Theoretical detection of triple star systems with figure-eight orbits

In recent years, researchers have discovered that various exotic orbits are theoretically possible for multiple star systems. The majority of these exotic orbits are unstable and unlikely to exist in the real universe.

However, one of these exotic orbits stands out from the others. It is the figure-eight orbit for three bodies. Amazingly this orbit has been proven to be stable (Simó). Therefore it is possible that real examples of figure-eight triple star systems may actually exist.

In the figure-eight orbit the three stars are of identical (or near-identical) mass, and all three travel along the same figure-eight path.

Now we examine how we might detect such a star system. If such a star system happened to be aligned so that its orbital plane intersected Earth, it might be detectable by us as an eclipsing variable. An eclipsing variable appears to us as a single point of light, whose brightness varies periodically. The variations in brightness occur whenever one of the stars passes in front of another of the stars, and blocks some or all of its light. A graph of the brightness plotted over time is called the light curve. What would the light-curve of a figure-of-eight trinary star system look like?

First we need to determine the sequence of eclipses. Let's assume that the figure-of-eight orbit is aligned with its long axis aligned at 90 degrees to a line joining the system to Earth. Diagrams 1 to 6 show that during each complete orbit of the trinary star system there would be 6 eclipses .....

The direction towards our solar sytem is marked by an arrow labelled "Sol". The other arrow shows the direction of motion of the stars around the orbit.The eclipse sequence is .....

A eclipses B ..... B eclipses C ..... C eclipses A ..... A eclipses B ..... B eclipses C ..... C eclipses A

This eclipse sequence is totally different from that which would be produced by an eclipsing trinary star system with conventional elliptical orbits. It consists of three eclipses, each of which occurs twice per orbit. Remarkably the other 3 eclipses, which we would expect from a more conventional system, never occur here. B never eclipses A, C never eclipses B, and A never eclipses C.

If the long axis of the star system is aligned exactly perpendicular to a line joining the system to Earth, then the six eclipses will occur at evenly spaced intervals (60 degrees phase). Also, eclipse 4 will be of exactly the same magnitude as eclipse 1, eclipse 5 will be of exactly the same magnitude as eclipse 2, and eclipse 6 will be of exactly the same magnitude as eclipse 3. So the light curve would consist of three minima in the first semi-cycle, followed by an identical set of three minima in the second semi-cycle.

If the three stars differ in luminosity, diameter, or other characteristics, then the three minima will be of three different magnitudes and widths.

In next diagram, the upper line shows an example of the light curve which this eclipse sequence might produce. The vertical axis repesents visual magnitude (observed brightness) of the star system, and the horizontal axis represents time. The timing of the six eclipses was computed by orbit simulation software. We assume that the 3 stars have equal sizes, but have different brightnesses. Therefore the minima are depicted with equal widths but different depths. The size and brightnesses of the stars have been arbitrarily chosen for this example. Therefore a real system might have minima with different widths and depths than depicted here.

The lower line in the diagram is the radial velocity curve of the brightest star. The vertical axis represents velocity away from us (up) or towards us (down), and the horizontal axis represents time. Velocities were computed by orbit simulation software. (The radial velocity curves for the other 2 stars would be identical but offset in time by by one-third of the period and by two-thirds of the period respectively).

The light curve has 6 minima per period. The radial velocity curve has 2 maxima per period. The shapes of both curves in the first semi-period are identical to their shapes in the second semi-period.

The next diagram shows the light curve and radial velocity curve for the same star system when viewed from the exactly opposite direction.

We have examined only the two scenarios where the star system's long axis is aligned at an angle of exactly 90 degrees to a line joining the system to Earth. Alignment at various other angles between 90 degrees and approximately 45 degrees produces light curves and radial velocity curves which are more complex. In particular, the light curve during the second semi-period is now similar but slightly different to the light curve in the first semi-period (the timing and depths of the minma are no longer identical). Also the radial velocity curve now exhibits a pronouncedly different shape in the two semi-periods. Alignment at angles between approximately 45 degrees and zero degrees produces even more complex results, some with 12 eclipses per period.

Note that it might be possible to detect a non-eclipsing figure-eight triple star system by its radial velocity curve alone, even if it is not eclipsing.

Here is one example of a real star system which has some peculiarities which indicate it might possibly merit further investigation.

W Serpentis is an eclipsing variable star system. It is generally considered to contain two stars. It has a highly unusual light curve consisting of a pattern of three minima. The pattern recurs every 14.1536 days. The light curve has puzzled investigators since its discovery. The pattern of three minima can not be explained in full by the eclipse cycle of any conventional-orbit binary (or multiple) star system. Therefore it is usually considered that only the deepest minimum is caused by eclipse. and that the other two minima are caused by some combination of non-eclipse causes, such as intrinsic variability of the stars, tidal distortion, and extrastellar material.

The next diagram shows the approximate light-curve of W Serpentis (based on diagram by Glasby) .....

The period of W Serpentis generally considered to be approximately 14.15 days, with three minima per period.

Some researchers (Zessewitsch, Fresa, McLaughlin), however, noticed variations in the shape of the light curve, with similarities between alternate durations of 14.15 days, and therefore proposed that the period might actually be double that figure, approximately 28.3 days, with 6 minima per period.

The radial velocity curve of W Serpentis has an interval of 14.15 days between maxima. In a binary star system, this interval is by definition the same as the orbital period. So it is understandable, given the natural assumption that this is an eclipsing binary, that the longer period is no longer favoured. However, in a figure-eight orbit, an interval of 14.15 days between radil velocity maxima would be consistent with a 28.3 day period.

Now let's look at some of the questions remaining to be solved for even a tentative figure-eight solution for W Serpentis.

The widths of the light curve minima show that the stars of W Serpentis are very close together (relative to their diameters). At this relatively close proximity there would be tidal interaction and distortion. Investigation is required into the stability of the figure-eight orbit in these circumstances.

The unusual shape and asymmetry of the W Serpentis radial velocity curve remains to be explained. The published analyses of radial velocity data have all been based on the assumption that the orbital period is 14.15 days. A re-analysis of the radial velocity data, examined over an orbital period of approx 28.3 days, may give a clearer picture.

So the peculiar characteristics of W Serpentis suggest that it might be an interesting candidate for further investigation as a possible figure-eight triple star system. But some important problems would need to be solved before this idea could seriously compete with the currently accepted solution that W Serpentis is an eclipsing binary.

The discovery of the amazing orbits of Janus and Epimetheus, and of Cruithne, has shown that there is no "elliptical orbits only" rule in this universe. A brief search through the vast quantity of variable star data available indicates that there are many star systems with peculiar light curves and radial velocity curves. It is probable that somewhere amongst them is a figure-eight orbit.

P.B.Etzel and H.van den Bergh, http://www.aas.org/publications/baas/v30n2/aas192/abs/S012008.html

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