Download ORBITAL MOTION

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WEEK 5 (17).
TEST: Tuesday 9:05 Week 19.
ASSIGNMENT 3: deadline today
ASSIGNMENT 4: deadline Week 18
The area under the Planck curve (integrating the Planck function) represents the
total energy flux, F (W/m2), emitted by a blackbody when we sum over all
wavelengths and solid angles:
Stefan-Boltzmann constant:

The temperature in this equation is the surface temperature of
the object.The object might be much hotter deep inside, but this
doesn't matter.
The strong temperature dependence of this formula was first deduced from
thermodynamics in 1879 by Josef Stefan (1835-1893) and was derived from
statistical mechanics in 1884 by Boltzmann. Therefore we call the expression
the Stefan-Boltzmann law. The brightness of a blackbody increases as the
fourth power of its temperature. If we approximate a star by a blackbody, the
total energy output per unit time of the star (its power or luminosity in watts)
is just
L = 4R2T4
since the surface area of a sphere of radius R is 4R2
Stellar Material
Our Sun is the only star for which I( has been accurately observed. Indeed, Lbol is
related to the solar constant: the total solar radiative flux received at the Earth’s orbit
outside our atmosphere (1370 W/m2).
The solar luminosity L (3.86 x 1026 W) is calculated from the solar constant in the
following manner. Using the inverse-square law, we find the radiative flux at the
Sun’s surface R. Then Lis just  times this flux. The solar energy distribution curve
may be approximated by a Planck blackbody curve at the effective temperature Teff,
defined as the temperature of a blackbody that would emit the same total energy as an
emitting body, such as the Sun or a star. Then the Stefan-Boltzmann law implies
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L = 4π R2 T4eff (e.g. J s-1 )
where is the Stefan-Boltzmann constant.
Example: A star with surface temperature T = 1.2 x 104K

The peak wavelength for this star is at
wavelength
= 0.0029 / (1.2 x 104) m
= (2.9/1.2) x 10-3-4 m = 2.4 x 10-7 m = 240
nm.
Physical Characteristics: statistical mechanics
Stellar Atmospheres
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The spectral energy distribution of starlight is determined in a star’s atmosphere, the
region from which radiation can freely escape. To understand stellar spectra, we first
discuss a model stellar atmosphere and investigate the characteristics that determine
the spectral features.
The stellar photosphere, a thin, gaseous layer, shields the stellar interior from view.
The photosphere is thin relative to the stellar radius, and so we regard it as a uniform
shell of gas. The physical properties of this shell may be approximately specified by
the average values of its pressure P, temperature T, and chemical composition µ
(chemical abundances).
We assume that the gas obeys the perfect-gas law:
P =nkT
where k is Boltzmann’s constant and n is the number of particles. This relationship is
also known as Boyle’s Law.
The Maxwell-Boltzmann Energy and Speed Distributions:
Boltzmann Equation
where Ni is the number of molecules at equilibrium temperature T, in a state i which
has energy Ei and degeneracy gi, N is the total number of molecules in the system and
k is the Boltzmann constant (denominator is called the partition function)
An important result that follows from it is that the (average) kinetic energy of a
particle, or assemblage of particles, is given by the relationship;
KE 
3
kT
2
Thus temperature is just a measure of the kinetic energy of a gas, or an assemblage of
particles. This equation applies equally well to a star as a whole, as to a single particle,
and later we will look at the comparison between a star’s kinetic and gravitational
(potential) energies.
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The kinetic energy is also a measure of the speed that atoms or molecules are moving
about at - the hotter they are, the faster they move. Thus, for a cloud of gas
surrounding a hot star of temperature T = 15,000 K, which consists of hydrogen atoms
(mass = 1.67 10-27 kg); the root mean square speed is given by:
3
1
kT  KE  mv2
2
2
3kT
 19 km s 1  50,000 mph
m
v
The particle number density is related to both the mass density (kg/m3) and the
composition (or mean molecular weight) µ by the following definition of µ:
1


