luna 10 -russian determination of the gravitational field of the moon by the motion of the ams...

8/8/2019 Luna 10 -Russian DETERMINATION OF THE GRAVITATIONAL FIELD OF THE MOON BY THE MOTION OF THE AMS LUNA-10

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National Aeronautics and Space Administrat ion --Goddard Space Flight Center

Co nt rac t No.NAS-5-12487

S T - C M - L P S -10532

DETERMINATION OF THE GRAVITATIONAL FIELD

OF TH E MOON B Y THE MOTION O F

THE AMs LUNA-10

by

E. L. Akim

(USSR)

3 $ 6 7 11 7 3 3 GPO PRICE $/ CFSTI PRICE(S) $e

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(NASA C R OR TM X OR AD N U M B E R ) Hard copy (HCIMicrofiche (M F) 6 9

ff 653 July85

2 NOVEMBER 1966


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CT - CM - L P S - 10532

D E T E W. I I N AT I O N O F TH E G R AV I TAT I O N A L FIELD

O F THE NOON BY THE MOTION O F

THE IlMS LUNA-10

Doklady A . N . S S S R ,Mekhanika,

Izdatel'stvo " N A U K A " , 1966Tom 170, No.4, 799-802

S U M M A R Y

by E. L. Akim

I v 3The motion of L U N A - 1 0 is stuu ieu in the noncentral gravita-

tional field of the Moon, taking into account the gravitationalinfluence of the Earth , Sun anu planets.

Eleven coefficients of expansion of Moon's gravitational po-tent ial are obtained from trajectory measurements.

Pear-like shape of the surface with elongation on the far s h eof the Moon is eviuent. I t is basically formed under the influence

of the terms in potential's expansion containing the cubic harmonicP (with the coefficient ~3~ ) and the zonal harmonic P20.3

** *

I f the gravitational field of the Moon were centra l, while theinfluence of the outer bodies is absent, the motion of the satellitewould take place dlonq an unperturDed orbit (Keplerian ellipse), ofwhich the shape and dimensions would remain invariable in absolutespace. The noncentrality of the Moon's grav itat iona l field and theaction of outer bodies, of which the principal ones are the Earthand the Sun, induce perturbations in the motion of the satellite.Under the action of the latter the satellite orbit is bound to evolvewit h time.

The perturbing action of the Earth ana the Sun on the motion oMoon's satellite is well known. The greatest interest is offeredby orbit evolution arising on account of an unknown noncentralityof the gravitationa l field of the Moon. The knowledge of this evo-luti on al lows to determine its parameters.

* O P R E D E L E N I Y E P O LYA TYAGCTZNIYA LUNY PO D V I Z I I E N I Y U I S L " L U N A - 1 0 "&-YdJu


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Trajectory measurements Of satellite motion, conducted inr

the entire period ofits existence (from 3 April to 30 May 1966)were used to ascer tain the evolution of LUNA-10 orbit. They weresubject to statistical processing with the view of joint determi-nation of parameters of Moon's gravitat ional field and of the ele-ments of its orbit. Laid a t the basis of the method of measurement

processinq was the analytical theory of motion of a Moon's satel-lite that would allow to encompass the entire two-month measur inginterval of satelli te motion.

The description of selenocentrical motion o f the satelliteand of the motion of the Moon around its proper mass cent er wasperformed in the Descart es rectangular selenocentrical system or'coordinates X Y Z . The plane XY of this system coincides with theplane of the mean equator of the Moon, and the plane XZ overlapswith the plane of its zero meridian of the t o epoch. The direc-tions of the axial system are fixed relative to stars. The axisX of the system is directed toward the Earth, the axis Z - to thenorthern pole of the Moon and the axis Y completes the right-hand=end system. The following elements of satellite orbit are usedfor the description of its motion: major semiaxis a, eccentrici-ty 5, inclination i, longitude of the ascending nodg Q , angulardistance from perizenter to the noae w , time of node passage TQ(The counting of the angular orbital elements i, a , w is made fromthe Moon's mean equator ial plane and its zero meridian of to epochby the method admi tted in celestial mechanics). It is assumed thatthe rotation of the Moon around it s proper mass center takes placeaccording t o Cassini laws, i. e., uniformly around the fixed axisOZ of the introduced system of coordinates. The motion of themass center of the Moon (origin of the system of coordinates) inthe geoequatorial system of coordinates with the mean equinox of thethe epoch 1960.0 is given by the Braun theory.

