Efficient Methanol Production on the Dark Side of a Prestellar Core

Cosmic-ray-induced desorption is supposed to sustain traceable gas-phase abundances of O- and C-bearing molecules also in the inner parts of cold, dense clouds (e.g., Leger et al. 1985; Hasegawa & Herbst 1993; Shen et al. 2004; Roberts et al. 2007; Hollenbach et al. 2009). The efficiency of cosmic-ray-induced sputtering of methanol embedded in water ice has been studied experimentally by Dartois et al. (2019). In their Figure 11, Dartois et al. (2019) present estimates for ${\mathrm{CH}}_{3}\mathrm{OH}$ sputtering rates as functions of the cosmic-ray ionization rate for two methanol fractions in the ice, 0.55 and 0.056, with respect to ${{\rm{H}}}_{2}{\rm{O}}$. The lower of these fractions is characteristic of interstellar ices (Boogert et al. 2015). Assuming that the cosmic-ray ionization rate of ${{\rm{H}}}_{2}$ is ${\zeta }^{{{\rm{H}}}_{2}}\sim 6\times {10}^{-17}\,{{\rm{s}}}^{-1}$ at the core boundary, one obtains from Figure 11 of Dartois et al. (2019) that the flux of sputtered ${\mathrm{CH}}_{3}\mathrm{OH}$ molecules is ${F}_{{\mathrm{CH}}_{3}\mathrm{OH}}^{\mathrm{sp}}\sim 0.14\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$ (here, we have multiplied the value indicated in the figure by two to account for the exiting cosmic-ray, as suggested in the caption). The adopted ${\zeta }^{{{\rm{H}}}_{2}}$ is consistent with the models of Padovani et al. (2009), in view of the fact that the ${{\rm{H}}}_{2}$ column density of the ambient cloud around H-MM1 is $\sim {10}^{22}\,{\mathrm{cm}}^{-2}$.

In order to estimate the desorption rate per H atom or per cubic centimeter in the interstellar medium, we adopt the MRN grain size distribution (Mathis et al. 1977). This is a power-law distribution of spherical grains with $\tfrac{{{dn}}_{{\rm{g}}}}{{da}}\propto {a}^{-3.5}$ between a_min = 0.005 μm (50 Å) and a_max = 0.25 μm, where a is the grain radius, and n_g is the number density of grains. Assuming that the dust-to-gas mass ratio is 0.01 and that the average grain material density is $3.0\,{\rm{g}}\,{\mathrm{cm}}^{-3}$, the cross-sectional area of dust grains is ${\sigma }_{{\rm{H}}}=1.64\times {10}^{-21}\,{\mathrm{cm}}^{2}$ per H atom. The cosmic-ray-induced sputtering rate of ${\mathrm{CH}}_{3}\mathrm{OH}$ from water ice is then ∼2.3 × 10⁻²² molecules ${{\rm{s}}}^{-1}$ per H atom. The rate per cubic centimeter and second can be obtained by multiplying this number by the total hydrogen density n_H, which is $\sim 2\times n({{\rm{H}}}_{2})$ in molecular clouds. The corresponding desorption rate per methanol molecule is $\sim 5.7\times {10}^{-17}$ mol⁻¹ ${{\rm{s}}}^{-1}$, assuming that the fractional methanol abundance on grains relative to H atoms is $n({\mathrm{CH}}_{3}\mathrm{OH},{\rm{s}})/{n}_{{\rm{H}}}\sim 4\times {10}^{-6}$. This abundance is derived using the composition of the ice mantles in quiescent clouds and cores listed in Table 2 of Boogert et al. (2015), assuming that the mantle constitutes 15% of the grain mass. The adopted ${\mathrm{CH}}_{3}\mathrm{OH}$ abundance agrees reasonably well with gas-grain chemistry models (e.g., Taquet et al. 2014; Vasyunin et al. 2017). Dartois et al. (2019) estimate that the flux of sputtered ${\mathrm{CH}}_{3}\mathrm{OH}$ from CO ice is 40 times higher than that from water ice, which gives a methanol sputtering rate of $\sim 9\times {10}^{-21}\,{{\rm{s}}}^{-1}$ per H atom or ∼2 × 10⁻¹⁵ mol${}^{-1}\,{{\rm{s}}}^{-1}$.

By way of comparison, the direct cosmic-ray-induced desorption rate of water in conditions described above can be estimated to be $\sim 4.6\times {10}^{-21}\,{{\rm{s}}}^{-1}$ per H atom or ∼9.3 × 10⁻¹⁷ mol${}^{-1}\,{{\rm{s}}}^{-1}$. Here, we have used the sputtering flux from water ice derived by Faure et al. (2019), ${F}_{{{\rm{H}}}_{2}{\rm{O}}}^{\mathrm{sp}}\,\sim {f}_{{{\rm{H}}}_{2}{\rm{O}}}\,0.8\,{\zeta }^{{{\rm{H}}}_{2}}/{10}^{-17}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$, where ${f}_{{{\rm{H}}}_{2}{\rm{O}}}\sim 0.6$ is the fraction of water molecules in the ice according to Table 2 of Boogert et al. (2015; see also Equation (7) of Faure et al. 2019). The fractional water abundance relative to H atoms, $n({{\rm{H}}}_{2}{\rm{O}},{\rm{s}})/{n}_{{\rm{H}}}\sim 5\times {10}^{-5}$, is estimated in the same manner as the fractional methanol abundance above. Because of similar binding energies of ${{\rm{H}}}_{2}{\rm{O}}$ and ${\mathrm{CH}}_{3}\mathrm{OH}$, the desorption rate estimate of Faure et al. (2019) should also be adequate for methanol when corrected for the methanol fraction, ∼0.08 with respect to ${{\rm{H}}}_{2}{\rm{O}}$. The resulting ${\mathrm{CH}}_{3}\mathrm{OH}$ desorption rate ($\sim 3.7\times {10}^{-22}\,{{\rm{s}}}^{-1}$ per H atom or ∼9.3 × 10⁻¹⁷ mol${}^{-1}\,{{\rm{s}}}^{-1}$) is ∼60% higher than that derived using data from Dartois et al. (2019). On the other hand, applying similar scaling to the cosmic-ray-induced sputtering rate per water molecule, 4.4 × 10⁻¹⁷ mol${}^{-1}\,{{\rm{s}}}^{-1}$, derived by Bringa & Johnson (2004), brings us very close to the result of Dartois et al. (2019), with a methanol sputtering rate of $\sim 1.8\times {10}^{-22}\,{{\rm{s}}}^{-1}$ per H atom or 4.4 × 10⁻¹⁷ mol${}^{-1}\,{{\rm{s}}}^{-1}$. Finally, we note that the cosmic-ray desorption rate per methanol molecule used by Hasegawa & Herbst (1993) is very low compared to the values quoted above, 6.3 × 10⁻²⁰ mol${}^{-1}\,{{\rm{s}}}^{-1}$ ($\sim 2.5\times {10}^{-25}\,{{\rm{s}}}^{-1}$ per H atom).

