109 | | For now we will assume that [[latex(\kappa_{0P})]] and [[latex(\kappa_{0R})]] are constant over the implicit update. In this case we can solve the radiation energy equation: |
110 | | |
111 | | || [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \mathbf{F} + \kappa_{0P} (4 \pi B-cE) = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E + \kappa_{0P} (4 \pi B-cE))]] || |
112 | | || [[latex(\frac{\partial e}{\partial t} = - \kappa_{0P} (4 \pi B-cE))]] || |
| 109 | For now we will assume that [[latex(\kappa_{0P})]] and [[latex(\kappa_{0R})]] are constant over the implicit update and we will treat the energy as the total internal energy ignoring kinetic and magnetic contributions. In this case we can solve the radiation energy equations: |
| 110 | |
| 111 | || [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E + \kappa_{0P} (4 \pi B(T)-cE) = \nabla \cdot \mathbf{F} + \kappa_{0P} (4 \pi B(T)-cE))]] || |
| 112 | || [[latex(\frac{\partial e}{\partial t} = - \kappa_{0P} (4 \pi B(T)-cE))]] || |
| 115 | |
| 116 | === Expanding about e,,0,, === |
| 117 | |
| 118 | Of course even if the opacity is independent of energy and radiation energy, the above combined system of equations is still non-linear due to the dependence on Temperature of the Planck Function [[latex(B(T))]]. |
| 119 | |
| 120 | If we ignore the changes in the Temperature due to heating during the implicit step which would feed back into the source function. We can improve this by writing |
| 121 | |
| 122 | [[latex(B(T) = B \left ( T_0+dT \right ) = B \left ( T_0 \right ) + \left . \frac{\partial B}{\partial T} \right | _{T_0} \frac{\partial T}{\partial e} de = \frac{c}{4 \pi} a_R \left ( T_0^4 + 4T_0^3\Gamma de \right ) = B_0 \left ( 1 + 4\Gamma \frac{e-e_0}{T_0} \right ) )]] |
| 123 | |
| 124 | where |
| 125 | |
| 126 | [[latex(\Gamma = \frac{dT}{dE} = \frac{(\gamma-1)}{n k_B})]] |
| 127 | |
| 128 | Then the system of equations becomes |
| 129 | |
| 130 | || [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E + \kappa_{0P} \left [4 \pi B_0 \left ( 1 + 4\Gamma \frac{e-e_0}{T_0} \right )-cE \right ] )]] || |
| 131 | || [[latex(\frac{\partial e}{\partial t} = - \kappa_{0P} \left [ 4 \pi B_0 \left ( 1 + 4\Gamma \frac{e-e_0}{T_0} \right )-cE \right ] )]] || |
153 | | |
154 | | === Expanding about e,,0,, === |
155 | | |
156 | | Of course the above system is only linear if we ignore the changes in the Temperature due to heating during the implicit step which would feed back into the source function. We can improve this by writing [[latex(B(T) = \frac{c}{4 \pi} a_RT^4 = \frac{c}{4 \pi} \left ( \frac{2}{n f k_B} \right ) ^4 a_R e^4 = \frac{c}{4 \pi} A_R e^4 = A_R e_0^4 \left ( 1 + 4 \delta + 6 \delta^2 + ... \right ))]] |
157 | | |
158 | | where [[latex(e = e_0 \left ( 1 + \delta \right ) )]] |
159 | | |
160 | | Then the system of equations becomes |
161 | | |
162 | | || [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E + c \kappa_{0P} \left [ A_R e_0^4 \left ( 1 + 4 \delta \right ) -E \right ] )]] || |
163 | | || [[latex(\frac{\partial \delta}{\partial t} = - \frac{c \kappa_{0P}}{e_0} \left [ \left (1+4 \delta \right )-E \right ] )]] || |
164 | | |
165 | | or equivalently |
166 | | |
167 | | || [[latex(\frac{\partial E}{\partial t} = \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E + \kappa_{0P} \left [ 4 \pi B \left ( T_0 \right ) \left ( 1 + 4 \delta \right ) -cE \right ] )]] || |
168 | | || [[latex(\frac{\partial \delta}{\partial t} = - \frac{ \kappa_{0P}}{e_0} \left [ 4 \pi B \left ( T_0 \right ) \left ( 1+4 \delta \right ) -cE \right ] )]] || |
169 | | |