Parameter distributions of Keplerian orbits

@article{Savransky2010ParameterDO,
  title={Parameter distributions of Keplerian orbits},
  author={Dmitry Savransky and Eric J. Cady and N. Jeremy Kasdin},
  journal={arXiv: Instrumentation and Methods for Astrophysics},
  year={2010}
}
Starting with just the assumption of uniformly distributed orbital orientations, we derive expressions for the distributions of the Keplerian orbital elements as functions of arbitrary distributions of eccentricity and semi-major axis. We present methods for finding the probability density functions of the true anomaly, eccentric anomaly, orbital radius, and other parameters used in describing direct planetary observations. We also demonstrate the independence of the distribution of phase angle… 

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