QUANTUM CALCULATION OF INELASTIC CO COLLISIONS WITH H. III. RATE COEFFICIENTS FOR RO-VIBRATIONAL TRANSITIONS

The CC calculations were performed using our scattering code described by Song et al. (2015), while the IOS calculations were carried out with the non-reactive scattering program MOLSCAT (Hutson & Green 1994). The scattering methods used to obtain cross sections are reported in detail in our previous paper (Song et al. 2015). Rate coefficients for specific ro-vibrational transitions were calculated by averaging the corresponding cross sections over a Maxwell–Boltzmann distribution of translational energies of the colliding particles,

$$\begin{eqnarray}&&r(T)={\left(\displaystyle \frac{8{k}_{{\rm{B}}}T}{\pi \mu }\right)}^{\displaystyle \frac{1}{2}}\displaystyle \frac{1}{{\left({k}_{{\rm{B}}}T\right)}^{2}}{\displaystyle \int }_{0}^{\infty }\sigma ({E}_{k})\mathrm{exp}\left(-\displaystyle \frac{{E}_{k}}{{k}_{{\rm{B}}}T}\right){E}_{k}{{dE}}_{k},\end{eqnarray} \tag{ 1 }$$

where μ is the reduced mass of H–CO, k_B is the Boltzmann constant and T is the gas temperature. Cross sections were calculated over an energy range from 0.1 to 15,000 cm⁻¹, see Song et al. (2015). The integral over collision energies was computed numerically with the trapezoidal rule after cubic spline interpolation of the cross sections on a logarithmic energy scale. The cross section σ(E_k) as a function of the collision energy E_k can be the state-to-state ro-vibrational cross section ${\sigma }_{v,j\to v^{\prime} ,j^{\prime} }({E}_{k}),$ the total vibrational quenching cross section ${\sigma }_{v,j\to v^{\prime} }({E}_{k})$ for the transitions from an initial v, j state to a final v' state, or the vibrational transition cross section ${\sigma }_{v\to v^{\prime} }({E}_{k}).$ The quantum numbers v and j refer to the vibration and rotation in the initial state, while v' and j' refer to the final state. The rate coefficients for the reverse transitions can be obtained by detailed balance

$$\begin{eqnarray}&&{r}_{v^{\prime} ,j^{\prime} \to v,j}(T)=\displaystyle \frac{2j+1}{2j^{\prime} +1}\mathrm{exp}\left(\displaystyle \frac{{\epsilon }_{v^{\prime} ,j^{\prime} }-{\epsilon }_{v,j}}{{k}_{{\rm{B}}}T}\right){r}_{v,j\to v^{\prime} ,j^{\prime} }(T),\end{eqnarray} \tag{ 2 }$$

where ${\epsilon }_{v,j}$ and ${\epsilon }_{v^{\prime} ,j^{\prime} }$ are the energies of the ro-vibrational levels. The state-to-state ro-vibrational cross sections ${\sigma }_{v,j\to v^{\prime} ,j^{\prime} }^{\mathrm{CC}}({E}_{k})$ are obtained from full CC calculations; the corresponding rate coefficients can be calculated from Equation (1). By summation of the state-to-state ro-vibrational cross sections over all final j' levels in the v' state, we obtain the total vibrational quenching cross section from a specific ro-vibrational initial state v, j to final state v'

$$\begin{eqnarray}&&{\sigma }_{v,j\to v^{\prime} }^{\mathrm{CC}}({E}_{k})=\displaystyle \sum _{j^{\prime} }{\sigma }_{v,j\to v^{\prime} ,j^{\prime} }^{\mathrm{CC}}({E}_{k}).\end{eqnarray} \tag{ 3 }$$

The corresponding rate coefficients are denoted by ${r}_{v,j\to v^{\prime} }^{\mathrm{CC}}(T).$ Averaging the rate coefficients ${r}_{v,j\to v^{\prime} }^{\mathrm{CC}}(T)$ over a thermal population of initial j states yields the vibrational transition rate coefficient based on the CC approach

$$\begin{eqnarray}&&{r}_{v\to v^{\prime} }^{\mathrm{CC}}(T)=\displaystyle \frac{{\displaystyle \sum }_{j}{g}_{j}\mathrm{exp}(-\displaystyle \frac{{\epsilon }_{v,j}}{{k}_{{\rm{B}}}T}){r}_{v,j\to v^{\prime} }^{\mathrm{CC}}(T)}{{\displaystyle \sum }_{j}{g}_{j}\mathrm{exp}(-\displaystyle \frac{{\epsilon }_{v,j}}{{k}_{{\rm{B}}}T})},\end{eqnarray} \tag{ 4 }$$

where g_j = 2j + 1 is the degeneracy of the rotational level j.

