An improved method to measure $\rm ^{12}C/^{13}C$ and $\rm ^{14}N/^{15}N$ abundance ratios: revisiting CN isotopologues in the Galactic outer disk

The variations of elemental abundance and their ratios along the Galactocentric radius result from the chemical evolution of the Milky Way disks. The $\rm ^{12}C/^{13}C$ ratio in particular is often used as a proxy to determine other isotopic ratios, such as $\rm ^{16}O/^{18}O$ and $\rm ^{14}N/^{15}N$. Measurements of $\rm ^{12}CN$ and $\rm ^{13}CN$ (or $\rm C^{15}N$) -- with their optical depths corrected via their hyper-fine structure lines -- have traditionally been exploited to constrain the Galactocentric gradients of the CNO isotopic ratios. Such methods typically make several simplifying assumptions (e.g. a filling factor of unity, the Rayleigh-Jeans approximation, and the neglect of the cosmic microwave background) while adopting a single average gas phase. However, these simplifications introduce significant biases to the measured $\rm ^{12}C/^{13}C$ and $\rm ^{14}N/^{15}N$. We demonstrate that exploiting the optically thin satellite lines of $\rm ^{12}CN$ constitutes a more reliable new method to derive $\rm ^{12}C/^{13}C$ and $\rm ^{14}N/^{15}N$ from CN isotopologues. We apply this satellite-line method to new IRAM 30-m observations of $\rm ^{12}CN$, $\rm ^{13}CN$, and $\rm C^{15}N$ $N=1\to0$ towards 15 metal-poor molecular clouds in the Galactic outer disk ($R_{\rm gc}>$ 12 kpc), supplemented by data from the literature. After updating their Galactocentric distances, we find that $\rm ^{12}C/^{13}C$ and $\rm ^{14}N/^{15}N$ gradients are in good agreement with those derived using independent optically thin molecular tracers, even in regions with the lowest metallicities. We therefore recommend using optically thin tracers for Galactic and extragalactic CNO isotopic measurements, which avoids the biases associated with the traditional method.

The production of metal elements and their isotopes depends on nucleosynthesis and stellar evolutionary processes (Burbidge et al. 1957;Meyer 1994).For example, the 12 C production is dominated by triple- reactions (Timmes et al. 1995;Woosley & Weaver 1995), while 13 C can be produced by the CNO-I cycle, 12 C-burning, or proton-capture nucleosynthesis (Meynet & Maeder 2002b;Botelho et al. 2020).The 14 N can be synthesized through the cold CNO cycle in the H-burning zone of stars (Karakas & Lattanzio 2014;Romano 2022), 12 C-burning in the low-metallicity fast massive rotators, and proton-capture reactions in the hot convective envelope of intermediate-mass stars (Marigo 2001;Pettini et al. 2002;Meynet & Maeder 2002a;Limongi & Chieffi 2018;Botelho et al. 2020).The production of 15 N and 13 C can happen in hot-CNO cycles in H-rich material accreting on white dwarfs (Audouze et al. 1973;Wiescher et al. 2010;Romano 2022).The Galactic 15 N is contributed by novae (Romano et al. 2017) and proton ingestion in the He shell of massive stars (Pignatari et al. 2015).
Despite extensive study, the isotopic ratio gradients in the lowmetallicity outer disk of the Milky Way remain poorly constrained.To date, measurements of only one target, WB89-391, at a Galactocentric distance  gc ≳ 12 kpc, have been made for the 12 C/ 13 C ratio using CN isotopologues (Milam et al. 2005).This single measurement (12 C/ 13 C ∼ 134) critically constrains the 14 N/ 15 N and 16 O/ 18 O gradients derived from C-bearing isotopologues.To fully constrain all CNO isotopic ratios in the outer Galactic disk, more data and more precise measurements of 12 C/ 13 C are needed.
The 12 C/ 13 C ratios in the ISM are determined using pairs of molecular isotopologues.Measurements of 12 C 18 O/ 13 C 18 O (e.g., Langer & Penzias 1990;Wouterloot & Brand 1996;Giannetti et al. 2014) or 12 C 34 S/ 13 C 34 S (Yan et al. 2023) may be limited to nearby strong targets because of the weak emission of 13 C 18 O or 13 C 34 S expected in the metal-poor outer disk clouds.Molecules such as H 2 CO/H 13 2 CO (Henkel et al. 1982(Henkel et al. , 1985;;Yan et al. 2019), CH + / 13 CH + (Ritchey et al. 2011), and CH/ 13 CH (Jacob et al. 2020) can be good tracers to derive 12 C/ 13 C through their absorption lines.H 2 CO isotopologues can be absorbed by the CMB but their absorption lines are still weak because of low abundance.The CH and CH + absorption lines are in rare cases of strong background continuum and/or high column density.The optically-thick line pairs, such as 13 CN and 12 CN  = 1 → 0, which require corrections to their optical depths that are mostly determined by fitting hyper-fine-structure (HfS) lines (Savage et al. 2002;Milam et al. 2005), could be adopted to measure 12 C/ 13 C in the outer disk regions.This method can also derive 14 N/ 15 N without any assumptions of 12 C/ 13 C (Adande & Ziurys 2012).
The current commonly used method for deriving the 12 C/ 13 C from CN isotopologues, with the HfS fitting, makes several assumptions and approximations.In order to derive the column density (i.e., Mangum & Shirley 2015), it assumes identical excitation temperature of 12 CN and 13 CN.However differential radiative trapping among spectral lines, due to different optical depths (if these are significant), can yield different excitation temperatures between the spectral lines.Also, a single gas phase is typically used (necessitated by the small number of transitions per species typically available, and lack of standard molecular cloud models), while a range of excitation conditions is present in molecular clouds impacting the volume-average excitation temperatures of the CN isotopologues.In addition, most studies (e.g., Savage et al. 2002;Milam et al. 2005) adopt the Rayleigh-Jeans (R-J) approximation, which becomes increasingly poor when ℎ ik / B  ex ≲ 1 starts approaching unity (and is of course inappropriate when ℎ ik / B  ex ≳ 1).In the cold ( kin ∼ 15 − 20 K), often sub-thermal line excitation conditions ( ex < kin ) prevailing in the bulk of molecular clouds in the quiescent ISM of the Galaxy, the R-J approximation is simply not good enough for abundance ratio studies conducted using the CN  = 1 → 0 and  = 2 → 1 at 112 GHz and 224 GHz.In addition, the cosmic microwave background (CMB) must be taken into account given that for the low-temperature molecular clouds where such isotopologue studies are conducted (and with the possible sub-thermal excitation of the lines utilized) , the line Planck temperature  ( ik ,  ex ) = ℎ ik / B  ℎ ik /   ex − 1 −1 can be so low that the  ( ik ,  CMB ) is no longer negligible for the frequencies used, even for the low  CMB,0 = 2.72 K) in the local Universe (see Zhang et al. 2016, for even more serious effects in the highz Universe where  CMB () = (1 + ) CMB,0 ).All these assumptions/approximations have been used for such studies conducted in widely different ISM environments, ranging from the very cold and quiescent ISM in the Galactic outer disk (Milam et al. 2005), where they become questionable, up to the warm, dense ISM in starburst galaxies (e.g., Henkel et al. 1993Henkel et al. , 2014;;Tang et al. 2019) where the aforementioned assumptions/approximations should be adequate.
In this work, we introduce an improved method to derive 12 C/ 13 C from CN isotopologues and compare it with current methods.We list the basic assumptions of the current methods of using CN isotopologues to derive 12 C/ 13 C and 14 N/ 15 N .We also present new 12 CN, 13 CN and C 15 N  = 1 → 0 observations in a small sample of molecular clouds with  gc > 12 kpc, which allows new constraints on the 12 C/ 13 C and 14 N/ 15 N outer gradients.Section 2 presents the observations and the data reduction.In Section 3, we list the current methods of deriving 12 C/ 13 C and 14 N/ 15 N with CN isotopologues and show their underlying assumptions and drawbacks.In Section 4, we introduce a new method to derive 12 C/ 13 C and 14 N/ 15 N .In Section 5, we present newly measured isotopic ratios and data derived from the improved traditional methods and the new method.In Section 6, we compare the Galactic 12 C/ 13 C and 14 N/ 15 N obtained by the different methods.We also compare CN isotopologues and the optically thin tracers 12 C 18 O/ 13 C 18 O, and we discuss physical and chemical processes that may bias the abundances.We present the main conclusion in Section 7.

Revision of distances
We update the Galactocentric distances, for both our targets and sources in the literature (Savage et al. 2002;Milam et al. 2005;Adande & Ziurys 2012;Giannetti et al. 2014;Jacob et al. 2020;Langer & Penzias 1990, 1993;Wouterloot & Brand 1996).For sources with direct trigonometric measurements (Reid et al. 2014(Reid et al. , 2019)), we adopt the measured values.For the others, we apply the up-to-date Galactic rotation curve model (Reid et al. 2019), which has been well calibrated with accurate trigonometric measurements, to obtain the kinematic distance.Specifically, we employed results based on the Parallax-  et al. 2002Milam et al. 2005Adande et al. 2012Giannetti et al. 2014Jacob et al. 2020Langer et al. 1990& 1993Wouterloot et al. 1996 This work The Sun Figure 1.Spatial distribution of sources in the Galactic plane.The pole is the Galactic center, and the red star shows the position of the Sun (Reid et al. 2019).Rings indicate the Galactocentric distances ( gc ).Red squares show our new observations.Black circles, blue multiplication signs, purple multiplication signs, green triangles, magenta thin diamonds, steel-blue rectangles, purple plus signs, and red diamonds, are positions of sources from Savage et al. (2002), Milam et al. (2005), Adande & Ziurys (2012), Giannetti et al. (2014), Jacob et al. (2020), Langer & Penzias (1990, 1993), and Wouterloot & Brand (1996), respectively.
Based Distance Calculator (Reid et al. 2016) 1 .The Galactocentric distances are then derived with the following equation: where  ⊙ = 8.15 kpc is the distance of the Sun from the centre of the Milky Way (Reid et al. 2019),  h is the heliocentric distance,  gc is the Galactocentric distance, and  is the Galactic longitude of the sources.
Figure 1 shows the locations of these sources.The Probability density functions (PDFs) generated by the distance calculator are presented in Appendix A. In Table 1 we list the coordinates, velocity at the frame of Local Standard of Rest (LSR) ( LSR ), estimated heliocentric distances, updated  gc , and observing time.
Planets, i.e., Saturn, Mars, and Venus, were used to perform the focus calibration, each after a prior pointing correction.In a few cases, we adopted PKS 2251+158, PKS 0316+413 and W3(OH) for the focus, when the planets were not available.These focus corrections were performed at the beginning of each observing slot and were repeated within 30 minutes after sunset or sunrise.We perform regular pointing calibration every 1-2 hours, with strong point continuum sources within the 15 • radius of targets, e.g., PKS 1749+096, PKS 0736+017, PKS 0316+413, NGC7538, and W3(OH).The typical pointing error is ∼ 3 ′′ (rms).
The observations were performed in two steps: we first performed an On-The-Fly (OTF) mapping towards each target, to get the spatial distribution of 13 CO  = 1 → 0 emission.Then we performed a single-pointing deep integration towards the emission peak position on the 13 CO  = 1 → 0 map.During the OTF mapping, we scanned along both right ascension (R.A.) and declination (Dec.)directions, with a spatial scan interval of 9.0 ′′ .Along the direction of each scan, it outputs a spectrum every 0.5 sec, which makes a 4.8 ′′ interval along the scan direction.Each OTF map covers an area of ∼ 2.4 ′ × 2.4 ′ .We adopted a positional beam-switch mode and performed deep integration at the peak position of 13 CO  = 1 → 0 for the extended targets.The OFF positions were set 10 ′ (in Azimuth) away from the target.For targets with compact 13 CO  = 1 → 0 emission (spatial FWHM < 1 ′ ), we use the wobbler switch mode.The beam switching used had a frequency of 2 Hz and a throw of 120 ′′ (in Azimuth) on either side of the target (to correct for any first-order beam-asymmetric beam effects between the two throw positions).
The beam sizes of the IRAM 30-m telescope are ∼ 22 ′′ and 14 ′′ at 110 GHz ( 13 CO) and 170 GHz (H 13 CN), with main beam efficiencies ( mb ) of ∼ 0.78 and ∼ 0.692 , respectively.We present the integration time of each target in Table 1.The noise levels of the final spectra are listed in Table 2. Particularly, we list the transitions of CN isotopologues and the main beam efficiencies ( mb ) of IRAM 30-m at the frequency of each transition in Table 3.

