30 JUNE 1967 05952-H223-RO-OO
TRW NOT E NO . 67- FM T- 52 1
PROJECT APOLLO
TASK MSC/T RW A-89
SITE ACCESSIBILITY ANALYSIS
FOR ADVANCED LUNAR MISSIONS
FINAL REPORT
VOLUME I
SUMMARY
Prepared for
Advanced Spacecraft Technology DivisionNational Aeronautics and Space Administration
Manned Spacecraft Center
Houston, TexasContract NAS 9-4810
TRWSYSTEMS
30 JUNE 1967
TRW NOTE NO .67-FMT-521
PROJECT APOLLO
TASK MSC/TRW A-89
SITE ACCESSIBILITY ANALYSIS
FOR ADVANCED LUNAR MISSIONS
SUMMARY
VOLUME I
05952-H223-RO-00
Prepared forMISSION PLANNING AND ANALYSIS DIVISION
NATIONAL AERONAUTICS AND SPACE ADMINISTRATIONMANNED SPACECRAFT CENTER
HOUSTON, TEXAS
Contract NAS 9-4810
Prepared by ~~H. Patapoff,Senior Staff EngineerMission Trajectory
Control Program
Approved by ~R. W. J son, ManagerMission Design and AnalysisMission Trajectory
Control Program
ACKNOW LEDG EMENT
Acknowledgement is given to the following per
sonnel: Mr. J. P. O'Malley, who was responsible for
the development of the two-impulse optimization pro
gram; Dr. P. A. Penzo, who formulated the mission
analysis graphical procedure; and Mr. S. W. Wilson,
who generated the translunar and transearth velocity
data.
ii
FOREWORD
This final technical report is subm.itted to NASA/
MSC by TRW System.s in accordance with Task A- 89
of the Apollo Mission Trajectory Control Program.,
Contract NAS 9-4810.
This report consists of two volum.es, each of
which is self- contained. Volum.e I sum.m.arizes the
results of the two-im.pulse study and presents a sim.
plified version of the graphical m.ethod for determ.ining
approxim.ate lunar areas of accessibility for m.ission
planning purposes. Volum.e II presents a com.plete
description of the two-im.pulse s cherne , including the
detailed graphical m.ethod of determ.ining lunar site
acces sibility.
In order to m.inim.ize the inclusion ofnon-essen
tial data in these two volum.es, several internal reports
were docum.ented under this task and are available on
request.
iii
CONTENTS
Page
4.2.24.2.34.2.44. 2. 54. 2.6
1. INTRODUCTION•••••••••
2. TRAJECTORY GEOMETRY.
2. 1 Physical Model •••••
2.2 Two-impulse Transfer
3. GROUNDRULES AND ASSUMPTIONS.
4. SITE ACCESSIBILITY ANALYSIS •••
4. 1 Acces sibility Georn.etrical Constraints ••
4. 2 .6.V Acces sibility Constraints ••••••
4. 2. 1 Translunar and Transearth .6.VRequirern.ents •••••••••••CSM Orbit Stay Tirn.e ••••••CSM Orbit Plane Changes •••Spacecraft Perforrn.ance Capability.CSM Continuous Abort Re qui r ernent ,Exarn.ple ~V .Const-rCl. int Curves .
1-1
2-1
2-2
2-2
3-1
4-1
4-1
4-8
4-94-104-114-134-154-19
5. BASIC MISSION ANALYSIS PROCEDURE • • • • 5-1
5. 1 Graphical Procedure ••••••••••••••••• 5-1
5.1.1 Specific Site. .. • • • • • • • • • • • • 5-15.1.2 Accessibility Contour Generation 5-6
5. 2 Mis sian Analysis Considerations. • • • • • 5-10
5.2. 1 Accessibility Contour Generation 5-105.2. 2 Specific Site Analysis. • 5-115.2. 3 Pararn.eter Optirn.ization 5-115.2.4 Mission Trade-offs. • •• 5-11
6. CONCLUSIONS AND REMARKS. • • • • • • • • • • • • • • • • • • • • 6-1
APPENDIX ••••
REFERENCES ••
v
A-1
R-1
ILLUST RA TIONS
Page
2-1 Earth-Moon Patched Conic Geometry
2- 2 Two-impulse Trajectory Profile
4-1 Lunar Orbit, Site Geometry .•.•
4- 2 Lunar Orbit, Plane Change Geometry ..
4-3 Example Geometric Constraint Curves
4-4 I-l, i and r2d Geometry ..........•..
4-5 Plane Change Geometry; 8 Exceededrn
4-6 ' Right Boundary Plane Change Geometry
4-7 Left Boundary Plane Change Geometry
4- 8 Zero LM Plane Change Geometry ....•
4-9 Zero LM Plane Change; i versus r2d
for Various SurfaceStay Time s. . • . • • . . . . . . . . . . . . . . . . . . .
4-10 Translunar and Transearth f::,.V for 96- and 72-HourFlight Times, Respectively ... .......•.•.•...
4-11 Lunar Orbit and Earth Moon Geometry at LOI and TEl.
4-12 CSM Plane Change Geometry ..
2-5
2-6
4-2
4-4
4-20
4-5
4-6
4-21
4-21
4-8
4-22
4-23
4-25
4-25
4-13
4-14
4-15
4-16
4-17
Translunar f::,.V versus k .
Ln X versus X .
Spacecraft Performance Capability ..
Sample Cases; or Continuous Abort Without CSM PlaneChange . . . . . . . . . . . . . . . . . . . . . .
Sample Cas e s ; or Continuous Abort With CSM PlaneChange .•.............•.... . .
4-26
4-27
4-28
4-29
4-17
4-18
4-19
Maximum Allowable Lunar Surface Stay Time versus SiteLatitude for Various LM Plane Change Capabilities. . . . . . A-1
Geometric Constraints for LM Plane Change Capabilityof 2 Degrees and Surface Stay Times of 1, 2, 4, 6, 10,and 12 Days . • . • . • . . . . . . . . . . . . . . . . . . . • • . . . . . A-2
.v i i
4-20
4-21
4-22
4-23
4-24
ILLUSTRATIONS (Continued)
Page
Geometric Constraints for LM Plane Change Capabilityof 4 Degrees and Surface Stay Times of 1, 2, 4, 6, 10,and 1 2 Days . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A - 3
Geometric Constraints for LM Plane Change Capabilityof 8 Degrees and Surface Stay Times of 1, 2, 4, 6, 10,and 1 2 Days . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . A-4
Geometric Constraints for LM Plane Change Capabilityof 12 Degrees and Surface Stay Times of 1, 2, 4, 6, 10,and 12 Days A-5
Geometric Constraints for LM Plane Change Capabilityof 20 Degrees and Surface Stay Times of 2, 4, 6, 10,and 12 Days . . • . . . . . . . . . . . . . . . . • . . . . . . . A-6
Translunar .6.V Requirements; 60-Hour Flight Time A-7
4-25 Translunar 1:::..V Requirements; 72-Hour Flight Time A-8
4-26
4-27
4-28
4-29
4-30
4-31
4-32
4-33
4-34
4-35
4-36
4-37
4-38
Translunar .6.V Requirements; 84-Hour Flight Time A-9
Translunar .6.V Requirements; 96-Hour Flight Time A-1O
Translunar .6.V Requirements; 108-Hour Flight Time .. A-11
Translunar .6.V Requirements; 120 -Hour Flight Time .. A-12
Translunar .6.V Requirements; 132-Hour Flight Time .. A-13
Transearth .6.V Requirements; 60-Hour Flight Time A-14
Transearth .6.V Requirements; 72-Hour Flight Time A-15
Transearth .6.V Requirements; 84-Hour Flight Time A-16
Transearth 6.V Requirements; 96-Hour Flight Time A-17
Transearth !:::..V Requirements; 108-Hour Flight Time .. A-18
Transearth .6.V Requirements; 120 -Hour Flight Time .. A-19
Transearth !:::..V Requirements; 132-Hour Flight Time .. A-2O
CSM Plane Change Angle versus Surface Stay Time forVarious Site Latitudes; 2-Day Total Stay Time for ZeroLM Plane Change Geometry . . . . . . . . . . . . . . . . . . . . . A- 21
viii
ILLUSTRATIONS (Continued)
Page
4-39
4-40
4-41
4-42
4-43
4-44
4-45
CSM Plane Change Angle versus Surface Stay Tim.e forVarious Site Latitudes; 4-Day Total Stay Tim.e for ZeroLM Plane Change Geom.etry .
CSM Plane Change Angle versus Surface Stay Tim.e forVarious Site Latitudes; 6-Day Total Stay Tim.e for ZeroLM Plane Change Geom.etry .. . .
CSM Plane Change Angle versus Surface Stay 'I'irne forVarious Site Latitudes; 8-Day Total Stay Tim.e for ZeroLM Plane Change Geom.etry .
CSM Plane Change Angle versus Surface Stay Tim.e forVarious Site Latitudes; 10-Day Total Stay Tim.e for ZeroLM Plane Change Geom.etry . . . . . . . . . . . . . . . . . .
CSM Plane Change Angle versus Surface Stay Tim.e forVarious Site Latitudes; 12-Day Total Stay Tim.e for ZeroLM Plane Change Geom.etry . . . . . . . . . . . . . . . ...
CSM Plane Change Effect Upon Inclination and Node ofCSM Orbit versus Surface Stay Tim.e for Various SiteLatitudes; 2-Day Total Stay Tim.e for Zero LM PlaneChange Geom.etry . . . . . . . . . . . . . . . . . . .. ". . ...
CSM Plane Change Effect Upon Inclination and Node ofCSM Orbit versus Surface Stay Tim.e for Various SiteLatitudes; 4-Day Total Stay Tim.e for Zero LM PlaneChange Geom.etry .
A-22
A-23
A-24
A-25
A-26
A-27
A-28
4-46
4-47
4-48
4-49
CSM Plane Change Effect Upon Inclination and Node ofCSM Orbit versus Surface Stay Tim.e for Various SiteLatitudes; 6-Day Total Stay Tim.e for Zero LM PlaneChange Geom.etry . . . . . . . . . . . . . . . . . . . . . . . . . . .. A-29
CSM P'l.ane Change Effect Upon Inclination and Node ofCSM Orbit versus Surface Stay Tim.e for Various SiteLatitudes; 8-Day Total Stay Tim.e for Zero LM PlaneChange Geom.etry . . . . . . . . . . . . . . . . . . . . . . . . . . .. A-30
CSM Plane Change Effect Upon Inclination and Node ofCSM Orbit versus Surface Stay Tim.e for Various SiteLatitudes; 10-Day Total Stay Tim.e for Zero LM PlaneChange Geom.etry . . . . . . . . . . . . . . . . . . . . . . . . . . .. A- 31
CSM Plane Change Effect Upon Inclination and Node ofCSM Orbit versus Surface Stay Tim.e for Various SiteLatitudes; 12-Day Total Stay Tim.e for Zero LM PlaneChange Geom.etry . . . . . . . . . . . . . . . . . . . . . . .. A- 3 2
ix
4-50
4-51
ILLUSTRATIONS (Continued)
Page
CSM Plane Change 6.V versus Plane Change Angle forSO-nautical mile Circular Orbit. . . . . . . . . . . . . . .. A-33
Possible Lunar Parking Orbits, Continuous Abort,14-Day Minimum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-34
5-1
5-2
5-3
5-4
5-5
5-6
6.V Margin versus Surface Stay Time; AristarchusExample 1 .
6.V Requirements versus Flight Time; AristarchusExample 2 ...••...•........•...••....
Procedure Example for Generating 6.V Constraint Curve
6.V Constraint Curve for Example Mission .
Procedure Examples for Generating Site AccessibilityContour .
Site Accessibility Contour for Example Mission
x
5-13
5-13
5-14
5-15
5-16
5-17
1. INTRODUC TION
This summary volume presents a simplified and relatively rapid
mission analysis procedure related to lunar site accessibility. It is a
graphical procedure oriented towards use by the mission planner at the
management level.
