The values of / and q have been determined as 

 3.353 X 10 3 and 3.468 X 10 ^ , respectively, 

 so that 



(2/3) ( / - q/2) = 1.0794 X 10 ^ ^ _ 



The value of A'2 Mis 3.986 X 10^ km= sec" 2. 



The gravitational acceleration is obtained by 

 differentiating (5) with respect to R to get 



dU 

 dR 



K^ M 

 R^ 



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

 R' 



•(6) 



This expression for gravitational acceleration due 

 to the mass of the earth could be equated directly 

 to the centripetal acceleration of an earth satel- 

 lite if it were traveling in a true circular orbit. 

 In an elliptical orbit, however, there is a devia- 

 tion from circular motion which can be identified 

 by substituting into (1) the following expression 

 from Kepler's third law for the orbital period: 



2tiA 



3/2 



K y/M 



The substitution gives 



(7) 



from which we obtain the following equation for 

 centripetal acceleration: 



earth's gravitational acceleration to the centri- 

 petal acceleration of an earth satellite as follows: 



K(<!^%^(jm 



+ 



/ RA\ R' 



K^Ma^ (f -q/2) (1-3 sm'-4>)\ 

 R' J 



and the angular motion is 



d_a_B_ / K^M 

 dt ^ R \ R^A 



K^M aM / - q/2) (1-3 sin^ (/>) > 



SUBPOINT LOCATION 



Satellite orbit predictions are given in terms 

 of subpoint locations at fixed intervals of time 

 following the northward equator crossing, re- 

 ferred to as the ascending node. Numerical inte- 

 gration of Equation (9), carried out with suitably 

 small time increments, can be made to yield suc- 

 cessive values of a requisite for subpoint compu- 

 tation. If a is taken to be zero when the satellite 

 is at perigee, then the plus sign preceding e 

 cos a in Equation (2) is applicable and R is given 

 by 



R 



A (1 -e') 

 1 +e cos a 



(10) 



R 



( I a 

 dt 



IP M B^ 

 R^ A 



(8) 



The right side of Equation (8) can be considered 

 as the product of {B^/RA) and the quantity 

 (K^M/R-). The latter is equivalent, except in 

 sign, to the first term on the right side of Equa- 

 tion (6) and represents the gravitational accel- 

 eration for undisturbed motion. The correction 

 for the ellipticity of an orbit is contained in the 

 factor B^/RA. Its value approaches 1 when an 

 orbit is nearly circular. By applying this factor to 

 the right side of Equation (6) we can relate the 



R can be computed from Equation (10) for each 

 iteration in the numerical integration of Equa- 

 tion (9) using the value of a from the preceding 

 iteration increased by one-half the average 

 angular displacement [360 ( A t)/2P\ during the 

 selected time increment, A^- In order to start 

 the numerical integration of Equation (9) at the 

 ascending node, an initial value of a must be de- 

 termined from the orientation of the orbital el- 

 lipse. The latter is expressed as the argument 

 of perigee, which is the geocentric angle of the 

 perigee measured in the orbital plane from the 

 ascending node in the direction of motion. The 



