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Up: Collisional rates for asymmetrical molecules


1 Introduction

Since in most regions of interstellar space the excitation of molecules (and atoms) deviates strongly from that under LTE conditions, a quantitative interpretation of observed interstellar molecular lines requires the simultaneous solution of the set of rate equations and the radiative transfer equation for a large number of lines. (In the rate equations one has to account for many more lines than are usually observed.) If all lines of the molecule under consideration are optically thin, the problem reduces to the solution of the - then linear - rate equations. The rate equations describe the interplay between radiative and collisional transitions and the solution yields the relative occupation numbers from which the line intensities follow. As input parameters one has to know the radiative and the collisional transition probabilities. While for purely rotational transitions the radiative transition probabilities are known for many molecules, the collisional rate coefficients present a problem in most cases. Since in interstellar molecular clouds hydrogen is predominantly in the form of H2, one usually considers only collisions with H2 molecules. To reduce the complexity of the problem one accounts in most cases only for H2 molecules in the rotational ground state (J=0). To simplify the problem even further, one often considers collisions with He atoms assuming that the rate coefficients for collisions with H2 may be obtained from the results for He by rescaling the latter accounting for the difference of the reduced mass. - In actual NLTE calculations one has to account for a large number of energy levels, which is possible only if the corresponding collisional rate coefficients are known. The goal of such computations is to determine the physical parameters in molecular clouds, in particular the temperature and the density. This requires that the rate coefficients are known for a sufficiently large range of temperatures.


 
Table 1: A) Wave-functions for ortho levels of C3H2
                   
No. Level E(cm-1) $\epsilon_{J \tau}$ $g_{J \tau}^1$ $g_{J \tau}^3$ $g_{J \tau}^5$ $g_{J \tau}^7$ $g_{J \tau}^9$ $g_{J \tau}^{11}$
1 101 1.633 - -0.707107          
2 110 2.245 + 0.707107          
3 212 4.480 - -0.707107          
4 221 6.315 + 0.707107          
5 303 8.387 - -0.676228 0.206677        
6 312 11.155 + 0.483241 -0.516216        
7 321 12.626 - -0.206677 -0.676228        
8 414 13.419 - -0.651603 0.274615        
9 330 13.530 + 0.516216 0.483241        
10 423 17.347 + 0.426209 -0.564221        
11 505 19.568 - -0.629935 0.316322 -0.055880      
12 432 20.204 - -0.274615 -0.651603        
13 441 22.394 + 0.564221 0.426209        
14 514 24.616 + 0.371915 -0.574031 0.179351      
15 616 26.834 - -0.611098 0.343724 -0.091722      
16 523 28.511 - -0.291406 -0.511000 0.392380      
17 532 31.083 + 0.344116 0.030181 -0.616987      
18 541 32.489 - -0.135148 -0.372585 -0.585590      
19 625 33.006 + 0.331431 -0.567582 0.260776      
20 550 34.045 + 0.493217 0.411797 0.295229      
21 707 35.217 - -0.594594 0.362415 -0.121998 0.015160    
22 634 38.044 - -0.309498 -0.423776 0.473946      
23 643 41.962 + 0.366852 -0.062054 -0.601306      
24 716 42.513 + 0.299686 -0.553473 0.317154 -0.057179    
25 818 44.717 - -0.580000 0.375444 -0.147684 0.028826    
26 652 44.976 - -0.175415 -0.449741 -0.516684      
27 661 47.731 + 0.505543 0.417133 0.265380      
28 725 48.675 - -0.318245 -0.345014 0.507768 -0.147840    
29 827 53.136 + 0.274154 -0.536222 0.356831 -0.099888    
30 734 53.688 + 0.336546 -0.148211 -0.521837 0.304065    
31 909 55.334 - -0.566976 0.384631 -0.169517 0.042949 -0.004117  
32 743 57.440 - -0.177015 -0.387188 -0.223015 0.518667    
33 752 59.706 + 0.261749 0.082995 -0.213490 -0.615647    
34 836 60.423 - -0.323060 -0.279038 0.514355 -0.230670    
35 761 61.383 - -0.117659 -0.315762 -0.421369 -0.457054    
36 770 63.933 + 0.477925 0.405972 0.285519 0.158914    
37 918 64.877 + 0.253141 -0.517977 0.384735 -0.138890 0.017558  
38 845 66.571 + 0.314342 -0.208485 -0.442017 0.402920    
39 101,10 67.068 - -0.555254 0.391121 -0.188132 0.056975 -0.008766  
40 854 71.575 - -0.202218 -0.391675 -0.095246 0.544634    
41 927 73.287 - -0.325373 -0.223583 0.505360 -0.293416 0.051595  
42 863 75.524 + 0.319063 0.063310 -0.303806 -0.549447    
43 1029 77.734 + 0.235518 -0.499801 0.404154 -0.173678 0.035017  
44 872 78.873 - -0.135426 -0.357449 -0.452279 -0.386437    
45 110,11 79.919 - -0.544627 0.395670 -0.204065 0.070554 -0.014334 0.001118
46 936 80.560 + 0.293570 -0.250434 -0.360834 0.454239 -0.120688  
47 881 82.372 + 0.473527 0.406192 0.291538 0.160579    



