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JPL PUBLICATION 82-43

Interplanetary Mission Design Handbook, Volume I, Part 2 to Ballistic Mission Opportunities, 1990-2005

Andrey B. Sergeyevsky Gerald C. Snyder Ross A. Cunniff

September 15, 1983

National Aeronautics and Space Administration

Jet Propulsion Laboratory

./ California Institute of Technology Pasadena, California

(I_AS_-CR- |7330b) INTE_YLA_E_ AEY MIsSIoN N8_.' 18226 DESIGN dkNDbUOK. VOLU_£ l, PAinT 2: EAETH TO MABZ BALLZS_'ZC hlSSiON CPYOI_TUNI_._S, |9_0-2005 _Jet Propu±sion ZaD.) 176 p Uncias _C AO9/MY AOI CSCL 22A G3/12 18z07

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TECHNICAl. REPORT STANDARD TITLE PAGE

1. Repot No. 2. Government Access;on No. 3. Recipient's Catalog No. 82-43, Vol. I, Pt. 2 4. Title and Subtitle 5. Report Date September 15, 1983 Interplanetary Mission Design Handbook, Volume I, Part 2: Earth to Mars Ballistic Mission 6. PerformingOrgan zot o"Code Opportunities, 1990-2005

7. Author_ 8. Performing Organization Report No A.B. SergeyevsKy, G.C. Snyder, R.A. Cunniff

| 9. _rforming Organization Name and Address 10. Work Unit No. JET PROPULSION LABORATORY 11. Contract or Gra_t No. California Institute of Technology 4800 Oak Grove Drive NAS 7-100 Pasadena, California 91109 1'3. Type of Report and Period Covered

12. Sponsoring Agency Name anct Address JPL Publication

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 14. Sponsoring Agency Code Washington, D.C. 20546 RD4/P- 186-30-03- ii-00

15. Supplementary Notes

16. A_tract This document contains graphical data necessary for the preliminary design of ballistic missions to Mars. Contours of launch-energy requirements, as well as many other launch and Mars-arrival parameters, are presented in launch-date/arrival-date space for all launch opportunities from 1990 through 2005. In addition, an extensive text is included which explains mission- design methods, from launch-window development to Mars-probe and orbiter-arrival design, utilizing the graphical data in this volume as well as numerous equations relating various parameters. This is one of a planned series of mission-design documents which will apply to all planets and some other bodies in the .

17. Key Wor_ (Selec_d by Author_)) 18. Distribution Statement Astronautics Astrodynamics Launch Vehicles and Space Vehicles Unclassified - Unlimited Lunar and Planetary Exploration

19. Security Cl_sif. (of this report) 20. Security Clmsif. (of this page) 2i. No. of Pages 22. Price

Unclassified Unclassified 176

JPL 0184 11f80

JPL PUBLICATION 82-43

Interplanetary Mission Design Handbook, Volume I, Part 2 Earth to Mars Ballistic Mission Opportunities, 1990- 2005

Andrey B. Sergeyevsky Gerald C. Snyder Ross A. Cunniff

September 15, 1983

National Aeronautics and Space Administration

Jet Propulsion Laboratory California Institute of Technology Pasadena, California The research described in this publication was carried out by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. Abstract

This document contains graphical data necessary for the preliminary design of ballistic missions to Mars. Contours of launch energy requirements, as well as many other launch and Mars arrival parameters, are presented in launch date/arrival date space for all launch opportunities from 1990 through 2005. In addition, an extensive text is included which explains mission design methods, from development to Mars probe and orbiter arrival design, utilizing the graphical data in this volume as well as numerous equations relating various parameters. This is one of a planned series of mission design documents which will apply to all planets and some other bodies in the solar system.

i11_ Preface

This publication is one of a series of volumes devoted to interplanetary trajectories of different types. Volume I deals with ballistic trajectories. The present publication is Part 2 and describes ballistic trajectories to Mars. Part 3, which was published in 1982, treated ballistic trajectories to . Part 4, which was published earlier in 1983, described ballistic trajectories to . Parts 1 and 5, which will be published in the near future, will treat ballistic trajectories to and Mars-to-Earth return trajectories, respectively.

iv Contents

I, Introduction ...... 1

II. Computational Algorithms ...... 1

A. General Description ...... 1

B. Two-Body Conic Transfer ...... 1

C. Pseudostate Method ...... 3

III. Trajectory Characteristics ...... 4

A. Mission Space ...... 4

B. Transfer Trajectory ...... 5

C. Launch/Injection Geometry ...... 9

1. Launch Azimuth Problem ...... 9

2. Daily Launch Windows ...... 10

3. Range Arithmetic ...... 15

4. Parking Regression ...... 16

5. Dogleg Ascent ...... 17

6. Tracking and Orientation ...... 17

7. Post-Launch State ...... 18

8. Orbital Launch Problem ...... 18

D. Planetary Arrival Synthesis ...... 18

1, Flyby Trajectory Design ...... 18

2. Capture Orbit Design ...... 24

3. Entry Probe and Lander Trajectory Design ...... 28

E. Launch Strategy Construction ...... 31

IV. Description of Trajectory Characteristics Data ...... 32

A. General ...... 32

B. Definition of Departure Variables ...... 32

C. Definition of Arrival Variables ...... 32

V. Table of Constants ...... 33

A. ...... 33

B, Earth/ System ...... 33

C. Mars System ...... 33 D. Sources ...... 33

Acknowledgments ...... 33

References ...... 33

Figures

1. The Lambert problem geometry ...... 2

2. Departure geometry and velocity vector diagram ...... 2

3. Pseudostate transfer geometry ...... 3

4. Mission space in departure/arrival date coordinates, typical example .. 5

5. Effect of transfer angle upon inclinationof trajectory arc ...... 6

6. Nodal transfer geometry ...... 6

7. Mission space with nodal transfer ...... 7

8. Broken-plane transfer geometry ...... 8

9. Sketch of mission space with broken-plane transfer effective energy requirements ...... 8

10. Launch/injection trajectory plane geometry ...... 9

11. Earth equator plane definition of involved in the launch problem ...... 10

12. Generalized relative launch time tRL T VS launch azimuth E_L and departure asymptote &= ...... 11

13. Permissible regions of azimuth vs asymptote declination launch space for Cape Canaveral ...... 12

14. Typical launch geometry example in celestial (inertial) Mercator coordinates ...... 12

15. Central range angle O between launch site and outgoing asymptote direction vs its declination and launch azimuth ...... 13

16. Typical example of daily launch geometry (3-dimensional) as viewed by an outside observer ahead of the spacecraft ...... 14

17. Basic geometry of the launch and ascent profile in the trajecto_ plane ...... 15

18. Angle from pedgee to departure asymptote ...... 16

19. Definition of cone and clock angle ...... 17

20. Planetary flyby geometry ...... 19

21. Definition of target or arrival B-plane coordinates ...... 20

22. Two T-axis definitions in the arrival B-plane ...... 21

23. Definition of approach orientational coordinates ZAPS and ETSP, Z.APE and ETEP ...... 22 vi 24. Phase angle geometry at arrival planet ...... 22

25. Typical entry and flyby trajectory geometry ...... 23

26. Coapsidal and cotangential capture orbit insertion geometries ...... 25

27. Coapsidal capture orbit insertion maneuver _V requirements for Mars ...... 26

28. Coapsidal and intersecting capture orbit insertion geometries ...... 27

29. Characteristics of intersecting capture orbit insertion and construction of optimal bum envelope at Mars ...... 29

30. Minimum 3.V required for insertion into Mars capture orbit of given apsidal orientation ...... 30

31. General satellite orbit parameters and precessional motion due to oblateness coefficient J2 ...... 31

32. Voyager (MJS77) trajectory space and launch strategy ...... 31

Mission Design Data Contour Plots, Earth to Mars Ballistic Mission Opportunities, 1990- 2005 ...... 35

1990 Opportunity ...... 37

1992 Opportunity ...... 49

1994 Opportunity ...... 61

1996/7 Opportunity ...... 73

1998/9 Opportunity ...... 85

200011 Opportunity ...... 109

2002/3 Opportunity ...... 133

2005 Opportunity ...... 157

I. Introduction but certainly not unrealistic using future orbital assembly techniques. The purpose of this series of Mission Design Handbooks is to provide trajectory designers and mission planners with A separate series of volumes (Ref. 3) is being published graphical trajectory information, sufficient for preliminary concurrently to provide purely geometrical (i.e., trajectory- interplanetary mission design and evaluation. In most respects independent) data on planetary positions and viewing/orienta- the series is a continuation of the previous three volumes of tion angles, experienced by a spacecraft in the vicinity of these the Mission Design Data, TM 33-736 (Ref. 1) and its predeces- planetary bodies. The data cover the time span through 2020 sors (e.g., Ref. 2); it extends their coverage to departures A.D., in order to allow sufficient mission duration time for all through the year 2005 A.D. Earth departures, up to 2005 A.D.

The entire series is planned as a sequence of volumes, each The geometric data are presented in graphical form and describing a distinct mission mode as follows: consist of 26 quantities, combined into eight plots for each calendar year and each target planet. The graphs display equa- Volume I: Ballistic (i.e., unpowered) transfers be- torial declination and of Earth and Sun (plan- tween Earth and a planet, consisting of etocentric), as well as those of the target planet (geocentric); one-leg trajectory arcs. For Venus and heliocentric () of the planet, its heliocentric Mars missions the planet-to-Earth return and geocentric distance; cone angles of Earth and Canopus, trajectory data are also provided. clock angle of Earth (when Sun/Canopu_-oriented); Earth-Sun- Volume II: Gravity-Assist (G/A) trajectory transfers, planet, as well as Sun-Earth-planet angles; and finally, rise and comprising from two to four ballistic set times for six deep-space tracking stations assuming a 6-deg interplanetary legs, connected by suc- mask. This information is similar to that in the second cessive planetary swingbys. part of each of the volumes previously published (Ref. 1).

Volume III: Delta-V-EGA (AVEGA) transfer trajec- tories utilizing an impulsive deep-space II. Computational Algorithms phasing and shaping bum, followed by a return to Earth for a G/A swingby maneu- A. General Description ver taking the spacecraft (S/C) to the The plots for the entire series were computer-generated. A eventual target planet. minimum of editorial and graphic support was postulated from the outset in an effort to reduce cost. Each volume consists of several parts, describing trajectory opportunities for missions toward specific target or swingby A number of computer programs were created and/or modi- bodies. fied to suit the needs of the Handbook production.

This Volume I, Part 2 of the series is devoted to ballistic The computing effort involved the generation of arrays of transfers between Earth and Mars. It describes trajectories transfer trajectory arcs connecting departure and arrival taking from 100 to 500 days of flight time for the 8 successive planets on a large number of suitable dates at each body. mission opportunities, departing Earth in the following years: Algorithms (computational models) to solve this problem can 1990, 1992, 1994, 1996/7, 1998/9, 2000/1, 2002/3, and vary greatly as to their complexity, cost of data generated, and 2005. resulting data accuracy. In light of these considerations, the choice of methods used in this effort has been assessed. Individual variables presented herein are described in detail B, Two-Body Conic Transfer in subsequent sections and summarized again in Section IV. Suffice it to say here that all the data are presented in sets of Each departure/arrival date combination represents a unique 11 contour plots each, displayed on the launch date/arrival transfer trajectory between two specified bodies, if the number date space for each opportunity. Required departure energy of revolutions of the spacecraft about the primary (e.g., the C3, departure asymptote declination and right ascension, Sun) is specified. The Lambert Theorem provides a suitable arrival V and its equatorial directions, as well as Sun and framework for the computation of such primary-centered Earth direction angles with respect to the departure/arrival trajectories, but it is of practical usefulness only if restricted asymptotes, are presented. two-body conic motion prevails.

It should be noted that parts of the launch space covered Restricted two-body motion implies that the dynamical may require launcher energies not presently (1983) available, system consists of only two bodies, one of which, the primary, ORIGINAL PAGE IS 3F POOR QUALITY is so much more massive than the other, that all of the system's gravitational attraction may be assumed as concentrated at a point-the center of that primary body. The secondary body of negligible mass (e.g., the spacecraft) then moves in Kep- lerian (conic) about the primary (e.g., the Sun) in such a way that the center of the primary is located at one of the foci of the conic (an ellipse, parabola, or hyperbola).

