Download ORBITAL MOTION

17 PH507
Multiwavelength
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Atomic Spectra - Absorption & Emission line series and continua
• Bohr
theory (last year's physics unit) adequately describes electron energy
levels in Hydrogen. Quantum mechanics is required for more massive atoms
to describe the dynamics of electrons. However, we are interested here only
in the energy levels of electron states rather than a detailed model or
description of atomic structure. We can therefore use ENERGY LEVEL
DIAGRAMS without worrying too much about the theory behind them.
• There are 3 basic photon absorption mechanisms related to electrons. Using
Hydrogen as the example, the electron energy levels are given by the
principal quantum number n, as:
E(n) = - 2 2 me e4 Z2 / n2 h2
The lowest energy level of H (n = 1) is about -13.6 eV.
The next energy level (n = 2) is
-3.4 eV.
The third (n = 3) is
-1.51 eV
from Bohr Theory
Opacity. We first introduced the concept of opacity when deriving the
equation of radiative transport. Opacity is the resistance of material
to the flow of heat, which in most stellar interiors is determined by all
the processes which scatter and absorb photons. We will now look at
each of these processes in turn, of which there are four:

bound-bound absorption

bound-free absorption

free-free absorption

scattering
The first three are known as true absorption processes because they
involve the disappearance of a photon, whereas the fourth process
only alters the direction of a photon. All four processes are described
below and are shown pictorially in figure 1.
Figure 1 : Schematic energy level diagram showing the four
microscopic processes which contribute to opacity in
stellar interiors.
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Multiwavelength
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bound-bound absorption
Bound-bound absorptions occur when an electron is moved from one
orbit in an atom or ion into another orbit of higher energy due to the
absorption of a photon. If the energy of the two orbits is E1 and E2, a
photon of frequency bb will produce a transition if
E2 - E1 = hvbb.
Bound-bound processes are responsible for the spectral lines visible in
stellar spectra, which are formed in the atmospheres of stars. In
stellar interiors, however, bound-bound processes are not of great
importance as most of the atoms are highly ionised and only a small
fraction contain electrons in bound orbits. In addition, most of the
photons in stellar interiors are so energetic that they are more likely
to cause bound-free absorptions, as described below.
bound-free absorption
Bound-free absorptions involve the ejection of an electron from a
bound orbit around an atom or ion into a free hyperbolic orbit due to
the absorption of a photon. A photon of frequency bf will convert a
bound electron of energy E1 into a free electron of energy E3 if
E3 - E1 = hvbf.
Provided the photon has sufficient energy to remove the electron
from the atom or ion, any value of energy can lead to a bound-free
process. Bound-free processes hence lead to continuous absorption in
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Multiwavelength
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stellar atmospheres. In stellar interiors, however, the importance of
bound-free processes is reduced due to the rarity of bound electrons.
free-free absorption
Free-free absorption occurs when a free electron of energy E3 absorbs
a photon of frequency ff and moves to a state with energy E4, where
E4 - E3 = h
ff.
There is no restriction on the energy of a photon which can induce a
free-free transition and hence free-free absorption is a continuous
absorption process which operates in both stellar atmospheres and
stellar interiors. Note that, in both free-free and bound-free
absorption, low energy photons are more likely to be absorbed than
high energy photons.
scattering
In addition to the above absorption processes, it is also possible for a
photon to be scattered by an electron or an atom. One can think of
scattering as a collision between two particles which bounce of one
another. If the energy of the photon satisfies
h
<< mc2,
where m is the mass of the particle doing the scattering, the particle
is scarcely moved by the collision. In this case the photon can be
imagined to be bounced off a stationary particle. Although this
process does not lead to the true absorption of radiation, it does slow
the rate at which energy escapes from a star because it continually
changes the direction of the photons.
Bound-Bound Transitions
• BOUND - BOUND transitions give rise to spectral lines.
• ABSORPTION LINE if a photon is absorbed, causing increase in energy of an
electron. Energy of absorbed photon:
h = E(nu) - E(nl)
(1)
where E(nu) and E(nl) are energies of upper and lower energy levels
respectively. This is RADIATIVE EXCITATION.
• Note energy can also be absorbed through collisions of a free particle
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Multiwavelength
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(COLLISIONAL EXCITATION) - no absorption line is seen in this case.
• Atom remains in excited state until
SPONTANEOUS EMISSION (photon is emitted typically after ~10-8 s)
or INDUCED EMISSION (Photon emitted at same energy and coherently
with incoming photon - as in lasers – stimulated emission).
Both produce EMISSION LINES.
(frequency/wavelength)
corresponding to difference in
energy levels.
 Narrow lines are seen since
transitions can only occur if
photon has energy
• Energy level diagram shows electron energy level changes for absorption of
a photon.
Lowest energy level set to zero energy. 1eV = 1.6 x 10-19 J.
13.6 eV
n=•
n=4
12.73 eV
n=3
12.07 eV
n=2
n=1
10.19 eV
Lyman
Series
Balmer Paschen
Series Series
0 eV
• Series of lines seen
-LYMAN SERIES transitions to/from n=1 lines seen in ultraviolet
-BALMER SERIES ""
n=2
""
visual
-PASCHEN SERIES""
n=3
""
infrared ...
Bound-free transitions
• If photon has energy greater than
that required to move an electron in
an atom from its current energy
level to level n=∞, the electron will
be released, ionizing the atom.
n=•
n=4
n=3
n=2
• Ionization potential for Hydrogen is
13.6 eV.
• Energy of absorbed photon is
n=1
2/2
1/2 m ev 2
13.6 eV
12.73 eV
12.07 eV
10.19 eV
0 eV
h = ( - E(nl)) + mev
(48)
• Since one of the states (free electron) can have any energy, the transition can
have any energy and the photon any frequency (above a certain value
determined by  and E(nl)).
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Astrophysics
Dr. S.F. Green
5
Thus BOUND-FREE transitions give an ABSORPTION CONTINUUM.
• RE-COMBINATION is a FREE-BOUND transition and results in an
EMISSION CONTINUUM.
• The
spectrum
produced
by
continuum • 
absorption from a single energy
level will therefore appear as a
series of lines of increasing energy


