A&A 471, 849-863 (2007)
DOI: 10.1051/0004-6361:20066608

A survey of [D₂CO]/[H₂CO] and [N₂D⁺]/[N₂H⁺] ratios towards protostellar cores

H. Roberts - T. J. Millar

School of Mathematics and Physics, Queens University Belfast, Belfast, BT7 1NN, UK

Received 20 October 2006 / Accepted 18 June 2007

Abstract
Aims. We use observations and models of molecular D/H ratios to probe the physical conditions and chemical history of the gas and to differentiate between gas-phase and grain-surface chemical processing in star forming regions.
Methods. As a follow up to previous observations of HDCO/H₂CO and DCN/HCN ratios in a selection of low-mass protostellar cores, we have measured D₂CO/H₂CO and N₂D⁺/N₂H⁺ ratios in these same sources. For comparison, we have also measured N₂D⁺/N₂H⁺ ratios towards several starless cores and have searched for N₂D⁺ and deuterated formaldehyde towards hot molecular cores (HMCs) associated with high mass star formation. We compare our results with predictions from detailed chemical models, and to other observations made in these sources.
Results. Towards the starless cores and low-mass protostellar sources we have found very high N₂D⁺ fractionation, which suggests that the bulk of the gas in these regions is cold and heavily depleted. The non-detections of N₂D⁺ in the HMCs indicate higher temperatures. We did detect HDCO towards two of the HMCs, with abundances 1-3% of H₂CO. These are the first detections of deuterated formaldehyde in high mass sources since Turner (1990) measured HDCO/H₂CO and D₂CO/H₂CO towards the Orion Compact Ridge.

Key words: ISM: molecules - ISM: clouds - ISM: abundances - stars: formation - astrochemistry

1 Introduction

Observations of deuterated molecules have become an important tool in the study of interstellar chemistry and star formation. Although the underlying (or cosmic) D/H ratio is low ($\sim$10^-5), the formation of deuterated molecules is preferred at low temperatures ($\leq$80 K) and leads to a high degree of fractionation in dark molecular clouds ( $T_{\rm kin} \sim10$ K, n(H $_2)\ga10^4$ cm^-3). The level of fractionation is also affected by the abundance of neutral species (e.g. CO, N₂, CS) in the gas, which depends on the balance between accretion onto and evaporation from dust grains. Observations of cold, centrally condensed cores (some of which may be the the birthplace of future stars) show "holes'' in the emission from typically abundant molecules such as CS and CO at the dust peak (e.g. Bergin et al. 2001; Caselli et al. 1999; Redman et al. 2002). This suggests that, at densities $\ga$10⁵ cm^-3, for heavier species, accretion is "winning''. Observations confirm that molecular D/H ratios do increase with increasing density and CO depletion (Bacmann et al. 2003,2002; Caselli et al. 1999).

Enhanced molecular D/H ratios are also observed in the envelopes surrounding newly born stars where the gas is $\ga$50 K and the material accreted onto the grains has evaporated. In fact, the highest levels of deuterium fractionation measured to date have been seen towards low-mass protostellar cores: NH₂D/NH ₃ > 0.1 (Saito et al. 2000; Hatchell 2003), ND₃/NH₃ = 0.001 (van der Tak et al. 2002; Lis et al. 2002), D₂CO/H₂CO = 0.05-0.4 (Loinard et al. 2000,2001; Ceccarelli et al. 2001), CH₂DOH/CH₃OH = 0.3, CHD₂OH/CH₃OH = 0.06, CD₃OH/CH₃OH = 0.01 (Parise et al. 2004,2002), D₂S/HDS = 0.1 (Vastel et al. 2003). Note that molecules have been observed with 3 deuterium atoms substituted for hydrogen - based on the cosmic D/H ratio we would expect such isotopologues to be $\sim$10¹⁵ times less abundant, so the ND₃/NH₃ and CD₃OH/CH₃OH ratios represent enhancements of $\ga$10¹² orders of magnitude.

In "hot molecular cores'' (HMCs) associated with high-mass star formation, which are clumps of dense gas with temperatures $\sim$100-200 K, the D/H ratios which have been measured are generally $\sim$10^-3 (e.g. Hatchell et al. 1999,1998a, and references therein), lower than the low-mass sources, yet still much higher than the cosmic value.

It is now generally accepted that the deuterium fractionation seen in warm gas was set during the colder prestellar core phase, when material was freezing onto grain surfaces. Once some heating event, such as the formation of the star or the passage of a shock, causes the grain mantles to evaporate the D/H ratios can survive for 10⁴-10⁶ yr in warm gas (Osamura et al. 2004; Rodgers & Millar 1996). Recent theoretical modelling by Doty et al. (2006) suggests that hot cores in both high and low mass protostellar sources will survive for only $\sim$a few $\times$10³-10⁴ yr after the formation of the protostar. This means that chemistry in the gas-phase would not have time to significantly alter the composition of the gas post-evaporation. Thus, observing the warm gas around young protostars can provide valuable information about the physical conditions in the cold, accreting gas, before the onset of star formation. For example, the lower fractionation towards HMCs may suggest that the gas from which these stars formed was initially warmer, or that the low temperature accretion phase did not last long enough to produce very high fractionation.

The composition of the gas surrounding a newly formed protostar is influenced by both gas-phase ion-molecule chemistry and reactions on the grain surfaces. Saturated species which do not form efficiently in the gas-phase at low temperatures (e.g. CH₃OH and H₂S) are readily observed, and the level of deuterium fractionation in these species (certainly for the low-mass sources, see above) indicates that the surface chemistry occurred during the cold, prestellar core phase. Recent observations of the envelopes surrounding low-mass protostars show jumps in abundance close to the protostar for several molecules, including H₂O, H₂CO, CH₃OH, SO and SO₂ (Ceccarelli et al. 2000b,a; Maret et al. 2004,2005; Schöier et al. 2002). Maret et al. (2004), who surveyed H₂CO abundances, found the best fit to their data if the temperature is higher than 50 K within $\sim$100 AU of the protostar.

Roberts et al. (2002a, hereafter Paper I) surveyed a number of low-mass protostellar sources in DCN and HDCO, concluding that the fractionation was consistent with gas-phase chemistry and accretion over a period of $\la$50 000 yr, but the chemical model used for comparison may have overestimated the formaldehyde fractionation (Osamura et al. 2005). By also obtaining D₂CO/H₂CO ratios, we can compare successive levels of deuteration (e.g. D₂CO/HDCO and HDCO/H₂CO) which will differ depending on whether the molecules have a gas-phase or grain-surface origin (Turner 1990; Rodgers & Charnley 2002).

Loinard et al. (2002) measured D₂CO/H₂CO ratios of 0.02-0.4 towards $\sim$20 low-mass protostellar sources, but did not detect D₂CO towards 6 more massive protostars (D₂CO/H₂CO < 0.01). Our survey of low-mass sources overlaps with theirs, but we also chose three different high-mass sources to survey. These are line-rich HMCs, drawn from those surveyed by Hatchell et al. (1998b), which show evidence of grain surface chemistry through large abundances of complex hydrogenated molecules and in which other molecular D/H ratios have been determined (Gensheimer et al. 1996; Hatchell et al. 1999,1998a).

We have also surveyed N₂D⁺ fractionation towards both the low and high mass protostellar cores in order to investigate the fractionation in the cold envelopes around the protostars. Molecular ions trace gas-phase chemistry: in dark clouds H₂ is ionised by cosmic rays and rapidly converted to H₃⁺. Due to its low proton affinity, H₃⁺ reacts with most other neutral species: e.g. CO and N₂, giving HCO⁺ and N₂H⁺. As N₂ appears to be less susceptible to freeze-out than CO, the N₂D⁺/N₂H⁺ ratio is a good tracer of deuteration in cold, dense, CO-depleted cores. Currently, it is unclear whether N₂H⁺ remains in the gas-phase after CO freezes out due to a difference in the surface binding energies of N₂ and CO (Bergin & Langer 1997) or, as recent experimental work shows no evidence for this (Öberg et al. 2005), to differences in their formation and destruction rates in the gas-phase (Vastel et al. 2006; Aikawa et al. 2001). By comparing N₂D⁺/N₂H⁺ and DCO⁺/HCO⁺ ratios we should be able to eliminate many of the uncertainties in the underlying chemical network and see how the different distributions of N-bearing species in prestellar cores can be explained. We have also measured N₂D⁺/N₂H⁺ ratios towards a number of starless cores and we present these data for comparison. The sources are listed in Table 1.

