Changes between Version 3 and Version 4 of u/johannjc/scratchpad22

Timestamp:

01/12/17 13:39:26 (8 years ago)

Author:

Jonathan

Comment:

—

u/johannjc/scratchpad22

@@ -1 @@
-== Scaling ==
+= Scaling =
 
-The equations in physical units look like
+== Cross section ==
+The equations for calculating the optical depth in physical units look like
 
 $d\tau = n \sigma dl$
-
-$\frac{dn}{dt} = n \sigma F$
 
 And in computational units
@@ -11 +10 @@
 $d\tau = n_c \sigma_c dl_c$
 
-$\frac{dn_c}{dt_c} = n_c \sigma_c F_c$
+Dividing the first equation by the second gives 
 
-Dividing the first equation gives $n_c \sigma_c dl_c = n \sigma dl = n_c N_s \sigma dl_c L_s$
+$n_c \sigma_c dl_c = n \sigma dl = n_c N_s \sigma dl_c L_s$
 
 where $N_s$ and $L_s$ are the number density and length scales in the code.
@@ -21 +20 @@
 $\sigma_c = \sigma N_s L_s$
 
-And dividing the second equation by the first, gives
+== Ionizing Flux ==
+The equation for calculating the ionization rate in physical units is
+
+$\frac{dn}{dt} = n \sigma F$
+
+and in computational units
+
+$\frac{dn_c}{dt_c} = n_c \sigma_c F_c$
+
+
+Dividing the first equation by the second, gives
 
 $N_s/T_s = N_s \frac{1}{N_s L_s} \frac{F}{F_c}$
@@ -28 +37 @@
 
 $F_c=F \frac{T_s}{L_s N_s}$
+
+== Recombination Rate ==
+
+The equation for recombination in physical units is
+
+$\frac{d n}{dt} = \alpha n_{HII} n_e$
+
+and in computational units
+
+$\frac{d n_c}{dt} = \alpha_c n_{c,HII} n_{c,e}$
+
+Dividing the two and solving for $\alpha_c$ gives
+
+$\alpha_c = \alpha  \frac{T_s}{N_s}$
+
+== Ionization Heating ==
+The equation in physical units is 
+
+$\frac{dE}{dt}=e_\Gamma \sigma n F$
+
+and in computational units
+
+$\frac{dE_c}{dt_c}=e_{c,\Gamma} \sigma_c n_c F_c$
+
+Dividing the two and solving for $e_{c,\Gamma}$ gives
+
+$e_{c,\Gamma} = \frac{\sigma}{\sigma_c} N_s \frac{F}{F_c}\frac{T_s}{E_s} = \frac{N_s}{E_s}$
+