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Astron. Astrophys. 331, 1130-1142 (1998)

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2. Numerical method

The code used is temporally and spatially second order accurate and takes account of cooling due to ionisation and radiative atomic transitions. It uses a MUSCL-type scheme (e.g. van Leer 1977) to integrate the hydrodynamic and ionisation equations. The advection terms are calculated in a straightforward upwind fashion, while all other differences are centered. The details and tests of this code are fully described in Downes (1996). The simulations are purely hydrodynamic and are performed in slab symmetry.

2.1. Equations

The equations solved are

[EQUATION]

where [FORMULA] is the mass density, [FORMULA] is the fluid velocity, P is the pressure, Q is the artificial viscosity (following von Neumann & Richtmyer 1950), I is the identity tensor, e is the total energy density, L is the energy loss, [FORMULA] is the number density of hydrogen atoms, f is the ionisation fraction of hydrogen, [FORMULA] is the rate of ionisation of atomic hydrogen, and [FORMULA] is a scalar introduced to track the jet material. Eqs.  1to 3are the conservation equations for mass, momentum and energy, respectively. Eq.  4describes the ionisation fraction of hydrogen, and Eq. 5is an equation for [FORMULA] which acts as a tracer for the jet material.

We assume the gas law to be

[EQUATION]

where n is the total number density of the gas and k is Boltzmann's constant. The total energy density is then given by the equation

[EQUATION]

where [FORMULA] is the heat capacity of the gas. The energy loss, L, is given by

[EQUATION]

where [FORMULA] is the energy loss due to radiative atomic transitions, [FORMULA] is the ionisation potential of atomic hydrogen, and H is the heating term used to maintain equilibrium at the initial temperature on the grid.

The rate of ionisation of atomic hydrogen J is given by

[EQUATION]

Here [FORMULA] and [FORMULA] are the collisional ionisation and radiative recombination coefficients in cm3 s-1 respectively and are given by

[EQUATION]

(Falle & Raga 1995).

2.2. Scheme

As an example of how we integrate Eqs. 1to 5let us consider a continuity equation for an arbitrary variable [FORMULA]. The scheme we use is a two-step scheme which steps to time [FORMULA] from [FORMULA] using an upwind first order scheme. The values of the variables at time [FORMULA] are then used to calculate fluxes which are second order accurate in space. These fluxes are used to integrate from time [FORMULA] to time [FORMULA]. This process leads to a scheme which is second order in time. Specifically, we define [FORMULA], the first order flux across the cell boundary [FORMULA] at time [FORMULA], as

[EQUATION]

where

[EQUATION]

and [FORMULA] is the advection velocity. To calculate the second order flux at time [FORMULA] we make an estimate of the gradient, [FORMULA], of the variable [FORMULA] within cell i. We do this using the equation

[EQUATION]

where the function av is a non-linear averaging function given by (van Leer 1977)

[EQUATION]

We can now define the second order flux at [FORMULA] and time [FORMULA] as

[EQUATION]

Note that the advection velocity [FORMULA] itself cannot easily be calculated in an upwind fashion and is approximated by

[EQUATION]

A temporally and spatially second order accurate scheme is then given by first solving

[EQUATION]

where [FORMULA]. Then, using the values [FORMULA] to calculate [FORMULA], we solve the equation

[EQUATION]

2.3. The cooling and heating functions

The cooling function used in this work is a combination of that used by Rossi et al. (1997) and that contained in Sutherland & Dopita (1993). At low temperatures ([FORMULA]) it is probable that the former is more accurate. However at higher temperatures this cooling function fails as it only calculates cooling due to at most singly ionised atomic species. The latter cooling function is calculated for a gas of cosmic abundances cooling from about [FORMULA] K and it does not take account of non-equilibrium ionisation (unlike the Rossi et al. (1997) cooling function). The resulting function is plotted in Fig. 1. Tests of our implementation of the cooling have shown that it performs very well, predicting reasonably accurately the stability limit as well as the amplitude of oscillation of an overstable radiative shock wave (see Downes 1996). Note that this test is for rather strong shocks ([FORMULA] km s-1) and so only tests the Sutherland & Dopita (1993) cooling function.

[FIGURE] Fig. 1. Plot of the basic cooling function used in these simulations. This is a mixture of two functions - that used by Rossi et al. (1997) for [FORMULA] K and that given by Sutherland & Dopita (1993) otherwise

Rossi et al. (1997) also used a constant volume heating function which cancels the cooling function at the equilibrium temperature, [FORMULA]. Here we compare the results of simulations which use this heating term with those which simply assume that, below [FORMULA] ([FORMULA] K in this work) the cooling is insignificant and therefore can be set to zero. Rather than simply cutting off the cooling at [FORMULA] we attenuate the cooling function shown in Fig. 1 as follows:

[EQUATION]

This effectively implies that the cooling is zero at [FORMULA] K. Note that this means that the cooling function is a very shallow function of temperature around [FORMULA]. The assumption of insignificant cooling below [FORMULA] is often used in simulations of purely atomic YSO jets as, below [FORMULA] K, the atomic cooling rate falls off rapidly (Blondin et al. 1990). We find that these two approaches to maintaining equilibrium in the initial configuration produce significantly different results. In addition we have run simulations where the heating term is proportional to the density, as might be the case if radiative transfer was occurring.

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© European Southern Observatory (ESO) 1998

Online publication: March 3, 1998
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