Sections 2 and 3 describe the observations and data reduction techniques, presenting the resulting column densities and molecular D/H ratios. In Sect. 4 we discuss these results and in Sect. 5 present comparisons with theory and with chemical models.

  
2 Observations

Table 1: Sources observed, along with velocities, bolometric temperatures and protostellar classes (where applicable) taken from the literature.

The low-mass protostellar sources include those in which we have already determined HDCO/H₂CO and DCN/HCN ratios using the University of Arizona's (UofA) 12 m radio telescope, at Kitt Peak, Arizona (Paper I). The lines we observed are listed in Table 2, along with associated molecular data (determined using the JPL molecular spectroscopy catalogue and the CDMS) and telescope FWHM beamwidths.

Initially, we set out to observe the 4_0,4-3_0,3 line of D₂CO at 231 GHz (Dec. 2001-Jan. 2002) with the James Clerk Maxwell Telescope (JCMT), on Mauna Kea, Hawaii, but the band in which the D₂CO line was observed also covered the 3-2 transition of N₂D⁺. As N₂D⁺ is believed to be a good tracer of gas-phase deuteration and depletion (see above) it was interesting that the N₂D⁺ line was strong ($\sim$1 K) in three of these sources, L1448NW, L1448mms and HH211, but absent (at a level of <0.1 K) in L1527 and L1551 IRS5. We wished to determine whether this apparent difference in abundance also corresponded to a difference in deuterium fractionation.

Table 2: Molecular line data.

In December 2002 we observed the ground state transitions of N₂H⁺ and N₂D⁺ and selected 2-1 lines of H₂¹³CO, HDCO and D₂CO (see Table 2) towards high and low mass star forming regions with the UofA 12 m telescope. The N₂D⁺ and N₂H⁺ J=1-0 lines towards the starless cores were observed in June 2002.

We used the IRAM 30 m telescope at Pico Veleta, Spain, in February 2005, primarily to observe the 3-2 transition of N₂H⁺, but we were also able to simultaneously re-observe the N₂D⁺ 3-2 line. Finally, in June 2005 we made further JCMT observations of the N₂D⁺ 3-2 multiplet and D₂CO 4_0,4-3_0,3 lines towards 6 additional low-mass protostellar sources.

The spectra are shown in Figs. 1-5. Spectra towards the starless cores are shown in Fig. 1, the spectra for the low-mass protostellar sources are shown in Figs. 2-4, while those observed towards the HMCs are shown in Fig. 5. In all cases, the observations are shaded grey, while the fits are shown as solid black lines. All spectra have beam scaled to the main-beam temperature scale, and the source velocity (see Table 1) has been subtracted from the velocity scale.

2.1 UofA 12 m telescope data

The 3 mm receivers were used to observe the N₂H⁺ and N₂D⁺ ground-state transitions and the D₂CO 2_1,2-1_1,1 transition, while the 2 mm receiver was used for the HDCO and H₂¹³CO lines. The observations were carried out in position-switching mode, using beam throws of 10-30' in azimuth (the off-positions were first checked for excess emission). The Millimeter AutoCorrelator (MAC) correlation spectrometer was used with 49 kHz resolution ($\la$0.2 km s^-1). In December 2002, typical system temperatures were 150-300 K. In June 2002 the system temperatures were 350-450 K for the N₂D⁺ lines and 200-250 K for the N₂H⁺ lines. Pointing was checked every two hours or so and errors found to be $\la$5-6''. The data were calibrated by the usual chopper wheel method and then corrected for $\eta_m^*$, the efficiency at which the source couples to the main diffraction beam, to give  $T_{\rm MB}$.

The N₂H⁺ 1-0 multiplet was detected towards all the starless cores, while the N₂D⁺ 1-0 multiplet was seen in only three out of the five (Fig. 1). For the five low-mass protostellar cores which we set out to observe, the N₂H⁺, N₂D⁺ and the D₂CO 2_1,2-1_1,1 line were seen towards all sources (Fig. 2). We also tried to increase the sample of low-mass protostellar sources from those in Paper I, but did not obtain a full set of data; the spectra observed towards these extra sources are shown in Fig. 4. H₂¹³CO and N₂H⁺ were detected towards all three of the HMCs, HDCO was detected in only two of them, and N₂D⁺ was not detected at all in the high-mass sources (Fig. 5).

2.2 JCMT data

The JCMT observations of the low-mass sources were carried out in beam-switching mode, using a chop-throw of -180'' in azimuth. The HMCs were observed in position-switching mode using off-positions from Hatchell et al. (1999). The Digital Autocorrelation Spectrometer (DAS) was used with a bandwidth of 760 MHz, giving a spectral resolution of $\sim$0.2 km s^-1. The data were corrected for $\eta_{\rm fss}\eta_{\rm mb}$ (the "forward scattering and spillover'' and "main beam'' efficiencies) to put the spectra on the main beam scale. This assumes that the source is extended, relative to the beam, which we believe to be the case for the low-mass protostellar envelopes (see later). In total, nine low-mass protostellar sources were observed (Figs. 2 and 3). The D₂CO 4_0.4-3_0,3 line was detected towards all but two (B5IRS1 and HH111) of these, while the N₂D⁺ 3-2 multiplet was weak or absent towards L1527, L1551IRS5, B5IRS1, HH111 and RNO43, but strong towards the other sources.

2.3 IRAM data

The observations were carried out in position switching mode, with an offset 20'E. Due to bad weather (snow) we were only able to observe three sources and the spectra we did obtain have rather bad baselines (see Fig. 2). These spectra have a resolution of 0.1 km s^-1 and a correction factor of $F_{\rm eff}$/ $B_{\rm eff}$ (the ratio of the forward efficiency to the main beam efficiency) was used to convert to the main beam scale. Again, this assumes the sources are extended relative to the beam.

  
3 Data reduction

In all cases we have assumed a filling factor of 1; the implications of this are discussed later, when we compare our results to other observations. As we are primarily interested in the ratios of our column densities, however, any errors introduced by this assumption should cancel out.

3.1 Formaldehyde

Gaussians fits to the detected lines are shown in Figs. 2-5. Based on our previous observations of different isotopomers of formaldehyde in these sources (Paper I) we expect that the H₂¹³CO, HDCO and D₂CO lines will be optically thin. We can, therefore, calculate the column density of molecules in the upper level of the transition ($N_{\rm u}$) from the integrated intensities of the lines using the standard formula:

$$\int T_{\rm MB} {\rm d}v = \frac{A_{\rm ul}hc^3}{8\pi k\nu^2} N_{\rm u}$$ (1)

(cf. Goldsmith & Langer 1999, where we use $T_{\rm MB}$ rather than T_a as appropriate for extended sources and for consistency between the different telescopes).

Table 3: Noise levels and integrated intensities for the formaldehyde lines, along with upper-level column densities calculated assuming optically thin lines.

The parameters of the Gaussian fits and the resulting upper level column densities are listed in Table 3. In the event that a line was not detected, we have calculated 3$\sigma$ upper limits on the column density using the linewidths that we measured for other formaldehyde lines in the same source.

Table 4: Excitation temperatures, formaldehyde column densities and molecular D/H ratios. N(H₂CO) and N(HDCO) come from Paper I, except for L1157 and NGC 1333 IRAS 4A where they come from the 2-1 line data in Table 3.

