Download Lecture 9: Radiation processes Almost all astronomical information

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Astrophysics
Dr Dirk Froebrich
-1-
Lecture 9: Radiation processes
Almost all astronomical information from beyond the Solar System comes
to us from some form of electromagnetic radiation (EMR).
We can now detect and study EMR over a range of wavelength or,
equivalently, photon energy, covering a range of at least 1016 - from short
wavelength, high photon energy gamma rays to long wavelength low
energy radio photons.
Out of all this vast range of wavelengths, our eyes are sensitive to a tiny
slice of wavelengths- roughly from 4500 to 6500 Å. The range of
wavelengths our eyes are sensitive to is called the visible wavelength
range. We will define a wavelength region reaching somewhat shorter (to
about 3200 Å) to somewhat longer (about 10,000 Å) than the visible as the
optical part of the spectrum.
Physicists measure optical wavelengths in nanometers (nm). Astronomers
tend to use Angstroms. 1 Å = 10-10 m = 0.1 nm. Thus, a physicist would
say the optical region extends from 320 to 1000 nm.)
All EMR comes in discrete lumps called photons. A photon has a definite
energy and frequency or wavelength. The relation between photon energy
(Eph) and photon frequency ν is given by:
Eph = hν
or, since c = λ ν
E ph =
hc
λ
where h is Planck’s constant and λ is the wavelength, and c is the speed of
light.
The energy of visible photons is around a few eV (electron volts). (An
electron volt is a non- metric unit of energy that is a good size for
measuring energies associated with changes of electron levels in atoms, and
also for measuring energy of visible light photons. 1 eV = 1.602 x 10-19
Joules.) An approximate value (1 in 104) for the energy of electromagnetic
radiation expressed in electron volts is given by 1234 / λ, where λ is the
wavelength in nanometres.
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Astrophysics
Dr Dirk Froebrich
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In purely astronomical terms, the optical portion of the spectrum is
important because most stars and galaxies emit a significant fraction of
their energy in this part of the spectrum. (This is not true for objects
significantly colder than stars - e.g. planets, interstellar dust and molecular
clouds, which emit in the infrared or at longer wavelengths - or
significantly hotter- e.g. ionised gas clouds, neutron stars, which emit in the
ultraviolet and x-ray regions of the spectrum. Another reason the optical
region is important is that many molecules and atoms have electronic
transitions in the optical wavelength region.
We will define the regions of the Electromagnetic Spectrum to have
wavelengths as follows:
•
•
•
•
•
•
Gamma-rays: < 0.1Å, highest frequency, shortest wavelength,
highest energy.
X-Rays: 0.1Å -- 100Å
Ultraviolet light: 100Å -- 3000Å
Visible light: 3000Å -- 10000Å = 1µ
µm (micrometer or micron)
Infrared Light: 1µ
µm -- 1mm (NIR, MIR, FIR, sub-mm)
Radio waves: >1mm, lowest frequency, longest wavelength,
lowest energy.
Wavelength Range
Radio
Infrared
Visible
Ultraviolet
X-rays
Gamma rays
wavelength > 10-4 m = 1 mm
700 nm < wavelength < 1 mm
400 nm < wavelength < 700 nm
20 nm < wavelength < 400 nm
0.1 nm < wavelength < 20 nm
wavelength < 0.1 nm
PH507
Astrophysics
Dr Dirk Froebrich
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Blackbody Radiation
Where then does a thermal continuous spectrum come from? Such a
continuous spectrum comes from a blackbody whose spectrum depends
only upon the absolute temperature. A blackbody is so named because it
absorbs all electromagnetic energy incident upon it - it is completely black.
To be in perfect thermal equilibrium, however, such a body must radiate
energy at exactly the same rate that it absorbs energy; otherwise, the body
will heat up or cool down (its temperature will change).
Ideally, a blackbody is a perfectly insulated enclosure within which radiation has come into thermal equilibrium with the walls of the enclosure.
Practically, blackbody radiation may be sampled by observing the
enclosure through a tiny pinhole in one of the walls.
