Here we derive the condition determining how much o - H2 is needed to slow down deuterium fractionation driven by Reaction (1). It is well established that the destruction of H2D+ by electrons and CO suppresses the fractionation as well as by H2. H2D+ also reacts with HD to form the doubly deuterated species D2H+, leading to further fractionation. Considering the balance of formation and destruction, the steady-state H2D+/H ratio can be calculated as (A.1)where n(i) is the number density of species i, kCO is the rate coefficient of H2D+ with CO, ke is the rate coefficient of dissociative electron recombination of H2D+, kHD is the rate coefficient of H2D+ with HD, and k1f and k1b are the effective rate coefficients for Reaction (1) in the forward direction and in the backward direction, respectively. Since the forward Reaction (1) is exothermic, k1f does not depend on the OPR(H2). We can say that the presence of o - H2 slows down the fractionation when k1bn(H2) is greater than kCOn(CO), ken(e), and kHDn(HD).
The endothermicity of Reaction (1) in the backward direction depends on the nuclear spin states of H2 and H2D+. The following set of reactions can occur: The rate coefficients of the reactions can be found in Hugo et al. (2009), who calculated them with the assumption that the reaction proceeds by a scrambling mechanism in which all protons are equivalent; results are different if long-range hopping is the effective mechanism. The effective rate coefficient k1b can be defined as follows (Gerlich et al. 2002; Lee & Bergin 2015): (A.6)From Eq. (A.6), we can express k1b as a function of ortho-to-para ratios of H2 and H2D+ (OPR(H2D+)), and gas temperature.
The OPR(H2D+) in the dense ISM is mainly determined by the following reactions (Gerlich et al. 2002; Hugo et al. 2009): Then, assuming steady state, the OPR(H2D+) is given as a function of OPR(H2) and gas temperature: (A.11)From Eqs. (A.6) and (A.11), we can express k1b as a function of OPR(H2) and the gas temperature. In Fig. 1, we show threshold abundances of CO and electrons as functions of the OPR(H2), below which Reaction (1) in the backward reaction is more efficient than the destruction by CO and electrons. Roughly speaking, the backward reaction rate exceeds the destruction rate by CO when OPR(H2) ≳ 20n(CO) /n(H2) at ≤20 K, while it exceeds the recombination rate with electrons when OPR(H2) ≳ 3 × 103n(e) /n(H2) at ≤20 K. Assuming the canonical HD abundance with respect to H2, 3 × 10-5, the condition k1bn(H2) >kHDn(HD) corresponds to OPR(H2) ≳ 6 × 10-4 at ≤20 K.
In this Appendix, we derive an analytical solution of OPR(H2) when abundances of ionic species and the H2 formation rate on grain surfaces are given. Let us consider the following set of differential equations, which describe the temporal variations of the abundances of o - H2 and p - H2 in the gas and solid phases:
where F, W, D are the adsorption rate, desorption rate, and rate of H2 destruction via e.g., photodissociation, respectively. ⟨ A ⟩ in Eqs. (B.3) and (B.4) is the population of species A in the active surface ice layers, and thus ⟨ A ⟩ ngr is the number density of species A in the surface ice layers per unit volume of gas. RH2 is the formation rate of H2 on grain surfaces, while bo is the branching ratio to form o - H2. In Eqs. (B.1) and (B.2), we neglect H2 formation in the gas phase for simplicity. In Eqs. (B.3) and (B.4), we do not consider nuclear spin conversion on the surface, though it is straightforward to include the spin conversion on the surface in the following analysis.
Combining Eqs. (B.1)−(B.4), we get (B.5)where we used n(o - H2) + n(p - H2) = n(H2) and n(H2) ≫ ⟨ H2 ⟩ ngr. The latter should be valid in molecular gases; the binding energy of H2 on a H2 substrate is only 23 K, corresponding to an sublimation timescale of ~10-10 s at T = 10 K (Vidali et al. 1991; Cuppen & Herbst 2007), while the adsorption timescale is ~109/nH yr. The H2 formation timescale, τH2, was defined as n(H2) /RH2. We also assumed that the rate coefficients of reactions to destroy o - H2 and p - H2 are the same, i.e., D(o - H2) /D(p - H2) = n(o - H2) /n(p - H2). Equation (B.5) describes the time evolution of the OPR(H2). The terms τo → p, τp → o, and τH2 are time-dependent in general. It is straightforward to solve Eq. (B.5) in the astrochemical simulations without nuclear spin state chemistry, and one may obtain a reasonable approximation of the temporal variation of the OPR(H2). In order to solve Eq. (B.5) analytically, we consider τo → p, τp → o, and τH2 as constant. This assumption is valid when the ortho-to-para spin conversion time scale is longer than the chemical (formation and destruction) timescale of hydrogen and light ions. The solution is given as follows: where τopr gives the characteristic timescale of OPR(H2) evolution, and OPR0(H2) is the initial OPR(H2). The steady-state value of OPR(H2) (OPRst(H2) ≡ OPR(H2)(t → ∞)) is given in Eq. (14), which was derived by Le Bourlot (1991) in a different manner.