In order to calculate total column densities we need some estimate of the excitation (rotation) temperature, $T_{\rm ex}$. We assume local thermodynamic equilibrium (LTE) holds, then:

$$N_{\rm u} = \frac{g_{\rm u}}{Q({T_{\rm ex}})} \exp(-E_{\rm u}/kT_{\rm ex}) N_{\rm TOT}.$$ (2)

The ortho and para forms of D₂CO can be treated as separate species, with different partition functions, $Q{(T_{\rm ex})}$. These have been calculated at different excitation temperatures using energy-level data in the JPL catalogue.

The high-temperature statistical o/p ratio for D₂CO is 2, but it is unclear what the actual value will be for interstellar molecules. For H₂CO, Dickens & Irvine (1999) found o/p ratios between 1.5 and 2 towards star-forming cores with outflows, rather than the high-temperature statistical value of 3, indicating that the formaldehyde formed at low temperatures (10-20 K; Kahane et al. 1984).

There are six source in which we observed both ortho- and para-D₂CO. Calculating $N_{\rm tot}$ with $T_{\rm ex} \sim18$ K (7 and 11 K for NGC 1333 4A and L1551IRS5, respectively) results in column densities of o-D₂CO being twice as high as p-D₂CO (o/p $\sim$ 2:1). An o/p ratio of 3:1 results from using excitation temperatures $\sim$2-6 K different, but the total N(D₂CO) values are changed by <10%. We have recalculated the HDCO and H₂CO column densities from Paper I using consistent  $T_{\rm ex}$ values; these are listed in Table 4. The uncertainty in the o/p ratio and $T_{\rm ex}$ gives overall uncertainties of $\sim$30% for N(H₂¹³CO). The ¹²C/¹³C ratio is not particularly well determined in molecular clouds: we have adopted a value of 60, but 75 might be more appropriate. This is an uncertainty of $\sim$25%, but it is smaller than the overall uncertainties quoted in Table 4, and, of course, varying the ¹²C/¹³C ratio would change the HDCO/H₂CO and D₂CO/H₂CO ratios by the same amount.

Those sources where we give a range of values for N(H₂CO) and the resulting fractionation are those in which we did not detect H₂¹³CO and so were unable to determine the optical depth of the main H₂CO line. For the other sources, the overall uncertainties in the D/H ratios resulting from the uncertainty in $T_{\rm ex}$ and the o/p ratio are $\sim$30%, similar to the uncertainty arising from the noise in the spectra. We, therefore, quote HDCO/H₂CO and D₂CO/H₂CO ratios with errors of 40-50%.

For the HMCs, where we believe that the bulk of the gas is warmer (Hatchell et al. 1998b,a) we have adopted a value of $T_{\rm ex}=50$ K. Increasing the excitation temperature from 50 to 150 K does increase the column density estimates by factors of 3-5, but reduces the calculated HDCO/H₂CO ratios by less than a factor of 2 and has little effect on the D₂CO/H₂CO limits.

For the low-mass sources, the range of HDCO fractionation is $\sim$0.02-0.07, while the D₂CO fractionation is $\sim$0.01-0.04. In those sources where we have good N(H₂CO) determinations, L1551 IRS5 has the highest fractionation, while HH211 has the lowest. There is a larger variation towards the HMCs, with the HDCO/H₂CO ratio in G31.41 being more than an order of magnitude lower than in G75.78.

We note that the D₂CO 2_1,2-1_1,1 and 4_0,4-3_0,3 lines were observed with different telescopes, whose beamwidths differ by a factor $\sim$3. Without knowing the spatial distribution of the formaldehyde emission around the source it is difficult to correct for this. If we assume that the region containing formaldehyde is roughly less than the area of the JCMT beam (the smaller beam) then we are underestimating the column density of D₂CO molecules in the the J=2_1,2 level (which was observed with the larger beam). For the sources where we observed the D₂CO 2-1 line with the UofA 12 m telescope, applying beam filling factors will have little effect on our estimated D₂CO/H₂CO ratios, since both the H₂CO and D₂CO column densities will be similarly affected. For the sources in which we have only observed one D₂CO line (the 4_0,4-3_0,3 line using the JCMT), applying a beam correction factor will lower the D₂CO/H₂CO ratios, since assuming a small source size will increase the H₂CO column density determinations from the UofA 12 m data. The D₂CO/H₂CO ratios towards B5IRS1, RNO43, HH111 and IRAS03282 would then be $\la$0.001.

While Maret et al. (2004) did find evidence for jumps in H₂CO abundance very close to the protostar, significant amounts of formaldehyde were also present in the larger envelope. Also, when Ceccarelli et al. (2001) mapped the D₂CO emission around the low-mass protostar IRAS 16293-2422 they found that it extends 40'' south of the protostar, so the D₂CO emission in the cores we have observed could be extended enough to fill even the 60'' beam of the UofA 12 m telescope.

3.2 Diazenylium

Table 5: CLASS fits to the multiplet transitions of N₂D⁺ and N₂H⁺ and resulting upper level column densities.

The 1-0 and 3-2 transitions of N₂H⁺ and N₂D⁺ have hyperfine structure, which we can use to get an estimate of the optical depths. We took the positions and relative intensities of the hfs components from Womack et al. (1992), Dore et al. (2004), Caselli et al. (2002a), and Gerin et al. (2001) for N₂H⁺ J=1-0, N₂D⁺ J=1-0, N₂H⁺ J=3-2 and N₂D⁺ J=3-2, respectively, and used the "HFS'' method of the data reduction program CLASS to fit the spectra. These fits are shown in Figs. 1-5 and the fit parameters are listed in Table 5. In most cases the transitions were optically thin, or had only moderate optical depth. This was corrected for, where necessary, in the calculation of the upper level column densities listed in Table 5.

Unfortunately, a lot of the IRAM data had bad baselines - almost sinusoidal (see Fig. 2, row 4) - which may be confusing the CLASS fitting algorithm when it looks for outlying HFS components. Comparing the JCMT and IRAM data for L1448mms and L1448NW, the height of the main lines are fairly consistent when all sets of data are converted to the $T_{\rm mb}$ scale, but the overall line shapes and CLASS fits differ significantly. The column densities based on the fits to the IRAM data have, therefore, been calculated assuming $\tau =0.1$ and are strictly lower limits.

Fits to the N₂D⁺ 3-2 multiplet observed at the JCMT show significant optical depth towards L1448NW and L1448mms. For HH211, the fit parameters had high uncertainties and so we have forced an optically thin fit which, again, gives only a lower limit on the column density.

For diazenylium, the uncertainty in the excitation temperature swamps the other observational uncertainties. In Table 6 we have adopted $T_{\rm ex} = 12$ K for the cold cores and other low-mass sources and 50 K for the HMCs. We note that varying $T_{\rm ex}$ between 5 and 30 K changes the column density determinations by less than a factor of a three and changes the D/H ratio determinations by <20%. We have, therefore, listed the N₂D⁺/N₂H⁺ in Table 6 with an overall error of $\pm$15%. Varying the excitation temperature in the HMCs from 50 to 150 K has a negligible effect on the tabulated upper limits.

  
4 Discussion of results

Table 6: Column densities for N₂D⁺ and N₂H⁺, calculated using the excitation temperatures shown (see text), along with resulting N₂D⁺/N₂H⁺ ratios.