-_d.. I The Lambert Theorem states that given a value of the gravi- EARTH tational parameter # (also known as GM) for the central body, !, I the time of flight between two arbitrary points in space, / R 1 and R2, is a function of only three independent variables: the sum of the distances of the two points from the focus, LANET IRII + IR2L, the distance between the two points C= SR2 - R I I, ' '// and the semimajor axis, a, of the conic orbital flight path I'=.f(R 1 +R 2 ,C.a) I between them (Fig. 1.)

Fig. I. The Lambert problem geometry Detailed algorithm descriptions of the Lambert method, in- eluding necessary branching and singularity precautions, are presented in numerous publications, e.g., Refs. 2 and 4. The computations result in a set of conic classical elements (a, e, i, the hyperbolic excess velocity, "V-infinity" or simply "speed" _2, co, ol) and the transfer angle, Ao12, or two equivalent (e.g., Ref. 4). The V** represents the velocity of the spacecraft spacecraft heliocentric velocity vectors, Vhs/c i- one at depar- at a great distance from the planet (where its gravitational ture. the other at the arrival planet. Subtraction of the appro- attraction is practically negligible). It is attained when the priate planetary heliocentric velocity vector, VhpLANETi , at spacecraft has climbed away from the departure planet, fol- the two corresponding times from each of these two space- lowing injection at velocity V1: craft velocity vectors results in a pair of planetocentric velocity states "at infinity" with respect to each planet (Fig. 2):

= _/V_ "9"1 (2) V**1 _'/ , kmts

V_i = VhS/C at i - VhpLANET i (1) or before it starts its fall into the arrival planet's gravity well, where i = 1 and 2 refer to positions at departure and arrival, where it eventually reaches a closest approach (periapse, C/A) respectively. The scalar of this V vector is also referred to as velocity Vp:

VhEAR Ttt

OUTGOING V

ASYMPTOTE h sic EARTH

INJECTION

_ _2, ASCENDING NODE

vl

Fig. 2. Departure geometry and velocity vector diagram ORIGINAL PAGE lg OF POOR QUALITY

Pseudostate theory is based on the assumption that for = + , n/s (3) modest gravitational perturbations the spacecraft conic motion Vp _/V_ 2 2#_rp about the primary and the pseudo-conic displacement due to a third body may be superimposed, if certain rules are followed. The variables rI and rp refer to the departure injection and arrival periapse planetocentric radii, respectively. Values for The method, as applied to transfer trajectory generation, the gravitational parameter/a (or GM) are given in subsequent does not provide a flight path-only its end states. It solves the Section V on constants. original Lambert problem, however, not between the true planetary positions themselves, but instead, between two com- The V. vectors, computed by the Lambert method, repre- puted "pseudostates." These are obtained by iteration on two sent a body center to body center transfer. They can, however, displacement vectors off the planetary positions on be translated parallel to themselves at either body without the dates of departure and arrival. By a suitable superposition excessive error due to the offset, and a great variety of realistic with a planetocentric rectilinear impact hyperbola and a departure and arrival trajectories may thus be constructed constant-velocity, "zero gravity," sweepback at each end of through their use, to be discussed later. The magnitude and the Lambertian conic (see Fig. 3), a satisfactory match is direction of V** as well as the angles that this vector forms obtained. with the Sun and Earth direction vectors at each terminus are required for these mission design exercises. Of the five arcs involved in the iteration, the last three (towards and at the target planet) act over the full flight time, Missions to the relatively small terrestrial planets such as AT_2, and represent: Mars are suited to be analyzed by the Lambert method, as the problem can be adequately represented by the restricted two- (1) Conic heliocentric motion between the two pseudo- body formulation, resulting in flight time errors of less than states R_ and R_ (capital R is used here for all helio- 1 day - an accuracy that cannot even be read from the contour centric positions), plots presented in this document. (2) The transformation of R_ to a planetocentric position, r 2 (lower case r is used for planetocentric positions), C. Pseudostate Method performed in the usual manner is followed by a "constant-velocity" sweepback in time to a point Actual precision interplanetary transfer trajectories, espe- r_ = r_ - V**2 X zkT12, correcting the planetocentric cially those involving the giant outer planets, do noticeably position r 2 to what it would have been at T1, had violate the assumptions inherent in the Lambert Theorem. there been no solar attraction during ATIj, and The restricted two-body problem, on which that theorem is finally, based, is supposed to describe the conic motion of a massless secondary (i.e., the spacecraft) about the point mass of a pri- (3) The planetocentric rectilinear incoming hyperbola, mary attractive body (i.e., the Sun), both objects being placed characterized by: incoming V-infinity V_2 , a radial in an otherwise empty Universe. In reality, the gravitational attraction of either departure or target body may significantly alter the entire transfer trajectory.

/ NO-GRAVITY Numerical N-body trajectory integration could be called upon to represent the true physical model for the laws of / SUN SWEEPBACK A % IR -":_ _ t motion, but would be too costly, considering the number of EAR- 'L - Voo2 X aT12 I r complete trajectories required to fully search and describe a . ra RECTI'LINEAR given mission opportunity. R2 HYPERBOLA,

The pseudostate theory, first introduced by S. W. Wilson (Ref. 5) and modified to solve the three-body Lambert prob- lem by D. V. Byrnes (Ref. 6), represents an extremely useful // PLANET LAMBERT CONIC - improvement over the standard Lambert solution. For the giant planet missions, it can correct about 95 percent of the AT12 //r_ = R_- R 2 three-body errors incurred, e.g., up to 30 days in flight time on a typical Jupiter-bound journey. Fig. 3. Pseudostate transfer geometry

3 target planet impact, and a trip-time ATI2 from ra payload mass. A "launch period" measured in days or even to periapse, which can be satisfied by iteration on the weeks, is thus definable: on any day within its confines the r_-magnitude and thus also on R_. capability of a given launch/injection vehicle must equal or exceed the departure energy requirement for a specified This last aspect provides for a great simplification of the payload weight. formulation as the R 2 end-point locus now moves only along the V,, 2 [1 r_ vector direction. The resulting reduction in com- In the course of time these minimum departure energy puting cost is significant, and the equivalence to the Lambert opportunities do recur regularly, at "synodic period" intervals point-to-point conic transfer model is attractive. reflecting a repetition of the relative angular geometry of the two planets. If 6oI and 6o2 are the orbital angular rates of the The first two segments of the transfer associated with the inner and outer of the two planets, respectively, moving about departure planet may be treated in a like manner. If the planet the Sun in circular orbits, then the mutual configuration of is Earth, the pseudostate correction may be disregarded (i.e., the two bodies changes at the following rate: R_ = Ra), or else the duration of Earth's perturbative effect may be reduced to a fraction of ATI2. It can also be set to col2 = coI -to 2 , rad/s (4) equal a fixed quantity, e.g., AT i = 20 days. The latter value was in fact used at the Earth's side of the transfer in the If a period of revolution, P, is det'med as data generation process for this document. 27r P =--, s (5) CO The rectilinear pseudostate method described above and in then Ref. 7 thus involves an iterative procedure, utilizing the 1 1 1 standard Lambert algorithm to obtain a starting set of values for V.. at each end of the transfer arc. This first guess is then = P-T - PS (6) improved by allowing the planetocentric pseudostate position where Ps, the synodic period, is the period of planetary vector r_ to be scaled up and down, using a suitable partial at geometry recurrence, while P1 and P2 are the orbital "sidereal either body, such that z_TREQ, the time required to fall along (i.e., inertial) periods" of the inner (faster) and the outer the rectilinear hyperbola through ra (the sum of the sweep- (slower) planet considered, respectively. back distance V**i × AT i and the planetocentric distance Ir_ I), equal the gravitational duration, AT/. Both r7 and Since planetary orbits are neither exactly circular nor co- V_,i, along which the rectilinear fall occurs, are continuously planar, launch opportunities do not repeat exactly, some reset utilizing the latest values of magnitude and direction of years being better than others in energy requirements or in V_, at each end, i, of the new Lambert transfer arc, as the other parameters. A complete repeat of trajectory character- iteration progresses. The procedure converges rapidly as the istics occurs only when exactly the same orbital geometry of hyperbolic trip time discrepancy, AAT = ATR_.O - ATi, falls departure and arrival body recurs. For negligibly perturbed below a preset small tolerance. Once the V_ vectors at each planets approximately identical inertial positions in space at planet are converged upon, the desired output variables can be departure and arrival imply near-recurrence of transfer tra- generated and contour plotted by existing standard algorithms. jectory characteristics. Such events can rigorously be assessed only for nearly resonant nonprecessing planetary orbits, i.e., As mentioned before, the pseudostate method was found for those whose periods can be related in terms of integer to be unnecessary for the accuracy of this Mars-oriented hand- fractions. For instance, if five revolutions of one body cor- book. The simpler Lambert method was used instead in the respond to three revolutions of the other, that time interval data-generation process. would constitute the "period of repeated characteristics." Near-integer ratios provide nearly repetitive configurations III. Trajectory Characteristics with respect to the lines of apsides and nodes. The Earth- relative synodic period of Mars is 779.935 days, i.e. about A. Mission Space 2.14 years. Each cycle of 7 consecutive Martian mission All realistic launch and injection vehicles are energy-limited opportunities amounts to 5459.55 days and is nearly repeti- and impose very stringent constraints on the interplanetary tive, driven by 8 Martian sidereal periods of 686.9804 days. It mission selection process. Only those transfer opportunities is obvious that for an identical mutual angular geometry Mars which occur near the times of a minimum Earth departure would be found short of its inertial position in the previous energy requirement are thus of practical interest. On either cycle by 36.3 days worth of motion (19.02 deg), while Earth side of such an optimal date, departure energy increases, first would have completed 5459.55/365.25 = 14.947 revolutions, slowly, followed by a rapid increase, thus requiring either a being short of the old mark by the same angular amount as greater launch capability, or alternatively a lower allowable Saturn. ORIGII',IALPAGE |_ OF poOR QUALITY

A much closer repetition of characteristics occurs at 17 full For this document, a 160-day departure date coverage span Martian revolutions (11678.667 days), after 15 Mars-departure was selected, primarily in order to encompass launch energy opportunities (11699.031 days = 32.0302 years), showing requirements of up to a C 3 = 50 km2/s 2 contour, where an inertial position excess of only 10.7 degs., 20.4 days C a = V 2 i.e., twice the injection energy per unit mass. worth of Mars motion beyond the original design arrival E I . Arrival date coverage was set at 400 days to point. display missions from 100--400 days of flight time. The matrix of departure and arrival dates to be presented comprises the "mission space" for each departure opportunity. A variety of considerations force the realistic launch period not to occur at the minimum energy combination of depar- B. Transfer Trajectory ture and arrival dates. readiness status, proce- dure slippage, weather anomalies, multiple launch strategies, As previously stated, each pair of departure/arrival dates arrival characteristics-all cause the launch or, more generally, specifies a unique transfer trajectory. Each such point in the the departure period to be extended over a number of days or mission space has associated with it an array of descriptive weeks and not necessarily centered on the minimum energy variables. Departure energy, characterized by C a , is by far the date. most significant among these parameters. It increases towards

EARTH - MARS 1990 , CSL TFL

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Fig. 4. Mission space in departure/arrival date coordinates, typical example ORIGINAL PAGE Illl OF POOR QUALITY the edges of the mission space, but it also experiences a dra- out of the ecliptic and forces a polar 180-deg transfer, in order matic rise along a "ridge," passing diagonally from lower left to pick up the target's vertical out-of-plane displacement. to upper right across the mission space (Fig. 4). This distur- bance is associated with all diametric, i.e., near-180-deg, The obvious sole exception to this rule is the nodal transfer transfer trajectories (Fig. 5). mission, where departure occurs at one node of the target planet orbit plane with the ecliptic, whereas arrival occurs at In 3-dimensional space the fact that all planetary orbits are the opposite such node. In these special cases, which recur not strictly coplanar causes such diametric transfer arcs to every half of the repeatability cycle, discussed in the preceding require high ecliptic inclinations, culminating in a polar flight paragraph, the transfer trajectory plane is indeterminate and path for an exact 180-deg ecliptic longitude increment be- may as well lie in the departure planet's orbit plane, thus tween departure and arrival points. The reason for this requiring a lesser departure energy (Fig. 6). The opposite strat- behavior is, as shown in Fig. 5, that the Sun and both trajec- egy (i.e., a transfer in the arrival planet's orbit plane) may be tory end points must lie in a single plane, while they are also preferred if arrival energy, V.,2, is to be minimized (Fig. 7). lining up along the same diameter across the ecliptic. The slightest target planet causes a deviation It should be noted that nodal transfers, being associated with a specific Mars arrival date, may show up in the data on = TRUE TRANSFER several consecutive Martian mission opportunity graphs (e.g., ANGLE 1992-1998), at points corresponding to the particular nodal -_1 = TRANSFER ANGLE arrival date. Their mission space position moves, from oppor- POLAR IN ECLIPTIC PLANE tunity to opportunity, along the 180-deg transfer ridge, by TRANSFER _ PLANET gradually sliding towards shorter trip times and earlier relative / ORB,T departure dates. Only one of these opportunities would occur at or near the minimum departure energy or the minimum

TARGET arrival V date, which require a near-perihelion to near- PLANET aphelion transfer trajectory. These pseudo-Hohmann nodal POSITIONS transfer opportunities provide significant energy advantages, f(_l ) but represent singularities, i.e., single-time-point missions, with extremely high error sensitivities. Present-day mission ECLIPTIC planning does not allow single fixed-time departure strategies; however, future operations modes, e.g., space station "on- time" launch, or alternately Earth (repeated) encounter at a specific time, may allow the advantages of a EARTH 41 nodal transfer to be utilized in full.

