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Appendix A: Water abundance problem: the point of view of observers and modellers

The goal of this appendix is to clarify the possible confusion of the meaning of “water abundance” between the observing and modelling communities. The rigorous comparison of observations to models requires the knowledge of constraints such as the length/age of the shock, as this section discusses now. We base this discussion on the model used to fit both the SiO and H₂O emission in the OF1 shock region of IRAS 17233–3606 with the following input parameters: pre-shock density n_H = 10⁶ cm^-3, shock velocity ν_s = 32 km s^-1, and magnetic field strength (perpendicular to the shock direction) B = 1 mG. Whether the radiative transfer of water is calculated along the shock equations in the model (so-called “s-s” in Fig. 3, “DRF” in Tables B.2–B.5) or a posteriori from the outputs of the shock model (“s-e” in Fig. 3, “AGU” in Tables B.2–B.5) does not change the thermal profile of the shock layer, nor the associated water abundances (e.g. Gusdorf et al. 2011). Everything stated in this appendix is therefore applicable to both “s-s” and “s-e” models.

In one-dimensional, stationary shock models (e.g., this work, Gusdorf et al. 2011; Draine et al. 1983; Kaufman & Neufeld 1996; Flower & Pineau Des Forêts 2010) the physical and chemical conditions are self-consistently calculated at each point of a shocked layer. The end product is a collection of physical (temperature, velocity, density) and chemical (abundances) quantities obtained at each point of the shocked layer. The position of each point is marked by a distance parameter with respect to a origin typically located in the pre-shock region. The position of the last point in the post-shock region then corresponds to the shock width. Typically, these shock models are used in a face-on configuration, so that the width one refers to is along the line-of-sight direction. Alternatively, the position of a point in the shock layer can be expressed through a time parameter: the time parameter for the last point in the post-shock region then corresponds to the flight time that a particle needs to flow through the total width of the shock. The correspondence between the time and distance parameters related to a neutral particle (t_n and z) is hence given by t_n = ∫(1/ν_n) dz, where ν_n is the particle velocity. While the shock width cannot be constrained by observations, an upper limit to the flow time is given by the dynamical age, which is inferred from mapped observations of spectrally resolved lines.

Fig. A.1 Upper panel: neutral temperature (black curve), total density (red dashed curve), water density (blue dashed curve), and fractional density (blue continuous curve). The so-called fractional density is the water density over the total density, locally defined at each point of the shock. Lower panel: neutral temperature (black curve), total column density (red dashed curve), water column density (blue dashed curve), and fractional column density (blue continuous curve). The so-called fractional column density is the water column density over the total column density. The column density (in cm-2) is the integral of the local density (in cm-3) along the shock width (in cm). In both panels, the three points labelled on each curve correspond to the distance parameter of 3.1 × 1015, 5.15 × 1015, 1016 cm, or to time parameters values of 500, 1000, and 2150 yr.

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Figure A.1 shows for this model the variation of the temperature of the neutral particles (K), as well as those of the water and total local densities (n(H₂O) and n_tot in cm^-3) and their ratio x(H₂O) = n(H₂O)/n_tot in the shock layer versus the distance parameter. To illustrate the relation between time and distance parameters through the shock layer, we have marked three points on each curve: 3.1 × 10¹⁵, 5.15 × 10¹⁵, 10¹⁶ cm, which correspond to 500, 1000, and 2150 yr, in our model. In our case, the highest value for the time parameter is constrained by the dynamical shock age of OF1, 500–1000 yr. Water abundance is often defined by modellers as the maximum fractional local abundance of water through the shock layer, that is, between the pre-shock region before the temperature rise and the maximum shock age (x(H₂O)_max = 1.4 × 10^-4 for our model, top panel of Fig. A.1). On the other hand, local quantities cannot be accessed through observations. Integrated quantities (against the width of the shock layer along the line of sight) such as column densities are measured by observers. Generally, “observational water abundances” are hence given in fractional column density units, that is, the ratio of the water column density divided by the total column density. This ratio is different the maximum fractional abundance of water that is generally provided and used by modellers. The difference between the two values is illustrated by comparing the upper panel of Fig. A.1 with its lower panel, which shows the evolution of the water and total column densities, N_H2O and N_tot, and of their ratio y(H₂O) = N(H₂O)/N_tot. In the modellers’ view, referring to the distance parameter as “z”, these column densities are defined by

where z_max is the total shock width, that is, the distance corresponding to the maximum value of the time parameter. In our case, the value of the fractional column density of water can be read in the bottom panel of Fig. A.1: y(H₂O) = 2.5 × 10^-5 (if the adopted dynamical age is 500 yr), = 1.2 × 10^-5 (if the adopted dynamical age is 1000 yr). We note that this value is about an order of magnitude lower than the maximum fractional abundance of water reached in the same shock layer.

We note that the decrease in the y(H₂O) curve is artificial and only due to the 1D nature of the model. Indeed, in the post-shock region, the total density of the gas is conserved (because it cannot escape sideways, for instance like in the case of a bow-shock), while the gas-phase water density decreases until all water molecules re-condensate on the interstellar grains because of the temperature decrease. The total column density hence increases (lower panel of Fig. A.1), while the water column density is constant, resulting in a decrease of the water column density ratio with the distance or time parameter. It is therefore essential to have a measurement of the dynamical time scale to stop the calculation at a realistic time to obtain a fractional column density of water as realistic as possible.

Appendix B: Additional tables and figures

Fig. B.1 Observed and modelled maximum brightness temperatures (circles), and integrated intensities (squares) for the blue-shifted emission. Data points (in black) are corrected for an emission region of 3 arcsec2 and for 60% of the emission due to OF1. Errorbars are ±20% of the observed value. Three models are shown: the model of Paper II with level populations in statistical equilibrium (“s-e” in red) with νs = 32 km s-1, one with a slower shock velocity (νs = 30 km s-1, blue), and a model in stationary-state (“s-s” in green).

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