3 | | Most of what follows is taken from [http://adsabs.harvard.edu/abs/2007ApJ...667..626K Krumholz et al. 2007] |

| 3 | Typically when we discuss the radiation field we use the spectral intensity [[latex(I \left ( \nu, \mathbf{x}, \Omega \right ) ]] which is a function of frequency, position, and direction. This is very similar to the phase space density used in deriving the fluid equations [[latex(f \left ( \mathbf{x}, \mathbf{v} \right ) )]] except that light always travels at 'c', so the velocity dependence is just a direction dependence. Furthermore, photons can have different frequencies, so there is an extra dimension to the phase space. And finally, instead of storing the phase space density of photons, the spectral intensity is the phase space density of energy flux... Going between photon number and energy flux just involves a factor of [[latex(\h \nu c)]] |

| 4 | |

| 5 | So we have |

| 6 | |

| 7 | [[latex(I \left ( \nu, \mathbf{x}, \Omega, \right ) = h \nu c n \left ( \nu, \mathbf{x}, \Omega, \right ) )]] |

| 8 | |

| 9 | and |

| 10 | |

| 11 | [[latex(dE = I \left ( \nu, \mathbf{x}, \Omega, \right ) d\nu d\Omega \dA \dt = h \nu n \left ( \nu, \mathbf{x}, \Omega, \right ) d\nu d\Omega dV)]] |

| 12 | |

| 13 | where the number of photons traveling normal to the surface dA that cross the surface dA in time dt is just the number of photons in the volume dV = dA c dt (assuming the photons are headed normal to dA)... |

| 14 | |

| 15 | so we have |

| 16 | [[latex(dE = I \left ( \nu, \mathbf{x}, \Omega, \right ) d\nu d\Omega \dA \dt = h \nu n \left ( \nu, \mathbf{x}, \Omega, \right ) d\nu d\Omega dA c dt)]] |

| 17 | |

| 18 | and we can identify |

| 19 | |

| 20 | [[latex(I \left ( \nu, \mathbf{x}, \Omega, \right ) = h \nu c n \left ( \nu, \mathbf{x}, \Omega, \right ) )]] |

| 21 | |

| 22 | Some of what follows is taken from [http://adsabs.harvard.edu/abs/2007ApJ...667..626K Krumholz et al. 2007] |