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

The total flux emitted by a black body over all wavelengths and in the half space is:

This is the Stefan-Boltzmann law, with

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:

Emissivity related to Stefan Boltzmann or graybody emissivity:

Kirchhoff’s law relates absorptivity and emissivity:

Typical infrared emissivities in % of various surfaces

Water 92 - 96

Fresh dry snow 82 - 99.5

Ice 96

Sand, dry 84 - 90

Soil, moist 95 - 98

Soil, dry plowed 90

Desert 90 - 91

Forest and shrubs 90

Skin, human 95

Concrete 71 - 88

Aluminium, polished 1 - 5

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:

In analogy an extinction efficiency is defined:

Why scattering is important

Particles of diameters less than approx. 1m 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.

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=1.00029+0i :

The distribution of the scattered radiation as a function of the scattering angle is given by the phase function

The determination of the phase function and the scattering efficiency is mathematically difficult. Closed theories are only available for the most simple cases.

Scattering regimes depending on particle size and wavelength

Mie scattering calculator

Daily average solar flux at the top of the atmosphere

The Clouds and the Earth's Radiant Energy System (CERES) - NASA

Summer solstice

The Clouds and the Earth's Radiant Energy System (CERES) - NASA

Winter solstice

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