ner_tags: O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Tag O O O O O O O O O O O O O O O O B-Model I-Model O B-Software I-Software B-Citation I-Citation I-Citation I-Citation O B-Citation O O O O O O O O O O O O O B-Software I-Software B-Citation I-Citation I-Citation I-Citation O O O B-Software I-Software I-Software I-Software I-Software B-Citation I-Citation O O O O O O O O O O O O B-ComputingFacility B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O B-Software I-Software O O O O O O O B-Software O O O O O O O O O O B-Software I-Software O O O O O O B-Software O O B-Software O O O O O O O O O O O O O O B-Software I-Software O O O O O O O O O O O O B-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software O O O O O O O O O O
tokens: from equation and L circ is determined by the radius of the circular orbit r circ , To avoid the singularity at L = L max , it is helpful to introduce a new variable s by α = (1 − s 2 ) 1/2 L max ( r )/ L circ . Equation ( 4 ) then reads: As zero-order approximation, Φ * was set to zero, and an approximation was obtained for ρ( r ). The normalization constant A was recalculated after each iteration to provide the total mass M * for the cluster. The next approximation for the potential Φ * was obtained from a numerical solution of the Poisson equation After obtaining self-consistent profiles ρ( r ) and Φ * ( r ), one can build a random realization of the DF. First, we find L of the particle using the standard von Neumann rejection technique for equation ( 3 ). Energy E = E 0 and angular momentum L determine the orbit. Secondly, we set the radius of the particle, which is uniformly distributed in the radial angle variable, w . The dependence r ( w; E, L ) can be obtained directly by orbit integration, if the number of particles is not too large, or interpolating between reference orbits from a library. The radius r immediately gives the square of radial velocity , and the absolute value of transversal velocity . The velocity components ( v θ , v φ ) are uniformly distributed over a polar angle ψ in the velocity polar coordinates ( v ⊥ , ψ). Finally, the spatial coordinates are obtained as a uniform distribution on a sphere. For our default model with N = 10 5 and the central point mass M = 100, this procedure gives a nearly spherically symmetric N -body realization with centre-of-mass offset 1.5 × 10 −3 . Due to numerical noise, the initial amplitudes of the spherical harmonics A l (see equation 12 ) for 1 ≤ l ≤ 4 are of the order of 10 −3 . 2.3 <italic>N</italic>-body simulations The simulations presented in this work have been carried out using the specially adapted direct N -body code φ- gpu (Berczik et al. 2011 , 2013 ). The basic concepts of the code are based on the publicly available φ- grape (Harfst et al. 2007 ) code, an Aarseth N -body 1-like code (Aarseth 1999 ) including an efficient mpi parallelization and support for the special-purpose hardware GRAPE (Fukushige et al. 1991 ) (currently Graphic Processing Units – GPU). The current version of the φ- gpu code has been fully rewritten in the C++ programming language for the most effective use of GPUs. The φ- gpu code is fully parallelized using the mpi library. The mpi parallelization was done in the same ‘j’ particle parallelization scheme as in the earlier φ- grape code. All particles are divided equally between the working nodes (using the MPI_Bcast() command) and in each node we calculate only the fractional forces for the particles in the current time-step, i.e. the so-called ‘active’ or ‘i’ particles. We get the full forces from all particles acting on the active particles after the global MPI_Allreduce() communication routine is applied. The code supports the use of