The simplified mission analysis procedure, in conjunction with the
graphical data included in this summary volume, will provide sufficient
accuracy to allow the mission planner to develop insight into the relation
ships between lunar site accessibility and mission requirements and con
straints. This allows the mission planner to coordinate these relationships
for effective mission design. Mission considerations may include
• The geometrical relationships and constraints betweensite accessibility and
LM surface stay time
LM plane change capability
LM abort requirements
CSM orbit requirements
CSM plane changes
• The 60V relationships between site accessibility and
CSM orbit stay time
Abort requirements
CSM plane changes
Translunar and transearth flight times
Spacecraft performance capability
• The optimization of various parameter s
Service module propellant (60V)
Mission duration
Translunar or transearth flight time
Surface stay time
1-1
The rni s ai on analysis procedure presented here consists of three
basic steps: (1) the dete rrrrina.ti on of the various g e ornet r i c a.I constraints
upon site accessibility, (2) the dete r rnina.tion of the D.-V constraints or
r e qui r errierrt s upon accessibility, and (3) the graphical procedure which
consists of the rna.nipuLati o n or interpretation of the results of the first
two steps to provide the answer or data for the specific rni s s i on considera
tion. These basic steps are discussed in Section 4. The detailed pro
cedure for two e xarripIe cases is presented in Section 5.
The translunar and transearth velocity data presented here and
used in the graphical procedure, .repre sent optimized two-impulse
transfers to and from the moon. Thes e transfers provide considerable
savings in SM fuel when compared with single-impulse transfers which
are presently planned for the Apollo mission. A complete description of
the optimization technique, the computer program, and the two-impulse
data generated with this program may be found in Volume II. The mission
analysis procedure, however, is independent of the mode of orbit transfer.
The only r e qu i r errient is velocity data in the proper format, whether it be
single -impulse or multi-impulse.
It is r e corrrrn e nde d that the mission analyst who will be concerned
with the more refined aspects treated in Volume II (Reference 1) read the
introductory discussion of site accessibility and description of the graphi
cal mission analysis procedure in this sUITlmary volurne prior to reading
Volume II.
1-2
2. TRAJECTORY GEOMETRY
The trajectory profile assumed for the lunar missions discussed
here is closely related to that of the Apollo mission. The significant
difference, which allows considerably more accessibility at the moon and
longer LM surface stay times, is the greater flexibility allowed for the
translunar and transearth transfer trajectories. For example, the free
return circumlunar constraint on the translunar phase has been removed. >:~
The effect of this constraint is to force the approach hyperbola at the moon
to lie near the rno on ' s equator (within 15 degrees) thus requiring lar ge yaw
penalties at lunar orbit insertion (LOI) to achieve higher latitudes. In
addition to this yaw penalty, the translunar flight times for circumlunar
trajectories are relatively short (65 to 85 hours) resulting in higher
approach velocities at the moon when compared with the longer flight times
(up to 132 hours) for the non-free return trajectories.
Additional reduction in the SM fuel requir ements is obtained by
allowing the CSM to perform a maneuver between translunar injection
(TLI) and LOr. This maneuver has been defined here as the two-impulse
transfer; the TLI is the first and LOI the second. A similar additional
impulse may be used after transearth injection (TEl) to also reduce the
fuel requirements to return to earth. The optimization of these two
impulse transfers have been performed and the data are presented in
Section 4. They will be briefly described in this section; however, a com
plete discussion may be found in Volume II.
All other Apollo trajectory constraints remain essentially unchanged,
including the launch and earth-orbit phase and the reentry phase. The
specific ground rules and assumptions have been listed in Section 3.
,,~
"It is possible to maintain the free-return constraint for a considerabletime after trans lunar injection along a high pericynthion circumlunar trajectory, and then utilize an SM impulse in the earth phas e to get on one ofthe optimum two-impulse trajectories described here. The degradation insite accessibility utilizing this "thr e e-dmpul s e " mode would be negligible.(Reference 2)
2-1
2. 1 PHYSICAL MODEL
The physical model assumes that the trajectory consists of patched
conics as depicted in Figure 2-1. Thus, the moon I s gravitational field
extends out to a distance of approximately 30, 000 nautical miles. This
limit is represented by a sphere whose center is at the moon and which
will move with the moon. Since the earth and s un ' s gravity is neglected
within this "sphere of action" (MSA), (Reference 3), all spacecraft free
flight motion can be repres ented by conic sections with the moon at one
focus. Thes e are called rno onpha s e conics. A similar situation exists
outside the MSA where it is assumed that only the e a.r ths gravitation is
important. Here, the conics will be earth centered and hence, called earth
phase conics, as indicated in Figure 2-1.
A complete translunar trajectory is generated by patching an earth
centered and a moon centered conic at the MSA so that they have the same
position and velocity at this point (Point B in Figure 2-1). The seeming
discontinuity at this point is caused by the relative motion of the MSA
(and hence, the conic within it) with respect to the earth centered conic.
Thus, in order to ensure continuity in the velocity vector at Point B, the
rnoon ' s velocity relative to the earth must be subtracted from the vehicle's
velocity relative to the earth to obtain the vehicle's velocity relative to the
moon. This is depicted in the velocity vector diagram shown in the lower
right-hand corner.
2. 2 TWO-IMPULSE TRANSFER
The two-impulse transfers to and from the moon are shown in Figure
2-2. It has been shown (References 1, 4) that minimum total t::.V require
ments will generally occur when both impulses are within the MSA. With
this restriction, the problem for the trans lunar transfer may then be
stated as follows:
Given a fixed day of launch (or lunar distance), trans lunarflight time from T LI to LOI, and a fixed inclination andnode of the 80-nautical mile CSM parking orbit, find theminimum total two -impuls e t::.V r equir ed in the MSA toenter this parking orbit.
In generating the two-impulse data, it was assumed that launch conditions
at the earth are a 90-degree azimuth from Cape Kennedy and translunar
2-2
injection from a 1OO-nautical mile parking orbit. The launch opportunity
chosen is the one resulting in the earth phase trajectory lying nearly in
the rno ons orbit plane. Reentry conditions are identical to those presently
planned for Apollo. That is, the trans earth trajectory is tar geted to
reentry at 400, 000 feet with a velocity path angle of -6.4 degrees.
Touchdown is as sumed to be at the center of the Apollo footprint. Also,
the earth phase conic is assumed to lie nearly in the rrio orrl s orbit plane. >:<
A planar view of the two-impulse transfers to and from the moon are
shown in Figure 2- 2. The patching point B, discus sed in Figure 2-1, is
also shown here. Considering drawing (A) first, the translunar injection
is targeted to a pericynthion altitude which may vary from 40 nautical
miles ( a lower limit constraint) to 26, 000 nautical miles. This altitude is
called the virtual pericynthion (Point C). The targeted moon phase incli
nation may also vary from a to 180 degrees. The first impulse within the
MSA (shown as B ') may be anywhere on this moon centered hyperbola,
including beyond virtual pericynthion. Also, this maneuver may be out-of
plane as required to intersect the desired parking orbit at Point C '. The
pericynthion altitude is restricted to lie between 40 and 80 nautical miles.
The second impulse (LOl) which may also be out-of-plane occurs at C '.
For a fixed flight time from TLl to LOl and a fixed CSM orbit, the
optimization to minimize the sum of the two impulses at Point B' and C'
is performed by varying the virtual pericynthion altitude at C, the inclina
tion of this hyperbola, the position of the first impulse (Point B') and the
position of the LOl on the orbit. Also, the flight time to virtual pericyn
thion is varied to ensure that a true minimum velocity is found. Thus,
this optimization represents a five parameter search.
A precisely mirror image situation occurs for the optimization of the
two-impulse trans earth transfer shown in drawing (B). The first impulse
-'-
"'This assumption, which has been made throughout this study, does notconsiderably degrade the two-impulse results which are presented. It canbe shown (Reference 1 ) that an out-of-plane launch or reentry perturbs theapproach (or return) hyperbolic moon centered asymptote by less than 5degrees. The actual effect of this perturbation on ~V may be found by atechnique presented in Volume II.
2-3
(TEl) will occur at B I and the second at C I. The optdrni z afion parameters
are identical with those for the translunar case. One variation is that if
TEl occurs past pericynthion, the 40-nautical mile altitude constraint on
this hyperbola need not be imposed. A similar argument applies to the
second or outer hyperbola from C I and on.
The complexity of this optimization problem requires that short
cuts be taken whenever possible and justified. One, mentioned above, is
that it is sufficient to consider that the earth centered conics lie ess en
tially in the moon's orbit plane. Two others are based on symmetry. The
first is that symmetry exists relative to the moon's orbit plane, so that
the two-impulse results for a given CSM orbit will be the same as the
results of a similar orbit where the ascending and descending nodes are
interchanged. Thus, the two-impulse 6.V requirements for a given orbit
will be the same for an orbit of the same inclination with the node dis-
placed 180 degrees.
Second, if the moon is at apogee (the results presented here are for
this situation), symmetry exists between translunar and trans earth trans
fers for a given flight time. The only difference in the earth phase conics
will be perigee distance; i. e., 100 nautical miles for the translunar and
approximately 20 nautical miles (vacuum) for the trans earth. The effect
of this variation on the two-impulse velocity requirements, however, does
not warrant completely reoptimizing the trans earth transfers. Using
symmetry, then, the translunar two-impulse results for a CSM orbit
whose node and inclination are ~* and i* (see footnote below), respectively,c c
will be equal to the transearth velocity requirements for a CSM orbit with
a node location of -~~< and inclination of i* (same) for the same flight time.c c
The two-impulse translunar and transearth optimized 6.V are pre
sented in a graphical form suitable for use in the mission analysis proce
dure.
~~ and i* are defined in Volume II to be the node and inclination in moono r bi t plcfne coordinates. However, for the simplified procedure, lunarlibrations and the inclinations of the moon's equator to the rno cn ' s orbitplane are neglected, so that ~~ and i>:c are equivalent to node longitude andinclination with respect to the lunar ~quator.
2-4
A.
TRA
NSL
UN
AR
INJE
CT
ION
POIN
TB
.IN
TE
RSE
CT
ION
WIT
HM
OO
N'S
SPH
ERE
OF
AC
TIO
N(M
SA)
C.
PER
ICY
NT
HIO
NO
FEX
TER
NA
LH
YPE
RB
OLA
N I U1
MO
ON
'SM
OT
ION
EART
HPH
ASE
CO
NIC
po
SEL
EN
OC
EN
TR
ICE
XT
ER
IOR
HY
PER
BO
lAM
OO
N'S
SPH
ERE
OF
AC
TIO
N
----.....
......
~'~~/
-:~\ ,
V(V
EH/E
AR
TH)
VE
LO
CIT
YD
IAG
RA
MA
TM
SA
Fig
ure
2-1
.E
art
h-M
oo
nP
atc
hed
Co
nic
Geo
rne
try
MSA
EART
HPH
ASE
CO
NIC
-t
VIR
TUA
LPE
RIC
YN
TH
ION
®U
NU
SED
POR
TIO
NIN
JEC
TIO
N-,
....
,O
FM
OO
NPH
ASE
/,
CO
NIC
PER
ICY
NT
HIO
N/,
/O
FTR
AN
SFER
/1S
T"
CO
NIC
~IMPUlSE'
--r-
-f_
/(T
EI)
\,
B'\ \ \
/'\
I
TRA
NSF
ER~C
TR
AJE
CT
OR
Y/
2ND
IMPU
LSE
LUN
AR
OR
BIT
DIR
EC
TIO
NO
FM
OO
N -
MSA (A
)(B
)TR
AN
SLU
NA
RTR
AN
SEA
RTH
VIR
TUA
L
UN
USE
DPO
RT
ION
/PE
RIC
YN
TH
ION
OF
MO
ON
PHA
SE©~----
CO
NIC
",\
---
~/'
"\
--~/
2ND
\PE
RIC
YN
"YH
ION
/IM
PULS
Er
OF
TRA
NSF
ERI
(LO
I).....
.........
,...~-
,......._
_C
ON
IC
~:~6
Ig~/
C'~
D;~',
__
_I I I
B'<
"'TR
AN
SFE
R'"
TRA
JEC
TOR
Y1S
TIM
PULS
E--------,
rLU
NA
RO
RB
IT~
~MOON
PHA
SEC
ON
ICN I 0
'
Fig
ure
2-
2.T
'wo
-drr
rpu
lse
Tra
jecto
ryP
rofi
le
3. GROUNDRULES AND ASSUMPTIONS
The data that are used or derived for use in the graphical procedure
are affected by the groundrule s and assurnptions listed below. However,
the accuracy achieved by simplifying the procedure is sufficient for mis -sion planning purposes. The groundrules and assumptions are listed as
follows:
a A Cape Kennedy launch at a 90-degree azimuth, with an injection into the translunar trajectory during the third earth parking orbit
a Earth phase of the translunar and transearth trajecto-ries lies nearly in the moon's orbit plane
Virtual pericynthion altitude between 40 to 26, 000 nautical miles
0 The moon is at maximum (and assumed constant) distance from the earth at the time of LO1 and TEI.
@ Time from translunar injection to lunar parking orbit and time from transearth injection to earth reentry are varied in i2-hour increments from 60 to 132 hours in generating the AV data.
