MEASUREMENT OF HDCO/H₂CO RATIOS IN THE ENVELOPES OF EXTREMELY COLD PROTOSTARS IN ORION

We calculate the column densities of H₂CO. Assuming that the line is optically thin and that the source size is similar to the main beam size, the column density in a lower state (defined as energy level $u\to l={(J+1)}_{{K}_{{\rm{a}}},{K}_{{\rm{c}}}+1}\to {J}_{{K}_{{\rm{a}}},{K}_{{\rm{c}}}}$) can be obtained by

$$\begin{eqnarray}&&{N}_{l}=93.5\displaystyle \frac{{g}_{l}}{{g}_{u}}\displaystyle \frac{{\nu }^{3}}{{A}_{{ul}}}\displaystyle \frac{1}{{T}_{{\rm{ex}}}[1-{\rm{exp}}(-h\nu /{{kT}}_{{\rm{ex}}})]}{\displaystyle \int }_{}{T}_{{\rm{mb}}}\;{dv},\end{eqnarray} \tag{ 1 }$$

where h is the Planck's constant, k is the Boltzmann's constant, ν is the frequency of the transition in GHz, ${T}_{{\rm{ex}}}$ is the excitation temperature, g is the statistical weight, A_ul is  the Einstein coefficient in s⁻¹, ${\displaystyle \int }_{}{T}_{{\rm{mb}}}\;{dv}$ is the integrated line intensity, and v is the velocity in km s⁻¹ (Rohlfs & Wilson 1999). The column density is in cm⁻², and the numerical factor in the equation corresponds to 8π/c³ and a conversion factor, when it is expressed in the units given above. All parameters for molecules were taken from the JPL databases (Pickett et al. 1998). Some sources may have moderate optical depths, as suggested in Section 3, and they are discussed in Section 4.4.

The total column density of ortho-H₂CO, ${N}_{{\rm{ortho}}},$ is related to the column density, ${N}_{l}=N({J}_{{K}_{{\rm{a}}},{K}_{{\rm{c}}}}),$ in the lower state ${J}_{{K}_{{\rm{a}}},{K}_{{\rm{c}}}}$ by

$$\begin{eqnarray}&&\displaystyle \frac{{N}_{{\rm{ortho}}}}{N({J}_{{K}_{{\rm{a}}},{K}_{{\rm{c}}}})}=\displaystyle \frac{Z}{g({J}_{{K}_{{\rm{a}}},{K}_{{\rm{c}}}})}{\rm{exp}}\left(\displaystyle \frac{E({J}_{{K}_{{\rm{a}}},{K}_{{\rm{c}}}})}{{{kT}}_{{\rm{ex}}}}\right),\end{eqnarray} \tag{ 2 }$$

where $g({J}_{{K}_{{\rm{a}}},{K}_{{\rm{c}}}})$ is the statistical weight ($3(2J+1)$ for ortho-H₂CO), $E({J}_{{K}_{{\rm{a}}}{K}_{{\rm{c}}}})$ is the energy of a ${J}_{{K}_{{\rm{a}}}{K}_{{\rm{c}}}}$ level, and Z is the partition function. If the molecules follow the Boltzmann distribution of a single ${T}_{{\rm{ex}}},$ the partition function is

$$\begin{eqnarray}&&Z=\displaystyle \sum \;z=\displaystyle \sum \;g({J}_{{K}_{{\rm{a}}},{K}_{{\rm{c}}}}){\rm{exp}}\left(-\displaystyle \frac{E({J}_{{K}_{{\rm{a}}}{K}_{{\rm{c}}}})}{{{kT}}_{{\rm{ex}}}}\right).\end{eqnarray} \tag{ 3 }$$

For ortho-H₂CO, ${K}_{{\rm{a}}}$ can only be odd. The total H₂CO column densities are calculated assuming an excitation temperature of 10 K and the statistical ortho to para ratio of 3:1 (Minh et al. 1995; Guzmán et al. 2013). The resulting column densities are given in Table 4. The column densities are not very sensitive to the assumed excitation temperature, when ${T}_{{\rm{ex}}}\;\gtrsim$ 10 K (Roberts et al. 2002). Using an excitation temperature of 30 or 50 K changes the inferred column densities by a factor of about 2.2 or 3.7, respectively. For example, if we calculate column densities of H₂CO adopting ${T}_{{\rm{ex}}}=30$ K for 090003 and HOPS 341, their column densities become $7.1\times {10}^{12}$ and $33.0\times {10}^{12}$ cm⁻², respectively. Indeed, we estimate ${T}_{{\rm{ex}}}\;\approx$ 14 K from those sources with detections of both HDCO transitions (Section 4.3). Therefore, the assumption of ${T}_{{\rm{ex}}}$ = 10 K is reasonable.

