Orbital stability

So far we have covered the case of a planet orbiting a single star. However, most stars in the galaxy are not of this type, in fact binary (and multiple) star systems are far more common: Up to 85% of stars may be bound in systems of two or more. Thus, when studying exoplanets, we also should investigate the case of binary systems.

While in a two-body problem the planet can be placed into almost any orbit around the star, this is not so in a three body problem - here many orbits turn out to be unstable and end up with one of the bodies (usually the planet) being ejected into space.

For the development of a life-bearing planet, an orbit should be at least stable for several hundred million years. An analysis for such a long time is way out of the scope of the worldbuilder code, but we can take a look at short-term stability issues.

Types of orbits in binary systems

There are two basic types of planetary orbits in binary star systems. In one (which we'll call the Janus-type) the two stars orbit close-by, and the planet is on an outer orbit around the barycenter of the stars. In the second type (which we'll call the Helliconia-type), the planet orbits around one of the stars, which in turn orbits around the second star.

In the first case, the planet may not come too close to the stars or its orbit will be distorted, in the second case it may not be too far from the single star it is orbiting. Conceptually, this has to do with the so-called Hill sphere, the region of gravitational influence around a body.

An unstable orbit

Let us start with the first case and the following configuration file:

config
orbital_sim_timestep 10.0

input
star
mass 0.96
T_surf 5600.0
name Janus-A

binary
mass 2.0
T_surf 8000.0
name Janus-B

planet
name Janus
mass 1.28
radius 1.0
semimajor_au 1.1
eccentricity 0.01
semimajor_binary_au 0.3
eccentricity_binary 0.01
axis_tilt 35
dec_offset 0.0
rot_period_h 12.0
mean_albedo 0.3
mean_heat_capacity 2.0
mean_depth 0.05
elements_lat 16
elements_lon 24

plot_orbit
file orbit_janus.dat
npoints 10000
timeframe_a 10.0

end

The keyword binary instructs the simulation to add a second star. Its orbit around the first star is defined by semimajor_au_binary and eccentricity_binary (for technical reasons these are defined in the planet block, although strictly speaking they do not refer to the planet). The The optional keyword apsis_angle can be used to rotate the periapsis of the binary system against the periapsis of the planet in the case of non-circular orbits. The optional keyword retrograde (taking the values 'true' and 'false') could be used to reverse the direction of the stellar orbit with respect to the planet.

Finally, the keyword timeframe_a instructs the simulation to run the orbital simulation not for one full orbit only but for 10 years.

Note that we have chosen a rather small timestep of 10 seconds for the orbital simulation. This is due to the fact that two massive stars orbiting close to each other reach high orbital velocities and experience strong accelerations - this necessitates high resolution to avoid too much numerical drift so that we can observe the actual physical instability.

Running the file gets this:

Unstable orbit around a binary star system.

For a time, the orbit meanders close to its original shape as it gets closer to one and then the other star, eventually it gets distorted more and migrates too far in, picks up energy from a star and goes into a higher eccentricity revolution, when it returns passes again close to a star and gets ejected.

Clearly, a planet a mere 1.1 au from the orbiting binaries will be on this position less than 10 years - so we can expect this zone to be completely clear of any orbiting asteroids, debris, rocks or spacecraft.

So - what about other orbits?

Resonances

One might think that going further out is beneficial, but for instance at a semimajor axis of 1.2 au, the situation is such that the period of the planet is pretty exactly eight times the period of the stars - so with good regularity, the apses line up and the planet orbit gets a periodical distortion. This may be a fairly small perturbation for one revolution, but eventually may add up to a catastrophic ejection scenario as well.

In the event, choosing 1.2 au semimajor axis, one can see the planet meander between an outer and an inner radius as it receives regular perturbations from the stars over the course of 50 years:

Unstable orbit around a binary star system.

However, outer and inner ring expand over time (albeit slowly, see lower right), and so the conclusion would be that this orbit too is unstable in the longer run, perhaps a few hundred years.

So, what orbits are stable? Generally the gravitational dipole field of the binary dies faster with distance than the monopole field, so quadrupling the semimajor axis pushes the perturbations down quite a bit more than the field itself. Further, if resonances are avoided, sufficiently far and non-resonant orbits may be stable. However, as has been stated above, while the simulation is fairly good in uncovering obviously unstable situations, it is not suited for real long-term stability analysis of orbits.

Continue with Plantes orbiting binary stars.

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