The ¹²C/¹³C Ratio in Sgr B2(N): Constraints for Galactic Chemical Evolution and Isotopic Chemistry

The five molecules chosen for this investigation, H₂CS, CH₃CCH, NH₂CHO, CH₂CHCN, and CH₃CH₂CN, contain carbon in differing functional groups. Table 1 summarizes the carbon isotopologues studied, the number of transitions detected (22–54 lines for the ¹²C species and 2–54 for ¹³C-substituted), the energy range of the lower state levels, typical line widths and LSR velocities, and the average rotational temperature and column density. As the table shows, each molecule exhibits two distinct velocity components near 62–64 (hot core Sgr B2(N1)) and 73 km s⁻¹ (hot core Sgr B2(N2)) with typical line widths of 9–14 km s⁻¹. Additional details will be given in a later paper on the entire survey.

A representative sample of the observational data is given in Table 2, which lists the complete data set for H₂CS and ${{{\rm{H}}}_{2}}^{13}\mathrm{CS}$. For H₂CS, 33 lines were observed that were uncontaminated; these transitions are shown in the table. The rest frequency, the upper state energy (E_u), and the product of the square of the dipole moment μ with the line strength (μ²S) of the transition are also given, along with the beam efficiency (η_c at 2 and 3 mm and η_b at 1 mm) and telescope beam size. The line temperature (${{T}_{R}}^{* }$ at 2 and 3 mm and ${{T}_{A}}^{* }$ at 1 mm), line width ΔV_1/2, and source velocity V_LSR are also given for each velocity component. At 3 mm in wavelength, these components are usually blended, producing a line with a width of ∼20 km s⁻¹. At 2 and 1 mm, the two components usually generate a distinct profile with a prominent feature at 64 km s⁻¹ and a weaker "shoulder" near 73 km s⁻¹.

Table 2.  Representative Data: Observations of H₂CS and ${{{\rm{H}}}_{2}}^{13}\mathrm{CS}$ in Sgr B2(N)

