The Probability and Effects of an Asteroid Impact with Earth

1
9. Stellar Atmospheres Goals 1. Develop the
basic equations of radiative transfer describing
the flow of light through stellar
atmospheres. 2. Examine how stellar continua and
spectral lines are affected by various
parameters, and how stellar abundances are
derived. 3. Derive some useful approximations
for describing the radiative flux from stars. 4.
Derive the fundamental equations describing the
equilibrium conditions for stellar atmospheres,
as used in stellar atmosphere models.
2
The Radiation Field In order to describe
radiation from a star (or nebula) it is necessary
to begin with some definitions of observable
parameters, the first being specific intensity.
Begin with radiation passing through an
infinitesimally small area of a star (or
nebulas) surface, dA, into an infinitesimally
small solid angle, dO, directed at an angle ? to
the surface normal.
The dimensions of the rectangle subtended by
solid angle dO are rd? and r sin ? df, so dO
sin ? d? df for r 1. The average intensity of
the light entering the solid angle dO originating
from the surface area dA amounts to the energy E?
d? per unit time dt projected in that direction,
i.e. dA cos ?.
3
The limit as dA, d?, dO, and dt ? 0 is referred
to as the specific intensity I?. Defined in such
fashion the intensity represents the amount of
energy per unit time present along the ray path,
which for dO ? 0 does not spread out as distance
increases (i.e. in comparison with flux).
Also, I? d? I? d?, so
4
The specific intensity may vary with direction,
so one defines a mean intensity ltI?gt (sometimes
referred to as J?) as If the radiation is
isotropic, i.e. the same intensity in all
directions, then ltI?gt I?. Black body
radiation is isotropic, in which case ltI?gt
B?. Now consider the energy carried by the
radiation field where we define dL in the
following illustration.
5
Consider the energy associated with the radiation
flow through a perfectly reflective measuring
cylinder (depicted) placed symmetrically about an
axis normal to the radiating surface. The
transit time for the radiation is So the
energy carried by the flow is given by
6
The energy density u? of the radiation flow is
found by integrating the energy of the flow over
all solid angles, i.e. And, for black body
radiation, which is isotropic, we
expect or The total energy density is
obtained through integration over all wavelengths
or frequencies, i.e.
7
For black body radiation we have Radiative
flux is a measure of the net energy flow across
dA. Thus, measures the flow of radiation
through the surface dA in the direction of the
z-axis.
8
Typically the radiation field is isotropic, i.e.
I? does not depend upon direction.
Then For a flow through only one
hemisphere, i.e. 0 ? ½p Sometimes an
astrophysical flux is defined. It is the true
flux F? divided by p, i.e. Be careful, since
the usage varies from one textbook to another!
9
The Difference Between Intensity and Flux? The
specific intensity of a source is independent of
distance from the source, whereas the radiative
flux varies with distance according to the
inverse square law, i.e. 1/r2. For a distant
source it is only possible to measure intensity
I? if the source is resolved, otherwise radiative
flux F? is measured. In the example shown one
measures specific intensity in (a) when the
source subtends an angle larger than the
resolution of the telescope/detector system,
otherwise radiative flux (b).
10
The Radiation Field, 2 A photon of energy E
carries a momentum p E/c, which means that it
can exert radiation pressure. For photons
incident on a reflecting surface (image at right)
the momentum exchange with the surface is simply
the change in momentum upon reflection
11
The radiation pressure Prad is equivalent to the
force exerted by the photons, i.e. the rate of
change of momentum per unit area. Integrati
on over one hemisphere gives the radiation
pressure exerted by the flow from the source,
i.e. a photon gas that does not reflect from
the surface For isotropic radiation the
formula becomes
12
But So The total radiation pressure is
found by integration over all wavelengths whic
h for isotropic black body radiation
becomes By way of comparison, the pressure
exerted by an ideal monotonic gas is 2/3 of its
energy density.
13
LTE The definition of temperature for a star
depends upon how T is derived For gas
in a box the various temperatures should match,
since thermodynamic equilibrium (TE) applies.
