A&A 436, 397-409 (2005)
R. Meijerink 1 - M. Spaans 2
1 - Sterrewacht Leiden, PO Box 9513, 2300 RA Leiden, The Netherlands
2 - Kapteyn Astronomical Institute, PO Box 800, 9700 AV Groningen, The Netherlands
Received 19 November 2004 / Accepted 22 February 2005
We present a far-ultraviolet (PDR) and an X-ray dominated region (XDR) code. We include and discuss thermal and chemical processes that pertain to irradiated gas. An elaborate chemical network is used and a careful treatment of PAHs and H2 formation, destruction and excitation is included. For both codes we calculate four depth-dependent models for different densities and radiation fields, relevant to conditions in starburst galaxies and active galactic nuclei. A detailed comparison between PDR and XDR physics is made for total gas column densities between 1020 and 1025 cm-2. We show cumulative line intensities for a number of fine-structure lines (e.g., [CII], [OI], [CI], [SiII], [FeII]), as well as cumulative column densities and column density ratios for a number of species (e.g., CO/H2, CO/C, HCO+/HCN, HNC/HCN). The comparison between the results for the PDRs and XDRs shows that column density ratios are almost constant up to for XDRs, unlike those in PDRs. For example, CO/C in PDRs changes over four orders of magnitude from the edge to . The CO/C and CO/H2 ratios are lower in XDRs at low column densities and rise at . At most column densities , HNC/HCN ratios are lower in XDRs too, but they show a more moderate increase at higher .
Key words: astrochemistry - galaxies: starburst - galaxies: active
Gas clouds in the inner kpc of many galaxies are exposed to intense radiation, which can originate from an active galactic nucleus (AGN), starburst regions or both. O and B stars dominate the radiation from starbursts, which is mostly in the far-ultraviolet ( 6.0 < E < 13.6 eV), turning cloud surfaces into Photon Dominated Regions (PDRs, Tielens & Hollenbach 1985). Hard X-rays (E > 1 keV) from black hole environments (AGN) penetrate deep into cloud volumes creating X-ray Dominated Regions (XDRs, Maloney et al. 1996). For each X-ray energy there is a characteristic depth where photon absorption occurs. So for different spectral shapes, one has different thermal and chemical structures through the cloud. Although one source can dominate energetically over the other (e.g., an AGN in NGC 1068 or a starburst in NGC 253), the very different physics (surface vs. volume) requires that both should be considered simultaneously in each galaxy.
In PDRs and XDRs, the chemical structure and thermal balance are completely determined by the radiation field. Therefore, PDRs and XDRs are direct manifestations of the energy balance of interstellar gas and their study allows one to determine how the ISM survives the presence of stars and AGN (Tielens & Hollenbach 1985; Boland & de Jong 1982; van Dishoeck & Black 1988; Le Bourlot et al. 1993; Wolfire et al. 1993; Spaans et al. 1994; Sternberg & Dalgarno 1995; Stoerrzer et al. 1998; Spaans 1996; Bertoldi & Draine 1996; Maloney et al. 1996; Lee et al. 1996; Kaufman et al. 1999; Le Petit et al. 2002, and references therein).
PDRs and XDRs have become increasingly important as diagnostic tools of astrophysical environments with the advent of infrared and (sub-)millimetre telescopes. PDRs emit fine-structure lines of [CI] 609, [CII] 158 and [OI] 63 m; rotational lines of CO; ro-vibrational and pure rotational lines of H2; many H2O lines as well as many broad mid-IR features associated with Polycylic Aromatic Hydrocarbons (PAHs). In PDRs, the bulk of H2 is converted into atomic hydrogen at the edge and CO to neutral carbon into ionised carbon. XDRs emit brightly in the [OI] 63, [CII] 158, [SiII] 35, and the [FeII] 1.26, 1.64 m lines as well as the 2 m ro-vibrational H2 transitions. The abundance of neutral carbon in XDRs is elevated compared to that in PDRs and the chemical transitions from H to H2 and C+ to C to CO are smoother (Maloney et al. 1996).
In this paper, we compare a far-ultraviolet and X-ray dominated region code. For the PDR and XDR, we discuss the cooling, heating and chemical processes. Then we show four models with different radiation fields and densities, for a semi-infinite slab geometry and irradiation from one side without geometrical dilution. We conclude with a comparison between the column densities, integrated line fluxes and abundance ratios. We will apply this tool to the centres of nearby active galaxies in subsequent papers. These codes can be used over a broad range of physical situations and scales, e.g., young stellar objects, planetary nebulae or gas outflow in galaxy clusters.
