Way Transfers to Mercury and Jupiter N66-16591

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

Technical Report No. 32-77

Design Parameters for Ballistic

Interplanetary Trajectories

Part II. One- Way Transfers

to Mercury and Jupiter

V. C. Clarke, Jr,

W. E. Bollman

P. H, Feitis

R. Y. Roth

N66-16591(ACCESSION NUMBER)

(THRU)

/(PAGES)

(N_A_CR OR rMX OR AD NUMBER)

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in!i

PROPULSION LABORATORY

CALIFORNIAINSTITUTE OF TECHNOLOGY

P A SA DF.J_A, CALIFORNIA

Januory 15, 1966

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

Technical Report No. 32-77

Design Parameters for Ballistic

Interplanetary Trajectories

Part II. One-Way Transfers

to Mercury and Jupiter

V. C. Clarke, Jr.

W. E. Bo//man

P. t1. Feitis

R. Y. Roth

T. W. Hamilton, Manager

Systems Analysis Section

JET PROPULSION LABORATORY

CALIFORNIA INSTITUTE OF TECHNOLOGY

PASADENA, CALIFORNIA

January 15, 1966

Copyright :_c 1966

Jet Propulsion Laboratory

California Institute of Technology

Prepared Under Contract No. NAS 7-100

National Aeronautics & Space Administration

JPL TECHNICAL REPORT NO. 32-77

CONTENTS

I.

II.

III,

Introduction. Analytical Model for Interplanetary Trajectories. 1

A. Heliocentric Motion .......... 1

1. Determination of Planar Orientation ...... 1

2. In-Plane Relations ........... 2

3. Lambert's Theorem 2

B. Launch-Planet Escape Hyperbola .......... 3

1. Assumptions ........... 3

2. Size and Shape of Escape Hyperbola ..... 3

C. Calculation of Differential Corrections . 4

Detailed Equations for Trajectory Computations .....

A. Heliocentric Phase ...........

1. Determination of Position at Launch and Arrival

2. Application of Lambert's Theorem ......

3. Calculation of Velocity Vectors for Probe and Planet .

4. Calculation of Various Trajectory Parameters

B.

C°

7

7

7

8

11

11

The Planetocentric-Conic Trajectories ...... 13

Discussion and

A. Introduction

1. Trajectory

2. Trajectory

Differential Corrections ..........

Explanation of Results.

Computations .......

Analysis ......

16

19

19

19

19

B.

C.

D.

E.

F.

Classification of Trajectories .

1. Type I and Type II ........

2. Class I and Class II ...........

3. Minimum-Energy Trajectories .......

Mission Payload ...........

Launch Period ......

General Characteristics of Trajectories .

1. Geocentric Parameters.

2. Heliocentric Parameters ..........

3. Planetocentric Parameters

Discussion of Earth-Mercury and Earth-Jupiter TrajectoryParameters ..............

1. Declination of the Geocentric Asymptote (1,,.

2. Angle Between the Sun-Earth Vector and the OutgoingGeocentric Asymptote L ..........

3.

4.

.

6.

19

19

29

29

33

33

.... 34

.... 34

34

38

38

38

38

Celestial Latitude of the Geocentric Asymptote "/i...... 39

True Anomaly at Launch and Arrival and Heliocentric

Transfer Angle ............. 39

Aphelion and Perihelion of the Transfer Orbit 39

Time of Flight ............ 39

III

JPL TECHNICAL REPORT NO. 32-77 =

CONTENTS (Cont'd)

IV.

V.

VI.

VII.

VIII.

IX.

7. Communication Distance ..........

8. Inclination of the Heliocentric Transfer Plane i and

Celestial Latitude at Arrival [3p .........

9. Asymptotic Approach Speed V_,p .........

10. Angle Between the Incoming Asymptote and theArrival Planet's Orbital Plane "/t, ..........

11. Angle Between the Arrival Planet-Sun V_ctor and the

Incoming Asymptote _r, ...........

12. Right Ascension ®j, and Declination (l,j,of the IncomingPlanet-Centered Asymptote ...........

13. Angle Between the Planet-Earth Vector and the

Incoming Asymptote _: ............

14. Angle Between the Planet-Canopus Vector and the

Incoming Asymptote _,. ............

G. Procedures for Utilization of Graphs in D_'sigl_ ot

Planetary Trajectories

X°

Mercury 1967: Trajectory-Parameter Graphs .....

Mercury 1968: Trajectory-Parameter Graphs .....

Jupiter 1968-69: Trajectory-Parameter Graphs .....

Jupiter 1969-70: Trajectory-Parameter Graphs .....

Jupiter 1970-71 : Trajectory-Parameter Graphs

Jupiter 1972: Trajectory-Parameter Graphs .......

Jupiter 1973: Trajectory-Parameter Graphs .......

Nomenclature ...............

References ................

40

40

41

41

41

41

41

43

43

47

80

113

150

182

217

251

288

289

TABLES

2-1. Mean planet elements .........

3-1. Characteristics of minimum-energy transfer

3-2. Mercury Type I and II transfer characteristics

3-3. Type I Jupiter transfer characteristics

3-4. Type II Jupiter transfer characteristics

12

....... 31

........ 35

........ 36

........ 37

IV

IPL TECHNICAL REPORT NO. 32-77

FIGURES

1--1.

1-2.

1-3.

1-4.

1-5.

1-6.

2-1.

2--2.

2-3.

2-4.

2-5.

2-6.

3--1.

3--2.

3--3.

3-4.

3m5.

3-6.

3B7.

3--8.

3--9.

3-10.

3-11.

3-12.

3-13.

Heliocentric-transfer geometry .

In-plane-transfer geometry .....

Determination of hyperbolic-excess velocity vector

Vehicle flight plane ........

Impact parameter.

The R, S, T target coordinate system

Geometrical configuration of heliocentric conic: (a) central angle

less than 180 deg; (b) central angle greater than 180 deg

Trajectory-plane-launch-site geometry

Trajectory-plane geometry .......

Ascent-trajectory geometry .........

Powered trajectory modified by coasting .....

Relation between longitude, right ascension, and time in

equatorial plane .........

Mercury 1964-1968: Minimum injection energy vs

launch date, Type I .....


launch date, Type I ........ 21


launch date, Type I ......... 22


launch date, Type II ...... 23


launch date, Type II ....... 24


launch date, Type II .........

Jupiter 1969-1973: Minimum injection energy vs

launch date, Type I ......... 26

Jupiter 1969-1973. Minimum injection energy vs

launch date, Type II ............. 27

Junipter 1968-1969: Time of flight vs launch date ..... 28

Jupiter 1968-1969: Minimum injection energy vs launch date . 30

Typical payload capability curves for Atlas/Centaur,

Saturn 1B, and Saturn 5 launch vehicles ...... 33

Permissible regions of the declination of the geocentric

asymptote ,1,_for Cape Kennedy launchings ...... 38

Inclination of the heliocentric orbital plane to ecliptic vs

heliocentric central angle ........ 40

9

14

14

15

15

16

2O

V


FIGURES (Cont'd)

3-14.

3-15.

3-16.

3-17.

4-1.

4-2.

4-3.

4-4.

4-5(I).

4-5(11).

4-6(I).

4-6(11).

4-7(I).

4-7(11).

4-8(I).

4-8(11).

4-9(I).

4-9(11).

4-10.

4-11(I).

4-11.(11).

Crescent orientations for typical trajectories to Mercury and

Jupiter as observed from approaching spacecraft

Near-Jupiter geometry for typical trajectories, viewed from

above ecliptic plane: (a} Type I trajectory, encountering

Jupiter before aphelion; (hi Type I-II trajectory, encountering

Jupiter after aphelion

Near-Mercury geometry for typical trajectories, viewed from

above ecliptic plane: (a) Type I trajectory, encountering

Mercury before perihelion; (b) Type II trajectory, encountering

Mercury after perihelion

Steps for construction of arrival date loci

Mercury 1967: Minimum injection energy vs launch date

Mercury 1967: Time of flight vs launch date ......

Mercury 1967: Heliocentric central angle vs launch date

Mercury 1967: Earth-Mercury communication distance vs

launch date

Mercury 1967: Declination of the geocentric asymptote vs

launch date, Type I


launch date, Type II .....

Mercury 1967: Right ascension of the geometric asymptote vs

launch date, Type I

Mercury 1967: Right ascension of the geocentric asymptote vs


Mercury 1967: Angle between outgoing geocentric asymptote

and launch planet's orbital plane vs launch date, Type I


and launch planet's orbital plane vs launch date, Type II

Mercury 1967: Angle between Sun-Earth vector and outgoing

geocentric asymptote vs launch date, Type I


geocentric asymptote vs launch date, Type II

Mercury 1967: True anomaly in transfer ellipse at launch

time vs launch date, Type I

Mercury 1967:

time vs launch

Mercury 1967:

time vs launch

Mercury 1967:

Mercury 1967:

True anomaly in transfer ellipse at launch

date, Type II

True anomaly in transfer ellipse at arrival

date

Perihelion of transfer orbit vs launch date, Type I

Perihelion of transfer orbit vs launch date, Type II .

42

43

44

46

49

50

51

52

53

54

55

56

O_

5_

59

6O

61

6z_

63

64

65

VI


FIGURES (Cont'd)

4-12(I).

4-12(11}.

4-13(I).

4-13(11).

4-14.

4-15(I).

4-15{11).

4-16(I).

4-16(11).

4-17.

4-18II).

4-18III).

4-19(I).

4-19111}.

5--1.

5-2.

5-3.

5-4.

5-5(I).

5-5(11).

5-6(I).

Mercury 1967: Aphelion of transfer orbit vs launch date, Type I

Mercury 1967: Aphelion of transfer orbit vs launch date, Type II

Mercury 1967: Inclination of the heliocentric transfer plane

vs launch date, Type I


vs launch date, Type II ...........

Mercury 1967: Celestial latitude at arrival time vs launch date .

Mercury 1967: Asymptotic speed with rescept to Mercury vs


Mercury 1967: Asymptotic speed with respect to Mercury vs

launch date, Type II .......

Mercury 1967: Angle between incoming hermiocentric

asymptote and arrival planet's orbital plane vs

launch date, Type I



launch date, Type II

Mercury 1967: Angle between Mercury-Sun vector and incoming

hermiocentric asymptote vs launch date

Mercury 1967: Declination of the hermiocentric asymptote vs

launch date, Type I


launch date, Type II .

Mercury 1967: Right ascension of hermiocentrlc asymptote vs


Mercury 1967: Right ascension of hermiocentric asymptote vs

launch date, Type II ..........


Mercury 1968: Time of flight vs launch date


Mercury 1968: Earth-Mercury communication distance vslaunch date


launch date, Type I ......


launch date, Type II .......

Mercury 1968: Right ascension of the geocentric asymptote

vs launch date, Type I .....

66

67

68

69

7O

71

7:2

73

74

76

77

7S

79

8:2

83

St

85

86

87

8S

VII


FIGURES (Cont'd)

5-6(11).

5-7(I).

5-7(11).

5-8(I).

5-8(11).

5-9(I).

5-9(11).

5-10.

5-11 (I).

5-1 1 (11).

5-1 2(I).

5-12(11).

5-13(I).

5-13(11).

5-14.

5-15(I).

5-15III).

5-16(I).

5-16(11).


vs launch date, Type II









Mercury 1968: True anomaly in transfer ellipse at launch time




Mercury 1968: True anomaly in transfer ellipse at

launch time vs launch date

Mercury 1968: Perihelion of transfer orbit vs

launch date, Type I



Mercury 1968: Apehelion of transfer orbit vs

launch date, Type I

Mercury 1968: Aphelion of transfer orbit vs



vs launch date, Type I ...........


vs launch date, Type II .....

Mercury 1968: Celestial latitude at arrival time vs launch date


launch date, Type I



Mercury 1968: Angle between the incoming hermiocentric


launch date, Type I


asymptote and arrival planet's obital plane vs


89

9O

91

92

9:3

94

95

96

97

98

99

100

lOl

102

103

104

105

106

107

VIII

JPL TECHNICAL REPORTNO. 32-77

FIGURES (Cont'd)

5-17.

5-18(I).

5-18(11).

5-19(I).

5-19(11).

6--1.

6-2.

6-3.

6-4(I).

6-4(11).

6-5(I).

6-5(11).

6-6(I).

6-6(11).

6-7(I).

6-7(11).

6-8(I).

6-8(11).

6-9(I).

6-9(11).


hermiocentric asymptote vs launch date, Type I





Mercury 1968: Right ascension of the hermiocentric asymptote



vs launch date, Type II ......

Jupiter 68-69: Minimum injection energy vs launch dale

Jupiter 68-69: Time of flight vs launch date

Jupiter 68-69: Heliocentric central angle vs launch date

Jupiter 68-69: Earth-Jupiter communication distance vs

launch date, Type I



Jupiter 68-69: Declination of the geocentric asymptote vs

launch date, Type I



Jupiter 68-69: Right ascension of the geocentric asymptote

vs launch date, Type I ......


vs launch date, Type II .....

Jupiter 68-69: Angle between the outgoing geocentric

asymptote and launch planet's orbital plane vs

launch date, Type I




Jupiter 68-69: Angle between Sun-Earth vector and the

outgoing geocentric asymptote vs launch date, Type I


outgoing geocentric asymptote vs launch date, Type II

Jupiter 68-69: True anomaly in transfer ellipse at

launch time vs launch date, Type I .......

Jupiter 68-69: True anomaly in transfer ellipse at

launch time vs launch date, Type II.

10S

109

110

111

ll2

115

lI6

117

119

12O

12I

]'_

123

12.t

125

126

127

19.8

129

IX


FIGURES (Cont'd)

6-10. Jupiter

arrival

6-11(I). Jupiter

Type I

6-11ill). Jupiter

Type II

6-12(I). Jupiter

Type I

6-12(11). Jupiter

Type II

6-13(I).

6-13(11).

6-15(11}.

6-16.

6-17.

6-18(I).

6-18(11).

6-19{I).

6-19(11).

6-20(I).

6-20(11).

6-21 (I).

6-21 ill).

68-69: True anomaly in transfer ellipse at

time vs launch date

68-69: Perihelion of transfer orbit vs launch date,


68-69: Aphelion of transfer orbit vs launch date,


Jupiter 68-69: Inclination of the heliocentric transfer

plane vs launch date, Type I ....


plane vs launch date, Type II

Jupiter 68-69: Celestial latitude at arrival time vs launch date

Jupiter 68-69: Asymptotic speed with respect to Jupiter vs

launch date, Type I .........



Jupiter 68-69: Angle between incoming zeocentric asymptote

and arrival planet's orbital plane vs launch date

Jupiter 68-69: Angle between Jupiter-Sun vector and incoming

zeocenlric asymptote vs launch date

Jupiter 68-69: Declination of the zeocentric asymptote vs

launch date, Type I



Jupiter 68-69: Right ascension of zeocentric asymptote vs




Jupiter 68-69: Angle between Planet-Earth vector and

incoming asymptote vs launch date, Type I


incoming asymptote vs launch date, Type II

Jupiter 68-69: Angle between the planet-Canopus vector

and incoming asymptote vs launch date, Type I


and incoming asymptote vs launch date, Type 11

130

131

132

133

134

135

136

137

1:38

139

l -t0

14l

1,12

143

144

145

146

147

148

1.t9

X


FIGURES (Cont'd)

7--1.

7-2.

7-3.

7--4(I).

7-4(11).

7-5(I).

7-5(11).

7-6(I).

7-6(11).

7-7(I).

7-7(11).

7--8.

7--9.

7-10.

7-11 (I).

7-11 (ll}.

7-12(I).

7-1 2(11).

7-13(I).

7-13(11).

7.-14.

Jupiter 69-70: Minimum injection energy vs launch date








launch date, Type I



Jupiter 69-70: Right ascension of the geocentric asymptote vs

launch date, Type I



Jupiter 69-70: Angle between outgoing geocentric asymptote




Jupiter 69-70: Angle between Sun-Earth vector and outgoing

geocentric asymptote vs launch date

Jupiter 69-70: True anomaly in transfer ellipse at launchtime vs launch date

Jupiter 69-70: True anomaly in transfer ellipse at arrival

time vs launch date

Jupiter 69-70: Perihelion of transfer orbit vs launch date,

Type I


Type II .

Jupiter 69-70: Aphelion of transfer orbit vs launch date,

Type I


Type II ..........

Jupiter 69-70: Inclination of the heliocentric transfer plane

vs launch date, Type I ....



Jupiter 69-10: Celestial latitude at arrival time vs launch date .

152

153

15.1

155

156

157

158

159

160

161

162

16:t

164

165

166

167

168

169

170

171

172

XI


FIGURES (Cant'd)

7-15(I).

7-15(11}.

7-16(I).

7-16(11).

7-17.

7-18(I).

7-18(11).

7-19(I).

7-19(11).

8--1,

8-2.

8-3.

8-4(I).

8-4(11).

8-5(I).

8-5(11).

8_6.

8h7.

8-8(1l.

8-8(11).

8-9(I).

Jupiter 69-70: Asymptotic speed with respect to Jupiter



vs launch date, Type II ......


and arrival planet's orbital plane vs launch date, Type I


and arrival planet's orbital plane vs launch date, Type II

Jupiter 69-70: Angle between Jupiter-Sun vector and

incoming zeocentric asymptote vs launch date


launch date, Type I



Jupiter 69-70: Right ascension of the zeocentric asymptote vs

launch date, Type I

Jupiter 69-70: Right ascension of the zeocentric asymptote vs





Jupiter 70-71 : Earth-Jupiter communication distance vs

launch date, Type I

Jupiter 70-71 : Earth-Jupiter communication distance vs



launch date, Type I

Jupiter 70-71 : Declination of the geocentric asymptote vs



launch date


and launch planet's orbital plane vs launch date


geocentric asymptote vs launch date, Type !



Jupiter 70-71: True anomaly in transfer ellipse at launch

time vs launch date, Type I ......

173

174

175

176

t77

178

179

180

lS1

1S.t

IS5

IS6

1,_7

18S

189

19O

191

192

193

194

195

XI!


FIGURES (Cont'd)

8-9(11).

8-10.

8-11(I).

8-11 (11).

8-12(I).

8-12(11).

8-13(I).

