Interstellar Nitrogen Isotope Ratios: New NH₃ Data from the Galactic Center out to the Perseus Arm

A total of 210 sources was chosen from previously studied strong NH₃ sources (e.g., Wyrowski & Walmsley 1996; Longmore et al. 2007; Rosolowsky et al. 2009; Lis et al. 2010; Cyganowski et al. 2013; Reid et al. 2014). Sources have accurate distance values, including 113 sources from trigonometric parallax measurements and 97 from the parallax-based distance calculator (Reid et al. 2014, 2019). Using a Bayesian approach, sources are assigned to arms based on their (l, b, v) coordinates with respect to arm signatures seen in CO and H i surveys. The most reasonable distance (near or far) can be derived through a full distance probability density function from the parallax-based distance calculator, considering a source's kinematic distance, displacement from the plane, and proximity to individual parallax sources. We believe that it is an important improvement to reveal radial variations of ¹⁴N/¹⁵N in an unbiased way. The heliocentric distance was used to calculate the galacocentric distance of targets (Roman-Duval et al. 2009),

$$\begin{eqnarray}&&{D}_{\mathrm{GC}}=\sqrt{{\left[{R}_{0}\cos (l)-d\right]}^{2}+{R}_{0}^{2}{\sin }^{2}(l)}.\end{eqnarray} \tag{ 1 }$$

l is the Galactic longitude, and R₀ and d are the distance of the Sun from the Galactic center (8.122 ± 0.031 kpc, from Gravity Collaboration et al. 2018) and of the targeted source from the Sun (Reid et al. 2014), respectively. The error in the distance to the Galactic center is so small that it is neglected in the following.

The sample includes star-forming regions at different evolutionary stages, including sources associated with infrared dark clouds (IRDCs), massive young stellar objects (YSOs), and H ii regions, ⁹ which are used to better constrain the radial trends of the Galactic ¹⁴N/¹⁵N isotope ratio. The source list is presented in the Appendix.

2.2.1. Tianma 65 m Observations

For our sample of 210 sources, we made observations of the (J, K) = (1, 1), (2, 2), and (3, 3) lines of ¹⁴NH₃ and ¹⁵NH₃ (see Table 1), first with the Shanghai Tianma 65 m radio telescope (TMRT) in 2019 April, May, November, and December, with a beam size of ∼50''. A cryogenically cooled K-band (17.9–26.2 GHz) receiver was employed, and the digital backend system, DIBAS, was used for recording (see Li et al. 2016). The DIBAS mode 22 was adopted for observations, with eight spectral windows, to cover the ¹⁴NH₃ and ¹⁵NH₃ lines simultaneously, each with a bandwidth of 23.4 MHz (16384 channels), supplying a spectral resolution of 1.43 kHz (∼0.02 km s⁻¹). The active surface system of the primary dish and a subreflector were used to improve the aperture efficiency. Observations were performed in position-switching mode. The system temperature was 100–200 K on an antenna temperature scale (${T}_{{\rm{A}}}^{* }$). The main-beam brightness temperature (T_mb) can be obtained from the antenna temperature scale by T_mb = ${T}_{{\rm{A}}}^{* }$/η_b , where η_b is the main-beam efficiency correction factor, with a mean value of ∼0.6 (Mei et al. 2020). The on-source integration time was about 0.1–3.0 hr for each of our sources.

Table 1. The Parameters of the ¹⁴NH₃ and ¹⁵NH₃ Transition Lines

Line	Frequency	${\mathrm{log}}_{10}({A}_{{ij}})$ a	Eu /kb	gu c
	(MHz)		(K)
14NH3(1, 1)	23694.5	−6.76650	24.35	6
14NH3(2, 2)	23722.6	−6.31125	65.34	10
14NH3(3, 3)	23870.1	−6.25203	124.73	28
15NH3(1, 1)	22624.9	−6.83680	23.82	6
15NH3(2, 2)	22649.8	−6.71062	64.93	10
15NH3(3, 3)	22789.4	−6.65097	123.91	28

Notes. The parameters are from the JPL Molecular Spectroscopy Catalog (Pickett et al. 1998).

^a Einstein coefficient for spontaneous emission. ^b Energy of the upper level above the ground state. ^c Upper state degeneracy.

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2.2.2. Effelsberg 100 m Observations

The continuum and line analysis single-dish software (CLASS) of the Grenoble Image and Line Data Analysis Software packages ¹¹ (GILDAS, e.g., Guilloteau & Lucas 2000) was used to reduce the spectral line data. After subtracting baselines and applying Hanning smoothing, the line parameters are obtained from Gaussian fits for detected lines (signal-to-noise ratio, S/N > 3σ), with a spectral resolution of ∼0.78 km s⁻¹ for TMRT and 0.70 km s⁻¹ for Effelsberg observations, respectively.

Toward the TMRT 65 m 210 targets, 141 sources were detected in at least one of the ¹⁴NH₃ lines. Of these, eight sources were also successfully detected in the ¹⁵NH₃(1, 1) line. The ¹⁵NH₃(2, 2) and (3, 3) lines are also detected in three sources (NGC 6334 I, W51D, and Orion-KL, see Figure 1).

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Of the 36 targets with strong ¹⁴NH₃ emission (Tianma flux density >1.5 Jy), the Effelsberg 100 m telescope successfully detected the ¹⁵NH₃(1, 1) lines toward 10 sources. The ¹⁵NH₃(2, 2) and (3, 3) lines are also detected in three of them (W51D, Orion-KL, and G10.47, see Figure 2).

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Combinations of the TMRT and Effelsberg observations lead to 15 sources with detections of ¹⁵NH₃ lines, including 3 sources (G081.7522, W51D, and Orion-KL) with detections by both telescopes. The spectral line parameters of these 15 sources are listed in Table 2. For G30.70 with an FWHM of ¹⁵NH₃(1, 1), which is much smaller than that of ¹⁴NH₃(1, 1), the integrated line intensity of ¹⁵NH₃(1, 1) was taken assuming that the ratio of integrated line intensities equals the ratio of the peak values of the main-beam brightness temperature. For the sources without a detection of ¹⁵NH₃, the upper limit of the line was estimated from the rms value, which was used for the subsequent analysis (see Section 3.3).

For the 15 sources with both ¹⁴NH₃ and ¹⁵NH₃ line detections, we determine the line and physical parameters here, including the optical depth, the total column density, and the rotation and kinetic temperatures.

3.2.1. Optical Depth

In view of the expected high nitrogen isotope ratios, there is reason to expect that in clouds with detected ¹⁵NH₃ emission, the ¹⁴NH₃ lines may be optically thick, leading to a nonlinear correlation between integrated intensity and molecular column density. We therefore determined the optical depth of the ¹⁴NH₃ inversion lines using two methods, the so-called intensity ratio method and the HF structure line fits. Spectroscpic information about the ¹⁴NH₃(1, 1) HF components is listed in Table 3.

