Diffusion time estimate

Starting from:

\frac{dE}{dt} = \nabla\cdot\frac{c \lambda}{\kappa_R\rho}\nabla E

We get:

\boxed{t_{diff}\approx \frac{L^2}{c}\frac{\kappa_R \rho}{\lambda}}

L is the size of our system, c is speed of light, kappa_R is the rosseland specific mean opacity, rho is density of the system, and lamba is a dimensionless parameter that seems to do with the length scale of gradients in radiative energy.

Offner et al '09 gives,

\kappa_R=.23 \frac{cm^2}{g}

for T=10 K gas. Assuming our system is a protostellar core, we have L~.1 pc, rho~1 solar mass/L³. In cgs these parameters work out to be:

c	3e+10
\rho	6.5e-20
L	½*3.08e+17
\kappa_R	.23

(For L, the radiation is leaving from the center of the volume, so is going approximately 1 half the length). I am not completely sure on \lambda, but from Offner's paper,

\lambda=\frac{1}{R}

where

R=\frac{1}{L}\frac{E}{\kappa_R \rho E}

and so I gather an estimate for lambda might be:

\lambda = L \kappa_R \rho

which using our values gives:

\lambda=.002

Using all of these values in the formula above for the diffusion time gives,

\boxed{t_{diff}\approx 7.5e+6 ~s}

or ~ 86 days.

That is a pretty quick diffusion time, considering the 'free streaming limit' gives:

\boxed{t_{fs}=\frac{L}{c}\approx\frac{3e+17}{3e+10}=10^7 ~s}

or 57 days. I am guessing lambda should be something much smaller than 1. From offner's paper, it seems lambda may be interpreted as 1/R. Any estimates on lambda?