* a Only retrograde orbits are considered.
a Lunar parking orbit altitude 80 nautical miles
e A patched conic trajectory model has been used.
0 No midcourse corrections are provided for the AV data.
e Inclination of moon's equator to the moon's orbit plane is assumed zero.
Lunar librations are neglected.
.I.'1%
Although a slight AV advantage may be attained by a choice of a retrograde or posigrade orbit for a given mission, posigrade orbits are not considered due to the fact that surface stay time is reduced because of nodal regres -sion. Also, this AV advantage is not as significant as the AV differences associated with the assumptions made in the simplified mission analysis procedure (i.e. , constant earth moon distance, no lunar librations, and zero inclination of moon's equator to moon's orbit plane).
0 The translunar AV for a given orbit is the same as that for the transearth AV required of an orbit whose inclination is the same and whose node is negative that of the translunar orbit (i.e m ,mirror image).
a CSM lunar orbit inclination and node variations due to the moon1s oblateness are neglected.
*< Only northern site latitudes are considered.
Modifications (for use with the analysis procedure) to the data to account
for variations in earth-moon distance and lunar libra,tions a re discussed
in Volume 11.
.I.'1.
A mirror image symmetry exists so that an analysis of a southern latitude site is made by assuming that it i s a northern latitude. All geometrical constraints, plane change magnitudes (the direction of plane changes, however, are reversed), and AV requirements are the same. However, for southern latitudes, the CSM orbit ascending node is displaced 180 degrees (the AV requirement is the same), from that for a site of the same longitude and a northern latitude of the same magnitude. This i s apparent from the diagram below:
N O R T H E R N SlTE
4 LUNAREQUATOR
~OUTHERNSlTE
4 . SITE ACCESSI BIL ITY ANA L YSIS
The re a re three l o g i c a l steps in site access ibility analysis. which i s
the basis for t h e graphical mission analysis procedure des cribed in
Section 5. They are as follows :
Step I T he determination of geometrica l constraints(o r requirements ) upon site ac cessibil ity
Step II The dete rmination of CSM .t:::.V constraints uponsite access ibi l ity
Step III T h e i nt e r p r eta t i on of t he r e sult s of Steps I and IIin d ete r minin g sit e a ccessibility for a given sitea nd mission, o r the generation of c on t ou r s oflu nar s u r face acces s ibility
Step I consists basically of determini ng the CSM o r b it s that w i ll
assure LM-CSM rendezvous capabil ity fo r a given s ite l a tit ud e, surface
stay t ime and L::.V capability.
Step II cons ists of de t e r m i n i n g what CSM o r bits are achievable fo r
a given rrri s s i cn profile a nd spacecraft f:J.. V cap abi lity.
Step III is the g raphical p rocedu re in which t he data f r- om Steps I and
II are interpreted t o dete r m ine accessibility.
B ef o r e d i s cus sing t h e graphical procedure i n detai l, it wi ll b e neces
sary (t o fully unders tand th e mts s ion analysis procedu r e ) to discuss th e
r e lation s hip s between rrri s e i on r e quire m e nt s or const ra i nts and Ste p s I and
II.
4 . 1 ACCESSIBILITY GEOMETRICAL CONS T RAI NTS
The re a re specific geornet r ical relationships between la ndin g site
la t itude, LM s tay time, CSM orbit inclination, L M a bort r e q u i r eme nts and
•p lane c ha n ge capability • so that ce rtain constraints exis t upon s ite acces-
s ibility. It i s ess e ntial, in unders tanding t he miss ion analysis procedure,
•A lthough L M plane c hange capability is actually a p e rfor-mance Hrntta t i on ,it is equivalent to a ge ornet r -ic accessibil ity c on s t ra int . s ince th e L M is aseparate stage a nd . therefore, is not i n clu de d in the CSM t:J.V optimization.
4 - 1
that the nature and reasons for these geometrical constraints be well
unde rs tood.
One bas ic geometrical relationship is that at the time of LM landing
(the LM may des cend to the lunar surface after a few revolutions of the
CSM orbit following LOl, or even several days afterwards), the landing
site lies in the plane of the CSM orbit. From this point on (unless the site
is at one of the poles, or the CSM orbit inclination is zero- - the site there
by being on the equator). the site will drift eastward out of the CSM orbit
plane. This relative drift is caused by the rotation of the moon about its
axis (13.2 degrees per day). *
The various geometrical relationships can be understood with the aid
of Figures 4- 1 and 4- 2. Figure 4- 1 shows a typical orbit- site geometry.
Points A and B correspond to the position of the landing site at LM arrival
and departure, respectively. Shown are four retrograde orbits passing
through the site at arrival.
A SITE AT ARRIVALB SITE AT DEPARTUREIJ SITE LATITUDE
Figure 4-1.
LUNAREQUATOR
Lunar Orbit-Site Geometry
*Nodal regression caused by the moon's oblateness is neglected in thesimplified mission analysis procedure.
4-2
Orbit a is the rrrinirnurn inclination CSM orbit in which the incLin
ation* i is equal in magnitude to the landing site latitude, p., Orbit b is
the maximum inclination (polar) orbit; whereas, c and d are orbits of
intermediate inclinations. Qd is the longitudinal displacement of the
ascending node of the CSM orbit relative to the landing site longitude mea
sured eastward from the site longitude. The angle .6.>"'s is the eastward site
longitude displacement corresponding to the surface stay time. Since all
poss ible CSM orbits correspond to a rotation about the about the initial .
site vector (the radius vector through the site at arrival), ~ is defined for
physical clarity to be the angle between the CSM orbit and the site meridian
at arrival (see Figure 4-1).
During the time that the LM remains on the moon, the d ihe d ral angle
8e between the vector through the site and the CSM orbit plane will vary.
Beginning with the time of descent, as stay time increases the site will
move eastward, and the plane change will initially increase and then vary
depending upon the geometry. e will be a function of latitude, inclinatione
and stay time, but not of site longitude.
The equation for ee
**can be shown to be:
where
= cos fJ. [ s in fJ. s in S (1 - cos.6.>'" ) - s in.6.>'" cos S ]s s(1)
sin S cos fJ. = cos i (2)
Figure 4- 2, which is a view of Figure 4- 1 looking down upon the
north pole region, depicts the geometrical relationships between the CSM
orbit, surface stay time and plane change capability for a given site
*The classical definition of inclination will not be used here, but willalways be taken between 0 and 90 degrees and the orbit indicated asretrograde.
**It should be noted that 8e can have negative values as well as positivevalues. This merely means that 8e is positive if the plane change as measured from the orbit to the site has a northward component.and is negativeif it is heading southward.
4-3
latitude. For a given stay tim.e (corresponding to a site longitudinal dis
placem.ent of ~A. ) and a m.axim.um. plane change capability e , it is seensm. .
from. Figure 4-2 that orbit 1 is the highest CSM orbit inclination possible
that will satisfy the continuous abort capability requirem.ent. This corre
sponds to point a in which the m.axim.um. plane change 8 occurs. On them.other hand, orbit 2 is the lowest inclination orbit that will satisfy the abort
requirem.ent. This corresponds to the m.axim.um. plane change e occur-m.
ring at the point of LM departure B. It becom.es apparent, then, that all
CSM orbits lying between 1 and 2 of Figure 4- 2 will satisfy the continuous
abort requirem.ent so that there will be, in general, a range of CSM orbits
that will satisfy the stay tim.e and plane change requirem.ents for a given
site latitude.
*Figure 4- 3 is a typical plot depicting the geom.etrical relationships
described above. These curves are generated from. Equations (1) and (2).
Figure 4- 3 can be related to Figure 4-2 as follows: First, it is noted that
i , fl, and 0d are related as shown in Figure 4-4, so that following a line of
constant fl (dotted curves of Figure 4- 3), i will vary as shown in Figure
4-4 as 0 d increases from. zero to 180 degrees. Consider point a of Figure
4- 3.SITE AT
B/DEPARTURE
Figure 4- 2. Lunar Orbit- Plane Change Geom.etry
)'C
. The geom.etrical constraints curves are presented in i- Od coordinatesfor use in the m.ission analysis graphical procedure described in Section 5.
4-4
LAND ING SITE AT ARRIVAL
,---CSM ORBITS
Figure 4 -4 . fJ. . i and 0 d G eom e t ry
T h is c o rre spon d s to the maximum CSM orbit inclinat ion that will prov ide
con t inuous abort capabil it y for a site latitude of 20 d e grees , plane c hange
capability of 4 de g r e es, and a surfac e stay time of 5 days. Point " a l l
c orresponds to orb it 1 of Figur e 4- 2 . Following t h e I-l = 20 -de g r e e curve
from point a to point b of Figure 4- 3, all CSM o rbits t h a t satisfy t h e 5-day
stay time requir e m ent are traversed. T his trav ersal c o r r e s p on d s to
increasing ~ from ~ 1 to ~ 2 as shown in F i gu r e 4-2. Referring to F i gu re
4 - 1, it i s se en t hat thi s change in ~ l owe r s the orbit in c lina ti on and shifts
the n ode s westward . This is also apparent from F igure 4-3 .
It is noted from F igure 4 - 3 t hat there ar e site latitudes which are
n ot obtainable for large r surface stay t imes. This is simply a result of
the f a c t that the total plane change variation exceeds the p lane chang e capa
bility for that give n stay time. This is d epicted in F igure 4- 5. For a
maximum plane chang e capability of 8 ,the continuous abort capability i sm
satisfied for t he stay time corresponding to the site trav ersal from point
A t o point b. However, the p lane change c apabil ity is exceeded from
point b to B. It is obvious that no CSM orbit w i ll satisfy this geom etry,
unless the p la n e chang e capability i s increased .
4-5
Figure 4 - 5.
B
Plane Change G eometry - 8 Exceededm
The right boundary c u r v e of Figur e 4-3 cor responds t o orbit 1 o f
F igure 4 -2. This b oundary curve will also r emain the s ame for sta y
times of 5 to 8 days . T his b ecom es a p p a r e n t by c onsidering F i g u re 4 -6.
Orbit 1 and point B corr e spond t o the stay time in which t he plane change
eq u a ls t h e maximum c a p a bilit y BIT} " It i s s e e n that this geo m e t r y rema i ns
f ixed f o r tha t r a n g e of stay t imes from po int B t o B ", The l eft boundarie s
of Figur e 4- 3 correspond to orbit 2 of F igure 4-2. Consider F igu re 4-7:
f or the stay time corresponding to point a , the range of allowabl e CSM
o r b i t s lies betwe en o rb its 1 and 2. F o r a l o ng e r stay line co rre s pon d in g
t o po int b, it i s seen t h at th e range b e comes smaller and boundary 2
a p p r oac he s boundary 1 to position 2 ' (the l e ft c u r v e of F igure 4 - 3 shifts
towards the right bou n da r y ) . As stay time increas es t o that cor res ponding
to point B, t hen ther e is only one CSM o r b it that will satisfy the geo me t r y .
T h i s corre s ponds t o t h e interse ction o f the b oundaries . L onger stay
times t hen become impossible for that geometry. Figur e 4 -18 i s a p lot
r e lating maximum allowable stay t imes a s a function of sit e l a t i t u d e f o r
various plane cha nge capa bilit y values .
4 -6
To eliminate the neces sity of generating the geometrical constraint
curves for the graphical mission analysis procedure, graphs * correspond
ing to Figure 4-3 have been constructed for various stay times and plane
change capabilities and appear in Figures 4-19 through 4-23. Figures
4-18 through 4-51 are located in the appendix of working graphs.
Consider the case in which the LM plane change capability is small
or zero. Essentially all necessary plane changes must then be made by
the CSM (CSM plane changes are discussed in Section 4. 2.3). Referring**to Figure 4-8, it is seen that the geometry for this simple case becomes
obvious. For a given site latitude l.l. and a given stay time ..Q.A , the sym-s
metry depicted in Figure 4- 8 must exist so that the site drifts into the
CSM orbit plane of the end of the desired stay time. All the CSM orbit***nodes must be coincident for a given stay time.