Table 3. Spectroscopic Information Related to the ¹⁴NH₃(1, 1) HF Components

Hyperfine	HFC	${F}^{{\prime} }\to F$	${F}_{1}^{{\prime} }\to {F}_{1}$	Frequency b	Relative	Velocity
Group a	Number			(kHz)	Intensities c	(km s−1)
osg.1	1	1/2, 1/2	(0,1)	−1568.49	1/27	−19.84
	2	1/2, 3/2	(0,1)	−1526.96	2/37	−19.32
isg.1	3	3/2, 1/2	(2,1)	−623.31	5/108	−7.89
	4	5/2, 3/2	(2,1)	−590.92	1/12	−7.47
	5	3/2, 3/2	(2,1)	−580.92	1/108	−7.35
mg	6	1/2, 1/2	(1,1)	−36.54	1/54	−0.46
	7	3/2, 1/2	(1,1)	−25.54	1/108	−0.32
	8	5/2, 3/2	(2,2)	−24.39	1/60	−0.31
	9	3/2, 3/2	(2,2)	−14.98	3/20	−0.19
	10	1/2, 3/2	(1,1)	5.85	1/108	0.07
	11	5/2, 5/2	(2,2)	10.52	7/30	0.13
	12	3/2, 3/2	(1,1)	16.85	5/108	0.21
	13	3/2, 5/2	(2,2)	19.93	1/60	0.25
isg.2	14	1/2, 3/2	(1,2)	571.79	5/108	7.23
	15	3/2, 3/2	(1,2)	582.79	1/108	7.37
	16	3/2, 5/2	(1,2)	617.70	1/12	7.81
osg.2	17	1/2, 1/2	(1,0)	1534.05	1/27	19.41
	18	3/2, 1/2	(1,0)	1545.05	2/27	19.55

Notes.

^a mg = main group of HF components, isg = group of inner satellite HF components, osg = group of outer satellite HF components. ^b The frequencies in column (5) are given relative to 23,694.5 MHz. ^c The HF intensities are taken from Mangum & Shirley (2015) and Wang et al. (2020). The sum of these intensities is 1.0.

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Intensity ratio method: Assuming the same excitation temperature and beam filling factor for all transitions under conditions of local thermodynamic equilibrium (LTE), the optical depth of the ¹⁴NH₃(1, 1) line can be estimated from the measured intensity ratio between its main and the satellite components (e.g., Ho & Townes 1983; Mangum et al. 1992),

$$\begin{eqnarray}&&\displaystyle \frac{{T}_{\mathrm{mb}}(1,1,m)}{{T}_{\mathrm{mb}}(1,1,s)}=\displaystyle \frac{1-{e}^{-\tau (\mathrm{1,1},m)}}{1-{e}^{-a\tau (\mathrm{1,1},m)}}\end{eqnarray} \tag{ 2 }$$

,where T_mb is the peak value of the main-beam brightness temperature, m and s refer to the main and satellite group components, respectively, τ(J, K, m) is the optical depth of the main group of HF components, a represents the expected intensity ratios of the satellite to the main group of HF components, and 0.278 and 0.222 for the inner and outer groups of HF features under conditions of LTE and optically thin line emission (see Ho & Townes 1983; Mangum & Shirley 2015; Zhou et al. 2020). For our sample, we used formula (2) to calculate the ¹⁴NH₃(1, 1) optical depth, according to the T_mb peak values of the main component and the inner and outer satellite components. The mean value of the four inner and outer calculated NH₃(1, 1) optical depths was taken for each source (Table 4).

Table 4. Observational Parameters of NH₃ Measured with the TMRT and the Efelsberg 100 m Telescope

Object	Telescope	Intensity Ratio Method			HF Fitting			Trot ir	Trot rd	Trot hf	Tk
		τ(1, 1)	τ(2, 2)	τ(3, 3)	τ(1, 1)	τ(2, 2)	τ(3, 3)	(K)	(K)	(K)	(K)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)
G032.04	TMRT	1.73(0.06)	0.54(0.02)	0.21(0.01)	1.83(0.16)	0.2(0.4)	0.1(0.3)	22(10)	11 (6)	19(3)	21(4)
G053.23	TMRT	1.44(0.04)	0.21(0.01)	...	1.05(0.06)	0.11 (0.04)	⋯	16(8)	9 (3)	11(5)	12(4)
G081.75	TMRT	1.35(0.01)	0.54(0.01)	0.12(0.01)	1.56(0.15)	0.47(0.06)	0.13(0.04)	27(9)	13 (5)	19(2)	22(3)
	Effelsberg	1.47(0.01)	0.53(0.01)	0.12(0.01)	1.65(0.12)	0.42(0.08)	0.15(0.04)	29(10)	13 (7)	19(3)	22(3)
G121.29	TMRT	1.02(0.01)	0.33(0.01)	0.16(0.01)	1.1(30.01)	0.48(0.03)	0.12(0.12)	20(9)	13 (6)	17(3)	20(4)
G30.70	TMRT	2.23(0.03)	1.38(0.02)	0.43(0.01)	3.27(0.06)	2.23(0.11)	0.12(0.04)	26(8)	14 (5)	20(2)	25(2)
NGC 6334 I	TMRT	1.75(0.02)	0.42(0.01)	0.42(0.01)	2.06(0.03)	4.72(0.19)	4.3(0.4)	25(11)	12 (6)	14(1)	17(1)
Orion-KL	TMRT	2.4(0.4)	4.8(0.8)	7.4(1.0)	⋯	⋯	⋯	36(17)	42 (11)	27(3)	40(4)
	Effelsberg	2.56(0.01)	5.52(0.02)	8.56(0.02)	⋯	⋯	⋯	35(14)	46 (10)	26(3)	37(4)
W51D	TMRT	2.15(0.05)	0.51(0.01)	0.86(0.01)	⋯	⋯	⋯	21(12)	14 (10)	18(4)	22(4)
	Effelsberg	2.1(0.2)	1.56(0.02)	1.61(0.02)	⋯	⋯	⋯	30(22)	20 (16)	24(4)	34(5)
G016.92	Effelsberg	1.14(0.03)	0.41(0.01)	0.12(0.01)	1.27(0.07)	1.0(0.3)	0.1(0.3)	27(13)	12 (6)	19(8)	23(9)
G10.47	Effelsberg	2.21(0.05)	1.35(0.03)	1.80(0.03)	⋯	⋯	⋯	29(8)	18 (6)	24(5)	32(6)
G188.79	Effelsberg	0.22(0.01)	0.16(0.01)	0.13(0.01)	0.23(0.05)	0.1(0.5)	0.10(0.03)	21(10)	14 (6)	21(17)	28(22)
G35.14	Effelsberg	1.16(0.02)	0.51(0.01)	0.26(0.01)	1.42(0.02)	1.12(0.19)	0.2(0.8)	29(19)	12 (4)	20(2)	24(2)
NGC 1333	Effelsberg	1.01(0.02)	0.12(0.01)	0.15(0.01)	1.1(0.2)	0.7(0.2)	0.13(0.11)	19(7)	10 (4)	12(5)	14(5)
G000.19	Effelsberg	1.12(0.10)	0.25(0.02)	0.11(0.02)	1.22(0.16)	2.6(1.1)	0.1(0.5)	17(5)	11 (3)	16(8)	18(9)
Barnard-1b	Effelsberg	2.25(0.02)	0.22(0.01)	0.11(0.02)	2.82(0.02)	0.33(0.10)	1.16(0.10)	14(4)	8 (3)	10(3)	11(3)

