NASA Th X.( 5 3Vl

X-552-70-289 PREPRI NT NASA Th X.( 5 3Vl

LUNAR GRAVITY MODELS FOR IMPROVED APOLLO ORBIT COMPUTATION

JAMES P. MURPHY THEODORE L. FELSENTREGER CARL A. WAGNER JAMES W. RYAN

JULY 1970

- - GODDARD SPACE FLIGHT CENTER GREENBELT, MARYLAND

o N - 7 444 ( {AGESS)_P_ (cooU) 2fPG2( 5 / (CODE O (NASA CROR TMX OR AD ReOpducodNUMBER)by (CATEGORY) NATIONAL TECHNICAL INFORMATION SERVICE Springoa.d, V. 22151 X-552-70-289 PREPRINT

LUNAR GRAVITY MODELS FOR IMPROVED APOLLO ORBIT COMPUTATION

James P. Murphy Theodore L. Felsentreger Carl A. Wagner Mission Trajectory Determination Branch Mission and Trajectory Analysis Division

James W. Ryan Data Evaluation Branch Manned Flight Planning and Analysis Division

July 1970

GODDARD SPACE FLIGHT CENTER Greenbelt, Maryland ABSTRACT

Three possible modifications of the operational lunar gravity model for Apollo missions are presented. One of these is recommended for use in future Apollo missions. This field, designated ML1.1, consists of the current oper ational model (LI) plus the following values for the (4,1) - 4 - 4 . harmonics: C4, = -. 1284 x 10 and S4, = .1590 x 10 The main benefit to be derived from the use of this model lies in its capability to predict the inclination and inertial node for all the Apollo orbits accurately. The current Li model fails to do this, especially for the low inclination cases. The (4,1) values were obtained by considering clas sical elements from Apollos 8, 10, 11, and 12 as observ ables in a least squares gravity retrieval program. The MLi.1 model performed as well or better than did the Li field in predicting the trajectories for Apollo and Lunar Orbiter spacecraft. The tracking data were fit just as well, also. The desirable properties of the Li model are all preserved with the MLl.1 model.

P1E AN; N07 FILMED;: CONTENTS

Abstract ...... iii

INTRODUCTION ...... 1

DATA...... _ 2

LUNAR DISTURBING POTENTIAL ...... 2

LAGRANGE PLANETARY EQUATIONS...... 5

ANALYSIS ...... 6

METHOD OF SOLUTION ...... 7

RESULTS ...... 8

TESTS ...... 9

DISCUSSION AND CONCLUSIONS ...... 10

References ...... 11

vr LUNAR GRAVITY MODELS FOR IMPROVED APOLLO ORBIT COMPUTATION

James P. Murphy, Theodore L. Felsentreger Carl A. Wagner and James W. Ryan GoddardSpace Flight Center

INTRODUCTION

The search for a siiple lunar gravitational model adequate for Apollo mission orbit predic tion has been continuing ever since the discovery of unexpectedly large anomalies in the tracking data of the Lunar Orbiter spacecrafts. These anomalies indicated that the moon's gravity field was somewhat more complex than had been thought.

For the Apollo 8 mission, the first manned flight to the moon, a tri-axial gravity model de rived by Jeffreys (Reference 1) was adopted both for mission control at Mission Control Center, Houston, and for operational support at Goddard Space Flight Center. However, this model proved to have rather poor orbit prediction capability for the mission, producing unacceptably large in plane and out-of-plane errors.

A slightly larger gravity model consisting of the C2 , C22, C30 , and C., spherical harmonic coefficients was developed by Risdal (Reference 2) and was adopted for operational use in the Apollo 10 and 11 missions. This model was derived from a study of the long-period variations in the orbital elements of the Lunar Orbiters, and was designated the "R2" model. While the R2 model provided improved orbit prediction capability, it still produced rather large downtrack errors and could not accurately predict orbit plane variations.

An adjustment to the R2 model was developed by Compton and To]son from an analysis of Apollo 8 data (Reference 3), and served to reduce downtrack errors significantly for Apollo 8.

This model, termed the Ll model, consists of the R2 field plus a value for the C33 coefficient. While having improved orbit prediction capability, the Ll model still failed to predict orbit plane variations (i.e., it could not adequately model the evolution of the inclination and longitude of ascending node).

This paper presents the results of a study designed to derive a simple lunar gravity model superior to the Li model in Apollo orbit prediction capability, with the particular property of be ing able to model inclination and longitude of ascending node histories. For this study, the data used were classical orbital elements at approximately one-orbit intervals for the Apollo 8, 10, 11,

I and 12 missions. The Lagrange Planetary Equations were numerically integrated to provide a reference trajectory, and the appropriate spherical harmonic coefficients were solved for by weighted least squares. A more detailed description of the method used and the data sets chosen follows in succeeding sections.

The previous operational Apollo lunar gravity models (i.e., the tri-axial, R2, and Li models) appear in Table 1.

DATA

The basic data used in the analysis consists of classical orbital elements for the Apollo 8, 10, 11, and 12 Command and Service Modules (CSM) as they orbited the moon. Each set of elements resulted from a one orbit solution for spacecraft position and velocity using Doppler data. Since, in each mission, the CSM was necessarily subjected to many orbit changes due to maneuvers, the data were divided into eight data sets, or "arcs," during each of which the spacecraft was free of such perturbations. A summary of these arcs indicating the "free flight" periods and orbit num bers involved is given in Table 2.

The orbital elements themselves for each arc, along with the Modified Julian Date (MJD) for each set of elements, are listed in Table 3. In the table,

a = semi-major axis (moon radii)

* = eccentricity

= angle of inclination to lunar equator (degrees)

= argument of perilune (degrees)

'N = inertial longitude of ascending node (degrees)

M= mean anomaly (degrees),

where the mean radius of the moon was taken to be 1738.09 km. For each arc, the "inertial" longi tude of ascending node is the selenographic node less the mean rotation of the moon, referred to the first time point in the arc. The reason for listing the inertial nodes rather than.the seleno graphic nodes will become clear in a later section when the data actually used as observables are discussed.

The elements for Apollos 8, 10, and 11 were taken from Reference 4, and are tabulated at perilune (i.e., mean anomaly - 0). The Apollo 12 elements 'Were determined at GSFC, and were obtained from Reference 5-these are tabulated very near perilune. All the elements are seleno graphic with the exception, as noted previously, of the inertial node.

LUNAR DISTURBING POTENTIAL

The universally recommended spherical harmonic expansion for the gravitational potential at a point with spherical coordinates (r, ,, )in a rotating coordinate system with origin at the center

2 of a primary body is (see Reference 6)

4 U = + P~ (s n ) [ q cosn + S .sin m •

If the moon is the primary body, then

2 = gravitational constant of the moon = 4902.778 km'/sec

= distance from the center of the moon

= selenographic latitude

X = selenographic longitude

a = radius of the moon = 1738.09 km

Pt. (sin 0) = associated Legendre function of degree t and, order m

Ct, s 4 = unnormalized spherical harmonic coefficients of degree t and order m.