(increasing
frequency, decreasing wavelength) up to a limit defined by -E(nl), with an
absorption continuum shortward of this limit. the characteristic of a bound-f
ree transition in a
spectrum is an edge: no absorption below some energy, then a sharp onset in
the absorption above that critical energy. As we’ll see, the absorption
decreases above the critical energy.
• For nl=1 the Lyman series (Lyman-, Lyman- etc) is observed together with
the Lyman continuum shortward of =91.2 nm. (Since interstellar space is
populated by very low density and low temperature hydrogen (ie with n=1),
photons with <91.2nm are easily absorbed so it is opaque in the near-UV).
For nl=2 the Balmer series (H, H etc) is observed together with the Balmer
continuum shortward of =364.7 nm.
Free-free transitions
• Absorption of a photon by a free electron in the vicinity of an ion.
Electron changes from free energy state with velocity v 1 to one with velocity
v2
i.e. h = 1/2 me v22 - 1/2 me v12 The term means
The inverse process “braking radiation” occurs when an electron is
accelerated by passage near an ion, and hence radiates. Bremsstrahlung and
free-free absorption are basic radiative processes that show up in many
contexts.
When X-rays and gamma-rays are considered, we’ll talk about the more
general process of Compton scattering (heating the electrons) and inverse
Compton cooling.
Cyclotron and Synchrotron Radiation When magnetic fields are present,
charges can interact with them and radiate or absorb radiation. For slowly
moving particles this happens at a single frequency, the cyclotron frequency.
For relativistically moving particles, the emission or absorption occurs over a
large range of frequencies, and is called in this case synchrotron radiation.
PH507
Astrophysics
Professor Michael Smith
6
Determination of 
• The actual spectrum of a star depends on the physical conditions (notably
temperature) and composition of the stellar atmosphere. The intensity is
produced at a physical level in the star where  ~ 2/3. In order to
determine the total spectrum, the value of  needs to be determined at all
wavelengths. The overall  is the sum of the contributions from each
atomic/molecular species in the atmosphere. Each component of  depends
on the number of atoms/molecules with a given energy state capable of
absorbing radiation at that frequency and the absorption efficiency. We will
deal with the energy state populations first:
Boltzmann's equation (Excitation equilibrium)
• Boltzmann's equation describes the population distribution of energy states
for a particular atom in a gas. The ratio of number of atoms per m3 in energy
state B to energy state A:
NB
g (EA - EB)/kT
= B e
(50)
N
g
A
A
where gA and gB are STATISTICAL WEIGHTS (number of different
quantum states of the same energy), k = Boltzmann const and T =
temperature of gas.
NB EB > EA so exponential power is -ve.
• The probability of finding an atom in an excited state decreases exponentially
with the energy of the excited state, but increases with increasing
temperature.
Saha Equation (Ionization Equilibrium)
• The Boltzmann eqn does not describe all the possible atomic states.
Excitation may cause electrons to be lost completely. There are therefore a
number of different ionization states for a given atom, each of which has one
or more energy states.
• The ratio of the number of atoms of ionization state i+1 to those of ionization
state i (i=I is neutral, i=II is singly ionized, etc) is given by
3/2
Ni+1
Ni
=
Ui+1 2
Ui Ne
2 me k T
2
h
-i /kT
e
where Ne is the electron density (number of electrons per m3), i is the
ionization potential of the ith ionization state, Ui+1 and Ui are PARTITION
FUNCTIONS obtained from the statistical
•
Ui = gi1 +
weights:
-Ein /kT
e
in
g
n=2
• The higher the Ionization Potential, i, the lower the fraction of atoms in the
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Astrophysics
Professor Michael Smith
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upper ionization state,
The higher the Temperature, , the higher the fraction of atoms in the upper
ionization state, (Collisional excitation is more likely to ionize atom),
The higher the electron density, the lower the fraction of atoms in the upper
ionization state (due to re-combination).
• The Boltzmann and Saha Equations give the fraction of atoms in a given
ionization state and energy level allowing (when combined with
absorption/emission probabilities)  and hence the line strengths to be
related to abundances.
Example - Abundances in the Sun
• In line forming regions in the Sun:
Gas
Hydrogen
Calcium
I
13.6 eV
6.1 eV
II
11.9 eV
T ~ 6000 K, Ne ~ 7x1019 m3.
UII/UI
2
~2
UIII/UII
~0.5
g1
2
1
g2
2
6
From Saha Equation for Hydrogen, the ratio of ionized to un-ionized
H atoms
NII/NI ≈ 6x10-5
i.e. most of Hydrogen is un-ionized.
From Boltzmann equation, ratio of number of atoms with electrons
in level n=2 to those in level n=1 (E1-E2 = -10.19 eV) is
N2/N1 ≈ 3x10-9
i.e almost all H atoms are in the ground state.
The H Balmer lines which originate from level n=2 are strong only because the H
abundance is so high.
From Saha Equation for Calcium,
NII/NI ≈ 600 and NIII/NII ≈ 2x10-3
i.e. most of Calcium is in the singly ionized state.
From Boltzmann equation, ratio of number of atoms with
electrons in
energy states which contribute to the H and K lines
to those in the ground state (E1-E2 = -3.15, -3.13 eV) is (NB/NA)II ≈ 10-2 i.e
most Ca atoms are in the ground state.
The H and K lines of Calcium are therefore strong because most Ca atoms in
the Sun are in an energy state capable of producing the lines.
• For stars cooler than the Sun more H is in the ground state so Balmer lines
will be weaker, for stars hotter than the Sun more H is in n=2 state so Balmer
lines will be stronger. (T ~ 85000 K needed for N2/N1 =1). But at this
temperature NII/NI = 105 so little remains un-ionized.
• Balmer line strength depends on excitation (function of T) and ionization
(function of T and Ne). Balance of effects occurs at T ~ 10,000 K so Balmer
lines are strongest in A0 stars.
PH507
Astrophysics
Professor Michael Smith
8
• A similar effect occurs for other species but at different temperatures.
Transition probabilities
• Once we know the population of all energy states for a given gaseous species
we need to know the transition probabilities for each energy state change
before the absorption coefficient can be determined.
• The transition probabilities must be calculated from atomic theory or
determined by experiment - much time has been invested in this major
problem in astrophysics.
• The EINSTEIN TRANSITION PROBABILITY (inverse of lifetime):
for spontaneous emission, A21  2
for stimulated emission
B21  -1
for absorption
A12  -1
Total 
• We can now calculate  for a given gaseous species.
(removing spectral line opacities for clarity):
falls off with decreasing 
Lyman continuum

