Ch. 5 - Basic DefinitionsCh. 5 - Basic Definitions

Specific intensity/mean intensityFluxThe K integral and radiation pressureAbsorption coefficient/optical depthEmission coefficientThe source functionThe transfer equation & examplesEinstein coefficients

DefinitionsDefinitions

Specific IntensitySpecific Intensity• Average Energy (Ed) is the amount of energy

carried into a cone in a time interval dt• Specific Intensity (ergs s-1 cm-1 cm-2 sr-1)• Intensity is a measure of brightness – the amount

of energy coming per second from a small area of surface towards a particular direction

• For a black body radiator, the Planck function gives the specific intensity (and it’s isotropic)

• Normally, specific intensity varies with direction

dddtdAd

dE

ddtdAd

dEI

coscos

Mean IntensityMean Intensity

• Average of specific intensity over all directions, divided by 4 steradians

• If the radiation field is isotropic (same intensity in all directions), then <I>=I

• Black body radiation is isotropic and <I>=B

ddIdIIJ sin4

1

4

10

2

0

Energy DensityEnergy Density• Energy Density (ud) – how much

energy is in the radiation field:• Consider a cylindrical volume dAdL• Energy density is Ed divided by the

volume dAdL, and integrated over all solid angles

• For an isotropic radiation field the energy density ud = (4/c)Id

• For blackbody radiation,

d

e

hcdB

cdu

kThc 1

/84 5

Radiative FluxRadiative Flux• Radiative flux is the rate at which

energy at a given wavelength flows through (or from) a unit surface area passes each second through a unit area in the direction of the z-axis (ergs cm-2 s-1)

• for isotropic radiation, there is no net transport of energy, so F=0

dddIddIF sincoscos2

0 0

Specific Intensity vs. Radiative Specific Intensity vs. Radiative FluxFlux

• Use specific intensity when the surface is resolved (e.g. a point on the surface of the Sun). The specific intensity is independent of distance (so long as we can resolve the object). For example, the surface brightness of a planetary nebula or a galaxy is independent of distance.

• Use radiative flux when the source isn’t resolved, and we're seeing light from the whole surface (integrating the specific intensity over all directions). The radiative flux declines with distance (1/r2).

LuminosityLuminosity

• Luminosity is the total energy radiated from a star, at all wavelengths, integrated over a full sphere.

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!)

Radiation Radiation PressurePressure

• Radiation Pressure – light carries momentum (p=E/c)

• Isotropic Radiation Pressure – force per unit area

• Blackbody Radiation Pressure –

dddIc

Psphererad

2

0cos

1

uPrad 3

1

The K Integral and Radiation The K Integral and Radiation PressurePressure

• Thought 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 Note that dI does NOT depend on I!

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 4

4/

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

Radiative Energy in a GasRadiative Energy in a Gas• As light passes through a gas, it is both

emitted and absorbed. The total change of intensity with distance is just

• dividing both sides by -kdl gives

dljdlIdI

j

Idl

dI

1

The Source FunctionThe Source Function

• The source function S is defined as 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 The Transfer EquationEquation

• We can then write the basic equation of transfer for radiation passing through gas, the change in intensity I is equal to:

dI = intensity emitted – intensity absorbeddI = jdl – Idl

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

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

SI

dl

dI

1

Special CasesSpecial Cases

• If the intensity of light DOES NOT VARY, then I=S (the intensity is equal to the source function)

• When we assume LTE, we are assuming that S=B

BI

d

dI

Thermodynamic Thermodynamic EquilibriumEquilibrium

• Every process of absorption is balanced by a process of absorption; no energy is added or subtracted from the radiation

• Then the total flux is constant with depth

• If the total flux is constant, then the mean intensity must be equal to the source function: <I>=S

4esurfacerad TFF

Simplifying AssumptionsSimplifying Assumptions

• Plane parallel atmospheres (the depth of a star’s atmosphere is thin compared to its radius, and the MFP of a photon is short compared to the depth of the atmosphere

• Opacity is independent of wavelength (a gray atmosphere)

dII

0

dSS

0

Eddington ApproximationEddington Approximation

• Assume that the intensity of the radiation (I) has one value in all directions toward the outward facing hemisphere and another value in all directions toward the inward facing hemisphere.

• These assumptions combined lead to a simple physical description of a gray atmosphere