Astron. Astrophys. 363, 1123-1133 (2000)

4. Discussion

The [CII] emission can originate in the neutral (e.g., Shibai et al. 1991) and ionized (e.g., Heiles 1994) phases of the ISM. When the neutral phase is considered, the nearly constant FIR [CII]-to-continuum ratio in the Galactic disk can be accounted for by a stable heating ratio of gas-to-dust as mentioned below (Sect. 4.1; also in Mochizuki & Nakagawa 2000). On the other hand, in the ionized phase, we do not find a reason for the stable line-to-continuum ratio because: (1) the [CII] emission is not the dominant coolant of the gas; (2) the gas is not heated via dust grains. Since the in the center of M31 is close to the nearly constant ratio in the Galactic disk, the [CII] emission there is likely to originate in the neutral phase. Thus, we discuss the different FIR [CII]/continuum flux ratios between the two galactic centers on the basis of Photon-Dominated Region (PDR) models, which are made for the neutral phase of the ISM.

4.1. Factors affecting the [CII]/100 µm ratio

When the ISM in a steady state is considered, the ratio of the [CII] flux to the FIR flux () integrated over wavelength can be written as:

where is the fraction of the [CII] cooling in the total gas cooling, and and are heating rates of the gas and dust, respectively. The gas heating is usually dominated by energetic photoelectrons from grain surfaces illuminated by stellar light (de Jong 1977) in the neutral phase of the galactic ISM. In this case, the heating ratio of can be replaced by the efficiency (Tielens & Hollenbach 1985), which is defined as the ratio of the energy carried away by the emitted photoelectrons to that absorbed by the grains, of the photoelectric heating as follows:

The nearly constant observed in the Galactic disk (Nakagawa et al. 1998) indicates that is nearly constant and that the [CII] emission dominates the gas cooling (), in wide ranges of physical conditions in the neutral phase of the Galactic ISM.

Although the ratio is stable, it can be affected by several factors:

1. The color of the stellar light illuminating the ISM. Grains can be heated by photons not sufficiently energetic to produce a photoelectron from the grains. Thus, soft stellar radiation decreases the effective for the whole wavelength range of the stellar light. This leads to a lower ratio (Paper I).

2. The hydrogen column density of a cloud. At a lower column density, the cloud is translucent for less energetic incident photons. This increases the emergent ratio to that of an opaque cloud illuminated by harder stellar radiation.

3. The charge of grains. When a grain is positively charged, further emission of a photoelectron requires more energy because of the opposite charges of the grain and electron (de Jong 1977). This decreases and consequently decreases the ratio. The grain charge is determined by the balance between the photoelectron emission and the grain-electron recombination. The emission rate of photoelectrons varies roughly as while the recombination rate does as , where is the intensity of the illuminating stellar radiation, and n is the gas density. Thus, the grains are positively charged at a high ratio of .

4. The gas cooling due to other lines. When the stellar radiation ionizing the carbon is weak, a sufficient amount of C⁺ ions is not produced. In this case, the gas must be cooled through other lines, such as [CI] fine-structure and CO rotational lines. On the other hand, the [OI] and the [SiII] fine-structure lines can cool the gas predominantly at a high gas temperature.

5. The temperature of grains. When the dust temperature is low () even near the cloud surface, where most of FIR continuum is emitted, does not trace accurately. This results in a higher ratio.

4.2. Comparison with PDR models

We compare the observed flux ratios with the luminosity ratios based on the PDR models of Mochizuki & Nakagawa (2000). The model cloud is spherical and immersed in isotropic stellar light with a wavelength range of . Equations for chemical equilibrium and thermal balance are solved at each radius in the cloud and then line and continuum luminosities are derived. Each of the models is characterized by three parameters: cloud-illuminating UV flux () relative to the solar neighborhood value; hydrogen number density (); mean hydrogen column density (). Constant gas density in the cloud is assumed instead of the density structure in the original models of Mochizuki & Nakagawa (2000), to simplify discussions on the density dependence. We also use instead of cloud mass (M) used in the original models, in order to show the dependence of luminosity ratios on the column density more clearly. The relation between the two parameters can be written as:

(e.g., Mochizuki & Nakagawa 2000) for a spherical cloud.

We modified the spectrum of the stellar radiation illuminating the model cloud in accordance with the soft radiation field in galactic centers. Nakagawa et al. (Paper I) estimated the Galactic fraction of the dust heating by the UV in that by the whole wavelength range: our Galactic center has a 3 times smaller fraction than our Galactic disk does. Accordingly, the flux at was enhanced in the present models by a factor of 3 relative to that at , compared to the solar neighborhood spectrum of Mathis et al. (1983). The photoelectric heating process adopted in the Mochizuki & Nakagawa (2000) models follows the formalism by de Jong et al. (1980): only far-UV photons with energies of are effective.