absorption
due to  -1 dependence
Log 
For Hydrogen
T~25000K (B star)
Balmer
continuum
absorption
Paschen
continuum
absorption
T~5000K (G star)
(nm)
• Similar diagrams exist for other species. The total  will be the sum for all
species in the star.
• The region of a star for which optical depth ~2/3 determines where
observed radiation originates. So if  is large, then = 2/3 at a high level
in the atmosphere and if  is low, = 2/3 deep in the atmosphere.
Solar photospheric opacity
• The solar atmosphere is dominated by hydrogen. The visible surface, the
photosphere, has a temperature ~5800 K. However, as can be seen from the
diagram above,  for hydrogen at low temperatures is low in the visible
region (~400-700nm). This is because the continuum absorption in the
visible is due to Paschen absorption (electrons originating in level n=3) and
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Astrophysics
Professor Michael Smith
9
most hydrogen is in ground state or n=2 level. We would therefore expect
the continuum to come from much deeper in the sun where temperatures are
higher. So what causes the high solar photospheric opacity?
The solar opacity comes from the H- ion. The ionization potential
For H- --> H + eis 0.75 eV (=1650nm).
From Boltzmann eqn
N3/N1 = 6 x 10-10
But from Saha eqn
N(H)/N(H-) ≈ 3 x 107
Therefore N(H-)/N3 ≈ 500.
Log 
T~25000K (B star)
H - bound-free H - free-free
T~5000K (G star)
(nm)
i.e. number of H- ions is greater than number of H atoms in level n=3, so
absorption of photons to dissociate H- to H dominates the continuum
absorption in the optical.
Limb darkening
• The Sun is less bright near the limb than at the centre of the disk.

