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By Giulio Avanzini

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7 permits the evaluation of the time of passage at the selected anomaly. The r and v components in the geocentric-equatorial (or heliocentric-ecliptic) frame can be obtained using a coordinate transformation. 7 Coordinate Transformation The more general problem of coordinate transformation is presented here for the specific case of passage from perifocal to geocentric–equatorial components, but the approach can be extended to change of reference between any pair of mutually orthogonal triads of unit vectors in a three– dimensional space.

In some cases the longitude of the ascending node is also assigned; the launch is possible twice a day when the launch site passes through the envisaged orbit plane, and the azimuth constraints can be fulfilled. The lift–off time is strictly fixed for maximum payload. However a fraction of the payload is traded for propellant, thus adding maneuvering capabilities and allwoign for an out–of–plane ascent. A launch window of several minutes is created around the theoretical launch time: the rocket departs as soon as all the requirements for a reliable launch are met.

The unit vectors from the center of the sphere to the vertices ˆA , e ˆB , and e ˆC , respectively. The length of the side BC, denoted by a, is seen as an angle are e from the center of the sphere (remember that a great circle on the unit sphere is a circle of unit radius, such that the length on an arc on the circle is equal to the angle at the center expressed in radians). The angle on the spherical surface at vertex A is the dihedral angle between planes AOB and AOC, and is denoted by α. 1 - Law of the cosines ˆB ) · (ˆ ˆC ) can be expressed in two ways: The dot product (ˆ eA × e eA × e ˆB ) · (ˆ ˆC ) = [(ˆ ˆB ) × e ˆA ] · e ˆC (ˆ eA × e eA × e eA × e 34 G.

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Astrodynamics by Giulio Avanzini

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