Transcript stellar atmospheres instructor notes

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.
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 nebula’s) surface, dA,
into an infinitesimally small
solid angle, dΩ, directed at an
angle θ to the surface normal.
The dimensions of the rectangle subtended by solid angle dΩ are
rdθ and r sin θ dφ, so dΩ = sin θ dθ dφ for r = 1. The average
intensity of the light entering the solid angle dΩ originating from
the surface area dA amounts to the energy Eλ dλ per unit time dt
projected in that direction, i.e. dA cos θ.
E d
i.e. I  
d dt dAcos d
The limit as dA, dλ, dΩ, and dt
→ 0 is referred to as the
specific intensity Iλ. Defined in
such fashion the intensity
represents the amount of
energy per unit time present
along the ray path, which for
dΩ → 0 does not spread out as
distance increases (i.e. in
comparison with flux). Also,
Iλ dλ = Iν dν, so
I 
c

2
I
The specific intensity may vary with direction, so one defines a
mean intensity <Iλ> (sometimes referred to as Jλ) as:
i.e.
I
1

4
1
 I  d  4
2 
  I  sin  d d
0 0
If the radiation is isotropic, i.e. the same intensity in all directions,
then
<Iλ> = Iλ.
Black body radiation is isotropic, in which case:
<Iλ> = Bλ.
Now consider the energy carried by the radiation field:
E d
Energy 
dAdL
where we define dL in the following illustration.
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:
dL cos 
dt 
c
So the energy carried by the flow
is given by:
I  d dt dAcos d
I  d dL cos dAcos d
Energy 

dAdL
dAdL c

I  d d
c
The energy density uλ of the radiation flow is found by integrating
the energy of the flow over all solid angles, i.e.:
1
1
u d   I  d d 
c
c
2
4
0 I  d sin  d d  c I  d
And, for black body radiation, which is isotropic, we expect:
4
4
8 hc 5
u d 
I  d 
B d  hc kT
d
c
c
e
1
or
4
4
8 h 3
u d 
I d 
B d  h kT
d
c
c
e 1
The total energy density is obtained through integration over all
wavelengths or frequencies, i.e.:

u   u d
0
For black body radiation we have:
4

c

4
0 B d  c
 T 4  4T 4

 
uBB
 aT 4
c
  
where a  7.566  1015 erg cm3 K 4
Radiative flux is a measure of the net energy flow across dA. Thus,
F d   I  d cos d
measures the flow of radiation through the surface dA in the
direction of the z-axis.
Typically the radiation field is isotropic, i.e. Iλ does not depend
upon direction. Then:
i.e. F  I 
2 

0 0
0
  sin  cos d d  2 I   sin  cos d  0,

since
 sin  cos d  0
0
For a flow through only one hemisphere, i.e. 0 ≤ θ ≤ ½π:

2
F  2 I   sin  cos  d  12  2 I    I 
0
Sometimes an “astrophysical flux” is defined. It is the true flux Fλ
divided by π, i.e.:
F 
F

and
F 
F

Be careful, since the usage varies from one textbook to another!
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 Iλ if the source is resolved,
otherwise radiative flux Fλ 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).
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:
dp d   p init ial  p finald
 E cos  E cos 
 

d

c
 c

2 E cos

d
c
2 I  cos2 

d dt dAd
c
The radiation pressure Prad is equivalent to the force exerted by
the photons, i.e. the rate of change of momentum per unit area.
dp d
2
Prad  d  
  I  cos2  d
dt dA
c
2

c
2 
2
I
d

cos
 sin  d d
 
0 0
Integration 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:
1
Prad  d   I  d cos 2  d
c
For isotropic radiation the formula becomes;
1
Prad d 
c
2 
2
I
d

sin

cos
 d d
 
0 0
But:
2
2
0 sin  cos  d  3
2
So:
Prad  d 
2 2
4
 I  d 
I  d
c
3
3c
The total radiation pressure is found by integration over all
wavelengths:

Prad 
P
rad 
d
0
which for isotropic black body radiation becomes:
4
Prad 
3c

4 T 4 4T 4 1 4 uBB
0 B d  3c    3c  3 aT  3
By way of comparison, the pressure exerted by an ideal monotonic
gas is 2/3 of its energy density.
LTE
The definition of temperature for a star depends upon how T is
derived:
Tex = temperature derived using the Boltzmann equation to
establish a match to the observed energy level populations
of atoms.
Tion = temperature 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.
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.
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:
T
HT 
dT dr
5770K

