DAVID G. SIMPSON 


BARKER'S EQUATIONBarker's equation in celestial mechanics is an equation giving the true anomaly f for a body in a parabolic orbit at any time t. It is essentially the parabolic counterpart of Kepler's equation. Barker's equation is ^{[15]}Here f is the true anomaly, G is the Newtonian gravitational constant, M is the mass of the central body, q is the pericenter distance, t is time, and T_{0} is the time of pericenter passage. Barker's equation is a cubic equation in the true anomaly f. It may be solved iteratively by standard numerical methods, by a direct method (described below), or by means of Barker's tables. Direct SolutionBarker's equation may be solved directly by means of the following equations: ^{[1]}Note that the righthand side of the first equation is 3/2 the absolute value of the righthand side of Barker's equation. In the last equation, sgn(x) is the signum function, which is +1 for positive x, 1 for negative x, and 0 if x=0. A software implementation of this direct solution is given by Simpson (Ref. [6]). References^{[1]} S.W. McCuskey. Introduction to Celestial Mechanics. AddisonWesley, Reading, Mass., 1963.^{[2]} K.P. Williams. The Calculation of the Orbits of Asteroids and Comets. Principia Press, Bloomington, Ind., 1934. ^{[3]} J.C. Watson. Theoretical Astronomy. Lippincott, 1868. ^{[4]} T.R. Oppolzer. Lehrbuch zur Bahnbestimmung der Kometen und Planeten I. Leipzig, Berlin, 1882. ^{[5]} http://www.davidgsimpson.com/ref/barkers.html ^{[6]} http://www.davidgsimpson.com/software/barkersoln_f90.txt
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