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Astron. Astrophys. 331, 347-360 (1998)

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4. Results

4.1. Hydrodynamic evolution

Given the agreement between CLOUDY and our time-dependent approach, we have calculated the evolution of an H II region long past the main-sequence lifetime of its ionizing source, using the same initial conditions as indicated in the previous section.

At [FORMULA] the ionizing flux is suddenly turned on and the IFs of hydrogen and singly ionized helium supersonically outrun the neutral gas. During this stage the hydrogen IF is of weak-R type, so the ionized and the neutral gas see it advancing at supersonic velocity. Fig. 3a shows the position of the IFs during the early formation phase, across the still unperturbed density distribution of the region. At [FORMULA] yr, the recombination timescale, the IF has almost reached the Strömgren radius and its velocity is about [FORMULA] (twice the sound speed in the ionized gas) due to the diminishing number of photons suffered both by geometrical dilution (the radiation flux is [FORMULA]) and recombination. The gas then begins to feel the pressure gradient across the IF, and as a consequence a shock (S1 in Fig. 3b) develops and the expansion phase begins. After this, the IFs of [FORMULA] and [FORMULA] cease to coincide with one another. The expansion phase (panels B) is characterized by a weak-D IF+shock configuration where the IF moves subsonically with respect to both the ionized and the shocked neutral gas. The expansion induces a rarefaction wave into the ionized volume which effectively begins to lower the density (RW1 in Fig. 3b). At time [FORMULA] yr, we have assumed that the stellar flux begins to decrease as [FORMULA] with [FORMULA]. This is meant to represent approximately the decreasing flux from an O star during its post-main sequence evolution (see BTY). The decrease in the flux forces the IFs to recede supersonically towards the stars (see Fig. 3-C), causing the recombination of a large volume of expanding gas. The IF width (Fig. 3d) is now given by the distance that an ion can travel before it recombines [FORMULA]. The gas speed flowing through the IF is about 10-18 km s-1, n is about [FORMULA] cm-3 caused by rarefaction and [FORMULA] cm-3 s-1, which gives a width of [FORMULA] pc. This extended IF produces an extended or more continuous pressure gradient which re-accelerates the newly recombined gas outwards, the development of a strong rarefaction wave (RW2 in Fig. 3d) driven by the receding IF then originating.

[FIGURE] Fig. 3. The evolution of H II regions. Density (solid line) and the ionization structure of hydrogen (dashed line; labelled X), and singly (dashed line labelled Y) and doubly ionized helium (dash-dotted line; labelled Z) at various evolutionary ages: a [FORMULA], b [FORMULA], c [FORMULA], d [FORMULA], e [FORMULA] and f [FORMULA] yr. The various shocks (S) and rarefaction waves (RW) that appear in the flow, as well as their direction of motion are indicated in the plots

When the outermost shell of shocked neutral gas stops feeling the pressure of the ionized gas, an `N' wave forms (see BTY). The pressure in the neutral shell is larger than in its immediate surroundings both ahead and behind it, and this causes a new shock, S2, moving inwards towards the star, and a rarefaction wave (RW3) which lowers the density of the leading shell. The shock S2 decelerates the newly recombined gas, previously accelerated by the receding IF and piles it up in an increasingly broader zone just behind the leading shell. Note that the zone between S1 and S2 is composed of two layers of different densities ([FORMULA] cm-3 and [FORMULA] cm-3). Inside the H II region the rarefaction waves RW1 and RW2 have lowered the density even further ([FORMULA] cm-3) and as a consequence the recombination rate is slower. This, together with the larger photon flux found nearer the star slows down the receding IFs. This happens up to a point when their recession velocity, with respect to the gas, [FORMULA], is less than [FORMULA] and the pressure gradient across the IF then has enough time to cause an outward acceleration. The pressure profile then steepens and a new shock (S3) develops. At the same time the shock formation generates a new rarefaction wave (RW4) moving inwards. This lowers the density and inhibits further the recombination. Consequently, a new supply of photons (despite the continuous decrease in the stellar photon flux) can reach the IF. In this way the IF stops its recession and turn back to reionize the recombined gas while driving the new shock, S3, outwards. The system IF + shock during this reionization stage is similar to that of the first expansion phase, except that the shock now travels into an expanding medium. The IF + shock system travels more quickly than the outer shocks because the density of the H II regions decreases faster than the stellar flux as a consequence of successive rarefactions. Eventually the shock S3 interacts with the reverse shock S2 producing a shock reflexion. Two new shocks S [FORMULA] and S [FORMULA] then form and pile up the low-density ionized gas against the layer of shocked neutral gas behind S1. The calculations were stopped at the point when the IF started pushed the shock S1 again. The further evolution is characterized by an extremely low-density H II region which finally recombines because of the assumed continuous drop in the ionizing flux.

