Planck’s law

1. 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.

2. 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,

3. 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.

4. 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:

5. 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

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

7. 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

8. 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:

9. Or, we can say that the energy received at some distance from a spherical source of energy is:

10. 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.

11. When a temperature of 5780oK is used the total energy per m2 closely approximates that measured at the top of the atmosphere by satellites.

12. 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.

13. 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).

14. 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.

15. 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.

16. 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,

17. 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.

18. 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:

19. The atmosphere is a selective absorber, allowing some wavelengths to be transmitted through, but absorbing and reflecting others.

20. 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.

21. 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.

22. 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.

23. 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

24. 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.

25. 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.

26. 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.

27. 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

28. 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

29. Problems • N1, N2, N5, N6, N16, N17, N19, N20

30. Earth-Moon by Voyager I