8-13(11).

8-14.

8-15(I).

8-15(11).

8-16.

8-17.

8-18(I).

8-18(11).

8-19(I).

8-19(11).

8-20(I).

8-20(11).

8-21 (I).

Jupiter

time vs

Jupiter

time vs

Jupiter

Type I

Jupiter

Type II

Jupiter

Type I

Jupiter

Type II

70-71: True Anomaly in transfer ellipse at launch


70-71 : True anomaly in transfer ellipse at arrival

launch date





Jupiter 70-71 : Inclination of the heliocentric transfer plane

vs launch date, Type I ....

Jupiter 70-71 : Inclination of the heliocentric transfer plane

vs launch date, Type II ....

Jupiter 70-71 : Celestial latitude at arrival time vs launch date





Jupiter 70-71 : Angle between incoming zeocentric asymptote


Jupiter 70-71 : Angle between Jupiter-Sun vector and incoming

zeocentric asymptote vs launch date

Jupiter 70-71 : Declination of zeocentric asymptote vs


Jupiter 70-71 : Declination of zeocentric asymptote vs

launch date, Type II ......


launch date, Type I .......


launch date, Type II ........

Jupiter 70-71 : Angle between planet-Earth vector and

incoming asymptote vs launch date, Type I ....

Jupiter 70-71 : Angle between planet-Earth vector and

incoming asymptote vs launch date, Type II ....

Jupiter 70-71: Angle between planet-Canopus vector and

incoming asymptote vs launch date, Type I ....

19(3

197

19S

199

200

2()1

202

203

201

2O5

206

207

20_%

:209

210

211

212

213

21-1

215

XIII


FIGURES (Cant'd)

8-21(11).

9_1.

9-2.

9-3.

9-4(I).

9-4(11).

9m5.

9m6.

9m7.

9-8(I).

9-8(11).

9-9(I).

9-9(11).

9-10.

9-11(I}.

9-11 (11).

9-12(I).

9-12(11).

9-13(I).

9-13(11).

9-14(I).

Jupiter 70-71 : Angle between planet-Canopus vector and


Jupiter 1972: Minimum injection energy vs launch date

Jupiter 1972: Time of flight vs launch date

Jupiter 1972: Heliocentric central angle vs launch date

Jupiter 1972: Earth-Jupiter communication distance vs

launch date, Type I



Jupiter 1972: Declination of the geocentric asymptote vs

launch date

Jupiter 1972: Right ascension of geocentric asympt,,te vs

launch date

Jupiter 1972: Angle between outgoing geocentric asymptote


Jupiter 1972: Angle between Sun-Earth vector and outgoing




Jupiter 1972: True anomaly in transfer ellipse at launch time



vs launch date, Type II .......

Jupiter 1972: True anomaly in transfer ellipse at arrival timevs launch date

Jupiter

Type I

Jupiter

Type II

Jupiter

Jupiter

Jupiter

launch

Jupiterlaunch

Jupiter

Type I

1972: Perihelion of transfer orbit vs launch date,


1972:

1972:

Aphelion of transfer orbit vs launch date, Type I

Aphelion of transfer orbit vs launch date, Type II

1972: Inclination of the heliocentric transfer plane vs

date, Type I

1972: Inclination of the heliocentric transfer plane vs

date, Type II

1972: Celestial latitude at arrival time vs launch date,

219

22O

221

223

224

225

226

227

22_

229

230

231

2:3:2

:2:33

2:3.1

2:35

236

2:37

23S

XIV


FIGURES (Cant'd)

9-14(11).

9-15(I).

9-15i11).

9-16.

9-17.

9-18.

9-19(I).

9-19(11).

9-20(I).

9-20(11).

9-21 (I).

9-21 fill

10-1.

10-2{I}.

I0-2(11).

10-3.

10-4(I).

10-4[11).

10-5(I).

10-5{11).

10-6(I).

Jupiter

Type II

Jupiterlaunch

Jupiter

launch


1972: Asymptotic speed with respect to Jupiter vs

date, Type I ......

1972: Asymptotic speed with respect to Jupiter vs

date, Type II

239

24O

241

Jupiter 1972: Angle between incoming zeocentric asymptote


Jupiter 1972: Angle between Jupiter-Sun vector and incoming

zeocentric asymptote vs launch date .... 243

Jupiter 1972: Declination of zeocentric asymptote vs

launch date .......... 244

Jupiter 1972: Right ascension of zeocentric asymptote vs

launch date, Type I .... 24.5


launch date, Typell . 246

Jupiter 1972: Angle between planet-Earth vector and incoming

asymptote vs launch date, Type I 247


asymptote vs launch date, Type II 248

Jupiter 1972: Angle between planet-Canapus vector and

incoming asymptote vs launch date, Type I 249

Jupiter 1972: Angle between planet-Canopus vector and


242

25O

253

254

255

256

257

258

Jupiter 1973: Minimum injection energy vs launch date .

Jupiter 1973: Time of flight vs launch date, Type I

Jupiter 1973: Time of flight vs launch date, Type II

Jupiter 1973: Heliocentric central angle vs launch date .




launch date, Type II ..........

Jupiter 1973: Declination of geocentric asymptote vs

launch date, Type I

Jupiter 1973: Declination of geocentric asymptote vs


Jupiter 1973: Right ascension of geocentric asymptote vs

launch date, Type I

259

260

261

XV


FIGURES (Cont'd)

10-6(11).

10-7.

10-8(I).

10-8(11).

10-9(I).

10-9(11).

10-10.

10-11(I).

10-11111L

10-12II).

10-12(11).

10-13(I).

10:13(11).

10-14(I).

10-14(11).

10-15(I).

10-15(11].

10-16.

10-17.






geocentric asymptote vs launch date, Type I ....


geocentric asymptote vs launch date, Type II ....





Jupiter 1973: True anomaly in transfer ellipse at arrival time

vs launch date

Jupiter 1973: Perihelion of transfer orbit vs launch date,

Type I .....


Type II .

Jupiter 1973: Aphelion of transfer orbit vs launch date,

Type I

Jupiter 1973: Aphelion of transfer orbit vs launch date,

Type II .....

Jupiter 1973: Inclination of the heliocentric transfer plane vs

launch date, Type I

Jupiter 1973: Inclination of the heliocentric transfer plane vs


Jupiter 1973: Celestial latitude at arrival time vs launch date,

Type I .....


Type II .

Jupiter 1973: Asymptotic speed with respect to Jupiter vs

launch date, Type I







262

263

264

265

266

267

26S

269

27O

271

272

273

274

275

276

277

278

279

280

XVI


FIGURES (Cont'd)

10-18.

10-19(I).

10-19(11).

10-20(I).

10-20(11).

10-21(I).

10-21 [11).

Jupiter 1973: Declination of zeocentric asymptote vslaunch date


launch date, Type I




asymptote vs launch date, Type I


asymptote vs launch date, Type II





281

2S2

283

2S4

285

2S6

.... 287

XVII


FOREWORD

This report presents, in graphical fo]'m_ the results of studies of

the characteristics of ballistic illtcrplanetarv [rztic(_'t()ries to _|el'('llrV

(launch dates: 1967-1968) and Jupiter (latH_c'h dates: 1968-1973).

Also included are a description of the physical model, the development

of the equations of the model, and a disct_ssi()l_ of the properties of

the trajectories.

XVII!


ABSTRACT I (.,' IThe general characteristics of ballistic interplanetary trajectories are

discussed, and detailed equations are developed for the analytical

model. Extensive data are presented ill graphical form for trajectories

to Mercury (launch dates: 1967-1968) and Jupiter (launch dates:

1968-1973). These graphs include: (1) curves of vis viva geocentric

energy vs launch date for minimmn-energy trajectories and (2) curves

of 19-21 different trajectory parameters vs launch date for various

vis viva geocentric energies. The trajectories were computed on the

IBM 7090 digital computer by numerical evaluation of the analytical

model, after which specific parameters of interest were automatically

plotted, carefully checked, and analyzed. Procedures are outlined for

use of these data by the trajectory engineer in the design and analysis

of interplanetary trajectories. _,L,It,t t_^ _t /

I. INTRODUCTION. ANALYTICAL MODEL FOR INTERPLANETARY TRAJECTORIES

The analytical model used in the generation of Mer-

cury and Jupiter trajectory parameters consists of three

distinct phases of two-body motion: (1) an escape hyper-

bola near the launch planet, (2) elliptical _ motion under

the attraction of the Sun, and (3) terminal hyperbolic

motion near the target planet.

A. Heliocentric Motion

Solution of the heliocentric elliptic motion is obtained

first under the following assumptions:

(1 The launch and target planets move in orbits about

the Sun as given in the national ephemerides.

Their velocity components are obtained by usingtwo-body conic formulas, mean orbital elements,

and their tabular positions, as listed in the ephem-erides.

(2 The launch and target planets are massless. Thus,

the only force acting on the probe is that of theSun.

(3 The position of the probe at launch into the helio-

centric orbit is the center of the massless launch

planet. Its position at arrival on the heliocentric

orbit is the center of the massless target planet.

_Hyperbolic heliocentric motion is not considered herein.

Thus, for solution of the heliocentric phase of motion,

the attractions of the launch and target planets are tem-

porarily disregarded. The primary result to be obtained

from the solution of the heliocentric-transfer problem is

determination of the hyperbolic-excess velocity vector

relative to the launch planet.

l. Determination of Planar Orientation

Since the launch and arrival positions of the probe

are assumed to be the centers of the launch and target

planets, they can immediately be determined, given the

launch and arrival "_times, by consulting the ephemeris.

Further, the orientation of the heliocentric transfer plane

can immediately be found. Let RL be the Sun-launch-

planet position vector at launch time TL, and let Rp be

the Sun-target-planet position vector at arrival time T,

(Fig. 1-1). Planar orientation is then found from the

unit normal W to the plane, as follows:

Rr_ X R,W - RL Rp sin ,I, ( 1-1 )

where the angle ge is defined below. The inclination i to

'Or, for convenience, the launch date and flight time can be specified.


the ecliptic plane _ can be found by

cos i = W" K' (1_.o)

K I

/I PLANET N _ _" _r, IPTIC /

CENTER

Fig. 1-1. Heliocentric-transfer geometry

where K' is a unit vector pointing in the direction of

the ecliptic north pole.

2. In-Plane Relations

The heliocentric central angle _I, (Fig. 1-1) is also

readily determined by utilizing the positions of the launch

and target planets. This angle may be obtained from

RL" I_ (1--3)cos,t = IRL!IRpl

sin ,I, = sgn [(RL X Rp)" K'] (1 -- cos ="I') '_ (1-4)

The velocity vector V of the spacecraft anywhere

along its path may be obtained from

V = V[(W X R) cos P + Rsin P] (1-5)

Here, R is the heliocentric position vector, R = IRI,

and V is the heliocentric speed, obtained from

qn this Report, the interest is only in transfers which have the same

rotational motion about the Sun as do the planets: thus, 0 -----i --_ _r/2.

and the path angle F is found from

[4 " ]sin 1-' = (1 -- e 2) (2a -- R) e sin v (1-7)

In Eqs. 1--6 and 1-7, GMs is the universal gravitational

constant times the mass of the Sun ( =2.959122083 X 10 -4

au_/day '-') ; a and e are the semimajor axis and eccentricity

of the transfer ellipse, respectively; and v is the true

anomaly of the probe, given by

a(1 - e =) - R(1-8)

cos v = ea

Now, there are two unknowns in Eqs. 1-5 to 1-S which

prevent th(-ir immediate evaluation: the semimajor axisa and the cc.ce]_tricity e. The determilmtion of these

quantities is the main problem. Battin (Ref. 1) has shown

that the eccentricity is actually a function of the semi-

major axis. Thus, it is first necessary to determine a.The semimajor axis is related to the time of flight Tr hy

Lambert's Theorem, which states: The transfer time be-

tween ally two l)oints on an ellipse is a fimction of the

sum of the distances of eact_ point from the [oerl.s', the

dista_ce between the points, and the semimajor axis of

the ellipse, t:nnctionally, the theorem is stated as

T,, = Tr (R,, + ap, C, a) ( 1-9 )

where the distance C between the launch planet at

launch time and the target planet at arrival time, as

shown in Fig. 1-2, is obtained from

C = IR, - R_I (1-10)

Since the time of flight Tr and the launch and arrival

positions Rr_ and Rp are known, only the semimajor axis

remains t() be found by iterative solution of Eq. 1-9.

After the semimajor axis a is obtained, the heliocentric

velocities of the probe at launch and arrival times V_

Rp

RL

Fig. 1-2. In-plane-transfer geometry

2


and Vp may be evaluated from Eq. 1-5 under the condi-

tions R =- RL and R = Rp. The path angles Pt,, Up and

true anomalies VL, Vp at launch and arrival times 4 may

also be evaluated from Eqs. 1-7 and 1-8 under the sameconditions.

Finally, the desired end result, the hyperbolic-excess

velocity Vhr, relative to the launch planet, may be found

( Fig. 1_3 ) by

VhL ----VL - V_ ( 1-11 )

where V1 is the velocity of the launch planet at launchtime.

Fig. 1-3. Determination of hyperbolic-excess

velocity vector

B. Launch-Planet Escape Hyperbola

The key result from tile solution of heliocentric trans-

fer is the hyperbolic-excess velocity vector VhL at launch.

The reason for the importance of this vector is that it

tells the direction in which the probe must be traveling

relative to the launch planet when on the point of leaving

the planet's gravitational influence. There are an infinite

number of escape trajectories (all hyperbolas) which can

have the same hyperbolic-excess velocity vector. How-

ever, only a portion of these are practical for use when

related to existing launch sites and boost-vehicle con-

straints. For example, it would be ridiculously costly in

payload--and impractical--to shoot a vehicle straight up.

Criteria for selection of a family of feasible escape tra-

jectories are given below.

1. Assumptions

The solution of the escape phase of motion is obtained

under the following assumptions: (1) The probe is acted

on only by the gravitational force of the launch planet,

and (2) the oblateness effects of the launch planet are

neglected.

_The details of quadrant choice for these angles are found in Ref. 3.

The direction of the asymptote of the escape hyperbola

is found by normalizing the hyperbolic-excess vector

V_L. The injection energy C3 of the escape hyperbola 5 is

found by squaring the hyperbolic-excess speed, or

C3 = V_L (1-12)

Thus, in contrast to the heliocentric problem, the launch

planet is now "massy," whereas the influence of the Sun

is neglected. However, the hyperbolic-excess velocity

vectors found by solving the heliocentric problem are

used as a starting point to solve the escape problem.

2. Size and Shape of Escape Hyperbola

As previously stated, only some of the infinite number

of escape trajectories are practical. Two of the practical

aspects of a set of trajectories are the sizes and shapes

of the hyperbolas.

Size is basically determined by the energy C._, which,

in turn, is a function of boost-vehicle capability. For

boost vehicles in use at this writing (or shortly to be

available), values of energy less than or equal to 25

km2/sec z are considered reasonable. The larger the value

of energy that the booster is required to deliver, the

smaller the payload and launch period over which the

vehicle may be fired.

The shape of the hyperbola is determined by its eccen-

tricity, which is a function of both the energy and the

perifocal distance, according to

e= 1+ RpC._ (1-13)GM

where R t, is the perifocal distance and GM is the uni-

versal gravitational constant times the mass of the lam.eh

planet. From Eq. 1-13, it can be seen that, for a fixed

perifocal distance, the eccentricity increases linearly with

the energy. The value of perifocal distance is not arbi-

trary, but depends strongly on the boost-vehicle trajec-

tory. It has been shown (Ref. 2) that, in the great

majority of cases, it is necessary and desirable to use a

circular parking orbit as part of the preinjection phase of

the escape trajectory. It is, further, an interesting fact

that the altitude of the parking orbit determines the

perifocal distance. If h is the parking-orbit altitude and

R0 is the launch planet's radius, then, to an extremely

close degree of approximation,

Rp = R0 + h (1-14)

5C_is actually twice the total energy per unit mass; i.e., the visviva

integral.

3


or the perifocal distance is equal to the launch-planet-centered radius of the parking orbit. In Bef. 2, it also

has been shown that the lowest possible parking orbit

(80-100 nm) allows greatest payload capability. Thus,

using 100 nm for the parking-orbit altitude, a practical

value of perifocal distance is 6560 kin. The perifocal

distance will vary only slightly about this value for other

parking-orbit altitaldes, or even for direct-ascent-typepreinjection trajectories. Therefore, both the size and

shape are essentially determined by the energy alone,which is found from Eq. 1-12.

Given the size and shape of the escape hyperbola, its

planar orientation must be determined. This can be done

by considering two vectors: (1) the direction of the

hyperbolic-excess vector, denoted by a unit vector S, and

(2) a unit vector R_ directed from the center of the

launch planet to the launch site. The vehicle's flight plane

will essentially be determined by these two vectors, as

shown in Fig. 1-4. A unit normal W to the launch-planet-

centered flight plane is determined by

Sw - × sl (1-15)

with the constraint that the Z component of W is always

positive.

Since R_ is a function of time, according to the rotationrate of the launch planet, the planar orientation must

continually change. In effect, this says that the launchazimuth is a continuous function of launch time.

A detailed description of the geometrical aspects of

the launch-planet ascent trajectory is not given here, trot

may be found in Ref. 2.

C. Calculation of Differential Corrections

The calculation of differential corrections for inter-

planetary trajectories may be accomplished in several

ways and depends on the choice of independent and de-

pendent variables. In this Report, a numerical differenc-

ing scheme is used. Basically, the independent variables--

the injection energy C_, declination _,s., and right ascen-

sion O_ of the outgoing asymptote S of the escape hyper-

bola- are varied, one at a time, to produce variations in

the dependent variables--the components of the impact

parameter B and the time-of-flight Tr.

The impact parameter B is defined as a vector origi-

nating at the center of the target planet and directed

perpendicular to the incoming asymptote of the target-

centered approach hyperbola (Fig. 1-5). The impact pa-

rameter B is resolved into two components which lie in a

plane normal to the incoming asymptote S. The orienta-

tions of the reference axes in this plane are arbitrary, but

one is usually selected to lie in a fixed plane. Thus, define

a unit vector T, lying in the ecliptic plane, according to

SXK'

T = IS X K'[ (1-16)

where K' is a unit vector normal to the ecliptic plane.

