J. Astrophys. Astr. (1996) 17, 41–52

Einstein A-coefficients for Rotational Transitions in Cyclopropenylidene

A. K. Sharma & SureshChandra* Department of Physics, University of Gorakhpur, Gorakhpur 273 009, India

Received 1994 October 6; accepted 1996 February 19

Abstract. Einstein Α-values are given for the electric dipole transitions in

the C3 H2- molecule between the rotational levels of the vibrational ground state up to 85cm–1. The mean radiation life-times of the levels are calculated from the Einstein Α-values. These values can be used as input parameters for analysing the spectra of C3H2.

Key words: C3H2-molecule—Einstein A-coefficients—molecular data.

1. Introduction

The spectral lines, at 85·338 GHz observed by Thaddeus et al. (1981); at 46·756 GHz observed by Suzuki et al. (1984); 18·343 GHz observed by Matthews & Sears (1985), and Matthews & Irvine (1985), remained unassigned for some time. Observations of Matthews and Irvine (1985) found that the transition at 18·343 GHz was easily detectable in a wide range of objects through out the galaxy. The sources where this transition was found included cold dust clouds, circumstellar envelopes, complex molecular clouds in the vicinity of HII regions, and the spiral arm clouds toward the supernova remnant cas A. However, the transition was never found in any hot IR nebula. The properties of emission and absorption profiles of the 18·343 GHz line indicated clearly that the radiation was due to the transition between two low-lying energy levels of a strong polar specie. Furthermore, its presence in the envelope of IRC +10216 suggested that it could be a neutral molecule likely to contain carbon, and not oxygen. Subsequently, Thaddeus et al. (1985) identified the 18·343 GHz line as the 110–101 transition of the three–membered ring molecule cyclopropenylidene (C3H2). Further, the two strong and previously-unidentified lines at 85·338 and 46·756 GHz were assigned to the transitions 212–101 and 211–202, respectively, in C3H2. The transition 110–101 in C3H2 is one of the strongest known interstellar lines observed in a wide range of objects. It suggests that C3H2 maybe potentially a useful molecule for probing the physical conditions in objects where it is observed. The molecular constants and ground rotational spectrum of C3H2 are accurately known, and there are a good number of spectral lines distributed throughout the observable microwave region. It is interesting to note that a number of groups or pairs of lines have similar frequencies, but correspond to different excitation energies. For example, the 110–101 and 220 – 211 lines at 18·343 and 21·587GHz, respectively, have

*Present address: School of Sciences, Indira Gandhi National Open University, New Delhi 110068.

41 42 Α. Κ. Sharma & Suresh Chandra excitation energies 0·9 and 9·7 K, respectively. The ratio of line strengths of such pairs should, in principle, be sensitive, through excitation effects, to the H2 number density in the clouds where the lines are observed. The advantage of using such a ratio for diagnostic purposes is that the two lines can be observed with the same telescope, simultaneously. Therefore, in forming the ratio of the line strengths, systematic errors involving calibration uncertainty, coupling efficiency and atmospheric absorption are obviously cancelled out. All these investigations, however, require Einstein Acoefficients for various transi- tions in the ground vibrational state as one of the important input parameters. Therefore, in the present investigation, we have calculated Einstein A-coefficients for electric dipole transitions between the rotational levels of the ground vibrational state up to 85cm–1. These Einstein A-values are used for calculating the mean radiative life-times of the levels.

2. The C3H2 -molecule

C3H2 (cyclopropenylidene) is a reactive molecule and is the first hydrocarbon ring molecule detected in space. According to ab intio calculations, it is one of the most stable molecules (Hehre et al. 1976; Lee et al. 1985) and its geometry is shown in Fig. 1.

The C3H2 is exceptionally polar hydrocarbon because of the two unpaired electrons on the bivalent carbon. The C3H2 is an asymmetric oblate top molecule (Ray parameter k = + 0·69). The equivalent off-axis Η-nuclei segregate the rotational levels into ortho and para symmetry species; radiative and collisional transitions between the two species are so highly forbidden that they can be treated as distinct molecules in the interstellar gas. Its rotational wave functions can be described by linear combinations of symmetric top wave functions:

(1) where α, β, γ are the Eulerian angles specifying the orientation of the molecule, J the

Figure 1. The geometry of the molecule C3H2. Einstein A-coefficients 43 j rotational quantum number, g j the expansion coefficients, D the Wigner D- ik MK functions, and the pseudo quantum number τ is defined by

(2)

The C3H2 has a large dipole moment µ = 3·325 Debye (Brown et al. 1987) along the b-axis of inertia. Thus, the allowed transitions are governed by the selection rules

J:∆J = 0, ±1.

K–1,K+1: even, odd ↔ odd, even even, even ↔ odd, odd.

3. Formation of C3H2 -molecule

Since the three-membered chain is less stable than the ring molecule, therefore, C3H2 should be in the form of a ring. It is formed via the dissociative recombination + of C3H3,

+ The C3H 3is a very stable ion, and according to Herbst et al. (1984) it can be produced from acetylene in only two steps:

(fast ion-molecule reaction).

(slow radiative association).

In steady state equilibrium, on the basis of a set of reations including these reactions, Herbst et al. (1984) predicted C3H2 to be one of the most abundant hydrocarbon molecules in diffuse molecular clouds.

4. Einstein A-coefficients

In the representation in which the axis of quantization is along the a-axis of inertia, the expression for the line strength is given by

(3) where the C’s are Clebsch-Gordan coefficients. The transition probabilities follow directly from the line strength (Chandra & Sahu 1993):

(4)

where the frequency ν corresponds to the energy difference of the two levels. 44 Α. Κ. Sharma & Suresh Chandra

J The expansion coefficients gtK are calculated by considering C3H2 as a rigid rotor. The molecular constants and distortional constants are adopted from Thaddeus et al. (1985) and are given in Table 1.

Table 1. Rotational and centrifugal distortion constants of C3H2 in MHz (Thaddeus et al. 1985).

5. Results and discussion

The computed values of Einstein Α-coefficients between the levels up to 85cm–1 are given in Tables 2 and 3 for ortho and para species, respectively. The A-valuesareused tocalculate the mean life times of the energy levels.The values of the life times of the levels along with energies are given in Table 4. Table 4 shows that the radiative life times of the levels go on decreasing with the increase of energy of the level, in general. However, there is an interesting feature that the radiative life time of the level 211 is 7 times larger than that of the level 202. Hence, the transition 211 –202 at 46·756 GHz may show stimulated emission phenomena in cold dark clouds, where collisional transitions are not dominant. The details would be worked out in the future. Einstein A-coefficients 45

46 A. Κ. Sharma & Suresh Chandra

Einstein A-coefficients 47

48 Α. Κ. Sharma & Suresh Chandra

Einstein A-coefficients 49

50 Α. Κ. Sharma & Suresh Chandra

Einstein Α-coefficients 51

Table 4. Energies and life times of the levels.

(Continued) 52 Α. Κ. Sharma & Suresh Chandra

Table 4. (Continued)

Acknowledgements

This work was partly done during the visit of S.C. to Inter-University Centre for Astronomy and Astrophysics (IUCAA), Pune. Financial support from the Council of Scientific and Industrial Research, New Delhi and the Indian Space Research Organi- zation, Bangalore is thankfully acknowledged.

References

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