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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 photon-electron 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 cross-section 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 random-walk 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 non-relativistic 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.
- Fall '05