Photodissociation and photodesorption by the external far-ultraviolet (FUV) radiation field are considered to be important near cloud boundaries up to an efficient visual extinction of A_V ∼ 5^mag (Hollenbach et al. 2009; Keto et al. 2014). Stellar radiation in the Ophiuchus complex is exceptionally strong. Hollenbach et al. (2009) assume that the FUV radiation field in the vicinity of the core Oph A in the western part of the complex is 300 times the average local interstellar radiation field in that band as that determined by Habing (1968; often denoted by G₀). Assuming that the FUV field is attenuated by 20 magnitudes in the ambient cloud (corresponding to an effective visual extinction of A_V ∼ 10^mag; e.g., Cardelli et al. 1989), the FUV flux at the core boundaries is $\sim 3\times {10}^{-6}\,{G}_{0}$, that is, ∼300 FUV photons ${\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$. We model the FUV field and its attenuation in the case of H-MM1 in Section 7.1 below.

Cosmic-ray protons and secondary electrons from cosmic-ray ionization sustain a flux of Lyman and Werner band photons (11.2–13.6 eV) through the collisional excitation of ${{\rm{H}}}_{2}$ to excited electronic states (Cecchi-Pestellini & Aiello 1992). According to the estimates of Cecchi-Pestellini & Aiello (1992) and Kalvāns (2018b), the flux of cosmic-ray-induced FUV photons is $\sim 3000\mbox{--}5000\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$ in the interiors of dense dark clouds. Often, a flux value of ${10}^{4}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$ or 10⁻⁴ G₀ is used in chemistry models (e.g., Keto et al. 2014).

The experimental results of Bertin et al. (2016) and Cruz-Diaz et al. (2016) indicate that the efficiency of FUV photodesorption of methanol is low, of the order of 10⁻⁵ molecules per incident photon. Assuming that dust grains are fully covered by methanol ice, the photodesorption rate corresponding to this efficiency is ∼3.3 × 10⁻²² methanol molecules ${{\rm{s}}}^{-1}$ per H atom. Using the composition of the ice mantles in quiescent clouds and cores listed in Table 2 of Boogert et al. (2015), with ∼5% methanol, the photodesorption rate becomes $\sim 1.6\times {10}^{-23}\,{{\rm{s}}}^{-1}$ per H atom (4.1 × 10⁻¹⁸ mol${}^{-1}\,{{\rm{s}}}^{-1}$). Here, we have adopted the FUV flux $5000\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$ from Cecchi-Pestellini & Aiello (1992) and used the total surface area of grains (4 × σ_H) from the MRN size distribution.

Öberg et al. (2009) measured an average loss rate of condensed methanol of ∼10⁻³/photon in irradiated ices. This loss rate is probably attributable to photodissociation, producing fragments such as ${\mathrm{CH}}_{3}$, OH, and ${\mathrm{CH}}_{3}{\rm{O}}$, that can remain in the ice mantle or be ejected into the gas phase (Bertin et al. 2016; Cruz-Diaz et al. 2016). The FUV photodissociation cross section of methanol ice is $\sim 2.7\times {10}^{-18}\,{\mathrm{cm}}^{2}$ (Öberg et al. 2009; Cruz-Diaz et al. 2016), so the photodissociation rate per methanol molecule caused by the cosmic-ray-induced FUV flux $5000\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$ is ∼1.3 × 10⁻¹⁴ mol${}^{-1}\,{{\rm{s}}}^{-1}$ (corresponding to $\sim 5.4\times {10}^{-20}\,{{\rm{s}}}^{-1}$ per H atom, assuming $X({\mathrm{CH}}_{3}\mathrm{OH},{\rm{s}})=4\times {10}^{-6}$).

Is cosmic-ray-induced photodissociation of mantle species an important source of chemical energy in dark cores? The ice mantle is supposed to constitute 15%–30% of the grain mass in dense dark clouds, and it is mainly composed of ${{\rm{H}}}_{2}{\rm{O}}$, $\mathrm{CO}$, ${\mathrm{CO}}_{2}$, ${\mathrm{CH}}_{3}\mathrm{OH}$, ${\mathrm{NH}}_{3}$, and ${\mathrm{CH}}_{4}$ (Boogert et al. 2015). Assuming that the mantle mass is 15% of that of the silicate core, one can calculate, adopting the same FUV flux as above, that the number of photons impinging onto a grain surface reaches 10% of the number of mantle molecules in ∼8000 yr. The 10% ratio of the number incident FUV photons to the number of molecules was considered crucial in the experiment of d'Hendecourt et al. (1982). The answer to the question asked above depends on the average number of heatings to the critical temperature 27 K during the quoted period of time.

Whole grain heating by cosmic-rays has been recently modeled by Kalvāns (2018a) and Kalvāns & Kalnin (2019). Using estimates of the energy spectrum of cosmic-rays and their fluxes in molecular clouds from Padovani et al. (2009) and Chabot (2016), Kalvāns has derived cosmic-ray heating rates that (for moderate extinctions) are clearly higher than earlier estimates (see Figure 9 in Kalvāns 2018a). According to this work, grains residing in an obscured cold cloud (A_V ∼ 10^mag) are heated to 27 K every ∼900 yr (Kalvāns 2018a; their Table 19). This interval is an order of magnitude shorter than the time needed for the accumulation of any substantial amount of radicals through cosmic-ray-induced photolysis.

