3. Results and discussion
3.1. Dynamical evolution
In this section we focus on the aspects related to the dynamical evolution of the gas affected by the expansion of the HII shell and the hot bubble contained in it into the diffuse intercloud gas, and the resulting structures. The simulations are continued until the size of the bubble expanding in the intercloud gas becomes approximately one order of magnitude greater than that of the initial bubble at the time of blowout.
3.1.1. O4 star,
Our first numerical simulation describes the expansion of the bubble generated by a O4 star located at from the boundary of the molecular cloud. In the initial conditions described in Sect. 2, pc. We follow the expansion along the first 400,000 years after breakout; the results are shown in Figs. 1 and 2, where the evolution of the values of the density and the thermal pressure are displayed in a logarithmic greyscale on the computational plane which contains the axis of symmetry.
The HII region initially surrounding the star is a thin, compressed shell. Very soon after the shell starts its expansion into the intercloud medium, the shell/bubble boundary follows it. As a consequence, the phase in which the expansion of the HII region into the intercloud medium is driven by the pressure difference between the dense ionized gas and the tenuous intercloud gas is very short, only a few times years in our initial conditions, and soon the HII shell as a whole is accelerating outwards pushed by the hot bubble interior. In these conditions, we expect the shell to become Rayleigh-Taylor unstable, and break up in fragments. This is difficult to follow in a 2-D simulation, given that the azimuthal components of the perturbations are not reproduced, and the growth of the instabilities is therefore inhibited as the distance to the axis of symmetry increases. However, near the axis we indeed see that the boundary between the shell and the bubble deforms. Finally, the bubble breaks out of the shell flanked by two denser blobs. This part of the bubble begins a fast expansion into the intercloud medium, which is now a diffuse HII region due to the release of the ionization front as the density of the blown out HII shell decreases.
The dense blobs lag behind the expanding shell, which after years has been disrupted by the bubble in other points too, and find themselves surrounded by a hot medium whose pressure decreases with time due to the growth of the bubble. Moreover, the large temperature difference produces a flow of energy into the blobs due to thermal conduction. The combined effect of expansion against the decreasing bubble pressure and evaporation due to thermal conduction ends up by diluting these fragments of the initial shell in the hot bubble interior.
A few shock structures form inside the bubble as the hot gas is accelerated, then shocked, then accelerated again by pressure gradients. In the frame corresponding to the evolution after years, three such shocks can be seen. The first one defines the boundary between the free flowing stellar wind and the bubble, and always keeps close to the star along the length of our simulations. The expansion of the bubble in the direction of the intercloud medium creates a pressure gradient which produces a fast stream of hot gas (with typical velocities around 1000 km s-1). When this stream reaches the vicinity of the expanding shell, a second shock is formed, separated from the shell by a broad layer of gas at a temperature of K and with density cm-3. The thermal pressure of this hot gas drives the expansion of the shell. In the region where the shell was first disrupted, the expansion of the bubble creates a new pressure gradient which in turn generates a third shock near the blown out shell.
At later times, the overall appearance of the HII region can be separated in: 1) a diffuse component, ionized by the release of the ionization front at blowout, slowly expanding by the factor of of pressure difference with its surrounding neutral medium, but still unaffected by the expanding bubble; 2) a dense but thin HII region surrounding the hot bubble, with some dense knots produced by the instability of the shell at blow out; 3) a compact component, contained in the molecular cloud; and 4) an intermediate density HII region formed by the flow of ionized gas from the compact component into the low density intercloud region. This latter component corresponds to the classical champagne flow. The denser knots described above react to the pressure of the hot bubble differently from the rest of the shell and, before they are evaporated by conduction into the bubble interior, they distort the shape of the shell. Although higher resolution and a full 3-D treatment is required to follow their development in detail, these knots are likely to cause some of the ubiquitous filamentary structures seen in HII regions.
