4. Blackbody radiation in a nutshell cont.
The Planck law can be expressed as the amount of photons per square meter and wavelength interval and solid angle. This is useful for photochemical applications:
By taking the derivative of B and setting to zero the maximum of the radiation can be determined. This leads to the Wien law:
CAVE!! When expressed in frequency, the maximum is at a different location. Remember that the Planck function is a density function
Teachers:Prof. Dr. K. Altwegg-von Burg, Prof. Dr. P. Wurz and Prof. Dr. Niklaus Kämpfer
University of Bern
Physics Institute
Climate and Environmental Physics
The total flux emitted by a black body over all wavelengths and in the half space is:
This is the Stefan-Boltzmann law, with

5. Taylor, Elem climate physics

6. Kirchhoff’s law
Planck’s function describes emission by a black body. This corresponds to the maximum possible emission from the object.
Real surfaces deviate from the ideal of a blackbody.
The ratio of what is emitted to what actually would be emitted is called emissivity.
There are two cases of interest: emissivity at a single wavelength and over a broad range of wavelengths
Monochromatic emissivity:
Broadband emissivity:
Climate and Environmental Physics
University of Bern
Emissivity related to Stefan Boltzmann or graybody emissivity:
Kirchhoff’s law relates absorptivity and emissivity:

7. Typical infrared emissivities in % of various surfaces
Water
Fresh dry snow
Ice 96
Sand, dry
Soil, moist
Soil, dry plowed 90
Desert
Forest and shrubs 90
Skin, human 95
Concrete
Aluminium, polished
Climate and Environmental Physics
University of Bern

8. Atmospheric absorption as measured horizontally at sea level.

9. Scattering of radiation
In addition to absorption light may also be scattered by air molecules, cloud dropletsand aerosols.
Scattering is a redistribution of radiation in different directions. Radiation in the original direction is diminished and shows up in other directions. This redistribution is characterized by the so called phase function.
In analogy to absorption a scattering cross section is introduced.
The combined effect of absorption plus scattering is called extinction.
In analogy to geometrical optics one would call the scattering cross section a kind of shadow. However this “shadow” can be much bigger than the actual geometrical cross section.
The ratio of the scattering cross section to the geometrical area A is called scattering efficiency:
Climate and Environmental Physics
University of Bern
In analogy an extinction efficiency is defined:

10. Why scattering is important
Particles of diameters less than approx. 1m are highly effective at scattering incoming solar radiation.
These particles reduce the amount of incoming solar energy as compared with that in their absence and consequently cool the Earth.
Mineral dust particles can scatter and absorb both incoming and outgoing radiation. In the visible part, light scattering dominates and they mainly cool. In the infrared region, mineral dust acts like an absorber and acts like a greenhouse gas, thus warms.
Sulfate aerosols and smoke of biomass burning are currently estimated to exert a global average cooling effect.
Aerosol concentrations are highly variable in space and time.
Greenhouse gas forcing operates day and night. Whereas aerosol forcing due to scattering operates only during daytime.
Aerosol radiative effects depend in a complicated way on the solar angle, relative humidity, particle size and composition and the albedo of the underlying surface.
For the interaction of solar radiation with atmospheric aerosols, elastic light scattering is the process of interest.
The absorption and elastic scattering of light by a spherical particle is a classical problem in physics, the mathematical formalism of which is called Mie theory.
Aerosols influence climate directly by scattering and absorption of solar radiation and indirectly through their role as cloud condensation nuclei.
Climate and Environmental Physics
University of Bern

11. Key parameters used in describing scattering
Key parameters are:
the wavelength
the particle size in relation to the wavelength
the complex index of refraction
The refractive index is normalized to the one of air N0= i :
The distribution of the scattered radiation as a function of the scattering angle is given by the phase function
Climate and Environmental Physics
University of Bern
The determination of the phase function and the scattering efficiency is mathematically difficult. Closed theories are only available for the most simple cases.

12. Scattering regimes depending on particle size and wavelength
Climate and Environmental Physics
University of Bern

14. Mie scattering calculator
Mie scattering calculator:
Mie scattering calculator

20. O2 and O3 absorb almost all wavelengths shorter than 300 nanometers
O2 and O3 absorb almost all wavelengths shorter than 300 nanometers. Water (H2O) absorbs many wavelengths above 700 nm, but this depends on the amount of water vapor in the atmosphere.

22. Taylor, Elementary climate physics

23. Daily average solar flux at the top of the atmosphere
Climate and Environmental Physics
University of Bern

24. The Clouds and the Earth's Radiant Energy System (CERES) - NASA
Summer solstice

25. The Clouds and the Earth's Radiant Energy System (CERES) - NASA
Winter solstice

26. The CERES data shown in this image are 14-day running average values of sunlight reflected back to space.
The lowest amount of sunlight reflected back to space, shown in blue, occurs over clear ocean areas. Green colors show gradually increasing amounts of reflected sunlight. The areas of greatest reflected solar energy, shown in white, occur both from the tops of thick clouds and from ice-covered regions on the Earth's surface during summer.The amount of incoming solar energy the Earth receives on June 21, the first day of summer, is 30 percent higher at the North Pole than at the equator. Just 6 months later in winter, the entire polar cap receives no energy since Earth's movement along its orbit has pointed the North Pole away from the Sun. This swing of illumination and reflection is shown dramatically in the CERES animation. Critical to understanding future climate are the subtle changes in reflected solar energy, such as changes in the surface area of the arctic ice cap or in cloud thickness. Ever-changing cloud cover or the seasonal retreat and advance of sea ice cause motion in this image.

27. For scientists to understand climate, they must also determine what drives the changes within the Earth's radiation balance. From March 2000 to May 2001, CERES measured some of these changes and produced new images that dynamically show heat (or thermal radiation) emitted to space from Earth's surface and atmosphere (left sphere) and sunlight reflected back to space by the ocean, land, aerosols, and clouds (right sphere).

31. NASA Ceres brochure

32. Intergovernmental Panel on Climate Change, Physical Sciences Group Report Ch. 1

33. Wells The Atmosphere and Ocean

34. Wells The Atmosphere and Ocean

35. NASA Ceres