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Astron. Astrophys. 326, 1195-1214 (1997)

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2. Initial conditions and numerical methods

In all the simulations presented in this paper, we will assume that the star is initially embedded in a dense cloud with [FORMULA] cm-3 and [FORMULA] K, in pressure equilibrium with a neutral diffuse medium with [FORMULA] cm-3 and [FORMULA] K. Such conditions may be commonly expected in molecular clouds (Blitz 1993). The distance from the star to the edge of the cloud is a free parameter expressed in multiples of the Strömgren radius in the dense medium. Heating of the cloud and intercloud medium is assumed to be proportional to the density in regions shielded from the ultraviolet radiation of the star. We have taken the expressions for cosmic ray heating given by Black 1987, but multiplied by a scaling factor so that the intercloud medium is thermally stable at the assigned temperature: a phase transition to a cold phase takes place when the density exceeds [FORMULA] cm-3. Although this is an arbitrary choice, the only relevant role played by such heating is to keep the temperatures of the unperturbed cloud and intercloud media at their initial values, which are little sensitive to the actual value of the heating rate. In regions where the Lyman continuum flux of the star is nonzero, we assume a blackbody spectral energy distribution of the ionizing radiation and a cross section for ultraviolet photon absorption by hydrogen atoms proportional to [FORMULA] (Spitzer 1978), with the excess photon energy going into heating of the gas. To simplify radiative transfer calculations, we neglect the hardening of the ultraviolet flux with increasing column density to the ionizing source. Only hydrogen photoionization is considered. Further simplifications include the on-the-spot and radial flux approximations (Bodenheimer et al. 1979, García-Segura & Franco 1996). Dust absorption is assumed to be proportional to the density, adopting an averaged opacity of [FORMULA] cm2 g-1 in the ultraviolet (Yorke et al. 1984). The transfer equation of ionizing radiation is thus

[EQUATION]

where [FORMULA] is the ionizing flux, [FORMULA] is the recombination coefficient excluding the ground level, [FORMULA] is the ionization fraction ([FORMULA] electron density, [FORMULA] proton density), [FORMULA] is the contribution of electrons from ionized metallic species to the ionization fraction (set to [FORMULA]), and r is the distance to the ionizing star. The numerical integration of Eq. (1) is carried out proceeding outwards from the star, making

[EQUATION]

where [FORMULA] is the right-hand term of Eq. (1), and [FORMULA] is the ionizing flux at [FORMULA]. [FORMULA] is interpolated from the fluxes at the positions of the nearest cells j, k as

[EQUATION]

The ionization fraction [FORMULA] is set at each point at the equilibrium value, given by the balance between photoionization, collisional ionization, and radiative recombination to levels other than the ground.

Cooling of the cool and warm gas is mainly by collisional excitation of fine structure metastable levels of neutral and ionized metals, for which expressions are taken from Dalgarno & McCray 1972. Cooling by collisional ionization and radiative recombination of hydrogen is also included, with relevant rate coefficients from Spitzer 1978. For [FORMULA] K, an interpolation of the cooling curve of Gaetz & Salpeter 1983 is used. No molecular gas has been explicitly considered, as this should be photodissociated by the stellar radiation with wavelengths longwards from the Lyman limit. Such radiation should also produce an increase in the temperature of the gas, which we have neglected. However, we have checked that this warming does not have any noticeable dynamical consequences on the process studied here.

The numerical simulations are carried out by means of a two-dimensional explicit Eulerian hydrodynamic code, using cylindrical coordinates on a computational grid of [FORMULA] cells. The hydrodynamic equations in cylindrical coordinates are used in the more convenient form for numerical integration:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

where z and x are respectively the axial and radial coordinates; [FORMULA] and e are the densities of matter and internal energy; p is the pressure ([FORMULA]); and u and v are the axial and radial components of the velocity.

