Theoretical Orbits of Planets in Binary Star Systems
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In table 1 the gravitational field of a single star is compared with the external gravitational field of a simple binary system with the same total mass. The distance between the two binary stars = 1 unit, so each star is 0.5 unit from the system centre. For clarity we use a unit of acceleration such that at distance = 8 the acceleration exerted by the single star = 1 unit.
| distance from system centre | single star field strength | binary field strength (average) | binary field strength (minimum) | binary field strength (maximum) |
| 8 | 1.000 | 1.003 | 0.944 | 1.012 |
| 7 | 1.306 | 1.311 | 1.296 | 1.326 |
| 6 | 1.777 | 1.787 | 1.759 | 1.815 |
| 5 | 2.560 | 2.579 | 2.522 | 2.638 |
| 4 | 4.000 | 4.048 | 3.910 | 4.192 |
| 3 | 7.111 | 7.263 | 6.825 | 7.732 |
| 2 | 16.00 | 16.80 | 14.61 | 19.34 |
| 1 | 64.00 | 79.72 | 45.79 | 142.2 |
The field strength of the single star at any given point is constant. However, the field strength of the binary system at any given point fluctuates cyclically over time. The period of the fluctuation in a simple binary (as defined above) = 0.5 (i.e. half the period of the star system). The magnitude of the fluctuation is relatively very small at distance = 8, but increases as we get closer to the star system. At distance = 1 the fluctuation is very large indeed.
At distance = 8, the average field strength of the binary system (averaged over a complete period of the binary system) is similar to the field strength of the single star. However, as we get closer to the stars, the binary average field strength increases more rapidly than the single star field strength. The gravitational field of the single star obeys the inverse square law. The average gravitational field of the binary system is the combination of the individual inverse-square fields of the two moving stars, but it is important to note that this combined field does not follow the inverse-square rule.
In this diagram a planet has an external orbit at distance d = 10 from the star system centre. At this relatively large distance, the combined gravitational field of stars A and B is similar to the field of a single star with their combined mass. Therefore the planet's orbit is very similar to an ellipse. (In this case we have shown an orbit with eccentricity = 0, so the orbit is circular). At large distances like this, retrograde and prograde orbits are both viable. We've shown a prograde orbit, The planet's orbital period p = 31.6.
In the next diagram, a planet has a retrograde orbit around the binary star system at distance d = 2. The fluctuating gravitational field causes deviations from a circular path but the orbit is perfectly stable. The period p = 2.77
Surprisingly, a prograde orbit at the same distance d = 2 would be unstable. Why is the retrograde orbit stable and the otherwise identical prograde orbit unstable?
It is because the gravitational field strength fluctuations experienced by the prograde planet have the same amplitude but a longer period than the fluctuations experienced by the retrograde planet. The longer period of fluctuation has a destabilizing effect.
In fact, at distances of less than d = 2, prograde orbits become even more unstable. The following examples of external orbits are possible only in a retrograde direction.
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Copyright 2001 S.Edgeworth