mH n

where mH = 1.67 x 10-27 kg is the mass of a hydrogen atom. For a star of pure atomic
hydrogen, µ = 1.
If the hydrogen is completely ionised, µ = 1/2 because electrons and protons
(hydrogen nuclei) are equal in number and electrons are far less massive than protons.
In general:
X = fraction of material by mass in form of hydrogen,
Y = fraction of material by mass in form of helium, and
Z = fraction of material by mass in form of heavier elements.
Hence,
X + Y + Z = 1.
Now, in a fully ionised gas,
 hydrogen gives
2 particles per mH
(a proton and an
electron),
 helium gives
3/4 particle per mH
(a nucleus containing 2
protons and 2 neutrons
= 4mH and two
electrons), and
 heavier elements give ~1/2 particle per mH (e.g. Carbon gives a
nucleus containing 6
protons and 6 neutrons
= 12mH and six
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electrons = 7/12,
Oxygen gives a nucleus
containing 8 protons
and 8 neutrons =
16mH and eight
electrons = 9/16, etc.)
Thus the number of particles per cubic metre due to
hydrogen = 2X
helium = 3Y
/ mH,
/ 4mH, and
heavier elements = Z
/ 2mH.
The total number of particles per cubic metre is then given by the sum of
the above, i.e.
n = (2X
/ mH) + (3Y
/ 4mH) + (Z
/ 2mH).
Rearranging, we obtain:
n=(
/ 4mH) (8X + 3Y + 2Z).
Now, X + Y + Z = 1, and hence Z = 1 - X - Y, giving:
n=(
/ 4mH) (6X + Y + 2).
Recalling that
= nmH
, we can write:
= 4 / (6X + Y + 2),
which is a good approximation to
except in the cool outer regions of
stars. For solar composition, X =0.747, Y =0.236 and Z =0.017, resulting
in ~ 0.6, i.e. the mean mass of the particles in the Sun is a little over half
the mass of a proton.
In general, stellar interior gases are ionised and

