Changes between Version 154 and Version 155 of FluxLimitedDiffusion

Timestamp:

04/01/13 19:43:09 (12 years ago)

Author:

Jonathan

Comment:

—

FluxLimitedDiffusion

@@ -27 +27 @@
 
 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.
-||  [[latex(\frac{\partial}{\partial t} I_\nu + c \mathbf{n} \cdot \nabla I_\nu = c \eta_\nu - c \xi_\nu I_\nu)]]  ||
+||  [[latex(\frac{\partial}{\partial t} I_\nu + c \mathbf{n} \cdot \nabla I_\nu = c \eta_\nu - c \chi_\nu I_\nu)]]  ||
 ||  [[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} )]]  ||
 
-and often the radiation transport equation is written 
-  [[latex(\frac{\partial I_\nu}{\partial \tau} = S_\nu - I \nu )]]
-
-where we have projected the transport equation along a charateristic.
-
-And there are various limits that are important to consider.
+If we solve the transport equation along a characteristic [[latex(\left \[ \mathbf{x}\left ( s \right ), t(s) \right \] = \left \[ \mathbf{x0} + \mathbf{n} s, \frac{s}{c
+} \right \] )]]
+
+we have
+
+[[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 )]]
+
+and then we can divide through by [[latex(\chi)]] we get
+
+[[latex(\frac{dI_\nu}{d\tau_\nu} = \frac{\eta_\nu}{\chi_\nu} - I_\nu = S_\nu - I_\nu)]]
+
+where [[latex(\tau_\nu = \int \kappa_\nu ds )]] is the optical depth
+
+
+There are various limits that are important to consider.
 
 || [[latex(\tau = l \kappa=\frac{l}{\lambda_p})]] || [[latex(\beta = \frac{u}{c})]] ||