ABSTRACT
The long-term orbital stability of small bodies near the central binary of the Alpha Centauri system. Test particles on circular orbits are integrated in the field of this binary for 32,000 binary periods or approximately 2.5 Myr. In the region exterior to the binary, particles with semi-major axes less than roughly three times the binary’s semi-major axis a are unstable. Inside the binary, particles are unstable if further than 0.2 ab from the primary, with stability closer in a strong function of inclination: orbits inclined near 90◦ are unstable in as close as 0.01 ab from either star.
Introduction
Though the formation of multiple star systems is possibly quite different from that of single stars like our Sun, it is plausible that multiple stars also host planetary systems. If this is the case, then the high frequency of binary and multiple systems implies that such planetary systems have been created in large numbers in our Galaxy. However, the question of whether planets might persist for long periods within such a system remains unanswered. Alpha Centauri, a triple system with two of the stars forming a close binary (semi-major axis 23 AU) and a third orbiting this pair at a much greater distance (12,000 AU), is extraordinary only in its proximity to the Sun (1.3 pc). For this reason, it is a prime place to prospect for planets and a logical starting point for our theoretical investigations of the stability of planetary orbits in multiple systems.
METHOD AND MODELS
We adopt a simple, empirical, observationally motivated criterion for stability. The term “stable” will be applied to test particles whose time-averaged semi-major axis does not vary from its initial value by more than 5% over the whole integration, the remainder being termed “unstable”. Thus, our definition of stability excludes planets that remain bound to the binary but migrate to larger or smaller orbits, encompassing only such planets as remain near their initial orbits. We also compute Lyapunov exponents, which measure the rate of exponential divergence of nearby orbits and are correlated with stability lifetimes.
The numerical integrations in this paper used the symplectic mapping for the N-body problem described by Wisdom and Holman (1991). This technique is typically an order of magnitude faster than conventional integration methods and has the additional advantage of showing no spurious dissipation other than that introduced by roundoff error. Lyapunov times were computed by evolving a tangent vector associated with each test particle during the orbit calculations (Mikkola and Innanen 1994). This procedure has two advantages over the common approach of measuring the Lyapunov time by evolving two nearby trajectories. First, using the tangent vector avoids the saturation and renormalization problems that accompany the two-trajectory technique. Second, the variational method is faster because the most expensive calculations required (the distances between the test particle and planets) do not need to be computed twice.
We approach the problem with a simple model which captures the overall dynamics. We ignore the distant third star, α Cen C (Proxima), as it appears likely that it is not bound to the central binary (Anosova et al. 1994), and because the perturbations it could inflict were it bound are extremely small. The orbit of the central pair is thus taken to be a fixed Kepler ellipse. The semimajor axis of the central binary ab is 23.4 AU, and its eccentricity is 0.52 and the inclination of its orbit to the plane of the sky is 79◦ (Heintz 1978). The primary, α Cen A, has a mass of 1.1 M⊙; the secondary, α Cen B, has a mass of 0.91 M⊙ (Kamper and Wesselink 1978). Their physical properties are outlined in Table 1. In the field of this binary we integrate a battery of massless test particles representing low-mass planets. As these particles do not interact with one another, this paper does not address the stability of multiple planet systems.
INITIAL CONDITIONS
Test particles are initially placed in circular orbits in two separate regions. The interior region is centered on the primary, and extends from 0.01 to 0.5 times the binary semi-major axis ab (0.23 to 11.7 AU). The exterior region is centered on the barycenter, and spans 1.5ab to 5ab (35 to 117 AU). Note that the mass fraction in the secondary (0.45) exceeds the maximum value (∼ 0.005 for a binary eccentricity of 0.52, Danby 1964) at which the L4 and L5 Lagrange points are stable, so no particles are expected to survive there. No separate study of the dynamics of orbits centered on α Cen B was performed due to the similarity of the masses of the primary and secondary. Such a study is expected to produce results qualitatively very similar to those obtained for orbits centered on α Cen A.
Thirteen different inclination values were examined, ranging from 0◦ to 180◦ in 15◦ increments. All particles were started on circular orbits, relative to the primary in the inner shell, and relative to the barycenter in the outer one.
SIMULATIONS
The inner region proves to be largely unstable over the integration time scale (Figure 1). Each cell of Figure 1 represents one of the test particles’ initial conditions. A white cell indicates a particle that was ejected or had a close encounter (defined to be a passage within 0.25 ab) with the secondary. Two other colours indicate those particles that survived for the entire simulation: those whose time-averaged semi-major axis deviated from its initial value by less than 5% are indicated in black, those which deviated by more than 5% are shown in grey.
Conclusions
As per this paper reveals that much of the region around the central α Cen binary is unstable. However, there are zones in which planets on circular orbits could be stable in the α Cen system on million-year time scales. These zones are located both far from (a ∼ > 70 AU) and near to (a ∼ < 3 AU) the primary. Stability is a strong function of the inclination for interior orbits, less so for exterior orbits. The inner stable region encompasses Hart’s (1979) habitable zone, however a planet orbiting in the more distant stable region would be inhospitable to life
References
The stability of planets in the Alpha Centauri system Paul A. Wiegert1, Matt Holman2 1 Department of Astronomy, University of Toronto, Toronto, Canada 2 Canadian Institute for Theoretical Astrophysics, Toronto, Canada
https://arxiv.org/pdf/astro-ph/9609106