The global properties of PDRs are determined by a number of physical processes:
The first few magnitudes of extinction of the PDR are usually referred to as the radical region since many carbon hydrides and their ions, e.g., CH, CH+, CN, HCN, HCO+ (and also CO+), reach their peak abundance there, caused by the presence of both C+ and H2 and the high ( 102-103 K) temperatures. Ion-molecule reactions take place that lead to the formation of a large number of different molecular species. Many of the atoms and molecules in the radical region of a PDR are collisionally excited at the ambient densities and temperatures, and emit brightly in the mid-IR, FIR, millimetre and sub-millimetre.
The global characteristics of any PDR are defined by a few key parameters:
The details about the thermal and chemical processes we use in the code are discussed in the appendices. In the rest of the paper we use G0, the Habing flux, as the normalisation in which we express the incident FUV radiation field, where G0=1 corresponds to a flux of .
In this section, we discuss the results for four PDR models in which we have varied the radiation field G0 and the density . The models are for a semi-infinite slab geometry, but the code also allows for two-sided slab geometries. The adopted model parameters are listed in Table 1. Models 2 and 4 will be shown in a paper by Röllig et al (in prep.), where they are used to compare 12 different PDR codes that are commonly used. The parameters are listed in Table 1. These values are typical for the high density, strong radiation field conditions we want to investigate in, e.g., a starburst.
Table 1: Adopted model parameters.
The fixed gas-phase and total abundances we use are given in Table 2. The total abundances are the average values of Asplund et al. (2004) and Jenkins (2004). To calculate the gas-phase abundances, we use the depletion factors calculated by Jenkins (2004).
Table 2: Abundances.
For both radiation fields and densities, the dominant source heating to a column density is photo-electric emission from grains. In the moderately low density, low radiation-field Model 1, viscous heating is about equally important in this range. For Model 2 where the radiation field is increased, it contributes somewhat more than 10 percent. For the low radiation-field, high density Model 3, carbon ionisation is the second most important heating source. In Model 4, where the radiation field is increased compared to Model 3, H2 pumping is the second most important. At high column densities ( ), [OI] 63 m absorption and gas-grain heating are important. For the low density PDRs Model 1 and 2, only [OI] 63 m dominates. When the density is increased in Models 3 and 4, gas-grain heating is equally important if not dominant. Other heating processes contribute less than 10 percent, but are sometimes important in determining the thermal balance.
In all models [OI] 63 m cooling dominates to . In the low density PDRs, [CII] 158 m cooling contributes more than ten percent of the cooling in this range, whereas at high densities, gas-grain cooling is the second most important coolant. In the high density, high radiation-field Model 4, this contribution can be almost forty percent. Deeper in the cloud, [CI] 610 m and CO line cooling become important. H2 line cooling can contribute up to 10 percent of the total cooling rate at some point, but is always a minor coolant.
The H H2 and C+ C CO transitions are quite sharp. Their actual location varies greatly, since this is strongly dependent on density and radiation field. Exposed to stronger radiation fields, the transitions occur deeper in the cloud, since the photo-dissociation rates are larger. At higher densities, the transitions occur closer to the surface of the cloud, since the recombination rates scale as n2. For the same reason, the H+ and O+ fractional abundances are systematically higher in the low density models. SiO and CS are more abundant and formed closer to the surface in the high density models, which is also the case for HCO+, HCN, HNC and C2H.
The edge temperatures (see Fig. 6) are affected most by the strength of the radiation field when the density is largest. At a density of , the difference is a factor of thirty for an increase from G0=103 to G0=105. In the low density case this is only a factor of two. Because of optical depth effects, CO cooling is less effective at high column densities. For this reason, temperatures rise again at in the low density models.
|Figure 1: Important heating processes for Model 1 ( top left), 2 ( top right), 3 ( bottom left) and 4 ( bottom right).|
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Compared to PDRs, the following processes play a role in XDRs (cf. Maloney et al. 1996), in part because of the production of UV photons as described above:
The global structure of any XDR is defined by a few key parameters: the density and the energy deposition rate (see Appendix E) per hydrogen atom. Because the heating in XDRs is driven by photo-ionisation, the heating efficiency is close to unity as opposed to that in PDRs where the photo-electric heating efficiency is of the order of (Bakes & Tielens 1994; Maloney et al. 1996). Unlike PDRs, XDRs are exposed to X-rays as well as FUV photons.
As one moves into the XDR, X-ray photons are attenuated due to atomic electronic absorptions. The lowest energy photons are attenuated most strongly, which leads to a dependence of the X-ray heating and ionisation rates at a given point on the slope of the X-ray spectrum. We assume, for energies between 0.1 and 10 keV, that the primary ionisation rate of hydrogen is negligible compared to the secondary ionisation rate and that Auger electrons contribute an energy that is equal to the photo-ionisation threshold energy (Voit 1991).
