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3 Collisional rate coefficients

Collisional rate coefficient for the rotational transition $J \tau \rightarrow
J' {\tau'}$ at kinetic temperature T, averaged over the Maxwellian distribution is given by


$\displaystyle C(J\tau \rightarrow J' \tau'\vert T) = \Big(\frac{8kT}{\pi \mu}\B...
...\Big(\frac{1}{kT}\Big)^2
\int_0^\infty\sigma(J\tau \rightarrow J' \tau'\vert E)$      
$\displaystyle E {\rm e}^{-E/kT} {\rm d}E$      

where the cross section $\sigma(J\tau \rightarrow J' \tau'\vert E)$ for the transition is given by


$\displaystyle \sigma(J\tau \rightarrow J' \tau'\vert E) = (2 J' + 1) \sum_{LMM'} S(J, \tau, J',
\tau'\vert L,M,M')$      
q(L,M,M'|E).      

Here, the sum is finite, limited by triangle inequalities on J, J' and L, and each of M and M' can independently assume the values ranging from -L to +L. The interaction potential is expressed in terms of spherical harmonics, and therefore, the scattering cross section is expressed here in terms of a parameter q(L,M,M'|E) (Hutson & Green 1995). The spectroscopic coefficients, $S(J, \tau, J',\tau'\vert L,M,M')$, depend on the wave-functions of the molecule and on angular momentum coupling factors:


$\displaystyle S(J, \tau, J',\tau'\vert L,M,M') = \sum_{p, p', q,q'} g_{J \tau }^p
g_{J \tau}^q g_{J' \tau'}^{p'} g_{J' \tau'}^{q'} (-1)^{p'+q'}$      
$\displaystyle \left( \begin{array}{ccc} J & L & J'\\  -p & M & p' \end{array} \right)
\left( \begin{array}{ccc} J & L & J'\\  -q & M' & q' \end{array} \right).$      

These spectroscopic coefficients are obviously independent of collision dynamics. Thus, the rate coefficient is given by


$\displaystyle C(J\tau \rightarrow J' \tau'\vert E) = (2 J' + 1) \sum_{LMM'} S(J, \tau, J',
\tau'\vert L,M,M')$      
Q(L,M,M'|T)      

where


$\displaystyle Q(L,M,M'\vert T) = \Big(\frac{8kT}{\pi \mu}\Big)^{1/2}
\Big(\frac{1}{kT}\Big)^2
\int_0^\infty q(L,M,M'\vert E)$      
$\displaystyle E {\rm e}^{-E/kT} {\rm d}E.$      

Now, owing to symmetries of $g_{J \tau }^K$, S(L, M, M') = S(L, M', M), so that only the real part of Q(L, M, M') is required, and the cross sections are real. - Green et al. (1987) calculated the Q(L,M,M'|T) for C3H2. These calculations were based on a potential energy surface describing the interaction of C3H2 with He and used the infinite order sudden approximation (IOSA) according to Green (1979). They derived the Q(L,M,M'|T) for $L\le25, M'\le M\le8$ and T=30, 60, 90, and 120 K. Despite the larger parameter space for which the Q(L,M,M'|T) were calculated, Green et al. give the state-to-state rate coefficients only for T=60 K and for transitions between the lowest 20 energy levels of ortho-C3H2 and the lowest 22 of para-C3H2. - Similar calculations were performed by Palma & Green (1987) for SiC2. They computed the Q(L,M,M'|T) for $L\le 25, M'\le M\le 6$ and T=25, 50, 75, 100, and 125 K. State-to-state coefficients were given only for T=100 K and for transitions between the lowest 18 energy levels.

As mentioned in the introduction, in astrophysical applications, i.e. for actual NLTE calculations, one needs to know the state-to-state rate coefficients for a large number of transitions in a sufficiently large range of temperatures. We therefore extended the work of Green et al. and Palma and Green, respectively, by using the Q(L,M,M'|T) computed by them to calculate state-to-state rate coefficients for an extended set of transitions and for all temperatures for which the Q(L,M,M'|T) were given.


 
Table 3: A) Collisional rate coefficient (cm3 s-1) for transitions in ortho-C3H2 at 30 K
Transition Rate l u
1(0, 1)==> 1(1, 0) 7.800D-12 1 2
1(0, 1)==> 2(1, 2) 3.041D-11 1 3
1(0, 1)==> 2(2, 1) 1.666D-11 1 4
1(0, 1)==> 3(0, 3) 3.489D-12 1 5
1(1, 0)==> 2(1, 2) 2.409D-12 2 3
1(1, 0)==> 2(2, 1) 1.143D-11 2 4
1(1, 0)==> 3(0, 3) 4.431D-11 2 5
2(1, 2)==> 2(2, 1) 1.217D-11 3 4
2(1, 2)==> 3(0, 3) 9.661D-12 3 5



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