Smoothed Particle Hydrodynamics Calculations of Stellar Interactions

Smoothed Particle Hydrodynamics is a multidimensional Lagrangian method of numerical hydrodynamics that has been used to tackle a wide variety of problems in astrophysics. Here we develop the basic equations of the SPH scheme, and we discuss some of its numerical properties and limitations. As an illustration of typical astrophysical applications, we discuss recent calculations of stellar interactions, including collisions between main sequence stars and the coalescence of compact binaries. 1 Smoothed Particle Hydrodynamics Smoothed Particle Hydrodynamics (SPH) is a Lagrangian method that was introduced specifically to simulate self-gravitating fluids moving freely in three dimensions. The key idea of SPH is to calculate pressure gradient forces by kernel estimation, directly from the particle positions, rather than by finite differencing on a grid, as in older particle methods such as PIC. SPH was first introduced by Lucy (1977) and Gingold & Monaghan (1977), who used it to study dynamical fission instabilities in rapidly rotating stars. Since then, a wide variety of astrophysical fluid dynamics problems have been tackled using SPH (see Monaghan 1992 for an overview). In addition to the stellar interaction problems described in §2, these have included planet and star formation (Nelson et al. 1998; Burkert et al. 1997), supernova explosions (Herant & Benz 1992; Garcia-Senz et al. 1998), large-scale cosmological structure formation (Katz et al. 1996; Shapiro et al. 1996), and galaxy formation (Katz 1992; Steinmetz 1996). Preprint submitted to Elsevier Preprint 1 February 2008 1.1 SPH from a Variational Principle A straightforward derivation of the basic SPH equations can be obtained from a Lagrangian formulation of hydrodynamics (Gingold & Monaghan 1982). Consider for simplicity the adiabatic evolution of an ideal fluid with equation of state p = Aρ , (1) where p is the pressure, ρ is the density, γ is the adiabatic exponent, and A (assumed here to be constant in space and time) is related to the specific entropy (s ∝ lnA). The Euler equations of motion, d~v dt = ∂~v ∂t + (~v · ∇)~v = − 1 ρ ∇p, (2) can be derived from a variational principle with the Lagrangian L = ∫ { 1 2 v − u[ρ(~r)] }