A NEW REFERENCE CHEMICAL COMPOSITION FOR TMC-1

Kaifu et al. (2004) used the Nobeyama 45 m dish between 1984 and 1996 to observe a full spectral survey of the Cyanopolyyne peak position of TMC-1 (TMC-1(CP). For the rest of the article, we will use TMC-1 as a shorthand for TMC-1(CP)) between 8.8 and 50 GHz with a spectral resolution of 37 kHz (this corresponds to a velocity resolution ranging from 1.26 to 0.22 km s⁻¹) with a typical sensitivity of 10 $\,\mathrm{mK}$ per channel.

The observational data from Kaifu et al. (2004) consists of integrated intensities associated with a single (or several in the case of unresolved hyperfine structure) line of a given molecule. These integrated intensities are computed by fitting Gaussian line profiles to the spectral data. No uncertainty is given for these integrated intensities but the authors give a measured line width ${\rm{\Delta }}V$ and a local noise level per channel σ. From these values, we compute the noise on the integrated intensity, assuming a Gaussian profile, using the following formula:

$$\begin{eqnarray}&&{\sigma }_{I}=\sqrt{\displaystyle \frac{2\pi }{8\mathrm{ln}2}}\,\sigma {\rm{\Delta }}V\simeq 1.064\,\sigma {\rm{\Delta }}V\end{eqnarray} \tag{ 1 }$$

,where the numerical term comes from the integral of a Gaussian function of $\mathrm{FWHM}={\rm{\Delta }}V$

The data from Kaifu et al. (2004) take into account the coupling with the source (they assume TMC-1 is a Gaussian source of FWHM $160^{\prime\prime}$). Thus, the column densities and abundances we compute correspond to this spatial scale.

Under the local thermal equilibrium hypothesis, integrated intensities are dependent on a set of microphysic parameters: the Einstein coefficients of spontaneous de-excitation between levels i and j: Aij, the degeneracy of the upper level i, g_i, the energy of the upper level i, E_i, and the partition function Q(T) measuring the number of occupied states at temperature T. These values are available from online databases, mainly the CDMS (Müller et al. 2005) and the JPL Spec databases (Pickett et al. 1998). The online databases often only give values of the partition functions down to a temperature of 9.75 K, but the measured excitation temperatures in dark clouds are often lower than that. Using the energy levels E_i and the corresponding level degeneracies g_i, we compute when necessary the values of the partition function at 2.375 and 5 K using the following formula

$$\begin{eqnarray}&&Q(T)=\displaystyle \sum _{i}\,{g}_{i}{\exp }^{-\displaystyle \frac{{E}_{i}}{{kT}}}.\end{eqnarray} \tag{ 2 }$$

We systematically check that the value obtained by this sum for 9.75 K is similar to the tabulated value of the databases.

Between these tabulated values, the logarithm of the partition function is interpolated linearly in temperature. The last two columns of Table 2 give the source of the spectroscopic parameters and the lowest temperature for which the partition function values are tabulated in the databases.

We determine the column densities and abundances in the local thermodynamical equilibrium (LTE) hypothesis. For a given molecule, we compute the integrated intensities for each line, using the hypothesis of Gaussian opacity profiles as a function of frequency. The modeled opacity is a function of the species column density, excitation temperature, and line width. The integrated intensity is derived from the opacity assuming a temperature of the cosmic microwave background of 2.73 K.

Some lines detected in Kaifu et al. (2004) are attributed to several blended transitions of a given molecule. In this case, we sum the computed intensity for the identified transitions. Following Ohishi & Kaifu (1998), the column density of a species X, noted $[{\text{}}X]$, are converted into abundance assuming a constant molecular hydrogen column density of $N({{\rm{H}}}_{2})={10}^{22}\,{\mathrm{cm}}^{-2}$:

$$\begin{eqnarray}&&[X]=N(X)/N({{\rm{H}}}_{2})\end{eqnarray} \tag{ 3 }$$

We devise a Bayesian approach which enables us to retrieve the LTE model parameter distributions. The advantages of a Bayesian approach over a more traditional optimization approach are threefold.