The gases in the interior of a star are opaque (highly absorbent)
to all radiation (otherwise, we would see the stellar interior at some
wavelength!); hence, the radiation there is blackbody in character. We
sample this radiation as it slowly leaks from the surface of the star - to a
rough approximation, the continuum radiation from some stars is
blackbody in nature.
Planck’s Radiation Law
After Maxwell's theory of electromagnetism appeared in 1864, many
attempts were made to understand blackbody radiation theoretically. None
succeeded until, in 1900, Max K. E. L. Planck (1858-1947) postulated that
electromagnetic energy can propagate only in discrete quanta, or photons,
each of energy E = hν. He then derived the spectral intensity relationship,
or Planck blackbody radiation law:



 2hν 3  
1
I(ν )dν =
 c 2   hkTν  
 e − 1 


2
where I(v)dν is the intensity (J/m /s/sr) of radiation from a blackbody at
temperature T in the frequency range between v and v + dv, h is
Planck's constant, c is the speed of light, and k is Boltzmann's constant.
Note the exponential in the denominator.
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Dr Dirk Froebrich
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Because the frequency v and wavelength λ of electromagnetic radiation are
related by λv = c, we may also express Planck's formula in terms of the
intensity emitted per unit wavelength interval:
This is illustrated for several values of T:
Note that both I(λ) and I(ν) increase as the blackbody temperature
increases - the blackbody becomes brighter. This effect is easily interpreted
when we note that I(ν)∆ν is directly proportional to the number of photons
emitted per second near the energy hν. The Planck function is special
enough so that its given its own symbol, B(λ) or B(ν), for intensity.
Long wavelengths: Rayleigh tail : B(λ ) ∝ 1 / λ4
Wien’s Law
A blackbody emits at a peak intensity that shifts to shorter wavelengths as
its temperature increases.
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Dr Dirk Froebrich
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• Wilhelm Wien (1864-1928) expressed the wavelength at which the
maximum intensity of blackbody radiation is emitted - the peak (that
wavelength for which dI(λ)/d λ = 0) of the Planck curve (found from
taking the first derivative of Planck's law) - by Wien's displacement
law:
λ max = 2.898 x 10-3 / T
where λ max is in metres when T is in Kelvin. Note that because λ maxT =
constant, increasing one proportionally decreases the other.
For example, the continuum spectrum from our Sun is approximately
blackbody, peaking at λ max ≈ 500nm. Therefore, the surface temperature
is near 5800 K.
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Dr Dirk Froebrich
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The Law of Stefan and Boltzmann
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:
The
formula
is:
•
The constant "sigma" is called the Stefan-Boltzmann constant and is
given
by:
•
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.
PH507
Astrophysics
Dr Dirk Froebrich
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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 = 4πR2σT4
since the surface area of a sphere of radius R is 4πR2
To summarise: A blackbody radiator has a number of special
characteristics. One, a blackbody emits some energy at all wavelengths.
Two, a hotter blackbody emits more energy per unit area and time at all
wavelengths than does a cooler one. Three, a hotter blackbody emits a
greater proportion of its radiation at shorter wavelengths than does a
cooler one. Four, the amount of radiation emitted per second by a unit
surface area of a blackbody depends on the fourth power of its temperature.
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Astrophysics
Dr Dirk Froebrich
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Stellar Material
Our Sun is the only star for which I(λ) 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 Lu (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 4π2 times this flux.
The solar energy distribution curve may be approximated by a Planck
blackbody curve at the effective temperature Teff (or bolometric
temperature Tbol), 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
L = 4π R 2 σ T4eff (e.g. J s-1 )
where σ is the Stefan-Boltzmann constant.
Example: A star with surface temperature T = 1.2 x 104K
•
The SED peak 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.
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Astrophysics
Dr Dirk Froebrich
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Physical Characteristics: statistical mechanics
Stellar Atmospheres
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)
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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.
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 2
kT = KE = mv
2
2
v=
3kT
= 19 km s −1 ≅ 50,000 mph
m
The particle number density is related to both the mass density ρ(kg/m3)
and the composition (or mean molecular weight) µ by the following
definition of µ:
1
µ
=
mH n
ρ
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,
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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)
heavier elements ~1/2 particle per mH
give
(e.g. Carbon gives a nucleus
containing 6 protons and 6
neutrons = 12mH and six
electrons = 7/12, Oxygen gives a
nucleus containing 8 protons and
8 neutrons = 16mH and eight
electrons = 9/16, etc.)
