Lecture 8

1. Lecture 8 Optical depth

2. Review: opacity • How does the intensity of radiation depend on opacity and distance travelled through a homogeneous medium? • 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 • 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 • The optical depth can be thought of as the number of mean free paths a photon takes between its initial position and the surface. • If r and klare approximately constant over the photon’s path, • where is the mean free path

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, 0.00022RSun.

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 log10r

15. Break

16. Low opacity High opacity Low opacity 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

17. Low opacity High opacity Low opacity 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.

18. Low opacity High opacity Low opacity 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

19. The Solar Atmosphere • 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~25000K • 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 • From an analysis of spectral lines, the following are the most abundant elements in the solar photosphere