Motion of a Satellite under the influence of an oblate EarthJune 27, 2001George H. BornFigure 1. Potential of an arbitrary shaped bodyAn expression forThe expression for the magnitude of the radius vector, , can be developed from the equation(39)This may be demonstrated as followsSubstituting eq.(47) into (46) yieldsVery near circular orbitsResultsAcknowledgementReferences1. Kaula, W., Theory of Satellite Geodesy, Dover, Mincola N.Y., 2000.Motion of a Satellite under the influence of an oblate EarthASEN 3200June 27, 2001George H. BornFigure 1. Potential of an arbitrary shaped bodyWe wish to derive the gravitational potential function for an arbitrary shape body as shown in figure 1. Consider a particle, , of mass, m. The gravitational force on P due to the differential mass dm given by Newton’s law of gravitation is3GmdmF (1)where dxdydzdm(2)1rdmy x zPand is the density. The acceleration of dm is given by mF i.e. the force per unit mass .3GdmmFa (3)We can express the acceleration as the gradient of a potential function, , where.Gdm(4)This is easily shown by taking the gradient of i.e.,kzjyixaˆˆˆ kzjyixGdmˆˆˆ2 kzjyixGdmˆˆˆ3 .3Gdm(5)To obtain the potential function for the entire body we would integrate eq.(4) over the volume of the body (Kaula, 2000). The result is 2 111sincossin1sin1llllllllllmmmmmSmCPrPrRJr(6)where the coordinates of P are now expressed in spherical coordinates ,,r, where is the geocentric latitude and is the longitude. Also, R is the equatorial radius of the primary body and sinmPl is the Legendre’s Associated Functions of degree and order m. The coefficients mCJll, and mSl are referred to as spherical harmonic coefficients. 2If m = 0 the coefficients are referred to as zonal harmonics. If 0mlthey are referred to as tesseral harmonics, and if 0ml, they are called sectoral harmonics.We will considered the effect of 2J, the second zonal harmonic, on the orbit of a satellite. 2Jis also known as the oblateness coefficient. Gravitationally, the earth can be modeled fairly accurately as an ellipsoid if revolution i.e. by the 2Jharmonic. In fact the next zonal harmonic,63105.2J, is thee orders of magnitude smaller than 321008.1J.+ indicates positive gravity anomaly- indicates negative gravity anomaly Figure 2. The Earth as described by 2JFigure 2 illustrates the equipotential surface representing the influence of 2J. The polar radius of the elliptical earth is 20 km smaller than the equatorial radius. This effect is primary due to the movement of mass to the equator cause by the centripetal force due to earth rotation. The value of 2Jfor the earth is 0.00108, for Mars 00196.02J, for the Moon 000195.02J, and for the asteroid Eros 11.02J. The value of 2Jfor the Moon is much lower for the Earth and Mars because its rotation rate is much less. If we evaluate eq.(6) for ,0,2 ml i.e. 2J, and drop the central force field contribution we obtain the perturbing potential, p3+ + --.)sin31(2222rRJrp(7)Equation (7) may be written in terms of the Kelper elements of a satellite’s orbit by replacing ,the geocentric latitude, with the inclination and argument of latitude of the satellite orbit.Figure 3. Orientation of the s/c orbitwhere, u argument of latitude, argument of perigee true anomalyi inclination right ascension of the ascending node geocentric latitudeFrom Figure (3) and spherical trig identitiesiusinsinsin Hence, eq.(7) may be written as )sinsin31(22222iurRJrp(8)4ruyzxiEquatororbitS/Cbutuu 2cos1sin22(9)and pmy be written as 22cossin21sin2131232222iiJrRrp(10)The differential equations which describe the variation with time of the orbit elements are called Lagrange's Planetary Equations (Roy, 1988), and are given byMnadtdap2 ppenaeMenaedtde222211ppienaienaidtdisin11sin1cos2222(11)iienadtdpsin1122eenaeiienaidtdpp22221sin1cosanaeenaendtdMpp2122In general the orbit element will experience secular and short period perturbations due to 2J. Secular perturbations grow linearly with time and short period perturbations are periodic in multiples of the true anomaly. The first order (i.e., to 2JO) secular terms may be obtained 5by averaging the short period terms out of pby integrating over the mean anomaly fromzero to 2, i.e.,2021dMpp(12)It has been shown by Tisserand (1889) that 202323121edMra(13)and .02cos212sin21320203 rara(14)Hence, .2232232sin2131123ieJaRp(15)Inspection of eqs.(11) shows that secular terms can only appear in , and M because p is only dependent on a, e, and i. If the partial derivatives of pare substituted into eqs.(11) the resulting secular rates are ineaRJscos12322222(16) ineaRJs222222sin252123(17) ineaRJnMs2232222sin231123 (18)Where2321an(19)and a is the mean value of awhich will be defined later.6A first order solution in 2J which ignores terms of order 2eJ may be developed by solving Lagrange’s planetary equations under the assumption that the reference orbit is a secularly precessing ellipse. This means that ,, ea and i are held constant on the right band side of Lagrange’s equations and , and M vary with linear rates given by equations (16) through (19) Consider for example the differential equation for the semimajor axis,.2Mnadtdap(20)If we assume we are dealing with near-circular orbits (e < 0.002), and write eq.(10) in terms of the mean anomaly M, and ignore term of 2eJO, by usingM Mear
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