State transition matrix of relative motion for the non-circular orbit. Relation with partial-derivative matrix in the satellite coordinate system
- Paper number
IAC-06-C1.6.08
- Author
Prof. Andrey Nazarenko, Space Observation Center, Russia
- Year
2006
- Abstract
The analytical formulas for the partial-derivative matrix U(t,t 0) RSW in the satellite coordinate system were derived by the Russian experts at the beginning of sixties (P.E. Elyasberg [1] and V.I. Charniy). This matrix includes the effect due to the reference orbit eccentricity. The relation between state vector variations at some time instant in geocentric celestial (x. y, z) and satellite coordinate system (R, S, W) is appeared by the orthogonal transformation matrix G
It is shown in this paper that the mentioned matrix U(t,t 0) RSW can be serve as the basis for calculating the state transition matrix of relative motion for a non-circular orbit. The derived formulas
U(t,t 0) rel= L(t)* U(t,t 0) RSW* L(t 0) −1, (1)
L(t)= F( G). (2)
being rather simple in appearance, represent generalization of the well known solution of the Clohessy-Wiltshire equations, suitable for circular orbits only.
This paper presents as well the results of the matrix U(t,t 0) rel calculation by using the developed method and their comparison with results of numerical integration and application of the solution of the Clohessy-Wiltshire equations for circular orbits.
That fact, that the formulas such as (1, 2) type are not mentioned in numerous modern scientific publications, testifies the insufficient attention to works of Astrodynamics classics, to which Pavel Elyasberg belongs, undoubtedly.
References.
1. P.E. Elyasberg. Introduction to theory of flight of the Earth artificial satellites. Moscow, Publishing house "Science", 1965.
- Abstract document
- Manuscript document
IAC-06-C1.6.08.pdf (🔒 authorized access only).
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