1. 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

2. Definitions

3. 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

4. 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

5. 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,

6. 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

7. 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).

8. Luminosity
Luminosity is the total energy radiated from a star, at all wavelengths, integrated over a full sphere.

9. 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

10. 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)

11. 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!)

12. Radiation Pressure Radiation Pressure – light carries momentum (p=E/c)
Isotropic Radiation Pressure – force per unit area
Blackbody Radiation Pressure –

13. The K Integral and Radiation Pressure
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?

14. 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

15. 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?

16. 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?

17. 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.
Note that dI does NOT depend on I!

18. 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

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

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

21. 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
dividing both sides by -klrdl gives

22. The Source Function
The source function Sn 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

23. The Transfer Equation
We can then write the basic equation of transfer for radiation passing through gas, the change in intensity In is equal to:
dIl = intensity emitted – intensity absorbed
dIl = jlrdl – klrIl dl
dIl /dtl = -Il + jl/kl = -Il + Sl
This is the basic equation which must be solved to compute the spectrum emerging from or passing through a gas.

24. 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

25. 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
If the total flux is constant, then the mean intensity must be equal to the source function: <I>=S

26. 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)

27. 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