The Jeans instability in smoothed particle hydrodynamics

We derive analytically the Jeans criterion for a gas simulated using an SPH code in which the number of neighbours Nneighb is held constant (approximately) and the gravity-softening length, ε, equals the smoothing length, h (approximately). We show that the Jeans criterion is reproduced accurately for resolved structures, i.e. those represented by > Nneighb particles. Unresolved structures are stabilized, as long as (i) the smoothing kernel W(u) is sufficiently centrally peaked, and (ii) the Jeans mass is resolved. Provided that these conditions are satisfied, then, in simulations of the formation of stars and galaxies, any fragmentation that occurs should be both physical and resolved. In particular there should be no creation of sub-Jeans condensations owing to numerical instability