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

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/ Wrubel COG from Aller and Chamberlin 1956

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  –Line strength depends (approximately) on the square root of the abundance

The Effect of Temperature on the COG Recall: –(under the assumption that F comes from a characteristic optical depth  ) Integrate over wavelength, and let l  =N  Recall  that the wavelength integral of the absorption coefficient is Express the number of absorbers in terms of hydrogen Finally,

The COG for weak lines Changes in log A are equivalent to changes in log gf, , or  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 gf, , or . A curve of growth for one line can be “scaled” to be used for other lines of the same species.

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?

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 g 1/3

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

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  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  that makes the abundance independent of line strength.

Microturbulence in the COG 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? 0 km/sec 5 km/sec

Determining Abundances Classical curve of growth analysis Fine analysis or detailed analysis –computes a curve of growth for each individual line using a model atmosphere Differential analysis –Derive abundances from one star only relative to another star –Usually differential to the Sun –gf values not needed Spectrum synthesis –Uses model atmosphere, line data to compute the spectrum

Jargon [m/H] = log N(m)/N(H) star – log N(m)/N(H) Sun [Fe/H] = -1.0 is the same as 1/10 solar [Fe/H] = -2.0 is the same as 1/100 solar [m/Fe] = log N(m)/N(Fe) star – log N(m)/N(Fe) Sun [Ca/Fe] = +0.3 means twice the number of Ca atoms per Fe atom

Solar Abundances from Grevesse and Sauval

Basic Methodology for “Solar-Type” Stars Determine initial stellar parameters –Composition –Effective temperature –Surface gravity –Microturbulence Derive an abundance from each line measured using fine analysis Determine the dependence of the derived abundances on –Excitation potential – adjust temperature –Line strength – adjust microturbulence –Ionization state – adjust surface gravity

Projects! You may work in teams (1, 2 or 3 students) Perform an analysis of the spectrum Confirm the atmospheric parameters (optional) Measure the abundance of the atomic species in homework 3 Use Moog: Chris Sneden – MOOG or use the computers in Rm 311 with Moog already installed

Data Select one of the data archives –FTS archive Wallace & Hinkle 1996, APJS, 107, 312 DPP: NOAO Digital Library –ELODIE archive Prugniel & Soubiran 2001, A&A, 369, 1048 The ELODIE archive –Others? –Work with published EQW data Select a sample of stars, at least one per team member

What’s known? Review the literature for your selected object extant photometry 2MASS, ISO data? radial velocity measurements? IUE/STIS spectra? previous atmospheric analyses? metallicity determinations? (see Catalogue of [Fe/H] (Cayrel de Strobel+, 1997)

Step 3 Measure equivalent widths/detailed COG Spectrum Synthesis? Use Thevenin line data –wavelength –e.p. –gf may work differentially to Arcturus (optical or IR) or the Sun if needed