DAVID G. SIMPSON
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BARKER'S EQUATION
Barker'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.
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 Solution
Barker's equation may be solved directly by means of the following equations: [1]
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. [6]).
References
[1] S.W. McCuskey. Introduction to Celestial Mechanics. Addison-Wesley, 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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