The vibrational transition cross sections ${\sigma }_{v\to v^{\prime} }^{\mathrm{IOS}}({E}_{k})$ are obtained directly from scattering calculations in the IOS approximation. The corresponding rate coefficients can again be calculated from Equation (1).

If we wish to calculate all ro-vibrational rate coefficients of interest for astrophysical modeling, the full CC method is prohibitively expensive. The incompleteness of the available ro-vibrational rate coefficients forced astronomers to use extrapolated data, commonly based on a complete decoupling of vibration and rotation (Faure & Josselin 2008; Thi et al. 2013; Bruderer et al. 2015). Here, we introduce a new extrapolation method for state-to-state ro-vibrational rate coefficients in which we assume the coupling in $v,j\to v^{\prime} ,j^{\prime}$ transitions with v' < v to be the same as in the $v=1,j\to v^{\prime} =0,j^{\prime}$ transition. Then, the state-to-state ro-vibrational rates are related to the corresponding rates for the $v=1,j\to v^{\prime} =0,j^{\prime}$ transitions as follows

$$\begin{eqnarray}&&{r}_{v,j\to v^{\prime} ,j^{\prime} }(T)={P}_{{vv}^{\prime} }(T){r}_{1,j\to 0,j^{\prime} }(T),\end{eqnarray} \tag{ 5 }$$

where the factor ${P}_{{vv}^{\prime} }(T)$ is defined as

$$\begin{eqnarray}&&{P}_{{vv}^{\prime} }(T)=\displaystyle \frac{{r}_{v\to v^{\prime} }(T){\displaystyle \sum }_{j}{g}_{j}\mathrm{exp}(-\displaystyle \frac{{\epsilon }_{v,j}}{{{k}}_{{\rm{B}}}T})}{{\displaystyle \sum }_{j}[{g}_{j}\mathrm{exp}(-\displaystyle \frac{{\epsilon }_{v,j}}{{{k}}_{{\rm{B}}}T}){\displaystyle \sum }_{j^{\prime} }{r}_{1,j\to 0,j^{\prime} }(T)]}=\displaystyle \frac{{r}_{v\to v^{\prime} }(T)}{{r}_{1\to 0}(T)}.\end{eqnarray} \tag{ 6 }$$

The difference with the extrapolation method used by Thi et al. (2013) and Faure & Josselin (2008) lies in the replacement of ${r}_{0,j\to 0,j^{\prime} }$ by ${r}_{1,j\to 0,j^{\prime} }.$ Although this change seems minor, the improvement in the estimated rate coefficients is substantial. We will show this in detail in Section 3, where we compare the extrapolated rates of $v=2,j\to v^{\prime} =1,j^{\prime}$ transitions with results obtained directly from CC calculations. The ro-vibrational rate coefficients ${r}_{1,j\to 0,j^{\prime} }(T)$ have been calculated using the full CC method, while the vibrational rate coefficients ${r}_{v\to v^{\prime} }(T)$ were obtained from the IOS approximation. All other ro-vibrational rate coefficients ${r}_{v,j\to v^{\prime} ,j^{\prime} }(T)$ can then be obtained from the extrapolation formula, Equation (5). Another advantage of our method is that it also yields the values for $v,j\to v^{\prime} ,j^{\prime} =j$ transitions; these could not be obtained with the previous extrapolation method since the data for rotationally elastic $j\to j$ transitions were not tabulated. And even if they had been available, they most likely would have given much too large extrapolated results.

