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

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

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:

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:

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).

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

Energy density Evaluate the energy density associated with radiation.

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

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)

Radiation pressure A photon of energy E carries momentum:

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:

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

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

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:

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.

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

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.

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

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