The treatment is described in the Appendices and follows, in part, the unpublished and little-known work by Yan (1997). Also, we extend the work of Maloney et al. (1996) in terms of depth dependence, H2 excitation and extent of the chemical network.
|Figure 2: Important cooling processes for Model 1 ( top left), 2 ( top right), 3 ( bottom left) and 4 ( bottom right).|
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In this section, we consider four models with the same energy inputs and densities as the PDRs in Table 1. The spectral energy distribution is of the form . The energy is emitted between 1 and 10 keV and should be replaced by in Table 1. This spectral shape and spectral range can be changed depending on the application. We take the parameters for the 1 keV electron to determine the electron energy deposition, since these parameters do not change for higher energies. When the spectral energy distribution is shifted towards higher energies, the X-rays will dominate a larger volume, since the absorption cross sections are smaller for higher energies. is the most important parameter for the chemical and thermal balance, where is the energy deposition rate per hydrogen nucleus. The abundances used are given in Table 2. The elements H, He, C, N, O, Si, S, Cl and Fe are included in the chemical network. The other elements listed are only used to calculate the photoelectric absorption cross section, .
In Fig. 1, the different heating sources are shown as a function of the total hydrogen column density, . All heating is done by X-rays, but the way it is transfered to the gas depends on the ionisation fraction. When the gas is highly ionised, , most () of the kinetic energy of the non-thermal electrons goes into Coulomb heating, which is the case in Models 1, 2 and 4 where is high to . For smaller ionisation fractions, , ionisation heating as discussed in Sect. B.2 is important or even dominant. In Model 3, ionisation heating and Coulomb heating are equally important at . In all models ionisation heating dominates especially at high column densities. When the excitation of H2 is dominated by non-thermal processes, collisional quenching of H2 can heat the gas. Naively, one would expect this dominance to occur where most of the X-rays are absorbed, but for high energy deposition rates , the temperature is high and thermal collisions dominate the population of the vibrational levels. Non-thermal excitation is dominant at low temperature, i.e., low .
In Fig. 2, the important cooling processes are shown as a function of total hydrogen column density, . At high temperatures (see Fig. 3), cooling by [CI] 9823, 9850 Å and [OI] 6300 Å metastable lines dominates, as is the case in the models with high radiation fields, Models 2 and 4. At lower temperatures, most of the cooling is provided by the fine-structure line [OI] 63 m (90), e.g., at the edge in the low-radiation field Models 1 and 3. In each model, gas-grain cooling dominates for low . In addition, specific cooling processes can be important in special cases. H2 vibrational cooling dominates at large depths in Model 2, but in Models 1, 3 and 4 it contributes no more than 10. H2 vibrational cooling is split into a radiative and a collisional part. When the excitation of H2 is dominated by non-thermal electrons, the gas is heated by collisional de-excitation of H2.
|Figure 3: Fractional abundances and temperature for Model 1 ( top left), 2 ( top right), 3 ( bottom left) and 4 ( bottom right).|
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|Figure 4: Fractional abundances for model 1 ( top left), 2 ( top right), 3 ( bottom left) and 4 ( bottom right).|
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|Figure 5: Comparison between the fractional abundances in the PDR ( left) and XDR ( right) for Model 4.|
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|Figure 6: Cumulative line intensities of [CII] 158 (solid), [SiII] 34.8 (dotted), [CI] 609 (dashed) and 369 m (dashed), for PDR ( left) and XDR ( right) models.|
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|Figure 7: Cumulative line intensities of [SI] 25.2 (solid), [OI] 63.2 (dotted), 145.6 (dashed), [FeII] 26.0 (dot-dashed) and 35.4 m (dotted-dashed), for PDR ( left) and XDR ( right) models.|
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|Figure 8: Cumulative column densities of C (dotted-dashed), CO (solid), C2H (dotted), H2O (dashed) and OH (dot-dashed), for PDR ( left) and XDR ( right) models.|
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|Figure 9: Cumulative column densities of CS (solid), HCN (dotted), HCO+ (dashed), HNC (dot-dashed) and SiO (dotted-dashed), for PDR ( left) and XDR ( right) models.|
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|Figure 10: Column density ratios CO/C (solid), CO/H2 (dotted), HCO+/HCN (dashed) and HNC/HCN (dot-dashed), for PDR ( left) and XDR ( right) models.|
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In Fig. 3, we show the temperature as a function of total hydrogen column density, . Variations in radiation field strength most strongly affect the high-density models. The temperature at the edge differs a factor of 30 in the high-density case. Since X-rays penetrate much deeper into a cloud than FUV photons, high temperatures are maintained to much greater depths into the clouds. is very important in determining the thermal balance. When is larger, this results in a higher temperature. Therefore, Model 2 has the largest temperature throughout the cloud. Density turns out to be important as well. Note that models 1 and 4 have similar incident and therefore have about the same temperature throughout the cloud.
