The sun as a blackbodyThe solar photosphere (where most of the light originates) has a temperature of approximately 6000 K.

core~107 K

photosphere~5760 K

•Understanding the spectra of hot, radiant objects was critical to modern physics. •Perfect emitters/absorbers are called “blackbodies”

corona~106 K

(hot, but tenuous)

Blackbody

•Visualize as a cavity with tiny hole.•Radiation entering a blackbody has low probability of escaping.•In equilibrium (constant T), the rate of energy entering must equal the rate of energy escaping

Perfect absorber

Blackbody in a cavity

T

T

Allow the blackbody to reach equilibrium with the cavity→ The BB must be radiating at the same rate it is absorbing→ Emitted spectrum must be independent of orientation of BB→ Spectrum of a BB must depend on T only.

PhotonsBased on Planck’s formulation, Einstein proposed that photons have energy:

E hf

For any wave: v f

hcE

So:

For light in vacuum: c f cf

Recall: 1240 eV nmhc e.g., 620 nm

2.0 eVE

A harmonic, traveling wave is described by its wave function,either a sinusoid: , cos 2 xx t A ft

or a complex exponential: , exp 2 xx t A i ft We can use the wave number: 1k

(sometimes )

2k

De Broglie’s hypothesis:hp

So, for photons: E pc

Blackbody radiation: approachAssume a cavity and radiation within it have temp T

To find the # of photons per unit photon energy, per unit volumewe need to know:i) density of statesii) distribution function

dndE

dn D E f E VdE

D E f E

NnV

dN D E f EdE

# of states/unit volume

Average # of photons in each state 

Blackbody radiation: DOS(I)

i) find D E

The waves are harmonic, i.e.,

2, e iftx t x

so we only need to consider the spatial part

In 1-D: 2e ikxx A

Trick: Assume periodic boundary conditions:

2e ik Lx L x

nk L n nnkL

nLn

Blackbody Rradiation: DOS (II)

1 20, , ,...kL L

1kL

g states per in wave number1L

The # of states with nk k 2 21

g kN k gkL

L

is

2

2 2

21

g kN k gk L

L

In 2-D:

In 3-D: 3

3 3

3

443

1 3

g kN k gk L

L

Blackbody radiation: DOS (III)

2

22 2gLN E E

h c

In 2-D:

In 3-D: 3

33 3

43

gLN E Eh c

In 1-D: 2gLN E Ehc

Change to energy basis: E hck Find D.O.S.: dND EdE

2

2 22 gLD E Eh c

In 2-D:

In 3-D: 3

23 3

4 gLD E Eh c

In 1-D: 2gLD Ehc

//constant

//linear

//quadratic

Blackbody spectrum: result

# of photons/photon energy/unit volume:

Set: 2g

3V L

2

3 38e 1E kT

dn D E f E EdE V h c

// Planck’s Radiation Law

//Two distinct polarizations of light

Energy/photon energy/volume :

3

3 38e 1E kT

du dn EEdE dE h c

// Blackbody spectrum

Blackbody Spectrum (II)Find total # of photons/volume :

2 3 3

23 33 30 08 8e 1E kTE E

dn E k Tn dE dE IdE h ch c

2 3 3 2! 1.202 2.4I 3 3

3 360.4 k Tn

h c

0

1 1e 1

k

k xxxI dx k k

1 !k k

1

111

k

nkn

where

3 4 4

33 33 30 08 8e 1E kTE E

du E k Tu dE dE IdE h ch c

Find total energy/volume :

4

43 4 4 3! 15

90I

5 4 4 44

3 3 3 38 163.215

k k Tu Th c h c

Blackbody radiation (I)

sd

ed

edA

sdA

s

e

R

r

dr

d

: # of photons per unit volumen

: energy per unit volumeu

sun earth

Blackbody radiation (II)The # of photons/volume/solid angle :

2A

r

2dAdr

Solid angle :

24tot

r

2r 4 sr steradians

4n 3

#4 m srn

Blackbody radiation (III)

2edV d r dr

2

4 4

ed rd dV nn

dt

2

cos e sdAdrr

cos4 e e sn dr d dA

dt dt

drcdt

2cos e sdAd

r

Area on surface ofBBPoint on surface of BB

Interior of BBInterior of BB

r

dr

c

sdA

re

1 1

2

2

Blackbody radiation (IV)

R

sdA2

cose se

dAdR

2 cos coss e e s

s e

dA dARd d

cos cose e s s s ed dA d dA

Relate solid-angle and area on sun to that on earth.

sun earth

cos4 4 s s en dV d n c d dA

dt

2coss e

sdAd

R

edAe s

2 3

Blackbody radiation (V)

cos cos4 4 s s e s s e

d n dV d d n c d dA E d dAdE dt dE

cos4 4s e

s s e en dV d n c d dA dA

dt

2

#s m

//photon flux densitycos4 s

s sn c d

4c dnE

dE

//spectral photon flux 2

#eV s m sr

E

cos4

s

e s s e ed n dV d d dA E d dA b E dA

dE dt dE

coss

s sdb E E ddE

//spectral photon flux density

2#

eV s mb E

1)

2)

3)

Blackbody radiation (VI) cos

ss sb E E d F E

cos cos sinF d d d

what range?