Cross sections for ro-vibrational $v=1,j=0-30\to v^{\prime} =0,j^{\prime}$ transitions from full CC calculations are reported in our previous paper (Song et al. 2015). We also concluded in that paper that the coupled states approximation is only suitable for collision energies above ≈1000 cm⁻¹ for the H–CO system and it is therefore not applied here. Table 1 lists the ro-vibrational transition rate coefficients for a gas temperature range of 10 K ≤ T ≤ 3000 K calculated from these cross sections with the use of Equation (1). The highest final j' = 27−42 values for which the rates are given in this table depend on the initial j quantum number. Transitions for even larger final j' are not reported, either because they are negligibly small, or because they were not completely converged with the largest number of partial waves, i.e., the highest total angular momentum J included. However, we add all final j' states, including those that may not be fully converged, to calculate the vibrational quenching cross sections ${\sigma }_{v,j\to v^{\prime} }^{\mathrm{CC}}({E}_{k})$ and the vibrational transition rate coefficients ${r}_{v\to v^{\prime} }^{\mathrm{CC}}(T).$ Hence, the CC vibrational quenching rate coefficients shown in Figure 1 are slightly different from the values that would be obtained by summing and averaging only the reported rates ${r}_{v,j\to v^{\prime} ,j^{\prime} }^{\mathrm{CC}}(T).$

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Table 1.  Rate Coefficients from CC Calculations for Ro-vibrational $v=1,j=0-30\to v^{\prime} =0,j^{\prime}$ Transitions in the Temperature Range of 10 K $\leqslant \;T\;\leqslant$ 3000 K (in Units of cm³ s⁻¹ with the Exponent Denoting Powers of 10)

		T (K)
j	${j}^{\prime }$	10	20	30	40	50	60	70	80	90	100	200	300	500	700	1000	1500	2000	2500	3000
0	0	1.949e-17	1.600e-17	1.585e-17	1.733e-17	2.019e-17	2.449e-17	3.032e-17	3.770e-17	4.674e-17	5.777e-17	8.020e-16	6.025e-15	4.334e-14	1.068e-13	2.096e-13	3.622e-13	4.999e-13	6.316e-13	7.573e-13
0	1	2.427e-17	1.627e-17	1.346e-17	1.281e-17	1.388e-17	1.679e-17	2.172e-17	2.881e-17	3.840e-17	5.135e-17	1.440e-15	1.111e-14	7.466e-14	1.793e-13	3.488e-13	5.965e-13	8.046e-13	9.865e-13	1.148e-12
0	2	2.768e-17	1.751e-17	1.338e-17	1.170e-17	1.152e-17	1.267e-17	1.514e-17	1.900e-17	2.455e-17	3.265e-17	1.297e-15	9.571e-15	5.504e-14	1.260e-13	2.465e-13	4.330e-13	5.900e-13	7.258e-13	8.462e-13

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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In order to test our proposed extrapolation scheme, we also calculated cross sections and rate coefficients for ro-vibrational $v=2,j=0-30\to v^{\prime} =1,j^{\prime}$ transitions with the CC method. We used a channel basis B4(75, 65, 54, 41, 20), where the notation Bn(j₀, j₁, j₂, ..., j_n) represents a basis with the highest vibrational level n and the highest rotational level j_i for vibrational level v = i (see Song et al. 2015). The state-to-state ro-vibrational rate coefficients were computed for a temperature range of 10 K $\leqslant \;T\;\leqslant$ 3000 K; they are listed in Table 2.

Table 2.  Rate Coefficients from CC Calculations for Ro-vibrational $v=2,j=0-30\to v^{\prime} =1,j^{\prime}$ Transitions in the Temperature Range of 10 K $\leqslant \;T\;\leqslant$ 3000 K (in Units of cm³ s⁻¹ with the Exponent Denoting Powers of 10)

		T (K)
j	${j}^{\prime }$	10	20	30	40	50	60	70	80	90	100	200	300	500	700	1000	1500	2000	2500	3000
0	0	3.814e-17	3.269e-17	3.445e-17	3.888e-17	4.467e-17	5.139e-17	5.892e-17	6.746e-17	7.789e-17	9.222e-17	1.654e-15	9.767e-15	5.489e-14	1.298e-13	2.628e-13	4.838e-13	6.837e-13	8.609e-13	1.016e-12
0	1	5.241e-17	3.537e-17	3.243e-17	3.475e-17	3.970e-17	4.645e-17	5.478e-17	6.516e-17	7.932e-17	1.011e-16	2.996e-15	1.754e-14	9.173e-14	2.081e-13	4.087e-13	7.353e-13	1.026e-12	1.283e-12	1.509e-12
0	2	7.974e-17	4.962e-17	3.823e-17	3.441e-17	3.412e-17	3.588e-17	3.920e-17	4.433e-17	5.256e-17	6.688e-17	2.499e-15	1.466e-14	6.790e-14	1.418e-13	2.623e-13	4.562e-13	6.375e-13	8.091e-13	9.696e-13

