Journey to the center of the common envelope evolution. Inner dynamics of the post-dynamical inspiral

Three-dimensional hydrodynamical simulations of common envelope evolution are often terminated soon after the initial dynamical plunge of the companion transitions into a long-lasting post-dynamical inspiral with slowly varying semi-major axis, $a_\text{b}$. This premature termination is often due to insufficient numerical resolution and challenges associated with the softening of the gravitational potential of the two cores. In this work, we use statically-refined 3D hydrodynamical simulations to study binaries orbiting inside a common envelope, exploring the effects of varying numerical resolution, $\delta$, gravitational potential softening prescriptions, and the associated softening lengthscale, $\epsilon$. We find that quantities such as the binary inspiral timescale or the volume-averaged shearing rate typically converge to asymptotic values only for $\epsilon \le 0.1 a_\text{b}$ and $\delta \le 6 \times 10^{-3}a_\text{b}$ with smaller $\epsilon$ requiring correspondingly smaller $\delta$. After a few tens of binary orbits, the two cores become surrounded by a corotating, nearly hydrostatic gas structure, resembling the shared envelope of a contact binary. We propose that this structure is responsible for the slowing down of the dynamical inspiral, leading to an asymptotic inspiral timescale of approximately $10^5$ orbital periods for a binary mass ratio $q=1/3$, and approximately $10^6$ orbital periods for a binary mass ratio $q=1$. By investigating kinetic helicity, we argue that the magnetic field is unlikely to organize into large-scale structures via the usual $\alpha$--effect during the post-dynamical phase. Even in the absence of magnetic fields, we observe intermittent polar outflows collimated by partially centrifugally evacuated polar funnels. (abridged)