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Appendix A: Shattering and vaporization: their implementation and validation

Appendix A.1: Shattering

Appendix A.1.1: Dynamical effects of shattering

The dynamical effects of shattering on the shock structure can be simulated by ensuring that the total grain cross section, ⟨ nσ ⟩, changes in accord with the results of Paper III. We found that it is possible to model the increase of ⟨ nσ ⟩ due to shattering by multiplying the total grain cross section, in the absence of grain-grain processing, by an additional factor, which varies with the spatial coordinate, z, through the shock wave. This factor is modelled as an intrinsic function of the compression of the ion fluid, normalized to the theoretical postshock compression at infinity, as predicted by the Rankine-Hugoniot relations. This normalized ion compression parameter constitutes a function, η(z), varying from 0 in the preshock to 1 in the postshock medium.

The preshock (medium 1) and postshock (medium 2) kinetic temperatures are approximately equal in the C-type shock models of our grid. Thus, the Rankine-Hugoniot continuity relations may be applied across the shock wave, replacing the relation of conservation of energy flux by the isothermal condition, T₁ = T₂. Then, setting the ratio of specific heats γ = 1, the expression for the compression ratio, ρ₂/ρ₁, across the shock wave may be derived (cf. Draine & McKee 1993). Under the conditions of our models, which are such that M_s = V_s/c₁ ≫   M_A = V_s/V_A ≫ 1, where M_s is the sonic Mach number of the flow, evaluated in the preshock gas, c₁ is the sound speed, and M_A is the Alfvénic Mach number, also in the preshock gas, the compression ratio reduces to , whence (A.1)where V_postshock is the flow speed in the postshock gas, m_H the proton mass and n_H   =   n(H) + 2  n(H₂) the proton density in the preshock gas. Using (A.2)(where V_s is the shock velocity and V_i the velocity of the ion fluid in the shock frame, respectively), the factor by which the total grain cross section is enhanced is given by (A.3)The extent of the increase of ⟨ nσ ⟩ due to shattering is given by the final value of the shattering-factor, . This value depends on the shock velocity and the magnetic field and is an external parameter, which needs to be extracted from the multi-fluid models. However, this simple functional form did not reproduce satisfactorily the onset of shattering in the shock. We therefore propose the following refined expression (A.4)that introduces another parameter, β, which needs to be extracted from the corresponding multi-fluid model. β describes the delay in the shattering feedback, relative to the ion compression, and is only weakly dependent on the shock velocity. The parameters, β and Σ_max, which correspond to the grid of models introduced in Sect. 3.1, are listed in Table A.1. Linear fits in the shock velocity V_s and the magnetic field parameter b are (A.5)and (A.6)The increase of the total grain collisional cross-section needs to be consistent with the mean square radius of the grains and with their total number density, following the compression of the ions. Analyzing the multi-fluid computations corresponding to Paper III, we find a reasonable approximation to the behaviour of the total grain number density, n_G, and the mean square radius, ⟨ σ ⟩ _G: the former increases as , the latter decreases as , relative to the corresponding values without shattering. These changes affect the rates of grain-catalyzed reactions, adsorption to the grain mantles, excitation of H₂ in collisions with grains, and transfer of momentum and thermal and kinetic energy between the neutral and the charged fluids.

Figure A.1 shows a comparison between the multi-fluid model of Paper III and our current model⁶ for a representative shock for which n_H = 10⁵ cm^-3, V_s = 30 km s^-1 and b = 1.5. The temperature profiles of the neutral fluid agree well. Furthermore, the variation of the total grain-core cross-section is reproduced by our simulation, as may be seen from the lower panel of Fig. A.1. The total grain cross-section (including mantles) increases somewhat later in our current model, due to small differences – independent of grain shattering – with the model of Paper III. The discrepancy in the value of the grain cross-section in the postshock medium arises because our simplified treatment of shattering tends to underestimate the final number density of small grains. However, this simplification introduces only small deviations in the hydrodynamic parameters, such that the values of both the peak temperature of the neutral fluid and the shock width (up to 100 K in the cooling flow) agree to within ±~15% for the entire grid of models incorporating shattering.

	Fig. A.1 Upper panel: temperature profiles of the neutral fluid for a shock with nH   =   105 cm-3, Vs = 30 km s-1 and b = 1.5 corresponding to Paper III (in blue) and our current model (in red). Lower panel: evolution of the total grain cross section, ⟨ nσ ⟩, for the same shock models, with and without taking into account the grain mantles and normalized to the preshock values of ⟨ nσ ⟩.
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Appendix A.1.2: Chemical effect of shattering

The change in the total grain cross-section, ⟨ nσ ⟩, owing to shattering, has consequences for the rates of grain-charging reactions. In order to model the effect of shattering on the abundances of charged grains, the corresponding chemical source term needs to be introduced.

