Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

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Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium

Transcript of Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

Page 1: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

Astro 300B: Jan. 26, 2011

Thermal radiation and Thermal Equilibrium

Page 2: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

Thermal Radiation, and Thermodynamic Equilibrium

Page 3: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

Thermal radiation is radiation emitted by matter in thermodynamic equilibrium.

When radiation is in thermal equilibrium, Iν is a universal function of frequency ν and temperature T – the Planck function, Bν.

Blackbody Radiation: BI

In a very optically thick media, recall the SOURCE FUNCTION

Ij

S

So thermal radiation has BjBS and

And the equation of radiative transfer becomes

)(or TBId

dIBI

dl

dI

Page 4: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

THERMODYNAMIC EQUILIBRIUM

When astronomers speak of thermodynamic equilibrium, they mean a lot

more than dT/dt = 0, i.e. temperature is a constant.DETAILED BALANCE: rate of every reaction = rate of inverse reaction on a microprocess level

If DETAILED BALANCE holds, then one can describe

(1) The radiation field by the Planck function(2) The ionization of atoms by the SAHA equation(3) The excitation of electroms in atoms by the Boltzman distribution(4) The velocity distribution of particles by the Maxwell-Boltzman distribution

ALL WITH THE SAME TEMPERATURE, T

When (1)-(4) are described with a single temperature, T, then the system is said to be in THERMODYNAMIC EQUILIBRIUM.

Page 5: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

In thermodynamic equilibrium, the radiation and matter have the same temperature, i.e. there is a very high level of coupling between matter and radiation Very high optical depth

By contrast, a system can be in statistical equilibrium, or in a steady state, but not be in thermodynamic equilibrium.

So it could be that measurable quantities are constant with time, but there are 4 different temperatures:

T(ionization) given by the Saha equationT(excitation) given by the Boltzman equationT(radiation) given by the Planck FunctionT(kinetic) given by the Maxwell-Boltzmann distribution

WhereT(ionization) ≠ T(excitation) ≠ T(radiation) ≠ T(kinetic)

Page 6: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

If locally, T(ion) = T(exc) = T(rad) = T(kinetic)

Then the system is in LOCAL THERMODYNAMIC EQUILIBRIUM, or LTE

This can be a good approximation if the mean free path for particle-photon interactions << scale upon which T changes

LOCAL THERMODYNAMIC EQUILIBRIUM (LTE)

Page 7: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

Example: H II Region (e.g. Orion Nebula, Eagle Nebula, etc)

Ionized region of interstellar gas around a very hot star

Radiation field is essentially a black-body at the temperature of the centralStar, T~50,000 – 100,000 K

However, the gas cools to Te ~ 10,000 K (Te = kinetic temperature of electrons)

H IH II

O star

Q.: Is this room in thermodynamic equilibrium?

Page 8: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.
Page 9: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

FYI, we write down the following functions, without deriving them:

(1) The Boltzman Equation

Boltzman showed that the probability of finding an atom with an electron, e-, in an excited state with energy χn above the ground state

decreases exponentially with χn and increases exponentially with temperature T

kTg

g

N

N nnn exp

11

WhereNn = # atoms in excited state n / volumeN1 = # atoms in ground state /volume

gn = 2n2 the statistical weight of level n = number of different angular momentum quantum numbers in energy level n

Page 10: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

(2) The Planck Function

1

12/2

3

kThec

hBI

Page 11: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

(3) The Maxwell-Boltzman distribution of speeds of electrons

e

e

kT

vm

e

e evkT

mvf

22

2/3 2

24)(

= fraction of electrons with velocity between v, v+dv

where me = mass of the electron Te = temperature of the electrons

Page 12: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

(4) The Saha Equation

kTe

m

m

m

me

m

eh

kTm

Z

Z

N

Nn

2/3

311 2

2

Where ne = number density of free electrons Nm = number density of atoms in the mth ionization state

Zm = partition function of the mth ionization state

1i

kTim

i

egZ

Page 13: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.
Page 14: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

Thermodynamics of Blackbody Radiation: The Stefan-Boltzman Law

Consider a piston containing black-body radiation:

Inside the piston: T, v, p uMove blue wall extract or perform work

First Law of Thermodynamics: dQ = dU + p dV where dQ = change in heat dU = total change in energy p = pressure dV = change in volume

Second Law of Thermodynamics: dS = dQ/T S = entropy

Page 15: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

Recall,

U = uV u = energy density energy/volumep = 1/3 u p = radiation pressure in piston

BJdJ

cu and

4

So…T

dQdS

T

dVp

T

dU (substitute dQ=dU+pdV)

T

dVu

T

uVd

3

1)( (substitute U=uV, p=1/3 u)

Page 16: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

T

dVudV

T

u

T

VdudS

3

1

dVT

u

dT

dTdu

T

V

3

4

dVT

udT

dT

du

T

V

3

4

So...