Stars cannot be in perfect TE since there is an
outward flow of energy producing a temperature
gradient in their atmospheres. If the distance
over which T changes significantly is small
relative to the distances traveled by atoms and
ions between collisions, then local thermodynamic
equilibrium (LTE) is a good approximation.
Tex temperature derived using the Boltzmann
equation to establish a match to the observed
energy level populations of atoms. Tion temperat
ure derived using the Saha equation to establish
a match to the observed ionization states of
atoms. Tkin temperature as inferred from the
Maxwell-Boltzmann equation to describe the
velocity distribution of particles. Tcolor or
TBB is the temperature established by matching
the unreddened flux distribution to a Planck
function.
14
In the Sun, T varies from 5650 K to 5890 K over a
distance of 27.7 km (1 K/0.1 km) according to the
Harvard-Smithsonian Reference Solar Atmosphere.
The resulting temperature scale height
is over which the temperature changes
by one scale factor and where T 5770 K has been
used as a typical region of the solar
atmosphere. Clearly it is safe to assume that
most regions of the solar atmosphere are in LTE.
Exceptions are restricted to regions where the
temperature changes rapidly.
15
Stellar Opacity The mean free path of particles
is calculated as follows Typical
densities in the solar atmosphere where T Teff
are of order ? 2.5 107 g/cm3. For pure
hydrogen gas, the corresponding number density is
n ?/mH 1.5 1017 /cm3. Two atoms will
collide if their centres pass within two Bohr
radii 2a0. In time t a single atom sweeps out a
volume given by p(2a0)2vt svt, where s
p(2a0)2 is the collisional cross-section.
16
There are nV atoms in the volume nsvt atoms.
The average distance traveled between collisions
is therefore The mean free path for a
hydrogen atom is therefore l 1/ns. For
hydrogen, a0 0.53 108 cm so s p(2a0)2
3.52 1016 cm2. The mean free path is much
smaller than the distance over which T changes by
1 K. Gas atoms in the solar atmosphere, and
typical stellar atmospheres, are therefore
confined to a reasonably isolated region within
which LTE can be assumed to be valid. The same
is not true for photons, since they are able to
escape freely into space.
17
Photon Absorption Absorption refers to
scattering and pure absorption of photons by
particles, anything that removes photons from a
beam of light. The amount of absorption dI? is
related to the initial intensity I? of the beam,
the distance traveled ds, the gas density ? and
the opacity of the gas as defined by its
absorption coefficient ?? The negative sign
indicates that the intensity of the beam
decreases in the presence of absorption. Note the
form of the relationship Integration of both
sides of the equation gives or
18
Because of the exponential drop-off, the
intensity decreases by a factor of 1/e 1/2.718
0.368 when the exponent is unity, i.e. over a
scale length of l 1/??? 1/ns?. In the case
of the Sun, for the parameters used earlier and
?5000Å 0.264 cm2/gm the implied scale length
for photons is which is comparable to the
temperature scale height. In other words, photons
travel through regions of different T. It is
convenient to introduce the term optical depth t?
such that for integration from the outermost
layer of a star inwards.
19
Application to Atmospheric Extinction Consider
the case of observations of stars made from
ground-based telescopes, where the light
traverses the Earths atmosphere and suffers
extinction. For starlight traversing
Earths atmosphere the distance traveled is ds
dt/cos z, where dt is the thickness of
the atmosphere, i.e. ds sec ? dt in the
diagram. So and or
20
Because of the curvature of the Earth, the value
of sec z is not quite equivalent to the air mass
X, which measures the total amount of atmospheric
extinction between the star and the observer. The
best formula for air mass X is X sec z (1
0.0012 tan2 z) The constant term k?
varies with wavelength ? and can vary from night
to night. i.e. m? m?(0) k? X
21
An example of an atmospheric extinction plot for
a standard star used for photometric
standardization, this time plotting F? versus sec
z rather than m? versus X.
22
The strong 1/?4 dependence of extinction in
Earths atmosphere means that blue stars fade
more rapidly than red stars with increasing air
mass X. It also gives rise to colour terms in the
extinction coefficients.
