(8.125) |

or

(8.126) |

where . The previous integral can easily be looked up in standard mathematical tables. In fact,

(8.127) |

(See Exercise 2.) Thus, the total power radiated per unit area by a black-body is

(8.128) |

This dependence of the radiated power is called the

(8.129) |

is called the

We can use the Stefan-Boltzmann law to estimate the temperature of the Earth from first principles. The Sun is a ball of glowing gas of radius km and surface temperature K. Its luminosity is

(8.130) |

according to the Stefan-Boltzmann law. The Earth is a globe of radius km located an average distance km from the Sun. The Earth intercepts an amount of energy

(8.131) |

per second from the Sun's radiative output: that is, the power output of the Sun reduced by the ratio of the solid angle subtended by the Earth at the Sun to the total solid angle . The Earth absorbs this energy, and then re-radiates it at longer wavelengths. The luminosity of the Earth is

(8.132) |

according to the Stefan-Boltzmann law, where is the average temperature of the Earth's surface. Here, we are ignoring any surface temperature variations between polar and equatorial regions, or between day and night. In steady-state, the luminosity of the Earth must balance the radiative power input from the Sun, so equating and we arrive at

(8.133) |

Remarkably, the ratio of the Earth's surface temperature to that of the Sun depends only on the Earth-Sun distance and the solar radius. The previous expression yields K or C (or F). This is slightly on the cold side, by a few degrees, because of the greenhouse action of the Earth's atmosphere, which was neglected in our calculation. Nevertheless, it is quite encouraging that such a crude calculation comes so close to the correct answer.