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Radiation transfer
The propagation of electromagnetic radiation through a medium is governed by
the equation of radiation transfer.
To talk of radiative transfer, we are
implicitly assuming that the conditions are such that transfer can occur - these
conditions are embodied in what is called the radiative equilibrium i.e.,
radiation is absorbed, re-emitted and scattered through the medium and can
propagate outwards to the observer. The radiation is modified by its transit and
this, in principle, permits us to infer something about the properties about the
medium (or media) through which the radiation has passed.
Schematically we
have:
IIN
Medium
IOUT
Towards
Observer
From
Source
The radiation that we detect is the sum total (perhaps zero!) of the original
radiation, modified by any emission and absorption that it has undergone when
transiting the medium. The transmission of radiation from one body (the source)
to us (the observer) is a case of energy transfer and is how we learn about the
Universe.
Measuring radiation
The two essential measures of energy flow are:
i)
the energy flow in a particular direction is called the specific intensity I
(sometimes also called the brightness), where I is the energy flow through
unit area in unit time per unit solid angle (i.e., a steradian, sr, where
there are 4  steradians in a sphere), with the unit area being
perpendicular to the direction chosen by the solid angle (which may not
coincide with the direction of radiation flow),
and:
ii)
the net energy flow summed over all directions (also called the flux, F) is
the energy flow through unit area in unit time, with the unit area fixed to
be perpendicular to the direction of net radiation flow.
Note that an
isotropic intensity will give zero flux as the positive and negative
1
directions of the radiation will cancel.
Strictly speaking, the definition
given here defines the bolometric flux since it measures the energy over
all
spectral
wavebands.
[Note:
sometimes
the
number
of
particles/photons flowing through an equivalent unit area is defined as a
flux, but the energy flux is the definition we will use here.]
In the diagram we consider the emission of radiation in a small cone around the line-of-sight to
the observer. The radiation is emitted from a region dA on the surface of the medium.
Since we are interested in spectra, and absorption and emission occur in a
frequency-dependent manner, we need to consider the intensity and flux per unit
wavelength or frequency (i.e., monochromatic quantities). It is the characteristic
spectral fingerprint, or signature, imprinted on the transmitted spectrum in the
form of such absorption or emission that enables us to disentangle the properties
of the gas traversed by the radiation.
Define I as the intensity of radiation passing perpendicular to the surface
element dA. In the MKS system, the units of I will be W m–2 Hz–1 sr–1. [Note
that I d= I d]
If the intensity also varies over time, then the change in
energy will be:
dE = dI dA d d dt
The total (bolometric) intensity can be found from simple integration over all
frequencies:
I   I dν

2
However, in terms of total energy output, we would integrate the spectrum over
all frequencies and directions to obtain the total (bolometric) emission of a star:
F   I cos θ dω

where the integral sign indicates that the integration is over all angles, and the
cos  term takes account of the projection of the area perpendicular to our line-ofsight relative to the normal to the source surface.
difference between intensity and flux:
It is important to note the
I is independent of the distance to the
source, whereas F obeys the standard inverse-square law. The specific intensity
can only be measured if we can resolve the emitting source – as a result it is the
flux that we use when measuring the radiation of stars.
For a spherical star of radius, R, we can obtain the total luminosity by integrating
over all directions to obtain:
L = 4  R2 Fout
where: Fout is the flux at the star's surface. Observations of point sources (such
as stars), yield the observed flux (L / 4  d2) for an object at distance, d.
For
extended objects (such as solar system objects, nebulae, and galaxies), we
measure the intensity received from various parts of the object (the surface
brightness, with the units of brightness per area, typically in terms of flux per
square arcsecond, or magnitudes per square arcsecond.
Inside a gas cloud or star, we are mainly concerned with the mean intensity, J,
which is found from integration of the intensity over all angles:
J
1
I dω
4π 
where the integral sign indicates that the integration is performed over the whole
unit sphere centered on the point of interest, and the units of J are the same as
those of I i.e., W m–2 sr–1.
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Emission and absorption coefficients
The volume emission coefficient (indicated by the symbol j), is the amount of
energy emitted into unit solid angle from unit volume per unit frequency per unit
solid angle per unit time. The units for j are therefore W m–3 Hz–1 sr–1
For a box of hot gas, of area dA and length dS, the energy emitted into unit solid
angle, d, over a small frequency interval, d, is:
dEemitted = j (dS dA) d d dt
Equivalently, the absorption of radiation of initial intensity I, such as by a box of
cold gas is:
dEabsorbed =  I (dS dA) d d dt
where  is the absorption coefficient (also called the opacity) of the medium
(which can change rapidly with frequency near absorption lines).
As energy is
quoted in units of joules, the units of intensity are in units of J m–2 Hz–1 sr–1 s–
1, and  has dimensions of length–1.