Thus
per
the
number
of
particles
cubic
metre
due
to
hydrogen = 2X ρ / mH,
helium = 3Y ρ / 4mH, and
heavier elements = Z ρ / 2mH.
The total number of particles per cubic metre is then given by the sum of
the above, i.e.
n = (2X ρ / mH) + (3Y ρ / 4mH) + (Z ρ / 2mH).
Rearranging, we obtain:
n = (ρ / 4mH) (8X + 3Y + 2Z).
Now, X + Y + Z = 1, and hence Z = 1 - X - Y, giving:
n = (ρ / 4mH) (6X + Y + 2).
Recalling that ρ = nmHµ, we can write:
µ = 4 / (6X + Y + 2),
which is a good approximation to µ except in the cool outer regions of stars.
For solar composition, Xu=0.747, Yu=0.236 and Zu=0.017, resulting in
µ~0.6, i.e. the mean mass of the particles in the Sun is a little over half the
mass of a proton.
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Astrophysics
Dr Dirk Froebrich
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In general, stellar interior gases are ionised and
µ=
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), µ ~ 0.5, and
for a white dwarf star (X = 0.0, Y = 1.0, Z = 0.0) µ ~ 1.33.
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:
λmaxT = 2.898 x 10-3 m . K, which states that the peak intensity of the
Planck curve occurs at a wavelength λmax that varies inversely with the
Planck temperature T. The value of λmax 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 = σT4 where
σ= 5.67 x 10-8 W/m2 . K4. Then the relation
L = 4πR2σT4eff
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 spectral-line 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.
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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.
Spectrophotometry: the Spectral Energy Distribution (SED)
The goal of the observational astronomer is 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
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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 λ = ergs s-1 cm-2 Å -1
if we measure per unit wavelength interval, or
fν = 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
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
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decreasing temperatures (see the effects of the Boltzmann and Saha
equations): OBAFGKM. 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, O3.5 . . . 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.”
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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Astrophysics
Dr Dirk Froebrich
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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.
Additional Spectral Types:
LTY: This is a continuation of the mass sequence for lower mass stars than
M-dwarfs, which became necessary due to the discovery of these very low
mass objects.
L: temperatures 1300-2000K, some stellar, some sub-stellar, metalhydrates and alkali metal in spectra, >900 known
T: cool brown dwarfs, 770-1000K, prominent methane lines in spectra,
~350 known
Y: ultra cool brown dwarfs, <700K, 14 detected by WISE,
WISE1828+2650 has T=298K
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Astrophysics
Dr Dirk Froebrich
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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 deter-mine them. (I.e. it might be better to talk of a mass
sequence, at least for the luminosity class V objects – the main sequence
stars.)
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 lowtemperature stars.
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Astrophysics
Dr Dirk Froebrich
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Dr Dirk Froebrich
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When the strengths of various spectral features are plotted against
excitation-ionisation (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 Planck-like 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.
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Astrophysics
Dr Dirk Froebrich
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SPECTROSCOPY
• 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 Iλ passing through a layer of gas:
Power passing into volume
Area
dA
Eλ = Iλ dω dA dλ
Power passing out
of volume
E λ + dE λ
where Iλ = intensity into
solid angle element dω
path length ds
NB in all these equations subscripts λ can be replaced by ν
In the volume of gas there is:
ABSORPTION - Power is reduced by amount
dEλ = - κλ ρ Eλ ds = - κλ ρ Iλ dω dA dλ ds
where κλ is the ABSORPTION COEFFICIENT or OPACITY
= the cross-section for absorption of radiation of wavelength λ (frequency
ν) per unit mass of gas.
Units of κλare m2 kg-1
The quantity κλ ρ is the fraction of power in a beam of radiation of
wavelength λ absorbed by unit depth of gas. It has units of m-1. (NB in
many texts κλ ρ is simply given the symbol κλ in the equations given here
- beware!)
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EMISSION - Power is increased by amount
dEλ = jλ ρ dω dA dλ ds
(1)
where jλ = EMISSION COEFFICIENT = amount of energy emitted
per second per unit mass per unit wavelength into unit solid angle.