In reactive desorption, a molecule forming on a grain escapes into the gas phase with the help of the exothermicity of the formation reaction, without any external agent such as a cosmic-ray or a photon (e.g., Garrod et al. 2006, 2007; Dulieu et al. 2013; Vasyunin & Herbst 2013; Minissale et al. 2016). The exothermicities of CO hydrogenation reactions are of the order of a few eV, exceeding thus clearly the binding energies of the products. The question is: which part of the released energy goes to the breaking of the chemical bond to the surface and how much goes to the heating of the grain (Garrod et al. 2006)?

One can derive an estimate for the production rate of gaseous methanol assuming that a certain fraction, f_CO, of CO molecules hitting grains will be completely hydrogenated to methanol and that a fraction, ${f}_{{\mathrm{CH}}_{3}\mathrm{OH}}$, of methanol molecules so formed will be ejected into the gas. Garrod et al. (2006) found that a low value of the hydrogenation factor, ${f}_{\mathrm{CO}}\sim 0.003$, reproduces the observed methanol abundances in their model. Another, much higher value of ${f}_{\mathrm{CO}}\sim 0.13$ can be derived based on the relative abundances of carbon-bearing ice molecules in dark clouds ices listed in Boogert et al. (2015).

The fraction of desorbed products, ${f}_{{\mathrm{CH}}_{3}\mathrm{OH}}$, also called the efficiency of reactive desorption, has been found to depend strongly on the substrate, so that it is much larger for a rigid surface than for amorphous water ice (Minissale et al. 2016 and references therein). Based on the theoretical description of the reactive desorption efficiency presented by Minissale et al. (2016), Vasyunin et al. (2017) assumed that the efficiency of methanol desorption upon formation on a CO-rich surface is of the order of 1%. Similar desorption efficiencies have been used in previous works (for example, Garrod et al. 2007 and Vasyunin & Herbst 2013). According to the model of Vasyunin et al. (2017), the formation of methanol is most efficient in the "CO freeze-out zone" at densities of $n({{\rm{H}}}_{2})\sim {10}^{4}\mbox{--}{10}^{5}\,{\mathrm{cm}}^{-3}$, where the gas-phase abundance of CO is still high, a few times 10⁻⁵ relative to ${{\rm{H}}}_{2}$. This model gives, therefore, an explanation for the fact that ${\mathrm{CH}}_{3}\mathrm{OH}$ seems to avoid the densest parts and sometimes shows a shell-like distribution (Tafalla et al. 2006; Bizzocchi et al. 2014; Punanova et al. 2018).

The density of CO molecules in the CO-freezing zone is similar to that of H atoms, that is ${n}_{\mathrm{CO}}\sim 1\,{\mathrm{cm}}^{-3}$ ($X(\mathrm{CO})={10}^{-5}\mbox{--}{10}^{-4}$), but their average thermal speed is a factor of five lower, ${\bar{v}}_{\mathrm{CO}}\sim 8700\,\mathrm{cm}\,{{\rm{s}}}^{-1}$. Using the cross-sectional area of grains from the MRN size distribution and the $\mathrm{CO}$ flux implied by the density and the speed quoted above, the methanol production rate by reactive desorption is ∼4.3 × 10⁻²² or 1.9 × 10⁻²⁰ molecules ${{\rm{s}}}^{-1}$ per H atom ($1.1\times {10}^{-16}$ or 4.6 × 10⁻¹⁵ mol${}^{-1}\,{{\rm{s}}}^{-1}$), depending on whether we adopt ${f}_{\mathrm{CO}}=0.003$ or ${f}_{\mathrm{CO}}=0.13$. The higher of these rates is comparable to the cosmic-ray-induced sputtering rate of methanol from CO ice estimated in Section 6.1, based on the results of Dartois et al. (2019).

As discussed by d'Hendecourt et al. (1982), low-velocity collisions between grains can lead to grain heating above the temperature threshold (∼30 K) that triggers explosive radical–radical reactions and the partial disruption of the grain mantle. Assuming that the grains consist of a silicate core and a mantle of water ice constituting ∼15% of the grain mass, one finds that the enthalpy change needed to raise the grain temperature from 10 to 27 K corresponds to a collision velocity of ∼30 ${\rm{m}}\,{{\rm{s}}}^{-1}$. Here, we have adopted the heat capacity functions from Leger et al. (1985; their Equation (1)) and Shulman (2004; their Equation (4)) for the grain core and the ice mantle, respectively. The collisional speed that is needed to heat these grains to 100 K is 180 ${\rm{m}}\,{{\rm{s}}}^{-1}$.

Collisions between grains require that the grain population has acquired an internal velocity dispersion. This can arise when grains are embedded in a turbulent gas (Voelk et al. 1978; Draine 1985). In the turbulent acceleration model presented by Draine (1985), the grain velocities are determined by turbulent velocity fluctuations occurring on a timescale comparable to that of the hydrodynamical drag. In Kolmogorov turbulence, the velocity is proportional to the square-root of the eddy turn-over time. Because the drag time is directly proportional to the grain radius, a, the velocity distribution of large grains has a square-root dependence on the grain radius, $v\propto {a}^{1/2}$. The smallest grains are also coupled to the magnetic field,²⁰ and their velocity dispersion is determined by turbulent fluctuations on a timescale comparable to the Larmor time (Lazarian & Yan 2002; Yan et al. 2004). This causes a $v\propto {a}^{3/2}$ dependence for the smallest grains. Overall, turbulent acceleration is more effective for large grains than for small grains. This gives rise to velocity differences between small and large grains and to an enhanced rate of grain–grain collisions.

