The Methods by Danjon and Rabe
The eccentricity e affects not only the position of the primary star in the orbit but also the rate of change of angular velocity over the revolution. Thus, e can in principle be determined from the shape of the curve θ(t) without using separations. Consider now two opposite points of the orbit. (θ2 – θ 1 = 180°) Then v2 – v1 — 180° also holds, as the projection effects are the same, but M2 — M1 does not equal 180° except if e = 0, or if the two points are on the major axis. Graph the function M2 - M1 - 180° for a number of pairs of opposite points over the orbit; then its zero points determine periastron and apastron, and the extremum values depend only on e = sin φ. In fact, they equal 2φ + sin 2φ The remaining elements are readily found, as corresponding pairs (θ, v) are now known.
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