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 [1-5]
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 T0 is the time of pericenter passage.
Direct SolutionBarker's equation may be solved directly by means of the following equations: 
Note that the right-hand side of the first equation is 3/2 the absolute value of the right-hand 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. ).
References S.W. McCuskey. Introduction to Celestial Mechanics. Addison-Wesley, Reading, Mass., 1963.
 K.P. Williams. The Calculation of the Orbits of Asteroids and Comets. Principia Press, Bloomington, Ind., 1934.
 J.C. Watson. Theoretical Astronomy. Lippincott, 1868.
 T.R. Oppolzer. Lehrbuch zur Bahnbestimmung der Kometen und Planeten I. Leipzig, Berlin, 1882.
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Copyright © 2006 David G. Simpson