T

u

dV

dS

dT

du

T

V

dT

dS

TV 3

4 and

Differentiate these….

Page 17: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

(Eqn.1) 12

dT

du

TdT

du

T

V

dV

d

dTdV

Sd

2) (Eqn. 1

3

4

3

4

3

42

2

dT

du

TT

u

T

u

dT

d

dVdT

Sd

Combining (1) and (2) dT

du

TT

u

dT

du

T

1

3

4

3

412

Multiply by TdT

du

T

u

dT

du

3

4

3

4

T

u

dT

du4

Page 18: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

T

dT

u

du4

aTu loglog4log a=constant of integration

4)( aTTu Energy density ~T4

u can be related to the Planck Function

J

cu

4 For isotropic radiation,

BJI

So… )(4

)(4

TBc

dTBc

duu

Page 19: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

Where B(T) = the integrated Planck function

4

4T

acdB

For a uniform, isotropically emitting surface, we showed that the flux

)(TBdBdFF

4

4T

ac

4

4T

ac

OR….

Page 20: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

4TF Stefan-Boltzmann Law

Where4

ac = 5.67x10-5 ergs cm-2 deg-4 sec-1

[flux] = ergs cm-2 sec-1 flux integrated over frequency, per area per sec

alsoc

a4

= 7.56x10-15 ergs cm-3 deg-4

Page 21: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

Blackbody Radiation; The Planck Spectrum

• The spectrum of thermal radiation, i.e. radiation in equilibrium with material at temperature T, was known experimentally before Planck

• Rayleigh & Jeans derived their relation for the blackbody spectrum for long wavelengths,

• Wien derived the spectrum at short wavelengths

• But, classical physics failed to explain the shape of the spectrum.

• Planck’s derivation involved the consideration of quantized electromagnetic oscillators, which are in equilibrium with the radiation field inside a cavity

the derivation launched Quantum Mechanics

See Feynmann Lectures, Vol. III, Chapt.4; R&L pp. 20-21

Page 22: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

Result:

Or in terms of Bλ recall

dIdI

1

12/3

3

kThec

hB

2 so

c

d

dc

1

12/5

2

kThce

hcB

ergs s-1 cm-2 Hz-1 ster-1

ergs s-1 cm-2 A-1 ster-1

Page 23: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

The Cosmic Microwave Background

The most famous (and perfect) blackbody spectrum is the “Cosmic Microwave Background.”

Until a few hundred thousand years after the Big Bang, the Universe was extremely hot, all hydrogen was ionized, and because of Thomson scattering by free electrons, the Universe was OPAQUE.

Then hydrogen recombined and the Universe became transparent.The relict radiation, which was last in thermodynamic equilibriumwith matter at the “surface of last scattering” is the CMB.

Currently the CMB radiation has the spectrum of a blackbody withT=2.73 K.

It is cooling as the Universe expands.

Page 24: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

The first accurate measurement of the spectrum of the CMBwas obtained with the FIRAS instrument aboard the CosmicBackground Explorer (COBE), from space:

See Mather + 1990 ApJLetters 354, L37

The smooth curve is the theoretical Planck Law. This plot was made using the first year of data; in subsequent plots the error bars are smaller than the width of the lines!

Page 25: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

Properties of the Planck Law

Page 26: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

Two limits simplify the Planck Law (and make it simpler to integrate):

Rayleigh-Jeans: hν << kT (Radio Astronomy) Wien hν >> kT

Page 27: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

Rayleigh-Jeans Law

kT

hkTh

1e so /kTh

so1

12)(

/2

3

kThec

hTI

becomes

kTc

TI RJ2

22)(

Page 28: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

The Ultraviolet Catastrophe

If the Rayleigh-Jean’s form for the spectrum of a blackbody held for all frequencies, then

as dI

And the total energy in the radiation field

Page 29: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

Wien’s Law

kThkTh eekTh

//

1

1

1 so

kThW ec

hTI /

2

32)(

Very steep decrease in brightness for peak

Page 30: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

Monotonicity with Temperature

If T1 > T2, then Bν(T1) > Bν(T2) for all frequencies

Of 2 blackbody curves, the one withhigher temperature lies entirely above the other.

1

12)(/2

3

kThec

h

dT

d

dT

TdB

2/

/

22

42

1

2

kTh

kTh

e

e

kTc

h

>0 always

Page 31: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

Wien Displacement Law

At what frequency does the Planck Law Bν(T) peak?

