4 Discussion

   
4.1 Conditions for extremely high [C II]/CO line ratios

The chemical balance between C⁺ ion and CO molecule depends on dust abundance (Maloney & Black 1988), because interstellar CO molecules are protected from photodissociation mainly by the dust extinction of the incident UV radiation (Wolfire et al. 1989). Accordingly, the high [C II]/CO J = 1-0 intensity ratios observed in irregular galaxies have been attributed to the low dust abundances due to the low metallicities in the galaxies (e.g., Mochizuki et al. 1994). For brief discussions on the C⁺-CO chemical balance, we consider a molecular cloud with a metallicity Xrelative to the solar neighborhood value. The dust abundance is assumed to be proportional to the metallicity. Chemical reaction rates in this cloud can be written as a function of hydrogen column density ($N_{\rm H}$) measured from the cloud surface into the cloud. The total hydrogen (mostly in H atom and H₂ molecule) number density, $n_{\rm H}$, is assumed to be uniform in the cloud.

The conversion of $\rm CO \rightarrow C^+$ is initiated mostly by the dissociation of CO molecule due to UV photons incident on the cloud (e.g., van Dishoeck & Black 1988). Since the photodissociation is followed quite quickly by the photoionization of $\rm C \rightarrow C^+$, the $\rm CO \rightarrow C^+$ conversion rate, $r_{\rm CO} ( N_{\rm H} )$, in the cloud can be written as

$$% r_{\rm CO} ( N_{\rm H} ) = G_0 \alpha_{\rm CO}^0 n_{\rm C... ...ft( - X \frac{ N_{\rm H} }{ N_{\rm H}^{\rm UV} } \right),$$ (1)

where G₀ is the incident UV flux relative to the solar neighborhood value, $\alpha_{\rm CO}^0$ is the CO photodissociation rate under the interstellar radiation field at the solar neighborhood, $n_{\rm CO} ( N_{\rm H} )$ is the number density of CO as a function of $N_{\rm H}$, and $N_{\rm H}^{\rm UV}$ is the hydrogen column density characteristic to the attenuation of the UV radiation effective to the CO dissociation at X = 1.

On the other hand, the conversion of $\rm C^+ \rightarrow CO$consists of two-body reactions in the gas phase. Thus, the $\rm C^+ \rightarrow CO$ conversion rate, $r_{\rm C^+}$, can be approximately written as

$$% r_{\rm C^+} ( N_{\rm H} ) = k_{\rm C^+} n_{\rm H} n_{\rm C^+} ( N_{\rm H} ),$$ (2)

where $k_{\rm C^+}$ is the total rate coefficient of the $\rm C^+ \rightarrow CO$ reactions and $n_{\rm C^+} ( N_{\rm H} )$ is the number density of C⁺ in the cloud. We assume $k_{\rm C^+}$ is constant for the present discussion, although it is actually a function of chemical composition as well as of gas temperature. Its dependence on X through the chemical composition is relatively weak because the $\rm C^+ \rightarrow CO$ conversion rate is often determined by reactions between carbon-bearing species and hydrogen species such as between hydrocarbon anions and hydrogen molecules.

When a steady state $r_{\rm CO} ( N_{\rm H} ) = r_{\rm C^+} ( N_{\rm H} )$ is assumed, the condition $n_{\rm CO} ( N_{\rm H} ) = n_{\rm C^+} ( N_{\rm H} )$ yields a certain value of $N_{\rm H}$, which we define as the transition column density ( $N_{\rm H}^{\rm transit}$) of C⁺-CO. The dominant form of gas-phase carbon is C⁺ at $N_{\rm H} < N_{\rm H}^{\rm transit}$, and CO at $N_{\rm H} > N_{\rm H}^{\rm transit}$. From Eqs. (1) and (2), $N_{\rm H}^{\rm transit}$can be written as

$$% N_{\rm H}^{\rm transit} = \frac{ N_{\rm H}^{\rm UV} }{ X ... ...rm CO}^0 }{ k_{\rm C^+} } \frac{ G_0 }{ n_{\rm H} } \right)$$ (3)

(e.g., Mochizuki et al. 1994).

