Spectroscopy – Chemical Analysis

• Curve of Growth

• Results

Differential Analysis

• The abundances depend on a variety of stellar parameters (effective temperature, gravity, etc) as well as oscillator strength f. In particular the the product of Af is obtained, the product of the abundance and the oscillator strength.

• The uncertainties in the f value is what limits you in practice. These depend on laboratory measurements, and for many lines poor values are known.

• A differential analysis is usually employed. That is the ratio of abundances between stars (best if they have the same effective temperature). In this way the oscillator strengths cancel.

• The chemical analysis holds only for the atmosphere of the star! E.g. chemical analyses of peculiar stars give abundances of rare earth elements 1000 – 100.000 greater than the Sun.

Dep

th (

km)

Log

The observed spectral lines come from a layer of the stellar atmosphere that is ≈ 500 km thick!

Caveat:

2 ∫–∞

∞

BE2) l+

dlog

log e=F

For direct computation we use the equation for the flux (LTE) and compute the flux for a series of points spanning the line

We then integrate across the line to get the equivalent width

W = ∫0

∞ Fc– F

Fc

dFc and F are the fluxes in the continuum and in the line, respectively

Given the line absorption coefficient, l, you adjust the abundance A until you match the observed equivalent width. Compters allow a direct computation. Old way was to use the curve of growth, i.e. the log-log plot of equivalent width and the abundance.

Scaling relations

Fc – F

Fc

≈

For weak lines:

Cl

W = ∫0

∞

dlC

The equivalent width of the line becomes:

l = N, is the mass density, N is the number of absorbers per unit volume, and is the absorption coefficient

= e2

mcf= 2

c

is the wavelength integrated absorption coefficient

W = C e2

mcf 2

c

N

Introduce the number abundance relative to hydrogen, A = NE/NH, and the fraction in the rth stage of ionization, Nr/NE (given by the Boltzmann equation), you can write N as

N = A NH gu(T)

Nr

NE

exp( ) kT–

NH ) (W

log ( ) = log + log A + log gf – – log

e2

mc u(T)

Nr/NE

= 5040/T and division by normalizes Doppler dependent phenomena

The equvialent width:

Depends on:

• Abundance

• gf

• Temperature and excitation potential

• Continuous opacity

A change in any one of these mimics a change in the abundance

→Depends on Teff, gravity, composition, etc.

This equation tell us that for a given star, the curve of growth for the same species where A is constant will differ only in displacements along the abscissa by individual values of gf, , and .

We chose a line, this fixes gf and , our stellar atmospheric model fixes and . We can then vary A and generate the curve of growth

Different lines of the same species have different gf and but these have to have the same abundance, A. This can be used to constrain the equation.

The scaling with is usually small, especially if lines are in the same wavelength region. For example, between 4000 and 6000 Å, ∂ log /∂ ≈ 0.1 cm2/gm per 1000 Å for T < 7500 K

The Curve of Growth

Weak lines: the Doppler core dominates and the width is set by the thermal broadening D. Depth of the line grows in proportion to abundance A

3 phases:

Saturation: central depth approches maximum value and line saturates towards a constant value

Strong lines: the optical depth in the wings become significant compared to . The strength depends on g, but for constant g the equivalent width is proportional to A½

The curve of growth shape looks the same, but is shifted to the right for higher values of the excitation potential. This is because fewer atoms are excited to the absorbing level when is higher. The amount of each shift can be interpreted as exc

Curve of Growth: Temperature Effects

It is difficult to determine the temperature of a star to better than 50–100 K. Temperature effects:

• Nr/NE (ratio of populated states to total number of atoms)

• continuous opacity)

• ex (i.e. Temperature)

And all of these effect the abundance

Curve of Growth: Gravity Effects

Gravity can effect line strength through

• Nr/NE

•

Since both of these can be sensitive to the pressure, For neutral lines the effects cancel

There is a linear relationship between log A and log g

l

≈ constant g–⅓

Teff Ca I Ca II Cr I Cr II Fe I Fe II

7200 0.02 –0.33 0.02 –0.33 0.01 –0.33

5040 0.00 –0.39 0.00 –0.40 –0.11 –0.45

3870 –0.06 –0.43 –0.26 –0.53 –0.35 –0.60

∂ log A/∂ log g

As long as an element is mostly ionized, lines neutral species are insensitive to gravity.

