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

2. Thermodynamic Equilibrium
Thermal Radiation, and
Thermodynamic Equilibrium

3. 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:
In a very optically thick media, recall the SOURCE FUNCTION
So thermal radiation has
And the equation of radiative transfer becomes

4. 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
The radiation field by the Planck function
The ionization of atoms by the SAHA equation
The excitation of electroms in atoms by the Boltzman distribution
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.

5. 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 equation
T(excitation) given by the Boltzman equation
T(radiation) given by the Planck Function
T(kinetic) given by the Maxwell-Boltzmann distribution
Where
T(ionization) ≠ T(excitation) ≠ T(radiation) ≠ T(kinetic)

6. LOCAL THERMODYNAMIC EQUILIBRIUM (LTE)
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

7. 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 central
Star, T~50,000 – 100,000 K
However, the gas cools to Te ~ 10,000 K
(Te = kinetic temperature of electrons)
O star
H II
H I
Q.: Is this room in thermodynamic equilibrium?

9. FYI, we write down the following functions, without deriving them:
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
Where
Nn = # atoms in excited state n / volume
N1 = # atoms in ground state /volume
gn = 2n2 the statistical weight of level n
= number of different angular momentum quantum numbers
in energy level n

10. (2) The Planck Function

11. (3) The Maxwell-Boltzman distribution of speeds of electrons
= fraction of electrons with velocity between v, v+dv
where me = mass of the electron
Te = temperature of the electrons

12. (4) The Saha Equation 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

14. Thermodynamics of Blackbody Radiation: The Stefan-Boltzman Law
Consider a piston containing black-body radiation:
Inside the piston: T, v, p u
Move 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

15. Recall,
U = uV u = energy density energy/volume
p = 1/3 u p = radiation pressure in piston
So…
(substitute dQ=dU+pdV)
(substitute U=uV, p=1/3 u)

16. So...
Differentiate these….

17. Combining (1) and (2) 
Multiply by T

18. a=constant of integration
Energy density ~T4
u can be related to the Planck Function
For isotropic radiation,
So…

19. Where B(T) = the integrated Planck function
For a uniform, isotropically emitting surface, we showed that the flux
OR….

20. Stefan-Boltzmann Law
Where
= 5.67x10-5 ergs cm-2 deg-4 sec-1
[flux] = ergs cm-2 sec-1
flux integrated over frequency,
per area per sec
= 7.56x ergs cm-3 deg-4
also

21. 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

22. Result:
ergs s-1 cm-2 Hz-1 ster-1
Or in terms of Bλ recall
ergs s-1 cm-2 A-1 ster-1

23. 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 equilibrium
with matter at the “surface of last scattering” is the CMB.
Currently the CMB radiation has the spectrum of a blackbody with
T=2.73 K.
It is cooling as the Universe expands.

24. The first accurate measurement of the spectrum of the CMB
was obtained with the FIRAS instrument aboard the Cosmic
Background Explorer (COBE), from space:
See Mather 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!

25. Properties of the Planck Law

26. Two limits simplify the Planck Law (and make it simpler to integrate):
Rayleigh-Jeans: hν << kT (Radio Astronomy)
Wien hν >> kT

27. Rayleigh-Jeans Law so
becomes

28. The Ultraviolet Catastrophe
If the Rayleigh-Jean’s form for the spectrum of a blackbody
held for all frequencies, then
And the total energy
in the radiation field 

29. Wien’s Law
 Very steep decrease in brightness for

30. Monotonicity with Temperature
If T1 > T2, then Bν(T1) > Bν(T2)
for all frequencies
Of 2 blackbody curves, the one with
higher temperature lies entirely
above the other.
>0 always

31. Wien Displacement Law
At what frequency does the Planck Law Bν(T) peak?
Bν(T) peaks at νmax, given by

32. Divide by exp(hν/kT), cancel some terms
Let
Need to solve
Solution is x= Need to solve graphically or iteratively.

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

35. NOTE: 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 Å
Why?
recall
So equal intervals in wavlength
correspond to very different intervals of
frequency across the spectrum
With increasing l, constant dl (the Il case)
corresponds to smaller and smaller dn
so these smaller dn intervals contain smaller energy, compared
to constant dn intervals (the In case)

36. Radiation constants in terms of physical constants
Recall the Stefan-Boltzman law for flux of a black body
So 

37. Also, since

38. As an example of the kind of things you can model with the
Planck radiation formulae, consider the following:
(see
(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 cm3 and a surface area of 8.54 cm 2.

