Transcript Basic Definitions - Case Western Reserve University

Basic Definitions
Terms we will need to discuss
radiative transfer.
r
n
Iν
dA




s$
Specific Intensity I
Iν(r,n,t) ≡ the amount of energy at position r (vector),
traveling in direction n (vector) at time t per unit
frequency interval, passing through a unit area
oriented normal to the beam, into a unit solid angle in
a second (unit time).
If θ is the angle between the normal s (unit vector) of
the reference surface dA and the direction n then the
energy passing through dA is
dEν = Iν(r,n,t) dA cos (θ) dν dω dt
Iν is measured in erg Hz-1 s-1 cm-2 steradian-1
Basic Definitions
2
Specific Intensity II

We shall only consider time independent properties of
radiation transfer


Drop the t
We shall restrict ourselves to the case of planeparallel geometry.



Why: the point of interest is d/D where d = depth of
atmosphere and D = radius of the star.
The formal requirement for plane-parallel geometry is that
d/R ~ 0
For the Sun: d ~ 500 km, D ~ 7(105) km


d/D ~ 7(10-4) for the Sun
The above ratio is typical for dwarfs, supergiants can be of order
0.3.
Basic Definitions
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Specific Intensity III

Go to the geometric
description (z,θ,φ) - polar
coordinates
θ is the polar angle
 φ is the azimuthal angle
 Z is with respect to the stellar
boundary (an arbitrary idea if
there ever was one).
Z
φ
Iν


θ
Z is measured positive
upwards


+ above “surface”
- below “surface”
Basic Definitions
4
Mean Intensity: Jν(z)

Simple Average of I over all solid angles
J 
 I ( z, , )d /  d
but
2

 d   
0
0
sin  d d  4
Therefore
2
J  (1/ 4 )  d

0


0
I ( z,  ,  )sin  d
Iν is generally considered to be independent of φ so

J  1/ 2  I ( z,  )sin  d
0
1
 1/ 2  I ( z,  )d 
1


Where µ = cosθ ==> dµ = -sinθdθ
The lack of azimuthal dependence implies homogeneity in the
Basic Definitions
atmosphere.
5
Physical Flux

Flux ≡ Net rate of Energy Flow across a unit area. It is a
vector quantity.
F 
P

I
(
r
,
n
)
nd



Go to (z, θ, φ) and ask for the flux through one of the plane
surfaces:
F P  F P  kˆ  I ( z,  ,  ) cos  d 




1
 2  I ( z,  )  d 
1

Note that the azimuthal dependence has been dropped!
Basic Definitions
6
Astrophysical Flux
F  (1/  ) F
A
θ
P
1
 2 I ( z,  ) d 
R
θ
r
To Observer
1
• FνP is related to the observed flux!
• R = Radius of star
• D = Distance to Star
• D>>R  All rays are parallel at the observer
• Flux received by an observer is dfν = Iν dω where
• dω = solid angle subtended by a differential area on
stellar surface
• Iν = Specific intensity of that area.
Basic Definitions
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Astrophysical Flux II
For this Geometry:
dA = 2πrdr
but R sinθ = r
dr = R cosθ dθ
dA = 2R sinθ Rcosθ dθ
dA = 2R2 sinθ cosθ dθ
Now μ = cosθ
dA = 2R2 μdμ
By definition dω = dA/D2
dω = 2(R/D)2 μ dμ
Basic Definitions
θ
R
θ
r
To Observer
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Astrophysical Flux III
Now from the annulus the radiation emerges at angle θ so the
appropriate value of Iν is Iν(0,μ). Now integrate over the star:
Remember dfν=Iνdω
R 2 1
f  2 ( )  I (0,  ) d 
D 0
R 2 P
R 2
 ( ) F  ( )  FA
D
D
NB: We have assumed I(0,-μ) = 0 ==> No incident radiation. We
observe fν for stars: not FνP
Inward μ: -1 < μ < 0
Outward μ: 0 < μ < -1
Basic Definitions
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Moments of the Radiation Field
1
M ( z, n )  1/ 2  I ( z,  )  n d 
1
Order 0 : ( the Mean Intensity )
1
J ( z )  1/ 2  I ( z,  )d 
1
Order 1: ( the Eddington Flux )
1
H ( z )  1/ 2  I ( z,  )  d 
1
Order 1: ( the K Integral )
1
K ( z )  1/ 2  I ( z,  )  d 
2
1
Basic Definitions
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Invariance of the Specific Intensity
• The definition of the specific intensity leads to
invariance.
P
dA
θ
s
s′
θ′
P′
dA′
Consider the ray packet
which goes through dA at P
and dA′ at P′
dEν = IνdAcosθdωdνdt = Iν′dA′cosθ′dω′dνdt
• dω = solid angle subtended by dA′ at P = dA′cosθ′/r2
• dω′ = solid angle subtended by dA at P′ = dAcosθ/r2
• dEν = IνdAcosθ dA′cosθ′/r2 dνdt = Iν′dA′cosθ′dAcosθ/r2
dνdt
• This means Iν = Iν′
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• Note that dEν contains Basic
theDefinitions
inverse square law.
Energy Density I





Consider an infinitesimal volume V into which
energy flows from all solid angles. From a specific
solid angle dω the energy flow through an element of
area dA is
δE = IνdAcosθdωdνdt
Consider only those photons in flight across V. If
their path length while in V is ℓ, then the time in V is
dt = ℓ/c. The volume they sweep is dV = ℓdAcosθ.
Put these into δEν:
δEν = Iν (dV/ℓcosθ) cosθ dω dν ℓ/c
δEν = (1/c) Iν dωdνdV
Basic Definitions
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Energy Density II

Now integrate over volume

Eνdν = [(1/c)∫VdVIνdω]dν

Let V → 0: then Iν can be assumed to be
independent of position in V.

Define Energy Density Uν ≡ Eν/V

Eν = (V/c)Iνdω

Uν = (1/c) Iνdω = (4π/c) Jν
 Jν
= (1/4π)Iνdω by definition
Basic Definitions
13
Photon Momentum Transfer

Momentum Per Photon = mc = mc2/c = hν/c




Mass of a Photon = hν/c2
Momentum of a pencil of radiation with
energy dEν = dEν/c
dpr(ν) = (1/dA) (dEνcosθ/c)
= ((IνdAcosθdω)cosθ) / cdA
= Iνcos2θdω/c
Now integrate: pr(ν) = (1/c)  Iνcos2θdω
= (4π/c)Kν
Basic Definitions
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Radiation Pressure
Photon Momentum Transport Redux



It is the momentum rate per unit and per unit solid angle
= Photon Flux * (”m”v per photon) * Projection Factor
dpr(ν) = (dEν/(hνdtdA)) * (hν/c) * cosθ where dEν =
IνdνdAcosθdωdt so dpr(ν) = (1/c) Iνcos2θdωdν
Integrate dpr(ν) over frequency and solid angle:
1 
pr    I cos2  d d
c 0 4
Or in terms of frequency
pr ( ) 

4
K
c
1
2
I
cos
 d


c 4
Basic Definitions
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Isotropic Radiation Field
I(μ) ≠ f(μ)
1
1
J ( z )  1/ 2  I ( z )d 
K ( z )  1/ 2  I  2 d 
1
 1/ 2 I (  | )
1
1
 1/ 2 I   2 d 
1
1
1
 1/ 2 I (1  ( 1))
 I
 1/ 2 I (
3
|11 )
3
 1/ 2 I (1/ 3  ( 1/ 3))
 I / 3
 J / 3
Basic Definitions
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