• 1.  
  The full distribution of parameters can be recovered. This means that upper abundance limits in the case of marginal detections and lower abundance limits in the case of highly optically thick lines are readily obtained.
• 2.  
  Prior knowledge can be included for some parameters. For example, with a single detected line, the abundance can only be constrained by assuming a prior value of the excitation temperature. A Bayesian approach enables us to include a prior distribution of excitation temperatures. The derived abundance then takes into account the prior uncertainty on the knowledge of the excitation temperature.
• 3.  
  Additional parameters can be added to the model, for example, to identify data points that are outlier to the physical model. These additional parameters are marginalized out in the end to obtain a posterior distribution of the parameters of interest; in our case, the molecular abundances.

For a few of the molecules, some of the integrated intensities were clearly outliers and could not be modeled consistently with the rest of the data set. This can arise when a line has either: (1) erroneous spectroscopic parameters, (2) been misattributed to the modeled species, or (3) when a line from another species is blended with one from the modeled species. The method we have developed automatically takes into account the probability of each data point being an outlier. This enables us to recover the parameters for the modeled species such as the abundance and excitation temperature in a robust way. Although the method is applied to radiative transfer modeling of molecular lines, it is quite general and can be readily applied to a wide range of problems where outlier detection is needed.

The model parameters are divided into two groups: the first concerning the LTE modeling, and the second concerning the modeling of outliers. The LTE model parameters are the column density N, the excitation temperature T_ex, the line width ${\rm{\Delta }}V$, and a scatter term σ that enables us to describe both the possible underestimation of the observational uncertainty and the fact that the LTE model is simpler than would be needed to perfectly describe the observations. The outlier model parameters are the two parameters (mean integrated intensity Y_o and dispersion around this mean ${\sigma }_{o}$) describing the Gaussian distribution from which the outliers are supposed to be originating. Additionally, each data point is attributed a binary parameter q_i which can take a value of either 0 if the point is an outlier or 1 if the point is not. There are thus as many q_i as there are data points. A final parameter is P_o, the probability that a point is an outlier considering the binomial distribution of the q_i parameters. It possible to analytically marginalize the q_i parameters while at the same time recovering the posterior probability of a point being an outlier. The full model thus has 10 free parameters and does not become more complicated to solve as the number of data points increase.

The mathematical details of this approach, in particular the expression of the likelihood used, are presented in Appendix A.

When only a single line is detected for a given species, the identification of outliers makes no sense. In this case, the number of parameters is limited to four: N, T_ex , ${\rm{\Delta }}V$, and $\mathrm{log}\sigma$. Since with only one line the additional scatter parameter is completely unconstrained, an informative prior consisting of a Gaussian function of mean −1.2 and standard deviation 0.4 was used. This is the median and sample standard deviation of the values of $\mathrm{log}\sigma$ determined from the 26 species with more than 2 detected lines.

The Bayesian approach requires us to choose prior distributions for all model parameters. Some of them can be rather uninformative. For example, the prior distribution of the logarithm of the column density is chosen to be uniform between 9 and 17 (with the column density in units of ${\mathrm{cm}}^{-2}$). Similarly, the prior probability of the outliers P_o is chosen as uniform between 0 and 1. Conversely, we use informative priors on two specific parameters. Following previous studies (Ohishi & Kaifu 1998) that found excitation temperatures around 5 K in TMC-1, we use a Gaussian prior on ${T}_{\mathrm{ex}}$ of mean 5 K and standard deviation 2 K along with a lower temperature limit of 2.73 K corresponding to the background temperature. Similarly, we choose a Gaussian distribution of mean 0.5 km s⁻¹ and standard deviation 0.2 km s⁻¹ for the prior distribution of line widths. Table 1 summarizes these choices.