Note. Column (1) source name; column (2) telescope; columns (3)–(5) the peak optical depths of the (J, K) = (1, 1), (2, 2), and (3, 3) main group of HF components of ¹⁴NH₃ from the intensity ratio method; columns (6)–(8) peak optical depths from the HF fitting procedure provided by CLASS; column (9) the rotational temperature T_rot ^ir from the intensity ratio method; column (10) T_rot ^rd from the rotation diagram method; column (11) T_rot ^hf from the improved HF fitting method (see Section 3.2.2) for 11 sources, that of NGC 6334 I from the RADEX calculation, and that of the remaining three sources with blended spectral features (G10.47, Orion-KL, and W51D) from the HFGR method (see details in Section 3.2.2); column (12) the kinetic temperature calculated from the empirical formula displayed in Appendix B of Tafalla et al. (2004).

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For the ¹⁴NH₃(2, 2) and (3, 3) transitions, there are only two sources with clearly separated satellite components (NGC 6334 I and G30.70). Here we also used the measured intensity ratio of the main and satellite components to determine their optical depth. For other sources, we could not use this method because the HF satellite components were too weak to be detected. However, the optical depth can be estimated from the intensity ratio of the main group of (2, 2) HF components to that of the (1, 1) transition, assuming equal excitation temperatures and beam filling factors (Mangum et al. 1992),

$$\begin{eqnarray}&&\displaystyle \frac{{T}_{\mathrm{mb}}(1,1,m)}{{T}_{\mathrm{mb}}(2,2,m)}=\displaystyle \frac{1-{e}^{-\tau (\mathrm{1,1},m)}}{1-{e}^{-\tau (\mathrm{2,2},m)}}.\end{eqnarray} \tag{ 3 }$$

HF fit method: The optical depth of the main HF component can also be determined by the HF fit method in CLASS ("method" command). Here, the excitation temperature and Gaussian opacity profiles are assumed to be the same for all HF components. τ_tot (the opacity summed over all the HF components) can also be retrieved from CLASS. The relation between τ_tot and τ(J, K, m) can be found in Equation (A8) by Mangum et al. (1992). Through adjusting parameters to fit the observed spectra, we derived the optical depths of the ¹⁴NH₃(1, 1), (2, 2), and (3, 3) transitions for 12 sources with ¹⁵NH₃ detections (see Table 4). For the other 3 sources in the sample (G10.47, Orion-KL, and W51D), the (1, 1), (2, 2), and (3, 3) optical depths could not be determined by the "method" fits in CLASS because the satellite HF components overlap their main group of HF components (see Table 4).

Comparisons of the opacities derived from the two different methods, the intensity ratio and HF fit methods, reveal consistency with those from the HF fits in CLASS. Therefore we chose to take the optical depth from the intensity ratio method, where results could be obtained from all sources with detected ¹⁵NH₃ emission for our analysis.

3.2.2. Temperature

Rotational temperature: NH₃ inversion lines have been widely used as tracers of the temperature in molecular clouds (Li et al. 2003; Tafalla et al. 2004; Mangum & Shirley 2015; Wang et al. 2020). Three main methods, either starting from observed or modeled spectra, are used to estimate the rotational temperature. These are the intensity ratio method, the rotation diagram method, and the improved HF fitting method. They are outlined below.

(a) Intensity ratio method: This method was described in Section 3.2.1 to determine the optical depth according to the intensity ratio of the main and satellite components from the observed spectra. According to the determined opacities and the measured brightness temperatures of the (1, 1) and (2, 2) main groups of HF components, the rotational temperature ${T}_{\mathrm{rot}}^{21}$ can be derived (Ho & Townes 1983; Mangum et al. 1992; Ragan et al. 2011) by

$$\begin{eqnarray}&&\begin{array}{l}{T}_{\mathrm{rot}}=-41.5\left[\mathrm{ln}\left(-\displaystyle \frac{0.283}{\tau (1,1,m)}\right.\right.\\ \quad {\left.\left.\times \,\mathrm{ln}\left[1-\displaystyle \frac{{T}_{\mathrm{mb}}(\mathrm{2,2},m)}{{T}_{\mathrm{mb}}(\mathrm{1,1},m)}(1-{e}^{-\tau (\mathrm{1,1},m)})\right]\right)\right]}^{-1}.\end{array}\end{eqnarray} \tag{ 4 }$$

The T_rot results for the 15 sources with ¹⁵NH₃ detections are listed in column 9 of Table 4.

(b) Rotation diagram method: For optically thin lines in LTE, the relation of the column density and energy above the ground state in the upper inversion doublet (the two states of a given inversion doublet are only about 1 K apart) with the corresponding values for the lower inversion doublet can be determined on the basis of the measured line temperatures. The rotation diagram, i.e., a plot of the upper level column density per statistical weight of a number of molecular energy levels, as a function of their energy above the ground state is frequently used to estimate the temperature and the total column density (e.g., Mangum et al. 1992; Goldsmith & Langer 1999). For the (J, K) = (1, 1) and (2, 2) transitions of ¹⁴NH₃, the column density in the upper state N_u for both transitions assuming optically thin emission can be written as

$$\begin{eqnarray}&&\begin{array}{l}{N}_{u}(1,1)=\displaystyle \frac{8\pi {{kv}}_{1}^{2}\displaystyle \int {T}_{\mathrm{mb}}^{11}{dv}}{{{hc}}^{3}{A}_{1}}\\ \quad =\displaystyle \frac{6.37\times {10}^{7}}{[{{\rm{K}}}^{-1}\ {\mathrm{cm}}^{-3}\ {\rm{s}}]}\times \displaystyle \frac{\displaystyle \int {T}_{\mathrm{mb}}^{11}{dv}}{[{\rm{K}}\ \mathrm{cm}\ {{\rm{s}}}^{-1}]}\end{array}\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&\begin{array}{l}{N}_{u}(2,2)=\displaystyle \frac{8\pi {{kv}}_{2}^{2}\displaystyle \int {T}_{\mathrm{mb}}^{22}{dv}}{{{hc}}^{3}{A}_{2}}\\ \quad =\displaystyle \frac{2.24\times {10}^{7}}{[{{\rm{K}}}^{-1}\,{\mathrm{cm}}^{-3}\,{\rm{s}}]}\times \displaystyle \frac{\displaystyle \int {T}_{\mathrm{mb}}^{22}{dv}}{[{\rm{K}}\,\mathrm{cm}\,{{\rm{s}}}^{-1}]},\end{array}\end{eqnarray} \tag{ 6 }$$

where N_u (1, 1) and N_u (2, 2) are the column density in the upper state for (J, K) = (1, 1) and (2, 2), respectively; k is the Boltzmann constant, c is the speed of light, and h is the Planck constant. A_ul is the Einstein coefficient for spontaneous emission, which was obtained from the JPL Molecular Spectroscopy Catalog (Pickett et al. 1998) and is listed in Table 1.