Data reduction
For data reduction, we used the Continuum and Line Analysis Singledish Software (CLASS) package from the Grenoble Image and Line Data Analysis Software (GILDAS, Guilloteau & Lucas 2000).For each sideband, three independent Fast Fourier Transform Spectroscopy (FTS) units cover a 4-GHz bandwidth, which sometimes causes different continuum levels on the same spectrum (so-called the platforming effect)3 .Therefore, we first split each spectrum into three frequency ranges (corresponding to the three units) and treat them independently.Then we locate the line-free channels and subtract a first-order baseline for each spectrum with command BASE.For spectra affected by apparent standing waves, which are less than 5% of the total, a sinusoidal function is adopted to fit and subtract the baseline.
Spectra at the same position are then averaged with the default weighting setup of TIME, by which the weight is proportional to the integrated time, frequency, and  −2 sys .In Table 2, we list the typical root-mean-square (RMS) of the antenna temperature in the  ★ A scale at 3-mm.For the IRAM 30-m telescope4 , the antenna temperature scale denotes one corrected only for atmospheric absorption5 .

Line intensities
We adopt the rest frequencies of molecular lines from NASA's Jet Propulsion Laboratory (JPL) 6 .We first fit a Gaussian profile to the 13 CO  = 1 → 0 spectra, which are all single peaked.Then, we use Δ= 8× FWHM/2 √ 2ln2 as the velocity range for other emission lines, by assuming that all lines of the same target have the same line width.The line-free channels are adopted as ∼ 4 × Δ at both sides of each line.Then we obtain the velocity-integrated-intensity in the velocity range of Δ, using the following equation: where Δ is the Full Width at Zero Intensity (FWZI) of the emission line, which is set to be 8× FWHM/2 is the main beam temperature; and  ★ A ,  eff , and  eff are the antenna temperature, the forward efficiency, and the telescope beam efficiency, respectively.
We derive the thermal noise error following Greve et al. (2009), with Eq. 3, which accounts for both the one associated with the velocity-integral of the line intensity over its FWZI and the one radio telescopes by Kutner & Ulich (1981),  ★ A designates a temperature scale corrected for atmospheric absorption and rearward beam spillover (see their Equation 14). 6https://spec.jpl.nasa.gov/ftp/pub/catalog/catform.htmlassociated with the subtracted baseline level (the latter becoming significant only if a wide line leaves little baseline "room" within a spectral window).
where   mb,chan is the channel noise level of the main beam temperature,  line is the number of channels covering the FWZI of the line,  base is the number of line-free channels as the baseline, and Δ res is the velocity resolution of the spectrum.The flux calibration and beam efficiency uncertainties (typically ∼ 10-15% in such single-dish measurements), are not included in our final line ratio uncertainties since all lines were measured simultaneously in our observations (the flux calibration and main beam efficiency factors are applied multiplicatively).
The two strongest satellite lines of 13 CN  = 1 → 0 ( rest at 108.780 and 108.782GHz) are blended, with a velocity separation of ∼ 6.0 kms −1 .We use the sum of their velocity-integrated intensities because they are very likely optically thin (see further discussion in Appendix B).The associated noise is obtained with Equation 3.

Upper limits of non-detected line fluxes
We define a line detection feature with the following three criteria: • I, More than three contiguous channels have  mb > 2  mb,chan , • II,  peak mb > 3  mb,chan , and • III,  line ≥ 3  line , i.e., S/N ≥ 3 For non-detected targets, we adopt 3  line as the upper limit of the velocity-integrated intensity.For blended lines, such as the satellite lines of 13 CN, we estimate the upper limits of the summed fluxes.First, we list the common assumptions in deriving 12 C/ 13 C and 14 N/ 15 N from emission lines of CN isotopologues.Here we take a simple example to obtain abundance ratios of 12 C/ 13 C and 14 N/ 15 N from 12 CN, 13 CN, and C 15 N with their  = 1 → 0 transition lines.For higher  levels, the method is essentially the same.To perform such derivations, several basic assumptions are needed for all models (details are shown in Appendix B) : (i) Column density ratios of isotopologues represent abundance ratios of isotopes, meaning that astrochemical effects are neglected.
(ii) In all regions, the populations at the energy levels that give rise to the HfS lines are assumed to have identical  ex among them.
(iii) Differences in the dipole moment matrix, the rotational partition function, the upper energy level, and the degeneracy of the upper energy between 12 CN, 13 CN, and C 15 N are ignored.
With these assumptions, ratios between column densities of 12 CN, 13 CN, and C 15 N (hereafter, we only consider the 12 CN and 13 CN pair, which is identical to the 12 CN and C 15 N pair.) equal their respective optical depth ratios, for  = 1 → 0: where 12 CN and 13 CN are column densities of 12 CN and 13 CN, respectively.12 CN and 13 CN are the total optical depths of 12 CN and 13 CN  = 1 → 0, respectively.
Ratios between the main beam temperatures of isotopologue lines would satisfy: where  ★ R, 13 CN is the line temperature measured from the observed spectra,  c, 13 CN is the beam efficiency,  ex, 12 CN is the excitation temperature of 12 CN  = 1 → 0 main component; and the factor 5/3 is a conversion factor from the column density ratio of the main components between 12 CN and 13 CN  = 1 → 0 to all components of 12 CN and 13 CN in this transition.
However, this Equation 6 and the corresponding Equations 2 and 3 in Savage et al. (2002), where  ★ R / c appears, contain two issues: First,  c is set as the antenna beam efficiency.This is incorrect since at the NRAO 12-m telescope the  ★ R scale is already corrected for both atmosphere and all telescope efficiency factors 7 , while  c stands for an irreducible (source-structure)-beam coupling factor, instead of a beam efficiency (see Kutner & Ulich 1981, for details).Second, the background correction is only considered in the denominator. ★ R / c =  R − bg , where  R and  bg are the source radiation temperature and background emission (the CMB), respectively.The excitation temperature,  ex in the nominator, on the other hand, does not subtract  bg .Unfortunately, the same problems exist also in Milam et al. (2005).
If both  ★ R and  c were set as their original definitions, i.e.,  ★ R is the observed source antenna temperature corrected for atmospheric attenuation, radiative loss, and rearward and forward scattering and spillover, and  c as the efficiency at which the source couples to the telescope beam, then Equation 6 still stands, as long as the background temperature is negligible, i.e. the target is warm enough compared to the CMB.However, the coupling factor,  c , is unknown because the source size and geometry are unclear, unless we assume that the sources are big enough to cover the whole forward beam. 7User's Manual For The NRA0 12M Millimeter-Wave Telescope, J.
Mangum, 01/18/00 Besides the aforementioned assumptions and problems the traditional method contains also the following unstated assumptions: • A beam filling factor of ∼1 for both 13 CN and 12 CN  = 1 → 0 lines.Equations 2 and 3 in Savage et al. (2002) can only be understood if the source geometric beam filling factor  s is set to ∼1 (i.e.extended targets fully resolved in both 12 CN and 13 CN), and the intrinsic (source structure)-beam coupling factor  c is also ∼1.
• The Rayleigh-Jeans approximation is adopted for expressing line radiation temperature, • Negligible contribution from the CMB emission, • The line optical depth is uniformly distributed within the beam size -a flat spatial distribution.
Most of these dense gas clumps are spatially compact within < 1 pc scales (e.g.Wu et al. 2010;Tafalla et al. 2002), especially for those targets from the outer Galactic disk.Most main-beam of single-dish telescopes could cover the emitting regions of 12 CN and 13 CN lines.Therefore, we update Equation 6 as follows, to accommodate the temperature definition by the IRAM 30-m telescope (with identical assumptions listed above): where  ★ A is the corrected antenna temperature, or, the forward beam brightness temperature (Wilson et al. 2013),  mb, 13 CN (=  eff  eff ) is the main beam efficiency of 13 CN  = 1 → 0, with  eff and  eff being the telescope beam efficiency and forward efficiency, respectively;  ex, 12 CN is the excitation temperature of 12 CN  = 1 → 0 main component; The factor 5/3 is a conversion factor from the column density ratio of the main components between 12 CN and 13 CN  = 1 → 0 to all components of 12 CN and 13 CN in this transition.
We re-organized this "traditional" method and present the detailed derivation in Appendix B. Note that a similar formula has also been adopted to derive 14 N/ 15 N (e.g., Adande & Ziurys 2012).

The hyper-fine structure of CN: fitting the optical depth
For 12 C 14 N and 12 C 15 N, the nuclear spin of 14 N and 15 N couples in the total angular momentum, which generates hyper-fine structures (here labeled by ; Skatrud et al. 1983;Saleck et al. 1994).For more complex 13 C 14 N, the angular momentum  first couples with the nuclear spin of 13 C atom to form an angular momentum  13 , which further couples with the nitrogen nuclear spin to form the total angular momentum  (Bogey et al. 1984).
For a  ex common among the various HfS CN satellite lines (which could be different from e.g., the rotational excitation temperature  rot , and  kin ), their corresponding optical depth ratios are fixed by the ratios of the corresponding  factors (line strengths) in the matrix element  ul (see Equations 62, 75 in Mangum & Shirley (2015), but also Skatrud et al. (1983)) that enters the expression of the Einstein coefficients  ul of the hyperfine lines.Should these lines be optically thin, the line optical depth ratios are also the ratios of line strengths (assuming a common  ex among the satellite lines involved).
One can derive the optical depths of 12 CN  = 1 → 0 lines from main beam temperature ratios between the nine components of the hyperfine structure lines (or, a subset of them, e.g.five components in Savage et al. 2002;Milam et al. 2005), using: where  mb,m, 12 CN and  mb,sat, 12 CN are the peak main beam temperature of the main component and the satellite component of 12 CN  = 1 → 0, respectively.We label the optical depths of the main component and the satellite component as  m, 12 CN and  sat, 12 CN , and  h is the intrinsic intensity ratio between the satellite line and the main component.This of course assumes that all these lines share the same  ex .This does not necessarily mean full local thermodynamic equilibrium (LTE).Only common excitation among HfS lines would work out, as  ex could be different from  rot or  kin ).We performed HfS fitting with the package developed by Estalella (2017), which shows better robustness and stability than the HfS fitting method provided in CLASS/Gildas (For details, see Appendix C).The fitted optical depths of the main line of 12 CN are shown in Table 4.