Table 1.  Prior Parameter Distributions

Parameter	Distribution	Expression	Lower Bound	Upper Bound
$\mathrm{log}N$	Uniform	$p(\mathrm{log}N)$ = ${[{(\mathrm{log}N)}_{\max }-{(\mathrm{log}N)}_{\min }]}^{-1}$	9	17
${T}_{\mathrm{ex}}$	Gaussian	$p({T}_{\mathrm{ex}})$ =  ${ \mathcal N }(5,2)$	2.73	20
${\rm{\Delta }}V$	Gaussian	$p({\rm{\Delta }}V)$ = ${ \mathcal N }(0.5,0.2)$	0	3
$\mathrm{log}\sigma$	Uniform	$p(\mathrm{log}\sigma )$  =  ${[{(\mathrm{log}\sigma )}_{\max }-{(\mathrm{log}\sigma )}_{\min }]}^{-1}$	−3	1
$\mathrm{log}\sigma$	Gaussiana	$p(\mathrm{log}\sigma )$  =  ${ \mathcal N }(-1.2,0.4)$	−3	1
Yo	Uniform	$p({Y}_{o})$ = ${[{({Y}_{o})}_{\max }-{({Y}_{o})}_{\min }]}^{-1}$	0	10
$\mathrm{log}{\sigma }_{o}$	Uniform	$p(\mathrm{log}{\sigma }_{o})$ = ${[{(\mathrm{log}{\sigma }_{o})}_{\max }-{(\mathrm{log}{\sigma }_{o})}_{\min }]}^{-1}$	−3	2
$\{{q}_{i}\}$	Binomial	$p(\{{q}_{i}\})$ = ${\prod }_{i=1}^{N}{[1-{P}_{o}]}^{{q}_{i}}{P}_{o}^{[1-{q}_{i}]}$
Po	Uniform	$p({P}_{o})$ = 1	0	1

Note.

^aThe Gaussian prior is used when only one line of a given molecule is detected.

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Figure 1 gives a graphical representation of the computed abundances for the main isotope-bearing molecules.

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Three main classes of outcomes can be identified. For some molecules, all observed lines have high opacities, and in these cases only lower limits can be placed on their abundances. This is the case for $\mathrm{CS}$ and $\mathrm{SO}$. For other molecules, the signal-to-noise ratio is not sufficient enough to obtain more than an upper limit on the abundances. This is the case for ${{\rm{C}}}_{2}{\rm{O}}$ and $\mathrm{OCS}$. For all other molecules, it is possible to determine an abundance along with an associated uncertainty.

Twenty isotopologues are also detected in the Nobeyama survey of TMC-1. We have computed their abundances relative to ${{\rm{H}}}_{2}$ and the ratio of the minor isotopologues to the main isotopologues for each molecule. The results are summarized in Figure 2 and Table 3.

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Table 3.  Best-fit Parameters by the Bayesian Approach for Rarer Isotope-bearing Species

