Description of radiation field

Qualitatively, we know that characterization should

involve

energy/time

frequency all functions of

direction

We also now that radiation is not altered by passing

through empty space.

Flux of energy varies in empty space and therefore

cannot be the fundamental quantity.

Ex.:

Energy per solid angle works.

Assume solar surface emits uniformly

Each unit of area on Sun emits

Fixed solid angle receives emission from area of r2.

For reference:

x,t.

Fs =L

4 r2

r

L

energy

time

r

energy flux

solid angle, time independent of r

A =1

r2

rr

2A

r( ) +1

r cosA cos( ) +

1

r cos

A

Intensity or radiance

I x,t, n( ) = 's 0 lim

E

t A

where

t = time

= wavelength

A = area n

= solid angle

With I defined this way, radiative transfer in empty space reduces to

dI

ds= 0, s = distance in n direction (along beam)

Energy flux In direction n, monochromatic flux is

Fn= I n( ) n n

4

d

Total flux

Fn= F

nd

0

Comments:

*Notice that if I n( ) = I n( ) then Fn= 0

* I n( ) n is a vector and Fn is a sum of vector components

F = Fe1

e1+ F

e2

e2+ F

e3

e3 is vector energy flux.

*Isotropic radiation

I x,t, n( ) I x,t( )No net flux by symmetry.

Flux in one direction across surface is of interest.

F += I n( ) n+ n d

2

n( )

= I cos 0

1

0

2

d cos( )d = I

Notice I < 2 I (whole hemisphere).

This result is of special interest because thermalemission from

a surface or isotropic scattering from a surface give this geometry.

Interaction with matter - Absorption

A medium can absorb, emit or scatter radiation.

Under absorption alone one expects

Energy loss along = Beam *absorption strength *n * s

beam distance s intensity per particle # particles

vol 1

Since n s =particles

area, need absorption strength in

I = I ˆa

n s Salby notation

+ direction

Thus

dI

I= ˆ

a n I

a I Units = length 1

The factor is the absorption coefficient. This attenuation

equation is called Lambort's Law. Its integral is

dI

I= ds

ln I = dss

+ const.

I = I s 0( )e ads

0

s

The "optical path" or "optical depth"

u s( ) =a

ds0

s

has no units.

I s( )I 0( )

= eu

s( ) Transmissivity

Comment: Absorption strength is ecpressed in a variety of

units. Fundamental combinaion is u.

dI

I= ds = du = ˆ

an ds

u is always (strength measure) (gas amount along path)

Strength might be area

particle,

area

mass,optical path

km.amagats. Be careful

Planck intensity spectrum

A radiation field inside a container with walls at

uniform temperature comes into thermodynamic

equilibrium.

(It is assumed that the walls have a continuous

spectrum of energy transitions that can happen.)

This spectrum is the Planck Function.

=2h c2

5 exp +hc

k T1

energy

(area)(time)(wave length)(steradian)

This is energy per unit wavelength. To evaluate ,energy per unit frequency use

E = =

=d

d, = c

=2h 3

c2 exp

h

k T1

In practice the first and second radiation constants are used to numerically

evaluate . For example:

=c

1

5

expc

2

T1

Max of is at T = 0.29 cm K.

Max of is at T = 0.51 cm K.

c1= 3.7419 10

16W m

2

c2= 1.4388 10

2m K

Brightness temperature

Thermal emission spectra are often resented in units

of brightness temperature.

Observe I ( ) radiance units

Set I equal to the Planck Function at unknown Tb( )

Tb,( ) = I

Invert, solve for Tb( ).

Tb( ) gives the temperature, at each wavelength,

that a black surface would need to have in order to

emit the observed intensity.

Stefan-Boltzmann Law

Many surfaces emit approximately

the same way that a port in a black

cavity would emit.

I = T ,( ) independent of direction

F =

F =0

d = T4

= 5.67 10 8 Wm2 K 4

Example: Estimate the temperature of a planet.

Power received from = a2

L

4 r2

1 A( )

= a2

Fs

rA( ), Fs

1368 W m 2

A 0.31

Power emitted by planet = 4 a2 T 4

T 4=

1 A

4F

s= 236 W m 2

T = 254 K

a

Equation of transfer

Thermal emission

In the presence of absorption and emission,

Where S is a source term due to emission by the

medium.

In a region many optical depths across, the exterior

boundaries will become unimportant. Then if the

region is isothermal, the radiation field will be in

thermodynamic equilibrium, with I = B .

Emission coefficient = absorption coefficient is Kirchoff’s Law.

Then dI

ds= 0, and I + S = 0

S = B

dI

ds= I B( )

dI

ds= I + S

Emission and scattering

Denote e

exxtinction coefficient. In general

e i

ni

i= monochromatic cross section (lenght)2

ni= number density of constituent i.

Let o

be fraction of extinction that scatters into a new

direction and is not absorbed.

o single scattering albedo. Governing equation

dI

ds=

eI + J( ),

Jscat( )

n( ) =o

P n, n( )4

I n( )d

4.

