1 Four macroscopic laws

1
1 - Four macroscopic laws

• Planck
• Boltzmann
• Saha
• Maxwell

A short summary of the properties of these
important functions which will enter in the
properties of gases in Local Thermodinamic
Equilibrium (LTE).
2
Local Thermodynamic Equilibrium

• In the study of stellar and planetary
  atmospheres, the assumption of Local
  Thermodynamic Equilibrium (LTE) is described by
• The source function is given by the Planck
  function.
• Excitation equilibrium given by Boltzmann
  equation
• Ionization equilibrium given by Saha equation
• Electron and ion velocity distributions are
  Maxwellian

3
Plancks law - 1
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 (energy per
unit area per unit time per unit bandwidth) of
such body.
4
Plancks law - 2
Changing from wavelength to frequency
This equation gives also the energy emittance,
because E h?. The peak of emissivity B? is at
frequency ?max 5.88x1010 T (Hz)
The corresponding wavelength is given by
5
Planck's law as photon emissivity
6
Graphical Representation of B?- 1
The Black-body curve for some temperatures of
astrophysical interest. In a linear scale Wien's
displacement law (?maxT const, see next
slides) is well visible. Notice (on the right)
that at high temperature and low temperature, the
peak of emissivity is outside the visible range.
7
Graphical representation - 2

• The Black Body function in a bi-logarithmic
  representation.
• The following properties can be seen
• The decrease at ? lt ?max is very steep, the flux
  radiated at ? lt ?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.

8
Wien's law
The wavelength of the maximum value of B?(T) is
given by
because ?max is not c/?max.
9
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.
10
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.

11
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
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.
12
Two asymptotic expressions - 1
The Planck function has two important asymptotic
expressions. 1 - At long wavelengths, or better,
when
then
,
which is the Rayleigh-Jeans law, valid in
particular in the radio-frequency domain. It
tells us that in this condition the radiation is
proportional to the possible density of
photons. Notice that in radio and infrared
astronomy the unit Jansky (Jy) is often
used 1 Jy 1 erg ? s-1?cm-2?Hz-1
13
Two asymptotic expressions - 2
2 - At high frequency, or better, when
then
,
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. Thus, at short wavelengths, the intensity
fall off exponentially, more sharply than in the
infrared when all photon states are
populated. The distribution of the higher energy
photons thus reflect the Boltzmann formula (see
next slides).
14
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
  8.6174x10-5 eV/K
• 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
• The statistical weights are the number of
  indistinguishable states of the same energy in
  each level, or else the number of electrons
  permitted in such level without violating the
  Pauli exclusion principle.

15
Statistical weights for H

• More than one quantum state may have the same
  energy.
• The number of these for orbital n is the
  statistical weight, or degeneracy, gn.
• For H, orbital n has a statistical weight of
  gn2n2.
• The various permutations for n 1 and n 2,
  with statistical weights 2 and 8 respectively,
  are shown in this table. The other 3 quantum
  numbers
• (l electron orbit, ml electron angular
  momentum with mlltlltml, ms electron spin
  angular momentum 1/2) will be explained in the
  Chapter on the H atom quantum mechanical
  structure. Notice that the energies of the
  different states are (approximately) equal.

16
Boltzmann law - 2

• An equivalent formulation is the following

(?Emn in eV), or else in logarithmic form
where ? 5040/T .
17
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).
18
Partition functions of selected elements
For most cases, U ? g1 independent of T, but
there are exceptions, e.g. for Na I, Ca, Fe.
Well consider again this topics after a
discussion of Sahas equation.
19
Exercise
Q. In the Solar photosphere, what is N2/N1 for
hydrogen?

• For every 2x108 H atoms in the ground state, just
  one is in the first excited state.
• Q. At what temperature will equal number of atoms
  have electrons in the ground (n 1) and first
  excited (n 2) states?