23
Opacity Sources in Stars 1. Bound-bound
transitions, involving photon absorption and
reemission in random directions resulting in a
net loss of light in the direction of the
original photon. 2. Bound-free transitions,
involving photoionizations from the ground state.
For hydrogen the cross-section for bound-free
absorption is 3. Free-free absorption, in
which free electrons passing near hydrogen atoms
absorb energy from photons. The process can occur
for all ranges of wavelengths, so ??(ff) is a
contributor to the continuous opacity along with
??(bf). The process is also referred to as
bremsstrahlung, or braking radiation. 4.
Electron scattering, or Thomson scattering, in
which photons are scattered by free electrons, a
wavelength-independent mechanism 2 109 larger
than sbf. The formula is
24
4. Electron scattering, part 2. Photons can also
be scattered by electrons that are loosely bound
to atoms. Compton scattering describes photon
scattering where ? lt size of the atoms. Since
most atoms and molecules have dimensions of 1 Å,
Compton scattering applies mainly to X-rays and
gamma rays. Rayleigh scattering describes photon
scattering where ? gt size of the atoms. The
latter process is highly wavelength dependent,
typically varying as 1/?4, as in atmospheric
scattering (below).
25
An example of various absorption sources in the
atmospheres of stars hydrogen and ionized
helium bound-free absorption (early-type stars),
and the H ion (late-type stars). The former is
highly ?dependent, the latter almost
?independent.
26
The continuum of the B7 V star Regulus (a Leo)
showing the signature of hydrogen bound-free
absorption in its spectral energy distribution.
Balmer Discontinuity 3647 A
27
Black body curves what the continuous energy
distributions of stars would look like in the
absence of continuous opacity sources in their
atmospheres.
28
Be careful how stellar spectral energy
distributions are plotted. They appear different
when different parameters are used.
29
Atomic bound-bound absorption by various metal
lines in the continuous spectra of late-type
stars, relative to H absorption.
30
Typical formulae are, for Rayleigh
scattering And for Thomson
scattering Atomic hydrogen absorption is
strongest where the population of the n 2 level
of hydrogen maximizes relative to all hydrogen
particles, i.e. near 9800 K. H opacity is the
dominant opacity source in cool stars. The
ionization potential for the H ion is only 0.754
eV, so any photon with ionizes it. Molecules
are also opacity sources (in cool stars) because
they can be dissociated and also give rise to
bound-bound and bound-free absorptions of
photons. Molecular absorptions produce large
numbers of closely-spaced lines, much like bands.
31
The total opacity in a star is the sum of the
various individual opacity sources, i.e. Where
the first three terms are wavelength
dependent. It is often useful to use an opacity
averaged over all wavelengths under
consideration, one that depends upon density,
temperature, and chemical composition. Such an
average opacity is known as the Rosseland mean
opacity, or simply the Rosseland mean. Although
there is no simple formula for the various
contributors, approximations have been developed,
namely
32
where X and Z are the fractional abundances of
hydrogen and the heavy elements, respectively, by
mass. Typical values for the Sun are X 0.75 and
Z 0.02. The terms gbf and gff are quantum
mechanical correction factors calculated by J. A.
Gaunt, hence their names Gaunt factors.
Generally gbf gff 1 for the visible and
ultraviolet regions of interest for stellar
atmospheres. The factor t is an additional
correction factor called a guillotine factor, and
describes the cutoff for ? once an atom or ion
has been ionized. Generally 1 lt t lt
100. Also The Rosseland mean opacity is
usually represented graphically.
33
From Rogers and Iglesius (1992) for X 0.70
and Z 0.02. Value of ?, in units of gm/cm3, are
indicated above each curve. The opacities are
calculated for a specific mixture of elements
known as the Anders-Grevesse abundances. Note 1
. ? ? as ? ?. 2. ? ? as T ? initially from the
ionization of H and He. 3. ? ? with further T ?
results from the 1/T3.5 dependence of Kramers
opacity. 4. ? flattens at large T as electron
scattering dominates.
