The deepest Chandra X-ray study of the plerionic supernova remnant G21.5$-$0.9

G21.5-0.9 is a plerionic supernova remnant (SNR) used as a calibration target for the Chandra X-ray telescope. The first observations found an extended halo surrounding the bright central pulsar wind nebula (PWN). A 2005 study discovered that this halo is limb-brightened and suggested the halo to be the missing SNR shell. In 2010 the spectrum of the limb-brightened shell was found to be dominated by non-thermal X-rays. In this study, we combine 15 years of Chandra observations comprising over 1~Msec of exposure time (796.1~ks with the Advanced CCD Imaging Spectrometer (ACIS) and 306.1~ks with the High Resolution Camera (HRC)) to provide the deepest-to-date imaging and spectroscopic study. The emission from the limb is primarily non-thermal and is described by a power-law model with a photon index $\Gamma = 2.22 \, (2.04-2.34)$, plus a weak thermal component characterized by a temperature $kT = 0.37\, (0.20-0.64)$ keV and a low ionization timescale of $n_{e}t < 2.95 \times 10^{10}$ cm$^{-3}$s. The northern knot located in the halo is best fitted with a two-component power-law + non-equilibrium ionization thermal model characterized by a temperature of 0.14 keV and an enhanced abundance of silicon, confirming its nature as ejecta. We revisit the spatially resolved spectral study of the PWN and find that its radial spectral profile can be explained by diffusion models. The best fit diffusion coefficient is $D \sim 2.1\times 10^{27}\rm cm^2/s$ assuming a magnetic field $B =130 \mu G$, which is consistent with recent 3D MHD simulation results.