The effect of a spherically symmetric stellar wind with mass loss rate [FORMULA] and terminal velocity [FORMULA] has been simulated in most cases by assuming a wind free-flowing region, defined by the computational cells lying at a radius less than ten times the cell size from the star. At the centers of these shells, the density is set to the corresponding value for a free-flowing wind, [FORMULA]. However, it is found sometimes that, when rescaling the computational grid to model the late stages of expansion (increasing the cell size), the shock front which defines the outer boundary of the free flowing wind lies inside the new ten cell-size radius from the star. In these cases, the effect of the stellar wind is simulated instead by assuming that all the kinetic energy of the wind is transformed into thermal energy in the vicinity of the star; the momentum and energy input is then simulated by increasing the material and energy densities in the cells surrounding the star (again within a ten cell-size radius) at a rate [FORMULA] and [FORMULA], respectively, so that their integrated values over the sphere volume equal [FORMULA] and [FORMULA]. This is taken into account when numerically integrating the hydrodynamic equations by adding the source terms [FORMULA], [FORMULA], [FORMULA], and [FORMULA] to Eqs. (4), (5), (6), and (7), respectively.

The particular form of Eqs. (4-7) is motivated by the convenience of expressing partial derivatives in a form suitable for finite difference equations, as stressed by Monaghan 1992; in our numerical scheme, the quantities assumed to vary linearly from cell to cell are the density of matter [FORMULA] and the components of the momentum density [FORMULA], [FORMULA], while we approximate the derivative of the energy density as

[EQUATION]

where l is one of the cylindrical coordinates x, z, and V is the component of the velocity in the direction of that coordinate. The subindex 0 denotes the local value of a quantity, and [FORMULA] and [FORMULA] the values one computational cell ahead and behind in the l direction, respectively. Assuming this form of the energy density gradient naturally introduces an artificial viscosity term in Eqs. (5), (6), and (7).

The inclusion in our calculations of high energy, low density gas enables the appearance of large accelerations in very short timescales due to its small inertia, especially near its interfaces with the dense gas. Updating the values of [FORMULA], u, v, and e by simply multiplying the right-hand sides of Eqs. (4)-(7) by [FORMULA] would then require exceedingly small time steps to avoid rapid oscillations in the direction of motion of small regions of the hot gas. This is particularly motivated by the dependence of the artificial viscosity term on the square of the velocity. To avoid this, Eqs. (4) to (7) may be written in the following form for a given computational cell:

[EQUATION]

where A is any of the quantities appearing in the left-hand sides of Eqs. (4)-(7); [FORMULA] and [FORMULA] contain combinations of values of the other variables and A in that cell and in neighbouring ones. Both [FORMULA] and [FORMULA] are assumed to remain constant along one computation step [FORMULA], so that numerical integration of (4)-(7) implies solving equations of the type

[EQUATION]

The value of [FORMULA] is then given by the allowed variation in A in a single computational step, together with the usual Courant condition (e.g. Arthur & Henney 1996).

After explicitly calculating the hydrodynamic evolution of the physical quantities of the fluid in a computational step, corrections are made on the energy density by multiplying the rates of heating and radiative losses by [FORMULA]. We also include the rate of energy exchange by thermal conduction between adjacent cells, for which we adopt the classical expression for unsaturated heat flux (Cowie & McKee 1977). The heat conduction flux between each pair of adjacent cells is calculated at their midpoint, and the value of [FORMULA] is estimated assuming that both matter and energy density vary linearly between them. The corrections are then recomputed with the new values of the energy density so that, if the sign of the correction is found to change, the equilibrium value of the energy density is adopted. Further corrections to the energy density are made as required after recomputing the ionizing flux distribution across the computational grid.

We have tested the reliability of the numerical methods employed here by simulating the time evolution of two cases with well-known analytical solutions: the expansion of a shell surrounding a bubble with constant injection of energy inside it, and the expansion of a Strömgren sphere powered only by photoionization, both in a homogeneous, low pressure medium. Simple analytical approximations to these problems exist (e.g. Dyson & Williams 1980, Shu 1993) which give the evolution of the radius, and other physical quantities, as a function of time. The results of one such comparison are shown in Fig. 1: the input parameters are those of a O4 star (see Sect. 2.1), embedded in a uniform medium of density [FORMULA] cm-3, and the evolution is followed for [FORMULA] yr, comparable to the span of our simulations. In the windless case, the initial radius is the Strömgren radius, with the velocity set to zero everywhere. For the case with wind, the simulation is started at the time when the expanding bubble reaches the Strömgren radius, with the velocity of the swept-up shell given by the analytical approximation discussed in Sect. 2.2.