The optical depths of the ¹⁴NH₃ transition lines of our sources are mostly large (≥1 for 14 out of 15 sources in NH₃(1, 1), see Table 4), so that the assumption of optically thin emission only provides lower limits to the ¹⁴NH₃ abundances and may underestimate the real abundance ratio ¹⁴NH₃/¹⁵NH₃. Thus an optical depth correction should be considered for the upper state column density (Goldsmith & Langer 1999; Mei et al. 2020), yielding ${N}_{u}^{{\prime} }$=${N}_{u}\tau (J,K,m)/(1-\exp (-\tau (J,K,m)))$. For the (1, 1) and (2, 2) lines, the relation between the opacity-corrected total column density N_t and ${N}_{u}^{{\prime} }$ in a Boltzmann distribution should be

$$\begin{eqnarray}&&\mathrm{ln}\displaystyle \frac{{N}_{t}}{Q({T}_{\mathrm{rot}})}=\mathrm{ln}\displaystyle \frac{{N}_{u}^{{\prime} }(1,1)}{{g}_{1}}+\displaystyle \frac{{E}_{1}}{{{kT}}_{\mathrm{rot}}}\end{eqnarray} \tag{ 7 }$$

and

$$\begin{eqnarray}&&\mathrm{ln}\displaystyle \frac{{N}_{t}}{Q({T}_{\mathrm{rot}})}=\mathrm{ln}\displaystyle \frac{{N}_{u}^{{\prime} }(2,2)}{{g}_{2}}+\displaystyle \frac{{E}_{2}}{{{kT}}_{\mathrm{rot}}}\end{eqnarray} \tag{ 8 }$$

,where g_u and E_u are the degeneracy and the energy of the upper state, respectively (see Table 1). Q(T_rot) is the partition function from the JPL Molecular Spectroscopy Catalog (Pickett et al. 1998).

Thus

$$\begin{eqnarray}&&\begin{array}{l}\mathrm{ln}\displaystyle \frac{{N}_{u}^{{\prime} }(1,1)}{{g}_{1}}-\mathrm{ln}\displaystyle \frac{{N}_{u}^{{\prime} }(2,2)}{{g}_{2}}\\ \quad =\,\mathrm{ln}\displaystyle \frac{{\tau }_{11}\displaystyle \int {T}_{\mathrm{mb}}^{11}{dv}}{1-\exp (-{\tau }_{11})}-\mathrm{ln}\displaystyle \frac{{\tau }_{22}\displaystyle \int {T}_{\mathrm{mb}}^{22}{dv}}{1-\exp (-{\tau }_{22})}\\ \qquad +1.56=\displaystyle \frac{{E}_{2}}{{{kT}}_{\mathrm{rot}}}-\displaystyle \frac{{E}_{1}}{{{kT}}_{\mathrm{rot}}}.\end{array}\end{eqnarray} \tag{ 9 }$$

For the measured (1, 1) and (2, 2) line intensities of our sample, we plotted the rotation diagram, i.e., $\mathrm{ln}({N}_{u}^{{\prime} }/{g}_{u})$ against E_u /k. The rotational temperature T_rot depends on the reciprocal value of the slope (see Figure 3 and Equation (9)). The T_rot results from this method for our sample are listed in column 10 of Table 4. The uncertainties on the rotational temperatures were derived applying error propagation based on Equation (9).

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(c) Improved HF fitting method: This is the updated version of Method in CLASS (Section 3.2.1), which can fit (1, 1) and (2, 2) lines simultaneously. It is included in the Python package Pyspeckit (Ginsburg & Mirocha 2011). Based on the model spectra produced by the radiative transfer function (T_mb(v) = η_f [J(T_ex) − J(T_bg)][1 − e^−τ(v)]), we can adjust the model parameters (the excitation temperature, the line width, etc.) to fit the observed spectra and determine the rotational temperature (Rodgers & Charnley 2008; Keown et al. 2017; Camacho et al. 2020). η_f is the beam filling factor assumed to be unity, J(T_ex) and J(T_bg) represent the radiation field corresponding to the excitation temperature and the cosmic microwave background temperature of 2.73 K, and τ(v) is the optical depth as a function of frequency. The following assumptions are adopted for the HF fitting (Ginsburg & Mirocha 2011):

(1) Gaussian profiles for the opacity as a function of frequency;

(2) the same excitation temperature for the ¹⁴NH₃(1, 1) and (2, 2) transitions;

(3) the lines all have the same width;

(4) the multiplet components do not overlap;

(5) LTE is prevailing.

Fixed values for the relative opacities and the frequency shift of each HF component were taken (Mangum & Shirley 2015). We used the package Pyspeckit to fit the spectra of sources, excluding the three sources with overlapping HF components (G10.47, W51D, and Orion-KL, see Section 3.2.1). With the exception of NGC 6334 I, the ¹⁴NH₃(1, 1) and (2, 2) groups of the spectra can be fitted well simultaneously (see Figure 4). For NGC 6334 I, the two groups of spectra could not be fitted simultaneously with the same value of the excitation temperature. To check possible non-LTE effects, we made RADEX ¹² calculations for this source and derived a T_rot value of ∼14.3 K with excitation temperatures of 17.1 and 7.3 K for the (1, 1) and (2, 2) lines. The rotation temperature is consistent with the T_rot result of 12.2 ± 6.1 K by the rotation diagram method (LTE), which is used in our analysis below.

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For the three sources with blended HF components in the spectra (G10.47, Orion-KL, and W51D), an improved method, the hyperfine group ratio (HFGR) method, was used to calculate the rotational temperature. This was developed recently by Wang et al. (2020), and it can effectively reduce the uncertainties related to spectral profiles because only the integrated intensity ratios of groups of HF components are used without spectral fitting. The T_rot results for these three sources by the HFGR method, NGC 6334 I by RADEX, and others by the HF fitting are listed in column 11 of Table 4.

Comparing the T_rot results from the three different methods mentioned above, we find that T_rot results from the rotation diagram method are systematically lower than those from the other two methods. The T_rot results from the intensity ratio method have systematically larger uncertainties, which may mainly be caused by uncertainties of the optical depth, estimated from the ratio of the peak of the main group of HF components and the HF groups giving rise to the inner and outer satellites. Based on the good fitting of the observed spectra in Figure 4, we therefore took the T_rot values that arise from the improved HF fitting method for the subsequent analysis. For the three sources with blended components, their T_rot values were taken from the HFGR recipe (see Table 4).

