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

Appendix A: gas cooling due to CO molecules

First we discuss conditions for a molecular cloud cooled predominantly by the CO rotational transitions, considering a simplified situation as follows. (1) The cloud has a sufficiently high column density to absorb the incident radiation over the wavelength range effective to the dust heating. (2) The dominant form of gas-phase carbon is C⁺ ion near the surface of the cloud, and CO molecule inside. We define the C⁺-CO transition column density, , as the hydrogen column density from the surface to the region where C⁺ and CO are equally abundant. (3) The gas is cooled by the [CII ] fine-structure line at and by the CO rotational lines at , where is the hydrogen column density from the surface. Under these assumptions, the cooling due to the CO lines exceeds that due to the [CII ] line for the whole cloud, when the energy input to the gas is larger in the CO region than in the C⁺ region: , where is the hydrogen column density characteristic for the attenuation of the gas-heating radiation.

The column density is determined by the chemical balance between the two cooling species. The conversion rate (; events per unit time per unit volume) of as a function of can be approximately written as:

where is the photodissociation rate of CO molecule at the unattenuated radiation field of , is the number density of CO molecule as a function of , and is the hydrogen column density characteristic for the attenuation of the CO-dissociating UV radiation. On the other hand, the rate () of conversion, which consists of two-body collisions in the gas, can be approximately written as:

where is the total rate coefficient of the reactions, and is the number density of C⁺ ion. The chemical balance is achieved by equating the two conversion rates: . At the C⁺-CO transition zone, where ,

is obtained (e.g., Mochizuki & Onaka 2000). As a result, the condition of CO-dominant cooling can be written as:

This condition can be satisfied by a small and/or a large . The latter requires (1) a small lower limit of photon energy capable of heating the gas and (2) soft cloud-illuminating radiation.

For more quantitative discussion, we calculated line and continuum ratios based on PDR models involving gas heating due to less energetic photons. In these models, we assumed an extreme case where photoelectric efficiency is a constant of , which represents the in the Galactic plane (Nakagawa et al. 1998), independent of photon energy. Except this assumption, the models are equivalent to those described in Sect. 4.2. The total cooling due to the CO rotational lines exceeds the cooling due to the [CII ] line, at , as shown in Fig. A.1 for and 10. In this case, the [CII ] emission does not trace the gas heating, and consequently is smaller than in the case of [CII ]-dominant cooling. According to the above models, the different ratios observed between the two galactic centers can be reproduced by the difference in gas density (Sect. 4.4).

Fig. A.1. Line and continuum luminosity ratios based on PDR models as a function of hydrogen number density, , of the model cloud. The assumed photoelectric efficiency is , independent of the energy of the incident photons. Otherwise, the models are equivalent to those for Fig. 5. The solid curves indicate the FIR luminosity ratio (, multiplied by ) of the [CII ] line to the continuum. The dashed curves indicate , where is the total luminosity of the CO rotational lines. Thin and thick curves are for and 10, respectively. The mean hydrogen column density of the cloud is assumed to be .

The gas-to-dust heating ratio can be insensitive to photon energy when the gas heating is dominated by the photoelectric effect through negatively-charged smaller particles (large molecules) such as Polycyclic Aromatic Hydrocarbon (PAH) anions rather than through grains, because small electron affinities of these anions allow a photoelectron to be emitted by a less energetic () photon. Lepp & Dalgarno (1988) estimated that the large molecules heat the gas predominantly when the abundance of these molecules exceeds relative to that of hydrogen. One fifth of this abundance was suggested for carbon chain anions in the Galactic ISM on the basis of diffuse infrared band (DIB) observations (Tulej et al. 1998). Since PAH molecules are more stable than these carbon chain molecules, the PAH anions may be abundant sufficiently in the ISM. Moreover, Uchida et al. (1998) found that the excitation of the infrared emission features (IEFs), which is often considered to be radiated from large molecules (e.g., Leger & Puget 1984), cannot be accounted for only by the absorption of UV photons; they proposed the absorption of visible photons for the additional excitation. The excitation due to visible photons may result from negative charge of the IEF carriers, because a large molecule can absorb a less energetic photon when it is ionized (Allamandola et al. 1989) than when neutral.

© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000
helpdesk@link.springer.de