1
3
1
2X  Y  Z
4
2
The mass fraction is the percentage by mass of one species relative to the total. Thus,
for a pure hydrogen star (X=1.0, Y = 0.0, Z = 0.0),  ~ 0.5, and for a white dwarf star (X
= 0.0, Y = 1.0, Z = 0.0)  ~ 1.33.
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Temperature Definitions
The continuous spectrum, or continuum, from a star may be approximated by the
Planck blackbody spectral-energy distribution. For a given star, the continuum
defines a colour temperature by fitting the appropriate Planck curve.
We can also define the temperature from Wien’s displacement law: maxT = 2.898 x
10-3 m . K which states that the peak intensity of the Planck curve occurs at a
wavelength max that varies inversely with the Planck temperature T. The value of
max then defines a temperature. Also note here that the hotter a star is, the greater
will be its luminous flux (in W/m2), in accordance with the Stefan-Boltzmann law: F
= T4 where = 5.67 x 10-8 W/m2 . K4. Then the relation
L = 4πR2T4eff
defines the effective temperature of the photosphere.
A word of caution: the effective temperature of a star is usually not identical to its
excitation (Boltzmann eqn) or ionisation temperature (Saha eqn) because spectralline formation redistributes radiation from the continuum. This effect is called line
blanketing and becomes important when the numbers and strengths of spectral lines
are large.
When spectral features are not numerous, we can detect the continuum between them
and obtain a reasonably accurate value for the star’s effective surface temperature.
The line blanketing alters the atmosphere’s blackbody character.
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Spectrophotometry: the SED
The goal of the observational astronomer to to make measurements of the EMR
from celestial objects with as much detail, or the finest resolution, possible.
There are of course different types of detail that we want to observe. These
include angular detail, wavelength detail, and time detail. The perfect
astronomical observing system would tell us the amount of radiation, as a
function of wavelength, from the entire sky in arbitrarily small angular slices.
We are always limited in angular and wavelength coverage, and limited in
resolution in angle and wavelength. If we want good information about the
wavelength distribution of EMR from an object (spectroscopy or
spectrophotometry) we have to give up angular detail. If we want good angular
resolution over a wide area of sky (imaging) we usually have to give up
wavelength resolution or coverage.
The ideal goal of spectrophotometry is to obtain the spectral energy
distribution (SED) of celestial objects, or how the energy from the object is
distributed in wavelength. We want to measure the amount of energy received
by an observer outside the Earth's atmosphere, per second, per unit area, per
unit wavelength or frequency interval. Units of spectral flux (in cgs) look like:
f  = ergs s-1 cm-2 Å -1
if we measure per unit wavelength interval, or
f = ergs s-1 cm-2 Hz -1
(pronounced f nu if we measure per unit frequency interval.
Classifying Stellar Spectra
Observations
A single stellar spectrum is produced when starlight is focused by a telescope onto a
spectrometer or spectrograph, where it is dispersed (spread out) in wavelength and
recorded photographically or electronically.
If the star is bright, we may obtain a high-dispersion spectrum, that is, a few mÅ per
millimetre on the spectrogram, because there is enough radiation to be spread broadly
and thinly. At high dispersion, a wealth of detail appears in the spectrum, but the
method is slow (only one stellar spectrum at a time) and limited to fairly bright stars.
Dispersion is the key to unlocking the information in starlight.
The Spectral-Line Sequence
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At first glance, the spectra of different stars seem to bear no relationship to one
another. In 1863, however, Angelo Secchi found that he could crudely order the
spectra and define different spectral types. Alternative ordering schemes appeared in
the ensuing years, but the system developed at the Harvard Observatory by Annie J.
Cannon and her colleagues was internationally adopted in 1910. This sequence, the
Harvard spectral classification system, is still used today. (About 400,000 stars were
classified by Cannon and published in various volumes of the Henry Draper Catalogue,
1910-1924, and its Extension, 1949.
At first, the Harvard scheme was based upon the strengths of the hydrogen Balmer
absorption lines in stellar spectra, and the spectral ordering was alphabetical (A
through to P). Some letters were eventually dropped, and the ordering was rearranged
to correspond to a sequence of decreasing temperatures (see the effects of the
Boltzmann and Saha equations): OBAFGKMRNS. Stars nearer the beginning of the
spectral sequence (closer to O) are sometimes called early-type stars, and those closer
to the M end are referred to as late-type.
Each spectral type is divided into ten parts from 0 (early) to 9 (late); for example, . . . F8
F9 G0 G1 G2 . . . G9 K0 . . . . In this scheme, our Sun is spectral type G2. In 1922, the
International Astronomical Union (IAU) adopted the Harvard system (with some
modifications) as the international standard.
Many mnemonics have been devised to help students retain the spectral sequence. A
variation of the traditional one is “Oh, Be a Fine Girl, Kiss Me Right Now, Smack.”
The next Figure shows exemplary stellar spectra arranged in order; note how the conspicuous spectral features strengthen and diminish in a characteristic way through the
spectral types.
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Comparison of spectra observed for seven different stars having a range of surface
temperatures. The hottest stars, at the top, show lines of helium and multiply-ionised
heavy elements. In the coolest stars, at the bottom, helium lines are not seen, but lines of
neutral atoms and molecules are plentiful. At intermediate temperatures, hydrogen
lines are strongest. The actual compositions of all seven stars are about the same.
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The Temperature Sequence
The spectral sequence is a temperature sequence, but we must carefully qualify this
statement. There are many different kinds of temperatures and many ways to determine them.
Theoretically, the temperature should correlate with spectral type and so with the
star’s colour. From the spectra of intermediate-type stars (A to K), we find that the
(continuum) colour temperature does so, but difficulties occur at both ends of the
sequence. For O and B stars, the continuum peaks in the far ultraviolet, where it is
undetectable by ground-based observations. Through satellite observations in the far
ultraviolet, we are beginning to understand the ultraviolet spectra of O and B stars.
For the cool M stars, not only does the Planck curve peak in the infrared, but
numerous molecular bands also blanket the spectra of these low-temperature stars.
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When the strengths of various spectral features are plotted against excitationionisation (or Boltzmann-Saha) temperature; the spectral sequence does correlate with
this temperature as seen below;
In practice, we measure a star’s colour index, CI = B - V, to determine the effective
stellar temperature.
If the stellar continuum is Planckian and contains no spectral lines, this procedure
clearly gives a unique temperature, but observational uncertainties and physical
effects do lead to problems:
(a) for the very hot O and B stars, CI varies slowly with Teff and small uncertainties in
its value lead to very large uncertainties in T;
(b) for the very cool M stars, CI is large and positive, but these faint stars have not
been adequately observed and so CI is not well determined for them;
(c) any instrumental deficiencies, calibration errors, or unknown blanketing in the B or
V bands affect the value of CI - and thus the deduced T. Hence, it is best to define the
CI versus T relation observationally.
SPECTROSCOPY
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• We have discussed stellar spectra and classification on an empirical basis:
Spectral sequence
O B A F G K M
Temperature
~40,000 K
---->
2500 K
Classification based on relative line strengths of He, H, Ca, metal, molecular
lines.
• We will now look a little deeper at stellar spectra and what they tell us about
stellar atmospheres.
Radiative Transfer Equation
• Imagine a beam of radiation of intensity I passing through a layer of gas:
Power passing into volume
Area
dA
E = I d dA d