Fig. 5 shows the luminosity ratio of as a function of , where is the [CII] line luminosity, and is the luminosity density (luminosity per unit frequency width) at a wavelength of , of the model cloud. The luminosity ratio is - when is sufficiently high, the grains are neutral and sufficiently warm, and the gas is predominantly cooled by the [CII] emission (e.g., and ; , , and ). These lower ratios are consistent with the flux ratios observed in our Galactic center (Fig. 4), as Nakagawa et al. (Paper I) discussed for - intensity ratio as a rough estimate. On the other hand, the higher M31 ratio can be produced when the following conditions are satisfied simultaneously:

1. . This keeps the grains neutral.

2. . This makes the cloud translucent for less energetic photons.

Fig. 5a-c. luminosity ratio as a function of the mean hydrogen column density of the cloud. The luminosity ratio is based on the PDR models by Mochizuki & Nakagawa (2000) with the following modifications: (1) the hydrogen number density is constant in the model cloud; (2) the flux of the incident stellar radiation is times the solar neighborhood value (Mathis et al. 1983) at , and times at . The solid, dotted, and dashed curves indicate , , and 10 models, respectively. a  The assumed density is . b  . c  .

Otherwise, the ratio becomes lower (Sect. 4.1). In particular, the small is crucial for producing the M31 ratio under the assumed spectrum of the incident stellar light, because is satisfied for typical galactic molecular clouds (Mochizuki & Nakagawa 2000).

The gas is predominantly cooled by [CI] fine-structure or CO rotational lines, when . This can produce lower ratios as observed in our Galactic center at a lower (Fig. 5c) than the cases of [CII]-dominant cooling. However, this results in a too weak FIR line and continuum emission relative to the CO (-0) emission, compared to the observations (Dame et al. 1987; for the CO emission) toward our Galactic center. Thus, the above models rule out cooling due to those lines in our Galactic center. This indicates that the difference in the ratio between the two galactic centers primarily results from the difference in . We discuss the possibility of CO-dominant cooling further below (Sect. 4.4 and Appendix A).

The grain temperature can be less than 20 K even near the surface of the model cloud, at . However, this effect does not account for the difference in the ratio between the two galactic centers because the UV flux is not extremely low in the bulge of M31 (of the order ; Bohlin et al. 1985).

The ratio and its dependence on is sensitive to the spectrum of the incident stellar light and to the photon-energy dependence of the efficiency , especially to the convolution of them. Recent models indicate that a photoelectron can be emitted also by a less energetic photon (Bakes & Tielens 1994) than classically expected. Nevertheless, the decreased contribution of less energetic photons at occurs independently of such details, as long as the stellar radiation is the heating source and the gas is heated by more energetic photons on average than the grains are. Thus, we discuss only the extreme case where the gas-to-dust heating ratio is independent of photon energy (Sect. 4.4 and Appendix A) in the present paper.

4.3. Molecular clouds in the M31 center

Loinard et al. (1995) observed the CO (-0) emission in the inner region of M31. The integrated main-beam temperature was at with a spatial resolution of , and no emission was detected at . Accordingly, we adopt for the average around . Since the [CII] flux in the LWS beam is there (Fig. 2a), the [CII]/CO (-0) line intensity ratio is . This roughly estimated ratio is higher than the in the inner region of our Galaxy (Nakagawa et al. 1998; Dame et al. 1987) and not lower than as found for starburst galaxies (Stacey et al. 1991) in spite of the less active recent star formation in M31. The high [CII]/CO (-0) line ratio is compatible with the low column density we proposed for a M31 cloud, because the low allows incident UV photons to dissociate CO molecules in a larger fraction of the gas contained in the cloud. We will discuss the [CII]/CO (-0) ratio further in a forthcoming paper based on CO observations with a better sensitivity.

The central region of M31 shows very low excitation of CO rotational transitions (; Loinard et al. 1995). This indicates that a large fraction of the molecular gas in this region has a low density () compared to that in our Galactic disk (). This decreases of M31 clouds relative to that of Galactic ones, if the typical mass of the clouds is similar between the two galaxies. When typical densities of and are adopted for M31 and Galactic clouds with the same M, respectively, the typical ratio of M31 to our Galaxy is using Eq. (3). Assuming in our Galaxy, we obtain in M31. These column densities are compatible with the observed difference in (Sect. 4.2).

The lower in M31 can result from a lower pressure of the ISM. The ISM pressure is likely to be lower in M31 than in our Galaxy, because the lower star-forming rate in M31 leads to a lower rate of supernova explosions.

A molecular cloud has a small opacity against the incident radiation field also in a galaxy with a low dust-to-gas abundance ratio, unless the low dust abundance affects the hydrogen column density of the cloud. This may account for the relatively large FIR [CII]/continuum intensity ratios observed (the Large Magellanic Cloud, Mochizuki et al. 1994; IC 10, Madden et al. 1997) in galaxies with low metallicities.

