Stellar Atmospheres

1
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
2
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 color-magnitude 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 non-equilibrium states.

3
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.
4
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.
5
Energy transport mechanisms

• A more physical approach to this subdivision
  would make use of the main energy transport
  mechanism in each region
• 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.
• in the chromosphere, there is also dissipation of
  waves (acoustic, magneto-hydrodynamic)
• 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.

6
Table 1- Solar Energy Output
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.
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.
7
Basics on Transport of radiation - 1
Consider a surface area of size , having normal
n, and an elementary solid angle d? in direction
(?, ?), where
The light passing through d? in the unit time, of
wavelength between (?, ?d?), carries an energy
E? that can be written as
(dimensions erg/s)
where the factor cos? comes from the surface
projection effect.
8
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,zl,m,nt) 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!

9
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.
10
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)
where ?? is a linear absorption coefficient
(cm-1) appropriate to that particular gas.
Emission On the other hand, each elementary
volume dVd? ?ds of gas will emit, contributing
some energy to the beam according to an
appropriate emissivity coefficient ??
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.
11
Transport of radiation - 5
Therefore, the total energy variation can be
expressed as
The intensity variation along the beam is then
Notice that the sign of dI?/ds is determined by
that of the difference
12
Optical depth and Source Function
Let us introduce now the a-dimensional variable
elementary optical depth
The previous equation for the intensity variation
becomes
where the function S? is called the source
function. The total optical depth of the column
is obtained by
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.
13
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
Under these conditions, the intensity of the
radiation (and so also the source function) is
expressed by Plancks function B?(T)
The second equality is another way of expressing
Kirchoffs 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.
14
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 Plancks 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.
15
Local thermodynamic equilibrium -2
The previous expression can be written as
which can be integrated over the interval
If moreover B? is assumed constant along the
path (as in laboratory conditions), the intensity
at the exit face of the column will be
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.
16
Case 1 no input radiation
No input radiation means I?(0) 0, the column of
hot gas shines with intensity given by
This case can be subdivided in two limiting
situations 1.1 when the optical depth ?? (s) is
very small (optically thin gas), then
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
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.
17
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) gtgt 0 (this
would be the case of a stellar atmosphere).
Again, two limiting cases can be considered 2.1
optically thin case
If the sign of the second term is negative (I?(0)
gt 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) lt B?), emission
lines, superimposed over the entrance continuum,
are observed where ?? is larger (see next
slide). 2.2 optically thick case
independent from the entrance spectrum.
19
Case 2.1 - Emission lines
In case 2.1, if the sign is positive (I?(0) lt
B?), emission lines, superimposed over the
entrance continuum, are observed. This is the
case for instance of the solar spectrum seen at ?
lt 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 doesnt come from the ionization, but
simply by the lines being in emission, instead
than in absorption as in the visible region.
20
Real stars
In the visible region, the stars usually show an
absorption line spectrum, they must therefore
correspond to the case I?,0 gt 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.
21
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.

22
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).
23
The radiative transfer equation - 2
As in the previous considerations, along the path
ds inside the atmosphere the following equation
will be satisfied
but
where
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).
where the quantity cos? is usually designated
with ?.. Notice the change in sign and the
presence of cos? with respect to previous
discussion.
25
The radiative transfer equation - 4
The radiative transfer equation for the plane
parallel case with azimuthal symmetry is then
In order to derive the intensity of the radiation
exiting the surface in a given direction ?,
namely I?(0,?), multiply both sides by
and obtain
26
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
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.
27
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
(this assumption can be seen as the result of the
usual technique of expanding a function in
Taylors series and considering the first order
only, but later on we will justify it on the
basis of Eddingtons approximation). We then get
the following result (using cos? for clarity)
which is also written as
As already said, in the LTE hypothesis S
coincides with the Plancks function B at a
z-dependent temperature
28
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.
29
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.

30
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
The radiation coming from the borders of the
solar disk leave the surface at ? 90, so that
The observations prove that we see less light
from the border, a fact named limb darkening. See
the following figures. We then conclude that
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.
31
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
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).
(Adapted from Pierce and Waddell, 1961)
33
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
where H ? 200 km. At about z 3H, the density of
matter becomes extremely small. If we remember
that at the Suns 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).
34
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)
Recalling the azimuthal symmetry
From the mathematical point of view, notice that
the flux is the first moment of the intensity
with respect to ? .
35
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 (? lt ?/2), and one for the
radiation going inwards (? gt ?/2)
In particular, at the surface of the star no
radiation will enter from above
(this would not be true for a planet, nor for a
close binary star!).
36
Average Intensity
In addition to the flux through a surface, we
can define, in a given point inside the
atmosphere, the average intensity
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
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
37
Average intensity, energy density and radiation
pressure
The average intensity is connected to the energy
density u? by
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
Therefore, the net impulse transferred by
radiation to the volume element is
From the mathematical point of view, the
radiation pressure is the 2-nd order moment of
the intensity with respect to ? .
38
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
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)
39
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 Rosselands
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)
the most important result is obtained
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.
40
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)

41
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.
42
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)
43
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
in the most transparent wavelength
in the most opaque wavelength
The treatment of the spectral lines would require
a much more careful discussion.
44
Temperature and Optical Depth - 1
Let us consider again the fundamental equation of
the radiative transfer in LTE
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).
45
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 ?.
46
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)
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.
47
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.
48
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 ? gt 1.5 the
data become very uncertain.
49
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.
50
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).
51
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 Wildts
  assumption of the importance of H- as main source
  of visible opacity.

52
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.
53
Temperature of the deeper photosphere
Finally, this figure shows theoretical results
for the deeper photosphere. Notice the
differences among the different authors.
54
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
The gas pressure can be expressed in terms of
density and temperature by the perfect gas law
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
Z1 particles, and we can assume their atomic
weight equal to 2Z, so that ? 2. Finally
55
The hydrostatic equilibrium condition - 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.
56
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.
57
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
• 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
• an optically thick volume of gas with the source
  function increasing toward the exterior, such as
  the solar spectrum below 1600 A.

58
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
59
The ionization of the solar upper atmosphere
60
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
61
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).
62
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).
63
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.
64
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 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.
65
Spectra of Planetary Nebulae
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-?.
66
The far IR (ISO) spectrum of NGC 7027
67
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 lt ? lt 10000 A.
68
Stellar AtmospheresLiterature

• 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