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Unformatted text preview: 9. Radiative transfer in moving media Reading: Shu, Vol.I, Ch.9 9.1 Radiative transfer for spectral lines In the preceding chapters we have separately derived equations and techniques to describe the dynamical evolution of matter and the transport of radiation. In practise, the two are coupled problems, because the radiating medium may move with different velocities or in different directions in different parts of the emission region. The effect will be most prominent for spectral lines. In the following we will discuss the (easier) case of pure emission and absorption and neglect scattering. Lines have a natural minimal width with a profile called a Lorentzian that can be understood using classical physics. Emission corresponds to an energy loss for the emitter or to a damping of its state. Consider a damped harmonic oscillator, whose energy decrease with a time constant γ . Then the amplitude is x ( t ) = x exp( ıω t ) exp(- γ 2 t ) t ≥ (9 . 1) We derive the line profile by a Fourier-Transformation ˜ x = 1 √ 2 π Z ∞-∞ x ( t ) exp(- ıωt ) dt = x √ 2 π 1 γ 2 + ı ( ω- ω ) (9 . 2) and the spectral power as P ( ω ) ∝ | ˜ x | 2 = 1 2 π x 2 γ 2 2 + ( ω- ω ) 2 (9 . 3) The line may be further broadened by the Doppler effect on account of the isotropic thermal motion of the emitters, described by a Gaussian, or on account of bulk motion. In some cases we may disentangle the effect of thermal motion and bulk motion. Suppose an observer, located at ~r , sees line radiation from gas at ~r . It is useful to introduce a path length coordinate along the line-of-sight. ~r = ~r + ~e k s (9 . 4) At each location a line is emitted (and absorbed) with a width, that is determined by the thermal velocity of the emitters, v th , and a Doppler shift on account of the bulk flow with velocity ~ V . ν = ν 1 + ~e k · ~ V c for u ¿ c (9 . 5) 1 If the thermal velocity is of the same order as or larger than the bulk velocity or change thereof, the bulk velocity will only moderately change the line profile. If, on the other hand, the change of bulk velocity if fast in comparison with the thermal velocity, i.e. l k = v th ~e k · ~ ∇ ( ~e k · ~ V ) ¿ L (9 . 6) where L is any other physical scale length of the system, then the total line profile will be dominated by the Doppler shift (8.5) and we may neglect the thermal line width. If the flow is purely expansive or purely contractive, one can show (textbook) that ∂ν ∂s > ∀ s or ∂ν ∂s < ∀ s (9 . 7) In other words, line emission emitted at some point on the line-of-sight will not suffer absorption during radiation transport, because the absorption frequencies anywhere on the line-of-sight are different from the emission frequency. In other words, there is a unique relation between emission frequency and position on the line-of-sight, so that we could unambiguously transform the line-of-sight integration of intensity to an integration over frequency.the line-of-sight integration of intensity to an integration over frequency....
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