In Figure 6, we show relative grain velocities as a function of grain radius according to the models of Draine (1985) and Lazarian & Yan (2002), for conditions characteristic of the outer envelope of a cold dense core, $n({{\rm{H}}}_{2})={10}^{5}$ cm⁻³, T = 10 K, $B=100\,\mu {\rm{G}}$. Here, it is assumed that the turbulent velocity field has a Kolmogorov-like spectrum, $v\propto {l}^{1/3}$. The absolute scale is set by assuming that the turbulent velocity is $0.7\,\mathrm{km}\,{{\rm{s}}}^{-1}$ on a scale of 0.034 pc. This corresponds to the velocity gradient across the H-MM1 core (Section 4; for discussion about the connection between velocity gradients and turbulence see Burkert & Bodenheimer 2000). Assuming the MRN grain size distribution, the small, slow grains with velocities below 10 ${\rm{m}}\,{{\rm{s}}}^{-1}$ comprise $\sim 50 \%$ of the total surface area of the grains, whereas the share of large, fast grains with v > 30 ${\rm{m}}\,{{\rm{s}}}^{-1}$ is only ∼5% of the surface area (albeit ∼50% of the dust mass).

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The frequency at which a "slow" grain collides with a "fast" grain is obtained from

$$\begin{eqnarray}&&{f}_{\mathrm{coll}}={n}_{{\rm{g}}}^{\mathrm{fast}}\ \pi {({\bar{a}}^{\mathrm{fast}}+{\bar{a}}^{\mathrm{slow}})}^{2}\,({\bar{v}}_{{\rm{g}}}^{\mathrm{fast}}-{\bar{v}}_{{\rm{g}}}^{\mathrm{slow}}),\end{eqnarray} \tag{ 1 }$$

where ${n}_{{\rm{g}}}^{\mathrm{fast}}$ is the number density of fast grains, ${\bar{a}}^{\mathrm{fast}}$ and ${\bar{a}}^{\mathrm{slow}}$ are the average radii of the fast and slow grains, respectively, ${\bar{v}}_{{\rm{g}}}^{\mathrm{fast}}$ and ${\bar{v}}_{{\rm{g}}}^{\mathrm{slow}}$ are their average speeds. Here, we have used the classical collisional cross section for hard spheres (e.g., Caselli et al. 1997). The MRN grain size distribution and the physical parameters mentioned above imply the following values for the average radii and speeds: ${\bar{a}}^{\mathrm{slow}}\sim 80\,\mathring{\rm A}$, ${\bar{a}}^{\mathrm{fast}}\sim 0.12\,\mu {\rm{m}}$, and ${\bar{v}}_{{\rm{g}}}^{\mathrm{fast}}-{\bar{v}}_{{\rm{g}}}^{\mathrm{slow}}\sim 34\,{\rm{m}}\,{{\rm{s}}}^{-1}$. The number density of "fast" grains is ${n}_{{\rm{g}}}^{\mathrm{fast}}\sim 8\times {10}^{-8}\,{\mathrm{cm}}^{-3}$. With these parameters, the collision frequency is very low, ${f}_{\mathrm{coll}}\sim 1.4\times {10}^{-13}\,{{\rm{s}}}^{-1}$, meaning that a slow grain collides with a fast one once in ∼200,000 yr. Using the actual grain size and speed distributions without averaging, one obtains ∼25% higher grain–grain collision rate at a minimum speed difference of 30 ${\rm{m}}\,{{\rm{s}}}^{-1}$.

When both particles are negatively charged, the kinetic energy overcomes the Coulomb barrier at this collision speed. We assume that a grain carries maximally one electron charge. The electrostatic repulsion can be important in collisions between the smallest grains with steep Coulomb potentials and small velocity fluctuations (Chokshi et al. 1993; Perez-Becker & Chiang 2011). In high-speed collisions, one of the grains is large, a ∼ 0.1 μm = 10⁻⁵ cm, and the minimum collision speed needed to overcome the Coulomb barrier is probably lower than the critical speed for sticking (Chokshi et al. 1993, Figure 8).

Despite the low collision frequency, the desorption rate can be high if every collision between a slow and a fast grain leads to a chemical explosion and the complete disruption of the ice mantle of the smaller, slow grain. In this case, the desorption rate of methanol would be ${f}_{\mathrm{coll}}\,{n}_{{\mathrm{CH}}_{3}\mathrm{OH}}^{\mathrm{surf},\ \mathrm{slow}}$ molecules ${\mathrm{cm}}^{-3}\,{{\rm{s}}}^{-1}$, where ${n}_{{\mathrm{CH}}_{3}\mathrm{OH}}^{\mathrm{surf},\ \mathrm{slow}}$ is the number density of methanol molecules residing on slow grains (corresponding in our example to $\sim 50 \%$ of their total number density). It is, however, questionable if mild grain–grain collisions that heat small grains to ∼27 K can trigger mantle explosions, when these grains are, according to the results of Kalvāns (2018a), heated to the same temperature much more often by cosmic-rays (see end of Section 6.2), and the accumulation of radicals in the mantle is thereby inhibited.

Therefore, it seems plausible that only energetic grain–grain collisions that heat them close to ∼100 K can lead to significant desorption of methanol through thermal evaporation. The maximum grain speed acquired in Kolmogorov turbulence depends on the density and temperature according to ${v}_{\max }\propto {n}^{-1/2}\,{T}^{-1/4}$ (Draine 1985). At densities above $\sim 3\times {10}^{5}\,{\mathrm{cm}}^{-3}$, the maximum speed drops below the critical value 30 ${\rm{m}}\,{{\rm{s}}}^{-1}$. Assuming that the temperature remains constant at 10 K, the maximum density where grain–grain collisions are energetic enough to heat them to 100 K (with a collisional speed of ∼180 ${\rm{m}}\,{{\rm{s}}}^{-1}$) is $n({{\rm{H}}}_{2})\sim 8\times {10}^{3}\,{\mathrm{cm}}^{-3}$. Grain–grain collision frequencies decrease toward lower densities, as is likely to happen to the methanol abundance on grains.

