# Ephemeris Formulae

For any time t, the coordinates 6, p or x, y are computed from the elements by means of the following formulae. The auxiliary circle has radius a. See Figure 7.7,

 / / X JiT i F - "x 1 X V Figure 7.7. Auxiliary circle, eccentric anomaly E, true anomaly v and radius vector r. The angle E is called the eccentric anomaly and has to be determined from the mean anomaly M: This equation is transcendental, i.e. it is not algebraic and has to be solved iteratively. A first approximation is given by the formula: This new E0 is used to calculate a new M0: M0 = E0 - e sin E0 A new E1 is obtained from M, M0 and E0: The last two formulae are iterated to the desired accuracy. Four iterations are sufficient for e < 0.95. Now the desired positions are calculated: Polar coordinates: 1 + e cos v tan (0 - Q) = tan (v + to) cos i p = r cos (v + t) sec (0 - Q). An alternative formula for the calculation of p, due to Michael Greaney,2 obviates the possibility of the formula becoming undefined, e.g. when 0 - Q = 90°: Rectangular coordinates: X = cos E - e; Y = V 1 - e 2sin E x = AX + FY; y = BX + GY. References 1 Heintz, W.D., 1990, Astron. Astrophys. Suppl., 82, 65. 2 Greaney, M.P., 1997, Calculating separation from binary orbits: an alternative expression, Webb Society Quarterly Journal, 107. This page intentionally left blank Chapter 8 