5890K  5650K  2.77106 cm
 6.66106 cm
 666km
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.
Stellar Opacity
The mean free path of particles is calculated as follows:
Typical densities in the solar atmosphere where T = Teff are of
order ρ ≈ 2.5 × 10–7 g/cm3. For pure hydrogen gas, the
corresponding number density is n = ρ/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:
π(2a0)2vt = σvt,
where σ = π(2a0)2 is the collisional cross-section.
There are nV atoms in the volume = nσvt atoms. The average
distance traveled between collisions is therefore:
distance traveled vt
1
l


number of atoms nvt n
The mean free path for a hydrogen atom is therefore l = 1/nσ. For
hydrogen, a0 = 0.53 × 10–8 cm so σ = π(2a0)2 = 3.52 × 10–16 cm2.
1
l
 1.89  10  2 cm  0.1 km, over which T  1 K.
n
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.
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 dIλ is related to the initial intensity Iλ of
the beam, the distance traveled ds, the gas density ρ and the
opacity of the gas as defined by its absorption coefficient κλ:
dI    I ds
The negative sign indicates that the intensity of the beam
decreases in the presence of absorption. Note the form of the
dI
relationship:
     ds
I
Integration of both sides of the equation gives:
ln I      s or I   I 0 e  s
or
I   I 0 e
s

    ds
0
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/ρκλ = 1/nσλ.
In the case of the Sun, for the parameters used earlier and κ5000Å
= 0.264 cm2/gm the implied scale length for photons is:
1
1
1
7
l



1
.
52

10
cm  152 km
7
n    0.264 2.5  10


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 τλ such that:
s
d       ds so that         ds
0
for integration from the outermost layer of a star inwards.
   1 defines opticallythick,and
   1 defines opticallythin
Application to Atmospheric Extinction
Consider the case of observations of stars made from groundbased telescopes, where the light traverses the Earth’s
atmosphere and suffers extinction.
For starlight traversing Earth’s
atmosphere the distance
traveled is ds = dt/cos z, where
dt is the thickness of the
atmosphere, i.e. ds = sec θ dt in
the diagram.
So:
s
t
t
       ds      sec z dt  sec z     dt    sec z
0
0
and:
or:
I   I 0 e
0
 o sec z
0
or
log I   log I 0  constant  sec z
m  mo  constant sec z
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 kλ varies
with wavelength λ and can vary
from night to night.
i.e. mλ = mλ(0) + kλ X
An example of an atmospheric extinction plot for a standard star
used for photometric standardization, this time plotting Fν versus
sec z rather than mλ versus X.
The strong 1/λ4 dependence of
extinction in Earth’s
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.
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
3
absorption is:
1
 bf  1.31  1015 5   o  cm2
n  5000 A 
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 κλ(ff) is a contributor to the
continuous opacity along with κλ(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 σbf. The formula is:
2
2
8  e 
 25
2


T 

6
.
655

10
cm
3  me c 2 
4. Electron scattering, part 2. Photons can also be scattered by
electrons that are loosely bound to atoms. Compton scattering
describes photon scattering where λ < 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 λ > size of the atoms.
The latter process is highly wavelength dependent, typically
varying as 1/λ4, as in atmospheric scattering (below).
An example of various absorption sources in the atmospheres of
stars: hydrogen and ionized helium bound-free absorption (earlytype stars), and the H– ion (late-type stars). The former is highly
λ–dependent, the latter almost λ–independent.
The continuum of the B7 V star Regulus (α Leo) showing the
signature of hydrogen bound-free absorption in its spectral
energy distribution.
Black body curves: what the continuous energy distributions of
stars would look like in the absence of continuous opacity sources
in their atmospheres.
Be careful how stellar spectral energy distributions are plotted.
They appear different when different parameters are used.
Atomic bound-bound absorption by various metal lines in the
continuous spectra of late-type stars, relative to H– absorption.
Typical formulae are, for Rayleigh scattering:
 8N e440  1
    2 4  4 cm3
 3m c  
And for Thomson scattering:
   6.655 1025 Ne
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
o
hc
o
12,400 eV A


 16,400A

0.754eV
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.
The total opacity in a star is the sum of the various individual
opacity sources, i.e.:
   bb    bf    ff    es
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:

25 g bf
 bf  4.34  10
Z 1  X  3.5 cm 2 /gm
t
T

 ff  3.68  10 22 gff 1  Z 1  X  3.5 cm 2 /gm
T
es  0.2 1  X  cm2/gm
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 κ once an atom or ion has been
ionized. Generally 1 < t < 100.
Not e t hat both  bf and  ff varyas
i.e.  bf   ff   0
Also:

T
3.5

T
3. 5
,
Kramer's Law, aft erH. A.Kramer
  bb  bf  ff  es
The Rosseland mean opacity is usually represented graphically.
From Rogers and Iglesius
(1992) for X = 0.70 and
Z = 0.02. Value of ρ, in units
of gm/cm3, are indicated
above each curve. The
opacities are calculated for a
specific mixture of elements
known as the AndersGrevesse abundances.
Note:
1. κ ↑ as ρ ↑.
2. κ ↑ as T ↑ initially from the
ionization of H and He.
3. κ ↓ with further T ↑ results
from the 1/T3.5 dependence of
Kramer’s opacity.
4. κ flattens at large T as
electron scattering dominates.
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:

d  l1  l2  l3    lN
The net displacement as an absolute value is given by:
     
     
d  d  l1  l1  l1  l2   l1  lN  l2  l1  l2  l2 
 
   
 
 l2  lN   lN  l1  lN  l2   lN  lN
 Nl 2  l 2 cos12  cos13    cos N 1N 
 Nl 2
since the term in brackets ≈ 0 for large N. Random angular
displacements generate an average value of θ ≈ π/2, i.e. cos θ = 0.
d 2  Nl 2 or d  l N
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/ρκλ = 1/nσλ and
1
ds
d
l N


so    d    
 N
   d   
l
So optical depth τλ = 1 implies a photon has suffered only one
scatter before escaping the star (actually τλ = ⅔ for light we see).
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, σ = π(2a0)2 = π(2 × 0.53 × 10–8)2 cm2 ≈ π × 10–16 cm2
So the mean free path is:
l = 1/nσλ = 1/(100 × π × 10–16) ≈ 3 × 1013 cm ≈ 2 A.U.
the root-mean-squared velocity is:
vRMS = (3kT/m)½ = (3 × 1.38 × 10–16 × 10,000/1.66 × 10–24)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
When viewing the Sun the light
originates from τλ = ⅔ 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.
Equation of Transfer
The emission of light by material along a specific line of sight is
proportional to the emission coefficient jλ of the material, the
density ρ, and the distance traversed ds, i.e.:
dIλ = jλρds
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:
dIλ = –κλρIλds
so for the combined processes we must have the equation of
radiative transfer:
dIλ = –κλρIλds + jλρds
or
 1 dI
j
 I 
 I   S
   ds

where
j

 S
is called thesourcefunction,
and the former equation is the equation of transfer.
The source function Sλ = jλ/κλ describes the proportionality
between the emitting and absorbing properties of the medium.
Clearly, Sλ has identical units to Iλ (cm s–3 steradian–1). The form
of the transfer equation leaves very simple expectations:
1 dI
 I   S
   ds
If dIλ/ds = 0, the light intensity is constant and Iλ = Sλ.
If dIλ/ds < 0, Iλ > Sλ, and with increasing s, Iλ → Sλ.
If dIλ/ds > 0, Iλ < Sλ, and with increasing s, Iλ → Sλ.
In other words, over any distance ds, the intensity of light
approaches the local source function. If the conditions for LTE
are satisfied, dIλ/ds = 0, so Iλ = Sλ. And Iλ = Bλ for black body
radiation, so Sλ = Bλ.
But Iλ ≠ Bλ unless τλ >> 1, i.e. the photons are able to interact
many times with matter in the star’s atmosphere.
Textbook Example:
Imagine a beam of light of intensity Iλ,o at s = 0 entering a volume
of gas of constant density ρ, constant opacity κλ, and constant
source function Sλ. What is Iλ(s)?
Solution (see textbook):
The result is:

I s  I ,o e  s  S 1  e  s

Equation of Transfer, 2
Recall that dτλ = –κλ ρds, if s is measured outwards radially in a
star but optical depth is measured inwards, so the equation of
transfer:
1 dI
 I   S
   ds
Can be rewritten as:
dI
 I   S
d 
Consider a plane parallel stellar
atmosphere and define dτλ in
terms of a reference direction,
z. Define:
0
  ,v z       dz
z
But:
  cos    ,v
for any direction s. Thus:
  ,v
d  ,v
 
