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Unformatted text preview: 1 Monday, November 28: Comptonization When photons and electrons coexist in the same volume of space, their scat tering interactions can transfer energy from photons to electrons (Compton scattering) or from electrons to photons (inverse Compton scattering). In general, therefore, when photons travel through a region containing free elec trons, their spectrum will be changed. In other words, the shape of the spe cific intensity I ν will be modified as photons are scattered to lower or higher ν . The change in the spectrum of light due to scattering from electrons is referred to as Comptonization . If light passes through a blob of material of finite size, the average change in a photon’s energy is given by the Compton y parameter. The magnitude of y is given by the relation y = Δ ² ² × N es , (1) where Δ ²/² is the average fractional change in the photon’s energy ² = hν from a single scattering, and N es is the average number of scatterings from electrons as the photon passes through the medium. If y ¿ 1, then the spec trum of light will only be slightly changed as it passes through the medium. If y À 1, however, the spectrum can be strongly modified. Computing N es is fairly easy. Suppose that the medium through which the light passes has a diameter L . The number density of free electrons in the medium is n e , and the typical Lorentz factor of the electrons is γ . If ² ¿ m e c 2 /γ , as we saw last Wednesday, the photonelectron interactions can be treated as Thomson scattering in the electron’s rest frame. In that case, the mean free path of the photons will be ‘ es = 1 / ( n e σ T ), where σ T is the Thomson crosssection of the electron. The optical depth of the blob is then τ es = L/‘ es = n e σ T L . (2) If τ es ¿ 1, the average number of scatterings will be N es = τ es ¿ 1. If τ es À 1, the photon will randomwalk through the medium, traveling an rms distance h R 2 i 1 / 2 ≈ √ N‘ es (3) after N scatterings. Traversing the medium requires traveling a distance h R 2 i 1 / 2 ≈ L , implying N es ≈ ( L/‘ es ) 2 ≈ τ 2 es . (4) 1 A good approximation for N es for all optical depths is N es = max( τ es , τ 2 es ) . (5) For optically thick media, with τ es = n e σ T L À 1, the photons undergo a great many scatterings before emerging from the medium. Computing Δ ²/² is a little more complicated, since it depends on the dis tribution of electron energies. One of the most useful cases is when the light passes through an ionized gas in which the electrons have a nonrelativistic thermal distribution, characterized by a temperature T < m e c 2 /k ∼ 6 × 10 9 K. In this case, the average fractional energy change of a photon under going a single scattering is Δ ² ² fl fl fl fl NR = 4 kT ² m e c 2 . (6) When 4 kT À ² , and the electron’s thermal energy is much larger than the initial photon energy, the photon gains energy on average. When 4 kT ¿ ² , and the electron’s thermal energy is much smaller than the initial photon...
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This note was uploaded on 07/17/2008 for the course ASTRO 822 taught by Professor Ryden during the Fall '05 term at Ohio State.
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