34
Radiative Transfer Consider the flow of photons
out of a star as a random walk problem. If a
photon has a mean free path ? average distance
traveled before absorption and reemission or
scattering from an atom ? of l, then a photon
undergoing a sequence of N random walks undergoes
a net displacement d where
35
The net displacement as an absolute value is
given by since the term in brackets 0
for large N. Random angular displacements
generate an average value of ? p/2, i.e. cos ?
0. i.e. d 10 l requires N 100 d 100 l
requires N 10,000 d 1000 l requires N
1,000,000 But d is also related to optical depth
since l 1/??? 1/ns? and So optical depth
t? 1 implies a photon has suffered only one
scatter before escaping the star (actually t? ?
for light we see).
36
Textbook Example What is the mean free path
length and average time between collisions for
atoms in the Orion Nebula where n 100
/cm3? Solution (see textbook) For hydrogen, s
p(2a0)2 p(2 0.53 108)2 cm2 p 1016
cm2 So the mean free path is l 1/ns?
1/(100 p 1016) 3 1013 cm 2 A.U. the
root-mean-squared velocity is vRMS (3kT/m)½
(3 1.38 1016 10,000/1.66 1024)½
1.6 106 cm/s and the average time between
collisions is t l/v (3 1013 cm)/(1.6
106 cm/s) 2 107 s 8 months
37
When viewing the Sun the light originates from t?
? for all parts of the visible disk. Near disk
centre the light originates from deeper, hotter
regions than for the solar limb, where the light
originates from shallow, cooler regions. The
result is an apparent limb darkening of the Sun.
38
Equation of Transfer The emission of light by
material along a specific line of sight is
proportional to the emission coefficient j? of
the material, the density ?, and the distance
traversed ds, i.e. dI? j??ds for photons
created by emission processes. The light beam is
also affected by the opacity of the gas, which
scatters and absorbs photons from the line of
sight. For absorption we have dI?
???I?ds so for the combined processes we must
have the equation of radiative transfer dI?
???I?ds j??ds or and the former equation
is the equation of transfer.
39
The source function S? j?/?? describes the
proportionality between the emitting and
absorbing properties of the medium. Clearly, S?
has identical units to I? (cm s3 steradian1).
The form of the transfer equation leaves very
simple expectations If dI?/ds 0, the light
intensity is constant and I? S?. If dI?/ds lt
0, I? gt S?, and with increasing s, I? ? S?. If
dI?/ds gt 0, I? lt S?, and with increasing s, I? ?
S?. In other words, over any distance ds, the
intensity of light approaches the local source
function. If the conditions for LTE are
satisfied, dI?/ds 0, so I? S?. And I? B?
for black body radiation, so S? B?. But I? ?
B? unless t? gtgt 1, i.e. the photons are able to
interact many times with matter in the stars
atmosphere.
40
Textbook Example Imagine a beam of light of
intensity I?,o at s 0 entering a volume of gas
of constant density ?, constant opacity ??, and
constant source function S?. What is
I?(s)? Solution (see textbook) The result is
41
Equation of Transfer, 2 Recall that dt? ?? ?
ds, if s is measured outwards radially in a star
but optical depth is measured inwards, so the
equation of transfer Can be rewritten
as Consider a plane parallel
stellar atmosphere and define dt? in terms of a
reference direction, z. Define
42
But for any direction s. Thus and the
transfer equation becomes A simple
approximation that can be made at this point is
to remove the wavelength dependence of the
opacity ?? . An atmosphere that is approximated
by a constant opacity ? as a function of ?, i.e.
the same opacity throughout the spectrum, is
referred to as a gray atmosphere, and is a good
approximation for some stars. Thus, t?,v becomes
tv and it is possible to generate values for
43
The equation of transfer then becomes The
resulting radiation field originating from such a
plane-parallel atmosphere can be established by
integration over all solid angles,
i.e. which reduces the transfer equation
to
44
The same equation of transfer can also be
multiplied by cos ? and integrated, resulting in
the first moment The same equation
in a spherical co-ordinate system is
45
The interpretation of the first moment equation
is that the net radiative flux is driven by the
natural gradient in radiation pressure within a
star. When LTE applies ltIgt S so that
Frad constant Fsurface sTeff4 The
situation requires flux conservation throughout
the stellar atmosphere
46
The Eddington Approximation The great
astrophysicist Sir Arthur Eddington took the same
equations a step further by adopting a simple
approximation for the flux from a star that
separated it into an outwards directed flux and
an inwards directed flux at each point in the
atmosphere. The intensity of the light
originating at each depth z in the atmosphere
therefore had two components Iin intensity
of the radiation in the inward
direction Iout intensity of the radiation
in the outward direction Note Iin 0 at the
top of the atmosphere. So, for any point in the
atmosphere
47
At the surface of the star the
situation is simplified by the fact that Iin 0.