3.3
The updated HfS method to derive 12 C/ 13 C and 14 N/ 15 N In this section, we consider the effect of adopting the Planckequivalent radiation temperature scale, and a non-zero CMB temperature, respectively.We then introduce our updated equation to derive 12 C/ 13 C and 14 N/ 15 N that combines both reformulations.

Corrections to the Planck's Equation
The R-J approximation gives deviation in the methods in Savage et al. (2002) and Milam et al. (2005) to derive 12 C/ 13 C and the method in Adande & Ziurys (2012) to derive 14 N/ 15 N .The 12 C/ 13 C derived from the Planck's radiation temperature will be smaller than that derived under the R-J approximation.As expected (details in Appendix B), the decrease of 12 C/ 13 C after revision will be larger when  ex is smaller, in a non-linear fashion.At the frequency of 12 CN  = 1 → 0 main component (113.490GHz), the decrease is ≤ 5 %, and ∼ 0.5 for  ex ≳ 53 K and  ex ∼ 4.3 K, respectively.

Corrections to the CMB contribution
Considering the CMB temperature  CMB = 2.73 K as the background temperature (e.g., Equation 2.3 in Zhang et al. 2016), the derived 12 C/ 13 C will also be lower than those derived without CMB contribution (see in Appendix B).In this case, the term  ex, 12 CN should be replaced by  ex, 12 CN −  CMB in Eq. 7.After the CMB correction, the derived 12 C/ 13 C would decrease by <5% for a  ex ≳54 K (i.e.negligible).However, for  ex ∼5.4 K, and ∼ 3.0 K, 12 C/ 13 C would decrease by ∼50% and ∼90%, respectively.Some targets in Milam et al. (2005) and Savage et al. (2002) show  12 CN N=1→0 ex ∼ (3-5) K. Therefore the CMB corrections must be included.
Starting from Equations 4 and 5, we consider the complete expression of the Planck's Equation (i.e., abandon the R-J approximation) and the radiative transfer contribution from the CMB: where  m/sum 12 and  m/sum 13 are line intensity ratios between the main HfS component and the sum of all HfS components, for 12 CN and 13 CN  = 1 → 0, respectively.In Table 3, we list the relative intensities of all components of 12 CN  = 1 → 0 and the two strongest components of 13 are the antenna temperatures of 12 CN and 13 CN  = 1 → 0 main components, respectively.The main beam efficiency at the frequency of 12 CN and 13 CN main components are  mb, 12 CN and  mb, 13 CN , respectively, listed in Table 3.
The 14 N/ 15 N can also be derived from the same formula by replacing the parameters of 13 CN  = 1 → 0 with parameters of C 15 N  = 1 → 0. The relative intensities of the two strongest components of C 15 N  = 1 → 0 and the IRAM main beam efficiencies at the frequencies of these transitions are also listed in Table 3.

Converting Flux Ratios to 𝑇 b ratios
We assume that all line components share the same Gaussian-like line profile and the optical depth broadening does not play a significant role.In this case, the ratios between the integrated line flux  main / sat can represent the ratios between line brightness temperature  b,main / b,sat (assuming nearly identical line frequencies).
For targets without 13 CN or C 15 N  = 1 → 0 detections, we estimate the upper limits of the total velocity-integrated intensity  of the two strongest lines.

Radiative transfer for multiple 𝑇 ex layers
In real molecular clouds, the excitation temperature often shows an inhomogeneous distribution inside molecular clouds, as multiple  ex layers (e.g., Zhou et al. 1993;Myers et al. 1996).To test for the possible effects of such excitation temperature differentials on the derivation of 12 C/ 13 C and 14 N/ 15 N , we set up a simple toy model with two different  ex layers (Fig. 2).In this model, the background layer has a high excitation temperature  ex,H (H for hot) while the foreground layer has a low excitation temperature  ex,C (C for cold).Detailed description and derivation are shown in Appendix B4.1.
Such a model indicates that the column density ratio and the brightness temperature ratio estimated from optical depth would systematically deviate from the intrinsic ratios with a simplified one-layer assumption.This is because the measured excitation temperature of 12 CN and 13 CN  = 1 → 0 will change with the optical depth with multiple  ex layers.In our toy model, for  ex,H = 30 K,  ex,C = 10 K,  H =  C =  and 12 C/ 13 C (intrinsic)=60, the intrinsic column density ratio and the intrinsic brightness temperature ratio will be ∼ 10% and ∼ 15% lower than the estimated ratios, respectively, when the optical depth of 12 CN main component  main eff ∼ 1 (i.e.∼ 10% and ∼ 15% deviations from Eq. 4 and Eq. 5, respectively).When To observer

𝜏 main eff
∼ 1, which is close to our measured  of 12 CN main components in our targets, such deviations will cause a ∼ 17% decrease of the derived 12 C/ 13 C compared to the intrinsic ratio (red dashed line in Fig. 3 and details in Appendix B4.1).
Multiple  ex layers are significant when velocity spread is limited within linewidths of micro-turbulence and thermal motion, or when more than one fore/background cloud has identical  LSR .Observational evidence of self-absorption in 12 CO (Enokiya et al. 2021), HCO + (Richardson et al. 1986), and even 13 CO (Sandell & Wright 2010), which indicates multiple  ex layers, are found in multiple sources of the CN sample (Wouterloot & Brand 1989;Savage et al. 2002).
Other issues can also influence the derivation of 12 C/ 13 C .For example, the excitation temperature of 12 CN may be higher than that of 13 CN because of large optical depths and radiative trapping thermalizing the lines of the most abundant isotopologue at densities n = n crit,thin (1 − e −  )/ < n crit,thin .Non-LTE will also affect the derivation by changing the relative intensities between 12 CN line components.These are beyond the scope of this work and wait for future investigation.

The method to derive 𝑅12 C/ 13 C in extragalactic targets
The observations of CN isotopologues in the extragalactic starburst galaxies provide 12 C/ 13 C derived from the following equation (Henkel et al. 2014;Tang et al. 2019): Here,  I, 12 CN and  II, 12 CN are the total integrated intensities of 12 CN  = 1 → 0  = 3/2 → 1/2 and 12 CN  = 1 → 0  = 1/2 → 1/2, respectively. I and  II are the total optical depth of the blended line components in  = 1 → 0  = 3/2 → 1/2 and  = 1/2 → 1/2, respectively.This method will have the same deviation to 12 C/ 13 C in Section 3.5 when the targets have complex excitation layers.Because of the large optical depth of 12 CN and the small optical depth of 13 CN, the effective  ex of 12 CN transitions will be smaller than that of 13 CN in our toy model, which causes the underestimation of 12 C/ 13 C .In addition, the optical depth is derived from an equation similar to Eq. 5 (e.g., Eq. ( 1) in Tang et al. 2019).In our toy model, the  I and  II derived in this way may be overestimated.The  ex layers will be more complex in reality than in our toy model, while differential excitation of the lines due to radiative trapping will only add to such complexity.Thus the 12 C/ 13 C ratio estimated in this way still contains highly unclear uncertainties.
4 NEW METHOD: DERIVING 12 C/ 13 C AND 14 N/ 15 N WITH
The 12 C/ 13 C can be derived from the following equation: Here, is the integrated intensity ratio between the sum of the two satellite lines and the sum of all the nine lines of 12 CN  = 1 → 0.  I/sum 13 is the ratio between the integrated intensities of the two strongest line components and that of all the components of 13 CN  = 1 → 0.  1 and  2 are the integrated intensities of 12 respectively.13 CN is the integrated intensity of the two strongest components of 13 CN  = 1 → 0. With the criteria in Section 2.3.2, if  1 +  2 is larger than the 3- value of the corresponding integrated intensity, we treat it as a detection.
This method still assumes a common  ex among the energy levels involved in the lines used (i.e., the CTEX assumption, Mangum & Shirley 2015, their Section 12).However, we sum up the integrated intensities of the two satellite lines to deduce the effect from hyperfine anomalies of 12 CN, which has been observed in several studies (e.g., Bachiller et al. 1997;Hily-Blant et al. 2010).
Similarly, we can derive 14 N/ 15 N with the same method by replacing the relative intensity ratio and the integrated line intensity of 13 CN  = 1 → 0 with those of C 15 N  = 1 → 0.

RESULTS
With the improved HfS method (the traditional method after corrections) listed in Section 3.3 and the new method (the satellite line method) in Section 4, we obtained the Galactic 12 C/ 13 C and 14 N/ 15 N gradients and add our new 12 C/ 13 C and 14 N/ 15 N results in the Galactic outer disk.We also discuss the differences between these gradients from different methods.

Line Detection and HfS fitting results
Among the total 15 sources, 11 targets have detections (S/N > 3) of more than two 12 CN  = 1 → 0 satellite lines.One target, SUN15 18, has a detection of the main component of 12 CN  = 1 → 0. The other three targets only have non-detections of 12 CN.
We list the antenna temperatures of 12 CN, 13 CN and C 15 N  = 1 → 0 in Table 4.The detected 12 CN, 13 CN and C 15 N spectra are shown in Fig. 4. We show the spectra of 12 CN and 13 CN  = 1 → 0 non-detected transitions in Appendix D.
For sources with more than two 12 CN satellite line detections, our HfS fitting (Fig. D8 in Appendix D) shows that  12,main < 1 in most of them (Table 4).Among them, three targets have large errors of  12,main , so we do not include them in the following analysis.

12 C/ 13 C gradient from the HfS method
With the antenna temperatures of 12 CN and 13 CN  = 1 → 0, the 12 C/ 13 C derived with Eq. 9 are listed in Table 4.We compare our newly measured 12 C/ 13 C with those from CN observations reported in Savage et al. (2002); Milam et al. (2005).We update the Galactocentric distances of their targets and re-derive their 12 C/ 13 C with Eq. 9.In Fig. 5 (a), we show the Galactic 12 C/ 13 C gradients based on 12 CN/ 13 CN derived from Eq. 9, which is systematically lower than the gradient reported by Milam et al. (2005).

14 N/ 15 N gradient from the HfS method
Fig. 6 (a) displays our Galactic 14 N/ 15 N results from the HfS method.Besides 14 N/ 15 N derived from the CN isotopologues with the HfS method (Adande & Ziurys 2012), we show the 14 N/ 15 N from other tracers together.We excluded targets for which 14 N/ 15 N was calculated by multiplying a fitted gradient with 12 C/ 13 C from the literature (e.g., Milam et al. 2005).
Our 14 N/ 15 N gradient is also consistently lower than gradients reported in previous studies (e.g., Wilson & Rood 1994;Adande & Ziurys 2012;Colzi et al. 2018).We do revise 12 CN/C 15 N in Adande & Ziurys (2012) as what we do also for 12 CN/ 13 CN in Savage et al. (2002) and Milam et al. (2005).In addition, Figure 6 shows the 14 N/ 15 N derived from H 13 CN/HC 15 N multiplying 12 C/ 13 C in two of our targets with both 12 C/ 13 C and detected H 13 CN and HC 15 N  = 2 → 1.More descriptions are in Appendix E. shows the Galactic 12 C/ 13 C gradient from CN satellite lines.This gradient from optically thin satellite lines is systematically higher than the one derived from optical depth correction with HfS fitting.
In Table 5, we also show the 14 N/ 15 N of our targets derived from optically-thin 12 CN satellite lines and 13 CN.Fig. 6 (b) shows the Galactic 14 N/ 15 N gradient where all the ratios from CN isotopologues have been revised to the optically-thin results.The 14 N/ 15 N derived from optically-thin CN satellite lines is higher than those from optical depth correction, which makes the Galactic 14 N/ 15 N gradient higher than the one in Fig. 6 (a).In addition, we also show 14 N/ 15 N derived from H 13 CN/HC 15 N multiplying 12 C/ 13 C , where 12 C/ 13 C is obtained from 12 CN optically-thin satellite lines, discussed in Section 6.2 and more details are in Appendix E.

