## Neighbour schemes

Direct N-body simulations on conventional computers benefit greatly from the use of the Ahmad-Cohen 1973 or AC neighbour scheme. Algorithms for both the divided difference method and Hermite formulation will therefore be discussed in the following sections. We also consider the implementation of the code nbody6++ on a popular type of parallel computer Spurzem, Baumgardt & Ibold, 2003 , since it seems that the future of large-N calculations is evolving in this direction at least for those who...

## Individual timesteps

Stellar systems are characterized by a range in density that gives rise to different time-scales for significant changes of the orbital parameters. In order to exploit this feature, and economize on the expensive force calculation, each particle is assigned its own time-step which is related to the orbital time-scale. Thus the aim is to ensure convergence of the force polynomial (2.1) with the minimum number of force evaluations. Since all interactions must be added consistently in a direct...

## Info

Cluster membership as function of time in Myr. Single stars and binaries are plotted separately. that a non-negligible fraction of the escapers have high velocities. Thus if we exclude the neutron star kick velocities, typically 0.4 of the terminal velocities exceed 10kms_1, compared with an initial rms velocity of 2.0kms_1 that declines by a factor of 2 at the half-life stage. Many of these velocities are due to energetic events connected with chain reg-ularization and can therefore...

## T 2 t2R3 T0 t0 R

T03) (3t0'2 R t03)) R4 . (11.5) The division by small values of R is not harmful because it does not affect the integration of the relative motion. Moreover, the c.m. approximation is used if the distance ratio, rj R, exceeds some prescribed value (i.e. the parameter A). Thus the inversion (11.4) is usually called for when the solution of (4.42) is obtained after predicting a nearby c.m. particle in connection with force evaluations. Note that all the quantities on the right-hand side of (11.5)...

## Accuracy and performance

The question of numerical accuracy has a long history and is a difficult one. We are mainly concerned with the practical matter of employing convergent Taylor series in order to obtain statistically viable results. At the simplest level, the basic integration schemes can be tested for the two-body problem, whereas trajectories in larger systems exhibit error growth on short time-scales. However, it is possible to achieve solutions of high accuracy for certain small systems when using...

## Rochelobe mass transfer

The stage of Roche lobe overflow follows naturally from tidal circulariza-tion and is full of astrophysical complications. All the relevant processes have been discussed in considerable detail elsewhere Tout et al., 1997 Hurley, 2000 Hurley, Tout & Pols, 2002 , and we therefore concentrate on some dynamically-related aspects here. The procedure is initiated upon completion of circularization and one of the stars filling its Roche radius, rR, defined by (9.20). We distinguish between the...

## Fewbody simulations

Even the general three-body problem hides an amazing variety of solutions which have inspired much numerical work. The intricate orbits displaying successive close encounters provide a serious test of integration methods. On a historical note, the first such numerical integration was carried out by Stromgren 1900, 1909 for a hierarchical triple over part of an outer orbit, with energy conservation to six figures.* Soon there followed a pioneering investigation by Burrau 1913 of a system with...

## Program organization

We now make an abrupt transition to a presentation of various algorithms utilized by the direct summation codes. Before proceeding further, it will be useful to include some practical aspects in order to have a proper setting for the subsequent more technical procedures. First we introduce the main codes that have been developed for studying different gravitational N-body problems. Where possible, the same data structure has been employed, except that the most recent versions are formulated in...

## Twobody regularization

Sooner or later during the integration of an N-body system close encounters create configurations that lead to difficulties or at best become very time-consuming if studied by direct methods. On further investigation one usually finds a binary of short period slowing down the calculation and introducing unacceptable systematic errors. Moreover, the eccentricity may attain a large value that necessitates small time-steps in the pericentre region unless special features are introduced. It can be...

## Ahmad Cohen method

The main idea of the AC scheme is to reduce the effort of evaluating the force contribution from distant particles by combining two polynomials based on separate time-scales. Splitting the total force on a given particle into an irregular and a regular component by we can replace the full N summation in (1.1) by a sum over the n nearest particles, together with a prediction of the distant contribution. Likewise, (3.1) may be differentiated once for the purpose of low-order prediction. This...

## CCTit 2 [4G2M2AA Bi3 817

Although this energy limit is less stringent compared with an isolated system, a star satisfying this condition may still take a long time to reach one of the Lagrange points, Li or L2, and some excess is usually required to ensure escape. We shall see later that there exist some orbits that return to the cluster after moving well outside the tidal radius Ross, Mennim & Heggie, 1997 . The tidal parameters are scaled to N-body units in the following way. First we choose the Oort's constants A...