Bν(T) peaks at νmax, given by 0max

ddB

01

12/2

3

kThec

h

d

d

0122

1 /2

3

2

3/

kThkTh e

d

d

c

h

c

h

d

de

Page 32: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

ehν / kT −1( )6hν 2

c 2

⎝ ⎜

⎠ ⎟=

2hν 3

c 2

⎝ ⎜

⎠ ⎟h

kTehν / kT

( )

Divide by exp(hν/kT), cancel some terms

kT

he kTh /13

Let kT

hx max

Need to solve xe x 13

Solution is x=2.82. Need to solve graphically or iteratively.

Page 33: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

kT

h max82.2

Page 34: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

110max deg 1088.5 HzT

Similarly, one can find the wavelength λmax at which Bλ(T) peaks

0max

ddB

deg cm 290.0max T

Page 35: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

NOTE: cmaxmax

That is to say, Bν and Bλ don’t peak at the same wavelength, or frequency.

For the Sun’s spectrum, λmax for Iλ is at about 4500 Å whereas λmax for Iν is at about 8000 Å

Why?

recall

dc

d2

So equal intervals in wavlength correspond to very different intervals offrequency across the spectrum

With increasing l, constant dl (the Il case) corresponds to smaller and smaller dnso these smaller dn intervals contain smaller energy, comparedto constant dn intervals (the In case)

Page 36: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

Radiation constants in terms of physical constants

Recall the Stefan-Boltzman law for flux of a black body

dB

TF

4

0 /

3

20 1

2 dec

hdB

kTh

then Let kT

hx

0

34

0 2 1

2dx

e

x

h

kT

c

hdB

x

0

432

444

0

3

15

2 so...

151T

hc

kdBdx

e

xx

32

45

15

2

hc

k So

Page 37: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

Also, since

4

0

4aTdB

cu

33

45

15

8

hc

ka

Page 38: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

As an example of the kind of things you can model with thePlanck radiation formulae, consider the following:

(see http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/nickel.html) (1) How much radiant energy comes from a nickel at room temperature per second?

Measured properties of the nickel are diameter = 2.14 cm, thickness 0.2 cm, mass 5.1 grams. This gives a volume of 0.719 cm3 and a surface area of 8.54 cm 2.

Page 39: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

The radiation from the nickel's surface can be calculated from the Stefan-Boltzman Law

F= σT4

The room temperature will be taken to be 22°C = 295 K.

Assuming an ideal radiator for this estimate, the radiated power is

P = σAT4 A=surface area of nickel = (5.67 x 10-8 W/m2K4)x(8.54 x 10-4 m2)x(295 K)4

= 0.367 watts.

So the radiated power from a nickel at room temperature is about 0.37 watts

Page 40: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

2. How many photons per second leave the nickel?Since we know the energy, we can divide it by the average photon energy.

We don't know a true average, but the wavelength of the peak of the blackbody radiation curve is a representative value which can be used as an estimate.

This may be obtained from the Wien displacement law.

lpeak = 0.0029 m K/295 K = 9.83 x 10-6 m = 9830 nm, in the infrared.

The energy per photon at this peak can be obtained from the Planck relationship.

Ephoton = hν = hc/λ = 1240 eV nm/ 9830 nm = 0.126 eV

Then the number of photons per second is very roughlyN = (0.367 J)/(0.126 eV x 1.6 x 10-19 J/eV) = 1.82 x 1019 photons

Page 41: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

Characteristic Temperatures for Blackbodies

1. BRIGHTNESS TEMPERATURE, Tb

Instead of stating Iν, one can state Tb, where

)( BTBI

i.e. Tb is the temperature of the blackbody having the same specific intensity as the source, at a particular frequency.

Page 42: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

Notes:

1. TB is often used in radio astronomy, and so you canassume that the Rayleigh-Jeans Law holds,

kTh so BkTcI

2

22

I

k

cTB 2

2

2or

2. The source need not be a blackbody, despite being describedas a source with brightness temperature TB.

3. Units of TB are easier to remember than units of Iν

Page 43: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

TB and the equation of Radiative Transfer:

)(TBId

dI

Assume Rayleigh-Jeans, kTc

I2

22

BB kTc

TBI2

22)(

BkTcd

d

d

dI2

22

So the equation of radiative transfer becomes:TT

d

dTB

B

Page 44: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

TTd

dTB

B materialtheofetemperaturT

IdescribingetemperaturbrightnessTB

eTeTT BB 1)0(

then,ith constant w is T If

BT then If The brightness temperature =The actual temperature at largeoptical depth

Otherwise, TTB

Page 45: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

TTd

dTB

B materialtheofetemperaturT

IdescribingetemperaturbrightnessTB

eTeTT BB 1)0(

then,ith constant w is T If

BT then If The brightness temperature =The actual temperature at largeoptical depth

Otherwise, TTB

Page 46: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

(2) Color Temperature, Tc

Often one can measure the spectrum of a source, and it is more orless a blackbody of some temperature, Tc.

