Ch. 5 - Basic Definitions
Specific intensity/mean intensity
Flux
The K integral and radiation pressure
Absorption coefficient/optical depth
Emission coefficient
The source function
The transfer equation & examples
Einstein coefficients
Definitions
Specific Intensity
• Average Energy (Eldl) is the amount of energy
carried into a cone in a time interval dt
• Specific Intensity (ergs s-1 cm-1 cm-2 sr-1)
• Intensity is a measure of brightness – the amount
of energy coming per second from a small area of
surface towards a particular direction
• For a black body radiator, the Planck function
gives the specific intensity (and it’s isotropic)
• Normally, specific intensity varies with direction
El dl
El dl
Il 

dldtdAcosd dldtdAcosdd
Mean Intensity
• Average of specific intensity over all
directions, divided by 4p steradians
• If the radiation field is isotropic (same
intensity in all directions), then <Il>=Il
• Black body radiation is isotropic and <Il>=Bl
Jl  Il
1
1

I l d 

4p
4p
2p
p
 
0
0
I l sin dd
Energy Density
• Energy Density (uldl) – how much energy
is in the radiation field:
• Consider a cylindrical volume dAdL
• Energy density is Eldl divided by the
volume dAdL, and integrated over all solid
angles
• For an isotropic radiation field the energy
density uldl = (4p/c)Ildl
• For blackbody radiation,
4p
8phc / l5
ul d l 
Bl dl  hc kT
d
c
e
1
Radiative Flux
• Radiative flux is the rate at which
energy at a given wavelength flows
through (or from) a unit surface area
passes each second through a unit
area in the direction of the z-axis
(ergs cm-2 s-1)
• for isotropic radiation, there is no
net transport of energy, so Fl=0
Fl   I l dl cosd  
2p
 0
p

0
I l dl cos sin dd
Specific Intensity vs. Radiative Flux
• Use specific intensity when the surface is
resolved (e.g. a point on the surface of the
Sun). The specific intensity is independent
of distance (so long as we can resolve the
object). For example, the surface
brightness of a planetary nebula or a
galaxy is independent of distance.
• Use radiative flux when the source isn’t
resolved, and we're seeing light from the
whole surface (integrating the specific
intensity over all directions). The radiative
flux declines with distance (1/r2).
Luminosity
• Luminosity is the total energy
radiated from a star, at all
wavelengths, integrated over a full
sphere.
Class Problem
• From the luminosity and radius of the
Sun, compute the bolometric flux, the
specific intensity, and the mean
intensity at the Sun’s surface.
• L = 3.91 x 1033 ergs sec
• R = 6.96 x 1010 cm
-1
Solution
• F= sT4
• L = 4pR2sT4 or L = 4pR2 F, F = L/4pR2
• Eddington Approximation – Assume I
is independent of direction within the
outgoing hemisphere. Then…
• F = pI 
• J = ½ I (radiation flows out, but not in)
The Numbers
• F = L/4pR2 = 6.3 x 1010 ergs s-1 cm-2
• I = F/p = 2 x 1010 ergs s-1 cm-2 steradian-1
• J = ½I= 1 x 1010 ergs s-1 cm-2 steradian-1
(note – these are BOLOMETRIC – integrated
over wavelength!)
Radiation Pressure
Prad
1 
2
  
I l dl cos ddl
c 0 sphere
• Radiation Pressure – light carries
momentum (p=E/c)
• Isotropic Radiation Pressure – force per
unit area
• Blackbody Radiation Pressure – Prad 
1
u
3
The K Integral and Radiation Pressure
K   I cos d
2
4s 4
PR 
T
3c
• Thought Problem: Compare the
contribution of radiation pressure to
total pressure in the Sun and in other
stars. For which kinds of stars is
radiation pressure important in a
stellar atmosphere?
Absorption Coefficient and Optical Depth
• Gas absorbs photons passing through it
– Photons are converted to thermal energy or
– Re-radiated isotropically
• Radiation lost is proportional to
–
–
–
–
absorption coefficient (per gram)
density
intensity
dI   
pathlength
 I dx
• Optical depth is the integral of the
absorption coefficient times the density
along the path
L
    dx
0
I ( )  I (0)e

Class Problem
• Consider radiation with intensity I(0)
passing through a layer with optical
depth  = 2. What is the intensity of
the radiation that emerges?
Class Problem
• A star has magnitude +12 measured
above the Earth’s atmosphere and
magnitude +13 measured from the
surface of the Earth. What is the
optical depth of the Earth’s
atmosphere?
Emission Coefficient
• There are two sources of radiation within a
volume of gas – real emission, as in the
creation of new photons from collisionally
excited gas, and scattering of photons into
the direction being considered. We can
define an emission coefficient for which
the change in the intensity of the radiation
is just the product of the emission
coefficient times the density times the
distance considered.
dI  j dx
Note that dI does NOT depend on I!
Pure Isotropic Scattering
• The gas itself is not radiating – photons only arise
from absorption and isotropic re-radiation
• Contribution of photons proportional to solid angle
and energy absorbed:
  I ddx
dj dx 
4p
S 
j

 J

j    I d / 4p 
I d   J

4p
J is the mean intensity
dI/d = -I + Jv
The source function depends only on
the radiation field
Pure Absorption
• No scattering – photons come only
from gas radiating as a black body
• Source function given by Planck
radiation law
Einstein Coefficients
• Spontaneous emission proportional to
Nn x Einstein probability coefficient
j = NnAulh
• Induced emission proportional to
intensity
 = NlBluh – NuBulh
Radiative Energy in a Gas
• As light passes through a gas, it is both
emitted and absorbed. The total change of
intensity with distance is just
dIl   l I l dl  jl dl
• dividing both sides by -kldl gives
jl
1 dIl

 Il 
 l  dl
l
The Source Function
• The source function S is defined as
the ratio of the emission coefficient
to the absorption coefficient
• The source function is useful in
computing the changes to radiation
passing through a gas
S  j / 
1 dIl


I

S
l
l
The Transfer Equation   dl
l
• We can then write the basic equation of
transfer for radiation passing through gas,
the change in intensity I is equal to:
dIl = intensity emitted – intensity absorbed
dIl = jldl – lIl dl
dIl /dl = -Il + jl/l = -Il + Sl
• This is the basic equation which must be
solved to compute the spectrum emerging
from or passing through a gas.
Special Cases
• If the intensity of light DOES NOT
VARY, then Il=Sl (the intensity is
equal to the source function)
• When we assume LTE, we are assuming
that Sl=Bl
dIl
 I l  Bl
d l
Thermodynamic Equilibrium
• Every process of absorption is balanced by a
process of absorption; no energy is added or
subtracted from the radiation
• Then the total flux is constant with depth
Frad  Fsurface  sT
4
e
• If the total flux is constant, then the mean
intensity must be equal to the source function:
<I>=S
Simplifying Assumptions
• Plane parallel atmospheres (the depth of a
star’s atmosphere is thin compared to its radius,
and the MFP of a photon is short compared to
the depth of the atmosphere
• Opacity is independent of wavelength (a gray
atmosphere)

I   I l dl
0

S   Sl dl
0
Eddington Approximation
• Assume that the intensity of the radiation
(Il) has one value in all directions toward
the outward facing hemisphere and
another value in all directions toward the
inward facing hemisphere.
• These assumptions combined lead to a
simple physical description of a gray
atmosphere