The remaining axis is then given by a unit vector R,

defined by

R = $ X T (1-17)

Figure 1-6 illustrates the orientation of the R, S, T

target coordinates.

The impact parameter B lies in the R-T plane and has

miss co,nponents B • T and B • R. The condition B" T =

B" R = 0 denotes vertical impact on the target. Thus,

B • T, B • R, and T_, are the three target dependent vari-ables. If Q_ represents a set of generalized independent

variables, such as injection position and velocity or otherconvenient wtriables, then the partial derivatives

rgB. T/OQ_, ?B" R/_Qi, _Tr/_Qi are first-order differen-

tial corrections or error coefficients relating miss at the

target and flight-time errors to the independent variables.

A convenient set of independent variables for inter-

planetary trajectories is the v/s viva injection energy C3,

the declination q,._, and the right ascension 0._ of the

asymptote of the escape hyperbola. These variables essen-

tially describe the launch hyperbolic-excess velocity vec-tor VhL.

4


NORTH POLE

!

LAUNCHER _--_

VERNALEQUINOX

K W

J

Fig. 1-4. Vehicle flight plane


PATH OF PROBE

CENTER OF

TARGET BODY S

INCOMING

ASYMPTOTE

Fig. 1-5. Impact parameter

0 T

R

i

Fig. 1-6. The R, S, T target coordinate system

6


II. DETAILED EQUATIONS FOR TRAJECTORY COMPUTATIONS

In Section I, a summary was given of the physical

model used to generate the parameters of Mercury and

Jupiter trajectories. The purpose of this Section is to pre-

sent, in full detail, the equations of the Jet Propulsion

Laboratory's Heliocentric Conic Trajectory Program ';, in-cluding the equations of Lambert's Theorem;, which were

developed for this investigation. The equations were

coded for the Laboratory's IBM 7090 digital computer.

This program has proved very useful over the past few

years in studying and analyzing interplanetary trajectories.

Some of the results of these studies are presented graphi-

cally in Sections III to X of this Report. The program

has been devised for calculation of trajectories from any

planet in the solar system to any other planet. In addition,

by adding the appropriate orbital elements, trajectories

to comets or to any body of known motion may becomputed.

The Heliocentric Conic Trajectory Program is divided

into four major sections:

(1) The heliocentric phase, in which basic computa-

tions are made for a unique conic trajectory, of

given flight time, that passes through the center

of two massless planets rotating around the Sun,

as defined in the national ephemerides.

(4) Computation of partial derivatives which relate

miss and time-of-flight errors at the target to vari-

ations in the hyperbolic-excess velocity vector of

the escape hyperbola near the launch planet.

A. Heliocentric Phase

I. Determination of Position at Launch and Arrival

For any given time, the positions of the planets can be

obtained from the ephemerides, referenced to a given

coordinate system and epoch. For the present purposes,

the heliocentric ephemerides of the planets, referenced

to the mean equator and equinox of 1950.0, were selected.

From these ephemerides, one may find the position vec-

tor R_. of the launch planet at lam]ch time T_., the posi-

tion vector R_, of the target planet at arrival time T,,, andthe position vector RS., of the lamach planet at arrival

time. For practical and computational purposes, it is con-venient to transform these coordinates to a heliocentric

ecliptic, mean-of-launch-date system. Thus, two rotation

matrices must be computed. First, a rotation is made from

heliocentric equatorial coordinates, mean of 1950.0, to

heliocentric equatorial, mean of launch date. This is ac-

complished by means of the matrix A, whose elements are

given below (as obtained from Ref. 3):

axl = 1 - 0.00029697T 2 - 0.0000001ST a

a12 = -- a2, = -- 0.02284988T -- 0.00000676T _ + 0.00000221T _

a,_ = - aa_ = - 0.00971711T + 0.00000207T _ + 0.00000096T a

a22 = 1 - 0.00024976T 2 - 0.00000015T a

a2a = aa2 = -- 0.00010859T 2 - 0.00000003T a

a_3 = 1 - 0.00004721T 2 + 0.00000002T s

(2-1)

(2) The near-launch-planet phase, in which a launch-

planet--centered conic is fitted to the heliocentric

conic for a given launch azimuth and launch site.

(3) The near-target-planet phase, in which the param-

eters of the planetocentric conic at the target arecalculated.

'Coded by W. J. Scholey and R. Y. Roth of the JPL Computer Appli-cations and Data Systems Section._Developed by E. Dobies, formerly of JPL, and coded by C. A. Sea-feldt, of the JPL Computer Applications and Data Systems Section.

where T is the number of Julian Centuries of 86,525 days

past the epoch 1950.0.

The second rotation transforms the coordinates from

heliocentric equatorial, mean of launch date, to helio-

centric ecliptic, mean of launch date. This is accomplishedby means of the matrix E:

1 0 0 ]

E = 0 cosE sin_] (2-2)0 -- sin E cos e

7


where E is the mean obliquity of the ecliptic on the launch

date and sin e and cos e are approximated by the follow-

ing equations from Re[, 4:

1 p2(T ) sin Eosin _ = sin _o + P(T) cos _o - -_-

1cos _ = cos eo - P(T) sin E0 -- _ P2(T) cos Eo

(e-a)

where

P(T) = - 0.000227111T - 0.0000000286T 2

+ 0._878T 3

and E0 is the mean obliquity of the ecliptic for 1950.0.

Now, in general, the components of the position vectors

RL, RLa, and Rp are found in mean-equinox, ecliptic-of-

launch-date coordinates by

RL

RLA = EA

Rp

11;

R;

The unit vectors R_, R_ may then be found:

R_- RL R_ = RpRL R---'_

where

(2-5)

(2-6)

RL = IRLI and R, = Ili_l (2-7)

The heliocentric central angle % i. e., the angle between

RL and Rp, is computed from

l_ L • Rpcos ,It --

RL Rp (2-8)

sin ,It = sgn [(RL × Rp)" K'] (1 - cos z ,I,) v'

where K' is the unit vector normal to the ecliptic plane

in the direction of the ecliptic north pole.

The unit vector W normal to the probe's orbit plane,

is found from the vector equation

W- RLXI_ (2-9)RL Rp sin ,It

Subsequently, the inclination of the orbit to the ecliptic

plane i may be found from

cos i = W' K' (2-10)

where 0 _-_ i _ _r/2 are the only cases of interest to this

program.

Essentially, calculation of the W vector determines the

planar orientation of the transfer conic.

2. Application of Lambert's Theorem

As shown in Section I, the semimajor axis a may be

found, given the flight time T_,, RL, and Rp. One usesLambert's Theorem, which states:

The transfer time between any two points on an

ellipse is a function of the sum of the distances

of each point from the focus, the distance be-

tween the points, and the semima/or axis of the

ellipse.

Or, functionally,

Te = Tr(RL + a,,C,a) (2-11)

where C is the chord distance from RL to Rp, or

C = II_, - l_ I (2--19.)

Now, since Tr, RL, Rp, and C are known, Lambert's The-

orem can be used to solve for a by an iterative processdescribed below.

First, the transfer time T,,, of the minimum-heliocentric-

energy tr@'ctory is computed from Eqs. 2-13 to '9..--22,

The minimum semimajor axis am for an ellipse can befound from

am = (RL + R, + C)/4 (2-18)

and e_, the eccentricity of the orbit with a minimum semi-

major axis, is expressed by

e,,, = .]_1 P" (2-14)a_

where p,, is the semilatus rectum obtained by

P- = 2(2am - RL)(2am -- R,) (2-15)C

The true anomaly at launch VL,,, is found from Eqs. 2-16

and 2-17. First, compute the angle 4:

cos q_ -- Rr. - am (1 - e_) 0_4_r (2-16)e,_ R_

where _ is an angle from the aphelion to Rz.

It can then be shown from the geometry of the mini-

mum-energy ellipse (Fig. 2-1) that

(2-17 )VL,, = _ + 4__ 7r < "_ < 2_

8


(a| Central angle less than 180 deg (b) Central angle greater than 180 deg

Fig. 2-1. Geometrical configuration of heliocentric conic

The true anomaly at arrival on the minimum-energy el-lipse vp,_ can be found from

v_ = VL,,, + * (2--18)

Now, the mean anomaly ML,_ at launch in the ellipse canbe found from

ML., = EL., -- e.. sin EL,,, (2--19)

where ELm, the eccentric anomaly, is given by the equations

em -t- cos t_LmCOS ELm= 1 + e,, cos vz_

and 0 _< EL,, _< 2at

sin Ez,= = _/1 -- e,J sin VLm1 + e,_ cos V_

(2-e0)

The mean anomaly at arrival M m can be obtained in a

similar manner. Then, the mean-anomaly difference is

AM,, : Mpm -- Mi.,_ (2-21)

By use of this difference, Tin, the time for a minimum-

energy trajectory from launch to target, can be found

from the equation

aras12AM,,,r. - (2-22)

where the units are years and astronomical units (au).

Next, the semimajor axis a, corresponding to the givenflight time Tr, is obtained. At this point, there can be

two types of trajectories, T_ > T,. or TF < T,_.

For TF > Tu, let a_+l : (i + 1)a,,, until TF(ai+_) > Tr

For Tr < T_, let ai+l = (i + 1)am, until TF(a,+_) < Tr

where i : 1, 2_ ... n, and al : am. Note that TF(ai) isthe time of flight corresponding to a_, and is calculated

from Eqs. 2-24 to 2-29 and 2-19 to 2-22; here, however,

the subscript i is used in place of m. When T(ai) < TF

< T(ai+_), then ai < a < ai+_, and a slope-intercept

method is used to obtain an approximation on a. Using

a_+, for a first estimate on aj, and using

a, :aj._- AaF TF- T(a"_)l[_ T(a_) - TF _J

aa = (aj.1 - a_) (2--23)

subscript j = 1, 2 .... n

for subsequent estimates, the value of a_ is used for a to

calculate a and/3.

[C 2 + RL 2- R/- 4a(Rp/3 : c°s-_ 2C(2a - a_)

IRL -- Rp cos "I'.]ct = COS-1 C

Now, the

(2-24)

linear eccentricity X can be found from

9


x, = 4,,, - _a_(Ca- a_) [_ + co_(i_ - #i)] (_25a)

x_ = 4a_- 2a_(_, - a_) [i + co_([_ + #i>] (Z-25b)

Equation 2--25a is used if TF > T,_, and • > rr; or if

Tp < Tin, and _z < 7r. However, Eq. 2--25b applies if

Tv > T.., and _z < 7r; or if Tr < T.., and _I. > _-.

Then, the true anomaly at launch is found from

rr -- "y

_.+ _,

a._ < a < ap

ar <a

a,,. < a < a.

Re > RL

Re < RL

Re > R_

Re < RL

a., < a < ap

a_ <a

am < a < aA

_-+_

_r--'/

(2--26)

and e is found from

X 2 + 4a RL -- 4a 2 (2-27)cos e = 2 X Rz

Also, aA and a_, the semimajor-axis limits, are found from

a A =

ap

Rp -- R_.l+Rp_ Rz, cos_zl

RL

R,_ - a_1+

RL -- Rp cos _I.

Rp > RL (2-28)

I0


If RL > Rp, aA and ap are interchanged. The true anomaly

at arrival vp is found from

%= vL+_I' 0<vp<2=

and the eccentricity e is found from

Xe = -- (2-29)

ga

Then, T(ai) is recalculated using Eqs. 2--19 to 2-22.

When ]TF -- T(a_)] < _, a predetermined conver-gence criterion, the conic is considered to be the one

desired, and a( = a,), e, VL, Vp are the parameters of theheliocentric transfer orbit.

At this point, the transfer orbit is completely deter-

mined. Other key quantities are calculated as shown in

the following paragraphs.

3. Calculation of Velocity Vectors for Probe and Planet

The heliocentric velocity of the probe V (with the

subscript L for launch time or p for arrival time), can be

calculated by using the equation

= V [(W X R) cos 1-' + R sin F] (2--30)Vlqt

where

The path angle F can be found by the expression

sinr = (I-- e 2)(2a- R) esinv (2--32)

where

o<r<_ if 0<v<_r

"IF

__<r_<o if ,,<v<_

The velocity vectors V1 and V2 of the launch and target

planets, respectively, are found in a similar manner. In

this calculation, however, the semimajor axis a, the path

angle r, and the eccentricity of the orbit e all refer to

the respective planet's orbit around the Sun. The true

anomaly v of the planet in its orbit at any time can be

found from Eqs. 2-33 to 2-36.

The unit vector W normal to the planers orbit plane is

given by

W = sin _ sin i, cos _ sin i, cos i 2-33)

where i is the inclination of the planet's orbital plane to

the ecliptic and f_ is the longitude of the planet's ascend-

ing node.

The unit vector P directed toward the perihelion is ex-

pressed by

Px = cos cocos f_ - sin to sin f_ cos i

Py = cos to sin f_ + sin to cos f_ cos i

Pz = sin to sin i

(2--34)

where ,o is the argument of perihelion of the planet.

The unit

from

Then

vector Q right-handed to P and W, is obtained

Q:w×P

cos v : R 1 • P_

sin v : R 1" Q l (2--85)

where, again, R 1 is the Sun-planet unit vector.

Numerical values for the planet elements a, e, i, f/, and

,,, (from Ref. 5) are presented in Table 2-1.

Finally, the hyperbolic-excess velocity vector at launchor arrival Vh can be found from

v,. = v - v,,o.., (2-,_6)

4. Calculation of Various Trajectory Parameters

To assist the trajectory engineer in selecting and de-

signing interplanetary trajectories, various key trajectory

parameters are computed. The formulas for these aregiven below.

The angle _, between the hyperbolic-excess velocity

vector and the planet's orbital plane can be found at

launch and arrival by

W JV h ./1- Tt-

sin ,/ - Vh 2 -< e _ 2 (2--87)

The right ascension 0 and the declination • of the

asymptote (launch or arrival) can be found by use of

the expressions

" (2-48)sin_= Sz --2 <'I'-<_"

!!


Table 2-1. Mean planet elements

Planet

Mercury

Venus

Earth

Mars

Jupiter

Saturn

Uranus

Neptune

Pluto

Semimajor axis a,

elg

0.387098

0.723331

1.000000

1.523679

5.2027

9.546

19.20

30.09

39.5

Eccenlricity •

0.205625 +0.000020T"

0.006793 -- 0.000050T

0.016729--0.000042T

0.093357 +0.000094T

0.048417q 0.000164T

0.055720 -- 0.000345T

0.0471 +0.0002T

0.00872 @0.00004T

0.247

Inclination to ecliptic i,

deg

7.003819_0.00175T

3.394264_0.00125T

0

1.849986--0.000639T

1.305875--0.005694T

2.490583--0.003889T

0.772792 _0.000639T

1.774486--0.00953T

17.140000--0.00556T

Longffude of ascending node _,

deg

47.737778+1.18500T

76.236389+0.90556T

0

49.173611+0.77389T

99.948611_1.01056T

113.226806+0.87306T

73.726528 +0.49861T

131.230833+1.09889T

109.633750_1.35806T

Argument of perihelion w,

deg

28.937778+0.36944T

54.619305 +0.50139T

102.078056+1.71667T

285.965000_1.06667T

273.577222_0.59944T

338.850694_1.08528T

96.129028_1.11250T

272.935934--0.43222T

113.860694+0.03083T

aMeasured in Julian Centuries from 1950.0

and

COS 0 --

sin 0 --

Sx

VsI: + s_,

0_<o_<2,_ (e-39)

where

v_E_ = (sx, sy, s_) (2-4o)s=G

Here, E is the rotation matrix given by Eq. 0-,--2; S is a

unit vector in the direction of the outgoing asymptote

when Vh is calculated at launch, and in the direction of

the incoming asymptote when Vh is calculated at the

target.

The angle ._I between the launch hyperbolic-excess

velocity and the Sun-launch-planet line at launch is de-

fined by

Vh_," R}. 0 G eL _< _" (2--43)cos G - Vh_

There are six other angles: _p, the Sun-probe-target

angle; 6:, the Sun-target-Earth angle; &, the probe-

target-Canopus angle; ns, the supplement of the angle

between the projection of the target-Sun veetor on the

R-T plane and the T direetion; _TE,the supplement of the

angle between the projection of the target-Earth vector

on the R-T plane and the T direction; no, the supplement

of the angle between the projection of the target--Canopus

vector on the R-T plane and the T direction. Here, T is

a unit vector lying in the ecliptic plane, given by

Other quantities which are of interest for the helio-

centric phase are given in Eqs. 9,-41 to 2-48, below.

The communication distance at arrival Rc is expressed

by

'x

a_ = IRd (Rc =Rp - RL.4

(2-41)

The arrival (or departure) angle a is obtained from

Vplanet ° Vh

COS _ -- Vplanet Vh 0 <_ a <_ _ (2--42)

and

R=SpXT

cos _p = - S_ 'R_

cos _ = -Sp" R_

cos _c = Sp" C

( 2-44 )

0_<¢<_,

(2-45)

12


where C is a unit vector to the star Canopus, and R_ is

obtained by normalizing Rc from Eq. 2-41.

R'R_sin 7/, -- sin _p

T.R_cos _, -- sin _p

R.R_sin_E-- sin_B

T'R_

cos_E- sin_

R.C

sin _]a = sin _e

T'C

cos _/_ -- sin _c(2-46)

The angle ap between the projection of the incoming

asymptote on the target planet's orbital plane and the

target-planet-Sun line at arrival time is defined by

cos _, = - R_"S_, t -,_ < _p < _ (2-47)

sin _p = - Sp," (W2 X R_) )

where Sp,, the projection of Sp on the target planet's

orbital plane, can be found by

s, - w_ (s,. w=) (2-48)s,, =is, w_ (s,. w.)-q

Here, Wz is a unit vector normal to the target planet's

orbital plane.

B. The Planetocentric-Conic Trajectories

The trajectory near the launch planet is found by

application of several conditions:

1) The injection energy C:, and the direction S of

the outgoing asymptote of the escape hyperbolaare obtained from the solution of the heliocentric-

transfer problem as given in Section II-A above.

2) The vehicle is launched from a given launch site,

specified by its latitude and longitude, into a

low-altitude circular parking orbit.

