A&A 400, 1129–1144 (2003) Astronomy DOI: 10.1051/0004-6361:20030065 & c ESO 2003 Astrophysics Symplectic integration of hierarchical stellar systems H. Beust Laboratoire d’Astrophysique de Grenoble, Universit´e J. Fourier, BP 53, 38041 Grenoble Cedex 9, France Received 30 September 2002 / Accepted 19 December 2002 Abstract. Symplectic integration has been used successfully for many years now for the study of dynamics in planetary systems. This technique takes advantage of the fact that in a planetary system, the mass of the central body is much larger than all the other ones; it fails if all massive bodies have comparable masses, such as in multiple stellar systems. A new symplectic integrator is presented that permits the study of the dynamics of hierarchical stellar systems of any size and shape, provided that the hierarchical structure of the system is preserved along the integration. Various application tests of this new integrator are given, such as the gap formation in circumbinary disks, the Kozai resonance in triple systems, the truncated circumbinary disk in the quadruple system GG Tauri, and the dynamics of the sextuple system Castor. Key words. methods: N-body simulations – methods: numerical – celestial mechanics – stars: binaries: general – stars: circumstellar matter 1. Introduction of the integration, although from a statistical point of view, the results are still valid (Levison & Duncan 1994). Chambers Symplectic integrators have become very popular in the last (1999), in his MERCURY integrator, incorporates the close en- decade in planetary dynamics problems, as they present two counter perturbation into the Keplerian term during close en- major advantages with respect to other n-body integrators: counters, but with the consequence of a significant decrease First, they exhibit no long-term accumulation of energy error, of the integration speed. Finally, Duncan et al. (1998), in their and second they provide a gain of at least one order of mag- SyMBA integrator, use a multiple time-step technique that han- nitude in computation speed, for equivalent accuracy, because dles close encounters while remaining symplectic. However, they allow one to adopt a much larger time-step than other inte- it fails in integrating close encounters with the central star. grators for the same result, leading to a much more rapid inte- A variant to this method is presented in Levison & Duncan gration. The most popular symplectic integrator is the second- (2000) that overcomes this difficulty but is more CPU-time order MVS (Mixed Variable Symplectic) one, first described consuming. by Wisdom & Holman (1991) and implemented by Levison & In planetary dynamics, the motion of the bodies is nearly Duncan (1994) in their SWIFT MVS integration package. Keplerian far from close encounters because the mass of the The general assumption for efficient symplectic integrators central star largely overcomes that of the other bodies. This is that the the motion of each body is separated into a dominant is no longer valid in multiple stellar systems, where several Keplerian part, solved to machine precision with an appropri- massive centers can be present. Multiple stellar systems can be ate algorithm, and a perturbation that is solved numerically. In classified into two sets: Trapezian and Hierarchical systems. In planetary dynamics, the dominant Keplerian part is due to the trapezian systems, all relative distances between the bodies are central star, and the perturbations to the planets. The MVS inte- more or less comparable. These systems are usually unstable grator is based on this assumption. However, the integration is on time-scales of a few Myrs or less, and it is very difficult in no longer valid whenever the perturbation becomes larger than, general to identify a dominant Keplerian part in the relative mo- or even of comparable order to the dominant Keplerian part. In tion of their components. Hierarchical systems are conversely planetary dynamics, this occurs when some of the bodies get characterized by nested orbits of very different sizes, and are close to each other, i.e., in the so-called case of close encoun- usually more stable. In that case, the motion of the bodies re- ters. A large effort was done in the past years to make sym- mains nearly Keplerian thanks to the very different size of the plectic integrators handle close encounters (Levison & Duncan nested orbits, although all massive bodies may have compara- 1994; Chambers 1999; Duncan et al. 1998): The basic idea is ble masses. to reduce the integration time-step during close encounters, but However, unless in specific cases, the MVS technique can- this necessarily no longer preserves the symplectic character not be used directly to compute the dynamics of hierarchical multiple stellar system, as it needs to hold one of the bodies e-mail: [email protected] as the “Sun”, and the other ones as the perturbing “planets”. Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20030065 1130 H. Beust: Symplectic integration of hierarchical stellar system An interesting new technique dedicated to the study of planets based on this scheme, but the way we split H into HA + HB in binary systems was recently presented by Chambers et al. will be different.Higherorder(fourth,sixth...)schemescanbe (2002), with application to the α Centauri system (Quintana build by combining more sub-steps (Yoshida 1990; Chambers et al. 2002). This integrator is however limited to binary sys- & Murison 2000), but they are less easy to implement, so that tems only. The purpose of this paper is to present a new MVS- the second order method has become by far the most commonly like integrator dedicated to the integration of the dynamics of used. As a matter of fact, it provides accurate results for most hierarchical systems of any size and structure, provided the hi- problems in planetary dynamics. For comparison, we never- erarchy is preserved. We give this new method the name HJS theless implemented for the HJS method two distinct integra- (Hierarchical Jacobi Symplectic). In Sect. 2, we briefly review tors: a second order one called HJS2 following the 3 sub-steps the theory of symplectic integrators and the MVS method, and scheme given above, and a fourth order one using 7 sub-steps develop the theory for the HJS method. Various integration (quoted as S4B by Chambers & Murison 2000), that we call tests of growing complexity are given in Sect. 3. Our conclu- HJS4. Whenever not specified, we use in the following the sec- sions are presented in Sect. 4. ond order method. Additional accuracy is reached in symplectic integration if we have now in addition H H ,i.e.,H = H with 1. 2. Theory of the Hierarchical Jacobi symplectic B A B A In that case we have integrator H˜ = H + O τ2 . (2) 2.1. Symplectic integration In practice, this last property allows us to integrate with much An integration of any Hamiltonian system is symplectic when longer time-steps than with conventional methods. HB HA it exactly preserves the generalized areas in phase-space. This is then a important condition to ensure for the validity of the is in particular the case if the integration exactly solves (within integration. In classical celestial mechanics problems, HA will computer round-off errors) a Hamiltonian. Symplectic integra- by the dominant Keplerian interaction and HB will be the per- tors do not actually strictly solve the real Hamiltonian of the turbation. problem, but do it for a surrogate Hamiltonian assumed to be close to the real one. Once this is achieved it can be shown that there is no long-term drift of the energy. This is not the case 2.2. The MVS method (Wisdom – Holman mapping) for more classical methods (Runge-Kutta, Burlish & Stoer...) Let us consider a general n-body gravitational system, with used in celestial mechanics, that are far from being symplectic. bodies labeled from 1 to n, masses m1,...,mn, positions The theory of symplectic integration is based on the x1,...,xn and impulsions p1,...,pn in the barycentric refer- Hamilton equations of motion that apply to any n-body prob- ential frame. Then the Hamiltonian of the system reads lem. Its background is for example described in Saha & n 2 Tremaine (1992) and Chambers (1999). The key idea of sym- p Gmim j H = i − , (3) plectic integration is to split the Hamiltonian H the problem 2m r i=1 i 1≤i< j≤n ij into pieces, say where G is the constant of gravitation and rij = ||xj − xi|| is the H = HA + HB, (1) distances between bodies i and j. Wisdom & Holman (1991) introduce the classical Jacobi coordinates x (i = 1 ...n), de- each of them being exactly integrable within computer round- i fined as follows: off errors. If this is possible, then a first order symplectic in- tegrator can be build if, starting from the generalized positions x 1 = rn;(4) t − τ t and momenta at time , we derive their values at time by x i = xi − ri−1 for 2 ≤ i ≤ n, (5) τ 1. advancing HB by a time step ; where ri is the location of the center or mass of bodies up to i. 2. advancing HA by the same amount. Wisdom & Holman (1991) rewrite the Hamiltonian (3) as 2 n 2 n Of course, we get another integrator if we first advance HA p1 pi Gmim1 Gmim j H = + − − , (6) and then HB. The integration is of first order because it exactly 2m 2m r r 1 i=2 i i=2 i1 2≤i< j ij solves a surrogate Hamiltonian H˜ that is close to H within O(τ).