A. 85400K a very high T is required for a
significant number of H atoms to have electrons
in the 1st excited states.
20
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 transient atmosphere (exosphere).
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.
21
Limiting cases - 2
If it happens that
then the level population is inverted with
respect to Boltzmann. Well see in a later
chapter that the linear 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 at 18 cm in
Orion. There are also infrared astrophysical
lasers, e.g. the CO2 laser around 9 and 10 ?m in
Mars and Earth atmospheres. Most likely, some
emission lines in the UV and Red region of the
spectrum of Eta Carinae are also excited by some
sort of astrophysical laser (APL) mechanism.
This possibility of having masers and lasers in
the visible domain is a very exciting one. It
could be the subject of the written part of the
final exam. See next slides for further details.
22
Masers and Lasers
A maser is a device that produces coherent
electromagnetic waves through amplification due
to stimulated emission. Historically the term
comes from the acronym "Microwave Amplification
by Stimulated Emission of Radiation", The
principle of the maser was described by N. Basov
and A. Prokhorov from Lebedev Institute of
Physics in 1952. Independently, C. H. Townes, J.
P. Gordon, and H. J. Zeiger built the first maser
at Columbia University in 1953. The device used
stimulated emission in a stream of energized
ammonia molecules to produce amplification of
microwaves at a frequency of 24 gigahertz. When
optical coherent oscillators were first
developed, they were called optical masers, but
it has become more common to refer to these as
lasers. The invention of the laser can be dated
to 1958 with the publication of the scientific
paper, Infrared and Optical Masers, by Arthur L.
Schawlow, then a Bell Labs researcher, and
Charles H. Townes, a consultant to Bell Labs. The
first successfully optical laser was (probably)
constructed by Maiman (1960).
23
Astrophysical Masers
The first astrophysical maser was discovered in
1965-68 in Orion Nebula, first the one form the
molecule OH at 18 cm, soon followed by masers in
H2O and SiO.
Two review papers M.Elitzur Masers in the Sky,
Scientific American, 272, No.2, 52 (Feb. 1995),
for radio masers C. H. Townes, Astronomical
masers and lasers, in Quantum Electron., 1997, 27
(12), 1031-1034
24
The OH radio maser - 1
The emission from the Hydroxyl molecule at 18 cm
wavelength is composed of four hyperfine lines at
1612, 1665, 1667 and 1720 MHz (see diagram).
25
The OH radio maser - 2
Why this OH emission is not a normal one? There
are 4 key elements 1 - the line ratio for a gas
in thermodynamic equilibrium is 1591, however
the observed intensity ratios are very different,
and even vary over time. 2 - A paradoxical
situation occurs if the gas is assumed in
thermodynamic equilibrium the emission profiles
of the OH lines as a function of wavelength
indicates a Doppler temperature of 50 K, however
the emission intensity is so strong that they
would correspond to a source temperature of
several 1015 Kelvin 3 - The emission originates
from extremely narrow point sources or highly
collimated beams. 4 - The emission lines are
polarized in a particular way, which depends on
time and which could not have been produced by
spontaneous emission. CONCLUSION Based on these
facts the radio astronomers conclude that the
microwaves are produced from amplification by
stimulated emission of radiation from an inverted
population. The molecules are pumped by infrared
light from nearby gas clouds that are collapsing
to form stars.
26
Infrared CO2 lasers onMars Earth
Non-thermal emission occurring in the cores of
the 9.4- and 10.4- ?m CO2 bands on Mars and Earth
(but also on Venus) have been identified as a
natural atmospheric laser. We use the acronym
APL to indicate astrophysical lasers. Notice that
APL ot necessarily due to a violation of
Boltzmann law, APL might exist without population
inversion.
27
CO2 lasers on Mars - 1
Spectra of Martian CO2 emission line as a
function of frequency difference from line center
(in MHz). The line is extremely narrow. The
emission peak is visible at resolutions R gt
1,000,000.
Blue profile is the total emergent intensity in
the absence of laser emission. Red profile is
Gaussian fit to laser emission line. Radiation is
from a 1.7 arc second beam (half-power width)
centered on Chryse Planitia. (adapted from Mumma
et al., 1981, 1983)
28
CO2 lasers on Mars - 2
Population inversion for the (0 00 1) and (1 00
0) states of the CO2 molecule. The emission is
excited by absorption of solar flux in the
near-IR CO2 bands, followed by collisional
transfer to the 001 state of CO2. Notice the
intervention of the N2 molecule (see also next
slide).
(Movie) science the construction of large-volume
radiation-pumped lasers, which utilize CO2
planetary mesospheres as a gain medium, is
theoretically possible (SETI ).
29
CO2 lasers on Earth
Vibrational energy states of CO2 and N2
associated with the natural 10.4 µm CO2 laser.
The principal source of the (0 00 1) state
pumping is the absorption of solar radiation in
the 4.3 µm CO2 band. The collisional transfer of
energy from O(1D) atoms to the (0 00 1) state
amplifies the inversion significantly.
Population inversion in the Earth's atmosphere
occurs near a height of 95100 km in
daytime. (G.M. Shved, V. P. Ogibalov, Natural
population inversion for the CO2 vibrational
states in Earth's atmosphere, J. Atmos.
Solar-Terrestrial Phys. 62, 993, 2000)
30
Infrared Heterodyne Spectrograph
Sketch of the infrared heterodyne spectrograph
used at the time. (Goddard Space Flight Center).
The CO2 laser beam from the bottom (vLO) is mixed
with the input signal (vIR) coming in from the
left. The sensitive IR photodetector produces an
electronic signal at the upper right (vLO-vIR)
which is further processed by an RF spectrometer
(not shown). Adapted from Kostiuk Mumma (1983).
31
Homodyne, heterodyne,..
In optical interferometry, homodyne signifies
that the reference radiation (the local
oscillator) is derived from the same source as
the signal before the modulating process. For
example, in a laser scattering measurement, the
laser beam is split into two parts. One is the
local oscillator and the other is sent to the
system to be probed. The scattered light is then
mixed with the local oscillator on the detector.
Heterodyne detection is commonly used in
telecommunications and astronomy for detecting
and analysing signals. The radiation in question
is most commonly either radio waves (see
superheterodyne receiver) or light (see
interferometry). The reference radiation is known
as the local oscillator. The signal and the local
oscillator are superimposed at a mixer. The
mixer, which is commonly a (photo-) diode, has a
non-linear response to the amplitude, that is, at
least part of the output is proportional to the
square of the input.
32
Emission-line lasers in Eta Carinae
Eta Carinae is one of most massive and energetic
(and enigmatic) stars in our Milky Way.

