Figure 1. — Right spherical triangle formed by the equator, 

 the orbital subpoint track and the meridian through the sub- 

 point ((/), A). 



initial value of a is obtained by subtracting the 

 argument of perigee from 360 . 



The gravity effect of the earth's equatorial 

 bulge causes the perigee to shift slightly during 

 each orbit. Consequently, it is necessary to know 

 the rate at which the orbital ellipse is shifting in 

 order to determine the correct initial values of 

 a for prediction of future orbits. 



The locus of orbital subpoints for an earth 

 satellite traces a great circle on a sphere concen- 

 tric with the earth. Each subpoint can be repre- 

 sented by a pair of coordinates (4>, A), where 4> 

 is the geocentric latitude and A is the longitudinal 

 displacement of the subpoint measured from the 

 longitude of the ascending node. Subpoint co- 

 ordinates corresponding to any point of the orbit 

 can be determined from its angular displacement, 

 measured along the orbital track from the as- 

 cending node. If a is the angular coordinate 

 (measured from perigee, as defined in the pre- 

 ceding section) and a,, is the value of a at the as- 

 cending node, the angular displacement is 

 ( «—«„). Figure 1 shows these quantities as com- 

 ponents of a right spherical triangle whose ver- 

 tices are defined by the subpoint ((/>, A), the co- 

 ordinates of the ascending node (0, 0) and the 

 point (0, A). The two sides forming the right angle 

 are (/) and A respectively. The opposite side is 

 ( a —a„) and is a segment of the subpoint track. 

 The angle opposite </>, denoted by /, is the inclina- 



tion of the orbital plane to the equatorial plane. 

 Application of the trigonometric formulas de- 

 rived from Napier's rules yields the following re- 

 lationships: 



Equations (11) and (12) give explicit formulas for 

 (}> and A as a function of (/) and ( a — a„). Equa- 

 tion (13) represents the great circle traced by the 

 orbital subpoints. 



ORBIT PREDICTION 



The concepts developed in the preceding sec- 

 tions deal with the computation of satellite sub- 

 points relative to the time and longitude of the 

 ascending node. Application of these concepts 

 to advance preparation of orbit schedules re- 

 quires, therefore, prediction of the times and 

 longitudes of ascending nodes for future orbits. 

 The APT Predict messages give orbital period 

 to the nearest second and longitudinal displace- 

 ment per orbit to the nearest hundredth of a 

 degree. This precision is adequate to extrapolate 

 a few orbits ahead. Greater precision is needed 

 for satisfactory extrapolation beyond a hundred 

 orbits and can be obtained from NESS or can be 

 gained by empirical adjustments. 



The orbital elements necessary for computa- 

 tion of subpoint locations for extrapolated orbits, 

 also obtainable from NESS, include the length 

 of the semimajor axis, eccentricity, orbital in- 

 clination, argument of perigee at a known time 

 or reference orbit, and the rate of change of 

 perigee. The latter is equal to the rate of change 

 of a,„ the value of the angular displacement a 

 at the ascending node, which can be computed 

 from the following expression (Runcorn, 1967): 



(/ a„/dt = 4.98(3M)3.^ (1 ^e2) 

 (degrees per day). 



^ (5 cos^ i—l) 



where e is the eccentricity of the orbital ellipse, 

 i is the inclination of the orbital plane to 



the earth's equatorial plane, 

 a is the earth's equatorial radius, and 

 A is the semimajor axis of the ellipse, as 



previously defined. 



