Depletion of Heavy Nitrogen in the Cold Gas of Star-forming Regions

One of the plausible scenarios of molecular cloud formation is that diffuse H i gas is compressed by super-sonic accretion flows (e.g., Inoue & Inutsuka 2012). To simulate the physical evolution of post-shock materials (i.e., forming cloud), we use the steady-shock model developed by Bergin et al. (2004) and Hassel et al. (2010). The model solves the conservation laws of mass and momentum with the energy equation, considering time-dependent cooling/heating rates and simplified chemistry in a plane parallel configuration. The cloud is assumed to be irradiated by the Draine field (Draine 1978), and the cosmic-ray ionization rate is set to be 1.3 × 10⁻¹⁷ s⁻¹. As time proceeds, the column density of post-shock materials increases, which assists molecular formation by attenuating the UV radiation. The column density of post-shock materials at a given time, t, after passing through the shock front is ${N}_{{\rm{H}}}\approx 2\times {10}^{21}\,({n}_{0}/10\,{\mathrm{cm}}^{-3})({v}_{0}/15\,{\mathrm{kms}}^{-1})(t/4\,\mathrm{Myr})\,{\mathrm{cm}}^{-2}$, where n₀ and v₀ are preshock H i gas density and velocity of the accretion flow, respectively. We convert N_H into visual extinction, A_V, by the formula A_V/N_H = 5 × 10⁻²² mag cm². The simulation is performed until A_V reaches 3 mag (i.e., ∼12 Myr). During most of the simulation period, the density and temperature of the cloud is ∼10⁴ cm⁻³ and 10–15 K, respectively (see Figure 2 of Furuya et al. 2015).

Our astrochemical model takes into account gas-phase chemistry, interactions between gas and (icy) grain surfaces, and grain surface chemistry. We adopt a three-phase model (Hasegawa & Herbst 1993b), assuming the top four monolayers (MLs) are chemically active; the rest of the ice mantles are assumed to be inert. Our chemical network is based on the gas-ice network of Garrod & Herbst (2006) and has been extended to include deuterium chemistry (up to triply deuterated species) and nuclear spin-state chemistry of H₂, H₃⁺ and their isotopologues. For this work, we excluded species containing chlorine, phosphorus, or more than three carbon atoms in order to shorten the computational time. In our model, the ortho-to-para ratio of H₂ is mostly determined by the competition between two processes; the ratio is three upon the formation of H₂ on grain surfaces (e.g., Watanabe et al. 2010), while the ratio is reduced via the proton exchange reactions between H₂ and H⁺ (or H₃⁺; e.g., Honvault et al. 2011). The reaction rate coefficients of the H₂–H₃⁺ system is taken from Hugo et al. (2009). Gas-phase nitrogen chemistry has been updated referring to Wakelam et al. (2013) and Loison et al. (2014). For this work, the chemical network is extended to include mono-¹⁵N species, relevant isotope exchange reactions (Roueff et al. 2015), and isotope-selective photodissociation of N₂ (Li et al. 2013; Heays et al. 2014). We therefore added 460 species and ∼26,000 reactions to our reduced deuterated network, resulting in the increase of the number of species from 1350 to 1810, and the number of reactions from ∼73,000 to ∼99,000.

The binding energy of each species is not constant in our simulations, but depends on the composition of an icy surface, following the method presented in Furuya & Persson (2018). The binding energy of species i, E_des(i), is calculated as a function of surface coverage of species j, θ_j, where j = H₂, CO, CO₂, or CH₃OH:

$$\begin{eqnarray}&&{E}_{\mathrm{des}}(i)=(1-{{\rm{\Sigma }}}_{j}{\theta }_{j}){E}_{\mathrm{des}}^{{{\rm{H}}}_{2}{\rm{O}}}(i)+{{\rm{\Sigma }}}_{j}{\theta }_{j}{E}_{\mathrm{des}}^{j}(i),\end{eqnarray} \tag{ 1 }$$

