Ch. 5 - Basic Definitions
Specific intensity/mean intensityFluxThe K integral and radiation pressureAbsorption coefficient & optical depthEmission coefficient & the source
functionScattering and absorptionEinstein coefficients
• Area • at an angle of
view from the normal to the surface
• through an increment of solid angle
Normal
A
to observer
We want to characterize the radiation fromSpecific
Intensity
Assume no azimuthal dependence
Specific Intensity• Average Energy (Ed) is the amount of energy
carried into a cone in a time interval dt• Specific Intensity in cgs (ergs s-1 cm-2 sr-1 Å-1)• Intensity is a measure of brightness – the amount
of energy coming from a point on the surface towards a particular direction at a given time, at a frequency
• For a black body radiator, the Planck function gives the specific intensity (and it’s isotropic)
• Normally, specific intensity varies with direction
dtddAd
dEI
cos
dtddAd
dEI
cos
I vs I
• The shapes of I and I are different because d and d are different sizes at the same energy of light:
d = -(c/2) d
• For example, in the Sun, I peaks at ~4500Å while I peaks at ~8000 Å
Mean Intensity
• Average of specific intensity over all directions
• If the radiation field is isotropic (same intensity in all directions), then <I>=I
• Black body radiation is isotropic and <I>=B
dIIJ
4
1
Flux• The flux F is the net energy flow across
an area A over time t, in the spectral range integrating over all directions
• energy per second at a given wavelength flowing through a unit surface area (ergs cm-2 s-1 Hz-1)
• for isotropic radiation, there is no net transport of energy, so F=0
dIF cos
On the physical boundary of a radiating sphere…
• if we define F =Fout + F
in
• then, at the surface, Fin is zero
• we also assumed no azimuthal dependence, so
• which gives the theoretical spectrum of a star
dIddIddIdF cossincossincossin2/
2
0
2/
0
2
00
2
0
dIF cossin22/
0
One more assumption:
• If Iis independent of , then
• This is known as the Eddington Approximation (we’ll see it again)
IdIF cossin22/
0
Specific Intensity vs. Flux
• 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).
The K Integral
• The K integral is useful because the radiation exerts pressure on the gas. The radiation pressure can be described as
dIF cos
dIJ
4
1
dIK 2cos
4
1
dKddIc
P cR
0
42
0cos
1
Radiation Pressure
• Again, if I is independent of direction, then
• Using the definition of the black body temperature, the radiation pressure becomes
dI
cPR
03
4
4
3
4T
cPR
Luminosity
• Luminosity is the total energy radiated from a star, at all wavelengths, 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= 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 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 Pressure
• 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
4T
cPR
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 (if no emission…)
dIdxIdI
L
dx0
eII )0()(
Class 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 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 at the wavelength corresponding to the measured magnitudes?
• 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 Coefficient
dxjdI Note that dI does NOT depend on I!
The Source Function
• The “source function” is just the ratio of the absorption coefficient to the emission coefficient:
jS
Sounds simple, but just wait….
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:
4
dxdIdxdj
JdIdIj 44/
• J is the mean intensity: dI/d = -I + Jv
• The source function depends only on the radiation field
For pure isotropic scattering
• Remember the definition of J
• So J = j/
• Hey! Then J = S for pure isotropic scattering
JdIdIj 44/
Pure Absorption
• No scattering – all incoming photons are destroyed and all emitted photons are newly created with a distribution set by the physical state of the gas.
• Source function given by Planck radiation law
• Generally, use B rather than S if the source function is the Planck function
Einstein Coefficients
• For spectral lines or bound-bound transitions, assumed isotropic
• Spontaneous emission is proportional to Nu x Einstein probability coefficient, Aul
j = NuAulh• (Nu is the number of excited atoms per
unit volume)• Induced emission proportional to intensity
= NlBluh – NuBulh
Induced (Stimulated) Emission
• Induced emission in the same direction as the inducing photon
• Induced emission proportional to intensity
I = NlBluIh – NuBulIh
True absorption Induced emission
Radiative 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 -kdx gives
dxjdxIdI
j
IdxdI
1
The 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 Equation
• 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 = jdx – Idx
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
dxdI
1
Special 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 Equilibrium
• Every process of absorption is balanced by a process of emission; 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 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 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