Figure 4- 8 is depicted in Figure 4- 9 in a form consistent with the
graphical mission analysis procedure in which the CSM orbit inclination i
is plotted as a function of the ascending node longitude displacement 0 d
from the landing site longitude. Figure 4-9 shows that the locus of points
satisfying the geometry of Figure 4- 8 is a vertical straight line for a given
stay time. For example, if a 6-day stay time is desired, then the CSM
orbit node must lie 130 degrees east of the site longitude. If a site lat
itude 20 degrees is also desired, then the orbit inclination must be
25 degrees (point A of Figure 4-9).
* .These graphs, which have been extracted from Volume II, also mcludenodal regression.):<):(
In fact this geometry would be desirable. It is not unreasonable toexpect that for any lunar mission, that CSM orbit will be selected whichintersects the landing site at the nominal time of LM ascent. This minimizes LM propellant requirements for ascent. Any plane changes, then,will be made by the CSM or LM only in the event of an abort or a non-nominal LM lift- off time.
***The longitudinal node displacement from the site longitude, 0d is givenby the expres sion 0d = 90 + A\s /2 (deg)
4-7
I:
-
• I
Figure 4-8. Specific CSM O r bit-Site Geomet ry
T h e r e l a tion s h ip be tween the geometrical accessibility constraints
and t;,.V c on s t r aint s now b e c om es a p pa r e nt. F o r a g iven miss ion profile
the a ch ie v e m e n t of a s pec if ic CSM orbit inclination and noda l long itude
will r e q ui r e a specif ic t ot a l CSM tN fo r t r a n s l u nar a nd t r ans e a r t h l'iV a nd
any C SM pla ne change s. 1£ thi s lies within the performance capabil ity of t h e
CSM and satisfies all m ission constraints, t h e n t h e s ite under cons ideration
f o r t h i s m is s ion p r o f ile is deemed accessible. If not , i t is inaccess ible.
Howeve r, accessib ility may possibly be achieved if t h e m i ssion p rofile or
t he capabi lity of t he CSM i s a ppropr iate ly modified.
4 .2 AV ACCESS IBILITY CONSTRAINTS
For a given m ission unde r con sideration. t he l:i.V const raints upon
s ite accessibility w i ll be dictate d b y
• T r a n s l una r t:N requirements
• T r a n s eart h injection I'!V r e q u i r em e n t s
• CSM orbit s tay t ime
4-8
• CSM pla n e change s
• CSM to t a l /iV c a pabi lit y
• CSM a bo r t r equir e m ent s
4 .2 . 1 Tra n s l unar and Transear th (\V R eq ui r e m ents
T h e t r a n s l una r a nd trans e a r t h CSM l\V r equire m e n t s w i ll depend
u pon the inc linatio n a nd node of th e CSM parkin g o rb it a nd th e t ime e l a psed
betwe e n L OI a n d TEl , neglecti ng a t th e moment . any o the r cons t ra i nt s or
r equireme nts .
F i g u r e s 4- 24 through 4 - 37 show the t ransluna r and tra n s earth 6.V r e
qui r ements displayed as inclination ve rsus node fo r va rious values of c on
s tant L:1V. The c u rves are ge ne r a ted for fligh t tim es f r om 60 to 13 2 h ou r s
at 12 -hour inte r vals for retrograde o rbits . The v e locity curves a re r e la
tive to the earth-moon p la ne c oordinates . which for th e ba sic p rocedure
a re as s umed c o i n cide n t with s e lenog raphic coo r d inates.
To u nde rs t a n d the r e la t ionsh i p between th e t r ans l una r .6.V r equire
ments of a given CSM park ing orb it t o the t r a nsea r th .6.V requirements of
the s ame orbit , c on s ide r th e veloc ity curves of F igure 4 - 10 , in wh ich th e
t r a n s l u nar and t r a n s e a rth L:1V curves a re s hown fo r flight tim e s of 96 and
72 h ours, r esp e ctiv ely . If, f o r e xample . it is de sired t o a c hieve a CSM
o rbit with a n inclin a tion of 30 deg rees a n d an as c e nding node l ong it ude of
62.5° Ea st (o r 117. 50 We st) , c o r re s pon ding t o point A , then t h e r equi r e d
t r a n sluna r.li.V wi ll be 38 00 feet pe r s eco nd. If t he CSM o r bi t stay time
i s z e r o (TEl o c cur s imm ediatel y a fte r LOI ), t h e geomet r y at LOI a nd
T E l i s t h e s a m e , so tha t t h e t r a n sea r th .li.V can b e found a t poi nt B (wh ich
cor respond s to t he same orbit as p oi nt A ) t o b e 3400 fe et per second.
T he tra n s luna r and tra n s ea r th 6.V fo r points A 1 a n d B I , whi ch co r res pond
t o a n orbi t inclina tion of 15 d e gre e s a nd as c en din g n ode l ongitu d e of
150 Ea s t (o r 1650 We st ) w ill b e 3300 and 3240 fe et p e r s e c ond , re s pe c
tiv e l y .
F or this s imple case , th e t w o curves can b e ove r layed with c oinci
de nt scales . a nd the t r a ns l una r a nd t r a n s ea r-th zs'V c a n b e read simu l ta
ne ous ly fo r a ny o rbit . Poi nts A and B (and Al w it h B." ] will b e c o incid ent.
Howeve r . for a g iven CSM orb it s t ay t i m e , th e earth- moo n geometry
changes so that the in t erpre t ation of t h e s e curve s m us t b e modified . T h i s
is d is cussed in t h e following section.
4 - 9
4. 2. 2 CSM Orbit Stay Time
During the CSM orbit stay time, the inclination will remain the
same':<; however, the earth moon line (the orbit plane will remain inerti-
ally fixed) will rotate eastward through some angle (or true anomaly, T] ).m
Figure 4- 11 depicts this motion for s orne given orbit stay time. The true
anomaly is the inertial angle that the moon rotates during the orbit stay
time (13.2 degrees per day).
Onc e T] has been determined, it is then possible to associate them
translunar and transearth velocity requirements. For example, assume a
true anomaly T] of 30 degrees corresponding to a CSM orbit stay time ofm
approximately 2.3 days. It is seen from Figure 4- 11 that the earth-moon
line has moved 30 degrees eastward (the orbit node has moved westward
30 degrees), so that at TEl the longitude of the ascending node correspond
ing to point A is now 32.5 degrees East (See Figure 4-10) corresponding
to point C. The trans earth velocity requirement is now 3150 feet per
second. For point A I, the ascending node has moved from 150
East to
150
West longitude (or from 1650
West to 1650
East Longitude), where
the trans earth 6.V required is now 3600 feet per second.
It becomes apparent, then, that if the trans earth velocity curves are
overlayed on the translunar curves with coincident scales, the translunar
and transearth 6.V requirements can be read off simultaneously for an CSM
orbit (any node-inclination combination) for a zero orbit stay time. As
CSM orbit stay time begins to increas e, the origin (which coincides with
the earth-moon line) of the transearth 6.V overlay shifts eastward (to the
right) relative to the translunar plot. For the case above (T] = 30m
degrees) the origin of the trans earth AV overlay is coincident with the 30-
degree longitude of the translunar plot. In Figure 4-10, points A and A '
will become coincident with points C and C 1, respectively. It is noted that
to read off values to the right of T] equal to 30 degrees on the trans lunarm
.,~
'I The inclination will change slightly relative to the earth moon plane as aresult of the moon's oblateness ; however, the change will be at most twodegrees for a 14-day orbit stay time, and is therefore neglected here.
4-10
plot, the left origin of the transearth scale is coincident with the 30-degree
longitude of the translunar plot; whereas, to read off values to the left, the
right origin of the transearth scale is placed coincident with the 30-degree
translunar longitude.
If continuous CSM abort is required for a mission, then the trans
earth i::J..V requirements must be investigated throughout the CSM orbit stay
time to find the maximum. i::J..V condition. This is discussed in Section
4.2.5.
4. 2. 3 CSM Orbit Plane Changes
The consideration of a CSM plane change during lunar orbit can sig
nificantly enhance site accessibility. This, in effect, increases the LM
plane change capability by the amount performed by the CSM. However,
any CSM plane change made while in the lunar parking orbit will not only
reduce the CSM fuel available for TEl but also will change the orbit incli
nation and node position thereby changing the transearth velocity require
ments. In addition, if continuous CSM and LM abort capability is r equired
for the mission, then the t rans ea r th zsV will also be a function of the time
at which the CSM plane change is made. The effects of abort requirements
are discussed in Section 4.2.5.
The basic mission analysis procedure described in Section 5 includes
CSM plane changes for the specific geometry depicted in Figures 4- 8 and
4- 9 in which the initial CSM orbit plane includes the landing site at arrival~c
and at the nominal time of departure. The typical plane change geometry
is depicted in Figure 4- 12. If a plane change is made at some time after*~:c
arrival (point a of Figure 4- 12), the required plane change is ee
resulting in a CSM orbit with a different inclination and node position. It
~c
See second footnote on Page 4-7.~:< >:c
For simplicity, it is assumed that all the plane change is performed bythe CSM. If a combined plane change is performed by the CSM and the LM,then the CSM plane change i::J..V and resulting changes in orbit inclination andnode position will be less than that presented in the graphical data of thisvolume (Satisfactory approximations can be made, however, with this data).Combined plane changes are treated in Volume II.
4-11
IS assumed that the plane change occurs 90 degrees befoe e (o r after ) th e
p o in t of clos e st approach of the CSM to the landing site correspond ing t o
p o int P' of Figure 4 - 12 . It is appa rent from t his figu re that as s tay t ime
increases from th e time of ar rival (a t which time 8 is ze ro and th e optt-e
mum poin t for a plane change is at p oint P ) t o one -half th e t ot a l s tay tim e ,
the plane change ee is a maximum (a nd equa l to i - ~), where
tan i_ tan ~
- cos >"s 72 (3 )
and the optimum p oint fo r a p lane change is on the equator (a t which time a
p lane change results in a cha nge in in clina tio n with no node shift ) . Sym
metry exists for the r e m a ind e r of t h e s tay tim e .
Figures 4 -38 through 4-43 show th e pla n e change a ngle e versuse
time and s i te l a t itud e fo r various stay tim e s . Figures 4-44 through 4-49
s how th e c hanges in the CSM orbit inclination and node longitude ve rsus
t i m e and site la tit ud e for various s ta y t imes . Figure 4- 50 shows the plan e
c hange 1:!.V versus plane change a n gle e fo r an 8 0 -nautical mile circu lare
orbit . The plane change t:N is also given by t h e expression:
Plane change 1:!.V = 10, 58 0 sin 6 12e
(4 )
It should be noted from th e s e figures that as the s tay time app roaches
13 .7 days cor responding to a site longitude displacement of 180 deg rees.
the geometry becomes somewhat unrea listic. A site displacement of 18 0
degrees requires a polar o rbit to satisfy the requirement that th e site a t
arrival a nd nominal departure be in the CSM orb it plane. This means that
a plane change up to 90 deg rees may be required for a n abort. On th e
other hand , if a site latitude of 20 deg rees. f or example, were cons ide red
for a stay time of 13. 7 d a y s . a m inimum inclination orbit of 20 degrees
would r e q u i r e a maximum plane change of 40 degrees .
The use of th e graphical d a t a of Figures 4- 38 through 4-50 is dis
cussed in Sections 4.2.5 .2 and 5. 1. 1.