The disturbingpotential R is equal to the potential u minus the central force term, or

, R = U

which can be written in terms of the classical orbital elements mentioned'previously. However; since the spacecraft orbit data resulted from one-orbit solutions, it is unnecessary to consider

- terms involving mean anomaly in R. Furthermore, in this study no terms involving spherical harmonic coefficients of higher degree than four were considered. Therefore, the truncated ex pression for R in terms of classical orbital elements is

2 1 a (3 sin' i - 2) R 4s 3 (1-_2)3/2 C20

2 2 3 a sin I

3 53Sn22 1 2 a (1i-e2)3/2 [C2 2 cos 2 1+

3, a e sin 1(5 5in2 i -4) 4 4 a (1 -e2)s/2 C30Sit

3 3 1,% e 4 16. (1-e2) 5/{[lstn s(i+ 3cos i)- 4(l+ cost)] [c31 Cos (W+K')+S3jsin(+)

2 + [5sin s(l-3cosi)-4(1-cosi)] [C3t COs(Wf)-S sin(o-f2)]}

3' 1 15Aa3 e sin 2 i) [C sin (w+ 20) - S3 COS (w+ 2fl)] + (i1 e )5/2 {(i -2 cosi -3 cos 3 2

+ (I1+ 2COSi-s3cos') [C 32 sin(w- 2Q)+ S32 COS(w- 2 )]}

/ 3 2 i 45 a e sin C os ( +3fl) 4 + +S 3sin(w +30)] 8 a (I e2)5/2 {( 1 )C 3 3 3

+ (1 -ss) S [C3 3Oco(w - 3Q) -S33 sin (w-32)]}

2 ) 2 4 2 2 2 + a3z {(2+ 3e (8 - 40 sin i + 35 sin i) + 10 e sin i (6 - 7 sin i) COs 2. 2 2 2 . 1+ a' i281 - e )7 C4 0[ /

COS (2sw+O)] + S1 ef~ e [7 i (1+2cos i)-6(1+cosi)] [C,, sin(2co~fl)-S 41

+ 2 (2+3e2) cos 1(4-7 sin2i) [C4 1 sinf0bS4 1 COSfl]

1COS + e2 [7 sin 2 i(1- 2cos i) -6(l-cos i)] [C4 1 sin (2w0-f2)+S4 (2c--fl)]}

2 7 (2w+ 2Q)+ S42 sin (2w+ 2n)] + 32 a(i-e2) '2 {3e2 (1+ cos i) (7 sin l COs i- (1+ COS )] [C 4 2 cos

2 2 2+ S sin 20] + (2 + 3e2) sin i(11 -21 cos i) [C 4 2 cos 4 2

20) S (2- 2)]} - 3e (1- cos i) [7 sin2 i OS i + (1 - Cons)] [C 42 eOS (2w- - 42 in

4 ( - 315g1.a Sin'i {° (15-a3ia(-2 i -2osoS3 i) [C,, sin . +3) -%S. oo(2 +3Q)]

2 s3n2] - 2(2+ 3e2) sin i COSi [C, 2sin30S 4 3 3

i) Sin (2w -3n) ens(2w-31)J} + e2 (1-3 cos2 + 2eCos~ [c4 3 +S43

4 2 315gga sin i

-i1+ a --e)'2{2 (1 + COS1)2 [C44 COS (2w+ 410)+s 4 in (2w+ 4D)

2 2 + (2+ 3e ) sin i [C44 Cos 40+ S44 sin 40]

2 2 + e (1 -cos i) [C44 cOS (2a- 41) -S 4 4 sin (2.-411)]} "

In this expression for a,the quantity Qis the selenographic longitude of ascending node.

LAGRANGE ;PLANETARY EQUATIONS

The Lagrange Planetary Equations for classical (i.e., Keplerian) orbital elements, which may be found in many textbooks on Celestial Mechanics, are as follows:

d. 2fa OF dt - c OM

2 de 1 -e OF l -e2OF

2 d - cOs i OF Y -e OF

di cOs i OF I 8F ) 2 /d . Ce sin i r,.#-(1- e ) si ON

d 12 ) OF dt dt -Cl-_ sinii O

2 (M l-e aF 2 fa8F

where F represents any disturbing function. For the equations, the reference coordinate system must be inertial (i.e., non-rotating); thus, for our purposes, the quantity N would be the inerial longitude of ascending node. The disturbing potential R defined previously will be used as the disturbing function F. Since R does not contain the mean anomaly M, the equations for da/dt and dN/dt, Iand that part of de/dt involving aFMa4I, are not needed. Therefore, the equations to be

5 considered are

de f-e2 aR dt - eOJ

2 d. cos i OR -e OR 7t - - f -( si + A e

di cos i R 1 OR dt -(1 ) sini a fl-) snl i ON

dN 1 OR 2) (I = e sin -

ANALYSIS

The problem now remains to determine which spherical harmonic coefficients should be solved for, and which data should be used in the solutions. As mentioned previously, it was noted that the Li model possessed better Apollo orbit prediction capabilities than both the tri-aidal and 12 models; its main shortcoming was its failure to predict orbit plane variations (i.e., variations in inclination and longitude of ascending node). As a primary "grand-rule," therefore, it was decided to seek an extension to the hl model rather than alter any of its coefficients, with emphasis placed on inclination and node prediction.

In addition, because of restrictions imposed by such things as the size of mission operation computer programs and the gravity field capacity wired into the spacecraft computer, and the de sire to keep trajectory computing time to a minimum, it was decided to consider only those spher ical harmonic coefficients of degree and order four or less. It was also decided toforego solving for the fourth degree zonal harmonic (C40)because zonals are best determined from secular and long period orbital element variations over considerably longer time spans than are covered by the data.

Now, the Apollo 8, 10, 11, and 12 orbits can be characterized as having relatively low eccen tricities and inclinations. Since the Lagrange Planetary Equations involve divisiors of e and sin i, those terms in the various partial derivatives of Rnot containing factors of e and sin i will pro vide perturbative effects enhanced by these relatively small divisors. This fact, coupled with the indication of high correlations among fourth degree coefficients from some preliminary computer runs, narrowed the choice of coefficients to be determined to the (3,2) and (4,1) harmonics. In addition, the analysis indicated that the (3,2) coefficients could best be determined from the e and data, and the (4,1) coefficients from the i and N data.

6 To summarize, then, the principal factors and guidelines considered in selection of the spherical harmonic coefficients to be solved for and the data to be used were the following:

1. The Li model would be held fixed. 2. No spherical harmonic coefficient with degree and order greater than four would be solved for. 3. No zonal harmonic coefficient (i.e., order = 0) would be solved for. 4. Only the (3, 2) and (4, 1) coefficients would be solved for, the former from e and - data, and the latter from i and N data.

METHOD OF SOLUTION

As was mentioned previously, solutions for the spherical harmonic coefficients were carried out by a weighted least squares procedure, and integration of the Lagrange Planetary Equations was performed numerically. The computer program used in obtaining the .solutions is called ROAD (Rapid Orbit Analysis and Determination Program), which essentially is designed to solve for common geodetic parameters and initial satellite orbital elements from the long term evolution of Kepler elements for a number of individual satellite arcs (Reference 7).

The program uses as an orbit generator the numerical integration of the Lagrange Planetary Equations. Partial derivatives of the observations (i.e., Kepler elements) with respect to solved for parameters (spherical harmonic coefficients and initial conditions) are obtained by simultaneous numerical integration of the "variation equations." These variation equations :are found-in the following manner:

Let the Lagrange Planetary Equations be written as

deI dt - f (e,,c.)

where

e. = Kepler elements (i = 1, 2, ... , 6)

c. = spherical harmonic coefficients (k = 1, 2, .-. ,no. of coefficients included) fi = function associated with e..

Then, the variation equations are

= dt / ae,1, 2, ,6no.of £- [ J (k 1, 2, no. of coefficients

'7 d Fe. 1 6 af1 [1e 1, d- LeJ e Li, 1

where

ej.0= initial value of .

These equations are numerically integrated to obtain the observation partial derivatives aei/acu and ae,/ae,,, which are needed for the first order differential corrections. The observation par tials and residuals are combined to form the observation equations. An estimate of the accuracy of each observation quantity is used as the weight for the corresponding observation equation. Fi nally, the normal equations for the parameter corrections are accumulated and solved by means of a standard weighted least squares process.

RESULTS

Three Modified Li (MLl) lunar gravity models were derived from the orbital element data in Table 3, and are listed in Table 4. In the MLI.1 model, the C4, and S4, coefficients were deter mined from i and N data only, with the Li model held fixed. For the ML1.2 model, the MLI.i model was held fixed, and the C32 and S32 coefficients were derived from e and wdata. To derive the MLi.3 model, the Li model again was held fixed, and the C32, S3 21 C41 and S4, coefficients were solved for using the e, . i, and N data. The results substantiate the previously stated conclusion that the (3,2) coefficients would be basically determined from the e and c data, and the (4,1) coef ficients from the i and Ndata-the (3,2) values differ very little between the MLi.2 and ML1.3 mod els,and the same holds true for the (4,1) values in the MLI.1 and ML1.3 models.