The equivalent width of ionized lines vary as g–⅓

As long as the element is mostly ionized, lines of neutral species are insensitive to gravity changes. Lines of ions are sensitive to gravity roughly as g–⅓

A separate and independent analysis can be done for the ions and neutrals of the same element. Both should have the same abundance, A. Gravity is a free parameter and you vary it until you force both ions and neutrals to give the same abundance.

Try to avoid strong lines in abundance analyses because of errors due to saturation

Microturbulence

When people first started doing abundance analyses the observed equivalent width of saturated lines was greater than the predicted values using thermal and natural broadening alone. An extra broadening was introduced, the micro-turbulent velocity . This is a „fudge factor“ introduced just to make the observed line strengths agree with the models. Its physical interpretation is that it arises form turbulent velocities in the atmosphere of the star.

Recall the combined absorption coefficient:

total) = (natural)*(Stark)*(v.d.Waals)*(thermal)

Which is a combination of the convolution of 4 broadening mechanisms. Now we have to add a 5th which is due to microturbulent broadening:

total) = (natural)*(Stark)*(v.d.Waals)*(thermal)*(micro)

Procedure for determining microturblent velocity:

• Fit the equivalent widths to the weakest lines where the line strength does not change with .

• This fixes A. You can now use the curve of growth for the saturated lines to compute .

• Also can just determine by trial and error until the derived abundance is independent of line strength.

But… the saturation portion of the curve of growth depends also on the temperature distribution….Doh!

Fitting the microturbulence

The temperature distribution can vary from star to star because of

• Line blanketing: so many lines that the line opacity affects the continuum opacity. This blocks flux which re-emerges in other regions of the spectrum

• Differences in the strength of convection

• Mechanical energy dissipation

This results in an ambiguity between T() and

Curve of Growth Analysis for Abundances

Advantage: Simple, you measure the equivalent width of a line and read the abundance off the log W versus log A plot

Disadvantage: Lots of calculation and the difficulty in dealing with microturbulence and saturation effects.

• Make an initial guess of

• The theoretical curves of growths are calculated for all measured equivalent widths of some element with lots of lines

• From each line an abundance A is obtained.

• Now plot A versus W

• We find that A is a function of W. must be wrong.

• Chose a new and start all over. Continue until you converge

Curve of Growth Analysis for Abundances

To simply things, we can use the scaling relations and just compute one reference curve of growth rather than many.

Simplified procedure:

• W is entered into the standard curve of growth taken for standard values ( = 0, log gf = 0, = 0)

• This abundance is valid for the standard curves parameters = A0

• The real abundance is obtained by:

log A = log (gf/gf1 )+ log (/0) – log(/1) – –0)

log A = log A0 – log A

So instead of plotting W versus A, we plot W versus log A

A reference curve-of-growth for a solar model

–6 –2–4 0 +2

+4

Log A

Reference curve-of-growth

log A = log (gf )+ log (/0) – log(/0) –

A

log A = log A1 – log A

Curve of Growth Analysis for Abundances

Abundance determinations with a graph and calculator

1. Plot observed log (W/) versus log gf – log (/) – exIf ex is wrong there will be a lot of scatter. The best value of ex minimizes the scatter.

= 5040/T

Curve of Growth Analysis for Abundances

Procedure:

2. Calculate the vertical shift between the observed and theoretical curves. The vertical shift is log T/c where

2T =

thermal + 2micro

3. Move horizontally to get the abundance

Vertical shift → turbulent velocities

Horizontal shift → abundances

Spectral Synthesis

In real life, one no longer does a curve-of-growth analysis, but rather a full spectral synthesis. This can be expanded to 3-D models and includes true velocity fields on the star.

ATLAS → http://kurucz.harvard.edu

Spectral synthesis programs can be obtained from the internet. Most popular are the ATLAS9 routines of Kurucz and MOOG from Sneden

MOOG → http://verdi.as.utexas.edu/moog.html

SME → Spectroscopy Made Easy: GUI based IDL routines for calculating

synthetic spectra (Valenti & Piskunov, A&A Supp, 1996, 118, 585

Tutorial:http://tauceti.sfsu.edu/Tutorials.html

All programs require a line list. This can be obtained from the VALD (Vienna Atomic Line Database): http://ams.astro.univie.ac.at/~vald/ or http://www.astro.uu.se/~vald/

Abundances: Nomenclature

[Fe/H] = the logarithm of the ratio of the iron abundance of the star to that of the sun.