The kinetic temperature: The conversion of the rotational temperature (T_rot) into the gas kinetic temperature (T_k ) is a critical part of the NH₃ inversion line analysis. Tafalla et al. (2004) provided a detailed T_k analysis in their Monte Carlo models using the collision coefficients of Danby et al. (1988) and T_rot results from an NH₃ line analysis and derived an expression for accurate gas temperature estimates,

$$\begin{eqnarray}&&{T}_{k}=\displaystyle \frac{{T}_{\mathrm{rot}}}{1-\tfrac{{T}_{\mathrm{rot}}}{{\rm{\Delta }}E}\mathrm{ln}[1+1.1\exp (-\tfrac{16}{{T}_{\mathrm{rot}}})]}.\end{eqnarray} \tag{ 10 }$$

Using this relation between T_k − T_rot and the T_rot results derived from the above analysis, we calculated the kinetic temperature values for the sample, which are given in column 12 of Table 4.

3.2.3. Column Density

As shown in the rotation diagram analysis in Section 3.2.2, with the assumption of LTE, the T_rot values were obtained from the ¹⁴NH₃(1, 1) and (2, 2) line intensities and corresponding optical depths for our sample. According to the derived T_rot values and Equations (7) or (8), we can further use the (1, 1) or (2, 2) line intensities and corresponding parameters of g_u , Q(T_rot) and E_u to determine opacity-corrected total column densities of ¹⁴NH₃.

For ¹⁵NH₃, we can carry out a similar analysis to determine the total column density for our sample, but in this case, with ¹⁴N/¹⁵N commonly in excess of 100, we can realistically assume that all lines are optically thin. We calculated the total column density of ¹⁵NH₃ from Equations (7) or (8), assuming the same T_rot value for ¹⁵NH₃ as for ¹⁴NH₃ and adopting the spectroscopic parameters of the ¹⁵NH₃ molecular species (see Table 1). For the two sources (NGC 6334 I and G10.47) with detections in both ¹⁵NH₃(1, 1) and ¹⁵NH₃(2, 2), we took the ¹⁵NH₃ total column density results from the ¹⁵NH₃(2, 2) line for the subsequent analysis because the quality of the ¹⁵NH₃(2, 2) spectra is better than those of the (1, 1) lines. For Orion-KL, with high-quality spectra in both transitions, we also used the rotation diagram method to determine T_rot (see Figure 3) and further determined its total column density. The results of the total column density without opacity corrections and the opacity-corrected values of both ¹⁴NH₃ and ¹⁵NH₃ of the 15 sources are listed in Table 5.

Table 5. Total ¹⁴NH₃ and ¹⁵NH₃ Column Densities with and without Opacity Corrections and Their Ratios

Object	Telescope	Nt (14NH3)	Nt (15NH3)	${N}_{t}^{\mathrm{Corr}}$(14NH3)	${N}_{t}^{\mathrm{Corr}}$(15NH3)	$\tfrac{{}^{14}{\mathrm{NH}}_{3}}{{}^{15}{\mathrm{NH}}_{3}}$	${\tfrac{{}^{14}{\mathrm{NH}}_{3}}{{}^{15}{\mathrm{NH}}_{3}}}^{\mathrm{Corr}}$	DSun	DGC	Classi-	References	Notes
		(cm−2)	(cm−2)	(cm−2)	(cm−2)			(kpc)	(kpc)	fication
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)
G000.19	Effelsberg	5.3E+14	2.2E+13	9.9E+14	2.5E+13	24 (10)	40 (13)	8.4(0.2)	0.31(0.15)	YSO	Par18	1
G10.47	Effelsberg	2.5E+15	3.2E+14	6.1E+15	4.5E+14	8 (5)	13 (6)	8.25(0.11)	1.53(0.15)	UCH II	Wyr96	2
G30.70	TMRT	4.3E+15	4.2E+13	1.1E+16	4.4E+13	102 (29)	253 (60)	5.0(0.7)	4.6(0.4)	YSO	Urq18	1
G032.04	TMRT	2.0E+15	4.3E+13	4.8E+15	5.0E+13	46 (21)	97 (31)	5.2(0.5)	4.8(0.3)	YSO	Coo13	1
W51D	TMRT	3.9E+15	3.2E+13	1.1E+16	3.6E+13	121 (46)	295 (81)	5.5(0.4)	6.2(0.3)	YSO	God15	1
			9.5E+13		2.0E+14	41 (25)	54 (27)					2
	Effelsberg	1.5E+15	2.2E+13	3.7E+15	2.2E+13	69 (42)	166 (62)					1
			4.7E+13		5.7E+13	33 (56)	64 (62)					2
G016.92	Effelsberg	5.4E+14	2.5E+13	9.8E+14	2.7E+13	21 (10)	35 (13)	1.81(0.13)	6.44(0.11)	H ii	Urq11	1
G35.14	Effelsberg	2.5E+15	2.9E+13	4.4E+15	3.1E+13	86 (39)	143 (50)	2.2(0.2)	6.5(0.3)	IRDC	Den84	1
NGC 6334 I	TMRT	1.2E+16	8.4E+13	2.5E+16	8.2E+13	145 (45)	301 (78)	1.34(0.11)	6.9(0.4)	IRDC	Wil13	1
			3.1E+14		6.0E+14	39 (9)	48 (11)					2
G053.23	TMRT	1.0E+15	6.2E+12	2.3E+15	7.5E+12	167 (143)	314 (172)	8.3(0.6)	7.3(0.5)	YSO	Urq18	1
G081.75	TMRT	1.8E+15	6.9E+12	3.5E+15	7.5E+12	259 (171)	467 (213)	2.35(0.12)	8.13(0.14)	YSO	Mau15	1
	Effelsberg	1.4E+15	4.2E+12	2.8E+15	4.6E+12	321 (157)	603 (213)					1
NGC 1333	Effelsberg	6.9E+14	4.8E+12	1.4E+15	5.7E+12	142 (75)	247 (96)	0.21(0.17)	8.3(0.4)	YSO	Lis10	1
Barnard-1b	Effelsberg	1.3E+15	1.7E+13	5.8E+15	2.5E+13	77 (32)	229 (62)	0.33(0.15)	8.4(0.5)	IRDC	Lis10	1
Orion-KL	TMRT	5.9E+15	5.6E+13	2.1E+16	7.7E+13	105 (39)	270 (72)	0.45(0.12)	8.5(0.6)	H ii	Kim08	1
			1.5E+14		1.5E+14	39 (10)	136 (45)					15NH3
	Effelsberg	7.9E+15	9.8E+13	3.1E+16	1.4E+14	80 (20)	212 (46)					1
			1.4E+14		1.4E+14	55 (12)	215 (47)					15NH3
G121.29	TMRT	1.4E+15	1.4E+13	2.4E+15	1.5E+13	100 (60)	159 (72)	1.7(1.3)	9(2)	IRDC	Ryg10	1
G188.79	Effelsberg	2.3E+14	7.6E+12	2.5E+14	7.7E+12	30 (17)	33 (17)	2.14(0.12)	10.3(1.2)	YSO	Cut03	1

