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θ
                                                                  φ










                             Figure 22: Thermal radiation from a surface of a black body.

                         Therefore, the total power F of the blackbody radiation from a spherical
                      star with radius R is equal to:
                                                                        4
                                                         2
                                                                    2
                                               F = 4πR S ? = 4πR σT .
                      The power flux density of the thermal radiation which we receive from a star
                      or a cloud of solid angle Ω, surface temperature T, and distance r, is from
                      equation (32):
                                                               ΩσT 4
                                                   S = Ω I =         ,
                                                                 π
                                                                                   2
                                                                               2
                      which, for the spherically symmetric case where Ω = πR /r , is reduced to
                                                  R 2        R 2       F
                                                        4
                                             S =     σT =       S ? =      ,
                                                   r 2       r 2     4πr 2
                      as expected.


                      9.7    Universality of the Relationship among Einstein’s
                             Coefficients

                      We derived equations (21) and (25), giving the following relationships among
                      Einstein’s coefficients:
                                                        n
                                                                   m
                                                    g m β m  = g n β ,
                                                                   n
                                                      α m  =   2hν 3  ,
                                                       n
                                                      β n m      c 3
                      or, equivalently,
                                                                   m
                                                   g m B n  = g n B ,
                                                                   n
                                                       m
                                                     A m       8πhν 3
                                                       n
                                                     B m   =     c 3  ,
                                                       n
                                                           25
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