Fig. 5. Effect of transfer angle upon inclination of The 180-deg transfer ridge subdivides the mission space trajectory arc into two basic regions: the Type I trajectory space below the ridge, exhibiting less than 180-deg transfer arcs, and the .PLANET ORBIT Type II space whose transfers are longer than 180 deg. In PLANE TRANSFER general the first type also provides shorter trip times.

__,_V_ 2 MIN.) Trajectories of both types are further subdivided in two parts-Classes 1 and 2. These are separated, generally hori- PLANET AT NODE _-J ------N AT ARRIVAL _ _- -._ zontally, by a boundary representing the locus of lowest C3 energy for each departure date. Classes separate longer duration missions from shorter ones within each type. Type I, Class 1 missions could thus be preferred because of their t SUN 7_ _ TECLIPTIC shorter trip times. _ /// ,_ TRANSFER Transfer energies become extremely high for very short _ (M_N.CaZl trip times, infinite if launch date equals arrival date, and of EARTH AT NODE course, meaningless for negative trip times. AT DEPARTURE

The reason that high-inclination transfers, as found along Fig. 6. Nodal transfer geometry the ridge, also require such high energy expenditures at depar- 98/0_tE2

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Fig. 7. Missionspace with nodal transfer ture is that the spacecraft velocity vector due to the Earth's orbital velocity must be rotated through large angles out of the ecliptic in addition to the need to acquire the required transfer C3NODAL = 172 - 1 _ 7.83 km2/s 2 trajectory energy. The value of C3 on the ridge is large but (8) finite; its saddle point minimum value occurs for a pseudo- Hohmann (i.e., perihelion to aphelion) polar transfer, requiring This is the lowest value of Ca required to fly from Earth to Mars, assuming circular planetary orbits. 2ap Ca = V2E \l+ap + 1] _ 1950km2/s 2 (7) Arrival V-infinity, V_A, is at its lowest when the transfer trajectory is near-coplanar and tangential to the target planet where Ve = 29.766 kmts, the Earth's heliocentric orbital orbit at arrival. velocity, and ap = 1.49 AU, Mars's (the arrival planet's) semi- major axis. By a similar estimate, it can be shown that for a Both C3 and V= A near the ridge can be significantly low- true nodal pseudo-Hohmann transfer, the minimum energy ered if deep-space deterministic maneuvers are introduced into required would reduce to the mission. The "broken-plane" maneuvers are a category of ,jRIGINAL PAGE iS OF poOR QUALITY such ridge-counteracting measures, which can nearly eliminate POLAR all vestiges of the near-180-deg transfer difficulties. TRANSFER

The basic principle employed in broken-plane transfers is to avoid high ecliptic inclinations of the trajectory by performing a plane change maneuver in the general vicinity of the halfway point, such that it would correct the spacecraft's aim toward the target planet's out-of-ecliptic position (Fig. 8).

Graphical data can be presented for this type of mission, but it requires an optimization of the sum of critical AV expenditures. The decision on which AVs should be included must be based on some knowledge of overall staging and arrival intentions, e.g., departure injection and arrival orbit insertion vehicle capabilities and geometric constraints or objectives ECLIPTIC __ili_____L/_ O pT i M A L contemplated. As an illustration, a sketch of resulting contours EARTH BROKEN- PLANE of C 3 is shown in Fig. 9 for a typical broken-plane oppor- TRANSFER tunity represented as a narrow strip covering the riuge area (Class 2 of Type I and Class 1 of Type II) on a nominal 1990 Fig. 8. Broken-planetransfergeometw

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O

48460.0 0 0 0

48420.0

483_0,0 --

48340.0

'48300.0

'48260.0 I 482200

48040 0 48080 0 48120.0 481_0.0 48200.0 LAUNCH J0-2400000,0

Fig. 9. Sketch of mission space with broken-plane transfer, effective energy requirements ORZGINAL PAGE Ig OF POOR QUALITY launch/arrival date C a contour plot. The deep-space maneuver magnitude Ca = [V=i 2, called out as C3L in the plots and AVBp is transformed into a C 3 equivalent by converting the representing twice the kinetic energy (per kilogram of injected new broken-plane C3B P to an injection velocity at parking mass) which must be matched by launch vehicle capabilities, orbit altitude, adding AVBe, and converting the sum to a new and the V-infinity direction with respect to the inertial Earth and slightly larger C3 value at each point on the strip. Mean Equator and equinox of 1950.0 (EME50) : the declination (i.e., ) of the outgoing asymp- C. Launch/Injection Geometry tote 5, (called DLA), and its right ascension (i.e., equatorial east longitude from vernal equinox, T) a** (or RLA). These The primary problem in departure trajectory design is to three quantities are contour-plotted in the handbook data pre- match the mission-required outgoing V-infinity vector, V, to sented in this volume. the specified launch site location on the rotating Earth. The

site is defmed by its geocentric latitude, 0z, and geographic 1. Launch azimuth problem. The first requirement to be east longitude, Xz. (Figs. 10 and 11). met by the trajectory analyst is to establish the orientation of the ascent trajectory plane (Ref. 8). In its simplest form this Range safety considerations prohibit overflight of popu- plane must contain the outgoing V-infinity (DLA, RLA) lated or coastal areas by the ascending launch vehicle. For each vector, the center of Earth, and the launch site at lift-off launch site (e.g., Kennedy Space Center, Western Test Range, (Fig. 10). As the launch site partakes in the sidereal rotation of or Guiana Space Center), a sector of allowed azimuth firing the Earth, the continuously changing ascent plane manifests directions EL is deffmed (measured in the site's local horizontal itself in a monotonic increase of the launch azimuth, Ez., with plane, clockwise from ). For each launch vehicle, the allowed sector may be further constrained by other safety lift-off time, tL, (or its angular counterpart, c_**- az. , measured in the equator plane): considerations, such as spent stage impact locations down the range and/or down-range significant event tracking capabilities. cos q_LX tan 5= - sin _L X cos (_= - eL) cotan Et. The outgoing V-infinity vector is a slowly varying function sin (_ - %) of departure and arrival date and may be considered constant for a given day of launch. It is usually specified by its energy (9)

ASYMPTOTE N

LAUNCH FLIGHT PATH MERIDIAN_

V_, DEPARTURE ASYMPTOTE

LAUNCH

_=0

VERNAL EQUINOX EQUATOR

TRAJECTORY PLANE

Fig. 10. Launch/injection trajectory plane geometry ORIGINALPAGE 19 OF POOR QUALITY'

The daily time history of azimuth can be obtained from Eq. GREENWICH (9) for a given _=, 8, departure asymptote direction by MERIDIAN following quadrant rules explained below and using the Oh GMT A GREENWICH following approximate expressions (see Fig. 11): EAST LONGITUDE OF'_, ((;HA) LAUNCH SITE _'L/ _ \ ClL = _L+GHADATE+(_EARTHX tL (I0)

I __Jl,_ _ ] VERNAL GHADATE = 100.075+0.9856123008Xdso (11) ( C'O'f/'._ll_L-e, X ] EQUINOX LAUNCH SITE_ • - / -- _)" "_J. where

/ _ DEPARTURE e L = Right ascension of launch site (¢z,' XL) at ANGLE TRAVERSED _ ASYMPTOTE tL (deg) BY LAUNCH SITE _ MERIDIAN SINCE Oh GMT GHA DA TE = Greenwich hour angle at 0n GMT of any OF LAUNCH DATE LAUNCH SITE date, the eastward angle between vernal AT LAUNCH, tL (GMT) equinox and the Greenwich meridian (deg), assumes equator is EME50.0 Fig. 11. Earth equator plane definition of angles involved t"OEAR TH = sidereal (inertial) rotation rate of Earth in the launch problem (15.041067179 deg/h of mean solar time)

dso = launch date in terms of full integer days tive) and 360.0 deg in the fourth quadrant (when cotan Z elapsed since 0 a Jan. 1, 1950 (days) is negative).

tL = lift-off time (h, GMT, i.e., mean solar time) A generalized plot of relative launch time tRL T VS launch azimuth Z L can be constructed based on Eqs. (9) and (12), if a fLxed launch site latitude is adopted (e.g., ¢L = 28.3 deg A relative launch time, [RLT, measured with respect to an inertial reference (the departure asymptote meridian's right for Kennedy Space Flight Center at Cape Canaveral, Florida). ascention a**), can be defined as a sidereal time (Earth's Such a plot is presented in Fig. 12 with departure asymptote rotation rate is 15.0 deg/h of sidereal time, exactly): declination 5=. as the contour parameter. The plot is applicable to any realistic departure condition, independent of _., date, or true launch time tL (Ref. 9). t_ - t_L tRL T = 24.0 15.0 , h (12) 2. Daily launch windows. Inspection of Fig. 12 indicates This time represents a generalized sidereal time of launch, that generally two contours exist for each declination value elapsed since the launch site last passed the departure asymp- (e.g., $. = -10 deg), one occurring at tRL T during the a.m. tote meridian, a**. hours, the other in the p.m. hours of the asymptote relative "day."

The actual Greenwich Mean (solar) Time (GMT) of launch, Since lift-off times are bounded by preselected launch-site- tL, may be obtained from tRL r by transforming it to mean solar time and adding a date, site, and asymptote-dependent dependent limiting values of launch azimuth _;L (e.g., 70 deg and 115 deg), each of the two declination contours thus con- adjustment : tains a segment during which launch is permissible-"a launch window." The two segments on the plot do define the two tRLTX 15.0 a** -GHADATE -_L (EAST) available daily launch windows. tL - + (13) ¢OEARTH Ca)EARTH As can be seen from Fig. 12, for 5.. = O, the two daily The expression for Z L (Eq. 9) must be used with computa- launch opportunities are separated by exactly 12 hours; with tional regard for quadrants, singular points, and sign conven- an increasing 15o,I they close in on each other, until at 15_ I = tions. If the launch is known to be direct (i.e., eastward), I_LI they merge into a single daily opportunity. For 18=1 > then Y'L' when cotan Z L is negative, must be corrected to I_zl, a "split" of that single launch window occurs, disallow- Z L = ZLNEG + 180.0. For retrograde (westward) launches, ing an ever-increasing sector of azimuth values. This sector is 180.0 deg must be added in the third (when cotan Z is posi- symmetric about east and its limits can be determined from lO ORIGINAL PACE l_t OF POOR QUALITY.

