3 Data reduction

3.1 Formaldehyde

The formulae which connect the integrated intensity of a rotational transition with the number of emitting molecules are, for an optically thin transition, neglecting background radiation;

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

and, assuming the system is in LTE;

$${N}_{\rm u} = \frac{g_{\rm u}}{Q(T_{\rm ex})}{\rm e}^{-E_{\rm u}/kT_{\rm ex}}{N_{\rm TOT}}$$ (2)

(see, for example, Goldsmith & Langer 1999). Where $N_{\rm u}$ is the column density of molecules in the upper level of the transition, $N_{\rm TOT}$ is the total column density of emitting molecules and $Q(T_{\rm ex})$ is the partition function.

For a line which has significant optical depth, a correction factor can be applied to the above formulae;

$$G(\tau) = \int{\frac{1-\exp(-\tau\phi(v))}{\tau\Delta(v)}{\rm d}v}$$ (3)

(Stutzki et al. 1989), an integral which has been approximated as;

$$G(\tau) \sim \frac{1-\exp(-\tau')}{\tau'}$$ (4)

where $\tau'=0.679\,\tau^{0.911}$, by Wyrowski et al. (1999) for a Gaussian line profile, with an error of 4% for $0.01<\tau<10$ (15% for $0.01<\tau<100$).

As formaldehyde is abundant in molecular clouds, the H₂CO2_1,1-1_1,0 transition is likely to have significant optical depth. In order to estimate this we have used the ratios of $\int{T_{\rm MB}{\rm d}v}$ for the transitions of the main isotopomer and of the ¹³C substitute.

Combining Eqs. (1) and (2), applying the correction factor, Eq. (4), for the H₂CO transition only, assuming a ¹²C/¹³C abundance ratio 60, and equal excitation and filling factors for the lines, then we expect that;

$$\frac{\int T_{\rm MB}\,{\rm d}v_{12}}{\int T_{\rm MB}\,{\rm d}v_{... ...\frac{A_{\rm ul(12)}}{A_{\rm ul(13)}}\left( \frac{1-\exp(-\tau')}{\tau'}\right)$$

$$\sim 65.7\ \left( \frac{1-\exp(-\tau')}{\tau'}\right)$$ (5)

where a subscript "12'' implies H₂CO, "13'' implies H₂¹³CO. Optical depths for the H₂CO2_1,1-1_1,0 transition can then be calculated.

The optical depths for the H₂CO transitions, listed in Table 5,

 

 

Table 5: Optical depths for the H₂CO2_1,1-1_1,0 and HCN1-0 transitions, along with excitation temperatures calculated for HCN and DCN and upper limits on $T_{\rm ex}$ for H₂CO.

Source	Optical Depth		Excitation temp. (K)
	$\tau_{{\rm H}_2{\rm CO}}$	$\tau_{\rm HCN}$	H2CO	HCN	DCN
B5IRS1	<11.5	10.7	<20	5	5
L1448mms	<4.6	8.6	<31	5	5
L1448NW	4.2	8.9	<27	8	5
HH211	5.8	9.0	<15	6	4
IRAS03282	<11.5	9.0	<27	5	5
L1527	10.8	7.5	<18	4	<5
L1551IRS5	2.4	16.9	<24	5	6
RNO43	<11.5	2.2	<30	4	>6
HH111	8.2	3.9	<33	5	--

are all less than 12, therefore, as we expect the isotope-substituted species to be between 10 and 100 times less abundant than the main species, it seems reasonable to assume that our H₂¹³CO and HDCO transitions will be optically thin. We, thus, use Eqs. (1) and (2), connecting $\int T_{\rm MB}\,{\rm d}v$ and upper-level column density, $N_{\rm u}$. We have also calculated the H₂CO column densities using these equations, but applying the correction factor in Eq. (4).

Limits on the excitation temperatures of H₂CO can be calculated, using the upper limits on the integrated intensity of the H₂CO5_1,4-5_1,5 transition, via

$$\frac{{N}_i}{{N}_j} = \frac{g_i}{g_j}\ \exp\left(\frac{E_j-E_i}{kT_{\rm ex}}\right)$$ (6)

and are listed in Table 5. In calculating total column densities for H₂CO, H₂¹³CO and HDCO we have, therefore, used a range of excitation temperatures between 5 and 40 K, though, in fact, for $T_{\rm ex}\geq 10$ K the column densities are not very sensitive to the assumed excitation temperature.

H₂CO and H₂¹³CO can exist in both ortho and para forms, depending on the alignment of the spins on the two hydrogens. In this study we have only observed transitions of ortho-H₂CO, therefore we need to correct our column densities for these molecules by some assumed ortho/para ratio. The high-temperature statistical value for this ratio is 3:1, however the actual ratio can depend on the temperatures at which the molecules formed, and so may be lower in cold clouds. Kahane et al. (1984), attempted to measure the H₂CO ortho:para ratio towards TMC-1. They obtained a best fit to their data of 1:1, but with large associated errors meaning that their observations could also be fit by a higher ratio.

