3 Four macroscopic laws

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3 - Four macroscopic laws

• Planck
• Boltzmann
• Saha
• Maxwell

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Planck law
The black body is an ideal body of uniform
temperature T(K) that perfectly absorbs all
incident radiation, and emits isotropically
unpolarized thermal radiation according to
Plancks law. It is customary to indicate with
B?(T) or B?(T) the specific intensity of such
body. In wavelength units, the emittance (over
the whole hemisphere) is
(ergcm-2s-1cm-1, ergcm-2s-1Hz), where h is
Plancks constant, h 6.57x10-27 erg?s-1 , and k
is Boltzmann constant, k 1.38x10-16 erg?s.
Expressing ? in cm 2?hc2 c1 3.742x10-2
erg?cm2?s-1, hc/k c2 1.439 cm?K   where c1
and c2 are the designations found on several
textbooks.
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Graphical Representation
The Black-body curve for some temperatures of
astrophysical interest. In a linear scale Wien's
displacement law (?maxT const) is well
visible. Notice that at high temperature and low
temperature, the peak of emissivity is outside
the visible range.
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Wien's law
The wavelength of the maximum value of B?(T) is
inversely proportional to the temperature and is
given by
The wavelength of the maximum value of B?(T) is
given by
because ?max is not c/?max. The flux at the peak
of emissivity is
which, for ? in microns and T 10000 K,
corresponds to
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The Sun as a Black Body
The overall spectrum of the Sun resembles that of
a Black Body at T 5800 K, but with a
significant decrease in the UV.
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Logarithmic representation

• The Black Body function in a logarithmic
  representation.
• The following properties can be seen
• The decrease at ?lt ?max is very steep, the flux
  radiated at ?maxlt?max is only 1/4 of the total.
• Below 0.17?max there is a fraction of only 10-10
  of the total.
• 98 of the energy is radiated between 0.5?max and
  8 ?max.

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Bolometric emittance
By integrating the law for ? from 0 to ? we
obtain the omni-directional emittance
(Stefan-Boltzmann law)

• We call more properly intensity B the emittance
  inside the unit solid angle. The constant becomes
  then ?/? (not 2?!). Some indicative values
• At 'normal' temperature T 290 K (17 C), ?max
  10 ?m, the intensity is B 128 Wm-2sr-1, and
  the emittance is F ?128 401 Wm2. The photon
  emissivity maximum is at ?max 31.1 ?m
• At 10 times higher temperatures (appropriate to
  cool stars), ?max 1 ?m, the intensity and
  emissivity raise 104 times to reach 4 and 1 MWm2
  respectively only 1 of the energy is emitted at
  ? lt 5000 A or at ? gt 8 ?m. Approximately 10 is
  radiated in the whole visible bands.
• By raising the temperature to 29000 K (hot stars,
  B type), ?max moves to 1000 A (below Lyman's
  head), and the emissivity goes to 1010 Wm2. The
  energy radiated in the visible is a small
  fraction indeed.

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Planck's law as photon emissivity
It is also useful to express Plancks
distribution as photon emissivity
(notice the exponent 3, not 4!) with
The wavelength of maximum photon emission, and
the corresponding photon flux are
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Two asymptotic expressions
The Planck function has two important asymptotic
expressions. At high frequency, or better, when
which is Wiens distribution, valid in the X- and
?-ray domain, but sufficiently correct also in
the visible for stars as hot or hotter than the
Sun. At long wavelengths, or better, when
which is the Rayleigh-Jeans law, valid in
particular in the radio-frequency domain.
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Practical realization of the black body
The black body is clearly an abstraction. It can
be realized with a thermostated cavity at
temperature T with a small opening (small with
respect to the radius of the cavity) to let the
radiation exit to the outside. The radiation
density inside the cavity is given by
where a 7.57x10-15 erg?cm-3?K-4.
If we would shine a beam of radiation inside the
cavity, we would not be able to see any detail of
the walls, nor any difference in color. Even if
we were inside the cavity, we still could not see
any detail, only an uniform radiation from any
direction, even using any arbitrary system of
lenses or mirrors. A thermometer placed inside
the cavity would always measure the same
temperature T whatever the medium inside the
cavity, gas, liquid or even vacuum.
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Boltzmann law - 1
Consider a large number of ions with two internal
energy levels. At thermal equilibrium, the
relative populations of the two levels is given
by Boltzmann's law
where k Boltzmann constant 1.38x10-16
erg/deg gn, gm are the statistical weights of
the two levels ?Enm Energy difference between
the two levels ? frequency of the photon
emitted in the transition
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Boltzmann law - 2

