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Orekit is a free low level space dynamics library written in Java. It provides basic elements (orbits, dates, attitude, frames, ...) and various algorithms to handle them. Homepage: http://www.orekit.org/
New Orekit website, based on Jekyll (a static website generator) and Bootstrap (a popular HTML, CSS, and JS framework).
The Orekit python wrapper allows using Orekit from a python environment. It was developed by SSC and provided to the Orekit project in May 2014.