Figures 1 through 32-show the fits to all the orbital element data for the Li, ML.1, ML1.2, and MLl.3 lunar gravity models. It is clear that the MLl models fit all the inclination and node data as well as, or considerably better than, the Li model does, with the single exception of the Apollo 8 inertial node. However, thisnode was much slower moving than the Apollo 10 and 11 nodes, so the fits are still quite comparable. In addition, the ML1.2 and ML1.3 models generally fit the e and . data better than the models not containing the (3,2) coefficients, although the differences are not so pronounced. It should also be noted that the e and - evolutions as predicted by the ML.Il model are comparable to those predicted by the Li model; at the same time, the MLI.l model pre dicts the inclinations and inertial nodes much better than does the Li model (with the previously noted minor exception). The "goodness of fits" are indicated in Table 5, which lists the root mean square of the observation residuals for each arc as produced by the Li, MLl.1, ML1.2, and ML1.3 lunar gravity models.

8 TESTS

In the initial stage of this work, the ROAD program was used to generate classical elements using the Li lunar potential model. These were found to be in good agreement with those obtained from a different program (References 8, 9) for all Apollo and Lunar Orbiter cases that were considered. It was noted that the non-central portion of the lunar gravity field dominated the'per turbations of not only the low altitude Apollo orbits but also many of the arcs of the more distant Lunar.Orbiters with semimajor axes of a thousand kilometers larger. In addition, numerical integration of Apollo state vectors were made including not only the terms that dominate the motion but also the short period terms. The integrations were performed with all the third body effects in cluded. These trajectories were in agreement with those published in Reference 8 and in this report.

The MLl.I field was obtained first (Reference 10). Some preliminary tests of it were performed after which it was forwarded to the Manned Spacecraft Center (Reference 11). These tests have since been expanded to include the MLl.2 and MLI.3 fields, and to involve the processing of multi revolution data arcs.

In order to determine the relative ability of the LI, the MLI.I, and the MLl.2 lunar gravity fields to model actual Apollo doppler tracking data, a series of tests were performed. Using the DEBTAP (Data Evaluation Branch Trajectory Analysis Program), a program which determines orbits of lunar satellites by means of weighted least squares, one revolution and two revolu tion arcs of doppler tracking data from Apollos 11 and 12 were processed. The list of standard-errors-of-fit presented in Table 6 is the result of this processing. Comparing standard errors-of-fit, which are simply the square roots of the sums of the weighted residuals, is the simplest and most comprehensive method of comparing the goodness of fit of two orbit determina tions. The results of the tests are quite clear cut. The Li and the MLI.l provide nearly identical data fitting capabilities. Based on the results presented in Table 6 it would be impossible to say which field was superior. The MLI.2 provides a somewhat degraded data fitting capability; the standard-errors-of-fit are perhaps 20% larger than for the LI and MLI.i in some cases. In other cases, some improvement was realized. The main conclusion to be drawn from this table is that the MLI.l fits Apollo 11 and Apollo 12 tracking data as well as the LI field.

In an attempt to show some significant superiority of one of the fields so far as fitting tracking data is concerned, an eight revolution, fifteen hour arc of two-way tracking data from Apollo 11 was processed. This data, from orbits 5 through 12, was processed with the Lungfish program. The standard deviations of fit for all four models appears in Table 7. Although the MLI.i field better predicts the out-of-plane variables than does the LI model, this fact does not seem to be reflected in these tracking data fits. The fact that the MLI.2 and MLI.3 fields better predict the in plane variables does manifest itself to some extent. Again, the main conclusion to be drawn here is that the MLI.i field is as good as the LI field so far as fitting tracking data is concerned.

The next test item was concerned with Lunar Orbiter trajectories. The Li, MLI.i, MLl'2, and MLI.3 models were used to generate classical elements at one day intervals for many Lunar Orbiter 2, 3, 4, and 5 arcs. As a result, several conclusions were drawn. First, the Li and MLI.i

9 models produced nearly identical e, . and N evolutions. Secondly, the MLl.1 was superior for inclination evolutions for the lowly inclined orbits when compared to the actual classical elements. For the highly inclined orbits, all four models performed equally as well. Thirdly, the ML1.2 and ML1.3 performed as well or better than the LI and MLI.l in all cases so far as e and w evolutions are concerned. Examples to substantiate these conclusions can be found in Figures 33 to 35.

Finally, Table 8 contains a list of previous determinations of the (4,1) and (3,2) tesseral harmonies together with the values for these coefficients appearing in the various ML1 fields. The conclusion reached upon inspection of this table might be that the values of these coefficients obtained from Apollo orbits are in reasonable agreement with those obtained elsewhere, especially the (4,1) terms. This is fairlyremarkable since the satellites used here were in verydifferent type orbits, some of the other determinations were made directly from the tracking data, and the fields differ greatly in size.

DISCUSSION AND CONCLUSIONS

Very few efforts to obtain knowledge of the lunar gravity field have been made to date using data from Apollo missions. Only one of these efforts has been successful (Reference 3). The analysis performed in this report has been successful so far as predicting out of plane variables only. In addition, the rather good agreement between the various determinations of the (4,1) co efficients (Table 8) indicates that there is a substantial amount of gravity information implicitly contained in the classical elements of the Apollo orbits. The even zonal harmonics of a primary are best determined by the secular effects on its satellites. The remaining zonals and all the non zonals must be determined from periodic perturbations of one kind or another. The only exceptions are cases of resonance. Therefore, it does not matter how long a satellite is in a particular orbit. The important question is concerned with how fast the angular variables that make up the arg ments in the long period perturbations are moving.

In the case of the Apollo orbits, the arguments of perilune are moving very fast. Further, this is not simply due to the fact that Apollo orbits are nearly circular since the determinations of the orbits with the epoch at perilune reveals this rapid motion. Mathematically, then, this rapid motion in the argument of perilune provides us with many samples of the long period effects of the lunar potential on the classical element in a relatively short period of time.

Initially, we restricted ourselves to consider only gravity coefficients within a (4,4) field due to limitations of hardware, software, and in the interests of rapid computations. Only the terms of third degree were independent of multipliers of eccentricit ' in the perturbations in eccentricity and divisors of eccentricity in argument of perilune perturbations. However, it was our desire not to alter any of the coefficients in the already proven Li model. This left only the (3,2) coefficient to consider so far as eccentricity and argument of perilune perturbations are concerned. The im provement with this single harmonic so far as eccentricity is concerned was for some cases dis appointing; the fits to tracking data were inconsisteht. However, this facet of the study was not without merit and the (3,2) coefficients may be of some practical application.

10 There seems to be little doubt, however, as to the merit of adopting the MLI.l lunar gravity field as the operational model for Apollo orbit computations. We recommend the MLi.1 model. for several reasons:

1. The MLI.I model vastly outpeiforms the LI model for predicting out of plane variables for Apollo missions, while preserving the proven capabilities of the Li field for in plane variables.

2. The MLI.l model also predicts these out of plane elements for the posigrade Lunar Or biters of moderate inclination as well or better than does the Li model, even though the MLI.l was derived from retrograde Apollo orbits.

3. The values for the (4,1) coefficients are consistent with values obtained from Lunar Or biter analyses.

4. The fits to Apollo tracking data for one, two, and eight revolution arcs are as good for MLi.i as they are for Ll.

The non zonal harmonics in the Ll field are all symmetric with respect to the reference longitude (that is, $22 = S = $33 = 0). The (4,1) harmonic in the MLi.l field is not. In fact, there is an accomodation between the proposed values for the (4,1) harmonic and the local gravity effects first observed on Lunar Orbiters (Referencel6), and which were called "mascons." In Figure 36, the shaded areas represent positive gravity anomalies due to the (4,1) harmonic. It is clear that this subdivision is in accord with many of the mascon efforts. This, in itself, does not justify adopting this harmonic. The justifications appeared above. It would, however, move the operational Apollo lunar gravity field one quantum jump closer to accomodation with the mascon results.

