1/30/11. Astro 300B: Jan. 26, Thermal radia+on and Thermal Equilibrium. Thermal Radia0on, and Thermodynamic Equilibrium

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1 Astro 300B: Jan. 26, 2011 Thermal radia+on and Thermal Equilibrium Thermal Radia0on, and Thermodynamic Equilibrium 1

2 Thermal radiation is radiation emitted by matter in thermodynamic equilibrium. When radiation is in thermal equilibrium, I ν is a universal function of frequency ν and temperature T the Planck function, B ν. Blackbody Radiation: In a very optically thick media, recall the SOURCE FUNCTION So thermal radiation has And the equation of radiative transfer becomes 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. 2

3 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) T(excitation) T(radiation) T(kinetic) given by the Saha equation given by the Boltzman equation given by the Planck Function given by the Maxwell-Boltzmann distribution Where T(ionization) T(excitation) T(radiation) T(kinetic) 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 3

4 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 K However, the gas cools to T e ~ 10,000 K (T e = kinetic temperature of electrons) O star H I H II Q.: Is this room in thermodynamic equilibrium? 4

5 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 χ n above the ground state decreases exponentially with χ n increases exponentially with temperature T and Where N n = # atoms in excited state n / volume N 1 = # atoms in ground state /volume g n = 2n 2 the statistical weight of level n = number of different angular momentum quantum numbers in energy level n (2) The Planck Function 5

6 (3) The Maxwell-Boltzman distribution of speeds of electrons = fraction of electrons with velocity between v, v+dv where m e = mass of the electron T e = temperature of the electrons (4) The Saha Equation Where n e = number density of free electrons N m = number density of atoms in the m th ionization state Z m = partition function of the m th ionization state 6

7 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 7

8 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) So... Differentiate these. 8

9 Combining (1) and (2) Multiply by T a=constant of integration Energy density ~T 4 u can be related to the Planck Function For isotropic radiation, So 9

10 Where B(T) = the integrated Planck function For a uniform, isotropically emitting surface, we showed that the flux OR. 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 also = 7.56x10-15 ergs cm -3 deg -4 10

11 Blackbody Radia0on; The Planck Spectrum The spectrum of thermal radia0on, i.e. radia0on in equilibrium with material at temperature T, was known experimentally before Planck Rayleigh & Jeans derived their rela0on 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 deriva0on involved the considera0on of quan0zed electromagne0c oscillators, which are in equilibrium with the radia0on field inside a cavity the deriva0on launched Quantum Mechanics See Feynmann Lectures, Vol. III, Chapt.4; R&L pp Result: Or in terms of B λ recall ergs s -1 cm -2 Hz -1 ster -1 ergs s -1 cm -2 A -1 ster -1 11

12 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. 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! 12

13 Proper0es of the Planck Law Two limits simplify the Planck Law (and make it simpler to integrate): Rayleigh-Jeans: hν << kt Wien hν >> kt (Radio Astronomy) 13

14 Rayleigh-Jeans Law so becomes 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 14

15 Wien s Law Very steep decrease in brightness for Monotonicity with Temperature If T 1 > T 2, then B ν (T 1 ) > B ν (T 2 ) for all frequencies Of 2 blackbody curves, the one with higher temperature lies entirely above the other. >0 always 15

16 Wien Displacement Law At what frequency does the Planck Law B ν (T) peak? B ν (T) peaks at ν max, given by ( e hν / kt 1) 6hν 2 = 2hν 3 h c 2 c 2 ( ) kt ehν / kt Divide by exp(hν/kt), cancel some terms Let Need to solve Solution is x=2.82. Need to solve graphically or iteratively. 16

17 Similarly, one can find the wavelength λ max at which B λ (T) peaks 17

18 NOTE: That is to say, B ν and B λ don t peak at the same wavelength, or frequency. For the Sun s spectrum, λ max for I λ is at about 4500 Å whereas λ max for I ν 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 I l case) corresponds to smaller and smaller dn so these smaller dn intervals contain smaller energy, compared to constant dn intervals (the I n case) Radiation constants in terms of physical constants Recall the Stefan-Boltzman law for flux of a black body So 18

