Computed radial velocity curves for the figure-eight orbit

Here are some computed radial velocity curves for the figure-eight orbit.

The shape of the curve is determined by angle at which the star system's orbit is oriented relative to us.

The first curve represents an orientation of the viewing point (Sol) relative to the star system exactly as shown in the orbit diagrams on the previous page. The second curve represents an orientation with the viewing point (Sol) rotated 22.5 degrees anti-clockwise in those orbit diagrams. Subsequent curves represent rotations of the viewing point by further increments of 22.5 degrees anti-clockwise. So the curves shown represent the star system viewed from the following angles in degrees...

0 22.5 45 67.5 90 112.5 135 157.5 180

Horizontal axis is time. Each curve shows approx 1.5 orbital periods. Note that there are 2 (or, for some curves, 3) maxima per orbital period. Vertical axis is radial velocity. Up represents motion away from us, and down represents motion towards us. Horizontal line represents zero radial velocity.

Complications such as distortion, reflection, eclipse, extrastellar matter, etc, are not shown. Each curve represents the curve of just one of the three stars. The curves for the other two stars are identical but shifted in time by one-third and two-thirds of the orbital period respectively.

Units have deliberately omitted so that the results are scalable to any perfect figure-eight orbit. Orbits on an orbital plane which is inclined relative to the viewing point would yield the same shape curves (but with amplitude reducing to zero if the relative inclination is perpendicular).

The remaining 7 curves in this series are not shown but can be obtained as follows... To get the 202.5 degree curve, invert the vertical axis of the 22.5 degree curve, To get the 225 degree curve, invert the vertical axis of the 45 degree curve, and so on.

The above curves represent the perfect figure-eight orbit. There are also some interesting variations of the orbit in which the starting parameters computed by Simó are amended as follows... the initially central star is given either a small Z position value or a small Z velocity. Some of these variations would give a a radial velocity curve which gradually changed over time. In particular, in one set of these variations, the orientation of the orbital shape oscillates back and forth through an angle of approx 90 degrees. This would give a radial velocity curve which gradually changed over time, oscillating back and forth through an approx 90 degree range of the above sample curves.

References...

http://burtleburtle.net/bob/physics/eight.html Excellent animations of some fascinating variations of the figure-eight orbit.

http:/orca.ucsc.edu/~rmont/annals.pdf A.Chenciner and R.Montgomery, mathematical description of the figure-eight orbit.

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