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. , p1. 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 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.