Download Presentation
## Planck’s law

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**Planck’s law**• Very early in the twentieth century, Max Karl Ernest Ludwig Planck put forth the idea of the quantum theory of radiation. • It basically says: It is not possible to increase the amount of radiation given off by a body by infinitely small increments. Instead, there are discrete stepwise increments involved in the emission of radiation.**There is a minimum-size parcel of energy that can be**radiated called a quantum. • The amount of energy contained in a quantum is equal to the product of Planck’s constant (h) and the frequency of emission (v) • Since: then,**Utilizing this general idea, he developed the law of**radiation which describes radiation as a function of wavelength and temperature. Where, El* = energy per unit time emitted from a unit surface area, per wavelength band centered on wavelength l. MONOCHROMATIC EMITTANCE k = Boltzman’s constant. T = Kelvin temperature.**If a maximum of 2% error is acceptable, then the**approximation exp(hc/lkT)>>1 for any value of the fraction greater than 4, then Planck’s equation can be written as:**If we take the derivative with respect to l, and let**we get: • Solving for x gives, and, Wien’s law • 1 mm = 1 x 10-6 m**Both Raleigh-Jeans and Wien’s equation can be derived from**Planck’s.**The total energy, in Watts/m2, the area under the line, is**given by: where, sSB = Stephan-Boltzmann constant = 5.67 x 10-8 W/m2 oK4**The total energy emitted by the sun is:**This energy passes through a sphere at Earth orbit radius. Total amount passing through sphere of Earth orbit radius is: So amount received at earth orbit per square meter is:**Or, we can say that the energy received at some distance**from a spherical source of energy is:**The quantity of energy per square meter passing through a**sphere of Earth orbit radius is the Solar Constant = 1368±7 W/m2 • It is not constant. • The Earth orbit radius changes. • The Solar output varies. • Dust particles between Earth and Sun reduces amount received.**When a temperature of 5780oK is used the total energy per m2**closely approximates that measured at the top of the atmosphere by satellites.**The Solar Constant energy is measured by satellites through**an area perpendicular to the solar radiation. • When this energy passes through the Earth’s atmosphere, • some is reflected back to space, • some scattered, • some absorbed by atmosphere, and some is • absorbed on the curved Earth’s surface.**Consider the figure to the right.**• The energy from the sun is passing through the square(AA) at the top of the atmosphere and striking the surface along the curved path B on an area AB (assuming no loss by east-west curvature).**All the energy that passes through AA per second will fall**on area AB, assuming no loss by the atmosphere. If Fsolar is the flux (energy per unit area per second, J/m2 s) passing through area AA and Fsurface is the flux falling on area AB, then the only loss is due to spreading across a curved surface and the ratio of the fluxes equals the ratio of the areas. Notice, the larger value is in the denominator.**But, A can cancel leaving: and A/B is just the sine of**the elevation angle. So, and, E = irradiance, (Solar flux at a particular time), or in kinematic form.**Since the distance from the Sun varies and a particular**place is not receiving radiation for 24 hours each day, the average daily insolation at any location is given by: where, So = 1368W/m2, = 149.6 Gm, R = actual Sun-Earth distance in Gm, ho = hour angle in radians. Ho is given by: f = latitude, ds = declination angle,**Absorption, Reflection Transmission**• Kirchoff’s law: Absorptivity and emissivity are equal at each wavelength. • Emissivity - fraction of blackbody radiation actually emitted, el. • Absorptivity - fraction of radiation striking surface that is actually absorbed, al. • Reflectivity - fraction of incident radiation which is reflected, rl. Includes scattering. • Transmissivity - fraction of incident radiation that is transmitted through a substance, tl.**Incoming solar radiation is either absorbed, reflected**(scattered) or transmitted. Albedo: ratio of reflected energy to total incoming energy. If there is no energy transmitted, then:**The atmosphere is a selective absorber, allowing some**wavelengths to be transmitted through, but absorbing and reflecting others.**Beer’s Law**• Shows the relationship between the amount of energy that will be transmitted across a layer of a substance to that incident on the surface of the layer. Where, dE = incremental energy change, • Eincedent = amount incident (not reflected) • n= concentration of absorbing particles in material. • b = cross section of an absorbing particle • ds = thickness of material. The taller the glass, the darker the brew, The less the amount of light that comes through.**Note: Stull is using Ds to represent the distance traveled**through the material. • This can also be written as: where, k = fraction of total material doing the absorbing, • r = density of material. So, kr is a measure of how much of the total is absorbing.**Surface Radiation budget**• To understand whether the earth and atmosphere system has a net gain or loss of energy over a period of time, both the incoming and outgoing fluxes of energy must be measured. • The earth receives most of its energy in the shortwave portion of the spectrum. • It radiates most of its energy in the longwave portion of the spectrum.**If F* is the net radiative flux, (positive upward and**perpendicular to the Earth’s surface), then: where, = downward solar radiation, = solar radiation reflected upward, = downward longwave radiation, = upward longwave emitted radiation**Downward solar radiation perpendicular to the Earth’s**surface which arrives at the Earth’s surface is given by: where, S = solar irradiance (Solar constant). The amount of solar energy at top of atmosphere. Y = elevation angle Tr = transmissivity (fraction of solar irradiance which gets transmitted. • Varies with absorbing particles, gases, path length.**An empirical formula for transmissivity is:**where: sH = fraction of high clouds (0-1), sM = fraction of middle clouds (0-1), sL = fraction of low clouds (0-1). • = solar radiation reflected from surface upward.**Longwave (IR)**• Any electromagnetic radiation of about 0.8 mm (some use 0.73 mm) up to about 100 mm. • Usually in meteorology it is considered to be (1) that energy emitted from the Earth’s surface upward. (2) The emitted radiation reflected back to the Earth. Thus, by the Stephan-Boltzmann equation: where, eIR = emissivity of the substance on the Earth’s surface. (Varies with substance) sSB = Stephan-Boltzmann constant.**IR radiation moving towards the Earth’s surface includes**Earth’s reflected radiation plus longwave radiation emitted by Sun. Total can be measured, but is difficult to calculate or separate. • Usually the net longwave radiation (flux) is determined, which can also be measured. Remember, radiation upward is positive. • Can be approximated by an empirical equation: sH, sM, sL are cloud cover fraction between 0 - 1**The net radiative flux (perpendicular to the Earth’s**surface) gained or lost by Earth’s surface is then: • Daytime: • Nighttime: where, A = Albedo of Earth surface material, S = Solar Constant 1368 W/m2 Tr = Net Sky Transmissivity Y = Elevation angle of Sun**Problems**• N1, N2, N5, N6, N16, N17, N19, N20