In Figs. 3 and 4, we show the fractional abundances of selected species. is not only important in the thermal balance, but also in the chemistry. Therefore Models 1 and 4 with about the same incident , show similar abundances. The most striking difference with the PDR models is that there is no longer a well-defined transition layer C+ C CO present. On the contrary, both C and C+ are present throughout most of the cloud having fractional abundances of 10-5-10-4. Only at very low , which results in a low temperature, is there a partial transition to CO. The transition from atomic to molecular hydrogen is much more gradual than in the PDR models. A considerable amount of OH is present in all models at all column densities. The temperature determined by is important. In Model 3, OH has the greatest abundance (>10-6) at all column densities. In other models such large fractions are seen only at very high depths into the cloud. The formation of CO and H2O is most efficient at high densities and low . Therefore, these species have large abundances throughout the high-density, low-radiation field Model 3. In Model 4, where the radiation field is somewhat higher, CO and H2O reach high abundances only at high . At low densities, they are only formed at large depths into the cloud (Models 1 and 2). Secondary ionisations are most important for the production of H+. Recombination is slower at lower densities. Therefore, the H+ fractional abundance is highest in Model 4. HCN, HCO+, HNC, C2H, CS and SiO have much larger abundances at high temperatures than in the PDR models.
We now perform a direct comparison between the PDR and XDR models. To emphasise that XDRs penetrate much deeper into cloud volumes than PDRs, we use the same scale for all models. Thus, it is also possible to distinguish between gradients in abundance, cumulative intensity, column density and column density ratios. XDR Model 3 is only plotted to , since becomes too small and no reliable results are obtained at higher column densities.
In Fig. 5, we show for Model 4 the abundances of selected species. At the edge, both neutral and ionised species are more abundant in the XDR models, and the relative abundances also differ with respect to one another. In the XDR for example, the neutral species CH and CH2 are more abundant than CH+ and CH2+, respectively. In the PDR, this is the other way around. CN and CN+ are almost equally abundant at the edge in the PDR, while CN exceeds CN+ by three orders of magnitude in the XDR. Although the amounts of CS+ and HCS+ are larger than those for CS and HCS, respectively, at the edge of the cloud in the XDR, the abundance difference is less than in the PDR. The abundance of He+ is five orders of magnitude larger in the XDR, due to secondary ionisations. H- is enhanced by three orders of magnitude, due to the higher ionisation degree. It is also easily seen that in PDRs the fractional abundances vary over many orders of magnitude, while the abundances in XDR Model 4 stay almost constant to a column density of , where the transition from H to H2 starts.
In Figs. 6 and 7, we show cumulative line intensities for fine-structure lines at all column densities, i.e., the emergent intensity arising from the edge of the cloud to column density :
Although the total [CII] 158 m line intensity is higher in the XDR, the flux originating from the edge to is higher in the PDR except when the XDR is characterised by very high values which is the case in Model 2. In all PDR models, all carbon is in C+ at the edge, while a large part of the carbon is neutral in XDR Models 1, 3 and 4. In all models, oxygen is mostly in atomic form. The [OI] 63 m line intensity to is larger in the low-density XDR models, which is possible due to higher electron abundances. The intensity is lower in the low radiation, high density XDR Model 3, since the temperature is higher in the PDR. For Model 4 they are about the same, since the density where the line becomes thermalised is almost reached. In the XDR, all line intensities increase approximately steadily with increasing column density. PDRs, however, primarily affect cloud surfaces causing more sudden changes. The line intensities of [CI] 609 m and 369 m arise from a more or less well defined part part of the cloud and start to increase at column densities cm-2. The line intensities of [CII] 158 m are higher than those of [SiII] 35 m in the PDRs except in Model 4. This is in contrast to the XDR models, where the [SiII] 35 m line intensity is always stronger. The fact that [SiII] 35 m lines are quite strong in XDRs was already noted by Maloney et al. (1996). The line intensities for [FeII] 26 m and 35 m are higher for the XDR models except again for Model 3.
In Figs. 8 and 9, we show cumulative column densities for selected species. They illustrate again that XDRs affect whole cloud volumes and PDRs create layered structures. In PDRs, the increase in column densities is very sudden for all species. For example, C and CO show this due to the very distinct C+/C/CO transition. In the XDRs, however, the increases in column density are much more gradual. The only sudden change in XDRs is where the H/H2 transition occurs.
