Basic Definitions Specific intensity/mean intensity Flux The K integral and radiation pressure Absorption coefficient/optical depth Emission coefficient The source function The transfer equation & examples Einstein coefficients

Specific Intensity/Mean Intensity Intensity is a measure of brightness – the amount of energy coming per second from a small area of surface towards a particular direction erg hz-1 s-1 cm-2 sterad-1 Jn is the mean intensity averaged over 4p steradians

Flux Flux is the rate at which energy at frequency n flows through (or from) a unit surface area either into a given hemisphere or in all directions. Units are ergs cm-2 s-1 Luminosity is the total energy radiated from the star, integrated over a full sphere.

Class Problem From the luminosity and radius of the Sun, compute the bolometric flux, the specific intensity, and the mean intensity at the Sun’s surface. L = 3.91 x 1033 ergs sec -1 R = 6.96 x 1010 cm

Solution F= sT4 L = 4pR2sT4 or L = 4pR2 F, F = L/4pR2 Eddington Approximation – Assume In is independent of direction within the outgoing hemisphere. Then… Fn = pI n Jn = ½ In (radiation flows out, but not in)

The Numbers F = L/4pR2 = 6.3 x 1010 ergs s-1 cm-2 I = F/p = 2 x 1010 ergs s-1 cm-2 steradian-1 J = ½I= 1 x 1010 ergs s-1 cm-2 steradian-1 (note – these are BOLOMETRIC – integrated over wavelength!)

The K Integral and Radiation Pressure Class Problem: Compare the contribution of radiation pressure to total pressure in the Sun and in other stars. For which kinds of stars is radiation pressure important in a stellar atmosphere?

Absorption Coefficient and Optical Depth Gas absorbs photons passing through it Photons are converted to thermal energy or Re-radiated isotropically Radiation lost is proportional to Absorption coefficient (per gram) Density Intensity Pathlength Optical depth is the integral of the absorption coefficient times the density along the path

Class Problem Consider radiation with intensity In(0) passing through a layer with optical depth tn = 2. What is the intensity of the radiation that emerges?

Class Problem A star has magnitude +12 measured above the Earth’s atmosphere and magnitude +13 measured from the surface of the Earth. What is the optical depth of the Earth’s atmosphere?

Emission Coefficient There are two sources of radiation within a volume of gas – real emission, as in the creation of new photons from collisionally excited gas, and scattering of photons into the direction being considered. We can define an emission coefficient for which the change in the intensity of the radiation is just the product of the emission coefficient times the density times the distance considered.

The Source Function The source function Sn is just the ratio of the emission coefficient to the absorption coefficient The source function is useful in computing the changes to radiation passing through a gas

The Transfer Equation For radiation passing through gas, the change in intensity In is equal to: dIn = intensity emitted – intensity absorbed dIn = jnrdx – knrIn dx dIn /dtn = -In + jn/kn = -In + Sn This is the basic radiation transfer equation which must be solved to compute the spectrum emerging from or passing through a gas.

Pure Isotropic Scattering The gas itself is not radiating – photons only arise from absorption and isotropic re-radiation Contribution of photons proportional to solid angle and energy absorbed: Jn is the mean intensity dI/dtn = -In + Jv The source function depends only on the radiation field

Pure Absorption No scattering – photons come only from gas radiating as a black body Source function given by Planck radiation law

Einstein Coefficients Spontaneous emission proportional to Nn x Einstein probability coefficient jnr = NnAulhn Induced emission proportional to intensity knr = NlBluhn – NuBulhn