DISCUSSION
6.1 The revision to the 12 C/ 13 C gradient from the HfS method In this section, we discuss the Galactic 12 C/ 13 C and 14 N/ 15 N gradients along the Galactic disc that are derived by applying the corrections mentioned above and our new method from CN isotopologues.

The updated Galactocentric distances
We employ the updated Galactic rotation curve from Reid et al. (2019).This choice leads to changes in the Galactocentric distances ( gc ) of most sources in our sample compared to the values reported in Savage et al. (2002) and Milam et al. (2005).For most targets in the Galactic inner (outer) disk,  gc increases (decreases) after applying the new rotation curve in Reid et al. (2019), with a typical difference of Δ gc ∼ 0 − 4 kpc.One of our targets, WB89 437, has had its trigonometric parallax measured using VLBI in Reid et al. (2014), which indicated a Galactocentric distance of  gc = 13.10 ± 0.38 kpc.This value is consistent with the value of  gc = 12.79 ± 0.38 kpc derived from the Parallax-Based Distance Calculator.
Most of our targets are located in the anti-center direction, where the distance determination is not affected by confusion at the tangent point curve.However, most  gc values in the literature (e.g., Brand & Wouterloot 1995;Savage et al. 2002;Milam et al. 2005;Wouterloot et al. 2008;Giannetti et al. 2014) were derived from the Galactic rotation curve measured more than twenty years ago (e.g., Brand et al. 1986Brand et al. , 1988;;Brand & Blitz 1993;McNamara et al. 2000).This could introduce a strong bias in the derived abundance gradients.Table 4. 12 C/ 13 C and 14 N/ 15 N ratios in our targets (The traditional HfS method).a. Failed to do HfS fitting for satellite lines of 12 CN  = 1 → 0. b.Huge error bars of  in these targets.
Table 5. 12 C/ 13 C and 14 N/ 15 N ratios in our targets (The optically-thin 12 CN satellite line method).In contrast, the distances derived from the updated rotation curve from Reid et al. (2019) not only benefit from more accurate measurements of the trigonometric parallax with VLBI, but also agree with the parallax-based distances of Galactic Hii regions that rely on Gaia EDR3 data (Méndez-Delgado et al. 2022) for nearby targets.The updated Galactocentric distance estimates lead to a systematic reduction in the number of targets with  gc >10 kpc, yielding a steeper slope for the fitted gradient than the one in the literature where the previous, larger distances were used.For example, the Galactocentric distance of WB89 391 is now set to ∼12 kpc.This raises concerns about the previously reported high 12 C/ 13 C value for this source (Milam 2007) as this would seem reasonable for  gc at ∼16 kpc, but not so for the updated distance of ∼12 kpc where a 12 C/ 13 C ∼134 seems rather high.After the update, the Galactocentric distance of the sample is  gc <12 kpc.Fig. B2 in Appendix B2 illustrates the changes in 12 C/ 13 C before and after our revision.

Revised 𝑅12 C/ 13 C after the Planck Equation and CMB contribution corrections
As expected, the 12 C/ 13 C values (for the transitions of 12 CN, and 13 CN =1-0) derived using Equation 9 are systematically lower than those obtained using the R-J approximation and without considering the CMB temperature for targets in Savage et al. (2002); Milam et al. (2005), see Fig. B2.As we illustrated in Section 3.3.1 and Section 3.3.2,if the  ex of 12 CN is lower, the revision to 12 C/ 13 C will be larger.The  ex of 12 CN  = 1 → 0 of targets in Milam et al. (2005) is relatively lower ( ex ∼ 3 − 7 K) than those ( ex ∼ 6 − 12 K) from targets in Savage et al. (2002), so the revisions of targets in Milam et al. (2005) are larger than those in Savage et al. (2002).We note that this analysis assumes a common  ex among the lines used.This CTEX assumption (see also Mangum & Shirley 2015) is less constraining than LTE, and is likely to hold for the optically thin (or modestly optically thick) lines from rare CN isotopologues (unlike the more abundant isotopologues of CO where radiative trapping of more optically thick lines can yield different excitation temperatures among the isotopologues used).

Constraint on the outer Galactic disk -WB89 391
WB89 391 is located in the outer Galactic disc and was previously observed by Milam et al. (2005); Milam (2007).However, we did not detect the 13 CN line for this target.As the original fluxes and their associated errors for the 12 CN and 13 CN lines were not reported in Milam et al. (2005); Milam (2007), we could only make a rough estimate of the noise level based on the  = 1 → 0 spectra of 12 CN and 13 CN presented in Milam (2007).
At the rest frequency of 13 CN  = 1 → 0 main component (108.78GHz), the beam sizes of IRAM 30-m 8 and ARO 12-m 9 are ∼ 22 ′′ and ∼ 59 ′′ , respectively.Assuming a point source, the noise levels of 13 CN (in flux density) are ∼ 0.02 Jy and ∼ 0.16 Jy for our measurement and the measurement in Milam et al. (2005), respectively.Therefore, our noise level is approximately eight times deeper than that reported in Milam et al. (2005).
The typical sizes of dense molecular cores have been shown to be about 0.1 pc (Wu et al. 2010), which is much smaller than our IRAM 30-m beamsize which corresponds to ∼ 1 pc.Thus, we assume 8 https://publicwiki.iram.es/Iram30mEfficiencies 9described in Milam et al. (2005) a point source distribution for 12 CN.In our study, we measured a 12 CN flux density of about 2.4 Jy, while in Milam (2007), the flux density was about 2.3 Jy.The consistency between the two measurements suggests that this source is a compact target and the pointing directions of the observations were not severely offset from each other.
We also revisit the 12 CN/ 13 CN ratio of WB89 391, using Equation 9 with the Planck expression and the CMB correction, based on the optical depth and the peak temperature reported in Milam et al. (2005) and Milam (2007).Our newly derived 12 C/ 13 C for WB89 391 is ∼5.3 ± 1.7, which is about twenty-five times lower than the previously reported ratio of ∼134.
Our IRAM 30-m non-detection of WB89 391 sets a 3- lower limit of > 36 for the 12 CN/ 13 CN ratio, a result also supported by our recent NOEMA observation (Sun et al. in prep).Therefore, we recommend that the 13 CN detection report in Milam et al. (2005) needs to be reconsidered for future analyses of the 12 C/ 13 C gradient.
The 12 CN/ 13 CN ratios derived from optically thin satellite lines are systematically higher than those derived from optical depth correction.It will lead to systematically higher 14 N/ 15 N when they are derived from multiplying 12 C/ 13 C in optically-thin condition, compared with those multiplying the 12 C/ 13 C from -correction method.
In particular, we derive 14 N/ 15 N of G211.59 from two methods: (a).Using H 13 CN/HC 15 N multiplying 12 C/ 13 C (b). Directly using 12 CN and C 15 N  = 1 → 0. Using the H 13 CN and HC 15 N data from G211.59, we find that H 13 C 14 N/H 12 C 15 N = 4.7 ± 2.5.Using method (a), if we adopt that 12 C/ 13 C = 52.2± 6.7 derived from the HfS method, we will get 14 N/ 15 N ∼ 242±40.Otherwise, if we adopt that 12 C/ 13 C = 72.2± 9.5 derived from the optically-thin satellite line method, we will have 14 N/ 15 N = 336±57, which is much higher than the first value of 14 N/ 15 N .Using method (b), we get 14 N/ 15 N = 166 ± 32 from the HfS method and 14 N/ 15 N = 222 ± 43 from the satellite line method.The 14 N/ 15 N from optical-thin satellite lines is also higher than the one from the HfS method.There is a discrepancy between the 14 N/ 15 N values obtained using the method that directly uses CN isotopes and the method that relies on 12 C/ 13 C as a conversion factor.This discrepancy persists regardless of whether we employ the HfS method or the optical-thin method, and it may be attributed to astrochemistry effects.
Recently, new 14 N/ 15 N measurements on the Galactic outer disk were reported by Colzi et al. (2022).They obtained the 14 N/ 15 N value by computing the abundance ratios of H 13 CN/HC 15 N and HN 13 C/H 15 NC, which were then multiplied by the 12 C/ 13 C gradient value provided in Milam et al. (2005).This work does not include corrections to astro-chemical effects on HCN isotopologues, and the 14 N/ 15 N ratios derived from their analysis are systematically higher than ours.Particularly, the Galactic chemical evolution model in Colzi et al. (2022) shows lower 14 N/ 15 N values in the Outer disk (14 N/ 15 N ∼250-400 at  gc ∼ 12 kpc) than those derived from their observations, but model predictions remain consistent with 14 N/ 15 N derived in this work.
However, WB89 391 induces a strong bias to the 12 C/ 13 C gradient in Milam et al. (2005) which then propagates to the 14 N/ 15 N gradient, making 14 N/ 15 N ratios highly overestimated in the Galactic outer disk.If instead, the H 13 CN/HC 15 N and HN 13 C/H 15 NC ratios were multiplied by the 12 C/ 13 C values obtained from optically-thin CN satellite lines in our work, the resulting 14 N/ 15 N would be more consistent with our measurement of 14 N/ 15 N derived from 12 CN/C 15 N, with a small discrepancy possibly due to astrochemical effects.

