Black Body - Radiative Cooling

Radiative Cooling

The integration of Planck's law over all frequencies provides the total energy per unit of time per unit of surface area radiated by a black body maintained at a temperature T, and is known as the Stefan–Boltzmann law:

where σ is the Stefan–Boltzmann constant, σ ≈ 5.67 × 10−8 W/(m2K4). To remain in thermal equilibrium at constant temperature T, the black body must absorb or internally generate this amount of power P over the given area A.

The cooling of a body due to thermal radiation is often approximated using the Stefan–Boltzmann law supplemented with a "gray body" emissivity ε ≤ 1 (P/A = εσT4). The rate of decrease of the temperature of the emitting body can be estimated from the power radiated and the body's heat capacity. This approach is a simplification that ignores details of the mechanisms behind heat redistribution (which may include changing composition, phase transitions or restructuring of the body) that occur within the body while it cools, and assumes that at each moment in time the body is characterized by a single temperature. It also ignores other possible complications, such as changes in the emissivity with temperature, and the role of other accompanying forms of energy emission, for example, emission of particles like neutrinos.

If a hot emitting body is assumed to follow the Stefan–Boltzmann law and its power emission P and temperature T is known, this law can be used to estimate the dimensions of the emitting object, because the total emitted power is proportional to the area of the emitting surface. In this way it was found that X-ray bursts observed by astronomers originated in neutron stars with a radius of about 10 km, rather than black holes as originally conjectured. It should be noted that an accurate estimate of size requires some knowledge of the emissivity, particularly its spectral and angular dependence.