We estimate $\alpha_{\rm CO}^0 / k_{\rm C^+} \sim 10^4\ \rm cm^{-3}$and $N_{\rm H}^{\rm UV} \simeq 7~10^{20}\ \rm cm^{-2}$according to the chemical network in the models of Hollenbach et al. (1991). Because of the $\ln[ ( \alpha_{\rm CO}^0 / k_{\rm C^+} ) ( G_0 / n_{\rm H} ) ]$dependence, $N_{\rm H}^{\rm transit}$ is insensitive to $G_0 / n_{\rm H}$at $G_0 / n_{\rm H} \gg ( \alpha_{\rm CO}^0 / k_{\rm C^+} )^{-1} \sim 10^{-4}\ \rm cm^3$. According to Eq. (3), $N_{\rm H}^{\rm transit}$ does not exceed $10^{22}\ \rm cm^{-2}$over the wide range of $G_0 / n_{\rm H} \leq 10\ \rm cm^3$. Thus, CO molecules survive inside the clouds with $n_{\rm H} \simeq 10^3\ \rm cm^{-3}$under the realistic conditions of $G_0 \leq 10^4$ ( G₀ = 10²-10⁴ in starbursts; Stacey et al. 1991), if the typical column density ( $N_{\rm H}^{\rm cloud}$) of the clouds is $N_{\rm H}^{\rm cloud} \simeq 2~10^{22}\ \rm cm^{-2}$($\simeq$ $1~10^{22}\ \rm cm^{-2}$ from the surface to the center). This makes the upper limit for the [C II]/CO J = 1-0 line ratios in normal and starburst galaxies (Stacey et al. 1991).

On the other hand, $N_{\rm H}^{\rm transit}$ varies as X^-1. When X = 1/4, typical in the low-metallicity galaxies observed in the [C II] line, $N_{\rm H}^{\rm transit}$ is $10^{22}\ \rm cm^{-2}$at $G_0 / n_{\rm H} \simeq 10^{-2}\ \rm cm^3$. Thus, if $N_{\rm H}^{\rm cloud} \simeq 2~10^{22}\ \rm cm^{-2}$, most of the CO molecules in the clouds are dissociated even at $G_0 \simeq 10$ and $n_{\rm H} \simeq 10^3\ \rm cm^{-3}$in the low-metallicity galaxies. This G₀ is close to the average ( $G_0 \simeq 5$) estimated for the clouds emitting the Galactic [C II] emission (Mochizuki & Nakagawa 2000). Hence, a typical G₀ and $n_{\rm H}$ of galactic molecular clouds can produce the extremely high [C II]/CO line ratios observed in the irregulars, beyond the upper limit for more luminous starbursts.

However, the $\ln[ ( \alpha_{\rm CO}^0 / k_{\rm C^+} ) ( G_0 / n_{\rm H} ) ]$dependence makes $N_{\rm H}^{\rm transit}$ more sensitive to $G_0 / n_{\rm H}$at than at . In the former case, a small $G_0 / n_{\rm H}$ can compensate a small X; (e.g., G₀ = 10 and ) provides $N_{\rm H}^{\rm transit} < 10^{22}\ \rm cm^{-2}$even at X = 1/4. As a result, a low-metallicity galaxy can show a normal [C II]/CO line ratio at a small $G_0 / n_{\rm H}$while a more luminous spiral galaxy cannot show a much higher ratio than the starburst limit even at a large $G_0 / n_{\rm H}$, under the assumption of $N_{\rm H}^{\rm cloud} \simeq 2~10^{22}\ \rm cm^{-2}$.

Figure 3: The ${\rm [C{\sc ii}]/^{12}CO}\ J = 1$-0 line luminosity ratio (solid curves) of a molecular cloud based on PDR models as a function of the hydrogen number density of the cloud. The relative metallicity and dust abundance are X = 1/4(corresponding to $\rm 12 + \log[ O / H ] = 8.1$) for the low-metallicity models, and X = 1 for the Galactic-abundance models. The incident UV flux and the mean hydrogen column density of the model cloud are G0 = 10and $\langle N_{\rm H} \rangle = 2~10^{22}\ \rm cm^{-2}$, respectively. The mean [C II]/CO J = 1-0 intensity ratios observed in the inner Galaxy and the LMC (dotted lines) and the upper limit obtained for I Zw 36 (dashed line) are also indicated

