Page 24 - 83 basic knowledge of astronomy
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of ordinary stars are fairly high (several thousands to several tens of thousand
                      Kelvin), their solid angles, which are inversely proportional to their squared
                      distances, are very small. For example, if we were to put the Sun at the
                      distance of the nearest star, which is about 3 light years, or 1 parsec, the
                      Sun’s angular diameter would be only ' 0.01 arcsecond! As a result, the
                                         6
                      flux density 5 × 10 Jy of the quiet Sun at 10 GHz (see Figure 2) would be
                      reduced down to 125 µJy at the distance of the nearest star. On the contrary,
                      the interstellar gas clouds, as cold as tens to hundreds of Kelvin, still clearly
                      shine in the radio sky, because their angular diameters are as large as minutes
                      to degrees of arc. For example, Figure 2 shows that the flux density of the
                      Orion Nebula is about 500 Jy at around 500 MHz.


                      9.5    Stefan–Boltzmann Law

                      The total intensity I(T) of the thermal radiation from a black body of tem-
                      perature T, over the entire frequency range, is given by
                                              ∞         ∞
                                                          2hν      1         1
                                             Z         Z      3
                                                                                  4
                                     I(T) =     I ν dν =               dν =    σT ,            (32)
                                                           c 2  e kT − 1     π
                                                                 hν
                                             0          0
                      where σ is the Stefan–Boltzmann constant:
                                              5 4
                                            2π k
                                                                               −4
                                       σ =         = 5.6697 × 10 −8  W m   −2  K .
                                               2 3
                                           15c h
                      Equation (32) can be derived using the integration formula:
                                                  ∞
                                                  Z  2x 2n−1      B n
                                                            dx =       ,                       (33)
                                                    e 2πx  − 1    2n
                                                  0
                      where B n is the n-th Bernoulli number, and B 2 = 1/30.
                      9.6    Total Blackbody Radiation from a Star or a Gas

                             Cloud

                      The power flux density S ? , at a surface of a blackbody, which is the power
                      over the entire frequency range through a cross section of unit area of the
                      surface (Figure 22), is given by the equation:

                                                       π
                                         ∞          ∞  2  2π
                                        Z          Z Z Z
                                  S ? =   S ν dν =          I ν cos θ sin θ dφ dθ dν
                                        0          0  0  0
                                            π
                                              2π
                                            2 Z Z
                                                                                 4
                                  = I(T)         cos θ sin θ dφ dθ = π I(T) = σT .             (34)
                                           0  0
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