Power passing out
of volume
E  + dE 
where I = intensity into
solid angle element d 
path length ds
NB in all these equations subscripts  can be replaced by 
In the volume of gas there is:
ABSORPTION - Power is reduced by amount
dE = -   E ds = -   I d dA d ds
where  is the ABSORPTION COEFFICIENT or OPACITY
= the cross-section for absorption of radiation of wavelength  (frequency )
per unit mass of gas.
Units of are m2 kg-1
The quantity  is the fraction of power in a beam of radiation of wavelength
 absorbed by unit depth of gas. It has units of m-1. (NB in many texts  is
simply given the symbol  in the equations given here - beware!)
EMISSION - Power is increased by amount
dE = j  d dA d ds
(1)
where j = EMISSION COEFFICIENT = amount of energy emitted per second
per unit mass per unit wavelength into unit solid angle.
Units of j (j) are W kg-1 µm-1 sr-1 (W kg-1 Hz-1 sr-1) or m s-3 sr-1
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(NB power production per unit volume per unit wavelength into unit solid
angle is  =j More confusion is possible here, since is also the
symbol used for total power output of a gas, units are W kg-1, - Beware!)
So total change in power is
dE = dI d dA d = -   I d dA d ds + j  d dA d ds
which reduces to
dI = -   I ds + j  ds
dI
ds
= - I + j 
(2)
(3)
This is a form of the radiative transfer equation in the plane parallel case.
Optical depth
• Take a volume of gas which only absorbs radiation (j = 0) at  :
dI = -   I ds
For a depth of gas s, the fractional change in intensity is given by
I (s)
s

dI
 =
-  ds

I
I (0) 
0



ln (
Integrating ==>
I (s)

I (0)
s
) = -

  ds
0

s
-
I (s) = I (0) e
==>

 ds
0

We define Optical Depth 
s

   ds

So


(4)
-

I (s) = I (0) e


(5)
• Intensity is reduced to 1/e (=1/2.718 = 0.37 ) of its original value if optical
depth = 1.
• Optical depth is not a physical depth. A large optical depth can occur in a
short physical distance if the absorption coefficient  is large, or a large
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physical distance if  is small.
Full Radiative Transfer Equation again
dI
= - I + j 
ds
divide by  
dI

j

= -I +

 ds



dI

d
= -I + S

(6)


As ds --> 0,  is constant over ds. 
This is the RADIATIVE TRANSFER EQUATION in the plane parallel case.
Define:
j
S = 
or
j =  S


 


where
S is the SOURCE FUNCTION.
Radiative transfer in a blackbody
• Remember definition of a blackbody as a perfect absorber and emitter of
radiation. Matter and radiation are in THERMODYNAMIC EQUILIBRIUM,
ie gross properties do not change with time. Therefore a beam of radiation in
a blackbody is constant:
dI
= 0 = - I + j 
ds
==> 0 =  (I - S),
i.e.
I = S.
but for a blackbody I = B
2
B =

2hc

the PLANCK FUNCTION
3
1
hc/kT
(e
from definition of source function, j =  S
- 1)
B =

2h
2
c
1
hkT
(e
- 1)
Summary: in complete thermodynamic equilibrium the source function equals the
Planck function,
i.e.
j =  B
(Kirchoff's Law).
(7)
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• In studies of stellar atmospheres we make the assumption of LOCAL
THERMODYNAMIC EQUILIBRIUM (LTE), i.e. thermodynamic equilibrium
for each particular layer of a star.
• Note that if incoming radiation at a particular wavelength (e.g. in a spectral
line) enters a blackbody gas it is absorbed, but emission is distributed over all
wavelengths according to the Planck function. All information about the
original energy distribution of the radiation is lost. This is what happens in
interior layers of a star where the density is high and photons of any
wavelength are absorbed in a very short distance. Such a gas is said to be
optically thick (see below).
Emission and Absorption lines
•  the absorption coefficient describes the efficiency of absorption of material
in the volume of gas. In a low density gas, photons can generally pass
through without interaction with atoms unless they have an energy
corresponding to a particular transition (electron energy level transition, or
vibrational/rotational state transition in molecules). At this particular
energy/frequency/wavelength the absorption coefficient  is large.
• Let's imagine the volume of gas shown earlier with both absorption and
emission:
I
I  (0)
path length s
dI

