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JPL PUBLICATION 82-43
Interplanetary Mission DesignHandbook, Volume I, Part 2Earth to Mars Ballistic Mission Opportunities,1990-2005
Andrey B. SergeyevskyGerald C. SnyderRoss A. Cunniff
September 15, 1983
./
National Aeronautics and
Space Administration
Jet Propulsion LaboratoryCalifornia Institute of TechnologyPasadena, 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 18z07i
https://ntrs.nasa.gov/search.jsp?R=19840010158 2018-07-14T20:53:15+00:00Z
1. Repot No.82-43, Vol. I, Pt. 2
4. Title and Subtitle
Interplanetary Mission Design Handbook, Volume I,Part 2: Earth to Mars Ballistic Mission
Opportunities, 1990-2005
TECHNICAl. REPORT STANDARD TITLE PAGE
2. Government Access;on No. 3. Recipient's Catalog No.
7. Author_A.B. SergeyevsKy, G.C. Snyder, R.A. Cunniff
|
9. _rforming Organization Name and Address
JET PROPULSION LABORATORY
California Institute of Technology4800 Oak Grove Drive
Pasadena, California 91109
12. Sponsoring Agency Name anct Address
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
Washington, D.C. 20546
5. Report DateSeptember 15, 1983
6. PerformingOrgan zot o"Code
8. Performing Organization Report No
10. Work Unit No.
11. Contract or Gra_t No.
NAS 7-100
1'3. Type of Report and Period Covered
JPL Publication
14. Sponsoring Agency CodeRD4/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 solar system.
17. Key Wor_ (Selec_d by Author_))
Astronautics
Astrodynamics
Launch Vehicles and Space Vehicles
Lunar and Planetary Exploration
19. Security Cl_sif. (of this report)
Unclassified
18. Distribution Statement
Unclassified - Unlimited
20. Security Clmsif. (of this page)
Unclassified
2i. No. of Pages
176
22. Price
JPL 0184 11f80
JPL PUBLICATION 82-43
Interplanetary Mission DesignHandbook, Volume I, Part 2
Earth to Mars Ballistic Mission Opportunities,1990- 2005
Andrey B. SergeyevskyGerald C. SnyderRoss A. Cunniff
September 15, 1983
National Aeronautics and
Space Administration
Jet Propulsion LaboratoryCalifornia Institute of TechnologyPasadena, California
The research described in this publication was carried out by the Jet PropulsionLaboratory, California Institute of Technology, under contract with the NationalAeronautics 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 launchand 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 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 isPart 2 and describes ballistic trajectories to Mars. Part 3, which was published in 1982,
treated ballistic trajectories to Jupiter. Part 4, which was published earlier in 1983,described ballistic trajectories to Saturn. Parts 1 and 5, which will be published in the
near future, will treat ballistic trajectories to Venus 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 Angle Arithmetic ..................................... 15
4. Parking Orbit Regression .................................... 16
5. Dogleg Ascent ............................................. 17
6. Tracking and Orientation .................................... 17
7. Post-Launch Spacecraft 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. Sun ............................................................ 33
B, Earth/Moon System .............................................. 33
C. Mars System ..................................................... 33
D. Sources ........................................................ 33
Acknowledgments ..................................................... 33
References ............................................................. 33
Figures
1.
2.
3.
4.
5.
6.
7.
8.
9.
The Lambert problem geometry ................................... 2
Departure geometry and velocity vector diagram .................... 2
Pseudostate transfer geometry .................................... 3
Mission space in departure/arrival date coordinates, typical example .. 5
Effect of transfer angle upon inclinationof trajectory arc .............. 6
Nodal transfer geometry .......................................... 6
Mission space with nodal transfer ................................. 7
Broken-plane transfer geometry ................................... 8
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 angles involved in thelaunch problem .................................................. 10
12. Generalized relative launch time tRL T VS launch azimuth E_Land departure asymptote declination &= ............................ 11
13. Permissible regions of azimuth vs asymptote declination launchspace for Cape Canaveral ........................................ 12
14. Typical launch geometry example in celestial (inertial) Mercatorcoordinates ..................................................... 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 ZAPSand 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
The purpose of this series of Mission Design Handbooks is
to provide trajectory designers and mission planners with
graphical trajectory information, sufficient for preliminary
interplanetary mission design and evaluation. In most respectsthe series is a continuation of the previous three volumes of
the Mission Design Data, TM 33-736 (Ref. 1) and its predeces-
sors (e.g., Ref. 2); it extends their coverage to departures
through the year 2005 A.D.
The entire series is planned as a sequence of volumes, each
describing a distinct mission mode as follows:
Volume I: Ballistic (i.e., unpowered) transfers be-tween Earth and a planet, consisting of
one-leg trajectory arcs. For Venus and
Mars missions the planet-to-Earth return
trajectory data are also provided.
Volume II: Gravity-Assist (G/A) trajectory transfers,
comprising from two to four ballistic
interplanetary legs, connected by suc-
cessive planetary swingbys.
Volume III: Delta-V-EGA (AVEGA) transfer trajec-
tories utilizing an impulsive deep-spacephasing and shaping bum, followed by a
return to Earth for a G/A swingby maneu-
ver taking the spacecraft (S/C) to the
eventual target planet.
Each volume consists of several parts, describing trajectoryopportunities for missions toward specific target or swingbybodies.
This Volume I, Part 2 of the series is devoted to ballistic
transfers between Earth and Mars. It describes trajectories
taking from 100 to 500 days of flight time for the 8 successive
mission opportunities, departing Earth in the following years:1990, 1992, 1994, 1996/7, 1998/9, 2000/1, 2002/3, and2005.
Individual variables presented herein are described in detail
in subsequent sections and summarized again in Section IV.Suffice it to say here that all the data are presented in sets of
11 contour plots each, displayed on the launch date/arrival
date space for each opportunity. Required departure energy
C3, departure asymptote declination and right ascension,arrival V and its equatorial directions, as well as Sun and
Earth direction angles with respect to the departure/arrivalasymptotes, are presented.
It should be noted that parts of the launch space covered
may require launcher energies not presently (1983) available,
but certainly not unrealistic using future orbital assembly
techniques.
A separate series of volumes (Ref. 3) is being published
concurrently to provide purely geometrical (i.e., trajectory-
independent) data on planetary positions and viewing/orienta-
tion angles, experienced by a spacecraft in the vicinity of these
planetary bodies. The data cover the time span through 2020A.D., in order to allow sufficient mission duration time for all
Earth departures, up to 2005 A.D.
The geometric data are presented in graphical form and
consist of 26 quantities, combined into eight plots for eachcalendar year and each target planet. The graphs display equa-
torial declination and right ascension of Earth and Sun (plan-etocentric), as well as those of the target planet (geocentric);
heliocentric (ecliptic) longitude of the planet, its heliocentric
and geocentric distance; cone angles of Earth and Canopus,
clock angle of Earth (when Sun/Canopu_-oriented); Earth-Sun-
planet, as well as Sun-Earth-planet angles; and finally, rise andset times for six deep-space tracking stations assuming a 6-deghorizon mask. This information is similar to that in the second
part of each of the volumes previously published (Ref. 1).
II. Computational Algorithms
A. General Description
The plots for the entire series were computer-generated. Aminimum of editorial and graphic support was postulated fromthe outset in an effort to reduce cost.
A number of computer programs were created and/or modi-
fied to suit the needs of the Handbook production.
The computing effort involved the generation of arrays oftransfer trajectory arcs connecting departure and arrival
planets on a large number of suitable dates at each body.
Algorithms (computational models) to solve this problem can
vary greatly as to their complexity, cost of data generated, andresulting data accuracy. In light of these considerations, thechoice of methods used in this effort has been assessed.
B, Two-Body Conic Transfer
Each departure/arrival date combination represents a unique
transfer trajectory between two specified bodies, if the number
of revolutions of the spacecraft about the primary (e.g., the
Sun) is specified. The Lambert Theorem provides a suitableframework for the computation of such primary-centered
trajectories, but it is of practical usefulness only if restricted
two-body conic motion prevails.
Restricted two-body motion implies that the dynamical
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) orbits 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).
The Lambert Theorem states that given a value of the gravi-
tational parameter # (also known as GM) for the central body,
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,
IRII + IR2L, the distance between the two points C= SR2 - R I I,and the semimajor axis, a, of the conic orbital flight pathbetween them (Fig. 1.)
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,
_2, co, ol) and the transfer angle, Ao12, or two equivalent
spacecraft heliocentric velocity vectors, Vhs/c i- one at depar-ture. the other at the arrival planet. Subtraction of the appro-
priate planetary heliocentric velocity vector, VhpLANETi , atthe two corresponding times from each of these two space-
craft velocity vectors results in a pair of planetocentric velocity
states "at infinity" with respect to each planet (Fig. 2):
V_i = VhS/C at i - VhpLANET i (1)
where i = 1 and 2 refer to positions at departure and arrival,
respectively. The scalar of this V vector is also referred to as
EARTH
-_d.. I
!,I /
LANET
' '//
I'=.f(R 1 +R 2 ,C.a) I
Fig. I. The Lambert problem geometry
the hyperbolic excess velocity, "V-infinity" or simply "speed"(e.g., Ref. 4). The V** represents the velocity of the spacecraft
at a great distance from the planet (where its gravitational
attraction is practically negligible). It is attained when thespacecraft has climbed away from the departure planet, fol-
lowing injection at velocity V1:
= _/V_ "9"1V**1 _'/ , kmts(2)
or before it starts its fall into the arrival planet's gravity well,
where it eventually reaches a closest approach (periapse, C/A)
velocity Vp:
VhEAR Ttt
EARTH
INJECTION
OUTGOING V
ASYMPTOTE h sic
_ _2, ASCENDING NODE
vl
Fig. 2. Departure geometry and velocity vector diagram
ORIGINAL PAGE lg
OF POOR QUALITY
_/V_ 2#_= + , n/s (3)Vp 2 rp
The variables rI and rp refer to the departure injection andarrival periapse planetocentric radii, respectively. Values for
the gravitational parameter/a (or GM) are given in subsequentSection V on constants.