Species	Transition $J{^{\prime} }_{{K}_{a},{K}_{c}}\to J{^{\prime\prime} }_{{K}_{a},{K}_{c}}$	Frequency (MHz)	${E}_{u}$(K)	μ2S(D2)	ηc/ηb	θb('')	VLSR (km s−1)	${{T}_{R}}^{* }/{{T}_{A}}^{* }$(K)	${\rm{\Delta }}{V}_{1/2}$ (km s−1)
H2CS	${2}_{\mathrm{0,2}}\to {1}_{\mathrm{0,1}}$	68699.38	4.95	5.43	0.94	92	63.7	0.375	14.0
							72.4	0.357	14.0
	${2}_{\mathrm{1,1}}\to {1}_{\mathrm{1,0}}$	69746.75	18.20	12.21	0.94	90	65.2	0.555	14.0
							73.8	0.523	14.0
	${3}_{\mathrm{1,3}}\to {2}_{\mathrm{1,2}}$	101477.81	22.92	21.70	0.88	62	64.4	1.030	14.0
							73.3	0.880	14.0
	${3}_{\mathrm{2,2}}\to {2}_{\mathrm{2,1}}$	103039.91a	62.59	4.52	0.87	61	62.5	0.691	14.0
	${3}_{\mathrm{0,3}}\to {2}_{\mathrm{0,2}}$	103040.45a	9.90	8.14	0.87	61	74.2	0.480	14.0
	${3}_{\mathrm{2,1}}\to {2}_{\mathrm{2,0}}$	103051.85	62.60	4.52	0.87	61	64.5	0.210	14.0
							73.2	0.145	14.0
	${3}_{\mathrm{1,2}}\to {2}_{\mathrm{1,1}}$	104617.04	23.23	21.70	0.87	60	62.1	0.962	14.0
							73.6	0.714	14.0
	${4}_{\mathrm{1,4}}\to {3}_{\mathrm{1,3}}$	135298.26	29.42	30.52	0.79	46	62.6	1.306	14.0
							73.7	0.760	14.0
	${4}_{\mathrm{3,2}}\to {3}_{\mathrm{3,1}}$	137369.49a	135.01	14.24	0.79	46	62.1	1.108	14.0
	${4}_{\mathrm{3,1}}\to {3}_{\mathrm{3,0}}$	137369.49a	135.01	14.24	0.79	46	73.0	0.647	14.0
	${4}_{\mathrm{0,4}}\to {3}_{\mathrm{0,3}}$	137371.21a	16.49	10.85	0.79	46
	${4}_{\mathrm{2,3}}\to {3}_{\mathrm{2,2}}$	137382.12	69.19	8.14	0.79	46	64.4	0.403	14.0
							73.2	0.346	14.0
	${4}_{\mathrm{2,2}}\to {3}_{\mathrm{2,1}}$	137412.01	69.19	8.14	0.79	46	64.2	0.310	14.0
							72.9	0.235	14.0
	${4}_{\mathrm{1,3}}\to {3}_{\mathrm{1,2}}$	139483.68	29.92	30.52	0.78	45	63.5	1.433	14.0
							72.1	0.979	14.0
	${5}_{\mathrm{1,4}}\to {4}_{\mathrm{1,4}}$	169114.08	37.54	39.06	0.70	37	63.9	1.712	14.0
							72.8	0.981	14.0
	${5}_{\mathrm{4,2}}\to {4}_{\mathrm{4,1}}$	171670.98a	235.31	4.88	0.69	37	62.0	0.141	14.0
	${5}_{\mathrm{4,1}}\to {4}_{\mathrm{4,0}}$	171670.98a	235.31	4.88	0.69	37	72.4	0.050	14.0
	${5}_{\mathrm{0,5}}\to {4}_{\mathrm{0,4}}$	171688.12	24.74	13.56	0.69	37	64.0	0.927	14.0
							72.7	0.569	14.0
	${5}_{\mathrm{2,3}}\to {4}_{\mathrm{2,2}}$	171780.17	77.44	11.39	0.69	37	64.0	0.483	14.0
							72.8	0.263	14.0
	${7}_{\mathrm{1,7}}\to {6}_{\mathrm{1,6}}$	236727.02	58.66	55.81	0.75	32	64.6	1.909	14.0
							73.4	1.106	14.0
	${7}_{\mathrm{5,3}}\to {6}_{\mathrm{5,2}}$	240261.99a	374.93	27.92	0.75	31	61.5	1.220	14.0
	${7}_{\mathrm{5,2}}\to {6}_{\mathrm{5,1}}$	240261.99a	374.93	27.92	0.75	31	72.8	0.420	14.0
	${7}_{\mathrm{0,7}}\to {6}_{\mathrm{0,6}}$	240266.87a	46.17	18.99	0.75	31
	${7}_{\mathrm{2,6}}\to {6}_{\mathrm{2,5}}$	240382.05a	98.88	17.44	0.75	31	64.5	1.355	14.0
	${7}_{\mathrm{3,5}}\to {6}_{\mathrm{3,4}}$	240393.04a	164.70	46.51	0.75	31	73.3	1.024	14.0
	${7}_{\mathrm{3,4}}\to {6}_{\mathrm{3,3}}$	240393.76a	164.70	46.51	0.75	31
	${7}_{\mathrm{2,5}}\to {6}_{\mathrm{2,4}}$	240549.07	98.90	17.44	0.75	31	65.8	0.644	14.0
							73.3	0.456	14.0
	${7}_{\mathrm{1,6}}\to {6}_{\mathrm{1,5}}$	244048.50	60.06	55.80	0.75	31	63.8	2.013	14.0
							73.7	0.925	14.0
	${8}_{\mathrm{1,8}}\to {7}_{\mathrm{1,7}}$	270521.93	71.65	64.09	0.75	28	64.1	1.904	14.0
							73.0	0.986	14.0
	${8}_{\mathrm{2,7}}\to {7}_{\mathrm{2,6}}$	274703.35	112.07	20.35	0.75	27	64.6	0.481	14.0
							73.3	0.290	14.0
	${8}_{\mathrm{3,6}}\to {7}_{\mathrm{3,5}}$	274732.98a	177.89	55.96	0.75	27	63.8	1.080	14.0
	${8}_{\mathrm{3,5}}\to {7}_{\mathrm{3,4}}$	274734.40a	177.89	55.96	0.75	27	73.7	0.459	14.0
	${8}_{\mathrm{2,6}}\to {7}_{\mathrm{2,5}}$	274953.74	112.10	20.35	0.75	27	63.9	0.769	14.0
							72.6	0.558	14.0
${{{\rm{H}}}_{2}}^{13}\mathrm{CS}$	${3}_{\mathrm{1,3}}\to {2}_{\mathrm{1,2}}$	97632.20	22.55	21.72	0.89	64	62.6	0.055	14.0
							71.8	0.033	14.0
	${5}_{\mathrm{1,5}}\to {4}_{\mathrm{1,4}}$	162706.57	36.62	39.10	0.71	39	63.1	0.061	14.0
							72.3	0.028	14.0