REFERENCES

1. Jeffreys, H., Monthly Notices of the Royal Astronomical Society, 122(5), pp. 431-432, 1961.

2. Risdal, R. E., "Development of a Simple Lunar Model for Apollo", Contract Report D2-100819-1, The Boeing Co., Seattle, Wash., 1968.

3. Compton, H. R., and Tolson, R. H., "Study of a Simple Lunar Gravitational Model for Applica tion to Apollo Orbit Determination and Prediction", Langley Research Center, June 11, 1969 (memo to J. C. McPherson, Manned Spacecraft Center, Houston).

4. "Lunar Model Analysis for Apollo 12", TRW Systems Group, Redondo Beach, Calif., Aug. 31, 1969.

5. "Apollo 12 MSFN Metric Tracking Performance", GSFC X-832-70-79, March 1970.

6. Hagihara, Y., Astronomical Journal, Vol. 67, p. 108, 1962.

11 7. "ROAD-Program Description and User's Guide", Wolf Research and Development Corp., Riverdale, Md. (prepared for NASA-GSFC under contract No. NAS-5-9756-182), 1970.

8. Diamant, L. S., "Explanatory Supplement to the Mean-Element Lunar Potential Model Catalogs", TRW Letter 5522.7-70-49, 26 March 1970.

9. Diamant, L. S., "Mean-Element Lunar Potential Model Catalog, Model L-1," TRW Letter 5522.7-70-76, 7 May 1970.

10. Murphy, J. P., "A Modified Li Field for Accurate Inclination and Inertial Node Predictions for Apollo Orbits", June 11, 1970 (memo to J. Barsky, Manned Flight Planning and Analysis Division).

11. Barsky, J., "An Augmented Li Lunar Potential Model Suggested by J. Murphy of Mission Trajectory Determination Branch of Goddard Space Flight Center", June 23, 1970 (memo to J. McPherson, Mathematical Physics Branch, Mission Planning and Analysis Division, Manned Spacecraft Center).

12. Lorell, J., "Lunar Orbiter Gravity Analysis", Technical Report 32-1387, Jet Propulsion Laboratory, Pasadena, Calif., June 15, 1969.

13. Tolson, R. H., and Gapeynski, J. P., "An Analysis of the Lunar Gravitational Field as Obtained from Lunar Orbiter Tracking Data", presented at the IQSY/COSPAR Assemblies, London, England, July 17-28, 1967.

14. Gapcynski, J. P., Blackshear, W. T., and Compton, H. R., "The Lunar Gravitational Field as Determined from the Tracking Data of the Lunar Orbiter Series of Spacecraft", presented at the AIAA-AAS Astrodynamics Specialists Conference, Grand Teton National Park, Wyoming, September 3-5, 1968.

15. Blackshear, W. T., "Progress on Lunar Gravitational Potential Determinations by Analysis of Lunar Orbiter Tracking Data" presented at the AGU 50th Annual Meeting, Washington, D. C., April 1969.

16. Muller, P. M. and Sjogren, W. L., Science, Vol. 161, page 680 (1968).

12 Table I

Apollo Operational Lunar Gravity Models.*

COEFFICIENT TRI-AXIAL R2 hl

C20 -2.0718677 -2.07108 -2.07108

C22 0.20239141 0:20716 0.207-15'

C3 0 0.21 0.21

C31 0.34 0.34

C3 3 0.02583 - 4 Multiplyall coefficients by 10 NASA-OSFC-T&DS MISSION&TRAJECTORYANALYSISDIVISION BRANCH 552 DATE July1970 BY J*P.Murphy PLOTNO. 2169

Table 2

Apollo Classical Element Data Arcs.

ARC MISSION ORBIT NUMBERS "FREE FLIGHT" PERIOD,

1 APOLLO'8 3, 4, 5, 6,7, 8, 9,10 CIRCULARIZATION TO TEl 2 APOLLO 10 3, 4, 5, 6,7, 8, 9,10, CIRCULARIZATION TO 11,13, 14, 15 EVASIVE MANEUVER 3 APOLLO 10 17,18,19,20,21,22,24, EVASIVE MANEUVER 25, 26, 27, 28, 29, 30, 31 TO TEl 4 APOLLO 11 3,4,5,6,7,8,9,10, CIRCULARIZATION TO 11,12,13 CSM/LM SEPARATION 5 APOLLO 11 14,15,16, 17,18,19,20, CSM/LM SEPARATION 21, 22, 23,24,26,27 TO TEl 6 APOLLO 12 3,4,5,6,7,8,9,10, CIRCULARIZATION TO 11,12 CSM/LMSEPARATION 7 APOLLO 12 20, 21, 22, 23, 24, 25, PLANE CHANGETO 26, 27, 28,29, 30,32 CSM/LM DOCKING 8 APOLLO 12 39, 40, 41,42, 43, 44, 45, PLANE CHANGE TO TEF

NASAGSFC-T&DS MISSION&TRAJECTORYANALYSIS DIVISION BRANCH 552 DATE Jul'y.1a7O BY J. P.Mmhy PLOTNO. 2170

13 Table 3

Apollo Classical Element Histories.

ARC 1 APOLLO TIME(MJD) a(R.R.) e A(EG.) .(OEG.) 0OEG.) 0(DG.) 40214.6117331 1.06434684 .0007980 167.6871 ?95.4667 50.3879 0.0 40214.6976670 1.06433545 .0010360 167.7067 310.2070 50.5119 0.0 40214.7812877 1.06428564 .0013120 167.7108 314.8446 50.6251 0.0 40214.8644853 1.06425793 .0015970 167.7001 317.6396 50.7262 0.0 40214.9474647 1.06422146 .0018930 167.7190 319.4810 50.8095 0.0 40215.0302992 1.06419200 .0021700 167.7444 320.8273 51.0147 0.0 40215.1128475 1.06412448 .0024310 167.7534 320.8720 51.1573 0.0 40215.1957676 1.06411903 .0027220 167.7465 322.4264 51.1694 0.0

ARC 2 APOLLO 10 TIME(MJD) a(M.R.) e (DEG.) w(DEG.) 0(DEG.) M(DEG.) 40363.0428336 1.06291014 .0008390 178.7469 35.0556 181.8886 0.0 40363.1360199 1.06291762 .0008660 178.8032 40.1027 184.2050 0.0 40363.2208300 1.06284642 .0010630 178.8184 50.7362 184.2,95 0.0 40363.3052676 1.06274243 .0012300 17S.8412 60.3651 184.9274 0.0 40363.3886009 1.06271069 .0014860 178.8890 63.9567 185.9713 0.0 40363.4710500 1.06271349 .0017470 178.8797 66.9879 187.1131 0.0 40363.5537603 1.06270823 .0020060 178.9009 69.4209 188.1167 0.0 40363.6361845 1.06264492 .0022040 178.9096 70.9939 189.5160 0.0 40363.7194735 1.06256917 .0024470 178.9422 74.5174 189.0592 0.0 40363.8837527 1.06242940 .0028970 178.9035 76.1084 192.4977 0.0 40363.9660240 1.0623238 .0031960 179.0009 76.4989 193.2508 0.0 40364.0481611 1.06238328 .0034650 178.9478 76.6095 194.3073 0.0 ARC 3 APOLLO 10 71E(MJO) a9M.R.) . ,e (0E.) 3(0E.) (DEG.) M(nEG.) 40364.2150285 1.06317023 .0043080 178.9779 88.0071 180.9614 0.0 40364.2976785 1.06340969 .0044660 179.0050 89.1406 181.2861 0.0 40364.3799058 1.06342670 *0047460 179.0232 89.575 182.7999 0.0 40364.4621410 1.06343144 .0050290 179.0559 89.8224 184.0953 0.0 40364.5443889 1.06343074 .0053150 179.0518 90.0000 185.2606 0.0 40364.6266194 1.06339514 .0000760 179.0651 90.4665 186.8180 0.0 40364.7911922 1.06343758 .0062280 179.1188 91.5016 189.4640 0.0 40364.8735225 1.06344354 .0065080 179.1128 91.1077 189.7110 0.0 40364.9558955 1.06347458 .0067870 179.1127 91.9069 190.9754 0.0 40365.0382494 1.06347896 .0070620 179.0947 92.6186 192.2488 0.0 40365.1205243 1.06345441 .0073550 179.1445 93.0802 193.6127 0.0 40365.2020104 1.06345108 .0076430 179.1507 93.1635 194.5558 0.0 40365.2851356 1.06347212 .0079370 179.1725 92.9556 195.0129 0.0 40365.3674532 1.06345599 .0082500 179.1904 92.8459 195.6209 0.0