3) After coasting in the parking orbit for a time t,,

as determined below, the final stage ignites and

propels the spacecraft to the final injection energy.

(4) The parking-orbit altitude plus the launch planet's

radius is equal to the perifocal distance of the

escape hyperbola. This is a good practical approxi-

mation, since it is most efficient to inject the space-

craft into the escape hyperbola near the perifocus.

(5) The launch planet is assumed to be spherical in

shape.

Given these conditions, the following formulae are used

to compute the parameters of the near-launch-planet

trajectory.

The eccentricity e of the launch-planet conic can be

computed from

.a_c_ (2--49)e = 1 + GML

where Rp is the perifocal distance of the near-launch-

planet conic, and C3 is the vis viva energy, defined as

c3 = vL (2-,50)

and GML is the universal gravitational constant times the

mass of the launch planet. The radius to injection R isfound from

P (2-51)R - 1 + eeosv

where v is the true anomaly of injection on the launch-

planet escape hyperbola, and p is the semilatus rectum

given by

P = -GMz, (1 - e 2) (2--52)C_

The path angle at injection F is found from

GMLcos P - ""VR

"it

0_< P_<-_-,if0 < v <,r

- _ < F < 0, ifTr < v < 2,r2 -- --

(2-53)

where V, the injection speed, is given by

] 2GM LV = C_ + -_- (2-54)

The vector W, perpendicular to the plane of the conic,

can be found by the solution of the two vector equations

W . S = 0 _ WxSx + WrSr + WzSz = O

W.W = 1 _ W2x + W2r + W2z = 1

(2-55)

13


where

and

w, = -(w,s, + wzsz) (_,-56)$I

-WzSrSz Sxwy- s_+sl, ± si,+s_, #l-S_-W_

(2-57)

It should be recalled that S is a unit vector in the direc-

tion of the outgoing asymptote of the escape hyperbola.

Thus, a condition is set on Wz:

wt _< 1 - sl (z-ss)

From the geometry (Fig. 2--2) of the launch azimuthEL and the launch latitude eL, it can be shown that

Wz = cos (I,Lsin EL (2.,-59)

From Eqs. 2-58 and 2--59, the Wz _ restrictions can becalculated as a restriction on EL, since q'L and S are fixed:

sin z X5 < 1 - S_______ (2--60)COSZCbL

A vector B, orthogonal to S and W, is calculated:

B = S × W (2--61)

Now, let P be a unit vector in the direction of the peri-

focus as shown in Fig. 2--3:

P = S cos v, + B sin vs (2-62)

FLIGHT 1

PATHNORTH

POLE

Fig. 2-2. Trajectory-plane-launch-site geometry

S

EARTH'S ,//_

Fig. 2-3. Trajectory-plane geometry

Then Q is the unit vector at right angles to P:

Q = S sin v, - B cos vs ( 2--63 )

The true anomaly v_ of the outgoing asymptote S is

given by

1cos v_ - 0 _< v, _< ,_ (2-64)

e

The right asc(_nsion OL of the launcher can be found from

cos O5 = Wx sin CI,n sin XL + Wr cos XLwl - 1

0<0__<2_

sin OL = Wv sin _L sin EL -- Wx cos ELw_ - 1

(2-65)

where (t,r. is the latitude of the launcher. A unit vector R ).in the direction of the launcher can be calculated in the

vernal-equinox equatorial system by

R_ = cos (b5 cos 05, cos _5 sin 0L, sin ¢L

(2-66)

The angle cb between the launcher and the perifocus of

the conic is given by

cos,_ = R_.PI 0<_<_ (2-67)

sin,I,=R_'Q ) -- --

The angle between launch and injection _, is defined by

¢, = 2_ - _ + v (2-68)

where, again, v is the true anomaly at injection into the

escape hyperbola.

14


A unit vector R' toward injection is seen in Fig. 2-4 to be

R 1 = P cos v + Q sin v = ( Rx, Rr, Rz ) (2-69)

The iniection latitude q, is given by

,r _ (2--70)sin_=Rz _ < _ < -_

The right ascension of injection 0 can be found from

cos O -

sin O --RxfRr

+ 1

0 < o _<2_ (2-71)

The longitude of injection 0 can be calculated from

0 = 0 - Ot - _tb + 0L 0 < 0 < 2_ (2-72)

where

OL

O9

tb

is the longitude of the launcheris the rotational rate of the earth

is the time from launch to injection

Here,

tb=t_+tz÷ [@1- (fix+%)] k,; (2-73)

where

tl

t2

13_1

q_2

k;

is the time of first burn (into the parking orbit )

is the time of second burn (into the escape

hyperbola )

is the angle of first burn

is the angle of final burn

is the inverse parking-orbit rate = 14.689 sec/deg

for a 100-nm Earth parking orbit

The time of coast t¢ is given by

to= [%-- (¢1+%)] k; (2-74)

The relations stated above are illustrated in Fig. 2--5.

INAL BURNINGINITIAL

BURNING -k COASTING .__E C TION

\\

/

Fig. 2-5. Powered trajectory modified by coasting

The azimuth at injection _ can be computed from

Sz - ks sin q_cos 2 = k4 cos ,P 0 < _ < 7r (2-75)

where

11ka- v_ sinv---cosv

e e

_/e _ - 1 1k, -- cosv +--sinv

e e

(2-76)

The unit vector to final-stage ignition RX2can be found ina manner similar to that used for R:

Fig. 2-4. Ascent-trajectory geometry 11_ = P cos ( v - % ) + Q sin ( v - % ) (2-77)

15


The latitude ,I,5 of final-stage ignition is computed from

sin ¢_ = az2 - _- _<_,2< g (2--78)

The right ascension 05 of final-stage ignition is given by

COS 02 --

sin 02 --

ax2

_/RL + RL

0 < 02 _ 2r (2-79)

and the longitude 02 of final stage ignition is defined by

05 = 0 + O_ - 0 + ,ot_ (9.-80)

The angle Av between injection and the outgoing asymp-

tote is expressed as

AI) : /gs -- 1)

The launch time Tz can be found by

TL =OL - OL - CHA

0 _(OL -- OL-- GHA)< 2r

(e--Sl)

where GHA is the Greenwich Hour Angle at Oh UT of the

launch day as shown in Fig. "9.2--6and is obtained from

GHA = 100707554260

+ 0.°9856473460 T,_

+ 2°.9015 × 10 -1_ T_

0 <_ GHA < 2_ (°--82)

Here, Td = days past Oh January 1, 1950.

VERNALEQUINOX

GREENWICHMERIDIAN AT

O h UT

/N OF INJECTION

MERIDIAN OF LAUNCH SITE

GREENWICH MERIDIAN AT LAUNCH

Fig. 2-6. Relation between longitude, right ascension, and

time in equatorial plane

The injection time T is calcnlated from

T = Tz + tb (2--83)

C. Differential Corrections

As outlined in Section I-C, the calculation of differen-

tial corrections, or partial derivatives relating variations

in the impact parameter B and flight time Tr to variations

in the hyperbolic-excess velocity vector Vt,L at launch, are

performed by a numerical differencing technique)

The basic idea in this technique is to compute a varied

or perturbed trajectory and then difference it with thereference case. A small variation AV_,L in the hyperbolic-

excess velocity vector is equivalent to a small variation

±VL in the launch heliocentric-velocity vector. Letting

primed quantities denote variables on the perturbed

trajectory, the launch heliocentric velocity on this trajec-

tory is, then,

v_ = v_ + ,,v,,_ (2--84)

where

±V_, = (C:)_ Aq% [ - sin q_scos Os, - sin _._.sin Os, cos ¢P._],

( C, )!_ ±0,. [ - cos _s sin Os, cos _s cos 0,, 0],- AC,

_iC,_ [ cos q's cos Os, cos _P,s.sin 0,<, sin 4,v]

where ±q,,<, ±0._ are small angular variations (0.2 deg),

and the energy variation is ACa = 0.005 Ca.

The semimajor axis a' is obtained from

a' - RL (2-85)2 -- V_ 5RL

GMs

The radial rate R_/is given by

h_ = v,. RL (Z-S6)RL

The semilatns rectum 19' and eccentricity e' are computed

p, = a[ (y_= - Rff)GMs

(2--87)

from

_This method was developed by William Kizner, Research Specialist,Systems Analysis Section, Jet Propulsion Laboratory.

16


The eccentric anomaly at launch E_ is expressed by

aL_sin E/_ =

e' (a'GMs) v*

cos E L = _- 1 -

The mean anomaly at launch M_, is obtained from

M'L = E_ - e' sin E L (2-90)

The mean orbital rate n' is given by

n' - (GM'_)_a,3/_ (2-91)

The mean anomaly at the target 3l_ is stated as

M; = n' TF + M_ (2-92)

The eccentric anomaly at the target E'_ is obtained from

the expansion

, ( 1 )E_=Ep+ 1-e'eosEp AM

1[(1 6" sin Ep ]

2 AM 2--e'c_s E_)_j

- e' cos Ep) _ aM3

if

cos Ep _ 0

(2-89)

However,

E_ = Ep

( 2-93 )

+ e cos E_ - 1 + x/(e cos Ep - 1) 2 + (2e sin Ep) AMe sin Ep

if (2-94)

where

cos Ep < 0

AM = M_ -- (Ep -- e' sin Ep)

' and v;The true anomalies at launch and at the target, v Lare found from

, p' - RL (2-95)cos v L - e' R_

0 < v;, < ,r if h i is positive

• - < v_ < 2= if h_ is negative

cosy' = cosE; -e'

P 1 - e' cos Ep"

, (1 - e'2) _ sin E_sin vp =

1 - e' cos E_

(2-96)

The heliocentric central angle g,' is given by

¢ __ I

• P _ t_ _L ( 2-97 )

The angular momentum h' is expressed as

h' = RL X V_ (2--98)

The heliocentric position vector at the target is given by

, (RL if,, h')KRL )R_ =Rp _cos + h'RL sin,I_' (2-99)

where

R; = a' ( 1 - e' cos E; ) (2-100)

A vector in the direction of perihelion with magnitude e'is computed from

E' = V% X h' RLGM-----7-- n--7 (2-101)

The heliocentrie velocity at the target is defined by

h' {a,_ )V'p = V × \ Rp + e' (2-102)

The hyperbolic-excess velocity at the target is expressedby

v_ = v; -v_ (2-103)

The difference between the heliocentric position vectors

on the perturbed and reference trajectories is given by

/',R;, = a;, - R,, (2-104)

The impact parameter B is computed from

B = - (AR;" V_,. ) V_,p + ag;v_

The flight-time error is stated as

ATv -- AR_ • V_pv_ (2-105)

The partial derivatives are formed by dividing ±Os,

A%, and AC_ into the miss components B" T, B • R, and

flight-time error ATe. In addition to the component

partials, the quantity OB/OQ_ is defined by

OB_ V(OB'T_z (0B'R_ 2-Iv"(.2-106)_' L\-_-;-Q,} + \--T07-Q,} j

17


The three partials, DB/_Os, DB/3,I,s, _BI_C:, are im-

portant measures of the error sensitMty of a trajectory.

The effect of uncertainty in the knowledge of the

astronomical-unit-to-kilometer conversion factor on target

miss and flight time may be determined by the following

formulae:

0B.T -2C3 0B'T

Oau au OC_

0B. R -2C:_ 0B.R

Oau au OC3

(_1o7)

whence

OB 2G OB (2--108)Oa-----u- au OC3

and

OTF -2C_ OTv

Oau -- au OCa(2-109)

where au is the astronomical-unit-to-kilometer conversion

factor

The effect of solar-radiation pressure acting on the

probe may also be evaluated as follows: In Eq. 2--84, let

,XVhL 0, but in Eqs. 2--85, 2-87, 2-89, "2,--91, and 2-101,

vary G3ls by adding an increment AGMs. This procedure

gives rise to a varied trajectory from which the impact

parameter B and flight-time error _xTr may be obtained.

The partials _B/OG31s and _Tr/cG; 1._ may then be cal-culated. Since the acceleration caused by solar-radiation

pressure acts in a direction opposite to the gravitationalattraction of the Sun, radiation pressure has the effect of

decreasing the Sun's gravitational attraction or

decreasing G._I,,.. A decrease, AGM,_ = -2.4 X 10 _ km3/

see _, corresponds to the solar-radiation pressure acting on

a ,300-kg spacecraft having a perfectly reflecting area of

8,6 mL Thus the miss, always being a positive number, is

obtained by ,.XB_ = 2.4 X 10" OB/OGMs, and the cor-

responding flight-time error is ±T,.p = - 2.4 X 10':OT_,/OG, Ls, which is sign-sensitive.

18


III. DISCUSSION AND EXPLANATION OF RESULTS

A. Introducfion

1. Trajectory Computations

Minimum (geocentric) energy trajectories from Earthto Mercury for the period 1964-197,5 and from Earth to

Jupiter for the period 1968-1973 were computed on the

7094 digital computer by numerically evaluating theanalytical model explained in Sections I and II. This

compntation resulted in curves of C:_ (cis vica geocentric

energy) vs launch date for these trajectories.

For each target planet, several 75-day launch intervals

were analyzed in greater detail: curves of 19-21 key tra-jectory parameters vs launch date for various t_is cica

geocentric energies (C:_) were automatically plotted.

2. Trajectory Analysis

Careful analysis of the results shows that as many as

four ballistic flight paths to the target planet exist per

lmmch date for a given (is vica geocentric energy (C_),assuming trips of less than one revolution arotmd the

Sun. A more extensive analysis might reveal more than

four flight paths. Actually, feasible launchings can occur

only for small time intervals (1-3 months), when the rela-

tive positions of the Earth and target planet are such

that the velocity requirements for ballistic transfers can

be rcasonahly achieved by modern boost vehicles. These

intervals occur once during each synodic period of the

planet. A synodic period is the time interval required for

the Earth and target planet to attain successive identical

relative angular relationships in heliocentric longitude,

e.g., the time t)etween Earth and target planet align-ments (116 clays for Mercury; 390 days for Jupiter).

Figures 3-1 to 3-6 show the minimum injection energy

required for transit to Mercury vs lmmch date. The dates

()f Earth-Mercury alignment are indicated by arrows

(T). Favorahle ]mmch opportmfities (i.e., relative minima

in injection energy) and planet alignments occur about

every 116 days. Two families of conic trajectories exist

(Types I and II), and will be descrihed in detail in Sec-

tion B. After three synodic periods (348 days), Mercury

and Earth again assume approximately the same space-

fixed position. This condition is also reflected in Figs.

,3-1 to ,3-3, in which the minimum energy curve for

Type I trajectories has very pronounced minima at 348-

day intervals. Type II trajectories exhibit this periodic

behavior also, although the curves are more complicated.

Within the time period considered, the minimum injec-

tion energy reaches an absolute minimum on November

:23, 1967, for Type I trajectories, and on November 3,

1968, for Type II trajectories. Only the launch oppor-tunities centered around these elates are discussed in

detail in Sections IV and V. The results are fairly repre-

sentative for all other launch opportunities.

If the synodic period of the Earth and target planet

were a rational fraction of a year, and if their orbits

remained unchanged, the minimmn energy curves would

be exactly periodic functions of tim('.

Figures 3-7 and 3-8 show the minimum injection energy

for transits to Jupiter vs launch (late for Type I and

Type II trajectories. The dates of Earth-Jupiter alignment

are indicated by arrows (T).

Favorable latmch opportunities, planet oppositions,

and approximately the same space-fixed geometry of

Earth and J_,piter occur every synodic period (390 days).

For latmeh opportunities, the absohlte minimum injec-

tion energy occurs on December 30, 1969, for Type II

transfers, and on January .3, 1970, for Type I trajectories.The launch opporhmities centered around these dates

are discussed in detail in Section VII. In addition, be-

cause of some interesting features that were discovered,

the launch opportunity extending from November 1968to January 1969 is treated in detail in Section VI.

B. Classification of Trajectories

1. Type I and Type II

In observing the variation of any trajectory parameter

vs launch date for fixed geocentric energies C:, it is noted

that two separate groups of closed contours, rather than

one, are described (see Fig. ,%9). These two sets of energycontours are designated as Type I and Type II trajec-

tories, where Type I trajectories are defined as having

heliocentric-transfer angles less than 180 deg, and Type II

trajectories are those having heliocentric-transfer angles

greater than 180 deg. A Type I trajectory traverses less

than half-way around the Sun from launch to planetencounter, whereas a Type II trajectory would traverse

more than half-way, but less than one full revolution,

around the Sun. For a given launch day and energy, then,

Type II trajectories require greater flight times than do

Type I. The existence of these two sets of energy contourscan be attributed to the fact that the orbits of Earth and

19


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'i _:;!: ::i!I[_;: :_ _ uJ ....

l _ _l--

_._ _,_. i _ .!: ._r-.......... _ i; ,_Z _._:

_i ':L_!_!:!!:_ ::,,_"-, i

'_ _ i_ "'_-_4,t'T ;1; :: 7 _. ....

t* I _ • i!i!_''-tf '' _1 'll'II=

..... t,..

+_+ -. +_4+ ...... i

I.....1_ !i ""' :_;:II-![_II - _......... : :.-

" f tt ...... lr TI:O O O O

o o d o

_ 13DO4bJ_

>--Z

_J

E:k13 G-

I'--

&

_ "oOaO3 _"

ID

=_-__

-° _ iE

o-

N

O"

>- I

',0O'

uA _

_4

e4o

o--M.

z_

24

.JPL TECHNICAL REPORT NO. 32-77

_J

00

0

0

0a0

oo¢.D

0

0

25


%

E

175

_65

155

145

135

i

i

NOTE: C 3 IS TWICE THE TOTAL ENERGY/UNIT

MASS WITH UNITS km2/sec z

l

I

I

(_) DATES OF EARTH--JUPITERALIGNMENTS

TYPE I

lIi

J

.... i ....

i 16

NOV FEB DEC MAR ,JAN APRIL FEB

1968 1969 1970 1971 1972

Fig. 3-7. Jupiter 1969-1973: Minimum injection energyvs launch date, Typel

MAY

26


tM

EJ¢

175

165

155

145

135

II5 ---- -

105

95

75

NOV

1968

l 1 INOTE: C 3 IS TWICE THE TOTAL ENERGY/UNIT

MASS WITH kmZ/sec 2

TYPE

(_) DATES OF EARTH--JUPITERALIGNMENTS

I

/

i i

| I

L/

16 I0 20

FEB DEC MAR JAN APRIL

1969 1970 1971

--It\ I

K14

FEB

1972

24

MAY

Fig. 3-8. Jupiter 1969-1973: Minimum injection energyvs launch date, Typell

27


0 0 o0 o o_ _- r.r) 04

0 0 0

o o o_-- 0

s_[rp 'Jl /HgI-I-I -I0 ":ll_ll

©c)QD

00ID

o00

,.1=i.ic"

=1,

T ,,.l-IDz

_J EI-,-

,0e,

I

i't=1

I

M..

