-160 -140 -120 -100 -60 -60 -g0 0 20 40 60 80 100 120 140 1«)


-160 -140 -120 -100 -60 -60 -g0 0 20 40 60 80 100 120 140 1«)


Flg. 6-5. Typical Ground Tracks. (A) Shuttle parking, (B) Low-altitude retrograde, (C) GPS, and (0) Molniya orbits. See text for orbital elements.

6.1.5 Time of Flight in an Elliptical Orbit

In analyzing Brahe's observational data, Kepler was able to solve the problem of relating position in the orbit to the elapsed time, t-tg, or conversely, how long it takes to go from one point to another in an orbit. To solve this, Kepler introduced the quantity M, called the mean anomaly, which is the fraction of an orbit period which has elapsed since perigee, expressed as an angle. The mean anomaly equals the true anomaly for a circular orbit. By definition, where Mn is the mean anomaly at time t0 and n is the mean motion, or average angular velocity, determined from the semimajor axis of the orbit:

where a is in km.

This solution will give the average position and velocity, but satellite orbits are elliptical, with a radius constantly varying in orbit. Because the satellite's velocity depends on this varying radius, it changes as well. To resolve this problem we define an intermediate variable called eccentric anomaly, E, for elliptical orbits. Table 6-3 lists the equations necessary to relate time of flight to orbital position.

s 36,173.585 a~™ degfc s 8,681,660.4 a~m rev/day = 3.125 297 7 x lO9«-^ deg/day

TABLE 6-3. Time of Flight In an Elliptic Orbit All angular quantities are in radians.




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