d
= S - I



Multiply both sides by e and re-arrange
dI
==>

d



e + I e = S e



==>
d
d


(I e ) = S e 



integrate over whole volume, i.e. from 0 to s, or 0 to 
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

I e

==>
=
0

S e

I e - I(0) = S e - S
==>
==>


I
I(0) e- +
radiation left
over from light
entering box.
=

0
assuming S = constant along path
S (1 - e- )
light from radiation
emitted in the
box.
(8)
 >> 1: OPTICALLY THICK CASE
If  >> 1, then e- --> 0, and eqn (8) becomes I = S
In LTE
S = B, the Planck function.
So for an optically thick gas, the emergent spectrum is the Planck function,
independent of composition or input intensity distribution. True for stellar
photosphere (the visible "surface" of a star).
• Case 1
• Case 2
(9)
 << 1 OPTICALLY THIN CASE
-
If  << 1, then e  ≈ 1 - 
(first two terms of Taylor series expansion)
eqn (8) becomes
I =
==>I = I(0) +  ( S - I(0) )
I(0) (1 - )
+
S (1 - 1 + )
• If I(0) = 0 : no radiation entering the box (from direction of interest):
From eqn (8)
I =  S
(=  B in LTE)
Since  = ∫  , then
I =   s S
If  is large (true at wavelength of spectral lines) then I is large,
we see EMISSION LINES. This happens for example in gaseous
nebulae or the solar corona when the sun is eclipsed.
• If I(0) ≠ 0 , let's examine eqn (8)
I = I(0) +  ( S - I(0) )
If S > I(0) then right hand term is +ve
when  is large (ie  is large) we see higher intensity than I(0)
==>
EMISSION LINES ON BACKGROUND INTENSITY.
If S < I(0) then right hand term is -ve
when  is large (ie  is large) we see lower intensity than I(0)
==>
ABSORPTION LINES ON BACKGROUND INTENSITY.
For stars we see absorption lines. This means I(0) > S,
(10)
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i.e. (intensity from deeper layers) > (source function for the top layers
Assuming LTE (S = B) the source function increases as temperature
increases:
I(0) = B(Tdeep layer) > S = B(Touter layer).
Therefore temperature must be increasing as we go into the star for
absorption lines to be observed.
• To summarise: 4 possibilities
- We see CONTINUUM RADIATION for an optically thick gas
(= PLANCK FUNCTION assuming LTE).
- We see EMISSION LINES for an optically thin gas.
- We see ABSORPTION LINES + CONTINUUM for an optically thick gas
overlaid by optically thin gas with temperature decreasing outwards.
- We see EMISSION LINES + CONTINUUM for an optically thick gas
overlaid by an optically thin gas with temperature increasing outwards.
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 radiation, 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
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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.
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.
38 PH507
Multiwavelength
20
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 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
38 PH507
Multiwavelength
21
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
(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.
n=•
n=4
n=3
13.6 eV
12.73 eV
12.07 eV
n=2
10.19 eV
n=1
Lyman
Series
Balmer Paschen
Series Series
0 eV
• Series of lines seen
-LYMAN SERIES transitions to/from n=1 lines seen in
ultraviolet
PH507
Astrophysics
-BALMER SERIES ""
-PASCHEN SERIES""
Dr. S.F. Green
n=3
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=2
""
infrared ...
n=•
n=4
n=3
n=2
• Ionization potential for Hydrogen is
13.6 eV.
• Energy of absorbed photon is
n=1
22
visual
1/2 m ev 2
13.6 eV
12.73 eV
12.07 eV
10.19 eV
0 eV
2/2
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)).
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- nm, Lyman- 102.57 nm 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nm, H 486.13 nm 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
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Astrophysics
Professor Michael Smith
23
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.
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 unit volume (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.
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.
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Astrophysics
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• 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:

n=2
-Ein /kT
gin e
• The higher the Ionization Potential, i, the lower the fraction of atoms in the
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 m-3.
UII/UI
2
~2
UIII/UII
~0.5
g1
2
1
g2
2
6
Hydrogen:
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.
Calcium:
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Astrophysics
Professor Michael Smith
25
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 con
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.
• 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
for absorption
B21  -1
A12  -1
Total 
• We can now calculate  for a given gaseous species.
(removing spectral line opacities for clarity):
For Hydrogen
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Lyman continuum
absorption
Professor Michael Smith
26
falls off with decreasing 

due to  -1 dependence
Log 
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
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, for H:
N3/N1 = 6 x 10-10
But from Saha eqn
N(H)/N(H-) ≈ 3 x 107
Therefore N(H-)/N3 ≈ 500.
PH507
Astrophysics
Log 
Professor Michael Smith
27
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.


We see that the brightness of the solar disk decreases from the centre to the
limb - this effect is termed limb darkening.
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Astrophysics
Professor Michael Smith
28
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.
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.
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Astrophysics
Professor Michael Smith
29
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.
At this point, you may have discerned an apparent paradox: how can the solar
limb appear darkened when the temperature rises rapidly through the
PH507
Astrophysics
Professor Michael Smith
30
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.
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

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Astrophysics
Professor Michael Smith
31
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
32
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
33
E
Star
B
A
o
D

