4 Processes of formation of the continuous spectrum

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4 - Processes of formation of the continuous
spectrum
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Processes of formation of the continuous spectrum

• Among the processes that contribute to the
  stellar we consider the following
• Photo-ionizazion by photon absorption from a
  bound to a free state (b-f) the inverse process
  is recombination
• Scattering by free electrons (Thomson)
• 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.

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The bound-free processes
The electron bound to the atom in level n,
absorbs a photon of energy h? greater than the
ionization energy from that level, and is freed
with a velocity vgt0. These are the so-called
bound-free (b-f) transitions, or photoionization,
for which the following relationship applies
where ?ion and ?n are the ionization energy of
the ground level and the energy of the n-th level
above the ground. If the inverse transition
occurs, the electron is captured by the ion, with
the emission of the corresponding photon
(free-bound process, f-b, or recombination). The
process is therefore essentially discontinuous,
photons cannot be absorbed unless their frequency
satisfies
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The b-f coefficient for H and He II
For the hydrogen-like atom, the bound-free
continuous absorption coefficient per atom was
calculated by Kramers (1923) on classical
grounds, and then by Gaunt in 1930 with quantum
mechanics. The result is  if ? ? ?n ,
?n 0 if ? gt ?n which is related to the
classical cross-section of the Bohr atom through
a correcting quantum-mechanical factor g
(Gaunt-factor), which is essentially 1 (there are
however cases where g gtgt1) ?n is the wavelength
of the edge of the Lyman, Balmer, Paschen, etc.
series. Numerically, we would expect that at the
Lyman limit cm2,
where a1 is the first Bohr radius. The correct
calculation shows that actually the value is 10
times smaller. For the second and higher levels,
the factor n5 reduces the cross sections by 32,
243, 1024, times respectively however, the
factor ?3 more than compensates for this decrease.
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The influence of Boltmann and Saha equations on
the b-f opacity of H and He
If we now wish to calculate the total opacity ??
due to the b-f mechanism at a specific wavelength
?, we have to sum over the number of H atoms in a
given level n. At this stage, we must take into
account that the population depends on the
temperature according to Boltzmann law, and
possibly also according to Saha law, in all cases
with a much smaller number of atoms in n 2, n
3, etc., than in n 1, as we have seen before.
Therefore, in a bi-logaritmic plot (log?n, log?)
we expect a linear increase of log?? with log?
with slope 3 until ?Lyman, then a discontinuous
jump to much lower values (say ten orders of
magnitude at ? 1.0, three orders of magnitude
at ? 0.2), followed by another linear increase
to ?Balmer, and so on. The same considerations
can be used for hydrogen-like atoms, with due
allowance of the Z value. Because the radii
decrease with Z, the absorption coefficients will
decrease with Z2. For He II the behavior of ??
will be similar to that of H I, with a lesser
dependence on T because of the much higher
excitation levels.
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b-f Opacity for H I and He II
? is the parameter 5040/T
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The (negative) H- ion
We have shown that at solar temperatures H I has
a very low absorption coefficient, essentially
due to H in n3, namely to Paschen continuum.
What other elements can contribute to the high
observed opacity, so evident for instance in the
strong limb darkening? Answer (Wildt, 1939)
the ion H-, namely a proton plus two bound
electrons. The second electron is very weakly
bound to the nucleus, and a little energy (red
photons) is sufficient to remove the second
electron and neutralize the H. No excited level
of H- is known. Both electrons are in n 1, one
with spin up and one with spin down, so that the
statistical weight is 1.
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The H- opacity
To compute the opacity of H- we must apply Saha
equation by interchanging the roles of ion and
neutral. Using Boltzmann equation we can show
that, in solar conditions, N(H in n3) ltlt N(H-)
In solar conditions therefore H- provides the
wanted opacity in the visible by means of
photo-neutralization in the IR it contributes
to the opacity by means of f -f - transitions.
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The b-f opacity of metals
Although the metal abundance is small,
nevertheless they are easily ionized, and each
atom provides many electrons. Let us consider
for instance Fe, whose abundance is about 10-4
that of H, and at solar temperatures is mostly
ionized. By taking into account the ionization
thresholds, we can easily compute that at short
wavelengths, say 2000 Å, Fe I is much more
efficient than H-, and so on. Therefore metals
in general will produce high opacities in the UV
of intermediate temperature stars, with