24

22

20

J_ 18

LU 16

-I- (o 14 z

J 12

F.- < .J 10 uJ n.- uJ Fm 0 8

C6

>- u') <

95 100 105 110 115 120

Fig. 12. Generalized relative launch time tRL T vs launch azimuth T-L and departure asymptote declination _**. Pair of typical example launch windows for $** = -10 deg shown by bold curve segments (reproduced from Ref. 9).

cos _= The angular equatorial distance between the ascending node = _+- (14) and the launch site meridian is given by sin _,LLIMIT COS _bL

As 5.. gets longer, the sector of unavailable launch azimuths sin _= X sin Y_L reaches the safety boundaries of permissible launches, and sin (a L - £2) = sin i (16) planar launch ceases to exist (Fig. 13). This subject will be Quadrant rules for this equation involve the observation that a addressed again in the discussion of "dogleg" ascents. negative cos E L places (a L - f2) into the second or third quad- Figure 14 is a sketch of a typical daily launch geometry rant, while the sign of sin (% - _2) determines the choice situation, shown upon a Mercator of the . between them. The two launch windows exhibit a similar geometry since the The range angle 0 is measured in the inertial ascent trajec- inclinations of the ascent trajectory planes are functions of tory plane from the lift-off point at launch all the way to the launch site latitude _L and azimuth Y_Lonly: departure asymptote direction, and can be computed for a given launch time t L or ct.. - aL(tL) and an azimuth Y_L cos i = cos _/_ x sin rz (15) already known from Eq. (9) as follows:

The two daily opportunities do differ greatly, however, in cos 8 = sin a= X sin $z, + cos a=. X cos ¢L X cos (% - %) the right ascension of the ascending mode _2 of the orbit and in the length of the-traversed in-plane arc, the range angle/9. (17)

11 ORIG|NAL PAGE !_1 OF POOR QUALITY

180 sin(% - %) x cos8® sin 8 = sin _L (18) 160

"A.OE',,ET, The extent of range angle O can be anywhere between 140 0 deg and 360 deg, so both cos 0 and sin O may be desired in

._ 120 its determination. The range angle O is related to the equa- torial plane angle, Act = tz=. - t_L, discussed before. Even "I" though the two angles are measured in different planes, they _- 100 LAUNCH both represent the angular distance between launch and depar- m REGION REGION ture asymptote, and hence they traverse the same number of N 80 "//"" r.,,-,,.,,///// < quadrants. "r c.} Z Figure 15 represents a generally applicable plot of central < 00NN..."...... range angle O vs the departure asymptote declination and --1 4O launch azimuth, computed using Eqs. (17) and (18) and a 20 __ JFi -- launch site latitude ¢c = 28.3 (Cape Canaveral). The twin daily launch opportunities are again evident, showing the LIMITING-LAUNCHAZIMUTH significant difference in available range angle when following a 0 I I I J vertical, constant 5=. line. -90 -70 -50 -30 -10 10 30 50 70 9O ASYMPTOTE DECLINATION 6oo , deg It is sometimes convenient to reverse the computational Fig. 13. Permissible regions of azimuth v= asymptote procedure and determine launch azimuth from known range declination launch space for Cape Canaveral angle O and tRLr, i.e., &_ = a= - %, as follows:

ASYMPTOTE RELATIVE LAUNCH TIME tRJLT , h BY DEFINITION 0 3 6 9 12 15 18 I 21 24/t_,T = oh= 24 h AT "'_ ...... LAU_ICH __:":':"" ...... -40 I I 2ND WINDOW Moo - MERIDIAN I_'L° I AZIMUTH I I AT I I _-,Lc = 110 ° ./ DAI LY TRACK _r OF LAUNCH SITE AT _L = 28"3o 1ST WINDOW "O- 2ND WINDOY LU OPENS AT / OPENS AT E3 Z/,, = 70 ° /1ST WINDOW -10 - -o / CLOSES AT _,Lo = 70 °

< //EASTER/Y _L = 110 ° ..J 0 _ LAUNCH -J < _1 ZL = 90 ° (_ SUN ASSUMED V,_, E 10 - (ASSUMED) 0 ASYMPTOTE 600 = - 15 ° LOCATION: < 2O - 6o0 = -15 ° O LU (x=o = 315 °

3O

eL 1 40 O(°° - _Z, 1 = 2850 3(°° 1 j--1 1 1 I 1 I -45 0 45 90 135 180 225 270 315 R I GHT ASCENSION ct, deg (EARTH EQUATORIAL CELESTIAL LONGITUDE)

Fig. 14. Typical launch geometry example in celestial (inertial) Mercator coordinates

12 ORIGINAL PAGE kill OF POOR QUALITY

400, D B

360

D

320 m _:,- 6od \ -_

m 280

"0 c_" _j\ _L=,2o 240

z < (_i" / .i.,f_J \ o='=='°" I \ (/.__ _.00,. I NOTF: _L IS LAUNCH AZIMUTH 200 z < rr o,os wo, -- 160

rr" " t i I I- Z tu 120 I ' 80

m

4O

,,,,,,,,,illilili ililillililililii' 'ilillili:lliihiliiilllilllllllliill lil__lllililil" liiliiiliiiililil 0 T,i,l,i,o -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80

DEPARTURE ASYMPTOTE DECLINATION (5oo, deg

Fig. 15. Central range angle 0 between launch site and outgoing asymptote direction vs its declination and launch azimuth. Pair of typical example launch windows for 5= = -10 deg shown by bold line segments (reproduced from Ref. 8).

cos _® X sin (a= - %) with the lighting conditions at lift-off and consequently allows sin Y'L = sin 0 (19) a lighting profile analysis along the entire ascent arc.

The angle ZALS, displayed in Fig. 16, is defined as the angle between the departure V= vector and the Sun-to-Earth sin 5=. - cos 8 X sin Cz, direction vector. It allows some judgment on available ascent cos zz. = sin O × cos_z. (20) lighting.

Figure 16 displays a 3-dimensional spatial view of the same The length of the range angle required exhibits a complex behavior-the first launch window of the example in Figs. 14 typical launch geometry example shown previously in map format in Fig. 14. The difference in available range angles as and 16 offers a longer range angle than the second, but the well as orientation of the trajectory planes for the two daily second launch window opens up with a range angle so short that into orbit is barely possible. Further launch launch opportunities clearly stands out. In addition, the figure illustrates the relationship between the "first" and "second delay shortens the range even further, forcing the acceptance daily" launch windows, defined in asymptote-relative time, of a very long coast (one full additional revolution in ) before transplanetary departure injection. A detailed tRLT, as contrasted with "morning" or "night" launches, defined in launch-site-local solar time. The latter is associated analysis of required arc lengths for the various sub-arcs of the

13 URIGINAL PAGE !8 OF POOR QUALITY

SUN W l N DOW (ASSUMED) CLOSES _/_, = 110 ° TRANSPLANETARY INJECTION CIRCLE

FIRST "DAILY" (tR], T ) • LAUNCH WINDOW SECOND "DAILY" OPENS (NOON LAUNCH) ) LAUNCH DOW (EVENING LAUNCH) WINDOW ]_L = 700

EARTH LAUNCH SITE _L = CONST.

SECOND DEPARTURE

ZALS SUN-TO-EARTH V ECTO R

FIRST OUTGOING DEPARTURE ASYMPTOTE

DEPARTURE -TRAJECTORY ENVELOPE

Fig. 16. Typical example of daily launch geometry (3-dimensional) as viewed by an outside observer ahead of the spacecraft

14 ORIGINAL PAGE rg OF POOR QUALITY

BOOSTER AS( ,LAUNCH SITE

/ PARKING ORBIT. /

COAST ARC _X_/// i I

I

FINAL STAGE BURN _._ _.

GEE OF

/

81 BURNING ARC OF BOOSTER VEHICLES INTO PARKING ORBIT

82 BURNING ARC OF FINAL STAGE THRUST e RANGE ANGLE BETWEEN LAUNCH SITE AND DEPARTURE RADIAL ASYMPTOTE _=0 ANGLE BETWEEN PERIGEE AND DEPARTURE RADIAL ASYMPTOTE u/ OF INJECTION

Fig. 17. Basic geometry of the launch and ascent profile in the trajectory plane (after Ref. 9)

r departure trajectory is thus a trajectory design effort of para- p = periapse , typically 6563 krn, for a horizontal mount importance (Fig. 17_ injection from a 185-kin (100-nmi) parking orbit

3. Range angle arithmetic. For a viable ascent trajectory I_ = GM, gravitational parameter of Earth (refer to the Table of Constants, Section V). design, the range angle 0 must first of all accommodate the twin burn arcs 01 and 02, representing ascent into parking The proper addition of these trajectory sub-arcs also requites orbit and transplanetary injection burn into the departure adjustment for nonhorizontal injection (i.e., for the flight hyperbola. In addition, it must also contain the angle from path angle 7l > 0), especially dgnificant on direct ascent periapse to the V.. direction, called "true anomaly of the miuions (no coast arc) and mt=sions with relatively low asymptote direction" (Fig. 18): thrust/weight ratio injection stages. The adjustment is accom- -1 plished as follows: (21) cos o= = c3 x r O = 01 + OCOAST + #2 + _ - i_l (22) l+_ ue where: where:

15 ORIGINAL PAGZ iS OF POOR QUALITY

140 L ! NOTE: 2 C3 = V_DE P IS TWICE

UJ THE TOTAL ENERGY n- \ PER UNIT MASS I--D 130

W oA ii o __. 120 >-

PERIGEE ALTITUDE ZOO 185 km (100 nmi) "' 110 370 km I- 1200 nmil _

100 I I I I I I I I 25 50 75 100 125 150 175 200 225 C3,km3/s2

Fig. 18. Angle from perigee to departure asymptote

= is the true anomaly of the injection point, usually icant role in the ascent orbit selection. A limit on maximum is near 0 deg, and can be computed by iteration coast duration allowed (fuel boil-off, battery life, guidance using: gyro drift, etc.) may also influence the long/short parking orbit decision. In principle, any number of additional parking orbit en sin _'z revolutions is permissible. Shuttle launches of interplanetary tan 7z - 1 + e/./cos v1 (23) missions (e.g., Galileo) are in fact required to use such addi- tional orbits for cargo bay door opening and payload deploy- 7r = injection flight path angle above local horizontal, ment sequences. In such cases, however, the precessional deg effects of Earth's oblateness upon the parking orbit, prirnarily the regression of the orbital plane, must be considered. en = eccentricity of the departure hyperbola:

4. Parking orbit regression. The average regression of the C3 xr e, = 1 + _ (24) nodes (i.e., the points of spacecraft passage through the equator plane) of a typical direct (prograde) circular parking orbit of 28.3-deg inclination with the Earth's equator, due to To make Eq. (22) balance, the parking orbit coast arc, Earth's oblateness, amounts to about 0.46 deg of westward 8coAs T, must pick up any slack remaining, as shown in nodal motion per revolution and can be approximately com- Fig. 17. A negative OcoAs T implies that the ZL-solution was puted from too short-ranged. A direct ascent with positive injection true anomaly vt (i.e., upward climbing flight path angle, 7i, at 540 °Xr 2 X,/2 Xcosi injection) with attendant sizable gravity losses, may be accept- = s , deg/revolution (25) /,2 able, or even desirable (within limits) for such missions. o Alternately, the other solution for Z L, exhibiting the longer where range angle 0, and thus a longer parking orbit coast, OcoAs T, should be implemented. An extra revolution in parking orbit r = Earth equatorial surface radius, 6378 km $ may be a viable alternative. Other considerations, such as desire for a lightside launch and/or injection, tracking ship ro = radius, typically 6748 km for an location and booster impact constraints, may all play a signif- orbital altitude of 370 km (200 nmi)

16 ORIGINAL PAG.Z E,._ OF POOR QUALITY

J2 = 0.00108263 for Earth 6. Tracking and orientation. As the spacecraft moves away from the Earth along the asymptote, it is seen at a nearly i = parking orbit inclination, deg. Can be computed for constant declination-that of the departure asymptote, 6** a given launch geometry from (DLA in the plotted data). The value of DLA greatly affects tracking coverage by stations located at various ; cos i = cos _t, × sin ZL (26) highest daily spacecraft elevations, and thus best reception, are enjoyed by stations whose latitude is closest to DLA. , using radio doppler data, is adversely affected This correction, multiplied by the orbital stay time of N by DLA's near zero degrees. revolutions, must be considered in determining a biased launch time and, hence, the fight ascension of the launch site at The spacecraft orientation in the first few weeks is often lift-off: determined by a compromise between communication (antenna pointing) and solar heating constraints. The Sun-spacecraft- %EFF = _L + (2 × N , deg (27) Earth (SPE) angle, defined as the angle between the outgoing V-infinity vector and the Sun-to-Earth direction, is very useful and is presented in the plots under the acronym ZALS. 5. Dogleg ascent. Planar ascent has been considered exclu- It was defined in the discussion of Fig. 16. This quantity has _avely, thus far. Reasons for performing a gradual powered many uses: plane change maneuver during ascent may be many. Inability to launch in a required azimuth direction because of launch (I) If the spacecraft is Sun-oriented, ZALS equals the site constraints is the prime reason for desiring a dogleg ascent Earth cone angle (CA), depicted in Fig. 19 (the cone profile. Other reasons may have to do with burn strategies or intercept of an existing orbiter by the ascending spacecraft, especially if its inclination is less than the latitude of the PLANET launch site. Doglegs are usually accomplished by a sequence of out-of-plane yaw turns during first- and second-stage burn, optimized to minimize performance loss and commencing as soon as possible after the early, low-altitude, high aerodynamic pressure phase of flight is completed, or after the necessary \ lateral range angle offset has been achieved.

