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Eccentricity Functions in the Higher Degree and Order Sectorial Gravitational Harmonic Coefficients

Ioannis Iraklis Haranas
Omiros Ragos
Michael Harney
Year: 2011 Pages: 4
In the study of an Earth orbiting satellite, the terms of the series expansion of the Earth's gravitational potential can be expressed as functions of the eccentricity of the satellite. These functions are also known as eccentricity functions. The series expansion of these functions given by Kaula [2] appears to result in instabilities at high eccentricities. When calculating the eccentricity functions, researchers resort to numerical integration techniques instead. The approach followed in this contribution bypasses the problem of instability at high eccentricities by using a Hansen coefficient definition. As a test, we first calculate analytical expressions for various known eccentricity functions and then we proceed with the calculation of the eccentricity functions associated with degree and order 20, 30, 40, 50 sectorial harmonic coefficient expansion of the gravitational potential. Our calculation demonstrates the efficiency of Hansen coefficient approach that differs from that given by Kaula. It is efficient, fast, and can easily be performed with the help of a personal computer, with no instabilities at higher eccentricities.