Mirage Sources and Large TeV Halo-Pulsar Offsets: Exploring the Parameter Space

Technical Summary: Mirage Sources and Large TeV Halo-Pulsar Offsets

Problem Statement
Recent observations by the Large High Altitude Air Shower Observatory (LHAASO) and HAWC have identified extended TeV gamma-ray emissions ("pulsar halos") around middle-aged pulsars. A significant challenge in interpreting these sources is the frequent observation of large spatial offsets between the pulsar position and the centroid of the TeV emission. Traditional models assuming isotropic diffusion of electron-positron ($e^\pm$) pairs in the interstellar medium (ISM) cannot naturally explain these offsets, as isotropic diffusion is only a valid approximation when the physical scale exceeds multiple magnetic coherence lengths. However, for ultra-high-energy electrons ($>100$ TeV) with rapid cooling rates, the propagation scale may be smaller than the coherence length of the turbulent ISM magnetic field. Consequently, particle transport may be highly asymmetric, following local magnetic field lines rather than diffusing uniformly. This paper investigates whether such asymmetric propagation can generate "mirage" sources—apparent emission peaks displaced from the accelerator—and significant spatial offsets.

Methodology
The authors employ a first-principles, GPU-accelerated test-particle simulation to track the trajectories of 100 TeV electrons injected into a turbulent magnetic field.

• Magnetic Field Generation: The turbulent magnetic field is modeled as a superposition of four independent Kolmogorov-spectrum ($n_{in} = -11/3$) fields generated via a Fourier-space method. This multi-scale approach (using grid spacings from $4 \times 10^{-9}$ to $5 \times 10^{-4}$ kpc) ensures a realistic inertial range and divergence-free fields ($\nabla \cdot B = 0$). The total field includes both regular ($B_r$) and turbulent ($B_t$) components.
• Particle Propagation: Electrons are propagated using a Boris pusher algorithm to solve the relativistic equation of motion under the Lorentz force. The simulation accounts for radiative energy losses via synchrotron radiation and inverse Compton (IC) scattering off the Cosmic Microwave Background (CMB) and the Interstellar Radiation Field (ISRF).
• Statistical Sampling: To efficiently sample the initial energy spectrum ($dN/dE \propto E^{-\alpha} \exp(-E/E_{cut})$), the authors use a weighted macro-particle approach. 8,192 macro-particles are injected, each representing a logarithmic energy bin with a weight proportional to the physical electron count in that bin.
• Observational Synthesis: The simulated electron distributions are converted into synthetic gamma-ray maps (25–40 TeV band) via an IC scattering matrix. These maps are convolved with a Gaussian Point Spread Function (PSF) of $\sigma_{PSF} = 0.43^\circ$ (mimicking LHAASO-KM2A) and added to a Poissonian background noise model.
• Source Identification: A Bayesian model comparison framework using Markov Chain Monte Carlo (MCMC) is applied. Models with increasing numbers of Gaussian sources are fitted to the synthetic data. A source is considered "identified" if the Test Statistic (TS) increases by more than 25 ($\Delta \log L > 12.5$) upon adding a component.

Key Contributions and Results
The study systematically explores the parameter space, varying the pulsar distance ($d$), electron injection spectral index ($\alpha$), total magnetic field strength ($B$), turbulence coherence length ($L_c$), and the ratio of regular to turbulent field strength ($B_r/B_t$).

1. Mirage Source Formation: The simulations confirm that asymmetric propagation can create multiple distinct gamma-ray sources ("mirage sources") from a single accelerator. This occurs when electrons propagate along magnetic field lines that, due to projection effects, appear to originate from a location distant from the injection point.
2. Source Multiplicity ($N$):
   • The number of identified sources is most strongly influenced by the magnetic field strength ($B$). Higher $B$ leads to faster cooling, confining electrons to smaller scales and reducing the likelihood of forming multiple distinct sources (lower $N$).
   • Source number is negatively correlated with distance ($d$) and coherence length ($L_c$), and positively correlated with $B^2$.
3. Source Separation and Flux Ratio:
   • Separation: The maximum separation between identified sources is positively correlated with $L_c$ and negatively correlated with $B^2$.
   • Flux Ratio ($\chi$): The ratio of the main source flux to the mirage source flux decreases with increasing $B^2$. In strong magnetic fields, the mirage source can sometimes be brighter than the main source.
   • Regular vs. Turbulent Fields: The separation distance exhibits a non-monotonic dependence on $B_r/B_t$. Maximum separation occurs at $B_r/B_t \approx 1$, where the regular field provides directional guidance while turbulence allows for sufficient scattering to create offsets. Pure turbulence or dominant regular fields result in smaller separations.
4. Spatial Offsets:
   • Offsets between the pulsar and the closest identified source are maximized when $B_r/B_t \approx 1$ and $L_c$ is large.
   • In regions with large coherence lengths (e.g., inter-arm regions or the Galactic halo), offsets can reach up to 50 pc.

Significance and Claims
The paper claims to provide a quantitative explanation for puzzling LHAASO sources where TeV halos appear spatially disconnected from their associated pulsars. Specifically, the authors suggest that sources such as 1LHAASO J0216+4237u, J0212+4254u, and J0206+4302u could be mirage halos powered by a single millisecond pulsar (PSR J0218+4232), rather than distinct accelerators.

The study emphasizes that the isotropic diffusion approximation is insufficient for modeling particle transport near accelerators in the ISM, particularly for high-energy leptons where cooling times are shorter than the magnetic coherence time. By demonstrating that projection effects in turbulent magnetic fields can naturally produce large offsets and multiple source morphologies, the paper offers a mechanism to reconcile the observed TeV sky with standard pulsar wind nebula models. The authors note that while their synthetic turbulence assumes isotropy (unlike the anisotropic Alfvénic turbulence in MHD), it qualitatively captures the large-scale filamentary nature of ISM turbulence responsible for these asymmetric emissions.