28


the destination planet are not coplanar. If the Earth and

target planet had coplanar circular orbits, only a single

group of energy" contours would exist. If the orbits ofEarth and the target planet were circular as well, this

single group of energy contours would include the well-known llohmann's transfer orbit. The ttohmann transfer

orbit is tangential to the orbits of Earth and target planet,

has a heliocentric transfer angle of 180 (leg, and requires

the least t;i.s'cica _eocentrie ('nerg_' (C:;) among all transfertraject()ries. The ttnhmann minimmn-energy trajectory

would remain constant for each synodic period, retaining

the same flight time and ci.s' cica geocentric energy. The

fact that tht" orbits of Mercury and Jupiter are slightly

inclin('d with respect to the ecliptic (7 and 1.3 deg, re-

spectively) strongly infll,ences the transfer trajectories

and causes the single group of trajectory contours to split

into two families, thus necessitating the designation of

Type I and Type 11 trajectories.

In Fig. 3-9 note the small family of closed contours

cent(red aromld January 1, 1969. This group has been

designated Type I-A. Type I-A trajectories exist for

Venus also. They are not attractive because their flight

times are longer for a given (;is t;ica geocentric energy.

2. Class I and Class II

As Fig. 3-9 shows, two trajectories usually exist for a

given type, energy, and launch (late. Trajectories corre-

sponding to the lower portions of the energy contours

are designated as (:lass I and art' connected by solid

lines and correspond to shorter flight times and smaller

heliocentric central angle. Trajectories corresponding to

the upper part of the energy contours are designated as

Class II and are collnected by dashed lines and corre-

spond to longer flight times and larger heliocentric angle.For a given Immeh (late and geocentric energy, the Type

I-Class I trajectory has the shortest flight time and the

Type I I--Class II trajectory has the longest flight time.

For example, Fig. 3-9 shows that for the launch (late

Novemlwr 26, 1969, and C: _ 12() kmVsec'-', the Type I-

Class I trajectory has a flight time of .520 days, whereas

the Type lI-Class II trajectory has a flight time of 1460

days. The flight time for the Type l-Class II trajectory is

880 days, and for the Type lI-Class I trajectory it is 982days.

3. Minimum Energy Trajectories

a. Minimum-energy loci. Further study of Fig. 3--9

shows that the geocentric energy C:, has a relative mini-

nmm of C::-::: 85.617 km:'/sec '-' on December 4, 1968,

where the flight time is 850 days. By moving away from

this point in any direction, C:_ increases. Type I trajec-

tories corresponding to C:, - 87 km'-'/sec'-' only exist from

November 29, 1968 (flight time 780 days) to December

10, 1968 (flight time 950 days). The geocentric energy

required is smaller inside the contour and larger outside

it. Consequently, the minimum possible energy on No-

vemt)er 29 and December 10, 1968, for Type I trajectories

is 87 km'-'/sec'-'. A curve of mininmm geocentric energy

vs Immch date may be constructed for Type I and Type II

trajectories by constructing the vertical tangents to the

contours of a given parameter such as launch time. By

connecting these points, a locus of minimum energy(Fig. 3-9) is obtained which separates Class 1 and Class II

trajectories within a given type.

Figure 3-10 was constructed in this way from Fig. 3-9and shows minimmn geocentric energy vs launch (late

from November 18, 1968, to January 7, 1969. The curve

corresponding to Type II trajectories was terminated on

December 28, 1968, because flight times corresponding

to later launch dates are in excess of four years. Tworelative minima, C:, - 85.617 km'-'/sec-' on December 4,

1968, and C:, --_89.410 km-'/sec'-' on January 1, 1969, exist

for Type I trajectories because of the noncoplanar and

noncircular nature of the orhits of the Earth and Jupiter.

The relative minimum for Type II trajectories is C:,77.832 km'-Tsec'-' and occurs on December 13, 1968.

b. Absolute minimum-energy traiectories. The abso-

lute minimum-energy trajectory for each type is defined

as the trajectory with the ]east geocentric energy within

each synodic period of the planet. Such trajectories are

useful as first approximations of launch dates, communi-

cation distances at planet encounter, and flight times for

missions within a given launch opportunity. Table .3-1

shows the characteristics of absolute minimum-energy

trajectories to Mercury and Jupiter. These trajectories

occur once every synodic period of the target planet, as

explained in Section III.A.2. For Mercury, a sharp de-

crease in the absolute minimum energy occurs every third

synodic period (348 days), as can be seen from Figs. 3-1

to 3-7. Table .3-1 shows that the lowest possible geo-

centric energy to Mercury is 41.2 km-'/sec: (launch date

October 31, 1968, Type II); to Jupiter it is 75.3 km'-'/see:

(launch date January 3, 1970, Type I).

Very recently Cutting and Sturms (Ref. 6) showed that

it is possible to reduce the energy requirements for a

Mercury mission by 70% by an encounter with Venus

before arrival at Mercury. The following is a typical

trajectory: Launch date August 14, 1970, injection energy13 km'-'/sec _, arrival date at Venus, November 26, 1970,

29

JPL TECHNICAL REPORT NO 32-77

51II0

105

I00

95

l L

j

NOTE: C3 IS TWICE THE TOTAL ENERGY/UNIT

MASS w/unit km z tsecZ

Li

I

II

Ii

I

16 21 26 6 II 16 21 26 31 I0 15

DECEMBER dANUARY, 1969

LAUNCH DATE

Fig. 3-10. Jupiter 1968-1969: Minimum injection energy vs launch date

3O


Table 3-1. Characteristics of minimum-energy-transfer

Planet Trajectorytype

Mercury I

I

I

II

II

II

I

I

I

I

II

II

II

II

I

I

I

II

II

II

I

I

I

II

II

II

I

I

I

II

II

III

I

I

I

It

I[

II

I

I

I

II

II

II

I

I

I

II

II

II

I

I

I

I

II

Launch

date

2/11/64

6/14/64

lO/6/643/1/646/_8/649/24/641/21/655/29/659/19/6512/31/652181655/27/65

911o1651111616551151669/3/66

12/_I/665/6/66

8125166

ii1816641261678117167111231674/25167

8/12/67

1117167

4/5/687/28/6811/12/684/12/68

7/28/68

11111683/15/69

7/11/69

11/I/693/27/69

7/13/69io/18/692/22/70

6/24/70

10116170

3110170

6/28/70

lOl317o2/1/71

6/7/719/29/712/23/71

6/9/7_9/19/711/13/72

5/22/77

9/11/72

12/22/72

1/27/72

Flight

time,

days

95

95

86

105

105

116

97

97

86

101

109

107

115

150

97

86

104

107

115

135

92

87

107

104

112

123

91

9O

103

102

110

115

92

92

9O

103

108

115

94

94

87

105

106

116

96

96

86

105

106

115

98

97

86

102

114

Geocentric

injection

energy,

m2/s 2 x lm

0.593

0.895

1.006

0.926

1.053

0.497

0,517

0.847

1.074

0.467

0.935

1.085

0,561

0.861

0.829

1,095

0.434

0.893

0.636

0.646

0.897

I .077

0.412

0.842

0.722

0.470

0.834

1.039

0.449

0.858

0.822

0.410

0.738

0,987

0.778

0,891

0.921

0.428

0.644

0,929

0.940

0,910

1.014

0.468

0.562

0,873

1.041

0.927

1.075

0.524

0.497

0.836

1.090

0.451

0.933

Heliocentric

central

angle/

deg

128.4

153.6

142.6

215.7

234.2

221.5

135.6

163.7

139.9

146.6

218.2

216.1

229.8

203.6

172.2

139.6

156.4

187.0

236.4

184.1

133.7

139.2

169.6

192.6

239.4

185.7

122.3

139.9

177,4

199.3

242.8

191.3

120.7

144.6

153.2

204.5

24t.9

203.0

124.7

149.4

145.9

211.4

239.2

215.6

129.9

154.4

139.9

217.0

226.6

223.9

138.8

163.7

136.0

149.8

221.8

Sun-planetdistance at

arrival,10 t_km

69.3

46.4

53.2

50.5

50.9

66.8

69.8

46.8

50.2

69.5

53.6

46.6

65.7

69.8

47.1

48.2

68.5

46.7

64.0

67.7

55.1

46.8

67.8

47.1

62.0

67.8

61.6

46.1

67.5

47.5

59.6

67.9

66.2

46.0

60.6

48.6

56.4

67.5

68.6

46.3

55.7

49.9

53.1

67.1

69.7

46.9

51.5

51.4

48.6

66.1

69.7

47.3

48.7

69.0

55.5

CelestialEaCh-planet latitudedistance at

arrival, of planet10 _ km at arrival/

deg

102 -4.69

135 1.04

126 6.70

178 --3.79

188 6.99

176 0.70

106 -3.25

142 0.43

125 7.04

112 --2.09

178 -5.52

174 5.60

184 1.40

121 -3.32

150 -0.24

126 6.65

119 --0.83

154 0.50

189 2.40

115 -0.11

120 --6.07

126 5.66

130 --0.16

161 --0.20

192 3.55

130 --0.16

108 6.98

126 4.12

143 0.10

168 --0.85

195 4.61

146 --0.24

102 --6.42

129 2.91

131 4.15

171 --2.15

193 5.86

159 0.10

101 --5.37

132 1.64

127 6.05

175 --3.24

191 6.75

171 0.38

102 --3.93

135 0.17

124 7.00

179 --4.25

182 6.73

179 1.10

108 --2.77

143 --0.46

123 6.84

114 --1.43

177 --6.16

31


Table 3-1 |Cant'd)

Planet

Mercury

Jupiter

Trajectory

type

I

li

I

IA

II

I

II

I

II

I

II

Launch

date

5/7/729/4/7211/121725/13/738/26/7312/3/735/4/738/20/7311/8/734/18/748/9/7411/16/744/22/748/6/7411/8/743/27/757/22/7511/13/754/5/757/24/75lo/28/75

1/2/7o12/31/6912/4/6812/31/6812/13/681/31/712/6/713/6/723/31/724/12/735/11/73

Flight

time,

days

113

114

146

93

87

105

103

113

129

91

88

108

101

111

116

92

9O

97

103

108

114

985

994

852

1349

1277

8O8

1180

744

1396

721

1404

aTwice the total energy per unit moss, or vls viva integral.

bThe angle subtended at the Sun between the Sun-Earth llne at

Measured from the ecliptic.

Geocentric

injection

energy/_

m2/s 2 x 10 _

0.961

0.592

0.782

0.858

1.098

0.423

0.858

0.672

0.555

0.874

1.067

0.412

0.842

0.762

0.419

0.798

1.023

0.555

0.873

0.862

0.414

0.752

0.753

0.856

0,884

0.778

0.777

0.833

0.812

0.857

0.841

0.839

Heliocentric

central

angle,

deg

1842

232.0

201.7

1733

140,7

161.3

188.1

237.4

183.6

128.2

139,9

176.3

194.6

241.2

184.3

121.4

139.4

177.0

200.6

243,2

195.9

178.8

181.4

165.8

180.4

191.3

167,8

194.8

162.0

198.8

159.9

196.3

Sun-planetdistance at

arrival,10_ km

46.8

65.1

69.7

47.2

47.6

68.1

46.8

63.2

67.7

57.8

46.4

67.7

47.2

61.2

67.7

63.9

46.1

67.4

48.1

58.8

67.8

779.0

778.7

804.4

779.2

784.1

767.4

750.0

751.6

742.1

741.7

754.4

Earth-planetdistance at

arrival,10 6 km

147

187

123

154

127

123

158

191

121

115

127

137

165

194

137

105

126

150

168

196

152

742

758

703

735

636

783

816

882

771

889

812

launch and the Sun-planet llne at attlval time.

Celestial

latitude

of planetat arrival,

deg

0.30

1.83

-- 2.79

.29

6.40

--0.45

0.46

2.85

--0.10

--6.66

5.18

--0.09

- 0.20

3.92

- 0.09

--6.92

3.54

0.18

-- 1.56

4.92

-0.17

!

0.00

--0.01

0.88

0.01

0.18

- 0.41

1.02

0.96

-- 1.23

-- 1.29

- 0.73

Venus perigee altitude 2676 km, .Mercury arrival dateJanuary 30, 1971, hyperbolic excess velocity at Mercury12.3 km/sec.

For transfers to Mercury, the geocentric energy issmallest if (1) Mercury is at aphelion (66.8 X 10" km) atencounter, (2) Mercury is in the ecliptic at encounter,(8) the heliocentric central angle is 180 deg, and (4) Earthis at perihelion at launch. Owl.'ng to the large inclination(i = 7 deg) and eccentricity (c := 0.2) of Mercury's orbit,

large variations in absolute minimuln geocentric energyoccur, as Table :3-1 shows: The trajectory (launch date,October 81, 1968) corresponding to the lowest geocentricenergy (C:, = 41.2 km_/see :) has a heliocentric central

angle of I-ICA 192.3 de_, the Stm-l)lanet dist_ilc'(' at

arriwd is SI)I ) : (57.9 X 10'; kin, all(1 the celestial l_ttitnth'

at arrival is I,AI ) : 0.46 deg. The trajectory \vith launch

date Nov('rnl)cr 24, 1967, has almost the same _'nergy

(41.3 kln_/_ec:), even though tilt' heliocentric transfer

angh' is sm_llt'r t)y 24 dcg (HCA 16S.6 de_). The

trajectory launched on May 15, 1966, shows the effect of

arrival near Mercury periheliml (SPP ; 47.6 X 10 '_ kin).

The geocentric energy is nearly doubled ('.C_ 84.8

km-'/see:). It is difficult to find a trajectory in Tal)le :3-1

which would show only the effect of reaching Mercury

out of the ecliptic, tlowever, several trajectories can be

found for which arrival occurs at Mercury aphelion and

when it is st]ongly inclined to the ecliptic. For example,

32


for a launch date of September 2, 1966, SPP - 47.6 × 10';

km, I,AP: 6.:39 deg and C:,--109.9 ktn'-'/see'-'. It is

interesting to note that these trajecto O" characteristics

are almost duplicated for launch date August 26, 1978.

Summarizing, it can he said that large variations in

celestial latitude and Sun-Mercury (tistance at encounter

strongly influence geocentric energy, whereas the effect

of changes in heliocentric central angle is small.

It should 1)e noted that within the time interval con- -_

sider('d in this report (1964-1975), Mercury aphelionoccurs very near the ec|il)tic st) that trajectories with very' ,_

(3-

favorable energy characteristics can he found. As Mer-

cnrv's apheli(m point moves away from the ecliptic, the

minimumienergy requirements incr('ase.

For Earth-Jupiter transfers, the geocentric energy, will

be smallest if (1) Jupiter is at perihelion (740 X 10 '; kin)

at encounter, (21) Jupiter is in the ecliptic at encounter,

(3) the heliocentric central angle is 180 deg, and (4) the

Earth is at aphelion (147 < 10; kin) at launch.

Variations in at)soh|te minimum geocentric energy are

caused by the inclination (i -- 1.3 (leg) and eccentricity

(e 0.048) of Jupiter's orbit an(t the elliptic orl)it of

the Earth.

The trajectory with the stnallest energy in Tahle ,3-I

(C: 75¢3 km-'/s('(,-') occurs for the launch date January

:3, 1970. Arrival occurs 985 days later, when Jupiter is

ch)se to the ecliptic (I,AP " 0.004 (leg) but relatively far

from perihelion (SPP 778.5 Y,, 10'; kin).

Further study of Table .3-1 shows that the strongest

variations in geocentric energy are caused by large

changes in the Stm-Jnpiter distance at arrival. In order to

reduce Earth-to-Jupiter flight times to two years, the

geocentric energy re(iuire(l is C:_ : 88 km"/sec-' (launch

(late December 1, 1968).

C. Mission Payload

The geocentric energy at hmnch C:_ determines the

payload that can be placed on an interplanetary trajec-

tory. Figure :3-11 shows approximate payload estimates

vs C:, for three launch-vehicle configurations: two-stage

Atlas�Centaur, three-stage Saturn 1B, and three-stage

Saturn 5. These curves arc only intended to give a gen-

eral idea of payload; they were obtained from the open

literature (Ref. 7).

The maximum payload that can be placed on an

Earth-Mercury trajectory using the Saturn 1B launch

100"000

90,000

8O'0OO

m/_oo

60,000

50,000

_.0,000

30,000

i

I

!

I SATUR/V 5

!

\ :

'x

"320,000 ..... _ /_ S,4/'U.._N Z8 - _

// /-- ATLAS/CENTAUR _1

,o,ooo 7-0-----._220 40 60 80 I00 120 140 160

GEOCENTRIC ENERGY AT LAUNCH C3, km2/sec 2

Fig. 3-1 1. Typical payload copobility curves for Atlas/

Centaur, Saturn 1B, and Saturn 5 launch vehicles

vehicle is :2500 lb. This payload is increased to 5:3,000 lh

if the Saturn .5 is used.

Using the, Saturn 5 vehich', the maxinmm payh)ad that

could be placed on an Earth-Jupiter trajectory is

26,000 ]h.

D. Launch Period

The practical design of a mission to Mercury and

Jupiter will involve the establishment of a launch period,

i.e., a period of consecutive days on which launch is

possible. On each day the trajectories would have to

satisfy numerous constraints, such as those placed on

energy, communication distance, flight time, etc. For

most missions, constraints on arrival conditions would

exist also.