 
Table 1: B) Wave-functions for ortho levels of C3H2
                   
No. Level E(cm-1) $\epsilon_{J \tau}$ $g_{J \tau}^0$ $g_{J \tau}^2$ $g_{J \tau}^4$ $g_{J \tau}^6$ $g_{J \tau}^8$ $g_{J \tau}^{10}$
1 000 0.000 + 1.000000          
2 111 1.729 + 1.000000          
3 202 4.468 + 0.827119 -0.397413        
4 211 6.027 - 0.000000 -0.707107        
5 220 6.747 + 0.562027 0.584861        
6 313 8.389 + 0.766296 -0.454307        
7 322 11.215 - 0.000000 -0.707107        
8 331 13.201 + 0.642488 0.541853        
9 404 13.419 + 0.718896 -0.479649 0.107382      
10 413 17.340 - 0.000000 -0.635216 0.310645      
11 515 19.568 + 0.683173 -0.490823 0.160406      
12 422 20.032 + 0.470746 0.222275 -0.582917      
13 431 21.441 - 0.000000 -0.310645 -0.635216      
14 440 22.621 + 0.511455 0.469606 0.385586      
15 524 24.617 - 0.000000 -0.584861 0.397413      
16 606 26.834 + 0.654761 -0.494988 0.199460 -0.029101    
17 533 28.538 + 0.487527 0.143014 -0.600588      
18 542 31.454 - 0.000000 -0.397413 -0.584861      
19 615 33.006 - 0.000000 -0.539835 0.445198 -0.101869    
20 551 33.902 + 0.543684 0.488507 0.336993      
21 717 35.217 + 0.631379 -0.495423 0.229268 -0.051692    
22 624 38.042 + 0.467641 0.064532 -0.571815 0.243966    
23 633 41.885 - 0.000000 -0.390129 -0.367525 0.461221    
24 726 42.513 - 0.000000 -0.501251 0.471081 -0.163799    
25 642 44.304 + 0.325435 0.207486 -0.110560 -0.625917    
26 808 44.717 + 0.611633 -0.493777 0.252481 -0.072993 0.007900  
27 651 45.793 - 0.000000 -0.237439 -0.408319 -0.52621    
28 660 47.814 + 0.496682 0.455820 0.347879 0.218773    
29 735 48.675 + 0.452248 0.007015 -0.530786 0.340519    
30 817 53.136 - 0.000000 -0.468041 0.483902 -0.213926 0.031800  
31 744 53.698 - 0.000000 -0.407106 -0.252299 0.520202    
32 919 55.334 + 0.594622 -0.490941 0.270828 -0.092671 0.015947  
33 753 57.623 + 0.386067 0.211407 -0.205920 -0.581705    
34 826 60.423 + 0.438239 -0.036789 -0.484223 0.400509 -0.087982  
35 762 60.778 - 0.000000 -0.288119 -0.463064 -0.450066    
36 771 63.886 + 0.497780 0.458054 0.351140 0.207351    
37 928 64.877 - 0.000000 -0.439270 0.488639 -0.254398 0.059624  
38 835 66.570 - 0.000000 -0.405374 -0.146845 0.525963 -0.193574  
39 100,10 67.068 + 0.579734 -0.487425 0.285498 -0.110612 0.024943 -0.002146
40 844 71.545 + 0.370380 0.158669 -0.278315 -0.445159 0.361396  
41 937 73.287 + 0.425919 -0.070707 -0.437671 0.436124 -0.150122  
42 853 75.155 - 0.000000 -0.261843 -0.328501 -0.0866310 0.562157  
43 862 77.309 + 0.262033 0.181832 -0.026777 -0.288719 -0.590365  
44 1019 77.734 - 0.000000 -0.414146 0.488263 -0.286888 0.087661 -0.009638
45 871 79.290 - 0.000000 -0.219197 -0.369272 -0.412424 -0.381442  
46 111,11 79.919 + 0.566537 -0.483534 0.297340 -0.126851 0.034461 -0.004794
47 946 80.561 - 0.000000 -0.399157 -0.061842 0.502292 -0.290778  
48 880 82.397 + 0.477503 0.443387 0.351575 0.229650 0.114276  