   ,v sec and d  
cos
cos
and the transfer equation becomes:
dI
cos
 I   S
d  ,v
A simple approximation that can be made at this point is to
remove the wavelength dependence of the opacity κλ . An
atmosphere that is approximated by a constant opacity κ as a
function of λ, i.e. the same opacity throughout the spectrum, is
referred to as a gray atmosphere, and is a good approximation for
some stars. Thus, τλ,v becomes τv and it is possible to generate


values for:
I   I  d and S   S d
0
0
The equation of transfer then becomes:
dI
cos
I S
d v
The resulting radiation field originating from such a planeparallel atmosphere can be established by integration over all
solid angles, i.e.:
d
I cos d   I d  S  d

d v
but
 d  4 ,
4 I   I d , and
which reduces the transfer equation to:
dFrad
 4  I  S 
d v
 I cos d  F
rad
The same equation of transfer can also be multiplied by cos θ and
integrated, resulting in the first moment:
d
2
I
cos
 d   I cos d  S  cos d

d v
but
d
d
2
 cos d  0 , and d v  I cos  d  d v Prad c 
which gives :
dPrad Frad

d v
c
[The same equation in a spherical co-ordinate system is:
dP rad   

Frad
dr
c
]
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.
dPrad Frad

d v
c
When LTE applies:
<I> = S so that Frad = constant = Fsurface = σTeff4
The situation requires flux conservation throughout the stellar
atmosphere:
dPrad Frad

 const ant
d v
c
1
so dPrad  Frad d v
c
1
or Prad  Frad  v  const ant
c
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:
I d
I 

4
 /2
1
2
 I out sin  d 
0
1
sin

d


2 I in

0
Frad   I cos d  2
1
2
 I
in
sin  d
/2
 /2
 12 I out


 sin  d 
I out  I in 
 /2
 /2
I
1
2
out
cos sin  d  2
0
 4   2 I  2 4    I
 2 I out 2
1
Prad 
c
in
2
2
 I cos  d  c
 
 /2

 I
in
/2
out
 I in 
2
2
0 I out cos  sin  d  c
 
cos sin  d

2
I
cos
 in  sin  d
 /2
2
2
2
4
1
1
I out  I in  

I out

I in

I
3
3
c
c
3c
3c
At the surface of the star the situation is simplified by the fact
that Iin = 0.
In such circumstances:
<I> = ½(Iout + Iin)
Frad = π(Iout – Iin)
Prad = 2π/3c(Iout + Iin) = 4π/3c <I>
The condition of flux constancy in
the atmosphere implies that
Iout > Iin at all levels of the
atmosphere.
1
From Prad  Frad  v  constant
c
it follows t hat
4
1
I  Frad  v  constant
3c
c
At the top of the atmosphere τv = 0 and Iin = 0, so:
<I> = ½Iout = ½(Frad/π) = Frad/2π
and the constant in the radiation equation is evaluated from:
4
1
4 Frad
2Frad
constant 
I  Frad  top 
0
3c
c
3c 2
3c
So:
4
F
2F
F
I  rad  v  rad  rad  v  23 
3c
c
3c
c
A simple substitution for the flux at the top of the atmosphere, i.e.
Fsurface = σTeff4 gives:
3Teff4
 v  23 
I 
4
For LTE:
T 4
SB

or
T 4
I 

Substitution then gives:
T 4 3Teff4
 v 


4
or:
T4 
3
4
2
3

Teff4  v  23 
An obvious result of the Eddington approximation is that, for the
gray atmosphere approximation, the temperature in a stellar
atmosphere is T = Teff when τv = ⅔, so can be thought of as the
point of origin for the light from a star (rather than, say, τv = 0 or
τv = 1).
The gray atmosphere approximation can be further tested using
the transfer equation:
dI
 I   S
d 
Multiplication of both sides by exp(–τλ) gives:
dI  
e  I  e     S e  
d 
or:
which becomes:


d  
e I    S e  
d 


d e  I   S e  d 
The equation can be integrated from an initial position of a ray of
light at an optical depth τλo to the top of the atmosphere, τλ = 0, to
give:
0
0
e   I     S e   d 
 o
which yields:
I  0   I  e  o 
 o
0
 
S
e
  d 
 o
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.
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:
     ,v sec
I  0  I o e
and
 v , o sec 
0

 v sec 
S
sec

e
d v

 v ,o
The observed intensity at the top of the
atmosphere is the result of all
contributions along the line of sight,
i.e. to τv,o = ∞. And e–∞ = 0, so:

