Transcript 6 - Stellar Atmospheres

Stellar Atmospheres
I Parte del
Corso di Astrofisica Stelle e Pianeti 2006-07
Approfondimenti sono contenuti in vari capitoli (in inglese, file
word) che non sono ancora in forma definitiva ma che possono
essere già utilizzati da chi fosse interessato
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Summary – stellar atmospheres theory
The atmosphere of a star contains less than 1x10-9 of its total mass,
but it is that what we can see, measure, and analyze.
Spectroscopic analyses provide elemental abundances and show us
results of cosmo-chemistry, starting from the earliest moments
of the formation of the Universe to present day.
Photometric analyses are used to put a star from the observed colormagnitude diagram (e.g. B-V, V) into the theoretical HR Diagram
(L,Teff) and, hence, to guide the theories of stellar structure and
evolution.
The study of stellar atmospheres is a very difficult task. The
atmosphere is that region, where the transition between the
thermodynamic equilibrium of the stellar interior into the almost
empty circumstellar space occurs. It is a region of extreme nonequilibrium states.
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Stellar Atmospheres
We call stellar atmosphere the ensemble of the outer layers to which the
energy, produced in the deep interior of the star, is carried, either by radiation,
convection or conduction. Interacting with the matter present in the outer
layers, this energy finally produces the observed electromagnetic radiation,
particle flux and magnetic field. By analogy with the terminology adopted for
the Sun, a typical atmosphere can be divided very schematically in several
regions, as in the figure.
The abscissa is the
outward radial
distance. The
ordinate is the
temperature.
Both scales vary
with the stellar
spectroscopic type.
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Temperature in the solar atmosphere
Schematics of the solar temperature profile (thick line) with height. The zero
level is the conventional surface. Notice the sudden increase of T in the transition
region between chromosphere and corona. The dotted line gives the matter
density profile.
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Energy transport mechanisms
A more physical approach to this subdivision would make use of the main
energy transport mechanism in each region:
1. in the photosphere the energy is transported by radiation (notice that this
assumption is equivalent to an outward decreasing temperature), and the
geometry is well approximated by plane-parallel stratification;.
2. in the chromosphere, there is also dissipation of waves (acoustic,
magneto-hydrodynamic);
3. above a very sharp transition layer, in the corona, the magnetic field
energy is very important. Large scale motions with velocities larger than
the escape velocity give origin to a loss of particles known as stellar wind,
and the plane parallel approximations is certainly untenable.
In the following, we will concentrate our attention essentially on the
photosphere and the chromosphere, and in the visible region of the
spectrum. This limitation can be justified a posteriori by the overall energy
coming out of the various layers, as schematically indicated in Table 1,
valid for the Sun.
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Table 1- Solar Energy Output
(ergcm-2s-1)
Radiative losses
Whole Sun
6.284x1010
Photosphere
Chromosphere
- Balmer series
- H- (negative H ion)
- Ly-
- Metallic lines (MgII, CaII)
Transition Zone
Corona (Quiet Sun)
6.2x1010
2-6x106
5x105
4x105
3x105
3-4x105
In the Sun, the
convective flux
(granulation) is of
the order of 1% and
the conductive flux is
of the order of 10-5 of
the radiative flux.
The solar wind
velocities range from
400 to 800 km/s.
4-6x105
6x105
The mass loss is
about 1036 particles
per second  1012 g/s
 10-14 M/y.
On other spectral type stars, the situation can be very different.
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Basics on Transport of radiation - 1
Consider a surface area of size , having normal n, and an elementary solid angle
d in direction (, ), where:
0    180 , 0    360
The light passing through d in the unit time, of wavelength between (, +d),
carries an energy E that can be written as:
E  I ( , )  d cos  d  d
(dimensions: erg/s)
where the factor cos comes from the surface projection effect.
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Basics on Transport of radiation - 2
The quantity I(, ) is called intensity of the radiation field: it is the energy
that flows each second through the unit area d in the wavelength interval d
into the unit solid angle d in direction  to the normal n.
The units of I are: ergcm-2s-1A-1sr-1 (in our mixed cgs system)
In general I = I(x,y,z,,,t) = I(x,y,z;l,m,n;t) where (l,m,n) are the direction
cosines.
For simplicity, in the following we shall assume:
- azimuthal symmetry of the radiation field, namely independence from ,
- stationarity, namely time constancy.
Furthermore, no account will be taken of a possible polarization of the beam.
The radiation field is said isotropic if the intensity is independent of direction
in that point, and homogeneous if it is the same in all points.
Caveat: in planetary atmospheres none of these simplifying assumptions is
fulfilled!
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Basics on Transport of radiation - 3
Consider now, in a idealized experiment which we could perform in the
laboratory, a beam of radiation of intensity I,0 which enters from the left in a
vessel of gas at a given temperature T. The cross-section of the vessel is , its
length is s. The shape of the column (here shown as a rectangular box) is
irrelevant, it could be a cylinder. We also suppose that the walls of the vessel are
transparent, in order not to have reflection effects.
We want to determine the intensity of the radiation exiting the vessel on the right.
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Transport of radiation - 4
Absorption: after a trajectory ds along the path of the beam, some energy of the
beam will be lost due to absorption (here, absorption has a loose significance, to
be specified later):
dE    E ds   ds  I  d  d  d
where  is a linear absorption coefficient (cm-1) appropriate to that particular
gas.
Emission: On the other hand, each elementary volume dV=d ds of gas will
emit, contributing some energy to the beam according to an appropriate
emissivity coefficient :
dE      d  ds  d  d
which we assume here independent from the incoming radiation field.
The units of  are ergcm-3sr-1. Notice that the emission is (assumed) isotropic.
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Transport of radiation - 5
Therefore, the total energy variation can be expressed as:
dE  dE   dE   ( I +  )d  ds  d
The intensity variation along the beam is then:
dI 
     I
ds
(  I + )ds  dI
Notice that the sign of dI/ds is determined by that of the difference:
(    I )
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Optical depth and Source Function
Let us introduce now the a-dimensional variable elementary optical depth:
d     ds
The previous equation for the intensity variation becomes:
dI 
dI

