D 955 LWEPIFANT PLAZA NORTH, S.W., WASHINGTON, D.C. 20024
TITLE- Effects of Physical Librations of the TM-71-2014-3 Moon on the Orbital Elements of a Lunar Satellite DATE-M~~~~29, 1971 FILING CASE NO(S)- 310 AUTROR(S)- A. J, Ferrari W. G. Heffron FILING SUBJECT(S) (ASSIGNED BY AUTHOR(S))- Moonl Selenodesy, PlanetaryI Librations
ABSTRACT
Physical librations of the moon, which cause seleno- graphic axes fixed in the true moon to have a different orientation than similar axes fixed in the mean moon, are small cyclic perturbations with periods of one month and longer, and amplitudes of 100 arc seconds or less.
These librations have two types of effects of present interest. If the orbital elements of a lunar satellite are referred to selenographic axes in the true moon as it rotates and librates, then the librations cause changes in the orientation angles (node, inclination and periapsis argu- ment) large enough that long period planetary perturbation theory cannot be used without compensation for such geometri- cal effects. As a second effect, the gravitic potential of the moon is actually wobbled in inertial space, a condition not included in the potential expression used in planetary perturbation theory e
The paper gives data on the magnitude of the physi- cal librations, the geometrical effects on the orbital elements and the equivalent changes in the coefficients in the potential. Fortunately, the last effect is shown to be small.
U m
SE E SI TM- 7 1- 20 14- 3
DI STRlBUTlON
COMPLETE MEMORANDUM TO- COMPLETE MEMORANDUM TO Jet Propulsion Laboratory CORRESPONDENCE FILES P, Gottlieb/233-307 OFFICIAL FILE COPY J. Lore11/156-217 plus one white copy for each P, Muller/156-251 additional case referenced W, Le Sjogren/180-304
TECHNICAL LIBRARY (4) University of California NASA Headquarters W. M. Kaula A. S. Lyman/MR Tokyo Astronomical Obs. L. R. Scherer/MAL W. E. Stoney/MAE Ye Kozai Goddard Space Flight Center Computer Science Corporation J. Barsky/554 D. H, Novak J, P. Murphy/552 USAF Cambridge Res. Labs. Lancrlev Research Center D, H. Eckhardt T. Blackshear/l52A W, H. Michael, Jr./152A Bellcomm, Inc. R, H. Tolson/l52A R. A. Bass Manned Spacecraft Center A. P. Boysen, Jr. J. 0. Cappellari, Jr. J. P. Mayer/FM D. A. De Graaf J. C. McPherson/FM4 F. El-Baz E, R, Schiesser/FM4 D. R. Hagner We R. Wollenhaupt/FM4 N, W. Hinners TO B. Hoekstra Aerospace Corporation M. Liwshitz Jo L. Marshall L. Wong . K, E. Martersteck We I, McLaughlin Mass. Institute of Technology J., Z, Menard P. E, Reynolds R. H. Battin P, S. Schaenman R, Ve Sperry Bell Telephone Labs-, Inc, Jo W. Timko R. L. Wagner - We M. Boyce/MH All Members Department 2014 F. To Geyling/WH Department 1024 File A. G. Lubowe/WH Be G, Niedfeldt/WH ABSTRACT ONLY TO Re A. Petrone - NASA/MA J. P, Downs De P. Ling M. P. Wilson date: March 29, 1971 955 L'Entant Plaza Washington, D. C. 20024 to: Distribution
from: A, J. Ferrari and W. G. Heffron subject: Effects of Physical Librations of the TM- 71- 2 0 14- 3 Moon on the Orbital Elements of a Lunar Satellite -- Case 310
TECHNICAL MEMORANDUM
INTRODUCTION The long-term selenodesy method for determining lunar gravity coefficients [l] uses a time history of orbital elements in conjunction with the long-period form of the Lagrange planetary equations. The disturbing function for spherical harmonics used in such work, typically as derived by Kaula [2], is referenced to a selenographic coordinate system, fixed in the moon. The x-y axes of this system are in the lunar equatorial plane and the z axis is along the lunar polar axis. The analy- tical derivation of the Lagrange perturbation equations requires that the osculating orbital elements be referenced to an inertial coordinate system. In order to make the inertial and selenographic systems simply relatable, it is assumed that they have a common z axis and