as, for example, in the transition from an optically thick to an optically thin regime in radiative-transfer calculations. Our approach is also an approximation and is similar to the one in Refs. [@Dionysopoulou:2012pp; @Palenzuela2013] in the sense that the matching from the stellar interior to the stellar exterior is achieved through a carefully chosen conductivity profile. More specifically, the conductivity profile adopted is directly related to the conserved rest-mass density (and hence to the rest-mass density) and given by $$\begin{aligned}
\sigma:=\sigma_0\max\left[1-\frac{2}{1+\exp\left[2 D_{\rm
        tol}{(D-D_{\rm rel})}/{D_{\rm atm}}\right]},\,0\right]\,,
\label{eq:bns:conductivityprofile}\end{aligned}$$ where $\sigma_0 \gg 1$ corresponds to a uniform scalar conductivity. The parameters $D_{\rm tol}$, and $D_{\rm rel}$ determine how sharp the transition to the exterior is. For the simulations reported here, we have chosen[^3], $\sigma_0 = 10^6 =
2.0\times10^{11}\,\sec^{-1}$, $D_{\rm tol}=0.01$ and $D_{\rm rel}=
100\,D_{\rm atm} = 100\,\rho_{\rm atm}$, where $D_{\rm atm}$ and $\rho_{\rm atm}$ are the values of the conserved and primitive rest-mass density in the “atmosphere.”

We recall that, as in other Eulerian hydrodynamics and MHD codes, also our `WhiskyRMHD` code makes use of a very low rest-mass density fluid to handle the evolution of the MHD equations in regions which are associated to the exterior of the stars. In such a 