3D from LAV*12D increases when the range angle tends toward 180~ for either the long transfer or the short transfer as shown in Fig. 25a, since the corresponding two-dimensional velocity-increment tends to decrease according to Fig. 22. Second, it is worth to note that in the case of 0~ or 180~ transfer, the base triangle defines no plane since it has degenerated into a line segment. Consequently, the optimum transfer plane is the one defined by the initial velocity vector and this line segment, and the case is always two-dimensional. Thus it seems curious that, while the three-dimensional effect tends to become more significant as * approaches 1800, as shown in the preceeding paragraph, it can be completely eliminated in the limiting case of' = 180~. Third, the reduction of VO to Vo cos( by tilting the initial velocity vector may effect the region in the hodograph plane where the point Q0 lies, and thereby effect the type of the optimum transfer trajectory. Thus it is quite possible that an initial velocity vector, which calls for an hyperbolic optimum when it lies in the plane of the base triangle, may call for an elliptic optimum instead when it is tilted up, though at a greater expense of the initial impulse. In general, the critical boundary and the unrealistic barrier both will be effected. In the case of the transfer from a circular orbit, an examination of the geometry of the hodograph shows that the effect of increasing the inclination angle co tends to increa.se the critical distance ratio n* for a fixed 4 between 0 and it (see Fig. 25b), and for a fixed n > n* it tends to shorten the critical range defined by the two critical angles. However, it is to be noted that such effects are not present in the case of 0~ or 180~ transfer, even though the transfer plane is taken to be different from the optimum one mentioned in the preceeding paragraph, since the critical distance ratio n* tends to infinity in both cases. 78

5; ) -w=O I I____ I=+10~ > 1.0 ~ =60o 3000 " 0.8 I.w 0.6 -QI ~ ~ ~ ~ ~ ~ ~~2 z circular orbit: Low distance ratio (n < n*). I - > -I/: 1200 2400 0 0.4 \ Iw \ 0.2 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 DISTANCE RATIO n = 4/,A 2 1 Fig. 25a Three-dimensional effect on the optimum transfer from a circular orbit: Low distance ratio (n < n*).

1.05 1.05 0,00 r.00 >* * 0.95 0.95.. O/'/ 10 o | / G o Critical point E 0.85 / 0.85 E /. 0.80 I I I I I I I I I 0.80 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 75 8.0 8.5 9.0 Distance Ratio n 2 A2/t1 Fig. 25b Three-dimensional effect on the optimum transfer from a circular orbit: High distance ratio (n > n*).

REFERENCES 1. Salmon, G., A Treatise on Conic Sections, 6th Ed., Longmans, Green and Co., 1879, pp. 337-338. 2. Burnside, W. S., and Panton, A. W., The Theory of Equations, Longmans, Green and Co., 7th Ed., 1912, pp. 144-145. 3. Yates, R. C., Curves, Department of Mathematics, U. S. Military Academy, 1946, pp. 84-90, 155-158. 4. Battin, R. H., "The Determination of Round Trip Planetary Reconnaissance Trajectories," J. Aero/Space Sci., Vol. 26, No. 9, September, 1959. 5. Godal, Th., "Conditions of Compatibility of Terminal Positions and Velocities," Proceedings XIth International Astronautical Congress, Stockholm, 1960, Springer Verlag, Vol. I, pp. 40-44. 6. Stark, H. M., "Optimum Trajectories Between Two Terminals in Space," ARS J., February, 1961, pp. 261-263. 7. Sun, F. T., "On the Hodograph Method for Solution of Orbit Problems," XIIth International Astronautical Congress, Washington, D.C., October, 1961, Academic Press, Inc., Vol. II, pp. 879-915. 8. Altman, S. P., and Pistiner, J. S., "Comment on the Correlation of Stark's'Two-Terminal Trajectory Optimization with an Orbital Hodograph Analysis," ARS J., 1962, Vol. 31, No. 11, November, 1961. 9. Sun, F. T., "Hodograph Analysis of the Free-Flight Trajectories Between Two Arbitrary Terminal Points," NASA CR-153, Washington, D.C., January, 1965. 81