the moon rotates about this z axis, But, in addition to its normal rotation about its polar axis, the selenographic system undergoes addi- tional rotations about all three axes due to precession and physical librations, These small rotations can affect long-term selenodesy calculations in two wayso First, if the selenographic axes are used as the reference system, some of the orbital elements will change simply because the axes librate and precess (geometrical effects). Second, since perturbation theory (in its present form) accommodates only polar rotations of the selenographic system, there are also physical effects induced by lunar librations and precession. This paper illustrates the geometrical and physical effects associated with the changing orientation -2-
of the selenographic axes. It is shown that geometrical effects can be accommodated either by using an inertial axes system or by compensating for the lunar librations and precession when the selenographic axes are used. " Further, it is shown that physical effects are small and negligible for all but the most exacting endeavors. MOTION OF THE MOON Standard angles [3,4,5,6] which yield the orienta- tion of the true selenographic of date (TSD) axes at any particular instant are referenced to the mean earth equator- equinox (MOE) axes of some chosen epoch. These two systems are relatable using a direction cosine matrix MT so that a position vector r measured in the TSD axes is trans- formed into the MOE axes as follows:
=M ~MOE T -TSDr
If the rotation matrices R1(0) I R2( 0) I and R3(0) are defined as
c = cose S = sine
the overall direction cosine matrix, MTF between the MOE axes and the TSD axes is given by the following cascade of rotation matrices:
MT = R (:-< ) R (0)R3 (-2-Z)7r R1(€) R3 (Q+180°+o) R1(I+p)R3 (Q+T-n-o) 32 0 1
(All angles are illustrated in Figure 1-1 -3-
8, z are earth precession angles and E is
The angles .Qt I, and C are the mean node, inclination, and anomaly of the moon. The rotational rates of the mean selenographic of date (MSG) axes are C-h(l-cos1) about the-polar axis (27.3 day period), and ii sin I cos( -a) and 0 sin I sin(&-S2) about the xoand y mean lunar axes respectively. I is constant and 6b measures the precession of the lunar node (18.6 year period)
The angles o8 p,and T are the physical librations of the moon, which distinguish the true from the mean lunar axes. They are cyclic (see Appendix for formulas) with principal components having periods of 27.5 days, 1 year, and 18.6 years in duration.
Figures 2 and 3 show the location of the true selenographic z and x axes relative to mean selenographic axes, both at the same instant. The differences can be given as three small (cyclic) angles
"3 - T - o(l-cos I)
a2 = 5 sin I cos(C-~+~-~)5 - p sin(d-G+-r-Z)5
5 5 = 5 sin sin( -Q+-C--) p cos((pQ+-C-2) "1 1 2 +
with direction cosines for mean-to-true axes of
Y
The figures are for the 28 day period following August 30, 1967, a date typical of the selenodesy phase of the Lunar Orbiter 111 satellite. -4-
The bias between the mean and true axes that appears in the figures is due to the components of physical libration which have a period longer than the 28 day time span in the figures The three small angles between true seleno- graphic axes of a fixed epoch (to) and true selenographic axes at a different instant (t)p are:
B3 = Ad + AT - (Abb+Ao) (1-cos(I+p))
(for example) applies at t and A(C is c(t) - ((to) The direction cosine matrix (to to t) is
Although the a's are all cyclic, the B's are notl because and 0 both monotonically change. B1 and B2 are treated as small angles, but if enough time passes B3 may be sizeable enough that small angle approximations are insufficient for it, The AQ - A.G(l-cos I) part of B3 does not show up in Figures 2 and 3 since in that figure the mean and true axes are always at the same instant. -5-
PERTURBATION THEORY
Lagrange s perturbation theory [ 23 app'lies for w .*= the basic vector differential equation
4 -r+- 3- ~=VR r
where R is now the disturbing potential function. To account forl sayB irregularities in density and shape of the moon, R can have the usual spherical harmonic function form:
R=k -Re mh -I- SLm r r PLm(sin+)[CLm cos sin mxl
where Re is the reference body radius, P the associated Legendre function, + is latitude, A is longitude and CLm and SLm are the specific constants which describe the body's potential. Using L=2 as the least index assumes the center of mass is at the origin of the coordinate axes In Lagrange's work appropriate ''orbital elements'' are found - quantities which describe the satellite orbit completely, which are constants if R = 0 and if the axes in which r is measured are inertial, The classical set of orbital eiements is suggested in Figure 4, S2 is the node on the planetss equator, i the inclination, w the argument of periapse, f the true anomalyp a the semi-major axis and e the eccentricityo All are constant except f, but Mo and n are constant in the equation for the mean anomaly M, and it is related to f through the eccentric anomaly E, The equations are -6-
23 na =H
M = E - e sin E = Mo + n(t-to)
f = E tan 2 tan -2
A typical equation in perturbation theory is
3n R~~ cos i dn + CLm SLm 2 2 a2 c20 (other and effects) dt = Z(1-e ) so if only C20 were non-zero, then accurate knowledge of dn/dt would yield accurate knowledge of C20e GEOMETRICAL EFFECTS The geometrical effects can be avoided completely if the axes used are inertial or have only a polar rotation. It is common, however, and suggested in many developments of perturbation theory 121 that the reference axes be the true planetary axes of a local epoch plus the planetary rotation after that epoch. If the axes used were the true selenographic axes at every instant, which is this same assumption except for the effects of lunar precession and physical libration, the changes in the orbit orientation angles would be, for a purely conic orbit, calculated by -7-
AR = -6 + arc tan (y2 cos iO/(sin io y1 cos io)) 3 -
Ai = -=Yl
bo = arc tan(-y2/(sin i cos io)) 0 - y1
Of these the mean lunar motion part of in AR would be 3 expected, but all the other parts would be due to physical librations.
As an idea of the magnitude of these changes, take orbital elements typical of the Lunar Orbiter I11 satellite during its selenodesy phase. Table 1 shows the elements and tabulates the element rates due to the currently used L-1 potential coefficients (Table 2) and due to geometrical effects, In the node, the geometrical effects are about Om6% of the L-1 effects, in inclination about 7%, and in argument of perilune about 3%. For other orbital elements, of coursed the percentages would be different, but they quite apparently are not trivial: potential coefficients deduced from the rates without com- pensating for the geometrical effects would be in error.
LIBRATION OF THE MOON'S POTENTIAL Suppose the reference axes used are inertial and are the true selenographic axes at a fixed epoch in the range of interest, Then the geometric effects of libra- tions are zero but the polar axis at the fixed epoch is not always coincident with the polar axis of the true moon (because of the Ba and B2 angles),
One way of coping with this problem is to refer the instantaneous lunar potential to the reference axes. That is, the coefficients which apply for the instantaneous moon are "rotated" through the angles BIP Bar B3# becoming time varying as B1# B2p B3 vary with time. Several authors [7,8,9] have shown how to calculate the rotated coef.Eicients, In particular Levie [9] shows that a coef- ficient with R=k can cause coefficients with R=k only, but with all values of me -8-
TABLE 1
GEOMETRICAL EFFECTS ON ORBITAL ELEMENTS
Rate due to Rate due to Quantity Value L-1 field Geometrical (deg/sec) Effects (deg/sec)
52 -11.693-6 0.0733-6
i 0 ., 0383-6
w 0.0783-6
a = 1965 km, e = 0.0436! Mo = 194f6, t = J.D. 2439733.37 -9-
As an example, using the L1 field (Table 2) one can show the effect of B1# B2 and B3 over the 28 day time period used in Figure 2, Table 2 shows characteristics " of the rotated field, using true selenographic axes at the initial time as reference axesp and with the mean motion part of the B3 rotation removed. 2. As another illustration, using the orbital elements used beforeg we have calculated the instantaneous rates due to the L-1 field - first as if the physical lunar axes were identical to the inertial axes and then as if the physical axes were displaced from the inertial axes by the amounts given by BIP B2 and B3 (less mean monthly motion) at a time 14 days after the initial epoch. For this particular time these angles are B1 = 0:062, B2 = 020039 and B3 - mean motion = -0H0116, Rotating the L-1 coefficients through these angles gives the "smudged" L-1 field. The rates are shown in Table 3: the differences are small but do occur in the third place of some of the rates Since 14 days is somewhat of a worst case for B2 and B38 it is also of interest to see the inte- grated effect of time varying smudging of the field. And so the Lagrangian long-period equations were integrated numerically for a 28 day period, In one case it was assumed the inertial and true selenographic axes were identical except for mean lunar motion. In the second the axes were identical at the start but the physical' librations and precession were also introduced, giving a time varying smudging to the E-1 potential coefficients. The orbital elements at the start were those given above and Table 4 shows them 14 and 28 days later, The changes are trivial and are far less than those due to the un- certainty in the coefficients in the L-1 field, In shortl the long period effects of the smudging seem ignorably small for practical work, CONCLUS IONS The geometrical effects of the lunar precession and physical libration are appreciable in selenodetic work if the elements of a satellite's orbit are referenced to true selenographic axes moving with the moon. But if true Selenographic axes of a fixed epoch in the period of interest are usedl then the geometrical effects are eli- minated and the physical effects which are introduced are negligible, - lo -
TABLE 2
VARTATION IN CQEFFICIENTS DUE TO VARIATION IN PHYSICAL LIBRATION $&ID LUNAR PRECESSION
Coefficient L-1 Field Value Variation
-0,207108~10-~ e 32x10-’ 9 c20 +O -0 e
1 0 +O. 139~10-~f -0 e 220x10-6
s21 0 +o. 111x10-6 -0.179~10-~
+O. 46x10-10 -10. c22 0.20716~10-~ , -0.49~10
0 +O. 902~10-~ -0 0
‘30 0.210~10~~ +O. 114~10-~I -0 e 180~10-~
‘31 0 e 340x10-‘ +O. 185~10~~(r -0.118~10-~
-0 ‘31 0 +O 301~10-~ 110X10-~
0 +O 818~10~~-0 e 517~10~~ 52
‘32 0 +O. 268x10-’ -0.167~10-~
0.2583~10-~ +O. 140x10-I1 -Oe79Ox10-11. c33
s33 0 +O - 169~10~~I -0 D - 11 -
TABLE 3
PHYSICAL EFFECTS OF LIBRATION AND PRECESSION ON RATE-OF-CHANGE OF ORBITAL ELEMENTS
~
Rate due to Change in Value L-1 Rate due to field "smudged" L-1 field
63?72 -11.693-6 O/S 0.000823-6
20?82 0,523-6 O/S 0.000233-6
354?59 2.193-6 O/S -0.0233-6
0.0436
a = 1965 km - 12 - - a rl oa, 0 c, -d M N w N rl 0 a, rl 0 W 0 74 0 0 rl 0 a1 0 0 0 0 I4 0 0 0 0 a, 0s OS Oe e ma 0 0 0 0 E=@ I I I + Kim ca u3 E m
__n_
In w * m a co a co 00 00 0. In t- 0 m 0 M N rl 0 0
E - ___. __.mp. x _pp_ _I__ - In a cn rl 0 II co 0 * M cn 0 Ki 0 0 In 0 0 0 co 0 Erc 0 0 0 0 0 0 0 0 0 00 00 OS @ 0 0 0 0 I I I I
_j___ql -
t- rl cn cn rl cn 0 t- 00 09 00 In W 0 N 0 N N B N 0
___.j
___4/ ___I_
C -d 3 a, - 13 -
ACKNOWLEDGMENT M. H, Gittes and S, Be Watson wrote the computer programs used to generate the data for this analysis. .
