1. Spectroscopy – Chemical Analysis
Abundance Analysis
Curve of Growth
Results
Next week:
Measurement of Stellar Parameters
Gravity
Radius
Temperature

2. 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 – greater than the Sun.

3. ∫ ∫ 2p Bn(tn) E2(tn) t0 ln + kn k0 dlog t0 log e = Fn W = Fc – Fn Fc
For direct computation we use the equation for the flux (LTE) and compute the flux for a series of points spanning the line 2p ∫ –∞ ∞
Bn(tn) E2(tn) t0
ln + kn k0
dlog t0
log e = Fn
We then integrate across the line to get the equivalent width
W = ∫ ∞
Fc – Fn Fc dn
Fc and Fn are the fluxes in the continuum and in the line, respectively
Given the line absorption coefficient, ln, 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.

4. ∫ Scaling relations Fc – Fn Fc ≈ C ln kn W = dn ln C kn
For weak lines:
Fc – Fn Fc ≈ C ln kn
The equivalent width of the line becomes:
W = ∫ ∞ dn ln C kn
lnr = Na, r is the mass density, N is the number of absorbers per unit volume, and a is the absorption coefficient

5. ( ) f f = W = C N = A NH g u(T) Nr NE exp kT – c pe2 mc a l2 c pe2 mc
a is the wavelength integrated absorption coefficient
W = C
pe2 mc f l2 c N kn
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 g
u(T) Nr NE
exp ( ) kT – c

6. ( ) = + log A + log gf l – qc – log kn NH W l log pe2 mc u(T) Nr/NE
The equvialent width: NH ) ( W l
log =
+ log A + log gf l – qc – log kn
pe2 mc
u(T)
Nr/NE
Abundance gf
Temperature and excitation potential
Continuous opacity
Depends on:
q = 1040/T and division by l normalizes Doppler dependent phenomena
→Depends on Teff, gravity, composition, etc.
A change in any one of these mimics a change in the abundance

7. 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 gfl, c, and kn.
We chose a line, this fixes gfl and c, our stellar atmospheric model fixes q and kn. We can then vary A and generate the curve of growth
Different lines of the same species have different gfl and c but these have to have the same abundance, A. This can be used to constrain the equation.
The scaling with kn is usually small, especially if lines are in the same wavelength region. For example, between 4000 and 6000 Å, ∂ log kn/∂l ≈ 0.1 cm2/gm per 1000 Å for T < 7500 K

8. The Curve of Growth 3 phases:
Weak lines: the Doppler core dominates and the width is set by the thermal broadening DlD. Depth of the line grows in proportion to abundance A
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 kn. The strength depends on g, but for constant g the equivalent width is proportional to A½

9. 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 c is higher. The amount of each shift can be interpreted as qexcc.

10. 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 kn
qex
And all of these effect the abundance

11. Curve of Growth: Gravity Effects
Gravity can effect line strength through
Nr/NE kn
Since both of these can be sensitive to the pressure, For neutral lines the effects cancel
There is a linear relationship between Dlog A and Dlog g ln kn
≈ constant g–⅓

12. ∂ log A/∂ log g Teff Ca I Ca II Cr I Cr II Fe I Fe II 7200 0.02 –0.33
0.01
5040
0.00
–0.39
–0.40
–0.11
–0.45
3870
–0.06
–0.43
–0.26
–0.53
–0.35
–0.60
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

13. 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

14. 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 x. 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.

15. a(total) = a(natural)*a(Stark)*a(v.d.Waals)*a(thermal)
Recall the combined absorption coefficient:
a(total) = a(natural)*a(Stark)*a(v.d.Waals)*a(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:
a(total) = a(natural)*a(Stark)*a(v.d.Waals)*a(thermal)*a(micro)

16. Procedure for determining microturblent velocity:
Fit the equivalent widths to the weakest lines where the line strength does not change with x.
This fixes A. You can now use the curve of growth for the saturated lines to compute x.
Also can just determine x 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!

