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Understanding Curve of Growth in Spectral Line Formation

Discover the theoretical relationship between line equivalent width, optical depth, and line formation in the context of spectroscopy. Learn how to analyze and interpret data using the classical Curve of Growth method.

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Understanding Curve of Growth in Spectral Line Formation

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  1. 11. Spectral Lines II • Rutten: 5.1, 5.2 • Line formation • Eddington-Milne • Curve of Growth

  2. Curve of Growth Curve of Growth is a theoretical plot of line equivalent width against line optical depth, usually plotted as log(Wl) – log(t). The Wl – t relation is seen in the Schuster-Schwarzschild model atmosphere: continuous spectrum is emitted from a deep layer, Ic = Bl(TR), second reversing layer at TL absorbs the spectral lines. Radiation we observe is (from L2): where tl is the optical depth of the reversing layer given by: where Ni is the column density along LOS (m-2) of particles in lower level i.

  3. The relative line depression is: Where Dmax = [Bl(TR) - Bl(TL)] / Bl(TR) the maximum depression for very strong lines. The equivalent width is showing how Wl relates to t and hence also Ni and f. The curve of growth can be split into three regimes depending on the optical depth (line strength): weak, saturated, and strong lines.

  4. 1. Weak Lines: For tl << 1, exp(- tl) ~ 1 – tl, so D ~ Dmaxtl. The Voigt profile is approximated by Doppler because opacity is too small to map damping wings into emergent spectrum. Replacing H(a,v) with exp(-[Dl/DlD]2) with area p½DlD gives: Wl ~ fNi. Part 1 of curve of growth is Doppler part and has slope 1:1. Linear increase due to optical thinness of reversing layer. 2. Saturated Lines: For tl > 1, line cannot be deeper than Dmax. The line width increases with tl0 and Where Q ~ 2 – 4. This is part 2 of curve of growth or the shoulder.

  5. 3. Strong Lines: For tl >> 1, the core doesn’t change any more. Line centre is fixed at Dmax, but line wings have tl < 1 and may grow in optically thin fashion to increase Wl. For large tl0 the wings contribute a lot because they map the damping part of A drop off with Dl that is less steep than the exponential decay of the Doppler core. In the damping part of H(a,v) we write: which gives: Thus part 3 or the damping part scales as Wl ~ (atl0)½ ~ (fNig)½ .

  6. log Wl 3 1 2 2 1 1 1 log tl0 Ic(TR) 0 1 2 3 Dl Il Bl(TL) Dmax 0 1 l0

  7. Classical Curve of Growth Fitting The strategy is to analyze lines via a curve of growth to determine abundances, damping parameter a, excitation temperature Texc, and microturbulence xmicro. Plot measured Wls as log(Wl/l) against where… Changing parameters changes the shape of theoretical curve of growth. By minimizing scatter of data can derive physical parameters.

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