Constraining the slow-diffusion zone size and electron injection spectral index for the Geminga pulsar halo

Measuring the electron diffusion coefficient is the most straightforward task in the study of gamma-ray pulsar halos. The updated measurements of the spatial morphology and spectrum of the Geminga halo by the High-Altitude Water Cherenkov (HAWC) experiment enable us to constrain parameters beyond the diffusion coefficient, including the size of the slow-diffusion zone and the electron injection spectrum from the pulsar wind nebulae (PWNe). Based on the two-zone diffusion model, we find that the slow-diffusion zone size ($r_*$) around Geminga is within the range of $30-70$~pc. The lower boundary of this range is determined by the goodness of fit of the model to the one-dimensional morphology of the Geminga halo. The upper limit is derived from fitting the gamma-ray spectrum of the Geminga halo, along with the expectations for the power-law index of the injection spectrum based on simulations and PWNe observations, i.e., $p\gtrsim1$. With $r_*$ set at its lower limit of $30$~pc, we obtain the maximum $p$ permitted by the HAWC spectrum measurement, with an upper limit of $2.17$ at a $3\sigma$ significance. Moreover, we find that when $r_*=30$~pc and $p=2.17$, the predicted positron spectrum generated by Geminga at Earth coincides with the AMS-02 measurement in the $50-500$~GeV range.