The ortho/para ratio can be used to provide information on the formation mechanisms of molecules. Minh et al. (1995), who observed H₂¹³CO in the quiescent cores TMC-1 and L134N, found an ortho/para ratio very close to the statistical value of 3. This suggests that these molecules formed in the gas-phase. Dickens & Irvine (1999) observed H₂CO towards star-forming cores, finding the ortho/para ratio to be between 1.5 and 2, indicating that it has been modified due to formation and/or equilibration of H₂CO on grains.

We have currently adopted the statistical ratio of 3:1, However we note that adopting a ratio of 1:1 would increase our H₂CO column densities by a factor of 3/2, and so reduce the D/H ratios by 2/3, while assuming a ratio of of 2:1, as seen towards other star-forming cores, would only increase the H₂CO column densities by a factor of 9/8.

The resulting column densities are given in Table 6.

 

 

Table 6: Column densities for H₂CO, H₂¹³CO and HDCO, calculated over a range of excitation temperatures.

Source	5 K	10 K	20 K	30 K	40 K
	N(H2CO)
	($\times$1013 cm-2)
B5IRS1	8.64	1.94	0.78	2.58	3.24
L1448mms	12.2	2.73	2.79	3.62	4.55
L1448NW	33.3	7.48	7.65	9.92	12.5
HH211	33.8	7.60	7.77	10.1	12.7
IRAS03282	10.5	2.35	2.41	3.13	3.92
L1527	20.5	4.60	4.71	6.11	7.68
L1551IRS5	16.2	3.64	3.73	4.83	6.07
RNO43	9.79	2.20	2.25	2.92	3.67
HH111	19.3	4.32	4.42	5.74	7.21
	N(H213CO)
	($\times$1012 cm-2)
B5IRS1	<2.14	<5.24	<5.61	<7.39	<9.35
L1448mms	<2.60	<0.64	<0.68	<0.90	<1.14
L1448NW	4.73	1.16	1.24	1.64	2.07
HH211	4.79	1.18	1.26	1.66	2.10
IRAS03282	<2.60	<0.64	<0.68	<0.90	<1.14
L1527	2.89	0.71	0.76	1.00	1.26
L1551IRS5	2.31	0.57	0.61	0.80	1.01
RNO43	<2.25	<0.55	<0.59	<0.78	<0.99
HH111	2.74	0.67	0.72	0.95	1.20
	N(HDCO)
	($\times$1012 cm-2)
B5IRS1	1.68	0.61	0.69	0.96	0.13
L1448mms	3.93	1.41	1.62	2.23	2.91
L1448NW	12.6	4.54	5.19	7.18	9.35
HH211	4.65	1.67	1.91	2.64	3.44
IRAS03282	2.28	0.82	0.94	1.29	1.68
L1527	8.52	3.06	3.50	4.84	6.30
L1551IRS5	6.55	2.35	2.69	3.72	4.85
RNO43	<3.37	<1.21	<1.39	<1.91	<2.49
HH111	3.43	1.23	1.41	1.95	2.54

In Table 7 we give [HDCO]/[H₂CO] ratios at $T_{\rm ex}=10$ K,

 

 

Table 7: [HDCO]/[H₂CO] for $T_{\rm ex}=10$ K.

Source	$\frac{N\rm (HDCO)}{N\rm (H_2^{13}CO) \times60}$	$\frac{N\rm (HDCO)}{N\rm (H_2CO)}$
B5IRS1	>0.019	0.066 ($\pm$0.050)
L1448mms	>0.037	0.069 ($\pm$0.027)
L1448NW	0.065 ($\pm$0.026)	0.061 ($\pm$0.013)
HH211	0.024 ($\pm$0.011)	0.022 ($\pm$0.004)
IRAS03282	>0.021	0.073 ($\pm$0.050)
L1527	0.072 ($\pm$0.021)	0.066 ($\pm$0.013)
L1551IRS5	0.069 ($\pm$0.017)	0.065 ($\pm$0.013)
RNO43	--	<0.117
HH111	0.031 ($\pm$0.021)	0.029 ($\pm$0.014)

which is consistent with the limits on $T_{\rm ex}$(H₂CO) in all sources. As the molecular D/H ratios are not very sensitive to temperature, the uncertainty in $T_{\rm ex}$ is swamped by the uncertainty arising from the noise in the spectra. The large uncertainties on the [HDCO]/[H₂CO] ratios towards B5IRS1, L1448mms and IRAS03282 are due to the fact that we only have an upper limit on the optical depth of the H₂CO 2-1 transition towards these sources.