	Figure 6: Top: a comparison of the D2CO/H2CO and N2D+/N2H+ ratios observed towards the low-mass protostellar sources. Bottom: a comparison of the HDCO/H2CO and N2D+/N2H+ ratios observed towards the low-mass protostellar sources and HMCs.
Open with DEXTER

Three of the cold cores and four of the low-mass protostellar sources have very high N₂D⁺/N₂H⁺ ratios: 0.16-0.3. These are comparable to the values seen towards the prestellar cores L134N (Gerin et al. 2001) and L1544 (Caselli et al. 2002b). The other two low-mass sources have ratios of 0.06 and 0.08, while the fractionation in the other two cold cores (L1517B and L1498) is <0.02. We note that Gerin et al. (2001) determined N₂D⁺/N₂H⁺ to be 0.15 towards B1, which is slightly lower than the value of 0.25 we obtained (though the two values agree to within stated uncertainties), and that Crapsi et al. (2005) did detect N₂D⁺ towards L1498 with N₂D⁺/N₂H $^+ \sim 0.04$. Both these studies used the IRAM 30 m telescope whereas L1498 was observed by us only with the ARO 12 m telescope, so the latter discrepancy is explained by beam dilution.

L1527 and L1551 IRS5, the low-mass protostellar sources with the lowest N₂D⁺/N₂H⁺ ratios, are the only two sources in our survey from the Taurus molecular cloud. Measurements of other molecular D/H ratios though, (e.g. HDCO, D₂CO, DCN) do not show systematic differences between clouds. Hatchell (2003) did measure NH₂D/NH₃ ratios towards these same sources in Taurus to be more than a factor of 2 lower than those in Perseus and Orion, but concluded that the NH₂D column densities were being underestimated because of beam dilution.

L1551 IRS5 is a Class I source and has the highest bolometric temperature of the sources we observed, so the lower N₂D⁺ fractionation could be due to its more evolved nature. L1527 has the narrowest line-widths of the protostellar sources, though, which indicates that its envelope is relatively undisturbed by the protostar. Interestingly, HH211, one of the least evolved Class 0 sources (by the, admittedly, crude measure of $T_{\rm bol}$) has the highest N₂D⁺/N₂H⁺ ratio observed in this survey, while also having one of the lowest formaldehyde fractionation of the sources surveyed (see Table 4). Overall, though, the N₂D⁺/N₂H⁺ and D₂CO/H₂CO ratios show no sign of a correlation with $T_{\rm bol}$.

In Fig. 6 we compare the N₂D⁺/N₂H⁺ ratios to both the D₂CO/H₂CO and HDCO/H₂CO ratios to see if the formaldehyde and diazenylium fractionation are, in fact, anti-correlated. Unfortunately, we do not have data in enough sources to make a convincing case. For four of the sources it does seem that the N₂D⁺/N₂H⁺ ratios decrease as the D₂CO/H₂CO ratios increase, but results for L1527 are anomalous, as both the N₂D⁺ and D₂CO fractionation are relatively low. Comparing the HDCO and N₂D⁺ fractionation, the HDCO/H₂CO ratios are very similar in most sources ($\sim$0.06), while the N₂D⁺ fractionation ranges from 0.06 to 0.2. Only HH211, which has the highest N₂D⁺/N₂H⁺ ratio measured in this survey and a HDCO/H₂CO ratio of only 0.02, suggests an anti-correlation.

We have considered various line width comparisons:

• the D₂CO and N₂D⁺ lines which were observed simultaneously with the JCMT have equal linewidths to within 25% (except L1448NW, where the D₂CO line was 1.5 times broader than the N₂D⁺ line);
• the linewidths of the N₂H⁺ and N₂D⁺ 1-0 transitions in each source are similar, indicating that these molecules are indeed located in the same region of each source;
• as we expect, the cold cores generally have narrower lines that the protostellar sources, since the protostar disrupts its surrounding envelope increasing turbulence;
• we, therefore, compared the N₂D⁺ fractionation to the N₂H⁺ linewidth but found no evidence for an evolutionary trend.

  
4.1 A comparison with previous formaldehyde surveys

Table 7: A comparison of our results with those from Maret et al. (2004) and Parise et al. (2006), using the IRAM 30 m telescope.

Using the IRAM 30 m telescope, Loinard et al. (2002) carried out a survey of D₂CO/H₂CO ratios towards protostellar sources, while, more recently, Parise et al. (2006) have determined methanol and formaldehyde fractionation towards a sample of low-mass protostellar cores (using data from Maret et al. 2004,2005).

For the sources where our survey overlaps with that of Loinard et al. (2002), we calculate higher H₂CO column densities but similar D₂CO column densities and, thus, lower D₂CO/H₂CO ratios. Loinard et al. only observed the main isotopomer of H₂CO, though, and assumed that the transitions were optically thin, noting that this may underestimate the H₂CO column densities. The fact that our D₂CO column densities based on UofA 12 m data are not systematically lower than those from Loinard et al. using the IRAM 30 m telescope supports our assumption that the D₂CO emission is extended enough to fill the 60'' beam of the 12 m telescope.

For the three sources where our survey overlaps with that of Maret et al. (2004) and Parise et al. (2006), the H₂CO column densities we have determined are consistent with theirs (see Table 7). For D₂CO, however, their column densities (averaged over a 10'' beam) are $\sim$5-20 times higher than ours. This leads to their D₂CO/H₂CO ratios also being significantly higher. The D₂CO rotational temperatures from Parise et al. (2006) are only 5-6 K, but these are based on assuming a source size of 10''. The beamsizes we used for the D₂CO observations were 20 and 60''. As discussed above, including a beam dilution factor would increase our H₂CO column density estimates more than the D₂CO column densities, giving even worse agreement for the fractionation.

It is possible that the abundance and fractionation of deuterated formaldehyde increases closer to the protostar, but more gradually and over a larger radius than the sharp jump in H₂CO abundance which occurs when $T_{\rm kin}>50$ K and the ice mantles evaporate (within $\sim$100 AU of the protostar; Maret et al. 2004). Parise et al. (2006) note that the linewidths of the deuterated formaldehyde transitions they observed are narrower than those of the main isotopomer, suggesting that the HDCO and D₂CO emission is dominated by the cold, outer envelopes of the protostars. If D₂CO is extended then Parise et al. could have overestimated its column density by correcting for a 10'' source. Alternately, if the H₂CO is more extended than the D₂CO, then we may be underestimating the D₂CO fractionation in the inner regions. Thus, (as Parise et al. also noted), further observations to map the formaldehyde emission at a higher angular resolution are required.

  
5 Theory and chemical modelling

Diazenylium forms via proton transfer from H₃⁺ and its deuterated analogues to N₂. The N₂D⁺/N₂H⁺ ratio, therefore, depends on the fractionation of deuterated H₃⁺. H₃⁺ is converted to its deuterated analogues via successive exchange reactions with HD (the interstellar reservoir of deuterium), a process which occurs more efficiently when other molecules that destroy H₃⁺ (CO, O, N₂, etc.) are depleted onto grains. At temperatures $\ga$25 K, deuterated H₃⁺ is destroyed by reaction with H₂, reducing the fractionation of these and related molecules such as DCO⁺ and N₂D⁺. The N₂D⁺/N₂H⁺ ratios of >0.1 we have measured towards several sources suggests, therefore, that the bulk of the gas is cold and heavily depleted in CO.

In the gas-phase, formaldehyde primarily forms via the reaction chain:

$${\rm CH_3^+ \chemlk{H_2} CH_5^+ \chemlk{e^-} CH_3 \chemlk{O} H_2CO},$$ (3)

which has deuterated analogues:

  

$${\rm CH_2D^+ \chemlk{H_2} CH_4D^+ \chemlk{e^-} CH_2D \chemlk{O} HDCO}$$ (4)

$${\rm CHD_2^+ \chemlk{H_2} CH_3D_2^+ \chemlk{e^-} CHD_2 \chemlk{O} D_2CO}.$$ (5)

The formaldehyde fractionation, therefore, depends on the fractionation of deuterated CH₃⁺. Like H₃⁺, CH₃⁺ is a "primary ion'' for deuterium fractionation since it is also converted to its deuterated analogues via successive reactions with HD and the reverse reactions with H₂ have energy barriers (Roberts et al. 2004; Smith et al. 1982). In this case, the barriers cannot be overcome unless the gas temperature is >50 K.

