1. 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

2. 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
Jn is the mean intensity averaged over 4p
steradians

3. Flux
Flux is the rate at which energy at frequency n flows through (or from) a unit surface area either into a given hemisphere or in all directions.
Units are ergs cm-2 s-1
Luminosity is the total energy radiated from the star, integrated over a full sphere.

4. 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 -1
R = 6.96 x 1010 cm

5. Solution F= sT4 L = 4pR2sT4 or L = 4pR2 F, F = L/4pR2
Eddington Approximation – Assume In is independent of direction within the outgoing hemisphere. Then…
Fn = pI n
Jn = ½ In (radiation flows out, but not in)

6. 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!)

7. The K Integral and Radiation Pressure
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?

8. 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
Pathlength
Optical depth is the integral of the absorption coefficient times the density along the path

9. Class Problem
Consider radiation with intensity In(0) passing through a layer with optical depth tn = 2. What is the intensity of the radiation that emerges?

10. 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?

11. 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.

12. The Source Function
The source function Sn 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

13. The Transfer Equation
For radiation passing through gas, the change in intensity In is equal to:
dIn = intensity emitted – intensity absorbed
dIn = jnrdx – knrIn dx
dIn /dtn = -In + jn/kn = -In + Sn
This is the basic radiation transfer equation which must be solved to compute the spectrum emerging from or passing through a gas.

14. 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:
Jn is the mean intensity
dI/dtn = -In + Jv
The source function depends only on
the radiation field

15. Pure Absorption
No scattering – photons come only from gas radiating as a black body
Source function given by Planck radiation law

16. Einstein Coefficients
Spontaneous emission proportional to Nn x Einstein probability coefficient
jnr = NnAulhn
Induced emission proportional to intensity
knr = NlBluhn – NuBulhn