ARC 4 APOLLO 11 TTME(MJD) a(M.R.) e i(DEG.) (DEG.) (DEG.) M(DEG.) 40421.9629387 1.06242502 .0059770 178.4394 249.5599 167.5323 0.0 40422.0454810 1.06247342 .000700 178.4649 250.5962 167.8274 0.0 40422.1201041 1.06241011 .0054790 178.4550 255.5073 171.4219 0.0 40422.2109644 1.06236627 .0052290 178.4656 259.5616 173.5490 0.0 40422.2936987 1.06238784 .0049900 178.4800 256.7675 172.1653 0.0 40422.3770300 1.06242940 .0047630 178.647 260.4040 171.1459 0.0 40422.4592083 1.06248219 .0045380 178.4915 259.8027 168.8719 0.0 40422.5420609 1.06251866 .0043210 178.217 264.7934 171.7437 0.0 40422.6248772 1.06236153 .0041610 178.5222 267.9222 172.8766 0.0 40&22.7062675 1.06255040 .0040040 178.5871 2654301 174.7143 0.0 40422.7900919 1.06238118 .0037890 178.5766 271.4727 174.3289 0.0 NAS14&S MISSION--R -ORANALYSISDIVISION BRANCH52 AEu 17 BY J p---h, FLOTNO. 11

14 Table 3 (Continued)

ARC 5 APOLLO 11 TIME(MJD) a(M.R.) e i{OEG.) w(DEG.) N(DEG.) RIDEG.) 40422.8718940 1.06231331 .0039640 170.5915 269.9769 163.2371 0.0 40422.95a903 1.06236750 .0037580 178.5875 271.7263 162.1272 0.0 40423.0379747 1.06234978 .0035680 170.9897 276.4500 163.5866 0.0 40423.1216486 1.06226403 .0034620 170.6448 284.7647 166.0358 0.0 40423.20462t2 1.06232022 .0033200 170.6532 288.3690 165.3336 0.0 40423.2885]18 1.06236978 .0032300 178.6658 294.3839 166.0860 0.0 40423.3721338 1.06262221 .0031600 178.6097 300.0971 166.6814 0.0 40423.4558624 1.06247009 .0031220 170.7099 307.1191 167.1189 0.0 40423.5397081 1.06250160 .0031280 178.7491 314.3773 167.7763 0.0 40423.6235247 1.06256075 .0031330 178.7525 321.5304 168.4644 0.0 40423.7073586 1.06260161 .0031810 178.7654 328.5203 168.9111 0.0 40423.878308 1.06262335 .0033830 178.7984 341.9411 170.0995 0.0 40423.9567812 1.06229490 .0035870 178.8136 341.8206 171.5603 0.0 ARC 6 APOLLO 12 TIEMJD) a().R.) e i(OG.) .(0EG.) 8108G.) M(DEG.) 409543.3961227 1.06277983 .0059077 164.8270 68.1980 337.1260 309.0230 40543.4786806 1.06278098 .0056158 164.7680 67.2900 337.2826 0.6490 405435607639 1.06277696, .0053223 16t.7840 65.9770 337.3070 0.4450 40543.6427454 1.06279939 .0050288 164.7360 64.6160 337.4881 0.0020 40543.7249537 1.06276890 .0047463 164.7460 63.6210 337.4952 0.0170 40543.8071759 1.06277523 .0041672 164.7410 62.0570 337.5724 0.7290 40543.8808889 1.06274244 .0041852 164.7350 60.6490 337.6050 309.0060 405t3.9711921 1.06278041 .0039200 164.7060 58.6020 337.7523 0.6310 40544.0036720 !.06270156 .0036729 164.6840 56.6510 337.8539 2.8950 40544.1348432 1.06276142 .0034020 164.6060 03.9370 338.1004 0.3180 ARC 7 APOLLO 12 T280)8J0) a0.R.) e i(EG0.) .(DEG.) N(OG.) MDG.) 40044.788a079 1.06363710 .0020915 165.6460 41.5060 334.5390 2.5070 4054 .8701736 1.06303068 .0018479 165.6040 36.3060 334.6430 2.9850 40544.9501273 1.06307211 .0016722 165.5900 28.6730 334.7336 359.7710 40545.0300852 1.06295013 .0014525 165.0630 21.7210 334.8238 359.8040 40545.1103935 1.0638890 .0014838 165.5430 7.9635 334.9517 0.3160 40545.1900926 1.06325104 .0014798 165.5400 305.4483 334.9864 0.7780 40545.2695718 1.06332583 .001552 165.4870 341.9360 335.1815 1.4350 40545.34P6690 1.06336956 .0016750 165.5230 328.4270 330.1336 0.1640 40545.4278935 1.06333734 .0017711 165.5110 317.0230 330.1912 2.5440 40545.5094144 1.06331433 .0018900 165.4960 309.0470 335.2486 1.4920 40545.5899306 1.06340868 .0021964 165.4130 303.1750 335.7792 359.2350 40545.7509028 1.06318085 .0024588 165.3940 28.2880 330.6828 359.6990 ARC 8 APOLLO 12 TIMEIMJO) a(M.R.) p (DEG.) .(DEG.) N(DEG.) M(DEG.) 40546.3253009 1.06345931 .0039980 16R.7580 282.6870 326.4160 0.1480 40546.4072456 1.06332411 .0041290 168.6980 2785620 326.2636 1.7830 40046.4087188 1.06329074 .0043590 168.6480 276.1120 326.4581 357.6920 40046.5701389 1.06329994 .0046261 168.6420 274.0240 326.4808 357.4490 405&6.6527546 1.06332906 .0248950 160.6360 272.1260 326.5662 0.0450 40546.7347222 1.06329362 .0051364 168.6050 270.4590 326.7211 359.5780 40546.8171759 1.06335518 .0004255 168.5820 269.3420 327.0103 0.8630

MIOON&TRAIE0T0--ANALYSISDIVISION BRANCH552 DADTJ,.1970 BY Iwph,0,y 51.. 012

15 Table 4

Modified Li (MLI) Lunar Gravity Models.*

COEFFICIENT MI1.1 ML1.2 ML1.3

C20 -2.07108 -2.07108 -2.07108

C22 0.20715 0.20715 0.20715 C30 0.21 0.21 0.21

C3 1 0.34 0.34 0.34 C32 0.1012 0.1040

S32 0.06790 0.07282

C33 0.02583 0.02583 0.02583

C41 -0.1284 -0.1284 -0.1083

S4 1 0.1590 0.1590 0.1460 -4 *Multiply all coefficients by 10

NASA-SFC-TBDS MISSION& TRAJECTORYANALYSISDIVISION BRANCH 552 DATE JuIy1970 BY J.P. MurIhy PLOTNO. 2172

16 Table 5

Root Mean Squares of Observation Residuals.