The V. vectors, computed by the Lambert method, repre-
sent a body center to body center transfer. They can, however,
be translated parallel to themselves at either body without
excessive error due to the offset, and a great variety of realisticdeparture and arrival trajectories may thus be constructed
through their use, to be discussed later. The magnitude and
direction of V** as well as the angles that this vector formswith the Sun and Earth direction vectors at each terminus are
required for these mission design exercises.
Missions to the relatively small terrestrial planets such as
Mars are suited to be analyzed by the Lambert method, as the
problem can be adequately represented by the restricted two-
body formulation, resulting in flight time errors of less than1 day - an accuracy that cannot even be read from the contour
plots presented in this document.
C. Pseudostate Method
Actual precision interplanetary transfer trajectories, espe-
cially those involving the giant outer planets, do noticeably
violate the assumptions inherent in the Lambert Theorem.
The restricted two-body problem, on which that theorem isbased, is supposed to describe the conic motion of a massless
secondary (i.e., the spacecraft) about the point mass of a pri-mary attractive body (i.e., the Sun), both objects being placed
in an otherwise empty Universe. In reality, the gravitationalattraction of either departure or target body may significantly
alter the entire transfer trajectory.
Numerical N-body trajectory integration could be called
upon to represent the true physical model for the laws of
motion, but would be too costly, considering the number ofcomplete trajectories required to fully search and describe a
given mission opportunity.
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
improvement over the standard Lambert solution. For the
giant planet missions, it can correct about 95 percent of thethree-body errors incurred, e.g., up to 30 days in flight time on
a typical Jupiter-bound journey.
Pseudostate theory is based on the assumption that for
modest gravitational perturbations the spacecraft conic motion
about the primary and the pseudo-conic displacement due to a
third body may be superimposed, if certain rules are followed.
The method, as applied to transfer trajectory generation,
does not provide a flight path-only its end states. It solves the
original Lambert problem, however, not between the true
planetary positions themselves, but instead, between two com-
puted "pseudostates." These are obtained by iteration on two
displacement vectors off the planetary ephemeris positions on
the dates of departure and arrival. By a suitable superposition
with a planetocentric rectilinear impact hyperbola and a
constant-velocity, "zero gravity," sweepback at each end of
the Lambertian conic (see Fig. 3), a satisfactory match isobtained.
Of the five arcs involved in the iteration, the last three
(towards and at the target planet) act over the full flight time,
AT_2, and represent:
(1)
(2)
Conic heliocentric motion between the two pseudo-
states R_ and R_ (capital R is used here for all helio-centric positions),
The transformation of R_ to a planetocentric position,
r 2 (lower case r is used for planetocentric positions),performed in the usual manner is followed by a
"constant-velocity" sweepback in time to a point
r_ = r_ - V**2 X zkT12, correcting the planetocentricposition r 2 to what it would have been at T1, had
there been no solar attraction during ATIj, andfinally,
(3) The planetocentric rectilinear incoming hyperbola,
characterized by: incoming V-infinity V_2 , a radial
/ NO-GRAVITY
/ SUN SWEEPBACK A %IR -":_ _ t
EAR- 'L - Voo2 X aT12 I r. ra RECTI'LINEAR
R2 HYPERBOLA,
// PLANETLAMBERT CONIC -
AT12 //r_ = R_- R 2
Fig. 3. Pseudostate transfer geometry
3
target planet impact, and a trip-time ATI2 from ra
to periapse, which can be satisfied by iteration on the
r_-magnitude and thus also on R_.
This last aspect provides for a great simplification of the
formulation as the R 2 end-point locus now moves only along
the V,, 2 [1 r_ vector direction. The resulting reduction in com-puting cost is significant, and the equivalence to the Lambert
point-to-point conic transfer model is attractive.
The first two segments of the transfer associated with the
departure planet may be treated in a like manner. If the planet
is Earth, the pseudostate correction may be disregarded (i.e.,
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
equal a fixed quantity, e.g., AT i = 20 days. The latter valuewas in fact used at the Earth's side of the transfer in the
data generation process for this document.
The rectilinear pseudostate method described above and in
Ref. 7 thus involves an iterative procedure, utilizing the
standard Lambert algorithm to obtain a starting set of valuesfor V.. at each end of the transfer arc. This first guess is then
improved by allowing the planetocentric pseudostate position
vector r_ to be scaled up and down, using a suitable partial at
either body, such that z_TREQ, the time required to fall alongthe rectilinear hyperbola through ra (the sum of the sweep-
back distance V**i × AT i and the planetocentric distance Ir_ I),equal the gravitational perturbation duration, AT/. Both r7 and
V_,i, along which the rectilinear fall occurs, are continuouslyreset utilizing the latest values of magnitude and direction ofV_, at each end, i, of the new Lambert transfer arc, as the
iteration progresses. The procedure converges rapidly as the
hyperbolic trip time discrepancy, AAT = ATR_.O - ATi, falls
below a preset small tolerance. Once the V_ vectors at each
planet are converged upon, the desired output variables can begenerated and contour plotted by existing standard algorithms.
As mentioned before, the pseudostate method was found
to be unnecessary for the accuracy of this Mars-oriented hand-
book. The simpler Lambert method was used instead in the
data-generation process.
III. Trajectory Characteristics
A. Mission Space
All realistic launch and injection vehicles are energy-limited
and impose very stringent constraints on the interplanetary
mission selection process. Only those transfer opportunities
which occur near the times of a minimum Earth departure
energy requirement are thus of practical interest. On either
side of such an optimal date, departure energy increases, first
slowly, followed by a rapid increase, thus requiring either a
greater launch capability, or alternatively a lower allowable
payload mass. A "launch period" measured in days or evenweeks, is thus definable: on any day within its confines the
capability of a given launch/injection vehicle must equal or
exceed the departure energy requirement for a specified
payload weight.
In the course of time these minimum departure energy
opportunities do recur regularly, at "synodic period" intervals
reflecting a repetition of the relative angular geometry of the
two planets. If 6o I and 6o2 are the orbital angular rates of theinner and outer of the two planets, respectively, moving about
the Sun in circular orbits, then the mutual configuration of
the two bodies changes at the following rate:
col2 = coI -to 2 , rad/s (4)
If a period of revolution, P, is det'med as
27rP =--, s (5)
CO
then
1 1 1
= P-T - PS (6)
where Ps, the synodic period, is the period of planetary
geometry recurrence, while P1 and P2 are the orbital "sidereal(i.e., inertial) periods" of the inner (faster) and the outer(slower) planet considered, respectively.
Since planetary orbits are neither exactly circular nor co-
planar, launch opportunities do not repeat exactly, some
years being better than others in energy requirements or in
other parameters. A complete repeat of trajectory character-
istics occurs only when exactly the same orbital geometry of
departure and arrival body recurs. For negligibly perturbed
planets approximately identical inertial positions in space atdeparture and arrival imply near-recurrence of transfer tra-
jectory characteristics. Such events can rigorously be assessedonly for nearly resonant nonprecessing planetary orbits, i.e.,
for those whose periods can be related in terms of integer
fractions. For instance, if five revolutions of one body cor-
respond to three revolutions of the other, that time interval
would constitute the "period of repeated characteristics."
Near-integer ratios provide nearly repetitive configurations
with respect to the lines of apsides and nodes. The Earth-
relative synodic period of Mars is 779.935 days, i.e. about2.14 years. Each cycle of 7 consecutive Martian mission
opportunities amounts to 5459.55 days and is nearly repeti-
tive, driven by 8 Martian sidereal periods of 686.9804 days. It
is obvious that for an identical mutual angular geometry Mars
would be found short of its inertial position in the previous
cycle by 36.3 days worth of motion (19.02 deg), while Earth
would have completed 5459.55/365.25 = 14.947 revolutions,
being short of the old mark by the same angular amount asSaturn.
ORIGII',IALPAGE |_
OF poOR QUALITY
A much closer repetition of characteristics occurs at 17 full
Martian revolutions (11678.667 days), after 15 Mars-departureopportunities (11699.031 days = 32.0302 years), showing
an inertial position excess of only 10.7 degs., 20.4 daysworth of Mars motion beyond the original design arrival
point.
A variety of considerations force the realistic launch period
not to occur at the minimum energy combination of depar-
ture and arrival dates. Launch vehicle readiness status, proce-
dure slippage, weather anomalies, multiple launch strategies,
arrival characteristics-all cause the launch or, more generally,
the departure period to be extended over a number of days orweeks and not necessarily centered on the minimum energydate.
For this document, a 160-day departure date coverage span
was selected, primarily in order to encompass launch energy
requirements of up to a C 3 = 50 km2/s 2 contour, where
C a = V 2 i.e., twice the injection energy per unit mass.
E I . Arrival date coverage was set at 400 days todisplay 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.
B. Transfer Trajectory
As previously stated, each pair of departure/arrival dates
specifies a unique transfer trajectory. Each such point in the
mission space has associated with it an array of descriptive
variables. Departure energy, characterized by C a , is by far the
most significant among these parameters. It increases towards
EARTH - MARS 1990 , CSL TFLw BALL I ST IC TRANSFER TRAJECTORY
91/ 12/30
.,,,,,. ,///A. ////,"////" I/Y//L_ _
""'°'" ///F///_ /,////,/)_"
'_ ////,'///_/ /1( _ _
_=_,,o°,,: 7,,//,,7'//.!t i _r/'j_l/VIIl'> 'll//Ik_t'_# .,_lll]SlllliL_l_ilL _i " _11111111111'__
9t/02/13
9[101104
90#11/2"5
_ _--- __ _1.I r I
LAUNCH DATE
4BO_0.O _BO_O.O _iD0.O _Bi_O.O _B_60.O_AUNCHJO-_O0000.O
Fig. 4. Mission space in departure/arrival date coordinates, typical example
.48620.0
•48.580.0
•48540.0
48500.0
0
• 48460.0 000o
ckl
48420,0 _3
.J
48380.0 --C_r¢
48340.0
48300.0
48260.0
48220.0
ORIGINAL PAGE Illl
OF POOR QUALITY
the edges of the mission space, but it also experiences a dra-
matic rise along a "ridge," passing diagonally from lower left
to upper right across the mission space (Fig. 4). This distur-
bance is associated with all diametric, i.e., near-180-deg,
transfer trajectories (Fig. 5).