48
In such circumstances ltIgt ½(Iout
Iin) Frad p(Iout Iin) Prad 2p/3c(Iout
Iin) 4p/3c ltIgt The condition of flux constancy
in the atmosphere implies that Iout gt Iin at all
levels of the atmosphere.
49
At the top of the atmosphere tv 0 and Iin 0,
so ltIgt ½Iout ½(Frad/p) Frad/2p and the
constant in the radiation equation is evaluated
from So A simple substitution for the
flux at the top of the atmosphere, i.e. Fsurface
sTeff4 gives For LTE
50
Substitution then gives or An obvious
result of the Eddington approximation is that,
for the gray atmosphere approximation, the
temperature in a stellar atmosphere is T Teff
when tv ?, so can be thought of as the point
of origin for the light from a star (rather than,
say, tv 0 or tv 1). The gray atmosphere
approximation can be further tested using the
transfer equation Multiplication of both
sides by exp(t?) gives
51
or which becomes The equation can be
integrated from an initial position of a ray of
light at an optical depth t?o to the top of the
atmosphere, t? 0, to give which
yields Namely, the intensity at the top of the
atmosphere intensity at any depth reduced by
attenuation less any further contribution along
the line of sight less attenuation.
52
If we next return to a discussion of the
intensity emerging at the surface of the
atmosphere from any direction, the optical depth
and intensity equation become The observed
intensity at the top of the atmosphere is the
result of all contributions along the line of
sight, i.e. to tv,o 8. And e8 0, so The
dependence of the source function S on optical
depth is unknown, but a reasonable first
approximation is
53
Next evaluate the integral using the above
approximation, in which case the integral
becomes But And which gives a
source function described by and an emergent
intensity described by the equation
54
The Solar Limb Darkening The assumption of a
gray atmosphere with the Eddington approximation
can now be used to generate a formula describing
the limb darkening of the Sun, i.e. So T
he solar limb darkening is best expressed
relative to the intensity at disk centre, i.e.
55
Substitution gives
56
Modeling Stellar Atmospheres The parameters for
the gas in all stellar atmospheres must obey
certain relationships with one another in order
to preserve equilibrium in the outer layers of
the star, namely A temperature distribution
T(t) to account for limb darkening. Flux
constancy, since there is no net loss or gain of
energy in a stars atmosphere,
i.e. Hydrostatic equilibrium
(right) which gives
57
Substitution gives For hydrostatic
equilibrium So
58
Structure of Spectral Lines Terminology F?
radiant flux at wavelength ? Fc continuum flux
expected ?o wavelength of line centre The
equivalent width of a line corresponds to the
width of a box in Å of continuum light absorbed
that is identical in area to the integrated area
of the spectral line. Full-width at half
maximum (??)½ width of a spectral line
measured between line depths corresponding to
one half the line depth at line centre ?o.
59
Line Broadening Mechanisms 1. Natural
Broadening According to the Heisenberg
uncertainty principle, ?E h/?t. Electrons spend
almost infinite time in ground states of atoms
and ions, so ?E 0 for n 1, but very little
time in excited levels.
60
Textbook Example The average time spent by an
electron in the 1st and 2nd excited levels of
hydrogen is ?t 108 s. What is the
corresponding expected half-width for the Balmer
Ha line, which corresponds to a transition
between levels n 2 and n 3? Solution (see
textbook) The calculated value for the natural
line width is which is much too narrow
relative to the actual observed line widths for
the hydrogen Balmer lines. It can be concluded
that natural broadening is not the source of line
broadening for the hydrogen lines, although it is
presumably important for the spectral lines of
heavy elements.