CLASSICAL ARC ROOT MEAN SQUARE ELEMENTS NO. L1 ML1.1 ML 1.2 ML 1.3

1 .0000894 .0000900 .000115 .000118 2 .000265 .000298 000545 .000499 3 .000116 .000124 .000129 .000132 4 .000257 .000253 .000145 .000143 5 .00101 .00101 .000761 .000779 6 .000401 .000399 .000107 .000123 7 .000443 .000546 .000161 .000153 8 .000136 .000163 .0000985 .000102

1 .0266 .00826 .00825 .00822 2 .112 .0286 .0285 .0371 i 3 .119 .0154 .0153 .0242 (degrees) 54 .135.0799 .0120.0222 .0120.0220 .0236, .0222 6 .0946 .0256 .0255 .0268 7 .117 .0235 .0235 .0258 8 .0844 .0205 .0203 .0259

1 10.4 10.4 2.48 2.60 2 5.49 8.43 2.99 2.62 3 1.60 5.70 3.86 2.97 'a 4 13.0 10.4 3.65 3.76 (degrees) 5 14.5 11.3 4.69 4,71 6 4.62 5.56 .406 .386 7 21.8 22.1 1.79 1.74 8 4.52 4.17 1.30 1.22 1 .0911 .224 .224 .199 2 6.49 .489 .486 1.02 j 3 9.01 .575 .597 1.20 1.77 (degrees) 4 4.35 1.73 1.73 5 3.88 .660 .660 .858 6 .100 .101 .101 .0968 7 .149 .155 .155 .151 8 .128 .125 .125 .124

NASA-GSFC-T&OS MISSION& TRAJECTORYANALYSISDIVISION BRANCH 552 DATE JuIy 19701 BY J P.Muphy PLOTNO. 2173

17 Table 6 Standard Errors of Fit to Apollo 11 and Apollo 12 Range and Range-Rate Tracking Data.

STANDARD ERROR OF FIT APOLLO ORBIT (dimensionless) MISSION NUMBER(S) L1 MLl.1 ML1.2

11 8 2.89 2.89 3.00 11 9 2.65 2.67 2.77 11 18 2.31 2.31 2.38 11 19 3.00 3.01 3.12 11 29 2.93 2.99 3.03 11 8 and 9 19.2 19.0 .17.7 11 18 and 19 15.9 15.1 14.3

12 9 18.4 18.1 23.4 12 10 17.8 17.9 24.6 12 16 18.8 18.8 24.5 12 17 21.4 20.7 26.0 12 40 15.4 15.2 18.6 12 41 14.5 14.3 17.2 12 42 15.9 15.8 20.3 12 9 and 10 33.2 28.2 28.0 12 40 and 41 31.6 32.1

NASA-GSFC-T&DS MISSION & TRAJECTORY ANALYSIS DIVISION BRANCH 552 DATE July 1970 BY J. P.Murphy PLOT NO. 2174

Table 7 Standard Deviations of Fit to a Fifteen Hour Arc of Apollo 11 Two-Way Doppler Observations.

LUNAR GRAVITY MODEL STANDARD DEVIATION (CPS)

L1 4.133

ML1.1 4.109

ML1.2 3.771

ML1.3 3.758

NASA-GSFC-T&DS MISSION& TRAJECTORYANALYSISDIVISION BRANCH 552 DATE July1970 BY J, P M.Mhy PLOTNO. 2175

18 Table 8

Values for(4,1) and (3,2) Coefficients.*

SOURCE REFERENCE C , 4 1 4 1 C3 2 S3 2 MLI1.1 10 -.1284 .1590 ML1.2 THIS REPORT -.1284 .1590 .1012 .0679 ML1.3 THIS REPORT -.1083 .1460 .1040 .0728

JPL-1 12 -.1237 .0564 -.0257 -.0200 JPL-2 12 -.1063 .0755 .0003 .0277 JPL-3 12 -.1607 .1099 .0076 .0283

LaRC (5x5) 13 -.1560 .0391 .1294 -.0147 LaRC (7x7) 14 .0813 -.0172 .0693 .0441 LaRC (13x13) 15 -.0573 .0680 .0502 .0203

-4 . *Multiply all coefficients by 10 NASA-GSFC-T&DS MISSION&TRAJECTORYANALYSISDIVISION BRANCH 552 DATE July1970 RY J.P. Muh" PLOTNO. 2176

19 .0028 .0026 .0024 ML 1.3

>- .0020 - Li 2 .0018 ML 1.1 .0016 zA.0014 . ,, .0012 ML 1.2. .0010 .0008 .0006 .0004 .0002 - CLASSICAL ELEMENT DATA

0 1 2 3 4 5 6 7 8 9 10 111213 ORBIT NUMBER

MISSION& TRACY0RYANALYS ISD IV]IS ION BRANCH552 DATEJVy1970 B JP l PL0TN0. 2133

Figure I-Arc I - Apollo 8 eccentricity vs orbit number.

20 167.760 167-750 S167;7:40-M _11 167.730 ML 1.2 "167-720 - .- ""ML 1.3' Z 167"710 1 167.700 L 167-690 167-680 " 167670 - - CLASSICAL ELEMENT DATA 1 67 66 .&0 . t I I I I I I I I I I 0 1 2 3 4 5 6 7 .8 9 10 11 12 13 ORBIT NUMBER

MION TRAJECTORYANA-LYS=SvsIoN .6AN-525 'ATe ,IT1000 DV 40p0007 PLOVISO.0134 "

Figure 2-Arc I - Apollo 8 inclination vs orbit number.

21 -35 - -40 - "MLIVL 1.31.2 '", ,,"

10 A SLII zU. -45 / I ~-503 / . /~" ML 1.1 LU -60 //-•CLASSICAL ELEMENT < -65 DATA 0 1 I 2 3I 4I 5I 6I 7 I 8I 9I I0 111213I

ORBIT NUMBER

MISSION1 -5-ETORANALYSIS DIVISION BRANCH552 DAT, BV J P.5 PLOTiO 135

Figure 3-Arc 1 - Apollo 8 argument of perilune vs orbit number.

22 51.4 51-3 " " 51-2

:o0 511

-U 509 o 508 ML 1.3 50-'7 507 ML 1.1 50-6M uj50"5 // Li ML1.2 --50.4 /

50-3 - CLASSICAL ELEM ENT DATA I I I I I I I I 0 1 2 3 4 5 6 7 8 9 10111213 ORBIT NUMBER

MISSONSTRAJECTORy NASA-SFC-O ANALYSIS DIVISION SY i.R5HS2 P SO P.OTNODATE111, LL'255

Figure 4-Arc I - Apollo 8 inertial node vs orbit number.

.,23 .0040. .0038 -/ .0036 / .0034 / .0032 .-• .0030 ML 1.2-e . E .0028 .oo2 L 1ML 1.1 S.0026 -Li 1L , .0024 & o .0022 .0020 .0018 ,. * .0016 .0014 0 .0012 * DATA .0010 ML 1.3' CLASSICAL ELEMENT DATA

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ORBIT NUMBER

I-ON & TR-E RYAAYS DVIN eRIICI52 1AEJl17

Figure 5-Arc 2 - Apollo 10 eccentricity vs orbit number.

24 179-04 179.02 179-00- 178-98 ML 1.1 - 178.96 ML 1.2 w 0 2 17894

: 178.92 178-90 1 - 178e-88 ML 1.3 z 878-86 : 178-84 178-82 L I 178-80 . - '...... 1.78-78 " 178.76 .CLA-SSICAL ELEMENT .DATA I 1 I I I 1 I I I ! 1 0 1 2 3-4 5 6 7 8 9 1011.1'213-1415'16 ORBIT NUMBER M-SSION&eT A=CTOHRY AIALYSmISIV SIION eYJPMu-h _LOTNO 21.S

Figure-6-Arc 2 - Apollo 10 inclination vs orbit-number.

