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4 [C II] 158 ${\mu }$m line emission from carbon RRL forming region

The [C  II] 158 ${\mu }$m line is due to the radiative decay of the fine structure transition $2{\rm P}_{3/2} \rightarrow 2{\rm P}_{1/2}$ in singly-ionized carbon. Recombination lines of carbon are a result of electronic transitions in a recombined atom in ionized gas. The excitation temperature of the fine structure transition ($\sim $91 K) is comparable to a subset of the temperatures which explain the observed low frequency carbon RRL emission (Kantharia & Anantharamaiah 2001). Moreover, dielectronic-like recombination (Watson et al. 1980) is a process involving the excitation of the fine-structure levels which modifies the electronic level populations in recombined carbon; thus modifying the observed line optical depths of the carbon recombination lines. Since the two emission mechanisms are intricately linked, it is interesting to study their correlation.

In this section, we estimate the expected [C  II] 158 ${\mu }$m  emission strength from low frequency carbon RRL-forming regions and compare the galactic distribution of the diffuse [C  II] 158 ${\mu }$m fine structure line with the carbon RRLs detected near 327 MHz.

4.1 The [C II] 158 ${\mu }$m emission from the 327 MHz carbon RRL forming regions

The [C  II] 158 ${\mu }$m line originates predominantly from three types of regions: photodissociation regions (PDRs), cold neutral medium (CNM) and extended low-density warm ionized medium (ELDWIM) (Petuchowski & Bennett 1993; Heiles 1994). As described by Hollenbach et al. (1991), carbon is mostly in singly-ionized state upto $A_{\rm V} <$ 4 mag in low-density PDRs (the dense PDRs have a relatively low volume filling factor and hence may not contribute largely to the global diffuse [C  II] 158 ${\mu }$m emission). Hence low-density PDRs, which for the present discussion are considered as regions associated with molecular clouds, are possible sources of the diffuse fine structure line emission as well as carbon RRL emission. The CNM is another source of singly-ionized carbon. The CNM is distinct from PDRs in that they are predominantly atomic clouds with neutral densities <103  cm-3 and typically $A_{\rm V} \le$ 1 mag (Heiles 1994). The [C  II] 158 ${\mu }$m line is the major cooling transition in the CNM and the low-density PDRs since the number density of the colliding particles is generally less than the critical density, which depends on the colliding particles and their temperature. For temperatures relevant for CNM and low-density PDRs ($\sim $20 to 500 K), the critical densities due to collision with atoms and molecules are $\sim $3000  cm-3 and $\sim $4000  cm-3 respectively (Launay & Roueff 1977; Flower & Launay 1977). For densities larger than these, the fine-structure level is collisionally de-excited. In the ELDWIM, which consists of both the warm ionized medium (WIM; Reynolds 1993) and low-density ( $n_{\rm e} \sim$ 1-10  cm-3) ionized gas in the inner Galaxy (Petuchowski & Bennett 1993; Heiles 1994), carbon is expected to be ionized. The critical density for collisions with electrons is $\sim $30  cm-3 (Hayes & Nussbaumer 1984) assuming a temperature of 7000 K for the low-density ionized component (Anantharamaiah 1985). Roshi & Anantharamaiah (2001b) calculated a contribution of $8.1\times 10^{-5}$ ergs s-1 cm-2 sr-1 from the low-density ionized regions (one of the components of ELDWIM) in the longitude range l = 0$^\circ $ to 20$^\circ $ (relevant for the comparison between carbon RL and far-infrared line emission). The diffuse [C  II] 158 ${\mu }$m emission within |b| < 2$^\circ $ obtained from the higher resolution far-infrared line observations is $\sim $1.5 $\times 10^{-4}$ ergs s-1 cm-2 sr-1(Nakagawa et al. 1998). Thus, the ELDWIM can contribute $\ge 54\% $ of the observed [C  II] 158 ${\mu }$m emission between l = 0$^\circ $ and 20$^\circ $. However, ELDWIM is not a dominant source of 327 MHz carbon RRL emission since its temperature is high (line optical depth $\propto T_{\rm e}^{-2.5}$) and carbon abundance is only depletion factor times the cosmic abundance ( $4~ \times~ 10^{-4}$; Spitzer 1978). Moreover, the ratio of the carbon to hydrogen line intensity detected in the 327 MHz survey is $\sim $0.5 which is much higher than what is expected from the abundance ratio, suggesting a distinct origin for the two lines. The CNM and PDRs with their relatively low temperatures are envisaged as likely sites of origin for the 327 MHz carbon RRLs. Hence we estimate the contribution of the carbon RRL forming CNM and PDRs to the observed [C  II] 158 ${\mu }$m line intensity.

