For additional details on the MACs and calculation of the gravitational acceleration see Wetzstein et al. (2008); Nelson et al. (2008). To determine the tree accuracy for the gravitational force calcula- tion in the following simulations, we use the tree opening angle criterion of Springel et al. (2001) withθ =5·10−3.
3.6
Boundary Conditions
Usually, only a small portion of the universe is modeled in one individual simulation. This re- quires conditions at the boundaries of the computational domain to be specified in order to model the influence of gas and matter outside it. For a cubic volume at least two types of boundary con- ditions are necessary to complete the set of equations: One for gravity and one for the dynamical or hydrodynamical evolution. For the purpose of individual prestellar core collapse it is best to assume that the sphere truely is in isolation - at least with respect to the gravitational potential. This assumption is reasonable, since the average density within a prestellar core is a factor 100 or more higher than within the surrounding molecular cloud. Thus, the freefall time of a prestel- lar core is also much shorter (105yr instead of 106−107yr) than the freefall time of the whole molecular cloud. Therefore, isolated gravity boundary conditions were employed in the follow- ing simulations.
For the SPH part, periodic boundaries are useful, as they assure the code’s stability by prevent- ing particles from accidentially leaving the box. As required for Bonnor-Ebert spheres, the cold and dense prestellar cores in the performed calculations are embedded in a warm (2000 Kelvin) and diffuse ambient medium. In order to avoid boundary effects the diameter of the sphere spans only half of the computational domain. Thus, warm ’boundary’ particles surrounding the collapsing core are not of particular dynamical interest, but are required in order to abide the external pressure confining the Bonnor-Ebert sphere. These ’boundary’ particles are allowed to move freely through the surfaces of the computational domain and also to feel each other in terms of neighbour contributions from across box boundaries. Periodic boundary conditions in SPH thus require the modification of the SPH neighbour search, such that a particle close to the box boundary will not see the volume beyond the boundary as a void without any neighbour particles. Instead the particle needs to find those particles as neighbours, which are close to the boundary at the opposite side of the box.
As this procedure is invoked for any neighbor search, not only quantities like density or velocity divergence are computed correctly, but also the pressure gradients and contributions to the derivative of the internal energy because a particle feels the pressure exerted by such periodic neighbors as originating from the volume outside of the simulation box.
3.7
Equation of State and Molecular Line Cooling
To close the set of equations describing the fluid flow, an additional equation is required: The Equation of State (EOS). An EOS is used to relate the internal energy of a particle to its density
and pressure. The most simple formulation is obtained for an ideal gas:
P=uρ(gamma−1), (3.44) where P is the gas pressure, u is the internal energy, γ the adiabatic index defining the stiffness of the EOS, andρ is the density within the gas element of interest. More precisely,γ is the ratio of specific heats Cp/Cv, where Cp is the specific heat at constant pressure and Cv is the specific
heat at constant volume. The internal energy u is directly proportional to the local sound speed
cs:
u= c
2
s
γ(γ−1) (3.45)
The local sound speed depends on the gas temperature T :
cs= s
γkBT
µmp
, (3.46)
where kB is the Boltzmann constant, mp is the mass of a proton and µ is the mean molecular
weight of an average gas molecule.
The adiabatic index can be related to the degrees of freedom f of a molecule by
γ= f+2
f . (3.47)
For a monoatomic gas with three degrees of freedom follows γ =5/3=1.67, whereas for a diatomic gas with five degrees of freedom we obtain γ =7/5=1.4. For a diatomic gas only translational and rotational degrees of freedom are considered, because vibrations are only ex- cited at very high temperatures. The most abundant molecule found in molecular cloud cores is molecular hydrogen (µ =2). Therefore the adiabatic indexγ=1.4 will be used in the following simulations. This EOS corresponds to a purely adiabatic behaviour for diatomic gas (e.g. Bate (1998)).
However, Inutsuka & Masunaga (2001) found that the collapse usually proceeds in an isother- mal fashion at low densities (ρ<ρcrit,0=10−13g cm−3). In this regime the cooling time scale
is short compared to the free-fall or dynamical time scale. Therefore the gas cools instanta- neously and keeps a constant temperature. The corresponding EOS for an isothermal gas can be approximated by settingγ =1 and u=c2s. At higher densities (10−8g cm−3>ρ >ρcrit,0) the
molecular gas within a dense core is typically assumed to be optically thick at all wavelengths. In this regime it behaves adiabatically. It is however not clear whether the transition from the isothermal to the adiabatic regime during prestellar core collapse is continuous or abrupt. Vari- ous other authors (Goodwin et al., 2004b; Bate, 1998; Matsumoto & Hanawa, 2003), introduce a rather abrupt transition. For instance Bate (1998) used a barotropic EOS to follow the collapse of a prestellar core up to stellar densities. Therefore he had to consider several different density regimes γ = 1.0 ρ≤1.0·10−13g cm−3, 1.4 1.0·10−13g cm−3 < ρ≤5.7·10−8g cm−3, 1.15 5.7·10−8g cm−3 < ρ≤1.0·10−3g cm−3, 1.67 ρ>1.0·10−3g cm−3, (3.48)