Units of jλ (jν) are Wkg-1µm-1sr-1 (Wkg-1Hz-1sr-1) or ms-3sr-1 (NB
power production per unit volume per unit wavelength into unit
solid angle is ελ=jλ ρ. More confusion is possible here, since ε is also
the symbol used for total power output of a gas, units are Wkg-1, Beware!)
So total change in power is
dEλ = dIλ dω dA dλ = - κλ ρ Iλ dω dA dλ ds + jλ ρ dω dA dλ ds
which reduces to
dIλ = - κλ ρ Iλ ds + jλ ρ ds
dIλ
ds
= - κ λ ρ Iλ + jλ ρ
(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 (jλ = 0) at λ :
dIλ = - κλ ρ Iλ ds
For a depth of gas s, the fractional change in intensity is given by
I (s)
λ
dI
∫ I
I (0)
λ
s
λ
λ
=
∫0 - κ
λ
ρ ds
s
I (s)
ln (
Integrating ==>
λ
I (0)
) = -
λ
s
-
==>
I (s) = I (0) e
λ
λ
∫ κλ ρ ds
0
∫0 κ
λ
ρ ds
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We define Optical Depth τ
s
τ
∫κ
=
λ
0
λ
ρ ds
(4)
-τ
λ
I (s) = I (0) e
So
λ
(5)
λ
• Intensity is reduced to 1/e (=1/2.718 = 0.37 ) of its original value if
optical depth τ= 1.
• Optical depth is not a physical depth. A large optical depth can occur in
a short physical distance if the absorption coefficient κ is large, or a large
physical distance if κ is small.
Full Radiative Transfer Equation again
dIλ
= - κ λ ρ Iλ + jλ ρ
ds
divide by ρ κλ:
dI
j
λ
κ ρ ds
= -I +
λ
λ
λ
κ
λ
dI
λ
dτ
= -I + S
λ
λ
(6)
As ds --> 0, κλ is constant over ds:
This is the RADIATIVE TRANSFER EQUATION in the plane parallel
case. Define:
λ
j
S =
λ
where
λ
κ
or
j = κ S
λ
λ
λ
λ
Sλ 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, i.e. gross properties do not change with time.
Therefore a beam of radiation in a blackbody is constant:
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dIλ
ds
Dr Dirk Froebrich
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= 0 = - κ λ ρ Iλ + jλ ρ
from definition of source function, jλ = κλ Sλ
==> 0 = κλ (Iλ - Sλ),
i.e. Iλ = Sλ.
but for a blackbody Iλ = Bλ the PLANCK FUNCTION
2
B =
λ
2hc
λ5
3
1
B =
hc/λkT
(e
- 1)
ν
2hν
2
c
1
hν/kT
(e
- 1)
Summary: in complete thermodynamic equilibrium the source function
equals the Planck function,
(7)
i.e.
jλ = κλ Bλ
(Kirchoff's Law).
• 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
• κλ 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
κλ is large.
• Let's imagine the volume of gas shown earlier with both absorption and
emission:
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Astrophysics
Dr Dirk Froebrich
- 24 -
Iλ
I λ(0)
path length s
dI
λ
= S - I
λ
dτ
λ
λ
Multiply both sides by eτλ and re-arrange
dI
==>
τ
τ
τ
eλ + I eλ = S eλ
λ
λ
dτ
λ
λ
d
==>
τ
τ
(I e λ) = S e λ
dτ
λ
λ
λ
integrate over whole volume, i.e. from 0 to s, or 0 to τλ
τ
τ
I eλ
λ
==>
τ
λ
=
0
τ
S eλ
λ
λ
0
assuming Sλ = constant along path
==> Iλ eτλ - Iλ(0) = Sλ eτλ - Sλ
==>
Iλ
Iλ(0) e-τλ +
radiation left
over from light
entering box.
=
Sλ (1 - e-τλ )
light from radiation
emitted in the
box.
