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, E * = energy per unit time emitted from a unit surface area, per wavelength band centered on wavelength. MONOCHROMATIC EMITTANCE k = Boltzman’s constant. T = Kelvin temperature.

 If a maximum of 2% error is acceptable, then the approximation exp(hc/ kT)>>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, and let we get:  Solving for x gives, and, Wien’s law  1  m = 1 x m

 Both Raleigh-Jeans and Wien’s equation can be derived from Planck’s.

 The total energy, in Watts/m 2, the area under the line, is given by: where,  SB = Stephan-Boltzmann constant = 5.67 x W/m 2 o K 4

 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/m 2  It is not constant. l The Earth orbit radius changes. l The Solar output varies. l Dust particles between Earth and Sun reduces amount received.

 When a temperature of 5780 o K is used the total energy per m 2 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, l some is reflected back to space, l some scattered, l some absorbed by atmosphere, and some is l 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 F solar is the flux (energy per unit area per second, J/m 2 s) passing through area AA and F surface 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, S o = 1368W/m 2, = Gm, R = actual Sun-Earth distance in Gm, h o = hour angle in radians. H o is given by:  = latitude,  s = declination angle,

Absorption, Reflection Transmission  Kirchoff’s law: Absorptivity and emissivity are equal at each wavelength. Emissivity - fraction of blackbody radiation actually emitted, e. Absorptivity - fraction of radiation striking surface that is actually absorbed, a. Reflectivity - fraction of incident radiation which is reflected, r. Includes scattering. Transmissivity - fraction of incident radiation that is transmitted through a substance, t.

 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,  E incedent = 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  s to represent the distance traveled through the material.  This can also be written as: where, k = fraction of total material doing the absorbing,   = density of material. So, k  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.  = elevation angle T r = transmissivity (fraction of solar irradiance which gets transmitted. l Varies with absorbing particles, gases, path length.

 An empirical formula for transmissivity is: where:  H = fraction of high clouds (0-1),  M = fraction of middle clouds (0-1),  L = fraction of low clouds (0-1).  = solar radiation reflected from surface upward.

Longwave (IR)  Any electromagnetic radiation of about 0.8  m (some use 0.73  m) up to about 100  m.  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,e IR = emissivity of the substance on the Earth’s surface. (Varies with substance)  SB = 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:  H,  M,  L 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: l Daytime: Nighttime: where, A = Albedo of Earth surface material, S = Solar Constant 1368 W/m2 T r = Net Sky Transmissivity  = Elevation angle of Sun

Problems  N1, N2, N5, N6, N16, N17, N19, N20

 Earth-Moon by Voyager I