3 Physical conditions of interstellar medium

Together with the [O I]63 $\mu$m and 145 $\mu$m lines, [C II]158 $\mu$m can be used to derive the physical conditions of the line-emitting regions based on PDR models (e.g., Tielens & Hollenbach 1985; Wolfire et al. 1990; Hollenbach & Tielens 1997; Kaufman et al. 1999), in which the major model parameters are the incident FUV radiation field flux $G_{\rm0}$ in units of the solar neighborhood value ( $1.6 \times 10^{-6}$ Wm^-2, Habing 1968) and the neutral hydrogen gas density n. However, the [C II] line could also originate from ionized regions and the fraction of the contribution cannot be estimated a priori.

To estimate the contribution to [C II]158 $\mu$m from PDRs, we take two approaches similar to Malhotra et al. (2001b). First we assume that all the emission of [O I]63 $\mu$m and far-infrared continuum ( $\lambda \ge 80~\mu$m) comes from PDRs. Since the temperature of sub-micron sized dust grains is determined by the intensity of the incident radiation (e.g., Onaka 2000), $G_{\rm0}$ can be estimated from the dust temperature $T_{\rm d}$ derived by Eq. (1). We used a semi-analytical equation of $G_{\rm0}$ and $T_{\rm d}$ given by Hollenbach et al. (1991) with A_V = 0.5 because a major fraction of [C II]158 $\mu$m and [O I]63 $\mu$m emissions stem from the region of $A_{V} \le 1$ (Kaufman et al. 1999). For M 82 we derive $G_{\rm0} = 10^{3.4}$, while Kaufman et al. (1999) estimated $G_{\rm0} = 10^{3.5}$ by taking account of several observed line intensities, suggesting that the present method provides a reasonable estimate of $G_{\rm0}$.

Then we compare the ratio of [O I] $/{\it FIR}$ with the PDR model of Kaufman et al. (1999) with the derived $G_{\rm0}$ to estimate n. Finally we estimate the intensity of [C II]158 $\mu$m from PDRs with the derived $G_{\rm0}$ and n. The current estimate of the flux uncertainty is 20% and there may be an uncertainty in the PDR model due to the assumed geometry. While FIR may be underestimated by a few tens % in the present analysis, it does not introduce a significant error compared to other uncertainties.

Figure 4 shows $G_{\rm0}$ and n against the color R(60/100).

Figure 4: a) FUV incident flux estimated by the dust temperature, $G_{\rm0}$, and the derived cloud gas density, n, against the far-infrared color R(60/100). Filled circles indicate $G_{\rm0}$ and open squares n [cm-3]. b) Ratio of the [C II] intensity from PDRs to the observed total [C II] intensity vs. the R(60/100).

The error in $G_{\rm0}$ comes from the uncertainty in $T_{\rm d}$, while that in n is estimated from the uncertainty in [O I] $/{\it FIR}$ to be about $\Delta \log n \simeq 0.3$ based on the PDR models. Both parameters, $G_{\rm0}$ and n, increase clearly with R(60/100) and n is found to be almost proportional to $G_{\rm0}$ ( $\log n = (0.8 \pm 0.25) \log G_{\rm0}+ 0.75_{-0.42}^{+0.26}$, see also Fig. 10). The trend mainly comes from the observed constancy of [O I] $/{\it FIR}$ against the color. In the PDR model the photoelectric heating efficiency is roughly a function of $G_{\rm0}/n$ (Bakes & Tielens 1994). For $G_{\rm0} \propto n$, the heating efficiency does not change and thus [O I] $/{\it FIR}$ stays constant. The [C II]158 $\mu$m $/{\it FIR}$, on the other hand, decreases because the collisional de-excitation becomes efficient as n increases: the [C II] line has a lower critical density for excitation ($\sim$ $3 \times 10^3$ cm^-3) than [O I]63 $\mu$m ($\sim$ $1\times 10^{7}$ cm^-3). In this situation, we expect an increase in the gas temperature. Because the gas temperature is already sufficiently high in the parameter range in question, the increase in the temperature does not affect the line intensities appreciably. Hence the decrease in [C II] $/{\it FIR}$ with color can be attributed to the increase in the collisional de-excitation of the [C II] transition for the present sample of the galaxies. The [C II]158 $\mu$m emission from PDRs is estimated to be $\sim$$50 \pm 30$% of the total observed [C II]158 $\mu$m emission in the present analysis and the fraction of the PDR component is indicated to decrease with R(60/100)(Fig. 4b). Therefore, the decrease in the total [C II]/FIR is attributed to the decrease in the PDR component due to the thermalization of the [C II] transition.

In order to examine the reliability of the present data reduction and analysis we can compare the present results with those of previous works for some individual galaxies. The comparison is summarized in Table 4.