In 3-dimensional space the fact that all planetary orbits are
not strictly coplanar causes such diametric transfer arcs to
require high ecliptic inclinations, culminating in a polar flight
path for an exact 180-deg ecliptic longitude increment be-
tween departure and arrival points. The reason for this
behavior is, as shown in Fig. 5, that the Sun and both trajec-
tory end points must lie in a single plane, while they are also
lining up along the same diameter across the ecliptic. The
slightest target planet orbital inclination causes a deviation
POLAR
TRANSFER _ PLANET/ ORB,T
EARTH 41
= TRUE TRANSFERANGLE
-_1 = TRANSFER ANGLEIN ECLIPTIC PLANE
TARGETPLANETPOSITIONS
f(_l )
ECLIPTIC
Fig. 5. Effect of transfer angle upon inclination of
trajectory arc
.PLANET ORBIT
PLANE TRANSFER
__,_V_ 2 MIN.)
PLANET AT NODE _-J -- ----NAT ARRIVAL _ _- -._
t SUN 7_ _ TECLIPTIC
_ /// ,_ TRANSFER
_ (M_N.CaZlEARTH AT NODEAT DEPARTURE
Fig. 6. Nodal transfer geometry
out of the ecliptic and forces a polar 180-deg transfer, in order
to pick up the target's vertical out-of-plane displacement.
The obvious sole exception to this rule is the nodal transfer
mission, where departure occurs at one node of the target
planet orbit plane with the ecliptic, whereas arrival occurs at
the opposite such node. In these special cases, which recur
every half of the repeatability cycle, discussed in the preceding
paragraph, the transfer trajectory plane is indeterminate and
may as well lie in the departure planet's orbit plane, thus
requiring a lesser departure energy (Fig. 6). The opposite strat-
egy (i.e., a transfer in the arrival planet's orbit plane) may be
preferred if arrival energy, V.,2, is to be minimized (Fig. 7).
It should be noted that nodal transfers, being associated
with a specific Mars arrival date, may show up in the data on
several consecutive Martian mission opportunity graphs (e.g.,
1992-1998), at points corresponding to the particular nodal
arrival date. Their mission space position moves, from oppor-
tunity to opportunity, along the 180-deg transfer ridge, by
gradually sliding towards shorter trip times and earlier relative
departure dates. Only one of these opportunities would occur
at or near the minimum departure energy or the minimum
arrival V date, which require a near-perihelion to near-
aphelion transfer trajectory. These pseudo-Hohmann nodal
transfer opportunities provide significant energy advantages,
but represent singularities, i.e., single-time-point missions,
with extremely high error sensitivities. Present-day mission
planning does not allow single fixed-time departure strategies;
however, future operations modes, e.g., space station "on-
time" launch, or alternately Earth gravity assist (repeated)
encounter at a specific time, may allow the advantages of a
nodal transfer to be utilized in full.
The 180-deg transfer ridge subdivides the mission space
into two basic regions: the Type I trajectory space below the
ridge, exhibiting less than 180-deg transfer arcs, and the
Type II space whose transfers are longer than 180 deg. In
general the first type also provides shorter trip times.
Trajectories of both types are further subdivided in two
parts-Classes 1 and 2. These are separated, generally hori-
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
shorter trip times.
Transfer energies become extremely high for very short
trip times, infinite if launch date equals arrival date, and of
course, meaningless for negative trip times.
The reason that high-inclination transfers, as found along
the ridge, also require such high energy expenditures at depar-
98/0_tE2
981041[2
98103103
9810U22
971121[3
1,1
G:O
.97/IU03>
97109124
97108115
97107106
97105127
97104/[7
LAUNCH DATE
_o32_._ _o3g_._ _o_d_._ _o,_._ _o,g_._LAUNCH JD-2400000.O
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
trajectory energy. The value of C3 on the ridge is large but
finite; its saddle point minimum value occurs for a pseudo-
Hohmann (i.e., perihelion to aphelion) polar transfer, requiring
2apCa = V2E \l+ap + 1] _ 1950km2/s 2 (7)
where Ve = 29.766 kmts, the Earth's heliocentric orbital
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
true nodal pseudo-Hohmann transfer, the minimum energy
required would reduce to
C3NODAL = 172 - 1 _ 7.83 km2/s 2
(8)
This is the lowest value of Ca required to fly from Earth toMars, assuming circular planetary orbits.
Arrival V-infinity, V_A, is at its lowest when the transfertrajectory is near-coplanar and tangential to the target planetorbit at arrival.
Both C3 and V= A near the ridge can be significantly low-ered if deep-space deterministic maneuvers are introduced into
the mission. The "broken-plane" maneuvers are a category of
,jRIGINAL PAGE iS
OF poOR QUALITY
such ridge-counteracting measures, which can nearly eliminate
all vestiges of the near-180-deg transfer difficulties.
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
contemplated. As an illustration, a sketch of resulting contours
of C 3 is shown in Fig. 9 for a typical broken-plane oppor-
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
POLAR
TRANSFER
ECLIPTIC __ili_____L/_ O pT i M A L
EARTH BROKEN-PLANE
TRANSFER
Fig. 8. Broken-planetransfergeometw
I
48040 0 48080 0 48120.0 481_0.0 48200.0
LAUNCH J0-2400000,0
-48620.0
48580.0
48_40.0
48_00.0
O
48460.0000
48420.0
483_0,0 --
48340.0
'48300.0
'48260.0
482200
Fig. 9. Sketch of mission space with broken-plane transfer, effective energy requirements
launch/arrival date C a contour plot. The deep-space maneuver
AVBp is transformed into a C 3 equivalent by converting the
new broken-plane C3B P to an injection velocity at parkingorbit altitude, adding AVBe, and converting the sum to a new
and slightly larger C3 value at each point on the strip.
C. Launch/Injection Geometry
The primary problem in departure trajectory design is to
match the mission-required outgoing V-infinity vector, V, to
the specified launch site location on the rotating Earth. The
site is defmed by its geocentric latitude, 0z, and geographic
east longitude, Xz. (Figs. 10 and 11).
Range safety considerations prohibit overflight of popu-
lated or coastal areas by the ascending launch vehicle. For each
launch site (e.g., Kennedy Space Center, Western Test Range,
or Guiana Space Center), a sector of allowed azimuth firing
directions EL is deffmed (measured in the site's local horizontalplane, clockwise from north). For each launch vehicle, theallowed sector may be further constrained by other safety
considerations, such as spent stage impact locations down the
range and/or down-range significant event tracking capabilities.
The outgoing V-infinity vector is a slowly varying function
of departure and arrival date and may be considered constant
for a given day of launch. It is usually specified by its energy
LAUNCH
MERIDIAN_
ORZGINAL PAGE Ig
OF POOR QUALITY
magnitude Ca = [V=i 2, called out as C3L in the plots andrepresenting twice the kinetic energy (per kilogram of injected
mass) which must be matched by launch vehicle capabilities,
and the V-infinity direction with respect to the inertial Earth
Mean Equator and equinox of 1950.0 (EME50) coordinate
system: the declination (i.e., latitude) of the outgoing asymp-
tote 5, (called DLA), and its right ascension (i.e., equatorialeast longitude from vernal equinox, T) a** (or RLA). These
three quantities are contour-plotted in the handbook data pre-sented in this volume.
1. Launch azimuth problem. The first requirement to be
met by the trajectory analyst is to establish the orientation of
the ascent trajectory plane (Ref. 8). In its simplest form this
plane must contain the outgoing V-infinity (DLA, RLA)vector, the center of Earth, and the launch site at lift-off
(Fig. 10). As the launch site partakes in the sidereal rotation of
the Earth, the continuously changing ascent plane manifests
itself in a monotonic increase of the launch azimuth, Ez., with
lift-off time, tL, (or its angular counterpart, c_**- az. , measuredin the equator plane):
cotan Et.cos q_LX tan 5= - sin _L X cos (_= - eL)
sin (_ - %)
(9)
N
ASYMPTOTE
MERIDIAN
FLIGHTPATH
V_, DEPARTUREASYMPTOTE
LAUNCH
_=0
VERNALEQUINOX EQUATOR
TRAJECTORY PLANE
Fig. 10. Launch/injection trajectory plane geometry
ORIGINALPAGE19OF POORQUALITY'
The daily time history of azimuth can be obtained from Eq.(9) for a given _=, 8, departure asymptote direction by
following quadrant rules explained below and using the
following approximate expressions (see Fig. 11):
ClL = _L+GHADATE+(_EARTHX tL (I0)
GHADATE = 100.075+0.9856123008Xdso (11)
where
GHA DA TE
t"OEAR TH
e L = Right ascension of launch site (¢z,' XL) attL (deg)
= Greenwich hour angle at 0n GMT of any
date, the eastward angle between vernal
equinox and the Greenwich meridian (deg),assumes equator is EME50.0
= sidereal (inertial) rotation rate of Earth
(15.041067179 deg/h of mean solar time)
dso = launch date in terms of full integer dayselapsed since 0 a Jan. 1, 1950 (days)
tL = lift-off time (h, GMT, i.e., mean solar time)
A relative launch time, [RLT, measured with respect to an
inertial reference (the departure asymptote meridian's rightascention a**), can be defined as a sidereal time (Earth's
rotation rate is 15.0 deg/h of sidereal time, exactly):
t_ - t_LtRL T = 24.0 15.0 , h (12)
This time represents a generalized sidereal time of launch,
elapsed since the launch site last passed the departure asymp-
tote meridian, a**.
