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

Yutaka Fujita, Rohta Takahashi, Norita Kawanaka

TL;DR

This work analyzes how PeV CRs from microquasars propagate through the surrounding medium, emphasizing the transition from ballistic to diffusive transport and its spectral signatures. It adopts a point-source injection with $Q(E_p) \propto E_p^{-2}$ and a diffusion coefficient $D(E_p)$ that may be suppressed near the source, $D(E_p) = 10^{28}\chi (E_p/10\mathrm{GeV})^{1/2}$. The key prediction is a spectral break around $E_p \sim 10$–$100$ TeV for $\chi \sim 1$, which would translate to $E_\gamma \sim 1$–$10$ TeV in gamma rays, though current VHE data show smooth spectra, implying $\chi \lesssim 0.01$–$0.1$ near microquasars. The findings constrain CR transport in the immediate environs of microquasars and guide future observations (e.g., with the CTA) to test diffusion suppression and ballistic-diffusive transition effects.

Abstract

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.

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

TL;DR

This work analyzes how PeV CRs from microquasars propagate through the surrounding medium, emphasizing the transition from ballistic to diffusive transport and its spectral signatures. It adopts a point-source injection with and a diffusion coefficient that may be suppressed near the source, . The key prediction is a spectral break around – TeV for , which would translate to – TeV in gamma rays, though current VHE data show smooth spectra, implying – near microquasars. The findings constrain CR transport in the immediate environs of microquasars and guide future observations (e.g., with the CTA) to test diffusion suppression and ballistic-diffusive transition effects.

Abstract

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.

Paper Structure

This paper contains 14 sections, 5 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Models
  3. Cosmic-Ray Injection
  4. Diffusion Coefficient
  5. Analytical Framework
  6. Results
  7. Almost Steady State ($t = 10^{10}$ yr)
  8. Early Phase ($t = 10^3$ yr)
  9. Typical Microquasar Age ($t = 10^6$ yr)
  10. Discussion
  11. Comparison with Observations
  12. Physical Mechanisms for Diffusion Coefficient Reduction
  13. Confinement of Cosmic Rays
  14. Conclusions

Figures (3)

  • Figure 1: The mean free path $\ell(E_p)$ of CRs. The lines represent $\chi=1$ (black solid), 0.1 (red dotted), 0.01 (blue dashed), and 0.001 (green dash-dotted).
  • Figure 2: The spatial distribution of ultrarelativistic particles emitted from a particle source at $r_*=0$ with continuous injection starting at $t_* = 0$. The number of particles emitted per unit time is one.
  • Figure 3: The spectra of CRs within $r_{\rm obs}$. The lines represent $\chi=1$ (black solid), 0.1 (red dotted), 0.01 (blue dashed), and 0.001 (green dash-dotted). (a) $r_{\rm obs}=10$ pc, $t=10^{10}$ yr, (b) $r_{\rm obs}=10$ pc, $t=10^{3}$ yr, (c) $r_{\rm obs}=10$ pc, $t=10^{6}$ yr, (d) $r_{\rm obs}=30$ pc, $t=10^{10}$ yr, (e) $r_{\rm obs}=30$ pc, $t=10^{3}$ yr, (f) $r_{\rm obs}=30$ pc, $t=10^{6}$ yr.