Transforming Mean and Osculating Elements Using Numerical Methods
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Transforming Mean and Osculating Elements Using Numerical Methods Todd A. Ely1
Published online: 24 June 2015 © The Author(s) 2015. This article is published with open access at Springerlink.com
Abstract Mean element propagation of perturbed two body orbits has as its mathematical basis the averaging theory of nonlinear dynamical systems. Mean elements define an orbit’s long-term evolution characteristics consisting of both secular and long-period effects. Using averaging theory, a near-identity transformation can be found that transforms between the mean elements and their osculating counterparts that augment the mean elements with short period effects. The ability to perform the conversion is necessary so that orbit design conducted in either mean elements or osculating can be effectively converted between each element type. In the present work, the near-identity transformation is found using the Fast Fourier Transform. An efficient method is found that is capable of recovering the mean or osculating elements to first-order. Keywords Mean elements · Averaging · Identity transformations · Dynamical systems
Introduction Mean element theories have proven useful for efficient determination of an orbit’s long-term characteristics; however, a mean orbit propagation often requires transforming a set of osculating initial conditions to mean to properly initialize the Prior version of this paper was presented at the AAS/AIAA Space Flight Mechanics Conference, San Diego, CA 14 – 17 February 2010 Todd A. Ely
[email protected] 1
Mission Design and Navigation Section, Jet Propulsion Laboratory, California Institute of Technology, MS 301-121, 4800 Oak Grove Drive, Pasadena, CA 91109-8099, USA
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J of Astronaut Sci (2015) 62:21–43
propagation. Conversely, once a mean orbit is obtained converting the results back to osculating elements requires the inverse transformation. Fortunately the mean element theory provides an available mechanism for finding a transformation that is consistent with the applied averaging technique. Indeed, the process to obtain the mean elements using averaging produces the transformation equations directly. This is a well-known result (cf. Sanders et al. [1], Chapter 7 on Averaging Over Angles) that will be reviewed later in this paper, and applied to the present context of perturbed two-body orbits. There is an extensive body of work applying averaging theory to satellite astrodynamics, which has traditionally focused on deriving analytic or semi-analytic theories to find satellite mean orbital elements and the associated mean-to-osculating or osculating-to-mean transformations. Two key contributions that have found wide use are Draper’s Semi-analytic Satellite Theory (DSST), documented extensively by Danielson et al. [2], McClain [3], McClain et al. [4], and the work by Guinn [5]. These prior works obtain the transformations by using analytical techniques to compute explicit formulae for the Fourier series that correspond to the short period terms of the orbit being examined. In the pre
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