Following Shen et al. (2004; see their Equation (29)), we assume that collisional energy in excess of heating a grain to 26 K is available for sublimation. In a totally inelastic collision at $180\,{\rm{m}}\,{{\rm{s}}}^{-1}$, this excess energy per unit mass is ∼16 J g⁻¹. Adopting the mantle composition from Boogert et al. (2015) and assuming that the mantle constitutes 15% of the grain mass, the total binding energy of the mantle molecules per one gram of dust is ∼230 J g⁻¹. Here, we have used the following binding energies: ${{\rm{H}}}_{2}{\rm{O}}$ 5700 K, CO 1150 K, CO₂ 2600 K, ${\mathrm{CH}}_{3}\mathrm{OH}$ 5500 K, ${\mathrm{NH}}_{3}$ 5500 K, and CH₄ 1300 K. So, it seems that the energy deposited by a grain–grain collision at 180 ${\rm{m}}\,{{\rm{s}}}^{-1}$ can evaporate, at most, ∼7% of the mantle material. The minimum collision speed required for the evaporation of the whole mantle is ∼680 ${\rm{m}}\,{{\rm{s}}}^{-1}$. Neglecting the differences in the binding energies, that is, assuming that the mantle species are evaporated in proportion to their relative abundances, we obtain a methanol desorption rate of $\sim 4\times {10}^{-21}\,{{\rm{s}}}^{-1}$ per H atom or $\sim 1\times {10}^{-15}$ mol${}^{-1}\,{{\rm{s}}}^{-1}$ at the density $n({{\rm{H}}}_{2})\sim 8\times {10}^{3}\,{\mathrm{cm}}^{-3}$ (by collisions at the minimum speed of 180 ${\rm{m}}\,{{\rm{s}}}^{-1}$). However, the presence of SO emission in the same region where we see ${\mathrm{CH}}_{3}\mathrm{OH}$ lines in H-MM1 suggests that the density methanol desorption layer is of the order of $\sim {10}^{5}\,{\mathrm{cm}}^{-3}$ (see Section 7). Kolmogorov turbulence at this density is unlikely to induce collisions leading to significant thermal evaporation.

The last mechanism discussed here is the release of methanol in grain–grain collisions induced by shear instability. This is motivated by the fact that the methanol distribution observed in H-MM1 (Figure 2) resembles Kelvin–Helmholtz clouds that sometimes develop in the atmosphere in flows with large vertical shears. As described in standard textbooks, such as Batchelor (1967), advection of vorticity amplifies the dominant sinusoidal disturbance in a vortex sheet and makes it roll up. The evolution of vortex sheets has been studied numerically and semi-analytically by Patnaik et al. (1976) and Corcos & Sherman (1976). According to their results, vorticity reaches its maximum values at places they call "braids" (near the troughs) and "cores" or cat's eyes (near the crests), separated by approximately half the wavelength of the dominant disturbance.

We conjecture that vorticity in the shear layer can accelerate dust grains to velocities deviating from the mean flow and induce grain–grain collisions, in the same manner as Kolmogorov-type turbulence discussed in the previous section. However, the energy spectrum and the corresponding velocity scaling law in a vortex sheet are different from those in fully developed turbulence (Kraichnan 1967, 1971; Gilbert 1988; Abid & Verga 2011).

The timescale of velocity fluctuations depends on the free-stream velocity and the size-scale of vorticity. The drag time for 0.1 μm grains is ∼100 yr at $n({{\rm{H}}}_{2})\sim {10}^{5}\,{\mathrm{cm}}^{-3}$. Assuming that the free-stream velocity is transonic, U ∼ 200 ${\rm{m}}\,{{\rm{s}}}^{-1}$, one obtains that only small vortices of size λ ≲ 5 au can give rise to significant velocity differences among the grain population.

In Figure 7, we plot the velocity field calculated from the stream function given by Corcos & Sherman (1976; their Equation (3.9)). Some of the streamlines are also plotted, including the so-called stagnation streamline, which outlines the cat's eyes. In this solution, the traverse time is not constant inside the cat's eyes but increases steeply from the center and reaches the maximum (∼300 yr in this example with U = 200 ${\rm{m}}\,{{\rm{s}}}^{-1}$, λ = 10 au) on the stagnation streamline. Beyond this streamline, the period of the smooth oscillatory motions attains quickly a constant value (∼230 yr).

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In the structure shown in Figure 7, the turn-over time of the flow is shorter than the drag time of 0.1 μm grains within the contour Ψ = 0. One could expect grain–grain collisions to be concentrated to these regions, where small grains should be advected by the swirling flow, but large grains will not respond to the velocity fluctuations. For still smaller vortices, stagnation points between the cat's eyes (at the origin of Figure 7) are likely places of vigorous grain–grain collisions. Because of the high speed of the flow, the collisions are more frequent and more energetic than those induced by Kolmogorov turbulence. Assuming an average collision speed of 200 ${\rm{m}}\,{{\rm{s}}}^{-1}$ and making the division between small and large grains at a = 0.1 μm, a small grain collides at an average frequency of $\sim 7\times {10}^{-13}\,{{\rm{s}}}^{-1}$ (every 45,000 yr) at a density of $n({{\rm{H}}}_{2})\sim {10}^{5}\,{\mathrm{cm}}^{-3}$. At this collision speed, the excess energy after heating the grains to 26 K is ∼20 J g⁻¹, and we assume that this is sufficient for evaporating 9% of the mantle material (see end of Section 6.4).

The resulting methanol desorption rate is ∼3 × 10⁻¹⁴ molecules ${\mathrm{cm}}^{-3}\,{{\rm{s}}}^{-1}$ or $\sim 2\times {10}^{-19}\,{{\rm{s}}}^{-1}$ per H atom, corresponding to ∼4 × 10⁻¹⁵ mol${}^{-1}\,{{\rm{s}}}^{-1}$. Here, we have again assumed that the fractional abundance of methanol ice is 4 × 10⁻⁶ with respect to H atoms. The desorption rate by shear-induced grain–grain collisions is comparable to the rates of reactive desorption and cosmic-ray-induced sputtering in their upper ranges. This mechanics is, however, confined to very small volumes near discontinuities in the velocity field. Assuming that the vortex sheet is linear with a thickness of 5 au, it fills approximately 1% of the 560 au beam of the present observations. A winding structure would increase the beam filling factor slightly. Nevertheless, taking the nonuniform beam filling into account, the contribution of grain–grain collisions in shear layers to the observed methanol abundance can be locally similar to those of the more widespread reactive desorption and cosmic-ray-induced sputtering. The estimated methanol desorption rates for different mechanisms are summarized in Table 2.