Notes. Source coordinates (B1950): α = 17^h44^m095, δ=−28°21'20''. Spectra were observed with the Millimeter Auto Correlator (MAC) and resampled at 1 MHz resolution using a standard cubic spline algorithm. Line parameters were derived from a least-squares fit of a Gaussian profile to the spectrum.

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Representative spectra from Sgr B2(N) for each isotopologue of the five respective molecules, both the ¹²C and ¹³C species, are shown in Figures 1(a) and (b). Here the black trace is the observed data, while the red line is the current model fit (see below). The features of interest are indicated with arrows. The top three panels in Figure 1(a) show one line of H₂CS and two for ${{{\rm{H}}}_{2}}^{13}\mathrm{CS}$ near 104.6, 97.6, and 162.7 GHz, respectively, while the next two graphs present data for NH₂CHO and ${{\mathrm{NH}}_{2}}^{13}\mathrm{CHO}$ near 140.6 and 169.0 GHz. The lower three panels in Figure 1(a) show various K components (K = 0–3 or 4) of transitions of CH₃CCH (102.5 GHz), ¹³CH₃CCH (83.1 GHz), and CH₃C¹³CH (99.4 GHz). The K = 0, 1, and 2 components are typically blended together into a single asymmetric feature for these species. Unfortunately, all lines for ${{\mathrm{CH}}_{3}}^{13}\mathrm{CCH}$ were contaminated either by those of the main isotopologue or by other molecules; accurate spectral parameters could not be determined for this species.

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The ${{{\rm{H}}}_{2}}^{13}\mathrm{CS}$ and ${{\mathrm{NH}}_{2}}^{13}\mathrm{CHO}$ spectra (second and fifth panels, Figure 1(a)) illustrate the evolution of the line shape as a function of frequency. The two velocity components are collapsed into one single feature in the 97.6 GHz line. For the formamide-¹³C transition at 169.0 GHz, a shoulder at higher velocity (∼73 km s⁻¹) can be seen on the "main" (64 km s⁻¹) line.

In Figure 1(b), the top four panels show representative spectra for the ¹²C and singly substituted ¹³C isotopologues of vinyl cyanide (CH₂CHCN: 94.9 GHz; ¹³CH₂CHCN: 110.9 GHz; ${{\mathrm{CH}}_{2}}^{13}\mathrm{CHCN}$:103.8 GHz; and CH₂CH¹³CN: 94.5 GHz). In the lower four panels, the ¹²C and singly substituted ¹³C species of ethyl cyanide (CH₃CH₂CN: 93.8 GHz; ¹³CH₃CH₂CN: 96.1 GHz; ${{\mathrm{CH}}_{3}}^{13}{\mathrm{CH}}_{2}\mathrm{CN}$: 98.1 GHz; and ${\mathrm{CH}}_{3}{{\mathrm{CH}}_{2}}^{13}\mathrm{CN}$: 112.1 GHz) are displayed. In each spectrum, typical multiple (a-type) transitions of the respective isotopologue are visible, as indicated again by arrows. Some transitions $J+1,{K}_{a}\to J,{K}_{a}$ lie at the same frequency, differentiated only by changes in the ${K}_{c}$ quantum number. In these cases, ${K}_{c}$ is substituted by an asterisk in the figure.