As a first approximation, launch periods can be de-

signed on the basis of geocentric energy, alone. Nat-

nrally, then, the minimum geocentric energy on every

launch day is chosen. For each trajectory type the curves

of minimum energy allow establishment of launch peri-

ods of arhitrary length centered around each of the

33


absolute minimum-energy launch dates given in Table3-1. This selection has been carried out in Tables 3-2,

3-,3, and :3-4 for several of the Earth-Mercury and

Earth-Jupiter opportunities tabulated in Table :3-1.

For each trajectory type and class, launch periods of

15, :30, and 4.5 days were determined. Each launch period

has a maximum geocentric energy associated with it.This maximum value is reached on the first and last dav

of each launch period and results in the lowest payload.

The minimnm injection energy during the launch period

can be found from the corresponding absolute mini]num-

energy trajectory in Table ,3-1. It is of interest for orbitermissions since more fuel can be added to the retro tanks

as the geocentric energy decreases during the launch

period, resulting in closer orbits at the target planet.Table :3--2 also shows the variations, during the launch

period, of the most important trajectory characteristics

on which constraints are usually placed.

The Earth-to-target-planet flight times Tv will affect

virtually every spacecraft sul_system. The communicationdistance at arrival iq, indicates the design of the com-

munications system. The planetocentric asymptotic speedV_, is a measure of the speed of the spacecraft at the

time of closest approach to the target and is important

for both orbiter and flyby missions.

The geocentric asymptotic declination is included inTable ,3-2 because it is an important parameter in de-

termining the injection location over the Earth's surface.

Acceptable values for this parameter are estimated tolie between -,'34 and +:34 deg. Values outside this range

result in severe restrictions for Capt" Kennedy launchings.

As an example, note how the 45-day launch period

November 21, 1968 to January 5, 1969, for Type II, Class

I trajectories to Jupiter was found from the curve ofmininmm geocentric energy, Fig. :3-10. Starting with the

launch date December 1:3, 1968 (C:_- 77.8 km_/see_),

C:_ was increased until the range of lanneh dates was 45

days. These limits then defined the launch period and

the corresponding maximum energy (C; = 86.2 km_/see 2)

over the period.

Sometimes it happens that the curves of minimum

geocentric energy corresponding to Type I and Type II

trajectories intersect within the launch period for a given

type. It then becomes possible to design launch periodsduring which both Type I and Type II trajectories would

be used. The corresponding geocentric energies are lower

than if either type were used alone.

E. General Characteristics of Trajectories

In order t(_ gain further insight into the nature of

Earth-Mercury and Earth-Jupiter trajectories, plots of

several key trajectory variables were prepared for each

of the lm,nch opportunities. The relevant trajectory

parameters can be categorized into three gronps: geo-centric, heliocentric, and planetocentrie. Sections IV-XV

show detailed plots of the following trajectory param-

eters, \v]li(,h _.lrt' italicized:

1. Geocentric Parameters

During the geocentric phase, the magnitude and direc-tion of the hyperIx)lic-excess velocity vector V_,p, or the

velocity vector of the spacecraft relative to the Earth atthe time of injection into heliocentric orbit, are of prin-

cipal interest.

The magnih,de of Vh. is represented by the geocentric

energy at lain,oh, C:, = V_,. All plots of trajectory param-eters show contours of constant and integral values of C:,

The directi_m of Vh,, also called the outgoing asymptote

S, is given relative to several coordinate systems. The

right ascension 0._ and declination _.,. of the outgoing

asymptote specify its direction relative to a geocentric

equatorial coordinate system. The celestial latitude ,/1. ofS or the an_le between the geocentric asymptote and the

launch planet's orbital plane and the angle bettceen theSun-Earth vector and outgoing geocentric asymptote _j,

specify the orientation of S relative to the heliocentric

ecliptic coordinate system.

Obviously, the energy has a direct influence on space-

craft weight. The declination of the outgoing asymptote

also severely restricts the interplanetary trajectories which

are possible in a more complicated way.

2. Heliocentric Parameters

The following properties of heliocentric transfer ellipse

are presented: aphelion distance R,,, perihelion distance

Rv, and inclination i with respect to the ecliptic. The

eccentricity c and semimajor axis a of the heliocentric

ellipse can be found from these. Sections IV-XV show

graphs of true atu)maly in the transfer ellipse at launchVt, and true anomaly in the transfer ellipse at arrival Vr,,

as well as of their difference, the heliocentric central

angle %

The Earth-to-target-planet time of flight T_. and com-munication distance Re are of direct interest to designers

of all spacecraft subsystems.

34


_ : ._==E

.__

: ._,

3

_' ": =E

E

= .=-

om.1-

"L-- _ _

.c :E_J

v) ._v nx

_ _:E_. E

0

E •

E"== _,

E

.=

c

o _ o

I -

O o. (_.

o,0 o Ko_4 _ c5

o o o

o K

_ coe4 _

b.

_ KO

O- _ 00 O_ CO

0- _. o. o. _.

q o q _ (R.CO

I _ I I I

_r _ CO

_r_ co

d c; o

-_. - o

o,

(5 o

o

o

T TCO

o

o. -. "9.("4 /% CO

U') CO CO

,_ ,6 t<

_ u_ co

e,i N N

° 7, _.u3

O O O

f'} "_t

I

co O O

I I q

_ _ co

_ _ co

o _ _

o _ _ _. _. :I I _ I I ]

0 0 0 _ ,_ 0CO _ aO CO

o _ ,..

O- CO h,.

o c_ c_

_ o _

_ _ o

o

K. o _

c_ o c_

_ o _

0 0

35


_n

i.

_a

oJ=_a

e-

.I-

D.

"1

m

Q.>.

i{,1{0

mJD0

I--

=Ei

o o o

_, _ _ _ K _. co

E o

=

:'= o

E • o., -

u.. :E r_ I_

.= ® '.= -_

.c ,-: d ..-:• _ _ I

co _ _

o _.

o o o

_. _ or_ co

c_ oK co o_

_ °

.,o ,,o ,,o

o o

co K r_

_. _. o

._ ". _ _ _ _ if-._. _ _._. • _. co 0_ _ co

E_EE. _ ._ _ .

K

_-,_ _ _ _ _

o o _ _ _ oO, O,

._o _ _= g _

E®.;

.-_-

Kco _. c>o o c_

I

- o _! !

o

co o, o_

_o

I i

o o oC_. C,'-

o o -0K _ t..co c_. o

o _-, _co _

o o .:. -:. _. "!

I I I I I

o o o o oco _

K co o: _. 06

t'_ _

_ ,6 ,6 .6 .d 4

o oo, co o_ co 00 00

o _ _co

o _ oco O_ 0

o _ _o

,q. _ r,,

'; I I

_. o o

I I

_ _ ocO 0 OI

co O_ 0

_. ° _,,o .,o ,.o

i t

_!. _ o

I I I

CO 0 ¢_1

0 0 0 0 0 0

0 0 co 0 0_ o _

o

co _. o

o _ _co 0'-

(5 c_ c_

_ o _ _ o _.

o o or',. P-..

_o ,o ,,o

o (5 c5

o

F. 7-.

co? ? 7

_ o _

o _ o

o

36


).

° _ ._=

.- .=o_ ® _EO_

: o _Eo.E

o a 3E

';0 c

u _

._E. _

"; =E°--

°-s_

e'- o

v ,c_

-- ®E_. E0. _ E. .E

"E; o •

=

I:. o " _E =E>. _ ;oc=_,

•-_ E •

.--r _

._o_,_E®_

_ .-®'_E

=l'E 0

o o0 '_ o ,o o o o

I I I I

I I _ I I I I

o

o o. _. ,q

CO O, O, O, 0- CO cO 0_

O O O O,

"O

(_ 0 0 0 0u'l K _t

o3 cO

K

_ o _ _ _ oo0 0 O, _ _ 0

I I

,-: 0

I

t'.. _

,d ,d ,d ,d _

o -_t o _ o, 0 o o

cO o. o_ 0.. o- 0_ o, 6,

0 0

o _ ocO 0 _0h_. cO CO

d d d

o _.('_ (.1

0- o,

t'x o

o o

o

oo I_

T "r

_ o _

o o _

b.

co _ _d d d

_ o _

? 9 ?(_ -%.

oco

(-1

I

r_

co

_ o

,0 co _ co_cO cO 0-

d d d d

o _

777 ?o- K

37

JPL TECHNICAL REPORT NO. 32-77,

The celestial latitude at arrival time /3v is referred to

the ecliptic plane.

3. Planetocentric Parameters

The direction of the approach hyperbolic-excess velocity

vector V1,_, or velocity, of the spacecraft with respect to

the target planet at arrival, is described in several coor-

dinate systems.

The right ascension 01 , and declination %, of the unit

vector along VJu,, also known as the approach or incom-

ing asymptote S, are referred to the Earth's equatorial

plane. The antic between the incoming asymptote and

the arrival planet's orbital plane "/i, and the angle between

the target planet-Sun vector and the incoming-planet-

centered asymptote ¢1, relate the incoming asymptote

direction to the orbital plane of the target planet. The

angle between the planet-Earth vector and the ineomin_

asymptote _: defines the motion of the spacecraft relative

to the Earth-planet line a few days prior to encounter.

The angle between the t_lanet-Canopus vector and the

incoming asymptote _,. is of importance because Canopus

will most likely be the second attitude reference for

these missions, in addition to the Sun.

The magnitude of V_u,, the asymptotic speed with re-

spect to the target planet Vh> in of great importance for

orbiter, lander, and multiplanet missions.

F. Discussion of Earth-Mercury and

Earth-Jupiter Trajectory Parameters _

1. Declination of the Geocentric Asymptote ,I,._

Equation (2-60) defined an inequality t)etween ,1,,,

El, (the launch azimuth), and ,l,z, (the latitude of the

launch site):

._ cos e _l_sin _ "_'_,

COS" (I)L

The regions within which launch from Cape Kennedy

can occur are shown in Fig. :3-12 as a function of El,.

Usually, the launch azimuth will be restricted to lie'

within certain bounds established by range safety con-

siderations (typically 90 to 114 deg).

For the two Mercury ot)portunities considered (1967,

1968), the declination constraint does not restrict the

_'A detailed discussion of the influence of the trajectory parameters

on trajectory design is presented in R('f. 8 (Section Ill.D). The

following material is int(,n(l(,d as supph'm_'ntary to that presentedin Ref. 8 and 9.

Z

O

I--

Z

COhi0

d

rr

0rr

t--D0

9O

iRANGE SAFETY J

AZIMUTH

9c

0 90 180NORTH EAST SOUTH

LAUNCH AZIMUTH IEL, deg

Fig. 3-12. Permissible regions of the declination of the

geocentric asymptote ,I,_ for Cape Kennedy launchings

Type l-Class I trajectories, which have the shortest flight

times. The constraint does eliminate a substantial part of

the other traj,'ctories (i.e., Type l-Class I and Type lI-

Class l and I!). This can be seen from the correspondin_

plots of ,I,, vs launch date (Figs. 4-5, 5-5).

For the 1968-69, 1969-70, and 1970-71 Jupiter oppor-

tunities, the declination constraint does not restrict the

possibh' Typc l-Class I trajectories (see Figs. 6-5, 7-5,

8-5) In 1972 and 1973 it becomes almost impossible to

use tiffs kind of trajectory (see Figs. ,9--5, and 10-5). In

fact, in 1972 and 1974 only the Type II-Class I trajec-

tories are not affected by the declination constraint. All

other traject_,ry types are severely, restricted by the

constraint.

2. Angle Between the Sun-Earth Vector and the

Outgoing Geocentric Asymptote ._L

This quantity, which is of importance for spacecraft

using the Earth and Sun as attitude references, takes on

values between 80 and 130 deg for the 1967 Mercury

Type I trajectories (45 to 110 deg for Type II trajectories;

see Fig. 4_). It takes on approximately the same range

of values during the 1968 Mercury opportunity owing

to the fact that the Earth-Sun-Mercury constellation is

about the same for both opportunities. The quantity

varies between 45 and 120 deg for the five Jupiter oppor-

tunities considered herein (1968-69, 1969-70, 1970-71,

38


1972, 197.'3) for energies less than 120 km2/sec '2. In all

cases, it decreases with increasing launch date.

3. Celestial Latitude of the Geocentric Asymptote _,;,

This quantity, referred to in the graphs as the angle

between the outgoing geocentric asymptote and the

launch planet's orbital plane, is strongly dependent on

the celestial latitude at arrival time /3;,. (See Ref. 1, page

35. Fig. ,3-14 of Ref. 8 applies for Earth-Mercury trajec-

tories as well as for Earth-Venns trajectories.)

Employing the same reasoning, low-energy trajectories

to the outer planets (Mars, Jupiter, etc.) have perihelion

distances on the order of 1 an and aphelion distances on

the order of the Sun-target phmet distance. Figures 6-9,

7-9, 8-9, and 10-9 show that for an Earth-Jupiter tra-

jectory, the true anomaly at hmnch remains within .'30

deg of perihelion (VI, - 0 deg). Figures 6-10, 7-10,

8-10, 9-10, and 10--10 show that the Jupiter encounter

can take phtee from about 40 (leg twfore to :30 deg after

aphelion.

For the Mercury opportunities in 1967 and 1968, the

target planet is relatively close to the ecliptic at encoun-

ter, which causes "//, to take on values on the order of

10 deg. For Jupiter, ,/;, can take on values between -45

to +45 deg for the energy range considered herein. Note

that the Type I-Class I trajectories, which have the

shortest flight times, all lie very close to the ecliptic. The

Type II-Class II trajectories, which correspond to thelongest flight time, also show this characteristic.

It might be asked how ,/_, can take on such high values

in spite of the fact that Jupiter's inclination is only 2.5

deg. In Ref. 8, page :34, it is shown that

V_, cos I'_ sin i

sin yt, _ V;,;,

where V;, is the heliocentric velocity at launch, V;,;, is the

hyperbolic-excess velocity at launch, 1";, is the heliocen-

tric path angle at launch, and i is the inclination of the

trajectory. The magnification factor V;,/V;;;, is on the

order of 4 for Earth-Jupiter trajectories.

4. True Anomaly at Launch and Arrival and

Heliocentric Transfer Angle

In general, low-energy elliptic heliocentric orbits from

the Earth to the inner phmets (Mercm'y, Venus) will

have aphelion distance on the order of 1 an (astronomical

unit) and perihelion distances on the order of the distance

between the target planet at the Sin1 (0.:3 au for Mer-

cury). Consequently, launch will occur near aphelion

(true anomaly V;, -- 180 deg) and arrival near perihelion

(true anomaly Vp - 0 deg). Figures 4-9 and 5--9 show

that the variation in V_, is small, 160 to 200 deg for Type

I and Type II Me]'eury trajectories. Note that latmeh

occurs before aphelion at the t)eginning of tim launch

period, and after aphelion at the end of the launch period,

as was the case for Earth-Vemls trajectories.

Figures 4- through 10-:3 show the variation in the

heliocentric transfer angle ,1% or the difference between

the true anomalies at launch and encounter, for Earth-

Mercury and Earth-Jupiter trajectories, respectively.

Note that none of the Type I or Type II contours inter-

sect the ,v = 180 deg line.

5. Aphelion and Perihelion of the Transfer Orbit

Figures 4-12 and 5-12 show that for Earth-Mercury

trajectories the aphelion distance (point farthest fi-om the

Sun) varies from 1 an by less than 10%. The perihelion

distances show stronger variations (Figs. 4-11 and 5.11).

As mentioned in Ref. 8, the probe passes aphelion after

hmneh if the true anomaly at launch is less than 180 deg;

it passes perihelion before arrival (i.e., passes within

Mercury's orbit) if the true anomaly tit arrival is positive.

For Earth-Jupiter transfer, the perihelion distance does

not change drastically (Figs. 6- to 10.11). Note, how-

ever, that tim aphelion distances can become as large as

2400 X 10': km, or over ,'3 times the Sun-Jupiter distance

for geocentric energies C,. = 120 km:'/sec 2 (Figs. 6-

through 10-12). The plots of true anomaly show tlmt

aphelion would take place after Jupiter encounter. Owing

to the exceptionally large mass of Jupiter, it will exert a

very strong perturt)ation on the trajectory during en-counter.

6. Time of Flight

Typical Earth-Mercury flight times range from 80 to

160 days, as is shown in Figs. 4-2 and 5--2. As the launch

date progresses, it is seen that flight times are shortened

for both Type I and Type II trajectories. Type II trajec-

tories offer lower energies and larger latmeh windows

for a given energy, with onh' a slight penalty in flighttime.

39


Flight times to Jupiter are on the order of i to 4 years

(Figs. 6- through 10-2). ,,ks la,mch date is delayed, flight

time incr('as('s sul)stantiallv for a _i\vn em'r_y.

7. Comnumication Distance

For Nh'rcurv trajt'ctorics, the c(mm.mication distance

at ('nc'ount,2r in('r('ast's as launch dat(' i_ dclay('d, wh('r('as

flight time decreases. The c())mmmication distance can

take on values betwc('n I00 ?.: 10' km to IS(} / 10" km

(see Figs..I-4 and 5--4).

Th(" communications distanc('s at Jupitur arrival ('xtend

from 650 / 10" km to 950 / 10" kin. Since thc Sun-

Jupiter distance is

)00

9O

740 10'; km and the Earth-Sun

distancv is 151) 10'; kin, it is clear that if the communi-

cations (listanc(' at ('ncounter is around 890 < 10'; kin,

thv Su. _ill li,. x('ry close to th(' Jupiter-Earth lint' (sec

}:i_s. 6- II)r(m_.'}l 10-4).