Rodgers & Charnley (2002) explain how the relative fractionation of HDCO and D₂CO can provide information about the formation mechanism of the species, and, in particular, whether they have a gas-phase or grain surface origin. They define the quantity F, where:

$$F = \frac{({\rm HDCO/H_2CO})^2}{{\rm D_2CO/H_2CO}},$$ (6)

and calculate that gas-phase chemistry will produce F values of 1.6 (for the rate coefficients used in our models we calculate 1.3), while surface chemistry models, based on successive reactions of CO with H and D, can give $F \geq4$ or $F \geq1$, depending on the exact mechanism.

For our low-mass sources F ranges from 0.03 to 0.2 (see Table 4). As a comparison, observations of formaldehyde towards the Orion Compact Ridge (OCR; Turner 1990) give F=6.7, while in IRAS 16293-2422 (Loinard et al. 2000), F=0.34. Thus, except for the OCR (which is a high-mass source), these results are not consistent with theoretical predictions for gas and grain synthesis, although Rodgers & Charnley also postulate a mechanism whereby grain surface chemistry over a long time period could produce lower F values.

  
5.1 Low-mass protostellar envelopes

	Figure 7: Observational determinations of the temperature (K) and density (cm-3) as a function of radius towards one prestellar core (L1544) and 2 low-mass protostellar envelopes (NGC 1333 IRAS 4A and L1448N).
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As explained above, the very high N₂D⁺/N₂H⁺ ratios we observed suggest that our observations are of the cold, depleted envelopes surrounding the protostars, rather than any "hot corino'' where large amounts of species have evaporated from grains. The 80'' beam of the ARO 12 m telescope equates to a diameter of 10 000 AU at 140 pc (Taurus) and 15 000 AU at 220 pc (Perseus). Figure 7 shows the temperature and density distribution as a function of radius in a typical prestellar core (L1544; data from Tafalla et al. 2002) together with two of the low-mass protostellar sources in our survey: NGC 1333 (H. Nomura, private communication) and L1448N (from Maret et al. 2004). The cold (<30 K) envelopes of NGC 1333 and L1448N extend from a few hundred to 10 000 AU where they are assumed to merge into the surrounding molecular cloud.

Initially, we have constructed a static protostellar core-envelope model, based on the structure of NGC 1333, shown in Fig. 7. The initial composition of the gas comes from running a series of 42 "prestellar core'' models which include reactions of species in the gas-phase and freeze-out of molecules onto grains. Each of these models represents a shell of gas, spaced at equal logarithmic intervals, using the temperature and density profiles of L1544 (see Fig. 7). A comparison of results from this prestellar core model with observations of L1544 has been presented in Roberts & Millar (2006). The chemistry is evolved for 10⁵ yr and then the output molecular abundances from each shell are used as inputs to a similar model which uses the temperature and density profiles determined from observations of NGC 1333. Obviously, this neglects the infall of material through the core, but assuming an infall velocity of 0.1 km s^-1, a parcel of gas would move only $\sim$2000 AU in 10⁵ yr. In the outer envelope this corresponds to a density increase of a factor $\la$2, which we would not expect to significantly affect the chemistry.

Each model considers 349 gas-phase species, 185 of which contain one or more deuterium atoms. There are $\sim$10 000 gas-phase reactions and $\sim$700 reactions of positive ions with negatively charged grains which result in dissociative recombination. Additionally, the neutral species can accrete onto and desorb from the grain surfaces. We assume a negatively charged grain population with a constant radius (0.1 $\mu$m). We assume that the cosmic-ray ionisation rate is the same everywhere and has the standard dark-cloud value: $1.3 \times 10^{-17}$ s^-1.

Table 8: Binding energies for selected surface species. The rate for thermal desorption at a given temperature, $T_{\rm kin}$, is $\sim$10 $^{12}\exp(-E_{\rm ads}/T_{\rm kin}$) s^-1. For N₂, $E_{\rm ads} = 380$ K generally gives better agreement with observations; the effect of using $E_{\rm ads} = 860$ K is discussed in Sect. 5.1.1.

The only mechanism we include to return material to the gas from the grains is thermal desorption. Each molecule is assigned a binding energy which is used to calculate a rate coefficient for desorption. Binding energies for selected species are listed in Table 8, most of these have been estimated based on experimental work with pure ices, rather than realistic interstellar grain analogues. The unknown desorption energies for many species, and the difficulties in incorporating realistic desorption mechanisms into large models in a systematic way, are major sources of uncertainty. At 10 K the thermal desorption rate for CO is $\sim$10^-30 s^-1, so slow as to be effectively zero, but once the temperature reaches 30 K the rate has increased to 10^-2 s^-1. This is orders of magnitude higher than the typical CO accretion rate, so all of the solid CO will evaporate almost instantaneously. Water ice has a much higher binding energy (see Table 8), meaning that the rate for thermal desorption is negligible even at 50 K. At 100 K the rate for thermal desorption of H₂O is $\sim$10^-9 s^-1 which is comparable with its accretion rate. We have assumed that the binding energy for N₂ is significantly lower than that of CO so that at 10 K the rate for N₂ desorption is $\sim$10^-5 s^-1. The N₂ which freezes out is, therefore, rapidly returned to the gas-phase and reacts with molecular hydrogen and deuterium, with their protonated ions (which also have low binding energies due to their low mass), and with cosmic rays and electrons to form e.g. N₂H⁺, N₂D⁺, and ammonia. The assumption of a relatively low binding energy for N₂ allows us to reproduce the observed distribution of molecules towards prestellar cores, where NH₃ and N₂H⁺ do not have the same (or have much smaller) central "holes'' due to depletion onto grains, seen for abundant molecules such as CO and CS (Bergin et al. 2001; Caselli et al. 1999; Tafalla et al. 2002, see also Fig. 8). As discussed in Sect. 1, however, the most recent experiments determined that solid-state processes for CO and N₂ are very similar, under astrophysically relevant conditions (Öberg et al. 2005; Bisschop et al. 2006). Results from chemical models using similar desorption energies for CO and N₂ are presented and discussed in Sect. 5.1.1, below.

	Figure 8: Predicted abundances of selected molecules as a function of distance from the protostar from the protostellar envelope model. Top: at the end of the prestellar core phase; Bottom: 105 yr after the formation of the protostar.
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We have not explicitly included chemical reactions on the grain surfaces once material freezes out, but, as discussed above, it is likely that the material which desorbs back into the gas will have been altered by chemical processing in the ices. Currently, we simulate the effects of surface hydrogenation in the shells where the bulk of the CO has frozen-out by converting a proportion of the CO ice to H₂CO and CH₃OH, S to H₂S, and O to H₂O. Table 9 (based on that in Osamura et al. 2004) shows the quantities of the molecules we have added in this way. The molecular abundances are based on those observed in warm gas towards protostellar sources. Most of the molecular D/H ratios come from theoretical calculations of grain surface chemistry (Stantcheva & Herbst 2003), although we have lowered the abundances of deuterated water based on recent observational results (Butner et al. 2007; Parise et al. 2003,2005). We note, though, that, for the models presented here, the exact composition of the gas in the central core is relatively unimportant due to its small size compared to the envelope (see below).

Figure 8 shows the initial distribution of species across the core input into the protostellar envelope model at t=0 yr (i.e. the output from the prestellar core model) and the distribution at t=10⁵ yr. The chemical evolution in the outer envelope surrounding the protostar is very much like that in a prestellar core. Gas-phase species continue to accrete onto grains and the "hole'' where CO is totally frozen out increases in size from $\sim$2000 to 4000 AU. There is a small "hot core'' region within a few hundred AU of the protostar where material has evaporated from the grains and a rich gas-phase chemistry is occurring.

