Online material

Appendix A: Treatment of the chemistry

In our previous papers (Godard et al. 2009, 2010, 2012), the chemical network implemented in the TDR model was built as a combination of the chemical network of the Meudon PDR code (Le Petit et al. 2006) and those extracted from the main online databases for astrochemistry⁷, UMIST (Woodall et al. 2007; McElroy et al. 2013), OSU (Hassel et al. 2010), and KIDA (Wakelam et al. 2012). To improve the usefulness of the model we break here from the previous approach and choose to adopt the chemical network of the Meudon PDR code⁸ except for a few special cases presented below.

Appendix A.1: Chlorine and fluorine chemistries

By default the chemical network of the Meudon PDR code contains 133 gas-phase atomic and molecular compounds − including hydrogen-, helium-, carbon-, nitrogen-, oxygen-, sulfur-, silicon-, and iron-bearing species − which interact with one another through 2631 gas-phase reactions. In order to obtain TDR predictions concerning the abundances of HF, HCl, and HCl⁺, all recently detected in the diffuse ISM with Herschel/HIFI (Neufeld et al. 2010b; Sonnentrucker et al. 2010; Monje et al. 2013; De Luca et al. 2012), the network has been expanded to 14 chlorine- and fluorine- bearing species: Cl, HCl, CCl, Cl⁺, HCl⁺, H₂Cl⁺, CCl⁺, H₂CCl⁺, F, HF, F⁺, HF⁺, H₂F⁺, and CF⁺. This additional network includes the 38 chemical reactions given in the recent review of experimental and theoretical studies performed by Neufeld & Wolfire (2009). Their rates are those of Neufeld & Wolfire (2009) except for the photodissociation of HF, HCl, and HCl⁺ and the photoionisation of Cl, the rates of which are taken from the UMIST database (McElroy et al. 2013) and the calculations of van Dishoeck (1988) and van Dishoeck et al. (2006). This additional network also includes the 38 ion-neutral and dissociative recombination reactions given by Anicich (2003) and the OSU database for astrochemistry (Hassel et al. 2010).

Appendix A.2: Photoreactions

Since the amounts of matter of prototypical diffuse and transluscent environments (Snow & McCall 2006) are insufficient to entirely absorb the mean interstellar UV radiation field, both photoionisation and photodissociation are known to play a major role in the destruction of atoms and molecules in the diffuse ISM. In the Meudon PDR code (Le Petit et al. 2006) both processes are treated self-consistently by solving the UV radiative transfer as a function of position in a plan parallel slab − including the absorption by molecular lines and the absorption and scattering by interstellar dust − and by computing at each point the photodestruction rates from the integration of the cross sections over the specific intensity of the UV radiation.

Such a detailed approach cannot, however, be applied to the TDR code since the radiative transfer is modelled with a single parameter, the extinction at visible wavelength A_V, assumed to be constant over the entire sightline (standard case, see Sect 3.3). The photo-reaction rates are thus computed as (A.1)where γ and β are constants taken from van Dishoeck (1988) and van Dishoeck et al. (2006) who performed fits of the photodestruction rates over the range A_V = 0 − 3 mag, assuming a slab of gas with constant density illuminated on one side and where the extinction is only the result of the absorption and scattering by 0.1 μm large interstellar grains.

While this expression is reliable for the photodissociation occurring though continuous absorption, it is however too simplistic to correctly describe the photodissociation by line absorption processes which strongly depends on the self-shielding, i.e. the column density of the molecule from the edge of the cloud up to A_V. We therefore apply Eq. (A.1) to all molecules except for H₂ and CO whose photodissociation is known to occur through line processes (Lee et al. 1996; Draine & Bertoldi 1996; Visser et al. 2009). For these two species we adopt the photo-reaction rates computed with the Meudon PDR code assuming a slab of gas with constant density illuminated on one side. The details of these results and their implementation in the TDR model are described in Appendix B.

Appendix A.3: Excitation of H₂

In addition to the chemical state of the gas the TDR model also follows the time-dependent evolution of the level populations of H₂ in the three different phases, the ambient medium, the active stage, and the relaxation stage. While the code was built to treat up to 318 different levels of molecular hydrogen, we only consider here the first 18 rovibrational levels (with an energy above the ground state smaller than 10⁴ K) in order to reduce the computation time.

Since the TDR model differs from PDR models by the absence of radiative transfer, neither the UV radiative pumping nor the near- and mid-infrared absorption and induced emission are taken into account. The populations of the excited levels of H₂ thus results from the combined effects of collisional excitation with H, He, and H₂ (Flower et al. 1998; Flower & Roueff 1998b,a; Le Bourlot et al. 1999) and spontaneous radiative decay. This latter process is included in the gas cooling function assuming that all the rovibrational lines of H₂ are optically thin, a hypothesis that holds in the diffuse ISM as long as N_H ≲ 4 × 10²⁴ cm^-2 for a gas velocity dispersion of 1 km s^-1.