Species	$\mathrm{log}N$ a	${T}_{\mathrm{ex}}$	PSNR	τ Range	Number of	Min Partition	Appendix
	(cm−2)	(K)			Transitions	Function Temp	Figure
${}^{13}{\mathrm{CCH}}_{2}$	$\lt 13.01$	${5.31}_{-1.49}^{+1.69}$	0.9	$\lt 0.04$	2	9.375	3.7
c-${{\rm{C}}}_{3}\mathrm{HD}$	${12.71}_{-2.03}^{+2.19}$	${5.23}_{-1.91}^{+2.13}$	2.0	$0.11\mbox{--}0.28$	4	9.375	3.8
${}^{13}\mathrm{CS}$	${12.26}_{-1.78}^{+0.27}$	${5.30}_{-1.66}^{+1.99}$	2.3	$0.00\mbox{--}0.51$	1	9.375	3.17
${{\rm{C}}}^{33}{\rm{S}}$	$\lt 12.42$	${5.22}_{-1.40}^{+1.52}$	1.2	$\lt 0.02$	1	9.375	3.19
${{\rm{C}}}^{34}{\rm{S}}$	${12.70}_{-0.14}^{+1.55}$	${4.94}_{-1.61}^{+1.91}$	6.9	$0.40\mbox{--}546.68$	1	9.375	3.20
$\mathrm{HDCS}$	$\lt 13.80$	${5.21}_{-1.57}^{+2.00}$	1.5	$\lt 0.41$	1	9.375	3.22
${}^{34}\mathrm{SO}$	$\lt 13.04$	${5.35}_{-1.55}^{+1.93}$	1.1	$\lt 0.08$	1	9.375	3.26
${\mathrm{DC}}_{3}{\rm{N}}$	${12.47}_{-0.24}^{+0.44}$	${5.53}_{-1.66}^{+1.90}$	4.3	$0.01\mbox{--}0.28$	5	9.375	3.35
${{\rm{H}}}^{13}{\mathrm{CC}}_{2}{\rm{N}}$	${12.25}_{-0.03}^{+0.03}$	${6.54}_{-1.07}^{+1.32}$	14.6	$0.01\mbox{--}0.15$	6	9.375	3.36
${\mathrm{HC}}^{13}\mathrm{CCN}$	${12.26}_{-0.02}^{+0.02}$	${5.38}_{-0.64}^{+0.57}$	20.6	$0.02\mbox{--}0.20$	6	9.375	3.32
${\mathrm{HC}}_{2}^{13}\mathrm{CN}$	${12.47}_{-0.03}^{+0.03}$	${5.90}_{-0.93}^{+1.29}$	11.5	$0.03\mbox{--}0.21$	6	9.375	3.34
${\mathrm{DC}}_{5}{\rm{N}}$	${12.05}_{-0.13}^{+0.16}$	${5.80}_{-0.85}^{+0.96}$	6.5	$0.02\mbox{--}0.04$	6	9.375	3.50
${{\rm{H}}}^{13}{\mathrm{CC}}_{4}{\rm{N}}$	${12.03}_{-0.23}^{+0.34}$	${6.31}_{-1.55}^{+1.66}$	3.8	$0.02\mbox{--}0.03$	3	9.375	3.51
${\mathrm{HC}}^{13}{\mathrm{CC}}_{3}{\rm{N}}$	${12.06}_{-0.20}^{+0.25}$	${4.03}_{-0.58}^{+0.99}$	6.9	$0.03\mbox{--}0.08$	4	9.375	3.53
${\mathrm{HC}}_{2}^{13}{\mathrm{CC}}_{2}{\rm{N}}$	${11.92}_{-0.11}^{+0.15}$	${5.49}_{-1.05}^{+1.09}$	4.8	$0.02\mbox{--}0.04$	6	9.375	3.52
${\mathrm{HC}}_{3}^{13}\mathrm{CCN}$	${12.25}_{-0.12}^{+0.14}$	${4.38}_{-0.45}^{+0.49}$	6.7	$0.03\mbox{--}0.12$	6	9.375	3.54
${\mathrm{HC}}_{4}^{13}\mathrm{CN}$	${12.01}_{-0.36}^{+0.23}$	${5.61}_{-1.33}^{+1.56}$	2.7	$0.03\mbox{--}0.04$	5	9.375	3.49

Note.

^aThe abundance [X] can be derived using $N(X)/N({{\rm{H}}}_{2})$ with $N({{\rm{H}}}_{2})={10}^{22}\,{\mathrm{cm}}^{-2}$ Spectroscopic data are from the CDMS database.

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Using the Bayesian approach, we have demonstrated that it is possible to identify outliers in the data set: observed line intensities that are not explainable by the global model best fitting the rest of the lines for a given species. Table 4 summarizes the properties of the outlier lines.

Table 4.  List of Lines that Have an Outlier Probability Larger than 0.75.