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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In addition, we obtained vibrational quenching cross sections from scattering calculations in the IOS approximation with a v = 0–14 basis for CO. This basis of 15 vibrational levels, with energies given in Table 3, is sufficiently large to converge all $v=1-5\to v^{\prime} \lt v$ transitions. In Table 4 we report the IOS vibrational transition rate coefficients for a temperature range of 10 K ≤ T ≤ 3000 K. The IOS approximation is not always suitable for cross sections at low collision energy and for rate coefficients at low temperature. We check this by comparing the vibrational rate coefficients produced with the IOS approximation and the results obtained by summing the state-to-state CC rate coefficients over final rotational j' levels and averaging over initial j levels. Figure 1 shows that the IOS rate coefficients agree well with the CC results, for both $v=1\to v^{\prime} =0$ and $v=2\to v^{\prime} =1$ transitions. The calculated results for the $v=1\to v^{\prime} =0$ transition also agree with the experimental data measured for this transition by Glass & Kironde (1982). All deviations between the IOS and CC rate coefficients in Figure 1 are less than 45%, which is sufficiently accurate for astrophysical applications. The IOS rate coefficients for $v=4\to v^{\prime} =1,2,3$ transitions in Table 4 at temperatures T = 40 and 50 K were obtained by interpolation of the rate coefficients at other temperatures. The original IOS data show a sharp peak around these temperatures due to resonance effects in the cross sections. However, the resonances in the IOS cross sections do not precisely coincide with the resonances in the cross sections from CC calculations (see Figure 2 of Song et al. 2015).

Table 3.  CO Vibrational Energy Levels

v	${\epsilon }_{v}({\mathrm{cm}}^{-1})$	v	${\epsilon }_{v}({\mathrm{cm}}^{-1})$	v	${\epsilon }_{v}({\mathrm{cm}}^{-1})$
0	1082	5	11534	10	21331
1	3225	6	13546	11	23213
2	5342	7	15531	12	25069
3	7432	8	17490	13	26899
4	9496	9	19424	14	28704

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First, we test the validity of the extrapolation method used by Thi et al. (2013) and Faure & Josselin (2008). Combining the IOS vibrational rate coefficient for the $v=1\to v^{\prime} =0$ transition with pure rotational (vibrationally elastic) rate coefficients from Walker et al. (2015), both of them based on the 3D H–CO potential of Song et al. (2013), we can obtain the state-to-state ro-vibrational rate coefficients for $v=1,j=0-30\to v^{\prime} =0,j^{\prime}$ transitions by extrapolation. Examples of the comparison of our CC rate coefficients, rate coefficients from extrapolation based on the method of Faure & Josselin (2008) and Thi et al. (2013), and the corresponding rate coefficients used by Thi et al. (2013) are illustrated in Figure 2. For transitions with final j' less than or close to the initial j, i.e., the $v=1,j=5\to v^{\prime} =0,j^{\prime} =0,6$ transitions, the CC and extrapolated rate coefficients agree quite well with each other, as shown in Figures 2(a) and (b). Large discrepancies appear, however, between CC and extrapolated rate coefficients for transitions with larger Δ$j=j^{\prime} -j$, especially at lower temperatures, see Figures 2(c) and (d). These large discrepancies result because the pure rotational (vibrationally elastic) transitions in v = 0 are endoergic for $j^{\prime} \gt j,$ while the ro-vibrational quenching processes $v=1,j\to v^{\prime} =0,j^{\prime}$ are exoergic for all j' < 33 even when j = 0. In addition, by analogy with Equation (6), the vibration-related scale factor ${P}_{{vv}^{\prime} }(T)(v\ne v^{\prime} )$ can be rewritten in the extrapolation method of Thi et al. (2013) and Faure & Josselin (2008) as ${r}_{v\to v^{\prime} }(T)/{r}_{0\to 0}(T).$ Clearly, the validity of scaling inelastic vibrational transition rate coefficients with vibrationally elastic data is questionable. The rate coefficients adopted in the modeling of Thi et al. (2013) are even more discrepant, as shown by the green lines with triangle markers in Figure 2. In their extrapolations, the pure rotational rates were adopted from Balakrishnan et al. (2002) who used the WKS potential, known to be inaccurate at long range. Moreover, the pure rotational rates for initial states with j > 7 and the vibrational rates for temperatures below 100 K were not available and had to be obtained by extrapolation.