Table 2.  Estimated ${\mathrm{CH}}_{3}\mathrm{OH}$ Desorption Rates and the Sections Where They Are Discussed

Mechanism	kdes (mol${}^{-1}\,{{\rm{s}}}^{-1}$)	Section
Cosmic-ray desorption	$6\times {10}^{-17}\mbox{--}2\times {10}^{-15}$	6.1
Photodesorption	4 × 10−18–8 × 10−17	6.2
Reactive desorption	1 × 10−16 − 5 × 10−15	6.3
Grain–Grain collisions
Kolmogorov turbulence	$\sim 1\times {10}^{-15}$ a	6.4
Shear instability	∼4 × 10−15b	6.5

Notes.

^aUnlikely to work at high densities. ^bLocalized to small areas.

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An asymmetric methanol distribution has been previously found in the prestellar core L1544 (Spezzano et al. 2016; Punanova et al. 2018). Spezzano et al. (2016) suggested that the methanol distribution in L1544 reflects asymmetric illumination, which hinders CO production on the more exposed side of the cloud. Intense radiation could also inhibit SO formation by ionization. The ionization potentials of SO and the S atom are relatively low, 10.29 eV and 10.36 eV, respectively; the limiting wavelength of ionizing radiation, $\lambda \lt 120\,\mathrm{nm}$, falls in the range where photons also can dissociate ${\mathrm{CH}}_{3}\mathrm{OH}$ efficiently (114 nm < λ < 180 nm; Cruz-Diaz et al. 2016).

The ambient cloud around H-MM1 continues toward the east, giving reason to believe that the interstellar radiation field is stronger on the western side where ${\mathrm{CH}}_{3}\mathrm{OH}$ and SO are weaker. This side faces the Upper Sco-Cen subgroup of the Scorpius-Centaurus OB association, including the luminous B-type double stars ρ Oph and HD 147889 (de Geus 1992; Preibisch & Mamajek 2008; Damiani et al. 2019). The blue subgiant binary HD 147889, about 1.2 pc west of H-MM1, is the dominant UV source in the region (Liseau et al. 1999; Casassus et al. 2008; Rawlings et al. 2013).

If the radiation field is stronger on the western side, this should be evident as a dust temperature difference between the two sides of the core because dust grains are primarily heated by absorption of starlight. We examined this hypothesis using mid- to far-infrared maps from Spitzer, Herschel, and SCUBA-2, and we found that the dust temperature reaches, indeed, its minimum on the eastern side of the core, close to the ${\mathrm{CH}}_{3}\mathrm{OH}$ maximum. The analysis of the dust continuum maps is presented in Appendix B. The analysis suggests a temperature drop from about 14–15 K on the western edge to about 11–12 K in the eastern cove of the core (see Figure B1).

We tested this result by simulating the effect of an asymmetric radiation field on a core model resembling H-MM1. The strength of the external field was adjusted until the model could reproduce the observed surface brightness maps at 850 μm. The details of the calculation are presented in Appendix B. The simulation shows that an anisotropic radiation field causes a displacement of the temperature minimum from the density maximum and that this displacement can be inferred by comparing 70 and 850 μm surface brightness maps. It also gives estimates for the strength of the FUV radiation field on the eastern and western edges of the core. The energy density of the FUV field along three cuts across the model core is shown in Figure B2. The strongest methanol emission in H-MM1 is found at the offsets 10''–20'' east of the density maximum (the map center). In the simulated core, the FUV energy density has a deep minimum in this range, and the average radiation flux originating from the external field is negligible. In the corresponding region on the western side of the core, the average FUV flux is approximately F_FUV = 80,000 photons ${\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$, that is, an order of magnitude higher than the flux of cosmic-ray-induced photons (${F}_{\mathrm{FUV}}=5000\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$) in dense dark clouds estimated by Cecchi-Pestellini & Aiello (1992).

The difference in the FUV fluxes at the western and eastern boundaries of the core is likely to affect the methanol abundances both in the gas and on grains. Assuming that ${\mathrm{CH}}_{3}\mathrm{OH}$ is formed on grains through CO accretion and removed by photodissociation and by some generic desorption mechanism, its equilibrium abundance on grains relative to H atoms can be written as

$$\begin{eqnarray}&&X({\mathrm{CH}}_{3}\mathrm{OH},{\rm{s}})=\displaystyle \frac{{f}_{\mathrm{CO}}\,{\sigma }_{{\rm{H}}}\,{\bar{v}}_{\mathrm{CO}}\,{n}_{\mathrm{CO}}}{{\sigma }_{\mathrm{ph},{\rm{s}}}\,{F}_{\mathrm{FUV}}+{k}_{\mathrm{des}}},\end{eqnarray} \tag{ 2 }$$

where f_CO is the efficiency factor for the conversion $\mathrm{CO}\to {\mathrm{CH}}_{3}\mathrm{OH}$ on grains, ${\bar{v}}_{\mathrm{CO}}$ is the average thermal speed of CO molecules, n_CO is their number density, ${\sigma }_{\mathrm{ph},{\rm{s}}}\,=2.7\times {10}^{-18}\,{\mathrm{cm}}^{2}$ is the FUV photodissociation cross section of methanol ice (Cruz-Diaz et al. 2016), and k_des is the desorption rate per molecule ${{\rm{s}}}^{-1}$. We assume here that the desorption rate is constant and represents the high end of the various estimates above, that is, k_des = 5 × 10⁻¹⁵ mol${}^{-1}\,{{\rm{s}}}^{-1}$. For simplicity, we also assume the number density and the average speed of CO molecules are constant in the core, ${n}_{\mathrm{CO}}=1\,{\mathrm{cm}}^{-3}$, ${\bar{v}}_{\mathrm{CO}}=8700\,\mathrm{cm}\,{{\rm{s}}}^{-1}$. Moreover, we assume that a constant cosmic-ray-induced FUV flux, $5000\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$, comes everywhere on top of the attenuated external field. We then adjust the efficiency factor f_CO so that the maximum fractional methanol abundance on grains is 4 × 10⁻⁶. The resulting factor is ${f}_{\mathrm{CO}}=0.05$.

