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Lecture 6

Lecture 6. The radiation field and opacity. Review. The main sequence is a mass sequence More massive stars are closer to the top-left (hot and bright). M=30M Sun. M=M Sun. M=0.2M Sun. Radiation intensity.

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Lecture 6

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  1. Lecture 6 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. Break

  10. 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:

  11. 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.

  12. 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.

  13. Example • Recall 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

  14. 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? • So it would be murky… The high opacity in the Sun (as we’ll see) is that the high temperature leads to many free electrons that are able to absorb photons

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