For estimating the intensity of the FIR line from carbon RRL forming regions, we considered typical parameters estimated for diffuse C  II regions. Kantharia & Anantharamaiah (2001) have modeled the diffuse C  II regions in a few directions in the inner Galaxy. They find that models with temperatures in the range 20 $\rightarrow$ 80 K can fit the observations depending on the angular extent of the line forming region. Even higher temperature ($\sim $150 K) models could fit the observations. Since the total H  I column density (hence H  I opacity) predicted by the higher temperature ($\sim $150 K) models are larger than that observed in the inner Galaxy, we use only models with temperatures in the range 20 $\rightarrow$ 80 K for the FIR line intensity calculation. The estimated electron density and path length corresponding to the observed integrated optical depth near 327 MHz of $\sim $0.01 km s-1  in the inner Galaxy for this temperature range are $0.1 \rightarrow 0.03$  cm-3 and $0.2 \rightarrow 20$ pc respectively. The temperatures and electron densities which explain the low frequency carbon RRLs are encountered in the CNM (Heiles 2001) as well as low-density PDR (Hollenbach et al. 1991). If the line emission is associated with the CNM then the neutral density is $\sim $500 $\rightarrow$ 150  cm-3, which is the atomic density in these clouds. We assumed a carbon depletion factor of 0.5 for these estimates and other calculations presented here. The thermal pressure of these regions are $10~000 \rightarrow 12~000$  cm-3 K, which are not unreasonable for the CNM (Jenkins et al. 1983). The above numbers translate to hydrogen column densities ranging from $\sim $ $3.1 \times 10^{20} \rightarrow
9.3 \times 10^{21}$ cm-2. Such column densities are not unreasonable in the inner Galaxy (Dickey & Lockman 1990). However toward the higher end, they cannot be reconciled with the observed width of the carbon lines (since they have to be shared by different CNM clouds). We discuss these issues in a later publication. Here, we consider the above possible physical conditions for diffuse C  II regions coexisting with CNM.

As described above, the physical properties of carbon RRL forming region is also encountered in low-density PDRs. The regions with $A_{\rm V} <$ 3 mag of low-density PDR models of Hollenbach et al. (1991) have temperature similar to the higher temperature ($\sim $80 K) models of carbon RRL forming regions. Hydrogen is mostly atomic in these regions of the PDR. The neutral density of these regions should be >150  cm-3 to produce the required electron density (>0.03 cm-3) needed for the carbon RRL forming region. Typical observed H  I column density of such regions associated with molecular clouds is $\sim $1020 cm-2 (Wannier et al. 1983), which means several such low-density PDRs are needed along a sight-line to produce the observed carbon RRL. The low-temperature ($\sim $20 K) "diffuse'' C  II regions could be zones with $A_{\rm V} \sim$ 4 mag of the PDR. For example, a low-density PDR model with $n_0 \sim 10^3$  cm-3 and incident FUV flux of $\sim $1.6 ergs cm-2 s-1can have gas temperature $\sim $20 K and electron density $\sim $0.1  cm-3 at $A_{\rm V} \sim$ 4 mag (Hollenbach et al. 1991). Hydrogen is mostly molecular in these regions of the PDR. For the estimation of FIR line emission from $A_{\rm V} \sim 4$ mag region, we use the above given parameters for the low-density PDR, which are typical values in the inner Galaxy. The H  I density for this model is $\sim $10  cm-3 and molecular density is $\sim $1000  cm-3 (see Fig. 4a of Hollenbach et al. 1991).

The intensity of the [C  II] 158 ${\mu }$m line from the neutral regions is given by (Bennett et al. 1994; Watson 1982)

\begin{displaymath}%
I_{{\rm C~{II}}} = 7.416\times 10^{-3}\frac{\frac{g_u}{g_l}...
...\left[ \Sigma \frac{n_i}{ncr_i} \right] ^{-1}} n_{{\rm C}^+}L,
\end{displaymath} (2)

where gu (=4), gl (=2) are the statistical weights of $2{\rm P}_{3/2}$ and $2{\rm P}_{1/2}$ states respectively, $h\nu = 1.26 \times 10^{-14}$ ergs is the energy of the 158 ${\mu }$m photon, k is the Boltzmann's constant, T is the gas temperature in K, $n_{\rm C^+}L$ is the column density of ionized carbon in  cm-3 pc. The above equation has been derived assuming a optically thin line from a two energy state atom. The population of energy states are determined by collisions and spontaneous emission in the optically thin case. In the above equation ni is the density of colliding particles. ncri is the critical density, which is defined as the ratio of the collision rate to the spontaneous emission rate. ncri depends on the temperature of the interacting particles. For the temperatures encountered in the carbon RRL forming region $ncr_i \sim$ 10  cm-3 for electron collision (Hayes & Nussbaumer 1984) and 3000 and 4000  cm-3 for neutral hydrogen and molecular hydrogen respectively as described earlier.