(8)
τλ >> 1: OPTICALLY THICK CASE
If τλ >> 1, then e-τλ --> 0, and eqn (8) becomes Iλ = Sλ
(9)
In LTE Sλ = Bλ, 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
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Dr Dirk Froebrich
τλ << 1 OPTICALLY THIN CASE
If τλ << 1, then e-τλ ≈ 1 - τλ
(first two terms of Taylor series expansion)
eqn (8) becomes Iλ = Iλ(0) (1 - τλ) + Sλ (1 - 1 + τλ)
==>Iλ = Iλ(0) + τλ ( Sλ - Iλ(0) )
- 25 -
• Case 2
(10)
• If Iλ(0) = 0 : no radiation entering the box (from direction of interest):
From eqn (8) Iλ = τλ Sλ (= τλ Bλ in LTE)
Since τλ = ∫ κλ , then
Iλ = κλ ρ s Sλ
If κλ is large (true at wavelength of spectral lines) then Iλ is large, we
see EMISSION LINES. This happens for example in gaseous nebulae
or the solar corona when the sun is eclipsed.
• If Iλ(0) ≠ 0 , let's examine eqn (8)
Iλ = Iλ(0) + τλ ( Sλ - Iλ(0) )
If Sλ > Iλ(0) then right hand term is positive
when τλ is large (i.e. κλ is large) we see higher intensity than Iλ(0)
==>
EMISSION LINES ON BACKGROUND INTENSITY.
If Sλ < Iλ(0) then right hand term is negative
when τλ is large (ie κλ is large) we see lower intensity than Iλ(0)
==>
ABSORPTION LINES ON BACKGROUND INTENSITY.
For stars we see absorption lines. This means Iλ(0) > Sλ,
i.e. (intensity from deeper layers) > (source function for the top layers
Assuming LTE (Sλ = Bλ) the source function increases as temperature
increases: Iλ(0) = Bλ(Tdeep layer) > Sλ = Bλ(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.
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Dr Dirk Froebrich
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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) = - me e4 / 32 π2 ε02 n2 ћ2
from Bohr Theory
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
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
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.
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Dr Dirk Froebrich
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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 νbb will produce a transition if
E2 - E1 = hνbb.
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.
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Dr Dirk Froebrich
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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 νbf will convert a bound
electron of energy E1 into a free electron of energy E3 if
E3 - E1 = hνbf.
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. Boundfree 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 νff and moves to a state with energy E4, where
E4 - E3 = hνff.
There is no restriction on the energy of a photon which can induce a freefree 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
hν << 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.
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Astrophysics
Dr Dirk Froebrich
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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:
hν = 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 ~10-8s)
or
INDUCED EMISSION (Photon emitted at same energy and coherently
with incoming photon - as in lasers – stimulated emission). Both produce
EMISSION LINES.
•
Narrow lines are seen since transitions can only occur if photon has
energy (frequency/wavelength) corresponding to difference in energy
levels
• 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
n=3
12.73 eV
12.07 eV
n=2
10.19 eV
n=1
0 eV
Lyman
Series
Balmer Paschen
Series Series
• Series of lines seen
-LYMAN-SERIES transitions to/from n=1 lines seen in UV
-BALMER-SERIES -"n=2
-“visual
-PASCHEN-SERIES -"n=3
-“.
infrared ...
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Dr Dirk Froebrich
30
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.
• Ionization potential for Hydrogen is X = 13.6 eV.
• Energy of absorbed photon is
hν = (X - E(nl)) + mev
2
/2
(48)
1/2 m ev 2
13.6 eV
n=∝
n=4
n=3
12.73 eV
12.07 eV
n=2
10.19 eV
n=1
0 eV
• 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 X 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 absorption from a single energy level will
therefore appear as a series of lines of increasing energy
continuum ∝ δ γ
ν
β
α
λ
(Increasing frequency, decreasing wavelength) up to a limit defined by XE(nl), with an absorption continuum shortward of this limit. The
characteristic of a bound-free 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.
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Dr Dirk Froebrich
31
• For nl=1 the Lyman series (Lyman-α 121.57nm, Lyman-β 102.57nm,
etc.) is observed together with the Lyman continuum shortward of
λ=91.2 nm. (Since interstellar space is populated by very low density
and low temperature hydrogen (i.e. with n=1), photons with λ<91.2nm
are easily absorbed so it is opaque in the near-UV).