Comparing CN with optically thin isotopic ratio tracers
Among the current tracers to derive 12 C/ 13 C , the abundance ratio of CO isotopologues (i.e., 12 C 18 O/ 13 C 18 O) have been adopted in plenty of Galactic targets, which gives a reliable sample size.In addition, C 18 O and 13 C 18 O lines are expected to be optically thin, which will significantly reduce the effect of different excitation layers (similar to CN isotopologues, see Section 3.5).
We select 12 C/ 13 C derived from 12 C 18 O/ 13 C 18 O with enough signal-to-noise ratios provided in the literature (Langer & Penzias 1990, 1993;Wouterloot & Brand 1996;Giannetti et al. 2014) and combine them with our 12 C/ 13 C derived from 12 CN/ 13 CN, shown in Fig. 7.The Galactocentric distances of all targets are derived with the same Galactic rotation curve model, following Reid et al. (2019).
Fig. 7 (a) overlays 12 C/ 13 C of Galactic outer disk clouds derived with the traditional method (the CN -correction method, 12 C/ 13 C re-calculated with data from Savage et al. (2002) and Milam et al. (2005)), and 12 C/ 13 C derived from 12 C 18 O/ 13 C 18 O collected in the literature.The 12 C/ 13 C ratios derived from 12 CN/ 13 CN (the traditional method) appear systematically lower than that derived from the C 18 O/ 13 C 18 O method.Fig. 7 (b) shows the same comparison as Fig. 7 (a), but now 12 C/ 13 C is derived with the Satellite-line method.The satellite line method seems to produce 12 C/ 13 C better matching the 12 C/ 13 C derived from C 18 O/ 13 C 18 O, compared with those from the traditional method.Some 12 C/ 13 C ratios have already been derived in the optically-thin condition in Savage et al. (2002), which do not change between Fig. 7 (a) and Fig. 7 (b).These 12 C/ 13 C ratios are consistent with 12 C/ 13 C from C 18 O/ 13 C 18 O.
In Fig. 7, we also show two previous linear fitted gradients.The previous 12 CN/ 13 CN gradient provided in Milam et al. (2005) is highly overestimated because of the R-J approximation and the neglect of the CMB temperature.The magenta one shows the fitted Galactic 12 C/ 13 C gradient with previous data points from multiple tracers in the literature, concluded in Jacob et al. (2020).This gradient is also influenced by the previously overestimated 12 CN/ 13 CN ratios.In the outer disk, both the 12 C 18 O/ 13 C 18 O and 12 CN/ 13 CN are expected to be lower than what the two previous gradients predict.
While there is significant scatter from each individual tracer, a general trend of increasing 12 C/ 13 C from the Galactic center to the outer disk is apparent.The results based on the CO isotopologues also show a dependence on the chosen −transition, i.e., 12 C/ 13 C derived from the  = 1 → 0 transition does not match with those obtained from the  = 2− → 1 lines (Wouterloot & Brand 1996), likely due to differential excitation from clouds (Jacob et al. 2020).The two new data points from our survey, G211.59, and WB89 380, match both increasing gradients fitted from CN and CO isotopologues.Recently, Yan et al. (2023) provide 12 C 34 S/ 13 C 34 S ratios also show an increasing 12 C/ 13 C gradient with 12 C/ 13 C ∼ 75 at  gc ∼ 10 kpc.Therefore, we would expect that the even further outer disk region ( gc > 15 kpc) may have even higher 12 C/ 13 C ratios if the low-metallicity fast rotators are not dominated in the chemical evolu-tion in this region.However, more observations are required to really constrain the chemical evolution in the further outer disk.In Appendix F, we show the linear fitted functions of the Galactic 12 C/ 13 C and 14 N/ 15 N gradients based on measurements from optically-thin conditions shown in Fig. 7 (b) and Fig. 6 (b), respectively.However, it is highly risky to use 12 C/ 13 C or 14 N/ 15 N from the linear fitting gradients instead of the direct measurement in individual targets (more details in Appendix F).

Astrochemical effects
Although the satellite method can reduce the deviation of abundance ratio measurements (Fig. 7 (b)), astrochemical effects may still bias the measured molecular abundances.This can introduce additional discrepancies between 12 C/ 13 C of the same target derived from different tracers, and that between different targets (thus in different chemical environments) but derived from the same tracer.
• UV self-shielding: When exposed to UV radiation, more abundant 12 C-bearing molecules can self-shield themselves in the inner clouds, making them less vulnerable to destruction by UV photons.On the other hand, the less abundant 13 C-bearing isotopologues would be more easily destroyed even in the dense interior of clouds.This self-shielding effect can lead to an overestimation of 12 C/ 13 C in emission line pairs such as C 18 O and 13 C 18 O, or CN and 13 CN (van Dishoeck & Black 1988;Röllig & Ossenkopf 2013).Given the high localization of strong Photon-Dominated Regions (PDRs) around O stars in molecular clouds (L ≤0.1 pc), such FUV-induced fractionization effects cannot affect globally-averaged isotopologue ratios over galaxy-sized molecular gas reservoirs, but they can affect local ISM regions like those observed in our study.
• Depletion: In dense and well-shielded regions, the temperature of dust can drop below the CO condensation temperature, typically around 17 K (e.g., Nakagawa 1980).As a result, CO isotopologues freeze out onto the surface of dust grains (e.g., Willacy et al. 1998;Savva et al. 2003;Giannetti et al. 2014;Feng et al. 2020).Observations suggest that the depletion factors of 12 C 18 O and 13 C 18 O vary with density or temperature, but no systematic differences have been found (Savva et al. 2003;Giannetti et al. 2014).
• Chemical fractionation: In addition, the difference between the zero-point energy of 12 CO and 13 CO causes the carbon isotopic fractionation reactions (e.g. Watson et al. 1976;Langer et al. 1984): The CN isotopologues have a similar effect (Kaiser et al. 1991;Roueff et al. 2015): 13 C + + 12 CN ↔ 13 CN + 12 C + + 31.2K (13) Both gas-phase and grain-phase chemical networks have been adopted to model such fractionation effects on CO and CN molecules (e.g.Smith & Adams 1980;Roueff et al. 2015;Colzi et al. 2020;Loison et al. 2020).The experiments in Smith & Adams (1980) show 13 C and 18 O fractionation with a temperature ≲ 80 K. Ritchey et al. (2011) suggest that 12 CO/ 13 CO and 12 CN/ 13 CN evolution have inverse trends in the fractionation model in diffuse gas.Gas-phase models in Roueff et al. (2015) show that the diversion of 12 CN/ 13 CN from original 12 C/ 13 C is not significant with a temperature ∼ 10 K.These models show stable C 14 N/C 15 N with an evolution time larger than 10 6 years.This is consistent with the fractionation work containing gas and dust (Colzi et al. 2018), where the 12 CN/ 13 CN becomes stable after an evolution time ∼ 10 7 years.The ratios derived from 12 CN/ 13 CN with the traditional method.Right: The ratios derived from 12 CN/ 13 CN with the optically-thin satellite line method.The grey dots, blue diamonds, and red squares are the 12 CN/ 13 CN of targets in Savage et al. (2002), Milam et al. (2005), and in this work, respectively.The spring-green squares, sea-green squares, and yellow-green triangles are C 18 O/ 13 C 18 O at their  = 1 → 0 transitions of targets in Langer & Penzias (1990, 1993), Wouterloot & Brand (1996) and Giannetti et al. (2014), respectively.The light-sea-green squares are C 18 O/ 13 C 18 O at their  = 2 → 1 transitions in Wouterloot & Brand (1996).The grey dashed line shows the previous 12 CN/ 13 CN gradient provided in Milam et al. (2005).The magenta dash-dotted line refers to the previous Galactic 12 C/ 13 C gradient fitted with ratios derived from multiple tracers, provided in Jacob et al. (2020).The green solid line and the blue dotted line refer to the linear fitting curves of 12 C/ 13 C from 12 C 18 O/ 13 C 18 O and 12 CN/ 13 CN, respectively.The color blocks show the 1- error range of corresponding linear-fitting curves.
Recent results in Colzi et al. (2022) further find that nitrogen fractionation partly contributes to the scatter in the 14 N/ 15 N measurements, while higher angular resolution observations are needed to disentangle local effects from nucleosynthesis contribution.
A detailed astrochemical study of the molecules used in such Galactic isotope ratio surveys, subjected to the varying far-UV radiation, cosmic ray, and pressure environments found in the sources used for such surveys will be particularly valuable.Should any  gcsystematic astrochemistry effects be found, such a study could yield valuable corrections to the (isotopologue)→(isotope) abundance ratio conversion and reduce the scatter in Figure 7.

Comparing with abundance ratios from stars
The 12 C/ 13 C ratio can also be measured from stars.Botelho et al. (2020) found local 12 C/ 13 C values of ∼ 70 − 100 in 63 Solar twins, higher than those derived from the interstellar medium (12 C/ 13 C ∼ 40 − 60 with  gc ∼ 8 kpc).Recently, Zhang et al. (2021) found an 12 C/ 13 C ratio of 97 +25 −18 in an isolated brown dwarf with an age of ∼ 125 Myr.This value is similar to the ratio found in the Sun (Ayres et al. 2013) and higher than the previously measured 12 C/ 13 C ratio in the Solar neighborhood (Milam et al. 2005).This is reasonable since abundance ratios measured from stars reflect the abundance of their parental ISM typically several billion years ago, which may be different from those in the local ISM because of stellar moving and the time evolution of 12 C/ 13 C .Recent studies based on Gaia and LAMOST data reveal that young (<100 Myr) stellar clusters in the Solar neighbourhood exhibit a variety of metallicity in their member stars, which indicates strong inhomogeneous mixing or fast star formation (Fu et al. 2022).These young clusters span a wide range of age and non-circular orbits, indicating multiple spatial origins (Fu et al. 2022).

Homogeneous mixing in the Solar neighbourhood
Optical spectroscopic observations of N and O towards Galactic Hii regions indicate well-mixed gas in the Galactic plane(e.g., Esteban et al. 2022), at least in the second and the third quadrants (Arellano-Córdova et al. 2020).Fig. G1 in Appendix G shows the location of those Hii regions and molecular clouds with 12 C/ 13 C in this work.Most targets are located in the same quadrants as those Hii regions, indicating similar abundance ratios at the same Galactic radii.
However, these studies only consider the major isotopes of carbon and nitrogen, and the behavior of the rare isotopes 13 C and 15 N may not be the same as that of the major isotopes.In fact, the production and enrichment of these rare isotopes often involve different mechanisms from those of the major isotopes, such as novae (e.g., Romano & Matteucci 2003;Cristallo et al. 2009;Romano 2022).Furthermore, molecular tracers can be affected by additional chemical biases that may increase the scatter in measured abundances compared to those obtained from stellar tracers.

Future prospects
Our analysis of the different methods used to derive 12 C/ 13 C and 14 N/ 15 N indicates that the radiative transfer conditions may play an important role in affecting the basic assumptions of those methods.In particular, deriving isotopic ratios from lines of two molecular transitions with significantly different optical depth values (e.g., 12 CN and 13 CN  = 1 → 0 main components, with ∼1 and ∼0.01,respectively) after performing optical depth corrections is highly non-trivial because of the expected non-uniform excitation conditions within the sources.
For the Galactic targets, parts of the 12 C/ 13 C and 14 N/ 15 N have been derived from 12 CN/ 13 CN/C 15 N (Savage et al. 2002;Milam et al. 2005; Adande & Ziurys 2012) or 14 NH 3 / 15 NH 3 (Chen et al. 2021), with optical depth correction.After correcting the results using the full Planck function approximation and including the  CMB , the results are still subjected to the uncertainties of the underlying line excitation structure of the sources.It is always preferable to use the optically-thin line components, such as C 18 O/ 13 C 18 O or the opticalthin satellite lines of 12 CN and 14 NH 3 with their isotopologues to derive 12 C/ 13 C and 14 N/ 15 N in Galactic targets.
For extragalactic sources the 12 C/ 13 C can be derived from the 12 CN/ 13 CN with optical depth correction (Henkel & Mauersberger 1993;Henkel et al. 2014;Tang et al. 2019) but this method has the same issues as those used in the Milky Way, and it is hard to separate the optically-thin satellite line from the blended 12 CN line components.It is better to use isotopologues with optically thin transitions (e.g., C 18 O/ 13 C 18 O) for deriving isotopic ratios, but this requires longer integration time with high sensitivity instruments.According to our 2/15 detection rate of 13 CN with an integration time ∼ 0.5 − 10 hours of our targets (see in Table 1, 4 and 5), we suggest that an observing time of more than six hours may be required to detect 13 C 18 O in the targets of the outer disk with a sensitivity similar to that of IRAM 30-m.
On the other hand, all the current methods are based on LTE assumptions.Non-LTE will not only cause different excitation between isotopologues but also change the relative intensities of hyperfine structures (e.g., Henkel et al. 1998).These issues will be examined by the non-LTE models in the future.

