Chapter 14 – Chemical AnalysisChapter 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 GrowthThe 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 GrowthCurves 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 The Effect of Temperature on the COGCOG

• Recall:

– (under the assumption that F comes from a characteristic optical depth )

• Integrate over wavelength, and let l=N

• Recallthat the wavelength integral of the absorption coefficient is

• Express the number of absorbers in terms of hydrogen

• Finally,

l

constant

c

c

F

FF

Nf

cmc

ew

22

constant

kTH

E

r eTu

gN

N

NAN

)(

logloglog)(

loglog2

2

gfAN

Tu

NN

mc

ewH

Er

The COG for weak linesThe COG for weak lines

logloglog)(

loglog2

2

gfAN

Tu

NN

mc

ewH

Er

Changes in log A are equivalent to changes in log gf, ,or

For a given star curves of growth for lines of the samespecies (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 beused for other lines of the same species.

A Thought ProblemA 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 The Effect of Surface Gravity on the COG for Weak Lineson 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

Effect of Pressure on the COG Effect of Pressure on the COG for Strong Linesfor 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 The Effect of MicroturbulenceMicroturbulence

• 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 COGMicroturbulence in the COG

-7

-6

-5

-4

-3

-13 -12 -11 -10 -9 -8 -7 -6

Log A + Log gf

Lo

g w

/la

mb

da

0 km/sec

1 km/sec

2 km/sec

3 km/sec

5 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?

0 km/sec

5 km/sec

Determining AbundancesDetermining 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

JargonJargon

• [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 Solar Abundances from Grevesse and SauvalGrevesse and Sauval

Eu

BaSr, Y, ZrSc

Li, Be, B

CNO

Fe

-1

2

5

8

10 20 30 40 50 60 70 80

Atomic Number

Lo

g e

(H

=12

)

Basic Methodology for “Solar-Type” StarsBasic 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!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

DataData

• 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?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 3Step 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