The actual Greenwich Mean (solar) Time (GMT) of launch,
tL, may be obtained from tRL r by transforming it to meansolar time and adding a date, site, and asymptote-dependent
adjustment :
tRLTX 15.0 a** -GHADATE -_L (EAST)tL - + (13)
¢OEARTH Ca)EARTH
The expression for Z L (Eq. 9) must be used with computa-tional regard for quadrants, singular points, and sign conven-
tions. If the launch is known to be direct (i.e., eastward),
then Y'L' when cotan Z L is negative, must be corrected to
Z L = ZLNEG + 180.0. For retrograde (westward) launches,180.0 deg must be added in the third (when cotan Z is posi-
GREENWICHMERIDIAN
Oh GMT A GREENWICH HOUR ANGLEEAST LONGITUDE OF'_, ((;HA)
LAUNCH SITE _'L/ _ \
I __Jl,_ _ ] VERNAL( C'O'f/'._ll_L-e, X ] EQUINOX
LAUNCH SITE_ • - / -- _)" "_J.
/ _ DEPARTUREANGLE TRAVERSED _ ASYMPTOTEBY LAUNCH SITE _ MERIDIANSINCE Oh GMT
OF LAUNCH DATE LAUNCH SITE
AT LAUNCH, tL (GMT)
Fig. 11. Earth equator plane definition of angles involvedin the launch problem
tive) and 360.0 deg in the fourth quadrant (when cotan Zis negative).
A generalized plot of relative launch time tRL T VS launchazimuth 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
for Kennedy Space Flight Center at Cape Canaveral, Florida).Such a plot is presented in Fig. 12 with departure asymptote
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).
2. Daily launch windows. Inspection of Fig. 12 indicatesthat generally two contours exist for each declination value
(e.g., $. = -10 deg), one occurring at tRL T during the a.m.hours, the other in the p.m. hours of the asymptote relative"day."
Since lift-off times are bounded by preselected launch-site-
dependent limiting values of launch azimuth _;L (e.g., 70 degand 115 deg), each of the two declination contours thus con-
tains a segment during which launch is permissible-"a launch
window." The two segments on the plot do define the two
available daily launch windows.
As can be seen from Fig. 12, for 5.. = O, the two dailylaunch opportunities are separated by exactly 12 hours; with
an increasing 15o, I they close in on each other, until at 15_ I =I_LI they merge into a single daily opportunity. For 18=1 >
I_zl, a "split" of that single launch window occurs, disallow-
ing an ever-increasing sector of azimuth values. This sector issymmetric about east and its limits can be determined from
lO
ORIGINAL PACE l_t
OF POOR QUALITY.
J_
LU
-I-(oz
J
F.-<.J
uJn.-
uJFm0
C6
>-u')<
24
22
20
18
16
14
12
10
8
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 _=
= _+- (14)sin _,LLIMIT COS _bL
As 5.. gets longer, the sector of unavailable launch azimuthsreaches the safety boundaries of permissible launches, and
planar launch ceases to exist (Fig. 13). This subject will beaddressed again in the discussion of "dogleg" ascents.
Figure 14 is a sketch of a typical daily launch geometry
situation, shown upon a Mercator map of the celestial sphere.The two launch windows exhibit a similar geometry since the
inclinations of the ascent trajectory planes are functions of
launch site latitude _L and azimuth Y_Lonly:
cos i = cos _/_ x sin rz (15)
The two daily opportunities do differ greatly, however, in
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.
The angular equatorial distance between the ascending node
and the launch site meridian is given by
sin _= X sin Y_Lsin (a L - £2) = sin i (16)
Quadrant rules for this equation involve the observation that a
negative cos E L places (a L - f2) into the second or third quad-rant, while the sign of sin (% - _2) determines the choicebetween them.
The range angle 0 is measured in the inertial ascent trajec-
tory plane from the lift-off point at launch all the way to the
departure asymptote direction, and can be computed for a
given launch time t L or ct.. - aL(tL) and an azimuth Y_Lalready known from Eq. (9) as follows:
cos 8 = sin a= X sin $z, + cos a=. X cos ¢L X cos (% - %)
(17)
11
ORIG|NAL PAGE !_1OF POOR QUALITY
180
160
140
._ 120
"I"_- 100
m
N
<"rc.}Z
<--1
"A.OE',,ET,
LAUNCHREGIONREGION
80 "//"" r.,,-,,.,,/////
00NN..."......4O
20 JFi --__LIMITING-LAUNCHAZIMUTH
0 I I I J
-90 -70 -50 -30 -10 10 30 50 70
ASYMPTOTE DECLINATION 6oo , deg
Fig. 13. Permissible regions of azimuth v= asymptote
declination launch space for Cape Canaveral
9O
sin(% - %) x cos8®
sin 8 = sin _L (18)
The extent of range angle O can be anywhere between
0 deg and 360 deg, so both cos 0 and sin O may be desired in
its determination. The range angle O is related to the equa-
torial plane angle, Act = tz=. - t_L, discussed before. Even
though the two angles are measured in different planes, they
both represent the angular distance between launch and depar-
ture asymptote, and hence they traverse the same number of
quadrants.
Figure 15 represents a generally applicable plot of central
range angle O vs the departure asymptote declination and
launch azimuth, computed using Eqs. (17) and (18) and a
launch site latitude ¢c = 28.3 (Cape Canaveral). The twin
daily launch opportunities are again evident, showing the
significant difference in available range angle when following a
vertical, constant 5=. line.
It is sometimes convenient to reverse the computational
procedure and determine launch azimuth from known range
angle O and tRLr, i.e., &_ = a= - %, as follows:
"O-LU
E3
<..J
-J
<E0
<
OLU
0 3I
-40
-10
0 _
10
2O
3O
40
-45
ASYMPTOTE RELATIVE LAUNCH TIME tRJLT , h
6 9 12 15 18
"'_ ........ LAU_ICH __:":':"" .......I II_'L° I AZIMUTH I I
I I
1ST WINDOW
OPENS AT /
Z/,, = 70 ° /1ST WINDOW- -o / CLOSES AT
//EASTER/Y _L = 110 °LAUNCH
_1 ZL = 90 ° (_ SUN
- (ASSUMED)
600 = - 15 °
2ND WINDOY
OPENS AT
_,Lo = 70 °
-
BY DEFINITION
21 24/t_,T = oh= 24 h ATMoo - MERIDIAN2ND WINDOW
AT
_-,Lc = 110 ° ./ DAI LY TRACK_r OF LAUNCH SITE
AT _L = 28"3o
eL 1O(°° - _Z, 3(°°1 = 2850j--1 1 1 1 I 1 I
0 45 90 135 180 225 270 315
R I GHT ASCENSION ct, deg(EARTH EQUATORIAL CELESTIAL LONGITUDE)
ASSUMED V,_,ASYMPTOTE
LOCATION:
6o0 = -15 °
(x=o = 315 °
Fig. 14. Typical launch geometry example in celestial (inertial) Mercator coordinates
12
ORIGINAL PAGE killOF POOR QUALITY
400,D
B
360
D
320m
m 280"0
c_"
240
z<
200z<rr-- 160
rr"I-Ztu 120
80
m
4O
0 T,i,l,i,o-70 -60
_:,- 6od \ -_
_j\ _L=,2o
(_i" / .i.,f_J \ o='=='°" I\ (/.__ _.00,. I NOTF: _L IS LAUNCH AZIMUTH
o,os wo," t i I
I '
,,,,,,,,,illilili ililillililililii' 'ilillili:lliihiliiilllilllllllliill lil__lllililil" liiliiiliiiililil
-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= - %)
sin Y'L = sin 0 (19)
sin 5=. - cos 8 X sin Cz,
cos zz. = sin O × cos_z. (20)
Figure 16 displays a 3-dimensional spatial view of the same
typical launch geometry example shown previously in map
format in Fig. 14. The difference in available range angles as
well as orientation of the trajectory planes for the two daily
launch opportunities clearly stands out. In addition, the figure
illustrates the relationship between the "first" and "second
daily" launch windows, defined in asymptote-relative time,
tRLT, as contrasted with "morning" or "night" launches,defined in launch-site-local solar time. The latter is associated
with the lighting conditions at lift-off and consequently allows
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
direction vector. It allows some judgment on available ascent
lighting.
The length of the range angle required exhibits a complex
behavior-the first launch window of the example in Figs. 14
and 16 offers a longer range angle than the second, but the
second launch window opens up with a range angle so short
that direct ascent into orbit is barely possible. Further launch
delay shortens the range even further, forcing the acceptance
of a very long coast (one full additional revolution in parking
orbit) before transplanetary departure injection. A detailed
analysis of required arc lengths for the various sub-arcs of the
13
URIGINAL PAGE !8
OF POOR QUALITY
SECOND "DAILY"
) LAUNCHDOW
(EVENING
LAUNCH)
W l N DOW
CLOSES
_/_, = 110 °
OPENS
TRANSPLANETARY
INJECTION
CIRCLE
SUN
(ASSUMED)
FIRST "DAILY" (tR], T )• LAUNCH WINDOW
(NOON LAUNCH)
WINDOW
]_L = 700
EARTH LAUNCH SITE
_L = CONST.
SECOND
DEPARTURE
SUN-TO-EARTH
V ECTO R
ZALS
FIRST
DEPARTURE OUTGOING
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
BOOSTER AS(,LAUNCH SITE
ORIGINAL PAGE rg
OF POOR QUALITY
/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/ TRUE ANOMALY OF INJECTION
Fig. 17. Basic geometry of the launch and ascent profile in the trajectory plane (after Ref. 9)
departure trajectory is thus a trajectory design effort of para-
mount importance (Fig. 17_
3. Range angle arithmetic. For a viable ascent trajectory
design, the range angle 0 must first of all accommodate the
twin burn arcs 01 and 02, representing ascent into parkingorbit and transplanetary injection burn into the departure
hyperbola. In addition, it must also contain the angle from
periapse to the V.. direction, called "true anomaly of the
asymptote direction" (Fig. 18):
-1(21)
cos o= = c3 x rl+_
ue
where:
r = periapse radius, typically 6563 krn, for a horizontalpinjection from a 185-kin (100-nmi) parking orbit
I_ = GM, gravitational parameter of Earth (refer to theTable of Constants, Section V).