Table 4.  Column Densities

Source	N(H2CO)	N(HDCO)	[HDCO]/[H2CO]
	(${10}^{12}\;{{\rm{cm}}}^{-2}$)	(${10}^{12}\;{{\rm{cm}}}^{-2}$)
119019	11.7 ± 1.3	0.79 ± 0.11	0.07 ± 0.01
HOPS 169	15.3 ± 1.6	1.76 ± 0.26	0.12 ± 0.02
082005	25.3 ± 2.6	1.00 ± 0.19	0.04 ± 0.01
090003	3.3 ± 0.4	1.00 ± 0.15	0.31 ± 0.06
HOPS 358	27.6 ± 2.9	2.33 ± 0.26	0.08 ± 0.01
HOPS 373	9.0 ± 1.0	1.90 ± 0.25	0.21 ± 0.04
093005	40.2 ± 4.1	1.82 ± 0.23	0.05 ± 0.01
HOPS 359	23.9 ± 2.6	2.61 ± 0.31	0.11 ± 0.02
HOPS 341	15.3 ± 1.6	1.09 ± 0.14	0.07 ± 0.01
097002	3.2 ± 0.4	0.60 ± 0.12	0.19 ± 0.04
HOPS 354	23.0 ± 2.4	1.65 ± 0.23	0.07 ± 0.01
HOPS 164	16.0 ± 1.7	1.25 ± 0.18	0.08 ± 0.01
HOPS 297	15.5 ± 1.7	1.00 ± 0.17	0.06 ± 0.01
HOPS 321	16.4 ± 1.7	0.54 ± 0.10	0.03 ± 0.01
HOPS 360	17.6 ± 1.8	1.52 ± 0.24	0.09 ± 0.02

Note. The column densities are calculated assuming ${T}_{{\rm{ex}}}$ = 10 K. The estimated errors come from the uncertainties of integrated intensities (Table 3), which include the statistical uncertainties and the calibration uncertainty of 10% (Lee et al. 2011).

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HDCO is a nearly symmetric top molecule. Therefore, the treatment for calculating column density is similar to the case of H₂CO. However, there is no H-pair, so there is no ortho/para modification. While H₂CO has an electric dipole moment along the A axis only, HDCO has an extra dipole moment along the B axis. As a result, transitions between K ladders are allowed, and all K ladders are radiatively connected. HDCO lines have hyperfine structures (Langer et al. 1979). For the HDCO ${2}_{\mathrm{0,2}}\to {1}_{\mathrm{0,1}}$ transition, the central group (2 components) dominates in intensity, and the velocity separation from the center to the satellites is $\lt 0.2$ km s⁻¹. The hyperfine components are not expected to be resolved, but they can make the line look wide. For the purpose of column density calculations, we ignore the effects of hyperfine structure. We use ${\mu }_{a}$ (2.324 Debye) for μ, since we are interested in the transitions in a K ladder (a type). The statistical weight g has a rotation contribution only, $g({J}_{{K}_{{\rm{a}}},{K}_{{\rm{c}}}})=2J+1.$ The HDCO transition is optically thin because deuterated species are 10–100 times less abundant than the main species.

The total HDCO column density is calculated by assuming an excitation temperature of 10 K and an optically thin line (Table 4). If we use ${T}_{{\rm{ex}}}=$ 30 or 50 K, the HDCO column densities change by factors of about 3.4 or 6.7, respectively. The [HDCO]/[H₂CO] ratios change by factors of about 1.6 or 1.8, respectively.

We calculated the excitation temperature using the line ratio of the HDCO ${J}_{{K}_{{\rm{a}}},{K}_{{\rm{c}}}}$ = ${2}_{\mathrm{0,2}}\to {1}_{\mathrm{0,1}}$ and ${2}_{\mathrm{1,1}}\to {1}_{\mathrm{1,0}}$ transitions, assuming that they have the same excitation temperature. Because the J quantum numbers are the same, factors depending on J are common factors in calculating the column density ratio. The column density ratio, ignoring the common factors, is

$$\begin{eqnarray}&&\displaystyle \frac{{N}_{{1}_{\mathrm{1,0}}}}{{N}_{{1}_{\mathrm{0,1}}}}={RT}\displaystyle \frac{4}{3}\displaystyle \frac{[1-{\rm{exp}}(-h\nu ({2}_{\mathrm{0,2}}-{1}_{\mathrm{0,1}})/{{kT}}_{{\rm{ex}}})]}{[1-{\rm{exp}}(-h\nu ({2}_{\mathrm{1,1}}-{1}_{\mathrm{1,0}})/{{kT}}_{{\rm{ex}}})]}\approx {RT}\displaystyle \frac{4}{3},\end{eqnarray} \tag{ 4 }$$