Left the big event from 1830 to 1900.
33
The structure of Eta Carinae
The three objects commonly referred to as blobs
B, C, and D) were first discovered using speckle
techniques as described by Weigelt Ebersberger
(1986) and Hofmann Weigelt (1988).
The separation between the blobs and the central
star has been successfully resolved using the
Hubble Space Telescope (HST) Wide Field Planetary
Camera 2 (WFPC2) (Morse et al. 1998). Combined,
these observations indicate that the three
objects all lie within 300 milliarcseconds (mas)
of the central star and are less than 50 mas in
diameter. Adopting a distance of 2.2 kpc (Allen
Hillier 1993), these correspond to a separation
of 660 AU and a maximum diameter of 110 AU. These
clumps of ejecta are thought to be compact knots
of gas ejected from the central star at some
point in the past (Davidson et al. 1995).
34
A possible laser effect in Eta Car
From S. Johansson and V. S. Letokhov , MNRAS
Volume 364 Issue 2 Page 731-737, December 2005.
.We explain the presence of the fluorescent OI
8446-Å and forbidden OI 6300-Å lines as well as
the absence of the allowed OI 7774-Å line in
spectra recorded with the Hubble Space Telescope
(HST)/STIS instrument (Gull et al.). From atomic
data and estimated stellar parameters we
demonstrate that there is a population inversion
and stimulated emission in the 3p 3P3s 3S
transition ?8446 due to photoexcitation by
accidental resonance (PAR) by H - Lyß radiation.
35
S. Johansson V.S. Letokhov, Astrophysical
lasers operating in optical Fe II lines in
stellar ejecta of Eta Carinae, AA 428, 497
(2004)
36
lasing lines
S. Johansson V.S. Letokhov, Astrophysical
lasers operating in optical Fe II lines in
stellar ejecta of Eta Carinae, AA 428, 497
(2004)
37
A more detailed model is presented in Johansson
and Lethokov, 2007
38
Speckle Interferometry

• Speckle interferometry is a technique used to
  improve the angular resolution of a telescope
  degraded by atmospheric seeing
• a very large number of short duration exposures
  are taken with very long focal length (say 100m)
  and narrow bandwidth (say 1 nm) in each exposure
  the seeing is frozen, each speckle represents the
  diffraction figure of the aperture
• Fourier Transforms allow the reconstruction of
  the true image
• The technique works well for simple structures
  (e.g. double or multiple stars, disks).

