Page 27 - 83 basic knowledge of astronomy
P. 27

Let the intensity I ν change by dI ν due to absorption and emission as
                      the radiation passes through an infinitesimal distance dl (Figure 23). The


                                               s
                                                   I  (s, l)        I  (s, l + dl)
                                                    ν
                                                                     ν
                                                             dl


                        Figure 23: Radiative transfer through a cylinder of infinitesimal length.

                      variation can be described as a sum of two contributions: one due to the
                      absorption −κ ν I ν dl, and one due to the emission  ν dl, where κ ν and  ν are
                      the following frequency–dependent coefficients:
                                                               −1
                                              :  opacity     m ,
                                         κ ν
                                                                             −1
                                              :  emissivity  W m  −3  Hz −1  sr .
                                          ν
                      We then obtain the radiative transfer equation in the following form:
                                                   dI ν
                                                       = −κ ν I ν +  ν .                      (36)
                                                    dl
                      10.2     Derivation of the Radiative Transfer Equation from

                               Einstein’s Elementary Quantum Theory of Radi-
                               ation

                      Equation (36) can also be derived from equation (15) obtained in the discus-
                      sion of Einstein’s elementary quantum theory of radiation:
                                       dn p (s, ν)
                                                                 n
                                                        m
                                                                                m
                                                 = (n n β − n m β )u ν (s) + n n α ,
                                          dt            n        m              n
                      where n p is the number density of photons per unit solid angle and per unit
                      frequency bandwidth, u ν is the spectral energy density per unit solid angle,
                      n m and n n are the number densities of the particles in states Z m and Z n ,
                                        n
                            m
                      and α , β n m  and β are Einstein’s differential coefficients. In fact, using the
                                        m
                            n
                      relations:
                                                 u ν (s) = hνn p (s, ν),
                                                 I ν (s) = cu ν (s),
                                                     dl = cdt ,

                      we can transform equation (15) to:
                                                                 hν
                                        dI ν          n       m                m
                                            = −(n m β − n n β )     I ν + hνn n α .            (37)
                                        dl           m        n  c             n
                                                           27
   22   23   24   25   26   27   28   29   30   31   32