A&A 423, 235-239 (2004)
DOI: 10.1051/0004-6361:20040161

Research Note

Suggestions for an interstellar C5H2 search[*]

S. Chandra[*] - S. A. Shinde

School of Physical Sciences, SRTM University, Nanded 431 606, India

Received 21 May 2003 / Accepted 5 May 2004

Abstract
Laboratory detection of four isomers of C5H2 molecule have been reported by Travers et al. (1997), McCarthy et al. (1997), and Gottlieb et al. (1998). They suggested for detection of the ring-chain isomer of C5H2 (c-C5H2) in cosmic objects, as it is the most stable one in comparison to the others. Two transitions 31 3 - 21 2 and 30 3 - 20 2 at 19.147 GHz and 19.606 GHz, respectively, of c-C5H2 have been detected in TMC-1. We suggest that the c-C5H2 may be identified in cool cosmic objects through its transition 31 3 - 40 4 at 4.3 GHz in absorption against the cosmic microwave background. Since in absence of availability of the collisional rates, we have used scaled values for them, we have checked the sensitivity of the lines on the rates by enhancing the rate for the transitions with $\Delta k_a = 0$ by a factor of 10. Though the transitions are not found sensitive, our results may be treated as qualitative in nature. This absorption line may play an important role for identification of c-C5H2 in cosmic objects.

Key words: ISM: molecules

1 Introduction

After detection of C4H2 and C6H2 molecules in the expanding shell of the evolved carbon star IRC +10216 and in the rich-molecular-source TMC-1, search for C5H2 molecule in cosmic objects is being carried out. Out of the four isomers of C5H2 molecule, detected in laboratory by Travers et al. (1997), McCarthy et al. (1997), and Gottlieb et al. (1998), the ring-chain isomer (c-C5H2) is the most probable candidate for its detection in cosmic objects, as it is the most stable one as compared to the others. This isomer is an asymmetric top, planar molecule having. a large electric dipole moment $\mu = 3.5$ Debye inclined with the axes of inertia so that its components along the a and b axes of inertia are $\mu_a$ = 2.04 Debye and $\mu_b = 2.89$ Debye. Thus, this isomer has both a-type and b-type radiative transitions, and therefore, the rotational energy levels cannot be separated into two different groups, as in case of an a-type or b-type molecule. Hence, the investigation of this molecule is quite complicated. The molecular data derived by Travers et al. (1997) for c-C5H2 are given in Table 1. Since the kinetic temperature in dark molecular clouds is rather low, only rotational transitions in the ground electronic and ground vibrational states take place. We propose to identify C5H2 in cosmic objects through its transition 31 3 - 40 4 in absorption against the cosmic 2.73 K background (also called the cosmic microwave background, denoted as CMB).

Table 1: Molecular data.

2 Einstein A-coefficients

Rotational wave functions for an asymmetric top molecule can be described by linear combination of symmetric top wave functions (Chandra et al. 1984a,b)
$\displaystyle \Psi_{J\tau M}(\alpha, \beta, \gamma) = \sqrt{\frac{2J+1}{8 \pi^2}} \sum_{K=-J}^J g_{\tau K}^J D_{MK}^J (\alpha, \beta, \gamma)$      

where $\alpha$, $\beta$, $\gamma$ are Euler angles specifying the orientation of the molecule, J the rotational quantum number, $g_{\tau K}^J$ the expansion coefficients, DMKJ the Wigner D-function and the pseudo quantum number $\tau$ is defined by

\begin{displaymath}\tau = k_a - k_c \end{displaymath}

where ka and kc are projections of J on the axis of symmetry in case of prolate and oblate symmetric tops, respectively. Rotational levels in an asymmetric top molecule are specified as Jka, kc or $J_\tau$. The rotational energy levels accounted in the present investigation are given in Table 2.

Table 2: Energy levels of ring-chain isomer of C5H2.