-25 ML 1.1 S80 -...... SL 1 70 -MML 1.3

L 60

- ML 1.2 250I--

~40 -

30 -CLASSICAL ELEMENT S I DATA I I I I I I I I I I I 0 1 2 3 4 5 6 7 8 9 10111213141516 ORBIT NUMBER

MI T-O -ORY MALYSISDIWSION 1y_1 P 1-1 PLOTIO _2-

Figure 7-Arc 2 - Apollo 10 argument of perilune vs orbit number.

26 -164

-166

S-168 ML 1.1 -170 ML 1.2

S-172 7- ML ML13

S-174

-176 Li

-178 @ oCLASSICAL ELEMENT DATA I I I I I I I I I ...... 0 1 2 3 4 5 6 7 8 9 10 11 1213 14 1516 ORBIT NUMBER

Figure 8-Arc 2 - Apollo 10 inertial node vs orbit number.

27 . 8 .0085 .0080 .

.0075 o

>- .0070

.0065 z

_U .0060 Li1 .0055 ML 1.1 ML 1.2 .0050 ML 1.3

.0045 oCLASSICAL ELEMENT DATA 1 I- j I 1 j j - ! L j L 0 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 ORBIT NUMBER

Figure 9-Arc 3 - Apollo 10 eccentricity vs orbit number.

28 179.20 179-18 179-16 a 179o14179 12 -...... 3.. :,179-10 ML 1.1 M 1 M 17908 179.16 zz 17902179 04 " .

179,00 -__L1 178.98 , 178,96 -CLASSICAL ELEMENT DATA S1819 20212223242526 27 28 29 30 3132

ORBIT NUMBER --SC DS MISSIN& TRAJECTOyANALYSS DIVISION BRANCH552 OA E '""'2-' B, P.-U.I 11-5N 212

Figure 10-Arc 3 - Apollo 10 inclination vs orbit number.

29 106 " 104 tb 102 100 - w 98 ML 1.1 S96 ,96 ML 1.3 0L 92 - 0 U_ = 88 .-. .. . . 86 84 ML1.2 < 82 -CLASSICAL ELEMENT DATA I~ I I 0 17 18I 19 20 21I 22 I 23-24I 25 26I 27 28 I 29 30 31I 32

ORBIT NUMBER

MISSION& TRXEORV ANALYSDVISION IH BRANCH &52 OA-E _Ju ,V j _ _Mu~PLOT NO 21-

Figure I]-Arc 3 - Apollo 10 argument of perilune vs orbit number.

30 -162

-164 , ML1 . -166 . a -168 ML 1.2

S-170 0O*,II 1.3 Z LLI 172 z

-176 Li1 -178

-180 -CLASSICAL ELEMENT DATA r I S1 I I I I I I 0 17 18 19 20 21 22 23 24 25 26 27 28 29 30 3132 ORBIT NUMBER

:YRANC-S IA., F.", ILI"

Figure 12-Arc 3 - Apollo 10 inertial node vs orbit number.

31 .0060 - .0058 " .0056 %% .0054 %,% .0052 N ML 1.3 .0050 / ML 1.2 >- .0048 2 .0046 .0044 L =.0042 ML 1.1 %0 .0040 .0038 .0036 .0034 .0032 .0030 'CLASSICAL ELEMENT DATA I I I I I I I I I I 0 1 2 3 4 5 6 7 8 910 11 12 13 1415

ORBIT NUMBER

MRSSION&TRAJECTORY ANALYSIS DIVISION BROTIH552 _ATEJubf12M0 BY JP 1 PLOT-O__145

Figure 13-Arc 4 - Apollo II eccentricity vs orbit number.

32 178.60 178.58 178-56 cu tb 178-54 - ML 1.1 : 178-52- ML 1.M L 1.3 0 178-50 ' 178.48 z 178-46- / L1 z 178-44 178-42' 178.40- -CLASSICAL ELEMENT DATA 0 1 2 3 4 5 6 7 8 9 1011 1213 14715 ORBIT NUMBER

IRNH By J P OS7PLOTSS.ATE11" S1AN

Figure 14-Arc 4--Apollo II inclination-vs orbit number.

33 -88 -90 0 -92 " -94 oS -96 ML 1.2 ", z -98 ML 1.3 -100 * -0 -102

o -104 * ML 1.1 ~ 106 S-108 /L -110 < -112 -114 -CLASSICAL ELEMENT DATA I I I I I I I I I I

0 1 2 4 5 J3 68 1 0 10112131415 ORBIT NUMBER ASA4SFC- S &ISO TUArR R ANALY5 SISvlON

Figure 15-Arc 4 - Apollo 11 argument of perilune vs orbit number.

34 175 ML 1.1 174 73ML12

172 o /..... oT : 170171 / .. z '... ML1.3

i.170 LU 169 L 1 168

167 *CLASSICAL ELEMENT DATA f I I I *1I I I t I I I I 0 1 2 3 4 5 6 7 8 9 10 11 12131415 ORBIT NUMBER

Y .MmPLOTO11_21 8

Figure 16-Arc 4 - Apollo 11 inertial node vs orbit number.;

35 .0040. .0038 - .0036 .0034 .0032 - * * > .0030 2 .0028 '. - .0026 z C .0024 ML 1.2 .0022 L I MML 1.3 .0020- ML 1.1 ML 1. .0018 .0016

.0012 -CLASSICAL ELEMENT DATA I I I I I I L L I I I I 0 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 ORBIT NUMBER

REOY- ALSS-IN•DIVISION ByJpM- LOTN 170

Figure 17-Arc 5 - Apollo 11eccentricity vs orbit number.

36 178.82 178"80 178"78 17876~ML ML111.1 .. 178-74 M ML13 ZS 178-72 o 178-70 < 178.68 _ 178-66 . -- 17864 - 178.62 L1 178.60 178,58 178-56 - CLASSICAL ELEMENT DATA L1 I I I I J I ! I IL I I 0 14 15 16 17 18 19 20 2122 23 24 -25 26 27 28 ORBIT NUMBER

...... TECTORYA LYSISIVISION ...ANCH -5 .- TS . 90

Figure 18-Arc 5 - Apollo 11 inclination vs orbit number.

37 -1.5 1 -20.

C" -25 / 'at -30 *, -35 -/ML 1 -40 ML 1.2 -45 ML1.3'\/ ,=-50 , 7 ML 1.1 ,.-55 / ,_ _-60 L -65 ./ / -70 -75 -. -80 - -" -85 * -90 -CLASSICAL ELEMENT DATA e 1 I I I 11I I I 1 1 1 1 0 14 15 16 17 18 19 20 21 22 23 24 25 26 27128 ORBIT NUMBER MISSION&T-E=rOR'y NULtYSISDIVISION .RANSH552 DATEMul1 ., J.P.M.,h, '_ ..O. 2151

Figure 19-Arc 5 - Apollo 11 argument of perilune vs orbit number.

38 171

170

S169 U0M'L 1.-1 ..... -u, 168 ML 1.2- 16 12..",

0 _ 167 ML 1.3 w 166

165 L

,164

163 -CLASSICAL ELEMENT DATA 7 I 9 I I I I I I 1 - -I -1 -- 0-'14 15 16 17 18 19 20 21 22 23 24 25 26 2,7 28 ORBIT NUMBER

SVJPMmfly PLOTNO 2162

Figure 20-Arc 5 - ApoIlo *11inertialnode vs ,orbit number.

39 .0060 .0058 .0056 .0054 .0052 -L o0050 - ML 1.1 ' .0048 z .0046 o .0044 Lu .0042 ML 1.2 .0040 ML 1.3 V .0038 , .0036 .0034 \ .0032 - CLASSICAL ELEMENT DATA' I I I I IIII I I I I L 0 1 2 3 4 5 6 7 8 9101112131415' ORBIT NUMBER

MI-ION&-TR-ECTORYANALYSISDIVISION BRNC152 DATE "L ._ 11-0PLTN212.

Figure 21-Arc 6 - Apollo 12 eccentricity vs orbit number.