noticeable discontinuities in the absorption
coefficient.
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Thomson scattering
If a free (non-relativistic) electron interacts
with a photon without the presence of a third
body, the incoming photon will simply be
deflected to another direction without changing
its energy (Thomson scattering). The
cross-section is given by
cm2
independent from the wavelength (Thomson
scattering by electrons does not modify the color
of the radiation). Averaging over all directions,
the cross-section becomes
cm2
Although in the solar atmosphere, electrons are
more abundant than H-, nevertheless the much
lower absorption cross-section makes them not
important in the UV, visible and even IR. The
situation reverses in high temperature stars,
where the H- are essentially non-existent, and
all matter is ionized. The same must occur in the
solar corona.
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Diffusion by free electrons
We now consider the interaction of free electrons
with radiation in the proximity of an ion.
Absorbing a photon, a free electron goes from a
state of velocity v1 to another one with velocity
v2 gt v1
There is an infinite number of those free-free
(f-f) transitions, and actually photons of the
same frequency can be absorbed by an infinite set
of starting velocities. An important point to
remember is that this processes must involve a
third body (namely an ion) in order to satisfy
the conservation of energy and momentum. The
inverse process, of emission occurring in a
transition of the electron between two free
states, is often called bremsstrahlung (radiation
by braking).
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The f-f process
Regarding f-f absorption processes, we have
already commented that they will be important
mostly at very long wavelengths, in particular in
the radio domain. Calculation show that the f-f
emissivity of the gas is given by     The
linear absorption coefficient can be obtained by
Kirchoff law   cm-1 (? in
?m)  
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Diffusion by bound electrons
In the classical picture of an H atom, we can
consider the electron as an oscillating charge
bound to the nucleus. This oscillating charge has
a set of natural frequencies (or better, of
eigenfrequencies) given by the
frequencies of the Lyman lines. Consider now an
incident electromagnetic field having frequency
the electron will be forced
to an oscillation with amplitude proportional to
By virtue of the fundamental relation between
acceleration and emitted power, it will radiate
an intensity given by     All resonant
frequencies will contribute to I? , according to
their respective oscillator strength fn (this is
the reason of the name).
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Rayleigh scattering
In the visible region , so
that the overall scattering coefficient can be
written as     which is called Rayleigh
scattering coefficient. The average resonant
frequency differs very little from
that of Ly?, because we have seen that the
oscillator strengths decrease very rapidly.
Therefore     where ?1 is approximately 1216 Å,
while ?(Ly?) 1215 Å. The efficiency is a very
steep function of ?. Rayleigh scattering is
important also for molecules, e.g. by H2 in cool
stars. We have discussed it in connection with N2
and O2 to explain why the Earth sky is blue.
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Resonant scattering
Eq. has another limiting regime,
when the emissivity becomes independent of the
wavelength, which is the case of the free
electron and Thomson scattering. Indeed, the
condition means that the
restoring force is essentially zero. If instead
there is a great increase in
efficiency, which becomes formally infinite at
the exact values of the eigenfrequencies, namely
at the resonant lines of the atom or molecule.
The precise treatment of the atomic spectral
lines or molecular spectral bands is however
outside the present scopes we are justified in
this by the small percentage of energy that
actually goes in the lines, compared to the
energy of the continuum. An important point we
have not discussed is the so-called radiation
damping which enters in the width of the line.
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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 meaning of the
horizontal line (Rosseland opacity) will be made
clear in a later paragraph it is an average
value of the opacity useful in calculating
stellar atmosphere models its value seems so low
in comparison with the true opacity because the
average is made in respect to the emergent flux
of radiation. Because the calculations make use
of the Saha formula, the values of ?? depend from
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. Right,
a B0 type, whose opacity in the visible is about
20 times larger than for a solar type star.
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Balmer discontinuity
The spectrum of ?2 Cet (top, the photoelectric
scan, bottom the photographic spectrum). Notice
the sharp drop in intensity at the Balmer limit.
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Exercises
1 apply Boltzmann and Saha formulae to compare
Nn3(H I) and N(H-) in solar conditions.