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Spectroscopy – Chemical Analysis
Abundance Analysis
• Curve of Growth
•
Results
Next week:
Measurement of Stellar Parameters
• Gravity
• Radius
• Temperature
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.
For direct computation we use the equation for the flux (LTE) and compute the
flux for a series of points spanning the line
∞
dlog t0
l n + kn
Fn = 2p Bn(tn) E2(tn)
t0
k0
log e
∫
–∞
We then integrate across the line to get the equivalent width
∞
W=
∫
0
Fc – Fn
dn
Fc
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.
Scaling relations
For weak lines:
Fc – Fn
ln
≈ C k
n
Fc
The equivalent width of the line becomes:
C
W= k
n
∞
∫
ln dn
0
lnr = Na, r is the mass density, N is the number of absorbers per unit
volume, and a is the absorption coefficient
a =
f
pe2 l2
mc c
W= C f
a is the wavelength integrated absorption
coefficient
pe2 l2
mc 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
c
N
g
–
r
N=A
NH
exp
kT
NE
u(T)
( )
The equvialent width:
log
W
l
( )
= log
(
pe2 Nr/NE
N
mc u(T) H
) + log A + log gf l – qc – log k
n
Depends on:
• Abundance
• gf
q = 1040/T and
division by l
normalizes Doppler
dependent phenomena
• Temperature and
excitation potential
• Continuous opacity
→Depends on Teff, gravity, composition, etc.
A change in any one of these mimics a change in the abundance
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
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
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.
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
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
–⅓
kn ≈ constant g
∂ log A/∂ log g
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
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 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.
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)
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!
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(t) and x
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
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
A reference curve-of-growth for a solar model
Reference curve-of-growth
DA
D log A = log (gf )+ log (l/l0) – log(kn/k0) – qc
–6
–4
–2
D Log A
0
+2
+4
log A = log A1 – D log A
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.
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
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.
Spectral synthesis programs can be obtained from the
internet. Most popular are the ATLAS9 routines of Kurucz
and MOOG from Sneden
ATLAS → http://kurucz.harvard.edu
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/
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!
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
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.
Comparison of the 1-D
abundances versus 3-D
abundances as a function of
equivalent width of the line.
The Solar Composition
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
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
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?
Abundances of Stars: Galactic Variations
Pop III stars only of Hydrogen and Helium.
Supernovae explosions pollute protogalactic 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
Globular clusters were
the first to form, thus
metal poor.
Disk stars are the
last to form, thus
metal rich
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
Frebel et al. 2007
Is this really one of the first stars?
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
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.
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
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.
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.
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.
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.
Abundances of Stars: Enigmas
Normal F0 star
Przybylski´s 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
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
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