Appendix B: Collision rates for CO

We used rate coefficents for collisions of CO with H₂ which are based on values found in the literature but which have been extended to rotational quantum numbers $J_{\rm u} = 40$, although extrapolations to higher J is not excluded.

For $J_{\rm u} \le 29$ and for low temperatures, (5-400) K, we used the recent rates of Flower (2001; ortho-H₂-CO and para-H₂-CO; downward rates). For the higher temperature range of (>400-2000) K, the calculations by Schinke et al. (1985; para-H₂-CO; upward rates) were used. The matching between these data sets is roughly acceptable, but there exist disagreements (Fig. B.1), which reflect the differences in assumptions and computational methods (see the discussion of resonances by Flower).

In order to arrive at a consistent set of collision rate constants for the hole range of temperatures, the Schinke et al. data (correctly transformed to de-excitation rates; see also: Viscuso & Chernoff 1988) were laterally shifted to fit the Flower data at 400 K. A satisfactory matching was, however, not really possible for the lowest transitions connecting to the ground state, see the lower panel of Fig. B.1. In the figure, these expanded rates are labelled $\gamma ^{\rm I}$.

Figure B.1: Downward rate constants for collisions of CO with para-H2: Upper panel: as a function of J, where the solid lines are for $T = 5,~10,~20,~40,~60,~100~{\rm and}~250$ K from Flower (2001), and the dashed lines for $T = 100,~250,~500,~750,~1000,~1500~{\rm and}~2000$ K from Schinke et al. (1985). Lower panel: as a function of $T_{\rm kin}$, with the data by Schinke et al. adjusted to fit those by Flower at 400 K, expanding the rates for the low temperatures to T=2000 K (see the text). These rates are labelled $\gamma ^{\rm I}$ in the figure.

These $\gamma ^{\rm I}$ data span J-values up to 20 and temperatures between 400 and 2000 K. For the same temperature intervall, McKee et al. (1982, MSWG) have published calculations (He-CO; downward rates) for J-values up to 32 (Fig. B.2 upper panel). These rates ( $\times 1.37$) were divided into the Schinke et al. rates and fit by a polynomial to correct the shape of the McKee et al. data (Fig. B.3 upper panel), viz.

$$% \begin{array}{lll} \ln{\gamma^{\rm I}}/\ln{\gamma_{_{\rm MSWG}}} = a + b~J & & 20 \leq J \leq 32 \end{array}$$ (B.1)

to obtain a new set of rate coefficients

$$% \gamma^{\rm II} = \gamma_{_{\rm MSWG}}^{a + b~J}.$$ (B.2)

Figure B.2: Rate constants for CO as a function of J: Upper panel: the solid lines refer to the rescaled data of Fig. B.1, i.e. $\gamma ^{\rm I}$, whereas the dashes are for $T = 500,~750,~1000,~1500~{\rm and}~2000$ K from McKee et al. (1982) (collisions with He, scaled to H2, and up to J = 32). Lower panel: the fitted ratios of these rates ( $J_{\rm u} \le 20$) as a function of $J_{\rm u}$ and, as an example, for the five selected temperatures.

For the expansion to J=40, the $\gamma ^{\rm II}$ were fit to a second order polynomial, viz.

$$% \begin{array}{lll} \ln{\gamma^{\rm II}} = c + d~J + e~J^2 & & 7 \leq J \leq ^{29}_{32} \end{array}$$ (B.3)

the result of which was used for the extrapolations of the para-H₂ rates up to J = 40, viz.

$$% \begin{array}{lll} \gamma^{\rm III} = \exp{({c + d~J + e~J^2})} & & J > ^{29}_{32}. \end{array}$$ (B.4)

This provides us finally with the full set of collision rate coefficents for $J_{\rm u} \rightarrow 0$ for $J_{\rm u}$ up to 40 (Fig. B.3).

So far, we have considered rates only connecting to the ground state, i.e. $\gamma_{J-0}$. Flower (2001) did provide rates for collision transitions between all level's, but for higher J-values and/or higher temperatures we don't have that information. If the kinetic energy of the collision partners on the other hand is large compared to the rotational energy spacing of the CO molecule, the other rate cofficients can be obtained from (Goldflam et al. 1977; McKee et al. 1982)

$$% \begin{array}{lll} \gamma_{J-J'} = \left(2J'+1\right) \sum... ... \end{array} \right)^{2} A_{L,J}^{2}~~\gamma_{L-0} \end{array}$$ (B.5)

where the Wigner 3-j symbol designates the Clebsch-Gordan coefficients, which were computed exactly (see, e.g., Zare 1986). Further, the factor A_L,J

$$% A_{L,J} = \frac {6 + (a L)^2}{6 + (a J)^2},~{\rm where}\hsp... ...B_0~l_{\rm sca} \left ( \frac {\mu}{T} \right )^{\frac {1}{2}}$$ (B.6)

given by McKee et al. (1982), should correct for the decreased accuracy when the energy splittings become larger. $B_0 = 1.9225~{\rm cm}^{-1}$is the rotational constant (Lovas et al. 1979), $l_{\rm sca} = 3$ Å is a typical scattering length, $\mu = 3.5$ amu is the reduced mass of the colliding CO and He particles and T is the kinetic gas temperature.

Figure B.3: Upper panels: applying the curvature corrections to the data by McKee et al. results in the rates named $\gamma ^{\rm II}$ and shown by the solid lines. The extrapolations of $\gamma ^{\rm II}$ to high J-values are indicated by the dashed lines. Lower panel: the finally adopted collision rate coefficients for CO, $\gamma ^{\rm III}$, for the hole range in temperature, T= (5-2000) K, and energy, J = 0-40.

Finally, the rates for the inverse transitions were obtained from the condition of detailed balancing, viz.

$$% (2J^{\prime} + 1)~\gamma_{J^{\prime}-J} = (2J + 1)~\gamma_{J-J^{\prime}}~\exp{ (-h~\nu_{J-J^{\prime}}/k~T). }$$ (B.7)

For ¹³CO, we used the same rate constants as for CO, with the proviso that the mass difference potentially introduces an additional error on the collision rates (3.5%).