\[ F \equiv \int_\text{beam} S(\vec{n}) P(\vec{n}) d\Omega. \]

Let \[ P(\vec{n}) \propto \exp\left[ -\frac{1}{2}\left( \frac{r}{\sigma} \right)^2 \right], \] and \[ S(\vec{n}) \propto r^\alpha, \] we have \[ F \propto \int r^\alpha \exp\left[ -\frac{1}{2}\left( \frac{r}{\sigma} \right)^2 \right] r dr \propto \Gamma\left(\frac{\alpha}{2}+1\right) \sigma^{\alpha+2}. \]

So we can scale the flux with \(\sigma^{\alpha+2}\).