Figure 9 shows evolution of the chemistry predicted in different regions of the protostellar envelope. At 10 AU (top panel) the temperature is high enough to evaporate the bulk of the grain mantle almost instantaneously, so CO, H₂O, CH₃OH and H₂S are abundant. The ionisation fraction is very low due to the high density so the N₂H⁺ abundance is also low. The high temperature means very little deuterium fractionation for the molecular ions (H₂D⁺, N₂D⁺, DCO⁺, etc.). We have assumed that the amount of H₂CO formed via hydrogenation of CO on the grain surface is two and a half times the amount which formed in the gas-phase and then accreted, so the formaldehyde fractionation in the post-evaporation gas is set by the surface chemistry.

Table 9: Abundances of molecules assumed to be synthesised by surface chemistry between the prestellar and protostellar phases. For the non-deuterated species the abundances are relative to n(H) while for the deuterated species the abundances are relative to the un-deuterated analogue.

	Figure 9: The predicted evolution of molecular abundances ( left panel) and D/H ratios ( right panel) after the formation of the central protostar in selected regions of the protostellar envelope. From top to bottom: 10 AU, $T_{\rm kin} = 340$ K, n(H $_2) = 3\times 10^{9}$; 100 AU, 40 K, 108 cm-3; 4000 AU, 10 K, $3\times 10^5$ cm-3; 10 000 AU, 10 K, $3\times 10^4$ cm-3.
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At 100 AU (second panel of Fig. 9) the temperature is $\sim$40 K so CO still rapidly desorbs from the grains. Water, formaldehyde, and methanol are more tightly bound, though, (see Table 8) so only a small fraction of these molecules desorb. After the temperature is raised, H₂D⁺ is rapidly destroyed by H₂ and its fractionation falls. It then takes only 10-20 yr for the DCO⁺/HCO⁺ and N₂D⁺/N₂H⁺ ratios to follow.

At 4000 AU (third panel of Fig. 9) the physical conditions are similar to those at the centre of a prestellar core. Carbon monoxide was already heavily depleted at the beginning of the protostellar phase and so the molecular D/H ratios are very high in this region, although the abundances of all but the nitrogen-bearing species are relatively low.

In the outer envelope at 10 000 AU (bottom panel of Fig. 9) the physical conditions are more like those found in a dark molecular cloud. Molecular D/H ratios increase as species accrete onto grains until, after a further $\sim$10⁶ yr, all the heavier species have frozen out.

Figure 10 shows cumulative column densities for selected molecules predicted by the model, at different times, moving from the outer to the inner region of the low-mass protostellar envelope. For the C- and O- bearing species, the inner envelope region between a few hundred and a few thousand AU doesn't contribute to the column density, as these species are mostly frozen onto the grains. The decrease in CO in the outer envelope over time, due to freeze-out, is apparent. The CO abundance increases by 2-4 orders of magnitude at radii <200 AU, but this is much smaller than the 20 000 AU total radius. There is a large jump in formaldehyde abundance at 100 AU at early times due to evaporation from grains. After $\sim$10⁴ yr, though, a significant proportion of the formaldehyde has been destroyed (see second panel of Fig. 9). For N₂D⁺ there is no central enhancement in the column density, as the ionisation fraction is very low at high densities, but the inner envelope region between 1000 and 10 000 AU does contribute to the column density until $\sim$10⁶ yr, when even the N-bearing species have frozen onto the grains.

In order to compare directly with the observations we have calculated beam-averaged column densities from the protostellar envelope model. Results for selected species as a function of time are shown in Fig. 11. These assume an 80'' FWHM beam and a source distance of 220 pc (appropriate for NGC 1333). The first thing that is apparent is that the predicted D₂CO/H₂CO ratio is more than 10 times less than we observed. The column densities are largely determined by the material in the outer envelope (where the abundance of D₂CO is 3 orders of magnitude less than H₂CO; see Fig. 8), since even though the the density of the "hot-core'' region is $\sim$3 orders of magnitude higher than that of the outer envelope, the envelope is so much larger than the core that it contains 10⁴ times the mass. The D₂CO/H₂CO ratio does reach 0.01 at $\sim$4000 AU (see Fig. 9), but the abundance of formaldehyde is this region is several orders of magnitude lower than at 10 000 AU.

We observed D₂CO column densities of $\sim$10¹² cm^-2, whereas the model predicts N(D₂CO ) < 10¹¹ cm^-2. For HDCO, however, the agreement is much better: most of the observed column densities were a few $\times$10¹² and the HDCO/H₂CO ratios were 0.02-0.07. At 10⁵ yr the model predicts N(HDCO $) = 2\times10^{12}$ cm^-2 and HDCO/H₂CO = 0.06.

This model predicts very high N₂D⁺ fractionation, since nitrogen bearing species remain relatively abundant closer to the protostar at 2000-4000 AU. After 10⁵ yr the predicted N₂D⁺/N₂H⁺ ratio is 0.3 and the column density of N₂D⁺ is $\sim$ $3\times10^{12}$ cm^-2. This fractionation is consistent with the highest ratios measured by our survey and the observed column densities towards these sources are within a factor of 2 of the model prediction.

In Paper I we observed HCN and DCN towards most of the same low-mass protostellar sources, finding DCN/HCN ratios of 0.03-0.06 and DCN column densities of $1{-}4\times$10¹² cm^-2. The model prediction of N(DCN $) = 2\times10^{12}$ cm^-2 agrees very well with these observations, although the predicted DCN/HCN ratio of 0.09 is slightly too high.

Figure 10: Cumulative column densities from the model at different times for ( from left to right): CO, H2CO, D2CO, N2D+. The extreme right-hand side of each plot shows the column density through only the outermost layer of the cloud; as the radius decreases, column densities through successive inner layers are added. Note: these only illustrate the relative contribution from different layers at different times, and do not represent what one would expect to observe by pointing a telescope away from the source centre.

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	Figure 11: Top: predicted molecular column densities evolving over time from the protostellar envelope model that we would expect to observe for a source at distance of 220 pc with an 80'' telescope beam. Bottom: average molecular D/H ratios from the same model.
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One major uncertainty in the deuterium chemistry models is the rate for the reaction of H₃⁺ with HD, which is a key reaction for deuterium fractionation at low temperatures. Experiments at 10 K (Gerlich et al. 2002) have suggested that this reaction might proceed $\sim$5 times slower than the Langevin rate normally assumed (Giles et al. 1992; Millar et al. 1989). A detailed consideration of the effects of slowing the fractionation reactions on one-point chemical models has been presented by Roberts et al. (2002b). The deuterium fractionation at low temperatures also depends on the ortho:para ratios of the reacting species, particularly H₂, which are not well known (Flower et al. 2006; Walmsley et al. 2004). For the models presented in this section, which involve averaging over regions with different physical conditions, using slower rates for the fractionation reactions does not have a drastic effect. The predicted column density ratios are reduced by factors of 2-4, but (again with the exception of D₂CO) this is not inconsistent with observations towards many of our sources.

  
5.1.1 The binding energy for N₂ and a comparison of N₂D⁺ and DCO⁺ fractionation

	Figure 12: As for Fig. 8, except that we assume CO and N2 have similar binding energies.
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We have also run the models using an N₂ binding energy of 860 K, which is consistent with the experimental result: $E_{\rm ads}$(N ₂) = 0.923  $E_{\rm ads}$(CO) (Öberg et al. 2005). Results are shown in Figs. 12 and 13. The nitrogen bearing species now freeze out on a similar timescale to other molecules so at the start of the protostellar phase (top panel of Fig. 12) there is a hole of radius $\sim$2000 AU at the centre where no heavy species remain in the gas-phase. This hole increases in diameter as the envelope surrounding the protostar evolves and a hot core region forms close to the protostar (bottom panel of Fig. 12). Except for the distribution of the N-bearing species, the overall picture is very similar to the previous model.

