Changes between Version 153 and Version 154 of FluxLimitedDiffusion
 Timestamp:
 04/01/13 10:29:38 (12 years ago)
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FluxLimitedDiffusion
v153 v154 1 1 [[PageOutline()]] 2 2 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 3 = Physics of Radiation Transfer = 4 5 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 6 * light always travels at 'c', so the velocity dependence is just a direction dependence. 7 * Furthermore, photons can have different frequencies, so there is an extra dimension to the phase space. 8 * Instead of storing the phase space density of photons, the spectral intensity is the phase space density of energy flux... 9 10 Going between photon number and energy just involves a factor of [[latex(h \nu)]] and going from energy density to energy flux density just involves a factor of [[latex(c)]] so we have 6 11 7 12 [[latex(I \left ( \nu, \mathbf{x}, \Omega, \right ) = h \nu c f \left ( \nu, \mathbf{x}, \Omega, \right ) )]] 8 13 9 and 14 This can also be seen by considering the differential energy 10 15 11 16 [[latex(dE = I \left ( \nu, \mathbf{x}, \Omega, \right ) d\nu d\Omega dA dt = h \nu f \left ( \nu, \mathbf{x}, \Omega, \right ) d\nu d\Omega dV)]] … … 13 18 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 19 15 so we have16 [[latex(dE = I \left ( \nu, \mathbf{x}, \Omega, \right ) d\nu d\Omega dA dt =h \nu f \left ( \nu, \mathbf{x}, \Omega, \right ) d\nu d\Omega dA c dt)]]17 18 and we can identify20 so we also have 21 [[latex(dE = h \nu f \left ( \nu, \mathbf{x}, \Omega, \right ) d\nu d\Omega dA c dt)]] 22 23 which gives 19 24 20 25 [[latex(I \left ( \nu, \mathbf{x}, \Omega, \right ) = h \nu c f \left ( \nu, \mathbf{x}, \Omega, \right ) )]] 21 26 22 27 23 There are also parallels between the radiation transport equation and the Boltzmann equation 24 28 There are also parallels between the radiation transport equation and the Boltzmann transport equation except that photons don't experience body forces, and photon's don't collide with each other, but scatter off of particles. 25 29  [[latex(\frac{\partial}{\partial t} I_\nu + c \mathbf{n} \cdot \nabla I_\nu = c \eta_\nu  c \xi_\nu I_\nu)]]  26 30  [[latex(\frac{\partial}{\partial t} f + \mathbf{v} \cdot \nabla f + \mathbf{a} \cdot \nabla_v f = \left ( \frac{\partial f}{\partial t} \right )_{coll} )]]  27 31 28 29 30 Some of what follows is taken from [http://adsabs.harvard.edu/abs/2007ApJ...667..626K Krumholz et al. 2007] 31 32 = Physics of Radiation Transfer = 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. 33 38 34 39  [[latex(\tau = l \kappa=\frac{l}{\lambda_p})]]  [[latex(\beta = \frac{u}{c})]]  … … 37 42  [[latex(\tau >> 1 \mbox{, } \beta \tau >> 1)]]  dynamic diffusion limit  38 43 44 39 45 [[CollapsibleStart(Equations of Radiation Hydrodynamics)]] 46 40 47 == Equations of Radiation Hydrodynamics == 48 Some of what follows is taken from [http://adsabs.harvard.edu/abs/2007ApJ...667..626K Krumholz et al. 2007] 41 49 42 50  [[latex(\frac{\partial \rho}{\partial t} + \nabla \cdot \left ( \rho \mathbf{v} \right ) = 0)]] 