Diffusive or Ballistic? Distributions and Spectra of PeV Cosmic Rays around Microquasars

In the standard Galactic cosmic-ray (CR) paradigm, protons are accelerated up to ~1 PeV by Galactic sources. While supernova remnants (SNRs) have been traditionally considered as the primary accelerators, recent observations by LHAASO and HAWC have detected very-high-energy (VHE) gamma rays exceeding 100 TeV from several microquasars, suggesting that these X-ray binaries can accelerate CRs beyond 1 PeV. We investigate the escape process of CRs from microquasars, focusing on the energy-dependent transport mechanisms. High-energy CRs are likely to have long mean free paths and move ballistically on scales smaller than their mean free path, while lower-energy CRs undergo diffusive propagation. This transition results in a spectral break in the CR distribution around the microquasar. We calculate CR energy spectra within a 10-30 pc radius for various diffusion coefficients and timescales. Our model predicts a spectral break and hardening at E_p ~10-100 TeV when the standard diffusion coefficient for the interstellar space is assumed. However, current VHE gamma-ray observations do not show clear spectral breaks, suggesting that the diffusion coefficient may be significantly reduced near microquasars, possibly due to magnetic field amplification by CR-driven turbulence.