Since the electric dipole moment of the molecule is inclined with its axes of inertia, it has both a-type as well as b-type rotational transitions. The a-type rotational transitions are governed by the selection rules

\begin{eqnarray*}&&J{:}~ \Delta J = 0,~ \pm 1 \\ &&k_a, k_c{:}~ {\rm even}, {\r... ...{\rm odd}, {\rm even} \longleftrightarrow {\rm odd}, {\rm odd}. \end{eqnarray*}


In the representation where the axis of quantization is along the a-axis of inertia, Einstein A-coefficient for the transition  $J'_{\tau'} \rightarrow J_\tau$ is (Chandra & Sahu 1993; Chandra & Rashmi 1998)
$\displaystyle A\left(J'_{\tau'} \rightarrow J_\tau\right) = \frac{64 \pi^4 \nu^... ...}\left[\sum_{K=-J}^J g_{\tau K}^J g_{\tau' K}^{J'} C_{J K 1 0}^{J' K} \right]^2$      

where $\mu_a$ is the electric dipole moment along the a-axis of inertia, and C...... the Clebsch Gordon coefficient. The b-type rotational transitions are governed by the selection rules

\begin{eqnarray*}&&J {:}~ \Delta J = 0,~ \pm 1 \\ &&k_a, k_c{:}~ {\rm even}, {\... ...\rm even}, {\rm even} \longleftrightarrow {\rm odd}, {\rm odd}. \end{eqnarray*}


In the representation where the axis of quantization is along the a-axis of inertia, Einstein A-coefficient for the transition $J'_{\tau'} \rightarrow J_\tau$ is (Chandra et al. 1984b; Chandra 2002)
$\displaystyle A\left(J'_{\tau'} \rightarrow J_\tau\right) = \frac{32 \pi^4 \nu^... ...C_{J K 1 1}^{J' K+1} + g_{\tau' K-1}^{J'} C_{J K 1, -1}^{J' K-1} \big)\right]^2$      

where $\mu_b$ is the electric dipole moment along the b-axis of inertia. Calculated values of Einstein A-coefficients for a-type as well as b-type rotational transitions between the levels up to 18 cm-1 are available in the electronic form in Table 3.

3 Collisional rate coefficients

Besides the radiative transition probabilities for optically allowed transitions between the rotational energy levels, data required as input for the present investigation are the rate coefficients for collisional transitions between the levels due to collisions with H2 molecules. Though the collisional transitions are not restricted through any selection rules, computation of them is quite cumbersome task. The required collisional rate coefficients are not available in the literature. In absence of them qualitative investigations can however be carried out by choosing some scaling for the rate coefficients which do not favour any anomalous behaviour from their own. As and when the collisional rate coefficients would be available, the investigation can be repeated for quantitative results. In the present investigation, the rate coefficients for downward transitions $J' k'_a k'_c \rightarrow J k_a k_c$ at a kinetic temperature T are taken as (Sharma & Chandra 2001)

$\displaystyle C\left(J' k'_a k'_c \rightarrow J k_a k_c\right) = 1 \times 10^{-11} \sqrt{T/30} / \left(2 J'+1\right).$      

These rate coefficients can be interpreted as the cross section times a thermal velocity. The factor (2 J'+1) is the degeneracy of the upper level of the transition. These rates have no selectivity and do not support any anomalous behaviour from their own. However, some transitions between the low lying levels may be sensitive to the collisional rates.

In order to investigate sensitivity of our results to the collisional rates, we enhanced the collisional rates for the transitions with $\Delta k_a$ = 0 by a factor of 10 (i.e., by one order of magnitude), which may be taken as an extreme case. The results with these enhanced rates are also given in this paper. Moreover, in absence of accurate collisional rates, our results can be treated as qualitative in nature.

For upward collisional rate coefficients, we accounted for the fact that downward and upward collisional rate coefficients are related through the detailed equilibrium (Chandra & Kegel 2000).