• An equivalent formulation is the following

(?Emn in eV), or else in logarithmic form
where ? 5040/T .
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Boltzmann law - 3
In particular, let us refer the population of the
i-th level to that of the ground state
Where now Ei is the excitation potential of the
i-th level above the ground state. By summation
over all i's we obtain the total population of
that atom
U(T) is called the partition function of that
atom (or ion).
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Some partition functions
For most cases, U? g1 independent of T, but
there are exceptions, e.g. for Na I, Ca, Fe.
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Limiting cases - 1
Consider for instance the closely spaced
D-doublet of Na I, the fundamental level is
single, the level 32P3/2 has statistical weight
4, the other one 32P1/2 has statistical weight 2,
so that we expect that the ratio between the
intensities of the two lines is 21. This case
applies for instance to the emission lines of Na
I in the lunar atmosphere. In the absorption
solar spectrum (here reflected by the Moon) the
ratio is approximately 1, because of the
saturation of the absorption lines, a first
example of the influence of the optical depth of
the gas. Notice also the telluric absorption
lines.
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Limiting cases - 2
If it happens that
then the coefficient of absorption becomes
negative, and the intensity increases along the
beam. This is the astrophysically important
situation of molecular masers emitting at
radiofrequency, e.g. the OH maser.
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Saha law
Suppose Ai1 and Ai are two successive states of
ionization of a given atom. The electrons of the
two ions will be essentially all in their ground
level. At the equilibrium we must have Ai1e- ?
Ai. Therefore the electron density must figure
explicitly in the equilibrium equation
where gi, gi1 are the statistical weights of
the fundamental levels of the 2 ions, and Ei is
the ionization potential of the i-th ion.
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Saha law - 2
Alternative formulations are
In the second equation, use has been made of the
perfect gas law applied to the electron component
The pressure is expressed in barye dyn?cm-2 (1
dyn?cm-2 10-6 bar 0.1 pascal)
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Saha law - 3
To obtain an intuitive demonstration of the
formula, consider the case neutral plus singly
ionized atom plus free electron. The electron
occupies the volume (2?mekT) in the impulse
space, and the volume 1/Ne in the Cartesian
space. In the phase space its volume is therefore
the product of the two expressed in units of h
and multiplied by 2 because of the two possible
spin states
The overall statistical weight is therefore that
of the free electron times that of the ground
level of the ion, g1. A better agreement can be
obtained by lowering the ionization potentials in
order to take into account the collisions in the
higher levels, but usually this is a small
correction.
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Ionizazion Potentials (eV)
21
Examples of Saha law
Ionization equilibrium between neutral and
ionized Calcium at solar photospheric conditions
of temperature and pressure. The inset shows the
situation of Iron in the solar corona.
22
An overall view of ionization in stars
Notice that at a given stellar temperature each
atom is essentially in one or at most two
ionization states.
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Extension to molecules
The extension to molecules is not
straightforward, because in a molecule the
internal energy is divided into three possible
forms, electronic, vibrational and rotationa. At
any rate, in simple cases and for a diatomic
molecule we can write an equation of the type
where A and B are the two atoms forming the
molecule, and D is the dissociation energy. More
in general, we have a law of mass action
(Guldberg and Wage).
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Diatomic Molecules
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Polyatomic Molecules
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Maxwell's law
In this thermal equilibrium conditions, the
speeds of each type of particles are distributed
according to the Maxwell function
where N nr. of particles per cm3 , and f(v)
gives the fraction with speed between (v, vdv).
The peak of the distribution (therefore, the most
probable speed) is at
Two other typical velocities are found useful
which are respectively the mean velocity, and the
equipartition velocity.
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Graph of Maxwell law - 1
The three different 'typical' velocites.
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Graph of Maxwell's law - 2
a is a parameter inversely proportional to
T. D(x) is the cumulative distribution, namely
the integral from 0 to x.
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Relationship Temperature - pressure
The equation of state of a stellar atmosphere
will be described by that of a perfect gas (an
assumption which has to be taken with great
caution in a planetary atmosphere)
where in Ni all ionization stages (including
neutral) are included. In a stellar atmosphere,
all matter is essentially composed by H and He,
with traces of other elements (collectively
indicated with Z) however the latter can
contribute an appreciable number of free
electrons, so that
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Two limiting regimes
all H is essentially ionized, so that
quite independent of the precise chemical
composition of the atmosphere.
only metals are ionized, so that
which is sensitive to the chemical composition
(although in precise considerations the
contribution of molecules should be included).
The Sun is therefore in an intermediate situation.
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Ionization and surface gravity
According to Saha equation, at any given
temperature the ionization degree depends only on
the electronic pressure. Let us consider two
stars having equal effective temperatures and
masses, but very different radii the smaller one
will be called a dwarf, the second a giant. The
first has a greater surface gravity (and
therefore greater pressure) than the second one.
In the dwarf, the ionization degree will be lower
than in the giant, so that if we judge
temperatures by ionization (namely by comparing
the relative intensities of lines of the same
elements in two different ionization stages), the
dwarf would need a higher temperature to reach
the same ionization degree. This fact introduces
a strong complication in the interpretation of
the stellar spectra, and we owe to Fowler and
Milne (around 1923), a clarification of the
situation. Table 19. 5 gives the electron
pressure (in barye) in the atmospheres of dwarfs,
giant and supergiant stars, respectively.
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Table of electron pressures
33
Ionization degree in the average stellar
atmosphere
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Variables in the spectral spectrum
We conclude that an accurate spectral
classification must depend on at least two
fundamental parameters, namely temperature and
pressure (or surface gravity, or radius, or
luminosity), so that it must be a two-dimensional
scheme (all other variables, such as chemical
composition, being ignored in this context), and
it must permit the determination of both
variables. This implies that for proper
classification, the lines of several elements,
having different ionization potentials, must be
simultaneously taken into account. Furthermore,
the knowledge of the pressures translate into the
knowledge of the radii, namely of the
luminosities, and of the distances through the
intermediate of the spectrum (spectral
parallaxes). Other variables might enter into the
classification, for instance the chemical
composition, the magnetic fields, the rotation,
but finer observations are needed.
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The solar spectrum in colors