4 - 12
4. 2.4 Spac e c raft Performance Capab il ity
In order to de te rmine whe the r t h e spacec raft is capab le of satisfying
t h e velocity requirements for a m i s s ion unde r c o n s i d e r a tion , i t i s conven
ient t o rela te the 6.V available for plane changes and trans ea r t h inject ion
to the 6.V required t o achi eve t h e des i red CSM orbit. T his r e lation s h i p
between t r a n s l u nar a nd t r a n s e a rth 6.V fo r a given spacec raft configurat ion
can be d e t e r m in e d f r-orn the followi ng exp ress ion:
whe re
( 5 )
k- ,;.v Ig I= e TL 0 sp
= t r-a ns ea r t h t::.V available (it / s e c )
=t r a ns luna r 6.V used (ft / s e c)
=g r avi tationa l cons t ant (32 . 174 ft2
/ s e c )
= specific i m pul s e (s e c )
=weight after TLI without spacec raft-launch veh ic l eadapter (S LA ) (l b )
W LM = Lunar modu le weight d i s c a r d e d (lb)
W CSM = C ommand and s ervice m odul e weight (lb )
The s pac ec raft c onfigurat ion used through this r eport ha s the follow
in g c haracteristics:
wo = 94 , 548 l b (without SLA )
WLM = 32 , 000 Ib
WCSM = 23, 562 Ib
IS p = 3 13 sec
Equation (5) t h e n becomes
';'VT E
= 10, 070 In (4 .01 3 k - 1.358 )
4 - 13
(6)
Using the VT L
v e r s us k and the I nx versus x c u rves of Figu re s 4-13 and
4-14. respe ctively . a table c an be c ons t ruct ed as f o llow s :
AVT L [ n (3 . 97k-1. 35 ) AV TEk (Fig 4 -14) 4 . 01 3k - 1. 3 58 (Fig 4- 15)
0. 60 5140 1. 050 0.049 490
O. 65 4 340 1. 2 50 0.2 2 3 2240
0.70 3590 1. 4 5 [ 0.372 3740
O. 75 2 900 1. 652 0. 502 5050
0. 80 225 0 1. 8 52 0. 6 [ 6 62 00
This table c an now be us e d to plot the performanc e cu rve sho wn in Figure
4 - 15 . If the trans lunar and trans earth midc ou rse c o rrection 6,V· s a re to
be includ ed in the gene ration of th e c apa bil ity cu rve of Figure 4-15 . it is
apparent tha t th is i s achie v ed by s imply s hifting the cur ve to th e left an
a mount e qual to the trans ea rth midcours e 6,V (r esulting in Curve AI ) and
th en low e r ing the re sulting c u rve an a mount e qual to the translunar mid
c our se 6,V (resulting in Curve B. ) Curve B will be used in the gene ration
of the 6,V constraint data in Section 5. whic h cor re sponds to the fo llowing
typical Apollo va lues:
T rans luna r m idcour s e 6,V = 162 ft / s e c
Transearth midcourse 6,V = 94 ft /se c
If a s pacec raft of diffe rent weight distributions than that s hown fo r
the ex am ple spac e craft of Figure 4- 15 i s c onside r ed for a mission, the n a
ne w capabil ity cu rve c an be generated as des cribed above with the a id of
F igures 4 - 13 and 4- 14 .
The proc edure in u sing the performanc e capability c urve of F igu re
4 -15 for a spe cific site analys is or an acce s s ibil i ty c ontou r generation i s
dis cuss ed in Se ction 5 .
4 - [ 4
4.2 . 5 Continuous Abort ReguireITlents
A c o nt inuous abort c a p a bilit y exists f o r a rn i s s i. on if there i s s uffici
e nt a v c a pa b i lit y r e m a fn tng after LOI for an earth return a t a ny time (i . e . ,
onc e per CSM revolution) during the l u na r orb it s tay time . Therefore. if
c o nt i n u ous a bort is r equired f o r a mi s s i on , it w i ll be necessary t o c ons ide r
t h e trans ea rth a v requi r e m e nt s not on ly at the e nd of the parking orbit stay
t i m e (at T El) b u t also throughout the stay tim e from LOI on. This result s
fr om t h e fact that the earth- m o on g e omet r y is c onti n ua lly changing wi t h
tim e so t ha t t rans earth veloc ity r e quirements are also c ha ngin g. In addi
tion, the effects of any r e q u ired CSM plane c ha n g e upon the t r a n s e a r th
..6. V r eq u i r ernenta m uat be c ons ide r e d.
Depending u p on t h e d es i r ed CSM orbit in c lina tion, noda l posit i on . and
o rbit stay t irne , the trans e a r th velocity requirements during CS M orb it
stay time can b ehave as de sc r ibed in the f ollowin g fou r c a s e s:
Cas e 1. Increas e to a m a x tmum and then decrease
Ca s e 2 . Con t i nu a lly increase afte r LOI
Cas e 3. C ontin ua lly d ec r ea s e after LOI
Cas e 4. D ecrease t o a rnin i rriurn and then increas e
T he c ontin u ou s a bort r e quire m ent w ith a nd without CSM plane c h ang e s will
be con s ide red.
4.2. 5 . 1 Continuous Abort Wi thout CSM Plane Change
The b e havior of t h e transearth a v r equir ements a s a func tion of
orbit stay t i m e i s r e adily de t e rmined from t h e tr ans eart h a v c u r v es.
F i gur e 4 - 16 depicts the fo u r c as e s li s t e d above for a tr ansearth fli ght tim e
o f 72 ho u rs . For re ference, t he trans lunar a v c u rve s for a fligh t a re a ls o
shown . An orbit sta y time of five days is a s s ume d , cor r e s po n d i ng to a
we stwar d no d e t r a v e r s a l of appr oximately 66 degre es . C ons idering
Cas e 1 o f F i gu r e 4- 15, point A 1 cor re s po n ds to a retr ograde CSM orbit o f
14 d egre e s in clina t i o n a n d as c ending n o d e lon gi tude o f 17 2 d e g r e e s East o r
8 de g r ee s West. The t r a n s lunar li.V r equired is 3 200 it/sec . If an abort
w ere r equired Lrnrnedi a te Iy after L OI. t he r equi r ed trans e a r t h li.V would
be 3 50 0 fe et per s econd. As o r bit stay t i m e incr eases, the a sc en di n g n ode
of t h e orbit rnove s westward, relative t o the earth moon lin e . and the
4 - 15
transearth t::..V continually increases to a rnaxirnum of 37€l0 feet per second
at point B 1 and then decreases to 3550 feet per second at the end of five
days (point C 1)' For this cas e 3700 feet per second would be budgeted for
the transearth t::..V for continuous abort. Case 2 depicts an orbit (with an
inclination of 45 degrees and ascending node longitude of 29 0 East or 151 0
West). The trans earth .6.V continually increases throughout the orbit stay
time so that the trans earth .6.V required (5300 feet per second) at TEl
. (point C 2) would be budgeted. Case 3 depicts an orbit in which the trans
earth .6.V continually decreases so that the maximum .6.V occurs immedi
ately after LOr. The trans earth .6.V that would be budgeted would then be
4350 feet per second, corresponding to point A 3. Case 4 depicts an orbit
in which the .6.V decreases to a minimum at point B4 and then continually
increases. The highest .6.V value of 3800 fee t per second, corresponding
to point C 4' would be budgeted for this case.
When the .6.V requirements are determined for a specific orbit, the
spacecraft capability curve of Figure 4-15 is used to determine whether
the velocity requirements are achievable.
The generation of .6.V constraint contours, which is the locus of CSM
orbits that consume all CSM propellant for a given mission with continuous
abort capability, is described in Section 5. 1. 2.
4.2.5.2 Continuous Abort with CSM Plane Change
To deterrrrine the maximum .6.V requirements for an earth return for
the case in which the CSM is to execute the plane change in an abort situ
ation, the time at which the plane change occurs must be considered, since
this affects the subsequent trans earth .6.V variation with time to TEL
The effect of a CSM plane change upon the subsequent transearth t::..V
requirements can be understood by considering the cases depicted in
Figure 4-17. Case (a) depicts Case 2 in the previous section, in which the
transearth .6.V continually increases after LOr. Point 2 corresponds to
half the surface stay time at which time the plane change angle is a maxi
mum. If a plane change is made at this time there is no node shift of the
CSM orbit, since the plane change is made when the CSM is over .t he
equator (see Figure 4-12). A specific example may be considered in discus s
ing plane changes at points 1 and 3. A surface stay time of 6 days and a
4-16
site l a t itu d e of 30 degrees a re a ssumed. T his is found from Figure 4 -9,
E quation 3, or from F igure 4 -40 which s hows t he plane c hange angle
versus stay t i m e (t h e maximum plane change angl e occurs aft er h a lf the
stay time has elapsed, and is equal to i - p.). Con s i d e r i n g p oint 1, it may
be assumed that t h i s corresponds to a plane c hange made after 1. 5 d a y s
have elaps ed. F r o m Figure 4 -40 it is seen that the r e qu i r e d CSM pla ne
ch ange is 5. 0 deg rees . The res ulting change in CSM or bit inclination an d
node longitude is foun d fr om F igure 4-46 i n which the inclination is low
ered 4. 8 degrees , and the ascending node ha s shifted e a s tw a r d 2. 8
degrees . If the pla n e c hange i s made at p oi nt 3, which is afte r half the
stay time h as e lapsed, the node s hi ft i s t h en we stwar d. The direction of
n ode shif~ is a lso apparent fr om in spection of Figur e 4 - 1 2. T he symmetry
in F igures 4 -38 through 4 -49 may be noted with r e s pe c t to the surface
stay time midpoint.
CONSTANT i:NCONTOURS
::.-.-----(b)
o
(0)
A•
Figu re 4 - 1 7. Sample C as es - Continuous Abort with C SMP lane C hange
Referring again to Figure 4-17 (a), the resulting !.:i.V requirements
at TEl, then, correspond to the end points 1 11, 2 1',or 3". However, the
worst !.:i.V condition for this case is that in which a LM abort is required
at the stay time midpoint, requiring the maximum CSM plane change, and
in which TEl occurs at the nominal time. This is due to the fact that any
reduction in transearth !.:i.V resulting from a CSM plane change will be less
than the !.:i.V required to perform the plane change. This is apparent by
inspection of the translunar and transearth !.:i.V curves of Figures 4-31 to
4-37, where the maximum !.:i.V gradient is approximately 67 feet per second
per degree plane change, corresponding to a flight time of 60 hours. The
!.:i.V gradients become smaller for longer flight times. The!.:i.V required
per degree plane change for a circular 80 nautical miles orbit is approxi
mately 92 feet per second. The larger the plane change, the larger the
difference, or net !.:i.V penalty will be. For the cas e depicted in Figure
4-17 (b), the worst !.:i.V abort cas e is the same as that stated above.
The following statements are apparent in determining the worst !.:i.V
case for continuous abort with CSM plane change.
• If the transearth !.:i.V continually increas es after LOl, thenthe worst !.:i.V case is one in which the maximum planechange is made (after half of the stay time has elapsed).
• If the transearth !.:i.V reaches a maximum well after halfthe stay time, then the worst case is one in which themaximum plane change is made.
For other cases, the worst !.:i.V abort case is determined by examining the
!.:i.V requirements throughout the stay time. Some cases may be apparent.
For example, if the transearth !.:i.V requirements continually decrease
throughout the stay time, then it should be determined whether the plane
change !.:i.V increase versus stay time or the transearth !.:i.V decrease versus
stay time is greater. If the transearth !.:i.V reduction versus stay time is
greater, then the worst abort cas e is at LO!. If not, then the worst cas e
if found by determining the !.:i.V requirements for several time points,
using the CSM plane change data in Section 8.
The CSM !.:i.V required versus plane change angle is shown in
Figure 4- 50 (which is obtained from Equation 4).
4-18
4. 2. 6 Example AV C ons traint Cur ves
Fig u r e 4-51 shows the envelope o f CSM i n cli n at i on s a nd n odal lo c a
tions that are achievable within the fra m ework o f t h e s pace c r aft c apabi lity
d e fined i n Figur e 4 -1 5. a 14 -day rna xlrnurn t otal mission t i m e. a nd a
co n tin uo u s abort r equir em ent wit hout C S M pla ne c hanges . The curv e s a r e
d r awn fo r thr e e l un a r stay t irne e (tirne e l a pse d b e tw e e n lunar orbit ins er t ion
a nd trans e a r t h injection) . For a giv en stay time, t h e rang e of po s s ib le
o r b its lies between t he co r r e s pon d ing s tay t trne boundaries.
T he r ight-hand boundary is t h e r esult of slow (ed 3Zh) trans ear t h
t imes . It r epre sents the locus o f cas e s w h e r e CSM a b ort follows immedi
a te l y a ft e r LOI. This cons traint is a function of the spac ecr a ft total AV
capabi li ty a nd is n ot a fu nc tion o f s tay tim e (for the values of stay t i rrre
conside red ). The boundari es on the left h an d are t he re s ult o f slow
(:::..132h) translunar fli ght tim e s with r eturn fli gh t t i rne no t being particu
l a rly cri ti c al. T h e conti nuous abor t r e qui rement (corresponding to z e r o
s ta y time) lies t o t he left of t h e 2, 3, an d 5- d ay s t a y t im e li ne s a nd i s n o t
s hown.