E.g. [Fe/H] = –2 → star has 1/100 solar abundance of iron

[Fe/H] = 0.5 → star has 3.16 x solar abundance of iron

And to complicate matters even further, most spectral synthesis is for 1-D plane parallel models with no true velocity fields. Work by Apslund and collaborators (Collet, Asplund, & Trampedach, 2007, A&A, 469, 687) indicated that when one uses a 3-D hydrodynamic modeling, that this can seriously affect the derived abundances.

1-D versus 3-D

Improvements:

• Better gf (oscillator strength values)

• Better treatment of convection

• Better opacities

• 3-D hydrodynamics

• Better observational data

The hydrodynamic simulations show that the abundance can have a strong effect on the velocity pattern of the star, and the velocity field has an effect on the derived abundances as well as the temperature structure of the star.

1.07

0.93

1.0

0.93

1.00

1.0

0.72

0.93

1.021.17

0.81

0.87

Recommended Values

3-D average /MARCS

Major differences of 3-D results

• Chmielewski, Brault & Müller (1975) reported that Beryllium was depleted by a factor of two. Be is now normal. This was because of poor UV opacities. Boron is also not depleted (UV opacities help)

• Carbon eventually revised down. Current abundance is a factor of 0.6 the earliest values

• Nitrogen is 0.6 – 0.8 earlier determinations

• Oxygen is the most abundant element not produced in the Big Bang, but its abundance is in dispute. In the past 20 years this value has dropped by 0.57.

• Magnesium abundance is consistent with meteoritic value, but gf-values of Mg are „notoriously uncertain“

The Solar Composition

Massive stars can burn elements up to iron in the core. Elements heavier than iron are formed by rapid and slow capture of neutrons

r-process: supernovae explosions

s-process: Asymptotic Giant Branch Stars

An old figure from Gray‘s book. Note that Be and B are depleted, but this is no longer the case with 3-D models

Uranium in Stars

Frebel et al. 2007

In this star Uranium is due to r-processing of elements

Abundances of Stars

Population I stars:

These are stars found in the galactic disk and in open clusters. Spectral studies have shown these to have abundances of „metals“ 0.5 – 2 x solar. These are relatively „young“ stars.

Population II stars:

These are stars found in the galactic halo and in globular clusters. Spectral studies have shown these to have „metal“ abundances of ~ 0.1 to 0.001 solar. These are presumably old stars.

Standard picture: Universe started out with Hydrogen and Helium, stars formed converting this to heavier elements → supernovae explosions pollute the interstellar medium with heavier elements. The next generation of stars have a higher abundance of metals

So with time the mean abundance of stars in the galaxy should increase.

Abundances of Stars: Galactic Variations

Disk: Pop I stars, metal rich

Bulge: Mostly Pop II stars, metal poor, some Pop I stars

Halo: Mostly Pop II stars, metal poor, globular clusters

Globular clusters

What does this tell us about the chemical evolution of the galaxy?

Abundances of Stars: Galactic Variations

Pop III stars only of Hydrogen and Helium. Supernovae explosions pollute proto-galactic cloud with some metals

t = 0

H, He, some metals

Formation of Pop II halo stars and globular clusters

t = t1

Abundances of Stars: Galactic Variations

Disk stars are the last to form, thus metal rich

Globular clusters were the first to form, thus metal poor.

t = t2

Abundances of Stars: Metal Poor

After the Big Bang the universe was entirely hydrogen and helium. This means that the first stars were pure hydrogen and helium. So where are all the Pop III stars (stars with no heavy elements)

Observational Cosmologists: Try to break the record for the highest redshift quasar. This pushes back the earliest time we can observe the universe. → z large (z is redshift)

Stellar Spectroscopists: Try to break the record for the lowest [Fe/H]. This pushes back to the earliest time that stars formed. → z small (z is metal content in this case)

Ultra Metal Poor Stars

And the current champion is HE 1327-2326 with an [Fe/H] = –5.4 or 0.000004 x solar metallicity

Is this really one of the first stars?

Frebel et al. 2007

Venn & Lambert (2008) have argued that this may not be the case. Peculiar stars such as post AGB stars and Boo stars have iron abundances as low as [Fe/H] ~ –5. These are thought to be due to the separation of gas and dust beyond the stellar surface followed by an accretion of the dust-depleted gas. Thus the iron abundances are artifically low, but the Carbon, Oxygen, and Nitrogen abundance is only about [X/Fe] ~ –2. So this may not be one of the first stars, rather a peculiar star like the Boo class of objects.

