We however do not employ this approximation because Eqn. (\[eq:centrallimit\]) is only valid for large $N_x$ and $N_x$ always becomes small near the target mesh. We thus only use the mean density in our supermesh approximation which is described by Eqn. (\[eq:mean\]). We will investigate the accuracy of this approximation in Section \[sec:tests\].

Now we have to determine on which supermeshes we perform the RT calculation. We chose to use the lowest level supermeshes whose angular size, $\theta$, is equal to or smaller than the angular size of a group of the sources, $\theta_{\rm s}$, since we assume plane-parallel radiation to construct the approximation. We define the luminosity-weighted rms projected radius as the effective projected size of the group of the sources[^2], i.e. if the target mesh is located along the $z$-direction from the centre of the luminosity, the projected size of the group is defined as $$r_{\rm rms}^2 = \frac{\sum_m L_m \left\{(x_m - \bar{x})^2 + (y_m - \bar{y})^2\right\}}{\sum_m L_m}, 
  \label{eq:rms}$$ where $\bar{x}$ and $\bar{y}$ are, respectively, the $x$ and $y$ components of the position vector of the luminosity centre and the subscript $m$ runs over all sources in the tree node in question. Practically, we calculate the following tensor 