Observer
C
B
A
D
E
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.
PH507
Astrophysics
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34
Radiation Mechanisms
1. 21 cm
Hydrogen gas is observed in a variety of states: in ionized, neutral
atomic, and molecular forms. The ionized hydrogen emits light in
the visible band as the electrons recombine with the protons and
the neutral atomic and molecular hydrogen emits light in the radio
band of the electromagnetic spectrum.
Most of the hydrogen in space (far from hot O and B-type stars) is
in the ground state. The electron moving around the proton can
have a spin in the same direction as the proton's spin (i.e.,
parallel) or spin in the direct opposite direction as the proton's
spin (i.e., anti-parallel). The energy state of an electron spinning
anti-parallel is slightly lower than the energy state of a parallelspin electron.
Remember that the atom always wants to be in the lowest energy
state possible, so the electron will eventually flip to the antiparallel spin direction if it was somehow knocked to the parallel
spin direction. The energy difference is very small, so a hydrogen
atom can wait on average a few million years before it undergoes
this transition.
The two levels of the hydrogen 1s ground state, slightly split by
the interaction between the electron spin and the nuclear spin.
The splitting is known as hyperfine structure.
Even though this is a RARE transition, the large amount of
hydrogen gas means that enough hydrogen atoms are emitting the
21-cm line radiation at any one given time to be easily detected
with radio telescopes. Our galaxy, the Milky Way, has about 3
PH507
Astrophysics
Professor Michael Smith
35
billion solar masses of H I gas with about 70% of it further out in
the Galaxy than the Sun. Most of the H I gas is in disk component
of our galaxy and is located within 720 light years from the
midplane of the disk.
What's very nice is that 21-cm line radiation is not blocked by
dust! The 21-cm line radiation provides the best way to map the
structure of the Galaxy.
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/h21.html
2. Thermal free-free or Bremsstrahlung emission
Another form of thermal emission comes from gas which has been
ionized. Atoms in the gas become ionized when their electrons
become stripped or dislodged. This results in charged particles
moving around in an ionized gas or "plasma", which is a fourth state
of matter, after solid, liquid, and gas. As this happens, the electrons
are accelerated by the charged particles, and the gas cloud emits
radiation continuously. This type of radiation is called "free-free"
emission or "bremsstrahlung".
3. Synchrotron radiation
Non-thermal emission does not have the characteristic signature
curve of blackbody radiation. In fact, it is quite the opposite, with
emission increasing at longer wavelengths.The most common form
of non-thermal emission found in astrophysics is called synchrotron
emission. Basically, synchrotron emission arises by the acceleration
of charged particles within a magnetic field. Most commonly, the
charged particles are electrons. Compared to protons, electrons
have relatively little mass and are easier to accelerate and can
therefore more easily respond to magnetic fields.
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As the energetic electrons encounter a magnetic field, they spiral
around it rather than move across it. Since the spiral is continuously
changing the direction of the electron, it is in effect accelerating, and
emitting radiation. The frequency of the emission is directly related
to how fast the electron is traveling. This can be related to the initial
velocity of the electron, or it can be due to the strength of the
PH507
Astrophysics
Professor Michael Smith
36
magnetic field. A stronger field creates a tighter spiral and therefore
greater acceleration.
For this emission to be strong enough to have any astronomical
value, the electrons must be traveling at nearly the speed of light
when they encounter a magnetic field; these are known as
"relativistic" electrons. (Lower-speed interactions do happen, and
are called cyclotron emission, but they are of considerably lower
power, and are virtually non-detectable astronomically).
As the electron travels around the magnetic field, it gives up energy
as it emits photons. The longer it is in the magnetic field, the more
energy it loses. As a result, the electron makes a wider spiral
around the magnetic field, and emits EM radiation at a longer
wavelength. To maintain synchrotron radiation, a continual supply of
relativistic electrons is necessary. Typically, these are supplied by
very powerful energy sources such as supernova remnants,
quasars, or other forms of active galactic nuclei (AGN).
It is important to note that, unlike thermal emission, synchrotron
emission is polarized. As the emitting electron is viewed side-on in
its spiral motion, it appears to move back-and-forth in straight lines.
Its synchrotron emission has its waves aligned in more or less the
same plane. At visible wavelengths this phenomenon can be viewed
with polarized lenses (as in certain sunglasses, and in modern 3-D
movie systems).
Synchrotron radiation is electromagnetic radiation, similar to cyclotron
radiation, but generated by the acceleration of ultrarelativistic (i.e., moving
near the speed of light) electrons through magnetic fields. This may be
achieved artificially by storage rings in a synchrotron, or naturally by fast
moving electrons moving through magnetic fields in space. The radiation
typically includes infrared, optical, ultraviolet, x-rays.
Synchrotron radiation is also generated by astronomical structures and
motions, typically where relativistic electrons spiral (and hence change
velocity) through magnetic fields. Two of its characteristics include (1)
Non-thermal radiation (2) Polarization.[
4. inverse Compton radiation
Inverse Compton scattering is important in astrophysics. In X-ray
astronomy, the accretion disk surrounding a black hole is believed to
produce a thermal spectrum. The lower energy photons produced from
this spectrum are scattered to higher energies by relativistic electrons in
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Astrophysics
Professor Michael Smith
37
the surrounding corona. This is believed to cause the power law
component in the X-ray spectra (0.2-10 keV) of accreting black holes.
The effect is also observed when photons from the Cosmic microwave
background move through the hot gas surrounding a galaxy cluster. The
CMB photons are scattered to higher energies by the electrons in this gas,
resulting in the Sunyaev-Zel'dovich effect.
\
The Inverse Compton process boosts up synchrotron photons by
means of scattering against the high energy electrons. Since that
the electrons that scatter against the synchrotron photons, belong
to the same seed of the electrons that have produced the
synchrotron photons, this process is also called ``Self
Synchrotron Compton'' or SSC
5. Masers
Another form of non-thermal emission comes from masers. A
maser, which stands for "microwave amplification by stimulated
emission of radiation", is similar to a laser (which amplifies radiation
at or near visible wavelengths). Masers are usually associated with
molecules, and in space masers occur naturally in molecular clouds
and in the envelopes of old stars. Maser action amplifies otherwise
faint emission lines at a specific frequency. In some cases the
luminosity from a given source in a single maser line can equal the
entire energy output of the Sun from its whole spectrum.
Masers require that a group of molecules be pumped to an
energized state (labeled E2 in the diagram at right), like compressed
springs ready to uncoil. When the energized molecules are exposed
to a small amount of radiation at just the right frequency, they
uncoil, dropping to a lower energy level (labeled E1 in the diagram),
and emit a radio photon. The process entices other nearby
molecules to do the same, and an avalanche of emission ensues,
resulting in the bright, monochromatic maser line. Masers rely on an
external energy source, such as a nearby, hot star, to pump the
molecules back into their excited state (E2), and then the whole
process starts again.
The first masers to be discovered came from the hydroxl radical
(OH), silicon oxide (SiO), and water (H2O). Other masers have been
discovered from molecules such as methanol (CH3OH), ammonia
(NH3), and formaldehyde (H2CO).
PH507
Astrophysics
Professor Michael Smith
38