Astron. Astrophys. 337, 517-538 (1998)

4. Physical processes in the neutral region

4.1. X-ray ionization and heating

The bulk of the Lyman continuum flux incident upon the neutral shell, especially those photons with h eV (the He⁺ ionization energy), are absorbed (and produce) in the ionized HII region (the planetary nebula) interior to the neutral shell, with density and thickness . However, a small fraction of the Lyman continuum photons, those with energies eV, significantly penetrate into the neutral gas to columns cm^-2, and partially ionize and heat the atoms and molecules there. As T_* rises to its maximum of K, this fraction rises to values of .

We follow the attenuation of the soft X-ray flux and the ionization of the neutral gas, using standard cross sections for H, He and H₂ (e.g., Maloney, Hollenbach, & Tielens 1996). At the relevant photon energies ( eV) hydrogen and helium dominate the X-ray opacity and the ionization is dominated by secondary electron collisions. An approximate analytic solution to the X-ray ionization rate (per H) as a function of effective shielding column (see Maloney et al. for a description of the method of solution) is given:

where cm^-2, , N is the column in the PDR, cm^-2, , and is a parameter proportional to the incident X-ray photon flux (see below). Eq. 8 is valid for , so that the ionization rate is dominated by photons with h significantly above the He⁺ ionization threshold. is furnished by the X-ray shielding column in the HII region and in the cm^-2 transition region between the totally ionized HII region and the partially ionized PDR. is given by the following expression for the incident photon flux (in photons cm^-2 s^-1 Hz^-1) for eV or :

where is the stellar radius and Hz is the threshold frequency for He⁺ ionization. Eq. 8 is derived assuming that 0.4 of the primary electron energy goes into secondary ionizations, appropriate for (Shull & van Steenberg 1985).

In our models the X-rays are mainly important as a heating mechanism for columns cm cm^-2 and when the central star temperature rises above K. The soft X-ray heating per hydrogen nucleus is given by (see Maloney et al. 1996)

where is the fraction of the primary electron's energy which is converted to heat, is the Lyman edge frequency and is the X-ray ionization rate given by Eq. 8. We adopt from Shull & van Steenberg (1985); when X-ray heating is important.

4.2. Chemistry in the photodissociation region (PDR)

Initially, the gas in the ejected shell or torus is assumed to be molecular, with all hydrogen H₂ and either all gas phase carbon (C/O) or oxygen (C/O) incorporated into CO. In this paper we assume a carbon-rich environment such that gas phase C/O . The grains are assumed to be identical to interstellar dust, with standard abundances and size distributions.

The HII region mass, the chemical abundances in the neutral gas, and the neutral gas temperature all then evolve with the rise and fall of the incident FUV, EUV (Lyman continuum) and X-ray fluxes and the falling gas density as the shell expands outwards. The EUV photons with 13.6 ev eV create an inner HII layer, with evolving mass (Fig. 3) and density . The FUV and soft X-rays dominate the chemistry and heating in the PDR. We note that the generalized definition of a PDR includes regions where the gas is entirely molecular H₂, but where the FUV or X-ray photons still dominate the heating or the dissociation of CO, OH or H₂O (Tielens & Hollenbach 1985).

4.2.1. H₂ chemistry

The neutral hydrogen is initially entirely molecular, but the onset of the FUV flux sends a wave of dissociation into the neutral shell. We adopt a simple expression for the photodissociation rate from Draine & Bertoldi (1996, their Eq.[37]),

where the term in brackets represents the self shielding, cm^-2, is the H₂ column density, cm^-2 s^-1 is the dissociation rate without self shielding (the exponential term is the dust shielding of the FUV), is the FUV flux in the interval 11 eV eV in units of the ISRF ( photons cm^-2 s^-1), and is the H₂ density.

H₂ is also dissociated by collisions with H, H₂ and electrons, by the ionization caused by non-thermal electrons from the soft X-rays (see Sect. 4.1), by reaction with O⁺ (Maloney et al. 1996), and by reaction with C⁺ and O at high ( K) PDR temperatures which allow activation energy barriers to be overcome. The O⁺ destruction mechanism is only important in the X-ray ionized regions in which H⁺ charge exchanges with O to form relatively high abundances of O⁺.