Appendix A.4: State-to-state chemical rates

During the past decades, the advances of the cross molecular beam experiments, the flowing afterglow apparatus, the ion trapping techniques, and the theoretical studies of chemical reaction dynamics (see, e.g. the reviews of the field by Casavecchia 2000; Levine 2005) have led to measurements and calculations of state-specific reaction rates for several neutral-neutral and ion-neutral reactions (e.g. Zanchet et al. 2013). In particular the internal energy of H₂ has been proven to systematically increase the reactivity of highly endothermic reactions (Hierl et al. 1997; Sultanov & Balakrishnan 2005; Zellner & Steinert 1981). To take this process into account, we have implemented in the TDR model a state-specific description of the reaction rates of

For the first reaction, we follow the approach of Agúndez et al. (2010) and adopt the state-specific rate constants of Gerlich et al. (1987) for H₂(υ = 0,J = 0...7) and the Langevin collision rate for higher energy levels (Hierl et al. 1997). Since there is no information concerning the state-to-state rate constants of reaction A.3, we assume (A.4)where E(υ,J) is the energy (expressed in K) of the level υ,J of H₂.

Comparing the predictions of the model with those obtained without including state-to-state chemistry (Godard et al. 2009), we find that these mechanisms have a little impact on the production of CH⁺ and SH⁺ in the diffuse gas.

Appendix A.5: Electron/ion recombinations on grains and PAHs

Electron transfers between ions and very small grains or polycyclic aromatic hydrocarbons (PAHs) are efficient in decreasing the ionisation state of atoms and molecules (Lepp et al. 1988). When implemented in astrochemical models, such processes are found to compete with (or even dominate) the radiative and dielectronic recombinations of several ions such as H⁺ or C⁺ (Bakes & Tielens 1998; Welty et al. 2003; Wolfire et al. 2008; Hollenbach et al. 2012) and thus to have a strong impact on the hydrogenation chains and the charge balance of carbon- and oxygen-bearing species (see Fig. C.2).

To take these mechanisms into account we have implemented in the TDR code a treatment of the charge of PAHs and very small grains using the formalism described by Bakes & Tielens (1994) and Weingartner & Draine (2001b). Large grains are neglected. Similarly to our treatment of photoreactions, the photoionisation rates of dust particles are modelled using Eq. (A.1) with coefficients inferred from the computations of the Meudon PDR code. The electronic attachment and ion recombinations on, respectively, the positively and negatively charged dust particles are modelled using the prescription of Draine & Sutin (1987) and taking into account the electron affinity, the escape length, and the ionisation potentials of the colliders in the computation of the rates (Weingartner & Draine 2001a,b).

Both PAHs and very small grains are described as spherical particles of radius 6.4 Å and 13 Å, respectively. Their abundances in the diffuse ISM are parametrised by the size distribution function of interstellar dust and the dust mass fraction with respect to the gas phase. In this paper we consider a dust-to-gas mass ratio of 0.01 and assume that the PAHs carry 4.6% of the total dust mass (Draine & Li 2007). The size distribution of PAHs is modelled as a log-normal function centered at 6.4 Å (Compiègne et al. 2011). The size distribution of very small grains is modelled as a power law with an index of −3.5. All these considerations lead to n(PAHs) /n_H = 4.2 × 10^-7 and n(VSGs) /n_H = 2.5 × 10^-8.

Appendix B: Photodissociation rate of H₂ and CO: self-shielding

	Fig. B.1 H2 photodissociation rate as a function of the extinction computed with the Meudon PDR code applied to PDRs with different densities illuminated on one side: from right to left nH = 10, 30, 50, 100, 300, 500, 1000, 3000, 5000, and 10 000 cm-3, successively.
Open with DEXTER

In the standard case of the TDR code, the radiative transfer is modelled with a single parameter A_V, the extinction of the radiation field in the visible photometric band (λ ~ 0.551μm). Such a prescription is, however, too simplistic to correctly describe the photodissociations of H₂ and CO which occur through line processes and therefore strongly depend on the self-shielding (van Dishoeck & Black 1986, 1988). Because these photodissociations drive the chemical transitions from H to H₂ and from C to CO, it is essential to take into account their dependence on A_V.