Although the bubble blowout rapidly decreases the pressure inside the compact component, which was driving the early expansion of the HII region previously to the blowout, the HII region continues to expand into the molecular cloud all along the time span covered by our simulations. In fact, the expansion rate can be enhanced by the blowout, in spite of the decrease in pressure, as noted by Franco et al. 1994: the flow of ionized gas toward the intercloud region decreases the column density between the ionization front and the star, and more photons are available to ionize fresh gas from the molecular cloud. On the other hand, the stellar wind provides ram pressure inside the compact component, keeping its volume density high. The continued flow of gas out of the compact component maintains a "wall" of moderately density gas which keeps the flow of hot gas weakly collimated during the whole span of our simulations. The importance of this collimating effect is expected to increase as the importance of ionization vs. stellar wind increases (i.e., as the parameter introduced in Sect. 2 decreases), as is confirmed by the simulations in Sect. 3.1.3.
At the end of our simulations, years after blow out, the shocked stellar wind, mass-loaded by thermal conduction with gas from the compact HII region, falls down to a density of 0.02 cm-3 and a temperature of K before being shocked again inside the bubble. The doubly-shocked gas driving the expansion of the bubble has now a density of 0.15 cm-3 and a temperature of typically K, which produces a pressure about 6 times larger than the pressure in the diffuse ionized gas against which it expands. This is to be compared with a pressure ratio exceeding 800 at blow out. The compact component has a density of 120 cm-3, which decreases in the champagne flow to 1.2 cm-3. Its expansion against the intercloud medium raises its density at the outer layers to about 6 cm-3, about a factor 2 larger than the density of the diffuse ionized intercloud medium. Typical densities in the new shell of swept-up gas, expanding at a velocity of about 50 km s-1, are of order of 20 cm-3.
3.1.2. O4 star,
We have repeated the calculations with the same stellar parameters, but now placing the star from the cloud-intercloud boundary. With the data of Table 1, Eq. (11) gives in this case pc, implying a minimum diameter of the embedding clump of over of 6 pc. This is rather large for a clump of this density (Williams et al. 1995), but still useful for comparison with the case outlined in Sect. 3.1.1. The essential difference in the initial conditions with respect to the case is that the dense shell containing the HII region is more massive, and is expanding more slowly at the time of blow out. The pressure of the hot gas is only that of the previously presented case. As a consequence, we may expect the initial HII shell to have a more important rôle in determining the dynamical evolution of the bubble.
The evolution of the density in this case is shown in Fig. 6. The initial evolution is much slower than in the case , due to both the smaller pressure from the shocked stellar wind and the larger inertia of the thicker HII layer. After years, the expansion of the ionized shell into the intercloud medium is still dominated by its own thermal pressure, rather than by the pressure of the hot gas in the bubble. The expansion does not release the ionization front until a later time.
In the frame showing the density distribution after years, it can be seen how the hot gas is escaping from the HII shell at different points. The ionization front is released in these points, remaining trapped in others, producing several beams of ionized gas in the intercloud medium. The denser blobs between the shell breakouts are still trapping the ionization front, shielding the gas behind them from the ionizing radiation of the star. This causes a pressure difference between these blobs and the hot gas surrounding them, which tends to further compress regions of high density. This kind of instability has been discussed in detail by Giuliani 1982 and García- Segura & Franco 1996. The azimuthal symmetry imposed in our simulations implies that the high density spikes appearing in the second frame of Fig. 6 are actually rings when they are not in the axis of symmetry, while only the central spike is a real high density column. As a result, the pressure of the hot gas over the dense protrusions produces a greater compression on the central one, which tends to grow much denser than the others. This is a well known feature of 2-D simulations in cylindrical symmetry; see for example the simulations by Róyczka & Tenorio-Tagle 1985a, or Blondin & Lundqvist 1993. Since real instabilities can also grow in the azimuthal direction, it can be expected that dense columns such as the one protruding the expanding bubble along the axis of symmetry will actually appear everywhere in the shell, the cylindrical symmetry actually damping the growth of the off-axis ones. The observed counterpart of these dense columns may be the elephant trunks commonly observed in HII regions.