	Figure 13: As for Fig. 11, except that we assume CO and N2 have similar binding energies.
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Adopting a higher binding energy for N₂ does cause the column density of D₂CO and the average D₂CO/H₂CO ratio to increase, but they are still significantly lower than the observations (see Fig. 13). In fact, many of the molecular D/H ratios have increased (although the N₂D⁺/N₂H⁺ ratio is lower) since there is a strong anti-correlation between the abundance of heavy species in the gas-phase (including N₂ and NH₃) and the level of deuterium fractionation (see e.g. Roberts et al. 2004). The beam-averaged column density of N₂D⁺ has fallen from a few $\times$10¹² cm^-2 at times <10⁵ yr to $\la$ $3\times10^{11}$ cm^-2. This is more than 10 times lower than the observation towards NGC 1333 IRAS 4A, but is similar to those towards L1527, B5IRS1, RNO43, and HH111. L1527 is the only one of these sources in which we also observed N₂H⁺, giving N₂D⁺/N₂H⁺ = 0.05. This observed ratio is lower than those predicted by either protostellar envelope model.

Figure 14: A comparison of the N2D+ and DCO+ fractionation observed towards a selection of sources compiled from our observations and results from the literature: Butner et al. (1995), Caselli et al. (2002b), Gerin et al. (2001), Tiné et al. (2000), Williams et al. (1998). The grey dashed line shows x=y; the solid black line indicates the range of values predicted by a chemical model of a low-mass protostellar envelope which assumes N2 can desorb efficiently from grain surfaces at lower temperatures than CO, while the dashed black line shows the prediction from the same model when N2 and CO are assumed to evaporate at the same rate.

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In Fig. 14 we compare N₂D⁺/N₂H⁺ and DCO⁺/HCO⁺ratios observed towards six different sources: dark clouds/starless cores (TMC-1, TMC-2, L1400G), prestellar cores (L1544 and L134N) and a protostellar envelope (L1551 IRS5). In all these sources the N₂D⁺ fractionation is 2 to 100 times higher than the DCO⁺ fractionation (there is one exception to this, not plotted in Fig. 14: Butner et al. (1995) measured DCO⁺/HCO ⁺ = 0.035 towards L1498, but we did not detect N₂D⁺, using the same 12 m telescope, in this source and calculate N₂D⁺/N₂H⁺<0.014). For a given set of physical conditions, models of deuterium chemistry predict that, even if HCO⁺ and N₂H⁺ have different abundances, DCO⁺ and N₂D⁺ will have the same fractionation (see Fig. 9), since they form directly from deuterated H₃⁺. The model presented in Sect. 5.1 predicts that the observed (i.e. based on beam-averaged column densities) N₂D⁺ fractionation will be 5-6 times higher than the DCO⁺ fractionation. Since N₂ resists freeze-out, N₂H⁺ and N₂D⁺ remain abundant in the inner envelope ($\sim$4000 AU), where fractionation is very high, for much longer than HCO⁺ and DCO⁺. The overall DCO⁺ fractionation, therefore, reflects that in the outer envelope, whereas emission from both the inner and outer envelope contributes to the N₂D⁺ fractionation.

If we assume, though, that the N₂ which collides with the grains remains frozen (as in the model presented in this section) then the distribution of N-bearing species across the core is very similar to the C- and O- bearing species and we predict N₂D⁺/N₂H$^+ \la1.5$ times higher than DCO⁺/HCO⁺. Incorporating the most recent experimental results for the binding energies of N₂ and CO on interstellar-analogue surfaces, therefore, gives us significantly worse agreement with observational results. Other authors (Vastel et al. 2006; Aikawa et al. 2001) have found that the difference in N₂D⁺ and DCO⁺ fractionation can be explained by the fact that CO is a major destroyer of N₂H⁺ and N₂D⁺ in the gas-phase, so the loss of CO onto grains causes an enhancement in their abundances. In our models, the major loss mechanism for N₂H⁺ is collision with negatively charged grains (we assume that this produces N₂ and H, which are returned to the gas-phase; see Roberts et al. 2004), so the depletion of CO does not have a significant effect.

5.2 Hot molecular cores

	Figure 15: Steady-state molecular D/H ratios as a function of temperature from gas-phase chemical models using n(H 2) = 106 cm-3, a cosmic ray ionisation rate of 1.3 $\times$ 10-17 s-1 and a visual extinction of 10 mag.
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The molecular D/H ratios previously determined towards our sample of HMCs are: HDO/H₂O $\sim$ 10^-4 (Gensheimer et al. 1996), DCN/HCN $\sim$ 10^-3 (Hatchell et al. 1998a), and HDS/H₂S < 10^-3 (Hatchell et al. 1999). The water observations were carried out using the IRAM 30 m telescope (11-12'' beam), while the other observations were made with the JCMT ($\sim$20'' beam). Figure 15 shows the equilibrium (steady-state) fractionation predicted by gas-phase ion-molecule chemistry for HDO, DCN, HDCO, D₂CO and N₂D⁺ for temperatures varying between 10 and 100 K. The H₂ density was assumed to be 10⁶ cm^-3, but the curves for n(H₂)=10⁴ cm^-3 are very similar. Before the onset of star-formation molecules accrete onto grain surfaces and the level of deuteration is "frozen in'' until the star forms and heats the gas, returning the molecules to the gas-phase. Molecular D/H ratios observed towards HMCs can, therefore, be significantly higher than the equilibrium values at 100-200 K. The exceptions to this are molecular ions, such as N₂D⁺ and N₂H⁺, which are expected to dissociate when they strike a negatively charged grain, so the D/H ratios for molecular ions set in the gas-phase are not preserved in the ice mantles. Once the gas is heated the N₂D⁺/N₂H⁺ ratio resets to its equilibrium value much more rapidly than the neutral species (see also the top two panels of Fig. 9). The upper limits on the N₂D⁺/N₂H⁺ ratios towards the HMCs suggest that the bulk of the gas is warmer than 30 K. Thus, even if the HMCs are much smaller than the telescope beam (a core with diameter 0.1 pc at a distance of 4 kpc subtends an angle of only 5''), they are not surrounded by a very cold envelope like the low-mass protostars.

Comparing the other curves in Fig. 15 with the ratios observed towards HMCs assumes that the fractionation was set in the gas-phase before the onset of star-formation, that there was not a long period of freeze-out onto grains at low temperatures, and that the grains merely acted as a repository for the molecules which accreted. The curve for HDS fractionation would closely follow that of HDO, but the actual abundance of HDS and H₂S formed in the gas phase at these temperatures is very low, so it is unlikely that the hydrogen sulphide observed towards HMCs could have been synthesised in the gas-phase. Instead, we have plotted the atomic D/H ratio and, if simple hydrogenation schemes for surface chemistry (e.g. Brown & Millar 1989; Turner 1990; Stantcheva & Herbst 2003) are a good approximation to reality, we expect the relative abundance of HDS to H₂S formed on the surface to be similar to the relative amount of D to H accreting onto the surface.

Of course, if hydrogen sulphide is forming on grain surfaces, then, clearly, we must allow the possibility that other molecules may be. Observations show that the bulk of the grain mantles consists of water ice (see e.g. Keane et al. 2001; Whittet et al. 1996; Gibb et al. 2004, and references therein) so H₂O must form efficiently. Experiments suggest that the CO which freezes out can be converted to formaldehyde and methanol (Watanabe & Kouchi 2002; Watanabe et al. 2003,2004). Figure 15 shows that we would expect the HDO forming on the grain surface (from atomic D) to have a similar fractionation to that formed in the gas-phase, but illustrates the point made by Turner (1990) and Rodgers & Charnley (2002) that surface chemistry could dramatically alter the HDCO/H₂CO and D₂CO/H₂CO ratios.

Assuming that H₂S is formed by surface chemistry, the upper limits on the HDS/H₂S ratios in the HMCs constrain the ice formation temperature to >30 K, while the observed HDO/H₂O ratios suggest temperatures of 50-60 K. The DCN/HCN ratios are reproduced at $T \sim60{-}70$ K, but it is possible that DCN fractionation is lowered in the gas-phase post-evaporation (Hatchell et al. 1998a), so these temperatures are upper limits.