 

 

Table 4: Comparison with previous studies.

object	ref.a	[O I]63 $\mu$m	[O III]88 $\mu$m	[N II]122 $\mu$m	[O I]145 $\mu$m	[C II]158 $\mu$m	$G_{\rm0}$	n
		(10-15 Wm-2)	(10-15 Wm-2)	(10-15 Wm-2)	(10-15 Wm-2)	(10-15 Wm-2)		(cm-3)
M 82	p	169 $\pm$ 34	91 $\pm$ 18	21 $\pm$ 4	15 $\pm$ 3	128 $\pm$ 26	103.4	103.6
	1	176 $\pm$ 5	86 $\pm$ 4	17 $\pm$ 3	12 $\pm$ 1	134 $\pm$ 1	102.8	103.3
Cen A	p	19 $\pm$ 4	6.3 $\pm$ 1.4		1.0 $\pm$ 0.2	29 $\pm$ 6	102.7	103.1
	2	19.6	7.0	1.5	1.1	29.1	$\sim$102	$\sim$103
NGC 4414	p	3.2 $\pm$ 0.6		0.9 $\pm$ 0.2		7.8 $\pm$ 1.6
	3	3.3 $\pm$ 1	1 $\pm$ 0.5	1.3 $\pm$ 0.4		10.6 $\pm$ 2
NGC 253	p	38 $\pm$ 7.5	11.4 $\pm$ 2.5			52 $\pm$ 10	103	102.5
	4	45 $\pm$ 6	6 $\pm$ 1			48 $\pm$ 2	104.3	104
NGC 3256	p	12.8 $\pm$ 2.6	6.0 $\pm$ 1.3			13.7 $\pm$ 2.7	103.1	103.2
	4	14.3 $\pm$ 2.6	4.8 $\pm$ 1.2			11.7 $\pm$ 2.5	103	103.9

^a References: p present work; 1) Colbert et al. (1999); 2) Unger et al. (2000); 3) Braine & Hughes (1999); 4) Carral et al. (1994).

Colbert et al. (1999) and Unger et al. (2000) analyzed the same LWS data and derived $G_{\rm0}$ and n by using the PDR models of Kaufman et al. (1999) for M 82 and Cen A, respectively. The present analysis provides fairly good agreement with their results. The line intensities derived in the present study also agree with those by Braine & Hughes (1999) within the measurement errors. Carral et al. (1994) reported the results of observations of NGC 253 and NGC 3256 by the KAO and obtained $G_{\rm0}$ and n based on the PDR models of Wolfire et al. (1990). While the line fluxes are in agreement with the present results within the errors except for [O III]88 $\mu$m of NGC 253, the present results indicate systematically low densities. The difference can be attributed to the relatively high gas temperature with the same $G_{\rm0}$ and n in the models of Kaufman et al. because their models include the additional gas heating due to polycyclic aromatic hydrocarbons (PAHs). The [C II] emission other than the PDR origin may be ascribed to the extended low density warm ionized medium (ELDWIM). A large fraction of [N II]122 $\mu$m line emission is thought to stem mostly from the ELDWIM (Wright et al. 1991; Heiles 1994; Bennett et al. 1994; Petuchowski et al. 1994). Figure 5 plots the ratio of the observed [N II]122 $\mu$m intensity to the non-PDR component of [C II]158 $\mu$m.

Figure 5: Ratio of [N II]122 $\mu$m intensity to the non-PDR component of [C II]158 $\mu$m against R(60/100).

The ratio shows a large scatter with a weak trend that the ratio decreases with R(60/100). Because of the uncertainties the trend may be spurious (see below).

All the data points are located within the range 0.1-0.7 in Fig. 5. The line ratio of [N II]/[C II] in the ionized gas depends on the electron density, but is insensitive to the temperature of the ionized gas. To estimate the line ratio expected from ionized gas we assume $T_{\rm e}=7000$ K and the abundance of [ C⁺]/[H⁺] = $4 \times 10^{-4} \delta_{{\rm C}^+}$ and [ N⁺]/[H $^+] = 1 \times 10^{-4} \delta_{{\rm N}^+}$ with the depletion factors as $\delta_{{\rm C}^+} = 0.65$ and $\delta_{{\rm N}^+} = 1$ in the following discussion (Heiles 1994). We adopted the collision coefficients for N⁺ from Stafford et al. (1994) and those for C⁺ from Heiles (1994). Recent HST observations indicate that the interstellar abundance of carbon and nitrogen in the gas phase is fairly constant on various lines of sight in our Galaxy as $\delta_{{\rm C}} = 0.35 \pm 0.05$ (Sofia et al. 1997) and $\delta_{{\rm N}} = 0.75 \pm 0.05$ (Meyer et al. 1997). Based on these values the relative abundance of N to C will increase by 30%. The following discussion thus has an uncertainty of this level associated with the assumed abundance.

