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National Administration Goddard Space Flight Center Contract No. NAS-5-12487

ST - CM - 10687

ON ?HE MOTION OF REMOTE ARTIFICIAL EARTH'S SATELLITES IN 'ME GRAVITATIONAL FIELDS OF THE ENITH AND OF "€EMOON

by V. P. Dolgachev

* (USSR)

368-187 10 3 (ACCESSION NUMBER) 4 q P (PXGES) 'COW c 20 (CATEGORY)

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ST-(34-10687

ON 'IHE MOTION OF REMOTE ARTIFICIAL EARTH'S SATELLITES IN THE G.MVITATI0NA.L FIELDS OF THE EARTH AND OF THE MOON

Vestnik Moskovskogo Universiteta by V. P. Dolgachev Astronomiya, No. 1, pp. 94 - 100, Izdatel'stvo MGU, 1968.

SUMMARY The expressions are presented of first order perturbations of AES' orbit elements due to Earth's oblateness, and of the second harmonic of Moon's attraction, the Moon being consicered as a material point moving along a circular orbit. The formulas derived are also valid for pert.urbations due to the Sun. * **

The expressions were derived in a preceding work [l] for secular varia- tions of remote AES in the gravitational fields of the Earth and of the Moon, whereupon perturbations from the second harmonic of the terrestrial potential and from the second and third harmonics of Moon's attraction were taken into account; it was moreover considered that the Moon is a material point moving along an elliptical orbit.

Remaining within the framework of the classical scheme proposed by Lagran- ge, we obtained perturbations in the Lagrange elements; it is then natural that the formulas obtained for the secular perturbations are valid for small eccentricities and inclinations. In the current work we consider the perturbations of the second harmonic of Earth's potential and from the second harmonic of Moon's attraction, con- sidering the latter as a material point moving along a circular orbit. Secular and long-period perturbations of first order in the motion of the ms are obtaine(j by- ..----r 4-----:4.: kqdy UL LldtlblLion to the independent variabie of the the unperturbed true anomaly of the satellite.

The formulas thus derived are valid for orbits with arbitrary inclination.

(*) 0 DVIZHENII DALEKIKH ISZ V BRAVITATSIONNOM POLE ZEMLI I LUNY 2

1. STATEMENT OF THE PROBLEM

Let us choose a rectangular system of coordinates Oxyz with origin at the Earth's center of masses in such a way that the plane xOy coincide with the orbit plane of the perturbing body (the Moon), and the axis Ox be directed into Moon orbit's perigee. Then, expanding the perturbing function of the problem in series by Legen- dre polynomials, and limiting ourselves to second harmonics of Earth's potential and Moon's attraction, we may write

+ Ni + COS 2v (cos 20 (Ni- N:) + 2N1N2sin 20) +-

+ sin 2v (si: 20 (N?- N;) -b 2N1N2cos 2 0)] .. where PL = fm, . PE = fm, ~~s~=~osucos(Q-~~)-sinucosisiri(Q-v~), N, = sin i cos E + cos i sln E cos 52, N2= sinEsin52, u =v+ 0,

-f is the gravitational constant, mE and mL are respectively the masses of the Earth and of the Moon, r and rL are the geocentrical radii of the satellite and of the Noon, v and ?L are the true anomalies of the satellite and of the Moon, J is the obzteness parameter, a. is the equatorial radius of the Earth, w is the perigee longitude, L is the inclination, Q is the longitude of satellite orbit plane, E is the angle between the Earth's equatorial plane and Moon's orbit plane.