61
2. Doppler Broadening As a result of the thermal
motions of atoms, they are moving relative to one
another at fairly large speeds, given by the
Maxwell-Boltzmann distribution. Recall the value
for the most probable speed which produces
Doppler shifts So we expect line widths to
vary as Where the factor of 2 is introduced
by the positive and negative velocity shifts. An
detailed analysis taking into account
contributions to the line across the stellar disk
and the true distribution of motions gives
62
Textbook Example What is the Doppler width for
spectral lines from hydrogen (m 1.6735 1024
gm) in the Sun, where Teff 5779 K? Solution
(see textbook) The calculated value for the
Doppler line width is which is much
larger than the natural broadening, although
still smaller than the observed line widths for
the hydrogen Balmer lines. It can be concluded
that Doppler broadening is not the main source of
line broadening for the hydrogen lines.
63
3. Doppler Broadening with Turbulent Motions If
there is an additional component of turbulence in
the gas, the actual velocities of the gas atoms
will be in excess of those predicted by the
Maxwell-Boltzmann equation, i.e. So we
expect line widths to vary as
64
Textbook Example What is the Doppler width for
spectral lines from hydrogen (m 1.6735 1024
gm) in the Sun, where Teff 5779 K, when the
turbulent velocities are 2 km/s? Solution (see
textbook) The calculated value for the Doppler
line width is Rotation and pulsation
also generate large-scale mass motions that
result in line broadening and add to the Doppler
broadening of spectral lines.
65
4. Pressure (and Collisional) Broadening An
additional source of line broadening is produced
by the perturbing actions of passing atoms and
ions. Effects can be produced by electric field
effects (Stark broadening) or pressure effects
(van der Waals broadening). The primary result
is a damping profile in the spectral line shape
that produces broadened line wings. For pressure
broadening the effect can be estimated using for
?t the average time between collisions The
expected line width should therefore vary
as Where n is the number density of the
atoms and s is the collisional cross-section.
66
Textbook Example What is the van der Waals
broadening for Balmer Ha lines in the Sun where
n 1.5 1017 cm3 and for hydrogen s p(2ao)2
p(2 0.5292 108 cm)2 3.5189 1016
cm2? Solution (see textbook) The calculated
value line width for van der Waals broadening
is which is approximately the same size
as the natural broadening discussed earlier.
67
5. Rotational Broadening Although the rotational
speed of the Sun is only 2 km/s, in some stars
that value can exceed 400 km/s! The effect on the
width of a spectral line resulting from the
smearing of gas motions on opposite hemispheres
of a star reaching 400 km/s can be estimated
using the Doppler effect, i.e. For a line at
5000 Å the resulting broadening is a very
significant amount. The effect of rotational
broadening of spectral lines dominates over all
other factors at high rotational speeds.
68
Rapid rotation affects mainly early-type stars.
In late-type dwarfs there is a break at
spectral type F5 that appears to mark the onset
of chromospheres in cooler stars.
69
Summary The effects of the various line
broadening mechanisms can be summarized as
follows Natural broadening 0.0001 Å Van der
Waals broadening 0.0002 Å Doppler
broadening 0.4 Å Turbulent broadening 0.4
Å Rotational broadening 13 Å Stark
broadening 550 Å The quadratic sum of all
natural broadening mechanisms the damping
profile for natural and pressure broadening, as
well as Doppler broadening is referred to as
the Voigt profile. In general, Voigt profiles for
spectral lines have Doppler cores and damping
wings.
70
For reference purposes, the cross-section for a
harmonic oscillator is generally given
as where the term in brackets is referred
to as the Lorentz profile.
71
Curves of Growth The strength of a spectral
line, as indicated by its equivalent width W, is
determined by a variety of factors, namely i.
the abundance of the element producing the line,
the greater the abundance the stronger the
line, ii. the transition probability for the
line, or the f-value, the higher the probability
f the stronger the line, iii. the population of
the energy level where the line originates, the
lower the population number for the energy level
the weaker the line, iv. the line broadening
mechanism, since mechanisms that produce strong
line wings do so at the expense of absorption in
the line core, v. the rotation velocity of the
star, and vi. the electron density Ne, since it
governs the damping portion of the line.