APPENDIX A GLOSSARY OF TERMS FOR TWO-TERMINAL TRAJECTORIESt (see Fig. A-l) Base Triangle The triangle formed by the initial terminal (Ql), the final terminal (Q2) and the center of the gravity field (0). Normal and Complementary Groups A two-terminal trajectory is said to be of the normal group or the complementary group according as its range angle is smaller or larger than 1800 corresponding to the so-called short and long transfers respectively. High and Low Classes A two-terminal trajectory is said to be of the high class or the low class according as its direction is inclined above or below the local minimum energy direction at the initial terminal. Conjugate Trajectories Two trajectories are said to be conjugate to each other if they have the same initial and final terminals, the same range angle, and the same speed at the initial terminal. Complementary Trajectories Two trajectories are said to be complementary to each other if they have the same initial and final terminals, the same initial speed and going in opposite directions around the field center. Complementary-Conjugate Trajectories Two trajectories are said to be complementary-conjugate to each other if one is the complementary of the conjugate of the other. tFor details see Section II, Ref. (9), in which these terms were introduced and discussed. 82

Realistic and Unrealistic Trajectories A two-terminal trajectory is said to be realistic if every point on the trajectory is at a finite distance from the field center; otherwise it is said to be unrealistic. Forbidden Region for the Direction of Departure For a fixed base triangle the forbidden region for the direction of departure is the angular region for such a direction along which no trajectory from the initial terminal Q1 can reach the final terminal Q2 whatever the departure speed. There are two such regions for the Keplerian trajectories, according to Ref. (9), as follows t The Outer Forbidden Region: the angular region included between the two directions of the conjugate pair of parabolic trajectories from Q1 to Q2. The Inner Forbidden Region: the angular region included between the two sides, OQ1 and Q1Q2 of the base triangle. "Similar regions exist for the direction of approach,, see Ref. (>), pp. 11-13. 85

(~H) _ Min. Energy Direction ^ 3\ /> ~ ~Min. Energy Direction ^~/ _- Q= I Normal, low I, 2 1 Normal, low Conjugate pair ~ ~ - (or I, 1':1E Normal, high I, I'.S.L~..... = -.-= l, Complementary pair (orn~,') I' Complementary, high I, Ie l Complementaryn1' Complementary, low,(o I' I ) Conjugate pair Outer forbidden region Inner forbidden region for departure direction 2 for departure direction Fig. A-i The two-terminal trajectories. 84

APPENDIX B THE INTERSECTING PROPERTY OF THE NORMALS OF A HYPERBOLA Statement of the Property Two normal lines at two distinct points on a hyperbola in the same quadrant will always intersect in the adjacent quadrant on the opposite side of the transversal axis of the hyperbola. An Analytical Proof Let the equation of the hyperbola be given by the parametric equations x = B tan w (B-l) y = A sec u Consider two normal lines at the points Q1 (l) and Q2 (c2) on the hyperbola, and let their point of intersection be P(xp,yp). For definiteness let us assume 0o<J <o^< - (B-2) 3 < 1l < 12 < K (B-2) so that Q1 and Q2 are distinct and in the same quadrant I. Then we are required to show that the point P is in the quadrant II (see Figure B-l.) Now the equation of the normal line at any point Q(w) on the hyperbola may be written B x sec w + A y tan c = C2tan w sec w (B-3) where c2 = A2 + B2 (B-4) 85

y p Normals QI HYPERBOLA x 0 Fig. B-1 Intersection of two normal lines to a hyperbola.. Thus for the point P we have B x sec c1 + A y tan CD = C' tan w sec CD (B-5) B xpsec C2 + A yp tan C2 = C tan c2 sec 2, Solving for Xp and yp we find (2 cos W2 - cos C1 x ='- tan O tan w Xp B tan tan 2 sin C2 -sin wc1 (B-6) c2 tan c2 - tan (1 P A sin oD2 - sin )1 86

from which we conclude under the assumption (B-2), that xp <, yp > 0 (B-7) In other words, P is in the quadrant II. 87

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