[FIGURE] Fig. 1. Comparison between analytical solutions and numerical simulations of the time evolution of the radius of a wind-blown bubble and a Strömgren sphere. In both cases, the initial radius is the Strömgren radius. The case of the wind bubble is represented by the solid line (analytical approximation) and open circles (numerical solution). The expansion of the Strömgren sphere is replresented by the dashed line (analytical approximation) and open triangles (numerical solution).

An acceptable agreement is found between the numerical solutions and the behaviour expected from the analytical approximations. In particular, at large times, when the initial transients have died out, the exponents of the power laws describing the dependence of the radius with time are very similar in both the numerical and analytical evolutions. The lag between the analytical and numerical evolutions in the case of the Strömgren sphere can be explained by the assumed zero expansion velocity at the start of the simulations, which assumes an ideal, instantaneous transition from the formation to the expansion phase; otherwise, the [FORMULA] expansion law is very well reproduced. It is not possible to perform in a similar way a direct comparison between the predicted and simulated pressures, as the analytical approach assumes a uniform pressure inside the HII region which does not correspond to the reality. However, the pressure at the end of our simulations just inside the ionization front is 10 % above the pressure predicted by the analytical approximation, and slowly falls inwards, as expected from qualitative considerations.

Concerning the case with stellar wind, the largest deviations from the analytical solution appear at the initial stages, in which the radius of the shell (defined numerically as the position of the density maximum as one proceeds away from the shell) grows more slowly than expected. We can identify two reasons for this behavior: first, the shell surrounding the bubble is very thin at the start of the simulation, and as it is evolved in time it spreads over a few cells and the maximum is slightly shifted towards the inside. Another reason seems to have to do with the fact that the gas in the bubble is initially assumed to be at rest, and large, chaotic motions induced by the bubble expansion quickly appear. A parto of the energy contents of the bubble may thus be expected to be spent in setting the hot gas in motion at the start of the simulations, rather than in producing work by expansion of the shell against the ambient medium. Nevertheless, this deviation does not last for long, and after less than [FORMULA] yr the bubble expansion closely follows the predicted [FORMULA] law (see Sect. 2.2). The lag between the numerical and analytical solutions remains at a constant value thereafter, and the relative difference accordingly decreases with time; at the end of the [FORMULA] yr period covered by the simulation, the numerically found value of the radius is slightly over 90 % of the analytical one.

2.1. Stellar parameters

In the present paper, we will focus our discussion on the effects caused by two types of star with very different ratios of ultraviolet luminosity to wind mechanical power, namely an O4 star and a B0 star. The input parameters used in the simulations for these two kinds of star are given in Table 1.

[TABLE]

Table 1. Adopted stellar parameters

The stellar parameters relevant to the problem under consideration are the ultraviolet flux of the star shortwards from the Lyman limit [FORMULA] (expressed here in photons per second), the effective stellar temperature [FORMULA], the mass loss rate [FORMULA], and the terminal wind velocity [FORMULA]. For the Lyman continuum flux, we have used the values tabulated by Hollenbach et al. 1994 (based on stellar models by Maeder & Meynet 1987), while the stellar wind parameters have been obtained from Leitherer et al. 1992, using zero-age main sequence mass-luminosity relations from Schaller et al. 1992. We have used wind parameters closer to the actual observations as presented by Leitherer et al. 1992 rather than the analytical expressions derived by them, which provide a good fit to the most massive end but are inappropriate for the B0 star.

2.2. Early evolution and initial conditions

Very soon after a star has formed, the ionized sphere around it reaches a radius close to the Strömgren radius [FORMULA]. This happens in a time much shorter than the expansion timescale of the Strömgren sphere (Dyson & Williams 1980). The stellar wind-blown bubble thus drives a shock and forms a dense shell of ionized gas inside the Strömgren sphere. The high density of the shocked shell increases the recombination rate inside it, thus reducing the flux of photons available to keep the outer HII region ionized, and eventually it grows thick enough so as to trap all the ionizing flux from the star. To calculate the conditions under which this happens, we use the expressions for the expansion of stellar wind-driven bubbles (Weaver et al. 1977), plus the strong shock approximation. The radius of the bubble, [FORMULA], at time t is

[EQUATION]

where [FORMULA] is the mechanical power of the stellar wind and [FORMULA] [FORMULA] is the density of the medium in which the bubble expands. The ionization front gets trapped in the expanding shell when the condition

