1Motion of a Satellite under the influence of an oblate Earth ASEN 3200 July 10, 2001 George H. Born Figure 1. Potential of an arbitrary shaped body We wish to derive the gravitational potential function for an arbitrary shape body as shown in figure 1. Consider a particle, ρ, of mass, m. The gravitational force on P due to the differential mass dm given by Newton’s law of gravitation is 3ρρGmdmF −= (1) where dxdydzdmγ= (2) rρdm y x z P2and γ is the density. The acceleration of dm is given by mF i.e. the force per unit mass .3ρρGdmmFa −== (3) We can express the acceleration as the gradient of a potential function, Φ , where .ρGdm=Φ (4) This is easily shown by taking the gradient of Φ i.e., kzjyixaˆˆˆ∂Φ∂+∂Φ∂+∂Φ∂=Φ∇= ⎥⎦⎤⎢⎣⎡∂∂+∂∂+∂∂−= kzjyixGdmˆˆˆ2ρρρρ []kzjyixGdmˆˆˆ3++−=ρ .3ρρGdm−= (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 () (){ }⎥⎥⎦⎤⎢⎢⎣⎡++⎟⎠⎞⎜⎝⎛−=Φ∑∑∑∞==+∞=2111sincossin1sin1llllllllllmmmmmSmCPrPrRJrλλφφµ (6) where the coordinates of P are now expressed in spherical coordinates ()λφ,,r , where φ is the geocentric latitude and λ is the longitude. Also, R is the equatorial radius of the primary body and ()φsinmPl is the Legendre’s Associated Functions of degree and order m.3The coefficients mCJll, and mSl are referred to as spherical harmonic coefficients. If m = 0 the coefficients are referred to as zonal harmonics. If 0≠≠ml they are referred to as tesseral harmonics, and if 0≠=ml , they are called sectoral harmonics. We will considered the effect of 2J , the second zonal harmonic, on the orbit of a satellite. 2J is also known as the oblateness coefficient. Gravitationally, the earth can be modeled fairly accurately as an ellipsoid of revolution i.e. by the 2J harmonic. In fact the next zonal harmonic, 63105.2−×−=J , is thee orders of magnitude smaller than 321008.1−×=J . + indicates positive gravity anomaly - indicates negative gravity anomaly Figure 2. The Earth as described by 2J Figure 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 2J for the earth is 0.00108, for Mars 00196.02=J , for the Moon 000195.02=J , and for the asteroid Eros 11.02=J . The value of 2J for 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 + + --4i.e. 2J, and drop the central force field contribution we obtain the perturbing potential, pΦ .)sin31(2222φµ−⎟⎠⎞⎜⎝⎛=ΦrRJrp (7) Equation (7) may be written in terms of the Kepler elements of a satellite’s orbit by replacing φ, the geocentric latitude, with the inclination and argument of latitude of the satellite orbit. Figure 3. Orientation of the s/c orbit where, u ≡ argument of latitude, υω+ ω ≡ argument of perigee υ ≡ true anomaly i ≡ inclination Ω ≡ right ascension of the ascending node φ ≡ geocentric latitude From Figure (3) and spherical trig identities iusinsinsin=φ ruφy z x iΩEquator orbit S/C5Hence, Eq. (7) may be written as )sinsin31(22222iurRJrp−⎟⎠⎞⎜⎝⎛=Φµ (8) but uu 2cos1sin22−= (9) and pΦ my be written as ()⎟⎠⎞⎜⎝⎛++−⎟⎠⎞⎜⎝⎛=Φυωµ22cossin21sin2131232222iiJrRrp (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 by Mnadtdap∂Φ∂=2 ω∂Φ∂−−∂Φ∂−=ppenaeMenaedtde222211 Ω∂Φ∂−−∂Φ∂−=ppienaienaidtdisin11sin1cos2222ω (11) iienadtdp∂Φ∂−=Ωsin1122 eenaeiienaidtdpp∂Φ∂−+∂Φ∂−−=22221sin1cosω anaeenaendtdMpp∂Φ∂−∂Φ∂−−=21226In general the orbit element will experience secular, long and short period perturbations due to 2J (Brouwer, 1959). Secular perturbations grow linearly with time; long and short period perturbations are periodic in multiples of the argument of perigee and true anomaly respectively. The first order (i.e., to ()2JO ) secular terms may be obtained by averaging the short period terms out of pΦby integrating over the mean anomaly from zero to π2 , i.e., ∫Φ=Φππ2021dMpp (12) It has been shown by Tisserand (1889) that ()∫−−=⎟⎠⎞⎜⎝⎛ππ202323121edMra (13) and .02cos212sin21320203=⎟⎠⎞⎜⎝⎛=⎟⎠⎞⎜⎝⎛∫∫υπυπππrara (14) Hence, ().2232232sin2131123⎟⎠⎞⎜⎝⎛−−=Φ−ieJaRpµ (15) Inspection of Eqs. (11) shows that secular terms can only appear in ω,Ω and M because pΦ is only dependent on a, e, and i. If the partial derivatives of pΦ are substituted into Eqs. (11) the resulting secular rates are ()ineaRJscos12322222−−=Ω (16)7 ()⎟⎠⎞⎜⎝⎛−−= ineaRJs222222sin252123ω (17) ()⎟⎠⎞⎜⎝⎛−−+=ineaRJnMs2232222sin231123 (18) Where 2321anµ= (19) and the overbar represents mean values that will be defined later. A first order solution in 2J 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 ω,Ω and M vary with linear rates given by equations (16) through (19) Consider for example the differential equation for the semimajor axis, .2Mnadtdap∂Φ∂= (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 using M=υ ()Mear cos1−= we obtain ().22cossin21sin2131232222⎟⎠⎞⎜⎝⎛++−⎟⎠⎞⎜⎝⎛=Φ MiiJaRapωµ (21)8 Thus, ().22sinsin23222MiJaRaMp+⎟⎠⎞⎜⎝⎛−=∂Φ∂ωµ (22) Substituting Eq. (22) into Eq. (20) and assuming a secularly precessing ellipse, i.e., a , e , and i are constant, and ()()[]()()()000222 ttMMtMtss−+++=+ωωω (23) we can integrate