The charge distribution of the fragments is essentially unknown and cannot be numerically integrated separately from that of grains already present in the medium. In the present paper⁷, the charge distribution of fragments is designed to ensure charge conservation. We fitted the fragment charge distribution by aligning the shock widths (see Fig. A.1, upper panel). This procedure yielded a charge distribution in which 1/2 of the fragments were neutral, and 1/4 were positively and 1/4 negatively charged, following shattering. Subsequently, the grain charge distribution evolved on a timescale that can be long – of the order of the flow time through the shock wave when the mean grain size becomes very small.

The source term for the creation of grains through shattering can be derived from the equation of conservation of the total flux of grains, in the absence of shattering, (A.7)where n_G0 is the number density of neutral grains, n_{G + ,G −} are the number densities of positively and negatively charged grains, respectively, V_G0 = V_n is the velocity of the neutral grains, in the shock frame, and V_{G + ,G −} = V_i is the velocity of the charged grains, in the shock frame. Differentiation of (A.7), subject to the charge distribution of the fragments, then yields (A.8)for the neutral grains, and (A.9)for the charged grains, where n_G,preshock is the total (charged and neutral) number density of grains in the preshock gas. The derivative of Σ is given by (A.10)where (A.11)and (A.12)With these parameters, our model is able to reproduce the main effects of the increase in the grain cross section, reported in Paper III: the effective rate of recombination of electrons and ions is enhanced; the fractional abundance of free electrons falls by three orders of magnitude; and dust grains become the dominant charge carriers, with equal numbers of positively and negatively charged grains being produced.

Appendix A.2: Vaporization

The effect of vaporization is modelled as an additional term in the creation rate of gas-phase SiO, from Si and O in grain cores (denoted by **), corresponding to a new type of pseudo-chemical reaction:

Si** + O** = SiO + GRAIN.Because vaporization sets in suddenly, when the vaporization threshold is reached, the function (A.13)can be used to approximate the rate of creation of SiO. The function Ω(z) is centred at the point where the compression of the charged fluid reaches 1/6 of its final value, as determined by our fitting procedure. Similarly, the factor 10³ in the exponent, which determines the steepness of the function, derives from a fit to the numerical results of Paper III. Using this function, the creation rate (cm^-3  s^-1) can be expressed as where use is made of the conservation of the flux of n_H, and where (A.17)The spatial change in number density of SiO, owing to vaporization, is then given by (A.18)The peak fractional abundance x(SiO)_peak needs to be computed with the multi-fluid model and constitutes a free parameter, given in Table A.1 for our grid of models. As can be seen in Table A.1, vaporization of SiO is relevant only for the shocks with high velocity and low magnetic field; it can be neglected for the shocks with V_s = 20 km s^-1 and b = 1.5, b = 2, as well as for the model with V_s = 30 km s^-1 and b = 2.5.

	Fig. A.2 Fractional abundance of SiO (full curves), as determined when including grain-grain processing, using the present model (in red) and the multi-fluid model of Paper III (in blue); the shock parameters are nH   =   105 cm-3, Vs = 30 km s-1 and b = 1.5. The temperature of the neutral fluid is shown also (broken curves; right-hand ordinate).
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Figure A.2 shows the result of our implementation of vaporization. We assume that, initially (in the preshock medium), there are no Si-bearing species, either in the gas phase or in the grain mantles; all the Si is contained in olivine (MgFeSiO₄) grain cores. This figure compares the fractional abundance of SiO, as computed with our current model (incorporating grain-grain processing) and as predicted by the multi-fluid model of Paper III. By construction, the peaks of the fractional abundance of SiO agree, whereas the timescale for its accretion on to grains differs between the two models, as is visible in the plot; this discrepancy relates to the imperfect agreement of the total cross-section, ⟨ nσ ⟩, in the postshock medium (see Fig. A.1). However, we have verified that the timescale for SiO accretion is not critical to our analysis: the complete neglect of accretion on to grains leads to increases in the integrated intensities of the lowest rotational transitions of SiO and CO, by factors of ~2 and ~3, respectively. We note that the intensities of these transitions are, in any case, affected by the foreground emission of ambient, non-shocked gas.

