Planckís law
There is a minimum-size parcel of energy that can be radiated called a quantum.
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 = Boltzmanís constant. T = Kelvin temperature.
If a maximum of 2% error is acceptable, then the approximation exp(hc/lkT)>ɭ for any value of the fraction greater than 4, then Planckís equation can be written as:
If we take the derivative with respect to l, and let we get:
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
The Solar Constant energy is measured by satellites through an area perpendicular to the solar radiation.
Consider the figure to the right.
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
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:
Beerís Law
Note: Stull is using Ds to represent the distance traveled through the material.
Surface Radiation budget
If F* is the net radiative flux, (positive upward and perpendicular to the Earthís surface), then: where, = downward solar radiation, = solar radiation reflected upward, = downward longwave radiation, = upward longwave emitted radiation
Downward solar radiation perpendicular to the Earthís surface which arrives at the Earthís 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.
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).
Longwave (IR)
IR radiation moving towards the Earthís surface includes Earthís reflected radiation plus longwave radiation emitted by Sun. Total can be measured, but is difficult to calculate or separate.
The net radiative flux (perpendicular to the Earthís surface) gained or lost by Earthís surface is then:
Problems
Earth-Moon by Voyager I
PPT Slide
Email: alcorn@ariel.met.tamu.edu
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