4 Basic formulation

In our investigation, NLTE occupation numbers of the energy levels of the molecule under investigation are calculated in an on-the-spot approximation by using the escape probability method (see, e.g., Rausch et al. 1996), where the external radiation field, impinging on a volume element generating the lines, is the CMB only. Besides the normal features, some lines of a molecule may show anomalous features: (i) absorption against the CMB (called, anomalous absorption) and (ii) maser action.

4.1 Anomalous absorption

Observation of a spectral line in absorption against the CMB is an unusual phenomenon. The transition 1 $_{10} \rightarrow 1_{11}$ of H2CO and 2 $_{20} \rightarrow 2_{11}$ of C3H2 are good examples of anomalous absorption. The intensity, $I_\nu$, of a line generated in an interstellar cloud, with homogeneous excitation conditions, is given by
$\displaystyle I_\nu - I_{\nu,\rm bg} = (S_\nu - I_{\nu,\rm bg}) (1 - {\rm e}^{-\tau_\nu})$     (1)

where $I_{\nu,\rm bg}$ is the intensity of the continuum against which the line is observed, $\tau _\nu $ the optical depth of the line, and $S_\nu$ the source function. For positive optical depth, observation of an interstellar line in absorption against the CMB (i.e., $I_\nu < I_{\nu,\rm bg}$), obviously, implies the excitation temperature $T_{\rm ex}$ of the line to be less than the CMB temperature $T_{\rm bg}$, but positive ( $0 < T_{\rm ex} < T_{\rm bg}$). It requires rather peculiar conditions in the molecule generating the line. Equation (1) may also be expressed as
$\displaystyle B_\nu(T_{\rm B}) - B_\nu(T_{\rm bg}) = \big[B_\nu(T_{\rm ex}) - B_\nu(T_{\rm bg})\big] (1 - {\rm e}^{-\tau_\nu})$     (2)

where $B_\nu$ represents a Planck's function corresponding to various temperatures and $T_{\rm B}$ is the brightness temperature of the line. (For absorption against the CMB, we have $T_{\rm B} < T_{\rm bg}$.) This obviously shows that for optically thin case, $\tau_\nu \approx 0$ and we have $T_{\rm B} = T_{\rm bg} \equiv 2.7$ K. Further, in the Rayleigh-Jeans limit [ $\nu(\mbox{GHz}) \ll 21 \ T(\mbox{K})]$, Eq. (2) can be written as
$\displaystyle T_{\rm B} = T_{\rm ex} + (T_{\rm bg} - T_{\rm ex}) {\rm e}^{-\tau_\nu}.$     (3)

For anomalous absorption, we have $T_{\rm ex} < T_{\rm bg}$ and $\tau_\nu > 0$, and therefore, $T_{\rm B} > T_{\rm ex}$. When $\tau _\nu $ is very large, then for the anomalous absorption, we have $T_{\rm B} = T_{\rm ex}$. It shows that for anomalous absorption, the brightness temperature of the line lies between $T_{\rm ex}$ of the line and $T_{\rm bg}$ ( $T_{\rm ex} \leq T_{\rm B} \leq T_{\rm bg}$).

4.2 Maser action

Maser action is another anomalous phenomenon shown by some molecules in the cosmic objects. The molecules OH, H2O and SiO are good examples of masing molecules. We can rearrange the Eq. (3) as
$\displaystyle T_{\rm B} = T_{\rm bg} {\rm e}^{-\tau_\nu} - T_{\rm ex} ({\rm e}^{-\tau_\nu} - 1).$      

For maser action, we have population inversion and thus $T_{\rm ex} < 0$ and $\tau_\nu < 0$, showing that $T_{\rm B} > T_{\rm bg}$. When $\vert\tau_\nu\vert$ is large (such that ${\rm e}^{-\tau_\nu} \gg 1$), then for the maser action, we have
$\displaystyle T_{\rm B} = (T_{\rm bg} - T_{\rm ex}) {\rm e}^{-\tau_\nu}.$      