where ${RT}={\displaystyle \int }_{}{T}_{{\rm{mb}}}({2}_{\mathrm{1,1}}-{1}_{\mathrm{1,0}}){dv}/{\displaystyle \int }_{}{T}_{{\rm{mb}}}({2}_{\mathrm{0,2}}-{1}_{\mathrm{0,1}}){dv}$ is the line ratio. Because the frequencies are very similar, the right exponential factor after ${RT}4/3$ is nearly 1. The column density ratio should be the same as the ratio of partition terms, and here the statistical weight g cancels out since g only depends on J.

$$\begin{eqnarray}&&{RT}\displaystyle \frac{4}{3}={\rm{exp}}\left(-\displaystyle \frac{E({1}_{\mathrm{1,0}})-E({1}_{\mathrm{0,1}})}{{{kT}}_{{\rm{ex}}}}\right).\end{eqnarray} \tag{ 5 }$$

Since E(1_1,0)/k $=\;11.18297$ K and E(1_0,1)/k $=\;3.09406$ K, the excitation temperature is,

$$\begin{eqnarray}&&{T}_{{\rm{ex}}}=\displaystyle \frac{-8.09}{{\rm{ln}}(4/3{RT})}.\end{eqnarray} \tag{ 6 }$$

In our observations, seven sources are detected in both transitions. When the line ratio is close to ∼0.75, ${T}_{{\rm{ex}}}$ diverges. The excitation temperatures are summarized in Table 5. We obtained an average excitation temperature of ∼14 ± 5 K. Therefore, the ${T}_{{\rm{ex}}}=10$ K used in the column density calculations is consistent with the measured ${T}_{{\rm{ex}}}.$

Table 5.  Excitation Temperatures

Source	RTa	${T}_{{\rm{ex}}}$ b
		(K)
119019	1.34	N
HOPS 358	0.21	6 (±1)
HOPS 373	0.24	7 (±3)
093005	0.58	31 (±18)
HOPS 359	0.39	12 (±4)
097002	1.41	N
HOPS 354	0.80	N

Notes.

^aThe line ratio of the HDCO ${2}_{\mathrm{0,2}}\to {1}_{\mathrm{0,1}}$ and ${2}_{\mathrm{1,1}}\to {1}_{\mathrm{1,0}}$ transitions. ^bN: ${T}_{{\rm{ex}}}$ cannot be determined because RT > 3/4. The qualities of their HDCO ${2}_{\mathrm{1,1}}\to {1}_{\mathrm{1,0}}$ spectra are relatively poor (Figure 1).

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The assumption of an optically thin line is reasonable for the lines of the deuterated molecules based on their low abundances. However, the optical depth of the H₂CO ${2}_{\mathrm{1,2}}\to {1}_{\mathrm{1,1}}$ transition can be high because H₂CO is abundant in molecular clouds. The optical depth of the H₂CO line can be estimated by comparing the fluxes between H₂CO and its isotopologue for the same transition. Unfortunately, our survey did not cover observations of the H${}_{2}^{13}$CO line. Though a direct measurement of the optical depth (τ) is not available, we have roughly estimated the line optical depths using the solution of the line radiative transfer equation.

$$\begin{eqnarray}&&\tau =-{\rm{ln}}(1-({T}_{{\rm{peak}}}/{J}_{\nu }({T}_{{\rm{ex}}})-{J}_{\nu }({T}_{{\rm{BG}}}))),\end{eqnarray} \tag{ 7 }$$

where ${T}_{{\rm{peak}}}$ is the peak temperature, and ${J}_{\nu }({T}_{{\rm{ex}}})$ and ${J}_{\nu }({T}_{{\rm{BG}}})$ are the equivalent Rayleigh–Jeans excitation and background temperatures, respectively. The optical depths estimated in this way are between 0.3 and 1.9. In other studies, the optical depths of 0.4–1.1 were estimated from low-mass protostars (e.g., Maret et al. 2004) and ∼1.6 from the dense core in the Horsehead PDR (Guzmán et al. 2011), based on the flux ratio of the H₂CO and ${{\rm{H}}}_{2}^{13}$ CO ${2}_{\mathrm{1,2}}\to {1}_{\mathrm{1,1}}$ transition. Larger line width is also known to be coupled with larger line optical depth (Phillips et al. 1979; Roberts et al. 2002; Ceccarelli et al. 2003). In Figure 3, we compare the line widths of H₂CO ${2}_{\mathrm{1,2}}\to {1}_{\mathrm{1,1}}$ and HDCO ${2}_{\mathrm{0,2}}\to {1}_{\mathrm{0,1}}$. Most sources with large H₂CO line widths do not show a good agreement, which is likely due to the optical depth in the H₂CO ${2}_{\mathrm{1,2}}\to {1}_{\mathrm{1,1}}$ transition. Considering the optical depth of the H₂CO estimated from our calculation, the column density of H₂CO should be corrected by a factor of $\tau /(1-{\rm{exp}}(-\tau )),$ then N(H₂CO) increases by a factor of 1.1 for $\tau =0.3$ and 2.2 for $\tau =1.9.$

