gives an explicit equation for the angular veloc- 

 ity, as follows: 



d a 

 dt 



2uAB 



(1) 



The values of A and B are related to the eccen- 

 tricity, e, of an ellipse by the expression 



_F_ 

 A 



where 



F =yJA^ -52 



and represents the distance from the center to 

 either focus of the ellipse. 



The relation between R and a can be obtained 

 from the basic equation of an ellipse in polar co- 

 ordinates, with the origin at one focus and the 

 polar axis coincident with the major axis of the 

 ellipse: 



R = 



A (1 



1 ± e cos a 



(2) 



DISTURBED ORBITAL MOTION 



Because the earth is not a concentrically homo- 

 geneous sphere, but rather an oblate ellipsoid 

 of revolution, the angular velocity is not exactly 

 defined by Equation (1). The elliptic motion is 

 governed by a force which can be defined in 

 terms of the Newtonian gravitational potential 

 U, as follows (Menzel, 1960): 



U = 



IG M K^ (I -J) (I -3sin^ 4>) 



R 



+ 



2R^ 



(3) 



where M is the mass of the earth, 



K^ is the constant of gravitation, 



I is the moment of inertia about the polar 



axis, 

 J is the moment of inertia about any dia- 

 meter in the earth's equatorial plane, 

 and 

 4) is the angle between the equatorial 

 plane and a line from the earth's cen- 

 ter to the satellite (i.e., cf) = latitude). 



The expression (I —J) can be related to the 

 earth's mass M, the equatorial radius a, the 



earth's flattening/, and the ratio q of the earth's 

 centripetal acceleration at the equator to gravity, 

 as follows: 



I -J =(2/3) Ma^ if -q/2). 



(4) 



Substituting Equation (4) in Equation (3), we 

 get 



U = 



K^ M 



R 



+ 



IP Ma^ {f -q/2) (1^3 sin^ (/>) 

 3R' 



(5) 



A more precise form of expression for the 

 earth's gravitational potential is given in terms 

 of spherical harmonics by 



U = 



IP M IP M 



R 



R 



QO 



2 J 



n 



-^ I Pn (sin </)) 



where the J„ are constants which can be deter- 

 mined from observations of earth satellites, and 

 the Pn (sin c^) are Legendre polynomials (Run- 

 corn, 1967). The first term in the series (corre- 

 sponding to )i =2) represents the main effect of 

 the earth's oblateness. The value of J, has been 

 determined as 1.0826 XIO' , which is 400 times 

 greater than J for any of the higher n. The 

 Legendre polynomial for w =2 is 



P, (sin 4>) = (3/2) sin^ </> - 1/2. 



If terms in higher n are omitted, the expression 

 for gravitational potential becomes 



U 



IP M IP M 



a2 J, (1 -3sin2 <J)) 



R 



2R' 



This expression is equivalent to Equation (5) 

 if terms of second and higher order are omitted 

 in the following relation between J. and the con- 

 stants / and q: 



J, = (2/3) (/ - q/2) 



-(1/3)/ 2 +(l/2)q2 +{2/2l)fq. 