The data sets for the five molecules across the 1, 2, and 3 mm bands were analyzed in a global fit using a radiative code. In this algorithm, observed spectra are simulated by varying the values of the line width and LSR velocity, as well as the rotational temperature, total column density, and source size. The "best fit" is established by a nonlinear least-squares evaluation. For some of the weaker isotopologues, certain free parameters had to be fixed to a given value, as described in the next section. The line shapes were assumed to be Gaussian. Rest frequencies, level energies, line strengths (μ²S), and partition functions were obtained from spectral-line databases.

In the analysis, a global fit was first carried out of the entire Sgr B2(N) data set, assuming the source fills the telescope beam. This fitting enabled the identification of obvious blended lines for a given species, which were then removed from the analysis. All uncontaminated transitions for a given molecule and its ¹³C isotopologues were then analyzed separately. For two species, H₂CS and CH₃CCH, the assumptions of low opacity and extended emission were excellent approximations, and the observed lines were well modeled. The extended emission of these two molecules is apparent in the 5' × 5' maps, centered on Sgr B2(N), made by the Mopra telescope (Jones et al. 2008). For the other three molecules (NH₂CHO, CH₂CHCN, and CH₃CH₂CN), many transitions in the main isotopologue were optically thick, and the emission appeared more confined with respect to the telescope beam. Because the source size and line opacities are very highly correlated, it was difficult to fit the optically thick transitions with sufficient confidence to determine accurate ¹²C/¹³C ratios.

For NH₂CHO and CH₃CH₂CN, this situation was remedied by fitting the numerous optically thin b-type lines of the main isotopic species, and adjusting the source size. Unfortunately, CH₂CHCN exhibits a small number of such transitions, which also have weak intensities. Consequently, only the ¹³C species were confidently modeled. Note that the line profiles of all five species were successfully fit with line widths between 9 and 14 km s⁻¹ and two LSR components near 64 and 73 km s⁻¹. To arrive at an estimate of the source sizes for NH₂CHO, CH₃CH₂CN, and CH₂CHCN, the line intensities were first compared to lines measured with the IRAM 30 m telescope. This value was used as an initial guess and then varied systematically until the best size was established by the least-squares analysis. Once determined, the source size was held fixed in the final fit, which established the rotational temperature and column density. This approach was necessary because the source size, rotational temperature, and column density are highly correlated (96%–99%) in the analysis. One parameter had to be held constant in order for the others to have significant values.

The results of this analysis are given in Table 1 and the simulated spectra for Sgr B2(N) are shown in red in Figure 1, as mentioned. Details for each molecule are outlined in the next section.

3.3.1. Thioformaldehyde (H₂CS and ${{H}_{2}}^{{13}}{CS}$)

The CDMS entries (046509 ver. 2 and 047505 ver. 1) were used to evaluate both the H₂CS and ${{{\rm{H}}}_{2}}^{13}\mathrm{CS}$ spectra (Endres et al. 2016). The database entries are based on the spectroscopy work of Maeda et al. (2008). Thirty-three lines of H₂CS were modeled, assuming the source fills the main beam and fixing line widths to ΔV_1/2 = 14.0 km s⁻¹. The velocity of one component was varied, yielding V_LSR = 63.4 km s⁻¹, while the other was fixed to V_LSR = 73.0 km s⁻¹. The rotational temperature of the 63.4 km s⁻¹ component was successfully modeled with T_rot = 71 K, while the other was fixed to T_rot = 75 K. The corresponding total column densities of the two velocity components were determined to be N_tot ∼ 6.9 × 10¹⁴ and 2.8 × 10¹⁴ cm⁻² (see Table 1).

Two distinct lines of ${{{\rm{H}}}_{2}}^{13}\mathrm{CS}$ were detected near 97.6 and 162.7 GHz. Consequently, the spectra were modeled with the line parameters fixed to those found for the ¹²C isotopologue (ΔV_1/2 = 14.0 km s⁻¹ and V_LSR = 64.0 and 73.0 km s⁻¹). The 64 km s⁻¹ component was found to have T_rot = 88 K and N_tot ∼ 3.7 × 10¹³ cm⁻², while the 73.0 km s⁻¹ feature exhibited T_rot = 86 K and N_tot ∼ 1.9 × 10¹³ cm⁻². The rotational temperatures are in reasonable agreement with those of the ¹²C species. Comparing column densities, ¹²C/¹³C ∼ 19 ± 6 and 15 ± 5 for the 64.0 and 73.0 km s⁻¹ components, respectively (see Table 3).

