Energy-dependent nebula extent and spatially resolved spectra of the pulsar wind nebula 3C 58

3C 58 is a pulsar wind nebula (PWN) that shows an interesting energy-dependent nebula extent and spatial variations of the photon index and surface brightness in the X-ray band. These observations provide useful information with which to study the spatially dependent radiative cooling of electrons and the energy-dependent transport mechanisms within the nebula. In this paper, the energy-dependent nebula extent and spatially resolved spectra of this PWN are investigated in the framework of a spatially dependent particle transport model. The observations of the nebula, including the photon spectral energy distribution, spatial variations of the X-ray spectrum, and measurements of the nebula extent, can be naturally explained in this model. Our results show that the energy-dependent nebula extent favours an advection-diffusion scenario with advection-dominated transport, and the variations of the nebula extent with energy in the X-ray band can be attributed to the cooling losses of high-energy electrons affected by synchrotron burn-off. Particle diffusion plays an important role in modifying the spatial variations of the photon index and surface brightness in the X-ray band. The radial extents of the nebula at radio, GeV and TeV energies are predicted by the model, indicating that the nebula extent of 3C 58 varies with energy in these bands. The analyses show that the dependence of the adiabatic cooling rate and synchrotron radiation on the spectral index of injected particles is important for changing the nebula extent at different energies.