We may not know Iν, but only Fν, if for example the source is unresolved.

Tc can be estimated fromλ(max), the peak of the spectrum,or the ratio of the spectrumat 2 wavelengths.

e.g. B-V colors of stars

Page 47: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

The solar spectrum vs. blackbody – from Caroll & Ostlie

Page 48: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.
Page 49: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

(3) Antenna Temperature, TA

A radio telescope mearures the brightness of a source,Often described by

A

SBA TT

Where η = the beam efficiency of the telescope, typically ~0.4-0.8

Ωs= solid angle subtended by the source

ΩA= solid angle from which the antenna receives radiation (“beam”)

Page 50: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

(4) Effective Temperature, Teff

If a source has total flux F, integrated over all frequencies

we can define Teff such that

4 effTF

Page 51: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

The Einstein Coefficients

Page 52: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

Einstein (1917) related αν and jν to microscopic processes,by considering how a photon interacts with a 2-level atom:

emission absorption

E2

E1

Level 2, statistical weight g2

Level 1, statistical weight g1

012 hEE

Absorption: system goes from Level 1 to Level 2 by absorbing a photon with energy hν0

Emission: system goes from Level 2 to Level 1 and a photon is emitted.

Three processes can occur: 1. Spontaneous Emission 2. Absorption 3. Stimulated Emission

Page 53: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

1. Spontaneous Emission

An atom in Level 2 drops to Level 1, emitting a photon, even in the absence of a radiation field

Einstein A coefficient

A21 ≡ transition probability per unit time for spontaneous emission[A21]= sec -1

Examples: permitted, dipole transitions A21 ~ 108 sec-1

magnetic dipole, forbidden transitions A21~103 sec-1

electric quadrupole, forbidden transitions A21~1 sec-1

2

1

Page 54: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

2. Absorption

An atom in level 1 absorbs a photon and ends up in level 2.

1

2

Due to the Heisenberg uncertainty principle, ΔE Δt > ħ, the energy levels are not precisely sharp

Each level has a “spread” in energy, called the “natural”Line width, a Lorentzian.

So let’s parameterize the line profile as φ(ν),Centered on frequency νo.

We define φ(ν) so that

φ(ν)

0

1)( d

Page 55: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

Einstein B-coefficients

B12 ≡ transition probability per unit time for absorption

Where

0

)( dJJ

J

Page 56: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

Stimulated Emission

The presence of a radiation field will stimulate an atom to go from level 2 level 1

JB21Transition probability, per unit time for stimulated emission

Page 57: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

Equation of Statistical Equilibrium

If detailed balance holds

Number of transitions/secfrom Level 1 Level 2

Number of transitions/secfrom Level 2 Level 1

Let n1 = # of atoms / volume in Level 1 n2 = # of atoms / volume in Level 2

Then:

JJ B n An Bn 212212121

Absorption Spontaneousemission

Stimulatedemission

=

Page 58: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

hence

121

12

2

1

21

21

BB

nn

BA

J

In thermodynamic equilibrium, the Boltzman equation gives n1/n2

kT

ho

gg

nn exp

2

1

2

1

1exp 0

212

121

21

21

kTh

BgBg

BA

J

So (1)

Page 59: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

In thermodynamic equilibrium, BJ

)( 0BJ

1exp

1

2

02

30

kT

hh

c

Since the Lorentzian is narrow, we can approximate

(2)

Page 60: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

Comparing (1) and (2), we getthe EINSTEIN RELATIONS

BgBg 212121

cBA

h2

3

2121

2

Page 61: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

Comments:• There’s no “T” in the Einstein Relations, they relate atomic constants

only. Hence, they must be true even if T.E. doesn’t hold.• Sometimes people derive the Einstein relations in terms of energy

density, uν instead of Jnu, so there’s an extra factor

of 4π/c:

BgBg 212121

h

cBA 0

8 3

32121

Page 62: Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium.

The Milne Relation

Another example of using detailed balance to derive relations which are independent of the LTE assumption

Relate photo-ionization cross-section at frequency nu, with cross-section for recombination for electrom with velocity v:

acmh

gg

v

v

222

22

2

1)(

See derivation in Osterbrock & Ferland