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Let us now discuss the results from our new extrapolation method based on vibrationally inelastic rates from quantum scattering calculations. Using the data in Tables 1 and 4, the ro-vibrational rate coefficients for $v=2-5,j=0-30\to v^{\prime} \lt v,j^{\prime}$ transitions can be extrapolated with Equation (5). Figure 3 illustrates some comparisons of ro-vibrational rate coefficients for transitions from v = 2, j to v' = 1, j'. It shows the full CC results, the rate coefficients from the new extrapolation method based on vibrationally inelastic data, and the results of Thi et al. (2013) based on vibrationally elastic data. For the four transitions shown, the new extrapolated rate coefficients agree very well with those from full CC calculations. The root mean square relative deviation of the rate coefficients in the temperature range of 10 K ≤ T ≤ 3000 K for all the transitions $v=2,j=0-30\to v^{\prime} =1,j^{\prime}$ is 40%. This is sufficiently accurate for astronomical applications. However, the data from the scaling approach of Thi et al. (2013) deviate dramatically from our results, especially for higher final j' values. The reasons why their data are less accurate were already discussed above. The deviation is largest in the temperature range of 10 K $\leqslant \;T\;\leqslant$ 100 K, probably because the vibrational transition rate coefficients at low temperature were not available at the time.

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ProDiMo is a 2D radiation thermo-chemical disk code (Woitke et al. 2009). The code solves the 2D continuum radiative transfer to obtain the dust temperature and radiation field throughout the disk and provides the self-consistent solution for the chemistry and gas heating/cooling balance. The gas temperatures can be used iteratively to find a vertical hydrostatic equilibrium solution. However, we use for this work a fixed parametrized gas scale height and assume that gas and dust are well mixed. Details of the numerical methods, the chemical network and a list of heating/cooling processes can be found in Woitke et al. (2009). The CO molecular data used are described in detail above and in Thi et al. (2013).

The code can be used in 1D slab mode, to calculate the emission emerging from a fixed total gas column with a constant gas temperature and constant volume densities of collision partners (H, H₂, He and e⁻). The non-LTE level populations are calculated using the escape probability method taking into account also IR pumping by thermal dust emission. Since we focus here on the effect of the new collisional rates, we minimize the effect of IR pumping by fixing the dust temperature at 20 K. The turbulent width is fixed to ${v}_{{\rm{turb}}}=1.0$ km s⁻¹ and the total broadening is given by $b=\sqrt{{v}_{{\rm{th}}}^{2}+{v}_{{\rm{turb}}}^{2}},$ where v_th is the thermal velocity of the CO molecules. The slab models provide a powerful means to exploit a very wide parameter space under which CO emission could arise in space.

In addition, we use the standard model for a disk around a 2.2 M_⊙ Herbig star (${L}_{*}=32$ ${L}_{\odot }$) taken from Thi et al. (2013). The disk has a radial size of 300 AU and a mass of 10⁻²${M}_{\odot }.$ More details on the dust opacities and vertical disk structure can be found in Table 3 of Thi et al. (2013). Such a disk model does not provide any freedom as to the conditions under which CO is emitting. Its abundance, excitation and emission are calculated self-consistently given the density distribution of gas within the disk and the irradiating stellar and interstellar radiation field.

We ran four series of models, each with three different total gas column densities of ${N}_{\langle {\rm{H}}\rangle }={10}^{19},$ 10²¹, and 10²³ cm⁻². The CO abundance is fixed at 10⁻⁴ with respect to the total hydrogen number density. Series 1 uses a fixed gas temperature T of 800 K and high densities of collision partners, $\mathrm{log}\;{n}_{{\rm{H}}}=\mathrm{log}\;{n}_{{{\rm{H}}}_{2}}=9,\;$ $\mathrm{log}$ n_He = 8, $\mathrm{log}$ n_e = 5 (in cm⁻³). Note that the gas in these slab models is not fully molecular, i.e., the H/H₂ abundance ratio is 1. Series 2 uses a lower gas temperature of 200 K and the same collision partner densities. Series 3 uses T = 800 K and a lower density of collision partners, $\mathrm{log}\;{n}_{{\rm{H}}}=\mathrm{log}\;{n}_{{{\rm{H}}}_{2}}=6,\;$ $\mathrm{log}\;{n}_{{\rm{He}}}=5,\;$ $\mathrm{log}\;{n}_{{\rm{e}}}=2$ (in cm⁻³). An additional series (4) was calculated to isolate the effect of H-collisions. In this series, the temperature and density of H are the same as in series 3, while the collision partner densities of H₂, He and electrons are assumed to be negligible.