40 164-84 164-82 L 1 164"80 , g 164-78 164-76

164-74 *0 164-72

< 164-70- ML 1.1 164.68 ML 1.20 164.66 ML 1.3 164-64 164,62 164-60 * CLASSICAL ELEMENT DATA

0 1 2 3 4 5 6 7 8 9 10 111213 1415 ORBIT NUMBER

BRANCH C- "ATE

Figure 22-Arc 6 - Apollo 12 inclination vs orbit number.

41 68 67 67 66 ~ ML 1.1 65 - 64 , 63 L z

61 \ 0- 60 - ' ML 1.2 059 1.3 z 58

57 56 \ 55 54 ° 53 - CLASSICAL ELEMENT DATA I I I I I I I I I I I 0 1 2 3 4 5 6 7 8 9 10 11 12 13 1415 ORBIT NUMBER O BT-OCrORY -LIS DIVISION BPANCHS A J Z7 BY J Pt- - PLOTNO 21SS

Figure 23-Arc 6 - Apollo 12argument of perilune vs orbit number.

42 -21-9 -22-0 -22-1 - O -22-2 -22*3 ML 1.1 -22,4 ML 1.2 -22-5 ML 1.3 _w-22-7 -226 - - / LL 1 z -22-8 -22-9 /

-23-0 - CLASSICAL, ELEMENT DATA I I I I I I I I I I I I 0 1 2 3 4 5 6 7 8 9101112131415 ORBIT NUMBER

_MINO&TRAJ-ORYANL'A-SD-~ ON _ NC, 6.Z -AEL BY J P M--W PLONO 2156

Figure 24-Arc 6 - Apollo 12 inertial node vs orbit number.

43 .o24 - CLASSICAL ELEMENT DATA I .0023 . .0022 • .0021 .0020 - ML 1.3M E5 .0019 o . / ML 1.2 -. 0018 Z* I ).0017 L uj .0016 *.iN. .0015 , .0014 -- ,," .0013 .0012 ML 1.1 I I I I I I I I I I 1 I 0 20 2122 23 24 25 26 27 28 29 30 31 32 33 ORBIT NUMBER

T- _SOCORY ANALYSISDIIIS101 BY J P.11 I'mh- 145T 215

Figure 25-Arc 7 - Apollo 12 eccentricity vs orbit number.

44 165.64 165-62 . 165-60 "L 165"58 L 165.56 165-54 165.52 0 - 165-50 ML 11 Z 165.48. ML 1.2 X 3 zo 165,46 - 165,44 165,42 165-40 165.38 o CLASSICAL ELEMENT DATA I I I I I I I I 0 20 21 22 23 242526 27 28 29 30 3132 33 ORBIT NUMBER S N-ASFC-DS = IO -- REOy- ANALYS1S1VIIO BC ASCAMhY 5L5550. CISC

Figure 26-Arc 7 - Apollo 12inclination vs orbit number.

45 3400 S20 Z10 0 - ML 1.2 ML 1.1 -10 ML 1.3 \, L I ,. ,2 -2 U- 30 - % _ -40 ,,d-50 " -60 " C-70 -80 - CLASSICAL ELEMENT DATA II !/1 1 I 0 2021 22 23 24 25 26 27 28 29 30 31 3233 ORB!T NUMBER M NsroNSR*-TaONtS~so~S sYJ 1APA, PLoPSo 2157

Figure 27-Arc 7 - Apollo 12orgument of perilune vs orbit number.

46 -24-1 / -24.2 -24-3 -24-4 L 1 -24-5 p -24-6 -oa -247 LU -24-8 S-249 ML 1.2 - ML 1.1 -25.0 ML 1.3 z -25-0 -25.1 - Uj -25.2

-254 -255 -256 - CLASSICAL -25o-6 o ELEMENT DATA -25-7 IL~ - - _L L L I- - L 0 20 2122 23 24 25 2 2 2 2 30 31 32 33 ORBIT NUMBER

MI-31N& TRAJEC-OYANA- -I DIVSIO BRANCH552 OATS1-I1910 PIV PLOTNOa 6

Figure 28-Arc 7 - Apollo 12 inertial node vs orbit number.

47 .0056.- ML 1.2 / Li .0054 ML 1.3 .0050 L _ .0048 ML 1.1 .0046 L .0044 .0042 .0040 , .0038 / .0036 -o CLASSICAL ELEMENT DATA I I I I I I I I 1.. I 0 39 40 4142 43 44 45 46 47 48 49 50 ORBIT NUMBER

..NIN . .. AJC-Y ANALYSISSIVISION Jd BYBRANCH JP.S7YYS2 PLOYSODATE 170 2167

Figure 29-Arc 8 - Apollo 12 eccentricity vs orbit number.

-48 16878 168.76 o 16874 L 1 o 168.72 . tao 168-70 :S- 168-68 " Z 168-66 .. ML 13

168-62 16860 ML 1.1 16858 ML 1.2. 168-56 16854 - CLASSICAL ELEMENT DATA I I I I I I I I 1 .1 1 0 39 40 41 42 43 44 45 46 47 48 49 50 ORBIT NUMBER N-SSF,-&OS MINION&TRAJECTORYANALYSISDIVISION 5RA5H0S OATE4010 m0 p PON__5p. 21.1

Figure 30-Arc 8 -Apollo 12 inclination vs orbit number.

49 -78 -79 -80 a' -81 -. -82 -83 L1 z =,-84 \ " -85 \ .,..ML 1.1 LO -85.7 U-_o -87-86 \ t- 87 -ML 1.2 z-88 - \ ML1.3 -89 ML1. -90 -91 -92 * CLASSICAL ELEMENT DATA

0 39 40 41 42 43 44 45 46 47 484950 ORBIT NUMBER MISSON&RAJECTORY ANALYSIS OIVISION BSANCHES2 DATEJY t7 BY L.~h PLOTNO. 1iD

orbit number. Figure 31-Arc 8 - Apollo 12 argument of perilune vs

50 -33-00 -33-10 a) -33-20 -33"30 L.1 U Li *> ML 1.1 33"40 ML 1.2 = -33-50 /ML 1.3 F- -33-60 w -33-70 -33-80 . CLASSICAL ELEMENT DATA

S39 40 4142 43 44 45 46 47 48 49 50 ORBIT NUMBER

MISSION -TAYAECTORYANALYSISDIVISION ByJMM YP AGANSr~2164

Figure 32-Arc 8 - Apollo 12 inertial node vs orbit number.

51 .068k / .066 -ML 1.2""1/ .064 ML 1.3 :062 Z3_ .058.060 -. / ,j - .054 N ML 1.1 .054 6 8 143 52052 " .050 - ' "

.046 - I .044 NOTE: DOTS INDICATE CLASSICAL EILEMENTS

0246810 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 TIME (days since 8/30/67)

Figure 33-Lunor Orbiter 3 eccentricify vs time.

52 21.0 20-9 - 20.8 20-7 ° 20.6 2 20.5 20.4 - ML 1.1 S20.3 * ML 1.2 2' 20.2 ML 1.3 7 20.1 20.0 L1 - 19.81199 \ / \\

19.7 \-• . 19'6 19.5 NOTE: DOTS ARE ACTUAL CLASSICAL ELEMENTS

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 TIME (days since 8/30/67)

34 Figure -Lunar Orbiter 3 inclination vs time.

53 .276

.275 * * o0 0

- .274 00 .0 ML 1.2/' " , .273 MiIL1.3 z MVill. , 0C- L 1 ,o.272

.271.

.270 NOTE: DOTS ARE ACTUAL CLASSICAL ELEMENTS S I I I I I I 0 1 2 3 4 5 6 7 8 9 10 TIME (days since 8/17/67)

-LJP .hv PL-W 26

Figure 35-Lunar Orbiter 5 eccentricity vs time.

54 rr

Figure 36-Gravltational effect of the (4, 1) coefficient in the ML.I field. (Note. ShadedareascomWneQto Petive 1iowaies)