As the bubble develops, its evolution is similar to that shown for the case. However, a morphological comparison between these two cases after years of evolution shows clear differences. The champagne flow out of the compact component is much more conspicuous in comparison to the hot bubble, and the pressure of the bubble gas is less than 3 times that in the ionized intercloud gas. The temperature in the bubble is below K in all its volume.
Radiative losses play an important rôle in the differences between the and cases. During the expansion of the bubble in the molecular cloud, the high temperature of the shocked stellar wind produces a high rate of evaporation by thermal conduction, which keeps the density, and therefore the cooling rate inside the bubble, relatively high. In the case, the bubble has been kept confined in the compact stage for much longer than in the case, and its further growth finds greater opposition by the thicker HII layer. Cooling by radiation of the gas inside the bubble along its history is thus much higher in the than in the case, thus reducing the ability of the hot gas to drive the expansion of the bubble. We estimate that the energy lost radiatively since the beginning of the expansion of the bubble amounts to % of the internal energy of the bubble at the time of blow out, as compared to less than 10 % for the case discussed in Sect. 3.1.1. This is mostly due to the greater age of the bubble at the time of blowout for the case as compared to the case (95,000 years and 30,000 years, respectively).
3.1.3. B0 star,
The evolution of the density of the gas surrounding a B0 star embedded in a molecular cloud at a distance from its edge is shown in Fig. 7. This case is different from the previously studied ones in that, although is relatively large, the condition implies now that the ionized compact shell is rather thin at the time of blow out, which takes place when the age of the shell is only 13,000 years. As a result, the pressure of the hot bubble at blow out is relatively high, 200 times that of the intercloud gas, and the original ionized shell is quickly disrupted.
The combination of a strong pressure discontinuity and a thin shell causes large velocities of the hot gas along the axis of symmetry, and therefore, given the small size of the bubble, large velocity gradients. Such velocity gradients have an important dynamical effect on the expansion: given that, for an adiabatically expanding monatomic gas, the sound speed , a linear expansion in the direction z causes the ratio of ram-to-thermal pressures vary as
where we have assumed a stationary flow, an acceptable approximation given that is well below the expansion timescal of the bubble. Eq. (22) thus indicates that the large velocity gradients generated by the bubble breakout dramatically decrease the importance of the thermal pressure in front of the ram pressure. In this case, therefore, the expansion of shell of swept-up gas in the intercloud medium is mostly ram-pressure dominated, and takes place predominantly in the z direction. This produces a fairly well collimated flow of hot gas, rather than a bubble. The temperature of the shocked gas at the head of the jet is somewhat below K, and about half that value along most of the jet length. Fig. 8 shows in detail the distribution of the ratio of the momentum in the z direction to the thermal pressure, comparing a thermal pressure-dominated case (that described in Sect. 3.1.1) with the present one.
In our simulations presented in Sects. 3.1.1 and 3.1.2, the expanding bubble decelerated and became close to pressure equilibrium long before it could overtake the ionization front released into the intercloud medium after blowout, i.e., the Strömgren radius in the diffuse intercloud gas. However, in the case described here the ionization front lies closer to the star, and is overtaken by the jet of hot gas in somewhat more than years. This can be clearly seen in the last two frames of Fig. 7, which include the outer boundaries of the diffuse HII region. Only in the direction of the jet, due to the much smaller column density, the ionization front can extend to a large distance from the star. On the other hand, the almost lack of expansion in the directions perpendicular to the jet allows the ionized shell to expand against the diffuse HII region, merging with the outflowing ionized gas from the compact component in the molecular cloud near the base of the shell. At the end of our simulation, 140,000 years after blow out, the diffuse HII region in the intercloud medium, the side walls of the jet, and the outer parts of the champagne flow all have densities and pressures within a factor of 2 from each other.