The lower boundary of the observed ratio 0.1 is then found to correspond to the low-density limit of the ratio in the ionized gas. The upper boundary 0.7 is obtained for a gas with $n_{\rm e}$ = 120 cm^-3. Petuchowski et al. (1994) reported a large [N II]122 $\mu$m to 205 $\mu$m line ratio in the central 850 pc of M 82 compared to the average ratio of the Milky Way (Wright et al. 1991), indicating that a fair fraction of the [N II] line emission comes from the ionized gas of $n_{\rm e} = 150$-180 cm^-3 in M 82. The ratio of [N II]122 $\mu$m to [C II]158 $\mu$m of non-PDR origin for M 82 is about 0.3 in the present analysis, suggesting that there may be a significant contribution to the [CII] emission from the low density diffuse ionized gas in the outer part (>850 pc) of the galaxy. The observed intensity is compatible with the interpretation that the non-PDR component of [C II]158 $\mu$m comes from the ionized gas that emits [N II]122 $\mu$m.

The ionized gas also emits a radio continuum. The intensity of free-free transition is written for $T_{\rm e} = 7000~ {\rm K}$ by

$$I(4.85~\hbox{GHz})\left[\hbox{mJy sr}^{-1}\right] = 3.12 \times 10^{-14}~ n_{\rm e}^2 l,$$ (3)

where $n_{\rm e}$ is the electron density in cm^-3 and l is the path length in cm (Spitzer 1978). The intensity of the [C II] line from the ionized gas is given by

$$I_{\hbox{[C{\sc ii}]}} ({\rm ELDWIM}) = (1/4\pi) L(T_{\rm e})n(C^+)n_{\rm e} l,$$ (4)

where $L(T_{\rm e})$ is the cooling function of [C II]158 $\mu$m (e.g., Hayes & Nussbaumer 1984). For a given [C II]158 $\mu$m intensity, the corresponding radio continuum intensity increases with $n_{\rm e}$ as can be estimated through Eqs. (3) and (4). As a conservative upper limit we assume that the non-PDR component of [C II] 158 $\mu$m emission comes from the ionized gas of density 200 cm^-3. Figure 6 shows the comparison between the predicted and observed flux densities.

Figure 6: The observed 4.85 GHz radio continuum flux density plotted against the 4.85 GHz radio continuum flux density predicted from the non-PDR component of [C II]158 $\mu$m emission (see text). The observed radio flux densities are taken from Gregory et al. (1991, 1994), Becker et al. (1991), and Griffith et al. (1994).

We take the 4.85 GHz radio continuum data from Becker et al. (1991), Gregory & Condon (1991), Gregory et al. (1994), and Griffith et al. (1994). The beam size of the radio observations was about $4\hbox{$^\prime$ }$ (FWHM). Observations of 1.49 GHz and 1.425 GHz indicate that most of the radio continuum emission comes from the area within about $1\hbox{$^\prime$ }$ of the center of the infrared emission (Condon et al. 1990, 1996). We assume that the observed 4.85 GHz emission also comes from the region within the LWS beam. The predicted value shows a positive correlation with the observed flux density except for Cen A. Cen A is a very strong radio source and most of the radio emission from Cen A is nonthermal (e.g. Sreekumar et al. 1999). The observed radio intensities are larger than the upper limits predicted from [C II]158 $\mu$m emission of the non-PDR origin for most of the sample galaxies. Thus the non-PDR component of [C II]158 $\mu$m emission is compatible with the observed radio continuum intensity when it arises mostly from the low-density ionized gas for the present sample of galaxies.

Figure 7 shows the ratio of [O I]145 $\mu$m to [O I]63 $\mu$m against R(60/100).

Figure 7: Ratio of the intensity of the [O I]145 $\mu$m line to the [O I]63 $\mu$m line against R(60/100).

The observed ratio is in the range 0.05-0.1 for most galaxies, which is in agreement with the prediction of the PDR model with the range of $G_{\rm0}$ and n estimated in the present analysis. The ratio does not exceed 0.1 within a reasonable range of the density and temperature (Watson 1984). However, a few galaxies clearly show higher ratios than the model. They are NGC 4945 and NGC 253, both of which have large inclination angles (for NGC 4945 and NGC 253, $i \simeq 85\hbox{$^\circ$ }$ and $80\hbox{$^\circ$ }$, respectively). NGC 520, which shows the third largest ratio, has $i=60\hbox{$^\circ$ }$. The absorption of [O I]63 $\mu$m in the interstellar medium may affect the ratio in these galaxies. For a majority of the galaxies, however, this effect is probably not significant.