39. 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 = σAT A=surface area of nickel
= (5.67 x 10-8 W/m2K4)x(8.54 x 10-4 m2)x(295 K)4
= watts.
So the radiated power from a nickel at room temperature is about 0.37 watts

40. 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 = 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 = eV
Then the number of photons per second is very roughly
N = (0.367 J)/(0.126 eV x 1.6 x J/eV) = 1.82 x 1019 photons

41. Characteristic Temperatures for Blackbodies
1. BRIGHTNESS TEMPERATURE, Tb
Instead of stating Iν, one can state Tb, where
i.e. Tb is the temperature of the blackbody having the same
specific intensity as the source, at a particular frequency.

42. Notes: TB is often used in radio astronomy, and so you can
assume that the Rayleigh-Jeans Law holds, so
The source need not be a blackbody, despite being described
as a source with brightness temperature TB.
3. Units of TB are easier to remember than units of Iν

43. TB and the equation of Radiative Transfer:
Assume Rayleigh-Jeans,
So the equation of radiative transfer becomes:

44. The brightness temperature =
The actual temperature at large
optical depth
Otherwise,

45. The brightness temperature =
The actual temperature at large
optical depth
Otherwise,

46. (2) Color Temperature, Tc
Often one can measure the spectrum of a source, and it is more or
less 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 from
λ(max), the peak of the spectrum,
or the ratio of the spectrum
at 2 wavelengths.
e.g. B-V colors of stars

47. The solar spectrum vs. blackbody – from Caroll & Ostlie

49. (3) Antenna Temperature, TA
A radio telescope mearures the brightness of a source,
Often described by
Where η = the beam efficiency of the telescope, typically ~
Ωs= solid angle subtended by the source
ΩA= solid angle from which the antenna receives radiation
(“beam”)

50. (4) Effective Temperature, Teff
If a source has total flux F, integrated over all frequencies
we can define Teff such that

51. The Einstein Coefficients

52. Einstein (1917) related αν and jν to microscopic processes,
by considering how a photon interacts with a 2-level atom: E2
Level 2, statistical weight g2
emission
absorption E1
Level 1, statistical weight g1
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

53. Einstein A coefficient
1. Spontaneous Emission 2
An atom in Level 2 drops to Level 1,
emitting a photon,
even in the absence of a radiation field 1
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

54. An atom in level 1 absorbs a photon and ends up in level 2.
2. Absorption
An atom in level 1 absorbs a photon and ends up in level 2. 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. 1
So let’s parameterize the line profile as φ(ν),
Centered on frequency νo.
We define φ(ν) so that
φ(ν)

55. Einstein B-coefficients
B ≡ transition probability per unit time for absorption
Where

56. Stimulated Emission
The presence of a radiation field will stimulate an atom to go from level 2  level 1
Transition probability, per unit time
for stimulated emission

57. Equation of Statistical Equilibrium
If detailed balance holds
Number of transitions/sec
from Level 1  Level 2 =
Number of transitions/sec
from Level 2  Level 1
Let n1 = # of atoms / volume in Level 1
n2 = # of atoms / volume in Level 2
Then:
Absorption
Spontaneous
emission
Stimulated
emission

58. hence In thermodynamic equilibrium, the Boltzman equation gives n1/n2
So 
(1)

59. In thermodynamic equilibrium,
Since the Lorentzian is narrow, we can approximate
(2)

60. Comparing (1) and (2), we get the EINSTEIN RELATIONS

61. 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:

62. 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:
See derivation in Osterbrock & Ferland