In Fig. 10, the cumulative column density ratios for CO/H2, CO/C, HNC/HCN and HCO+/HCN are shown as a function of total hydrogen column density. The ratios for the XDRs are almost constant up to , unlike those in PDR models. In PDRs, CO/C ratios increase by approximately four orders of magnitude from the edge (10-4) to (1). In XDRs, this ratio is constant to and then increases slowly. For each cloud size, while keeping the energy input the same, CO/C ratios increase at higher densities. The ratios go down for higher radiation fields. For the same density and energy input, CO/C is lower when the cloud is irradiated by X-ray photons, with the exception of Model 3 where this is only valid at . CO/H2 is somewhat more complex. When only the energy input is increased in PDRs, this ratio is higher when . For , the ratios are about the same. There is also a minimum where the H/H2 transition occurs. This minimum is more prominent for higher radiation fields. In XDRs, the CO/H2 ratio is lower when the radiation field is higher. In PDRs and XDRs, the CO/H2 ratios are higher when the density is increased. When the cloud is irradiated by X-ray photons, CO/H2 ratios are lower, again with the exception of Model 3 at . In PDR Models 1, 2, and 3, significant column densities for HCN, HNC and HCO+ are reached between and . Therefore, the HNC/HCN and HCO+/HCN ratios discussed are for column densities . In PDRs, HNC/HCN is lower when the density is higher. No significant changes are seen for different radiation fields in these columns. HNC/HCN is generally lower for high in XDRs. At high column densities, where is low, HCN/HNC ratios are equal or somewhat higher than those for the PDR. HCO+/HCN and HNC/HCN are of the same order of magnitude in PDRs, but in XDRs HCO+/HCN is higher in most cases.
We are indebted to Frank Israel for initiating this project and for his helpful suggestions and remarks. We are greatful to Ewine van Dishoeck for useful discussions on PDR and XDR physics. We thank the anonymous referee for a careful reading of the manuscript and constructive comments.
In PDRs the photo-electric emission from (small) dust grains and PAHs is the dominant heating source. We use the analytical expression given by Bakes & Tielens (1994) which is given by
where is the radiation field attenuated by dust absorption (Black & Dalgarno 1977) and the heating efficiency is given by
Note that the efficiency depends on the ratio , which is the ratio of the ionisation and recombination rates. is the visual extinction at optical wavelengths caused by interstellar dust. Bohlin et al. (1978) relate the total column density of hydrogen, to colour excess, E(B-V):
The visual extinction then follows consequently: and . Note that the results of Savage et al. (1977) are often used, but in this paper only H2(and not H I ) is taken into account.
At the edge of the cloud, most of the carbon is singly ionised. The photo-electron energy released in an ionisation is eV. The ionisation rate, at a certain point in the cloud, is given by . The heating rate due to the ionisation of carbon is then given by
After substitution of numerical values we get the following heating rate for the local radiation field :
where this time is the radiation field attenuated by dust absorption (Black & Dalgarno 1977), carbon self-absorption (Werner 1970) and H2 (de Jong et al. 1980).
Absorption of Lyman-Werner band photons leads to the excitation of H2. About 10% of the excitations leads to decay into the continuum of the ground electronic state (Field et al. 1966; Stecher & Williams 1967). The heating related to this dissociation is given by
where is the mean kinetic energy of the H atoms and is set to 0.4 eV (Spaans 1996). The excitation rate of H2 is given by , where is the local radiation field given by
Self-shielding is explicitly taken into account for the excitation of H2, by the introduction of the shielding factor (see Sect. D.2.3). After substitution of numerical values we get a heating rate of
FUV excitation is followed by decay to ro-vibrational levels in the ground state. Collisional de-excitation leads to gas heating. This cascade process is very complicated, but we simplify this process by using a two-level approximation (see Sect. D.2.2). The resulting heating rate is given by
where the coefficients are given by Hollenbach & McKee (1979)
Both of the above expression are in units of .
When gas and grains differ in temperature they can transfer heat through collisions. The heating rate of the gas is given by (Hollenbach & McKee 1989,1979)
The minimum grain size is set at Å and the dust temperature is given by
based on the results of Hollenbach et al. (1991).
Radiation pressure accelerates grains relative to the gas and the resulting drag contributes viscous heating to the gas. Grain acceleration time scales are short compared to other time scales, and therefore the grains may be considered moving at their local drift velocity, . All the momentum is transferred to the gas, predominantly by Coulomb forces. For drift velocities cm s-1 (Spitzer 1978), no significant gas-grain separation takes place. In the following we take cm s-1. The heating rate is given by
where is the grain volume density, is the grain charge, and are the respective and electron volume densities and the functions and G(y) are given by
where and the thermal velocity of C+ ions and electrons. The error function is given by
At large column densities, cosmic ray heating can become important. Glassgold & Langer (1973) and Cravens & Dalgarno (1978) calculated that the amount of heat deposited in a molecular gas is about 8 keV per primary ionisation. Then, Tielens & Hollenbach (1985) find for the total heating rate, including helium ionisation
where is the cosmic ray ionisation rate per H2 molecule.