The absorption coefficient per unit
mass, , is important when considering radiation pressure, and is defined by:
dI / I = –   ds
where:
dI is the change in radiation intensity along distance element ds,  is the
gas density.  also has a specific name, and is called the absorption
coefficient per unit volume.
The negative sign indicates that the intensity
decreases with distance into the medium.
Expressed in words, the absorption
coefficient per unit volume is the fractional decrease in intensity per unit distance.
The absorption cross-section, sigma, is given by:
 =   / n
where: n is the number of absorbing atoms per unit volume. For absorption that
is uniform along the path under consideration, we can therefore express the
absorption of radiation in a simple form relative to the incident intensity I(0), and
the path length, s, as:
I = I(0) e–(s )
or:
I = I(0) e–
where we have combined the exponential terms into a single variable  which is
called the (frequency-dependent) optical depth of the medium, and is unitless.
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Note that for  = 1 (an optical depth of unity), the transmitted intensity is 1/e (or
approximately 37%) of the incident intensity.
Aside from the simplicity of the
optical depth relationship, the virtue of the use of optical depth is that the
apparent visible surface of the Sun, or any semi-transparent object, occurs for an
optical depth of unity.
In a generalised form, the incremental optical depth is given by:
d =  dz = –  ds
where: s is measured in the direction of radiation travel (i.e., from source to
observer), and z is measured from observer to source:
z=0
=0
s
z
Schematic diagram showing the relationship between the source and observer co-ordinate
systems.
For realistic media,  varies with position and the optical depth is given by
integrating over the entire path:
z
τ ν   σ (z) dz
0
i.e., we measure the optical depth into the object from the direction of the
observer. Note that this definition enables us to define the optical depth into the
surface of a thick medium (such as the Sun) from the outside, even though we
can not penetrate the entire body of gas. Note also that the optical depth
contains a combination of the absorption coefficient and the gas column, both of
which are both important astrophysical quantities.
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The net difference between the emission and absorption along the cylinder is:
 E = dEem – dEabs
= (j–  I) ds dA d d dt
and this represents a change in the intensity:
= dI dA d d dt
which can be rearranged as:
dI / ds = j –  I
which is the equation of radiative transfer in its most basic form.
If we divide this equation by , we obtain:
dI υ
j
 υ  I υ  Sυ  I υ
σ υ ds σ υ
where: S represents the ratio of the emission to the absorption coefficients, and
is called the source function.
From the equations for optical depth given previously, we can restate this
dI
 I  S
dτ
equation in optical depth form:
Note that, for
dI / d = 0, the emitted and absorbed radiation are in equilibrium.
When this happens, radiation interacts strongly with the matter, and the gas is
said
to
be
in
thermodynamic
equilibrium
(or
local
thermodynamic
equilibrium, if the approximation is only good locally – this is frequently
assumed in stellar models). This is Kirchoff’s Law, and a sample of the interior
of a star is a good example of a near-equilibrium system. Consequently, for a
perfect absorber of radiation, the emitted radiation is described by the Planck
equation for blackbody radiation at the gas temperature, T:
S
 B (T ) 
υ
υ
2hυ3
1
c 2 e h kT  1
This equation is important for the derivation of stellar atmospheric properties. In
general terms, when material is in thermodynamic equilibrium it is in mechanical,
thermal, and chemical equilibrium.
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LTE and non-LTE in the case of stars
In studying stellar structure we are interested in the interplay of radiation and
matter in stars. For photons in the interior region, the simplifying assumption is
that all constituents of the gas (plasma) are in their most probably microstate as
a result of their random or uncorrelated collisions.
As a consequence of this,
matter is in thermodynamic equilibrium and all aspects of the plasma can be
determined from the temperature alone, which represents the mean energy of the
plasma, together with the equilibrium distribution function for the particular type
of particle under consideration.
The basis of this assumption is that randomising collisions occur on a fast enough
timescale (or, equivalently, in a small enough region) that the state variables can
be considered constant. Such is clearly true in the depths of a star, but at some
point this must break down as the surface is approached. The breakdown occurs
because the photons in equilibrium with the plasma obey different statistics
(Bose) from those pertaining to matter (Maxwell-Boltzmann), and the mean free
path for photons is much longer than that for matter. As a result, photons are
expected to be the first particles to detect the boundary with space. Since the
probability of escape of a photon depends on its frequency and the atomic physics
of the opacity source, not all photons escape with equal ability. Thus the photon