8. Inclination of the Ileliocentric Transfer Plane i and

Celestial l,alitude at Arrival/3,

The incli],ation i of the heliocentric transf('r orbit to

tl_t' cc']iptic pJau(' is essentially a function of two param-

('ters, (1} tl,v 1,4ioccutrie central angle q,, amt (2)the

<'elcstial l:ttitu(!c ft. of the planet at arrival, sine:(:

/_p IS CELESTIAL LATITUDE OF i: "

MEASURED FROM ECLIPTIC

sini = sin,),

IO

o0 40 80 )20 160 200 240 280 520 560 400

HELIOCENTRIC CENTRAL ANGLE, deg

Ei8.3-13, In¢linolion of the helio(enlricorbifol ploneto e¢lipficvs heliocentric cenlralangle

4O


If thc heliocentric central angle is fixed, and the absolute

magnitu(lc of the celestial latitude at arrival is increased,the inclination also increases. If the celestial latitude of

the planet at arrival is zero, the inclination is equal to

zero, no matter what the heliocentric central angle may

be (with the exception of a central angle of 180 deg).

ttowever, as shown in Fig. :3-1:3, if the celestial latitude

fl_, of the planet at encounter is fixed at an al)solutevalue

greater than zero, and the central angle is varied, the

inclination will reach its minimum value (equal to fl,) at

central angles of 90 anti 270 deg, or its maximum value of

90 deg at the central angles equal to h_,, 180 /3_,,180 + /3,,

and :360 fl,. It is because of the fact that the inclination

increases near the central angles of 180 +-fi, that Type l-

Class II and Type II-Class I trajectories come into exist-

enee. The increasing helioccntric inclination eventually

results in it sharp increase in energy C:_as the central angle

approaches 180 dog -'-fl,, theret)y bringing the two groups

of energy contours into existence. It may now 1)ecome

apparent that the inclination of the heliocentric orbital

plane may take on values greater than the celestial lati-

tude of the planet at arrival because of the varying helio-

centric central angle.

For the Mercury and Jupiter trajectories considered

herein the inclination can be as high as 12 deg. By deft-

nition, inclination is positive, whether cncom_ter takes

phtce at)ore or 1)eh)w the ecliptic. Figures 4-1:3 and ,%1:3

show inclination vs humch date for .Mercury. Figures 6-

through 10-1:3 show this parameter for Earth-Jupitertransfers.

9. Asymptotic Approach Speed V_,_,

Figures 4-15 and 5-15 show that the variation in

asymptotic speed with respect to Mercury is from 12 to

24 km/sec for both Type I and Type II trajectories. Type

1-Class I trajectories (shortest flight times) have higher

approach speeds than Type I-Class II trajectories. It will

be recalled that approach speeds to Venus were on the

order of ,3.75-10.7 km/sec.

Approach speeds to Jupiter lie in the range of 5.5-14

km/see. The lowest asymptotic approach speeds are ob-

tained using Type I-Class II or Type II-Class I trajec-

tories, shown in Figs. 6-- through 10-15.

10. Angle Between the Incoming Asymptote and the

Arrival Planet's Orbital Plane y,

For Mercury, Type I-Class I and Type II-Class II

trajectories have values of "/i, in the range --10 to +,30

deg. This angle can hecome as large as 60 deg for the

other trajectories, as is explained in Ref. 8, page 51. (See

Figs. 4-16 and 5-16.) For Jupiter trajectories, -/_ remains

on the order of 15 deg or less for all opportunities. It

takes on values on the order of 5 deg for Type I-Class I

and Type II-Class II trajectories. (See Figs. 6- through

1(_16.)

11. Angle Between the Arrival Planet-Sun Vector and

the Incoming Asymptote G

For Mercury Type I trajectories, the angle G is acute,

whereas it varies between 60 and 160 deg for Type II

trajectories. (See Figs. 4-17 and 5-17.) This angle is of

considerable importance in the analysis of postencounter

Jupiter trajectories (see Section Ill.F). As was shown for

the Mars case in Ref. 8, page 51, G values for Type I-

Class I trajectories are obtuse at the beginning of thelaunch windows and become acute at the end of the launch

windows. The angle is smaller for the other Jupiter trajec-tories as is shown in Figs. 6 through 10-17.

Figure ,3-14 shows the illuminated crescent orientations

at arrival for various values of G. Figures 3-15a and 3-15b

show near-planet geometries for Type I and Type IIapproach trajectories to Mercury and Jupiter, respectively.

Figures 3-16a and 3-16b show these trajectories as

viewed by an observer in inertial space.

12. Right Ascension Oj, and Declination _, of the

Incoming Planet-Centered Asymptote

For Mercury trajectories, 0_, takes on values from 300

to 355 deg. Right ascension O_, -- 360 deg corresponds to

the vernal equinox direction, or direction of the Sun-Earth

vector at the beginning of spring. The declination _measured relative to the Earth's equator, varies from 40

to -t-40 deg.

For the Jupiter trajectories, there is a considerable

variation in 0t, for different launch opportunities: 160 to

185 deg (1968--69), 290 to 320 deg (1973). The correspond-

ing variations in % are -8 to 22 dug (1968-69) and -36

to -8 deg (1973), respectively.

13. Angle Between the Planet-Earth Vector and the

Incoming Asymptote _:

Plots of this parameter are not available for the Mercury

trajectories at this time. The angle eL. is equivalent to the

Earth-probe-target-planet angle a few days before arrival.

The angle _: - 180 deg corresponds to motion along the

Earth-target-planet line, away from the Earth, during the

41


(a) MERCURY TYPE T TRAJECTORY ( 7"p > Odeg)

ENCOUNTER BEFORE PERIHELION

(c')MERCURY TYPE l-r TRAJECTORY ( 7"p > O deg)

ENCOUNTER AFTER PERIHELION

( b) MERCURY TYPE I TRAJECTORY ( X# < Odeg)

ENCOUNTER BEFORE PERIHELION

(d) MERCURY TYPE TI TRAJECTORY ( 3Yp < O deg)

ENCOUNTER AFTER PERIHELION

(e) JUPITER TYPE I TRAJECTORY ( 7p • O deg)

ENCOUNTER BEFORE APHELION

(f) JUPITER TYPE I TRAJECTORY ( Xp < 0 deg)

ENCOUNTER BEFORE APHELION

(g) JUPITER TYPE I TRAJECTORY ( 7',0> 0 deg)

ENCOUNTER AFTER APHELION

(h) JUPITER TYPE I TRAJECTORY ( ),p

ENCOUNTER AFTER APHELION

Fig. 3-14. Crescent orientations for typical trajectories to Mercury and Jupiter

as observed from approaching spacecraft

< 0 deg)

42


DIRECTIONTO SUN

! ASYMPTOTE.x.x_x._ _.

DIRECTION _ _. "Y'II_TOSUN = -'%.L.IIL\ .I

cp \ "-\/ \

(a)

./

/DIRECTION DIRECTION //'_J /

TO SUN _ TO SUN ,4 _p _/'/_'_-_ /

I(b)

DIRECTIONOF APPROACHASYMPTOTE

Fig. 3-15. Near-Jupiter geometry for typical trajectories, viewed from above ecliptic plane: (a) Type I trajectory,

encountering Jupiter before aphelion; (b) Type I-II trajectory, encountering Jupiter after aphelion

last few days before target-planet encounter; (, :: 180 deg

means that the probe will t)e occulted from view of the

Earth l)('cause of bending of the trajectory by the target

planet. Figures 6- through 10-20 show that this param-

eter varies lwtween 60 and 160 deg for Earth-Jupiter

trajectories.

14. Angle Between the Planet-Canopus Vector and

the Incoming Asymptote ,L

Plots of ,L. are not available for Earth-Mercury transfers

at this time. For Earth-Jupiter transfers, the ranges of ¢,.are 75-150 deg (196S--69), 00-50 deg (1970-71), 0-20 deg

(1972), 90-118 deg (1973), respectively; ¢, - 0 deg indi-

cates that the spacecraft will approach Jupiter along the

spacecraft-Jupiter-C.anopus line. (See Figs. 6- through10-:21.)

G. Procedures for Utilization of Graphs in

Design of Planetary Trajectories

For a trajectory design, a source of information must

1)e availal)lc which can be quickly scanned to determine

the range of feasihle trajectories for a mission. The graphs

presented in Sections IV to X of this report constitute

such a source. The order and procedure for actual use of

these graphs hy the trajectory engineer in the design and

analysis of trajectories arc now reviewed.

(1) Both Type I and Type II transfers should bescanned. Those trajectories should be selected for

which the declination of the outgoing geocentric

asymptote lies roughly between -,34 and -t 34 deg.

These are the feasible trajectories for launchings

from AFETR. The algehraic value of the declina-

43


DIRECTIONTO SUN

DIRECTIONTO SUN

/

DIRECTIONTO SUN

(a)

\ /\ I

'\

\I

r..p

\\-i-\ \

\ \\\ \

\ DIRECTIONOF APPROACH

ASYMPTOTE

DIRECTIONTO SUN

(b)

i1 /

/t/

//"_

/ / /// / /

# // /

// //24 DIRECTION/ OF APPROACH/ ASYMPTOTE

Fig. 3-16. Near-Mercury geometry for typical trajectories, viewed from above ecliptic plane: (a) Type I trajectory,

encountering Mercury before perihelion; (hi Type II trajectory, encountering Mercury after perihelion

tion reveals vital information concerning the pre-

injection trajectory (see Section llI.E.]).

(2) For tile trajectories of paragraph (1), tile energy

requirements (C:) for various firing periods should

])e ol)served. If it is found that booster payload

capability and the desired payload weight match

the required injection energy for a given firing

1)eriod (for example, :30 days) and also satisfy para-

graph (1) for Type I and Type II trajectories, a

decision must then be mad(, to utilize either Type 1

or Type I1 trajectories or. perhaps, t)oth (see

Section Ill.D).

(3) In making the decision to utilize either the Type I

or the Type II trajectory, or }}oth, the curves of

flight time and Earth-prolw communication dis-

tance vs latmch date are most hellfful. In general,

Type I1 trajectories have longer flight times and

4)

l'7,arth-l)r,,be distances at (.llC()tlll[i.,l tlhtl) d() "l.x pc 1.

Thv a(.t,ml difl('rem:es in magnitude (h'l)t'nd ()ll the

mission mid range of injection ( '1_('_"(L_>"I Ill g,'fteral,

tt,' hmvcr the tlight time the ,_reat('r is tt,' sc,si-

tixitx _)( tllc trajectory to injt'ction errors.

Nt,x(. t].' 1)arameter C_., th(* analt' t.'lw,'('n th,'

outt_,_in_ asymptote and the Sun-Eartl, xcct,,r at

helioc,'l,tric injection, shouhl tw studied. It will 1,'

rtx'alh'd that this is equivalent to the l'_'arth-l_ro]w--

Sun angle at a few days after launch. Sim'v Ih('r(!

a]'p usually many limitations on a Sl)acccraft xxhich

is stabilb, ed and contro]h'd in attitude lw ,)plical

rt'fcr(mct'_, (such as the Earth. Sun. and CaTu)ptls).

the F,avlh-probe-Sun angle may lw restricted m'arthe Earth and. perhal)s, throughout the tli_ht. The

Ixtra]neter ¢/, is most ht'll)ful in trajectory ch'si_l

tot d,'t('rmining the constraint near the Earth.

44


(5) If the vahw ()f ._:, is h'ss than 9() (]('_ for Mercury

traj('('t()rics, the pr(/he is iJsually being inje('ted into

the ]l(.liot't'llhi('-trall,Sf('r orl)it l)efore aphe]ion and

will trav('l o,,tsi(h' the Earth's (irl}it before "fallinf'

in toward Xl(']cury:. To be precise, one can obs('rve

the curves ()f hue anomaly at launch. Because (if

t(,lllp(,rtltllr(,-('()lltro[ pr()l)lelns, th('rt' I/1_.1y 1)(" a

restriction ()l_ the maximum Sun-probe distance (hu-

ina flight. T(I determine this maximum distance

dming ttight, one simply trin(ls the aph('lion distance

()t: t]w pr()h(' f](m_ tlw graphs. If ,G, is greater than

,()0 (h'_ l]l('ll |]1(' nl:.tximlull distancu (hiring flight ttl

?xlcrc_nv is ,'ss('nlially the Sm_-prob(" distance at

la,mch.

(6) It th(' value of _.:, is gr('at('r than 90 (leg for .lupiter

trajectories, the prol)e is usual]v lwing injected into

the heliocentric-transf('r orl)it before perihelion and

will "fitll" inside the Earth's orbit heft)re traversin_

(rid t().lllpit('r. T('nq)('r:tturc-c(/ntrol consi(l('rati()ns

may rustrict the minimum Sml-l)rol/e distance (]ur-

hl_ flighl. To d(.t('rmin(' tht' minimum (]istam'u

simply find the p('rihelion of the l)]oli("s transfl'r

(irl)it f()r t]w (h'sir('d traj('('tory or ran_(' of traj('('-

lorius. If G. is h'ss than 90 (h'_. the minimlml S_m-

pr()])c dishmcc (hlritlL_ flight to Julfit('r is the

(]islan('(' at ]aun(']l.

(7) For a spac('craft stabilized and c,()]flr()ll('d in ;tttitu(t(,

]ly Sun and Earth optical rufu]'t'nc('s, a constraint

may exist which will rustric't th(' Earth-pr()be-Sun

angle t() a vahle g]'_'at(']" than 0 (l(' R hy a t:('\v degrees

(n ]('ss than 180 d('_ by a f('w de_r('('s. This immu-

(tiat('ly i]np]i('s that a trajectory must l)(' ('h()s('n that

has an itwlinalion ()f the heli()c('ntric m']lital 1)lan( '

to th(' ecliptic whi(.h is _ruat('r than 0 (h'_. To satisfv

th(' above constraint, th(' inclinati(m may have t()

he 0.o5, 0.50, or even ].0 (leg, depending on the

r('sh'i('titm and on t]w trajectory itself. Th(' values

of th(, ()rbital inclinations for Sl)('t'ific trajt'('torics

can t)t" t_/_l_l(] fr()m tlw graphs in Suctions IX' t(I X.

(8) F(ir ess(,ntially all ft'asil)le Type II traject(/ri('s to

Xh'r('ln'y an(l a few of Type 1, Me]('my (,n(.(luntcr

filr some missions will take place after p(,]ih('lion of

tlw transfer orbit. This means that the probe will

pass closest to the Sm_ several days or, perhaps,

s('veral weeks before encounter, depending on the

trajectory, l_;ecause of tt'nlperatm('-control require-

merits, there may 1)e a restriction on the minimum

Sun-probe distance during ttight. To dctermine_ the

value of this parameter, one finds the perihcli(m

distance of the prot)e for the desired trajectory from

(9)

(lO)

(ll)

(t o)

the _raphs. Tiffs parameter is the minimum S_m-

p,ol)c (]istan('(' dHl'ill_ flight to Nh'rcu]v if th(' trlic

ano]nalv of arrival is in the first or sc('tmd quadrant.

For Type I trajectories t(i .]upitcr. uncountt'r usually

takes place before apht'lion of the transfl'r orbit.

For those trajectories having true anomali('s at

arrival in the third or fourth quadrant, howcv('r,

encounter will take place Mter aplleli(m. This mvans

that the maximum Slm-1)rolw distance' durin( Z flight

(l('('urs })(']()r(' ('lw(/|mtt'l'. "]'('lllpt'ratur('-(.:ontr()] c()t/-

si(lurati(ms may r('strict th(' maximum S_m-pr()])('

distance during fli:aht. To dctcrmim' this maximum

for the ll('('('sStllV t]ajt'ctorit's, find th(' value (if lh{'

aphelion distance from tlw graphs in Sc(.ti(ms VI

to X.

Igor l)]ant'tary missions such its ]a_l(lcrs (w orllih'rs,

it will pr()hably h(,com(' ncc('ssarv t(/ utilize' trajec-

tories for which the h vpurholic-cxccss speed at the

plant't is m'arlv minimum. In fact, in order to

maximizu the scic],tific-1)ayhm(t weight, a "'tradcofl:'"

must l)(' made in minimizin_ the gu(/ct'ntric cncr_v

(Ir l)]ant't()c('ntric ('n('rgv,.. Such tr:tj('ctories will

alh)w the heaviest payl()ads to 1)(, ]an(h'(t ()n the

planet or inj('cted in a th'sir('d ulliptical rehit ai'o_md

the planet with the use ()f rt,tro-maneuvcrs. The

traj('t't(wv designur can fin(] tlw values of th('

hypcrbolic-cxcuss Slwcds at the planet h/r vari()us

traj('c'torit's Jrom the graphs in S('cli()_s IV to N, and

can dcturmine tlw feasibility of thc trajcctori('s from

the l)roce(hlres in paraRral)hs (1) to (9).

l:()r a spacecraft which is stabiliz('d and controlh'd

in attitude by optical rt'terenccs, there may he

ccrtain restrictions on the Sun-probe-planet ;tnRle

as the pr()l)t' apl)r()aches the planet. Such restricti()ns

may 1)(' nt'ct'ssarv ])('c'a_se ()f approach-_ui(lanc(_

consi(h'rations. The paramctt,r g:,, the an_lu tlt'-

t\v('u_, the approach ;tSVml)tOtc and th(' tare, ct-

l)]iuwt-S_m recto|, is ('(l_fiv:tl('nt to the S_ln-pr()l)t'-

planet angle a f('w da\'s he|ore ctwounter. The

parameter can 1)t' most helpful in analyzing per-

missibh' traj(,('t_)ri('s near th(' planet.

l{egar(ling the design of trajectories near the planet,

it is n('edh'ss t(I sax that, by altering the paramet('rs

in the prcinjection trajectory near Earth Iiy sma]l

in('rem('nts, the probe can ])t' made to pass on anysidc of the planet, ll(Iwever, the inclinations of the

planetcwentric ]_yperl)o]ic orbit (or ('lliptical orbit,

assuming that the r('quir('d retro-man('uv('r is made

45


in the orbital plan(' of the hvpurh(/ht) to the planc't's STEPS• I AND 2

orbital l)]anu dupcnds on the aiming point at the

l)lanct,as well as on flw l)aranwtur 7_i, th('an_h'

I)(_t\v(_CII the approa(']l asymptote and the planet's-o

()rl)itaIplan< The minimum inclination which can

lu' attained is u(tua] t() 7 I' 1t: "/;, is z('ro, the inclina- __b..

tion ¢)f the .rlfit to thu plallct7 nrlfita] plam, will

range fr()m () re) l_0 (h'_ dul)undin _ on the aiming o

point, hl('linati()llS {4 0 t() 90 (h'_ imply direct _-

re(Ilion (in Ill(' dir('ction of tlw phmct's orbital rota-

tion1, \%hUl'Ca_ im'lillati(m_ t]'cun 90 to 180 dc_ imply

l'ctr()i_l'ttdc motion ((}i)l)()sitc t<) t]w planet's (}rbita]

r()tati()n). If y,, '-I,5 de!& th(" inc]ilmtion may

ran_u from ,1,5 to 135 c|c_ f()r all aiming points. STEP 5

[[ -/_, ' _}() dug, the im'lilmti(m will equal 90 d('g

f()r any aimin_ point xxlfich i_ (,h()_(,n at the planet, v%LO

l" ( )1" %, i I , "_ ) (} ( ] ( ' _, t } J { ' l) I'( } I ){ ' will apl)r(xlch the planet ?:,,

h(lm a path tx,l()w tlw l)]itllct's orhita_l t)lam': for

7_,< 0 (h'_, th(" al)l)roach will h(' from al)ove. In

manv cases, it may l)(" (h'siral)h' t() design the near- 5k--

p]a.n('t trajuct(wy \_ith a 1)r('scril)cd inclination in

mind su',::h that the llr()l)( ' ",',ill pass thc target t)ody

in the l)hnwt's i)rhital (/r, l)('rhap_, (,quatl)rial plane.

