A Warm Up Exercise. A Warm Up Exercise. A Warm Up Exercise. A Warm Up Exercise. The Solar Flux
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1 When you compare gamma ray photons with photons of radio waves, which of the following is true? Gamma rays have a shorter wavelength and less energy Gamma rays have a shorter wavelength and same energy Gamma rays have a longer wavelength and less energy Gamma rays have a shorter wavelength and more energy Gamma rays have a longer wavelength and more energy Temperature and Black Bodies When you compare gamma ray photons with photons of radio waves, which of the following is true? Gamma rays have a shorter wavelength and less energy Gamma rays have a shorter wavelength and same energy Gamma rays have a longer wavelength and less energy Gamma rays have a shorter wavelength and more energy Gamma rays have a longer wavelength and more energy How does the apparent brightness of a light source depend on it distance from you? a) It varies directly as the distance b) It varies inversely with the distance c) It varies as the inverse square of the distance d) It does not vary with distance at all How does the apparent brightness of a light source depend on it distance from you? a) It varies directly as the distance b) It varies inversely with the distance c) It varies as the inverse square of the distance d) It does not vary with distance at all The Solar Flux The flux from the Sun at distance D is simply the of the Sun spread over the surface of a sphere of radius d : Luminosity Brightness or Flux Area AU 1400 Watts/m D Luminosity 4 D luminosity Planets further from the Sun should be colder since the receive less solar heating 1
2 Solar Power At Earth The energy a planet receives (per second) is the incident flux (energy per unit area per second) times the projected area of the planet Incident flux = (solar luminosity)/(area of sphere radius 1 AU) = L/4 d d=1au=distance of Earth from Sun Projected area of the Earth = R R=radius of the earth So the Earth receives (incident flux)(projected area)=lr /4d = Watts of solar radiation = 177 million 1 Gigawatt power plants Energy Balance That energy has to go somewhere if it simply stayed on the Earth, the Earth would get steadily hotter with time But the Earth s temperature is roughly constant, so the Earth must be losing the energy again But how? Temperature We would like to relate the solar heating to the temperature of the Earth. Temperature is a measurement of the internal energy content of an object. Solids: Higher temperature means higher average vibrational energy per atom or molecule. Gases: Higher temperature means more average kinetic energy (faster speeds) per atom or molecule. Cool Gas Slow Average Speeds Hot Gas Faster Average Speeds Absolute Temperature The higher the temperature, the more rapidly the atoms or molecules in a gas move As we reduce the temperature, they move more and more slowly If it gets cold enough, all motion stops! Corresponds to 73º Celsius ( 459º F) Realistically, this absolute zero is an ultimate limit that you never quite reach. Current laboratory limits are about 10-9 degrees (Celsius or Kelvin) above absolute zero! Kelvin Temperature Scale Most scientists use an absolute temperature system developed by the English physicist Lord Kelvin (1800s). The size of the degrees is the same as Celsius, but 0 Kelvin correponds to absolute zero Absolute Kelvin Scale (K): T= 0 K = Absolute Zero (all motion stops, 73º Celsius) T=73 K = pure water freezes (0º Celsius) T=373 K = pure water boils (100º Celsius) Advantage: total internal energy is directly proportional to the temperature in Kelvins.