distribution departs increasingly from Planckian as the boundary is approached,
and strict thermodynamic equilibrium breaks down.
Exercise: calculate the mean temperature change over 1 cm of solar
material, assuming that the core is at 15 million K, and the photosphere
is at 6,000 K. Take the solar radius as 700,000 km.
Because of the relatively long free path for photons, the state variables for the
plasma over a “typical” mean free path will vary significantly relative to the value
of the variables themselves.
Hence the radiation field at the boundary will be
made up of photons arising in very different environments and the characteristics
of the radiation field will no longer be determined by the values of one set of state
variables, but will depend on the solution for the entire atmosphere traversed by
the photons.
An important aspect of equilibrium is the timescale of equilibrium is the timescale
of relevant phenomena. In the case of stars we need to consider the timescales
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for changes in nuclear energy production, chemical
changes (abundance
variations due to nuclear or convective motions), mechanical changes (convection
or shocks), and compare these with the timescale for thermal equilibrium.
We
find that: tnuclear > tchemical > tmechanical > tthermal so once a star tends to achieve
thermal stability quite rapidly even if conditions are changing. This is, together
with the small changes in conditions over small volumes, is the justification for
the assumption of LTE.
Interpretation of the equation of radiative transfer
The formal solution of the radiative transfer equation can be derived by
multiplying the previous equation:
dI
 I  S
dτ
Multiplying both sides by
e  we can solve the integral:
Setting the optical depth at the input face to be zero, and the total optical depth
to be τ , we therefore derive the formal solution to the optical depth equation,
dI 
e  I e   S e 
dτ
or
dI 
e  I e   S e 
dτ
d
(I e  )  S e 
dτ
 I e    S e  dτ
I ( a )e  a - I ( b )e  b   S e  dτ
I ( a )  I ( b )e
( b  a )
b
  S e  dτ
a
which yields the observed intensity of the radiation:
I υ  I υ (0) e
τυ
-τυ
  Sυe- τ υ dτ υ
The first part of this solution represents
0
the exponential decrease in the
transmitted intensity of the radiation incident on the medium, whereas the
second term represents the emission (and subsequent absorption) of radiation
from within the medium.
The frequency dependence of the emission and
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absorption components leads to the formation of emission and/or absorption
features.
The whole radiative transfer equation may be expressed in words as: The
radiative energy emerging from a medium is equal to that entering it, plus any
energy emitted by the medium, minus any energy absorbed by the medium.
Spectral lines and optical depth
For the case of an homogenous medium, the absorption cross-section () is
constant throughout and, as noted previously, the optical depth is given by the
simple relationship  = s . Two regimes of optical depth are important: one is
where  << 1 (the optically thin regime), and the other is where  >> 1 (the
optically thick regime). Note that we retain the explicit frequency dependence of
optical depth, to allow for a difference between the behaviour between the core of
emission lines (where optical depth may be high), and their wings (where the
optical depth may be much smaller).
Let us review the behaviour of a medium in the two regimes for the case of no
background source, so only the source term component is important:
i)
for the optically thin regime, a photon stands a high probability of
escaping the medium and the transmitted intensity depends more or less
linearly on the length of the optical path. Integrating the right-hand-side
of the above equation yields:
I(s) = S(1 – e–τ)
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Since the optical depth is low, we can expand the exponential using the
first terms of a Taylor series:
I(s) = I(0) e–s ~ I (0) (1 – s)
~ I (0) (1 – )
From this simplification we then can obtain:
 = s  ~ { I (0) – I (s)} / I (0)
We can also rewrite the intensity as:
I ~ S (1 – 1 + ) =  S
Or, in LTE:
I ~  B
if  is constant along the path. In other words, the intensity of the beam
leaving our box will be large when the absorption coefficient is large, and
small when the absorption is small. An example of this is emission from a
thin gas: the emission will be highest where  is highest i.e., at the
frequencies of emission lines so these lines will be much stronger than
emission in the surrounding continuum.
A practical example of this
limiting case is interstellar gas clouds for which the emission is found to be
nearly linear with density from ~ 106 particles m–3 up to 1010 particles m–3
and a further example is the solar corona.
Optical depth is low in the
continuum and line wings of such objects, although it is high close to the
line centres, resulting in an emission line spectrum.
Two illustrative cases: low optical depth with no background source (left),
and low optical depth together with a background source (right).
10
ii)
for the optically thick regime when there is no background source, the