• The continuum from the blue to the red, with the
  strongest absorption lines.

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Some identifications on the solar spectrum
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Abundances of the elements in the Sun (by number)
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Abundances per number and mass

• Name number mass electrons
• H 100 100 100
• He 9.8 39 20
• C,N,O,Ne 0.15 2.2 1.1
• Other 0.01 0.4 0.21

Abundance by mass X (fraction of H) 0.71 Y
(fraction of He) 0.27 Z (all the rest)
0.02
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Temperature effect along the Main Sequence(from
A0-V to F5-V)
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Temperature effect along the Main Sequence(from
K5-V to M4.5-V)
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MKK Luminosity Effect at A0
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Luminosity effect at A0 (Gray Atlas)
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The Inglis-Teller law
the number of visible Balmer lines in the spectra
of A (and also B) stars is connected to the
electronic density by the Inglis-Teller law
where n is the number of the last visible line.
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Luminosity Effects at G0 (MKK)
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Luminosity Effect at G0 (Gray Atlas)
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Luminosity effect at M2
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Exercises
1 Apply Boltzmanns excitation formula to the H
I atoms in the solar layer having T 5800 K to
calculate N2/N1. Solution The numerical values
are ?12 10.16 eV, g1 2, g2 8, therefore
N2/N1 ? 5.9x10-9. Most of the H I atoms remain at
the fundamental state, nevertheless the Balmer
series can be observed in absorption, because of
the high number of H I atoms in the column. 2
Apply Sahas formula to determine Na II / Na I if
T 5800 K and Pe 10 barye. Solution from the
relevant tables, we can derive Na II / Na I ?
2460, only a very small fraction of the Na in the
solar atmosphere is therefore neutral, and yet
the Na-D doublet is one of the strongest
features. 3 Apply Boltzmann and Saha formulae
to compare Nn2(H I) and N(H II) at several
temperatures (e.g. T 50000 K , T 10000 K, T
4000 K). Check with T ??. 4 - Calculate the
number of HI atom in a column of 1 cm2 section
and 10 kpc long. How many of them will radiate
the 21cm line every second if the life time is
107 years?