Interstellar dust grains also serve as important coolants. In this case, collisions with gas atoms and molecules lead to lattice vibrations, which decay through the emission of infrared photons. As we have seen, the same grains are also heated through the absorption of optical and ultraviolet photons. Thus, the grain temperature, Td, is generally different from the gas temperature T
Let us first consider Ad, the volumetric cooling rate due to emission from the grains themselves. Note that a volumetric rate is appropriate here since the cloud is usually optically thin to these infrared photons. We may determine Ad from the general formula for thermal emission in equation (2.30), after replacing pnuMJs by the equivalent ndadQv,abs. Integrating (j)therm over all solid angles and noting that the radiation is isotropic, we find
For a typical grain temperature of 30 K, Bv (Td) peaks at a wavelength of about 100 pm, much greater than grain dimensions. As we noted in Chapter 2, the absorption efficiency Qv,abs at long wavelengths tends to the quadratic form
Using this last relation in equation (7.36), along with nd ad = £d nH, we find
Vmax nH 2
A 4nQVmax Sd nH f 2 D frp W
To evaluate Ad numerically, we also recall from Chapter 2 that Qvmax may be written in terms of the corresponding opacity as ¡imH KVmax/£d. Here, n is the mean mass per particle relative to the atomic hydrogen value mH (see Equations (2.2) and (2.38)). For the opacity, we employ the theoretical result that k = 0.34 cm2 g"1 at a wavelength of 100 pm (vmax = 3.0 x 1012 s"1). Using n = 1.3 for an HI gas and noting that the nondimensional integral has a value of 122, we find
In writing the last equation, we have implicitly assumed that grains of all size have the same temperature. This cannot generally be true. A grain bathed by ultraviolet light, for example, has an absorption Q that is independent of the grain radius a, as long as the grain diameter exceeds the incident wavelength. On the other hand, this same grain emits in the infrared, where the associated Q varies as a. Smaller grains therefore need to be warmer in order to compensate for their lower intrinsic emission rates. In practice, however, the range in temperature is modest enough that an average Td is still a useful concept.
Finally, we discuss gas cooling by collisions with grains. We stress that this process represents the transfer of energy between two components within a cloud, rather than a direct loss to interstellar space. Knowledge of this rate, which we denote Ag^d, is often essential to establishing the temperature of both gas and grains. A single grain is struck by a hydrogen molecule once in the time tcoll given in equation (5.9), where we now replace each HI subscript by H2. The impacting molecule brings with it translational kinetic energy (3/2)kBTg, which it imparts to the grain lattice.3 Assuming there is time for the molecule to reach thermal equilibrium with the lattice before departing, it leaves with energy (3/2)kBTd. The net cooling rate for the gas is therefore
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