The proper addition of these trajectory sub-arcs also requitesadjustment for nonhorizontal injection (i.e., for the flight
path angle 7l > 0), especially dgnificant on direct ascent
miuions (no coast arc) and mt=sions with relatively low
thrust/weight ratio injection stages. The adjustment is accom-
plished as follows:
O = 01 + OCOAST + #2 + _ - i_l (22)
where:
15
140
UJn-
D 130I--
W
oAii
o __. 120>-
ZOO
"' 110
I-
10025
L
\
ORIGINAL PAGZ iS
OF POOR QUALITY
I I I50 75
370 km
PERIGEE
ALTITUDE
185 km (100 nmi)
1200 nmil _
!
NOTE: 2
C3 = V_DE P IS TWICETHE TOTAL ENERGYPER UNIT MASS
I I I100 125 150
C3,km3/s2
I I
175 200
Fig. 18. Angle from perigee to departure asymptote
225
= is the true anomaly of the injection point, usuallyis near 0 deg, and can be computed by iteration
using:
en sin _'z
tan 7z - 1 + e/./cos v1 (23)
7r = injection flight path angle above local horizontal,deg
en = eccentricity of the departure hyperbola:
C3 xre, = 1 + _ (24)
To make Eq. (22) balance, the parking orbit coast arc,
8coAs T, must pick up any slack remaining, as shown in
Fig. 17. A negative OcoAs T implies that the ZL-solution was
too short-ranged. A direct ascent with positive injection true
anomaly vt (i.e., upward climbing flight path angle, 7i, atinjection) with attendant sizable gravity losses, may be accept-
able, or even desirable (within limits) for such missions.
Alternately, the other solution for Z L, exhibiting the longerrange angle 0, and thus a longer parking orbit coast, OcoAs T,
should be implemented. An extra revolution in parking orbit
may be a viable alternative. Other considerations, such asdesire for a lightside launch and/or injection, tracking ship
location and booster impact constraints, may all play a signif-
icant role in the ascent orbit selection. A limit on maximum
coast duration allowed (fuel boil-off, battery life, guidance
gyro drift, etc.) may also influence the long/short parking orbit
decision. In principle, any number of additional parking orbit
revolutions is permissible. Shuttle launches of interplanetarymissions (e.g., Galileo) are in fact required to use such addi-
tional orbits for cargo bay door opening and payload deploy-ment sequences. In such cases, however, the precessional
effects of Earth's oblateness upon the parking orbit, prirnarily
the regression of the orbital plane, must be considered.
4. Parking orbit regression. The average regression of the
nodes (i.e., the points of spacecraft passage through the
equator plane) of a typical direct (prograde) circular parkingorbit of 28.3-deg inclination with the Earth's equator, due to
Earth's oblateness, amounts to about 0.46 deg of westward
nodal motion per revolution and can be approximately com-puted from
where
540 °Xr 2 X,/2 Xcosi= s , deg/revolution (25)
/,2o
r = Earth equatorial surface radius, 6378 km$
ro = circular orbit radius, typically 6748 km for an
orbital altitude of 370 km (200 nmi)
16
ORIGINAL PAG.Z E,._
OF POOR QUALITY
J2 = 0.00108263 for Earth
i = parking orbit inclination, deg. Can be computed fora given launch geometry from
cos i = cos _t, × sin ZL (26)
This correction, multiplied by the orbital stay time of N
revolutions, must be considered in determining a biased launch
time and, hence, the fight ascension of the launch site atlift-off:
%EFF = _L + (2 × N , deg (27)
5. Dogleg ascent. Planar ascent has been considered exclu-
_avely, thus far. Reasons for performing a gradual powered
plane change maneuver during ascent may be many. Inability
to launch in a required azimuth direction because of launch
site constraints is the prime reason for desiring a dogleg ascent
profile. Other reasons may have to do with burn strategiesor intercept of an existing orbiter by the ascending spacecraft,
especially if its inclination is less than the latitude of the
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 rotatedthrough the same angle, but they may on occasion be opera-
tionally preferable.
As already discussed, a special geometric situation develops
whenever the departure asymptote declination magnitudeexceeds 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
early and/or late in the mission's departure launch period, and
is frequently associated with dual hunches, when month-long
departure periods are desired.
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.
6. Tracking and orientation. As the spacecraft moves away
from the Earth along the asymptote, it is seen at a nearly
constant declination-that of the departure asymptote, 6**
(DLA in the plotted data). The value of DLA greatly affects
tracking coverage by stations located at various latitudes;
highest daily spacecraft elevations, and thus best reception, are
enjoyed by stations whose latitude is closest to DLA. Orbit
determination, using radio doppler data, is adversely affected
by DLA's near zero degrees.
The spacecraft orientation in the first few weeks is often
determined by a compromise between communication (antenna
pointing) and solar heating constraints. The Sun-spacecraft-
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.
It was defined in the discussion of Fig. 16. This quantity hasmany uses:
(I) If the spacecraft is Sun-oriented, ZALS equals the
Earth cone angle (CA), depicted in Fig. 19 (the cone
PLANET
\
CONEANGLE
TOOBJECT
STAR(CANOPUS)
Fig. 19. Definition of cone and clock angle
17
(2)
angle of an object is a spacecraft-fixed coordinate,
an angle between the vehicle's longitudinal -Z axis and
the object direction).
The Sun-phase angle, cbs (phase angle is the Sun-object-spacecraft angle) describes the state of the object's
disk lighting: a fully lit disk is at zero phase. For the
Earth (and Moon), several days after launch, the sun-phase angle is:
6ps = 180- ZALS (28)
(3) The contour labeled ZALS = 90 ° separates two cate-
gories of transfer trajectories-those early departures
that first cut inside Earth's orbit, thus starting out atnegative heliocentric true anomalies for ZALS >90 °,
and those later ones that start at positive true anoma-
lies, heading out toward Jupiter and never experiencing
the increased solar heating at distances of less than1 AU for ZALS < 90 °.
7. Post-launch spacecraft state. After the spacecraft has
departed from the immediate vicinity of Earth (i.e., left the
Earth's sphere of influence of about 1-2 million km), it moveson a heliocentric conic, whose initial conditions may be
approximated as
Vs/c = VEARTt4 + V. (29)
Rs/c = REART_ + VS/c X At (30)
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
be constructed using @_3L = V , DLA = 8., and RLA = c_in three components as follows:
V = (V × cos_ × cos 8**, V X sina. X cosS.,
V** × sin 6.*) (31)
8. Orbital launch problem. Orbital launch from the Shuttle,from other elements of the STS, or from any temporary or
permanent orbital space station complexes, introduces entirelynew concepts into the Earth-departure problem. Some of
the new constraints, already mentioned, limit our ability
to launch a given interplanetary mission. The slowly regressing
space station orbit (see Eq. 25) generally does not contain
the V-inffmity vector required at departure. Orbit lifetime
or other considerations may dictate a space station's orbital
altitude that may be too high for an efficient injection burn.
Innovative departure strategies are beginning to emerge,
attempting to alleviate these problems - a recent Science
Applications, Inc. (SAI) study (Ref. 10) points to some of
the techniques available, such as passive wait for natural
alignment of the continuously regressing space station orbit
plane (driven by Earth's oblateness) with the required
V-infinity vector, or the utilization of 2- and 3- impulse man-
euvers, seeking to perform spacecraft plane changes near the
apogee of a phasing orbit where velocity is lowest and thus
turning the orbit is easiest. These two approaches can becombined with each other, as well as with other suitable
maneuvers, such as:
(1) Deep space propulsive burns for orbit shaping and
phasing,
(2) Gravity assist flyby, including Earth return AVEGA,
(3) Aerodynamic turns at grazing perigees or at inter-
mediate planetary swingbys, and
(4) Multiple revolution injection bums, requiring several
low, grazing passes, combined with apogee plane changemaneuvers, etc.
All of these devices can be optimized to permit satisfactory
orbital launches, as well as to achieve the most desirable condi-
tions at the final arrival body. In general, space launch advan-
tages, such as on-orbit assembly and checkout of payloads and
clustered multiple propulsion stages, or orbital construction of
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
the space station's inherent orbital orientation incompatibilitywith departure requirements.
D. Planetary Arrival Synthesis
The planetary arrival trajectory design problem involves
satisfying the project's engineering and science objectives at
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,
orbiter, atmospheric probe, and, to a small extent, landermissions.
1. Flyby trajectory design. In this mission mode, the arrival
trajectory is not modified in any deterministic way at theplanet-the original aim point and arrival time are chosen to
satisfy the largest number of potential objectives, long before-hand.
This process involves the choice of arrival date to ensuredesirable characteristics, such as the values of the variables
VHP, DAP, ZAPS, etc., presented in plotted form in the datasection of this volume.
18
ORIG,,_AL PAGE l_
OF POOR QUALITY
DAP, the planet-equatorial declination, 6**, of the incoming
asymptote, i.e., of the V-infinity vector, provides the measure
of the minimum possible inclination of flyby. Its negative isalso known as the latitude of vertical impact (LVI).
The magnitude of V-infinity, VHP = IV._L, enables one to
control the flyby turn angle A_ between the incoming and
outgoing V** vectors by a suitable choice of closest approach
(C/A) radius, rp (see Fig. 20):
A_ = 180-2p, deg
where p, the asymptote half-angle, is found from:
(32)
1 1cos p - - (33)
e V: r** pl+--/ap
VHP also enables the designer to evaluate planetocentric
velocity, V, at any distance, r, on the flyby hyperbola:
2 _p V:V = x/--- 7- + ** ,km/s (34)
In the above equations,/ap (or GMp), is the gravitational param-eter of the arrival body.
