Issue 
A&A
Volume 541, May 2012



Article Number  A76  
Number of page(s)  17  
Section  Interstellar and circumstellar matter  
DOI  https://doi.org/10.1051/00046361/201118126  
Published online  01 May 2012 
Online material
Appendix A: Other recent updates to the PDR code
Progress have been achieved in computing fine structure and rotational excitation due to collisions with H and/or H_{2} as reported in the BASECOL (Dubernet et al. 2006) and LAMDA (Schöier et al. 2005) databases. We have updated and/or implemented the collisional excitation rates that play an active role in the cooling processes and computed explicitly their emission spectrum by solving the statistical equilibrium equations, including radiative pumping by the cosmic background radiation field and dust infrared emission (Gonzalez Garcia et al. 2008). We point out that the new finestructure excitation collision rates of atomic oxygen computed by Abrahamsson et al. (2007) have a significant impact on the temperature at the edge of PDRs. As an example, for a typical proton density of 10^{4} cm^{3} and a radiation scaling factor of 10, the temperature at the edge is 90 K with the old Launay & Roueff (1977) atomic oxygen collision rates and only 67 K with the values displayed in Abrahamsson et al. (2007). At the present stage, the emission spectra of millimeter and submillimeter transitions of HCN, OH, CH^{ + }, and O_{2} are readily computed and the implementation of other molecules is straightforward and depends only on the availability of the collision rates by the relevant perturbers. Another significant issue is the inclusion of the thermal and charge balance of the grains in the overall ionization fraction. The thermal balance is obtained through a coupling with the DustEM program Compiègne et al. (2011), which can be switched on through the F_Dustem parameter (F_Dustem=1). If not (F_Dustem=0), the temperature of the various grain bin sizes is obtained from the formula given in Eq. (5) of Hollenbach et al. (1991), where the actual value of the radiation field is introduced. The determination of the grain charge is obtained from the balance between photoelectric effect and recombination on dust particles, expanding on the treatment of Draine & Sutin (1987) and Bakes & Tielens (1994).
The code also computes the photodissociation rates from the integration of the photodissociation crosssections, when available, with the interstellar radiation field. The attenuation by dust particles is then directly obtained from the dust properties considered in the model, i.e. their absorption and extinction coefficients, which depend on the size and the nature of the dust particles. Different options are proposed depending on the treatment of the grain temperatures. If F_Dustem=1, we use the absorption and scattering coefficients computed by the DustEM code (Compiègne et al. 2011). If F_Dustem=0, we derive the albedo and dust properties from the extinction curve given by the Fitzpatrick and Massa analytic expansion (Fitzpatrick & Massa 2007), extended towards longer wavelengths by the data from Weingartner & Draine (2001)^{4}. It is remarkable that we recover the dustfree photodissociation rates displayed in van Dishoeck (1988) for the Mathis or Draine incident radiation field. The A_{V} dependence of the photodissociation rates then directly reflects the appropriate dust environment.
Appendix B: LangmuirHinshelwood mechanism
Upon landing on a grain, most heavy species may build an ice mantle. In this case, it is possible to account for the total number of physisorbed molecules by integrating over the grain size distribution. This is the subject of a forthcoming paper on surface chemistry (Le Petit et al., in prep.). However, this is most probably not the case for the lightest species (H, H_{2}, D, HD, ...). We assume here that they only build a single monolayer above either the grain surface or the ice mantle.
In that case, two effects must be taken into account:

upon landing on a site already occupied by a light species, theimpinging species is rejected to the gas phase;

binding depends on the (sizedependent) temperature of the grain and thus the steady state depends on the size (and characteristics) of the grain.
Hence, we must compute the number of physisorbed particles of type X on a grain of size a. This is N_{X:}(a) in the following (in particles per grain, and not in particles per cubic centimeter). The total amount of X on all grains, that follows by integration, is where the last expression is for a MRN size distribution with dn_{g} = A_{gr} n_{H} a^{ − α} da. A_{gr} is a normalization factor, n_{H} is the gas density (in cm^{3}), and a the grain radius (in cm). In the following, we use that case as an example, but it is easy to generalize to any distribution. Numerical integration is performed by discretizing the size. When needed, we use where the weights w_{i} and abscissae a_{i} are chosen according to the distribution. The number of abscissae is npg. Table B.1 displays the abscissae and weights computed for the parameters of the MRN distribution given in Table 1. These coefficients must be computed anew whenever one changes the range of sizes^{5}. Here Remark that the power a^{3.5} does not appear in the discrete sum.
Abscissae and weights for the Gaussian integration of a MRN size distribution with npg = 12, α = 3.5, a_{min} = 3 × 10^{7} cm, and a_{max} = 3 × 10^{5} cm.
Three types of reactions must be considered:

Adsorption;

Ejection;