When OPRst(H2) ≪ OPR0(H2) ≪ 1, Eq. (B.6) can be simply rewritten as OPR(H2)(t) ≈ OPRst(H2) + OPR0(H2)exp(−t/τopr). Then, it takes a greater time than ln [ OPR0(H2) / OPRst(H2) ] τopr to reach the steady state value.
Evolution of molecular abundances have often been investigated via pseudo-time dependent models in which hydrogen is assumed to be in H2 at the beginning of the calculation. In such models, the molecular D/H ratio depends on the initial OPR(H2), which is treated as a free parameter (e.g., Flower et al. 2006). In this appendix we analytically derive the OPR(H2) when the conversion of hydrogen into H2 is almost complete.
During the H/H2 transition, ortho-para spin conversion occurs through Reaction (17). H+ is primarily formed via cosmic-ray/X-ray ionization of atomic hydrogen, while it is mainly destroyed via recombination with electrons, and charge transfer to other species, such as atomic deuterium and oxygen (e.g., Dalgarno et al. 1973). The former is valid when n(H) /n(H2) >ξH/ (bH+ξH2) ≈ 0.05, where ξH and ξH2 are the ionization rates of atomic hydrogen and H2, respectively. bH+ is the branching ratio to form H+ for the ionization of H2. At steady state, the number density of H+ can be given as (C.1)where (C.2)where k(X + Y) is the rate coefficient of reaction X + Y. We assumed that the number density of electrons is equal to that of carbon ions. The carbon ion is the dominant form of carbon in the H/H2 transition region, while oxygen and deuterium are predominantly in atomic form. With Eq. (C.1), we can evaluate β1 and β2 (Eqs. (15) and (16)) by the following: where x(i) is the abundance of species i with respect to hydrogen nuclei, nH is the number density of hydrogen nuclei, vth is the thermal velocity of atomic hydrogen, and k17f and k17b are the rate coefficients of Reaction (17) in the forward direction and the backward direction, respectively. We used ξH2 = 2.2ξH. We assumed that the H2 formation rate is half of the accretion rate of atomic hydrogen onto dust grains. By substituting Eqs. (C.3) and (C.4) into Eq. (14), we can get the OPR(H2) in the H/H2 transition regime under the steady state assumption as a function of ξH2/nH and gas temperature. In Fig. C.1, we show the OPR(H2) when the the conversion of hydrogen into H2 is (almost) complete, i.e., x(H2) = 0.5. Above the thick dashed line, where τo → p is smaller than τH2, the steady-state assumption is justified.
It is clear that the OPR(H2) is greater than unity only when H2 formation occurs at low ionization (ξH2/nH< 10-22 cm3 s-1) and/or warm conditions. In our fiducial model, this occurs at Tgas> 50 K, but the exact value depends on ξH2/nH. In those regions with such warm gas temperatures, the dust temperature would also be warm. At Tdust ≳ 20 K, the H2 formation rate may drop considerably, depending on characteristics of chemisorption sites on grain surfaces (e.g., Hollenbach & Salpeter 1971; Cazaux & Tielens 2004; Iqbal et al. 2014), which is not considered here.
