Basic DefinitionsBasic 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 IntensitySpecific 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

dAdwdtdv

dEI

cos

dIJ4

1

J is the mean intensity averaged over 4 steradians

FluxFlux

• Flux is the rate at which energy at frequency 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.

dIF cos 2/

0

cossin2

dIF

Class ProblemClass 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

SolutionSolution

• F= T4

• L = 4R2T4 or L = 4R2 F, F = L/4R2

• Eddington Approximation – Assume Iis independent of direction within the outgoing hemisphere. Then…

• F = I • J = ½ I(radiation flows out, but not in)

The NumbersThe Numbers

• F = L/4R2 = 6.3 x 1010 ergs s-1 cm-2

• I = F/ = 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 The K Integral and Radiation PressurePressure

• 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?

dIK 2cos 4

3

4Tc

PR

Absorption Coefficient and Optical Absorption Coefficient and Optical DepthDepth

• 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

dxIdI

L

dx0

eII )0()(

Class ProblemClass Problem

• Consider radiation with intensity I(0) passing through a layer with optical depth = 2. What is the intensity of the radiation that emerges?

Class ProblemClass 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?

• 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.

Emission CoefficientEmission Coefficient

dxjdI

The Source FunctionThe Source Function

• The source function S 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

/jS

The Transfer EquationThe Transfer Equation

• For radiation passing through gas, the change in intensity I is equal to:

dI = intensity emitted – intensity absorbed

dI = jdx – Idx

dI /d = -I + j/ = -I + S

• This is the basic radiation transfer equation which must be solved to compute the spectrum emerging from or passing through a gas.

Pure Isotropic ScatteringPure 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:

4

dxdIdxdj

JdIdIj 44/

Jj

S J is the mean intensitydI/d = -I + Jv

The source function depends only on the radiation field

Pure AbsorptionPure Absorption

• No scattering – photons come only from gas radiating as a black body

• Source function given by Planck radiation law

Einstein CoefficientsEinstein Coefficients

• Spontaneous emission proportional to Nn x Einstein probability coefficient

j = NnAulh

• Induced emission proportional to intensity

= NlBluh – NuBulh