The continuum spectrum of the entire solar disk defines a Stefan-Boltzmann
effective temperature of 5800 K for the photosphere, but how does the
temperature vary in the photosphere? A clue is evident in a white-light
photograph of the Sun.
PH507
Astrophysics
Professor Michael Smith
10


We see that the brightness of the solar disk decreases from the centre to the
limb - this effect is termed limb darkening.
Limb darkening arises because we see deeper, hotter gas layers when we look
directly at the centre of the disk and higher, cooler layers when we look near
the limb.
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Astrophysics
Professor Michael Smith
11
Assume that we can see only a fixed distance d through the solar atmosphere. The
limb appears darkened as the temperature decreases from the lower to the upper
photosphere because, according to the Stefan-Boltzmann law (Section 8-6), a cool gas
radiates less energy per unit area than does a hot gas. The top of the photosphere, or
bottom of the chromosphere, is defined as height = 0 km. Outward through the
photosphere, the temperature drops rapidly then again starts to rise at about 500 km
into the chromosphere, reaching very high temperatures in the corona.
Formation of solar absorption lines. Photons with energies well away from any atomic
transition can escape from relatively deep in the photosphere, but those with energies
close to a transition are more likely to be reabsorbed before escaping, so the ones we see
on Earth tend to come from higher, cooler levels in the solar atmosphere. The inset
shows a close-up tracing of two of the thousands of solar absorption lines, those
produced by calcium at about 395 nm.
PH507
Astrophysics
Professor Michael Smith
12
At this point, you may have discerned an apparent paradox: how can the solar
limb appear darkened when the temperature rises rapidly through the
chromosphere? Answering this question requires an understanding of the
concepts of opacity and optical depth. Simply put, the chromosphere is almost
optically transparent relative to the photosphere. Hence, the Sun appears to
end sharply at its photospheric surface - within the outer 300 km of its 700,000
km radius.
Our line of sight penetrates the solar atmosphere only to the depth from which
radiation can escape unhindered (where the optical depth is small). Interior
to this point, solar radiation is constantly absorbed and re-emitted (and so
scattered) by atoms and ions.
PH507
Astrophysics
Professor Michael Smith
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Y
Length of each solid bar is
approximately the same,
i.e. depth for which =2/3
R
y
Observer
X
Rx
Since R y > R x, radiation from the edge
of the disk, Y, originates from a higher
(cooler) region than at the centre of the
disk, X.
Assuming LTE, the continuum radiation
is described by the Planck function since
Y is at lower temperature, radiation is of
lower intensity
Spectral line formation
• Lines form higher in atmosphere than continuum. For optical lines this
corresponds to lower temperature than continuum and therefore lower
intensity (absorption lines) (see p21 where S < I).
small
~2/3 low in
atmosphere
6500
T (K)
high
~2/3 high in
atmosphere
4500
0
200
400 km
Height above photosphere
F