//Geometric factor

2

sincos

12

xdx d

x dx x

2 2 2 200 0 0

1cos sin sin sin2

ss

sF d d

For a spherical source:

srF

9149.6 10 msD

91.392 10 msd

//earth-sun distance

//sun diameter

32tan 4.65 10ss

s

dD

4.6 mrad = 0.267s //semi-angle of sun, viewed from earth

Blackbody radiation (VII)

2

2sinsins s

fXf

2 maxsins s sF F f

Geometric factor is maximized at the surface of the sun: max 90s maxF

At the surface of the earth:

2 5

maxsin 2.2 10s s

FfF

Concentration increases this factor: sf X f

max

max 42

1 1 4.6 10sins s s

fXf f

maxXf

X

max 2sin 90 1f

Blackbody Radiation (cont’d.)

//spectral irradiance L E E b E

cos4 4 s s eu dV d u c d dA

dt

cos cos cos4 4 4s s e s s e s s e

d u dV d d u c c dud dA d dA E E d dAdE dt dE dE

1)

du dnEdE dE

cos4 4s s

s s e eu dV d u c d dA P dA

dt

2

Wm

P //power flux densitycos4 s

s su cP d

2)

cos4

s

e s s e e ed u dV d dP dA E E d dA E b E dA L E dA

dE dt dE

3)

2

WeV m

L E

Consider radiated energy:

2W

eV m srE E

E E //spectral energy flux

Blackbody radiation (IX)

max

2maxW1300 m

sPPX

5760 KsT

5 48

3 2 22 W5.67 10 15 m K

sk

h c

// Stefan constant

// Total power density from sun on earth

Power flux:

max 42

MW62 m

sP T On sun’s surface:

4f cP u

5 44

3 3815

ku Th c

4 44 4

3 2215

skP f T f T

h c

Sun:

4max1 1

4.6 10sf

X

On earth’s surface:

max 1f

Blackbody Radiation (cont’d.)

b E F E 4cf 4

dn f c dndE dE

L E E b E F E E 4cf 4

dn f c duEdE dE

Notice that a flux density (or spectral flux density) is proportional to a density (or spectral density) within the BB:

4f c n

4f cP u

4f cJ q n

4

f c dnj E q b E qdE

j E q b E //spectral equivalent current flux

//equivalent current flux4f cJ q q n

2A

eV mj E

2

Am

J

Blackbody Radiation (cont’d.)

2

11 2 1 2, ,

4E

E Ef cE E b E dE n E E

2

11 2 1 2, ,

4E

E Ef cP E E L E dE u E E

2

11 2 1 2, ,

4E

E Ef cJ E E j E dE q n E E

We may also want to consider a limited energy range. Define:

2

11 2,

E

E Eduu E E dEdE

2

11 2,

E

E Ednn E E dEdE

Then:

SummarySpectral densities and flux densities

Spectral photon density 3#

eV m

Spectral photon flux density 2

#eV s m

dn D E f EdE V

4f c dnb E

dE

Spectral energy density 3J

eV m

Spectral irradiance 2W

eV m

du dnEdE dE

4

f c duL EdE

Densities and fluxes

Photon density 3#

m

Photon flux 2#

s m

2

11 2,

E

E Ednn E E dEdE

1 2 1 2, ,4

f cE E n E E

Energy density 3J

m

Power flux 2Wm

2

11 2,

E

E Eduu E E dEdE

1 2 1 2, ,4

f cP E E u E E

Bose-Einstein thermal distribution function

Wavelength Representations

4E f c dnb

dE

dn dE dnd d dE

du dE dud d dE

4f c dnb

d

4E f c duL

dE

4f c duL

d

hcE

2

2hc EdE

hc

2

EhcL L

2

Ehcb b

Real vs. calculated solar spectra

1sin shh

h 1h air mass1

sin s

n

42s AM1.5→

Angle of sun from horizon

s

s

lat

lat

Noon, summer solsticeNoon, winter solstice

solar radiation

90 23.5 cos 2365 dayss lat

N

# of days since summer solsticeN

90 23.5s lat 90 23.5s lat

23.5° 23.5°

44.1lat Rapid City, SD:

22.6s 69.4s

air mass 2.6n air mass 1.07n