When X-rays are absorbed, fast electrons are produced. These fast electrons lose part of their energy through Coulomb interactions with thermal electrons, so the X-ray heating is given by
where is the heating efficiency, depending on the H2/H ratio and the electron abundance x. We use the results of Dalgarno et al. (1999). Their calculated heating efficiency in an ionised gas mixture is given by
where . and are the heating efficiencies for the ionised pure He and H2 mixture and the He and H mixture, respectively. Both are parametrised through
The values of , c and are given in Table 7 of Dalgarno et al. (1999), and x is the electron fractional abundance. It has to be modified when the H2-He mixture is considered:
H2 ionisation can lead to gas heating (Glassgold & Langer 1973). When H2 is ionised by a fast electron and subsequently recombines dissociatively, about 10.9 eV ( erg) of the ionisation energy can go into kinetic energy. H2+ can also charge transfer with H. This is an exothermic reaction, with an energy yield of 1.88 eV, of which we assume half, 0.94 eV ( erg), to go into heating. H2+ can also react to H3+, and subsequently recombine dissociatively or react with other species. Glassgold & Langer (1973) argued that for every H3+ ion formed 8.6 eV ( erg) goes into gas heating. The H2 ionisation rate cooling is then given by
where , and are the rates of dissociative recombination, charge transfer with hydrogen and the reaction to H3+, respectively.
We use the results of Sect. A.5. The dust temperature was found by Yan (1997):
where is the grain abundance and in erg s-1.
When the vibrational levels of H2 are populated by non-thermal processes, thermal collisional quenching and excitation can result in a net heating despite downward radiations. When non-thermal reactions are not important, H2 can be an important coolant. The resulting collisional vibrational heating or cooling is given by
Where is the total collision rate from level vj to v'j' in units of s-1. Radiative cooling due to downward decay of the vibrational levels is given by
The population of the vibrational levels is discussed in Sect. D.3.4.
Since most of the gas is atomic in the radical region, the dominant coolants are the atomic fine-structure lines. The most prominent cooling lines are the [CII] 158 m and [OI] 63 m and 146 m lines. For the calculation of the thermal balance we also take into account Si+, C, Si, S, Fe and Fe+. We use a compilation for the collisional data from Sternberg & Dalgarno (1995), Hollenbach & McKee (1989), Sampson et al. (1994), Dufton & Kingston (1994), Johnson et al. (1987), Roueff & Le Bourlot (1990), Schröder et al. (1991), Mendoza (1983), Chambaud et al. (1980) and Jaquet et al. (1992). We take into account collisions with electrons, H+, H and H2 (ortho and para) for the excitation of the species to different levels. In the PDRs, collisions with H+ are not the dominant excitation source but in XDRs the ionised fraction of hydrogen can be as large as ten percent and become important for the excitation of some levels.
We included the metastable cooling lines of C, C+, Si, Si+, O, O+, S, S+, Fe and Fe+. All the data is taken from Hollenbach & McKee (1989) except for Si+ (Dufton & Kingston 1994), C+(Sampson et al. 1994) and O+ (McLaughlin & Bell 1993).
At temperatures higher than 5000 K, cooling due to recombination of electrons with grains (PAHs) is important. The cooling depends on the recombination rate which is proportional to the product . The cooling rate increases when goes up, due to an increase in charge and hence Coulomb interaction. Bakes & Tielens (1994) calculated numerically the recombination cooling for a variety of physical conditions. An analytical fit to the data is given by
where and .
For the rotational and vibrational cooling of H2, CO and H2O, we use the fitted rate coefficients of Neufeld & Kaufman (1993) and Neufeld et al. (1995). They present a cooling rate for species i through:
where and n(xi) are the densities of H2and species xi, respectively. L is given by
We interpolate in the tables given by Neufeld & Kaufman (1993) and Neufeld et al. (1995), to find the values L0, n1/2 and and . L0 is the cooling rate coefficient in the low density limit and n1/2 is the H2 density where L has fallen by a factor of two below L0. is chosen to minimize the maximal fractional error in the fit at other densities. L0 is a function of temperature, and , n1/2, and are functions of temperature and the optical depth parameter , which is given by the gradient . N(xi) is the column density of the species xi. To take into account collisional excitation by electrons and atomic hydrogen, we follow Yan (1997) and replace by and . For H2 rotational and vibrational cooling, and are given by
For rotational cooling by CO, is given by
where , and . For H2O rotational cooling, is given by
where and are the H2 and electron impact excitation rate coefficients, respectively. for the excitation from level in units of is given by
where b is the rotational constant in , dthe dipole moment in Debye, , and C is given by
For CO vibrational cooling, is given by
For H2O vibrational cooling, is given by
The cooling due to the excitation of hydrogen is important at temperatures T > 5000 K. The cooling rate is given by Spitzer (1978):
For most of the chemical reaction rates, we make use of the UMIST database for astrochemistry by Le Teuff et al. (2000). In de PDR model we use a network containing all the species with a size up to 6 atoms. For the XDR model we use all species with sizes up to 3 atoms and some of 4 atoms. These species are taken from Yan (1997). Below we discuss the additional reactions.
