An analytical model for the magnetic field in the thick shell of Galactic bubbles with uniform initial conditions

Bubbles and super-bubbles are ubiquitous in the interstellar medium and influence their local magnetic field. Starting from the assumption that bubbles result from violent explosions that sweep matter away in a thick shell, we derive the analytical equations for the divergence-free, regular magnetic field in the shell. The explosion velocity field is assumed to be radial but not necessarily spherical, making it possible to model various-shaped bubbles. Assuming an explosion center, the magnetic field at the present time is fully determined by the initial uniform magnetic field, the present-time geometry of the bubble shell, and a radial vector field that encodes the explosion-induced displacement of matter, from its original location to its present-time location. We present the main characteristics of our magnetic-field model using a simple linear model for the radial displacements. Next, we use our analytical prescription, informed by a three-dimensional dust density map, to estimate the expected contribution of the shell of the Local Bubble, the super-bubbles in which the Sun resides, to the integrated Faraday rotation measures and synchrotron emission and compare these to full-sky observational data. We find that, while the contribution to the former is minimal, the contribution to the latter is very significant at Galactic latitudes $|b|>45^\circ$. Our results underline the need to take the Local Bubble into account in large-scale Galactic magnetic field studies.