During the late evolutionary stages, the radius of the external shock does not evolve following the classical formula

[EQUATION]

because by then it is not the pressure of the ionized gas that drives it but rather its own inertia. Even when the IF and the new shock, S3, catch up with the external layer, the density of the H II region is so low that the pressure of the ionized gas just equals the external pressure. Thus one can assume that all the gas of the nebula is contained in a thin layer and since the momentum conservation

[EQUATION]

which implies

[EQUATION]

where [FORMULA] yr) is the time when the star leaves the main sequence. When the velocity of this layer drops below the sound velocity in the neutral medium, [FORMULA], the shock S1 will decay into a sound wave and the dynamical effect of the original H II region will have finished. Setting [FORMULA] we can obtain the time [FORMULA] when the evolution ends:

[EQUATION]

Substituting values, it is found that [FORMULA] yr for a final radius of [FORMULA].

Our calculations confirm the earlier results from BTY and TBBY. In particular, during the formation and expansion phases our results matches the well-known theory of the evolution of H II regions exactly. We have also calculated the short recombination and reionization phases, generated during the massive star excursion away from the main sequence. During recombination we find very extended IFs (or rather recombination fronts) with a width [FORMULA] (i.e. dependent on the gas velocity), which drive further rarefaction waves into the ionized volume. Our method thus recovers all dynamical events in the evolution of H II regions and furthermore, given the more complete approach to the problem, offers many more possibilities for direct comparison with the observations.

4.2. Optical appearance of evolving H II regions

To compare with the observations we have selected two basic observational magnitudes, the surface brightness with respect to H [FORMULA], i.e. the luminosity in a given line as a function of radius, which supplies information about the physical conditions in the nebula (density, ionization, temperature, etc), and line profiles, which are related to the kinematic state of the nebula. It is important to notice that this is only possible from the numerical output of a hydrocode that calculates both the hydrodynamics and the ionization structure of the nebula (see appendix B for details on the line transfer calculations). The surface brightnesses are referred to H [FORMULA] ; the usual presentation of observational results. The lines selected were: (L1) [N I ] [FORMULA] 5200, (L2) [N II ] [FORMULA] 6584, (L3) [O I ] [FORMULA] 6300, (L4) [O II ] [FORMULA] 3727, (L5) [O III ] [FORMULA] 5007, (L6) H [FORMULA] and (L7) H [FORMULA]. Several other lines from C, N, O and Ne were calculated but those listed above are among the most frequently used in observational studies of H II regions. Although our models are one-dimensional, they can supply important trends or traces of the possible stage of evolution of an H II region. Most of the observational information about H II regions is in form of integrated or global quantities (except in cases with excellent spatial resolution), hence deviations from spherical symmetry and those caused by inhomogeneities may not be so important. We regard our results as an excellent standard reference frame for future observational and theoretical work.

