Abstract
In a 1778 publication in Latin, titled Nova Methodvs Motvm Planetarvm Determinandi (New method to determine the motion of planets), Euler derives an equation of the center, which, apparently, has been forgotten. In the present work, the developments that led to Euler’s equation of the center are revisited, and applied to three planets of the Solar System. These are then compared with results obtained from an equation of center that has been proposed, showing good agreement for planets with not so high eccentricities. Nonetheless, Euler’s derivation was not influential, and since then, the resulting equation of the center has been neglected by scholars and by specialized publications alike.
Highlights
Since antiquity, the problem of predicting the motions of the heavenly bodies has been simplified by reducing it to one of a single body in orbit about another
In a 1778 publication in Latin, titled Nova Methodvs Motvm Planetarvm Determinandi (New method to determine the motion of planets), Euler derives an equation of the center, which, apparently, has been forgotten
Euler’s derivation was not influential, and since the resulting equation of the center has been neglected by scholars and by specialized publications alike
Summary
The problem of predicting the motions of the heavenly bodies has been simplified by reducing it to one of a single body in orbit about another. In calculating the position of the body around its orbit, it is often convenient to begin by assuming circular motion. This first approximation is a constant angular rate multiplied by an amount of time. The ancient Greeks, in particular Hipparchus, knew the equation of the center as prostaphaeresis, their understanding of the geometry of the planets’ motion was not the same It was specified and used by Kepler, as that variable quantity determined by calculation which must be added or subtracted from the mean motion to obtain the true motion. A common practice is to develop the equation of the center as a series in mean anomaly whose coefficients are given in terms of powers of the eccentricity of the orbit. Modern computer power has replaced the need for such tables and procedures
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