Thermodynamics of Objects in Space Inverse Square Law If an object emits a total power (energy per unit time) L (L for luminosity), this radiation will spread outward into space, becoming more spread-out and diffuse as it goes. If it spreads out evenly in all directions, then by the time is it a distance r from the source, the energy per unit area per second (the Flux F) is given by the inverse square law:
F=
L 4πr 2
Not all this energy may be absorbed by an object. For typical planets, about 80% of the radiation that falls on them is absorbed, and the other 20% is reflected back out into space. The fraction that is reflected depends, of course, on the nature of the planet’s surface. This fraction is called the “Albedo”.
Heat Capacity If an object receives more heat than it dissipates, then its temperature will rise. By how much? This depends on the mass of the object M, and on its specific heat capacity c. The specific heat capacity is a measure of how much energy is required to heat 1 kg of some substance by one degree K. If the thermal energy of some system increases by a net amount ΔH, then its temperature will rise by
ΔT =
ΔH Mc
The product of M and c is also known as the “Thermal Mass”. An object with a high thermal mass is hard to heat up, but once it’s hot it stays hot for a long time! Architects try to design houses with large thermal masses, so that they store daytime heat and release it at night. Astronomers try and design telescopes with low thermal masses, so they rapidly cool down to the temperature of the surrounding air at night. This prevents thermals from forming on the mirror, and blurring the images.
Radiation Any object at a temperature above absolute zero will radiate energy. The hotter it is, the more it radiates. If this object is perfectly black, we call it a “black body” (imaginative, huh?). In this case, there is a simple equation for how much power this object radiates.
L = AσT 4
This is called the Stefan-Boltzmann Equation. A is the surface area of the black body, T is its temperature, and σ = 5.67×10-8 Wm-2K-4 (the Stefan-Boltzmann constant). No real object is ever a true Black Body, however. How much a real object falls short of being a true black body is a function of wavelength, and is normally written as ξ. The “efficiency factor” is between 0 and 1, and tells you what fraction of the true black body
radiation a given object actually emits. For a polished silver ball, ξ might be 0.01, while for a matt black box, it might be as high as 0.999. Helpfully, ξ not only tells you how well an object emits radiation, but also how well it absorbs it. What is the spectrum of the “Black Body Radiation”? This is given by the famous Planck Equation:
2πhc 2
L = hc λ λ5 (e λkT − 1) where Lλ is the energy emitted per unit area, per unit wavelength λ. h is Planck’s constant, and is equal to 6.626×10-34 Js. c is the speed of light, and is equal to 3.0×108 m s-1. k is Boltzmann’s constant, and is equal to 1.38×10-23J K-1. For an object with the temperature of the Sun’s surface (6000K), this spectrum peaks at optical wavelengths. Cooler objects emit most of their radiation in the red or IR, while hotter objects emit in the UV or even in the X-rays.
