lecture notes2

2 h ktd 2 3 h ktd d c e 1 2 c e 1 2 2 the

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Unformatted text preview: appearing around a bright source, an effect observed in X ­rays. As λ increases, the halo grows in size, € until at λ~a the diffraction pattern and geometrically refracted/reflected pattern merge into the dipole scattering pattern described previously. C. THERMAL EMISSION The radiation absorbed by a grain is re ­emitted in the infrared. We are now in a position to understand the spectrum of emitted radiation. We use the principle of detailed balance (again!) to compare the emission of a dust grain to absorption. If the grain has temperature Td, then in a blackbody of temperature Td it must absorb an amount of radiation per unit frequency: dPabs 8πhν 3 1 = σ abs (ν ). 2 hν / kTd dν c e −1 For practical cases the grains will have sizes up to ~0.5 μm and the radiation will emerge at wavelengths of many microns (due to the exponential factor); so we are € in the long ­wavelength limit and may write: dPabs 8πhν 3 1 12πha 3 Imε 192π 3 ha 3ν 4 1 Imε = = . 2 hν / kTd 2 3 hν / kTd dν c e −1 |ε + 2 | c e − 1 | ε + 2 |2 The luminosity of the grain per unit frequency, Lν, must be equal to this: € 192π 3 ha 3ν 4 1 Imε Lν = . 3 hν / kTd c e − 1 | ε + 2 |2 The grain will reach an equilibrium temperature when its total luminosity L = ∫ Lν dν is equal to the amount of radiation absorbed. € 3. Grain Properties Next we consider the composition of dust grains and their likely dielectric constants. Our attention will then turn to their thermal balance. A. MATERIALS 4 Clearly dust grains must be made primarily out of the abundant elements. The most abundant 10 elements (for solar composition) by number are:1 H 1 He 0.085 O 0.00049 C 0.00027 Ne 0.000085 N 0.000068 Mg 0.000040 Si 0.000032 Fe 0.000032 S 0.000013 While...
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