Appendix B: Molecular line emission

	Fig. B.1 Computed H2 excitation diagrams for rovibrational levels with energies Ev,J ≤ 10  000 K and for shocks with nH   =   105 cm-3, Vs = 30 km s-1 and b = 1.5 (red), b = 2.0 (green) and b = 2.5 (blue). Full lines: model M1; dotted lines: model M2.
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	Fig. B.2 Integrated intensities of the rotational transitions Jup → Jup − 1 of CO for shocks with nH   =   105 cm-3, Vs = 30 km s-1 and b = 1.5 (red), b = 2.0 (green) and b = 2.5 (blue). Full lines: model M1; dotted lines: model M2.
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	Fig. B.3 Integrated intensities of selected rotational transitions of ortho-H2O plotted against the excitation energy of the emitting level, expressed relative to the energy of the J = 0 = K ground state of para-H2O. Results are shown for shocks with nH   =   105 cm-3, Vs = 30 km s-1 and b = 1.5 (red), b = 2.0 (green) and b = 2.5 (blue). Full circles: model M1; open circles: model M2.
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	Fig. B.4 Integrated intensities of the rotational transitions of OH for emitting levels of negative parity, plotted against the excitation energy of the upper level. Results are shown for shocks with Vs = 30 km s-1 and b = 1.5 (red), b = 2.0 (green) and b = 2.5 (blue). Full circles: model M1; open circles: model M2.
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The introduction of shattering leads to a reduction of the shock width, and hence to lower column densities of shocked material, and to higher peak temperatures, which affect the chemistry and enhance the fractional abundances of molecules in excited states. Which of these effects prevails is determined by the chemical and spectroscopic properties of the individual molecular species.

Appendix B.1: H₂

The intensities of pure rotational and ro-vibrational lines of H₂ contain key information on the structure of C-type shock waves, as was demonstrated, for example, by Wilgenbus et al. (2000). These lines are optically thin, and their intensities are integrated in parallel with the shock structure, neglecting radiative transfer.

As may be seen from Fig. B.1, the introduction of shattering leads to a reduction in the computed column densities of the lowest rotational levels of H₂, by approximately an order of magnitude. With increasing energy of the emitting level, the effect of the decrease in the shock width is compensated by the higher peak temperature, and the column densities predicted by the models that include shattering eventually exceed those calculated neglecting shattering. The change-over occurs at energies of the emitting level of ≳6000 K for V_s = 20 km s^-1, ≳7000 K for V_s = 30 km s^-1, and ≳10 000 K for V_s = 40 km s^-1, with the exact values depending on the magnetic field strength. The computed intensities of selected lines of H₂ are given in Table B.1, together with the intensities of forbidden lines of atomic oxygen (63 μm and 147 μm), of atomic sulfur (25 μm) and of [C I] (610 μm and 370 μm).

Appendix B.2: CO, H₂O and OH

Figure B.2 shows the integrated line intensities, TdV, of the rotational transitions of CO, plotted against the rotational quantum number, J, of the emitting level for shocks with n_H   =   10⁵ cm^-3 and V_s = 30 km s^-1. The line intensities, which are listed in Tables B.2–B.4, are lower when shattering is included, owing to the reduction in the shock width; this effect is most pronounced at low magnetic field strengths. While the intensities computed with models M2 peak at around J_up = 7, those of models M1 peak at higher values of J_up and exhibit a plateau extending to J_up ≈ 12. These differences reflect the corresponding excitation conditions. As in the case of SiO (see Sect. 4.2), the peak temperature shows a stronger dependence on the strength of the magnetic field in models that include grain-grain processing. Accordingly, the integrated intensities of highly excited transitions vary with b for models M1.

Similarly to CO, the intensities of lines of H₂O also become weaker when shattering is included, owing to the reduced shock width. Figure B.3 shows the computed intensities of the lines of ortho-H₂O as a function of the excitation energy of the emitting level. The intensities of all the lines of ortho- and para-H₂O that fall in the Herschel/PACS/HIFI bands are listed in Tables B.6−B.11.

	Fig. B.5 Rate of cooling by the principal molecular coolants, H2 (mauve), H2O (green), CO (blue) and OH (red) for Vs = 40 km s-1 and b = 2 for model M1 (left panel) and M2/M3 (right panel), which are shown together because the presence of SiO in grain mantles does not affect the cooling of the shock wave. Note the different distance intervals on the x-axes.
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Contrary to the behaviour of CO and H₂O, the line intensities of OH become stronger for the 30 km s^-1 and 40 km s^-1 shocks when shattering is included, as may be seen for V_s = 30 km s^-1 in Fig. B.4. The lines displayed fall within the Herschel/PACS band; their emitting levels have negative parity. The intensities of all transitions of OH observable with Herschel are listed in Table B.5. The increase in the integrated intensities in models M1 is due to the higher peak temperatures, which favour the conversion of the gas-phase oxygen that is not bound in CO into OH. Again, we see a variation of the integrated line intensities with the magnetic field strength in models M1, associated with the temperature-dependent rate of OH formation.

On the basis of these findings, it is interesting to ask how the allowance for grain-grain processing affects the radiative cooling

of the medium. Although the narrower shock width, and hence larger velocity gradients, in scenario M1 might be expected to modify the optical thickness of the lines, and thereby their rate of cooling, we do not detect such an effect in our models, as is demonstrated by Fig. B.5. Instead, we see an increase in the contribution of OH to the rate of cooling, owing to the enhanced abundance of gas-phase OH in the hot shocked medium (see the lower panel of Fig. 4).

© ESO, 2013