39
Schematic of a Stromgren boundary
Recall the ionization of HI any photon with a
wavelength less than that of the Lyman limit
which hits the neutral Hydrogen atom is capable
of ionising it, whatever n-level the electron is
in (but it is essentially n 1). Recombination
to n 1 will produce another UV photon, and the
tow process balance each other. So we can
determine the dimension of the ionized sphere by
setting two things equal the number Nuv of
ionizing photons emitted by the star, per second,
and the number of recombinations into all states
other than n 1.
Adapted from http//www.astro.rug.nl/ndouglas/te
aching/InleidingStk2/stromgren.html
40
Saha Equation - 1
For a gas at a high enough temperature, a
fraction of the atoms will be ionized. One or
more of the electrons normally bound to the atom
will be ejected from the atom and will form an
electron gas that co-exists with the gas of
atomic ions and neutral atoms. This state of
matter is called a plasma. The Saha equation
describes the degree of ionization of this plasma
as a function of the temperature, density, and
ionization energies of the atoms. The Saha
equation only holds for weakly ionized plasmas
for which the Debye length is large (see later).
This means that the "screening" of the coulomb
charge of ions and electrons by other ions and
electrons is negligible. The subsequent lowering
of the ionization potentials and the "cutoff" of
the partition function is therefore also
negligible.
41
Saha equation -2
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 ground levels of the two ions, and (Ei1- Ei
) is the ionization potential of the i-th
ion. This is Sahas equation, known also as
Saha-Langmuir equation.
42
Saha equation - 3
Alternative formulations are
The pressure is expressed in barye dyn?cm-2 (1
dyn?cm-2 10-6 bar 0.1 pascal). High
temperature favours ionization, high pressure
favours recombination. In stellar atmospheres, Pe
lies in the range 1 dyn/cm2 (cool stars) to 1000
dyne/cm2 (hot stars). Note that 1dyn/cm20.1N/m2
(SI units), so for SI calculations the constant
is -1.4772 instead of -0.4772.
43
The Fe ionization in the solar corona
The very low density of the solar corona favors a
very high ionization. The temperature estimated
by the ionization degree reaches millions of K.
44
Another expression for partition functions
(adapted from Gray App D2)
45
Saha equation - 4
To obtain an intuitive demonstration of the
formula, consider the case neutral singly
ionized atom free electron. The electron
occupies the volume (2?mekT) in the impulse
space, and the volume 1/Ne in the Cartesian
space. Therefore, in the phase space, its volume
is the product of the two, expressed in units of
h and multiplied by 2 because of the two possible
spin states
Then, the overall statistical weight is 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. The quantity
is known as thermal deBroglie wavelength of the
electron.
46
Debye length
The Debye length ?D is the scale over which
mobile charge carriers (e.g. electrons) screen
out electric fields in plasmas. In other words,
the Debye length is the distance over which
significant charge separation can occur. An
approximate expression for ?D is
where qe is the electron charge. In astrophysical
plasmas, where the electron density is relatively
low, the Debye length may reach macroscopic
values, such as in the magnetosphere, solar wind,
interstellar medium and intergalactic medium (see
table in next slide).
47
Debye lengths in astrophysical plasmas
48
Ionization Potentials (eV)
49
Exercise on Saha equation - 1

• Q. What is the H ionization in Solar
  photosphere?
• For neutral hydrogen, essentially all the
  particles are in the ground state, with g0u02.
  When H is ionized, g1, u1.
• Assuming T5770K, Pe15 dyn/cm2 for the
  photosphere

i.e. H/H07x10-5 less than one out of 104 H
atoms is ionized in the solar photosphere.
Recall now that the Balmer lines (involving an
upward transition from n 2) reach a peak
strength at subtype A (T?9000K) so why do the
Balmer lines diminish in strength at higher
temperatures? Repeat the calculation for T 104
K, and show that Saha equation answers the
question (next slide).
50
Exercise on Saha equation - 2

• We can use Saha equation to study the degree
  of ionization of H in general in stellar
  photospheres. The fraction of ionized hydrogen to
  the total is shown in the figure below H
  switches from mostly neutral below 7000K to
  mostly ionized above 11000K.

51

Combining Saha and Boltzmann equations - 1

• We can now use the Boltzmann and Saha
  equations to measure H(n2)/H(total) as a
  function of T. The results are shown here, with
  the highest value around 10,000K.