Species	Transition	Observed	Outlier	Modeled	Observed
		Frequency	Probability	Intensity	Intensity
		($\,\mathrm{MHz}$)		(K km s−1)	(K km s−1)
${\mathrm{HC}}_{3}{\rm{N}}$	$J^{\prime} -J^{\prime\prime} ,F^{\prime} -F^{\prime\prime}$ = 1–0, 2–1	9098.329	0.90	1.3 ± 0.1	2.638 ± 0.020
${\mathrm{HC}}_{3}{\rm{N}}$	$J^{\prime} -J^{\prime\prime} ,F^{\prime} -F^{\prime\prime}$ = 5–4, 6–5 and 5–4 and 4–3	45490.332	0.93	7.2 ± 0.4	5.239 ± 0.020
${\mathrm{HC}}_{7}{\rm{N}}$	$J^{\prime} -J^{\prime\prime}$ = 11–10	12407.995	0.999	0.56 ± 0.25	0.228 ± 0.009
${\mathrm{HC}}_{7}{\rm{N}}$	$J^{\prime} -J^{\prime\prime}$ = 12–11	13535.999	0.999	0.58 ± 0.24	0.212 ± 0.011
${{\rm{C}}}_{4}{\rm{H}}$	$N^{\prime} -N^{\prime\prime} J^{\prime} -J^{\prime\prime} F^{\prime} -F^{\prime\prime}$ = 4–3, 9/2–7/2, 4–3	38049.629	0.94	0.58 ± 0.24	0.212 ± 0.011

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Two molecules stand out in the outlier detection scheme, ${{\rm{H}}}_{2}\mathrm{CCN}$ and ${{\rm{C}}}_{3}{\rm{N}}$. In both cases, about half of the observed intensities are not compatible with the best-fit model (see Figure Set 3). With so many outlier lines, the problem cannot arise solely from observational effects (possible blended or misattributed lines). The origin of the issue is possibly the values of the spectroscopic catalogs such as erroneous normalized intensities or level degeneracies (the spectroscopy data for ${{\rm{H}}}_{2}\mathrm{CCN}$ is only available from CDMS, while the same data set is present and identical in both the CDMS and JPL catalogs for ${{\rm{C}}}_{3}{\rm{N}}$).

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In these two cases, the outlier detection scheme is not used. The simpler model with just the three LTE parameters (N, T_ex and ${\rm{\Delta }}V$) and a scatter term is used as described in Section 3.2 for the case when only a single line of a molecule is detected. The sampling method adjusts by increasing the scatter error added in quadrature to the observational uncertainty to encompass all observed points at the expense of a larger uncertainty on the other parameters, in particular the column density and thus the abundance.

The derived column densities and abundances for these molecules were published in Ohishi et al. (1992) and Ohishi & Kaifu (1998), which have been used since as references for abundances in TMC-1 and by extension for dark clouds in general. In Ohishi & Kaifu (1998), the derived abundances have no associated uncertainty estimates and the authors, while discussing the advantage of having detected lines for some species spanning a wide range of opacities, do not discuss the possible effect of high opacity or low signal-to-noise detection on the molecular abundances, especially for species where the number of detected lines is low (some species have only one detected line).

Ohishi & Kaifu (1998) have used the data later published in Kaifu et al. (2004) to compute the abundances of observed species. Since the local thermal equilibrium approach is used in both our study and theirs, the main difference should arise from the spectrometric data used and from the fact that our Bayesian approach enables us to compute uncertainties on the abundances and even upper or lower limits when the signal-to-noise ratio is too low or the opacity is too large for all detected lines of a given species. In Figure 1, the values from Ohishi & Kaifu (1998) are shown as large white circles. Agúndez & Wakelam (2013) made a review of the literature to identify molecular abundances in TMC-1. The results they compiled appear as a small circle in Figure 1. Most of the previously determined abundances fall within the 95% (2σ) confidence interval of our Bayesian method. We now discuss notable differences, i.e., species where the old values fall outside our 95% confidence interval. Most of the abundance determination in Ohishi & Kaifu (1998) were carried out without taking into account the hyperfine splitting of the observations. This can lead to biased results when opacities vary across the hyperfine components.

5.2.1. Polyynes

${{\rm{C}}}_{6}{\rm{H}}$ has a similar abundance as the one derived by Ohishi & Kaifu (1998). ${{\rm{C}}}_{5}{\rm{H}}$ has a new abundance that is lower by about an order of magnitude from previous values. The value we compute for linear ${{\rm{C}}}_{3}{\rm{H}}$ falls between the values of Ohishi & Kaifu (1998) and Agúndez & Wakelam (2013). The value reported for this species in Agúndez & Wakelam (2013) comes from IRAM 30 m millimeter observations (Fossé et al. 2001). The difference of a factor of two in beam size between the observations could explain the abundance difference. Cyclic ${{\rm{C}}}_{3}{\rm{H}}$ is found to be about three times more abundant.