3.1.4. O4 star, , without thermal conduction
Thermal conduction between the hot gas and the warm dense gas in the shell is an essential ingredient in the evolution of wind blown bubbles in the interstellar medium (Weaver et al. 1977; Mac Low & Mc Cray 1988; Van Buren et al. 1990; Shull & Saken 1995). Most of the hot gas inside a bubble comes from material evaporated from its walls, rather than from the stellar mass carried by the stellar wind. It also plays an indirect but fundamental rôle in the cooling of the bubble interior, by determining its density and therefore its radiative cooling rate. For the reasons indicated at the end of Sect. 2.2, the analytical approximation of Weaver et al. 1977 for bubble expansion in a uniform medium is not substantially modified by the inclusion of thermal conduction in the early stages of bubble expansion, given the initial set of parameters with which we are dealing here (although this may not be true in other scenarios; see for example Van Buren et al. 1990). In particular, the internal pressure has a similar evolution in both cases, due to the generally small radiative losses in the bubble interior. However, the temperature inside the bubble varies by nearly two order of magnitude depending on whether only the mass yielded by the star is considered as contributing to the density inside the bubble, or whether the evaporated mass from the shell is also included. This has very important consequences for the high energy emission from the bubble interior, as will be seen in Sect. 3.2.1. On the other hand, we may expect that the dynamical evolution of the bubble after blow out will also be sensitive to the inclusion or not of a thermal conduction term in the energy equation, since the cooling rate becomes important in later evolutionary stages,
To assess the importance of thermal conduction in the dynamical evolution of our model HII regions, we have reproduced the simulations discussed in Sect. 3.1.1, but now without allowing conductive energy exchange between cells, and assuming that the mass initially inside the bubble comes entirely from the stellar wind. The resulting evolution of the density in the computational plane is shown in Fig. 9, which is to be compared to Figs. 2 and 3.
In the initial stages, the larger density contrast between the intercloud medium and the hot gas at blowout enhances the instabilities in the expanding shell, as is visible in the first frame of Fig. 9. The initial expansion is thus faster in this case. As the expansion proceeds, the large thermal pressure pushes the outflow of dense gas from the compact HII region inside the molecular cloud sideways. Later on, a large scale fountain-like flow of hot gas fills the whole bubble: the upward flowing gas undergoes a moderately strong shock (due to its temperature of K) well inside the bubble, creating a thick buffer of gas with typical temperatures of K which expands in all directions. The initial champagne flow is pressed against the surface of the molecular cloud. However, gas continues to flow from the compact component, creating a plume of warm ionized gas completely enclosed within the bubble. A part of it is dragged up by the hot gas, tracing its turbulent flow inside the bubble in the last two frames of Fig. 9. In the case without thermal conduction, the champagne flow does not play any rôle in collimating the hot bubble gas. The final density of the compact component is 40 % greater than in the case which includes thermal conduction, due to the larger confining pressure generated by the shocked stellar wind. The ionization front expanding into the molecular cloud has nevertheless the same radius in both cases.
3.1.5. O4 star, , without stellar wind
Our final simulation corresponds to the classical champagne flow scenario, which we now follow using the same parameters as in the cases discussed in Sects. 3.1.1 and 3.1.4, but suppressing the stellar wind. The evolution of the density is presented in Fig. 10. The absence of the hot gas bubble simplifies the dynamics of the gas, which now follows the behaviour outlined by Tenorio-Tagle et al. 1979. The outflow of ionized gas from the compact component reaches velocities of up to 50 km s-1 bounded by a broad, weakly shocked layer which expands into the intercloud medium at a velocity of about 20 km s-1 in the last frame. The volume affected by the expansion of the HII region after blow out is thus much smaller than in the case including the stellar wind, as could be guessed by comparing the relative sizes of the champagne flow and the overall extent of the bubble in the last frame of Fig. 3. The decreased pressure due to the absence of the stellar wind also reduces the expansion rate of the compact ionized component in the molecular cloud; 400,000 years after blowout, its radius is 3.1 pc, as compared to 4.2 pc for the compact HII component in the case including the stellar wind.