Table 3.  Summary of Derived ¹²C/¹³C Ratios

12C Species	13C Species	VLSR (km s−1)	12C/13C
H2CS	${{{\rm{H}}}_{2}}^{13}\mathrm{CS}$	64.0	19 ± 6
		73.0	15 ± 5
CH3CCH	13CH3CCH	64.0	24 ± 7
		73.0	18 ± 5
CH3CCH	CH3C13CH	64.0	19 ± 5
		73.0	22 ± 6
NH2CHO	${{\mathrm{NH}}_{2}}^{13}\mathrm{CHO}$	64.0	25 ± 8
		73.0	31 ± 8
CH3CH2CN	13CH3CH2CN	62.1	21 ± 9
		73.0	20 ± 5
CH3CH2CN	CH3${}^{13}$CH2CN	62.7	33 ± 13
		73.0	24 ± 6
CH3CH2CN	${\mathrm{CH}}_{3}{{\mathrm{CH}}_{2}}^{13}\mathrm{CN}$	62.1	29 ± 12
		73.0	28 ± 8
Average		64.0	24 ± 9
		73.0	23 ± 6

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In their survey of Sgr B2, Belloche et al. (2013) observed lines of H₂CS, but did not have a definitive detection of ${{{\rm{H}}}_{2}}^{13}\mathrm{CS}$. They modeled their main isotope data at 3 mm with two very similar velocity components with 8 km s⁻¹ line widths, and a third, less certain feature near 54 km s⁻¹. We do not see this lower velocity gas in our data. These authors fit their spectra with both hot and cold components and with a source size of a few arcseconds, but commented that their resulting line parameters were not well constrained. Our data were well modeled with an extended source and a single temperature component, which may simply reflect our larger beam size.

3.3.2. Methylacetylene (CH₃CCH, ¹³CH₃CCH, and CH₃C¹³CH)

3.3.3. Formamide (NH₂CHO and ${{{NH}}_{2}}^{{13}}{CHO}$)

3.3.4. Vinyl Cyanide (CH₂CHCN, ¹³CH₂CHCN, ${{{CH}}_{2}}^{{13}}{CHCN}$, and CH₂CH¹³CN)

3.3.5. Ethyl Cyanide (CH₃CH₂CN, ¹³CH₃CH₂CN, ${{{CH}}_{3}}^{{13}}{{CH}}_{2}{CN}$, and ${{CH}}_{3}{{{CH}}_{2}}^{{13}}{CN}$)

Based on 4 different molecules, 7 varying isotopologue combinations, and 2 separate gas components, 14 individual estimates of the ¹²C/¹³C ratio in the Galactic Center have been obtained. The average carbon isotope ratio based on these data was found to be ¹²C/¹³C ∼ 24 ± 9 for the 64 km s⁻¹ component and 23 ± 6 for the 73 km s⁻¹ feature, see Table 3. The values for the two components are identical within the uncertainties, and there is no quantitative deviation with molecule choice. Previous estimates of the gradient in the Sgr B2 region were based on ratios obtained from CO, H₂CO, and CN (¹²C/¹³C ∼ 24 ± 1, 10, >18; see Milam et al. 2005). These new measurements, as well as others described in the Introduction, provide additional constraints for the ratio in the Galactic center.

With these new values, coupled with the CO and CN data from Milam et al. (2005), a revised Galactic gradient can be computed. (The value of ¹²C/¹³C = 10 for H₂CO in Sgr B2(OH) was excluded since it is most likely optically thick). A least-squares fit to this combined data set yields

$$\begin{eqnarray}&&{}^{12}{\rm{C}}{/}^{13}{\rm{C}}=5.21(0.52){D}_{\mathrm{GC}}+22.6(3.3).\end{eqnarray} \tag{ 1 }$$

In comparison, Milam et al. (2005) calculated ¹²C/¹³C = 6.21(1.00) D_GC + 18.71(7.37). The new gradient therefore has a slightly shallower slope, but higher precision. A graph of the ¹²C/¹³C ratio as a function of distance from the Galactic center D_gc is shown in Figure 2. The newly measured ¹²C/¹³C ratios also reinforce the existence of the isotope gradient itself.