72
Oscillator strength, f, represents the effective
number of electrons per atom participating in a
transition between energy levels. For abundance
studies of stars we want A, the number of
absorbing atoms per unit area of a stars surface
that have electrons in the proper energy level
for producing a photon at the wavelength ? of the
spectral line we are measuring. As more and more
atoms contribute to the shape and area of an
observed spectral line, the normalized equivalent
width W/? of the line changes in a specific
fashion, referred to as the curve of growth.
73
The three portions of the curve of growth (COG)
are i. linear portion, where W/? increases with
A ii. plateau, where W/? is proportional to (ln
A)½ iii. damping, where W/? increases in
proportion to A½
W (ln A)1/2
plateau
damping portion
W A1/2
W A
linear portion
74
The variation of log W/? with A relative to
the shape of the spectral line.
75
The effect of different amounts of
microturbulence ? (in km/s).
76
Matching observations of solar titanium I lines
to theoretical COG predictions to establish the
amount of microturbulence ?.
77
The iron I lines in HD 219134 indicate the
presence of measurable microturbulence ? in its
atmosphere.
78
A stars surface gravity g affects the density of
atoms in its atmosphere, affecting the damping
portion of the COG directly.
79
The effect of different excitation energies ? (?
here) for spectral lines from the same element.
80
Recall the Boltzmann equation So For
atomic spectral lines originating from the
ionization state, location of log W/? for the
lines on the curve of growth depend upon the
excitation potential of the level of origin ?m,
the excitation temperature Tex, and the
oscillator strengths f for the lines. A plot of
horizontal shift ?log A versus excitation
potential ?m for a series of lines then specifies
Tex.
81
A plot of the values for different parameters can
often be used to establish the effective
temperature and surface gravity in individual
stars.
82
Accurate estimates of stellar abundances are now
done using spectral synthesis rather than curve
of growth methods, but both depend upon a
reliably established temperature distribution,
like that for the Sun shown here.
83
Textbook Example ?(Å) W(Å) f log W/? log
(f?/5000 Å) 3302.38 0.088 0.0214 4.57 1.85 588
9.97 0.730 0.6450 3.91 0.12 Given the data for
solar sodium lines above (Na I), what is the
abundance of sodium in the Sun?
Solution (see textbook) From the solar curve of
growth, the corresponding values for the
abundance of Na I atoms producing the lines is
given below, along with derived values for the
relative abundance of Na II to Na I, and the
overall abundance of sodium. ?(Å) log
(fNa?/5000Å) log (f?/5000Å) log Na log
Nion/Nn log N(Na) 3302.38 13.20 1.85
15.05 3.3856 18.386 5889.97 14.83 0.12
14.95 3.3856 18.386 The inferred abundance of
sodium is 2.43 1016 cm2, which implies a mass
of 9.3 105 gm cm2, (5.4 105 gm cm2 from a
more detailed analysis), relative to 1.1 gm cm2
for hydrogen.
84
How one uses a model curve of growth with
measurements of equivalent width to infer element
abundance.
85
Textbook Example mks ?(Å) log W/? log
(f?/5000 Å) log (Nf?/5000 Å) log
N 3302.38 4.57 1.85 17.20 19.05 5889.97 3.91
0.12 18.83 18.95
?(Å) log N Correction log (Natot
I) Correction log N(Na) 3302.38 19.05 0.0002 19.0
5 3.386 22.44 5889.97 18.95 0.0064 18.97 3.386 22
.36
The derived abundance corresponds to 2.5 1022
atoms of sodium (Na) per square meter of
atmosphere, or 9.6 104 kg of sodium (Na) per
square meter of atmosphere, compared with 11
kg/m2 for hydrogen (H). The values (13.20
rather than 17.20 18.83) on the previous slide
refer to the abundance in cgs units (i.e. cm2)
rather than mks units (i.e. m2).