By contrast, powered plane change maneuvers out of parking orbit or during transplanetary injection are much less efficient, as a much higher velocity vector must nowbe rotated CONE through the same angle, but they may on occasion be opera- ANGLE tionally preferable.

As already discussed, a special geometric situation develops whenever the departure asymptote declination magnitude exceeds the latitude of the launch site, causing a "split azi- muth" daily launch window. Figure 13 shows the effects of asymptote declination and range safety constraints upon the launch problem. As the absolute value of declination increases. it eventually reaches the safety constraint on azimuth, prevent- ing any further planar launches. The situation occurs mostly TO early and/or late in the mission's departure launch period, and OBJECT is frequently associated with dual hunches, when month-long departure periods are desired. (CANOPUS) The Shuttle-era Space Transportation System (STS), includ- hag contemplated upper stages, is capable of executing dogleg operations as well. These would, however, effectively reduce the launch vehicle's payload (or 6"3) capability, as they did on expendable launch vehicles of the past. Fig. 19. Definition of cone and clock angle

17 angle of an object is a spacecraft-fixed coordinate, Applications, Inc. (SAI) study (Ref. 10) points to some of an angle between the vehicle's longitudinal -Z axis and the techniques available, such as passive wait for natural the object direction). alignment of the continuously regressing space station orbit plane (driven by Earth's oblateness) with the required (2) The Sun-phase angle, cbs (phase angle is the Sun-object- V-infinity vector, or the utilization of 2- and 3- impulse man- spacecraft angle) describes the state of the object's euvers, seeking to perform spacecraft plane changes near the disk lighting: a fully lit disk is at zero phase. For the apogee of a phasing orbit where velocity is lowest and thus Earth (and Moon), several days after launch, the sun- turning the orbit is easiest. These two approaches can be phase angle is: combined with each other, as well as with other suitable maneuvers, such as: 6ps = 180- ZALS (28) (1) Deep space propulsive burns for orbit shaping and (3) The contour labeled ZALS = 90 ° separates two cate- phasing, gories of transfer trajectories-those early departures (2) Gravity assist flyby, including Earth return AVEGA, that first cut inside Earth's orbit, thus starting out at negative heliocentric true anomalies for ZALS >90 °, (3) Aerodynamic turns at grazing perigees or at inter- and those later ones that start at positive true anoma- mediate planetary swingbys, and lies, heading out toward Jupiter and never experiencing the increased solar heating at distances of less than (4) Multiple revolution injection bums, requiring several 1 AU for ZALS < 90 °. low, grazing passes, combined with apogee plane change maneuvers, etc.

7. Post-launch spacecraft state. After the spacecraft has departed from the immediate vicinity of Earth (i.e., left the All of these devices can be optimized to permit satisfactory Earth's sphere of influence of about 1-2 million km), it moves orbital launches, as well as to achieve the most desirable condi- on a heliocentric conic, whose initial conditions may be tions at the final arrival body. In general, space launch advan- approximated as tages, such as on-orbit assembly and checkout of payloads and clustered multiple propulsion stages, or orbital construction of Vs/c = VEARTt4 + V. (29) bulky and fragile subsystems (solar panels, sails, antennas, radiators, booms, etc.) will, it is hoped, greatly outweigh the significant deep-space mission penalties incurred because of Rs/c = REART_ + VS/c X At (30) the space station's inherent orbital orientation incompatibility with departure requirements. where R and V of Earth are evaluated from an ephemeris at time of injection and At represents time elapsed since then (in seconds). The V vector in EME50 cartesian coordinates can D. Planetary Arrival Synthesis be constructed using @_3L = V , DLA = 8., and RLA = c_ in three components as follows: The planetary arrival trajectory design problem involves satisfying the project's engineering and science objectives at V = (V × cos_ × cos 8**, V X sina. X cosS., the target body by shaping the arrival trajectory in a suitable manner. As these objectives may be quite diverse, only four illustrative scenarios shall be discussed in this section-flyby, V** × sin 6.*) (31) orbiter, atmospheric probe, and, to a small extent, lander 8. Orbital launch problem. Orbital launch from the Shuttle, missions. from other elements of the STS, or from any temporary or permanent orbital space station complexes, introduces entirely 1. Flyby trajectory design. In this mission mode, the arrival new concepts into the Earth-departure problem. Some of trajectory is not modified in any deterministic way at the the new constraints, already mentioned, limit our ability planet-the original aim point and arrival time are chosen to to launch a given interplanetary mission. The slowly regressing satisfy the largest number of potential objectives, long before- space station orbit (see Eq. 25) generally does not contain hand. the V-inffmity vector required at departure. Orbit lifetime or other considerations may dictate a space station's orbital This process involves the choice of arrival date to ensure altitude that may be too high for an efficient injection burn. desirable characteristics, such as the values of the variables Innovative departure strategies are beginning to emerge, VHP, DAP, ZAPS, etc., presented in plotted form in the data attempting to alleviate these problems - a recent Science section of this volume.

18 ORIG,,_AL PAGE l_ OF POOR QUALITY

DAP, the planet-equatorial declination, 6**, of the incoming 1 1 cos p - - (33) asymptote, i.e., of the V-infinity vector, provides the measure e V: r ** p of the minimum possible inclination of flyby. Its negative is l+-- also known as the latitude of vertical impact (LVI). /ap

The magnitude of V-infinity, VHP = IV._L, enables one to VHP also enables the designer to evaluate planetocentric control the flyby turn angle A_ between the incoming and velocity, V, at any distance, r, on the flyby hyperbola: outgoing V** vectors by a suitable choice of closest approach

(C/A) radius, rp (see Fig. 20): 2 _p V: V = x/--- 7- + ** ,km/s (34)

A_ = 180-2p, deg (32) In the above equations,/ap (or GMp), is the gravitational param- where p, the asymptote half-angle, is found from: eter of the arrival body.

PLANET EQUATOR

PLANET VERNAL EQUINOX

OUTBOUND ASYMPTOTE V_IN B

\ \

L VI = -

ZAPE i / \

VERTICAL IMPACT TURN ANGLE POINT = 180-2p SUN \ \ EARTH \ ='RIAPSE _2

)ING NODE

INBOUND ASYMPTOTE

Fig. 20. Planetary flyby geometry

19 ORIGINAL PAGE IS OF POOR QUALITY

Another pair of significant variables on which to base TARGET B-PLANE, arrival date selection are ZAPS and ZAPE-the angles between 1 TO INCOMING PARALLEL TO F-infinity and the planet-to-Sun and -Earth vectors, respec- tively. These two angles represent the cone angle (CA) of the planet during the far-encounter phase for a Sun- or an Earth- oriented spacecraft, in that order. ZAPS also determines the phase angle, q_s, of the planet's solar illumination, as seen by the spacecraft on its far-encounter approach leg to the planet:

_'s--180- zAes (3s) I ..,..% OJ"

Both the cone angle and the phase angle have already been defined and discussed in the Earth departure section above. \ - " -TjC' B IHYPERBouc The flyby itself is specified by the aim point chosen upon \ _ W I PATH OF the arrival planet target plane. This plane, often referred to as the B-plane, is a highly useful aim point design 'tool. It is a plane passed through the center of a celestial body normal to V, the relative spacecraft incoming velocity vector at infinity. / .... The incoming asymptote, i.e., the straight-line, zero-gravity extension of the V**-vector, penetrates the B-plane at the aim V_ "TRAJECTORY PLANE point. This point, defined by the target vector B in the B-plane, is often described by its two components B • T and B • P,, where the axes T and R form an orthogonal set with V. The B MISS PARAMETER, B_ T-axis is chosen to be parallel to a fundamental plane, usually (TARGET VECTOR) the ecliptic (Fig. 21) or alternatively, the planet's equator. The 0 AIM POINT ORIENTATION magnitude of B equals the semi-minor axis of the flyby hyper- '_ PARALLEL TO INCOMING ASYMPTOTE, Moo bola, b, and can be related to the closest approach distance, T PARALLEL TO ECLIPTIC PLANE also referred to as the periapse radius, rp, by AND ±TOS" R =SXT

Fig. 21. Definition of target or arrival B-plane coordinates IB[ Vi,((1 + v:r):**#pv - 1 )l,:, km (36) or The two systems of B-plane T-axis definition can be recon- ciled by a planar rotation. -AO, between the ecliptic T- and R-axes and the T'- and _,'-axis orientations of the equator % = ''_'**V + \_V_] + IBl2 ,km (37) based system

The direction angle 8 of the B-vector, B, measured in the sin(% - %p) target plane clockwise from the T-axis to the B-vector position tan AO = It can easily be related to the inclination, i, of the flyby trajec- cos 5= × tan tSEp - sin 5= X cos (a - aEp) tory, provided that both 5** (DAP) and the T-axis, from which (39) 0 is measured clockwise, are defined with respect to the same fundamental plane to which the inclination is desired. For a where system based on the planet equator (Fig. 22): aEp and 5Ep are right ascension and declination of the cos%e = cosoee¢ × cos5 (38) ecliptic pole in planet equatorial coordinates. For Mars using **PE Q constants in Section V: which assumes that Ovv,Q is computed with the T-axis parallel o_£e= 267.6227, BE? = 63.2838, deg to the planet equator (i.e., TeE Q = V** × POLEeEQ), not the ecliptic, as is frequently assumed (TEc L = V × POLEEcL). a= and 5** are RAP and DAP, the directions of incoming Care must be taken to use the 0 angle as dofined and intended. V, also in planet equatorial coordinates. ORIGINAL PAGE I_l OF POOR QUALITY

ECLIPTIC POLE (EP) PLrCL PEQ POLE DECLINATION PPE, OF EP, _EP

PEQ RIGHT ASCENSION OF EP,

- 90 °

ECLIPTIC PLANE (ECL)

PLANET EQUATOR (PEQ) o_,_- 180 B-PLANE(±TOV=) TECL = V=X Pecl.