A drawback to t he for m in which the data are sh ow n in F i gur e 4-5 1 is
that no i n fo r m a t io n i s s how n for the combination of outbou n d a n d r eturn
t r aj ec to r y fl ight t i m e s that will y i e ld the des ired inclination and nod e .
F or this i nfo r-rnat.i on the supporting data has been made available.
4 - 19
-1---
I?,,=
80--
I<
,V
--I
---
'--"
=70
II-
I--
-"=
60-
I--
---
-"=
50-
II
-'--
-,,=
40
--
-
II
-'I\-
-/"=
30--
1'1
I~
--b
,,=20
"-
IV
1---~
,,=10
--
,,-
5-l-
I--
--f-
---
---
--
RETR
OGRA
DE
MA
X.
PL
AN
EC
HA
NG
E=
4D
EG
SUR
FACE
STAY
TIM
E=
5DA
YS90 80 - o w 0 Z6
00 .... « z -' u
...Z
40, N
....0
m '" 0 :;: ~ u20 o
o20
406
080
100
120
140
160
180
LON
GIT
UD
ED
ISPL
AC
EM
EN
TO
FC
SM
OR
BIT
ASC
EN
DIN
GN
OD
ERE
LAT
IVE
TOSI
TELO
NG
ITU
DE
,0
d
Fig
ur
e4
-3.
Ex
am
ple
Ge
om
etr
icC
on
str
ain
tC
ur
ve
s
CSM ORBIT1
B'
Figu re 4 -6. Right Boundary Plane Change Geometry
A
CSM ORBIT1
Figure 4 - 7. Left Boundary Plane Change Geomet ry
4 -Z 1
180
160
140
120
100
8060
4020
~~r-_
.-t:
:=':
~\;\
['>1'
,-
-,,
-80
t-.-
-,. -
k-'/
1/;
--
-1\l\
1',
-,,=1
70.
'J~
I1/
-,-
II'
1\1'
.-
Ir-
~,-
i.-l-
/1:I
---
I\
"<
,--.
d60
//
"-
/,-
\\
<,
<,
I"1
I-t-
--1--
.-I-
//
!~-
--
\1"
\-,
<,
-'"~
56,- -
/1/
/I
I\
\-,
"
<,
1I-
t-.
-/
"I
1III
-<
,-
"-
1/"~
40'.
"I
T-
l-e--
'-
....-
--
\\
"'-
",;
30'
-~-
1/1/
I
\'.
-I-
,-1
/i
\1-
-.'-
-1
'.~
1--
\"
~20
I-/
/'-
!--
--
-,1'
-l.-~
/
-<
,'-
,,=10
!-.l
---
/'-
!-
'-,,~5
--l
-.-
--
-I
o o
20
RET
ROG
RAD
E90 80
>- -c zZ
60o- o w C
l
~ U Z >-
40...
a5,,
,,~O ~ u
CSM
ORB
ITN
OD
ELO
NG
ITU
DE
DIS
PLA
CE
ME
NT
FR
OM
SIT
ELO
NG
ITU
DE
,0
d(D
EG
)
Fig
ure
4-
9.
Ze
ro
LM
Pla
ne
Ch
an
ge
;i
ve
rs
us
(2d
for
Va
rio
us
Su
rfa
ceSt
ay
Tim
es
5250 TRANSLUNAR96 HOURS RE TROGRADE0 ., I -; I I I I II ~ i III q I I
"I , \- ,
\ 1 V / J I / I I I I Ii \ ! \ \ 1 \I ,
6 \ \ \
e. \
1"'-.~5000 V / I / IJ / II I I I I \ \ \z 60,
I I: ~ \ \ I'0 <, ~ / 1// 1/ / 1/ I I iT ) [\ \ '\ \ ,;:: 1- _-c <, - - "" / II \ "
"
Z::J 1"- ...
1'_ - - ~'I V / '/ 7 rT l \" \ \ " ~~-c-
u I ~ I /~ -. '- - ,/ i II ' .......-to- - V v/ 1/ 11 I f\ 1\ \ -, ,,- ....j"-.iO / \0 3
I--- - - - - ~ -~ A / Io 4000 - '-~ --- I-- r----- V V ' '/ /I
\ \ <, <, to- .....u t- _ _ _ - I / I <, -1- ., ___ - I-=:l---V
II I -, <;1'---,;;0- I- I ~ - - ~ I- t / 1 I
1- _
j""v I ~ I-- -..
I'r f--_ - t--- 1-1- - f '- - - I I --
00 20 40 60 80 100 120 140 160 180
0 30 60 EAST 90 120 ISO 180- 180 - ISO - 120 WEST - 90 -6 0 -30 0
LONGI J OE O F ASCENDING NO DE
30- ISO
60 EAST 90- 120 WEST -90
LONGITUDE OF ASCEND ING NO DE
120-6 0
150- 30
180o
Figure 4- 10 . T ran s lunar a n d Tran s earth 6.V for 96 - a nd7 2 - H our Flight Time s , R e spec t i v ely
4 - 23
CSM ORBIT
LUNAREQUATOR ·
EARTHAT TEl
Figure 4 -1 1. Lunar Orbit and Earth Moon G eometry a t L OI a n d TEl
4 - 24
LU NAR
EQUATOR
F i gur e 4-12. CSM Plane C ha n g e Geometry
4 - 25
0.800.750.700.650.60
2500~ ""' ......Io..- """'__~ ~_-J
0.55
3500
5000
3000
6000~-------------------------.
5500
U 4500LUVI
~L&.-
..J
;;-<1
4000
k
Figure 4 - 13. T r a n s lunar 6.V ver sus k
4 - 26
LN X
0.7
0.6
0.5
0.4
0.3
0.2
o. 1 t:::·; , ~: t::..'M: " '1.1' i · . '. ' ) .. ..... " •. •,.... . , . ., H""'. It '- I.. • · .. ~7tt' · L.• ~J , ~ J.j ~ r: ;... f.4. of ~ J I ;- i ~ ; I ' ~ I " • y..;t:-: - ~ '.. ". .I., ~. •I ~++ 1~l ;; t 1 .t .. ~, : ~'. L;" .. ~ .. + ·Fr ' ," . , i ~ ,., : ol ". , t "-t ;.j..-t ~~ f Ii"': ' ,.1 " - . ' .
h--+tt·:·tr:lii~ .t-i-. : : I' ~ .:: 1:: l ;j .: :'l' c j ..-::-I-t.::,.. --I- j,' I tRt 1p:~~ -- ... U·:t :::• • 4 . _ ; :t. " 1" . ;.;....;.:: t · · l ... J • ._ ' ';" "" t J_.... f i:r. I " l r ~ ' ~ . . 1
+' J. : , ; +- .... .; ..+. 1.+i.. . I.... ~ , ..... I '! ~ ~ • ...-!- ••~ .~ -f 1 40 t o -• • ! . ,.~ .. . . . -t- . .. .
E~ !] ; . ~ Wi ffil :in; ;1';" .~:lili-"..;:: ;.:! 4-:-. : -:-: ••~~ ~ .~ , H.I- :+'; ':i ~ L ~h ~+~~ -f::- : ?"" : ..Ifft .'t . ... I ,.LIT .-;:i l·::;:tt::t· 1<- - • ., Jj' ~ " " . . l:j:::H:t ,+ . :'1: T1 . i i , ' . +-t-'+ ._ ., . ' - - - ,.• ,
rt;'.j::: ~ ;,1;-11;!'U' ;.: t t. +~ :.; ': ;',' r~ : ' ii~ mr 11- ' . tI1. , if~ii fl;·· :::-' rl 1:ffi trrr ..I ·-; ; I • . ,,1 . ~ I .,: t~ ., ", ' ! .: ; 1, ... ,i' ,· I , ;.t ·" i ·!. '-I ,. . ' 1:.i:r: .. . [.: , ." ~ , '-r-l . " ,.t-t t- .. · , 1- I· · f . , --..+;.. ...~- . . _." " l-t+i · 1"' - l , , - - I f.+. • • • .• L- · t - -t~o
1.0 1.2 1.4 1.6 1.8 2.0
Figure 4-14. Ln X versus X
4-27
- u w V1 ~ u..
'-" -J
J-
~ .. ~ 0::: «
.t>-
ZI
:::>N ex
>-J
V1 Z ~ J-
4000
3500
3000
3000
3500
4000
4500
5000
5500
TRA
NSE
AR
THb.
VR
EM
AIN
ING
,b.
VT
E(F
T/S
EC
)
Fig
ur
e4
-15
.S
pacecra
ftP
erf
orm
.an
ce
Cap
ab
ilit
y
,C~
2"
.....
CASE
1
\\
"
\ \ \ \ \~--5000---
.\
I268
0I I
--"
---
5000
"
A2
I / // "
.CA
SE4
-----
-30
00
30~-----.:==.....---.-..4000--l=-----+--------.:::l~--+---I:----:--t-----\----+---------t
60I------I---+---f----+--\------rt---+--------:=---l--t-----\--=-..:...~---+_\'-----__+_t
90r---,-..---""T"'""--r----,-----r-"T""T'---....,...----,r-----r---r-~__,_~----r_.,..._-r_---_.,
-I U Zz o ~ z.-....
V)
w w ~ o w o ........
180 o
150
-30
6090
120
-120
-90
-60
LO
NG
ITU
DE
OF
AS
CE
ND
ING
NO
DE
30
-150
SOLI
DC
UR
VES
:TR
AN
SEA
RTH
72h
hD
OTT
EDC
UR
VES
;TR
AN
SLU
NA
R96
O'-
----
"'__
-.L
---I
..--'-
.L.-
..J
....
..J
...-
-"'
o -180
Fig
ure
4-1
6.
Sam
ple
Cases;
or
Co
nti
nu
ou
sA
bo
rtW
ith
ou
tC
SM
Pla
ne
Ch
an
ge
5. BASIC MISSION ANALYSIS PROCEDURE
With the graphical data contained in this report and a full under
standing of the geometrical and 6.V acces sibility constraints, an inexhaust
ible variety of missions can be analyzed with relative ease. The procedure
may be apparent from the discussion of Section 4; however, specific cases
will be cited in the following sections to review the procedure details.
5. 1 GRAPHICAL PROCEDURE
Two types of rni s sions will be considered in the example cas es. The
first will be an accessibility analysis of a specific site, and the second will
be the generation of an accessibility contour for an example mission.
5.1.1 Specific Site Analysis
The specific site selected for the example is Aristarchus, which is
located at 47.5 0 W. longitude and 23.80 N. latitude. Two example cases
will be considered for this site; Example 1 is the determination of a max
imum allowable stay time, and Example 2 is the determination of a mini
mum achievable total mission duration.
Example 1: It is desired to determine the maximum surface stay
time achievable for Aristarchus with the following mission require
ments:
• Maximum mission duration of 14 days
• Continuous abort capability with plane change to beperformed by the CSM
• No LM plane change
• LM descent to occur 12 hours after LOI
• Site to be in CSM orbit plane at nominal (i. e., no abortoccurrence) time of LM ascent
• TEl to nominally occur 12 hours after LM ascent
• Translunar and transearth flight times to be within therange of 60 to 132 hours
• Spacecraft configuration and rni.dcou.r s e 6.V requirementsas shown in Curve B of Figure 4-15
5-1
The approach taken here will be first to assume a surface stay time
and determine the optimum (minimum fuel) flight profile satisfying
the mission requirements. If this is within the capability of the
spacecraft, then a longer stay time is assumed, and the optimum
profile determined; if not, then a shorter stay time is as sumed.
The maximum stay time is then determined by interpolation (or
extrapolation) of the above data by finding that surface stay time
which just depletes the spacecraft propellant for the minimum fuel
flight profile. In accordance with the three basic steps of the graph
ical procedure, this is accomplished as follows:
Step I: Determination of Geometrical Constraints
The CSM orbit which satisfies the required geometry can be obtained
from Figure 4-4; or more accur ate Iy", from Equation (3) on page
4-12 and the equation in the footnote on page 4-7. The results of
interest are listed in Table 5-1 for various surface stay times. The
longitude of the CSM orbit ascending node is adjusted to account for
the required 12-hour wait between LOI and LM descent (since the
site must lie in the CSM orbit plane at LM descent). A 12-hour wait
requires placing the ascending node at LOI 6.6 degrees further east
ward.