Note. Column (1) source name; column (2) telescope; columns (3)–(4) ¹⁴NH₃ and ¹⁵NH₃ column densities neglecting opacity corrections; column (5) column densities ${N}_{t}^{\mathrm{Corr}}$(¹⁴NH₃) accounting for opacity effects; column (6) the corrected column density ${N}_{t}^{\mathrm{Corr}}$(¹⁵NH₃) was obtained with the assumption of the same T_rot as T_rot(¹⁴NH₃), which can be derived taking into account the optical depth correction on ¹⁴NH₃ in the rotation diagram method; and column (7) the ratios of the column densities neglecting opacity corrections. Errors (in parentheses) include standard deviations from the line fitting procedure; column (8) opacity-corrected values of ¹⁴NH₃/¹⁵NH₃; column (9) heliocentric distance with error, from the parallax-based distance calculator; column (10) galactocentric distance with error from the heliocentric distance; column (11) source classification. IRDC and YSO, H ii: associated with an H ii region; UCH II: associated with an ultracompact H ii region; column (12) references for the classification from the literature. Par18: Parsons et al. (2018); Wyr96: Wyrowski & Walmsley (1996); Urq18: Urquhart et al. (2018); Coo13: Cooper et al. (2013); God15: Goddi et al. (2015); Urq11: Urquhart et al. (2011); Wil13: Willis et al. (2013); Mau15: Maud et al. (2015); Lis10: Lis et al. (2010); Kim08: Kim et al. (2008); Ryg10: Rygl et al. (2010); Cut03: Cutri et al. (2003); and Den84: Dent et al. (1984). Note 1: the total column densities of ¹⁵NH₃ were obtained from the ¹⁵NH₃(1, 1) line intensity, assuming the same T_rot value for ¹⁵NH₃ as for ¹⁴NH₃. Note 2: the total column densities of ¹⁵NH₃ were obtained from the ¹⁵NH₃(2, 2) line intensity, assuming the same T_rot value for ¹⁵NH₃ as for ¹⁴NH₃. Note on ¹⁵NH₃: the total column densities of ¹⁵NH₃ were obtained using the T_rot of ¹⁵NH₃ from its ¹⁵NH₃(1, 1) and (2, 2) line intensities.

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As shown in Section 3.2.3, we obtained the column densities of ¹⁴NH₃ and ¹⁵NH₃ of 15 sources. Based on these results, we estimate the ¹⁴N/¹⁵N isotope ratios (see Table 5). For the 2 sources (NGC 6334 I and G10.47) with a ¹⁵NH₃ (2, 2) detection, the results from their NH₃(2, 2) lines (with higher quality than their (1, 1) lines, see Section 3.1) are used for later analysis because the S/Ns are higher. For the sources that were measured by both the Effelsberg and TMRT telescope (G081.75, W51 D, and Orion-KL), the mean value of their ¹⁴NH₃/¹⁵NH₃ ratios was taken. For the sources without a detection of ¹⁵NH₃, we estimate the lower limit of ¹⁴NH₃/¹⁵NH₃, according to the peak temperature of ¹⁴NH₃ and the 3 rms value of the ¹⁵NH₃ line (shown as gray points with arrows in Figure 5(a)). The lower limit is mostly around 11, with a mean value of 13, which is lower than all ratios derived from our 15 detections. A comparison with previous studies is presented in Section 4.1, and possible contaminating effects affecting the abundance ratios are discussed in Sections 4.2 and 4.3.

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Of the 15 sources with measured ¹⁴N/¹⁵N ratio, 5 (Orion-KL, Barnard-1b, NGC 1333, W51 D, and G000.19 in the Galactic center region) were also measured in previous studies, which therefore provide ¹⁴N/¹⁵N abundance ratios from a variety of molecular species, namely CN, HNC, HCN, NH₂D, and N₂H⁺. Comparisons show that the measured ratios considering opacity effects are basically consistent with the previous results within the uncertainties (see details in Table 6).

Table 6. Comparisons with ¹⁴N/¹⁵N Ratios from the Literature

Object	Species	α(2000)	δ(2000)	$\tfrac{{}^{14}N}{{}^{15}N}$	Beam Size	References
(1)	(2)	(3)	(4)	(5)	(6)	(7)
Orion-KL	NH3	05:35:14	−05:22:29	241 ± 71	40''	2, This paper
	NH3	05:35:14	−05:22:29	100 ± 51	40''	25, Gong et al. (2015)
	NH3	05:35:14	−05:22:46	170${}_{-80}^{+140}$	40''	5, Hermsen et al. (1985)
	HNC	05:32:46	−05:24:23	159 ± 40	63''	Adande & Ziurys (2012)
	CN	05:32:46	−05:24:23	234 ± 47	63''	Adande & Ziurys (2012)
Barnard-1b	NH3	03:33:20	+31:07:34	229 ± 62	40''	2, This paper
	NH3	03:33:20	+31:07:34	334 ± 50	33''	2, Lis et al. (2010)
	NH3	03:33:20	+31:07:34	300 ± 50	33''	2, Daniel et al. (2013)
	NH2D	03:33:20	+31:07:34	230${}_{-55}^{+105}$	29''	Daniel et al. (2013)
	CN	03:33:20	+31:07:34	290${}_{-80}^{+160}$	21''	Daniel et al. (2013)
	HCN	03:33:20	+31:07:34	330${}_{-50}^{+60}$	29''	Daniel et al. (2013)
	N2H+	03:33:20	+31:07:34	400${}_{-60}^{+100}$	27''	Daniel et al. (2013)
	HNC	03:33:20	+31:07:34	225${}_{-45}^{+75}$	28''	Daniel et al. (2013)
	NH2D	03:33:20	+31:07:34	470${}_{-100}^{+170}$	29''	Gerin et al. (2009)
NGC 1333	NH3	03:29:11	+31:13:26	247 ± 95	40''	2, This paper
	NH3	03:29:11	+31:13:26	344 ± 173	33''	2, Lis et al. (2010)
	NH2D	03:29:12	+31:13:25	360${}_{-110}^{+260}$	240''	Gerin et al. (2009)
W51 D	NH3	19:23:39	+14:31:07	230 ± 102	40''	2, This paper
	NH3	19:23:39	+14:31:10	660 ± 300 a	40''	6, Mauersberger et al. (1987)
	NH3	19:23:39	+14:31:10	400 ± 200 b	40''	13, Mauersberger et al. (1987)
G000.19	NH3	17:47:52	-28:59:59	40 ± 13	40''	2, This paper
Sgr A	NH3	17:45:52	-28:59:59	∼1000	40''	2, Güsten & Ungerechts (1985)
	HCN	17:45:52	−28:59:59	∼510	124''	Wannier et al. (1981)
Sgr B2	NH3	17:47:19	−28:22.08	210 ± 90	33''	12, Mills et al. (2018)

Note. Column (1) source name; column (2) species; column (3) R.A. (J2000); column (4) decl. (J2000); column (5) resulting nitrogen isotope abundance ratio; column (6) applied beam size of the telescope; column (7) references and the number of transitions (in case NH₃ has been used) that were considered for the determination of the column density.

