Impact of cosmic-ray acceleration on the early evolution of bow shocks around massive runaway stars

Bow shocks generated from the interaction of winds from massive runaway stars with the interstellar medium have been shown to be prominent particle accelerators through recent γ-ray and radio synchrotron observations. They are ideal candidates for exploring the space- and time-dependence of particle acceleration due to their axisymmetric morphology and short dynamical timescales (∼ 100 kyr). Current studies on the particle acceleration from bow shocks investigate the cosmic-ray (CR) transport only in the test-particle limit. We aim to bridge this gap by conducting ideal cosmic-ray magnetohydrodynamic simulations in the advection-diffusion limit that accelerate CRs at spatially resolved shocks at each timestep. We qualitatively compared with current observations through the expected γ-ray and synchrotron emission. We performed 3D ideal cosmic-ray magnetohydrodynamic simulations with the Eulerian grid-based code FLASH, where stellar winds are injected through tabulated wind velocities and mass-loss rates. We implemented a gradient-based shock-detection algorithm to resolve the shocked regions where the CRs are injected dynamically. Simulations were performed for different values of the CR diffusion coefficient and star velocities within an ISM-like environment up to 500 kyr to show the impact of dynamical CR injection on the early evolution of the wind-driven bow shock. With a simplified spectral model in post-processing, we calculated the expected upper limits of γ-ray and synchrotron emission and compared the results with those from current observations. Variations of CR diffusion rates can strongly dictate the morphology of the bow shock and the overall γ-ray and radio synchrotron luminosity due to the balance between the CR injection efficiency and diffusion. Our results are underestimated compared to those from current observations, even with an overestimated particle acceleration, primarily due to the weaker wind kinematic luminosities we used and the lack of a stellar magnetic field in our model. The morphology of wind-driven bow shocks and their non-thermal emission can be substantially affected by CR acceleration and diffusion, if there is efficient particle acceleration at shocks.