Step II: Determination of 6.V Requirements
The graphical data neces sary for the determination of the 6.V
requirements are
• Translunar 6.V data (Figures 4-24 through 4-30)
• Transearth 6.V data (Figures 4-31 through 4-34)
• CSM plane change data (Figures 4- 38 through 4- 50)
• Spacecraft capability curve (Figure 4-15)
~:~The required CSM orbit inclination can also be obtained from the CSMplane change data.
5-2
Table 5-1. CSM Orbit Parameters, Aristarchus Example 1
Surface Stay Time (day)2 4 5 6
D.As/2 (deg) 13. 2 26.4 33. 0 39. 5
i (deg) 24.4 26.2 27.7 29.8
r.!d (deg) 103. 2 116. 4 123. 0 129.5
Longitude of Ascending 55.70E 68.9OE 75.5OE 82.00E
Node at LM Descent
Longitude of Ascending 62.30E 75.50E 82. 10E 88.6°ENode at LOI
Longitude of Ascending 22.80E 9.6°E 3.00E 3.70W
Node at Nominal TEl
Maximum Plane Change (deg) 0.6 2.4 3. 9 6.0- From Figures 4- 38through 4-40, or Figures4-44 through 4-46, orEquation (3)
D.V fo r Maximum Plane 55 220 360 550Change (ftl sec)From Figure 4- 50 orEquation (4)
The optimum (minimum D.V) flight profile is to be found for each
value of surface stay time considered. This is accomplished by
assuming a translunar flight time and determining the trans earth
D.V requirements for various transearth flight times to obtain the
minimum. This is repeated for other translunar flight times to
determine the over-all minimum D.V flight profile.
For example, consider a surface stay time of 2 days. From
Table 5-1, it is seen that the required CSM orbit is inclined 24.4
degrees with an ascending node longitude at LOI of 62. 30 East. The
ascending node longitude at TEl is 22. 8 0 East corresponding to a
CSM orbit stay time of 3 days. A translunar flight time of 132 hours
is assumed. From Figure 4 - 30 it is found that the translunar D.V
required to achieve this CSM orbit is 3080 feet per second. From ·
5-3
Figure 4-15 it is found that the transearth AV available is 4320
feet per second. The trans earth AV r e qui r e rne nt s for various flight
t irrie s are now considered to d ete r mi.ne the m inimum transearth AV
for the 132-hour translunar flight fi rne , A 60-hour trans earth flight
t'irne is considered. It is seen f'r orn Figure 4- 31 (by overlaying upon
the translunar curve) that the transearth AV i rnrnedia.te Iy after LOI
is 3880 feet per second, then decreases to a rni.ni.murn of 3500 feet
per second, and then increases to 3580 feet per second at the end of
the 3-day CSM orbit stay t.i.rrie , It is noted that after LOI, the trans
earth AV decreases at a faster rate than the plane change AV is
increasing. This is ascertained by corrrpa.r ing Figure 4- 31 with
Figures 4-38 and 4-50 (or Equation (4)). The worst abort case*,
then, is i rnrnediate Iy after LOI requiring 3880 feet per second.
This gives a AV rnar gin (AV r erriaining) or 440 feet per second. A
72-hour transearth flight ti.rne (Figure 4-32) is considered. It is
seen that the transearth AV Irrirnedi a te Iy after LOI is 3300 feet per
second, decreases to a rninimurn of 3000 feet per second, and
increases to 3300 feet per second at the end of the 3-day CSM orbit
stay t'irne , The worst abort c a s e ", then, is that in which a LM abort
is required at the rriaximurn plane change angle and in which TEl
occurs at the norninal ti.rne of 3 days after LOr. The rriaximurn plane
change angle is 0.6 degree, requiring a CSM AV of 55 feet per
second. This also reduces the CSM orbit inclination by O. 6 degree,
so that the transearth AV is now reduced f'r o rn 3300 to 3290 feet per
second. The rriaxirnurn AV for this case is then 3290 + 55 = 3345
feet per second. The AV rria.r gin is now 995 feet per second. The
above steps are repeated until the rnirii.rnurn trans earth AV is found
for the a s aurrie d translunar flight ti rne of 132 hours.
Other translunar flight ti.rne s are as s urned for the given stay till1e of
2 days, and the over-allll1inill1ull1 AV is then d e te r rrii.ned for that
flight ttrne. The optirnurn flight profile for the 2-day surface stay
t'irne corresponds to translunar and trans earth flight time s of 132 and
,,~
"'See Section 4. 2. 5
5-4
84 hours, respectively, resulting in a AV ma r gin of 1000 feet per
second.
The above is repeated for surface stay tirne s of 4, 5, and 6 days (in
which the AV rna r gi.n for the rni.ni rnurn AV flight profile b ecorne s
negative, so that 6 days cannot be achieved).
Step III: Interpretation of Results
The results of the above cornputat.ions are pres ented in Figure 5-1
in which the AV rna r gi.n for the m ini.mum AV flight profile is plotted
versus surface stay time, It is seen that the rna.xirnurn surface stay
time is 5. 7 days which is that corresponding to a zero AV rnar gi.n
(total propellant depletion). The trans lunar and trans earth flight
fi.rrie s for the 5.7 -day surface stay t irn e are approxirn ate.ly 111 and
64 hours, respectively, resulting in a total rrri s s i on duration of
appro.xirnate Iy 14 days. The rapid decrease in AV rna r gin after 4
days is caused by the 14-day total rni s s ion duration constraint.
ExaITlple 2: It is desired to de te r mi.ne the rnini.rnurn total rni s s i on
duration flight profile for a 2-day surface stay t irne at Aristarchus
and for the s arne rni s s i on r equi r ernent s of Exarripl.e 1.
The rni.nirnurn total rrri s s i on duration flight profile is defined here to
rnean the rnini.rnurn c ombi.ne d translunar and transearth flight ti rne s
for the norni.nal rni s s i on in which no abort occurs. However, con
tinuous abort capability is still required.
The approach taken is to dete r rrrine the transearth AV available after
LOI versus translunar flight t.irne , This is done using the translunar
AV curves (Figures 4-24 through 4-30) to de t e r rni.ne the AV required
(versus flight ti.rne ) to attain the required CSM orbit (frorn Table 5-1:
inclination of 24.4 degrees and ascending node longitude of 62. 30 E. )
and the spacecraft capability curve of Figure 4- 15. The results are
plotted in Figure 5-2. In addition, the t r ari s e a r th AV at norni.na.l TEl
(node longitude = 22. 8 0 E. ) versus trans earth flight time with and
without abort is plotted in Figure 5- 2. For this rni s s i on, the worst
abort case for transearth flight t.i rne s of 72 to 120 hours is one in
which a LM abort is required for the rnaximum CSM plane change
5-5
and in which TEl occurs at the nominal time. The two curves for
abort and no abort in Figure 5 - 2 are vertically displaced from 35 to
45 feet per second, which corresponds to the difference between the
maximum plane change ~V of 55 feet per second and the reduction in
transearth ~V at nominal TEl of 10 to 20 feet per second.
The curves of Figure 5- 2 are interpreted as follows. For a trans
lunar flight time of 97.5 hours, for example, the~V available after
LOI is 3400 feet per second. This means that a continuous abort is
always possible and that abort (worst case) transearth flight times
between 67.5 and 97.5 hours can be achieved. For the nominal case
of no abort, the minimum trans earth flight time that can be achieved
with 3400 feet per second ~V is 65.5 hours giving a combined trans
lunar plus minimum transearth flight time of 165 hours.
It is noted from Figure 5 - 2 that the minimum allowable translunar
flight time is 95.5 hours, which results in a transearth ~V availabil
ity of 3320 feet per second. This corresponds to the minimum
allowable ~V to provide continuous abort capability. This case
corresponds to a transearth abort flight time of 80 hours. For the
nominal TEl time, 3420 feet per second gives a transearth flight
time of 69. 5 hours giving a combined total of 165 hour s ,
The minimum total flight time can be found by plotting the sum of the
translunar and trans earth flight times versus the translunar flight
time (for translunar flight times greater than 95.5 hours to assure
continuous abort capability) as done above. A minimum combined
flight time of 163 hours is achieved corresponding to a translunar
flight time between 98 and 99 hours and a transearth flight time
between 65 and 64 hours, respectively. However, the accuracy of
the curves within the region of 60 to 72 hours is questionable, so
that the determination of the minimum flight time by inspection of
Figure 5 - 2 is adequate.
5.1.2 Accessibility Contour Generation
An accessibility contour is the locus of points that separate the
accessible and inaccessible areas of the lunar surface. The generation
5-6
of this contour requires the generation of the geometrical (Step I) and the
C:1V (Step II) constraint curves.
Two important features concerning these constraint curves will now
be re-stated. A geometrical constraint curve (Figures 4-19 through 4-23)
shows all CSM orbits in the form of inclination versus node longitude dis
placement (relative to the site longitude) that satisfy the stay time, LM
plane change capability, and continuous abort requirement. These geo
metric constraint curves are independent of site longitude, but can be
interpreted as those corresponding to site longitudes of zero. For
example, if a CSM orbit node displacement of 120 degrees is considered
for a site longitude of zero, the ascending node of the orbit is then 1200
East*. On the other hand, if a site longitude of 20 0 West is considered,
then the orbit node will be 120 0 east of the site corresponding to a longi
tude of 1000 East.
Now the question arises: What CSM orbits that satisfy the geoTI1et
ric constraints can be achieved by the spacecraft? The flV constraint
curve answers this question, since it shows the locus of all CSM orbits
(i versus ~) that deplete all available CSM propellant.
The generation (Step II) of the C:1V constraint curve and the manipula
tion (Step III) of the geometric and ~V constraint curves to obtain the site
accessibility contour will now be described for the following example.
Consider the following mis sion:
• Total mission duration = 14 days
• Translunar flight t.irne = 96 hours
• Transearth flight time = 72 hours
~ Time in lunar orbit (retrograde) = 7 days
• Surface stay time = 5 days
• LM descent 1 day after LOI
, ....
"'It is recalled that only retrograde orbi ts are considered, so that the CSMorbit ascending node will always be east of the site.
5 -7
• LM plane change capability = 4 degrees
• No CSM plane changes
• ,C on t i n u ou s abort capability
• Translunar and transearth midcourse 6.V of 162 and94 feet per second, respectively
• Spacecraft configuration of Figure 4- 15
Step I: Geometric Constraint Curve
Figure 4- 3 is the geometric constraint curve to be us ed for this
mission.
Step II: Generation of the 6.V Constraint Curve
The graphical data needed to generate the 6.V constraint curve are
• Translunar!::t..V curve for 96-hour flight time(Figure 4- 27)
• Transearth 6.V overlay curve for 72-hour flight time(Figure 4- 32)
• Spacecraft capability curve (Figure 4-15)
The 72-hour transearth !::t..V curve is now overlayed on the 96-hour
translunar 6.V curve with scales coincident as shown in Figure 5- 3.
To determine the locus of CSM orbits that consume all available
propellant, a translunar 6.V of 3000 feet per second is assumed.
From the spacecraft capability curve of Figure 4-15, the transearth
!::t..V available is 4470 feet per second. A 7-day orbit stay time
corresponds to a westward node shift of 92.3 degrees, so that CSM
orbit is to be found in which the maximum trans earth 6.V during this
time interval if 4470 feet per second. This is conveniently done by
cutting or marking the edge of a piece of cardboard or paper equal
to 92.3 degrees on the longitude scale. The right edge of this paper
is shifted along the translunar 6.V contour equal to 3000 feet per
second until the orbit is found in which the maximum transearth 6.V
for that CSM corresponding to point A will vary as the node shifts
from A to C in the 7 days. The maximum !::t..V corresponding to point
B is seen to be less than the required 4470 feet per second. This
5-8
line is now shifted upward, keeping the right edge en the 3000-foot
per second translunar AV contour until the maximum transearth AV
is 4470 feet per second, corresponding to point 13 I. That orbit with
inclination and node corresponding to point A I, then, is a point on
the AV acces sibility constraint curve. It is convenient to place a
vellum on the overlays and mark these points. This process is
repeated by as suming other values of AVT L until the curve of Figure
5 -4 is obtained. The interpretation of this curve is that all CSM
orbits in the region above the curve cannot be achieved for the space
craft of Figure 4-15.