Figure 4: Same as Fig. 3 but with variation in G0 for X = 1/4

   
4.2 Comparison with photon-dominated region models

For more quantitative investigation, we compare the observed [C II]/CO J = 1-0 line ratio with the photon-dominated region (PDR) models by Mochizuki & Nakagawa (2000). In their models, the luminosities of emission lines emergent from the model cloud are derived from a given set of the incident UV flux G₀ (Sect. 4.1), the mean number density of total hydrogen in the cloud $\langle n_{\rm H} \rangle$, and the cloud mass M. We assume a uniform hydrogen density in the cloud for simple discussion of the density dependence of the line ratio, and thus use the hydrogen density $n_{\rm H}$ instead of $\langle n_{\rm H} \rangle$. In addition, the mean hydrogen column density $\langle N_{\rm H} \rangle$ is used as in Mochizuki (2000) instead of M, in accordance with the discussion on $N_{\rm H}^{\rm transit}$in Sect. 4.1.

We can estimate G₀ in I Zw 36 from UV observations with an accuracy limited by the uncertainty (see below) in extinction. The flux density observed toward I Zw 36 is $F_{2000} = 1.0~10^{-13}\ \rm ergs\ s^{-1}\ cm^{-2}$ Å^-1at $\lambda = 2000$ Å after the correction for an extinction of 0.7 mag (Donas et al. 1987). Assuming a uniform distribution of UV sources in a sphere with diameter equal to that in the optical ( $a = 0 \hbox{$.\mkern-4mu^\prime$ }92$; de Vaucouleurs et al. 1991), we obtain the volume-averaged flux density, $\langle F_{2000} \rangle$, in the sphere: $\langle F_{2000} \rangle = 12 F_{2000}^0$, where $F_{2000}^0 = 1.0~10^{-6}\ \rm ergs\ s^{-1}\ cm^{-2}$ Å^-1is the 2000 Å flux density in the solar neighborhood by Mathis et al. (1983). Since a is close to the [C II] and CO beam sizes, we adopt $G_0 \simeq 10$ as an average for I Zw 36 on the scale seen in the [C II]/CO line ratio.

Figure 3 shows the calculated [C II]/CO J = 1-0 line ratios at G₀ = 10plotted as a function of $n_{\rm H}$, along with the observed ratios. The metallicity and the dust abundance in the low-metallicity models are X = 1/4, while X = 1 in the original models (Mochizuki & Nakagawa 2000) for our Galaxy. The former metallicity corresponds to $\rm 12 + \log[ O / H ] = 8.1$, typical in the low-metallicity galaxies observed in the [C II] line. The Galactic-abundance models of X = 1 with the same G₀ are also plotted for comparison. We adopted $\langle N_{\rm H} \rangle = 2~10^{22}\ \rm cm^{-2}$for both the X = 1/4 and X = 1 models, on the basis of the rough estimate (Sect. 4.1) that accounts for the much higher [C II]/CO J = 1-0 line ratios in irregulars than in starbursts (see also below).

At X = 1 and $n_{\rm H} \simeq 10^3\ \rm cm^{-3}$, which represent Galactic molecular clouds, the calculated line ratio is ${\rm [C{\sc ii}]/CO}\ J = 1$- $0 \simeq 1~10^3$, close to those observed in the inner Galactic plane. On the other hand, at X = 1/4, the model with the same $n_{\rm H}$has an extremely high ratio of ${\rm [C{\sc ii}]/CO}\ J = 1$-0 > 10⁴as observed in irregular galaxies. This indicates that the difference in X can account for the observed difference in the [C II]/CO ratio between our Galaxy and the irregulars. With increasing $n_{\rm H}$, the calculated line ratio decreases as expected from Eq. (3). The ratio becomes consistent with the upper limit observed in I Zw 36 at $n_{\rm H} > 10^4\ \rm cm^{-3}$. These calculations indicate that a higher gas density in I Zw 36 is required to reproduce the observed difference in the [C II]/CO ratio between I Zw 36 and the irregular galaxies.