The distinctive dependences of the photodissociation rates of H₂ and CO on A_V, N(H₂), and N(CO) have been studied analytically and numerically by Draine & Bertoldi (1996) and Lee et al. (1996). In their approaches Draine & Bertoldi (1996) consider a directional UV radiation field propagating perpendicularly to an infinite plan-parallel slab of gas, while Lee et al. (1996) apply the approximation proposed by Federman et al. (1979) to solve the self-shielding of H₂ and CO. We have decided to part from these previous studies and to compute the photodissociation rates of H₂ and CO assuming an isotropic radiation field and taking into account the impact of the line broadening on the self-shielding.

The Meudon PDR code was therefore run along a grid of models defined by the following range of parameters: cm^-3, (in Mathis’s units), and s^-1. The code was set to solve the exact transfer within the electronic lines of H₂ and CO assuming a Doppler broadening of 3.5 km s^-1. While the code is designed to compute self-consistently the formation rate of H₂ on grain surfaces using both the Eley-Rideal and Langmuir-Hinshelwood mechanisms (Le Bourlot et al. 2012; Bron et al. 2014), we switched off these processes and set the H₂ formation rate to its observed value of 3 × 10^-17(T/ 100 K)^1/2n_Hn(H) cm^-3 s^-1 where n(H) is the density of atomic hydrogen. The obtained photodissociation rates of H₂ and CO and the predicted molecular fractions of the gas are shown in Figs. B.1–B.3 as functions of A_V and for the subset of models χ = 1 and ζ_H2 = 3 × 10^-16 s^-1.

	Fig. B.2 CO photodissociation rate as a function of the extinction computed with the Meudon PDR code applied to PDRs with different densities illuminated on one side: from top to bottom nH = 10, 30, 50, 100, 300, 500, 1000, 3000, 5000, and 10 000 cm-3, successively.
Open with DEXTER

	Fig. B.3 Molecular fraction defined as fH2 = 2n(H2) /nH as a function of AV computed with the Meudon PDR code for several densities: from right to left nH = 10, 30, 50, 100, 300, 500, 1000, 3000, 5000, and 10 000 cm-3, successively.
Open with DEXTER

The obtained photodissociation rates of H₂ differ from those derived by Draine & Bertoldi (1996) by less than a factor of three for N(H₂) ≲ 2 × 10²¹ cm^-2 and by more than an order of magnitude for N(H₂) ≳ 3 × 10²¹ cm^-2. The former difference comes from the effect of the line broadening on the self-shielding while the latter is ascribed to the different prescriptions (directional or isotropic) used for the radiative transfer. These interpretations are confirmed with the comparison of our results with those computed by Lee et al. (1996). In this case the photodissociation rates of H₂ are found to differ by less than a factor of three over the whole range of N(H₂) depending on the values of the density and the cosmic ray ionisation rate ζ_H2.

Appendix C: PDR and TDR chemical networks

The chemical compositions of photodissociation regions and turbulent dissipation regions are driven by very different chemical patterns. This occurs because the heating of PDRs is dominated by the interaction of UV photons with interstellar matter while that of TDRs is controlled by turbulent dissipation, but also because the state-of-the-art PDR models systematically neglect the gas dynamics and thus solve the at-equilibrium chemical state, while TDR models focus on the non-equilibrium effects, i.e. on the coupling between the dynamics and the chemistry as a function of time. To illustrate these differences we show in Figs. C.1 and C.2 the main production and destruction routes of 27 species including the carbon, sulfur, and oxygen hydrogenation chains as given by the PDR and TDR models. In both cases we show the production pathways obtained locally assuming a diffuse molecular gas of density n_H = 50 cm^-3, a shielding A_V = 0.4 mag, and a cosmic ray ionisation rate ζ_H2 = 10^-16 s^-1. For the TDR model, the results are extracted at the r₀ radius (see Paper I) of an active vortex set by the following parameters: u_θ,M = 3.5 km s^-1 and a = 3 × 10^-11 s^-1. All these figures are simplified: for each species shown, only the processes that together contribute to more than 70 percent of the total destruction and formation rates are shown.

Fig. C.1 Carbon and sulfur chemical networks driven by the UV radiation field (top panel) and the turbulent dissipation (bottom panel) in the diffuse ISM assuming nH = 50 cm-3 and AV = 0.4 mag. This figure is simplified: for each species, only the processes that together contribute to more than 70 percent of the total destruction and formation rates are displayed. The endothermicities and energy barriers of endoenergic reactions are indicated in red.

Open with DEXTER

Fig. C.2 Carbon and oxygen chemical networks driven by the UV radiation field (top panel) and the turbulent dissipation (bottom panel) in the diffuse ISM assuming nH = 50 cm-3 and AV = 0.4 mag. This figure is simplified: for each species, only the processes that together contribute to more than 70 percent of the total destruction and formation rates are displayed. The endothermicities and energy barriers of endoenergic reactions are indicated in red.

Open with DEXTER

© ESO, 2014