The H₂ reforms on grain surfaces, and we assume the grains to have interstellar properties so that the formation rate of H₂ is given

with cm³ s^-1, n the gas density and the atomic hydrogen density (cf., Tielens & Hollenbach 1985).

H₂ can also be formed by the reaction of H^- with H. When X-rays are important, the electron (and therefore H^-) abundance is elevated so that this rate can become comparable to grain formation. This rate can be written

where the rate coefficient (in cm³ s^-1) is given by (Hollenbach & McKee 1989)

with K, is the fractional abundance of atomic hydrogen, that of protons, represents the photodissociation term for H^-, s^-1, and cm^-3.

In summary, we treat the H₂ chemistry in detail, numerically integrating the time dependent equations as the density and radiation fields evolve and as the ionization front and dissociation front move with respect to the PDR gas. Time dependent calculations are needed, for example, when the timescale for H₂ formation, yrs (see Eq. 13), exceeds the yr timescale for changes in the FUV flux (see Fig. 2) or the timescale for molecular gas to advect through the PDR as the dissociation front advances.

4.2.2. Carbon and oxygen chemistry

In this initial paper, we have greatly simplified the carbon and oxygen chemistry in order to maintain a relatively simple and efficient computer code. As discussed below, we analytically estimate the columns of C⁺ and O, and estimate the possible contribution of OH and H₂O to the cooling. We defer a careful treatment of the carbon and oxygen chemistry to a subsequent paper (Latter et al, in preparation), in which the results of this paper will be used as input to a separate chemical code which will provide, for example, CO, CO⁺, and HCO⁺ line intensities and a careful calculation of OH and H₂O abundances and the C⁺/CO boundary layer.

The PDR results of Tielens & Hollenbach (1985), Wolfire, Tielens, & Hollenbach (1989), and Hollenbach, Tielens & Takahashi (1991) show that carbon is ionized to hydrogen columns of roughly cm^-2 and the oxygen is entirely atomic to this same column. Beyond for C/O, the oxygen is in CO, and all carbon not in CO is assumed to be atomic. For the case C/O1, all the carbon is in CO and all oxygen not in CO remains atomic beyond . In this initial paper, we use this prescription for the abundances of C⁺ and O and calculate the temperature structure and the CII 158µm and OI 63µm, OI 145µm intensities.

At columns , and when the temperature of the neutral gas is K and significant H₂ is present, neutral-neutral reactions with activation energies K can proceed at a sufficiently rapid rate to produce large quantities of OH and H₂O. The OH and H₂O are destroyed by C⁺ and by FUV photodissociation.

The rate coefficients for these reactions, the reverse reactions, and the FUV photodissociation reactions are given, for example, in Hollenbach & McKee (1989). Although most of the oxygen is atomic, the cooling by OH and H₂O in some cases may be important. We include this reaction sequence to provide an estimate of OH and H₂O abundances in the extreme cases where their cooling may dominate in the PDR.

4.2.3. Ionization balance

If the star has not reached its "X-ray emitting" temperatures, the ionization balance in the PDR zone is simply given by the FUV ionization of C to C⁺. However, when X-rays are important, H, H₂, and He are ionized by X-rays to form H⁺, H and He⁺. These ions are destroyed by recombination with electrons, reactions with atoms and molecules, and collisions with small grains and PAHs. Maloney et al. (1996) detail this chemistry and give approximate analytic solutions for the electron abundance . We use these analytic results, appropriately modified to include the effect of the external FUV field. The resultant electron abundance sets the level of X-ray heating (see Sect. 4.1) and helps to set the formation rate of H₂ via H^-.

4.3. Thermal balance and radiative transfer

4.3.1. Heating mechanisms

X-ray heating, which is important at low columns and high stellar temperatures, has been discussed in Sect. 4.1. The neutral gas is also heated by grain photoelectric heating (Bakes & Tielens 1994), FUV photodissociation of H₂ and photoionization of C, and FUV pumping or formation pumping of the H₂ molecule, followed by collisional deexcitation of the excited vibrational levels (cf., Tielens & Hollenbach 1985, Hollenbach & McKee 1989, Burton, Hollenbach & Tielens 1990). These processes are well known and are often applied in the context of equilibrium chemistry. The time dependent H₂ chemistry tends to produce higher H₂ abundances than an equilibrium calculation, and, as a result, enhances the significance of FUV pump heating by H₂.

