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

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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

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4 - 23

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LUNAREQUATOR ·

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Figure 4 -1 1. Lunar Orbit and Earth Moon G eometry a t L OI a n d TEl

4 - 24

LU NAR

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F i gur e 4-12. CSM Plane C ha n g e Geometry

4 - 25

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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--------=~-----.,

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o 246

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. . ,

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~:"'¢ ~ - :"'-+="': '=-"

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60

Figure 5- 2. ~V Requirements versus Flight Time - Aristarchus,Example 2

5 -13

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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