 
Table 2: Wave-functions for ortho levels of SiC2
No. Level E(cm-1) $\epsilon_{J \tau}$ $g_{J \tau}^0$ $g_{J \tau}^2$ $g_{J \tau}^4$ $g_{J \tau}^6$ $g_{J \tau}^8$ $g_{J \tau}^{10}$
1 000 0.000 + 1.000000          
2 101 0.787 + 1.000000          
3 202 2.357 + 0.999583 0.020428        
4 303 4.701 + 0.997932 0.045452        
5 221 7.788 - 0.000000 -0.707107        
6 220 7.793 + 0.028889 -0.706812        
7 404 7.805 + 0.993927 0.077808 0.000857      
8 322 10.148 - 0.000000 -0.707107        
9 321 10.171 + 0.064278 -0.705644        
10 505 11.654 + 0.986385 0.116259 0.002504      
11 423 13.292 - 0.000000 -0.707030 -0.010411      
12 422 13.359 + 0.110043 -0.702736 -0.010391      
13 606 16.230 + 0.974474 0.158655 0.005368 0.000040    
14 524 17.216 - 0.000000 -0.706812 -0.020428      
15 523 17.370 + 0.164448 -0.697183 -0.020343      
16 707 21.517 + 0.958119 0.202263 0.009672 0.000136    
17 625 21.917 - 0.000000 -0.706371 -0.032245 -0.000273    
18 624 22.218 + 0.224483 -0.688316 -0.032014 -0.000273    
19 726 27.389 - 0.000000 -0.705615 -0.045896 -0.000732    
20 808 27.502 + 0.938122 0.244385 0.015471 0.000330 1.87D-06  
21 725 27.913 + 0.286321 -0.675978 -0.045430 -0.000733    
22 441 29.574 - 0.000000 -0.010411 0.707030      
23 440 29.574 + 0.000413 -0.010422 0.707030      
24 542 33.512 - 0.000000 -0.020428 0.706812      
25 541 33.512 + 0.001238 -0.020478 0.706810      
26 827 33.629 - 0.000000 -0.704441 -0.061322 -0.001491 -9.08D-06  
27 909 34.179 + 0.915870 0.282980 0.022653 0.000664 6.90D-06  
28 826 34.463 + 0.346182 -0.660609 -0.060603 -0.001503 -9.24D-06  
29 643 38.240 - 0.000000 -0.032246 0.706306 0.009593    
30 642 38.240 + 0.002769 -0.032403 0.706296 0.009593    
31 928 40.631 - 0.000000 -0.702740 -0.078416 -0.002640 -0.000030  
32 100,10 41.543 + 0.892835 0.316956 0.030999 0.001178 0.000018 -2.89D-08
33 927 41.868 + 0.401203 -0.643029 -0.077622 -0.002690 -0.000031  
34 744 43.761 - 0.000000 -0.045900 0.705384 0.018054    
35 743 43.761 + 0.005302 -0.046300 0.705348 0.018054    
36 1029 48.387 - 0.000000 -0.700406 -0.097022 -0.004268 -0.000072 -3.44D-07
37 110,11 49.594 + 0.870196 0.346057 0.040250 0.001905 0.000040 3.47D-07
38 845 50.076 - 0.000000 -0.061334 0.703903 0.027556 0.000210  
39 844 50.078 + 0.009185 -0.062220 0.703795 0.027554 0.000210  
40 1028 50.125 + 0.449816 -0.624074 -0.096670 -0.004429 -0.000076 -4.11D-07


The calculation of the collisional rate coefficients is a difficult task, and the situation becomes even more difficult for the case of an asymmetrical top molecule colliding with H2 molecules. In the present investigation, we have accounted for the collisional transitions between the rotational energy levels in C3H2 and SiC2 molecules, extending previous work by Green et al. (1987) and Palma & Green (1987).


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