PH507
Astrophysics
Professor Michael Smith
14
Spectral line strength
Spectral lines are never perfectly monochromatic. Quantum mechanical
considerations govern minimum line width, and many other processes cause
line broadening :
Shape of absorption line — line profile.
Natural broadening — consequence of uncertainty principle.
Doppler broadening — consequence of velocity distribution.
Pressure broadening — perturbation of energy levels by ions.
• For abundance calculations we want to know the total line strength. Total
line strength is characterised by EQUIVALENT WIDTH.
� Equivalent width: measure strength of lines.
� Rectangle with same area as line, i.e. same amount of absorption.
� EW is width in °A across rectangle
� Need EW to determine number of absorbing atoms
Stellar composition
• Derived from spectral line strengths in stellar atmospheres. In the solar
neighbourhood, the composition of stellar atmospheres is:
Element H
He
C,N,O,Ne,Na,Mg,Al,Si,Ca,Fe, others
% mass 70
28
~2.
Spectral line structure
• NATURAL WIDTH: Due to uncertainty principle, E=h/t, applied to
lifetime of excited state. For "normal" lines the atom is excited (by a photon
or collision) to an excited state which has a short lifetime t ~ 10-8 s. The
upper energy level therefore has uncertain energy E and the resultant
spectral line (absorption or emission) has an uncertain energy (wavelength).
The line has a Lorentz profile,  ~ 10-5 nm for visible light.
• COLLISIONAL/PRESSURE BROADENING:
Outer energy levels of atoms affected by presence of neighbouring charged
particles (ions and electrons). random effects lead to line broadening since
the energy of upper energy level changes relative to the unexcited state
energy level. This is the basis of the Luminosity classification for A,B stars.
Gaussian profile.  ~ 0.02 - 2 nm.
• DOPPLER BROADENING:
Due to motions in gas producing the line. Doppler shift occurs for each each
photon emitted (or absorbed) since the gas producing the line is moving
relative to the observer (or gas producing the photon).
Thermal Doppler broadening due to motions of individual atoms in the gas.
~0.01 - 0.02 nm for Balmer lines in the Sun. Gaussian profile.
PH507
Astrophysics
Professor Michael Smith
15
Bulk motions of gas in convection cells. Gaussian profile.
• ROTATION:
If there is no limb darkening, then lines have hemispherical profile due to
combination of radiation from surface elements with different radial
velocities. Effect depends on rotation rate, size of star and angle of polar tilt.
In general, V sin i is derived from the profile.
_
V -1
(km s )
200
Receding
+V
A
F

C
B
A
C
B
Approaching
-V
100

o
0
O B A F G K
• ATMOSPHERIC OUTFLOW:
Many different types.
Star with expanding gas shell (result of outburst) gives P-CYGNI PROFILE.
Continuum (+ absorption lines) from star, emission or absorption lines from
shell:
F
Expanding
gas
shell
D
C
Star
D
B

D
A
o

Observer
B
C
A
C
B
Radiation from star, A, passes through cooler cloud giving absorption line
due to shell material which is blue shifted relative to star. Elsewhere,
emission lines are seen.
Be STARS: Very rapid rotators with material lost from the equator:
Radiation from star, A, passes through cooler cloud giving absorption line.
Overall line structure is hemispherical rotation line (B,D). Emission lines
seen due to shell material (C,E).
PH507
Astrophysics
Professor Michael Smith
C
F
Rotating
gas
shell
E
Star
B
A
o
Observer
C
16
B
A
D
E
D

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Astrophysics
Professor Michael Smith
17
Forbidden lines
• Only certain transitions are generally seen for two reasons:
1) Outer energy levels are far from the nucleus so in dense gases, levels are
distorted or destroyed by interactions.
2) Selection rules for change of quantum numbers restrict possible
transitions.
• In fact forbidden transitions are not actually forbidden. However, the
probability of a forbidden transition is very low, so an allowed transition will
generally occur. The lifetimes in an excited state for which there are no
allowed downward transitions are ~10-3 - 109 seconds (ie very low transition
probability). These are called METASTABLE STATES.
• De-excitation from a metastable state can be by:
1) Collisional excitation, or absorption of another photon to higher energy
state allowing another downward transition to the equilibrium state,
2) FORBIDDEN TRANSITION producing a FORBIDDEN LINE. Usually
denoted with [], e.g. [OII 731.99].
• Forbidden lines are usually much fainter than those from allowed transitions
due to low probability.
• In interstellar nebulae excited by UV from nearby hot stars, some elements'
excited states have no allowed downward transitions to the ground state. In
the absence of frequent collisions (due to low density) or high photon flux, a
forbidden transition is the only way to the ground state.
• These lines were not understood for a long while. A new element Nebulium
was invented to account for them.