where ${E}_{\mathrm{des}}^{j}(i)$ is the binding energy of species i on species j. The set of binding energies on water ice, ${E}_{\mathrm{des}}^{{{\rm{H}}}_{2}{\rm{O}}}$, is taken from Garrod & Herbst (2006) and Wakelam et al. (2017). In particular for this study, ${E}_{\mathrm{des}}^{{{\rm{H}}}_{2}{\rm{O}}}$ are set to be 550 K for H i and H₂, 720 K for N i, 1170 K for N₂, and 5500 K for NH₃ (Collings et al. 2004; Fayolle et al. 2016; Minissale et al. 2016). There is no laboratory data or estimate for most ${E}_{\mathrm{des}}^{j}(i)$ in the literature. In order to deduce ${E}_{\mathrm{des}}^{j}$ for all species, where j is either H₂, CO, CO₂, or CH₃OH, we assume scaling relations (cf. Taquet et al. 2014),

$$\begin{eqnarray}&&{E}_{\mathrm{des}}^{j}(i)={\epsilon }_{j}{E}_{\mathrm{des}}^{{{\rm{H}}}_{2}{\rm{O}}}(i),\end{eqnarray} \tag{ 2 }$$

where _j is ${E}_{\mathrm{des}}^{j}(j)/{E}_{\mathrm{des}}^{{{\rm{H}}}_{2}{\rm{O}}}(j)$. We adopt ${\epsilon }_{{{\rm{H}}}_{2}}=23/550$, _CO = 855/1300, ${\epsilon }_{{\mathrm{CO}}_{2}}=2300/2690$, and ${\epsilon }_{{\mathrm{CH}}_{3}\mathrm{OH}}=4200/5500$ (e.g., Öberg et al. 2005; Cuppen & Herbst 2007; Noble et al. 2012). For example, the binding energy of N₂ becomes 770 K in our simulations when an icy surface is fully covered by CO (Öberg et al. 2005; Fayolle et al. 2016). For sticking probabilities of gaseous species onto (icy) grain surfaces, we use the formula recommended by He et al. (2016, their Equation (1)), which leads to the sticking probability of around unity for all species for the relevant temperature range in our physical model. For ¹⁵N- and/or D-bearing species, we use the same binding energies and sticking probabilities as for normal species. The exception is the binding energy of atomic deuterium, with a binding energy that is set to be 21 K higher than that of atomic hydrogen (Caselli et al. 2002). Adsorption rates are inversely proportional to the square root of the mass of species, so that the adsorption of gaseous species tends to make the bulk gas enriched in heavier isotopes, D and ¹⁵N. This effect is not important especially for ¹⁵N, because the difference in the rates is only a few percent.

As nonthermal desorption processes, we consider stochastic heating by cosmic-rays, photodesorption, and chemical desorption (e.g., Hasegawa & Herbst 1993a; Westley et al. 1995; Garrod et al. 2007). We assume that species formed by surface reactions are desorbed by chemical desorption with the probability of roughly 1% (Garrod et al. 2007). Photodissociation rates of icy species are calculated in the same way as Furuya et al. (2015). Photodissociation occurs only in the surface layers (i.e., the top four MLs) in our model, as the rest of ice mantle is assumed to be chemically inert. In other words, photofragments are assumed to immediately recombine in the bulk ice mantle in our model (Furuya et al. 2017). According to molecular dynamics (MD) simulations, there are several possible outcomes after a UV photon dissociates water ice: (i) the photofragments are trapped on the surface; (ii) either of the fragments is desorbed into the gas phase; (iii) the fragments recombine and the product is either trapped on the surface or desorbed into the gas phase, etc. (Andersson & van Dishoeck 2008; Arasa et al. 2015; see also e.g., Hama et al. 2010 for experimental studies). Note that desorption of the photofragments or the recombination product occurs from the top several MLs only (Andersson & van Dishoeck 2008; Arasa et al. 2015). To the best of our knowledge, similar MD simulations for other molecules have not been reported in the literature. For water ice photodissociation, we consider all of the possible outcomes and their branching ratios found by the MD simulations of Arasa et al. (2015). For photodissociation of other icy species, we assume the same branching ratios as in the gas-phase photodissociation, and that all of the photofragments are trapped on the surface for simplicity. For those species, the photodesorption rate is given as the following, separately from photodissociation rate:

$$\begin{eqnarray}{R}_{\mathrm{phdes},i} & = & \pi {a}^{2}{n}_{\mathrm{gr}}[{F}_{\mathrm{ISUV}}\exp (-{\gamma }_{i}{A}_{V})+{F}_{\mathrm{CRUV}}]\\ & & \times {\theta }_{i}{Y}_{i}\min ({N}_{\mathrm{layer}}/4,\,1),\end{eqnarray} \tag{ 3 }$$

where n_gr is the number density of dust grains, a is the radius of dust grains (0.1 μm), F_ISUV and F_CRUV are the interstellar and cosmic-ray induced FUV photon fluxes, respectively, γ_i is the parameter for the attenuation of interstellar radiation field by dust grains, Y_i is the photodesorption yield per incident FUV photon for thick ice (≥4 MLs), and N_layer is the number of MLs of the whole ice mantle. The parameter θ_i is the surface coverage of species i, which is defined as a fractional abundance of species i in the top four MLs. The photodesorption yield for N₂ ice per incident FUV photon is given as an increasing function of the surface coverage of CO, varying from 3 × 10⁻³ to 8 × 10⁻³ (Bertin et al. 2013). We assume that NH₃ ice is photodesorbed intact (as NH₃) with the yield of 10⁻³. This value is similar to the recently measured photodesorption yield for pure NH₃ ice in laboratory (${2.1}_{-1.0}^{+2.1}\,\times {10}^{-3}$; Martin-Domenech et al. 2018), but the yield would depend on ice compositions and the FUV spectrum adopted in experiments (Ligterink et al. 2015; Martin-Domenech et al. 2018). The dependence on the NH₃ photodesorption yield is discussed in Section 3.2. We assume the same photodesorption yields for ¹⁵N- and/or D-bearing species as for normal species.

Elemental abundance ratios for H:He:C:N:O:Na:Mg:Si:S:Fe are 1.00:9.75(−2):7.86(−5):2.47(−5):1.80(−4):2.25(−9):1.09(−8):9.74(−9):9.14(−8):2.74(−9), where a(−b) means a × 10^−b (Aikawa & Herbst 1999). Elemental abundance ratios, [D/H]_elem and [N/¹⁵N]_elem, are set to be 1.5 × 10⁻⁵ and 300, respectively (Linsky 2003; Ritchey et al. 2015). All of the elements, including hydrogen, are initially in the form of either neutral atoms or atomic ions, depending on their ionization energy.

Figure 1(a) shows abundances of N-bearing species as functions of A_V. With increasing A_V, N i is gradually converted to NH₃ ice and N₂. At the final simulation time in our model (∼12 Myr), which corresponds to A_V = 3 mag, most nitrogen is locked in molecules, and the fraction of elemental nitrogen in ${\rm{N}}\,{\rm{I}},{{\rm{N}}}_{2}$ (gas+ice) and NH₃ ice are 0.05%, 20%, and 80%, respectively. Note that ∼90% of nitrogen is in ice at A_V = 3 mag. In our models, the partitioning of elemental nitrogen among the three species depends on the photodesorption yield of ammonia ice as shown in Furuya & Persson (2018) and briefly discussed in Section 3.2. Observationally, the primary nitrogen reservoir in dense star-forming regions is not well constrained, although it has been expected to be either $N\,{\rm{}}i$, gaseous ${{\rm{N}}}_{2}$, or icy N-bearing species, such as ${{\rm{N}}}_{2}$ and NH₃ (e.g., Maret et al. 2006; Daranlot et al. 2012).

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The N/¹⁵N abundance ratios in selected species are shown in Figure 1(b). In our model, ¹⁵N fractionation is predominantly caused by isotope-selective photodissociation of N₂ rather than isotope exchange reactions. At A_V ∼ 1.8 mag, the primary reservoir of nitrogen in the gas phase changes from N i to N₂. Around this N i/N₂ transition, N i is significantly enriched in ¹⁵N (a factor of up to two compared to [N/¹⁵N]_elem), while N₂ is significantly depleted in ¹⁵N. Isotope-selective photodissociation of N₂ is at work only in the limited regions (1 mag ≲ A_V ≲ 2.5 mag), where the interstellar UV radiation field is not significantly attenuated (see also Heays et al. 2014). On the other hand, it is seen that the N/¹⁵N ratio of the bulk gas starts to increase around the N i/N₂ transition. This is due to the freeze-out of ¹⁵N-enriched N i and subsequent NH₃ ice formation, which result in the ¹⁵N depletion in the bulk gas. Even at A_V ∼ 3 mag, where N₂ photodissociation is negligible, the bulk gas is depleted in ¹⁵N with the N/¹⁵N ratio of ∼400. The freeze-out of N₂ is less efficient than that of N i in our fiducial model, because adsorbed N₂ does not react with other species in contrast to N i, and adsorbed N₂ partly goes back to the gas phase via photodesorption and stochastic heating by cosmic rays. The binding energy of N₂ is much lower than that of NH₃, while the photodesorption yield of N₂ is higher than that of NH₃ in our fiducial model (>3×10⁻³ versus 10⁻³). Then, most nitrogen lost from the gas phase is in the form of N i rather than N₂ around the N i/N₂ transition in our fiducial model, leading to the ¹⁵N depletion in the bulk gas.