The case of ${{\rm{C}}}_{4}{\rm{H}}$ is particular because of the electronic structure of this radical. Previous studies (Woon 1995; Mazzotti et al. 2011) have established that ${{\rm{C}}}_{4}{\rm{H}}$ presents two low-lying electronic states of ${}^{2}{{\rm{\Sigma }}}^{+}$ and ${}^{2}{\rm{\Pi }}$ symmetries. The experimental spectra of ${{\rm{C}}}_{4}{\rm{H}}$ are consistent with the ground state ${}^{2}{{\rm{\Sigma }}}^{+}$  (Dismuke et al. 1975; Gottlieb et al. 1983; Shen et al. 1990). Recently, Senent & Hochlaf (2010) confirmed that ${{\rm{C}}}_{4}{\rm{H}}$ radical presents two electronic states, ${}^{2}{{\rm{\Sigma }}}^{+}$  and ${}^{2}{\rm{\Pi }}$, which are near degenerate. Their computation confirms that the ground electronic state of ${{\rm{C}}}_{4}{\rm{H}}$ radical is ${}^{2}{{\rm{\Sigma }}}^{+}$ , which agrees with the findings of electron paramagnetic resonance by Dismuke et al. (1975). Their computation also confirms the short gap between the two low-lying electronic states. They have found that they are around 9 cm⁻¹ and 2800 cm⁻¹ at the (R)CCSD(T) and CASSCF levels of quantum chemical theory. The dipole moment is specific to each electronic state but since the two electronic states are very close to each other and the molecule is not static, the motion will cause electronic states to mix. However, since the concept of an average dipole moment for a mixed state is not well defined, this issue needs more studies from both theoretical and experimental points of view.

The CDMS database gives a dipole value of 0.870 $\,\mathrm{Debye}$,which corresponds to the electronic ground state of the molecule. The computed dipole value for the ${}^{2}{\rm{\Pi }}$ electronic state is 4.3 $\,\mathrm{Debye}$ (Senent & Hochlaf 2010). Nevertheless, in Table 2 and Figure 1 we show the full range of abundances permitted by any value of the dipole moment between the values computed for the ${}^{2}{{\rm{\Sigma }}}^{+}$  and ${}^{2}{\rm{\Pi }}$  states.

In the optically thin regime, the column density scales as the square of the dipole. Thus, once the correct dipole moment of ${{\rm{C}}}_{4}{\rm{H}}$ has been determined, an updated value of the abundance can be approximated using the following expression:

$$\begin{eqnarray}&&N{({{\rm{C}}}_{4}{\rm{H}})}_{D}={\left(\displaystyle \frac{D}{4.3}\right)}^{2}N{({{\rm{C}}}_{4}{\rm{H}})}_{4.3}\end{eqnarray} \tag{ 4 }$$

where $N{({{\rm{C}}}_{4}{\rm{H}})}_{D}$ is the column density considering a dipole moment of value D and $N{({{\rm{C}}}_{4}{\rm{H}})}_{4.3}=3\,\times \,{10}^{13}\,{\mathrm{cm}}^{-2}$ the column density considering a dipole moment of value 4.3 Debye.

Of the detected molecules which posses distinct ortho and para states, only ${{\rm{C}}}_{4}{{\rm{H}}}_{2}$ has a number of detected lines (12) that is large enough that the abundances of the ortho and para forms can be determined independently. The median ortho-to-para abundance ratio is found to be around 6, but this ratio is highly uncertain with a one sigma confidence interval ranging from 1.7 to 70. The two ortho and para abundances bracket the abundance previously determined by Ohishi & Kaifu (1998).

5.2.2. Cyanopolyynes

5.2.3. Other Molecules

5.3.1. Sulfur

5.3.2. Carbon

5.3.3. Hydrogen