CONCLUSIONS
We examine the assumptions and shortcomings of three different methods used to derive 12 C/ 13 C from 12 CN/ 13 CN namely: (a) the one using HfS fitting to do optical-depth correction of 12 CN, adopted in the literature, (b) the improved -correction method incorporating the Planck radiation temperature and the CMB contribution, (c) the method deriving 12 C/ 13 C from the ratio of optically-thin 12 CN satellite lines and 13 CN lines.We also point out the similar issues of the methods of deriving 14 N/ 15 N from 12 CN and C 15 N. We observed 12 CN, 13 CN, and C 15 N  = 1 → 0 towards 15 molecular clouds on the outer Galactic disk ( gc ∼ 12 kpc − 24 kpc).The Galactic 12 C/ 13 C and 14 N/ 15 N gradients obtained from different methods are shown by combining 12 C/ 13 C and 14 N/ 15 N derived from our new observations and those in the literature revised by our improved methods.Our results are as follows: (i).The current method for deriving 12 CN/ 13 CN in the literature adopts the Rayleigh-Jeans approximation and neglects the CMB, which then highly overestimates 12 C/ 13 C when the excitation temperature of 12 CN lines is <10 K.The improved -correction method using the Planck equation and considering the CMB avoids these systematic overestimates, however, it still adopts the assumption of uniform excitation conditions.We show the latter still yields the under-estimate of 12 C/ 13 C using a simple 2-phase excitation model.Scaling the intensity ratio of optically thin 12 CN satellite lines and 13 CN to the column density ratio of 12 CN/ 13 CN avoids the short-comings of the other two methods and is a better method for deriving reliable 12 C/ 13 C from CN isotopologues.
(ii).Our method requires long integration time.For most targets, we can only set lower limits, but for the objects with the longer integration time (G211.59and WB89 380), we measure 12 C/ 13 C ∼ 60.WB89 391, the target located at the outermost Galactic radius in the literature, shows no detection of 13 CN from our IRAM 30-m data, which is inconsistent with the previous result reported in Milam et al. (2005).With ∼ 8× better sensitivity, our new data sets a lower limit of 12 C/ 13 C ≳ 36 in WB89 391 in the optically thin condition.We also give 14 N/ 15 N ∼ 220 from 12 CN/C 15 N for one target and 14 N/ 15 N ∼ 300 from H 13 CN/HC 15 N with 12 C/ 13 C for two targets, at  gc ∼ 12 kpc.These ratios are much lower than 14 N/ 15 N derived from H 13 CN/HC 15 N or HN 13 C/H 15 NC multiplying 12 C/ 13 C gradient in Milam et al. (2005), in better agreement with predictions of chemical evolution models.
(iii).The updated Galactic gradient of 12 C/ 13 C (derived from 12 CN/ 13 CN with -correction) yields systematically lower values than previous results.The updated 12 C/ 13 C gradient from CN optically-thin satellite lines is systematically steeper than the one from optical-depth correction.Such changes will regulate the Galactic 14 N/ 15 N deriving from other ratios (e.g., H 13 CN/HC 15 N) multiplying 12 C/ 13 C .In addition, the 12 C/ 13 C obtained from 12 CN/ 13 CN in the optically thin condition is more consistent with 12 C/ 13 C from 12 C 18 O/ 13 C 18 O than the one derived from the -corrected method.

ACKNOWLEDGEMENTS
ZYZ and YCS acknowledge Prof. Rob Ivison and Dr. Xiaoting Fu for helpful discussions about this work.We also acknowledge the very helpful comments of our reviewer.This work is based on observations carried out under project numbers 031-17 and 005-20 with the IRAM 30-m telescope.IRAM is supported by INSU/CNRS (France), MPG (Germany), and IGN (Spain).This work is supported by the National Natural Science Foundation of China (NSFC) under grants No. 12173016, 12041305, 12173067, and 12103024, the fellowship of China Postdoctoral Science Foundation 2021M691531, the Program for Innovative Talents, Entrepreneurs in Jiangsu, and the science research grants from the China Manned Space Project with NO.CMS-CSST-2021-A08 and NO.CMS-CSST-2021-A07.DR acknowledges the Italian National Institute for Astrophysics for funding the project "An in-depth theoretical study of CNO element evolution in galaxies" through Theory Grant Fu.Ob. 1.05.12.06.08.

DATA AVAILABILITY
The 12 C/ 13 C and 14 N/ 15 N ratios from our new observations are listed in Table 4 and Table 5.The revised isotopic ratios from the targets in the literature can be derived with data provided in the corresponding works (Savage et al. 2002;Milam et al. 2005;Adande & Ziurys 2012;Colzi et al. 2018;Langer & Penzias 1990, 1993;Wouterloot & Brand 1996;Giannetti et al. 2014), via Equations in this paper.The original IRAM 30-m data underlying this work will be shared on reasonable request to the corresponding author.

APPENDIX A: THE PDFS OF ESTIMATED DISTANCES FOR OUR TARGETS
We show the PDFs of the distances generated by the Parallax-Based Distance Calculator (Reid et al. 2016) for our targets in Fig. A1, and  A2.Two of our targets, G211.59 and WB89 437, have direct parallax measurements of the water masers (Reid et al. 2019).So we use the parallax measurements to derive the Galactocentric distances for the two sources and do not show the PDFs of these two sources here.In this section, we illustrate our derivation of 12 C/ 13 C .We assume that the element abundance ratio 12 C/ 13 C equals the column density ratio between 12 CN and 13 CN (i.e.ignore the astrochemistry effects).The column density can be described as (Mangum & Shirley 2015):

APPENDIX B: THE DERIVATION OF A MORE COMPLETE THE 𝑅12
Here,  tot means the total molecular column density of all energy levels.ℎ is the Planck constant and  lu represents the dipole moment matrix element. tot represents the rotational partition function. u and  u represent the energy level and the degeneracy of the upperlevel u, respectively.In this work, we assume the molecular structures of 12 CN and 13 CN are similar so the difference between  lu ,  tot ,  u and  u can be ignored.We also assume a common excitation temperature  ex 10 .Then the column density ratio of 12 CN and 13 CN  = 1 → 0 can be described as: Here  12 and  13 represent the optical depth of 12 CN and 13 CN  = 1 → 0, respectively.We can derive the column density ratio of the main components of 12 CN and 13 CN  = 1 → 0 and then convert this ratio to the column density ratio of all the energy levels.We can set  12 and  13 as conversion factors from the main component to all states of 12 CN and 13 CN, respectively.That means: The  m,12 and  m,13 are the optical depth of the main component of 12 CN and 13 CN, respectively.
On the other hand, the equation of radiative transfer for a uniformexcitation source is (e.g., Mangum & Shirley 2015): Here  R is the beam-averaged source radiation temperature with  the source beam-filling factor and  bg the background temperature here assumed to be the CMB at 2.73 K.The beam filling factor: f=Ω s /Ω mb , with Ω s = ∫  n , with  d the beam forward-pattern (typically Gaussian for the IRAM 30m telescope at its operating frequencies, but not necessarily so for high-frequency sub-mm telescopes).
The   () expression gives the Rayleigh-Jeans equivalent temperature, but after using the Planck function for the black body radiation: If the line is optically thin, Equation B5becomes Insert Equation B7 into B1 and simplify the B1 as: Here the (mol,  ex ) refers to all the parameters in Equation B1 to the left of the integral sign.Then we get: as the source-averaged column density, with ⟨ tot ⟩ beam =   tot being the beam-averaged column density.Here it is important to note that while line ratio excitation analysis can constrain the  ex values, this cannot be done for the source beam filling factor unless independent source size estimates can be had (e.g. via past interferometric observations of the same molecules or concomitant line and/or continuum emitters (e.g.dust emission).Thus typically in the literature ⟨ tot ⟩ beam is typically reported.
If the line has some modest optical depths,  tot can still be derived after multiplying a -correcting factor  1− −  (Goldsmith & Langer 1999;Mangum & Shirley 2015), yielding: , but such corrections become unreliable for large line optical depths (≥2-3).For the small velocity range of the lines in this work (FWHM∼ 1.5 − 4 km s −1 ), the optical depth correction factor, (mol,  ex ), and  (,  ex ) factors can all be considered constant across the line width.Thus we can compute the velocity-averaged column density across the entire line profile (which yields the highest possible S/N ratio for spectral line observations) from Equation B10 as: For the 12 C/ 13 C derivation, with 12 CN  = 1 → 0 optically thick and the 13 CN  = 1 → 0 optically thin (thus a -correction factor only applies to 12 CN), and(mol,  ex ) and the source beam-filling factors  identical between the two lines, we can write: Now we consider the formula to calculate 12 C/ 13 C used by Savage et al. (2002) and Milam et al. (2005).The formula they used is: for all telescope efficiency factors (and of course atmospheric absorption).Here,  main is the optical depth of the main component of 12 CN and  ★ R, 13 CN is the measured radiation temperature of the 13 CN main component and 3/5 is the conversion factor from the ratio of main components to the ratio of all levels.
To deduce Equation B13, we need to obtain a formula with  ex, 12 CN ,  main and  ★ R, 13 CN .We replace  ★ R, 12 CN by inserting Equation B7 into Equation B12: Assuming the optical depth is nearly constant across the narrow line profile, we can further write: as the observed corrected source antenna temperature 11 of the 13 CN main component as reported by the (former) NRAO 12m-telescope we will have: Assuming the 13 CN beam filling factor  = 1, and converting the column density at one transition to the total molecular gas column density, we obtain: Using the Rayleigh-Jeans approximation ( ()≈ ) and ignoring CMB temperature  bg , yields the same equation as Equation B13, but without the erroneous  c factor. 1211 Corrected for atmospheric attenuation, radiative loss, rearward and forward scattering, and spillover, according to the NRAO 12-m manual. 12Here is a good opportunity to mention that an  c factor with  ★ R = c  R is used to denote a remaining non-trivial (source-structure)-beam coupling factor that exists even for resolved sources (Kutner & Ulich 1981).In such cases (where  d < s ) the source beam filling factor  , is longer  ∝ 1/Ω mb , but it converges to a value ∼  c than can still be <1 (assuming only the Gaussian part of the forward beam pattern coupling to the extended source).This factor cannot be corrected without strong assumptions about the source structure underlying the main beam pattern (an impossibility for molecular clouds), and thus it is assumed as ∼1 in line studies of resolved molecular clouds when an absolute value of  (∼  c ) is needed.For line ratio studies conducted for resolved molecular clouds, the less constraining assumption of a common  c factor among the various lines used is only needed.On the question of how we actually know that  c < 1, even for well-resolved molecular clouds (unless one observes especially compact regions such as Bok globules with high angular resolution), the answer is given from both theory and observations.The fractal-like structures of molecular clouds, with is assumed due to the unresolved mapping data, thus yielding a minimum  ex value.b.Huge errors are shown for SUN15 21, SUN15 56, and SUN15 57, which means that the ratio limit results for these three sources are not credible.