We also find differences in the sizes of the regions reached by the champagne flows. Although the one in the case with stellar wind can be calculated only roughly, due to its shape being distorted by the bubble, we estimate it to be about 22 pc, while the outer radius of the expanding flow in the windless case is 15 pc. This is due to the faster initial champagne flow in the case including wind, due to the greater pressure (by a factor of nearly 4) at the time of blow out, which produces a faster initial growth. The evolution of the pressure inside the compact component differs remarkably between the two cases, with the expansion of the bubble producing a faster decline. At the end of the time span covered by the simulations, the pressure in the case with stellar wind has dropped below 90 % of the corresponding value in the windless case. The more intense initial outflow, plus the drop in pressure, make the density in the compact component to be 120 cm-3 in the case with wind, and twice that value in the windless case at the end of the 400,000 years. The decline of the pressure contributed by the shocked stellar wind inside the compact component is fast after blowout. At late stages, the low density shocked stellar wind quickly leaves the compact component in the direction of the intercloud medium, leaving the stellar wind-contributed pressure in the compact component near its minimum value, given by the ram pressure of the stellar wind (). The pressure in the compact component thus becomes determined by the balance between the rate at which cloud material is ionized, and the rate at which it can escape in the champagne flow.
The different time evolutions of the pressure and the density inside the compact component have an influence on the rate at which material from the cloud is photoionized, as already outlined in Sect. 3.1.1, and ultimately determine the survival time of the cloud after photoionization has set in. Obviously, this also determines the final extent of the champagne flow and its morphological relevance with respect to the stellar wind-bolwn bubble. The stellar wind plays only an indirect rôle by regulating the pressure inside the compact component and, to a lesser extent, by evaporating warm ionized gas from the inner walls of the compact HII region into the bubble. The photoionization rate of the cloud can be expected to be sensitive to many parameters, such as the density, the ionizing flux, the degree of clumpiness, or the depth of the ionizing star inside the cloud. Including the stellar wind only adds more free parameters to the problem, without it being likely to change the essential fact that photoionization is the ultimate agent driving the destruction of the cloud. This conclusion stands even if the star explodes as a supernova while still partially embedded in the cloud (Yorke et al. 1989).
3.2. Simulated maps
Observations in different windows of the electromagnetic spectrum permit useful diagnostics on aspects such as the physical conditions of the gas across the HII region, its internal dynamics, the parameters of the exciting star, or the metal abundances. Our simulations allow us to proceed inversely: having calculated the values of physical variables as they evolve in the neighbourhood of a star with a given set of parameters, we can derive simulated maps in different spectral domains. Such simulated maps in visible wavelengths were produced by Yorke et al. 1983, 1984 for the windless champagne phase.
Here we present maps of high energy and centimeter emission which probe well differenced components of our model HII regions: the hot gas inside the bubble, and the moderately dense ionized gas properly defining the HII region.
3.2.1. X-ray emission
Recently, HII regions in our Galaxy and the Magellanic Clouds have been extensively observed in X-rays thanks to the information collected by the Einstein and ROSAT satellites (see the Introduction for references), and they will surely be a primary target for future missions such as AXAF and XMM. X-ray observations are generally presented in the form of maps of the intensity emitted in different broad bands, allowing one to define a hardness ratio primarily related to the temperature of the emitting region. The galactic interstellar medium is relatively transparent to X-ray bands with energies around 1 keV, where intense emission is expected from gas at the temperatures of wind-blown bubbles or young supernova remnants. This makes these observations a very useful tool for the study of both the internal processes of these objects and the galactic structure as traced by them.
