Rudolphine Tables and instructive examples for computing planetary positions. For most stars these tables were accurate to within one arc minute, and included corrective factors for atmospheric refraction. The tables were sufficiently accurate to predict a transit of Mercury observed by Pierre Gassendi in 1631 and a transit of Venus observed by Jeremiah Horrox in 1639. Rudolphine Tables The Rudolphine Tables () consist of a star catalogue and planetary tables published by Johannes Kepler in 1627, using some observational data collected by Tycho Brahe (1546–1601). The tables are named as "Rudolphine" in memory of Rudolf II, Holy Roman Emperor. Star tables had
Stellar dynamics from the collisionless Boltzmann equation. This was originally developed by Ludwig Boltzmann to describe the non-equilibrium behavior of a thermodynamic system. Similarly to statistical mechanics, stellar dynamics make use of distribution functions that encapsulate the information of a stellar system in a probabilistic manner. The single particle phase-space distribution function, formula_16, is defined in a way such that formula_17 represents the probability of finding a given star with position formula_18 around a differential volume formula_19 and velocity formula_20 around a differential volume formula_21. The distribution is function is normalized such that integrating it over all positions and velocities will equal