Ephemeris Formulae

Abstract

For any time t, the coordinates θ, ρ or x, y are computed from the elements by means of the following formulae:

$$\mu \left( {t - {\rm T}} \right) = {\rm M} = {\rm E} - e\sin {\rm E}$$

((10))

$$\tan v/2 = \sqrt {\left( {1 - e} \right)\left( {1 - e} \right)} \tan {\rm E}/2$$

((11))

$$r = a\left( {1 - {e^2}} \right)/\left( {1 - e\cos v} \right)$$

((12))

$$\tan \left( {\theta - \Omega } \right) = \tan \left( {v + \omega } \right)\cos i$$

((13))

$$\rho = r\cos \left( {v + \omega } \right)\sec \left( {\theta - \Omega } \right)$$

((14))

$$\begin{gathered}X = \cos {\rm E} - e \hfill \\Y = \sqrt {1 - {e^2}} \sin {\rm E} \hfill \\\end{gathered} $$

((15))

$$\begin{gathered}x = {\rm A}{\rm X} + FY \hfill \\y = {\rm B}{\rm X} + GY \hfill \\\end{gathered} $$

((16))

The process leads from the mean anomaly M via the auxiliary angle E (the eccentric anomaly) to the polar coordinates (v = true anomaly, r = radius vector) or to the normalized rectangular coordinates X, Y in the true orbit (Figure 8).