I  0   S sec  e  v sec  d v
0
The dependence of the source function S on optical depth is
unknown, but a reasonable first approximation is:
S  a  b  v
Next evaluate the integral using the above approximation, in
which case the integral becomes:
0
0
0
 v ,o
 v ,o
 v ,o
 v sec 
 v sec 
 v sec 
S
sec

e
d


a
sec

e
d


b
sec


e
d v
v
 
v
 
v

0
 sec e
But:
0
 v ,o
 v sec 
d v   e
 v sec  
0
 0  e0  1

1
1
 v sec 
x
x
x 
d v 
x e dx 
 xe  e
And:  sec   v e

0
sec  0
sec 
 v ,o
1

0  0  0  1  cos
sec 
which gives a source function described by:
S  a  b cos
and an emergent intensity described by the equation:
I  0  a  b cos
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 4 3Teff4
 v 
S I 


4
2 3Teff4 Teff4
a 

3 4
2
3Teff4
and b 
4
The solar limb darkening is best
expressed relative to the intensity
at disk centre, i.e.:
I  
I  

I   0
Io
2
3
  a  b  v
Substitution gives:
I   a  b cos


Io
a  b

i.e.,
5
4

2
Teff4  34 Teff4 cos
4
4

3
T

T
2 eff
4 eff
Teff4 25  53 cos  2 3
  cos
4 2
5
3
5 5
4 Teff  5  5 
I  
 0.4  0.6 cos
Io
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(τ) to
account for limb darkening.
Flux constancy, since there is no net
loss or gain of energy in a star’s
atmosphere, i.e.:

F
rad
d  constant Fo
0

dFrad
or 
d  0
d 
0
Hydrostatic equilibrium (right):
which gives:
Substitution gives:

dPrad
σTeff4
0 d  d  2c
For hydrostatic equilibrium:
dP
g

d   
So:
dPgas
d 

g
o
dP
from
 g
dz

dPrad Teff4

d 
c
Teff4
c
if
o  
Structure of Spectral Lines
Terminology:
Fλ = radiant flux at wavelength λ
Fc = continuum flux expected
λo = wavelength of line centre
Fc  F
Fc  F
 depth of the line
W 
d  equivalent width
Fc
Fc
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:
(Δλ)½ = width of a spectral line
measured between line depths
corresponding to one half the
line depth at line centre λo.
Line Broadening Mechanisms
1. Natural Broadening
According to the Heisenberg uncertainty principle, ΔE = ħ/Δt.
Electrons spend almost infinite time in ground states of atoms and
ions, so ΔE ≈ 0 for n = 1, but very little time in excited levels.
Since Ephoton 
 ΔE photon
hc

hcΔ
h
 2 

2 Δt
2  1
1 
 Δ 


2c  Δt i Δt f 
2 1
or  Δ  
2c Δt o
1
2
Textbook Example:
The average time spent by an electron in the 1st and 2nd excited
levels of hydrogen is Δt = 10–8 s. What is the corresponding
expected half-width for the Balmer Hα 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:

656310 cm
1
Δ  
 2.3 10
2 2.997910 cm/s 10 s
2
8
1
2
4
10
8

A
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.
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 :
Δ
vr
 2kT 


vmp  


c
 m 
So we expect line widths to vary as:
2  2k T 
Δ 


c  m 
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:
 Δ  1 2
2  2kT ln 2 



c  m 
Textbook Example:
What is the Doppler width for spectral lines from hydrogen (m =
1.6735 × 10–24 gm) in the Sun, where Teff = 5779 K?
Solution (see textbook):
The calculated value for the Doppler line width is:
Δ 
1
2 10
 2 1.3811016  5779 ln 2 gm cm2 / s2 
cm


2.99791010 cm/s 
1.67351024 gm

5
2

 0.356A
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.
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.:
2
vtrue

2kT
2
 vturb
m
So we expect line widths to vary as:
Δ 12
2  2kT
2 


v

turb  ln 2
c  m

Textbook Example:
What is the Doppler width for spectral lines from hydrogen (m =
1.6735 × 10–24 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:
16

