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Celest Mech Dyn Astr (2014) 118:111
DOI 10.1007/s10569-013-9521-8
ORIGINAL ARTICLE
Daniele Mortari Antonio Elipe
Received: 21 June 2013 / Revised: 2 September 2013 / Accepted: 23 September 2013 / Published online: 9 November 2013 Springer Science+Business Media Dordrecht 2013
Abstract A new approach to solve Keplers equation based on the use of implicit functions is proposed here. First, new upper and lower bounds are derived for two ranges of mean anomaly. These upper and lower bounds initialize a two-step procedure involving the solution of two implicit functions. These two implicit functions, which are non-rational (polynomial) Bzier functions, can be linear or quadratic, depending on the derivatives of the initial bound values. These are new initial bounds that have been compared and proven more accurate than Serans bounds. The procedure reaches machine error accuracy with no more that one quadratic and one linear iterations, experienced in the tough range, where the eccentricity is close to one and the mean anomaly to zero. The proposed method is particularly suitable for space-based applications with limited computational capability.
Keywords Kepler equation Optimal starter Bzier functions Root nder
1 Introduction
Keplers equation (KE) for the elliptical case,
f (E) = E e sin E M = 0 (1)
D. Mortari (B)
Department of Aerospace Engineering, Texas A&M University, 746C H.R. Bright Bldg., College Station, TX 77843-3141, USA e-mail: [email protected]
A. Elipe
Centro Universitario de la Defensa, 50090 Zaragoza, Spain e-mail: [email protected]
A. Elipe
Dpto. Matemtica AplicadaIUMA, Universidad de Zaragoza, 50009 Zaragoza, Spain
Solving Keplers equation using implicit functions
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2 D. Mortari, A. Elipe
is an important example of root nder problem. E is the unknown eccentric anomaly, e is the
orbit eccentricity, and M, called mean anomaly, is linearly related to time, M =
2
T t, where
T is the orbital period, and t is the elapsed time from pericenter. Given M and e solving KE is an extremely important problem that must be executed to perform orbit propagation, that is, to compute where an object orbiting around a gravitational body is at different values of time, t. This is done as the eccentric anomaly is directly related to true anomaly () identifying the angular displacement in the orbit.
This nonlinear simple transcendental equation has kept busy many...