3 Some Cold Things Absolute Kelvin Scale (K): T=195 K = Freezing point of Carbon Dioxide T= 90 K = Liquid Oxygen T= 77 K = Liquid Nitrogen T= 4 K = Liquid Helium This is at Earth atmospheric pressure Black Body Radiation Anything with a temperature emits radiation A Blackbody is an object that absorbs all light. Absorbs at all wavelengths. As it absorbs light, it heats up. Characterized by its Temperature. It is also the perfect radiator: Emits at all wavelengths (continuous spectrum) The total energy emitted depends only on the temperature. The wavelength where most of the radiation is emitted depends only on the temperature. The Sun and the Planets are not perfect black bodies, but they are close enough. Stefan-Boltzmann Law The total Energy emitted per second per area by a blackbody with Temperature (T), otherwise known as the flux, is F= T 4 where = Watts/m K 4 is the Stefan-Boltzmann constant Hotter objects are Brighter (at All Wavelengths) Wien s Law The emission from a black body peaks at a wavelength set by the temperature max =(0.009/T)m =( /T)nm for T in Kelvin Hotter objects are bluer (i.e. have spectra that peak at shorter wavelengths) Cooler objects are redder (i.e. have spectra that peak at longer wavelengths) Hotter objects are brighter at all wavelengths Hotter objects are bluer (peak at shorter wavelengths) Infrared Example The Light Bulb The filament in a light bulb is about 0.6m long and m in diameter, giving it an area of area A=(length)(circumference)= (length)(diameter)=10-4 m Using electricity we then heat the filament to about T=00K, so it emits flux F= T 4 = Watts/m Giving a total luminosity of L=(flux)(area)=FA=130 Watts max =( /T)nm=1300nm (it peaks in the infrared) nm Incandescent bulbs produce more heat than light, which is why fluorescent bulbs (which are not black bodies) are more efficient 3
4 You have a surface area of about area A= (.0m 0.5m)=m Example - You Infrared Camera Image Wavelength ~000 nm ( microns) Your body temperature is about T=310K, so it emits flux F= T 4 =500 Watts/m Giving a total luminosity of L=(flux)(area)=FA=1000 Watts max =( /T)nm=10000 nm (in the infrared) But everything around you is about the same temperature, so the NET energy loss (or gain) is much smaller The Sun The surface temperature of the Sun is T=5800 K, so the flux emerging from the surface of the Sun is F= T 4 = Watts/m = 64 megawatts/m So a patch 4m 4m puts out the power of a 1 gigawatt electrical plant! The total solar luminosity is the flux times the surface area of the sun L =4 R F= Watts The peak wavelength is max =( /T)nm=500 nm (it peaks in the green) The Earth The mean surface temperature of the Earth is T=93 K (0 o C). Assuming it is a black body, the emitted flux is F= T 4 =40 Watts/m The total luminosity is the flux times the surface area of the Earth L=4 R F= Watts max =( /T)nm=10000 nm (the infrared) Your eyes work where the Sun puts out most of its light. It doesn t look green because of the details of how the eye determines color and the shape of the black body spectrum we see black bodies in the sequence red, yellow, blue-white, white as they get hotter (just like the light bulb experiment). The Earth This emitted luminosity is almost exactly equal to the amount coming from the Sun this is not a coincidence Heating by the Sun has to be exactly balanced by cooling if the Earth s temperature is to be (roughly) constant The Expected Temperature of a Planet For a planet of radius R at distance D, the solar heating rate (Watts) is L=(solar flux at the planet)(projected area of planet) =(L /4 D )( R )=L R /4D The only way the Earth can cool is to emit radiation into space. Thus, the average temperature of a planet is a competition between the radiation coming from the Sun and the radiation emitted by the planet with the temperature of the planet is determined by where the two balance. For a planet of radius R and temperature T, the cooling rate (Watts) is L=(black body flux)(surface area of planet) =( T 4 )(4 R )= 4 R T 4 4
5 The Expected Temperature of a Planet In equilibrium, these two must be equal (or the temperature would either rise or fall to compensate) T 4 = L /16 D Note that the radius of the planet cancels and the equilibrium temperature depends only on the luminosity of the sun and the distance of the planet from the sun T=79(AU/D) 1/ K For the Earth, this is close to freezing (73K) and colder than the actual mean temperature (93K) Does T=79(AU/D) 1/ K work? Planet D/AU actual T predicted T Mercury K 450K rotation~orbit Venus K 38K huge green house effect! Earth K 79K Mars 1.5 0K 6K Jupiter K 1K radiates more than it gets Saturn K 90K Uranus K 64K Neptune K 51K Pluto K 44K About right, but we missed some details planets reflect some sunlight, and clouds modify things (greenhouse effect). 5
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