Double Stars pp 34-35 | Cite as
Ephemeris Formulae
Chapter
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).
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© D. Reidel Publishing Company, Dordrecht, Holland 1978