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Astron. Astrophys. 338, 273-291 (1998)

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3. Numerical simulations

To obtain a more realistic picture of wind bow shocks produced by fast moving OB stars, we have performed two-dimensional hydrodynamical simulations of the gas flow towards a star with a strong stellar wind. The simulations have been carried out with a somewhat improved version of the hydrodynamical code used by Comerón (1997) to model the expansion of H II regions taking into account the presence of a stellar wind and a density discontinuity in the surrounding medium. The code explicitly solves the Eulerian hydrodynamical equations in a computational grid representing an axial plane of a cylindrical coordinate system, with symmetry around the axis. The matter, energy, and momentum densities are defined at the center of each computational cell, and evolved with time according to the hydrodynamical equations, schematically written as

[EQUATION]

where w is any of the physical variables characterizing the fluid, and A and B are coefficients containing combinations of those physical variables and their spatial derivatives. In each computational step, [FORMULA] and [FORMULA] are treated as constants, so that after a time increase [FORMULA]

[EQUATION]

As a refinement of the method used by Comerón (1997), the evolution of [FORMULA] is actually calculated in two steps: in the first one, a first set of values of [FORMULA], [FORMULA] is calculated for each variable w, and provisional values of [FORMULA] are found using Eq. (25). These values are used to compute a new set of [FORMULA], [FORMULA], and finally the definitive [FORMULA] are calculated with Eq. (25), now using the average of the first and second set of [FORMULA], [FORMULA] values. This procedure provides a second-order accuracy, and practically eliminates some small-amplitude oscillations occasionally appearing near the axis of symmetry in the first-order approach (see, e.g., the comparison between the first and second order approximations in Rózyczka 1985). We note, however, that the use of the approximation given by Eq. (25) already results in a higher accuracy than a simple finite differences approach.

The hydrodynamical equations are written in a form adequate for the assumption that the matter and momentum densities vary linearly from cell to cell. The spatial derivatives of the internal energy density are assumed to have a form that naturally introduces a Von Neumann-Richtmyer artificial-viscosity term in the momentum and energy equations, as explained in Comerón (1997). Non-hydrodynamic heating and cooling are not included in the internal energy equation (cf. Eq. (24)). Instead, after evolving the physical quantities over [FORMULA] using Eq. (25), the matter and energy densities at each point are used to calculate the rates of heating and cooling due to radiative losses, heating proportional to the density (such as cosmic ray heating), and photoionization. Also calculated is the rate transfer of energy by thermal conduction between each cell and its neighbours. All these contributions are added together and multiplied by [FORMULA], and a new value of the energy density at each cell is calculated. With this new value, we recalculate the net variation of the energy density due to these processes. If its sign is found to be different from the one calculated in the first place, the energy density is simply replaced by its thermal equilibrium value. The reader is referred to Comerón (1997) for a more detailed explanation of the numerical methods summarized here, as well as for a description of the physical ingredients of the model of the interstellar medium. Finally, radiative transfer is not explicitly calculated; we simply assume that the ionizing flux from the star at each point is sufficient to keep hydrogen completely ionized. This is justified if the flux exceeds the value given by the right-hand side of condition (21).

We have used a grid of [FORMULA] cells in our simulations. We set the scale of the grid by taking its side to be 4 times the standoff distance [FORMULA], given by Eq. (1). With the resulting resolution, the densest, thinnest parts of the bow shock are always well resolved by the grid, an essential condition for the proper treatment of instabilities. For some selected cases, we produced higher resolution simulations using a [FORMULA] grid which yielded the same results regarding structure and stability of the bow shock, what makes us confident that the resolution used here is sufficient for our purposes. The reference frame is chosen to be moving with the star, which is placed at the axis of symmetry. The ambient medium flows parallel to this axis towards the star, which is placed at a convenient distance from the center of the grid. This distance is chosen such that the obvious constraint that the head of the bow shock never overflows the computational grid, is satisfied; most of the simulations presented here have the star placed at 3 times the distance [FORMULA] from the edge of the grid. The stellar wind is assumed to flow freely within a radius of 10 computational cells from the star. This is a valid boundary condition, as the freely flowing wind actually reaches distances from the star exceeding the size of these 10 cells by far. The gas is allowed to flow out of the grid through its sides, with the exception of the side where the ambient medium "enters" the computational grid.

3.1. Choice of initial parameters

A number of simulations have been carried out in the present study, each one characterized by a set of initial conditions and by the ingredients considered in tracking the thermal evolution of the gas. The basic parameters of each simulation are summarized in Table 1. Given the allowed wide range in each of these parameters, we have opted for choosing a fiducial set representative of the "typical" characteristics of an OB runaway star moving in a diffuse medium (the reference case A in Table 1). The other five cases are then defined in order to focus on the effects that some changes have on the structure of the bow shock. Finally, the last two cases are chosen to illustrate the effects of the gas physics on the macroscopic structure of the bow shock. The first six cases considered here include both a finite cooling time and thermal conduction. Case G assumes a constant gas temperature of 8,500 K (equivalent to the instantaneous cooling case discussed in Sect. 2), and in case H we have excluded energy transfer by thermal conduction, corresponding to the physics in the Raga et al. 1997 model. Note that, due to the procedure used to select the cell size described in the previous paragraph, the computational grids correspond to widely different physical areas, as noted in the sixth column of Table 1.


[TABLE]

Table 1. Characteristics of the simulations.
Note: The column "gas physics" refers to the treatment of cooling in the simulations:
a: both thermal conduction and finite cooling time have been included in the simulations.
b: the gas has been treated as isothermal.
c: finite cooling time considered, but thermal conduction not included in the simulation.


In all the models whose evolution is followed here, the simulation starts with the star surrounded by a small expanding shell having a radius of [FORMULA]. The expansion velocity is given by the well-known expansion law of a thin shell powered by the thermal pressure of a shocked stellar wind (e.g. Weaver et al. 1977). At this stage, the expansion velocity is much greater than [FORMULA], and the shell expands nearly isotropically. The dependence on a particular set of initial conditions quickly vanishes as the front part of the shell approaches [FORMULA].

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© European Southern Observatory (ESO) 1998

Online publication: September 8, 1998
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