TpE Q = VooX PPEQ Ao¢ = 90 - [(aEp - 90) - (%0 - 180)] = o¢_ - aEp

A0= ARCTAN [SIN Aot/(COS _5,_x TAN 6Ep - SIN 5,_ x COS An,)]

Fig. 22. Two T-axis definitions in the arrival B-plane

The correction dx0 is applied to a 0 angle computed in the periapse, at the entry point of a probe, or generally at any ecliptic system as follows (Fig. 22): position r (subscript S = Sun, could be replaced by E = Earth, if desired), (see Fig. 24):

OpE Q = OEC L + AO (40) cos '_s = -cos flo. X cos ZAP s - sin 13®X sin ZAP S A The ecliptic T, RECL axes, however, have to be rotated by -A0 (clockwise direction is positive in the B-plane) to obtain × cos (ETsP - Os/c) (41) planet equatorial T, RPEQ coordinate axes. The B-magnitude of an aim point in either system is the same. where

The projections of the Sun-to-planet and Earth-to-planet Os/C = the aim point angle in the B-plane, must be with vectors into the B-plane represent aim point loci of diametric respect to the same T, as ETSP. Sun and Earth occultations, respectively., as defined in Fig. 23. /3® = the arrival range angle from infinity (a position far The B-plane 0-angles (with respect to TF.CL axis) of these vari- out on the incoming asymptote) to the point of ables are presented and labeled ETSP and ETEP, respectively, interest r (Fig. 25). in the plotted mission data. In addition to helping design or else avoid diametric occultations, these quantities allow The computation of the arrival range angle, t3=, to the computation of phase angles, _i, of the planet at the spacecraft position of the desired event depends on its type, as follows:

21 ORIGINAL P,_QE I_ OF POOR QUALITY

SUN (OR EARTH) TO ARRIVAL l. At flyby periapse, (3. ;) 90 deg): PLANET B-PLANE -I cos #= = cos (-v=) - (42) V 2 r == p l+-- SHADOW gp

(SUN OR (rp is periapse radius). EARTH) / . At given radius r, anywhere on the flyby trajectory / / (see Fig. 25):

3= = -v + v (43) PLANET '(OR v= should be computed from periapse equation, I I Eq. (42) v r at r can be obtained from (-v= < v r _; + I I v=, v r has negative values on the incoming branch):

ETSP T r 2+ p = -1 (OR ETEP) r V2 ) r /./p cos v = (44)

(1 +rpV----_2_/

TO SUN 3. At the entry point having a specified flight path angle (OR EARTH) R "rE (Fig. 25):

Fig. 23. Definition of approach orientational coordinates ZAPS and ETSP, ZAPE and ETEP _. = -v,. + vE (45)

PECL / /x / \

I v=

INCOMING ASYMPTOTE _)s = 90 °

180-ZAPS

/ r, RADIUS TO A POINT ON TO SUN TRAJECTORY PHASE ANGLE, dab PROJECTED ONTO TRAJECTORY ARC, _]_ RECL UNIT-SPHERE

A=ETSP- 180 °- Fig. 24. Phase angle geometry at arrival planet

22 OF POOR QUALI,'_

ARBITRARY TRAJECTORY OUTGOING POINT AT RADIUS r ASYMPTOTE

AND TRUE ANOMALY vr_

FICTITIOUS ENTRY PERIAPSE

RADIUS, rp / / FLYBY PERIAPSE

PLANETARY RADIUS, rpF ENTRY ENTRY INTERFACE

TURN ANGLE = 180-2p

VERTICAL IMPACT

ENTRY POINT AT RADIUS

rE = rSURF + h E , AND FLIGHT PATH ANGLE _/'E FLYBY

INCOMING ASYMPTOTE

ENTRY TRAJECTORY

TRAJECTORY

IBFI bE FLYBY MISS PARAMETER

Fig. 25. Typical entry and flyby trajectory geometry

23 ORIGINAL PAGE I_ OF POOR QUALITY

where vE is the true anomaly at entry, should always occultations, require close control of arrival time. The number be negative, and can be computed if entry radius and of satellites passed at various distances also depends on the altitude, rE = rsuRP + hE and 7E, the entry angle, are time of planet C/A. These fine adjustments do, however, known: demand arrival time accuracies substantially in excess of those provided by the computational algorithm used in this effort, r 2+ -1 which generated the subject data (accuracies of 1-5 min for rE Up events or 1-2 h for encounters would be required vs uncertain- cos v_. = (46) ties of up to 1.5 days actually obtained with the rectilinear impact pseudo-state theorem). Numerically searched-in inte- (l+----_prpV2 ]t grated trajectories, based on the information presented as a first guess input, are mandatory for such precision trajectory whereas the fictitious periapse radius rp for the entry, work. to satisfy 7E at rE, is equal to Preliminary design considerations for penetrating, grazing, or avoiding a host of planet-centered fields and particle struc-

¥ tures, such as magnetic fields, radiation belts, plasma tori, ring P I,>E._ Iand debris structures, occultations by Sun, Earth, , or satellites, etc., can all be presented on specialized plots, e.g., (47) the B-plane, and do affect the choice of suitable aim point and arrival time. All of these studies require the propagation The B value corresponding to this entry point can be com- of a number of flyby trajectories. Adequate initial conditions puted from Eq. (36), while the 0 angle in the B-plane would for such efforts can be found in the handbook as: VHP (V) depend on the desired entry latitude inclination, Eq. (38), or and DAP (8..) already defined, as well as RAP (a), the phase angle, Eq. (41). planet equatorial right ascension of the incoming asymptote (i.e., its east longitude from the ascending node of the planet's The general flyby problem poses the least stringent con- mean orbital plane on its mean equator, both of date). The straints on a planetary encounter mission, thus allowing designer's choice of the aim point vector, either as B and 0, or optimization choices from a large list of secondary parameters, as cartesian B - T and B • R, completes the input set. Suitable such as satellite viewing and occultation, planetary fields and programs generally exist to process this information. particle in situ measurements, special phase.angle effects, etc. 2. Capture orbit design. The capture problem usually A review of the plotted handbook variables, required in the involves the task of determining what kind of spacecraft orbit phase-angle equation (Eq. 41), shows that the greatest magni- is most desired and the interconnected problem of how and at tude variations are experienced by the ZAPS angle, which is what cost such an orbit may be achieved. A scale of varying strongly flight-time dependent: the longer the trip, the smaller complexity may be associated with the effort envisioned-an ZAPS. For low equatorial inclination, direct flyby orbits, this elliptical long period orbit with no specific orientation at the implies a steady move of the periapse towards the lit side and, trivial end of the scale, through orbits of controlled or opti- eventually, to nearly subsolar periapses for long missions. This mized lines of apsides (i.e., periapse location), nodes, inclina- also implies that on such flights the approach legs of the tra- tion, or a safe perturbed orbital altitude. Satellite G/A-aided jectory are facing the morning terminator or even the dark capture, followed by a satellite tour, involving multiple satel- side, as trip time becomes longer, exhibiting large phase angles lite G/A encounters on a number of revolutions, each designed (recall that phase is the supplement of the ZAPS angle on to achieve specific goals, probably rates as the most complex the approach leg). This important variation is caused by a grad- capture orbit class. Some orbits are energeticaUy very difficult ual shift of the incoming approach direction, as flight time in- to achieve, such as close circular orbits, but all require signifi- creases, from the subsolar part of the target planet's leading cant expenditures of fuel. As maneuvers form the background hemisphere (in the sense of its orbital motion) to its antisolar to this subject a number of useful orbit design concepts shall part. be presented to enable even an unprepared user to experiment with the data presented. The arrival time choice on a very fine scale may greatly depend on the desire to observe specific atmospheric/surface The simplest and most efficient mode of orbit injection is features or to achieve close encounters with specific satellites a coplanar burn at a common periapse of the arrival hyperbola of the arrival planet. Passages through special satellite event and the resulting capture orbit (Fig. 26). The maneuver A V zones, e.g., flux tubes, wakes, geocentric and/or heliocentric required is:

24 ORIGINAL PAGE i_ OF POOR QUALITY

A plot of orbit insertion AV required as a function of rp and AV = V_=+ -- - , km/s (48) P (using Eq. 50) is presented in Fig. 27. The apoapse radius of J 2/Zp J_/_p X ra rp % + r) such an orbit of given period would be

The for such an orbit, requiring knowledge 2#p XP 2 (51) of periapse and apoapse radii, rp and ra, is rA = 3 rt2 rp , km X/

An evaluation of Eq. (50) (and Fig. 27) shows that lowest (49) P = 2_" IFL--"Y--_ J , s orbit insertion AV is obtained for the lowest value of rp, the longest period P, and the lowest V. of arrival. If on the other hand, a known orbit period P (in seconds) is desired, the expression for AV is Of some interest is injection into circular capture orbits, a special case of the coapsidal insertion problem. It can be shown (Ref. 9) that an optimal A V exists for insertion into 2/_p 2/ap 2/_pX capture orbits of constant eccentricity, including e = 0, i.e., av = /v: + -- - / j(e ,(5o) rp rp circular orbits, which would require a specific radius:

COAPS I DA L COTANGENTIAL CAPTU R E CAPTU R E ORBIT ORBIT

v. / ASYMPTOTE

\ / _rCA \ ,,_\ APPROACH \v H \ HYPERBOLA ./" TT T'

COTANGENTIAL / ORBIT PERIAPSE _VE_- \ \ COAPSIDAL BURN AND PERIAPSE OF HYPERBOLIC PERIAPSE \ ITS CAPTURE ORBIT

Fig. 26. Coapsidal and cotangential capture orbit insertion geometries

25 ORIGINAL PACI_ I_ OF POOR QUALITY

I A60 , one can solve for the hyperbolic periapse r and the I I I I P CA MARS SURFACE-SYNCHRONOUS bum radius r using selected values of true anomaly at hyper- CIRCULAR ORBIT (COl bolic burn point vn and its capture orbit equivalent P = 24.62 H,rp = B.012tl RM PERIOD

DEIMOS SYNCHRONOUS CO

P - 30.30 H,rp = 6.93 RM

O utilizing the following three equations (where E and H stand PHOBOS SYNCHRONOUS CO > P = 7.65 H,rp = 2.77 RM for elliptic and hyperbolic, respectively): Z £ l-

rp - 1.0883 RM Z (hp - sinuH_ rcA[(_. _;v_ +2) X(rA-r ) ff (56) O rp = 2 RM. 24 H ELL sinve

< 2r'IrP ( 1+ _rc'tV_)lap p = 3 RM, 24 H ELLIPTICAL

ORBIT {ELL) and rp = PERIAPSE (COAPSIDAL INSERTION) RADIUS (1 RM = 3397,5 kin)

I I l l l l I 1 2 4 6 8 10 rca _ 2) ARRIVAL V_ krn/$ = (57) Fig. 27. Coapsidal capture orbit insertion maneuver AV requirements for Mars (using Eq. 50) (z ---_/cos_.+l

2 lap rco- (52) 2r. v_ 1) (58) while the corresponding optimal value for A V would be r . p - cosve

v® avco - _ (53) The procedure of obtaining a solution to these equations is iterative. For a set of given values for rA, rp , and an assumed Frequently the orbital radius obtained by use of Eq. (52) is Aco, a set of vH and vE, the hyperbolic and elliptical burn incompatible with practical injection aspects or with arrival point true anomalies which would satisfy Eqs. (55-58) can planet science and engineering objectives. be found. This in turn leads to r, the maneuver point radial distance, and hence, A V (Eq. 54). A plot of A V cost for a set A more general coplanar mode of capture orbit insertion, of consecutive A6op choices will provide the lowest AVvalue requiring only tangentiality of the two trajectories at an for this maneuver mode. arbitrary maneuver point of radius r common to both orbits, Fig. 26, requires a propulsive effort of For an optimal insertion into an orbit of an arbitrary major axis orientation one must turn to the more general, still co- planar, but intersecting (i.e., nontangential burn point) man- AV= _+_ - (54) euver (see Fig. 28). It provides sufficient flexibility to allow V 2 rlap _//2 lapr(r(r A ++rp)re - r) numerical optimization of A V with respect to apsidal rotation, Aco. P It can be clearly seen that by performing the burn at periapse the substitution r = r brings us back to Eq. (48). A more appropriate way to define apsidal orientation is to measure the post-maneuver capture orbit periapse position The cotangential maneuver mode provides nonoptimal con- angle with respect to a fixed direction, e.g., a far encounter trol over the orientation of the major axis of the capture orbit. point on the incoming asymptote, -V**, thus defining a cap- If it is desired to rotate this line of apsides clockwise by ture orbit periapse range angle OF POOR QUALIFY

INTERSECTING COAPSIDAL CAPTU R E CAPTU R E ORBIT ORBIT

v. 2ND BURN POINT r 2 OPTION

v= ASYMPTOTE

\ APPROACH \ NOTE : I v HYPERBOLA E = ELLIPTIC H = HYPERBOLIC

-_'_\ HYPERBOLIC FLYBY INTERSECTING CAPTURE ORBIT / COP-T----- PERIAPSE, rCA, CAN

PERIAPSE, rp I VARY: rA >t rCA >t O, I PROVIDED r s IS AVOIDED. I OPTIMAL BURN AM 1ST BURN POINT REQUIRES FCA >1 rtp. OPTION

Fig. 28. Coizpsidal and intersecting capture orbit insertion geometries

where

+ Q = lira XrpXrcA [21_+rcA V 2 ]

= Aeop + arc cos (59)

J[21av(r-rcA)+ V_ (r2-4A)] (r-rp)(r A -r) I

(60) Taken from Ref. 9, the expression for the intersecting burn AVis: planet-centered radius at burn, rA _ r _ tc, 4, km av _ = v_+2._ _ ;$ - _ +_ rA and rp = apoapse and periapse radii of capture ellipse = closest approach radius of flyby hyperbola where