Reaction.
We do not consider reactions with heavy atoms nor molecules (including ices) which will be the subject of a followup paper.
B.1. Adsorption
If the outer layer of the grain is populated by light species (H, H_{2}), then any one of them may lead to the rejection of an impinging atom. We consider n_{j} of these species. The number of accretion events of a species X per unit time interval on a single grain of size a is then where s(X) is the sticking coefficient of species X, d_{s} the mean distance between adsorption sites (assumed to be identical for all grains), and the term in parentheses takes into account rejection by any species Y_{j} that is already on the grain. The term is the total number of adsorption sites on a grain of size a. Formally, this equation may be divided into a firstorder formation reaction of rate , and n_{j} different secondorder destruction reactions of rate . The relevant creation and destruction equations are thus (B.1)or (B.2)where S_{gr} is the total surface of grains per unit volume. One can see that, although the form of the reaction terms is preserved (first or second order polynomials in the variables) the total number of individual contributions becomes large (one accretion leads to (n_{j} × npg + 1) reactions)^{6}.
B.2. Desorption processes
Ejection can occur spontaneously (thermal evaporation) or by either photodesorption or cosmicray ejection. All processes are similar in the sense that they only involve a single variable N_{X}(a) for a grain of size a.
If the vibration frequency of the adsorbed particle is ν_{0}, the temperature of the grain is T_{gr}(a), and the binding energy is T_{b}(X), then the number of evaporation per unit time is If the flux of photons (respectively cosmic rays) is F_{ph} (respectively F_{CR}) and the number of particles desorbed by impact is η_{ph} (respectively η_{CR}), then the number of desorption is (for a photon) Writing , and , we have (B.3)(B.4)
B.3. Surface reactions
We seek to compute the number of encounters per grain and per unit time. We adopt first the point of view of X:. On a single grain of size a, the number of encounter per s is proportional to 1/t_{X} the inverse hoping time of X:, the probability of finding a Y: upon landing and the number of X: During the same time, from the point of view of Y:, the number of encounters made is So the total number of encounters is where the factor of takes care of each encounter having been counted twice. Thus, for two surface species, we can write , and (B.5)The production rate of Z in the gas phase, occurring directly after the encounter of two adsorbed atoms, is obtained after integration on the grain size distribution (B.6)
B.4. Approximate H_{2} formation rate
For a single grain size, and negligible photodesorption and cosmicray desorption, we can derive an analytic approximation to the H_{2} formation rate in the spirit of the discussion of Biham & Lipshtat (2002). If H is the only atom sticking to a grain of size a with a sticking probability of 1, then In a steady state, this leads to (B.7)where is the maximum number of H on the grain, and the two critical densities [H] _{a} and[H] _{b} are defined as Table B.2 gives the values of the critical densities for different grain temperatures.
Critical densities in the LH formation rate of H_{2} for amorphous carbon and a mean distance between physisorbed sites of 2.6 Å , where T is the gas temperature in K.
The H_{2} formation rate per grain is then (with n_{g} the number of grains per cubic centimeter of gas = ) and can be given analytically from the previous formulae. Both [H] _{a} and[H] _{b} vary slowly with the gas temperature, but very strongly with the grain temperature. Thus, we can define two limiting regimes for the gas phase atomic hydrogen density, namely
This is possible only for cold grains (typically below 15 K). Hence, it requires both a high density (or pressure) and a weak radiation field. This is the case for all grain sizes as soon as T_{g} is higher than about 25 K. Hence, it applies to all strong radiation field models. In that case, the formation rate increases as the square of the density of H.The usual expression for the formation rate R_{H2} in cm^{3} s^{1} follows from Since n_{g} is proportional to n_{H}, we see that in the first case R_{H2} ∝ 1/[H] , whereas in the second R_{H2} ∝ [H] . These relations apply only for the approximations made here.
Appendix C: EleyRideal mechanism
C.1. Formalism
We consider the impact of a fast atom (hot gas) on a grain. Since a fully detailed description (taking into account all possible kinds of surfaces) is far beyond the capabilities of our model, we search for an approximate mechanism that takes into account the following requirements:

it is efficient in “hot” gas and on “hot” grains, hence the impinging Hatom must eventually reach a chemisorbed site on the grain;

it leads to H_{2} formation rates that are consistent with observational constraints;

