Changes between Version 2 and Version 3 of u/erica/RadHydro
- Timestamp:
- 03/29/16 16:37:23 (9 years ago)
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u/erica/RadHydro
v2 v3 1 1 '''Radiative Hydrodynamics''' 2 2 3 Here is how the equation for internal energy is modifieddue to radiation:3 Here is how the internal energy in the grid can change due to radiation: 4 4 5 5 [[latex($\frac{\partial (\rho e)}{\partial t} = - \kappa_R \rho (4\pi B-cE)$)]] 6 6 7 where [[latex($B= \sigma T^4$)]] is the bolometric Planck function for a blackbody (BB), and [[latex($4 \pi B=a T^4$)]] is the energy output due to BB radiation. [[latex($cE$)]] is the radiative energy density in the grid. This equation shows that when [[latex($4 \pi B > cE$)]],7 where [[latex($B=B(T)=\sigma T^4$)]] is the bolometric Planck function for a blackbody (BB), and [[latex($4 \pi B=a T^4$)]] is the energy output due to BB radiation. [[latex($cE$)]] is the radiative energy density in the grid. This equation shows that when [[latex($4 \pi B > cE$)]], 8 8 9 9 [[latex($\frac{\partial \rho e}{\partial t}<0$)]] 10 10 11 which is interpreted as the matter losing energy via BB radiation. That is, the internal energy of that zone will decrease, thereby producing a concomitant increase in the radiation field's energy, or E, as we will see next. This equation also tells us that when [[latex($4 \pi B<cE$)]], there is more energy in the radiation field than in the BB, and so it gets absorbed by the matter. This causes the internal energy to increase,11 which is interpreted as the matter losing energy via BB radiation. That is, the internal energy of that zone will decrease, having been transferred into radiative energy. This equation also tells us that when [[latex($4 \pi B<cE$)]], there is more energy in the radiation field than in the BB, and so it gets absorbed by the matter. This causes the internal energy to increase, 12 12 13 13 [[latex($\frac{\partial \rho e}{\partial t}>0$)]] 14 14 15 Since the matter and the radiation are coupled in this way, the equation that governs the total radiative energy in the grid is the inverse of the internal energy, but with an added term for diffusion. Thus, E, can change in the grid either by acquiring energy from the blackbody radiation, and/or, by diffusion.15 Changes to the internal energy lead to changes in temperature, and thus the next time step the amount of radiative energy from the BB will change (recall B=B(T)). 16 16 17 17 Now, since the matter and the radiation are coupled in this way, the equation that governs the radiative energy in the grid is the inverse of the internal energy. Additionally, the radiative energy can diffuse through the grid, so there is an extra term for diffusion: 18 18 19 19 [[latex($\frac{\partial E}{\partial t}= \nabla \cdot (\frac{c \lambda}{\kappa_R \rho}) \nabla E + \kappa_p \rho(4\pi B-cE)$)]] 20 20 21 21 for the internal energy, and: