https://ntrs.nasa.gov/search.jsp?R=19660021250 2020-03-16T20:50:47+00:00Z View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by NASA Technical Reports Server 4 \I NATIONAL AERONAUTIC5 AND SPACE ADMINISTRATION Technical Memorandum No. 33-287 AppJication of Double-A veraged Equations of SateMite Mition to a Lunar Orbiter GPO PRICE $ CFSTI PRICE(S1 $ Hard c?py (HC) //029 Microfiche (MF) /dm ff 653 Jutv 65 ET PROPULSlON LABORATORY CALIFORNIA lNSTlTUTE OF TECHNOLOGY PASADENA,CALIFORNIA June 30, 1966 NATIONAL AERONAUTICS AND SPACE ADMINISTRATION Technical Memorandum No. 33-287 Application of Double-Averaged Equations of Satellite Motion to a Lunar Orbiter J. Lore11 T. Hamilton, Manager Systems Analysis Section JET PROPULSION LABORATORY CALIFORNIA INSTITUTE OF TECHNOLOGY PAs A D EN A. CALIF o R N IA June 30, 1966 - JPL TECHNICAL MEMORANDUM NO. 33-287 CONTENTS I I.lnhwluction.f.. , . 1 II. Derivation of the Double-Averaged Equations . , . 2 111. Double-Averaged Equations With Oblateness . 5 IV. Implications for Lifetime of a Lunar Satellite . 8 V. Conclusions . , . 13 References . 14 1. Semimajoraxis . 9 2. Eccentricity . 10 3. Longitude of ascending node . 10 4. Inclination . 11 5. Argument of pericenier . 11 6. Radius of closest approach . 12 7. Effect of Earth and Sun on radius of closest approach . 13 ill IPL TECHNICAL MEMORANDUM NO. 33-287 ABSTRACT Equations of motion for satellites disturbed by both third-body and oblateness effects are presented in their averaged and double-averaged forms. These equations are appropriate for long-term studies of satel- lite orbits. In particular, they should serve as the starting point for study and classification of orbits of close lunar satellites. 1. INTRODUCTION This Report presents La,eresults of some o1 LA,eresearch study to a numerical analysis and a classification of the done in connection with determining the effects of third- motion. Reference 4 contains an analysis of Earth satellite body perturbations on satellite motion. In particular, the orbits perturbed by the Sun and Moon, and Lidov, in material contained herein supplements work already re- Ref. 5, presents a general treatment of the subject. ported in Refs. 1, 2, and 3. The work discussed in this Report is aimed specifically at artificial satellites of the Moon and is with The motion of a satellite with periodic perturbation concerned the applicability of the double-averaged equations. may be described mathematically in any one of three In Part 11, the double-averaged equations are derived di- ways: (1) by the complete equations of motion, (2) by the rectly from the complete equations of motion. This averaged equations of motion, or (3) by the double- derivation and the form of the result should be compared averaged equations of motion. with the single-averaged equations as given in Ref. 3. It is included here for completeness, since it was not given in the referenced papers. In the double-averaged equations, the averages are taken with respect to both the satellite period and the third-body period. The resulting differential equations In Part 111, the implications for satellite lifetimes are are comparatively simple, lending themselves to analytical interpreted in terms of specific orbits. Here, the complete and simple numerical interpretation. Williams and Lorell equations of motion were integrated, showing both short- (Ref. 1) used these equations in an analytical interpreta- term and long-term effects. In addition, the effects of the tion of the motion, while Lorell (Ref. 2) extended the Sun and the Earth are compared. * JPL TECHNICAL MEMORANDUM NO. 33-287 II. DERIVATION OF THE DOUBLE-AVERAGED EQUATIONS The motion of a satellite perturbed only by a distant third body has been studied recently using averaging methods (Refs. 15). In Ref. 3, the basic single-averaged equations are derived. (Single average refers to average with respect to the satellite motion only; double average refers to average with respect to both the satellite motion and the third- body motion.) In Refs. 15, the single-averaged equations are presented for the third body in an arbitrary orbit. This model is applicable to general situations (e.g., Moon satellite perturbed by Earth and Sun). The position of the third body with respect to the reference coordinate system is arbitrary. On the other hand, the model for the double-averaged equations assigns the third body to an elliptic orbit in the x-y plane. The double-averaged equations are of doubtful value for simulation purposes. However, for general study of long- term orbit behaviors, they are very useful. References 1,2, 4, and 5 include studies and classification of orbits using these