Online Material

Acknowledgements

The authors are grateful to many funding agencies and the HIFI-ICC staff who has been contributing for the construction of Herschel and HIFI for many years. HIFI has been designed and built by a consortium of institutes and university departments from across Europe, Canada and the United States under the leadership of SRON Netherlands Institute for Space Research, Groningen, The Netherlands and with major contributions from Germany, France and the US. Consortium members are: Canada: CSA, U.Waterloo; France: CESR, LAB, LERMA, IRAM; Germany: KOSMA, MPIfR, MPS; Ireland, NUI Maynooth; Italy: ASI, IFSI-INAF, Osservatorio Astrofisico di Arcetri- INAF; Netherlands: SRON, TUD; Poland: CAMK, CBK; Spain: Observatorio Astronómico Nacional (IGN), Centro de Astrobiología (CSIC-INTA). Sweden: Chalmers University of Technology - MC2, RSS & GARD; Onsala Space Observatory; Swedish National Space Board, Stockholm University - Stockholm Observatory; Switzerland: ETH Zurich, FHNW; USA: Caltech, JPL, NHSC.

Appendix A: Radex model

Figure A.1 shows the CO 6-5/10-9 line ratios for a slab model with a range of temperatures and densities.

Appendix B: Abundance profiles for IRAS 2A

Among the three sources, IRAS 2A has been selected for detailed CO abundance profile modeling because more data are available on this source, and because its physical and chemical structure has been well characterized through the high angular resolution submillimeter single dish and interferometric observations of Jørgensen et al. (2005a,2002). The physical parameters are taken from the continuum modeling results of Jørgensen et al. (2002). In that paper, the 1D dust radiative transfer code DUSTY (Ivezic & Elitzur 1997) was used assuming a power law to describe the density gradient. The dust temperature as function of radius was calculated self-consistently through radiative transfer given a central source luminosity. Best-fit model parameters were obtained by comparison with the spectral energy distribution and the submillimeter continuum spatial extent. The resulting envelope structure parameters are used as input to the Ratran radiative transfer modeling code (Hogerheijde & van der Tak 2000) to model the CO line intensities for a given CO abundance structure through the envelope. The model extends to 11000 AU from the protostar, where the density has dropped to cm^-3. The CO-H₂ collisional rate coefficients of Yang et al. (2010) have been adopted.

The C¹⁸O lines are used to determine the CO abundance structure because the lines of this isotopologue are largely optically thin and because they have well-defined Gaussian line shapes originating from the quiescent envelope without strong contaminations from outflows. Three types of abundance profiles are examined, namely ``constant'', ``anti-jump'' and ``drop'' abundance profiles. Illustrative models are shown in Fig. B.1 and the results from these models are summarized in Table B.1.

Table B.1:   Summary of CO abundance profiles for IRAS 2A.

B.1 Constant abundance model

The simplest approach is to adopt a constant abundance across the entire envelope. However, with this approach, and within the framework of the adopted source model, it is not possible to simultaneously reproduce all line intensities. This was already shown by Jørgensen et al. (2005c). For lower abundances it is possible to reproduce the lower-J lines, while higher abundances are required for higher-J lines. In Fig. B.2 the C¹⁸O spectra of a constant-abundance profile are shown for an abundance of , together with the observed spectra of IRAS 2A. Based on these results, the constant-abundance profile is ruled out for all three sources.

B.2 Anti-jump abundance models

The anti-jump model is commonly adopted in models of pre-stellar cores without a central heating source (e.g., Tafalla et al. 2004; Bergin & Snell 2002). Following Jørgensen et al. (2005c), an anti-jump abundance profile was employed by varying the desorption density, , and inner abundance in order to find a fit to our observed lines. Here, the outer abundance X₀ was kept high at corresponding to a ¹²CO abundance of for ¹⁶O/¹⁸O = 550 as was found appropriate for the case of IRAS 2A by Jørgensen et al. (2005c). This value is consistent with the CO/H₂ abundance ratio determined by Lacy et al. (1994) for dense gas without CO freeze-out.

The best fit to the three lowest C¹⁸O lines (1-0, 2-1 and 3-2) is consistent with that found by Jørgensen et al. (2005c), corresponding to  cm^-3 and (CO abundance of ). In the fits, the calibration uncertainty of each line (ranging from 20 to 30%) is taken into account. These modeled spectra are overplotted on the observed spectra in Fig. B.2 as the blue lines, and show that the anti-jump profile fits well the lower-J lines but very much underproduces the higher-J lines.

The value of X₀ was verified a posteriori by keeping at two different values of and  cm^-3. This is illustrated in Fig. B.3 where the contours show that for both values of , the best-fit value of X₀ is , the value also found in Jørgensen et al. (2005c). The contours have been calculated from the lower-J lines only, as these are paramount in constraining the value of X₀. Different plots were made, where it was clear that higher-J lines only constrain , as expected. The effect of is illustrated in Fig. B.4 for the two values given above.

B.3 Drop-abundance profile

In order to fit the higher-J lines, it is necessary to employ a drop-abundance structure in which the inner abundance increases above the ice evaporation temperature (Jørgensen et al. 2005c). The abundances and X₀ for are kept the same as in the anti-jump model, but is not necessarily the same as X₀. In order to find the best-fit parameters for the higher-J lines, the inner abundance and the evaporation temperature were varied. The plots (Fig. B.5, left panel) show best-fit values for an inner abundance of and an evaporation temperature of 25 K (consistent with the laboratory values), although the latter value is not strongly constrained. These parameters fit well the higher-J C¹⁸O 6-5 and 9-8 lines (Fig. B.2). The C¹⁸O 5-4 line is underproduced in all models, likely because the larger HIFI beam picks up extended emission from additional dense material to the northeast of the source seen in BIMA C¹⁸O 1-0 map (Volgenau et al. 2006).

Because the results do not depend strongly on , an alternative approach is to keep the evaporation temperature fixed at 25 K and vary both and by fitting both low- and high-J lines simultaneously. In this case, only an upper limit on of is found (Fig. B.5, right panel), whereas the inferred value of is the same. This figure conclusively illustrates that , i.e., that a jump in the abundance due to evaporation is needed.

The above conclusion is robust within the context of the adopted physical model. Alternatively, one could investigate different physical models such as those used by Chiang et al. (2008), which have a density enhancement in the inner envelope due to a magnetic shock wall. This density increase could partly mitigate the need for the abundance enhancement although it is unlikely that the density jump is large enough to fully compensate. Such models are outside the scope of this paper. An observational test of our model would be to image the N₂H⁺ 1-0 line at high angular resolution: its emission should drop in the inner 900 AU (4 ) where N₂H⁺ would be destroyed by the enhanced gas-phase CO.