Assuming that accretion onto grains and photodissociation dominate the destruction of gas-phase methanol, its equilibrium abundance can be obtained from

$$\begin{eqnarray}&&X({\mathrm{CH}}_{3}\mathrm{OH},{\rm{g}})=\displaystyle \frac{X({\mathrm{CH}}_{3}\mathrm{OH},{\rm{s}}){k}_{\mathrm{des}}}{{n}_{{\rm{H}}}\,{\sigma }_{{\rm{H}}}\,{\bar{v}}_{{\mathrm{CH}}_{3}\mathrm{OH}}+{\sigma }_{\mathrm{ph},{\rm{g}}}\,{F}_{\mathrm{FUV}}},\end{eqnarray} \tag{ 3 }$$

where ${\bar{v}}_{{\mathrm{CH}}_{3}\mathrm{OH}}$ is the average thermal speed of methanol molecules ($\sim 8100\,\mathrm{cm}\,{{\rm{s}}}^{-1}$ at 10 K), and ${\sigma }_{\mathrm{ph},{\rm{g}}}$ is the FUV photodissociation cross section of methanol in the gas. For the latter, we adopt the value ${\sigma }_{\mathrm{ph},{\rm{g}}}\sim 1\times {10}^{-17}\,{\mathrm{cm}}^{2}$ based on the experimental results of Person & Nicole (1971) and Cheng et al. (2002). The hydrogen density, ${n}_{{\rm{H}}}$, comes from our core model. The grain-surface and gas-phase abundances of methanol calculated from Equations (2) and (3) along a horizontal line that goes through the density maximum of the core model are plotted in Figure 8. The abundance profile in the gas phase is double-peaked, with the stronger peak residing on the dark side of the core, whereas on grains, the abundance profile is flat-topped. The eastern and western gas-phase peaks, with $X({\mathrm{CH}}_{3}\mathrm{OH},{\rm{g}})\sim 4\times {10}^{-9}$ and $X({\mathrm{CH}}_{3}\mathrm{OH},{\rm{g}})\sim 3\times {10}^{-9}$, occur at the densities $n({{\rm{H}}}_{2})\sim 1.8\times {10}^{5}\,{\mathrm{cm}}^{-3}$ (east) and $n({{\rm{H}}}_{2})\sim 2.0\times {10}^{5}\,{\mathrm{cm}}^{-3}$ (west). The fractional grain-surface methanol abundances at these locations are $X({\mathrm{CH}}_{3}\mathrm{OH},{\rm{s}})\sim 1.3\times {10}^{-6}$ (east) and $X({\mathrm{CH}}_{3}\mathrm{OH},{\rm{s}})\,\sim 2.6\times {10}^{-7}$ (west). The stronger methanol peak lies much farther out from the core center than in the observed core, but we cannot exclude the possibility that the difference is caused by a projection effect.

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This simple model can be envisaged to give a qualitative explanation for the methanol distribution observed in H-MM1, even though Figure 8 shows the methanol abundance along a line lying in the plane of the sky, while we are measuring line-of-sight averages. However, the contrast between the peak methanol abundances on the eastern and western sides, ≳10:1, cannot be reproduced by this model. Increasing the external FUV flux would shift the methanol peaks inward, toward higher densities, but the abundance ratio would remain the same, approximately 4:3. Likewise, one can increase or decrease the gas-phase abundances by adjusting the desorption rate with the same effect on the abundance ratio.

Other than indicating asymmetry in the radiation field, the dust temperature drop itself can contribute to the high abundances of ${\mathrm{CH}}_{3}\mathrm{OH}$ on the eastern side. Methanol production is probably favored by efficient accretion and hydrogenation of CO on grains at low temperatures. It should be noted, however, that the cooler region does not cover the whole eastern side of the core where strong ${\mathrm{CH}}_{3}\mathrm{OH}$ emission is found, and it does not extend to the SO maximum. At the northern and southern ends of the integral-shaped ${\mathrm{CH}}_{3}\mathrm{OH}$ and SO emission region, the dust temperature is similar to that on the western side where these species show only weak emission.

The estimates and considerations presented above suggest that in addition to the fact that the H-MM1 core is exposed to a strong, asymmetric radiation field that hinders methanol formation on its western side, desorption is particularly effective at its eastern boundary. Of the mechanisms discussed in Section 6, photodesorption is very unlikely to cause this effect. Also, it is not likely that the cosmic-ray flux shows substantial anisotropy on this scale or that the efficiency of reactive desorption is higher in the east, except perhaps in the vicinity of the temperature minimum (Appendix B). In what follows, we therefore consider desorption related to gas dynamics at the core boundaries.

Both ${\mathrm{CH}}_{3}\mathrm{OH}$ and SO are known to increase in shocks, and they have been used to probe outflows and the accretion process associated with star formation (Bachiller & Pérez Gutiérrez 1997; Podio et al. 2015; Oya et al. 2016). Strong enhancement of SO and ${\mathrm{SO}}_{2}$ is predicted by models of magnetized molecular C-shocks, as a result of neutral–neutral reactions in the shock-heated gas and the erosion of S-rich icy grain mantles owing to bombardment by heated gas particles (Pineau des Forêts et al. 1993; Flower & Pineau des Forets 1994). Sputtering of the grain mantles associated with shocks can also increase the ${\mathrm{CH}}_{3}\mathrm{OH}$ abundance substantially in the gas phase (Jiménez-Serra et al. 2008). Previous observations of mid-J CO rotational lines toward Perseus and Taurus complexes suggest low-velocity shocks associated with dissipation of turbulence and core formation in molecular clouds (Pon et al. 2014; Larson et al. 2015). Judging from the fact that the nonthermal velocity dispersion experiences an abrupt change at the core boundary (Auddy et al. 2019; see Section 7.3), low-velocity shocks caused by accreting material are also possible in the case of H-MM1. However, the present data consisting of low-lying rotational lines ${\mathrm{CH}}_{3}\mathrm{OH}$, SO, and ${\mathrm{NH}}_{2}{\rm{D}}$ do not show any evidence of shock heating or velocity gradients that would be large enough (${\rm{\Delta }}v\gtrsim 10\,\mathrm{km}\,{{\rm{s}}}^{-1}$) to give rise to significant shock-induced sputtering (Caselli et al. 1997; Jiménez-Serra et al. 2008).