We shall denote the first term of expression (l), having the multiplier pt, by RL, and the term containing the multiplier YE, by RL. If in place of cos JI we introduce into RL its expression by the Kepler elements of the satellite and of the Moon, RL will take the following form:

RL = -nlr22 (3 cos2 i - 1) + -nlr2 sin2 i [cos 2 (52 - v~)+ 8 '38 = { i cos':! (v o)] 2 sin4 -cos2 (v ut o -Q) -+ + + 2 + + 4-

where

If we assume the unperturbed true anomaly of the satellite for the new independent variable, the Lagrange equation for the osculating elliptical elements, valid for the computations of first order, will take the form [2] ../. . .*

If we take the unperturbed true anomaly of the satellite for the new independent vari.able, the Lagrange equation for the osculating ellitpical elements, valid for the computations of first order, will take the form [2]: --da 2 a1 dR *’ * dv -+) ,f ’ dMo

Here a is the major semiaxis, e is the eccentricity, n is the average motion and MT is the mean satellite anomaly in the epoch. - The time -t is linked with the unperturbed true anomaly by the following rela t ion :

in which ‘I is the time of satellite passage through the perigee.

2. DEPENDENCE BETWEEN THE TRUE ANOMALIES OF THE MOON AND OF THE SATELLITE For the computation of AES’ and Moon’s mean anomalies at a given moment of time -t we resort to the well known formulas

M = M, +nf, ML= M! + nLt, (4) where n and nL are the mean anomalistic motions of the satellite and of the Moon, %d Mo and MLo are the mean anomalies in the initial epoch.

Let us denote the ratio of the mean motions n and nL by and m = n /n. - L Eliminating t from Eqs. (4), we shall obtain ML = M1 + mM, where it is postular ted M, = MLoImM, . Making use of the equation of the center and neglecting the eccentricity of the orbit of the Moon, we shall obtain ../. . 4

.VL = M, + m v-2e sinu + -e2shi2v3 + . . . = M,+ rnv + rnef(u), ( 4 1 where f(v) is a limited periodical function of v.- Neglecting in the expressions cos 2(v * v,+ w - R ) by small quantities -em, we shall obtain the final expresion for the perturbing function R,: 1 RL = n:r2 (3 cos2i - 1) + sin2i [cos 2 (SZ - MI - mrr) + 7 8

+ COS 2 (u + 0)] +- 2sin4T cos 2(v + mu+ MI+ 0 -9) +

i 2cos - cos 2(u- rnv - M, + 0 Q)}. + 2 + (5)

3. FORMULAS FOR PERTURBATIONS Having computed the corresponding derivatives from R, and R by elements and substiting theur expressions into Eq.(3), we shall obtain a system of dif- ferential equations of motion of the satellite, in which the right-hand parts are functions of the true anomaly v. Taking advantage of the expansion of expressions of the form (:)" into Fourier series by multiples of the true anomaly v, we shall have

k=I where M,(4 (e) = - (1 .i-e cos COS 2n kvdv.

Integration of system (3) in the assumption that the orbit elements are replaced in the right-hand parts by constant quantities gives perturbations of first order a = Q, + ala, e = e, + ale, ...... / M, = Mi + 6,M0, where a09 en, -- - 'Mi are integration constants; ala, 6,e, . . . , alM0 are perturbations of the respective elements.

When integrating system (3) we preserved the secular perturbations and also the periodical perturbations, of which the total period with respect to v- is ZIT /m.

Such periodical perturbations will correspond to those terms in R,, for which k = n and they will be called in the following long-period perturbations, The final expressions for the secular and long-period perturbations 5 from the Moon for all the six elements have the form

61aL = 0, dleL =-3 (57(P)” LqL, 4n ae = - cos i - a,&+ s,c,,

4iL=3 (+>”{-ctg ~G,BL+ - . - M\”sini ;os 2 (Q - 4 (+y ’ ’ [ 2 m

- M~-mu) ~i~)sin2 L tg L cos 2 (mu + O - Q + M~)+ + 22 + Mi2’cos2- i ctg- i cos 2 (mu -- o - Q + 2 2

&Q L= -?8 (+y ($>”5 (cos i [ -2rnM\’)v + 21nM!,*) u cos 20 + +M~’sln2(mv-Q+MM,) Ml)-

Here

Following are the coefficients of Fourier series entering into the expressions for elements perturbations : 6

Making use of the recurrent formulas for the coefficients dk), obtained by E. P. Aksenov [3] , we find at', Mio), MY', MY).,

As to the expressions for the perturbations of AES orbit elements from the second harmanic of Earth's potential, we may obtain closed expressions relative to eccentricity as well as relative to orbit inclination, to which Kozai pointed for the first time [4].

