Free Access
Volume 548, December 2012
Article Number L4
Number of page(s) 5
Section Letters
Published online 22 November 2012

Online material

Appendix A: DCO+–H2 collisional coefficients

Hyperfine-resolved DCO+–He rate coefficients have been published recently by Buffa (2012). It is generally assumed that rate coefficients with He multiplied by a factor of 1.4 can provide an estimate of rate coefficients with H2(J = 0) (Lique et al. 2008). However, for a molecular ion like DCO+, this approximation may be invalid, since the electrostatic interaction of an ion with H2 differs significantly from that with He. We have thus decided to determine hyperfine DCO+–H2 rate coefficients from close coupling (CC) HCO+–H2 rate coefficients () of Flower (1999) using the Infinite order sudden (IOS) approximation described in Faure & Lique (2012). Here, the isotopic substitution of HCO+ has been ignored since it is expected to have a little impact on the magnitude of the rate coefficients (Buffa 2012).

In DCO+, the coupling between the nuclear spin (I = 1) of the deuterium atom and the molecular rotation results in a weak splitting (Alexander & Dagdigian 1985) of each rotational level J, into three hyperfine levels (except for the J = 0 level which is split into only 1 level). Each hyperfine level is designated by a quantum number F (F = I + J) varying between |I − J| and I + J.

Within the IOS approximation, inelastic rotational rate coefficients can be calculated from the “fundamental” rates (those out of the lowest J = 0 level) as follows (e.g. Corey & McCourt 1983): (A.1)Similarly, IOS rate coefficients among hyperfine structure levels can be obtained from the rate coefficients using the following formula (e.g. Corey & McCourt 1983): (A.2)

where and {   } are respectively the “3 − j” and “6 − j” Wigner symbols.

The IOS approximation is expected to be inaccurate at low temperature. However, it is also expected to correctly predict the relative rates among hyperfine levels within a rotational J → J′ transition. Propensity rules are indeed properly included through the Wigner coefficients. As a result, Neufeld & Green (1994) have suggested to compute the hyperfine rates as (A.3)using the CC rate coefficients kCC(0 → L) of Flower (1999) for the IOS “fundamental” rates () in Eqs. (A.1) and (A.2).

In addition, the fundamental excitation rates were replaced by the fundamental de-excitation rates using the detailed balance relation: (A.4)This procedure is found to significantly improve the results at low temperature due to important threshold effects.

Then, from the HCO+–H2 rotational rate coefficients of Flower (1999), we have determined the IOS DCO+–H2 hyperfine rate coefficients using the computational scheme described above. The complete set of (de-)excitation rate coefficients is available online from the LAMDA1 and BASECOL2 websites.The present approach has been shown to be accurate, even at low temperature, and has also been shown to induce almost no consequence on the radiative transfer modeling compared to a more exact calculation of the DCO+–H2 rate coefficients (Faure & Lique 2012). However, we note that with the present approach, some hyperfine rate coefficients (those from the J = 1,F = 0 level to a J = F level) are strictly zero. This selection rule is explained by the “3 − j” and “6 − j” Wigner symbols that vanish for these kinds of transitions. Using a more accurate approach, these rate coefficients will not be strictly zero but will generally be smaller than the other rates. In addition, Faure & Lique (2012) have shown that it should imply almost no consequences for astrophysical modeling.

© ESO, 2012

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