The star 55 Cancri hosts a system of five known planets of diverse nature, ranging from the small 55 Cancri e close to the star (3% of Jupiter’s mass, around 10 times Earth’s, at 0.016 AU) to the massive and distant 55 Cancri d (4 times Jupiter’s mass at 5.7 AU). When many planets are detected in a system, it is important to verify that the planets are in a stable configuration that will not be disrupted on relatively short time-scales. While this is true for 55 Cancri there is a hitherto overlooked complication in this system: the existence of a binary companion to the primary star. While distant, the companion could still potentially disrupt the planetary system. A recent paper by Kaib, Raymond & Duncan attempts to determine whether the planetary system is stable to perturbations from this companion.
The particular dynamical effect that could destabilise the planets is known as the Lidov-Kozai effect, after its discoverers. In the simplest case, this occurs in systems comprising a star, a companion such as a planet or binary star, and a massless test particle such as a comet or asteroid, when the ratio of semi-major axes is very large, and the mutual inclination of their orbits exceeds a certain critical value. If the conditions are met, the test particle experiences very large oscillations in eccentricity and inclination. There is a constant of motion, , where is the inclination and is the eccentricity, so the eccentricity and inclination oscillations are in phase. The time-scale for them to occur is approximately , where is the orbital period of the planet, and are the masses of the primary star and binary companion, and and are the semi-major axes of the planet and binary companion.
In the 55 Cancri system, there is a massive binary companion star, and the planets, being much less massive, would effectively behave as test particles. Since there is no reason to assume that the companion’s orbit is in the same plane as the planets’, their mutual inclination could be large, and the planets would then undergo Lidov-Kozai cycles. Such cycles would be devastating to the 55 Cancri planetary system, since the planetary eccentricities would increase to very large values, their orbits would intersect, and the planets could collide or scatter each other onto very different orbits. In contrast, in the system as currently observed, the planets’ eccentricities are all fairly low, less than 0.1 , and the system has remained stable for around 10 billion years. This suggests that the Lidov-Kozai effect is not at work in this system.
The authors investigate why this is so, integrating the equations of motion for the planets and the companion star interacting gravitationally. Since the companion star’s orbit is unknown (only the distance from the star projected onto the plane of the sky can be measured), they integrate many different systems with different orbits of the binary companion. They also reduce the five planet system to a four or two planet system in order to follow the evolution for longer, since the time that the equations can be integrated for is limited by finite computing power. The qualitative behaviour is, however, the same in all cases.
They find that the system is stable in the majority of runs (84%), and the planets do not undergo the large eccentricity oscillations associated with the Lidov-Kozai effect. However, there are still large inclination oscillations, as shown below:
The planets’ inclinations vary enormously, but all change in phase: effectively, the system behaves as a rigid body, locked together. The inclined binary companion excites inclination oscillations in the planets. If each planet were experiencing Lidov-Kozai cycles alone, the periods of the oscillations would all be different, since the period depends on the planet’s semi-major axis (see the equation above). Here, however, the planets are sufficiently close that they transfer their change in inclination to each other on much shorter time-scales, and thus all behave the same way under the companion’s perturbations. The eccentricity oscillations are suppressed because the Lidov-Kozai effect depends on a sensitive resonance between the precession rates of the orbital plane and the pericentre, which is broken by the additional perturbations from the other planets. For a detailed analysis of this stabilisation mechanism, see this paper.
While the inclination shown in the above figure reaches a maximum of around 120 degrees, this is dependent on the unknown orbital parameters of the companion star. The authors therefore integrate 500 systems with different binary orbits, to determine statistically the distribution of planetary inclinations that would result. The results are shown below:
This paper therefore makes a startling prediction: the planetary system is overwhelmingly likely to be misaligned with its host star. This is in contrast to our own Solar System, where all the planets’ orbits are within a few degrees of the Sun’s equator. For 55 Cancri, the median value is around 60 degrees, and the planets can often be driven onto retrograde orbits, going backwards relative to their star, as shown above.
Since planet e transits the star, it is possible to determine the inclination observationally, by means of the Rossiter-Mclaughlin effect. Thus, the authors’ prediction that the system is misaligned with the host star will be able to be tested in the near future.