Step III: Generation of Site Acces sibility Contour
Site accessibility is determined by overlaying the AV constraint
curve (vellum), Figure 5-4, on the geometrical constraint curve,
Figure 4- 3, as shown in Figure 5 -5. Figure (a) corresponds to a
site longitude .of 30 East (the position of the zero of the, geometrical
constraint curve on the scale of the AV constraint curve indicates
the site longitude). It is noted in Figure (a) that site latitudes from
zero to 36 d e gr .ee s (point A) can be achieved fora site longitude of
30 East. Figure (b) shows a site longitude of 20 0 East (or 160
d e gre e s West), in which all latitudes are accessible up to a maxi
mum of 26 degrees. For a longitude of 50 West (or 1 75 0 East) the
maximum latitude is 34 degrees (Figure c). For 20 0 West, or 1600
East, the maximum latitude is 26 degrees.
As the overlays are displaced relative to each other, and the lati
tudes , recorded" the resulting acces sibility contour of Figure 5 -6 is
obtained. It should be pointed out that the resulting curve of Figure
5 -6 was constructed with the assumption that LM descent occurred
at LOI. For the example case, then, in which the LM descends
one day after LOI, the curve vi' Figure 5 -6 must be shifted to the
left 13.2 degrees. For example, if there is no waiting period
.b e tw e eri LOI and LM descent, the maximum achievable site latitude
for a 50 -degree East longitude is 20 degrees (from Figure 5 -6). For
a one day wait, the 20 maximum latitude corresponds to a longitude
of 63.2 degrees East.
5-9
5.2 MISSION ANALYSIS CONSIDERATIONS
A lunar accessibility analysis will, in general, fall under one of the
following categories:
• Accessibility Contour Generation
• Specific Site Analysis
• Parameter Optimization
• Mission Trade-offs
5.2.1 Accessibility Contour Generation
In general, an accessibility contour is generated for a given space
craft capability to determine just what portion of the lunar surface is
accessible to satisfy given mission requirements. Such a profile may
appear like that shown in Figure 5-6. The generation of this contour is
discussed in Section 5.1. 2. The following questions concerning the char
acteristics of this accessibility contour may arise.
1) How does a given change in LM plane change capability,orbit stay time, surface stay time, spacecraft capability, ~V budget, mission duration, translunar flighttime, transearth flight time, etc., change the accessibility contour?
2) How does a change in abort requirements affect thecontour?
3) How doe s a CSM plane change maneuver affect thecontour?
4) How does positioning the surface stay time intervalwithin the CSM orbit stay time interval affect thecontour?
These questions can be answered by use of the graphical procedure with
appropriate changes in values or requirements.
The generation of various contours are extremely useful to the
mission analyst in that he can develop an understanding of the relation
ships between accessibility and mission profile modifications.
5-10
5. 2. 2 Specific Site Analysis
If a mission planner is concerned with designing a mission with
respect to a given site, then three basic questions become apparent.
1) Is this site attainable for a given spacecraftcapability?
2) What is the minimum total ti.V required to attainaccessibility for a given mission profile?
3) How does the total ti.V vary with mission changesor parameter variations?
Some of the above questions may possibly be answered by any con
tours that may have been previously generated. For example, if an
accessibility contour were generated based upon spacecraft performance
alone, then question (1) can readily be answered. If accessibility contours
were generated for various values of propellant margin for the mission
profile and spacecraft capability, then question (2) can be answered
directly. Many specific questions concerning a given site are readily
answered by the graphical procedure.
5. 2. 3 Parameter Optimization
Several optimization considerations which may cause concern for a
given mission are
1) Maximization of stay time for a specific site
2) Minimization of total ti.V
3) Minimization of translunar, transearth or totalmission duration
The se optimizations can be performed with the basic procedur e,
although several iterations may be necessary to achieve optimization.
The disadvantage of any additional time that may be necessary to perform
a specific optimization would be offs et by the advantage of gaining insight
into the relationships between the parameters varied and site accessibility.
5.2.4 Mission Trade-offs
Although minimum total ti.V is one of the goals of a mission design,
the re are several trade- offs to be considered for a mission under consid-
5-11
eration, some of which may be
• Surface stay time versus I:1V penalty
• 1:1V gained by performing a CSM plane change versusdesirability of the additional SPS burn
• Accessibility enhancement versus relaxation of continuous abort requirements (such as intermittent abort)
5-12
1000 r--------=~-----.,
500
-300
150
-V)~
~w
100 :Ei=~
:cC>::;U-
50
o 246
Figure 5- L ~v Margin versus Surface Stay T'i me Aristarchus,Example 1
. . ,
iiiIJIl;i~=t~~ll~;i~JJ~j~~lE. 0' 0- " "'0 - . ;,,··- t~ - O J:1-" '~' T C·'tO'O '--:AFT ERlOIj . :"'!"- L~:.: r:':jJ%F~:"'¢ ~ - :"'-+="': '=-"
\~'~:~-!lt'i8(-2~t~~123oCRf~~~ ..: "",,,,---- -~ ' TE tN WITH- .
, ;.. ' : ~- "';' ~- ' " OUT ABORT :
:~~-l:-i:1~f~)~·~--~fJN;fill~. .... -' . • .:....._- ,. .-- . __. j ' _.. ;-.- :
I ' • I
..._.: - - FLIGHT TIME (HRS) ..
3000
4000
3500
4500 r---------~--~~-~-...
140120100802500 ~----i----"""----+-----f
60
Figure 5- 2. ~V Requirements versus Flight Time - Aristarchus,Example 2
5 -13
,
I / I/ "
5000
"60
II
II
'I
,'
I}t
.I
I\
}I
•I
II
------
30t'
--ea
t40
00
-,-C
I
3000
I
C",
."",
--------
--
J..
30
00
-'-
-----
90
'iii
ii
jiiiiii
iii
ii'
,
,........
V") w w ~ 0 w Q ......... Z 0 .- -c Z --J
U Z - .- coU
1~
I0
..... ~~ V
") u
180 o
150
-30
6090
120
-120
-90
-60
LO
NG
ITU
DE
OF
ASC
EN
DIN
GN
OD
E
30
-15
0
SOLI
DC
UR
VES
:TR
AN
SEA
RTH
72h
DO
TTED
CU
RV
ES:T
RA
NSL
UN
AR
96h
O'
II
!!
I!
I
o -18
0
Fig
ure
5-
3.
Pro
ced
ure
Ex
arn
ple
for
Gen
era
tin
g~V
Co
nstr
ain
tC
urv
e
90
r'----_
--..:...._
--------------------
,
Z' J \0-
IIN
AC
CE
SS
IBLE
INA
CC
ES
SIB
LE« Z
RE
GIO
N.R
EGIO
N-I u Z - l- ce ~ 0
U'1~
I'
V')
......u
U'1
~\\\
\\\\
\\\\
\\\\
\\,,
\......
-
~\~
00
-180
180 a
LO
NG
ITU
DE
OF
AS
CE
ND
IN9
NO
D.E
'(D
EG
)
.F'i.
gu
re5
-4•
.6.
VC
on
stra
int
Cu
rve
for
Ex
am
ple
Mis
sio
n
.INA~CEss
jBLe
RE
Gfo
N1
\\\\\
W
.-I
_~=
/uI
~,:
,'i I i
-30
-60
WES
T-9
0-1
20-1
50
I
-~
6--
k:.
--
-~
I
'"--
Iv:--
-~
=70
_\--
I-I ~=
60
~~f5~
SIBL
EI
I~cel
SSIB
LE~
-1
GIO
t~
=50
l~\IJ
----
--1'1"\~J--t---
1\1"
.."
1.."\"\I\I\~~~
l=
20
~\\I
~u_
.-
~,
IV
h-
~=
10'
----
_-
S=
sH
---t
--
5,6
,7,8
~J
--
-f--
---
I1f
t'"
Ln
on
t:A
CT
1')
(\''''
A.n
-18
0
90
;;; o ~ uz o ~ z a ~
150
-30
120
-60
EAST
WES
T-9
0
60 -12
0'"=
sl
i
:ti I•
,i
~30
-IS
O
.\\\\
1
z o ~ z ~ u ~ ;;; o ~ u
LO
NG
ITU
DE
OF
ASC
EN
DIN
GN
OD
EL
ON
GIT
UD
EO
FA
SCE
ND
ING
NO
DE
(a)
(b)
o o-3
0-6
0W
EST
-90
-12
0-1
50
-180
- ~l-
-b
--f-
--~
I1'
\v
--I-
~=
70--
-I
1\I/~
z--
~=
60
~I"~CE~
J:>IaL
EI
---l~~~
SSIB
LEZ
IE~IOI'l
j!=
so'
-~
II
--
~~
--~=
40--
,......
II
~~
~\\I
0-~
~=
30--
~
I"wU
\~~~
.-~
~\I\\"
..,U
~\I\".""
-'-/
~-
j!=
10-
--~=s
..J:..:
:::::.-
---
-~r-
-5
,6,7
,8
-s
----
---
-0
3060
90EA
STlA
ll15
ll18
1~0 o
rso -30
120
-60
EAST
WES
T90 -9
060 -12
0
30 -15
0o
-18
0
·u
~~
r--
V-
--
::;-
I
'"/'
II
--I-
~=
70--
-i
I-
\_- II
-i
~=
60I
INA
CES
SIBl
EI
---IN
Afc
CESS
lrL
E•
RE
GH
NI
-R
EG
ON
,;
~=
50
1I
I-~
II
---
~=
40-
I~=
301~~~~\
\\111'I
!I
I
J,~~
l~\
-.-~~
'I\lL
,-
!.\!""
"""
"l'I'
,1\1\
'"=
20!
--
:..,
1l".
III".
\\lll
"~.::~
\X~\
",u
m\\
\I
I
l=
101
/e---~
jI
1._s
~=
s.
__
_~S
,6,7,8
-,--
----
-Ii
-"
"-..
nn
i:A
rT
,.,,1'\
1~
!"I
z o ~ z u ~ ;;; o ~ u
U1 I ..... 0'
LO
NG
ITU
DE
OF
ASC
EN
DIN
GN
OD
E
(c)
LO
NG
ITU
OE
OF
ASC
EN
DIN
GN
OD
E
(d)
Fig
ure
5-
5.
Pro
ced
ure
Ex
arn
p'l
es
for
Gen
era
tin
gS
ite
Acces
sib
ilit
yC
on
tou
r
90 60w Q => l- I- « ..J
w l- V')
U1 I .....
30-.
]
INA
CC
ES
SIB
LER
EG
ION
~\\\\\\\\~
~~~
.\\\
~~
~
~-
-~-
- -~~--
00
-180
30 -150
60 -120
90EA
ST
-90
WES
T
SITE
LON
GIT
UD
E
120
-60
150
-30
180 o
Fig
ure
5-
6.S
ite
Accessib
ilit
yC
on
tou
rfo
rE
xam
ple
Mis
sio
n
6. CONCLUSIONS AND REMARKS
The usefulness of the basic graphical site accessibility analysis pro
cedures discussed above for m.ission analysis and planning purposes is
apparent. The accuracy, considering the assum.ptions upon which this
sim.plified procedure is based, is sufficiently good to allow the m.ission
planner to develop insight into the nature and extent of the effects of the
m.any m.ission requirem.ents and constraints upon lunar site accessibility.
If, however, m.ore accuracy is desired, then the data and procedures pre
sented in Volum.e II can be used.
In addition, it is expected that com.puter program. developm.ent
activity will be necessary for accurate m.ission analysis and planning for
future Apollo and AAP m.issions. Insight gained from. a thorough know
ledge of the lunar accessibility analysis technique described in these two
volum.es including the as sociated lim.itations will be valuable in determ.ining
what these program. developm.ent requirem.ents should be.
6-1
REFERENCES
1. "Site Accessibility Analysis for Advanced Lunar Missions, Volume II,Site Accessibility Handbook," TRW 05952-H214-RO-00, 30 April 1967.
2. "Evaluation of Alternate Translunar Flight Plans for the LunarLanding Mission," TRW 66-FMT-232, 15 July 1966.
3. P. A. Penzo, "An Analysis of Moon-to-Earth Trajectories, " ARSPreprint 2606- 62, November 1962.
4. M. D. Kitchens, C. W. Messer, and R. B. Bristown, "A PreliminaryEvaluation of Apollo Capabilities for Lunar Orbit Coverage and LunarSurface Landings Utilizing Multiple-Impulse Techniques forTransearth Injection from Highly Inclined Lunar Orbits, ". NASA/MSCWorking Paper 1199, 11 April 1966.
R-1
Top Related