the number of free parameters remains at the lowest possible number.
Since this process takes place at the edge of the cloud, we assume that the grains are essentially bare (without ice coating) and that the process does not depend on the grain temperature. This approximation is justified since, at the edge of PDRs, grain temperatures (at most 100 K) are much lower than either the gas temperature or chemical binding energies on grain surfaces.
On impact, the gas phase H can find either a free chemisorpsion site or an already chemisorbed H. In the second case, since the formation of H_{2} releases 4.5 eV, an energy far higher than the chemisorbed well, we assume that a newly formed H_{2} is immediately released in the gas phase. Hence where [H::] _{max} is the maximum number of chemisorbed H atoms (saturated grains) and [H::] the corresponding abundance. In addition v_{th} is the thermal velocity of the gas phase H and we consider the geometrical crosssection to compute the total amount of grain surface per unit volume ⟨ nσ_{gr} ⟩ . If the mean distance between chemisorption sites is d_{s} and is the same on all types of grains, one can see (from purely geometric considerations) that This is true for any grain size distribution. If the gas phase atom impacts a free chemisorption site, we have to estimate the probability that it sticks to the grain. The simplest hypothesis requires that it be proportional to the number of collisions of H with grains per unit of time (v_{th} ⟨ nσ_{gr} ⟩ ), possibly with a barrier to cross (where T is the gas temperature and T_{1} the threshold), with a temperature dependent sticking coefficient α(T) and proportional to the “free room” . Hence we write This equation is split in the code into two: a direct formation reaction and a “pseudo” rejection reaction. The corresponding rates are
C.2. Sticking coefficient and choice of T_{1}
There is not much information on how to define the sticking function α(T), but we expect that it goes to 0 for very high temperatures (the atom just bounces on the grain without there being any time to evacuate the excess kinetic energy). We introduce the empirical form (C.1)In this expression, the index β controls the steepness of the decrease in α(T), and T_{2} defines the temperature such that .
We may constrain the value of β with the following considerations:

an estimate of the velocity v_{2} above which theatom bounces back to the gas is given by

the sticking coefficient is approximated as the fraction of gas phase atom with velocity lower than v_{2}. Using a Maxwell distribution at temperature T, we have
Given α(T), we may investigate which barrier T_{1} gives a “standard” formation rate of 3 × 10^{17} cm^{3} s^{1} at a given temperature T. We find that Postulating that where grains are warm, the gas is warm too, we may require that this standard rate is reached for a temperature in the range [150:450] K. This translate into a range [100:800] K for T_{1}. Our choice of T_{1} = 300 K favors an efficient formation, and reflects the idea that chemisorption is easy (but not instantaneous) on grains with plenty of surface defects. This leads to a higher formation rate at high gas temperature as found observationally by Habart et al. (2004).
C.3. Analytical approximation
Fig. C.1
Left axis: variation of k_{ER} with gas temperature T (relative to the one at 100 K). Right axis: chemisorption efficiency κ (see text). 

Open with DEXTER 
If hydrogen is the only chemisorbed species, the abundance of H:: can be analytically derived at a steady state which leads to an H_{2} formation rate of (C.2)with In this expression, v_{th} ⟨ nσ_{gr} ⟩ [H] refers to a purely geometric collisional process. Figure C.1 displays the variation in the chemisorption rate as a function of gas temperature (relative to the one at 100 K). The rate is negligible at low temperature owing to the exponential barrier. It grows as the square root of T where the barrier is negligible, then is quenched by the sticking cutoff. The chemisorption efficiency, κ(T), is displayed in Fig. C.1 (right axis). It peaks at a few hundred Kelvin and remains significantly high up to a few thousands of Kelvin.
This behavior is qualitatively very similar to results found by Cuppen et al. (2010, their Fig. 2) from Monte Carlo simulations of H_{2} formation including both physisorption and chemisorption. They also found that the formation efficiency increases with temperature for gas temperatures around a thousand K.
Fig. C.2
Effect of the variations in T_{1} and T_{2} on the intensity of H_{2} 1–0 S(1) line for the model P = 10^{7} cm^{3} K and χ = 1000. Values plotted in the plane T_{1},T_{2} are I(T_{1},T_{2})/I(300,464). 

Open with DEXTER 
In Sect. 4.2.2, we present several line intensities computed from models in which H_{2} is formed by the ER and LH mechanisms. In these models, we adopted β = 1.5, T_{1} = 300 K, and T_{2} = 464 K. Figure C.2 presents the effect of variations in T_{1} and T_{2} on the intensity of one line of H_{2}. We note that T_{1} is the most important parameter and that the line intensity can be reduced by
a factor ≃ 5 if this parameter is increased from 300 K to 1000 K. For other lines, such as 0–0 S(0) this decrease can reach a factor of ten. As mentioned above, for real interstellar grains, we can expect to have a large range of T_{1} depending on the nature and structure of the grains surfaces. Even if most chemisorbed sites have high thresholds, it only requires a few low thresholds sites for the ER mechanism to be efficient.
Appendix D: A_{V} to size conversion
For constant dust properties along the line of sight, it is possible to convert optical depth in the visible to a distance (in pc) analytically. We define C_{d} to be the total proton columndensity to color index ratio, and R_{V} the usual extinction to color index ratio: Then, using A_{V} = 2.5 log _{10}(e) τ_{V}, we have hence, with , we have We note that , where κ_{V} + σ_{V} is the extinction (sum of absorption plus scattering) per H atom by grains at the wavelength of the photometric band V.
© ESO, 2012
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