This work determined column densities of various ¹³C isotopomers of methyl acetylene (¹³CH₃CCH, and CH₃C¹³CH), vinyl cyanide (¹³CH₂CHCN, ${{\mathrm{CH}}_{2}}^{13}\mathrm{CHCN}$, and CH₂CH¹³CN), and ethyl cyanide (¹³CH₃CH₂CN, ${{\mathrm{CH}}_{3}}^{13}{\mathrm{CH}}_{2}\mathrm{CN}$, and ${\mathrm{CH}}_{3}{{\mathrm{CH}}_{2}}^{13}\mathrm{CN}$), based on a wide range of transitions. Within the uncertainties, the resulting column densities and therefore abundances of the corresponding isotopologues for each of these three species were found to be the same, independent of the location of the ¹³C substitution. This result indicates rather clearly that the formation mechanism of these species does not produce any significant preferential isotopic enhancement. Such chemical fraction effects are removed through "isotopic scrambling" of the molecules themselves or their precursors, which might be expected given the warmer (T ≥ 100 K) gas in Sgr B2(N). The ¹²C/¹³C ratios observed in this cloud thus represent the true isotope gradient arising from stellar nucleosynthesis.

Chemical fractionation occurs because of differences in zero-point energies. Ion-molecule isotopic exchange reactions are often significantly exothermic, such as

$$\begin{eqnarray}&&{}^{13}{{\rm{C}}}^{+}+{}^{12}\mathrm{CO}\to {}^{13}\mathrm{CO}+{}^{12}{{\rm{C}}}^{+}+{\rm{\Delta }}{\rm{E}}\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{}^{13}{{\rm{C}}}^{+}+{}^{12}\mathrm{CN}\to {}^{13}\mathrm{CN}+{{}^{12}{\rm{C}}}^{+}+{\rm{\Delta }}{\rm{E}}.\end{eqnarray} \tag{ 3 }$$

Here ΔE is the zero-point energy difference between the ¹²C and ¹³C isotopologues, estimated to be 35 K for CO and 34 K for CN. However, the carbon atoms in the case of CH₃CCH, CH₂CHCN, and CH₃CH₂CN cannot be easily exchanged by these types of simple reactions, as they all are bonded to more than one other atom. Any ¹³C exchange must occur through precursor molecules, requiring knowledge of formation pathways.

A possible gas-phase formation mechanism for ethyl cyanide in the gas phase involves a radiative association reaction, followed by dissociative electron recombination, as shown in Figure 3 (UDFA 2012 database: McElroy et al. 2012: http://udfa.ajmarkwick.net/index.php):

$$\begin{eqnarray}&&{{\mathrm{CH}}_{3}}^{+}+{\mathrm{CH}}_{3}\mathrm{CN}\to {\mathrm{CH}}_{3}{\mathrm{CH}}_{2}{\mathrm{CNH}}^{+}+h\nu \end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{\mathrm{CH}}_{3}{\mathrm{CH}}_{2}{\mathrm{CNH}}^{+}+{{\rm{e}}}^{-}\to {\mathrm{CH}}_{3}{\mathrm{CH}}_{2}\mathrm{CN}+{\rm{H}}.\end{eqnarray} \tag{ 5 }$$