5 Numerical results and discussion

In our model, the free parametres are the hydrogen density $n_{\rm H_2}$ and $\gamma \equiv n_{\rm mol}/({\rm d}v_r/{\rm d}r)$, where $n_{\rm mol}$ is density of the molecule, and ${\rm d}v_r/{\rm d}r$ the velocity gradient in the object. In order to include a large number of cosmic objects where c-C5H2 may be found, numerical calculations are carried out for wide ranges of physical parametres. We have taken two values 10-5 and 10-4 cm-3 (km s-1)-1 pc for $\gamma$. For each value of $\gamma$, the molecular hydrogen density $n_{\rm H_2}$ is varied over the range from 103 to 106 cm-3, and calculations are performed for two kinetic temperatures 10 and 20 K, as the temperature in the clouds cannot be larger than this. In the present investigation, a set of 64 linear equations coupled with 274 equations of radiative transfer is solved through the iterative procedure for given values of $n_{\rm H_2}$ and $\gamma$.

5.1 Observations

For the observed two transitions 31 3 - 21 2 and 30 3 - 20 2 at 19.147 GHz and 19.606 GHz, respectively of c-C5H2 detected in TMC-1, Dickens et al. (2001) obtained $\Delta v = 0.15$ km s-1, and the upper limits for the relative column density of c-C5H2 was obtained as $2 \times 10^{12}$ cm-2 and $4 \times 10^{12}$ cm-2, respectively. (Frequencies of these transitions given in Table 1 of Dickens et al. (2001) suffer from misprint as we discussed with them.) For these values of Dickens et al. (2001), the corresponding values of the free parametre $\gamma$ are $4.5 \times 10^{-6}$ and $9 \times 10^{-6}$ cm-3 (km s-1)-1 pc. This higher value $9 \times 10^{-6}$ is closed to our value 10-5, and our second value is larger by one order of magnitude.


  \begin{figure} \par\includegraphics[height=15cm,width=14.5cm,clip]{c5h2fig1.eps}\end{figure} Figure 1: Variation of brightness temperature $T_{\rm B}$ (K) versus hydrogen density $n_{\rm H_2}$ of the lines written at the top of each column for kinetic temperatures 10 K (rows 1 and 3) and 20 K (rows 2 and 4). Solid line is for $\gamma =$ 10-5 cm-3 (km s-1)-1 pc, and the dotted line for $\gamma =$ 10-5 cm-4 (km s-1)-1 pc. Lower rows (3 and 4) are for the case where collisional rates of the transitions with $\Delta k_a = 0$ are increased by a factor of 10.
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For the observed lines 31 3 - 21 2 and 30 3 - 20 2, variation of brightness temperature $T_{\rm B}$ versus the hydrogen density $n_{\rm H_2}$ for T = 10 K (row 1) and 20 K (row 2) is shown in Fig. 1. We found $T_{\rm B} > T_{\rm bg}$, although not very large for $\gamma = 10^{-5}$, but showing emission feature for both the lines. Since both the lines are detected in emission, the reason for low value of ( $T_{\rm B} - T_{\rm bg}$) in our calculations can be assigned to the qualitative nature of our investigations. When $\gamma$ (i.e., $N(c\mbox{-C}_5\mbox{H}_2)/\Delta v$) is increased by one order of magnitude, $T_{\rm B}$ shows a remarkable increase where optical depth $\tau _\nu $ goes up to 0.4 and 0.5 at T = 10 K, and 0.06 and -0.1 at T= 20 K for 31 3 - 21 2 and 30 3 - 20 2 transitions, respectively. The results with the enhanced collisional rates for the lines are shown in rows 3 and 4 of Fig. 1. With the enhanced rates, a little increase in $T_{\rm B}$ is found; which is more remarkable on the two sides of the peak. The peak in $T_{\rm B}$ is found to shift towards the low density region, and at the peak, the optical depth is found to decrease.