PLANETVERNAL EQUINOX
L VI = -
VERTICALIMPACTPOINT
SUN
PLANET EQUATOR
\\
ZAPEi
/ \
EARTH
\\\
_2
V_IN B
='RIAPSE
OUTBOUNDASYMPTOTE
TURN ANGLE
= 180-2p
INBOUNDASYMPTOTE
Fig. 20.
)ING NODE
Planetary flyby geometry
19
Another pair of significant variables on which to basearrival date selection are ZAPS and ZAPE-the angles between
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 bythe spacecraft on its far-encounter approach leg to the planet:
_'s--180-zAes (3s)
Both the cone angle and the phase angle have already been
defined and discussed in the Earth departure section above.
The flyby itself is specified by the aim point chosen upon
the arrival planet target plane. This plane, often referred toas 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 aimpoint. 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
T-axis is chosen to be parallel to a fundamental plane, usually
the ecliptic (Fig. 21) or alternatively, the planet's equator. Themagnitude of B equals the semi-minor axis of the flyby hyper-bola, b, and can be related to the closest approach distance,
also referred to as the periapse radius, rp, by
or
,(( v:r):)l,:IB[ Vi 1 + **#pv - 1 , km (36)
% = V''_'**+ \_V_] + IBl2 ,km (37)
The direction angle 8 of the B-vector, B, measured in the
target plane clockwise from the T-axis to the B-vector position
can easily be related to the inclination, i, of the flyby trajec-
tory, provided that both 5** (DAP) and the T-axis, from which
0 is measured clockwise, are defined with respect to the same
fundamental plane to which the inclination is desired. For asystem based on the planet equator (Fig. 22):
cos%e = cosoee¢ × cos5 (38)**PE Q
which assumes that Ovv,Q is computed with the T-axis parallel
to the planet equator (i.e., TeE Q = V** × POLEeEQ), not theecliptic, as is frequently assumed (TEc L = V × POLEEcL).Care must be taken to use the 0 angle as dofined and intended.
ORIGINAL PAGE IS
OF POOR QUALITY
TARGET B-PLANE,1 TO INCOMING PARALLEL TO
I ..,..% OJ"
\ - " -TjC' B IHYPERBouc\ _ W I PATH OF
/ ....V_ "TRAJECTORY PLANE
B MISS PARAMETER, B_
(TARGET VECTOR)
0 AIM POINT ORIENTATION
'_ PARALLEL TO INCOMING ASYMPTOTE, MooT PARALLEL TO ECLIPTIC PLANE
AND ±TOS"
R =SXT
Fig. 21. Definition of target or arrival B-plane coordinates
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
based system
tan AO =sin(% - %p)
cos 5= × tan tSEp - sin 5= X cos (a - aEp)
(39)
where
aEp and 5Ep are right ascension and declination of the
ecliptic pole in planet equatorial coordinates. For Mars usingconstants in Section V:
o_£e= 267.6227, BE? = 63.2838, deg
a= and 5** are RAP and DAP, the directions of incoming
V, also in planet equatorial coordinates.
It
PEQ POLE
PPE,
ORIGINAL PAGE I_lOF POOR QUALITY
ECLIPTIC POLE (EP)
PLrCLDECLINATION
OF EP, _EP
PEQ RIGHTASCENSIONOF EP, <XEp
- 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
ecliptic system as follows (Fig. 22):
OpE Q = OEC L + AO (40)
A
The ecliptic T, RECL axes, however, have to be rotated by-A0 (clockwise direction is positive in the B-plane) to obtain
planet equatorial T, RPEQ coordinate axes. The B-magnitudeof an aim point in either system is the same.
The projections of the Sun-to-planet and Earth-to-planetvectors into the B-plane represent aim point loci of diametric
Sun and Earth occultations, respectively., as defined in Fig. 23.The B-plane 0-angles (with respect to TF.CL axis) of these vari-
ables are presented and labeled ETSP and ETEP, respectively,
in the plotted mission data. In addition to helping design orelse avoid diametric occultations, these quantities allow
computation of phase angles, _i, of the planet at the spacecraft
periapse, at the entry point of a probe, or generally at any
position r (subscript S = Sun, could be replaced by E = Earth,
if desired), (see Fig. 24):
cos '_s = -cos flo. X cos ZAP s - sin 13®X sin ZAP S
where
× cos (ETsP - Os/c) (41)
Os/C = the aim point angle in the B-plane, must be withrespect to the same T, as ETSP.
/3® = the arrival range angle from infinity (a position farout on the incoming asymptote) to the point of
interest r (Fig. 25).
The computation of the arrival range angle, t3=, to the
position of the desired event depends on its type, as follows:
21
ORIGINAL P,_QE I_
OF POOR QUALITY
ARRIVAL
B-PLANE
SUN (OR EARTH) TO
PLANET
//
/
SHADOW
(SUN OR
EARTH)
'(ORIIII
PLANET
ETSP
(OR ETEP)
T
TO SUN
(OR EARTH) R
Fig. 23. Definition of approach orientational
coordinates ZAPS and ETSP, ZAPE and ETEP
l. At flyby periapse, (3. ;) 90 deg):
.
cos #= = cos (-v=) --I
V 2 r== p
l+--
gp
(42)
(rp is periapse radius).
At given radius r, anywhere on the flyby trajectory(see Fig. 25):
3= = -v + v (43)
v= should be computed from periapse equation,
Eq. (42) v r at r can be obtained from (-v= < v r _; +
v=, v r has negative values on the incoming branch):
3.
r V2 )r 2+ p = -1
r /./pcos v = (44)
(1 +rpV----_2_/
At the entry point having a specified flight path angle
"rE (Fig. 25):
_. = -v,. + vE (45)
//x
/ \
I v=
INCOMING
ASYMPTOTE
180-ZAPS
/TO SUN
TRAJECTORY ARC, _]_ RECL
PECL
PHASE ANGLE, dab
Fig. 24. Phase angle geometry at arrival planet
_)s = 90 °
r, RADIUS TO
A POINT ON
TRAJECTORY
PROJECTED ONTO
UNIT-SPHERE
A=ETSP- 180 °-
22
OF POOR QUALI,'_
PLANETARY
ENTRY
INTERFACE
ARBITRARY TRAJECTORY
POINT AT RADIUS r
AND TRUE ANOMALY vr_
FICTITIOUSENTRY PERIAPSE
RADIUS, rp //
ENTRY
OUTGOING
ASYMPTOTE
FLYBY PERIAPSE
RADIUS, rpF
TURN ANGLE
= 180-2p
INCOMING
ASYMPTOTE
VERTICAL
IMPACT
FLYBY
ENTRY POINTAT RADIUS
rE = rSURF + h E ,
AND FLIGHT PATH
ANGLE _/'E
ENTRYTRAJECTORY
IBFI bEFLYBY MISS
PARAMETER
TRAJECTORY
Fig. 25. Typical entry and flyby trajectory geometry
23
where vE is the true anomaly at entry, should alwaysbe negative, and can be computed if entry radius and
altitude, rE = rsuRP + hE and 7E, the entry angle, areknown:
r 2+ -1
rE Upcos v_. = (46)
(l+----_p ]rpV2 t
¥P
whereas the fictitious periapse radius rpto satisfy 7E at rE, is equal to
for the entry,
I,>E. <v'->I_ /dp
(47)
The B value corresponding to this entry point can be com-
puted from Eq. (36), while the 0 angle in the B-plane would
depend on the desired entry latitude inclination, Eq. (38), or
phase angle, Eq. (41).
The general flyby problem poses the least stringent con-
straints on a planetary encounter mission, thus allowing
optimization choices from a large list of secondary parameters,such as satellite viewing and occultation, planetary fields and
particle in situ measurements, special phase.angle effects, etc.
A review of the plotted handbook variables, required in the
phase-angle equation (Eq. 41), shows that the greatest magni-
tude variations are experienced by the ZAPS angle, which is
strongly flight-time dependent: the longer the trip, the smallerZAPS. For low equatorial inclination, direct flyby orbits, this
implies a steady move of the periapse towards the lit side and,
eventually, to nearly subsolar periapses for long missions. This
also implies that on such flights the approach legs of the tra-jectory are facing the morning terminator or even the dark
side, as trip time becomes longer, exhibiting large phase angles
(recall that phase is the supplement of the ZAPS angle on
the approach leg). This important variation is caused by a grad-
ual shift of the incoming approach direction, as flight time in-creases, from the subsolar part of the target planet's leading
hemisphere (in the sense of its orbital motion) to its antisolar
part.
The arrival time choice on a very fine scale may greatly
depend on the desire to observe specific atmospheric/surface
features or to achieve close encounters with specific satellites
of the arrival planet. Passages through special satellite event
zones, e.g., flux tubes, wakes, geocentric and/or heliocentric
ORIGINAL PAGE I_
OF POOR QUALITY
occultations, require close control of arrival time. The number
of satellites passed at various distances also depends on the
time of planet C/A. These fine adjustments do, however,
demand arrival time accuracies substantially in excess of those
provided by the computational algorithm used in this effort,
which generated the subject data (accuracies of 1-5 min for
events or 1-2 h for encounters would be required vs uncertain-
ties of up to 1.5 days actually obtained with the rectilinear
impact pseudo-state theorem). Numerically searched-in inte-
grated trajectories, based on the information presented as a
first guess input, are mandatory for such precision trajectorywork.
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, ringand debris structures, occultations by Sun, Earth, stars, or
satellites, etc., can all be presented on specialized plots, e.g.,the B-plane, and do affect the choice of suitable aim point
and arrival time. All of these studies require the propagation
of a number of flyby trajectories. Adequate initial conditions
for such efforts can be found in the handbook as: VHP (V)
and DAP (8..) already defined, as well as RAP (a), the
planet equatorial right ascension of the incoming asymptote
(i.e., its east longitude from the ascending node of the planet's
mean orbital plane on its mean equator, both of date). The
designer's choice of the aim point vector, either as B and 0, oras cartesian B - T and B • R, completes the input set. Suitable
programs generally exist to process this information.
