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Normalization and Implementation of Three Gravitational Acceleration ModelsUnlike the uniform density spherical shell approximations of Newton, the consequence of spaceflight in the real universe is that gravitational fields are sensitive to the asphericity of their generating central bodies. The gravitational potential of an aspherical central body is typically resolved using spherical harmonic approximations. However, attempting to directly calculate the spherical harmonic approximations results in at least two singularities that must be removed to generalize the method and solve for any possible orbit, including polar orbits. Samuel Pines, Bill Lear, and Robert Gottlieb developed three unique algorithms to eliminate these singularities. This paper documents the methodical normalization of two of the three known formulations for singularity-free gravitational acceleration (namely, the Lear and Gottlieb algorithms) and formulates a general method for defining normalization parameters used to generate normalized Legendre polynomials and Associated Legendre Functions (ALFs) for any algorithm. A treatment of the conventional formulation of the gravitational potential and acceleration is also provided, in addition to a brief overview of the philosophical differences between the three known singularity-free algorithms.
Document ID
Acquisition Source
Johnson Space Center
Document Type
Technical Publication (TP)
Eckman, Randy A.
(NASA Johnson Space Center Houston, TX, United States)
Brown, Aaron J.
(NASA Johnson Space Center Houston, TX, United States)
Adamo, Daniel R.
(Independent astrodynamics consultant)
Gottlieb, Robert G.
(Odyssey Space Research, LLC Houston, TX, United States)
Date Acquired
September 14, 2016
Publication Date
June 1, 2016
Subject Category
Report/Patent Number
Distribution Limits
Public Use Permitted.
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