We calculate the expected intensity of the [C  II] 158 ${\mu }$m emission from the CNM and PDR at temperatures of 20 K to be $5.2 \times 10^{-7}$ ergs s-2 cm-2 sr-1and $6.2 \times 10^{-7}$ ergs s-2 cm-2 sr-1 respectively. For temperatures of 80 K, the expected intensity of the fine-structure line from CNM and PDR is found to be $1.4 \times 10^{-4}$ ergs s-2 cm-2 sr-1. Comparing these estimates with what the Balloon-borne Infrared Carbon Explorer (BICE) observed in the inner Galaxy for |b| < 1$^\circ $ (Nakagawa et al. 1998), it appears that for temperatures near 20 K, the contribution to the total observed [C  II] 158 ${\mu }$m intensity is a negligible 0.4% whereas if the temperatures of the C  II regions are near 80 K, then 95% of the total observed [C  II] 158 ${\mu }$m intensity can arise in the diffuse C  II  regions coexistent with CNM or low-density PDRs. Thus, if the temperature of the carbon RRL forming regions is low ($\sim $20 K), then most of the fine-structure line emission is likely to arise elsewhere - either in the ELDWIM or CNM and low-density PDR that do not produce observable carbon RRLs. If the temperature is high ($\sim $80 K), then most of the fine-structure emission is likely to come from the PDRs and CNM that form the same family of diffuse C  II regions which give rise to the low frequency carbon RRLs. In that case a more accurate estimate of the physical properties of the carbon RRL forming region is required to determine the relative importance of ELDWIM and PDRs/CNM to the global contribution of [C  II] 158 ${\mu }$m  line emission. This will be attempted in future with multi-frequency carbon RRL data.

We note that in the inner Galaxy the assumption that the [C  II] 158 ${\mu }$m emission is optically thin is not entirely true. The opacity of the [C  II] 158 ${\mu }$m line is $\sim $0.9 for a typical carbon RRL width of 14 km s-1 (Heiles 1994) arising in a cloud with temperature 80 K in the inner Galaxy. However, for simplicity and to get a first order estimate, we have considered the optically thin case which gives us the interesting results discussed above.

4.2 Longitudinal distribution of the carbon FIR line and radio line

We also attempted a comparison of the longitudinal distribution of the two tracers of ionized carbon. This is relevant since, as discussed in the previous subsection, a considerable fraction of the fine-structure line can be accounted for by the diffuse C  II regions observed in low frequency carbon RRLs under certain physical conditions. If the longitudinal distributions of the two tracers are similar, it would support the higher temperature ($\sim $80 K) models for the carbon RRL forming regions and a substantial fraction of the observed [C  II] 158 ${\mu }$m emission is likely to arise in the carbon RRL forming region.

Wright et al. (1991) and Bennett et al. (1994) have presented the galactic distribution of the [C  II] 158 ${\mu }$m line with an angular resolution of $\sim $7$^\circ $ using the data from the Far-Infrared Absolute Spectrophotometer (FIRAS) aboard the Cosmic Microwave Background Explorer (COBE). Bennett et al. (1994) report strong [C  II] 158 ${\mu }$m emission in the galactic plane with a half-intensity longitude range of $\sim $320$^\circ $ $\rightarrow$ 40$^\circ $. A peak in the FIR emission is seen near l=80$^\circ $  which matches with a peak seen in our 327 MHz RRL data. Bennett et al. (1994) cautioned against over-interpreting this peak due to few measurements in that region. Although the angular resolutions of the two datasets are different, a comparison of the gross distribution shows that the [C  II] 158 ${\mu }$m emission is more widespread than the carbon RRLs near 327 MHz (see Fig. 2). Nakagawa et al. (1998) have used the data from BICE with a much finer angular resolution of 15'. Their survey covers the region from l = $350^{\circ} \rightarrow 25^{\circ}$. They detect [C  II] 158 ${\mu }$m emission in this longitude range from both "compact'' and "diffuse'' regions. The "diffuse'' [C  II] 158 ${\mu }$m emission is observed to extend almost uniformly till the longitude limits of their observations in the galactic plane. In slight contrast, carbon RRLs near 327 MHz have been detected almost contiguously between l = 0$^\circ $ to 20$^\circ $. We do not detect carbon RRLs in the fourth quadrant (up to $l \sim -15$$^\circ $) due to reduced sensitivity of the equatorially-mounted Ooty Radio Telescope. Nakagawa et al. (1998) also observed reduced [C  II] 158 ${\mu }$m emission in regions adjacent to the galactic center up to about longitudes $\sim $$\pm 4$$^\circ $. This is a behavior distinct from our low-resolution survey carbon RRL data which shows comparable integrated optical depths in the galactic plane from l=0$^\circ $  till l=+10$^\circ $  (see Fig. 2).

In summary, intense FIR emission in the galactic plane is observed in the longitude range where carbon RRL near 327 MHz is detected. But the FIR emission seems to be more widespread in the galactic plane. Note that the comparison is, however, limited by (1) the large difference in the sensitivity of the FIR and carbon RRL observations; (2) poor velocity resolution of the existing FIR data. A comparison of the LSR velocities of the two tracers is essential in further establishing any connection between the two spectral lines. We therefore conclude that the existing data does not rule out the possibility that the "diffuse'' C  II regions can significantly contribute to the [C  II] 158 ${\mu }$m emission in the inner region of the Galaxy.


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