He (assume H/He10 by number) complicates this
calculation, since ionized He provides excess
electrons with which H ions can recombine, so it
takes higher temperatures to achieve the same
degree of ionization.
52
Saha-Boltzmann for CaI and CaII
For Calcium, ?ion 6.1 eV, u0 1.32, u 2.3 ,
so Saha equation at solar temperature implies
Essentially all Calcium is singly ionized (CaII).
The number of Ca ions in the first excited
state relative to the ground state (g1 2, g24,
? 3.12eV) is 1/265, so nearly all Calcium in
the Suns photosphere is in the ground state of
Ca II. Combining these results N(Ca)/N(H)N(Ca
II g.s.)/N(H n2) (1/5x105)(0.99/5x10-9)
400 There are 400 times more Caii ions with
electrons in the ground-state (which produce the
Ca II HK lines) than there are neutral H atoms
in the first excited state (which produce
Balmers lines) therefore the H and K Ca II
lines in the Sun are so strong due to T
dependence of excitation and ionization, not high
Ca/H.
53
Exercise on Saha equation - 3
This figure shows the ionization equilibrium
between neutral and ionized Calcium at solar
photospheric conditions of temperature and
pressure. Complete with a discussion of Ca III.
54
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.
55
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
56
Two limiting regimes
all H is essentially ionized, so that
1)
,
quite independent of the precise chemical
composition of the atmosphere.
only metals are ionized, so that
2)
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.
57
Dependence of ionization from pressure - 1

• Another observational effect that can be
  understood using Sahas equation is that
  supergiants and giants have lower temperatures
  than dwarfs of the same spectral type
• Spectral classes are defined by line ratios of
  different ions, e.g. He II 4542A/He I 4471 for O
  stars. At higher temperatures the fraction of He
  II will increase relative to He I, so the above
  ratio will increase.
• However, supergiants have lower surface gravities
  (or pressure) than main-sequence stars, so from
  Saha equation a lower Pe at the same temperature
  will give a higher ion fraction, N/N0.
• Assigning a given spectral class corresponds to
  fixing N/N0
• a star with lower pressure can have a lower Teff
  for the same ratio and spectral class.

58
Dependence of ionisation from pressure - 2
In other words according to Saha equation, at
any given temperature the ionization degree
depends 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.
59
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 (spectroscopic
parallaxes). Other variables might enter into
the classification, for instance the chemical
composition, the magnetic fields, the rotation,
but finer observations are needed. The
following Tables give the electron pressure (in
barye) in the atmospheres of dwarfs, giant and
supergiant stars, respectively, and the
ionization degree in the average stellar
atmosphere.
60
Table of electron pressures
61
Ionization degree in the average stellar
atmosphere
62
Exercizes
1 Apply Boltzmanns excitation formula to the H
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 atoms remain at
the fundamental state, nevertheless the Balmer
series can be observed in absorption, because of
the high number of H 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
above 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 ??.
63
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 rotational.
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, UA and UB are their partition
functions, D is the dissociation energy of the
molecule, QAB Qrot?Qvib?Qel is the molecular
partition function, and
is the reduced mass of the molecule. More in
general, we have a law of mass action (Guldberg
and Wage).
64
Diatomic Molecules
65
Polyatomic Molecules
Notice that dissociation potentials are much
lower than ionization potentials. Pay also
attention to CO and CO2.
66
Exercises on molecules
1 find the dissociation temperatures of
selected molecules 2 complete the tables with
the values of molecules seen in K and M type
stars
67
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.
68
Graph of Maxwell law - 1
The three different 'typical' velocites.
69
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.
70
Gas Dynamics
According to Maxwells law, the dynamical state
of the gas is characterized by the vector
v(x,y,z) in each point of the atmosphere. Such
vector might be due to a systematic component
(e.g. a current, or a wind) plus disordered
velocity fields, such as turbulent motions on
different spatial scales (e.g. the terrestrial
atmosphere). In addition, if the atmosphere is
ionized, its motions can be largely controlled by
the presence of magnetic fields. The diagnosis
of the dynamical state of the gas is therefore e
very complex matter.
71
Solar and stellar winds
Moreover, Maxwells equation implies that,
whatever the escape velocity and the temperature
at the surface of the body, a fraction (even if a
small one) of the gaseous particles will have a
velocity larger than the escape velocity. Such
fraction will therefore be lost to the outer
space. It is like having a wind leaving the
star (or even the planet, e.g. from our
atmosphere there is a continuous loss of
Hydrogen, originating a geo-coronal Lyman alpha
cloud surrounding the Earth). The phenomenon
of mass loss is however much more complex than
that due to Maxwells equation (a loss well
call a thermal one), in particular if the
matter is ionized and couples to the magnetic
field (e.g. the solar corona). For a discussion
of the solar wind based on SoHO data, see New
insights into solarwind physics from SoHO, di S.
R. Cranmer, arXivastro-ph/0409260 v1. Stellar
winds from massive stars are discussed in the
lectures by Yael Nazè on the web site
http//www.astro.unipd.it/planets/barbieri/didatti
ca.html