Space environment and magnetospheric Poynting fluxes of the exoplanet $\tau$ Bo\"otis b

Context: The first tentative detection of a magnetic field on the Hot Jupiter type exoplanet $\tau$ Bo\"otis b was recently reported by Turner et al. (2021). The magnetic field was inferred from observations of circularly-polarized radio emission obtained with the LOFAR telescopes. The radio emission is possibly a consequence of the interaction of the surrounding stellar wind with the planet's magnetic field. Methods: We perform magnetohydrodynamic simulations of the space environment around $\tau$ Bo\"otis b and its interaction with the stellar wind using the PLUTO code. We study the magnetospheric energy fluxes and effects of different magnetic field orientations in order to understand the physical processes which cause energy fluxes leading to the observed radio emission given the proposed magnetic field strength in Turner et al. (2021). Furthermore we study the effect of stellar wind density and pressure on magnetospheric energy fluxes given the uncertainty of extrasolar stellar wind predictions. Results: We find in our simulations that the interaction is most likely super--Alfv\'enic and energy fluxes generated by the stellar wind--planet interaction are consistent with the observed radio powers. Magnetospheric Poynting fluxes are of the order of 1-8 $\times10^{18}$ W for open, semi-open and closed magnetospheres. The Poynting fluxes are energetically consistent with the radio powers in Turner et al. (2021) for a magnetospheric Poynting flux-to-radio efficiency $> 10^{-3}$ when the magnetic fields of the planet and star are aligned. In case of lower efficiency factors the magnetospheric emission scenario is according to the parameter space modeled in this study not powerful enough. In case of a magnetic polarity reversal of the host star towards an anti-aligned field configuration, expected radio powers in the magnetospheric emission scenario fall below the observable threshold.