2kT
2

1
.
381

10
 5779
2
5
2
2


 vturb  

2

10
cm
/
s

 24
m
1.6735

10
gm


 9.9381011 cm2 / s2

Δ 
1

2
 ~ as before  0.363A
Rotation and pulsation also generate large-scale mass motions that
result in line broadening and add to the Doppler broadening of
spectral lines.
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 Waal’s 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 Δt
the average time between collisions:
l
1
m
Δto  
v n 2kT
The expected line width should therefore vary as:
Δ 12
2 1 2 n  2kT 




c Δto c   m 
Where n is the number density of the atoms and σ is the
collisional cross-section.
Textbook Example:
What is the van der Waal’s broadening for Balmer Hα lines in
the Sun where n = 1.5 × 1017 cm–3 and for hydrogen σ = π(2ao)2 =
π(2 × 0.5292 × 10–8 cm)2 = 3.5189 × 10–16 cm2?
Solution (see textbook):
The calculated value line width for van der Waal’s broadening is:

6563 10
 Δ  
8
1
2
cm 1.5  1017 cm- 3  3.5189 1016 cm2 
 2.99791010 cm/s
2
12
 95.38  10 cm/s  2.36  10
10
4

cm  2.36  10 A
which is approximately the same size as the natural broadening
discussed earlier.
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.:
Δ
2vR


c
For a line at 5000 Å the resulting broadening is:


2v R
2  400km/s
Δ  
 5000A 
 13.3 A
5
c
2.9979 10 km/s
a very significant amount. The effect of rotational broadening of
spectral lines dominates over all other factors at high rotational
speeds.
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.
Summary
The effects of the various line broadening mechanisms can be
summarized as follows:
Natural broadening:
~0.0001 Å
Van der Waal’s broadening: ~0.0002 Å
Doppler broadening:
~0.4 Å
Turbulent broadening:
~0.4 Å
Rotational broadening:
≤13 Å
Stark broadening:
~5–50 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.
For reference purposes, the cross-section for a harmonic oscillator
is generally given as:



e2 

   
when   o

2
mc 

  o 2    2  


 
where the term in brackets is referred to as the Lorentz profile.
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.
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 star’s surface that have electrons in the
proper energy level for producing a photon at the wavelength λ 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/λ of the line changes in a specific
fashion, referred to as the curve of growth.
The three portions of the curve of growth (COG) are:
i. linear portion, where W/λ increases with A
ii. plateau, where W/λ is proportional to (ln A)½
iii. damping, where W/λ increases in proportion to A½
The variation
of log W/λ
with A relative
to the shape
of the spectral
line.
The effect of different amounts of microturbulence ξ (in km/s).
Matching observations of solar titanium I lines to theoretical
COG predictions to establish the amount of microturbulence ξ.
The iron I lines in HD 219134 indicate the presence of measurable
microturbulence ξ in its atmosphere.
A star’s surface gravity g affects the density of atoms in its
atmosphere, affecting the damping portion of the COG directly.
The effect of different excitation energies ξ (χ here) for spectral
lines from the same element.
Recall the Boltzmann equation:
Nm
gm
log
  m  log
N
u T 
So:
log N total f  log N total  log f
gm
 log N m  m  log
 log f
u T 
5040 m
 const ant
 log f
T
For atomic spectral lines originating from the ionization state,
location of log W/λ for the lines on the curve of growth depend
upon the excitation potential of the level of origin ξm, the
excitation temperature Tex, and the oscillator strengths f for the
lines.
A plot of horizontal shift Δlog A versus excitation potential ξm for
a series of lines then specifies Tex.
A plot of the values for different parameters can often be used to
establish the effective temperature and surface gravity in
individual stars.
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.
Textbook Example:
λ(Å)
W(Å)
f
log W/λ log (fλ/5000 Å)
3302.38
0.088
0.0214
–4.57
–1.85
5889.97
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.
λ(Å) log (fNaλ/5000Å) log (fλ/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 cm–2, which
implies a mass of 9.3 × 10–5 gm cm–2, (5.4 × 10–5 gm cm–2 from a
more detailed analysis), relative to 1.1 gm cm–2 for hydrogen.
How one uses a model curve of growth with measurements of
equivalent width to infer element abundance.
Textbook Example:
λ(Å)
3302.38
5889.97
log W/λ
–4.57
–3.91
λ(Å)
3302.38
5889.97
log N
19.05
18.95
mks
log (fλ/5000 Å) log (Nfλ/5000 Å)
–1.85
17.20
–0.12
18.83
log N
19.05
18.95
Correction log (Natot I) Correction log N(Na)
0.0002
19.05
3.386
22.44
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 × 10–4 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. cm–2) rather than
mks units (i.e. m–2).