In the analysis described above it is difficult to properly evaluate the uncertainties in the comparison with the model. In order to examine how robust the derived conclusions are, we take another approach to estimate the PDR contribution to the [C II]158 $\mu$m line emission. Figure 5 indicates that there seems no strong trend in the ratio of [N II]122 $\mu$m to the non-PDR origin of [C II]158 $\mu$m. We thus simply assume that the contribution from the ionized gas to [C II]158 $\mu$m emission is proportional to the [N II]122 $\mu$m intensity. We take a mean value of Fig. 5 as

$$[C{\sc ii}]158~\mu{\rm m}({\rm ELDWIM}) = 3.5 \times [N{\sc ii}]122~\mu{\rm m}.$$ (5)

This relation corresponds to a gas of $n_{\rm e} = 35 ~{\rm cm}^{-3}$ for $T_{\rm e}=7000$ K. We estimate the intensity of [C II]158 $\mu$m from the ionized gas based on the [N II]122 $\mu$m intensity by using Eq. (5) and attribute the rest to that coming from PDRs. We then estimate $G_{\rm0}$ and n from [O I]/[C II] and ([C II]+[O I]) $/{\it FIR}$. Figure 8a shows $G_{\rm0}$ and n estimated in this method.

Figure 8: The results of the analysis based on the assumption that the contribution from the ionized gas to [C II]158 $\mu$m is proportional to [N II]122 $\mu$m as given by Eq. (5) (see text). a) FUV incident flux, $G_{\rm0}$ (filled circle) and the gas density, n (open square) derived from [O I]/[C II] and ([C II]+[O I]) $/{\it FIR}$ of the PDR model by Kaufman et al. (1999) against R(60/100). b) The ratio of the estimated [C II] intensity from PDRs to the observed total [C II] intensity against R(60/100).

The values of $G_{\rm0}$ derived in this method are in agreement with those obtained in the first approach within the estimated errors. Thus the parameters derived in the second method are still compatible with the continuum spectrum of the LWS spectra. We have obtained the same trend as in the first approach: both n and $G_{\rm0}$ increase with the color and n is roughly proportional to $G_{\rm0}$ (see also Fig. 10). We conclude that the linear increase of n with $G_{\rm0}$ is a rather secure result for the present sample galaxies. The weak trend seen in Fig. 5 is not necessarily real. Figure 8b shows the fraction of the PDR component of [C II] emission derived in this analysis, suggesting that the PDR component does not vary with R(60/100) in contrast to Fig. 4b. Thus in this analysis the decrease in [C II] $/{\it FIR}$ can be interpreted in terms mainly of the decrease in the ionized component relative to FIR as indicated in the decrease in [N II] $/{\it FIR}$ (Fig. 2a), though the decrease in [C II](PDR) $/{\it FIR}$ due to the thermalization also contributes partly.

The ratio of [C II]158 $\mu$m to ¹²CO (J=1-0) line intensity is another measure for the diagnosis of PDRs.

Figure 9: Ratio of the [C II]158 $\mu$m intensity to the CO (J=1-0) intensity against R(60/100). The CO emission has not been detected for NGC 6824 and a lower limit for the ratio is plotted.

Because C⁺ is converted to CO as the gas shielding to prevent CO from dissociation becomes efficient, the thickness of C⁺ layer is a function of $G_{\rm0}/n$ (Mochizuki & Onaka 2001). The intensity ratio increases with $G_{\rm0}$ and decreases with nand it is a function of $G_{\rm0}/n$ in the range of $G_{\rm0}$ and n in question (Pierini et al. 1999; Kaufman et al. 1999; Mochizuki & Nakagawa 2000). Metallicity also plays an important role in the C⁺ to CO conversion (Mochizuki et al. 1994). Hence, unless there is an appreciable variation in the metallicity in the sample galaxies, the ratio of [C II]/CO approximately varies with $G_{\rm0}/n$. Figure 9 plots the ratio against the color. The CO data are taken from Young et al. (1995), Elfhag et al. (1996), Aalto et al. (1995), Curran et al. (2001), Stacey et al. (1991), and Eckart et al. (1990). The CO observations had the beam size of 45-56 $^{\prime\prime}$. We assume that the CO emission comes mostly from the central part of the galaxies and did not make any corrections for the beam size. Except for NGC 6824, the ratio stays almost constant, supporting $n \propto G_{\rm0}$. The constancy of $G_{\rm0}/n$ is also confirmed by the [C II]/CO ratio. The CO (J=1-0) emission in NGC 6824 was not detected (Young et al. 1995). With the upper limit the ratio of [C II]/CO is estimated to be larger than 10⁴, which is in a similar range to those found in quiet spirals (Smith & Madden 1997). The values of the observed ratio of the other galaxies $8 \times 10^2 - 3 \times 10^3$ are in agreement with those observed in Galactic PDRs (Stacey et al. 1991). It is slightly smaller than the values predicted in the PDR model.