The formaldehyde fractionation remains relatively high until the temperature reaches $\sim$60 K, due to the fact that the deuterated CH₃⁺ ions have a higher energy barrier to destruction by H₂ than the deuterated H₃⁺ ions (see Sect. 5). The HDCO/H₂CO ratios of 0.01 and 0.03, along with D₂CO/H₂CO upper limits of 0.001 measured towards G34.26 and G75.78 are also consistent with formation temperatures of 50-60 K, if the bulk of the formaldehyde was formed in the gas-phase. Towards G31.41, however, the HDCO/H₂CO ratio was only 0.002, which would imply gas-phase temperatures >90 K. This is inconsistent with the fractionation of DCN and HDO, so it seems likely that in this source the HDCO/H₂CO ratio set in the gas-phase was then diluted by the formation of additional formaldehyde on the grains. Assuming that both gas-phase and grain surface formation are, therefore, also important in G34.26 and G75.78, all we can say is that the gas-temperature was <60 K.

These molecular D/H ratios are all consistent with the HMCs having formed out of relatively warm (50 K) gas. They suggest that surface chemistry has affected the composition of the gas, but that there was not a long period of accretion at very low temperatures, such as is implied by the results for the low-mass protostars.

6 Discussion

We have measured DCN/HCN, HDCO/H₂CO, D₂CO/H₂CO and N₂D⁺/N₂H⁺ ratios towards a number of low-mass protostellar sources at different evolutionary stages and in different molecular clouds. The N₂D⁺/N₂H⁺ ratios are all >0.01 which implies that the bulk of the emission comes from a region where the gas temperature is <25 K; i.e. from the extended envelope surrounding the protostars. In four of the sources the N₂D⁺ fractionation was >0.1, suggesting that the surrounding envelopes are depleted in CO. A static protostellar core-envelope model can explain the deuterium fractionation in most of these molecules, but the D₂CO/H₂CO ratio remains a problem. The physical model we have adopted is relatively simple, but this does not appear to be the main problem, since the average ratios from the model are close to those predicted by theory and the theory does not reproduce the relative abundances of HDCO and D₂CO which are observed. It seems more likely that there is a chemical process missing from both the theory and models. A candidate for this is formation of D₂CO on grain surfaces, followed by a non-thermal desorption mechanism which is efficient even at 10 K, but if this mechanism also returns CO to the gas-phase, it will act to suppress the deuterium fractionation in molecular ions like N₂D⁺ and DCO⁺.

This study finds no compelling evidence for an evolutionary trend where the formaldehyde fractionation increases as the N₂D⁺ fractionation decreases, but we plan to measure N₂D⁺/N₂H⁺ ratios in more of the sources in which we have D₂CO/H₂CO ratios. We are also developing models of collapsing protostellar core-envelope systems (Roberts et al. in preparation) which will explicitly include grain-surface chemistry along with time-dependent heating of the gas and desorption. These should provide better predictions for the evolution of molecular D/H ratios in star-forming regions and may explain why two of the low-mass protostellar envelopes we observed have N₂D⁺ fractionation <0.1, which does not require CO to be heavily depleted.

We have also measured N₂D⁺/N₂H⁺ ratios towards five cold, starless cores. The only systematic difference between these observations and the low-mass protostars is that the lines from the cold cores are narrower. Three of the starless cores have N₂D⁺/N₂H⁺ ratios >0.1, similar to the ratio observed towards the prestellar core L1544, suggesting that these sources also have significant, extended CO depletion. The ratios of <0.03 towards the other two starless cores are more consistent with a dark cloud gas-phase chemistry (see e.g. Roberts & Millar 2000).

Towards the hot molecular cores, associated with high mass star formation, we have measured HDCO/H₂CO ratios and calculated upper limits on the D₂CO and N₂D⁺ fractionation. Comparing these, along with DCN/HCN, HDO/H₂O, and HDS/H₂S ratios determined by other authors, to the predicted equilibrium fractionation as a function of temperature, we find that the observations are all consistent with the molecules having formed when the gas temperature was 50-60 K. The current temperature at the centre of the cores appears to be >80 K and the observations of water and hydrogen sulphide (among others, see Hatchell et al. 1998b) confirm that grain surface chemistry has affected the composition of the gas. Thus, the fractionation of the deuterated molecules that are observed was most likely set during an earlier stage of evolution, before the protostar formed, and preserved in the grain mantles. It does not appear that these cores underwent long periods of accretion at low temperatures, or that they are currently surrounded by cold, depleted envelopes, like the low-mass protostars. This suggests that the star formed out of relatively warm gas and dust (50-60 K), which would support current theories concerning the differences between low and high-mass star formation: high mass stars form more rapidly and in clusters, whereas low mass stars form in isolation after a relatively long prestellar core phase.

The upper limits on the N₂D⁺/N₂H⁺ ratios suggest that the envelopes surrounding the HMCs have temperatures >30 K. Recently, though, Fontani et al. (2006) detected N₂D⁺ in the vicinity of other high-mass protostellar candidates, sources which are probably younger than ours, finding N₂D⁺/N₂H⁺ ratios of $\sim$10^-2. This indicates the presence of cold (<25 K) gas, but not necessarily heavy molecular depletion (in fact, Fontani et al. measure CO depletion factors of $\sim$3). They conclude that this cold gas could either be the remnant of a prestellar core phase, or could be located in nearby starless cores.

Clearly, further observations of both low and high-mass protostars at higher spatial resolution are required in order to determine the chemical history of the gas and dust.

Acknowledgements

The 12 m Telescope is a facility of the National Science Foundation currently operated by the University of Arizona Steward Observatory under a loan agreement with the National Radio Astronomy Observatory. The James Clerk Maxwell Telescope is operated by The Joint Astronomy Centre on behalf of the Particle Physics and Astronomy Research Council of the United Kingdom, the Netherlands Organisation for Scientific Research, and the National Research Council of Canada. The authors would like to thank Albrecht Sievers and Manolo Ruiz at IRAM for carrying out the IRAM 30 m observations. Much of this work was carried out while the authors were employed by the University of Manchester, UK. Astrophysics at Queens University Belfast and The University of Manchester is supported by PPARC.

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Online Material

	Figure 1: Spectra of the 1-0 transitions of N2H+ ( left panel) and N2D+ ( right panel) towards cold, starless cores from the UofA 12 m telescope.
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Figure 2: Spectra observed towards 5 of the low-mass protostellar sources. ROW 1: D2CO 21,2-11,1 from the UofA 12 m telescopes; ROW 2: D2CO 40,4-30,3 from the JCMT; ROW 3: N2H+ 1-0 multiplets from the UofA 12 m telescope; ROW 4: N2H+ 3-2 multiplets observed with the IRAM 30 m telescope (the uncertainty in the CLASS fits for these spectra is illustrated for N2H+ in L1448mms: the solid black line shows a fit with $\tau =8$, while the dashed line shows an alternate fit with $\tau =0.1$); ROW 5: N2D+ 1-0 multiplets from the UofA 12 m telescopes; ROW 6: N2D+ 3-2 multiplets from the JCMT; ROW 7: N2D+ 3-2 multiplets from the IRAM 30 m telescope.

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	Figure 3: Spectra observed towards a further 4 low-mass protostellar sources using the JCMT. UPPER ROW: D2CO 40,4-30,3 line; LOWER ROW: N2D+ 3-2 multiplet.
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	Figure 4: Spectra observed towards an extra 4 low-mass protostellar sources using the UofA 12 m telescope and the JCMT; lines as described in the top right-hand corner of each spectrum.
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	Figure 5: Spectra observed towards the HMCs using the UofA 12 m telescope. ROW 1: H213CO 21,2-11,1 line; ROW 2: HDCO 21,1-11,0 line; ROW 3: N2H+ 1-0 multiplet; ROW 4: searches for the N2D+ 1-0 multiplet.
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