= flag: 5 = +1 if injection occurs on same leg , mean orbital motion, rad/s (63) (inbound or outbound) of both hyperbola and capture ellipse, 8 = -I if not. _A+re a -- 2 , semi-major axis of eUipitical orbit, km (64) It should be pointed out that Eq. (57) and (58) still apply in the intersecting insertion case, while Eq. (56) does not 2rA9 p -- , semi-latus rectum of elliptical orbit, km (65) (as it assumes orbit tangency at burn point). rA+ 9

R S = Equatorial surface radius of Mars, km The evaluation of intersecting orbit insertion is more "/2 = Oblateness coefficient of Mars (for values see Sec- straightforward than it was for the cotangential case. By tion V on constants). assuming V and orbit size (e.g., V = 3 km/s, rp = 1.0883 ILVI.P = 24 hours, a typical capture orbit) and stepping through For the example orbit (1.0883 × 10.733 RM) used in Figs. a set of values for vE. the capture orbit burn point true anomaly, 29 and 30, if near equatorial, the regression of the node and one obtains, using Eqs. (57-60), a family of curves, one for advance of periapse, computed using Eqs. (61-65), would each value of RCA, the hyperbolic closest approach distance. amount to -0.272 and +0.543 deg/day, respectively. For As shown in Fig. 29, the envelope of these curves provides the contrast, for a grazing (a = 3697.5 km) low-inclination, near- optimal insertion burn AV for any value of apsidal rotation circular orbit, the regression of the node would race along Aw desired. The plot also shows clearly that cotangential and at -11.34 deg/day, while periapse would advance at 22.68 deg/ apsidal insertion burns are energetically inferior to burns on day: i.e., it would take 15.87 days for the line of apsides to do the envelope locus. For the same assumed capture orbit, a a complete turn. family of optimal insertion envelopes, for a range of values of arrival I,', is presented in Fig. 30. It should be noted that _2 = 0 occurs for i = 90 deg, while = 0 is found for i = 63.435 deg. Sun-synchronism of the The location of periapse with respect to the subsolar point node is achieved by retrograde polar orbits, e.g., 1.0883 RM is of extreme importance to many mission objectives. It can be circular orbit should be inclined 92.649 deg. controlled by choice of departure and arrival dates, by AV expenditure at capture orbit insertion, by an aerodynamic 3. Entry probe and lander trajectory design. Entry tra- maneuver during aerobraking, by depending on the planet's jectory design is on one hand concerned with maintenance of motion around the Sun to move the subsolar point in a manner acceptable probe entry angles and low relative velocity with optimizing orbital science, or by using natural perturbations respect to the rotating atmosphere. On the other hand, the and making a judicious choice of orbit size, equatorial inclina- geometric relationship of entry point, subsolar point and tion, i, and initial , _o, such as to cause Earth (or relay spacecraft) is of paramount importance. regression of the node, _, and the advance of periapsis, _, both due to oblateness to move the orbit in a desired manner Lighting during entry and descent is often considered the or at a specific rate. Maintenance of Sun-synchronism could primary problem to be resolved. As detailed in the flyby and provide constant lighting phase angle at periapse, etc., by some orbital sections above, the choice of trip time affects the value of the ZAPS angle which in turn moves the entry point for or all of these techniques. For an elliptical capture orbit (Fig. 31) longer missions closer to the subsolar point and even beyond, towards the morning terminator.

Landers or balloons, regardless of deceleration mode, - 3 s....._._R 180 prefer the morning terminator entry point which provides a ---- __ X p2 COS i × _,rr deg/s (61) better chance for vapor-humidity experiments, and allows a longer daylight interval for operations following arrival.

3 Rg n J2 The radio-link problem, allowing data flow directly to (2 - (5/2) sin 2 i) X 18.__0,deg/s =TX-- ff Earth, or via another spacecraft in a relay role, is very complex. p2 It could require studies of the Earth phase angle at the entry (62) locations or alternately, it could require detailed parametric OR'GINP, L P'?_3_ _ OF POOR QUALII3f

0

o 0

0 -8

-r (11 --k 0 ZZ "T m. 0,_ o Q. O m

CN m .J zO 0,_ _c ZuJ oco. Z 0 oon"

E u._-m OZZ _ILl 0 8" _6 u ..I Z O_ = w ._o " 2 0 U.I _" (3 o if) :3tr II o_ ,, O0 Z_ ::) , d (_ ',.9 " ,q- Z o 0 uJ ._

II -< a mr Zz ,,, ._ o W 0'_ Z > 0 n- o B ...I oh > H U O.J -3-m -- zO -..¢: >. == 0 0 m U.,I {J ",, LU n" _z Zw i. "-,>. o-a. u...r _EI" 0_.0 z m. c.)< "6 w p- .t,

0

o e- o0 0 0 oi cO 0 0 _-

0 p.

cO I I I I 1 1 I $ o 00 co ,-- o d

S/W)I 'IOI_A_ 'i::I3AN::INVlAI NOIIEI:::ISNI IIBEIO SEI_'IAI

29 ORIGINAL PAGE Ig OF POOR QUALITY

¢-q \

3O ORIGINAL PAGE i4 OF POOR QUALITY

ADVANCE OF A different choice of strategy could be to follow a contour PERIAPSE line of some characteristic, such as DLA or ZAP. One could _J also follow the minimum value locus of a parameter, e.g., CaL (i.e., the boundary between Class 1 and 2 within Type I

RUEANOMALY LINE OF or II) for each launch date, throughout the launch space. APSIDES Fundamentally different is a launch strategy for a dual o, ELLIPSE multiple spacecraft mission, involving more than one launch, either of which may possibly pursue divergent objectives. As PLANET an example, Fig. 32 shows the Voyagers 1 and 2 launch strat- EQUATOR egy, plotted on an Earth departure vs Saturn arrival date plot. A 14-day pad turnaround separation between launches was f to be maintained, a 10-day opportunity was to be available for each launch, and the two spacecraft had substantially different objectives at Jupiter and Saturn-one was to be Io-intensive and a close Jupiter flyby, to be followed by a close Titan encounter at Saturn, and the other was Ganymede- OFNODES and/or Callisto-intensive, a distant Jupiter flyby, as a safety (ASCENDING precaution against Jovian radiation damage, aimed to continue NODE) past Saturn to and . Here, even the spacecraft departure order was reversed by the strategy within the S/C MOTION launch space. REGRESSION OF NODE Launch strategies for orbital departures from a space station in a specific orbit promise to introduce new dimensions into Fig. 31. General satellite orbit parameters and precessional mission planning and design. New concepts are beginning to motion due to oblateness coefficient '/2 (from Ref. 9) emerge on this subject e.g., Refs. 10 and 11.

OCALL_STO INTENSIVE _ GAnYMEdE ANO I0 INTENSIVE studies involving relative motions of probe and relay spacecraft throughout probe entry and its following slow descent.

Balloon missions could also involve consideration of a JAN 1 lg82 variety of wind drift models, and thus, are even more complex as far as the communications problem with the Earth or the spacecraft is concerned.

E. Launch Strategy Construction

The constraints and desires, briefly discussed above, may be displayed on the mission space launch/arrival day plot as being limited by the contour boundaries of"C3L, the dates, DLA, VHP, ZAP, etc., thus displaying the allowable launch space.

Within this launch space a preferred day-by-day launch strategy must be specified, in accordance with prevailing objectives. The simplest launch strategy, often used to maintain a constant arrival date at the target planet, results in daily launch points on a horizontal line from leftmost to rightmost maximum allowable C L boundary for that arrival date. DEC 1 Such a strategy makes u3e of the fact that most arrival charac- 19B0 teristics may stay nearly constant across the launch space. 16 20 24 28 1 S g 13 17 21 AUGUST 1977 SEPTEMBER Lighting and satellite positions in this case are fixed, thus LAUNCH DATE allowing a similar encounter, satellite G/A, or satellite tour. Fig. 32. Voyager (MJS77) trajectory space and launch strategy

31 IV. Description of Trajectory Characteristics vector obtained by vectorial subtraction of the Data heliocentric planetary orbital velocity from the spacecraft arrival heliocentric velocity. It repre- A. General sents planet-relative velocity at great distance from The data represent trajectory performance information target planet, at beginning of far encounter. Can plotted in the departure date vs arrival date space, thus clef'ru- be used to compute spacecraft velocity at any ing all possible direct ballistic transfer trajectories between the point r of flyby, including C/A (periapse) dis- two bodies within the time span considered for each oppor- tance rp : tunity. Twelve individual parameters are contour-plotted. The first, C3L, is plotted bold on a Time of Flight (TFL) back- V= V_ +_ ,km/s ground; the remaining ten variables are plotted with bold con- r touring on a faint CsL background. Eleven plots are presented for each of eight mission opportunities between 1990 and where 2005.

The individual plots are labeled in the upper outer corner lAp(MARS = gravitational parameter GM of by bold logos displaying an acronym of the variable plotted, SYSTEM) the arrival planet system - Mars the mission's departure year, and a symbol of the target planet. plus all satellites. (For values, These permit a quick and fail-safe location of desired informa- refer to Section V on con- tion. stants.) DAP: B. Definition of Departure Variables 80.A , planetocentric declination (vs mean planet equator of date) of arrival V_. vector. Defines CsL: Earth departure energy (km2/s2); same as the lowest possible flyby/orbiter equatorial inclination square of departure hyperbolic excess velocity (deg). V 2 = CsL = V_ - 2 tIE/R t, where RAP: a_A, planetocentric right ascension (vs mean V/ = conic injection velocity (kin/s). planet equator and equinox of date, i.e., RAP is measured in the planet equator plane from ascend- R, = R S + hp injection radius (km), sum of ing node of the planet's mean orbit plane on the surface radius RSpt.ANET and injection planetary equator, both of date). Can be used altitude ht,. where RSEaRrH refers to together with VHP and DAP to compute an Earth's surface radius. (For value, see initial flyby trajectory state, but requires B-plane Section V on constants.) aim point information, e.g., B and O (deg).

lAe = gravitational constant times mass of the ZAPS: Angle between arrival V**vector and the arrival launch body (for values, refer to Section V planet-to-Sun vector. Equivalent to planet-probe- on constants). Sun angle at far encounter; for subsolar impact C3L must be equal to or exceeded by the launch would be equal to 180 deg. Can be used with vehicle capabilities. ETSP, VHP, DAP, and 0 to determine solar phase DLA: 6**L, geocentric declination (vs mean Earth equator angle at periapse, entry, etc. (deg). of 1950.0) of the departure _ vector. May im- ZAPE: Angle between arrival V_. vector and the planet-to- pose launch constraints (deg). Earth vector. Equivalent to planet-probe-Earth RLA: e'_.L, geocentric right ascension (vs mean Earth angle at far encounter (deg). equator and equinox of 1950.0) of the departure ETSP: Angle in arrival B-plane, measured from T-axis*, vector. Can be used with CaL and DLA to clockwise to projection of Sun-to-planet vector. compute a heliocentric initial state for trajector_ Equivalent to solar occultation region centerline analysis (deg). direction in B-plane (deg).

ZALS: Angle between departure V** vector and Sun-Earth ETEP: Angle in arrival B-plane, measured from T-axis*, vector. Equivalent to Earth-probe-Sun angle several clockwise, to projection of Earth-to-planet vector. days out (deg). Equivalent to Earth occultation region centerline C. Deflr tion of Arrival Variables direction in B-plane (deg).

VHP. V_A, planetocentric arrival hyperbolic excess *ETSP and ETEP plots are based on T-axis def'med as being parallel to velocity or V-ird'mity (km/s), the magnitude of the ecliptic plane (see text for explanation).

32 V. Table of Constants Direction of the Martian planetary equatorial north pole (in Earth Mean Equator of 1950.0 coordinates): Constants used to generate the information presented are summarized in this section. a e = 317.342 deg, 6_, = 52.711 deg

A. Sun

GM = 132,712,439,935. km3/s 2 Mean RSURFAC E = 696,000. km Planet Surface Orbit and GM, Radius, Radius,* Period, Satellites km3/s 2 km km hours B. Earth/Moon System Mars 42,828.286 3397.5 - 24.6229621 GMsYsTEM = 403,503.253 kma/s 2 (alone) Phobos 0.00066 13.5 X 10.7 × 9.6 9374 7.6538444 GMEART H = 398,600.448 073 km3/s 2 Deimos 0.00013 7.5 X 6.0 × 5.5 23457 30.2985774 J2 = 0.00108263 *Computed from period and Mars GM, rounded. REART H = 6378.140 km S UR FACE D. Sources

C. Mars System The constants represent the DE-118 planetary ephemeris (Ref. 12) and /Viking trajectory reconstruction data. GMsYsTEM = 42,828.287 kma/s 2 Definition of the Earth's equator (EME50.0) is consistent with Refs. 13-14, but would require minor adjustments for J2MAR s = 0.001965 the new equator and equinox, of J2000.O (Ref. 15).