At 150–200 K, the first reaction has a rate of ∼10⁻¹⁰ cm³ s⁻¹ , while the second reaction occurs at ∼10⁻⁶ cm³ s⁻¹ (McElroy et al. 2012). These rates seem plausible for creating ethyl cyanide, and radiative association reactions, similar to reaction 4, have been incorporated into astrochemical models to successfully produce complex organic species (Vasyunin & Herbst 2013). The precursor species, methyl cyanide (CH₃CN), can be produced by the following processes involving CH₃CNH⁺ (McElroy et al. 2012; Wakelam et al. 2015):

$$\begin{eqnarray}&&{{\mathrm{CH}}_{3}}^{+}+\mathrm{HCN}\to {\mathrm{CH}}_{3}{\mathrm{CNH}}^{+}+h\nu \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{{\mathrm{CH}}_{4}}^{+}+\mathrm{HCN}\to {\mathrm{CH}}_{3}{\mathrm{CNH}}^{+}+{\rm{H}}\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{\mathrm{CH}}_{3}{\mathrm{CNH}}^{+}+{{\rm{e}}}^{-}\to {\mathrm{CH}}_{3}\mathrm{CN}+{\rm{H}}.\end{eqnarray} \tag{ 8 }$$

Reaction 6 has a rate of k₆ ≈ 10⁻⁸ cm³ s⁻¹, reaction 7 proceeds at k₇ ≈ 3 × 10⁻⁹ cm³ s⁻¹, and reaction 8 has k₈ ≈ 2 ×10⁻⁷ cm³ s⁻¹ (Herbst 1985; Anicich 1993; Loison et al. 2014). The carbon–carbon bonds in CH₃CH₂CN are therefore likely formed from HCN and ${{\mathrm{CH}}_{3}}^{+}$ and/or ${{\mathrm{CH}}_{4}}^{+}$. As a consequence, these precursors in Sgr B2(N) likely exhibit ¹²C/¹³C ratios comparable to those in ethyl cyanide. Unfortunately, it is not possible to reliably measure the ¹²C/¹³C ratios in these precursor species from the survey data because of self-absorption (HCN) or lack of a permanent dipole moment (${{\mathrm{CH}}_{3}}^{+}$ and ${{\mathrm{CH}}_{4}}^{+}$).

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Ethyl cyanide could also be formed on grain surfaces, produced by the successive hydrogenation of C₃N or HC₃N (Caselli et al. 1993). The two precursor species are principally produced by gas-phase reactions, and are indeed observed toward Sgr B2(N). In this case, the ¹³C would be first "scrambled" in C₃N or HC₃N in the gas phase and the resulting isotopic composition of the carbon backbone "locked" on grain surfaces. Hydrogenation leading to ethyl cyanide presumably would then not alter the ¹³C tagging.

A possible source of vinyl cyanide is ethyl cyanide itself. The reaction of ethyl cyanide with H⁺ or ${{{\rm{H}}}_{3}}^{+}$ leads to H₂ elimination and the formation of CH₂CHCNH⁺ (Caselli et al. 1993). Dissociative recombination of this ion creates vinyl cyanide, in which case the ¹³C distribution in ethyl cyanide is simply preserved. Another route for the formation for vinyl cyanide is the neutral-neutral reaction between ethene and the CN radical; see the scheme in Figure 3 (KIDA database: Wakelam et al. 2015: http://kida.obs.u-bordeaux1.fr):

$$\begin{eqnarray}&&{\mathrm{CH}}_{2}{\mathrm{CH}}_{2}+\mathrm{CN}\to {\mathrm{CH}}_{2}\mathrm{CHCN}+{\rm{H}}.\end{eqnarray} \tag{ 9 }$$

This reaction has a maximum rate near 50 K of k₉ ≈10⁻¹⁰ cm³ s⁻¹, which diminishes by a factor of 2 approaching 300 K (Sims et al. 1993; Wakelam et al. 2015). The carbon-13 isotope could then be incorporated into vinyl cyanide via the following reactions:

$$\begin{eqnarray}&&{}^{13}{\mathrm{CH}}_{2}{\mathrm{CH}}_{2}+\mathrm{CN}\to {{}^{13}\mathrm{CH}}_{2}\mathrm{CHCN}+{\rm{H}}\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{}^{13}{\mathrm{CH}}_{2}{\mathrm{CH}}_{2}+\mathrm{CN}\to {\mathrm{CH}}_{2}{}^{13}\mathrm{CHCN}+{\rm{H}}\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{\mathrm{CH}}_{2}{\mathrm{CH}}_{2}+{}^{13}\mathrm{CN}\to {\mathrm{CH}}_{2}{\mathrm{CH}}^{13}\mathrm{CN}+{\rm{H}}.\end{eqnarray} \tag{ 12 }$$