  \begin{figure} \par\includegraphics[width=14cm,clip]{c5h2fig2.eps}\end{figure} Figure 2: Variation of brightness temperature $T_{\rm B}$ (Col. 1), excitation temperature $T_{\rm ex}$ (Col. 2), optical depth $\tau _\nu $(Col. 3) and brightness temperature with enhanced collisional rates (Col. 4) versus hydrogen density $n_{\rm H_2}$ of 31 3 - 40 4transition. Solid line is for $\gamma = 10^{-5}$ cm-3 (km s-1)-1 pc, and the dotted line for $\gamma =$ 10-4 cm-3 (km s-1)-1 pc. For the brightness temperature $T_{\rm B}$ in Col. 4, the collisional rates for the transitions with $\Delta k_a = 0$ are increased by a factor of 10.
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5.2 The 70 7 - 61 6 and 60 6 - 515 transitions

A number of lines of c-C5H2 were found showing negative value for the excitation temperature (maser action). Though it was not large in magnitude, we have however investigated two transitions 70 7 - 61 6 and 60 6 - 515 at 17.4 GHZ and 10.1 GHz, respectively. Other lines showing negative value for $T_{\rm ex}$ were insignificant. Variation of $T_{\rm B}$ versus $n_{\rm H_2}$ for these two lines is also shown in Fig. 1. The optical depth, at the peak, goes up to -0.5 and 0.2 at T = 10 K, and -0.8 and 0.1 at T= 20 K for 70 7 - 61 6 and 60 6 - 515, respectively. The effect of the enhancement of collisional rates for these lines is found more prominent than those observed ones. With the enhanced collisional rates, at the peak, the optical depth goes up to -1.0 and 0.3 at T = 10 K, and -1.4 and 0.06 at T= 20 K for 70 7 - 61 6 and 60 6 - 515, respectively.

5.3 The 31 3 - 40 4 transition

A number of lines of c-C5H2 are found in absorption against the cosmic 2.7 K background. However, the transition 31 3 - 40 4 at 4.3 GHz has shown reasonably good absorption phenomenon. Einstein A-coefficient for this transition is $1.74\times10^{-9}$ s-1. For this transition, we have plotted brightness temperature $T_{\rm B}$ (Col. 1), excitation temperature  $T_{\rm ex}$ (Col. 2), optical depth $\tau _\nu $ (Col. 3) and again $T_{\rm B}$ where the enhanced collisional rates are used (Col. 4). Anomalous absorption is found to increase with $\gamma$. A moderate increase is found with the increase of kinetic temperature T, where the position of the minimum value of $T_{\rm B}$ is found to shift towards the low density region.

It is interesting to have a note that the observed transition 31 3 - 21 2( $A = 1.30\times10^{-7}$ s-1) is detected in emission whereas the transition 31 3 - 40 4, having the common upper level 31 3 shows absorption against the CMB. The process of absorption can be understood in the following manner. The level 31 3, besides to 40 4, decays radiatively to the levels 21 2 and 20 2 ( $A = 3.49\times10^{-6}$ s-1). The lower level 40 4 has only one radiative decay to 30 3 ( $A = 3.84\times10^{-7}$ s-1). The radiative life time of the level 31 3 ( $2.8~\times~10^5$ s) of the transition is one order of magnitude smaller than that of the level 40 4 ( $2.6\times10^6$ s). Thus, the molecule in the lower level exists longer and can absorb the radiations corresponding to the transition 31 3 - 40 4 coming from the background.

6 Conclusion

Here, we have used scaled values of collisional rates, and therefore, our results are qualitative in nature. We have obtained emission feature of the observed lines qualitatively. We found that detection of c-C5H2 is likely in cool cosmic objects through anomalous absorption of the transition 31 3 - 40 4. In absence of accurate collisional rates, our investigation provides information that this transition may play important role for identification of c-C5H2 in cool cosmic objects.

Acknowledgements
We are grateful to Prof. Jayant V. Narlikar, Prof. Dr. W. H. Kegel and Prof. S. A. Suryawanshi for their encouragement. Thanks are due to learned referee for constructive and valuable comments which improved the manuscript a lot.

References

 
Copyright ESO 2004