For nl=2 the Balmer series (Hα 656.28nm, Hβ 486.13nm, etc.) is
observed together with the Balmer continuum shortward of λ=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 v1 to one with
velocity v2
i.e. hν = 1/2 me v22 - 1/2 me v12
The term means that 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 κλ
• 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 τλ ~ 2/3. In
order to determine the total spectrum, the value of κλ needs to be
determined at all wavelengths. The overall κλ is the sum of the
contributions from each atomic/molecular species in the atmosphere.
Each component of κλ depends on the number of atoms/molecules with
a given energy state capable of absorbing radiation at that frequency and
the absorption efficiency. We deal with the energy state populations first:
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Dr Dirk Froebrich
32
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
NA
gB (EA - EB)/kT
e
gA
=
(50)
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 negative.
• 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
2π me k T
h
2
-Χi /kT
e
where Ne is the electron density (number of electrons per m3), Xi is the
ionization potential of the ith ionization state, Ui+1 and Ui are
PARTITION FUNCTIONS obtained from the statistical weights:
∞
Ui = gi 1 +
∑
-Ei n /kT
gi n e
n =2
• The higher the Ionization Potential, Xi, the lower the fraction of atoms in
the upper ionization state.
The higher the Temperature, T, 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).
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33
• 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) κλ 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
T ~ 6000 K, Ne ~ 7x1019 m-3.
ΧI
ΧII
UII/UI UIII/UII
13.6 eV
2
6.1 eV 11.9 eV
~2
~0.5
g1
2
1
g2
2
6
Hydrogen:
From Saha Equ. 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:
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.
PH507
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Dr Dirk Froebrich
34
• 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 ∝ ν2
for stimulated emission
B21 ∝ ν-1
for absorption
A12 ∝ ν-1
Total κλ
• We can now calculate κλ for a given gaseous species. For Hydrogen
(removing spectral line opacities for clarity):
Lyman continuum
absorption
κ falls off with decreasing λ
λ
due to ν -1 dependence
Log κλ
T~25000K (B star)
Balmer
continuum
absorption
Paschen
continuum
absorption
T~5000K (G star)
30
100
300
1000
λ(nm)
• Similar diagrams exist for other species. The total κλ will be the sum for
all species in the star.
• The region of a star for which optical depth τ~2/3 determines where
observed radiation originates. So if κλ is large, then τλ= 2/3 at a high
level in the atmosphere and if κλ is low, τλ=
 2/3 deep in the atmosphere.
PH507
Astrophysics
Dr Dirk Froebrich
35
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, κλ for hydrogen at low temperatures is low in the
visible region (λ~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 (λ=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 κλ
Dr Dirk Froebrich
36
T~25000K (B star)
H - bound-free H - free-free
T~5000K (G star)
30
100 300
1000 λ(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 StefanBoltzmann 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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Dr Dirk Froebrich
37
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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Dr Dirk Froebrich
38
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
Dr Dirk Froebrich
39
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.
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 p18 where Sλ < Iλ).
κ small
τ~2/3 low in
atmosphere
6500
T (K)
κ high
τ~2/3 high in
atmosphere
4500
Fλ
λ
0
200
400 km
Height above photosphere
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.
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Astrophysics
Dr Dirk Froebrich
40
• 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, ∆λ ~ 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. ∆λ ~ 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. Bulk motions of
gas in convection cells. Gaussian profile.
PH507
Astrophysics
Dr Dirk Froebrich
41
• 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
(km s -1)
200
Receding
+V
A
F
λ
C
B
A
C
100
B
Approaching
-V
λ
λo
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
λ
D
A
λo
λ
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.
PH507
Astrophysics
Dr Dirk Froebrich
C
Fλ
Rotating
gas
shell
42
E
Star
B
A
λo
D
λ
Observer
C
B
A
D
E
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).
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 (i.e. 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.
• “Forbidden lines are allowed in 99.999% of the Universe!”
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Astrophysics
Dr Dirk Froebrich
43
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 parallel-spin electron.
Remember that the atom always wants to be in the lowest energy state
possible, so the electron will eventually flip to the anti-parallel 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 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
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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 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,
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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 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
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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).