The degree of ¹⁵N depletion in the bulk gas depends on how much of ¹⁵N i is frozen out and converted to ¹⁵NH₃ ice around the N i/N₂ transition. The dominant destruction process of NH₃ ice in our fiducial model is photodesorption, assuming a yield of 10⁻³ per incident FUV photon. It is expected that the degree of ¹⁵N depletion in the bulk gas depends on the NH₃ photodesorption yield $({Y}_{\mathrm{pd}}^{{\mathrm{NH}}_{3}})$. Note that photodissociation of NH₃ ice, the products of which are assumed to be NH_2ice + H_ice or NH_ice + H_2ice, is included in our model with a much higher rate than its photodesorption (by a factor of >10). A significant fraction of the photofragments, however, is hydrogenated to reform NH₃ ice. The late limiting step of NH₃ ice formation is adsorption of N i onto grain surfaces in our model, and the reformation of NH₃ ice via surface hydrogenation reactions is efficient enough to compensate for its photodissociation. Consequently, the dominant destruction process of NH₃ ice is photodesorption rather than photodissociation. In our fiducial model, the rates of photodissociation and photodesorption of NH₃ ice are calculated separately, while the MD simulations have shown that, at least for water ice, photodesorption is one of the possible outcomes of photodissociation (e.g., Andersson & van Dishoeck 2008). In order to investigate the dependence of our results on the assumptions on photodissociation and photodesorption of NH₃ ice, we ran an additional model, in which photodissociation of NH_n ice (n = 1, 2, 3) leads to several outcomes, including desorption of NH_n or NH_(n − 1), assuming that the probability of each outcome is the same as that of water ice photodissociation obtained by the MD simulations of Arasa et al. (2015). The most significant difference between this modified model and our fiducial model is that in the former, photodissociation of NH_n ice most likely leads to NH_{(n − 1)ice} + H_gas rather than NH_(n−1)ice + H_ice. In the modified model, the total photodesorption yield of NH₃ ice (desorbed as NH₂ or NH₃) per incident photon is ∼6 × 10⁻⁴. We confirmed that our result is robust; the bulk gas is depleted in ¹⁵N with the N/¹⁵N ratio of 440 at A_V = 3 mag in the modified model. The evolution of the abundances and the N/¹⁵N ratios of the major nitrogen species is also similar in the fiducial and modified models. In the rest of this paper, we use our original treatment of photodissociation and photodesorption of NH₃ ice, but varying ${Y}_{\mathrm{pd}}^{{\mathrm{NH}}_{3}}$.

Figure 2 depicts variations of the N/¹⁵N ratio in the bulk gas (panel (a)) and in the bulk ice (panel (b)) when ${Y}_{\mathrm{pd}}^{{\mathrm{NH}}_{3}}$ is varied in the range between 3 × 10⁻⁴ and 10⁻². In general, the model with lower ${Y}_{\mathrm{pd}}^{{\mathrm{NH}}_{3}}$ shows the larger level of ¹⁵N depletion in the bulk gas. The bulk gas ¹⁵N depletion is most significant in the model with ${Y}_{\mathrm{pd}}^{{\mathrm{NH}}_{3}}=5\times {10}^{-4};$ the N/¹⁵N ratio is ∼500, which is comparable to some of the observed molecular N/¹⁵N ratios (see Section. 4.1). Compared to the degree of the bulk gas ¹⁵N depletion, the enrichment of ¹⁵N in the bulk ice is modest, because more nitrogen is present in the ice than in the gas phase.