B2 The effects of the R-J approximation and the 𝑇 CMB emission
In order to quantify the effect of the R-J approximation we derive the excitation temperature  ex of 12 CN  = 1 → 0 for our targets, according to the following equation: We assume that  bg = CMB =2.73 K, with  ★ A and   of 12 CN  = 1 → 0 obtained from HfS fitting.The  ex of 12 CN is then derived from Eq. B18.The results are listed in Table B1.The mapping data of 13 CO is used to estimate the filling factor  , with the assumption that the shape of targets is round with an area including all pixels having  mb, 13 CO > 0.5  peak mb, 13 CO .We define the ratios between the revised 12 C/ 13 C and the previous 12 C/ 13 C , labeled as  compare .Then we consider three conditions: (a) Using the Planck equation and without  CMB : their supersonic gas motions, contain compact velocity-coherent clumps that are extremely small (∼10 −3 pc, e.g., Falgarone et al. 1998).These will certainly fail to completely fill in the angular area of any typical spectral line observations of molecular clouds, thus keeping the respective  c < 1, even if the cloud structures themselves appear well-resolved by the beams used.Multi- CO observations that include the easily thermalized optically thick CO  = 1 → 0 line, when modeled via various radiative transfer codes yield also expected emergent radiation temperature of the  = 1 → 0 line  R,J=1→0 .For cold clouds in the Galaxy this is typically ∼ (10 − 15) K, similar to the  kin and  dust of such clouds, yet the typical brightness temperature observed for well-resolved such clouds are  R ∼ (5 − 6) K (i.e.indicating an  c ∼1/3-1/2 factor).
2.0 2.7 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0  (b) With R-J approximation and with  CMB : Using the Planck equation and with  CMB : Fig. B1 shows the change of  compare with  ex .For all the conditions  compare will decrease when  ex decreases.In condition (a),  compare > 0.95, which means the decrease of 12 C/ 13 C after revision is ≤ 5 %, for  ex ≳ 53 K.  compare becomes ∼ 0.5 for  ex ∼ 4.3 K.In condition (b),  compare > 0.95 and ∼ 0.5 for  ex ≳ 54 K and ∼ 6.3 K, and ∼ 3.1 K, respectively.For case (c), used in this work, it is  compare >0.95, ∼ 0.5, and ∼0.1, for  ex ≳70 K, ∼6.3 K, and ∼ 3.1 K, respectively.We conclude that if  ex ≤10 K, the Planck equation and the CMB must be considered to derive 12 C/ 13 C from 12 CN/ 13 CN.
In Fig. B2, we show how the effects of the R-J approximation and the CMB temperature will revise the previous Galactic 12 CN/ 13 CN gradient derived from the HfS method.After the Galactocentric distance revision, the targets have a change of  gc at ∼ 0 − 4 kpc, and the  gc are limited to ≲ 12 kpc.The values of 12 C/ 13 C systematically decrease after the revision.Especially, the previous out-most constraint, 12 C/ 13 C in WB89 391, has been in an unphysical region after the revision, which means the previous measurement of this target is highly doubtful.

B3
The derivation of a -corrected 12 C/ 13 C formula Equation B17 contains the assumption that the filling factor  = 1.However, in this work, we take a more simple consideration, namely we assume  ex is the same for both 12 CN and 13 CN  = 1 → 0 main components.Because the corresponding frequencies are so similar for these two components the now frequency-dependant expressions  ( ex ) −  ( bg ) will also be very similar for the two lines (and of course the main beam efficiency factors  mb as well).We assume the spatial distributions of these isotopologues are similar so that  (or  c for resolved-source observations) is the same.According to Eq. B18, we then have: Here  b, 12 CN and  b, 13 CN are the peaks of the brightness temperature of the main components of 12 CN and 13 CN.We set  b = b, 12 CN / b, 13 CN , which is the brightness temperature ratio of the main components of 12 CN and 13 CN.With the assumption that the filling factor of 12 CN ( 12 CN ) equals the filling factor of 13 CN ( 12 CN ), we have  b = mb, 12 CN / mb, 13 CN . m,12 and  m,13 are the op-tical depths of the 12 CN and 13 CN main components, respectively.We also set: with  1213 = main, 12 CN / main, 13 CN being the column density ratio between the main components of 12 CN and 13 CN  = 1 → 0. Then we have  m,13 =  m,12 / 1213 .Inserting this into B22 yields: ) which, solving for  1213 , it simply yields: That is: We then convert the column density ratio of the main components to that of all levels of 12 CN and 13 CN  = 1 → 0. That is: Equation B27 is the one used in this work to derive 12 C/ 13 C with optical-depth correction.There is no need to consider particular  ex and filling factor  (as long as they are common for the lines/species used) once we convert the measured antenna temperature  ★ A to the main beam temperature  mb with the main beam efficiencies  mb at the frequency of 12 CN and 13 CN main components.This derivation takes into account the CMB and abandons the simple Rayleigh-Jeans approximation.It is thus more appropriate than Equation B13 for the low  ex often found in these sources (and Eq.B27 is unaffected by the error in Eq.B13 regarding the  c factor).
We now compare 12 C/ 13 C derived from Eq. B17 (labeled as Equation I) and Eq.B27 (labeled as Equation II) in Fig. B3.We use the  ex , ,  ★ A, 12 CN and  ★ A, 13 CN of targets in Savage et al. (2002); Milam et al. (2005) and derive 12 C/ 13 C in these targets with the two equations.
Because of the incorrect treatment of the irreducible beam coupling factor  c in Savage et al. (2002) and Milam et al. (2005) (Mentioned in Section 3), the excitation temperature they derived should be revised as follows: ex,revis.= ( ex,pre.−  CMB ) *  mb,12−m +  CMB (B28) Here  ex,pre.and  ex,revis.are the previous excitation temperature in Savage et al. (2002) and Milam et al. (2005), and the revised values of excitation temperature. mb,12−m = 0.82 is the main beam efficiency of NRAO 12-m telescope at the rest frequency of 12 CN  = 1 → 0 main component, which is incorrectly used in Eq.( 2) and (3) in Savage et al. (2002) with a misunderstanding symbol  c .The revised  ex are listed in Table .B2.
In Fig. B3, the 12 C/ 13 C ratios derived from Eq. B17 and Eq. 9 are mostly consistent within the errorbar.However, the ratios from Eq.B17 are slightly but systematically lower than those from Eq. 9, which is highly likely because the actual filling factor  ≤ 1.This comparison means the revision of the previous 12 C/ 13 C derivation is essential no matter whether we consider the  ex and the filling factor  or not.
Another possibility is that in Savage et al. (2002) and Milam et al. (2005) they derive  ex correctly by considering  c as the irreducible beam coupling factor, but express it wrongly as beam efficiency.In this condition, the ratios from Eq. B17 and 9 will be even more consistent than what we show in Fig. B3. 12 CN and 13 CN B4.1 A two  ex layer toy model In Fig. 2 we show a toy model cloud with two layers having different  ex for the same molecule/transition (e.g., 12 CN).From the right to the left, there is the background, a layer with a higher  ex , a layer with a lower  ex , and the observer.Indicatively, we set the high excitation temperature  ex,H =30 K and the lower excitation temperature  ex,C =10 K.The optical depth of the high  ex layer and the low  ex layer is  H and  C , respectively.The intensity at a certain frequency on the background, on the interface of the two layers, and the intensity observed by us are labeled as  ,0 ,  ,1 and  ,2 , respectively.

B4 Different excitation conditions for
We use this toy model in order to explore, in a simple manner, the possible effects of underlying excitation variations (i.e. a nonuniform  ex ) on the extracted abundance ratios Assuming uniform  ex conditions within each layer the emergent line brightnesses will be (e.g., Mangum & Shirley 2015): where   is the Planck function.Here, the spectral line optical depths are strong functions of frequency:  H,C = 0 ( −  0 ), with ( −  0 ) a normalized emission line profile, strongly peaked at some central frequency, and a width determined by the gas motions within each layer.The assumption of a common line profile function ( −  • ) =  C ( −  0 ) =  H ( −  0 ) for both gas layers is equivalent to the assumption of micro-turbulent and/or thermal gas velocity fields dominating both gas layers.Indeed only under this assumption, the radiative transfer formalism that follows is applicable 13 (with front layer gas able to absorb line radiation emanating from the back layer).
Thus we can combine the two previous equations to find: In spectroscopic observations, we often measure the line intensity against a non-zero continuum radiation field.This may emanate from the source itself (e.g., dust continuum, synchrotron) and/or from a background source (e.g., the CMB).This (line)-(continuum) differential measurement is obtained spectrally by subtracting the continuum as it is measured at neighbouring frequencies that are not line dominated.If the source itself does not emit any continuum, the subtraction of the OFF-source measurement (as done in single-dish observations) also yields such a (line)-(continuum) differential (with the OFF-line part of such spectrum ∼0).The intensity measured then is: Setting the background emission to be a blackbody means  ,0 =   ( bg ), and we will have: where  bg is the background source temperature which here we set to be the CMB temperature ( bg =  CMB = 2.73 K).Eq.B33 is the same as Eq. 2 in Myers et al. (1996), showing that the two-layer  ex model benefits understanding the line profiles of a collapsing cloud (e.g., Zhou et al. 1993;Myers et al. 1996).
An observer that assumes a uniform  ex single-layer source to analyze its line emission from the two layers, he/she will then assume an effective optical depth and an effective excitation temperature as  eff and  ex,eff , defined from: From Eq. B33 and Eq.B34, we then have: 13 For the macroturbulent/supersonic gas velocity fields of molecular clouds, the velocity gradients are so steep that  C ( −  0 ) ≠  H ( −  0 ) even for neighbouring gas cells.This is why line absorption occurs only locally within such cells, and line optical depths are not added across a line of sight (the LVG approximation used to solve radiative transfer in molecular clouds is based on this, with  =  ( ì  ) being a very strong function of the velocity field.) and reordering the Equation above, we obtain: We can find that  +  ≡ 1, it means that the effective   ( ex,eff ) is actually a weighted-averaged value of   ( ex,H ) and   ( ex,C ), with two weighting factors  and .The weighting factors  and  are both functions of  H and  C , which means the effective  ex of 12 CN and 13 CN emission lines should be different because of their different optical depth.
In Fig. B4 (a), we show the change of  and  with optical depth , assuming  H =  C = .The  and  have monotonous and opposite trends to change with .The effective temperature of 12 CN main component will be much lower than the effective temperature of the optically thin 13 CN main component.
If the linewidth is dominated by macro-turbulence or bulk motion, instead of thermal/micro-turbulence, the two-layer  ex model may not play a dominant role.This might also affect the case in this work, in which the observed linewidths are ∼ 1.5−4 km • s −1 .The radiation between gas layers may not be well-coupled.However, such radiative coupling effects would still bias the isotopic ratios for all methods that need optical-depth corrections.

B5 The deviation from original assumptions
Deriving 12 C/ 13 C from CN isotopologues has the basic assumptions including Eq. B22 and Eq.B23.However, considering the different effective  ex between 12 CN and 13 CN  = 1 → 0, and back to Eq. B1 and Eq.B5, we have: We have assumed that the filling factors 12 CN = 13 CN in Eq.B40, so that the brightness temperature ratio equals the main beam temperature ratio.Here we simplify Equations B39 and B40 with  1 and  2 : .We still ignore the difference between other molecular parameters in Eq.B1 of 12 CN and 13 CN, except for  u and  ex .
Assuming that the column density ratio of 12 CN and 13 CN  = 1 → 0 equals 60, and  H =  C = , we quantify the deviation from the basic assumptions in Eq. 4 and Eq. 5 with the factors  1 and  2 varying with the effective optical depth of 12 CN  = 1 → 0 main component  main eff .In Fig. B4 (b) and (c), we show the change of  1 and  2 as a function of  eff main .In our toy model with  ex,H = 30 K and  ex,C = 10 K, the intrinsic column density ratio will be smaller than the estimated column density ratio.If the optical depth is larger, the discrepancy will be larger.It means using Eq. 4 and Eq. 5 will overestimate the column density ratio and the brightness temperature ratio between 12 CN and 13 CN  = 1 → 0, respectively.When  main eff =  H + C ∼ 1, which is similar to the measured optical depth of 12 CN  = 1 → 0 main component in our targets, the intrinsic column density ratio will be ∼ 15 % lower than the estimated ratio, and the intrinsic brightness temperature ratio will be 10 % lower than the estimated one.
In Fig, 3, we quantify the total effects of these two discrepancies on the derived 12 C/ 13 C .Based on our toy model assumptions and adopting Eq. 4 and Eq. 5, using the HfS fitting method with the 12 CN main component will let the derived 12 C/ 13 C ∼ 17 % lower than the intrinsic 12 C/ 13 C when the optical depth of the 12 CN main component ∼ 1.However, if we use the 12 CN optical-thin satellite line (e.g.,  = 3/2 → 1/2,  = 1/2 → 3/2 at 113.520 GHz), the derived 12 C/ 13 C will be closed to the intrinsic column density ratio of 12 CN and 13 CN.

