Four classical methods for determining planetary elliptic elements: A comparison

The discovery of the asteroid Ceres by Piazzi in 1801 motivated the development of a mathematical technique proposed by Gauss, (Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections, 1963) which allows to recover the orbit of a celestial body starting from a minimum of three observations. Here we compare the method proposed by Gauss (Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections, New York, 1963) with the techniques (based on three observations) developed by Laplace (Collected Works 10, 93-146, 1780) and by Mossotti (Memoria Postuma, 1866). We also consider another method developed by Mossotti (Nuova analisi del problema di determinate le orbite dei corpi celesti, 1816-1818), based on four observations. We provide a theoretical and numerical comparison among the different procedures. As an application, we consider the computation of the orbit of the asteroid Juno.