The noncentrality of the qravitational field of the Moon is theesse ntia l fact determining the evolution of the orbit of LUNA-10.The perturbations of satellite orbit arising on account of the no?-centrali ty of the graviati onal field of the Moon are particularlyclearly manifest in the evolution of the longitude 52 of the ascend-ing node o f the satellite ana of the angular distance w from peri-selion to the node, This evolutisi? of the elements Q , (ri during thetime of its ex is t- mc e is represented in Fig.1 as a function of thenumb er of satelli te convolutions. The evolution of parameters n ,wcont ains a visible secu lar offset leading to regression of both parameters. A noticeable periodical perturbation is superimposed onthe secular offset of parameter w . For 460 satellite convolutionsthe perturbations of parameters s2,w , conditioned by the noncentra-lity of the gravit ational field of the Moon attained the valuesA P 7.7;Aw=-11.80. The perturbation of satellite orbit inclina-tion i and of its eccentricity e bear fundamentally a periodicalcharaEte r and have an amplitude T i = 0.15'; AeG 0.003.

The satellite orbit perturbation on account of field noncentra-lity leads to perturbations of its coordinates, constituting for asingle satellite convolution a value Ihr I s .75 km.


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The above comparative estimates show that for the orbit ofLUNA-10 the perturbations on account of Moon's gravit ationa lfield noncentrality exceed by a factor of 5 - 6 the perturba-tions caused by the gravitational influence of the Earth and theSun. The planetary perturbations of satellite motion are smal lover the considered time interval and t his is why they were notconsidered.

Fig.1. P.erturbations in the longitude Q o f the ascendingnode and in the angular distance wf rom the pericenter tobrbit node. Heavy lines refer to the noncentrality ofMoon's gravita tional field, the thin lines point to per-turbations on account of gravitational influence of the

Earth and the Sun

The processing o f trajectory measurements, containing theirunique dynamic tie over the flight interval of LUNA-10 for twolunar months allowed to determine the quantitative characte ris-tics of the noncentrality of Moon's gravit ationa l field.

The expres sion for the gravitational potential U of the Moonis accept ed in the form of expansion in series by spheri cal func-tion$

where P is the mass of the Moon, R is its mean radius; r,$,A ar ethe spheri cal coordin ates of the point: r is the polar radius ,JI is t he latitude counted from the mean Equato r of the Moon, X is

the longitude counted from the zero meridian of the epoch to;P,m(sin$$ are the connected Legendre functions. We uti lized forfor the determined parameters of Moon's gravitational field thecoefficients c , d of this expansion.

m r u n


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c

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A s a result of this processing we obtained the numericalvalues for eleven expansion coefficients of the g ravit ationa l po-tential of the Moon. Together

rors these values are presentedbe low :

with their maximum possible er- I

cz0 = (-O,SOG rt 0,022) .10-3,

czi = (0,157 i .059) . O-&,

C L O = (0,333 5 0,270) . O-'.toward

I EarthThere i s a substanti al cor-

relation between the errors inthe determination of parameters

and cb O . The correlationfactor is k = -0.99. The mutual

correlation between the errorsof the remaining determined para-meters does not exceed 0.4.

c20

The above numerical valuesof parameters cz0 and c22 agreewell with their known values ob-tained by libration measurements111

The nonzero value of thecoeffi cient s cnm with odd indexm and of coeffi cient s d withan even index m sugges t syha tthe gravitatioKa1 field on thevisible and the far sides of theMoon is asymmetrical.

-

For' the illus tration of aspecific gravitational paten-tial o f Moon we considered thelevel surface passing throughthe point with spherical coor-dinates r = 1/38 km, JI= 0 and

A , = 0.

Fig.2. Cross-sec tions of thelevel surface of the Moon'sgravitational potential (radialdefle ctions from the circumferencex 1000 times). A) Equatorial,6 ) meridional ( A = 0) , B ) meridional

( A = g o o )


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The shape of the potential surface is clearly seen as pear-likewith elongation on the far side of the Moon. This pear-like shapeis formed mainly under the influence of t e terms in the potential

and the zo nal harmonic P20.

trajectory measurements on LUNA-10 and the analysis of subsequentAMs will permit the refining of the obtained parameters and comple -ment them furth er, and it will be also possible to refine the motionof its cen ter of masses.

expansion containing the cubic harmonic P 3 with the coefficient c 3 1

The above resul ts are preliminary. The furthest processing of

Received nn13 September 1966

R E F E R E N C E

1. C. L. G O U D A S . , Icarus, 3, 375, 1964.

Con tract No.NAS-5-12487 Translated by A N D R E L. BRICHANTVOLT T ECHNI CAL CORPORATION1145 19th st NW on 1 November 1966D.C. 20036. Tel: 223-6700


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