The maps of X-ray emission presented here use the curves of integrated cooling efficiency in broad bands calculated by Raymond et al. 1976, which include free-free, recombination, and line emission. The emissivity per unit volume is thus . To directly use the results of Raymond et al. 1976, we have considered two broad bands covered by the ROSAT channels, namely those between 0.53 and 0.87 eV (the "soft" band) and between 0.87 and 1.56 keV (the "hard" band). The effects of interstellar extinction can be noticeable in these bands. Given the low density of the intercloud medium where the bubble expands, extinction of X-rays from most of the bubble volume can be expected to be produced almost entirely in the intervening interstellar medium, unrelated to the HII region. We do not consider this distance-dependent interstellar extinction in our maps. In the case of the X-ray emission produced inside the compact component, however, the line of sight will generally pass through a considerable column of dense gas, unless it is seen nearly pole-on (where pole-on is defined as the situation in which the line of sight is perpendicular to the cloud boundary). We will consider this extinction as internal, in the sense that it is produced in the molecular cloud + HII region complex. Given typical densities and lenghtscales of our simulations, internal extinction dominates over the interstellar extinction for HII regions located within a few kiloparsecs from the Sun. In order to take into account extinction in our calculations, we have used the cross section values of Morrison & McCammon 1983. An accurate evaluation of attenuated X-ray emission maps would require to consider the variation of the extinction over the broad bands considered, as well as the precise spectral energy distribution of each X-ray emitting volume. To simplify, we have used instead the value of the extinction averaged over the wavelength interval covered by each band. We have not considered in our maps the contribution of the X-ray emission arising from the immediate neighbourhood of the exciting star itself (Chlebowski & Garmany 1991).
Our results are presented in Fig. 11 for the case discussed in Sect. 3.1.1. The bubble is shown from two different vantage points, forming angles of 200 and 700 with the axis of symmetry, at the end of the interval covered by our simulations. Fig. 11 displays the integrated emission in the soft band and the hardness ratio H defined as
where L is the luminosity in the corresponding band.
A characteristic feature of the dynamics of the hot gas in our models is the production of two peaks in the X-ray luminosity, which become more detached as we proceed to higher energies. Such a distribution is reminiscent of that observed in some HII regions such as RCW 49 (Belloni & Mereghetti 1994). We refer to those peaks as the compact and the extended components, arising from the vicinity of the star and from the bubble expanding in the intercloud medium, respectively. As could be expected, the most intense X-ray emission comes from the compact component, both in soft and hard rays. However, when the HII region is seen at a high angle from the axis of symmetry, the soft radiation from the compact component is greatly atenuated by the intervening dense gas. The effects of this can be appreciated in the bottom left panel of Fig. 11, where the spot of large hardness ratio (nearly 1) lies near the position of the star, but further inside the cloud. This is almost entirely due to extinction, rather than to the intrinsic hardness of the radiation in that region; the more pole-on view in the right side panels minimizes the effects of the extinction. The intrinsic soft X-ray emission from the compact component has a peak approximately 40 times more intense than that of the extended component. The intrinsic hardness ratio reaches a maximum of -0.46 in the compact component and -0.78 in the extended one, dropping to -0.93 in the chimney connecting the compact component with the shocked gas in the extended one.
3.2.2. Free-free emission
Centimeter-wavelength observations are widely used in studies of HII in regions (see e.g. Spitzer 1978, Scheffler & Elsässer 1987, Gordon 1987, Osterbrock 1989, for introductions to the field). The continuum emission is dominated by thermal bremsstrahlung, and high-level transitions of hydrogen and helium (the so-called recombination lines) are abundant in this wavelength range. The high resolution presently available at these wavelengths in velocity and, with aperture synthesis, also in position, make these observations a fundamental tool for the study of the physical conditions and internal kinematics of HII regions.