OPR(H2) when the conversion of hydrogen into H2 is almost complete under the steady-state assumption. Above the thick dashed line, where τo → p is smaller than τH2, the steady-state assumption is justified. Above the thin dashed line, where β1 is larger than β2, the OPR(H2) is similar to the low-temperature thermalized value of 9exp(−170.5 /Tgas). See Appendix C.
|Open with DEXTER|
Appendix D: Scaling relation of the HDO/H2O ratio in the surface ice layers with the atomic D/H ratio
Here we derive the scaling relation of the HDO / H2O ratio in the chemically active surface ice layers with the atomic D/H ratio, which is applicable under the UV irradiation conditions where photodesorption regulates the growth of ice mantles. Figure 9 shows the important reactions for H2O ice and HDO ice in our models, and they are considered in the following analysis. Let us consider the following set of differential equations which describe temporal variations of the abundances of H2O, HDO, OH and OD in the active surface ice layers: where kph and kphdes are the rate coefficients of photodissociation and photodesorption in the active surface ice layers, respectively, of H2O ice and HDO ice. bOH is the branching ratio to form OH + D for HDO ice photodissociation (~0.3, Koning et al. 2013). Combining Eqs. (D.1)−(D.4), the evolutionary equation of the HDO / H2O ratio in the active ice layers (fHDO, s = ⟨ HDO ⟩ / ⟨ H2O ⟩) can be written as follows: (D.5)where we used the inequalities ⟨ OH ⟩ ≪ ⟨ H2O ⟩, ⟨ OD ⟩ ≪ ⟨ HDO ⟩, and fHDO, s ≪ 1, which are verified by our numerical simulations. We defined fD, s = ⟨ D ⟩ / ⟨ H ⟩. We also used the relations, δhop ≡ khop(D) /khop(H) ~ kO + D/kO + H ~ kOH + D/kOH + H, which should be valid because of the much higher hopping rates of atomic H and D compared to those of atomic O and OH radical. The energy barrier difference against hopping between atomic D and H is ~10 K on amorphous water ice (Hama et al. 2012). When the right hand side of Eq. (D.5) is less (more) than zero at given fD, s, the ratio fHDO, s decreases (increases) on a timescale shorter than bOHkph. Since the photodissociation timescale in gas with low extinction is much shorter than the dynamical timescale (>1 Myr in the case of our cloud formation model), let us assume steady state, which corresponds to the minimum (or maximum) of fHDO, s/fD, s. Then we obtain (D.6)To produce water ice mantles under UV irradiation, the rate of OH formation through the hydrogenation of atomic oxygen (i.e., kO + H ⟨ O ⟩ ⟨ H ⟩) should be larger than the photodesorption rate of H2O (see Fig. 9). When the dynamical timescale is longer than the timescale of the OH formation, the chemistry evolves as the OH formation rate and the H2O photodesorption rate are almost balanced: (D.7)This relation is confirmed in our simulations. The population of OH can be evaluated from the following equation: (D.8)Then (D.9)where we used kph ⟨ H2O ⟩ ≫ kphdes ⟨ H2O ⟩ ≈ kO + H ⟨ O ⟩ ⟨ H ⟩. From Eqs. (D.6), (D.7), and (D.9), we get the scaling relation, (D.10)where Pphdes is kphdes/kph ~ 0.02 (Andersson et al. 2006; Arasa et al. 2015). The term δhopfD, s/ (1 + Γ) corresponds to pOH → HDO which is discussed in the main text. In the case with Γ ≪ 1 (or, in other words, OH + H is the dominant formation pathway of H2O ice), we get fHDO, s ≈ fD, s. In another extreme case Γ ≫ 1, we get fHDO, s ≈ (Γ-1 + 0.02)fD, s, i.e., fHDO, s is much smaller than fD, s. For example, Γ is ~104 at AV = 1 mag in our fiducial simulation, and it further increases with the increase of AV. The minimum of the fHDO, s/fD, s ratio is ~0.02 in our fiducial simulation, which is well reproduced by Eq. (D.10).
As discussed in Sect. 5.1, the evolution of the OPR(H2) depends on the assumed heavy metal abundances. The goal of this appendix is to determine which case, the LM abundances (fiducial case) or the HM abundances, gives the better predictions on the OPR(H2) evolution in the ISM.