equations. References 2 and 5 suggest in addition the inclusion of oblateness effects in such studies, and pre- sent the appropriate equations. Performance of these studies, however, remains €or the future. The perturbing function due to a third body is well known, and may be found in any text on celestial mechanics. A form suitable to the present use is given by Clemence and Brouwer (Ref. 6, p. 310): )...I+ in which the expansion is in powers of r/r3,where' R = disturbing function r = distance from satellite to primary r3 = distance from disturbing third body to primary $ = satellite-primary-third-body angle In the case of an artificial satellite of the Moon disturbed by the Earth, the ratio r/r3 is of the order 1/10. There- fore, it is reasonable as a first approximation to consider the truncated disturbing function The derivation of the single-averaged equations is given in Refs. 1 and 2 in matrix form, and will not be repeated here. The derivation of the double-averaged equations follows. - Let a bar denote a single average and a double bar a double average; e.g., R, E, and 6 are the perturbing function and its single and double averages. The procedure is to represent R first in Cartesian coordinates and then in Kepler elements, assuming the third body in a Kepler ellipse about the primary. Thus, 'The units here are arbitrary. However, for use in conjunction with the equations for oblateness, the unit of length must be the lunar radius (see footnote 5). 2 JPL TECHNICAL MEMORANDUM NO. 33-287 * where Y - = cosusinR cosisinucoss2 r + in which t(=o+f f 1 _-- p l+ecosf and p=a(l-e*)=semilatusredum a, e, i, o,R = usual Kepler elements f = true anomaly With subscript 3 on each variable, the notation described by the above equations applies to the third-body motion. Choose the coordinate axes so that the third-body orbit is in the x-y plane, and the x-axis is in the direction of peri- apsis of the third-body orbit. Then, x3 - = coSf3 r3 Y3 - = sinf3 (4) f-3 and After expanding and averaging twice, JPL TECHNICAL MEMORANDUM NO. 33-287 The next step is to evaluate the averages. To start with, it is understood that averaging over the satellite variable means an average with respect to the mean anomaly of the satellite orbit, and for the third body, the average is with respect to the mean anomaly of its orbit. Averages are obtained as follows: where M is the mean anomaly and Thus, the last two terms of Eq. (6) may be combined to give But, Furthermore, 2= sinfcosf df = 0 (1 + emsf)' and Hence, the expression in Eq. (10) becomes - Finally, substituting and combining terms in Eq. (6), the expression for is found to be2 *The double bar ( =) is omitted from the Kepler elements in these equations because there is no ambiguity; i.e., all variables are aver- aged variables. 4 JPL TECHNICAL MEMORANDUM NO. 33-287 The double-averaged equations of motion are then derived by substituting this disturbing function in the Lagrange planetary equation, giving da - =O dt de p3e (1 - e’)% - - --15 - , - e2 3/2 sin 20 sin2i dt 8 flu;, ( 3) do, 3 p3 (1 - e2)% 5 sin2w (e2- sin2i) _-- - dt 2 na; (1 - [I+ 2(1- e.) 1 and if x is the sixth Kepler element representing time of pericenter passage, ez - sin?i [(l- e2)cos2 o 2 (3 2e2)sin2 01 dt + + + 111. DOUBLE-AVERAGED EQUATIONS WITH OBLATENESS To account for both oblateness of the primary and third-body effects, it is necessary to consider the relative orien- tation between the primary’s equator and the third-body orbits. A convenient way to identify the important angles and the relations between them is to set up three right-handed coordinate systems as follow^:^ 1. x, y, z, with the xy-plane the plane of the third-body orbit 2. Z, ij, Z, with the @-plane the plane of the satellite orbit, and the x-axis in the direction of periapsis 3. x’, y‘, z‘, with the x’y’-plane the plane of the equator and the x-axis in arbitrary direction (to be specialized later) Let the coordinate transformations be given in terms of Euler angles: (i)=A( ;) = B( 3) (;;)=c(;) ’The bar (-) notation here is not to be confused with its previous meaning as an average. 5 ~~ JPL TECHNICAL MEMORANDUM NO. 33-287 Thus, B = AC-I (24) Here the matrices A, B, and C can be written in terms of the respective sets of Euler angles: with inverse A-’ given by --- The matrices B and B-’ are of the same form, but with barred angles;‘ i.e., 0, O, i. The matrices C and C-’ are of the same form, but with primed angles; i.e., Q’, or, i’. Spec%cally, de Euler angles 0, m, i relate the satellite orbit to the third-body orbit; the angles 5, 0, :relate the satellite orbit to de equator; and the angles a’, a‘, i‘ relate the equator to the third-body orbit.