B5.1 More general conditions
The real excitation conditions of a molecular cloud may be much more complex than our toy model.The detailed deviation of the effective optical depth and effective excitation temperature of 12 CN and 13 CN is beyond this work.Here we free the values of  ex, 12 CN and  ex, 13 CN and analyze the deviation on the 12 C/ 13 C ratios if the excitation temperature of 12 CN and 13 CN  = 1 → 0 are different.
Considering the 1 and 2, Equation B27 should become: In Figure B5, we quantify the effect on our derived 12 C/ 13 C when  ex of 12 CN and 13 CN  = 1 → 0 are different.Especially, when  ex, 12 CN is much higher than  ex, 13 CN ,  2 will be high and cause a The ratio ( 1 ) between the intrinsic column density ratio and the estimated column density ratio, varying with the effective optical depth of 12 CN main component ( main eff =  H +  C ). (c).The ratio ( 2 ) between the intrinsic brightness temperature ratio and the estimated brightness temperature ratio, varying with  main eff .
large deviation.The total effect is also sensitive to the difference of  ex, 12 CN and  ex, 13 CN .The derived 12 C/ 13 C is not equal to the intrinsic abundance ratio when  ex, 12 CN equals  ex, 13 CN , because the upper energy levels  u of 12 CN and 13 CN have a slight difference.We ignore it in our 12 C/ 13 C derivation.

APPENDIX C: COMPARISON OF TWO HFS FITTING PROCEDURES
In this section, we test the performances of two procedures that can be used for the HfS fitting.We generate 12 CN  = 1 → 0 spectra artificially.Then we use the HfS procedure in CLASS and the procedure developed by Estalella (2017) to fit these spectra and test the accuracy and scatter of the results.We assume the line widths of 12 CN  = 1 → 0 components are the same.We also assume that both the intrinsic line and the optical depth  have Gaussian profiles.If  = 0, the profiles of the line components can be described as: Here  = FWHM/2 √ 2ln2 and we assume that  = 2 km s −1 . refers to the intrinsic intensity of a line component, so the total profile can be described as: Here   indicates the intensities of different line components.We assume  = 5 K for the main component.
For the line components with the optical depth, we have the following equation: Here  hf and  ★ A,hf are the central optical depth and the peak antenna temperature of a certain HfS line component, respectively, while  m and  ★ A,m refer to the same properties of the main component.Because the difference between the main beam efficiency at the frequency of the main component and the satellite lines is very small (< 0.1% for IRAM 30-m), we replace the  mb in Eq. 8 with the antenna temperature  ★ A .We have  hf =  hf *  m .Here,  hf is the column density ratio between an HfS line component and the main component.So line profiles can be described as follows: The basic equation HfS fitting procedures used is C3.With the known  hf and the observed antenna temperature  ★ A,hf of multiple HfS line components, the optical depth of the main component ( m ) and  can be derived by the fitting of line profiles.Then we can derive the excitation temperature  ex with Equation B18: Here we consider  ex in the Rayleigh-Jeans approximation because the procedure in CLASS output  ex under R-J approximation.We use this equation to derive  ex from the procedure of Estalella (2017) in our test.
We generated a list of values ranging from 0 to 5 to be the list of  m , with an interval at 0.1, to generate ideal 12 CN spectra.There are 500 spectra, which is enough for the test.Figure C1 shows the fitting results of the two procedures and the comparison with ideal values.The procedure from Estalella (2017) has a slightly larger RMS compared with the CLASS procedure, and the accuracy of the  ex is not as good as in CLASS.However, the stability of the procedure from Estalella ( 2017) is much better than the procedure in CLASS while the results from CLASS include some points that differ a lot from the ideal values.To avoid the risk that the fitting to the real spectra is too bad, we choose the procedure from Estalella (2017) to fit our observed 12 CN  = 1 → 0 spectra.

APPENDIX D: SPECTRA OF LINES
In this section, we show the spectra of the lines we used to derive 12 C/ 13 C and 14 N/ 15 N .In Fig. D1, we show the 13 CO  = 1 → 0 lines and the Gaussian fitting to these lines which derives the  LSR of our targets.In Figures D2 and D3 we show the 12 CN and 13 CN  = 1 → 0 spectra without 13 CN detections.We show the 12 CN and C 15 N  = 1 → 0 spectra in Figures D4.The

APPENDIX E: 14 N/ 15 N DERIVED FROM HCN ISOTOPOLOGUES
For targets with detected H 13 CN and HC 15 N  = 2 → 1 at 2-mm band, the 14 N/ 15 N can also be derived from their 12 C/ 13 C , as the following equation: Here  H 13 CN and  HC 15 N are the integrated intensities of the H 13 CN and HC 15 N  = 2 → 1, respectively.Because the integrated intensity we measured is the integral of antenna temperature and the velocity, we revise the difference between the main beam efficiency at the frequency of H 13 CN and HC 15 N  = 1 → 0. For IRAM 30-m, we have  mb,H 13 CN = 0.687 and  mb,H 13 CN = 0.688.
We have detected both H 13 CN and HC 15 N  = 2 → 1 in three targets, two of them have measurements of 12 C/ 13 C to get 14 N/ 15 N .The line intensities and derived 14 N/ 15 N from H 13 CN and HC 15 N  = 2 → 1 are shown in Table E1.The spectra of H 13 CN and HC 15 N  = 2 → 1 are shown in Figures D6 and D7.
In Table E1 and Table E2, we show the 14 N/ 15 N derived from multiplying 12 C/ 13 C from the -correction method and multiplying 12 C/ 13 C from optically-thin satellite lines, respectively.Adopting 12 C/ 13 C from the optically-thin satellite line method, the 14 N/ 15 N value in G211.59 will be ∼ 100 higher than the one adopting 12 C/ 13 C from the -correction method.For other targets, the change is not significant.(Savage et al. 2002;Milam et al. 2005;Langer & Penzias 1990;Wouterloot & Brand 1996;Giannetti et al. 2014;Jacob et al. 2020) and the red squares represent the targets in this work.The blue stars represent the location of H II regions in Arellano-Córdova et al. (2020).

Figure 2 .
Figure 2. The toy model of molecular clouds with two  ex layers. .

Figure 3 .
Figure 3.The theoretical values of derived 12 C/ 13 C compared with the intrinsic values of 12 C/ 13 C for the traditional HfS method and the satellite line method, varying with the optical depth of 12 CN main component.We assume that  ex,H = 30 K,  ex,C = 10 K,  H =  C =  and 12 C/ 13 C (intrinsic) = 60.

Figure 4 .
Figure 4. Spectra of 12 CN, 13 CN and C 15 N .The upper panels show spectra of 12 CN.The bottom panels are spectra of 13 CN in the left panel, the middle panel, and C 15 N in the right panel.Blue histograms show the spectra.Red solid lines show the relative location of hyperfine-structure transitions, labeled by red arrows.The gray dashed lines show the baseline of each spectrum.Left: The spectra of G211.59 with no smoothness, with the velocity resolution at ∼ 0.53 km • s −1 Middle: The spectra of WB89 380 after smoothing four channels into one, with the velocity resolution at ∼ 2.15 km • s −1 Right: The spectra of G211.59, showing the C 15 N detection with the velocity resolution at ∼ 0.53 km • s −1 .

Figure 7 .
Figure 7.The 12 C/ 13 C ratios obtained from different tracers.Left: The ratios derived from 12 CN/ 13 CN with the traditional method.Right: The ratios derived C/ 13 C FORMULA B1 The derivation of 12 C/ 13 C formula in the literature s  n ()Ψ s ()  , where  n is the beam pattern function, Ψ s the normalised source brightness distribution function (see Kutner & Ulich 1981, Equation 3) and  s the solid angle range subtended by the source, while Ω mb = ∫  d

Figure A1 .
Figure A1.The probability distribution functions of the target distances.The red dotted curve, green dashed curve, blue dash-dot curve, magenta dash-dot curve, and purple dash-dot curve refer to the distance PDFs from Spiral arm distribution, kinematic methods, parallax,  and , respectively.The black solid curve is the combined PDF from all these PDFs.The green dashed vertical line shows the distance estimated only by kinematic PDF and the back dotted vertical line shows the final estimated distance.

Figure A2 .
Figure A2.The probability distribution functions of the target distances (Continued).

Figure B1 .Figure B2 .
Figure B1.The effects of Rayleigh-Jeans approximation and background temperature.The solid lines show the change of  compare with the excitation temperature  ex , in the condition a (blue), condition b (green), and condition c (orange).The dashed lines show the  ex of 12 CN  = 1 → 0 in our sources and colorful blocks show the error range of  ex .

Figure B3 .
Figure B3.The comparison with 12 C/ 13 C results derived by Eq.B17 (Equation I) and Eq.B27 (Equation II).The blue dashed line refers to where the x-value equals the y-value.The color map shows the Galactocentric distances of the targets.
Figure B4.(a).The variation of the weighting factor  (blue solid line) and  (red dashed line) with optical depth.(b).The ratio ( 1 ) between the intrinsic column density ratio and the estimated column density ratio, varying with the effective optical depth of 12 CN main component ( main eff =  H +  C ). (c).The ratio ( 2 ) between the intrinsic brightness temperature ratio and the estimated brightness temperature ratio, varying with  main eff .

Figure B5 .
Figure B5.(a). the variation of  1 with  ex of 12 CN and 13 CN.(b): the variation of  2 with  ex of 12 CN and 13 CN.The contours show the values of  1 and  2 on the upper two figures.(c): assuming the optical depth of the main component of 12 CN ( m, 12 CN = 10 and  m, 12 CN = 0.1, respectively), the ratios between the derived 12 C/ 13 C from Equation B27 and 12 C/ 13 C from Equation B43.The contours show the value of these ratios.

Figure C1 .
Figure C1.Results for the two HfS fitting procedures.The black hollow squares and solid blue circles show the results from the HfS fitting procedure in CLASS and the procedure developed by Estalella (2017), respectively.The grey line indicates where the fitted values equal ideal values.(a).The comparison between the ideal  m value and the fitting values of the two procedures.The grey line refers to the ideal  m we generate.(b).The comparison of the RMS of the fitting to the 12 CN line profile by these two procedures.(c). ex derived from the two procedures.(d).Same meaning as (c) but zoom in on the y-axis.

Table 2 .
The RMS of 3-mm spectra in our targets.

Table 3 .
Transitions of CN isotopologues and IRAM 30-m main beam efficiencies.
main, 12 CN 10While strictly speaking LTE is when  ex = kin , it is unlikely that all the rotational energy levels would have a common excitation achieved without the aid of frequent collisions.

Table E1 .
14N/ 15 N ratios derived from HCN isotopologues in our targets (The traditional HfS method).

Table E2 .
14N/ 15 N ratios from HCN isotopologues with 12 C/ 13 C from 12 CN satellite lines in our targets.