32 | | and often the radiation transport equation is written |

33 | | [[latex(\frac{\partial I_\nu}{\partial \tau} = S_\nu - I \nu )]] |

34 | | |

35 | | where we have projected the transport equation along a charateristic. |

36 | | |

37 | | And there are various limits that are important to consider. |

| 32 | If we solve the transport equation along a characteristic [[latex(\left \[ \mathbf{x}\left ( s \right ), t(s) \right \] = \left \[ \mathbf{x0} + \mathbf{n} s, \frac{s}{c |

| 33 | } \right \] )]] |

| 34 | |

| 35 | we have |

| 36 | |

| 37 | [[latex(\frac{d I_\nu}{d s} = \frac{\partial I_\nu}{\partial x^i} \frac{\partial{x^i}}{\partial s} + \frac{\partial I_\nu}{\partial t}\frac{\partial t}{\partial s} = \mathbf{n} \cdot \nabla I_\nu + frac{1}{c}\frac{\partial I_\nu}{\partial t} = \eta_\nu - \chi_\nu I_\nu )]] |

| 38 | |

| 39 | and then we can divide through by [[latex(\chi)]] we get |

| 40 | |

| 41 | [[latex(\frac{dI_\nu}{d\tau_\nu} = \frac{\eta_\nu}{\chi_\nu} - I_\nu = S_\nu - I_\nu)]] |

| 42 | |

| 43 | where [[latex(\tau_\nu = \int \kappa_\nu ds )]] is the optical depth |

| 44 | |

| 45 | |

| 46 | There are various limits that are important to consider. |