[EQUATION]

is fulfilled: [FORMULA] is the density of the unperturbed medium outer to the shell, [FORMULA] is the density of the shocked gas in the shell, and [FORMULA] is the total recombination rate excluding recombinations to the ground level. In this expression, we neglect dust absorption and recombinations inside the hot, low density gas of the bubble, and take the mass of the shell [FORMULA] to be the same as the mass originally inside the sphere of radius [FORMULA], thus assuming that only a small fraction of the shell has evaporated by conduction. The value of [FORMULA] is estimated from the strong shock approximation, [FORMULA], where [FORMULA] is the sound speed in the ionized surrounding gas. Using Eq. (11) to obtain [FORMULA] as the time derivative of [FORMULA],

[EQUATION]

Combining this with Eqs. (11) and (12) gives us the time [FORMULA] when Eq. (12) is fulfilled:

[EQUATION]

It is useful to relate the characteristic quantities of the bubble at the trapping of the ionization front to quantities associated to the expansion of the classical Strömgren sphere or to the unperturbed surrounding medium. In this way, using the values of the Strömgren sphere radius

[EQUATION]

and the expansion timescale of the Strömgren sphere defined as

[EQUATION]

the ratios of the bubble expansion velocity to the ionized gas sound speed ([FORMULA]), the bubble radius to the Strömgren radius ([FORMULA]), the trapping time to the expansion time of the Strömgren sphere ([FORMULA]) and the pressure inside the bubble to the pressure in the Strömgren sphere ([FORMULA]) take the following simple forms:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

where

[EQUATION]

For the stellar parameters listed in Table 1 and the adopted density of the molecular cloud, we obtain [FORMULA] for an O4 star, and [FORMULA] for a B0 star. Consequently, [FORMULA] for a O4 star and 0.31 for a B0 star, this is, the trapping happens when the bubble is still inside the initial Strömgren sphere. Moreover, [FORMULA] for the O4 star and 0.03 for the B0 star, meaning that the Strömgren sphere has not had time to expand significantly before the trapping of the ionization front in the expanding shell. Finally, the expansion of the shell takes place at [FORMULA] for the O4 star and 5.7 for the B0 star, thus justifying the strong shock approximation to the shell expansion.

Once the ionization front becomes trapped in the expanding shell, the material in the initial HII region outside it recombines in a time [FORMULA], which is only [FORMULA] years in such a dense medium. Further expansion of the shell therefore takes place in a neutral medium, and the structure of the HII region is as depicted in Weaver et al. 1977: proceeding outwards from the star, one finds successively layers of free-flowing wind, shocked wind, dense ionized gas, shocked neutral gas shielded from the UV radiation of the star by the HII layer, and finally the neutral outer medium.

We start our simulations at the time when the expanding bubble reaches the cloud-intercloud interface, using as initial values for the variables those derived from the analytical solution to the early stages of the expansion as outlined above. The dense shell containing the HII region is assumed to be totally ionized, due to the impossibility of resolving the very thin layer of shocked neutral gas surrounding it in our computational grid; this is an acceptable approximation, as long as the surface density of the shell is dominated by the ionized component. For the interior structure of the bubble, we integrate the evaporation rate due to thermal conduction as a function of time using the analytic expressions of the radius and internal pressure from Weaver et al. 1977. In this way, one can obtain an analytic expression for the average density and temperature as a function of time, whose corresponding expressions can be found, for instance, in Shull & Saken 1995. The evaporated mass is found to be only a small fraction of the swept-up mass building up the shell, as assumed above, while an estimate of the radiative losses inside the bubble shows that in general they are small during its early evolution. The initial distributions of density and temperature are then found by assuming conductive equilibrium inside an isobaric bubble (Weaver et al. 1977,Mac Low & McCray 1988, Shull &Saken 1995). Nevertheless, this configuration is changed by the combined action of shell expansion and mechanical power injection from the central star very soon after the start of the numerical simulation.

The distance of the star to the cloud-intercloud interface is a free parameter of our simulations, which we express as a multiple d of [FORMULA]. We find it convenient to use a length of [FORMULA] and [FORMULA] in the z and x directions respectively as the initial size of the computational grid, with the cloud-intercloud interface located at the middle of the grid. The grid is scaled up, and the center repositioned along the axis of symmetry, when the expanding HII region approaches its boundaries.

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© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998
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