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Astron. Astrophys. 345, 977-985 (1999)

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2. Numerical model

2.1. Equations and numerical method

The equations solved are


where [FORMULA], [FORMULA], P, e and I are the mass density, velocity, pressure, total energy density and identity matrix respectively. [FORMULA] and [FORMULA] are the number densities of atomic and molecular hydrogen, x is the ionization fraction of atomic hydrogen, T is the temperature, [FORMULA] is the ionization/recombination rate of atomic hydrogen, [FORMULA] is the dissociation coefficient of molecular hydrogen, and [FORMULA] is a passive scalar which is used to track the jet gas. We also have the definitions


where [FORMULA] is the specific heat at constant volume, [FORMULA] is Boltzmann's constant, [FORMULA] is the ionization energy of hydrogen and [FORMULA] is the dissociation energy of H2. So L is a function which denotes the energy loss and gain due to radiative and chemical processes. [FORMULA] is the loss due to radiative transitions and is made up of a function for losses due to atomic transitions (Sutherland & Dopita 1993), and one for losses due to molecular transitions (Lepp & Shull 1983). The second term in L is the energy dumped into ionization of H, and the third is that dumped into dissociation of H2. The dissociation coefficient [FORMULA] is obtained from Dove & Mandy (1986) and the ionization rate, J, is that used by Falle & Raga (1995).

These equations are solved in a 2D cylindrically symmetric geometry using a temporally and spatially second order accurate MUSCL scheme (van Leer 1977; Falle 1991). The code uses a linear Riemann solver except where the resolved pressure differs from either the left or right state at the cell interface by greater than 10% where it uses a non-linear solver (following Falle 1996, private communication). Non-linear Riemann solvers allow correct treatment of shocks and rarefactions without artificial viscosity or entropy fixes. Applying them only in non-smooth regions of the flow means that, while the benefits are the same, the computational overhead is minimised. This code is an updated version of that described in Downes & Ray (1998).

Sometimes negative pressures are predicted by simulations involving radiative cooling. Typically these are overcome by simply resetting the calculated pressure to an arbitrary, but small, positive value. However, this involves injecting internal energy into the system and this is undesirable. A fairly reliable way of overcoming this problem is discussed in Appendix A.

2.2. Initial conditions

Initially the ambient density and pressure on the grid are uniform and defined so that the ambient temperature on the grid is [FORMULA] K. The jet temperature is set to [FORMULA] K. The function L is set to zero below this latter temperature as the data used in the cooling functions becomes unreliable and cooling below this temperature is not dynamically significant anyway. In most cases the ratio of jet density to ambient density ([FORMULA]) is set to 1 (see Table 1). The ratio [FORMULA] both inside and outside the jet, unless otherwise indicated (again, see Table 1). In all cases the gas is assumed to be one of solar abundances. The boundary conditions are reflecting on [FORMULA] (i.e. the jet axis) and on [FORMULA] except where the jet enters, and gradient zero on every other boundary. The computational domain measures [FORMULA] cells (but larger in the [FORMULA] simulations), with a spacing of [FORMULA] cm. We find that the efficiency of momentum transfer is sensitive to the grid spacing. We performed a number of simulations with different spacings and concluded that this is the absolute minimum necessary to get reliable results. This length should be reduced with increasing density. Incidentally this means that examining this property at the densities used by, for example, Smith et al. (1997) is impractical.


Table 1. A list of the simulations performed in this work. Unless otherwise stated [FORMULA]

The jet enters the grid at [FORMULA] and [FORMULA] and the boundary conditions are set to force inflow with the jet parameters. R is set at [FORMULA] cm or 50 grid cells. The jet velocity is given by


with [FORMULA] km s-1 corresponding to a Mach number of 65 and [FORMULA] are chosen so that the corresponding periods are 5, 10, 20 and 50 yrs. Here [FORMULA] is effectively an amplitude for the velocity variations where present. The jet is initially given a small shear layer of about 5 cells ([FORMULA] cm) in order to avoid numerical problems at the boundary between the jet and ambient medium. In this layer the velocity decays linearly to zero.

Nine simulations were run with varying values of density and velocity perturbation. These are listed in Table 1 which also gives the key we will use to refer to the simulations. In addition simulations of a purely atomic jet, and of a jet with a wide shear layer of almost 25 cells ([FORMULA] cm) were run. The velocity of the jet with the wide shear layer is given by


where [FORMULA] is given by Eq. 10. Each system was simulated to an age of 300 yrs. Although this is very young compared to the observed age of stellar jets, it was felt that the qualitative behaviour of the system at longer times could reliably be inferred from these results. The densities chosen are rather low to ensure adequate resolution of the system as described above. Unfortunately this precludes the use of the data of McKee et al. (1982) for calculations of the emissions from CO as their calculations are only valid for gases of much higher densities. As a result we present our findings in terms of the mass of molecular gas rather than its luminosity.

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© European Southern Observatory (ESO) 1999

Online publication: April 28, 1999