The formation of H2 is very efficient over a wide range of temperatures. It was already shown by Gould & Salpeter (1963) that H2 is not formed efficiently in the gas phase. Most of the formation, which is still not very well understood, takes place on grain surfaces (Hollenbach & Salpeter 1971). Recently, Cazaux & Tielens (2002,2004) developed a model for the formation of hydrogen under astrophysically relevant conditions. They compared their results with the laboratory experiments by Pirronello et al. (1999) and Katz et al. (1999). They find a recombination rate of
where and are the volume density and cross section of dust grains and , and are the volume density, thermal velocity and thermally averaged sticking coefficient of hydrogen atoms. We use the sticking coefficient given by Hollenbach & McKee (1979)
where is the dust temperature. Equation (D.2) is the same as Eq. (4) in Tielens & Hollenbach (1985), except for the term , the recombination efficiency, which is given by
where is the H2 fraction that stays on the surface after formation, and are the desorption rates of molecular hydrogen and physisorbed hydrogen atoms, respectively, F is the flux of hydrogen atoms and is the evaporation rate from physisorbed to chemisorbed sites. These three terms dominate in different temperature regimes. See Cazaux & Tielens (2002,2004) for a more detailed discussion.
Collisions of electrons and ions with grains can become an important recombination process in dense clouds of low ionisation. We include reactions with PAHs following Sect. 5 of Wolfire et al. (2003) in the PDR models. The C+/C transition occurs at larger column densities when PAHs are included. Deep into the cloud, the electron abundance is reduced by several orders of magnitude.
In PDRs, molecular hydrogen can be excited by absorption of FUV photons in the Lyman-Werner bands. Fluorescence leads to dissociation in about in of the cases (see Field et al. 1966; Stecher & Williams 1967), and in the remaining of the cases to a vibrationally excited state of the ground electronic state (Black & Dalgarno 1976). To simplify matters, we treat the electronic ground state as having a vibrational ground state and a single excited vibrational state. London (1978) found that the effective quantum number for this pseudo-level is v = 6, and the effective energy is K. We treat excited molecular hydrogen, H2V, as a separate species in our chemistry. H2V can be destroyed by direct FUV dissociation, radiative decay or collisional de-excitation, and chemical reactions with other species. Since vibrational decay is a forbidden process, a large abundance of H2V can be maintained. H2V can react with other species with no activation barrier or a reduced one. In the UMIST database, the rates for a reaction between two species are parameterised as
For reactions with H2V, is replaced by = max(0.0, ). When reactions have an activation barrier lower than 2.6 eV, the barrier is set to zero. When the barrier is larger than 2.6 eV, the barrier is reduced by 2.6 eV. Tielens & Hollenbach (1985) state that for important reactions such as
this is a good approximation since the activation barrier of 0.5 eV is a lot smaller than the vibrational excitation energy of 2.6 eV. For reactions with barriers of the same order or larger one can overestimate the reaction rates.
In PDRs, the photo-dissociation rate of both H2 and CO is influenced by line as well as continuum absorption. The dissociation rate of H2 is decreased by self-shielding. For an H2 line optical depth , we adopt the self-shielding factor given by (Shull 1978). When the line absorption is dominated by the Doppler cores or the Lorentz wings (i.e., ), we use the self-shielding factor as given by de Jong et al. (1980). The CO photo-dissociation rate is decreased by both CO self-shielding and H2 mutual shielding. We use Table 5 of van Dishoeck & Black (1988), to determine the shielding factor as a function of column densities and .
In the XDRs we do not use the photo-ionisation rates from UMIST. X-rays are absorbed in K-shell levels releasing an electron. An electron from a higher level may fill the empty spot and with the energy surplus another so called Auger electron is ejected. This process leads to multiply ionised species. Due to charge transfer with H, H2 and He, they are quickly reduced to the doubly ionised state. We therefore assume that the ionisation by an X-ray photon leads to a doubly ionised species, as does absorption of an X-ray photon by a singly ionised species. When rates for charge transfer with H and He are very fast, elements are quickly reduced to singly ionised atoms, which is the case for O2+, Si2+ and Cl2+. Therefore, we add only O2+ to the chemical network to represent them. We assume that Si and Cl get singly ionised after absorbing an X-ray photon. We also include C2+, N2+, S2+ and Fe2+. The direct (or primary) ionisation rate of species i at a certain depth z into the cloud is given by
where the ionisation cross sections are taken from Verner & Yakovlev (1995).