4.4. Gas heating due to less energetic photons

We discuss another case where the molecular clouds in M31 have a column density sufficiently high () to absorb the soft interstellar radiation field. In this case, the heating ratio of gas-to-dust must be insensitive to the energy of incident photons over the energy range effective for dust heating because the FIR ratio observed in M31 is not decreased by the soft radiation field. Since the photon-energy dependence of the gas heating is not so well-determined as that of the dust heating, we consider in this subsection that a photoelectron can be emitted from a grain surface by a less energetic photon as well as by a UV photon.

When the photon energy effective for gas heating decreases on average, the gas-heating radiation penetrates a molecular cloud more deeply. This increases the contribution of CO rotational transitions to the gas cooling. When the cooling due to the CO emission exceeds that due to the [CII] emission, the [CII] emission does not trace the gas cooling, and consequently decreases. This CO cooling is more effective at a higher gas density because the high density enables CO molecules to survive even close to the cloud surface where the gas-heating rate is higher than inside (Appendix A).

In order to examine the influence of an energy-insensitive gas-to-dust heating ratio on the CO cooling quantitatively, we carried out calculations with PDR models under the assumption that photoelectric efficiency is independent of photon energy (Appendix A). These simulations indicate that the ratio is so low as observed toward our Galactic center at because of CO-dominant cooling. On the other hand, at lower gas densities (), the models provide nearly constant ratios as high as observed in the central kiloparsec of M31 and in the general Galactic plane because of [CII ]-dominant cooling. As a result, the observed difference in between the centers of M31 and our Galaxy can be reproduced by a difference in gas density.

However, the models of CO-dominant cooling are incompatible with observations of our Galactic center (Bennett et al. 1994) with the Far-Infrared Absolute Spectrophotometer (FIRAS). We averaged the line fluxes of the two FIRAS pixels centered on , , to compare them with the CO-dominant cooling model described in Appendix A with parameters of , , and reproducing the FIR ratio observed toward our Galactic center (Fig. 6). Each of the FIRAS pixels represents a line flux averaged over a region (Bennett et al. 1994) while we discuss CO cooling at the scale height displayed in Fig. 2 and Fig. 4, (corresponding to the LWS beam at the distance of M31). Since the [CII] emission may be extended beyond the regions where the CO emission dominates the gas cooling, we consider two limits in the distribution of emission. In one limit, the [CII] and CO emissions have the same scale heights. In the other, the [CII] emission is uniformly distributed in the FIRAS pixels, while the CO emission is confined to . Fig. 6 shows that the observed mid-J CO lines are too weak compared to the model of CO-dominant cooling independently of the assumed emission distribution. Thus, the CO-dominant cooling is unlikely in our Galactic center, at least on the scale of kpc.

Fig. 6. Flux of the [CII] fine-structure and CO rotational lines normalized to that of the [CII] line v.s. line wavelength. The thick curve indicates FIRAS observations toward the Galactic center (Bennett et al. 1994). The error bars () and the upper limits () represent statistic uncertainties. The thin curves indicate calculations based on the PDR model described in Appendix A with , , and . The upper and lower thin curves are for the cases where the [CII] and CO emissions originate in the same regions and where the [CII] emission fills a scale height of 370 pc (the FIRAS resolution at the distance of our Galactic center) while the CO emission is confined to that of 230 pc (the LWS resolution at the distance of M31), respectively.

The exclusion of the CO-dominant cooling is insensitive to compared models, because the observed line fluxes directly restrict the energy carried away by the CO lines. We therefore conclude that the difference in between the two galactic centers is due to the difference in typical column density of clouds.

4.5. [CII] self-absorption

The edge-on view of our Galaxy may result in a large optical depth of the [CII] line toward our Galactic center. On the other hand, the [CII] opacity is unlikely to be large toward the M31 center, which contains a much smaller amount of the neutral ISM than the Galactic counterpart does. This suggests that the self-absorption of the line may cause the difference in the ratio between the two galaxies. However, Nakagawa et al. (Paper I), who estimated the line opacity toward our Galactic center, concluded that this effect is insignificant. In the following, we discuss a few points supporting Paper I.

1. Nakagawa et al. (Paper I) compared the Galactic [CII ] emission also to the CO - emission (Dame et al. 1987). The CO emission does not show a deficit toward our Galactic center unlike the [CII] emission, although the former line generally has a much larger optical depth.

2. Sauty et al. (1998) simulated the [CII] line and the continuum emission for the edge-on galaxy NGC 891, assuming distributions of the molecular clouds and the UV sources in the galaxy. Their results showed a [CII] optical depth of , not large enough compared to the deficit in the ratio (a factor of 2-3) toward our Galactic center.

3. Madden et al. (1993) observed the distribution of the [CII] emission in NGC 6946. This galaxy also shows a [CII] deficit relative to the FIR continuum toward its center in spite of its face-on view, although the limited spatial resolution (2.7 kpc when a distance of 10.1 Mpc is adopted) prevents a quantitative comparison to our Galactic center.

© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000
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