Figure 4 illustrates the change in the non-LTE fluxes of the v = 1−0 band resulting from the old and new collisional rates for H–CO in the slab models. We note that changes are typically smallest for the slab with the largest line optical depth $\mathrm{log}{N}_{\langle {\rm{H}}\rangle }=23$ ($\mathrm{log}{N}_{{\rm{CO}}}=19$). For slabs at T = 800 K, the line fluxes with the new rates are smaller than those with the old rates. Exceptions are the smaller and the highest j levels; the flux changes are generally largest for the lowest four levels. It is also evident from these comparisons that the implementation of the new rates will have an effect on the shape of the vibrational band structure. In most cases of high T (800 K) slabs, the band structure (flux versus wavelength) will get flatter for low j and drop faster at high j compared to the band structure with the old collision data. The total cooling rate of the v = 1−0 band (j ≤ 30) is typically 20%–30% lower with the new collisional rates. This will have an impact on the gas temperature whenever CO cooling dominates the energy balance. Examples are (1) the surface layers of the inner ≈10 pc of active galactic nucleus (AGN) disks (Meijerink et al. 2013), (2) the inner ≈10 AU of protoplanetary disks (Woitke et al. 2009) and (3) intermediate density (${n}_{\langle {\rm{H}}\rangle }\ \approx \;{10}^{5}$ cm⁻³) J-type shocks (Hollenbach & McKee 1989). If H collisions are omitted entirely in slab series 1, the fluxes in the most optically thick case are lower by a factor of 100; hence H collisions are more important than H₂ collisions and they are key in modeling the CO ro-vibrational emission.

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The gas temperature T is a free parameter and we can explore the effect of the new collision rates at lower temperatures. The comparison between series 1 and 2 (top left and right panel of Figure 4) clearly shows that the changes in the line fluxes increase substantially for T = 200 K. However, at these low temperatures, the populations in the CO vibrational levels are very low and hence also the emerging line fluxes will be low. An exception may be the case of fluorescent excitation, where the vibrational levels are populated by an external radiation field to non-LTE values. Examples are the inner disks of Herbig Ae stars, where the stellar radiation field can substantially pump the vibrational levels to vibrational temperatures of a few thousand K (Brittain et al. 2009; van der Plas et al. 2015).

The bottom panels of Figure 4 illustrate the effects for low density environments (non-LTE cases). Here, line-to-line differences are larger than in the high density environment. The overall picture does not change very much if we neglect the other collision partners, H₂, He, and electrons.

Besides the application to protoplanetary disks discussed in the next section, these new rate coefficients will also be important for proper modeling of CO infrared emission from other interstellar regions with large H/H₂ transition zones such as dissociative J-type shocks (Neufeld & Dalgarno 1989; González-Alfonso et al. 2002) or dense X-ray irradiated gas found near AGNs.

Figure 5 shows the change in the CO ro-vibrational emission of the v = 1−0 band at 4.7 μm resulting from the old and the new (this work) H–CO collision rate coefficients. The collision rate coefficients with H₂, He, and electrons remain unchanged. Differences are small, on the order of ±5%. The line fluxes with the new rates show a clear pattern with respect to the old ones: the new rates lead to lower line fluxes at low j, higher fluxes in the intermediate range and then again lower fluxes for j ≳ 25. The upturn at the highest j is a boundary effect, since the rotational states of CO are artificially cut off in the model at j = 30. As noted in Thi et al. (2013), the CO emission arises from a hot surface layer with T ≈ 1000 K, where the gas is predominantly atomic (see their Figure 12). At these high temperatures, differences between the old and new collision rates are smaller than at low temperatures (Figure 2) and this partially explains the relatively small changes observed in the disk model. In addition, IR pumping is competing with collisions, especially in the inner disk, where dust continuum emission peaks in the near-IR ($1000{\rm{K}}\lt {T}_{{\rm{dust}}}\lt 1500$ K).

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