1. Spectral Line Strength and Chemical Abundance: Curve of Growth
Assumptions
Equivalent Width
Scaling Models with Observations
Spectrum Synthesis with Atmospheres

2. Simplifying Assumptions
lines form by same mechanism: suppose no scattering:
lines form in a layer characterized by T, P (poor, lines form over different levels)
all lines have the same profile shape
damping parameter is a free variable
Milne-Eddington approximation for atmosphere with constant

3. Emergent Flux Source function = Planck function
Recall second exponential integral
Write in terms of continuum optical depth

4. Flux for Linear Planck Function
Source function:
Expression for emergent flux

5. Line Depth where A0 = depth for strongest lines Grey case
Grey, weak line case (same as last time)

6. Equivalent Width Integral over normalized Doppler shift
Express profile as Voigt function
Form “reduced equivalent width” Integral depends on # absorbers (in β0)
(profile symmetrical about v=0)

8. Curve of Growth weak: increases with # absorbers (linear)
opaque: line reaches A0 limit (saturation, flat)
strong: line wings important (damping/square root)

9. Comparing Observed and Model Curves of Growth
y-axis
x-axis

10. Strategy for Temperature
measure (Wλ / λ) for lines of given ion/element
for each line determine gijk , fij , λ , Χijk
plot in curve of growth
find excitation temperature that will shift individual points with Χijk into a single curve by shifting by
amount –Χijk θexc

11. Shift (in x,y) observed to match theoretical
vertical shift gives Doppler broadening microturbulence vturb delays saturation
horizontal shift gives constant
get ne , kc from atmosphere model
Saha eqtn. to get total number of atoms of element
damping part related to line broadening

14. LTE Spectrum Synthesis with Model Atmospheres
model follows depth dependence of
accounts for T changes, ionization-excitation, line broadening with depth
for discrete set of frequency points find

15. ATLAS - WIDTH6 Select model parameters
Solve for line flux and integrate across profile
Iterative approach: trial abundance  Fν  Wλ  compare to obs  adjust abundance ⏎
Correct for trends of abundance and ... Χijk  fix Teff ... ionization  fix log g (and Teff) ... Wλ  fix microturbulence
Spectrum synthesis (moog) if lines are blended