Scaling

The equations in physical units look like

d\tau = n \sigma dl

\frac{dn}{dt} = n \sigma F

And in computational units

d\tau = n_c \sigma_c dl_c

\frac{dn_c}{dt_c} = n_c \sigma_c F_c

Dividing the first equation gives 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.

This gives \sigma_c = \sigma N_s L_s

And dividing the second equation by the first, gives

N_s/T_s = N_s \frac{1}{N_s L_s} \frac{F}{F_c}

which gives F_c=F \frac{T_s}{L_s N_s}

Test problem

2D version of Tripathi without atmosphere

\rho_c = 6\times 10^8 cm^{-3}

R_c=1.5\times 10^{10} cm

F=2\times 10^{14} cm^{-2} s^{-1}

T= 10^3 K

embedded in a lower density pressure balanced ambient with a density ratio

\chi=100

With a cross section for photo-ionization of 6.3e-18 and our clump's column density of 9e18, we would expect the clump to be optically thick

And with a flux of 2e14, we would expect photo-ionization of the column to take 4.5e4 seconds