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Lecture 8: Stellar Atmosphere 1 - Radiation Field and Opacity

This lecture covers the main sequence of stars, radiation intensity, mean intensity, energy density, radiative flux, radiation pressure, mean free path, local thermodynamic equilibrium, and opacity. Examples and calculations are provided to illustrate these concepts.

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Lecture 8: Stellar Atmosphere 1 - Radiation Field and Opacity

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

  2. 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. 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).

  5. Energy density • Evaluate the energy density associated with radiation.

  6. 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)

  7. Radiation pressure • A photon of energy E carries momentum:

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

  9. 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:

  10. Local Thermodynamic equilibrium • LocalThermal 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.

  11. Opacity • How does the intensity of radiation depend on opacity and distance travelled through a homogeneous medium? • 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.

  12. 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

  13. 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

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