Simple Earth Climate Model - Additional Concept Explanations

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Black Body Radiation: Stefan-Boltzmann Law.

A black body is an idealized object that is a perfect absorber as well as a perfect emitter of electromagnetic (EM) radiation. Although idealized, many objects come close to this ideal state and so it is a useful concept. Planck's radiation law describes the spectral energy density of such an object in detail but here we are only interested in the total radiated energy, which is described by Stefan-Boltzmann's Law (that follows from Planck's law):

$ P = \sigma A \epsilon T^4 $

Where:

P is the power radiated isotropically, or the amount of energy per second (units: Watts, W)

σ is the Stefan-Boltzmann constant, equal to 5.6696x10-8 W/(m2 ∙ K4)

A is the area of emission (units: square metres, m2)

ε is the emissivity of the object, or the fraction of EM radiation a surface absorbs (0 < ε < 1)

T is the temperature of the object (units: Kelvins, K)


Solar Constant Calculation

Using the Stefan-Boltzmann Law, and assuming the Sun is a blackbody (ε = 1), it can be found that

<br />
\begin{eqnarray}<br />
   P_s & =&  4 \pi r s^2 \sigma T_s^4 \nonumber \\<br />
   & \approx &  3.9 \times 10^{26} \textnormal{ W} \nonumber<br />
   \end{eqnarray}<br />

The incident solar radiation, S, on the surface of Earth's atmosphere that the sunlight shines on is

<br />
\begin{eqnarray}<br />
   S & =& \dfrac{P_s}{4 \pi r_{se}^2} \nonumber \\<br />
   & \approx &  1367 \textnormal{ W/m}^2 \nonumber<br />
   \end{eqnarray}<br />


Effective Area

Figure 1. A representation of the sunlight emitted from the Sun hitting the Earth. Photo taken by Claire Wheeler.

Sunlight, or solar radiation, includes the total spectrum of electromagnetic radiation given off by the Sun. This solar radiation is emitted isotropically in all directions, and a tiny fraction will hit the Earth. It can be noted that essentially zero solar power is absorbed by interplanetary space. As the relative size of Earth is incredibly tiny in relation to its orbital radius around the Sun, the ratio of its projected 2D area (the disk in Fig. 3) on the 3D surface area (a sphere, whose cross section is represented by the dotted line in Fig. 2) of the solar radiation distribution is equal to the fraction, ƒ, of the solar power incident on the Earth.

$ f = \dfrac{ \pi r_e^2}{4 \pi r_{se}^2} $ 

Figure 2. The solar radiation, emitted isotropically by the Sun, coming into contact with Earth. Image not to scale. Values were found here1.

 

 

Figure 3. The projected area of Earth on the spherically distributed solar radiation emitted by the Sun. Above, Earth is a disk with a radius, re, of 6.37x106 m. The dashed lines indicate the slice of incident solar radiation on Earth. Image not to scale.  Values were found here1.

In simpler terms, if a blackbody light bulb was placed in a closed spherical container in the same conditions as above, the blackbody container would receive 100% of the power emitted by the bulb; in a open hemispherical container with the bulb in the centre of the open face, the container would receive 50% of the power emitted by the bulb. In the case of the example with the Sun and the Earth, the solar radiation can be thought of as a balloon, while the projected area of the Earth can be thought of as a dime pressed onto the side of the balloon.


Albedo

The Earth is a bright object; it is visible when viewed from space. Therefore a fraction of solar radiation is reflected straight back into space without ever warming the Earth. This reflective property is called the albedo, A. For Earth, A ~0.3, and is mainly due to clouds, haze and ice. In fact, 0.2 reflects off the clouds, 0.06 off the air, and 0.04 off Earth's surface2


The Greenhouse Effect

The simple model used in this article assumes that Earth lacks an atmosphere. Our planet, however, has an atmosphere, or else it could not support life and water. Earth's atmosphere is mostly transparent to solar radiation (37% visible, 51% near infrared (IR), 12% ultraviolet (UV), see Plancks_Law_Spreadsheet.xls). Therefore, most of Earth's incident solar radiation is not reflected out to space, but rather gets through the atmosphere and warms us. On the other hand, Earth's atmosphere absorbs much of its own radiation (longer wavelength IR). The atmosphere acts like one way glass, allowing solar radiation to enter, but preventing the Earth's radiation from exiting. This is called the Greenhouse Effect because glass behaves in a similar fashion, and is why glass is used in greenhouses.

  • 1. a. b. Knight, Randall D. Astronomical Data. In: Physics for Scientists and Engineers, A Strategic Approach (2), edited by Martha Steele. San Francisco: Pearson Education, Inc., 2008, inside back cover.
  • 2. Aubrecht GJ. Solar Energy: Wind, Photovoltaics, and Large-Scale Installatons. In: Energy - Physical, Environmental, and Social Impact (3), edited by Erik Fahlgren. Upper Saddle River, NJ: Pearson Education Inc., 2006, chapt. 16, pp. 334,336.

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