    I  +    I  +S
  ds d 

where the function S is called the source function.
The total optical depth of the column is obtained by:
s
  ( s )     dx
0
which clearly shows that the same geometrical thickness can correspond to
very different optical depths.
Notice that these definitions are appropriate to the laboratory experiment,
where the radiation comes into the volume at x = 0 and exits at x = s.
Later on, we shall reverse the direction, the radiation from the stellar surface
will come from the deep interior and will exit at z = 0.
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The Source Function at equilibrium
Let us assume that the gas is in perfect thermodynamic equilibrium, and that the
passage of the radiation field does not alter this condition. The intensity of the
beam cannot change either, so that:
dI 
dI 
 
   I  +     I  +  S    ( I  +S )  0
ds
d 
I   S
Under these conditions, the intensity of the radiation (and so also the source
function) is expressed by Planck’s function B(T):
I   S  B (T ) 
2hc 2

5
1
e
hc / k T
1
    B (T )
The second equality is another way of expressing Kirchoff’s law: the ratio of the
emissivity to the absorption coefficient is independent from the chemical
composition of the gas.
Notice that the Planck function is isotropic: we shall maintain isotropy of S (and
) even when the identification of S with B is not justified.
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Local thermodynamic equilibrium -1
Indeed, the assumption S = B is very convenient and provides very useful
indications, but it is not always justified.
In the general case, the source function must be derived by the detailed
knowledge of the physical conditions of the atmosphere. In particular, in a
stellar atmosphere, the strict condition of thermodynamic equilibrium never
applies; following Milne, we shall assume its local validity (a condition
known as LTE), with a temperature T having a well determined value at any
depth z, but changing along the column; the source function along the path is
equal to Planck’s function, but with changing T:
S=B (T).
The consequence is not entirely intuitive: even if the absorption would take
place in only one absorption line, nevertheless the emissions would be
distributed over all wavelengths according to B. Furthermore, the emission
will be isotropic.
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Local thermodynamic equilibrium -2
The previous expression can be written as:
  
   dI 

e 
 I  
e  B (T )e 
 d 
 
 


B
e
d 


d  I  e


which can be integrated over the interval 0       (s)
 (s)

0
 (s)
z
z
B e  dz  I  e  
0
If moreover B is assumed constant along the path (as in laboratory conditions),
the intensity at the exit face of the column will be:
  ( s)
I  (  ( s))  I  (0)e
 B (1  e
  ( s)
)
The first term on the right-hand side is the percentage of energy that
entered the volume at x = 0, and left from the front face at x = s (whose optical
depth is  (s)). The second term is the contribution of the emissivity of the gas.
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Case 1: no input radiation
No input radiation means I(0) = 0, the column of hot gas shines with intensity
given by:

I   B (T ) 1  e  ( s )