Fig. 4 shows the results for the H II region during its formation phase. The surface brightness of [FORMULA] 5007 present a core distribution with the maximum at the centre and a smooth declination towards the edge. However, [FORMULA] 6584 and [FORMULA] 3727 present radial distributions with a central depression and a sharp rise towards the edge. This is due to the emissivity of these lines which peaks at the IF where the temperature reaches a maximum (see Fig. 2). The difference between the centre and the edge is about a factor of 10 in [FORMULA] 6584 and [FORMULA] for [FORMULA] 3727. The line profiles (shown in Fig. 4) at different impact parameters all show the thermal width corresponding to gas at rest.

[FIGURE] Fig. 4. Surface brightness relative to H [FORMULA] and line profiles (inner squares) for model A (see Fig. 3). The line codification is as follows: (L1) [N I ] [FORMULA] 5200, (L2) [N II ] [FORMULA] 6584, (L3) [O I ] [FORMULA] 6300, (L4) [O II ] [FORMULA] 3727, (L5) [O III ] [FORMULA] 5007, (L6) H [FORMULA] and (L7) H [FORMULA]. The line profiles are integrated at different impact parameters across the nebula indicated by the arrows on the x -axis. The numerical values of the impact parameters is indicated in pc on the arrows. The line profiles of H [FORMULA] have not been included in the plot since they are nearly identical to those of H [FORMULA]. The thick black line is the surface brightness of H [FORMULA] normalized to its maximum value at that evolutionary time.

Fig. 5 depicts the expansion phase. [FORMULA] 3727 and [FORMULA] 6584 now increase smoothly towards the edge and [FORMULA] 5007 presents a very rounded form. These features are produced by the lower density and the flatter temperature distribution obtained as the H II region expands. This is quite clear in the surface brightness distribution of H [FORMULA] with a gradient change over 9-10 pc corresponding to the head of the RW position. Another important feature is that [FORMULA] 6300 emerges at the edge of the region. This is caused by the development of a high-density shocked layer ahead the IF. Between the IF and this layer there is a zone where oxygen is mostly neutral but the electronic density is high enough to excite the [FORMULA] 6300 line. Mallik (1975) found that lines of neutral species (O0 and N0) can be emitted only in the transition region since they require the presence of both neutrals and electrons for their formation. Note that in equilibrium models of H II region (Rubin 1968) the [O I ] and [N I ] lines are extremely weak. Regarding the line profiles, [FORMULA] 5200, [FORMULA] 6584 and [FORMULA] 6300 show splitting with profiles centred at [FORMULA] km s-1. The [FORMULA] 6584 line is thus an important kinematical probe due to its high luminosity and its ability to split even at low gas velocities ([FORMULA] km s-1).

[FIGURE] Fig. 5. Surface brightness relative to H [FORMULA] and line profiles for the model B (see Fig. 3). The symbols and arrows have the same meaning as in the Fig. 4.

Fig. 6 shows results during the recombination phase. The intensities of [FORMULA] 3727 and [FORMULA] 6584 now present a flatter distribution and the lines of the neutral species [FORMULA] 6300 and [FORMULA] 5200 are enhanced at the leading edge of the IF. The profiles of [FORMULA] 6300 and [FORMULA] 5200 present two components across the centre of the nebula, one at the centre with [FORMULA] km s-1 and the other at [FORMULA] km s-1, while other lines show only one maximum. This two components corresponds to a velocity profile that grows nearly linear from the centre to the edge of the nebula where it reaches a velocity of [FORMULA] km s-1. The lines of the higher ions do not present this double component because with the recession of their Strömgren spheres they are confined to the centre of the nebula.

[FIGURE] Fig. 6. Surface brightness relative to H [FORMULA] and line profiles for the model D (see Fig. 3). The symbols and arrows have the same meaning as in the Fig. 4.

Finally, Fig. 7 displays the phase of reionization. The [FORMULA] 6300 and [FORMULA] 5200 lines now form an extended ring bordering the H II region. The intensity of H [FORMULA] is however very low in this part of the region and the possibility of observing this enhancement of the neutral lines at the edge of old H II regions would be scarce.

