Impact of Cosmic Ray Acceleration on the Early Evolution of Bow Shocks around Massive Runaway Stars

TL;DR

This work addresses how cosmic ray (CR) acceleration at bow shocks around massive runaway stars influences the early evolution of wind-driven bubbles. It introduces a 3D ideal CR-MHD framework in FLASH with advection-diffusion CR transport and an on-the-fly, shock-driven CR injection scheme based on diffusive shock acceleration (DSA), including an anisotropic diffusion tensor with $κ_{\parallel}$ and $κ_{\perp}$. The authors demonstrate robustness through standard shock tests and apply the model to multiple bow-shock setups, showing that CR diffusion rates strongly modulate bow-shock morphology and the associated non-thermal emission, yielding gamma-ray and radio synchrotron upper limits that are broadly consistent with current observations. While the simplified spectral treatment yields qualitative agreement with observations, the study highlights the need for spectrally resolved CR modelling and additional physics (primary electrons, CR streaming, radiative shocks) to refine predictions and interpretation of high-energy bow-shock signals.

Abstract

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. Here, we study particle acceleration from bow shocks by conducting 3D ideal cosmic ray magnetohydrodynamic simulations in the advection-diffusion limit. We use the Eulerian grid-based code FLASH, where stellar winds are injected through tabulated wind velocities and mass loss rates. We implement a gradient-based shock detection algorithm to resolve the shocked regions where the CRs are injected dynamically. Simulations are performed for different values of the CR diffusion coefficient and star velocities within an ISM-like environment up to 180 kyr to showcase 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 calculate the expected upper limits of $γ$-ray and synchrotron emission and compare with those from current observations. We observe that 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 yield qualitatively comparable results with current observations, primarily attributed to the high-energy protons and electrons contributing to non-thermal emission from efficient acceleration at the forward shock through the approximations and assumptions in the injection algorithm. We conclude that CR acceleration, with varying CR diffusion rates, may substantially affect the morphology of wind-driven bow shocks and their non-thermal emission, if there is efficient particle acceleration in the forward shock. [abridged]

Figures (26)

• Figure 1: Schematic of the implementation of the determination of the pre-shock (blue), post-shock (orange), and shock surface cells (green) for a shock candidate cell (black circle) within the shock zone.
• Figure 2: Schematic of the implementation of CR injection within our framework. The "injection zone" (in pink), the numerically broadened shock surface used for energy injection, starts from the shock surface cells (green) and extends beyond the post-shock cells (yellow) up to a user-defined width $N_\mathrm{inj, max} = 4$. The pre-shock cells (blue) and the shock zone as defined in \ref{['fig:prepostshock_schematic']} are also shown.
• Figure 3: The dependence of the CR acceleration efficiency $\eta(\mathcal{M}_1, \theta_B)$ on the pre-shock Mach number $\mathcal{M}_1$ and pre-shock magnetic obliquity $\theta_B$ at 15, 30, 45, and 60 degrees. Here, we use the prescription from Kang2007 and Pais2018 for the Mach number and magnetic obliquity dependence, respectively. The dashed and solid lines indicate the efficiency calculate with and without pre-existing CRs, respectively.
• Figure 4: Results from the 2D Sod shock tube test without (left) and with (right) CR injection at $t = 0.35$. We show the density, velocity, pressure, and the upstream Mach number averaged over the $y$-direction for both cases. We also plot the individual contributions of the pressure ($P_\mathrm{th}$ without CR injection, $P_\mathrm{th}$ and $P_\mathrm{CR}$ with CR injection) for the pressure distribution. The analytical solution obtained from Sod1978 and Pfrommer2017 for the case without and with CR acceleration (black dotted line) are also shown.
• Figure 5: Pressure distribution of the total pressure $P_\mathrm{tot}$ (blue), thermal pressure $P_\mathrm{th}$ (red), and CR pressure $P_\mathrm{CR}$ (green) from the 2-D Sod shock tube test with magnetic obliquities of $\theta_B = 30^\circ, \, 45^\circ, \,$ and $60^\circ$ at $t = 0.35$. The distribution is zoomed-in to the shocked region to better visualise the CR pressure at each obliquity. The dashed lines indicate the respective analytical solution for each pressure contribution as provided in Pfrommer2017. The magnetic pressure contribution is not shown as they are negligible compared to the other pressure terms.