The UV flux of $G_0 \simeq 10$ adopted above was estimated on the basis of the extinction derived from the H I 21 cm line intensity, H I flux divided by a² (Donas et al. 1987). This method can underestimate the extinction by up to 1 mag (Donas et al. 1987) because of oversimplified assumptions on the distribution of H I gas (Donas & Deharveng 1984). By comparison the extinction based on a line ratio toward the dominant H II region (for a smaller aperture of $10\hbox{$^{\prime\prime}$ }\times 20\hbox{$^{\prime\prime}$ }$) in I Zw 36 is 1.4 mag at $\lambda = 1909$ Å (Viallefond & Thuan 1983). Accordingly, we calculated ratios also for G₀ = 10^1.5as well as G₀ = 10^0.5(Fig. 4). At G₀ > 10, the line ratio in the figure is too high compared to the I Zw 36 observations. Thus, G₀ cannot be so large on the galactic scale unless $\langle N_{\rm H} \rangle$ is substantially larger than $2~10^{22}\ \rm cm^{-2}$. On the other hand, at $G_0 \simeq 6$ (no extinction at $\lambda = 2000$ Å), Fig. 4 shows that a high gas density of is still required. As a result, the observed low line ratio in I Zw 36 indicates a high gas density of .

For the above discussions, we assumed $N_{\rm H}^{\rm cloud} \sim 10^{22}\ \rm cm^{-2}$in every galaxy, on the basis of the difference in the line ratios between irregulars and starbursts (Sect. 4.1). However, this assumption is difficult to confirm observationally. When a constant cloud mass is assumed, instead of the constant $N_{\rm H}^{\rm cloud}$ above, $N_{\rm H}^{\rm cloud}$ would increase with $n_{\rm H}$ (e.g., Mochizuki 2000). In addition, Pak et al. (1998) concluded that $N_{\rm H}^{\rm cloud}$ increases with decreasing metallicity, on the basis of large-scale observations of H₂ vib-rotational lines. These suggest that $N_{\rm H}^{\rm cloud}$ may be larger in I Zw 36 than in the irregulars observed in the [C II] emission. Since a larger $N_{\rm H}^{\rm cloud}$ can also contribute to a lower [C II]/CO line ratio, we expect that $n_{\rm H}$ in I Zw 36 is between that in the irregulars and that estimated above for the constant $N_{\rm H}^{\rm cloud}$ case: . Otherwise, a large amount of neutral gas in I Zw 36 would have a very high column density of .

   
4.3 Interstellar medium in I Zw 36

The [C II]/CO J = 1-0 line ratio varies from place to place in a galaxy. For the low ratio of I Zw 36 on the galactic scale, a large fraction of neutral interstellar gas should have a high density. However, the presently available observations do not allow us to investigate density distribution of molecular gas within I Zw 36. Instead we discuss distributions of the starburst (young) stellar population by considering that a starburst results from a high density of molecular gas.

Papaderos et al. (1996a) decomposed the optical spatial profiles of BCDGs into starburst and underlying stellar components, and then derived the area ratios of the starburst components to the underlying components. They found that the area ratio increases with decreasing galactic luminosity (Papaderos et al. 1996b), which is generally correlated with metallicity. A fraction as large as about a half of the optical area is occupied by the starburst component in I Zw 36, which lies close to the low-luminosity end in their sample. Hence, I Zw 36 is likely to have physical conditions producing high-density gas in a larger fraction of its optical area than more luminous BCDGs and irregulars are. This supports a higher average gas density within the optical area in the galaxy.

The present [C II]/CO observations place a limit only for the gas density in the CO beam (Sect. 3), which has a similar size to the optical area of the galaxy. Thus, the proposed high density does not conflict with the presence of a diffuse H I halo (Viallefond & Thuan 1983) extended to a diameter of $\simeq$ $150\hbox{$^{\prime\prime}$ }$, where the gas density is possibly lower.

Since irregular galaxies have extremely high [C II]/CO line ratios generally (Fig. 2), lower [C II]/CO line ratios as well as expected higher gas densities may be distinctive characteristics of a certain class of BCDGs among low-metallicity galaxies: galactic morphology may be one of the crucial factors. This implies that the gravitational potential may change on a large scale with galactic evolution, if BCDGs and irregular galaxies have evolutionary links (e.g., Davies & Phillipps 1988). Such a change is more likely to occur in a dwarf galaxy than in a more massive one, because a dwarf galaxy has a large mass fraction of ISM (Huchtmeier & Richter 1988). For investigation of difference in the ISM properties between BCDGs and irregulars, more samples of BCDGs with variation in metallicity (luminosity) would be helpful.