It is tlnls appar('nt that the param('t('r 7_, may i)r

may not permit the s('h'ctd pass. To find th(' vil]u('

()I" ;_r t'()l" _[ _i _'('} } trai('cl{)rv, ol)_crv(' the c'm'vu o1:711

",_,hunwh date f,, llw de_ird mission.

(1_)'1"o comt)r('humt full,," the _i,_nificanc'c' oI; each

l)aramct('r and its variation \vith launch (late and

arrival dat(', it may b(' a(|'<ant;tff.t'olls to construct

the loci ()f constant arrival tt.:ttt,s on (,itc,]l pertinent

_4rat)]l ill Suctions IV to X. This lm).v I)(" done ill

thr('e st('ps, its sh()_n in Fi_. 3-17.

(ilt) Construct the loci ()t d('sir('d arrival dates (mth('

era'yes of flight tim(, vs ]aunch (late.

(b) \lark the 1ram,oh dates and _('oc'('ntric ('n('r_ics

for which int('rn('ution (){ t]w arrival loci and the

vari()Hs ('nt'rg}" ('(nit(mrs ()(:('ur.

\\

POINTS OF INTERSECTION

LOCI OF DESIRED ARRIVAL DATE_

LAUNCH DATE

................

LOCIOF DESIRED ARRIVAL DATE'Xx

OPOINTS OF INTERSECTION

TAKEN FROM LOCI ABOVE

LAUNCH DATE

Fig. 3-17. Steps for construction ofarrival date loci

(c) Tak,' the int('rsection points (ff (1)) and construct

tlw :uTival-(lat(" loc'i on the _raph of inter('st.

Exl),'rimK'u has shown that traj('c'tory parameters tend

to t'xhitfil _tl_ il_x'ariance if tlw arrival date is huld fixed

whil(, tl,' laun('h dat(' is varid. This fact has provd very

us('ful in trajectory design.

46


IV. MERCURY 1967: TRAJECTORY PARAMETER GRAPHS

Figure

4-1.

4-2.

4-3.

4-4.

4-5(I).

4-5(11).

4-6(I).

4-6(11).

4-7(I).

4-7111).

4-8(I).

4-8(11).

4-9(I).

4-9(11).

4-10.

4-1111).

4-11111).

4-12(I).

4-12(11).

4-13(I).

4-13(11).

4-14.

Mercury 1967: Minimum iniection energyvs launch date




launch date


launch date, Type I




launch date, Type I














time vs launch date, Type II

Mercury 1967: True anomaly in transfer ellipse at arrival

time vs launch date

Mercury 1967: Perihelion of transfer orbitvs launch date, Type I

Mercury 1967: Perihelion of transfer orbit vs launch date, Type II

Mercury 1967: Aphelion of transfer orbitvs launch date, Typel

Mercury 1967: Aphelion of transfer orbit vs launch date, Type II





Mercury 1967: Celestial latitude at arrival timevs launch date

47


IV. MERCURY 1967: TRAJECTORY PARAMETER GRAPHS (Cont'd)

Figure

4-15(I).

4-15(11).

4-16(I).

4-16(11).

4-17.

4-18(I).

4-18(11).

4-19(I).

4-19(11).


launch date, Type I





launch date, Type I





hermiocentric asymptote vs launch date


launch date, Type I




launch date, Type I



48


,.2

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082 [:; . :.::

!i;i,_!i: ..............079 i' J

:ii i\: 1 iili .......

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tt''t't_' " ' " _ ........

£L:+_-L*,2 :::_, ................. _L::-:-

:_;:t:''r ...._:i:? ":!'+....

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-_!:_!: TYFiE I"1"i U_:;I::I:;_ ;i. ::: _ , - T _-

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t-T- a;_:_, : ; F;

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• _ft_., c.,._.;.• ::{ :rI:? :[t!::ll;f: it:; "+J [ l 'T*?I ' '

tt l t.-r -,*- '_'t

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_l_ _T_';II 21 31

OCT

0.70

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0.49

0.46

0.43

0.40

ii

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I

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:i;

i::

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!! ............................Ii i: i::I,:•i ...........

:; :i! ::;:i::: :::: :::

I_l I I 1 i i

'ifii i;': !:fi: ::::tf:i :::: iT:?'-

; _"" l:_ :1I: ...............

.......:_ :::: :_:: ::::ti:! :::: :::,f ..... . ...... _,.. .,{ ....

' T'Tl I I , I I , I ' ' 4 " + : _

"t "?F' _I .... t' "t'''i .... *;_',**

-*;, ._.; .,_ ...... , .... ++..++.

,* ;{-_ ,.., -+4- ,l ........ ++.,++_

I! t!;; ::{: :t!: t........ :hil!_:::

.... '41 .,,.! _i'. :;:: .... ._*t_-,-_.

:t:_.... :I:: !::f.... ]!!]

:[ ,,,, ................_.. 214" -.+. .+ .......

II I II _lJ III I I I ;i li

,t,.I?;I........*__ !i :ii i !l :i*, Ii J i

+*-+

_:_ . TYPE I !ii ::;I ;i; ,_i•[

I;,4

NT 11!? :: : ....

_? -P_- lift! '+* ...... T:T :T] +........... llll

I!"1_??" _-?? 1771-.jf:-_4 ....

T: : ...... _ ( .........

:i i t!7 ::T1 :i:1 : _:, i I ,_i.i

_* +*-+ .+,t [.ti:.,_ t, .....

NOTE: C3 IS TWICE THE TOTAL

ENERGY/UNIT MASS WITH

UNITS m2/sec2 X I06

...... [ ] l [" l 4_"4-- "_'_ :

Io 20 30DEC

i

::T

9

JAN

Fig. 4-1. Mercury 1967: Minimum injection energy vs launch date

49


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launch date


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Mercury 1968: True anomaly in transfer ellipse at

launch time vs launch date


launch date, Type I




launch date, Type I


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V. MERCURY 1968: TRAJECTORY PARAMETER GRAPHS (Cont'd)

Figure

5-13(11).

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launch date, Type I





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Mercury 1968: Right ascension of the hermiocentricasymptote




81


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VII. JUPITER 1969-70: TRAJECTORY PARAMETER GRAPHS

Figure

7-1. Jupiter 69-70:

7-2. Jupiter 69-70:

7-3. Jupiter 69-70:

7-4(I). Jupiter 69-70:

7--4(11).

7-5(I).

7-5(11}.

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7-12(I).

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7-13(11).

Minimum injection energy vs launch date

Time of flight vs launch date

Heliocentric central angle vs launch date

Earth-Jupiter communication distance vs

launch date, Type I

Jupiter 69-70: Earth-Jupiter communication distancevs


Jupiter 69-70: Declination of the geocentric asymptotevs

launch date, Type I

Jupiter 69-70: Declination of the geocentric asymptotevs


Jupiter 69-70: Right ascension of the geocentric asymptotevs

launch date, Type I








geocentric asymptote vs launch date


time vs launch date


time vs launch date


Type I


Type II

Jupiter 69-70: Aphelion of transfer orbitvs launch date,

Type I


Type II





150


VII. JUPITER 1969-70: TRAJECTORY PARAMETER GRAPHS (Cont'd)

Figure

7-14.

7-15(I).

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7-16(I).

7-16(11).

7-17.

7-18(I).

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7-19(I).

7-19(11).

Jupiter69-70: Celestial latitude at arrival timevslaunch date






and arrival planers orbital plane vs launch date, Type I


and arrival planet's orbital plane vs launch date, Type II

Jupiter 69-70: Angle between Jupiter-Sun vector and

incoming zeocentric asymptote vs launch date

Jupiter

launch

Jupiterlaunch

Jupiterlaunch

Jupiter

launch

69-70: Declination of the zeocentric asymptote vs

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69-70: Declination of the zeocentric asymptote vs

date, Type II

69-70: Right ascension of thezeocentricasymptotevs

date, Type I

69-70: Right ascension of thezeocentricasymptotevs

date, Type II

151


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VIII. JUPITER 1970-71: TRAJECTORY PARAMETER GRAPHS

Figure

8-1. Jupiter

8-2. Jupiter

8-3. Jupiter

8-4(I). Jupiter

launch

8-4(11). Jupiter

launch

8-5(I). Jupiter

launch

8-5(11). Jupiter

launch

8-6. Jupiter

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8--7.

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70-71: Minimum injection energy vs launch date

70-71: Time of flight vs launch date

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70-71: Earth-Jupiter communication distancevs

date, Type I

70-71: Earth-Jupiter communication distancevs

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date, Type I

70-71: Declination of the geocentric asymptotevs

date, Type II

70-71: Right ascension of the geocentric asymptotevs

date



Jupiter 70-71: Angle betweenSun-Earth vector and outgoing




Jupiter 70-71 : True anomaly in transfer ellipse at launch



time vs launch date, Type II


time vs launch date


Type I


Type II

Jupiter 70-71: Aphelion of transfer orbitvs launch date,

Type I


Type II





182


VIII. JUPITER 1970-71: TRAJECTORY PARAMETER GRAPHS (Cont'd)

Figure

8-14.

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8-2011).

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8-21 (11).


Jupiter 70-71: Asymptotic speed with respect to Jupitervs

launch date, Type I

Jupiter 70-71: Asymptotic speed with respect to Jupitervs


Jupiter 70-71: Angle between incoming zeocentricasymptote




Jupiter 70-71: Declination of zeocentricasymptotevs

launch date, Type I

Jupiter 70-71: Declination of zeocentricasymptotevs


Jupiter 70-71: Right ascension ofzeocentric asymptote vs

launch date, Type I

Jupiter 70-71: Right ascension of zeocentricasymptote vs


Jupiter 70-71: Angle between planet-Earth vector and


Jupiter 70-71: Angle between planet-Earth vector and


Jupiter 70-71: Angle between planet-Canopusvector and


Jupiter 70-71: Angle between planet-Canopus vector and


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IX. JUPITER 1972: TRAJECTORY PARAMETER GRAPHS

Figure

9-1.

9-2.

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9-4(I).

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9_5.

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Jupiter 1972:

Jupiter 1972:

Jupiter 1972:


Time of flight vs launch date


Jupiter 1972: Earth-Jupiter communication distancevs

launch date, Type I

Jupiter 1972: Earth-Jupiter communication distancevs


Jupiter 1972: Declination of the geocentric asymptotevs

launch date

Jupiter 1972: Right ascension of geocentric asymptotevs

launch date











Jupiter 1972: True anomaly in transfer ellipse at arrival time

vs launch date


Type I

Jupiter

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Jupiter

Jupiter

launch

Jupiter

launch

Jupiter

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1972: Aphelion of transfer orbit vs launch date, Typel

1972: Aphelion of transfer orbit vs launch date, Typell

1972: Inclination of the heliocentric transfer planevs

date, Type I

1972: Inclination of the heliocentric transfer planevs

date, Type II



217


IX. JUPITER 1972: TRAJECTORY PARAMETER GRAPHS (Cont'd)

Figure

9-15(I).

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9-16.

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9-19(I).

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launch date, Type I

Jupiter 1972: Asymptotic speed with respect to Jupitervs






Jupiter 1972: Declination of zeocentric asymptote vs

launch date

Jupiter 1972: Right ascension ofzeocentricasymptotevs

launch date, Type I

Jupiter 1972: Right ascension ofzeocentric asymptotevs



asymptote vs launch date, Type I


asymptote vs launch date, Type II



Jupiter 1972: Angle between planet-Canopusvector and


218


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X. JUPITER 1973: TRAJECTORY PARAMETER GRAPHS

Figure

10-1.

10-2(I).

10-2(11).

10-3.

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10-4(11).

10-5(I).

10-5(11}.

10-6(I).

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10-9(11).

10-10.

10-11(I).

10-1 1(11).

10-12(I).

10-12(11).

Jupiter

Jupiter

Jupiter

Jupiter

Jupiter

launch

Jupiter

launch

Jupiter

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Jupiter

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Jupiter

1973:

1973:

1973:

1973:

1973:


Time of flight vs launch date, Type I

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Earth-Jupiter communication distance vs

date, Type I

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date, Type II

1973: Declination of geocentric asymptotevs

date, Type I

1973: Declination of geocentric asymptotevs

date, Type II

1973: Right ascension of geocentric asymptotevs

launch date, Type I













Jupiter1973: True anomaly in transfer ellipse at arrival time

vs launch date

Jupiter

Type I

Jupiter

Type II

Jupiter

Type I

Jupiter

Type II

1973: Perihelion of transfer orbit vslaunchdate,


1973: Aphelion of transfer orbit vs launch date,

1973: Aphelion of transfer orbitvs launch date,

251


X. JUPITER 1973: TRAJECTORY PARAMETER GRAPHS (Cont'd)

Figure

10-13(I).

10-13(11).

10-14(I).

10-14(11). Jupiter

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10-15(1). Jupiter

launch

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10-16.

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launch date

10-19(11. Jupiter 1973:

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10-19(11). Jupiter 1973:

launch date,

10-20(I). Jupiter 1973:

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10-20(11}. Jupiter 1973:

asymptote vs

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10-21 (11).

Jupiter 1973: Inclination of the heliocentric transfer planevs

launch dale, Type I

Jupiter 1973: Inclination of the heliocentric transfer planevs



Type I

1973" Celestial latitude at arrival time vs launch date,

1973: Asymptotic speed with respect to Jupitervs

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1973: Asymptotic speed with respect to Jupitervs

date, Type II

Jupiter 1973: Angle between incoming zeocentricasymptote




Declination of zeocentric asymptote vs

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Type I

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Type II

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NOMENCLATURE

C:_ vis viva energy (injection energy, ()f escape

hyperbola)

i inclination of heliocentric transfer orbit, deg

R,, aphelion of transfer orbit, millions of km

R(_ Earth-Mercury (or-Jupiter) communicationdistance, millions of km

R_, perihelion of transfer orbit, millions of km

T_, time of flight, days

vL true anomaly in transfer ellipse at launch time,

deg

v_ true anomaly in transfer ellipse at arrival time,

deg

Vhp asymptotic speed (or hyperbolic-excess speed)

with respect to Mercury (or Jupiter), km/see

13_, celestial latitude of Mercury (or Jupiter) at

arrival time, deg

yL angle between outgoing geocentric asymptote

and launch planet's orbital plane, deg (also

defined as celestial latitude of outgoingasymptote)

yp angle between incoming hermiocentric t_or

zeocentric" asymptote and arrival planet's

orbital plane, deg

{_ angle between Sun-Earth vector and outgoing

geocentric asymptote, deg (also defined as

Earth-probe-Sun angle)

_, angle between Mercury-Sun (or Jupiter-Sun)

vector and incoming hermiocentric (or

zeocentric) asymptote, deg (also defined as

Sun-probe-target angle)

0_, right ascension of hermiocentrie (or zeocentric)

asymptote, deg

0,_ right ascension of geocentric asymptote, deg

• _, declination of hermiocentric (or zeocentrie)

asymptote, deg

qb_ declination of geocentric asymptote, deg

,I, heliocentric central angle, deg

aThis table of Nomenclature presents only the notation used in thegraphs of Sections IV to X. All other terms are defined at pointof first mention in Sections I to III.

6Hermiocentrie refers to Mercury.

"Zeocentric refers to Jupiter.

288


REFERENCES

1. Battin, R. H., "The Determination of Round-Trip Planetary Reconnaissance Trajec-

tories," Journal of the Aero/Space Sciences, Vol. 26, No. 9, September 1959.

2. Clarke, V. C., Jr., Design of Lunar and Interplanetary Ascent Trajectories, Technical

Report No. 32-30, Rev. 1, Jet Propulsion Laboratory, Pasadena, March 15, 1962.

3. Holdbriclge, D. B., Space Trajectories Program for the IBM Computer, Technical

Report No. 32-223, Rev. 1, Jet Propulsion Laboratory, Pasadena, September 1,

1962.

4. The American Ephemeris and Nautical Almanac for the year 1960, p. 498, U. S.

Government Printing Office, Washington.

5. Allen, C. W., Astrophysical Quantities, The Athlone Press, University of London,

London, 1955.

6. Cutting, E., and Sturms, F. M., "Trajectory Analysis of a 1970 Mission to Mercury

via a Close Encounter with Venus," AIAA Paper 60-90, AIAA 2nd Annual Aero-

space Sciences Meeting, New York, New York, January 25-27, 1965.

7. Baker, A. B., "Approximate Payload Capabilities of Boosters for Planetary

Missions," Journal of Spacecraft and Rockets, Vol. 1, No. 4, p. 439, 1964, RCA,

Princeton, New Jersey.

8. Clarke, V. C., Jr., Bollman, W. E., Roth, R. Y., and Scholey, W. J.,Design Param-

eters for Ballistic Interplanetary Trajectories, Part I. One-way Transfers to Mars

and Venus, Technical Report No. 32-77, Jet Propulsion Laboratory, Pasadena,

California, January 16, 1963.

9. W. E. Bollman, One-way Type-II Transfers to Mars, Addendum 1 to Ref. 8, Jet

Propulsion Laboratory, Pasadena, California (to be published).

ACKNOWLEDGMENT

Tile authors wish to acknowledge the valuable assistance of

M. Beckwith, who carried out the production of tile computations and

automatic ph)tting of the graphs.

289

Way Transfers to Mercury and Jupiter N66-16591

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Transcript of Way Transfers to Mercury and Jupiter N66-16591