17. Fitting the microturbulence

18. 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(t) and x

20. 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 x
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. x must be wrong.
Chose a new x and start all over. Continue until you converge

21. 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 (c = 0, log gf = 0, l = l0)
This abundance is valid for the standard curves parameters = A0
The real abundance is obtained by:
D log A = log (gf/gf1 )+ log (l/l0) – log(kn/k1) – q(c–c0)
log A = log A0 – D log A
So instead of plotting W versus A, we plot W versus D log A

22. A reference curve-of-growth for a solar model

23. DA D Log A Reference curve-of-growth
D log A = log (gf )+ log (l/l0) – log(kn/k0) – qc –6 –4 –2 +2 +4
D Log A
log A = log A1 – D log A

24. Curve of Growth Analysis for Abundances
Abundance determinations with a graph and calculator
1. Plot observed log (Wl/l) versus log gfl – log (kn/k0) – qexc. If qexc is wrong there will be a lot of scatter. The best value of qex minimizes the scatter.

25. Curve of Growth Analysis for Abundances
Procedure:
2. Calculate the vertical shift between the observed and theoretical curves. The vertical shift is log xT/c where
x2T = x2thermal + x2micro
3. Move horizontally to get the abundance
Vertical shift → turbulent velocities
Horizontal shift → abundances

26. 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.

27. Spectral synthesis programs can be obtained from the internet
Spectral synthesis programs can be obtained from the internet. Most popular are the ATLAS9 routines of Kurucz and MOOG from Sneden
ATLAS →
MOOG →
SME → Spectroscopy Made Easy: GUI based IDL routines for calculating
synthetic spectra (Valenti & Piskunov, A&A Supp, 1996, 118, 585
Tutorial:
All programs require a line list. This can be obtained from the VALD (Vienna Atomic Line Database): or

28. 1-D versus 3-D
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. For example, in the sun the metallicity values decrease by a factor of 2!

29. 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

30. 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.

31. Comparison of the 1-D abundances versus 3-D abundances as a function of equivalent width of the line.

32. The Solar Composition

33. 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

34. Uranium in Stars
Frebel et al. 2007
In this star Uranium is due to r-processing of elements

35. 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 solar. These are presumably old stars.

36. 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.

37. Abundances of Stars: Galactic Variations
Halo: Mostly Pop II stars, metal poor, globular clusters
Globular clusters
Disk: Pop I stars, metal rich
Bulge: Mostly Pop II stars, metal poor, some Pop I stars
What does this tell us about the chemical evolution of the galaxy?

38. 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

39. Abundances of Stars: Galactic Variations
Globular clusters were the first to form, thus metal poor.
Disk stars are the last to form, thus metal rich
t = t2

40. 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)

41. Ultra Metal Poor Stars

42. Is this really one of the first stars?
And the current champion is HE with an [Fe/H] = –5.4 or x solar metallicity
Frebel et al. 2007
Is this really one of the first stars?

43. Venn & Lambert (2008) have argued that this may not be the case
Venn & Lambert (2008) have argued that this may not be the case. Peculiar stars such as post AGB stars and l 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 l 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

44. Abundances of Stars: Super Metal Rich
These are stars with metallicity [Fe/H] ~ +0.3 – +0.5
Valenti & Fischer
There is believed to be a connection between metallicity and planet formation. Stars with higher metalicity tend to have a higher frequency of planets.

45. Endl et al. 2007: HD 155358 two planets and..
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
…[Fe/H] = –0.68. This certainly muddles the metallicity-planet connection

46. 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.

47. Abundances of Stars: Lithium
Li in the sun
Li : 6708 Å

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

49. 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.

50. Maybe pollution can explain the metallicity-planet connection
Giant hosting planet stars do not show a metallicity enhancement such as the planet hosting stars on the main sequence. Luca et al. hypothesize that the high metal content is due to pollution by planets. When the stars evolve to giants they have deeper convection zones which mixes the chemicals.

51. Abundances of Stars: Enigmas
Normal F0 star
Przybylski´s star
50% of the spectral lines are unidentified
Abundance of Lantanides x solar
Presence of radioactive elements, including Pm with a half life of 17.7 years

52. Origin of the anomalous abundances
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
Explanations: abnormal model atmosopheres, accretion of planetesimals, interior nuclear processes with mixing, surface nuclear processes, or magnetic accretion.
Most accepted hypothesis: Diffusion

53. 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