We have produced simulated maps of free-free continuum emission in the centimeter wavelengths where the HII region is optically thin. We estimate the continuum optical depth using the approximation of Altenhoff et al. 1960:
where is the frequency in GHz, is the electron temperature in K, and EM is the emission measure in cm-6 pc. For the set of conditions of our simulations, the maximum value of reaches only in the compact component for the commonly used wavelength cm, being much smaller elsewhere. The intensity I depicted in our maps is the emissivity integrated along the line of sight l:
where x is the ionization fraction. The proportionality factor depends weakly on the frequency of the observations. The weak dependence of Eq. (25) also on the temperature makes our maps to trace essentially the emission measure of the ionized gas.
The results are presented in Fig. 12. Like for the maps of X-ray emission, the HII region is shown from two vantage points, with lines of sight forming angles of 200 and 700 with the axis of symmetry. The figures depict the case discussed in Sect. 3.1.1 at the end of the period covered by the simulations. For the sake of comparison, a free-free continuum map is also shown for the windless case in Fig. 12.
Free-free emission is dominated by the compact component, and limb brightening can be appreciated in the extended component both in the cases with and without wind. In the case with wind, this is due simply to the fact that the HII region is the envelope of the bubble of hot gas, while in the windless case the limb brightening outlines the weak shock caused by the expansion of the ionized gas in the cloud and intercloud medium. The compact component is also limb-brightened in the case with wind, but not in the windless case, in which the intensity decreases by a factor of 2.2 going from the center to the limb (although this is difficult to appreciate in Fig. 12, due to its dynamic range of two orders of magnitude to account for the differences between the compact and the extended components). In both cases, there is an extended halo of emission produced in the intercloud medium, ionized by the release of the ionization front shortly after the bubble blowout. The actual intensity of this halo depends on whether the diffuse HII region extending in the intercloud medium is density or ionization bounded. For our O4 star, the Stromgren sphere in the intercloud medium would extend out to a distance of nearly 50 pc from the star and, assuming it to be hemispherical, would contain about M of gas. This is of order of the total HI contents of some typical giant molecular clouds in the solar neighbourhood (Blitz 1993), so the most likely scenario for this diffuse component is the density bounded one. Therefore, in our simulated maps we have not added the contribution of diffuse ionized gas lying between the limits of our computational grid and the outermost ionization front.
In spite of the differences discussed in Sect. 3.1.5, the peak intensities in the compact and extended components are similar in the cases with and without wind in the evolutionary stage presented here. In the compact component, the lower density in the case with wind is compensated by its larger extent, while in the extended component the higher density and larger size of the expanding shell is compensated by its thinness. This leaves its peak emission measure similar to that of the weakly shocked outer boundary of the champagne flow.
3.2.3. Low frequency recombination line emission
Our last set of maps show the ratio of intensities between a hydrogen recombination line and the underlying free-free continuum at a fixed frequency. The choice of a single frequency, rather than the value integrated over the line, is intended to put emphasis on the kinematics, rather than on the physical conditions of the emitting gas.
To calculate the line to continuum intensity ratio at a given frequency , , we first consider its LTE value:
where n is the principal quantum number of the final state of the transition, is the normalized line profile function, and A depends on , n, the difference between levels in the transition , the oscillator strength of the transition, and the helium to hydrogen ratio. In the centimeter range considered here lie many of the recombination lines () with values of n around 100; therefore, the exponential term in the numerator of Eq. (26) has a value very close to unity across the ionized region.
Eq. (26) should however be modified in case of NLTE. For transitions to values of n around 100, as those considered here, the population factors describing the departure from LTE are expected to be well below unity for all the range of physical conditions covered by our simulations (Sejnowski & Hjellming 1969), so we cannot directly apply Eq. (26). For NLTE in optically thin conditions (, which applies here as discussed above), the line intensity is modified (Gordon 1987) by:
where is the line intensity in LTE, is the departure level of the transition, and is
Curves giving the value of as a function of n can be found in Sejnowski & Hjellming 1969 for typical conditions in HII regions, ranging from diffuse to compact. For transitions to levels with at frequencies around 5 GHz (6 cm), we find , which multiplied by our estimate of the maximum value of in our simulations, , makes the factor in parentheses in Eq. (26) deviate from unity by not more than about 4%. In good approximation, therefore, our line intensities are reduced with respect to the LTE values by a factor . This factor is somewhat dependent on the temperature and density of the medium but, from Figs. 2 and 3 of Sejnowski & Hjellming 1969, we find that lies always between 0.7 and 0.8 for the conditions found in our simulations. Taking all this into account, we can use Eq. (26) to produce our maps, simply reducing it by a factor 0.75, without introducing an error greater than 10 %. We have thus engulfed this factor in the constant A and, since the latter depends on the precise characteristics of the transitions to be considered, we have left our maps of line to continuum ratio uncalibrated for the sake of generality.