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The last column of Table 4 lists [HDCO]/[H₂CO] ratios calculated assuming ${T}_{{\rm{ex}}}$ = 10 K. This ratio is not very sensitive to ${T}_{{\rm{ex}}}$ (Section 4.2). The [HDCO]/[H₂CO] ratio ranges from 0.03 to 0.31, which is consistent with values derived from the low-mass Class 0 protostars by Parise et al. (2006). The average [HDCO]/[H₂CO] ratio and standard deviation of the total target sample are 0.105 and 0.075, respectively. Figure 4(a) shows the distribution of the [HDCO]/[H₂CO] ratios for the full sample. Considering that the ratio is always positive and that the distribution is asymmetric, the distribution can be characterized better with the mode rather than the average. The mode value and FWHM are ∼0.07 and ∼0.02, respectively. The [HDCO]/[H₂CO] ratios of the three PBRSs on the high end (090003, HOPS 373, and 097002) are significantly larger than the mode value. They may represent either a long tail of a unimodal distribution or a secondary peak of a bimodal distribution. More extensive studies are necessary to understand the detailed shape of the distribution.

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Figure 4(b) shows the [HDCO]/[H₂CO] distributions for the PBRS and non-PBRS samples separately. About 30% of PBRSs (3/11) have high [HDCO]/[H₂CO] ratios ($\gt 0.15$), and 70% of PBRSs (8/11) have ratios similar to non-PBRSs. Both PBRSs and non-PBRSs have the same mode (∼0.07) as the full sample.

Since the number of non-PBRS sources is relatively small, the [HDCO]/[H₂CO] ratios of Class 0 sources measured by Roberts et al. (2002) may be helpful in understanding the distributions. Figure 4(c) shows the [HDCO]/[H₂CO] distribution of non-PBRS sources, including seven sources from Roberts et al. (2002). The general properties of the distribution remain the same. The mode is ∼0.07, and the range of values is less than ∼0.12, which reinforces the finding that the PBRS distribution has a relatively large dispersion and that the three PBRSs on the high end are outliers. However, note that five sources in the sample of Roberts et al. (2002) are in regions closer than the Orion cloud, which could introduce some unexpected effects to the distribution. Therefore, a more extensive survey with a larger sample size will be helpful in understanding the distributions better.

Figure 5(a) shows a comparison of the [HDCO]/[H₂CO] ratios with and without optical depth correction. Although the optical depth corrections for the H₂CO observations make the [HDCO]/[H₂CO] ratios decrease for all sources, changes of the [HDCO]/[H₂CO] ratios are not much. In Figure 5(b), the overall distributions of the τ-corrected [HDCO]/[H₂CO] ratios show similar characteristics compared with Figure 4(b). The three outliers with high deuterium fractionation are still far from the mode of the distribution. Even if we consider the optical depth, the [HDCO]/[H₂CO] ratios of 090003, HOPS 373, and 097002 are still high (>0.15). Therefore, in the subsequent discussions below, we will consider only the [HDCO]/[H₂CO] distribution derived with the assumption of optically thin lines.

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The [HDCO]/[H₂CO] ratios are plotted against ${T}_{{\rm{bol}}}$ and ${L}_{{\rm{bol}}}$ in Figure 6. When plotted against ${L}_{{\rm{bol}}}$ in Figure 6(b), no obvious correlation was found. The [HDCO]/[H₂CO] ratios for low-mass Class 0 protostars reported by Roberts et al. (2002) are also plotted in Figure 6(a). As suggested by Roberts et al. (2002), there is no marked correlation between the [HDCO]/[H₂CO] ratios and the bolometric temperature (a proxy of the evolutionary stage of protostars). Though the PBRS and non-PBRS samples show similar mode values, the PBRS sample shows a significantly larger scatter. This difference suggests that either PBRS have an intrinsically large variation in the deuterium fractionation or the three outliers are chemically distinct from the majority of Class 0 protostars.

