Changes between Version 25 and Version 26 of u/johannjc/scratchpad
- Timestamp:
- 04/22/14 12:46:13 (11 years ago)
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u/johannjc/scratchpad
v25 v26 74 74 [[latex($\Delta t < \xi \frac{T_0}{4 \Gamma \kappa_{0P} \left | 4 \pi B_0 - c\left(E_0+\partial_tE\Delta t\right ) \right | }$)]] which gives us a quadratic for [[latex($\Delta t$)]] 75 75 76 [[latex($4 \Gamma \kappa_{0P} c \left | \partial_tE \right | \Delta t^2 + \left(4 \Gamma \kappa_{0P} \left | 4 \pi B_0 - cE_0 \right | \right ) \Delta t < \xi T_0 $)]] 76 77 77 [[latex($-4 \Gamma \kappa_{0P} c \partial_tE \Delta t^2 + \left(4 \Gamma \kappa_{0P} \left ( 4 \pi B_0 - cE_0 \right ) \right ) \Delta t - \xi T_0 < 0$)]] 78 where we have conservatively assumed that the diffusion always causes the gas to be more out of equilibrium. 78 79 79 [[latex($4 \Gamma \kappa_{0P} c \partial_tE \Delta t^2 - \left(4 \Gamma \kappa_{0P} \left ( 4 \pi B_0 - cE_0 \right ) \right ) \Delta t - \xi T_0 < 0$)]] 80 Now if we recognize that there are three time scales at play here: 80 81 81 yucky... 82 || Diffusion Time || [[latex($\tau_D=\frac{T}{4 \Gamma \left | \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E \right | }$)]] || 83 || Coupling Time || [[latex($\tau_C=\frac{T}{ 4 \Gamma \kappa_{0P} \left |4 \pi B_0-cE \right |}$)]] || 84 || Absorption Time || [[latex($\tau_A=\frac{1}{c \kappa_{0P}}$)]] || 82 85 83 How about we assume that the gas is close to equilibrium? Then we get 86 then this quadratic simplifies to 84 87 85 [[latex($\ Delta t < \xi \frac{T_0}{4 \Gamma \kappa_{0P} \left | \partial_tE\Delta t \right | }$)]]88 [[latex($\frac{\Delta t^2}{\tau_A\tau_D} + \frac{\Delta t}{\tau_C} < \xi$)]] 86 89 87 [[latex($\Delta t < \sqrt{\xi \frac{T_0}{4 \Gamma \kappa_{0P} c \left | \nabla \cdot \frac{c\lambda}{\kappa_{0R}} \nabla E \right |} }$)]] which is just the geometric average of the coupling time and the diffusion time 90 [[latex($\Delta t = \sqrt{\left ( \frac{\tau_A \tau_D}{2\tau_C}\right)^2+ \xi{\tau_D\tau_A}}-\frac{\tau_A \tau_D}{2\tau_C} $)]] 91 92 [[latex($\Delta t = \frac{\tau_A \tau_D}{2\tau_C} \left ( \sqrt{1+ \frac{4\xi \tau_C^2}{\tau_D\tau_A}} - 1 \right) $)]] 93 94 95 So we can choose 96 * the diffusion time if we assume the gas and radiation are strongly coupled - optically thick, 97 * the coupling time if we assume that the gas is optically thin (which should imply the radiation is fairly diffused) 98 * Or we can solve the quadratic
