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,

auxiliary circle

/ /

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.

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Chapter 8

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