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To focus on the large variation of the deuterium fractionation in the PBRS sample, we calculate the fractional abundances of the PBRS sample. The fractional HDCO and H₂CO abundances were calculated by dividing the total HDCO and H₂CO column densities by the H₂ column density i.e., X(HDCO) = [HDCO]/[H₂] and X(H₂CO) = [H₂CO]/[H₂]. Using the 870 μm flux density in Table 5 of Stutz et al. (2013), the beam-averaged column density of PBRS can be computed with:

$$\begin{eqnarray}&&{N}_{{{\rm{H}}}_{2}}=\displaystyle \frac{{F}_{\nu }R}{{B}_{\nu }({T}_{D}){\rm{\Omega }}{\kappa }_{\nu }\mu {m}_{{\rm{H}}}},\end{eqnarray} \tag{ 8 }$$

where Ω is the beam solid angle, μ is the mean molecular weight of the interstellar medium, which we assume to be 2.8 (Kauffmann et al. 2008), ${m}_{{\rm{H}}}$ is the mass of the hydrogen atom, ${\kappa }_{\nu }$ is the dust absorption coefficient (1.85 cm² g⁻¹, Schuller et al. 2009), and R is the gas-to-dust mass ratio (assumed to be 100). We assume a dust temperature of 20 K (Stutz et al. 2010). The observed abundances of HDCO and H₂CO range from 2 × 10⁻¹¹ to 7 × 10⁻¹¹ and 8 × 10⁻¹¹ to $1\times {10}^{-9},$ respectively. The H₂CO abundances of PBRS are similar to the outer H₂CO abundances toward low-mass protostars reported by Maret et al. (2004). They have explained that the outer regions of the envelopes of Class 0 sources reflect the pre-collapse conditions based on the similarity of the values between the abundances of prestellar cores studied by Bacmann et al. (2002, 2003) and the outer H₂CO abundances of low-mass protostars.

The HDCO and H₂CO abundances relative to H₂ are plotted in Figure 7. As the H₂CO abundance increases, the HDCO abundance increases for the PBRS with the ratio of $\gt 0.1.$ The highly deuterated PBRSs ([HDCO]/[H₂CO] > 0.15) are significantly different from the other PBRSs in that they have small H₂CO abundances and their H₂CO line widths are quite narrow (<1 km s⁻¹, Figure 3). The degree of deuterium fractionation is suggested to increase with increasing CO depletion (Bacmann et al. 2003). The CO and H₂CO molecules are depleted by a similar factor in the outer envelope of Class 0 protostar (Maret et al. 2004). The low H₂CO abundance of highly deuterated PBRS is likely to be explained by the depletion of H₂CO. The narrow line widths imply that these cores are quiescent, which is associated with early phases of core evolution. Therefore, the PBRSs with high [HDCO]/[H₂CO] ratios are probably in the very earliest stage of star formation.

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Figure 8 shows the relation between [HDCO]/[H₂CO] and the mass-to-luminosity ratio that is expected to decrease with time. The PBRSs with high [HDCO]/[H₂CO] ratios have relatively large mass-to-luminosity ratios, indicating that they are likely in the earliest stage of star formation. However, two PBRSs with the largest mass-to-luminosity ratios have low [HDCO]/[H₂CO] ratios. Therefore, it is difficult to find a simple relation between the two quantities. If the mass-to-luminosity ratio is a good tracer of evolution, [HDCO]/[H₂CO] seems to have a large dispersion at early stages and converge to a typical value of Class 0 sources at later stages.

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The large variation of deuterium fractionation in the whole PBRS sample leads us to suggest that the PBRSs form in diverse conditions or have diverse formation histories. Indeed, the degree of deuterium fractionation of H₂CO is sensitive to the initial D/H ratios of gaseous molecules determined before the collapse phase (Turner 1990; Charnley et al. 1992; Roberts et al. 2002; Cazaux et al. 2011; Aikawa et al. 2012).

PBRS 097002 and 090003 are among the highly deuterated PBRSs and have flat visibility amplitude profiles in the 2.9 mm dust continuum observations by Tobin et al. (2015). They interpret that the PBRSs with flat visibility amplitude profiles are the youngest in a brief phase of high infall/accretion. Since PBRSs 093005 and 082005 have the lowest [HDCO]/[H₂CO] ratios and also have flat visibility amplitude profiles, it is unclear if the variation of deuterium fractionation in H₂CO is due to the core evolution. Additional observations in pure gas-phase species, e.g., N₂D⁺ and N₂H⁺, can help our understanding of the chemical evolutionary sequence of PBRS.