An optimum time-stepping scheme for N-body simulations

We present a new time-stepping criterion for N-body simulations that is based on the true dynamical time of a particle. This allows us to follow the orbits of particles correctly in all environments since it has better adaptivity than previous time-stepping criteria used in N-body simulations. Furthermore, it requires far fewer force evaluations in low density regions of the simulation and has no dependence on artificial parameters such as, for example, the softening length. This can be orders of magnitude faster than conventional ad-hoc methods that employ combinations of acceleration and softening and is ideally suited for hard problems, such as obtaining the correct dynamics in the very central regions of dark matter haloes. We also derive an eccentricity correction for a general leapfrog integration scheme that can follow gravitational scattering events for orbits with eccentricity e -> 1 with high precision. These new approaches allow us to study a range of problems in collisionless and collisional dynamics from few body problems to cosmological structure formation. We present tests of the time-stepping scheme in N-body simulations of 2-body orbits with eccentricity e -> 1 (elliptic and hyperbolic), equilibrium haloes and a hierarchical cosmological structure formation run.Comment: 15 pages, 10 figures, replaced with version that matches published versio

The Formation of the First Stars. I. The Primordial Star Forming Cloud

To constrain the nature of the very first stars, we investigate the collapse and fragmentation of primordial, metal-free gas clouds. We explore the physics of primordial star formation by means of three-dimensional simulations of the dark matter and gas components, using smoothed particle hydrodynamics, under a wide range of initial conditions, including the initial spin, the total mass of the halo, the redshift of virialization, the power spectrum of the DM fluctuations, the presence of HD cooling, and the number of particles employed in the simulation. We find characteristic values for the temperature, T ~ a few 100 K, and the density, n ~ 10^3-10^4 cm^-3, characterising the gas at the end of the initial free-fall phase. These values are rather insensitive to the initial conditions. The corresponding Jeans mass is M_J ~ 10^3 M_sun. The existence of these characteristic values has a robust explanation in the microphysics of H2 cooling, connected to the minimum temperature that can be reached with the H2 coolant, and to the critical density at which the transition takes place betweeb levels being populated according to NLTE, and according to LTE. In all cases, the gas dissipatively settles into an irregular, central configuration which has a filamentary and knotty appearance. The fluid regions with the highest densities are the first to undergo runaway collapse due to gravitational instability, and to form clumps with initial masses ~ 10^3 M_sun, close to the characteristic Jeans scale. These results suggest that the first stars might have been quite massive, possibly even very massive with M_star > 100 M_sun.Comment: Minor revisions. 26 pages, including 24 figures and 5 tables. ApJ, in press. To appear in the Dec. 20, 2001 issue (v563