Part of the kinetic energy of fast photoelectrons is lost by ionisations. These secondary ionisations are far more important for H, H2 and He than direct ionisation. Dalgarno et al. (1999) calculate the number of ions produced for a given species i. For a given electron energy E, is given by
where W is the mean energy per ion pair. Dalgarno et al. (1999) calculated W for pure ionised H-He and H2-He mixtures and parameterised W as:
where W0, c and are given in Table 4 of their paper. The corrected mean energies for ionisation in the H-H2-He mixture are given by
The ionisation rate at depth z into the cloud for species i is then given by
We rewrite this to a rate dependent on the fractional abundance of the species xi:
where xi is the fraction of species i. Since we integrate over the range 1-10 keV and W goes to a limiting value, we use the parameters applicable to the 1 keV electron. The ionisation rate then simplifies to:
We also include secondary ionisations for C, N, O, Si, S, Cl, Fe, C+, N+, O+, S+ and Fe+. We scale the ionisation rate of these species to that of atomic hydrogen by
We integrate over the range 0.1-1.0 keV to get an average value of the electron impact ionisation cross section . Using the experimental data fits of Lennon et al. (1988). The scaling factors for C, N, O, Si, S, Cl, Fe, C+, N+, O+, S+, and Fe+ are 3.92, 3.22, 2.97, 6.67, 6.11, 6.51, 4.18, 1.06, 1.24, 1.32, 1.97, and 2.38, respectively.
When energetic electrons created in X-ray ionisations collide with atomic and molecular hydrogen, H2 Lyman-Werner and H Lyman photons are produced, which can significantly affect the chemistry. The photoreaction rate Ri per atom or molecule of species i is given by
The values of pa are taken from Table 4.7 of Yan (1997) and values of pm are the rates for cosmic-ray induced reactions from Le Teuff et al. (2000). There is an exception for CO, however, where we take the rate, corrected for self-shielding, given by Maloney et al. (1996):
Vibrationally excited H2 can enhance reactions with an activation barrier and also be an important heating or cooling source. To calculate the populations of the vibrational levels of H2, we take into account:
We use the results of Dalgarno et al. (1999) to calculate the
X-ray induced excitation to the vibrational levels v=1 and v=2. The ratio of the yields Y(v=2)/Y(v=1) is about 0.070. Excitation to higher levels is not taken into account, since
the yield to higher levels decreases very rapidly. First we calculate
the mean energy for excitation, W, in the H2-He mixture. The
parameters are listed in Table 5 of Dalgarno et al. (1999). The function W has the same form as Eq. (D.7). The mean
energy for excitation also depends on the abundances of H and H2. The yield has to be corrected with a factor C(H,H2), which
is given by
a(x) = 0.5 (x/10-4)0.15. The rates
for excitation by thermal electrons are taken from Yan (1997),
who finds that the transitions rates for H2(v=0) to H2(v=1,2)are given by
The excitation rate for the transition
is taken to be v times the
rate. Excitations with
are not taken into account. The quenching rates are
calculated through detailed balance. The quenching rates from
by atomic hydrogen are given in Table 4.2 of
Yan (1997) and are of the form:
The excitation rates are obtained by detailed balance. For
the molecular excitation and quenching rates we use the results of
Tine et al. (1997). Collisions where either before or after one of the
H2 molecules is in the v=0 state are considered. The rate
coefficients are of the form:
and are given in Table 1 of Tine et al. (1997), who also
considered collisions with He. They give a rate coefficient for the
For the other transitions with
the same rates
are used. The upward transitions can be obtained by detailed balance.
Yan (1997) also calculated the dissociation and ionisation rates
by thermal electrons and since the ionisation threshold is much higher
than the vibrational energies one rate is used for all vibrational energies:
The dissociation rates by atomic hydrogen are given in Table 4.3 of Yan (1997), which are of the same form as Eq. (D.19). For the dissociation rates by H2 we use the results
of Lepp & Shull (1983), which are given by
the dissociative attachment reaction:
we use the results of Wadehra & Bardsley (1978) and the reaction
rates have the same form as Eq. (D.19). Vibrationally
excited H2 can be destroyed in chemical reactions. Endothermic
reactions with vibrationally excited H2 can lower the activation
barrier, by using the energy of the vibrational level. The barrier is
reduced, but cannot become negative:
When H2 is formed in chemical reactions which are exothermic, part
of the formation energy goes into the excitation of the vibrational
levels. Formation of H2 on grains can play a very significant
role. H2 has a binding energy of 4.48 eV. Following
Sternberg & Dalgarno (1989), we assume one third of this energy to be
distributed statistically over all the vibrational levels:
The photon energy absorbed per hydrogen nucleus, ,
is given by
The interval [
is the spectral
range where the energy is emitted. The photoelectric absorption cross
section per hydrogen nucleus,
is given by
Morrison & McCammon (1983) state that the X-ray opacity is
independent of the degree of depletion onto grains. Therefore, we take
the total (gas and dust) elemental abundances,
as given in Table 2 to calculate
The X-ray absorption cross sections, ,
are taken from Verner & Yakovlev (1995). The flux F(E,z) at depth z into
the cloud is given by
where is the total column of hydrogen nuclei and F(E,z=0) the flux at the surface of the cloud.