Where are the Pop III stars? Current wisdom says that pure H/He stars have to be very massive and thus have very short lifetimes. They have long since vanished

These are stars with metallicity [Fe/H] ~ +0.3 – +0.5

Abundances of Stars: Super Metal Rich

There is believed to be a connection between metallicity and planet formation. Stars with higher metalicity tend to have a higher frequency of planets.

Valenti & Fischer

Endl et al. 2007: HD 155358 two planets and..

…[Fe/H] = –0.68. This certainly muddles the metallicity-planet connection

Hyades stars have [Fe/H] = 0.2 and according to V&F relationship 10% of the stars should have giant planets, but none have been found in a sample of 100 stars

Abundances of Stars: Lithium

Abundance variations can also be caused by evolutionary changes in the stellar composition. An example is Lithium

Lithium is destroyed at temperatures of T ≈ 2 x 106 K. The convection zone of the star brings Li to the deeper, hotter layers of the star where it is destroyed by conversion to He. It is used as an indication of age, although it depends on the depth of the convection zone, temperature profile (convection zone), and age of star.

In the Sun Lithium is depleted with respect to meteoritic composition by a factor of 150

Abundances of Stars: Lithium

Li in the sun

Li : 6708 Å

Abundances of Stars: Lithium

The Lithium abundance depends on both the temperature (depth of convection zone and temperature profile) and age.

Lithium is „messy“ and can only be used as an approximate age indicator

Old Pop II stars of roughly the same age

Abundances of Stars: Lithium

Surprise, surprise even cool giant stars can have high lithium:

The presence of Li in giant stars is a mystery. These stars have deep convection zones and are old stars. They should have destroyed their lithium long ago. One hypothesis: pollution due to swallowing a binary or even planetary companion.

Abundances of Stars: Enigmas

Przybylski´s star

Normal F0 star

• 50% of the spectral lines are unidentified

• Abundance of Lantanides 100.000 x solar

• Presence of radioactive elements, including Pm with a half life of 17.7 years

Origin of the anomalous abundances

• Explanations: abnormal model atmosopheres, accretion of planetesimals, interior nuclear processes with mixing, surface nuclear processes, or magnetic accretion.

The Ap phenomenon must be a surface phenomenon since the overabundance of rare earth elements (e.g. Eu is overabundant by a factor of up to 104 ) is so great that a signficant fraction of the supply of such elements in the Universe would be contained in Ap stars if this abundance extended throughout the star

• Most accepted hypothesis: Diffusion

The Diffusion Theory of Michaud (1970)

• A stars have high effective temperatures (high radiation field)

• A stars have an outer radiative zone (stable). Magnetic field further stabilizes the atmosphere

• If an element has many absorption lines near flux maximum radiation pressure drives it outwards where it can accumulate and become overabundant

• If an element has few absorption lines near flux maximum radiation it sinks under its own weight and can become underabundant

A bit of History

Cecilia Payne-Gaposchkin (1900-1979).

At Harvard in her Ph.D thesis on Stellar Atmospheres she:

• Realized that Saha‘s theory of ionization could be used to determine the temperature and chemical composition of stars

• Identified the spectral sequence as a temperature sequence and correctly concluded that the large variations in absorption lines seen in stars is due to ionization and not abundances

• Found abundances of silicon, carbon, etc on sun similar to earth

• Concluded that the sun, stars, and thus most of the universe is made of hydrogen and helium.

„There remains one very much more serious discrepancy, namely, that for hydrogen, helium and oxygen. Here I am convinced that there is something seriously wrong with the present theory. It is clearly impossible that hydrogen should be a million times more abundant than the metals, and I have no doubt that the number of hydrogen atoms in the two quantum state is enormously greater than is indicated by the theory of Fowler and Milne. „

Henry Norris Russell in a letter to CPG

Otto Struve: „undoubtedly the most brilliant Ph.D thesis ever written in Astronomy“

Youngest scientist to be listed in American Men of Science !!!

Later Russell reversed his position after seeing new analysis of the sun an wrote a paper “On the Composition of the Sun’s Atmosphere” where he states that the sun is mostly hydrogen. He acknowledged the work of Payne-Gaposchkin, but Russell had the ‚stature‘ to convince the community. A case where the person who persuades the community is the one getting the credit!