4.3.2. Cooling mechanisms and radiative transfer

Significant coolants in the neutral region include vibrational and rotational transitions of H₂, rotational transitions of CO, OH, and H₂O, fine structure transitions of O, C⁺ and C⁰, grain cooling of the gas, and cooling due to adiabatic expansion. We also include collisional excitation of Ly, OI 6300Å, SII 6730Å, and FeII 1.26µm, FeII 1.64µm, which are significant coolants if the neutral region is driven to K. The cooling rates for these processes have been taken from Hollenbach & McKee (1979,1989) and Tielens & Hollenbach (1985). The OI 63µm, CII 158µm, CO, OH, and H₂O lines are treated with the escape probability formalism described in Hollenbach & McKee (1979), because their line opacities can be large and self-absorption followed by collisional deexcitation may be important.

The emergent intensities in the lines are found by integrating the (escape probability-corrected) thermal emissivities through the thickness of the neutral slab, and dividing by 4. For the case of the vibrational emission from H₂, we add to the thermal component a "nonthermal" contribution to the 1-0S(1) and 2-1S(1) intensities caused by the FUV pumping and formation pumping of the vibrational states (see Burton et al. 1990 for details of this procedure). Our model for H₂ does not assume LTE, but calculates the statistical equilibrium of the vibrational levels as they are populated by collisions and FUV pumping and depopulated by spontaneous emission and collisions. We have checked in some cases our model results with those of Draine & Bertoldi (1996) and find good agreement (to within a factor of 2) with their more detailed treatement which included more up to date collisional rates and 299 bound states of the H₂ molecule. We also include the excitation caused by the collisions of H₂ with nonthermal electrons produced by X-rays (Voit 1991).

4.4. Shock processes

The shell, moving at 25 km s^-1, overtakes and shocks the red giant wind ( km s^-1) at a relative speed of km s^-1. Relative speeds of this order probably provide the maximum output of H₂ 2µm shock emission. Slower shocks do not heat the gas sufficiently to produce the vibrational emission (which originates 6000 K above ground); faster shocks ( km s^-1) dissociate the H₂ (e.g., Burton, Hollenbach & Tielens 1992).

The preshock density (in cm^-3) is given by the ambient density of the red giant wind at distance R from the central star

where is the rate of mass-loss in units of yr^-1, is the filling factor of the red giant wind and cm s^-1. The intensity and (in erg cm^-2 s^-1 sr^-1) of the 1-0S(1) and 2-1S(1) H₂ lines emerging normal to the shock is taken from an analytic fit to the J shock model described in Burton, Hollenbach & Tielens (1992).

where is the pre-shock density in units of cm^-3. These intensities are calculated from shock models by integrating the H₂ line emissivity through the shock. The population in the upper state of the transition is determined from a statistical equilibrium calculation which includes excitation by collisions and depopulation by collisions and spontaneous emission (see Hollenbach & McKee 1989). Eqs. 17 and 18 are good to within a factor of 2 for where cm s^-1. We treat the shock as a J shock, as opposed to a C shock (Draine 1980), because the magnetic field in the red giant wind is presumably very small. In any event, our 10-20 km s^-1 J shocks produce more H₂ 2 µm emission than a 10-20 km s^-1 C shock, which is cooler, so our treatment provides at least an upper limit to the H₂ 2 µm shock emission. The luminosity in the lines (in units of ) is given

for cm^-3 and . With these expressions, the ratio for =10 km s^-1 and 0.2 for =17 km s^-1, a value we have used in most of our calculations.

It should be noted that in cases where the shell is driven by a fast ( km s^-1) wind from the central star, there are two shocks: a wind shock on the inside of the shell where the wind overtakes and drives the shell, and the "ambient" outer shock we have modelled above where the shell overtakes the red giant wind. Because the wind shock is highly dissociative, it is not a strong source of H₂ emission (Hollenbach & McKee 1989). It could, however, be a source of X-rays which penetrate and radiatively heat the neutral shell.

© European Southern Observatory (ESO) 1998

Online publication: August 17, 1998
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