This case can be subdivided in two limiting situations:
1.1 when the optical depth  (s) is very small (optically thin gas), then:
I    (s)  B (T )    s  B (T )
The intensity will be large only at the wavelengths where  is large, namely at
the resonance lines typical of the gas at that temperature, lines which we see
in emission.
1.2 when the optical depth  (s) is very large (optically thick gas), then:
I  B (T )
The intensity becomes totally independent from the length of the column and
also from the chemical composition of the gas (namely from ). We observe a
black body of a given temperature T.
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Case 1.1 : no input radiation and very thin gas
Case 1.1 is typical of many astrophysical situations, such as
emission and planetary nebulae, or the solar corona
observed outside the solar limb (e.g. during an eclipse, or
with a coronagraph occulting the disk).
These gases are all very hot, as it can be judged by the high
ionization, but we see emission lines because their optical
depth is small, not because they are hot.
Notice also that the ‘small optical thickness’ condition
certainly prevail in the continuum, and in the wings of the
line. However, at the very peak of the line the depth can
become high. The brightness will then approach that of
the black body having the temperature of the gas.
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Case 2: Appreciable input radiation
Appreciable input radiation I(0) >> 0 (this would be the case of a stellar
atmosphere). Again, two limiting cases can be considered:
2.1 optically thin case:

B  I  (0) 
I  (  ( s))  I  (0) 1    s

I
(0)



If the sign of the second term is negative (I(0) > B), we observe the spectral
distribution at the entrance of the column minus a fraction which is larger where
 is larger, namely absorption lines superimposed on the entrance spectrum.
The interpretation is fairly straightforward assuming that I and B are both black
body functions: the temperature of the entrance radiation is higher than the
temperature of the gas in the column.
If the sign is positive (I(0) < B), emission lines, superimposed over the entrance
continuum, are observed where  is larger (see next slide).
2.2 optically thick case:
I  B (T )
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independent from the entrance spectrum.
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Case 2.1 - Emission lines
In case 2.1, if the sign is positive (I(0) < B), emission lines,
superimposed over the entrance continuum, are observed.
This is the case for instance of the solar spectrum seen at 
< 1600 A: all lines are in emission, not in absorption.
Evidently, the UV absorption coefficient becomes so large
(large optical thickness) that we only see the upper layers of
the atmosphere (the chromosphere), that must have a source
function (namely a temperature) increasing toward the
exterior, therefore higher than that of the visible photosphere
(say 12.000 K instead of 6.000 K).
Notice that this conclusion doesn’t come from the
ionization, but simply by the lines being in emission, instead
than in absorption as in the visible region.
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Real stars
In the visible region, the stars usually show an absorption
line spectrum, they must therefore correspond to the case I,0 > S
(the intensity coming from the interior is higher than the source
function of the external layers, like having a reversing layer on top
of a hotter surface). In the LTE assumption, this also means that the
temperature of the external layers is smaller than the temperature of
the interior layers (outwards decreasing temperature). However, for
real stars, the discussion is much more complex, even assuming
LTE and radiative transport only, because nor the density nor the
source function are constant inside the atmosphere.
Although the concept of a reversing layer maintains a considerable
intuitive validity, a deeper physical and mathematical analysis is
therefore warranted, as done in the following.
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Absorption line spectra
Concluding this fairly approximate discussion, an absorption line spectrum
is formed in:
•
•
A deep optically thick gas
surmounted by a thinner
layer, with source function
decreasing outwards, as in
the solar photosphere; in
the present simplified
interpretation, source
function means
temperature.
Absorptions can also form in an optically thin gas penetrated by a
background radiation whose intensity is larger than the source function of
the column. This can be the case of a thin shell around a star, or of the
interstellar medium between us and a hot star.
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The radiative transfer equation - 1
The previous approximate discussion has shown that a variety of cases are
possible, even in laboratory conditions.
To describe and understand the stellar (and planetary) atmospheres, we must
put the above considerations on firmer physical and mathematical grounds.
We shall assume that the energy coming from the interior of the star is
transported through the atmosphere by radiation only, an assumption which
is not always justified, because other mechanisms, such as convection and
conduction are possible, but not treated at present.
Another simplification is the assumption of a plane parallel atmosphere. In
the case of the Sun, this assumption is well justified in the visible domain;
indeed, the geometrical thickness of the solar atmosphere to visible radiation
(photosphere) is much smaller than the solar radius.
As already done before, the radiation field is assumed stationary and
unpolarized, with azimuthal symmetry (dependence on  only, independence
from  . We have already commented that planetary atmospheres are more
complex).
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The radiative transfer equation - 2
As in the previous considerations, along the path ds inside the atmosphere the
following equation will be satisfied:
dI  ( )
  I  ( )+  /     I  ( )+S
  ds
where:
I  I (  , )
but
S  S (  )
because of the assumed isotropy of the source function.
Let us assume now a given plane as surface of the stellar atmosphere; the
geometrical position of this surface is at moment immaterial. The unit vector n
indicates the outward normal to the plane atmosphere, and  the angle of a
given radiation pencil with n.
It is convenient at this point to introduce a change of perspective, because the
observer sees the radiation from the outside, as explained in the following
figure:
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The radiative transfer equation - 3
Be z a linear coordinate (say, in km) increasing from the surface inward, and 
the perpendicular optical depth, also increasing inwards along the perpendicular
to the surface. The coordinate s instead increases outwards, so that along the
beam of radiation, s increases outwards at an angle  (see figure).
d    dz   cos  d s    d s
where the quantity cos is usually designated with .. Notice the change in
sign and the presence of cos with respect to previous discussion.
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The radiative transfer equation - 4
The radiative transfer equation for the plane parallel case with azimuthal
symmetry is then:

dI  (  ,  )
 I  (  ,  ) - S (  )
d 
In order to derive the intensity of the radiation exiting the surface in a given
direction , namely I(0,), multiply both sides by:
e   / 
and obtain:

dI  (  ,  )   / 
e
 ( I  (  ,  )  S (  ))e  / 
d 
dI  (  ,  )e  / 
 S e  / 
d(  /  )
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The radiative transfer equation - 5
Integrating in  from 0 to , and taking into account that for  going to  the
exponential term vanishes, we finally get:

I  (0,  )   S (  )e  /  d(  /  )
0
Notice that in this integral equation,  =cos is not a variable, but a
parameter; the equation provides a family of solutions, one for each direction
with respect to the normal n to the surface of the star.
The interpretation is straightforward: the intensity leaving the surface
at an angle  results from the summation of all the contributions of the volume
elements along the path of the light.
If we can measure I(0,cos) , as is possible for the Sun, then by inverting the
previous equation we could derive S. However, mathematically the inversion is
always a difficult task, not necessarily single-valued and very sensitive to
measurement errors. Here we treat only the direct problem, by assuming a
particular functional form for S, and deriving I.
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A first approximation for S()
Let us make the simple assumption that the unknown source function S is a linear
function of the optical depth:
S (  )  a  b 
(this assumption can be seen as the result of the usual technique of expanding a function
in Taylor’s series and considering the first order only, but later on we will justify it on the
basis of Eddington’s approximation).
We then get the following result (using cos for clarity):

I  (0, cos  )   (a  b  )e  sec d(  sec  )  a  b cos   S (   cos  )
0
which is also written as:
I (0, )   w  (1  w )cos  I (0,0)
As already said, in the LTE hypothesis S coincides with the Planck’s function
B at a z-dependent temperature:
S (  ( z))  B (T (  ( z)))
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Limb Darkening
The previous equation tells us that at the center of the disk we see the source
function S at a depth  = 1, at the border of the disk we see S at the surface;
but remembering that in the plane parallel approximation z/s = cos , we see that
 = cos when s = 1; in conclusion, at any point on the stellar disk we always
see down to a depth corresponding to s = 1, as in the Figure:
Because in the photosphere the temperature increases inwards, the temperature
measured at the center of the disk must be higher than at the limb.
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Two reasons for the photospheric limb darkening
Therefore, we have
identified two reasons for
the limb darkening:
1 - optical depth
2 - temperature gradient in
the photosphere
This figure shows two
possible polar diagrams:
-On the left a = b = 1
-On the right a = 0.5, b = 2
The larger the ratio b/ a ,
the more the radiation is in
forward direction.
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The solar limb darkening - 1
Let us observe the Sun, which is effectively at infinite distance but resolved as
a disk. The radiation coming to the observer from the center of the disk leaves
the star perpendicular to the surface, so that:
I (0,0)  a  b  S (   1)  B (T (   1))
The radiation coming from the borders of the solar disk leave the surface at  =
90°, so that:
I (0,90)  a  S (   0)  B (T (   0))
The observations prove that we see less light from the border, a fact named limb
darkening. See the following figures. We then conclude that:
B (T (   0))  B (T (   1))
Therefore, in the solar photosphere, the temperature indeed decreases outwards.
The solar limb darkening gave indeed the first convincing proof ot the validity
of the previous assumptions, in particular of radiative transport of the energy
through the photosphere.
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Solar Limb Darkening - 2
The Sun is darker (cooler, redder) at the limb. Notice how well the
observations are described by the first approximation. The temperature is
6050 K at the center, and 4550 K at the limb (the effective temperature
being intermediate,  5800 K).
In the visible at 5010A, I(0,0)= 4x108 ergcm-2s-1A-1.
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Solar limb darkening 3
(Adapted from
Pierce and
Waddell, 1961)
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The solar limb darkening observed at
wavelengths from 3000 A to 2.6 m, for
several values of cos from 0.2 ( = 79°)
to 0.9 ( = 25°). The previous first
approximation requires a correction. For
instance, at  = 5010, the observations
provide: a = 0.26, b = 0.87 plus a smaller
term, which can be expressed as = –0.13
cos2 , which we will consider further at a
later stage.
Notice the irregularities in the curve (e.g.
at 3600 A and 13000) due to sharp
variations in the absorption coefficient due
to H and H-. The limb darkening becomes
smaller in the near IR because the larger
optical depth allows to observe the regions
of minimum temperature.
From these observations we can determine
 (see also later).
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The radius of the Sun
As a consequence, the optical depth enters in the definition of the
‘radius’ of the star. For the Sun, a good approximation for the
decrease of density  with the height z above the surface is:
 ( z )  e z / H
where H  200 km. At about z = 3H, the density of matter becomes
extremely small. If we remember that at the Sun’s distance, 1 arcsec
corresponds approximately to 700 km, we see that only very good
seeing conditions will permit to detect a difference in solar radius
according to the wavelength.
It is also clear that the ‘radius’ of the Sun can have very different
values at wavelengths where the absorption coefficient is very
different from that at visible wavelength, for instance in the radio
domain the Sun has a much larger radius (approximately 1.8
times).
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The radiation flux - 1
In addition to the intensity, we wish to determine the flux, namely the total
amount of radiation leaving the unit area per unit time per unit bandwidth in all
directions, a quantity usually indicated with F (notice the factor  usually
entering in the definition, however not all authors have it, indeed this flux is
often referred to as astrophysical flux):
 F (  ) 
 I  (  ,  )    d
4
Recalling the azimuthal symmetry:
 F (  ) 
2