For CH₂CH¹³CN to have a comparable ¹²C/¹³C ratio to ¹³CH₂CHCN and ${{\mathrm{CH}}_{2}}^{13}\mathrm{CHCN}$ in this scheme, the ratios in CH₂CH₂ and CN must be very similar. Carbon-13 cannot be easily substituted on ethene, but must arise from its precursors, CH₄ + CH. Therefore, CH${}_{4}$, CH, and CN must all have similar isotopic compositions. Once again, it is difficult to extract the degree of ¹³C enrichment in these molecules from the survey; CN in Sgr B2(N) shows self-absorption, CH₄ lacks a dipole moment, and CH has spectra in the submillimeter (ν > 300 GHz). Based on past observations of CN by Milam et al. (2005), however, ¹²C/¹³C ≥ 18 toward nearby Sgr B2(OH), consistent with the proposed scenario.

Formation on dust grains provides another synthetic route. Vinyl cyanide is created by successive hydrogenation of C₃N or HC₃N on grain surfaces, in analogy to ethyl cyanide (Caselli et al. 1993). Again, the ¹³C distribution in the carbon backbone of vinyl cyanide therefore arises from gas-phase molecules, as for ethyl cyanide.

In contrast to vinyl cyanide, there are multiple gas-phase pathways to form CH₃CCH, as displayed in Figure 3. Hickson et al. (2016) suggest that the most likely gas-phase route to this molecule is dissociative electron recombination from protonated methylacetylene (allyl cation), ${\mathrm{CH}}_{2}{{\mathrm{CHCH}}_{2}}^{+}$. This reaction occurs at a rate of ∼3 × 10⁻⁷ cm³ s⁻¹ between 50 and 100 K (Wakelam et al. 2015). There are multiple ion-molecule routes to synthesize ${\mathrm{CH}}_{2}{{\mathrm{CHCH}}_{2}}^{+}$ from ethane, ethene, methane, or methyl cyanide, involving ${{\mathrm{CH}}_{3}}^{+}$, ${{\rm{C}}}_{2}{{{\rm{H}}}_{3}}^{+}$, or ${{\rm{C}}}_{2}{{{\rm{H}}}_{3}}^{+}$ (Figure 3). These reactions all form carbon–carbon bonds, and have relatively fast rates ranging from 2.2 × 10⁻¹⁰ to 1.1 × 10⁻⁹ cm³ s⁻¹ (Wakelam et al. 2015; Hickson et al. 2016). Two neutral-neutral reactions involving ethene or ethyl radical are also possible pathways, although the branching ratios are not certain (Hickson et al. 2016):

$$\begin{eqnarray}&&{\mathrm{CH}}_{2}{\mathrm{CH}}_{2}+\mathrm{CH}\to {\mathrm{CH}}_{3}\mathrm{CCH}+{\rm{H}}\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&{\mathrm{CH}}_{3}{\mathrm{CH}}_{2}+{\rm{C}}\to {\mathrm{CH}}_{3}\mathrm{CCH}+{\rm{H}}.\end{eqnarray} \tag{ 14 }$$

The rate for reaction 13 is k₁₃ ≈ 3 × 10⁻¹⁰ cm³ s⁻¹ between 50 and 100 K, while reaction 14 proceeds with k₁₄ ≈ 2 ×10⁻¹⁰ cm³ s⁻¹ (Canosa et al. 1997; McElroy et al. 2012).

Methylacetylene could also be produced on the surface of dust grains by sequential hydrogenation of ${\rm{C}}{}_{3}$. Again, ${\rm{C}}{}_{3}$ is first formed in the gas phase, freezing in the ¹³C tagging, and is subsequently incorporated onto grain surfaces (Hickson et al. 2016). This mechanism, however, could be disrupted by the presence of atomic oxygen on the grain surface. It is clear that there are a sufficient numbers of routes leading to the incorporation of carbon-13 into CH₃CCH to dilute any site specificity from a given precursor.