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Figure 3 shows fractions of elemental ¹⁴N (panels (a)–(c)) and ¹⁵N (d)–(f) locked in N i, N₂, and NH₃ as functions of A_V in the models with ${Y}_{\mathrm{pd}}^{{\mathrm{NH}}_{3}}=5\times {10}^{-4}$ (left panels), 10⁻³ (middle, our fiducial model), and 3 × 10⁻³ (right). With increasing ${Y}_{\mathrm{pd}}^{{\mathrm{NH}}_{3}}$, the fraction of nitrogen in NH₃ ice is reduced, while that in N₂ is enhanced (Furuya & Persson 2018). In the models with higher ${Y}_{\mathrm{pd}}^{{\mathrm{NH}}_{3}}$, the self-shielding of N₂ is more efficient, which makes the N i/N₂ transition sharper, and enhances the maximum level of ¹⁵N enrichment in N i (Figure 3, panels (g)–(i)). Nevertheless, with increasing ${Y}_{\mathrm{pd}}^{{\mathrm{NH}}_{3}}$ the level of the bulk gas ¹⁵N depletion is lowered as discussed above. The ¹⁵N fractionation between the gas and the ice mostly occurs between the ¹⁴N i/¹⁴N₂ transition and the ¹⁵N i/¹⁴N¹⁵N transition. With increasing ${Y}_{\mathrm{pd}}^{{\mathrm{NH}}_{3}}$, the accumulation of NH₃ ice is slowed down, while the ¹⁴N₂/¹⁴N i abundance ratio in the gas phase becomes higher between the two transitions. Then the freeze-out of ¹⁵N-depleted N₂ becomes more important with increasing ${Y}_{\mathrm{pd}}^{{\mathrm{NH}}_{3}}$, which (partly) cancels out the bulk gas ¹⁵N depletion through the freeze-out of ¹⁵N-enriched N i. In the model with ${Y}_{\mathrm{pd}}^{{\mathrm{NH}}_{3}}={10}^{-2}$, where the photodesorption yield of NH₃ ice is higher than that of N₂ ice, the bulk gas becomes slightly enriched in ¹⁵N, because more N₂ than N i is frozen out.

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Table 1 compares the observed and model N/¹⁵N ratios of N₂H⁺, gaseous NH₃, and nitriles (HCN and CN). Note that the observed N/¹⁵N ratios for the nitriles are derived from the observations of the corresponding ¹³C isotopologues, assuming that the molecular C/¹³C ratios are the same as the elemental C/¹³C ratio. The observationally derived N/¹⁵N ratio for the nitriles may be considered as lower limits (see Roueff et al. 2015). As a general trend in our models, the N/¹⁵N ratio decreases in the order, N₂H⁺, NH₃, and the nitriles. As our model in the previous section is more appropriate for molecular clouds than for prestellar cores, we further run our astrochemical models under prestellar core conditions (2 × 10⁵ cm⁻³, 10 K, and 10 mag) for 10⁶ year, using the molecular abundances at A_V = 3 mag as the initial abundances. Figure 4 shows the temporal evolution of abundances (panel (a)) and the N/¹⁵N ratios (panel (b)) of selected species under the prestellar core conditions in the model with ${Y}_{\mathrm{pd}}^{{\mathrm{NH}}_{3}}={10}^{-3}$ (see also Table 1). The abundances of CO and N₂ decrease with time due to freeze-out, while the drop of CO abundance is more significant than that of N₂. The N₂H⁺ abundance is relatively constant, because their formation and destruction rates depend on the abundances of N₂ and CO, respectively. The gaseous NH₃ abundance increases with time, because the CO freeze-out leads to the enhanced conversion rate from N₂ to NH₃ (e.g., Aikawa et al. 2005). It is confirmed that the ¹⁵N depletion in the bulk gas is almost preserved, while the differences in the N/¹⁵N ratios among the molecules become smaller with time. The latter point indicates that ¹⁵N fractionation by isotope exchange reactions is not efficient in the prestellar core conditions, which is consistent with the model of Roueff et al. (2015). Although the rate of cosmic-ray-induced photodissociation of N¹⁵N is higher than that of N₂, this process does not contribute to ¹⁵N fractionation in prestellar cores; destruction of N₂ by He⁺, which is the product of cosmic-ray ionization of atomic He, is much faster than cosmic-ray-induced photodissociation of N₂ (Heays et al. 2014). Deuterium fractionation, which is driven by hydrogen-isotope exchange reactions, becomes more efficient with time under the prestellar core conditions (Figure 4(c)). The different time dependence (or no clear correlation) between ¹⁵N and deuterium fractionation has been reported by Fontani et al. (2015) and De Simone et al. (2018), based on the observations of N₂H⁺, N₂D⁺, and ¹⁵N isotopologues of N₂H⁺ toward a sample of high-mass and low-mass star-forming cores.