0
0
 I  cos  d   d  I  sin  cos  d 
4
1
1
1
1
 2  I  cos  d(cos  )  2  I  (  ,  )    d
From the mathematical point of view, notice that the flux is the first moment
of the intensity with respect to  .
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The radiation flux - 2
If the intensity were a strictly isotropic function, independent of  , the integral
would vanish: no net flux would be observed, in any direction (this is the case for
instance inside a cavity in thermodynamic equilibrium). Therefore the flux
measures the anisotropy of the radiation field.
To see this more clearly, let us split the integral in two parts, one for the radiation
going outwards ( < /2), and one for the radiation going inwards ( > /2):
1
0

 F (  )  2  I  (  )    d  2         F out  F in 
1
0 
1
1
In particular, at the surface of the star no radiation will enter from above:
1
 F (0)  2  I  (0,  )    d
out
0
(this would not be true for a planet, nor for a close binary star!).
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Average Intensity
In addition to the flux through a surface, we can define, in a given point inside
the atmosphere, the average intensity:
J 
1
4
 I  d

where the integral is extended to the effective solid angle of the source. Inside the
stellar atmosphere this angle  is 4. for an isotropic radiation field it would be :
J  I
This condition is fairly well satisfied in the deep interiors of the star, where the
temperature gradient is very small, but only very approximately so in the
photosphere.
From the mathematical point of view, the average intensity is the zero-th order
moment of the intensity with respect to  . Notice that J can be defined even
outside the atmosphere. For instance, the average intensity of the solar
radiation at the Earth is:
1
J 
 I 
4
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Average intensity, energy density and radiation
pressure
The average intensity is connected to the energy density u by:
4
1
u 
J    I  d
c
c
The energy density in its turn is connected to the radiation pressure, because
any photon of frequency h has an associated momentum p= h/c, and the
arrival frequency of photons on the walls of the column is:
nph  (I / h )cos dd
Therefore, the net impulse transferred by radiation to the volume element is:
1
2
P   I cos2  d 
c 4
c
1
2
I


  d
1
From the mathematical point of view, the radiation pressure is the 2-nd order
moment of the intensity with respect to  .
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Exit Flux and Temperature in LTE
At this point, we wish to determine how the exit flux through the surface is
connected to the effective temperature of the star. Let us introduce again the
hypothesis that S is a linear function of the optical depth. The intensity is then a
linear function of , and we obtain:
 F
1
out
2 
2


(0)  2   a  b      d   a  b    S    
3 
3


0
a most important result known as Eddington-Barbier relation: the flux that exits
the surface at each wavelength, equals the source function at an optical depth 2/3
at that wavelength.
In particular, in the LTE hypothesis S = B (T):


2
 
 
2 
 F out (0)   S       B T    
3
3
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
38
The gray atmosphere
If in addition, the absorption coefficient  could be assumed independent of 
(namely, if the stellar atmosphere could be considered a gray atmosphere), the
resulting outward flux would be that of a black body with the temperature at 
=2/3, and each linear coordinate z would have the same optical depth. A
particular way of defining an average absorption coefficient is Rosseland’s mean
(see next three slides). Since by definition the integral of the outward flux over all
wavelengths is proportional to the 4-th power of the effective temperature
(Stefan-Boltzmann law):