photon is very unlikely to escape the medium and, in the limit, we have:
e -τυ  0
and the observed spectrum will show absorption at that frequency. The
limiting emission is I ~ B and hence the emitted intensity will be
independent of the exact value of .
For a blackbody  is large at all frequencies, so the emission is that predicted by
the source function – in this case the Planck function.
An example of the high optical depth case is to be found in the emission of stars,
although exact agreement with the Planck function is disturbed because the
source function is not constant along the path of the from the interior to the
surface.
The case of moderate optical depth (left) and high optical depth (right). In the
moderate optical depth case we see absorption due to the line opacity of the
medium. For high optical depths we see only the outer layers of the medium facing
us.
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Thermalisation
When we speak of thermal emission we mean that the source function is the
Planck function.
The Planck function for temperature relevant to the gas in
question sets a fundamental upper limit on the amount of radiation which can be
emitted at a particular frequency – if emission lines are emitted from material
which is optically thick at the line centre, then the lines peak intensity will be
capped, whereas the wings of the line will not be similarly limited (see figure):
The effect of increasing overall optical depth on the intensity of lines and continuum,
and the shape of the emission lines.
Note that low optical depth material can consist of a small amount of a strongly
absorbing material, or a larger amount of weakly absorbing material.
The
subscript also serves as a reminder that properties can change dramatically with
frequency e.g., around the frequency of line atomic or molecular line or
bandhead frequencies.
Opacity, line formation and limb darkening
For the high optical depth case (such as the photosphere of our Sun), the
temperature will decrease with radius from the centre of the star since the energy
flux will be directed down a temperature gradient.
(Conversely, a matter-
temperature gradient will develop whenever there radiation flux is non-zero
throughout some optically thick region.)
In an optically thick line, viewing the
material in the line core will show us material which is closer to the surface and,
hence, it will also be of lower temperature. Hence observation of a spectral line
will show a dark core (arising in the cooler material closest to the surface) sitting
on a brighter background emitted by material further below the surface.
12
τ increasing
Formation of absorption lines in the solar atmosphere. At left is the run of temperature
with respect
to height
(notetangentially
units are km),
right is
the we
location
regions
Similarly,
when
we look
to and
the atSun’s
limb
will of
seevarious
the upper
(cooler)
layers
of line
theprofile
atmosphere
and the
they
willH and
appear
less bright
theline
contributing
to the
(in this case
CaII
K doublet).
Note than
that the
material
centre comes
of the from
solar adisk.
core hasviewed
a highertowards
opacity,the
therefore
higher (i.e. cooler) region.
Scattering
Note also that high optical depth regions usually affect photons by both
scattering and absorbing them. When scattered, a photon’s direction of motion is
changed, and it no longer carries information about its origin, and when
absorbed, a photon will no longer be able to convey information such as the
temperature in the medium in which it originated. An example of the difference
between these effects is provided by considering the effects of clouds and smoke:
when overcast, scattering in the clouds produces roughly uniform illumination
and we can tell night from day (though we can not tell the location of the Sun
with accuracy), but when under a heavy pall of absorbing smoke, we can not
even tell night from day, and illumination is governed by local conditions alone.
13
The plane-parallel approximation
Recall from before that the thickness of the line-producing portion of the Sun’s
atmosphere is relatively small (< 1,000 km or so) and so is much less than the
solar radius. Under these conditions, we may consider the atmosphere of the Sun
(or any other star, or planetary atmosphere with a similar ratio of atmospheric
thickness to object size) to be plane-parallel.
We are familiar with the
appearance of the “flat Earth” in which the horizon appears – to a good
approximation – to be flat due to the limited range of our vision.
Consider the following representation of a plane-parallel atmosphere:
A
z
B
θ
l
As noted before, when we look into the atmosphere we view to an optical depth of
roughly unity. However, it is apparently from the figure that there the height in
the atmosphere of this unity optical depth point depends on the viewing angle (in
this case, observing from B gives a sightline where unity optical depth is reached
higher in the atmosphere).
When viewed vertically from A we have:
z
      z  dz 
0
Where the integration is over dummy variable z’ in the vertical (z) direction.
From B, however, the optical depth seen by observation at the slant angle, θ,
through a longer atmospheric path, l, will be increased by 1/(cos θ) (=sec θ):
  ( ) 