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In the models with ${Y}_{\mathrm{pd}}^{{\mathrm{NH}}_{3}}\leqslant {10}^{-3}$, all of the species in Table 1 are depleted in ¹⁵N, which is consistent with the observations in L1544 qualitatively, but the models still underestimate the degree of ¹⁵N depletion, in particular, in N₂H⁺. The results presented in this paper are based on a specific physical model. More numerical studies, and varying physical parameters, are necessary to better understand the ¹⁵N observations.

Additional comments on each species are as follows. N₂H⁺ is a daughter molecule of N₂, and thus the N/¹⁵N ratio of N₂H⁺ follows that of N₂. The N¹⁵NH⁺/¹⁵NNH⁺ ratio is around unity throughout our simulations, which is consistent with the observations in L1544, but inconsistent with those in B1. The only known process so far that could deviate the N¹⁵NH⁺/¹⁵NNH⁺ ratio from unity is the isotope exchange reaction between N¹⁵N and N₂H⁺ (Adams & Smith 1981; Roueff et al. 2015). This pathway would be inefficient in the ISM, as the exchange reaction competes with destruction of N₂H⁺ by CO, which would be much faster. Gaseous NH₃ is mainly produced by photodesorption of NH₃ ice with some contribution from the sequential ion-neutral reactions, starting from N₂ + He⁺. Then the N/¹⁵N ratio of gaseous NH₃ follows that of N i. Wirström et al. (2012) found that gaseous NH₃ can be significantly depleted in ¹⁵N (by a factor of around two compared to [N/¹⁵N]_elem) in some conditions as a result of competition among the three reactions, ¹⁵N⁺ + N₂, ¹⁵N⁺ + CO, and ¹⁵N⁺ + H₂. The necessary conditions are 200x($\mathrm{CO}$) ≲ OPR(H₂) ≲ 60x(N₂), where x(i) is the abundance of species i with respect to hydrogen nuclei and OPR(H₂) is the ortho-to-para ratio of H₂. In our models, in which adsorption and desorption of N i, N₂, and CO are considered, N₂ is less abundant than CO in the gas phase, even though the binding energy of N₂ is set to be lower than that of CO. Then the mechanism found by Wirström et al. (2012) does not work in our models. The critical assumptions in the models of Wirström et al. (2012) and Wirström & Charnley (2018) are that N i and N₂ do not freeze out onto grain surfaces, while CO does, leading to x(N₂) > x($\mathrm{CO}$) after several freeze-out timescales. CN and HCN are produced from N i. Their N/¹⁵N ratios basically follow that of N i, but more enriched in ¹⁵N, due to the isotope exchange reaction between CN and ¹⁵N (Roueff et al. 2015).