^a The ¹⁴N/¹⁵N ratio from the rotation diagram method with the populations of nonmetastable levels. ^b The ¹⁴N/¹⁵N ratio from the rotation diagram method without the populations of nonmetastable levels.

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Orion-KL: This source has been observed in different species to measure the isotope ratio ¹⁴N/¹⁵N. Assuming LTE conditions, Hermsen et al. (1985) derived a ¹⁴NH₃/¹⁵NH₃ ratio of ${170}_{-80}^{+140}$. Subsequently, Adande & Ziurys (2012) performed observations of HN¹³C, H¹⁵NC, CN, and C¹⁵N toward this source. They obtained a ¹⁴N/¹⁵N ratio of 234 ± 47 from the ratio of the brightness temperatures of the strongest HF components of CN and C¹⁵N, weighted by the HF relative intensities. From their observations of HN¹³C and H¹⁵NC using the double isotope method, they derived a reasonably consistent ratio of 159 ± 40. Recently, a relatively lower value of 100 ± 51 was reported from NH₃ observations, but without considering optical depth corrections (Gong et al. 2015). The measurements of this source are within the uncertainties consistent with our result of 241 ± 71.

Barnard-1b, NGC 1333: The ¹⁴N/¹⁵N of Barnard-1b, NGC 1333 was measured by Gerin et al. (2009) and Lis et al. (2010) from observations of NH₂D and NH₃, respectively. The results from these two tracers are consistent within the uncertainties, with ¹⁴N/¹⁵N = 334 ± 50 and 344 ± 173 (from NH₃, Lis et al. 2010) and ${470}_{-100}^{+170}$ and ${360}_{-110}^{+260}$ (from NH₂D, Gerin et al. 2009) for Barnard-1b and NGC 1333. For Barnard-1b, the isotope ratio from NH₂D in Gerin et al. (2009) appears to be higher than that from NH₃ in Lis et al. (2010). Daniel et al. (2013) performed observations of multitracers (NH₃, NH₂D, CN, HCN, and N₂H⁺) to investigate its ¹⁵N fractionation. Assuming non-LTE conditions, they derived similar ¹⁴N/¹⁵N abundance ratios for all the tracers, independent of the chemical family. And they found a strong dependence of the column density of ¹⁵NH₂D on the excitation temperature. Using the same observational data for NH₂D as Gerin et al. (2009), they made a model analysis to obtain the excitation temperature of ¹⁵NH₂D instead of assuming the same excitation temperature for ¹⁵NH₂D and ¹⁴NH₂D (Gerin et al. 2009). Therefore they obtained a relative accurate ¹⁴N/¹⁵N value of ${230}_{-55}^{+105}$ for this source with respect to ${470}_{-100}^{+170}$ in Gerin et al. (2009). This is consistent with our ratio of 229 ± 62 from the rotation diagram method (LTE), which should reflect unsignificant non-LTE effects in our analysis.

W51D: Using Effelsberg data of 13 emission lines of NH₃ for a rotation diagram analysis (LTE), Mauersberger et al. (1987) derived a ¹⁴N/¹⁵N result of 660 ± 300 toward W51D, which is larger than our result of 230 ± 102 from both Effelsberg and TMRT data using the rotation diagram method. This large difference is probably caused by the fact that many NH₃ lines from levels with high energy above the ground levels were used in their analysis, instead of only the metastable (1, 1) and (2, 2) lines of NH₃ as in our analysis. Using only the metastable (1, 1) and (2, 2) lines of NH₃ and ¹⁵NH₃ in Mauersberger et al. (1987), we performed a consistent analysis and and obtained lower T_rot and ¹⁴NH₃/¹⁵NH₃ values of 24 ± 4.1 and 95 ± 36, which are consistent with our new results. In addition, the ¹⁴NH₃/¹⁵NH₃ ratio also critically depends on the population of the NH₃ nonmetastable ammonia levels (Mauersberger et al. 1987). The transition lines from high levels with different excitation conditions should trace denser regions (Goddi et al. 2015). As for our other sources, the measurements from the (1, 1) and (2, 2) lines, presumably representing the bulk of the gas due to their low excitation, are taken for our subsequent analysis.

Galactic center region: As mentioned before (see Section 1), the only direct measurements toward the Galactic center region obtained a very high value of ¹⁴N/¹⁵N of ∼1000 (Güsten & Ungerechts 1985), while extrapolations of the trend with galactocentric distances, extrapolated from the disk, indicate much lower values (Adande & Ziurys 2012; Colzi et al. 2018b). Recently, Mills et al. (2018) performed VLA mapping of ¹⁴NH₃ and ¹⁵NH₃ toward Sgr B2 (N) and measured ¹⁴N/¹⁵N ratios of ∼450 and a lower value of ∼200 for the resolved two hot cores N1 and N2, respectively. Our observations toward G000.19, about 30' away from Sgr A, provide a ¹⁴N/¹⁵N value of ∼40, which is much lower than previous results toward the Galactic center. Actually, nonuniform ratios for other isotopes were reported toward the Galactic center region. Zhang et al. (2015) mapped typical molecular clouds (six sources, including Sgr A, Sgr B2, Sgr C, and Sgr D) in the J = 1 − 0 lines of C¹⁸O and C¹⁷O and obtained different ratios of ¹⁸O/¹⁷O toward these sources, while all their ratios are lower relative to the molecular clouds in the Galactic disk. This indicates a chemical differentiation of the region, either due to a different origin of the gas, due to different degrees of nuclear processing inside the central molecular zone, or due to fractionation effects (Zhang et al. 2015; Loison et al. 2019). In addition, low isotope ratio values of ¹²C/¹³C (∼13) have recently been reported toward Orion-KL and other star formation regions from mid-IR observation data, which are believed to be not biased by chemical effects. As previously mentioned, the Galactic center region is not covered in current Galactic chemical evolution models (e.g., Romano et al. 2017, 2019). Unlike other isotope ratios that all report low (but nonuniform) ratios in the Galactic center region, both high and low ¹⁴N/¹⁵N ratios may be found in this region. This makes nitrogen "special" in this sense: it could imply strong effects due to both nucleosynthesis and chemical fractionation, in spite of the rather high kinetic temperatures in the Galactic center region (e.g., Ginsburg et al. 2016), which needs more measurements and modeling work.