19 Also, since 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 cm 3 and a surface area of 8.54 cm 2. 19

20 The radiation from the nickel's surface can be calculated from the Stefan-Boltzman Law F= σt 4 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 4 A=surface area of nickel = (5.67 x 10-8 W/m 2 K 4 )x(8.54 x 10-4 m 2 )x(295 K) 4 = watts. So the radiated power from a nickel at room temperature is about 0.37 watts 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. l peak = 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. E photon = 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 photons 20

21 Characteristic Temperatures for Blackbodies 1. BRIGHTNESS TEMPERATURE, T b Instead of stating I ν, one can state T b, where i.e. T b is the temperature of the blackbody having the same specific intensity as the source, at a particular frequency. Notes: 1. T B is often used in radio astronomy, and so you can assume that the Rayleigh-Jeans Law holds, so 2. The source need not be a blackbody, despite being described as a source with brightness temperature T B. 3. Units of T B are easier to remember than units of I ν 21

22 T B and the equation of Radiative Transfer: Assume Rayleigh-Jeans, So the equation of radiative transfer becomes: The brightness temperature = The actual temperature at large optical depth Otherwise, 22

23 The brightness temperature = The actual temperature at large optical depth Otherwise, (2) Color Temperature, T c 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 I ν, but only F ν, if for example the source is unresolved. T c 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 23

24 The solar spectrum vs. blackbody from Caroll & Ostlie 24

25 (3) Antenna Temperature, T A A radio telescope mearures the brightness of a source, Often described by Where η = the beam efficiency of the telescope, typically ~ Ω s = solid angle subtended by the source Ω A = solid angle from which the antenna receives radiation ( beam ) (4) Effective Temperature, T eff If a source has total flux F, integrated over all frequencies we can define T eff such that 25

26 The Einstein Coefficients Einstein (1917) related α ν and j ν to microscopic processes, by considering how a photon interacts with a 2- level atom: E 2 emission E 1 absorp0on Level 2, sta0s0cal weight g 2 Level 1, sta0s0cal weight g 1 Absorp+on: system goes from Level 1 to Level 2 by absorbing a photon with energy hν 0 Emission: system goes from Level 2 to Level 1 and a photon is emiked. Three processes can occur: 1. Spontaneous Emission 2. Absorp0on 3. S0mulated Emission 26

27 1. Spontaneous Emission An atom in Level 2 drops to Level 1, emilng a photon, even in the absence of a radia0on field 2 1 Einstein A coefficient A 21 transi0on probability per unit 0me for spontaneous emission [A 21 ]= sec - 1 Examples: permiked, dipole transi0ons A 21 ~ 10 8 sec - 1 magne0c dipole, forbidden transi0ons A 21 ~10 3 sec - 1 electric quadrupole, forbidden transi0ons A 21 ~1 sec Absorp+on An atom in level 1 absorbs a photon and ends up in level 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 ν o. φ(ν) We define φ(ν) so that 27

28 Einstein B- coefficients B 12 transi0on probability per unit 0me for absorp8on Where S0mulated Emission The presence of a radia0on field will s0mulate an atom to go from level 2 level 1 Transi0on probability, per unit 0me for s8mulated emission 28

29 Equa0on of Sta0s0cal Equilibrium If detailed balance holds Number of transi0ons/sec from Level 1 Level 2 = Number of transi0ons/sec from Level 2 Level 1 Let n 1 = # of atoms / volume in Level 1 n 2 = # of atoms / volume in Level 2 Then: Absorp+on Spontaneous emission S+mulated emission hence In thermodynamic equilibrium, the Boltzman equa0on gives n 1 /n 2 So (1) 29

30 In thermodynamic equilibrium, Since the Lorentzian is narrow, we can approximate (2) Comparing (1) and (2), we get the EINSTEIN RELATIONS 30

31 Comments: There s no T in the Einstein Rela0ons, they relate atomic constants only. Hence, they must be true even if T.E. doesn t hold. Some0mes people derive the Einstein rela0ons in terms of energy density, u ν instead of Jnu, so there s an extra factor of 4π/c: The Milne Rela0on Another example of using detailed balance to derive rela0ons which are independent of the LTE assump0on Relate photo- ioniza0on cross- sec0on at frequency nu, with cross- sec0on for recombina0on for electrom with velocity v: See deriva0on in Osterbrock & Ferland 31