By o definition,

P n, n( )4

d

4= 1

What is Jemission( )

? Expect C B , if LTE.

dI

ds=

eI +

0P n, n( ) I n( )

d

4+ C B

Deep inside homogeneous region I B (isotropic)

and dI

ds= 0. Thus

-B +0

B P n, n( ) I n( )d

4+ C B = 0

n

n

Ex. Forward scattering

C = 10

Kirchoff again

Summary:

1

e

dI

ds= I + 1

0( )B +0

P n, n( ) I n( )d

4

Formal integral in absence of scattering

Let e

ds = dX optical thickness

s = 0 s x

x=0

I

dI

dX= I B T( ) (no scattering)

ex d

dXe

x

I( ) = B

ex

I0

x

= ex

0

x

B X( )dX

ex

I I X = 0( ) = ex

0

x

B X( )dX

I X = 0( ) = I X( )e x

+ ex

0

x

B X( )dX

Intensity change

Extinction Emission Scattering

Intensity emerging

Attenuated incident intensity

Accumulated emission along path

Absorption spectrum

Transitions

In planetary atmospheres transitions between discrete energy levels are of most importance. In stars free-free

or bound-free transitions give continuum absorption.

absorption lines dominate.

Exception: UV and EUV absorption in upper atmosphere.

Transitions are

Electronic visible

Vibrational IR & combinations

Rotational far IR

Examples of spectra are shown in Salby and in

viewgraphs.

Units

Spectrscopists usually work in frequency units expressed

in wave numbers

h = E

= c

=c

s1( )

=c=

1 (lenght 1,usually cm 1)

wave number

Handy identity: microns( ) =10,00

cm1( )

Lorentz line profile In the lower atmosphere, collisional broadening is the most

important factor producing frequent dispersion of absorption.

Phase coherence of emitted photon is destroyed by

collisions.

t = time between collisions

t

Spectrum is broadened by 1

t.

1

c

1

t wave number units

To estimate ,

lmfp

A n = 1

1

t=

lmfp

= An

A ~ 4a0( )

2

, a0= Bohr radius ~ 0.5 A

o

10 19 m2

n ~ 2.69 1025 m 3 Loschmidt's #

~ 300 m s (sound speed)

1

t

1

c~

An

c~

3 102 10 19 2.7 1025

3 108~ 3 m 1

= .03 cm 1

Typical IR frequency = 10 microns = 1000 in 1

.03 cm-1

1000 cm-1

Narrow. Since n,

Width pressure.

The shape is the Lorentz profile

fL= L

0( )2

+L

2, f

Ld = 1

0

0

The cross section is

= S fL ( ), S line strength.

The strength is a function of temperature S(T ), and L is usually taken to be

L=

L0

p

p0

T0

T

n

,

where p0,T

0 are STP, and n

1

2 and

L0 are tabulated in line atlases.

Doppler line broadening •

~c

~350m s 1

3 108 m s 1~ 10 6

~ 10 6 ~ 10 6 103 ~ .001 cm 1

Important in upper atmosphere where p small and Lorentz width is small.

Doppler shape:

D= 0

c

2K3T

m

fD=

1

D

exp 0

0

2

Voigt line profile

Convolve Doppler & Lorentz broadening

There are efficient algorithms to evaluate it See Grody and Yung.

Line atlases

See the description of the HITRAN, GEISA atlases.

Line strength S(T) involves lower state energy. This is because of state population dependence on T.

From LTOU,

f0

( ) = fL 0( ) f

D ( )d

Si

T( ) Si

Tr

( )T

Tr

n

exphcE

i

KB

1

T

1

Tr

n = 2 linear molecule

n =5

2 nonlinear molecule

Ei= lower state energy for line i

Tr= 296 K = reference T.

Plane parallel atmosphere, emission and

absorption

Transfer eq. in terms of optical depth.

d = dz

No scattering.

dI

ds= I B( )

Geometry of ray

ds=dz

cos

dz

μ

Plane parallel is idealized case

= z( ), B = B z( ). Then I = I z( ),

1 dI

ds=

μ dI

dz= μ

dI

d

μdI

d= I B

Solution for upward intensity

I z,μ( ) = Is

,μ( )es μ

z( )μ

+ B

z( )

s

( )eμ

z( )μ

d

μ

\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

z ds

\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

z

Z = 0

Usually Is

,μ( ) B Ts

( ). In some cases,

Is

,μ( ) B Ts

( ).

Where the emissivity varies spectrally but with wide Bands

rather than narrow lines. Mars surface mineralogy Is an example.

For now, assume B (Ts).

Average over a band

In practice a finite is of interest.

• For heating rates can average over spectrum.

• For remote sensing the instrument has spectral resolution

limitations.

Average over small enough so that the Plank function ~

constant.

I z,μ( ) =1

B Ts

( )es

μd +

1B

s

( )e μd

μ

d

B smooth B Ts

( )1

e

s

μ d B

s

( ) 1e μ

d

μ

= B z,0,μ( ) + B

s

z( )d z, z ,μ( )

dzdz

where

z, z ,μ( ) =1

e μ d

N.B. Sign changes from d

dz and from reversal of integration range.