3.2 Hydrogen cyanide

For the HCN1-0 transition we obtained $\tau$ from the "HFS'' method of CLASS. We then calculated the radiation temperature, $\Delta T_{\rm R}$, of each triplet by correcting $T_{\rm MB}$ for the main beam efficiency, $\eta_{\rm mb}$, ($\sim$0.61 at 88.6 GHz) and estimated excitation temperatures, $T_{\rm ex}$, from the equation;

$$\Delta {T_{\rm R}} = \left[ J_\nu({T_{\rm ex}}) - J_\nu({T_{\rm bg}})\right]\left(1-{\rm e}^{-\tau}\right)$$ (7)

where $T_{\rm bg}$ is the temperature of the cosmic background radiation, 2.7 K. Optical depths and excitation temperatures for HCN are listed in Table 5.

Column densities for HCN were calculated using;

$${N} = \frac{8\pi\nu_{\rm ul}^3}{c^3}\frac{Q(T_{\rm ex})}{g_{... ...rm u}/kT_{\rm ex}}}{{\rm e}^{h\nu/kT_{\rm ex}}-1}\tau_{\rm ul}$$ (8)

(Tiné et al. 2000), where $\nu_{\rm ul}$ is the frequency of the unsplit transition, $Q(T_{\rm ex})$ is the rotational partition function and $\Delta v$ is the average FWHM linewidth, all other symbols have their usual meanings.

We calculated column densities for H¹³CN and DCN assuming optically thin lines (Eqs. (1) and (2)). As we observed two transitions of DCN, excitation temperatures could be calculated via Eq. (6). Values for $T_{\rm ex}$(DCN) are listed in Table 5.

We note that effects such as beam dilution and/or self reversal in the HCN lines, may lead us to underestimate excitation temperatures for HCN. However, these temperatures are in good agreement, within the uncertainties arising from the spectral noise, with the excitation temperatures of DCN.

Column densities for HCN, H¹³CN and DCN, along with [DCN]/[HCN] ratios, are given in Table 8.

 

 

Table 8: Column densities for HCN, H¹³CN and DCN and resulting [DCN]/[HCN] ratios.

Source	N(HCN)	N(H13CN)	N(DCN)	${\frac{N\rm (DCN)}{N\rm (HCN]}}$	${\frac{N\rm (DCN)}{N\rm (H^{13}CN]\times60}}$
	($\times$1013 cm-2)	($\times$1012 cm-2)	($\times$1012 cm-3)
B5IRS1	2.85 ($\pm$0.37)	--	1.03 ($\pm$0.15)	0.036 ($\pm$0.007)	--
L1448mms	8.54 ($\pm$0.68)	1.26 ($\pm$0.24)	3.52 ($\pm$0.39)	0.041 ($\pm$0.006)	0.047 ($\pm$0.010)
L1448NW	6.77 ($\pm$0.40)	1.93 ($\pm$0.29)	4.35 ($\pm$0.44)	0.064 ($\pm$0.007)	0.038 ($\pm$0.007)
HH211	4.65 ($\pm$0.32)	0.85 ($\pm$0.15)	1.93 ($\pm$0.25)	0.042 ($\pm$0.006)	0.038 ($\pm$0.008)
IRAS03282	3.32 ($\pm$0.30)	0.54 ($\pm$0.15)	1.37 ($\pm$0.16)	0.041 ($\pm$0.006)	0.042 ($\pm$0.013)
L1527	2.51 ($\pm$0.45)	0.40 ($\pm$0.10)	0.93 ($\pm$0.12)	0.037 ($\pm$0.008)	0.039 ($\pm$0.011)
L1551IRS5	8.42 ($\pm$0.93)	0.51 ($\pm$0.10)	1.57 ($\pm$0.20)	0.019 ($\pm$0.003)	0.051 ($\pm$0.012)
RNO43	2.02 ($\pm$0.42)	<0.58	0.75 ($\pm$0.21)	0.037 ($\pm$0.013)	>0.022 ($\pm$0.006)
HH111	5.34 ($\pm$1.39)	<0.54	<0.82	<0.015	--

In calculating N(H¹³CN) we have assumed excitation temperatures equivalent to $T_{\rm ex}$(DCN).

L1448NW and L1551IRS5 are the only sources in which [DCN]/[HCN] ratios calculated from observations of HCN do not agree well with those calculated using N(H¹³CN). This is most likely due to errors in our estimation of the integrated intensity and/or optical depth of the HCN1-0 transitions due to self absorption, and so we prefer the values derived from N(H¹³CN).

3.3 Cyclopropenylidene

The 3_1,2-2_2,1 transition of c-C₃H₂ has previously been observed towards most of the sources in our survey (Buckle & Fuller 2001) and minimum column densities calculated. We have tentative detections of the 2_2,0-1_1,1 transition of the deuterated counterpart of this molecule, c-C₃HD, towards 4 sources, B5IRS1, L1448mms, L1448NW and L1527. Column densities and D/H ratios have been calculated, assuming optically thin transitions and an excitation temperature of 10 K, and are listed in Table 9. The upper limits on the [C₃HD]/[C₃H₂] ratios are consistent with observations made in L134N and TMC-1 (Bell et al. 1988), which found [C₃HD]/[C₃H $_2]\sim 0.1$.