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# The smaller the grains the hotter they can get and

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Unformatted text preview: B. SMALL GRAINS AND THERMAL SPIKES If a dust grain is very small the above analysis does not apply. To see this, we consider the thermal energy in a dust grain. The vibrational energy density in a material at low temperature (Td<<TDebye) is given by π 2 ( kTd ) 4 U= V , 10(c s ) 3 where cs is the sound speed in the grain. If we take Td = 20 K, this can be written as € 3 ȹ Td ȹ 4 ȹ 4 km/s ȹ ȹ a ȹ 3 U = 0.002ȹ ȹ ȹ ȹ eV. ȹ ȹ ȹ 20K Ⱥ ȹ c s Ⱥ ȹ 1nm Ⱥ A typical far ultraviolet photon that is absorbed by a dust grain has an energy of ~10 eV. This is equal to the equilibrium thermal energy of the grain if a ~ ac = 17 € nm (with the critical size depending somewhat on the intensity of the radiation field, G0). Thus for grains larger than ac the grains should be treated as being in equilibrium at a constant temperature. In contrast, grains smaller than ac can be treated as having discrete heating events every time they absorb a UV photon. Following such absorptions, they rapidly cool to below their equilibrium temperature. These discrete events are called thermal spikes. The maximum temperature reached during a thermal spike can be obtained by taking the Debye formula, setting the total energy to ~10 eV, and solving for the temperature: ȹ c ȹ 3 / 4 ȹ a ȹ−3 / 4 Tmax = 170ȹ s ȹ ȹ ȹ K. ȹ 4 km/s Ⱥ ȹ 1nm Ⱥ 3 Reach et al, ApJ 451, 188 (1995). € 6 For the very smallest grains (the PAHs, which may have only several tens of atoms and a ~ 0.4 nm) the maximum temperature may reach ~1000 K. Thermal spikes are important because they enable dust grains to radiate at frequencies larger than ~10k(20K)/h ~ 4 THz. This enables us to distinguish the small tail of the size distribution: we look beyond the Wien tail of the thermal dust emission and look for an extension that is inconsistent with any kind of equilibrium blackbody. The smaller the grains, the hotter they can get, and the farther into the mid ­IR one will observe their emission. Furthermore, one can observe the vibrational modes of these smallest grains in emission and hence directly probe their composition. For example, the non ­detection of the Si−O stretch band at 10 μm in emission limits the abundance of small silicate grains. C. POLYCYCLIC AROMATIC HYDROCARBONS The major emission features seen in the mid ­IR are the PAH bands. The strongest of these (and their identifications) are: 3.3 μm C−H stretch 6.2, 7.7 μm C−C stretch 8.6 μm C−H in ­plane bend 11.3, 12.7 μm C ­H...
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