2
 F (0)    B (T (  ))d  
3
0
the most important result is obtained:
4
2 

4
T
(


)


T
eff

3 
2

Teff  T   
3

Although we have reached this conclusion using drastic approximations, however
the observations prove that the spectral energy of the Sun is reasonably similar to
that of a black body at 5800 K, which is therefore the temperature at  = 2/3.
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The Rosseland mean
We recall the main processes that give rise to the opacity in the continuum
and in the lines (details are available on word files, for interested
students):
•Photo-ionizazion by photon absorption from a bound to a free state (b-f);
the inverse process is recombination
•Scattering by free electrons (Thomson), atoms and molecules (Rayleigh)
•Absorption of a photon by an electron transition between two free
levels (free-free). It can take place only with the presence of a ion
(conservation of energy and momentum). The inverse process is also
called thermal bremsstrahlung.
•The negative H ion H•Resonant scattering in absorption lines of atoms and molecules
•Mie scattering by large particles (usually not considered in stellar
atmospheres)
•Raman scattering (inelastic) by molecules (again, usually not considered
in stellar atmospheres)
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The total absorption coefficient
Summing up all the contributions of the different processes applied for each
chemical species, and with proper weights that take into account the relative
abundances, one finally obtains the overall opacity of the gas having a given
chemical composition, e.g. the solar composition, a given temperature and a
given electron pressure. The process is legitimate, because opacities sum up,
and the total coefficient is simply the sum of the partial ones. However, the
calculation certainly it is not simple, especially if molecules have to be taken into
account. The figure shows examples for two different temperatures, one slightly
cooler than solar and one much hotter. The horizontal line is Rosseland mean
opacity, namely an average value of the opacity useful in calculating stellar
atmosphere models.
Because the calculations make use of Saha formula, the values of  depend on
the electronic pressure in the gas; the solar curve was computed with log Pe =
0.5, the hot star with log Pe = 3.5.
It is to be expected therefore that the importance of the several discontinuities
(e.g. at the Balmer limit) will be different for different luminosity classes. These
expectations are born out by the observations.
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Two graphs of the continuous absorption
coefficient
Left, a star slightly cooler than the Sun. Notice the importance of the negative
H ion (H-) in the visible and near IR. Right, a B0 type, whose opacity in the
visible is about 20 times larger than for a solar type star (from Unsold)
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The continuous absorption coefficient
The previous slide shows that the solar continuous opacity has a minimum in the
near IR at 16 micrometers (the matter is more transparent), while it is maximum
at radio frequencies and in the near UV going towards the Lyman limit. Notice
how different is the situation for hotter stars. We can also say that photons come
out of the photosphere from very different regions, according to their wavelength
. If at that  the matter is transparent, we see deep in the atmosphere, if opaque,
we see only the outer layers. Roughly speaking, photons of a given  are the
result of absorption and emission processes taking place in regions extending
from   100 to   0.001, the value   1 being a very useful indicative value.
The geometrical radial coordinates z1 and z2 of the atmosphere therefore are such
that:

      dx  100
in the most transparent wavelength
z1

      dx  0.001
in the most opaque wavelength
z2
The treatment of the spectral lines would require a much more careful discussion.
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Temperature and Optical Depth - 1
Let us consider again the fundamental equation of the radiative transfer in
LTE:
dI  ( ,  )
 I  (  ,  )  S (  )
d / 
which we want to solve in order to determine the source function at each
optical depth.
To do so, let us examine again the consequences of the initial assumptions
that energy is transported by radiation only, and that ETL is satisfied, namely
that each layer maintains a well defined T, and that the temperature must
decrease outwards.
We shall simplify furthermore the problem assuming that S = B (so that
source function is equivalent to temperature), and that the atmosphere is gray.
(This approach is of course a very drastic approximation of the real
atmosphere).
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The radiative equilibrium
These requirements amount to say that the (bolometric) energy flux must remain
constant with the depth, or else:
Notice two points:
- the condition of thermal (or radiative) equilibrium is not equivalent to
thermodynamic equilibrium: only the overall flux remains constant with z, not
the temperature, proceeding outwards the radiation color becomes redder and
redder.
- Each depth z has the same temperature T, but its optical depth depends on the
wavelength. Only for the grey atmosphere each z has a unique optical depth .
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Temperature and Optical Depth - 2
At this point, multiply the fundamental equation by  , and integrate over all ’s
and over all ’s. This procedure gives (as shown by Eddington, see file word for
the function K and the demonstration):
J SB
Therefore, in the gray atmosphere, the source function equals the mean intensity.
But it is also:
so that finally:
a result which is is known as Milne-Eddington equation.
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The Milne-Eddington equation
To determine the value of the constant, consider again the equation:
where T0 is the value of the temperature at the boundary  = 0.
The total emergent flux is twice that of a black body at temperature T0 (because
the inward flux must be 0):
so that finally:
With more general assumptions, one finds:
with q slowly varying between q(0) = 0.58 and q() = 0.71.
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Solar Photospheric Source Function S()
The solar photosphere is more complex that the simplified model discussed so
far, because the observations prove that the limb darkening requires at least a
second order term in  . However, it can be demonstrated (see Exercises) that
if:
Therefore, the main result remains valid, that
by measuring the limb darkening law we can
derive the source function at each optical
depth. The figure shows the result for the
continuum at 5010 A taking into account the
second-order term (in2 ) in Pierce and Waddell
data.
For  > 1.5 the data become very uncertain.
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Solar Photospheric Temperature: T()
In the LTE assumption, source
function is equivalent to
temperature, which can thus be
derived by the same procedure.
Notice that at this stage we
have only T(), not T(z), in
this particular figure in the
continuum at  = 5010 A.
Our assumptions though
require that each layer z in the
parallel stratified atmosphere
has a unique temperature.
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Solar Photospheric temperature (T)
Therefore, we calculate a family
of curves for each measured
wavelength, as in this example.
A given temperature (in this case
6300K) must belong to the same
height z.
This method gives us the
empirical observational mean to
determine the different optical
depths corresponding to the same
geometrical position. The optical
depth at 5010 is slightly smaller
than at 3737 A, and decidedly
smaller than at 8660 (see the
abscissae).
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Determination of the absorption coefficients 
Recalling that the optical depth is the integral of the linear absorption
coefficient  over the geometrical depth z, we can therefore derive:
- the relative ratios of all linear absorption coefficients  /0
-the function  (T) , as
was done for the first
time by Chalonge and
Kourganoff in 1946 (see
figure), confirming the
validity of Wildt’s
assumption of the
importance of H- as main
source of visible opacity.
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Temperature of the outer photosphere
The real limit of the previous
discussion is reached when we
encounter the chromosphere.
Going from high to small optical
depths (right to left), the
temperature decreases until it
reaches a minimum aroud 4300
K, and then increases again
toward the chromospheric
values (above 10000 K).
Notice the differences in the
different theoretical models.
To explore the upper
photosphere we can use the far
infrared (100-200 micron), or
the UV around 1600 A.
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Temperature of the deeper photosphere
Finally, this figure
shows theoretical
results for the deeper
photosphere. Notice
the differences
among the different
authors.
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The hydrostatic equilibrium condition - 1
To derive other values of the physical conditions in the solar photosphere, let
impose a condition of hydrostatic equilibrium for a spherical, non rotating
gaseous star: the variation of pressure P with depth r will equal the gravitational
attraction of the matter inside r, through the differential equation:
dP ( r )
M (r )
 G  (r ) 2
dr
r
The gas pressure can be expressed in terms of density and temperature by
the perfect gas law:
Pg  NkT  k
T
T
 0.825 108
 mH

where mH is the Hydrogen atom mass, and  is the mean molecular weight. If
the gas were composed only by ionized H,  would be equal to 0.5; for He II,
 = 4/3; heavy metals of charge Z produce Z+1 particles, and we can assume
their atomic weight equal to 2Z, so that  = 2. Finally:
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The hydrostatic equilibrium condition - 2

