By Guochang Xu
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3) and is called the argument of perigee. The argument of perigee defines the axis direction of the ellipse related to the equatorial plane. 3 Keplerian Equation Up to now, five integration constants have been derived. They are inclination angle i, right ascension of ascending node Ω, semi-major axis a, eccentricity e of the ellipse, and argument of perigee ω . 3 Keplerian Equation 31 RO O F Fig. 4 Orbital geometry π ab = 1 2h 2π ab μ a(1 − e2 ) The average angular velocity n is then 2π = a−3/2 μ 1/2 .
5). S is the vertical Fig. 5 Mean anomaly of satellite 32 3 Keplerian Orbits F projection of the satellite S on the circle with a radius of a (semi-major axis of the ellipse). The distance between the geometric centre O of the ellipse and the geocentre O is ae. 33) RO O y = r sin f = b sin E = a 1 − e2 sin E, where the second equation can be obtained by substituting the first into the standard ellipse equation (x2 /a2 + y2 /b2 = 1) and omitting the small terms that contain e (for the satellite, generally, e << 1), where b is the semi-minor axis of the ellipse.
Fig. 19), it turns out to be d(r2 ϑ˙ ) = 0. 20) RO O Because rϑ˙ is the tangential velocity, r2 ϑ˙ is the two times of the area velocity of the radius of the satellite. 20) and comparing it with the discussion in Sect. 21) r2 ϑ˙ = h. 21), one gets and dr dϑ d dr = = dt dϑ dt dϑ TE dϑ = hu2 dt DP h/2 is the area velocity of the radius of the satellite. 19), the equation has to be transformed into a differential equation of r with respect to variable f . 24) EC d2 u d2 u dϑ d2 r = −h2 u2 2 . 25) μ , h2 where d1 and d2 are constants of integration.
Keplerian solutions by Guochang Xu