Chapter 14 – Chemical Analysis

1. Chapter 14 – Chemical Analysis • Review of curves of growth • How does line strength depend on excitation potential, ionization potential, atmospheric parameters (temperature and gravity), microturbulence • Differential Analysis • Fine Analysis • Spectrum Synthesis

2. The Curve of Growth • The curve of growth is a mathematical relation between the chemical abundance of an element and the line equivalent width • The equivalent width is expressed independent of wavelength as log W/l Wrubel COG from Aller and Chamberlin 1956

3. Curves of Growth Traditionally, curves of growth are described in three sections • The linear part: • The width is set by the thermal width • Eqw is proportional to abundance • The “flat” part: • The central depth approaches its maximum value • Line strength grows asymptotically towards a constant value • The “damping” part: • Line width and strength depends on the damping constant • The line opacity in the wings is significant compared to kn • Line strength depends (approximately) on the square root of the abundance

4. The Effect of Temperature on the COG • Recall: • (under the assumption that Fn comes from a characteristic optical depth tn) • Integrate over wavelength, and let lnr=Na • Recallthat the wavelength integral of the absorption coefficient is • Express the number of absorbers in terms of hydrogen • Finally,

5. The COG for weak lines Changes in log A are equivalent to changes in log gfl, qc, or kn For a given star curves of growth for lines of the same species (where A is a constant) will only be displaced along the abcissa according to individual values of gfl, c, or kn. A curve of growth for one line can be “scaled” to be used for other lines of the same species.

6. A Thought Problem • The equivalent width of a 2.5 eV Fe I line in star A, a star in a star cluster is 25 mA. Star A has a temperature of 5200 K. • In star B in the same cluster, the same Fe I line has an equivalent width of 35 mA. • What is the temperature of star B, assuming the stars have the same composition • What is the iron abundance of star B if the stars have the same temperature?

7. The Effect of Surface Gravity on the COG for Weak Lines • Both the ionization equilibrium and the opacity depend on surface gravity • For neutral lines of ionized species (e.g. Fe I in the Sun) these effects cancel, so the COG is independent of gravity • For ionized lines of ionized species (e.g Fe II in the Sun), the curves shift to the right with increasing gravity, roughly as g1/3

8. Effect of Pressure on the COG for Strong Lines • The higher the damping constant, the stronger the lines get at the same abundance. • The damping parts of the COG will look different for different lines

9. The Effect of Microturbulence • The observed equivalent widths of saturated lines are greater than predicted by models using just thermal and damping broadening. • Microturbulence is defined as an isotropic, Gaussian velocity distribution x in km/sec. • It is an ad hoc free parameter in the analysis, with values typically between 0.5 and 5 km/sec • Lower luminosity stars generally have lower values of microturbulence. • The microturbulence is determined as the value of x that makes the abundance independent of line strength.

10. Microturbulence in the COG 5 km/sec 0 km/sec Questions – At what line strength do lines become sensitive to microturbulence? Why is it hard to determine abundances from lines on the “flat part” of the curve of growth?