1. Lecture 8
Optical depth

2. Review: opacity
Opacity (k) is a cross-section per unit mass (units m2/kg) of material for absorbing photons of a specific wavelength.
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.

3. Optical depth
The optical depth is a unitless quantity that relates to how much light is absorbed along its path s.
The difference in optical depth between the initial and final position of a light ray is then given by:
Note the optical depth is greater at the initial position than at the final. i.e. the optical depth decreases as the photon moves toward the observer.

4. Optical depth For a stellar atmosphere, we can set at the surface.
Thus, for a photon which traveled a distance s through the stellar atmosphere, the initial optical depth is
and the intensity of the light ray can be expressed as:

5. Optical depth
If r and kl are approximately constant over the photon’s path,
where is the mean free path
The optical depth can be thought of as the number of mean free paths a photon takes between its initial position and the surface.

6. Random walk
As a photon moves through a dense gas, it interacts with atoms being scattered, absorbed, and re-emitted.
If each scattering process is random, how far does the photon move (on average)?

7. Random walk
Recall that the optical depth is approximately the number of mean free paths traveled:
The number of interactions a photon has on the way to the surface is
Thus a photon can escape to the surface when
and the number of steps required increases as the square of the distance travelled
A more careful calculation (coming soon) will show that the average optical depth from which photons originate is

8. Optical depth: example
Using the numbers we used before for the solar photosphere, approximately how far into the Sun corresponds to an optical depth of t=1?
The photons we see represent the conditions very close to the Sun’s surface, RSun.

9. Sources of opacity
The opacity is determined by the physical processes that can remove photons from a beam of radiation. In general this includes true absorption processes, as well as scattering.
Bound-bound transitions
Bound-free transitions
Free-free absorption
Electron scattering

10. The Balmer jump
The energy of an electron in the n=2 orbit of a hydrogen atom is
So the atom can be photoionized by any photon with wavelength
This gives rise to the Balmer jump in stellar spectra
Note the jump is not present in hotter stars, where most of the hydrogen is already ionized

11. The Lyman limit
Similarly, the energy of an electron in the n=1 orbit of a hydrogen atom is
So the atom can be photoionized by any photon with wavelength
This is a very efficient process, and it takes very little neutral hydrogen to absorb all photons with this energy.
For stars more than a few pc away, there is sufficient neutral hydrogen between them and us to almost all the light at these UV wavelengths. We therefore know little about the far-UV flux of hot stars.

12. Typical sources of opacity
In most stellar atmospheres, the primary source of continuum opacity is the photoionization of H- ions
In cool stars, molecules can form. These complex arrangements of atoms can make many bound-bound and bound-free transitions which greatly increase the opacity

13. Rosseland mean opacity
The total opacity is the sum of all these sources. It depends on the wavelength of the light being absorbed, as well as the composition, density and temperature of the gas.
A weighted average over all wavelengths gives the Rosseland mean opacity

14. Rosseland mean opacity
Opacity increases with density at fixed T
At fixed density and low temperatures, opacity rises steeply as T increases
After the peak, the opacity decreases like
The bump at T~40,000 K is due to the second ionization of He
At the highest temperatures, the opacity is due to electron scattering
log10 r

15. Break

16. Absorption lines and Kirchoff’s laws
Photons we see originate (on average) from an optical depth of 2/3.
Recall the optical depth is related to the opacity along the photon path by
If the opacity is much higher at some wavelength l0 (for example due to bound-bound absorption) the photons we see will originate from closer to the surface
High opacity
Low opacity

17. Absorption lines and Kirchoff’s laws
This doesn’t make any difference if the temperature of the gas is uniform: the luminosity radiated from all points in the gas is the same, so it doesn’t matter where the photon originates from
However – if the temperature is lower near the surface, the luminosity will be much lower:
Thus the luminosity will be lowest at wavelengths where the opacity is the highest: this gives rise to absorption lines.
High opacity
Low opacity

18. Emission lines
What would be the result if the star’s temperature increased outward?
In this case – light from regions of high opacity would arise from higher in the atmosphere, where the temperature (and therefore luminosity) is higher. We would therefore see an emission line
High opacity
Low opacity

19. The Solar Atmosphere Core
The solar atmosphere extends thousands of km above the photosphere (from which the optical radiation is emitted)
The temperature increases above the photosphere (in the chromosphere and corona)
We can therefore see emission lines from these regions
T~106 K
T~25000 K
HeI emission from the chromosphere (~20000K)
T~5770 K
Core
T~107 K

20. Line profiles
The region near the central wavelength is the core of the line
The sides which join up with the continuum are the wings.
The area of the line is related to the amount of absorption
wings
core
The central regions of the line must be formed at higher (and cooler) regions of the atmosphere. The wings probe deeper into the atmosphere.

21. Line profile
Absorption lines are due to bound-bound atomic transitions of fixed energy. Why are the lines not infinitely narrow?
Natural broadening
Pressure or collisional broadening
Doppler broadening

22. Voigt profile
The absorption line profile is due to the Doppler and damping (pressure and natural) broadening. The resulting profile is known as the Voigt profile.

23. Pressure broadening: example
For supergiant stars, the atmospheres are ~1 million times less dense than in the sun. So the pressure broadening is weaker by this factor, which is why more luminous stars have much narrower lines (the cores are still dominated by Doppler broadening).

24. Reminder: Spectroscopic parallax
In principle, you can identify both the spectral class and the luminosity class from the spectrum.
This can be used to determine the distance to the star. This method is known as spectroscopic parallax

25. Solar abundances Element Atomic Log Relative Column Density Number
From an analysis of spectral lines, the following are the most abundant elements in the solar photosphere
Element
Atomic
Log Relative
Column Density
Number
Abundance
kg m-2
Hydrogen 1 11
Helium 2
-1.01 43
Oxygen 8
-3.07
0.15
Carbon 6
-3.4
0.053
Neon 10
-3.91
0.027
Nitrogen 7 -4
0.015
Iron 26
-4.33
0.029
Magnesium 12
-4.42
0.01
Silicon 14
-4.45
0.011
Sulfur 16
-4.79
0.0057