[FIGURE] Fig. 7. Surface brightness relative to H [FORMULA] and line profiles for the model E (see Fig. 3). The symbols and arrows have the same meaning as in the Fig. 4.

Summarizing evolving H II region in a low constant medium lead through their expansion to an enhancement at the outer edge of lines of neutral species [FORMULA] 6300 and [FORMULA] 5200. As the evolution proceeds and more gas is accumulated into the shocked layer, the intensity of these lines increases. Moreover, they can show double and multiple components during this phase. When the star leaves the main sequence the narrow outer rings of [FORMULA] 6300 and [FORMULA] 5200 become broader but their surface brightness decreases. The detection of these broadened rings in [FORMULA] 6300 and [FORMULA] 5200 would be a clear trace of very evolved H II regions. However, the [FORMULA] 6584 line emitted in the fully ionized zone is a good indicator of expansion; it splits even at low expansion speeds [FORMULA] km s-1.

4.3. Diagnostic diagrams

Baldwin, Phillips & Terlevich 1981(hereafter BPT) explored the parameter space of several emission-line quotients to find that different excitation type objects (H II regions, PNe, shocked ionized sources, etc.) occupy well defined zones in what they called diagnostic diagrams. Fig. 8 shows three of these frequently used diagnostic diagrams where the corresponding line ratios of the models calculated in the preceding sections have been plotted. In these diagrams both evolution and spatial sequences are shown. The line quotients integrated over the whole spherical nebula are displayed with large black squares labelled with the corresponding capital letter of the model presented in Sect. 4.1. The sequence given by these line ratios begins in the high-excitation part of the diagrams and evolves towards the low-excitation part. Fig. 8 also displays the envelope of the area filled by the set of 80 Galactic and extragalactic H II regions compiled by BPT (area delimited by a dashed line) and their fit made with an excitation model to these regions (dash-dotted line). The sequence of large squares follows the general trend of the fit but falls slightly outside the area defined by the observations. Fig. 8 also shows the spatial sequences of the line quotients integrated at different impact parameters for each model. The beginning of the spatial sequence is traced by a small black square, labelled with small letters representing every model, and correspond to the line ratios obtained across the centre of the region, i.e. with an impact parameter equal to 0. The rest of the impact parameters are traced with circles of smaller size as we move towards the edge of the nebula and each spatial sequence is joined by a dotted line. Note that all spatial sequences resembles the form of the fit by BPT with the lowest excitation obtained at the edge of the nebula, except for model A(a). This departure is caused by the temperature peak at the IF in the initial model. The collisionally excited lines are strongly temperature dependent while the recombination lines of hydrogen dependent weakly on T. An increment in temperature tends therefore to increase the quotient between the collisionally excited lines and the recombination lines of hydrogen. Finally, note that the evolution or sequence followed by the small squares falls well into the observational area of H II regions.

[FIGURE] Fig. 8. Diagnostic diagrams. Large black squares indicate the line quotient corresponding to the whole spherical nebula for each model (labelled A, B, C, D, E and F). The small black squares indicate the line quotient corresponding to the line of sight across the center of the nebula, i.e. impact parameter equal to 0, for each model (labelled a, b, c, d, e and f). The line quotient corresponding to other impact parameters are plotted with circles of smaller size for larger impact parameters. The large and small black squares have been joined by a continuous line in order to indicate evolution, and the circles have been joined by a dotted line in order to indicate the spatial sequence in each model. The dashed-dotted line is a fit to a set of 80 Galactic and extragalactic H II regions (BPT) and the area delimited by a dashed line is the approximated envelope to [FORMULA] % of these H II regions.

The diagnostic diagrams are a powerful method of determining the physical conditions of ionized nebula and probably also their evolutionary stage. Here we have drawn the sequence for a fixed set of initial conditions (density, ionizing flux and abundances). The use of other initial conditions and their impact on the diagnostic diagrams will be the subject of a forthcoming communication.

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© European Southern Observatory (ESO) 1998

Online publication: February 4, 1998
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