The expressions presented below for the perturbations of elements differ from Kozai's formulas only in that the satellite's orbit elements refer to Moon's orbit plane, while Kozai took the equatorial elements. The expressions obtained by us are brought out here for the sake of uniformity and possibility of conduct- ing the analysis of the joint influence of the Moon and Earth's oblateness on the motion of a remote AES.

e - N, COS 20j (e sin u + sin 20 + sin 3v) - \ 3 /

. 'e ' -(N, ms2o + N, sin2w) e cos v + ms2v + - cos3u)], ( 3 7

1 - - [(IV: - N:) cos2a) + ~N,N,sin 201 (e sin v + sin 2v + -e sin 3v 2 3

[ (N: -Ni) sin 20 t 2N,N, cos 201 (e cos v + cos 2v + -ecos where e

+(I + $ e2) sin2v +-e(3 1 + +)sin33 + -e2sin1v3 + -sine3 5v , 2 4 8 3 BE = - I(1: - N;) sin 20 + ~NJ,cos 201 e sin v j- sin 20 + -e sin 3v) + 3 +'[(N: - N;) cos 20 - 2N,N2 sin 201 e cos v -+ cos 2v + -e cos 3v), 3

dlCE= J($)a{~l--((N:i-N:)j~~u+(i+_ie2)sinv+3 3 , 2

+ -c sin 20 + -ea sin 3v J - I [ (~2'- N:) cos 2; + ~N,N,51 201 x 2. 12

x (2e2- 1)sinv+3esin2v+- (74- -e2)sin3v+11 -esin4u+3 i 3 4 2. + $ sln 5v + -11 (N: - N:) sin20 + ~N,N,cos 20 x 34 3 x [ (-$es- I)cosv+3ecosZv+- 74- -e2 cos3v+-ees4v+3 3'( l14 1 2 4- -e, cos 5.]}. 4 N, = sin 2i cos2 e + cos 2i sin 2.5 cos Q -sin 2isin2e cos2 Q N, = cos i sin Q sin 28 - sin i sin2 E sin 29, N, = sin2&sin 252 (1 + cos2 i) + -1 sin 2i sin 212 sin Q, 2 N, = sia iesin i cos 9 + 2 sin2 E cos 20cos i.

It is possible to obtain by the above-expounded method the expressions for long-period perturbations, taking into account eL and em. Such formulas have been obtained; however, the scope of the present paper does not allow their presentation here. , ..- ,.

In conclusion it must be noted that having taken the ecliptic plane for the basic readout plane, we may write by analogy the expression for the secular and long-period perturbations from the Sun.

**** T H E E N D **;r;*

Moscow State University Manuscript received on 28 March 1967 Chair of Celestial Mechanics and Gravimetry

REFERENCES , [l] . V. P. DOLGACHEV. Soobshcheniya GAISI , P.a.157, 19 7 * [Z]. G. A. CHEBOTAREV. Analiticheskiye i kachstvennyye metody nebesnoy mekha- niki (Analytical and qualitative methods of celestial mechanics). M., NAUKA, 1965. [3]. E. P. AKSENOV. Trudy DAISH, t.36, 1967. [4]. Y. KOZAI. Astronom. J. -64, 9, 1959.

* 'IGAISH" stands for Gosudarstvennyy Astronomicheskiy Institut Shternberga or State Astronomical Institute in the name of Shternberg.

CONTRACT N0.W-5-12487 Translated by ANDRE L. BRICHANT VOLT TECHNICAL CORPORATION 1145 - 19th St.NW on 9 - 10 March 1968 WASHINGTON D.C. 20036 Telephone: (202j - 223-6700 (kt.36j.