cos 
   sec 
The transfer equation is therefore modified to:
cos 
dI 
 I   S
d 
14
As a result, the emergent intensity is given by:

I (0, )   j z  exp( [  ( z ) / cos  ] dz / cos 
0
where we have extended the integral to infinity as the star is effectively infinite
from the point of view of the atmosphere’s thickness.
If the material is in TE, then conditions do not change with position, and the
material must be optically thick, and the intensity is given by the ratio of the
emissivity to absorptivity (i.e., the source function)):
j    B (T )
Since the atmosphere is assumed to be in TE, then the emission coefficient is
equal to the absorption coefficient at the same depth times the Planck function of
the local temperature. Substituting in to the previous equation gives:

I (0, )   B T  exp( [  ( z ) / cos  ]   ( z ) dz / cos 
0
We can use the relationship between optical depth and absorption to get an
equivalent representation:

I (0, )   B T  exp( [  / cos  ] d  / cos 
0
The choice of equation used will depend on the nature of the problem to be
solved, but note that the angle, θ, is constant of integration for a plane-parallel
atmosphere as curvature can be neglected.
Either equation can be used to
calculate the intensity of emitted radiation coming from a star in any given
direction, provided that the temperature and absorption are known with depth.
Note that the equations link the intensity of the emergent ray to the emissivity of
the radiating layers and the optical depth
Limb-darkening
Unlike most other stars, our Sun shows a well resolved disk and observations can
be made of intensity across the surface. It is readily seen that the solar disk is
non-uniform in brightness in the visible, with distinct shading towards the solar
limb – this is called limb darkening and is illustrated in the figure below,
together with the example of the supergiant Betelgeuse:
15
Limb darkening as observed for our Sun (left), and as derived for the M2 supergiant
Betelgeuse (right).
The explanation of this effect is related to the discussion above, and is illustrated
schematically in the figure below where the difference in height in the solar
atmosphere probed to unity optical depth is shown:
A more careful analysis of the photon travel paths shows that, when averaged
over the whole of the solar (or stellar) disk, the average level to which we can see
in the solar or stellar continuum is
τ ~ 2/3.
We can model the limb darkening using our plane-parallel model atmosphere by
the use of an angular dependent intensity term, called the Eddington
approximation:
I ( )  I (  0) [1  u (1  cos  )]
where x is called the limb-darkening coefficient.
Note that u = 0 corresponds to
no limb darkening, whereas u = 1 corresponds to the case where the brightness
falls to zero at the limb.
Combining with the angular-dependent intensity equation from before gives:

I (0,0) [1  u  u cos  ]   B T  exp( [  / cos  ] d  / cos 
0
Note that, since the Planck function depends on temperature, it also has a depth
dependence.
It is easily shown that the above equation is satisfied if:
B T   I (0,0) [1  u  u   ]
In other words, if we can measure the intensity at the centre of the disk and the
limb-darkening coefficient at some frequency, then we can determine the Planck
function at any depth. Since the Planck function depends on temperature, this
means that we can determine the temperature stratification as a function of
optical depth. Using this information, we can develop a model to fit the
observations – this is called the semi-empirical method.
16
Observation of limb-darkening for the Sun at three colours: red (top curve),
green (middle) and blue (bottom). Note the stronger limb darkening with
increasing frequency.
Note that limb darkening has been inferred for a number of stars through
estimates of light-curve changes in eclipsing binary systems, by interferometric
means, or by means of micro-lensing. It is also important to take limb-darkening
into account when determining stellar diameters by means of these any effects.
Comparison of observed limb-darkening in the visible
with theory
Eddington
cos θ
Observed
approximation
intensity
1.00
1.000
1.000
0.90
0.940
0.944
0.80
0.880
0.898
0.70
0.820
0.842
0.60
0.760
0.788
0.50
0.700
0.730
0.40
0.640
0.670
0.30
0.580
0.602
0.20
0.520
0.552
0.10
0.460
0.450
More advanced point:
By using the assignment μ = cos θ, and by making t a dummy variable for optical
depth, the solution of radiatiive transfer we derived in the plan-parallel case
above can be represented as:

I (  ,0)   S (t ) e t 
0
17
dt


L( S(t))

i.e. the intensity can be viewed as the Laplace Transform of the source function.
Hence the determination of the angular dependence of the emergent intensity is
equivalent to determining the behaviour of the source function with depth. Since
the source function is determined by temperature, determination of the depth
dependence of the source function is equivalent to determining the depth
dependence of the temperature. This is of considerable importance as it enables
stellar models to be directly checked (if only for the outer layers).
Note that for a source function which can be represented as a linear function:
S (t )  a(t )  b
The resultant intensity is then given by:
I (  ,0)  a  b
Hence the coefficient a is a measure of the source function gradient, and the
constant b gives the value of the source function at the boundary. Unfortunately,
the worst representation of stellar models is usually near the surface, but this
corresponds to limb measurements on the limb which is precisely the point where
measurements become the most difficult.
18