1
2X  3Y / 4  Z / 2
(where X, Y, Z are the number density of H, He and metals respectively),
which for a typical fully ionized stellar mixture, like the solar corona, or the
deep interior, provides   0.6.
Let us introduce the surface gravity
where the scale height H is function of  and T. In the solar photosphere at 6000
K, the matter is certainly not fully ionized, in particular H and He are essentially
all neutral, and only metals provide electrons, so that:
  1, and H  200 km (solar photosphere)
The density therefore must be   1017.5 atoms/cm3, and the electron density Ne 
10-4  1013.5 electrons/cm3.
These must be roughly the values at optical depth  = 1.
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Solar temperature and density as function of
geometric depth in the photosphere
These two figures give another (and very
schematic) representation of the behavior of
temperature and density with the linear
coordinate z . Zero is the conventional
bottom of the visible photosphere.
Using an average optical depth, the
geometric thickness of the photosphere is
approximately 400 km between 6000 and
4400 K (photosphere). A thickness of 500
km give an optical depth vis  10.
The density is obtained by the average
opacity per gram of matter.
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Examples of emission line spectra
We give now some examples of spectra differing from those
usually encountered on normal stars. The previous
discussion has shown that there are two limiting cases
leading to the formation of an emission line spectrum,
namely:
1. A gas optically thin in the continuum, with no
background light, such as the solar chromosphere and
corona seen outside the disk, or emission nebulae
2. an optically thick volume of gas with the source function
increasing toward the exterior, such as the solar spectrum
below 1600 A.
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The chromosphere
The thickness of the chromosphere is
approximately 3000 km from T = 4300 K to T
= 30000 K as derived from direct spatial
resolution during a solar eclipse (upper figure).
The main limitation of spatial resolution from
ground observations can be derived by the
following considerations: at the distance of the
Sun, 1 arcsec equals approximately 700 km,
which is more or less the geometrical
thickness of the photosphere. Therefore, very
good atmospheric seeing conditions, and very
good optical quality of instruments, are
required.
The figure on the left shows the steady
decrease of matter density with the height h,
and the sudden increase in ionization
(temperature) at h 500 km
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The ionization of the solar upper
atmosphere
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The spectrum of the solar corona
Some
visible
lines:
4471 He I
4686 He II
4713 He I
H-beta
5303 Fe XIV
5876 He I
6374 Fe X
H-alpha
6678 He I
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The solar spectrum below 1600 A - 1
Strong lines:
Ly- 1216 A
C II 1336 A
Si IV 1406 A
C IV 1550 A
O III 1660 A
From a photographic rocket spectrum (the emission lines are black).
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The solar spectrum below 1600 A - 2
At 6000 K the photosphere does not radiate much UV, the entire
emission below 1500 A is 1/20 of that in 1 A at 5000 A. Therefore,
we see only chromosphere and corona when we look at the Sun in
those wavelengths (the lines seen there are resonance lines, the most
important in the spectrum).
The opacity increases as we go into the UV, so our line of sight
terminates higher in the photosphere, until at 1800 A we reach the
temperature minimum of about 4000 K (which is the color
temperature of the radiation at 1800 A).
The emission lines seen at  1800 A come from higher, hotter
regions: the exponential increase of excitation with temperature
overweighs the falloff in density, resulting in emission lines
(absorption lines could be present but have not been detected below
1500 A).
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The solar spectrum below 1600 A - 3
The intensities of the UV lines are determined by excitation
conditions, abundances, and atomic peculiarities.
HI Ly- 1216 A is as strong as all the other UV lines put together
He II Ly- 304 A is as strong as all UV lines below 500 A
Because of the low density, collisional ionization is not balanced
by their counterparts, namely triple collisions. As a result, the most
common ions have ionization potentials five or ten times higher
the thermal energy.
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Solved exercize on the solar corona
Visual observations taken during solar eclipses show a faint corona starting at the
solar limb (conventionally, at 1.003 solar radii R0 from the center of the disk) with
an intensity of 10-5 that of the disk, and dropping to 10-8 after 1 solar radius.
Assuming that the main
mechanism of opacity is Thomson scattering, determine
R
the volume and column density of free electrons.
Solution: in a drastically simplified discussion, the light we see is scattered to  
90°, the Thomson cross-section is 3.3x10-25 cm2 , the total column is about 1R0 
7x1010 cm, so that:
We can also derive the scale height H by the condition of hydrostatic equilibrium:
where T  1.2x106 K,  is the mean molecular weight ( 0.6), mH the mass of
the proton, and g the surface gravity.
Further notions will be given on the chapter on the Sun.
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Spectra of Planetary Nebulae
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Short- ward of the
Balmer limit, starts a
faint blue continue which
has its peak around 2400
A. This continuum is due
to two-photon emission,
an H I mechanism
discovered by M.
Goppert-Mayer in 1932:
between n =1 and n = 2
there might be a
'phantom' level giving
rise to two UV
continuum photon instead
of one Ly-. The sum of
energies of the two must
equal that of Ly-.
65
The far IR (ISO) spectrum of NGC 7027
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Exercises
1 - Calculate the term
for I,0 = B (5900 K), and S = B (4500 K) and S = B
(10000 K) respectively, in the interval 3000 A <  < 10000 A.
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Stellar Atmospheres
Literature
•S. Chandrasekhar, Radiative Transfer, Dover
•D. Mihalas, Stellar Atmospheres, W.H. Freeman, San Francisco
•A. Unsöld, Physik der Sternatmosphären, Springer Verlag (in
German), and The New Cosmos
•E. Bohm-Vitense, Introduction to stellar astrophysics, vol. 2,
Cambridge Un. Press
•L. Gratton, Introduzione all'Astrofisica, 2voll. (in Italian),
Zanichelli
•R. Rutten, Lecture Notes Radiative Transfer in Stellar Atmospheres
http://www.fys.ruu.nl/~rutten/node20.html
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