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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 N₂ 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  cm³   s^-1 was adopted for these ion-neutral reactions. The zero-point energy differences are taken from TH00. The reactions and their rate coefficients are listed in Table A.2.

Appendix B: Collisional rate coefficients

The hyperfine rate coefficients for ¹³CN+H₂ and C¹⁵N+H₂ were derived from the fine-structure rate coefficients computed by Kalugina et al. (2012) for CN+H₂. The latter coefficients were determined from fully quantum close-coupling (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 ¹³C (with nuclear spin I = 1/2) and ¹⁵N (I = 1/2) substitutions in CN (in which ¹⁴N has a nuclear spin I = 1) significantly modify the hyperfine structure of the molecule. Yet, the fine-structure rate coefficients can be considered nearly identical for the three isotopologues. To derive the ¹³CN and C¹⁵N hyperfine rate coefficients, we used the infinite-order-sudden (IOS) approximation applied to the CN fine-structure rate coefficients of Kalugina et al. (2012). Within this approximation, where the fine-structure energy spacings are ignored compared to the collision energy, the rate coefficients between fine-structure levels (k_{Nj → N′j′}(T) for a ²Σ state molecule) can be obtained directly from the “fundamental” fine-structure 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 differ¹.

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 C15N(1–0) hyperfine multiplet. The boxes (full line) delineate the solutions for the C15N (cyan) and the 13CN lines (white). The dashed box shows the C15N solutions when the C15N column density is multiplied by factors of 7.5 and 7 for L1544 and L1498 respectively.

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For C¹⁵N, which possesses a single non-zero nuclear spin, the IOS rate coefficients among hyperfine-structure levels can be obtained from the rate coefficients as (B.2)where I = 1/2 is the nuclear-spin of ¹⁵N. In practice, the CN fine-structure 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 C¹⁵N were considered, corresponding to rate coefficients for all 210 transitions among levels with N ≤ 5.

For ¹³CN, which possesses two non-zero nuclear spins, the IOS rate coefficients among hyperfine structure levels can be obtained similarly, including an additional coupling: (B.5)where I₁ = 1/2 and I₂ = 1 are the nuclear spins of ¹³C and ¹⁴N respectively. The propensity rule is in this case Δj = ΔF₁ = ΔF and the scaling formula writes (B.6)In practice, the first 62 hyperfine levels of ¹³CN were considered, corresponding to rate coefficients for all 1676 transitions among levels with N ≤ 5.

© ESO, 2013