Transcript Basic Definitions - Indiana University

Basic Definitions
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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
Specific Intensity/Mean Intensity
• Intensity is a measure of brightness –
the amount of energy coming per
second from a small area of surface
towards a particular direction
• erg hz-1 s-1 cm-2 sterad-1
dE
I 
cos dAdwdtdv
1
J 
Id

4
J is the mean intensity averaged over 4
steradians
Flux
• Flux is the rate at which energy at
frequency  flows through (or from) a unit
surface area either into a given hemisphere
or in all directions.
• Units are ergs cm-2 s-1
F   I cos d
F  2
 /2
 I sin  cosd
0
• Luminosity is the total energy radiated
from the star, 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 = 4R2sT4 or L = 4R2 F, F = L/4R2
• Eddington Approximation – Assume I
is independent of direction within the
outgoing hemisphere. Then…
• F = I 
• J = ½ I (radiation flows out, but not in)
The Numbers
• F = L/4R2 = 6.3 x 1010 ergs s-1 cm-2
• I = F/ = 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!)
The K Integral and Radiation Pressure
K   I cos d
2
4s 4
PR 
T
3c
• Class 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    I dx
Pathlength
 
• 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
The Source Function
• The source function S is just 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 / 
The Transfer Equation
• For radiation passing through gas, the
change in intensity I is equal to:
dI = intensity emitted – intensity absorbed
dI = jdx – I dx
dI /d = -I + j/ = -I + S
• This is the basic radiation transfer
equation which must be solved to compute
the spectrum emerging from or passing
through a gas.
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 
4
S 
j

 J

j    I d / 4 
I d   J

4
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