1. Lecture 8: Stellar Atmosphere
1. The radiation field and opacity

2. The main sequence is a mass sequence
Review
The main sequence is a mass sequence
More massive stars are closer to the top-left (hot and bright)
M=30MSun
M=MSun
M=0.2MSun

3. Radiation intensity
The intensity of radiation is defined as the amount of energy carried by the light of wavelength between l and l+dl in time dt through area dA into a solid angle dW:

4. Example: Radiation intensity
The blackbody radiation formula we have used is an example of an intensity distribution:
The total energy rate emitted by a blackbody is just:
We did this integration before and found the Stefan-Boltzmann law:

5. Mean intensity
In general, Il depends on direction. The mean intensity is defined to be the average intensity radiated in all directions (i.e. over all solid angles dW).

6. Mean intensity
If Il is isotropic (i.e. independent of direction f and q) then
This is true of a blackbody.

7. Energy density
Evaluate the energy density associated with radiation.

8. Energy density In the special case of blackbody radiation:
Evaluate the energy density associated with radiation.
In the special case of blackbody radiation:
where a=7.566x10-16 J/m3/K4 is the radiation constant

9. Radiative flux
The radiative flux is the net energy with wavelength between l and l+dl that passes through a unit area in unit time.
For isotropic radiation there is no net flux (an equal amount passes through the unit area in opposing directions)

10. Radiation pressure
A photon of energy E carries momentum:

11. Radiation pressure A photon of energy E carries momentum:
Light therefore exerts a radiation pressure
Consider a beam of radiation with energy Eldl hitting a surface dA at angle q:

12. Example: blackbody radiation pressure
At room temperature,
And the pressure is
However, at T=107K, the pressure is
times larger, or

13. Summary of Definitions
Mean intensity (sometimes written Jl):
Energy density:
Radiative flux:
Radiation pressure:

14. Mean free path
How far does an atom move before interacting with another, in an ideal gas with number density n?
Collisional cross section:
Mean Free path:

15. Local Thermodynamic equilibrium
Local Thermal Equilibrium (LTE) holds if the distance matter and radiation can travel between interactions is much smaller than the distance over which temperature changes.
Compare the mean free path of a hydrogen atom in the solar photosphere (where the temperature gradient is about 8.7 K/km) to the temperature scale height.

16. Opacity
Opacity (k) is a cross-section per unit mass (units m2/kg) of material for absorbing photons of a specific wavelength.
where m is the mean molecular weight.
The quantity
is the number of mean free paths traveled.
so we define

17. Opacity is the mean free path
How does the intensity of radiation depend on opacity and distance travelled through a homogeneous medium?
After the photon has traveled one mean free path its intensity will have decreased by a factor e-1=0.37.

18. Example in the Sun’s photosphere,
Assuming it is pure hydrogen, the density is:
The opacity in this region of the atmosphere, at the wavelength of visible light (500 nm) is
The photon mean free path is
So photons can travel a very long way before the intensity decreases appreciably. The atmosphere is not in LTE – photons in a given place in the atmosphere originated somewhere with a different temperature

19. Example The density of Earth’s atmosphere at sea level is
What would the photon mean free path be if the atmosphere had the same opacity as the Sun?
The high opacity in the Sun is that the high temperature leads to many free electrons that are able to absorb photons