Our observations may be biased emphasizing bright sources, with possibly systematically higher ¹⁴NH₃ opacities, which could lead to uncertain opacity corrections when trying to determine ¹⁴NH₃/¹⁵NH₃ ratios. In addition, sources at different distance imply that different linear beam sizes are covered by the telescope. A larger linear size of sources at larger distances may include more relatively diffuse low-density gas of different kinetic temperature, which could affect the isotope ratio results. In order to assess possible observational effects on the abundance ratios, we plot the abundance ratio against the heliocentric distance in Figure 6. The plot shows no systematic dependence on the ratio and the distance, which indicates that observational bias related to beam dilution is not significant. This is supported by the comparison of the abundance ratios for the three sources (Orion-KL, W51D, and G081.75) detected by both telescopes, which gives consistent ratio values within the uncertainties (see Table 5).

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To determine accurate isotopic abundance ratios from observed ¹⁴NH₃/¹⁵NH₃ line intensities, the possibility of chemical nitrogen fractionation should also be briefly discussed. Although there is a number of dedicated papers (Rodgers & Charnley 2008; Lis et al. 2010; Roueff et al. 2015; Colzi et al. 2018a; Wirström & Charnley 2018; Loison et al. 2019; Viti et al. 2019), the N fractionation is still a matter of debate.

The main mechanisms assumed to cause nitrogen fractionation are isotope-exchange reactions. N isotope-exchange reactions normally occur at low temperatures, with ¹⁵N enhancement in CO-depleted dense gas at low temperatures of <10 K (Adams 1981; Terzieva & Herbst 2000; Charnley & Rodgers 2002; Rodgers & Charnley 2008; Fontani et al. 2015; Colzi et al. 2018a; Loison et al. 2019). However, these reactions at low temperature should be inhibited by an entrance barrier, and thus the ¹⁴N/¹⁵N ratios do not change with time (Roueff et al. 2015; Wirström & Charnley 2018), which has also been demonstrated observationally by Fontani et al. (2015) and Colzi et al. (2018a).

Another possible mechanism for the N fractionation, isotope-selective photodissociation, was proposed by Heays et al. (2014), Visser et al. (2018), and Furuya & Aikawa (2018). ¹⁴N/¹⁵N fractionation is believed to be predominantly caused by isotope-selective photodissociation of N₂ rather than isotope-exchange reactions (Furuya & Aikawa 2018).

All our sources have known kinetic temperatures higher than 10 K, which may imply that the ¹⁴NH₃/¹⁵NH₃ ratios are not seriously affected by the N-fractionation effect. The plot of ¹⁴NH₃/¹⁵NH₃ against the kinetic temperature of sources (Figure 7) shows no significant correlation, which may indicate that fractionation effects are not a decisive factor affecting measurements. However, the nitrogen fractionation may be scale dependent, possibly representing a local effect, and observations with highly different beam sizes might provide different ¹⁴N/¹⁵N values (Colzi et al. 2019). Our measurements from single-dish telescopes with a relatively larger beam size may include more relatively diffuse low-density gas, which could be affected by the interstellar radiation field. Observations with high resolution should be helpful to probe the N-fractionation effect in both the molecular cores and outskirts and to determine accurate ratio values of ¹⁴N/¹⁵N.

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Figure 5 plots the measured ¹⁴N/¹⁵N isotope ratios from ¹⁴NH₃/¹⁵NH₃ (inverted triangles) against galactocentric distance. Our measurement suggests that the isotope ratio increases with galactocentric distance. Our sources belong to different stages of massive star formation, including four sources in IRDCs (green inverted triangles), eight sources associated with YSOs (red inverted triangles), and three sources next to H ii regions (black inverted triangles). Comparisons show that both measured ratios and the ¹⁴N/¹⁵N gradient with galactocentric radius are independent of the evolutionary stage. An unweighted linear fit was used to fit data (solid red line) in order to avoid biasing the results toward low values with small error bars. Our data provide a weak radial gradient of $\tfrac{{}^{14}{\mathrm{NH}}_{3}}{{}^{15}{\mathrm{NH}}_{3}}=(17.50\pm 13.14){D}_{\mathrm{GC}}+(53.91\pm 91.74)$, with a Pearson rank correlation coefficient of R = 0.35. ¹³

For comparison, previous measurements from HCN and/or HNC in Adande & Ziurys (2012) and Colzi et al. (2018b) are added as empty gray circles and diamonds, respectively. Fits to both data sets are shown as dashed and dotted lines in Figure 5, respectively . We find that our ratios tend to be slightly lower than the previous results, although a similar trend can be found. However, these results may suffer from uncertainties, using relatively early measurements of ¹²C/¹³C (Milam et al. 2005). The most recent results related to carbon isotope ratios, the only ones leading to a self-consistent interpretation of sulfur isotope ratios through the use of double isotope ratios involving ¹²C/¹³C (Yu et al. 2020), are reported by Yan et al. (2019). The authors presented observations of the K-doublet lines of H₂CO and ${{\rm{H}}}_{2}^{13}$ CO at the C (∼5 GHz) and Ku (∼15 GHz) bands toward a large sample of Galactic molecular clouds. Thus we modify these previous results (Adande & Ziurys 2012; Colzi et al. 2018b), taking the most recent ¹²C/¹³C ratios (Yan et al. 2019) and distance values from trigonometric parallax measurements into account (Reid et al. 2014, 2019). The modified previous results are plotted together with our results shown in Figure 5(b). We find that their ratios decrease by about 12%–15%, and the difference between their results and ours becomes smaller. This is visualized by the smaller gap between the two fitted (solid red and dashed blue) lines. However, the difference does not vanish. This may suggest that (1) our approach using exclusively the lowest metastable inversion lines of ammonia leads to too low ratios, (2) that fractionation plays a role, and/or (3) that the use of double isotope ratios adds uncertainties in the determination of nitrogen isotope ratios.

Theoretical models for GCE are important tools for understanding the isotopic ratio evolution in the Galaxy. Recently, new GCE models were developed to track the cosmic evolution of the CNO isotopes in the ISM of galaxies, yielding powerful constraints on their stellar IMF (Romano et al. 2017, 2019; Zhang et al. 2018). The theoretical ¹⁴N/¹⁵N gradient across the Milky Way disk is shown in the dash–dotted magenta curves (Model-5 in Figure 5 in Romano et al. 2017) and the dash–dotted yellow curves (Model-11 in Figure 6 in Romano et al. 2019) in Figure 5. Nucleosynthesis prescriptions in Model 5 (Romano et al. 2017) adopted the yields for low- and intermediate-mass stars, massive stars, super-AGB stars, and nova, while different initial rotation velocities for low-metallicity massive stars were also considered in Model-11 (Romano et al. 2019). The trend that the measured ¹⁴N/¹⁵N isotope ratios increase with galactocentric distance is consistent with the predictions of both models. And it is interesting that measurements show a "tentative indication" of the trend to decrease from 8 up to 10 kpc (but based on only three sources), which is similar to the predictions from both models. More data from the Galactic center and the sources at a large distance (>8 kpc) as well as more Galactic disk values with smaller uncertainty would still be highly desirable to better constrain this gradient.