The interior parts of the H-MM1 core have a very low level of nonthermal motions. This is evident from the velocity dispersion of the ortho-${\mathrm{NH}}_{2}{\rm{D}}$ lines illustrated in Figures 4(a) and (5). In a study based on ${\mathrm{NH}}_{3}(1,1)$ and (2, 2) inversion line observations from the GAS (Friesen et al. 2017), Auddy et al. (2019) show that the nonthermal velocity dispersion increases suddenly at the boundaries of this and several other cores in Ophiuchus. Turbulent acceleration of grains and grain–grain collisions can be envisioned to lead to an enhanced desorption of frozen molecules in the transition zone. The process should affect all atoms and molecules residing in the CO-rich outer layers of grain mantles, and this would explain why ${\mathrm{CH}}_{3}\mathrm{OH}$ and SO have similar distributions.

In this scenario, the asymmetric distributions of ${\mathrm{CH}}_{3}\mathrm{OH}$ and SO around H-MM1 would either indicate that turbulence is stronger on the eastern side of the core or that dust grain surfaces are richer in CO on that side. The latter suggestion is supported by the fact that the ${\mathrm{CH}}_{3}\mathrm{OH}$ peak coincides with the dust temperature minimum (Appendix B). The former condition cannot be properly tested using the present data because the boundaries are not probed with any other lines than methanol and SO; the ${\mathrm{NH}}_{2}{\rm{D}}$ line emission is confined to the inner regions where the lines are narrow. The angular resolution of the Green Bank ${\mathrm{NH}}_{3}$ maps (32''; Friesen et al. 2017), which cover both the core and the ambient cloud, is not sufficient for detailed comparison of the velocity dispersions at the eastern and western boundaries. Nevertheless, a similar analysis as performed by Auddy et al. (2019), but averaging over semicircles shows that the nonthermal velocity dispersion (measured along the line of sight) grows more slowly on the eastern side than on the western side, contradicting the supposition of a more vigorous turbulence on that side. A diagram illustrating this difference is shown in Figure C1.

One remarkable feature of the brightest ${\mathrm{CH}}_{3}\mathrm{OH}$ emission region is that it closely follows the eastern boundary of the core as marked by ${\mathrm{NH}}_{2}{\rm{D}}$ emission. The wavy shape and the rolling-up seen in the north (Figure 2) are the hallmarks of Kelvin–Helmholtz instability (KHI), which can occur in sheared flows with density stratification. The channel maps shown in Figure 3 give the impression that the dense gas is flowing along a loop. If the ambient gas does not share this motion, the core is surrounded by a shear layer. The presence of shear in a gas flow with a high Reynolds number would imply small-scale vorticity, and the undulating methanol emission region would, in this case, consist of a chain of secondary billows that are unresolved in the present map. In this scenario, the methanol peak can be identified with a "braid" with crowded isopycnic surfaces and the overturning billow in the north with a "core" where these surfaces roll up (Patnaik et al. 1976; see Section 6.5). The apparent wavelength of the methanol feature is approximately 60'', corresponding to 8400 au or 0.041 pc. The conceived structure is illustrated in Figure 9, which shows the maps of Figure 2 rotated in such a way that the suggested shear layer is approximately horizontal and the density increases downward. The directions of the relative velocities of the denser and thinner gas components are shown with arrows, and the positions of the "braid" and "core" are indicated.

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In what follows, we attempt to estimate the flow properties, assuming that, analogously to atmospheric billow clouds, the largest wavelength corresponds to the so-called internal gravity waves, which oscillate at the Brunt–Väisälä frequency. This frequency, also known as the buoyancy frequency, is defined by $N=\sqrt{\tfrac{-g}{\rho }\tfrac{\partial \rho }{\partial z}}$, where g is the gravitational acceleration (directed to the negative z-direction), and ρ is the gas density. We estimated the density and the density gradient in the supposed shear layer by fitting a Plummer-type function to the ${{\rm{H}}}_{2}$ column density profile from the IRAC 8 μm absorption. The inversion method is explained by Arzoumanian et al. (2011).

The fit was made to the cross-sectional profile along an axis going through the methanol peak at R.A.16:27:59.5, decl. −24:33:30 (J2000). The axis is tilted with respect to the R.A. axis by 25°. According to this fit, the number densities at the methanol peak and at the spine of cloud are ∼3.8 × 10⁵ cm⁻³ and ∼9.2 × 10⁵ cm⁻³, respectively. The separation between these points is approximately 10'' (1400 au). We assumed that the gravitational field at the methanol peak is dominated by the mass contained in a circular region of a radius of 10'', centered at the crossing of the cloud spine and the cross-sectional axis. This mass is 0.11 M_⊙. The Brunt–Väisälä frequency obtained is N ∼ 1.3 × 10⁻¹² Hz (period 150,000 yr). The multiplication of N by the apparent wavelength gives a phase velocity of ∼260 m s⁻¹. This should correspond to the average speed of the sheared flow.

In the case that the flow is subject to KHI, a condition is imposed to the velocity shear: the Richardson number, defined by $\mathrm{Ri}=\tfrac{{N}^{2}}{{(\partial v/\partial z)}^{2}}$, is less than one-fourth. The implied minimum shear is $\sim 2.6\times {10}^{-12}\,{{\rm{s}}}^{-1}$ or $\sim 80\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{pc}}^{-1}$. This value exceeds the north–south velocity gradient derived from the molecular line maps by a factor of four. Assuming that the velocity at the outer boundary of the shear layer does not exceed $400\,{\rm{m}}\,{{\rm{s}}}^{-1}$, which is the typical nonthermal velocity far from the core, the maximum thickness of the layer is approximately 1000 au. This estimate agrees with the thickness of the methanol layer.

In view of the uncertainties concerning the "vertical" scale and the structure of the vorticity, we do not attempt to estimate the methanol production rate in the suggested vortex sheet. We merely state that the present observations indicate that the formation of methanol is more efficient in this layer than elsewhere in the core envelope. We suggest, based on the theoretical and numerical studies quoted above, that this is caused by the fact that two-dimensional turbulence can have larger velocity fluctuations than full three-dimensional turbulence on a timescale comparable to the drag time of dust grains.