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

Astro 300B: Jan. 26, 2011 Thermal radiation and Thermal Equilibrium. Thermal Radiation, and Thermodynamic Equilibrium. Thermal radiation is radiation emitted by matter in thermodynamic equilibrium.

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

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

  2. 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?

  8. 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

  9. (2) The Planck Function

  10. (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

  11. (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

  12. 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/TS = entropy

  13. Recall, U = uVu = 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)

  14. So... Differentiate these….

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

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

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

  18. 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 = 7.56x10-15 ergs cm-3 deg-4 also

  19. 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

  20. 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

  21. 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.

  22. 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 + 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!

  23. Properties of the Planck Law

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

  25. Rayleigh-Jeans Law so becomes

  26. 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 

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

  28. 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

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

  30. Divide by exp(hν/kT), cancel some terms Let Need to solve Solution is x=2.82. Need to solve graphically or iteratively.

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

  32. 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 8000 Å Why? recall So equal intervals in wavlength correspond to very different intervals of frequency across the spectrum With increasing l, constant dl (the Ilcase) corresponds to smaller and smaller dn so these smaller dn intervals contain smaller energy, compared to constant dn intervals (the In case)

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

  34. Also, since

  35. As an example of the kind of things you can model with the Planck 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.

  36. 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

  37. 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 roughly N = (0.367 J)/(0.126 eV x 1.6 x 10-19 J/eV) = 1.82 x 1019 photons

  38. 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.

  39. 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

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

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

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

  43. (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

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

  45. (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 ~0.4-0.8 Ωs= solid angle subtended by the source ΩA= solid angle from which the antenna receives radiation (“beam”)

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

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