Thermal emission of EM radiationLecture overviewEarth’s energy budgetKirchhoff’s LawBlackbody radiationSlide 6EM wave modes in a cavitySlide 8Rayleigh-Jeans LawThe Ultraviolet catastropheIn what energy levels do molecules reside?Planck’s FunctionSlide 14Slide 15Planck’s Function (version 4)Planck curves for blackbodies at various temperaturesThe SunPlanck curves – 6000 KThe Sun at different wavelengthsDeparture from the blackbody assumptionPlanck curves – light bulbPlanck curves – ‘red hot’Planck curves – humans, terrestrial materialsPlanck curves – cosmic backgroundWien’s LawWien’s Law in wavelength domainBroadband fluxesStefan-Boltzmann LawStefan-Boltzmann Law derivationSlide 32Slide 33Slide 34In-band radiancePlanck’s function at longer wavelengthsRayleigh-Jeans ApproximationEmissivity (monochromatic)Emissivity (graybody)Inverting the Planck functionBrightness temperatureInfrared emission spectrumWhen does thermal emission matter?Exception: sun glintApplications: IR imaging from spaceSlide 47Slide 48Slide 49Slide 50IR ‘split-window’ techniqueSlide 52Slide 53Slide 54Slide 55Mt. Spurr volcanic ash cloudSlide 57Applications: radiation balanceSlide 59Slide 60Slide 61Applications: nighttime radiative coolingSlide 63Slide 64Thermal emission of EM radiation‘The birth of photons’Lecture overview• Shortwave (solar) radiation• Principle source is the sun• Distribution governed by absorption, transmission and reflection• Longwave radiation • Principle source is thermal emission by the Earth and atmosphere• Conversion of internal energy to radiant energy• Blackbody radiation• Radiation Laws• EmissivityEarth’s energy budget• Earth’s radiation budget is the balance between incoming solar radiation and outgoing (reflected and emitted) energy from the Earth• Governs Earth’s surface temperature and the greenhouse effectEmissionKirchhoff’s Law• Proposed by Gustav Kirchhoff in 1859• At thermal equilibrium, the emissivity of a body equals its absorptivity• ‘A good absorber is a good emitter’• ‘A good reflector is a poor emitter’ (and vice versa), e.g., an emergency thermal blanket• If a material is capable of absorbing a particular frequency, it follows that it will be a good emitter at the same frequency • NB. Nearly always an approximation for real surfaces – absorptivity may depend on both wavelength and direction€ ελ(θ,φ) = aλ(θ,φ)Blackbody radiation• All objects above absolute zero (0 K or -273°C) radiate EM energy (due to vibration of atoms)• EM emission dependent on temperature (atomic/molecular vibrations)• What is the theoretical maximum amount of radiation that can be emitted by an object?• Blackbody : a (hypothetical) body that absorbs all EM radiation incident on it• Absorptivity (a) = 1 (and hence ε = 1, from Kirchhoff’s Law)• The emissivity of a material is the ratio of what it emits to what would be emitted if it were a blackbody (ε = 0 for a whitebody, ε = 1 for a blackbody, 0 < ε < 1 for a graybody)• Blackbodies are used to calibrate remote sensing instrumentsBlackbody radiation• Blackbody radiation is isotropic and anything but black…• The name comes from the assumption that the body absorbs at every frequency and hence would look black (but snow is almost a blackbody in the IR)• Also called ‘equilibrium radiation’, ‘cavity radiation’ or ‘radiation in equilibrium with matter’• Foundation for development of radiation lawsPhotons enter a cavity through a small hole. With each additional reflection from the cavity wall, the chance of absorption increases. If the hole is small and the cavity large, the absorptivity of the hole tends towards unity. The result is blackbody radiation at the temperature of the walls of the cavity.What is the intensity of radiation emitted by a blackbody?EM wave modes in a cavity• The radiated energy from a cavity can be considered to be produced by standing waves or resonant modes of the cavity• You can fit more modes of shorter wavelength (higher frequency) into a cavity• Work by Lord Rayleigh and Sir James Jeans (in 1900-1905) showed that the number of modes was proportional to the frequency squaredBlackbody radiation• The amount of radiation emitted in a given frequency range should be proportional to the number of modes in that range• ‘Classical’ (pre-quantum) physics suggested that all modes had an equal chance of being produced, and that the number of modes went up as the square of the frequencyRayleigh-Jeans Law• Classical prediction of radiance from a blackbody at temperature T• c: speed of light (2.998×108 m s-1)• kB: Boltzmann’s constant (1.381×10-23 J K-1)• Blackbody emission proportional to temperature, and inversely proportional to the fourth power of wavelengthB(T) 2ckB4TThe Ultraviolet catastrophe• The Rayleigh-Jeans law predicted a continued increase of radiated energy with increasing frequency and an ‘ultraviolet catastrophe’• Max Planck solved this (in 1901) by assuming that the EM modes in a cavity were quantized with energy (E) equal to hν (h = Planck’s constant)In what energy levels do molecules reside?Ni is the number density of molecules in state i (i.e., the number of molecules per cm3).T is the temperature, and kB is Boltzmann’s constant - relates energy at the particle level to temperature.Boltzmann’s factor: exp[-E/kT] – expresses the probability of a state of energy E relative to a state of zero energy.The probability of a given energy state decreases exponentially with energy.[ ]exp /i i BN E k T� -EnergyPopulation densityN1N3N2E3E1E2Boltzmann distributionPlanck’s Function• Radiation formula derived by Max Planck in 1901 • Gives the spectral radiance emitted by a blackbody at temperature T (K) in the direction normal to the surface• Units?• c: speed of light (2.998×108 m s-1)• h: Planck’s constant (6.626×10-34 J s)• kB: Boltzmann’s constant (1.381×10-23 J K-1) – relates energy at the particle level to temperature• Note that this form of the Planck function uses wavelength (λ)BT 2hc25(ehc kBT 1)Planck’s Function• Collect constants together• c1 = 1.19×10-16 W m2; c2 = 1.44×10-2 m K• Temperature in Kelvin (K), wavelength in meters for these constants• Need to be careful with units!BT c15(ec2T 1)Planck’s Function• Planck’s law as a