There are some estimates of OPR(H2) in cold dense clouds/cores from observations of molecules other than H2 in the literature. We do not use them for the constraint here, because how to estimate OPR(H2) from the observations is not well-established, and because the evolution of OPR(H2) in the ISM can be in the non-equilibrium manner and may vary among sources depending on their past physical evolution. Instead, we compare observations of sulfur-bearing molecules toward diffuse/translucent/molecular clouds in the literature with our model results. We focus on the total abundance of selected sulfur-bearing molecules in the gas phase (CS, SO, and H2S) and the HCS+/CS abundance ratio. The former can probe the total abundance of elemental sulfur in the gas phase, though the main form of gaseous sulfur is likely to be the neutral atom S or S+ especially in gas with low extinction. The latter, the HCS+/CS ratio, can probe the ionization degree of the gas, which is related to the S+ abundance (and abundances of the other heavy metal ions). The HCS+/CS ratio is anticorrelated with the electron abundance in our models, because CS is formed by dissociative electron recombination of HCS+ and destroyed by photodissociation. Although the HCS+/CS ratio also depends on the photodissociation timescale of CS, it is common between the fiducial model and model HM at given AV. The rate coefficient and the branching ratios for the electron recombination of HCS+ are taken from Montaigne et al. (2005).
Turner (1996) derived the molecular abundances of H2S, CS, and SO in translucent clouds with line of sight visual extinction of up to 5 mag. He found that the total abundance of these three species with respect to hydrogen nuclei is typically 10-8–10-7. In the dense molecular clouds, TMC-1 and L134N, with higher line of sight visual extinction than the samples in Turner (1996), the total abundance is 10-9–10-8 (Ohishi et al. 1992; Dickens et al. 2000). The lower total abundance in the dense molecular clouds implies that depletion of gaseous sulfur is significant in the dense clouds (Joseph et al. 1986). It is not obvious which AV in our model can be compared with the observations towards the translucent and dense molecular clouds. Towards TMC-1(CP) and L134N, there is evidence of CO freeze-out; the gaseous CO abundance with respect to hydrogen nuclei (4 × 10-5) is less than the canonical value of 10-4 and infrared absorption by CO ice is detected in the line of sight towards the vicinity of them (Whittet 2007, 2013). We assume that our results at AV> 2 mag, where the gaseous CO abundance decreases to less than ~4 × 10-5 because of the freeze-out, can be compared with the observations in TMC-1 and L134N, while the results at AV< 2 mag are compared with the observations in clouds with lower line of site extinction.
Figure 11 shows the total abundance of H2S, CS, and SO as a function of visual extinction in our models. Model HM better reproduces the total abundance of the S-bearing molecules observed in the translucent clouds, while the fiducial model better reproduces the total abundance observed in the dense molecular clouds. This result implies that again, the depletion of sulfur from the gas phase occurs during the formation and evolution of molecular clouds, and that model HM underestimates the degree of the sulfur depletion.
The HCS+/CS ratio has been derived toward diffuse clouds (0.08, Lucas & Liszt 2002), clouds with line of sight visual extinction of up to 5 mag (0.01−0.1, Turner 1996), and the dense molecular clouds, TMC-1 and L134N (0.06, Ohishi et al. 1992). Roughly speaking, the ratio is in the range of 0.01−0.1 in the diffuse and dense ISM. Figure 12 shows the HCS+/CS ratio and the electron abundance in the fiducial model and model HM. The fiducial model reproduces the HCS+/CS ratio much better than model HM, though the both models tend to underestimate the HCS+/CS ratio compared with observations, i.e., overestimate the ionization degree of the gas.
Considering the HCS+/CS ratio is reproduced by the fiducial model much better than by model HM, we conclude that the fiducial model gives a better prediction for the S-bearing species and thus for the OPR(H2). In particular, model HM seems to overpredict the S-bearing species in the gas phase. The non-success of the model HM probably means that a non-negligible fraction of sulfur is incorporated into dust grains (Keller et al. 2002) and/or the non-thermal desorption rates of icy sulfur are overestimated in the current model; the partitioning of elemental sulfur between gas and ice in our model depends on the assumed non-thermal desorption rates, especially chemisorption probabilities, which remain uncertain at the current stage. In our models, the dominant sulfur reservoirs in ices are HS and H2S, though there has been no detection of H2S ice in the ISM. HS ice is formed from H2S ice via the hydrogen abstraction reaction, H2S + H, with an activation energy barrier of 860 K (Hasegawa et al. 1992). HS ice is hydrogenated to form H2S ice again, desorbing ~1% of the product H2S through chemisorption. This loop is the main mechanism of the desorption of icy sulfur in our model as in Garrod et al. (2007). If we set the chemisorption probability for the hydrogenation of HS ice to be 0.1%, only 3% of elemental sulfur remains in the gas phase at AV = 3 mag in model HM, though gaseous sulfur is still more abundant than in the fiducial model.
© ESO, 2015