Astrodynamics by Giulio Avanzini

By Giulio Avanzini

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This can be evaluated as the sum of the right ascension of the Greenwich meridian GM plus the longitude of GS, α = αG + Lo where αG = αG0 + ω⊕ (t − t0 ) The angle αG0 at t0 is often provided as the Greenwich sidereal time at 0:00 of January, 1st of a specified year. In such a case, αG0 is close to 100 deg, with less than 1 deg oscillations due to the non–integer number of the days in a year. As the equatorial plane is the base plane of the geocentric equatorial frame, the declination of GS is equal to its latitude, that is δ =Lat.

Three-Dimensional Analysis of Motion 27 In a similar way one evaluates the alternate parameters • ˆ cos u0 = ˆi · n (u0 > π, if i3 < 0) • ˆ1 · g ˆ1 cos Π = p (P i > π, if e2 < 0) • ˆ1 cos ℓ0 = ˆi · g (ℓ0 > π, if i2 < 0) The last two equations hold only for zero orbit inclination (i = 0). 6 Determining Spacecraft Position and Velocity After the orbital elements have been obtained from the knowledge of r and v at a specified time, the problem of updating the spacecraft position and velocity is solved in the perifocal reference frame using the closed–form solution of the equation of motion.

The tesseral and sectorial harmonics causes the deviation in the East–West direction; the former are dependent on both latitude and longitude, the latter on longitude only. The terms with n = 1 are absent, as the center of the frame and the center of mass of the Earth coincide. The even numbered zonal harmonics are symmetric about the equatorial plane and the odd numbered harmonics antisymmetric. 082637 10−3 , is three order of magnitude greater than any other; J3 describes the Earth as pear–shaped.

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