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Appendix A: Chemical model
We implemented the fractionation reactions of TH00, assuming a symmetry factor f(B,m) of unity for all reactions, unless N_{2} appears as a reactant or as a product. In these cases, f(B,m) = 0.5 or 1 respectively. A constant Langevin rate of 10^{9} cm^{3} s^{1} was adopted for these ionneutral reactions. The zeropoint energy differences are taken from TH00. The reactions and their rate coefficients are listed in Table A.2.
Elemental abundances (taken from Flower & Pineau des Forêts 2003), except for O. Numbers in parentheses are powers of 10.
Fractionation reactions and rate coefficients (k(T) = α(T/300)^{β}exp(−ΔE/T) cm^{3} s^{1}) implemented in our chemical network (from TH00).
Appendix B: Collisional rate coefficients
The hyperfine rate coefficients for ^{13}CN+H_{2} and C^{15}N+H_{2} were derived from the finestructure rate coefficients computed by Kalugina et al. (2012) for CN+H_{2}. The latter coefficients were determined from fully quantum closecoupling (CC) calculations based on a highly correlated potential energy surface. Rate coefficients were deduced for temperatures ranging from 5 to 100 K. Full details can be found in Kalugina et al. (2012).
The ^{13}C (with nuclear spin I = 1/2) and ^{15}N (I = 1/2) substitutions in CN (in which ^{14}N has a nuclear spin I = 1) significantly modify the hyperfine structure of the molecule. Yet, the finestructure rate coefficients can be considered nearly identical for the three isotopologues. To derive the ^{13}CN and C^{15}N hyperfine rate coefficients, we used the infiniteordersudden (IOS) approximation applied to the CN finestructure rate coefficients of Kalugina et al. (2012). Within this approximation, where the finestructure energy spacings are ignored compared to the collision energy, the rate coefficients between finestructure levels (k_{Nj → N′j′}(T) for a ^{2}Σ state molecule) can be obtained directly from the “fundamental” finestructure rate coefficients (those out of the lowest N = 0 level) as follows: (B.1)where ϵ is equal to +1 if the parity of initial and final rotational Nj level is the same or –1 if the parity of initial and final rotational Nj level differ^{1}.
Fig. B.1
LVG predictions at T = 10 K towards L1544 (left) and L1498 (right). In each panel, the grey scale shows the predicted intensity of the 110024.590 MHz component of the C^{15}N(1–0) hyperfine multiplet. The boxes (full line) delineate the solutions for the C^{15}N (cyan) and the ^{13}CN lines (white). The dashed box shows the C^{15}N solutions when the C^{15}N column density is multiplied by factors of 7.5 and 7 for L1544 and L1498 respectively. 

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For C^{15}N, which possesses a single nonzero nuclear spin, the IOS rate coefficients among hyperfinestructure levels can be obtained from the rate coefficients as (B.2)where I = 1/2 is the nuclearspin of ^{15}N. In practice, the CN finestructure energy spacings are not negligibly small and the IOS approximation is expected to fail at low temperature (T < 100 K). However, since it correctly predicts the relative rates among hyperfine levels (because the propensity rules are properly included through the Wigner coefficients), a simple method to correct the low temperature results is to scale the IOS results, as originally suggested by Neufeld & Green (1994): (B.3)In this approach the CC rate coefficients k^{CC}(0,1/2 → L,L + 1/2) must be employed as the IOS fundamental rates in both Eqs. (B.1) and (B.2). The scaling procedure thus ensures that (B.4)It should be noted that the propensity rule Δj = ΔF predicted by the recoupling approach is also properly reproduced by IOS approximation, as discussed in Faure & Lique (2012). We note that for CN there is also a strong propensity for transitions with even ΔNKalugina et al. (2012). In practice, the first 22 hyperfine levels of C^{15}N were considered, corresponding to rate coefficients for all 210 transitions among levels with N ≤ 5.
For ^{13}CN, which possesses two nonzero nuclear spins, the IOS rate coefficients among hyperfine structure levels can be obtained similarly, including an additional coupling: (B.5)where I_{1} = 1/2 and I_{2} = 1 are the nuclear spins of ^{13}C and ^{14}N respectively. The propensity rule is in this case Δj = ΔF_{1} = ΔF and the scaling formula writes (B.6)In practice, the first 62 hyperfine levels of ^{13}CN were considered, corresponding to rate coefficients for all 1676 transitions among levels with N ≤ 5.
© ESO, 2013