2. Capture orbit design. The capture problem usuallyinvolves the task of determining what kind of spacecraft orbit
is most desired and the interconnected problem of how and at
what cost such an orbit may be achieved. A scale of varying
complexity may be associated with the effort envisioned-anelliptical long period orbit with no specific orientation at the
trivial end of the scale, through orbits of controlled or opti-
mized lines of apsides (i.e., periapse location), nodes, inclina-
tion, or a safe perturbed orbital altitude. Satellite G/A-aided
capture, followed by a satellite tour, involving multiple satel-
lite G/A encounters on a number of revolutions, each designed
to achieve specific goals, probably rates as the most complex
capture orbit class. Some orbits are energeticaUy very difficult
to achieve, such as close circular orbits, but all require signifi-
cant expenditures of fuel. As maneuvers form the backgroundto this subject a number of useful orbit design concepts shall
be presented to enable even an unprepared user to experiment
with the data presented.
The simplest and most efficient mode of orbit injection is
a coplanar burn at a common periapse of the arrival hyperbola
and the resulting capture orbit (Fig. 26). The maneuver A Vrequired is:
24
ORIGINAL PAGE i_
OF POOR QUALITY
J 2/Zp J_/_p X raAV = V_=+ -- -
rp % + r), km/s (48)
A plot of orbit insertion AV required as a function of rp and
P (using Eq. 50) is presented in Fig. 27. The apoapse radius of
such an orbit of given period would be
The orbital period for such an orbit, requiring knowledge
of periapse and apoapse radii, rp and ra, is
IFP = 2_" L--"Y--_ J , s(49)
If on the other hand, a known orbit period P (in seconds)
is desired, the expression for AV is
/ / j(2/_p 2/ap 2/_pX
av = v: + -- - e ,(5o)rp rp
2#p XP 2rA = 3 rt2 rp , km
X/
(51)
An evaluation of Eq. (50) (and Fig. 27) shows that lowest
orbit insertion AV is obtained for the lowest value of rp, the
longest period P, and the lowest V. of arrival.
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
capture orbits of constant eccentricity, including e = 0, i.e.,
circular orbits, which would require a specific radius:
COAPS I DA L
CAPTU R E
ORBIT
COTANGENTIAL
CAPTU R E
ORBIT
v.ASYMPTOTE
/
APPROACH
HYPERBOLA
COTANGENTIAL /ORBIT PERIAPSE
\
/ _rCA \ ,,_\
\v H \./" TT T'
_VE_- \ \ COAPSIDAL BURNAND PERIAPSE OF
HYPERBOLIC PERIAPSE \ ITS CAPTURE ORBIT
Fig. 26. Coapsidal and cotangential capture orbit insertion geometries
25
ORIGINAL PACI_ I_
OF POOR QUALITY
O
>
Z
£l-
Z
ffO
<
I I IMARS SURFACE-SYNCHRONOUS
CIRCULAR ORBIT (COl
P = 24.62 H,rp = B.012tl RM
DEIMOS SYNCHRONOUS CO
P - 30.30 H,rp = 6.93 RM
PHOBOS SYNCHRONOUS CO
P = 7.65 H,rp = 2.77 RM
I
PERIOD
I I2 4
rp - 1.0883 RM
(hp -
rp = 2 RM. 24 H ELL
p = 3 RM, 24 H ELLIPTICAL
ORBIT {ELL)
rp = PERIAPSE (COAPSIDAL
INSERTION) RADIUS
(1 RM = 3397,5 kin)
l l l l I 16 8 10
ARRIVAL V_ krn/$
Fig. 27. Coapsidal capture orbit insertion maneuver AV
requirements for Mars (using Eq. 50)
2 laprco-
v_(52)
while the corresponding optimal value for A V would be
v®avco - _ (53)
Frequently the orbital radius obtained by use of Eq. (52) isincompatible with practical injection aspects or with arrival
planet science and engineering objectives.
A more general coplanar mode of capture orbit insertion,
requiring only tangentiality of the two trajectories at an
arbitrary maneuver point of radius r common to both orbits,
Fig. 26, requires a propulsive effort of
V 2 lap /2 lap (rA + re - r)AV= _+_ - (54)r _/ r(r A +rp)
It can be clearly seen that by performing the burn at periapse
the substitution r = r brings us back to Eq. (48).
The cotangential maneuver mode provides nonoptimal con-
trol over the orientation of the major axis of the capture orbit.
If it is desired to rotate this line of apsides clockwise by
I A60 , one can solve for the hyperbolic periapse r and theP CA
bum radius r using selected values of true anomaly at hyper-
bolic burn point vn and its capture orbit equivalent
utilizing the following three equations (where E and H stand
for elliptic and hyperbolic, respectively):
(_. v_ )sinuH_ rcA[ _; +2 X(rA-r )sinve
2r'IrP ( 1+ _rc'tV_)lap
(56)
and
rca _ 2)
= (57)
( </z ---_/cos_.+l
2r.
1)r . p - cosve
(58)
The procedure of obtaining a solution to these equations is
iterative. For a set of given values for rA, r , and an assumedpAco, a set of vH and vE, the hyperbolic and elliptical burn
point true anomalies which would satisfy Eqs. (55-58) canbe 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 setof consecutive A6o choices will provide the lowest AVvaluepfor this maneuver mode.
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-euver (see Fig. 28). It provides sufficient flexibility to allow
numerical optimization of A V with respect to apsidal rotation,Aco.
P
A more appropriate way to define apsidal orientation is to
measure the post-maneuver capture orbit periapse positionangle with respect to a fixed direction, e.g., a far encounter
point on the incoming asymptote, -V**, thus defining a cap-ture orbit periapse range angle
COAPSIDAL
CAPTU R E
ORBIT
OF POOR QUALIFY
INTERSECTING
CAPTU R E
ORBIT
r 2
v.2ND BURN
POINT
OPTION
v=ASYMPTOTE
APPROACH
HYPERBOLA
INTERSECTINGCAPTURE ORBIT /
PERIAPSE, rp
\\
I vNOTE :
E = ELLIPTIC
H = HYPERBOLIC
-_'_\ HYPERBOLIC FLYBY
COP-T----- PERIAPSE, rCA, CAN
VARY: rA >t rCA >t O,
PROVIDED r s IS AVOIDED.
OPTIMAL BURN AM
REQUIRES FCA >1 rtp.
III
1ST BURN POINT
OPTION
= Aeop + arc cos (59)
Fig. 28. Coizpsidal and intersecting capture orbit insertion geometries
where
Q = lira XrpXrcA [21_+rcA V 2 ]
Taken from Ref. 9, the expression for the intersecting burn
AVis:
av _ = v_+2._ _ ;$ - _ +_
+
J[21av(r-rcA)+ V_ (r2-4A)] (r-rp)(r A -r) I
(60)
rA and rp =
planet-centered radius at burn, rA _ r _ tc, 4,
km
apoapse and periapse radii of capture ellipse
= closest approach radius of flyby hyperbola
= flag: 5 = +1 if injection occurs on same leg
(inbound or outbound) of both hyperbola
and capture ellipse, 8 = -I if not.
It should be pointed out that Eq. (57) and (58) still apply
in the intersecting insertion case, while Eq. (56) does not
(as it assumes orbit tangency at burn point).
The evaluation of intersecting orbit insertion is more
straightforward than it was for the cotangential case. By
assuming V and orbit size (e.g., V = 3 km/s, rp = 1.0883ILVI.P = 24 hours, a typical capture orbit) and stepping through
a set of values for vE. the capture orbit burn point true anomaly,
one obtains, using Eqs. (57-60), a family of curves, one for
each value of RCA, the hyperbolic closest approach distance.As shown in Fig. 29, the envelope of these curves provides the
optimal insertion burn AV for any value of apsidal rotation
Aw desired. The plot also shows clearly that cotangential andapsidal insertion burns are energetically inferior to burns on
the envelope locus. For the same assumed capture orbit, a
family of optimal insertion envelopes, for a range of values
of arrival I,', is presented in Fig. 30.
The location of periapse with respect to the subsolar point
is of extreme importance to many mission objectives. It can be
controlled by choice of departure and arrival dates, by AV
expenditure at capture orbit insertion, by an aerodynamic
maneuver during aerobraking, by depending on the planet'smotion around the Sun to move the subsolar point in a manner
optimizing orbital science, or by using natural perturbations
and making a judicious choice of orbit size, equatorial inclina-
tion, i, and initial argument of periapsis, _o, such as to causeregression of the node, _, and the advance of periapsis, _,both due to oblateness to move the orbit in a desired manner
or at a specific rate. Maintenance of Sun-synchronism could
provide constant lighting phase angle at periapse, etc., by some
or all of these techniques. For an elliptical capture orbit
(Fig. 31)
- 3 Rs....._._ 180
---- __ X p2 COS i × _,rr deg/s (61)
3=TX--
Rg n J2
p2
(2 - (5/2) sin 2 i) X 18.__0,deg/sff
(62)
where
, mean orbital motion, rad/s (63)
_A+rea --
2 , semi-major axis of eUipitical orbit, km (64)
p --
2rA9
rA+ 9, semi-latus rectum of elliptical orbit, km (65)
R S = Equatorial surface radius of Mars, km
"/2 = Oblateness coefficient of Mars (for values see Sec-tion V on constants).
For the example orbit (1.0883 × 10.733 RM) used in Figs.29 and 30, if near equatorial, the regression of the node and
advance of periapse, computed using Eqs. (61-65), would
amount to -0.272 and +0.543 deg/day, respectively. For
contrast, for a grazing (a = 3697.5 km) low-inclination, near-circular orbit, the regression of the node would race along
at -11.34 deg/day, while periapse would advance at 22.68 deg/
day: i.e., it would take 15.87 days for the line of apsides to do
a complete turn.
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
node is achieved by retrograde polar orbits, e.g., 1.0883 RMcircular orbit should be inclined 92.649 deg.