A, Je Ferrari
2014-WGH-k~~AJF We G. Hkfkron
Attachment Appendix Figures 1-4 REFERENCES
1. Blackshear, W. TOPCompton, He Re, and Schiess, J, R', "Preliminary Results on the Lunar Gravitational Field from Analysis of Long-Period and Secular Effects on Lunar Orbiter I," Presented at NASA Seminar on Guidance Theory and Trajectory Analysis, Electronics Research Center, Cambridge, Massachusettsp May 1967.
2. Kaula, W. MeI "Theory of Satellite Geodesy, 'I Blaisdell Publishing Co,, Waltham, Massachusetts, 1966. 3. "Explanatory Supplement to the Astronomical Ephemeris and Nautical Almanac," Anon. Her majesty's Stationery Office, London, 1961,
4. Schiesser, E. ReI et al, "Basic Equations and Logic for the Real Time Ground Navigation Program for the Apollo Lunar Landing MissionsI" NASA Manned Spacecraft Center Internal Note 68-FM-100, April 15, 1968.
5. Eckhardt, D. H. I "Lunar Libration Tables," The Moon, Vol. 1, No. 2, February 1970. 6. Melbourne, W. G., et al, "Constants and Related Informa- tion for Astrodynamic Calculations, 1968," JPL Report 32-1306, July 15, 1968. 7, Courant, R. and Hilbert, D., "Methods of Mathematical Physics," Interscience Publications, Inc,, New York, New York, 1953, 8. James, R. W., "Transformation of Spherical Harmonics Under Change of Reference," Geophys. Je R. Astr, Soc, , 17, pp. 305-316, 1969, 9. Levie, S. L., Jr., "Transformation of a Potential Function Under Coordinate Rotations," Bellcomm TR-70-310-1, February 18, 1970, APPENDIX
R Formulas for the physical librations are adapted from those of Eckhardt [SI although the angular arguments used here are different from those given in the reference., Fur- ther, Eckhardt gives 11 terms in the Io and p expansions and 18 for T [4]; the expansions given below (and used in this study) carry only those terms with amplitudes greater than lo", following a JPL suggestion [6] The formulas used are
Io = -1001'63 sin g' + 231'75 sin(g'+2w') - lot58 sin(2gf+2w')
p = -98'.'36 COS g' + 231'84 cos(g'+20~) - 101'77 c0s(2g'+2wt).
T = -161'87 sin g' + 911'57 sin g - 151'32 sin 2w'
+10:'0 sin(2g+2~-2w')
+ 14','27sin(36O0(O,53733431 - l.O104982~10"~(36525T))
where
g' = mean anomaly of the moon = d - I?'
g = mean anomaly of the sun = L - r - A2 -
w = argument of periapsis of the sun measured from the ascending node of the orbit of the moon = r - R
w' = argument of periapsis of the moon = r - R
I = 5521115 w 0z 0 0 0 0 0 -0.026 -0.024 -0.022 -0.020 -0.018
0 120 0.004
80
0 0.0 ! 2 X Q 40 X z 2 Zi usZK 0 I?ag WO 0.0 zc- W
05 2 -40 n 2 n 0 -0.002
-80
0 -120 -0.004
-1 60 I 1 I -800 -700 -600 -500 ( RS) ENT ALONG MEAN V AX
ON) AND IN DEGREES. 0 0 0 0 0 0 0 -0.026 -0.024 -0.022 -0.020 -0.018 -0.016 -0.014
0 0.004
100
0 (A 0.002 x Q N 2 5 I- ZZ0 0 ski 0.0 QE I-- s2 Lu 0 4 0 n -0.002 (A
-100
0 -0.004
0 -0.006 21 Y --200 -
I I I I 1 1 1 -800 -700 -600 -500 -400 1 MEAN Y AXIS X