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Appendix A: Derivation of gas to ice ratios

The gas to ice ratio for a particular species in a protostellar envelope can be estimated by assuming a steady-state between photodesorption and freeze-out:

		$\displaystyle Y_{\rm pd}\times I_{\rm UV} \times \sigma_{\rm gr}\times f_{\rm x... ...rac{T}{m_{\rm x}}\right)^{\frac{1}{2}}\times \sigma_{\rm gr} \times n^g_{\rm x}$	(A.1)
		$\displaystyle f_{\rm x} = \frac{n^{\rm i}_{\rm x}}{n^{\rm i}}$	(A.2)

where $Y_{\rm pd}$ is the photodesorption yield set to be (1-3) $\times$ 10^-3 photon^-1 from our experiments, $I_{\rm UV}$ is the cosmic-ray-induced UV field of 10⁴ photons cm^-2 s^-1 and $\sigma_{\rm gr}$ is the grain cross section. The cosmic-ray-induced UV flux assumes a cosmic ray ionization rate of 1.3 $\times$ 10^-17 s^-1. Because photodesorption is a surface process, the photodesorption rate of species x depends on the fractional ice abundance $f_{\rm x}$ , which is defined to be the ratio of the number density of species x in the ice, $n^{\rm i}_{\rm x}$ , to the total ice number density, $n^{\rm i}$ . The freeze-out rate of species x depends on the gas temperature T, which is set to 15 K, the molecular weight $m_{\rm x}$ , and the gas number density $n^{\rm g}_{\rm x}$ . For an average molecular weight of 32, this results in a gas phase abundance $n^{\rm g}_{\rm x}/n_{\rm H}$ of (3-9) $\times$ $10^{-4}f_{\rm x}/n_{\rm H}$ . From this and an average total ice abundance $n^{\rm i}/n_{\rm H}$ of 10^-4, the predicted gas to ice phase abundance ratio is:

$\begin{displaymath}% \frac{n^{\rm g}_{\rm x}}{n^{\rm i}_{\rm x}}\sim \frac{(3{-}... ...x}}{n^{\rm i}/n_{\rm H}\times f_{\rm x}}\sim (3{-}9)/n_{\rm H} \end{displaymath}$

(A.3)

For a typical envelope density of 10⁴ cm^-3, ice photodesorption hence predicts a gas to ice ratio of 10^-4-10^-3. The derivation of a gas to ice ratio from observed cold gas emission lines and ice absorption features in the same line of sight is complicated by the fact that different regions can contribute by varying amounts. The emission features trace gas in the envelope and cloud both in front and behind the protostar, while the ice absorption features only trace envelope material directly in front of the protostar. The column is hence twice as long for the gas observations. This is probably more than compensated for by using beam averaged gas column densities, as is done in this study, because the large beam traces on average less dense material compared to the pencil beam of the ice absorption observations. Note also that the CH₃OH ice abundance may vary between lower and higher density regions. To quantify a conversion factor between the observed and true gas to ice ratio requires detailed modeling of each source which is outside the scope of this study. Instead we here assume that the observed ratio is a lower limit to the true ratio.