Figure 5 shows the fractional composition (panel (a)), the ¹⁵N enrichment (panel (b)) and the deuterium enrichment (panel (c)) in NH₃ and HCN ices as functions of cumulative number of ice layers formed in our fiducial model. Around 75 MLs of ice in total are formed by A_V = 3 mag. ${{\rm{H}}}_{2}{\rm{O}}$ is the dominant component in the lower ice layers (≲60 MLs), while CO becomes more abundant than ${{\rm{H}}}_{2}{\rm{O}}$ in the upper ice layers. The CO/${{\rm{H}}}_{2}{\rm{O}}$ ratio in the whole ice mantle is 40% in our model, which is similar to the observed median ice composition toward background stars (31%, Öberg et al. 2011). The NH₃/${{\rm{H}}}_{2}{\rm{O}}$ and HCN/${{\rm{H}}}_{2}{\rm{O}}$ ratios in the whole ice mantle are ∼17% and ∼0.2%, respectively, in our fiducial model. To the best of our knowledge, no measurements are available for NH₃ ice and HCN ice toward background stars in the literature. The NH₃/${{\rm{H}}}_{2}{\rm{O}}$ ice ratio in our model is higher than the observed median ratio toward low-mass and high-mass protostars (Öberg et al. 2011, and references therein) by a factor of ∼3, as is often reported in published astrochemical models (e.g., Chang & Herbst 2014; Furuya et al. 2015; Pauly & Garrod 2016). If the NH₃ and NH₄⁺ ice abundances toward the protostellar sources are summed up (see Table 3 of Öberg et al. 2011), the discrepancy still remains, but becomes smaller (within a factor of 2).

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The NH₃ and HCN in the whole ice mantles are enriched both in deuterium and ¹⁵N compared to the elemental ratios, but the degrees of the enrichment are significantly different. In our fiducial model, the NH₂D/NH₃ ratio and the DCN/HCN ratio in the whole ice mantles are 8 × 10⁻⁴ and 3 × 10⁻⁴, respectively, while the N/¹⁵N ratios of NH₃ ice and HCN ice are 270 and 240, respectively (Table 1). The level of ¹⁵N enrichment in the icy species weakly depends on ${Y}_{\mathrm{pd}}^{{\mathrm{NH}}_{3}}$. For example, in the model with ${Y}_{\mathrm{pd}}^{{\mathrm{NH}}_{3}}=3\times {10}^{-3}$, the N/¹⁵N ratios of NH₃ ice and HCN ice in the whole ice mantles are 250 and 190, respectively, which are lower than those in the fiducial model. This trend can be understood, recalling that the maximum level of ¹⁵N enrichment in N i is higher with increasing ${Y}_{\mathrm{pd}}^{{\mathrm{NH}}_{3}}$ (Figure 3, panels (g)–(i)), and that N₂ ice does not react with other species to form NH₃ ice or HCN ice. Another factor is the timing of NH₃ ice formation; in the models with lower ${Y}_{\mathrm{pd}}^{{\mathrm{NH}}_{3}}$, the amount of NH₃ ice formed prior to the N i/N₂ chemical transition, which is not fractionated in ¹⁵N, is higher (Figure 3, panels (a)–(c)). This early-formed unfractionated NH₃ ice can overshadow later-formed ¹⁵N-enriched NH₃ ice. Then, stronger ¹⁵N depletion in the bulk gas does not necessarily mean stronger ¹⁵N enrichment of icy species in our proposed scenario. On the other hand, regardless of ${Y}_{\mathrm{pd}}^{{\mathrm{NH}}_{3}}$, our model predicts that the N/¹⁵N ratios of icy HCN and icy NH₃ are lower than those of corresponding gaseous molecules. This prediction can be observationally tested by comparing a molecular N/¹⁵N ratio in the warm gas (>100 K) around protostars, where all ices sublimate, with that in the cold outer envelope; the molecular N/¹⁵N ratio in the warm gas should be lower than that in the cold gas.

The fractionation pattern within the ice mantles is different between deuterium and ¹⁵N. The concentration of ¹⁵N is nonmonotonic within the ice mantles, while that of deuterim increases from the lower layers to the upper layers. This is a direct consequence that ¹⁵N fractionation is triggered by isotope-selective photodissociation of N₂, while deuterium fractionation is caused by isotope exchange reactions; the former is most efficient at certain A_V, while the latter becomes more efficient with time. The signatures of the different fractionation mechanisms are preserved in the ice-layered structure. This prediction could be observationally tested by comparing the NH₃/¹⁵NH₃ ratio with the NH₂D/¹⁵NH₂D ratio in the warm gas (>100 K) around protostars, where all ices sublimate; the NH₃/¹⁵NH₃ ratio should be lower than NH₂D/¹⁵NH₂D. Note that if nitrogen fractionation occurs by isotope exchange reactions as deuterium fractionation, the NH₃/¹⁵NH₃ ratio should be larger than NH₂D/¹⁵NH₂D (see Figure 4 of Rodgers & Charnley 2008b).