For the line profile function we have used a Gaussian shape, considering only thermal width, Doppler shifted by the bulk motion of the emitting volume. Writing it explicitly as a function of the macroscopic LSR velocity v rather than the frequency,
being the hydrogen atom mass. For transitions to levels of in the range of densities found in our simulations, thermal pressure broadening increases the line width over the thermal component by only a 2 % at most (Brocklehurst & Seaton 1972), and we have not considered it here.
The results are presented in Fig. 13. The cases considered are the same as for Figs. 11 and 12. The upper panels are centered at a radial velocity of 30 km s-1 away from the observer, while the bottom ones are centered at the same velocity but with opposite sign. The grey scales in Fig. 13 are adjusted to cover the appropriate dynamic range for each panel, i.e., a given intensity of grey does not correspond to the same value of the line-to-continuum ratio in all the panels.
As can be expected, the aspect of these velocity maps strongly depends on the angle between the axis of symmetry and the line of sight. In the nearly edge-on view displayed in the upper left panel, the brightest areas correspond to the far side of the expanding shell receding from the observer. In the upper right panel, since the bulk of the material is moving towards the observer, the ratio remains very small over the whole area, and is in fact dominated by the wings of the nearly zero-velocity gas of the diffuse ionized intercloud medium. In the bottom panels, the wings of the lines produced in the parts of the expanding shell approaching the observer dominate (this is, most of the gas in the nearly pole-on view of the last panel). The gas in the compact component also expands, but much more slowly than the 30 km s-1 at which our maps are centered. The high density of the compact component provides a bright continuum background to the line emission from the shell in the regions that are seen projected against it. This is the reason why its position is marked by a dark circle in the bottom panels of Fig. 13.
A common feature to the cases with and without wind is that most of the high density gas is contained in the compact component and the champagne flow. These regions are thus those which dominate the emission measure integrated over the whole area of the HII region; as a consequence, one should not expect appreciable differences in the integrated line profiles between the cases with and without wind in observations of distant, unresolved HII regions. To illustrate this point, we have integrated the line profiles over the whole projected area of the simulation seen pole-on. This should be the most favourable case to make out the differences between the cases with and without wind. The results show that differences in the integrated line profile can be noticed only in the high velocity wings, primarily the blueshifted one. Moreover, if the lines of light elements are considered, these differences are almost completely masked by the thermal wings of the much more intense emission from the gas moving at slower velocities. To enhance the differences between both cases, we present in Fig. 14 the results of the calculations with the thermal linewidth used in Eq. (29) reduced by a factor of 100. Note that the scale in Fig. 14 is logarithmic, i.e., significant differences begin to appear only at emission levels about four orders of magnitude fainter than the peak. The main departure from the broad, asymmetric profile of the windless champagne phase (which is clearly different from the typical profiles produced by expanding shells; Tenorio-Tagle et al. 1996) is produced at velocities larger than 40 km s-1, arising in the expanding shell. Larger velocities at very low emission levels come from the boundary layers between the shell and the hot gas. This low level emission probably has an important contribution from numerical effects, due to the sharp gradients in density, temperature and velocity in these layers, which are smoothed over a few computational cells.
© European Southern Observatory (ESO) 1997
Online publication: April 8, 1998