3. Entry probe and lander trajectory design. Entry tra-
jectory design is on one hand concerned with maintenance of
acceptable probe entry angles and low relative velocity with
respect to the rotating atmosphere. On the other hand, the
geometric relationship of entry point, subsolar point and
Earth (or relay spacecraft) is of paramount importance.
Lighting during entry and descent is often considered the
primary problem to be resolved. As detailed in the flyby and
orbital sections above, the choice of trip time affects the valueof the ZAPS angle which in turn moves the entry point for
longer missions closer to the subsolar point and even beyond,
towards the morning terminator.
Landers or balloons, regardless of deceleration mode,
prefer the morning terminator entry point which provides a
better chance for vapor-humidity experiments, and allows a
longer daylight interval for operations following arrival.
The radio-link problem, allowing data flow directly to
Earth, or via another spacecraft in a relay role, is very complex.
It could require studies of the Earth phase angle at the entry
locations or alternately, it could require detailed parametric
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RUEANOMALY
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f
ADVANCE OF
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_J
LINE OFAPSIDES
S/C MOTION
REGRESSION
OF NODE
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(ASCENDING
NODE)
Fig. 31. General satellite orbit parameters and precessional
motion due to oblateness coefficient '/2 (from Ref. 9)
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
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 maintaina constant arrival date at the target planet, results in daily
launch points on a horizontal line from leftmost to rightmostmaximum allowable C L boundary for that arrival date.Such a strategy makes u3e of the fact that most arrival charac-
teristics may stay nearly constant across the launch space.
Lighting and satellite positions in this case are fixed, thus
allowing a similar encounter, satellite G/A, or satellite tour.
A different choice of strategy could be to follow a contourline of some characteristic, such as DLA or ZAP. One could
also follow the minimum value locus of a parameter, e.g.,
CaL (i.e., the boundary between Class 1 and 2 within Type I
or II) for each launch date, throughout the launch space.
Fundamentally different is a launch strategy for a dual o,
multiple spacecraft mission, involving more than one launch,
either of which may possibly pursue divergent objectives. As
an example, Fig. 32 shows the Voyagers 1 and 2 launch strat-egy, plotted on an Earth departure vs Saturn arrival date plot.
A 14-day pad turnaround separation between launches was
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 aclose Titan encounter at Saturn, and the other was Ganymede-
and/or Callisto-intensive, a distant Jupiter flyby, as a safety
precaution against Jovian radiation damage, aimed to continue
past Saturn to Uranus and Neptune. Here, even the spacecraft
departure order was reversed by the strategy within the
launch space.
Launch strategies for orbital departures from a space station
in a specific orbit promise to introduce new dimensions into
mission planning and design. New concepts are beginning to
emerge on this subject e.g., Refs. 10 and 11.
OCALL_STO INTENSIVE _ GAnYMEdE ANO I0 INTENSIVE
JAN 1
lg82
DEC 1
19B0
16 20 24 28 1 S g 13 17 21
AUGUST 1977 SEPTEMBER
LAUNCH DATE
Fig. 32. Voyager (MJS77) trajectory space and launch strategy
31
IV. Description of Trajectory CharacteristicsData
A. General
The data represent trajectory performance information
plotted in the departure date vs arrival date space, thus clef'ru-
ing all possible direct ballistic transfer trajectories between the
two bodies within the time span considered for each oppor-
tunity. Twelve individual parameters are contour-plotted. The
first, C3L, is plotted bold on a Time of Flight (TFL) back-
ground; the remaining ten variables are plotted with bold con-
touring on a faint CsL background. Eleven plots are presented
for each of eight mission opportunities between 1990 and2005.
The individual plots are labeled in the upper outer cornerby bold logos displaying an acronym of the variable plotted,
the mission's departure year, and a symbol of the target planet.
These permit a quick and fail-safe location of desired informa-tion.
B. Definition of Departure Variables
CsL: Earth departure energy (km2/s2); same as thesquare of departure hyperbolic excess velocity
V 2 = CsL = V_ - 2 tIE/R t, where
V/ = conic injection velocity (kin/s).
R, = R S + hp injection radius (km), sum of
surface radius RSpt.ANET and injection
altitude ht,. where RSEaRrH refers toEarth's surface radius. (For value, see
Section V on constants.)
lAe = gravitational constant times mass of thelaunch body (for values, refer to Section V
on constants).
C3L must be equal to or exceeded by the launchvehicle capabilities.
DLA: 6**L, geocentric declination (vs mean Earth equator
of 1950.0) of the departure _ vector. May im-pose launch constraints (deg).
RLA: e'_.L, geocentric right ascension (vs mean Earthequator and equinox of 1950.0) of the departure
vector. Can be used with CaL and DLA tocompute a heliocentric initial state for trajector_
analysis (deg).
ZALS: Angle between departure V** vector and Sun-Earth
vector. Equivalent to Earth-probe-Sun angle several
days out (deg).
C. Deflr tion of Arrival Variables
VHP. V_A, planetocentric arrival hyperbolic excessvelocity or V-ird'mity (km/s), the magnitude of the
vector obtained by vectorial subtraction of the
heliocentric planetary orbital velocity from the
spacecraft arrival heliocentric velocity. It repre-
sents planet-relative velocity at great distance from
target planet, at beginning of far encounter. Can
be used to compute spacecraft velocity at any
point r of flyby, including C/A (periapse) dis-
tance rp :
V= V_ +_ ,km/sr
where
DAP:
RAP:
ZAPS:
ZAPE:
ETSP:
ETEP:
lAp(MARS = gravitational parameter GM ofSYSTEM) the arrival planet system - Mars
plus all satellites. (For values,refer to Section V on con-
stants.)
80.A , planetocentric declination (vs mean planetequator of date) of arrival V_. vector. Defineslowest possible flyby/orbiter equatorial inclination(deg).
a_A, planetocentric right ascension (vs meanplanet equator and equinox of date, i.e., RAP ismeasured in the planet equator plane from ascend-
ing node of the planet's mean orbit plane on the
planetary equator, both of date). Can be used
together with VHP and DAP to compute aninitial flyby trajectory state, but requires B-plane
aim point information, e.g., B and O (deg).
Angle between arrival V**vector and the arrival
planet-to-Sun vector. Equivalent to planet-probe-
Sun angle at far encounter; for subsolar impact
would be equal to 180 deg. Can be used with
ETSP, VHP, DAP, and 0 to determine solar phase
angle at periapse, entry, etc. (deg).
Angle between arrival V_. vector and the planet-to-
Earth vector. Equivalent to planet-probe-Earth
angle at far encounter (deg).
Angle in arrival B-plane, measured from T-axis*,
clockwise to projection of Sun-to-planet vector.Equivalent to solar occultation region centerline
direction in B-plane (deg).
Angle in arrival B-plane, measured from T-axis*,
clockwise, to projection of Earth-to-planet vector.
Equivalent to Earth occultation region centerline
direction in B-plane (deg).
*ETSP and ETEP plots are based on T-axis def'med as being parallel to
ecliptic plane (see text for explanation).
32
V. Table of Constants
Constants used to generate the information presented are
summarized in this section.
A. Sun
GM = 132,712,439,935. km3/s 2
RSURFAC E = 696,000. km
B. Earth/Moon System
GMsYsTEM = 403,503.253 kma/s 2
GMEART H = 398,600.448 073 km3/s 2
J2 = 0.00108263
REART H = 6378.140 km
S UR FACE
C. Mars System
GMsYsTEM = 42,828.287 kma/s 2
J2MAR s = 0.001965
Direction of the Martian planetary equatorial north pole
(in Earth Mean Equator of 1950.0 coordinates):
a e = 317.342 deg, 6_, = 52.711 deg
Mean
Planet Surface Orbit
and GM, Radius, Radius,* Period,
Satellites km3/s 2 km km hours
Mars 42,828.286 3397.5 - 24.6229621
(alone)
Phobos 0.00066 13.5 X 10.7 × 9.6 9374 7.6538444
Deimos 0.00013 7.5 X 6.0 × 5.5 23457 30.2985774
*Computed from period and Mars GM, rounded.
D. Sources
The constants represent the DE-118 planetary ephemeris
(Ref. 12) and Mariner 9/Viking trajectory reconstruction data.
Definition of the Earth's equator (EME50.0) is consistent
with Refs. 13-14, but would require minor adjustments for
the new equator and equinox, epoch 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 Mercury 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 RectilinearImpact Pseudostate Method to Modeling of Third-Body Effects on Interplanetary
Trajectories," AIAA Paper 83-0015, presented at the AIAA 21 st Aerospace SciencesMeeting, 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 DataContour Plots
Earth to Mars BallisticMission Opportunities 1990- 2005
35
Earth to Mars
1990
Opportunity
ENERGY MINIMA
DEPARTURE ARRIVALVALUE 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
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DEPARTURE ARRIVALVALUE TYPE (YEAR/MONTH/DAY) (YEAR/MONTH/DAY)
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Earth to Mars
1996/7
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ENERGY MINIMA
DEPARTURE ARRIVAL
VALUE TYPE (YEAR/MONTH/DAY) (YEAR/MONTH/DAY)
C3L 9.0071 I 96/11/30 97/07/19
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ENERGY MINIMA
DEPARTURE ARRIVALVALUE TYPE (YEAR/MONTH/DAY) (YEAR/MONTH/DAY)
C3L 9.0164 I 99/01/16 99/07/30
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ORIGINAL PAGE ISOF POOR QUALITY,
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Earth to Mars
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ENERGY MINIMA
DEPARTURE ARRIVALVALUE TYPE (YEAR/MONTH/DAY) (YEAR/MONTH/DAY)
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C 3 L 7.8538 II 2001/04/16 2002/01/27
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VHP 3.7766 II 2001/01/13 2001/10/08
PRECEDING PAGE BLANK NOT FILMED
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Earth to Mars
2002/3
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ENERGY MINIMA
DEPARTURE ARRIVALVALUE TYPE (YEAR/MONTH/DAY) (YEAR/MONTH/DAY)
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C3L 12.564 II 2003/05/10 2003/12/29
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