Acknowledgments

The contributions, reviews, and suggestions by members of the Handbook Advisory Committee, especially those of K. T. Nock, P. A. Penzo, W. I. McLaughlin, W. E. BoUman, R. A. Wallace, D. F. Bender, D.V. Byrnes, L.A. D'Amario, T. H. Sweetser, and R. E. Diehl, graphic assistance by Nanette Yin, as well as the computational and plotting algorithm development effort by R. S. Schlaifer are acknowledged and greatly appreci- ated. The authors would like to thank Mary Fran Buehler, Paulette Cali. Douglas Maple, and Michael Farquhar for their editorial contribution.

References

1. Sergeyevsky, A. B., "Mission Design Data for Venus, Mars, and Jupiter Through 1990," Technical Memorandum 33-736, Vols. I, II, III, Jet Propulsion Laboratory, Pasadena, Calif., Sept. 1, 1975.

2. Clarke, V. C., Jr., Bollman, W. E., Feitis, P. H., Roth, R. Y., "Design Parameters for Ballistic Interplanetary Trajectories, Part II: One-way Transfers to and Jupiter," Technical Report 32-77, Jet Propulsion Laboratory, Pasadena, Calif., Jan. 1966.

33 3. Snyder, G. C., Sergeyevsky, A. B., Paulson, B. L., "Planetary Geometry Handbook," JPL Publication 82-44, (in preparation).

4. Ross, S., Planetary Flight Handbook, NASA SP-35, Vol. III, Parts 1,5, and 7, Aug. 1963 - Jan. 1969.

5. Wilson, S. W., "A Pseudostate Theory for the Approximation of Three-Body Trajec- tories," AIAA Paper 70-1061, presented at the AIAA Astrodynamics Conference, Santa Barbara, Calif., Aug. 1970.

6. Bymes, D. V., "Application of the Pseudostate Theory to the Three-Body Lambert Problem," AAS Paper 79-163, presented at the AAS/AIAA Astrodynamics Confer- ence, Provincetown, Mass., June 1979.

7. Sergeyevsky, A. B., Bymes, D. V., D'Amario, L. A., "Application of the Rectilinear Impact Pseudostate Method to Modeling of Third-Body Effects on Interplanetary Trajectories," AIAA Paper 83-0015, presented at the AIAA 21 st Aerospace Sciences Meeting, Reno, Nev., Jan, 1983.

8. Clarke, V.C., Jr., "Design of Lunar and Interplanetary Ascent Trajectories," Tech- nicalReport 32-30, Jet Propulsion Laboratory, Pasadena, Calif., March 15, 1962.

9. Kohlhase, C. E., Bollman, W. E., "Trajectory Selection Considerations for Voyager Missions to Mars During the 1971-1977 Time Period," JPL Internal Document EPD-281, Jet Propulsion Laboratory, Pasadena, Calif., Sept. 1965.

10. Anonymous, "Assessment of Planetary Mission Performance as Launched From a Space Operations Center," Presented by Science Applications, Inc. to NASA Head- quarters, Feb. 1, 1982.

1 I. Beerer, J. G., "Orbit Change Requirements and Evaluation," JPL Internal Document 725-74, Jet Propulsion Laboratory, Pasadena, Calif., March 9, 1982.

12. Newhall, XX, Standish, E. M., and Williams, J. G., "DE- 102 : A Numerically Integrated Ephemeris of the Moon and Planets, Spanning 44 Centuries," Astronomy andAstro- physics, 1983 (in press).

13. Explanatory Supplement to the Ephemeris, Her Majesty's Stationery Office, London, 1961.

14. Sturms, F. M., Polynomial Expressions for Planetary Equators and Orbit Elements With Respect to the Mean 1950.0 Coordinate System, Technical Report 32-1508, pp. 6-9. Jet Propulsion Laboratory, Pasadena, Calif., Jan. 15, 1971.

15. Davies, M. E., et al., "Report of the IAU Working Group on Cartographic Coordinates and Rotational Elements of the Planets and Satellites: 1982," CelestialMechanics. Vol. 29, No. 4, pp. 309-321, April 1983. Mission Design Data Contour Plots

Earth to Mars Ballistic Mission Opportunities 1990- 2005

35

Earth to Mars

1990

Opportunity

ENERGY MINIMA

DEPARTURE ARRIVAL VALUE TYPE (YEAR/MONTH/DAY) (YEAR/MONTH/DAY)

C3L 17.780 I 90/08/29 91/03/18

C3L 14.389 II 90/09/10 91/10/05

VHP 2.3281 I 90/08/27 91/05/24

VHP 2.3958 II 90/07/13 91/05/17

PRECEDING PAGE BLANK NOT FILMED"

37 ,

C3L ORIGINAL PAGE IS OF POOR QUALITY

1990 EARTH MARS 1990 , C3L , TFL )# BALL IST IC TRANSFER TRAJECTORY

911117./30 I - 48620.0

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38 1

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4O 1 ZALS ORIGINAL PAGE 14 OF POOR QUALITY d

ERRTH HRRS 1990 , C3L , ZRLS 1990 BALL IST IC TRANSFER TRAJECTORY

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EARTH MARS 1990, C3L , ZAPS 1990 BALL I ST I C TRANSFER TRAJECTORY

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48 10.

O'RIG]NAL PAGE ;,'_ ETSl OF POOR QUALITY

EARTH - HAAS 1990 , C3L , ETSP 199( BALL [ ST I C TRANSFER TRAJECTORY

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48 ORIG.,'_,IALPAC.Z ;_ OF POOR QUALITY

Earth to Mars

1992

Opportunity

ENERGY MINIMA

DEPARTURE ARRIVAL VALUE TYPE (YEAR/MONTH/OAY) (YEAR/MONTH/DAY)

C3L 18.341 I 92/10/06 93/05/09

C3L 11.732 II 92/09/30 93/09/19

VHP 2.3844 I 92/10/29 93/07/25

VHP 2.3987 II 92/09104 93/08/04

49 = C3L ORIGINAL PAGE IS OF POOR QUALITY

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52 4_

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63 ORIGINAL PAGE iS OF POOR QUALITY

1992 EARTH HARS 1992, C3L , VHP BALL [ ST l C TI:_iNSI:'F.J_ TI_:kTECTORY

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54

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56 i ZAPS ORIGINAL PACIE _1 OF POOR QUALITY

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6o ORIGINAL P,o,GZ _'3 OF POOR QUALITY

Earth to Mars

1994

Opportunity

ENERGY MINIMA

DEPARTURE ARRIVAL VALUE TYPE (YEAR/MONTH/DAY) (YEAR/MONTH/DAY)

C3L 13.766 I 94/11/06 95/07/05

C3L 9.4674 II 94/10/24 95/08/27

VHP 2.5727 94/11/12 95/08/21

VHP 2.5142 94/11/03 95/09/06

61 1

C3L ORIGINAL PAG_ [_ OF POOR QUALI='i_"

1994 EARTH - MARS 1994 , C3L . TFL , BALL [ST IC TRANSFER TRAJECTORY

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72 ORIGINAL PAGE [9 OF POOR QUALITY

Earth to Mars

1996/7

Opportunity

ENERGY MINIMA

DEPARTURE ARRIVAL VALUE TYPE (YEAR/MONTH/DAY) (YEAR/MONTH/DAY)

C3L 9.0071 I 96/11/30 97/07/19

C3L 8.9323 II 96/11/21 97/09/29

VHP 3.4813 I 97/01/06 97/08/31

VHP 2.8683 II 96/11/16 97/09/15

73 ,

C3L ORIGINAL P/',_E;_ c; OF POOR QUALITY 1996/7 EARTH - MARS 1996/? , C3L TFL BALL [ST IC TRANSFER TRAJECTORY

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74 . DLA ORIGINAL PAc2_ _3 OF POOR QUALITY d EARTH - HAAS 1996 / ? , C3L , DLR 199617 - BILL I ST IC TRRNSFER TRt:kTECTORY

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75 , 3. RLA ORIGINAL PAGE IS d OF POOR QUALITY 199617 EARTH - MARS 199617 , C3L RLA BILL I ST I C TRt_SI:'ER TRI_ECTORY

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76 ,

ORIGINAL PAGE 14 ZALS OF POOR QUALITY d

EARTH - HAAS 199617, C3L , ZALS 199617 B4:ILLISTIC TRRNSF'ER TRAJECTORY

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77 MHP ORIGINAL PAGE |S d OF POOR QUALI"rY 199617 EARTH - HAAS 19961"7 , C3L , VHP _:ILLISTIC TRANSFER TRI_ECTORY

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ORIGINAL PAGE i,_ OF POOR QUALITY d EARTH - P1ARS 1996/7 , C3L , ZAPS 199617 B:ILLISTIC TI:_IN_ER TI:_::_ECTORY

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81 zAPE ORIGINAL PAGE Ig d OF PoOR QUALrR' 199617 EARTH - MARS L996 / 7 , C3L , ZAPE BI:ILL [ ST I C TRRNS/ER TRRJECTORY

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82 10. ETSP ORIGINAL PAGE 19 OF POOR QUALITY

EARTH - MARS 199617, C3L , ETSP 1996/7 , BALLISTIC TRANSFER TRAJECTORY

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83 11.

ETEP ORIGINAL PAGE Ig d OF POOR QUALITY 199617 EARTH - MAAS 19961"-/, C3L ETEP BR_LISTIC TRt:IN_ER TRAJECTORY

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1998/9

Opportunity

ENERGY MINIMA

DEPARTURE ARRIVAL VALUE TYPE (YEAR/MONTH/DAY) (YEAR/MONTH/DAY)

C3L 9.0164 I 99/01/16 99/07/30

C3L 8.4403 II 99/02/07 99/12/31

VHP 3.7826 I 99/02/19 99/09/20

VHP 3.3391 II 98/12/08 99/09/23

85 ,

C3L oRIGINAL PAGE _9 OF POOR QUALITY

1998/9 EARTH - MARS 1998/9 , C3L TFL * BALL IST [C TI_ANSFEI_ TRAJECTORY

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86 ORIGINAL PAGE I_' OF POOR QUALITY

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88 ORIGINAL PAGE |$ OF POOR QUALITY

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107

Earth to Mars

2000/1

Opportunity

ENERGY MINIMA

DEPARTURE ARRIVAL VALUE TYPE (YEAR/MONTH/DAY) (YEAR/MONTH/DAY)

C3L 8.6342 I 2001/03/19 2001/09/10

C 3 L 7.8538 II 2001/04/16 2002/01/27

VH P 3.4757 I 2001/04/08 2001 / 10/24

VHP 3.7766 II 2001/01/13 2001/10/08

PRECEDING PAGE BLANK NOT FILMED

109 1

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131

Earth to Mars

2002/3

Opportunity

ENERGY MINIMA

DEPARTURE ARRIVAL VALUE TYPE (YEAR/MONTH/DAY) (YEAR/MONTH/DAY)

C 3 L 8.8102 I 2003/06/07 2003/12/25

C3L 12.564 II 2003/05/10 2003/12/29

VHP 2.6975 I 2003/06/13 2003/12/31

VH P 2.7652 II 2003/05/10 2003/12/30

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150 ORIG!NAL PAGE !_i OF POOR QUALITY

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154 ORIGINAL PAGE iS 3F POOR QUALITY

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155

Earth to Mars

2005

Opportunity

ENERGY MINIMA

DEPARTURE ARRIVAL VALUE TYPE (YEAR/MONTH/DAY) (YEAR/MONTH/DAY)

C3L 15.883 I 2005/08/10 2006/02/22

C3L 15.445 II 2005/09/02 2006/10/08

VHP 2.3602 I 2005/09/08 2006/04/20

VHP 2.4668 II 2005/06/21 2006/04/04

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