d’Alembert-type scheme with a chain regularization for -body problem
We design an accurate orbital integration scheme for the general -body problem preserving all the conserved quantities but the angular momentum. This scheme is based on the chain concept (MA) and is regarded as an extension of a d’Alembert-type scheme (Betsch2005) for constrained Hamiltonian systems. It also coincides with the discrete-time general three-body problem (Minesaki-2013a) for particle number . Although the proposed scheme is only second-order accurate, it can accurately reproduce some periodic orbits, which generic geometric numerical integrators cannot do.
To find periodic orbits in the -body problem (), we use methods (e.g., Baltagiannis-b; Broucke-1969) consisting of two procedures: (1) We introduce a rotating-pulsating frame where a few primaries are fixed. (2) We use the Runge–Kutta–Fehlberg method and set the allowable energy variation and errors of the positions, or the Steffensen method with recurrent power series so that we compute the periodic orbits in one period. However, these methods are not suitable for reproducing the orbits in this problem for a long time interval because of the following two drawbacks: (a) A rotating-pulsating frame, in which the methods described in (2) are applied, has to be altered in accordance with periodic orbits. (b) The methods in (2) cannot accurately compute periodic orbits for a long time interval because they do not preserve any conserved quantities.
On the other hand, for any initial condition, including the conditions of some periodic orbits, numerical integration methods are applied to the -body problem in the barycentric inertial frame. If we use a non-geometric integration method, this method cannot reproduce periodic orbits for a long time interval because of drawback (b). In addition, even if we used each of the geometric integration methods (e.g., the symplectic and energy-momentum methods), they cannot necessarily reproduce periodic orbits. Both the symplectic and energy-momentum methods cannot illustrate elliptic orbits in the two-body problem (Minesaki-2002; Minesaki-2004) and elliptic Lagrange orbits in the three-body problem (Minesaki-2013a). To overcome drawbacks (a) and (b), the author already proposed the discrete-time general three-body problem (d-GBP) (Minesaki-2013a) and the discrete-time restricted three-body problem (d-RBP) (Minesaki-2013c) for the general three-body problem (GBP) and restricted three-body problem (RBP) in the barycentric inertial frame, respectively. These schemes (Minesaki-2013a; Minesaki-2013c) are given by an extension of a d’Alembert-type scheme (Betsch2005). The d-GBP retains all the conserved quantities but the angular momentum, and the d-RBP preserves all the conserved quantities but the Jacobi integration. In this paper, we design an accurate orbital integration scheme like the d-GBP and d-RBP for the general -body problem (GBP). The new scheme is based on a d’Alembert-type scheme (Betsch2005) and a chain regularization (MA). It keeps all the conserved quantities except the angular momentum and can accurately compute some periodic orbits.
This paper is organized as follows. In Section 2, after labeling the masses according to the chain concept (MA) and using the Levi-Civita transformation (Levi-Civita), we express the general -body problem as a constrained Hamiltonian system without Lagrangian multipliers. Further, we rewrite this problem using only the vectors related to the chained ones. In Section 3, we apply the same discrete-time formulation adopted for the GBP in (Minesaki-2013a) to the resulting problem, so we have a discrete-time problem. We prove that the discrete-time problem preserves all the conserved quantities of the GBP except the angular momentum. In Section 4, we check that the discrete-time problem ensures such preservation of the general -body problem numerically. Moreover, we show that it correctly calculates some periodic orbits.
2 Regularization of General -body Problem
For an arbitrary number of masses , we give the transformation formulae, equations of motion for the GBP, and selection of a chain of interparticle vectors such that the close encounters requiring regularization are included in the chain. This formulation includes the same transformation formulae and selection of a chain as in (MA). It has the advantage that its computational cost is far lower than that of Heggie’s global formulation (Heggie) for a large number of masses .
In Section 2.1, we briefly review the GBP in the barycentric frame and how to form a chain of interparticle vectors and label masses using the chain algorithm in (MA). In Section 2.2, using the Levi-Civita transformation (Levi-Civita), we rewrite the GBP, which is similar to the problem given by Heggie’s global regularization (Heggie). For a large number of masses , the rewritten problem involves many redundant variables. In Section 2.3, using some constraints, we express the problem in terms of only the chained position and momentum vectors to reduce the number of redundant variables.
2.1 Labeling Particles Using Chain Concept
The small distance between two bodies experiencing a close encounter is represented as a difference between large numbers in straightforward formulations of the -body problem. Thus, round-off easily becomes a significant source of error. To avoid this, we use the chain concept of (MA) introduced for regularization algorithms.
In this chain method, a chain of interparticle vectors is constructed so that all the particles are included in this chain. Note that small distances are part of the chain. We begin by searching for the shortest distance, which is taken as the first piece of the chain. Next, we find the particle closest to one or the other end of the presently known part of the chain. Then, we add this particle to the end of the chain that is closer. This process is repeated until all the particles are involved. After every integration step, we check whether any non-chained vector is shorter than the smallest of the chained vectors that are in contact with one or the other end of the vector under consideration, namely, if any triangle formed by two consecutive chain vectors has the shortest side non-chained. If this is the case, a new chain is formed. Hereafter, suppose the masses are relabeled , , , along the chain.
We assume that is the position vector of a point with mass in the barycentric frame. We also define as a momentum conjugate to . We set the gravitational constant equal to one for simplicity. In addition, position vectors satisfy the following constraints:
The equations of motion in the barycentric frame are given by the Hamiltonian:
The dynamical system corresponding to this Hamiltonian is
However, for two-body close encounters, we need to simultaneously use two position vectors in the barycentric frame. Therefore, the barycentric frame is not useful for computing close encounters between two masses.
2.2 General -body Problem with Redundant Variables
The GBP in the relative frame is much more symmetric than that in the barycentric frame. It also has a significant advantage in investigating such properties as periodic orbits and close encounters (e.g., (Broucke-Relative; Broucke-Periodic; Explicit-Symplectic)). It can be integrated numerically without catastrophic errors after the Levi-Civita or Kustaanheimo–Stiefel transformation (KS; Levi-Civita; SS).
In Section 2.2.1, we rewrite the GBP in the relative frame. The resulting problem involves very many gravitational force terms for a large number of masses . Thus, we deform this system to reduce the number of force terms. In Section 2.2.2, we rewrite the system using the Levi-Civita variables.
2.2.1 General -body Problem in Relative Frame
We introduce a relative frame to consider two-body close approaches easily. We use the relative position vectors defined by
and the momentum conjugate to as
where is the total mass, , of the GBP. These position and momentum vectors also satisfy the following constraints: