Download Chapter 14 – Chemical Analysis

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/l
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 kn
– Line strength depends
(approximately) on the square
root of the abundance
The Effect of Temperature on the COG
• Recall:
Fc  Fn
ln
 constant 
Fc
kn
– (under the assumption that Fn comes from a
characteristic optical depth tn)
• Integrate over wavelength, and let lnr=Na
• Recall that the wavelength integral of the absorption
coefficient is
e 2 l2 N
w  constant 
mc c
f
kn
• Express the number of absorbers in terms of hydrogen
• Finally,
Nr
g 
NA
NH
e
NE
u (T )
kT
 e 2 N r N E

log  log  2
N H   log A  log gfl    log kn
l
 mc u(T )

w
The COG for weak lines
 e 2 N r N E

log  log  2
N H   log A  log gfl    log kn
l
 mc u(T )

w
Changes in log A are equivalent to changes in log gfl, ,
or kn
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 gfl,
, or kn.
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
g1/3
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 x 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
x that makes the abundance independent of line
strength.
Microturbulence in the COG
-3
5 km/sec
Log w/lambda
-4
0 km/sec
-5
0 km/sec
1 km/sec
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2 km/sec
3 km/sec
5 km/sec
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-13
-12
-11
-10
-9
-8
-7
Log A + Log gf
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?
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