We present spectral analysis of the Crab Nebula obtained with the Chandra X-ray observatory .
The X-ray spectrum is characterized by a power-law whose index varies across the nebula .
The variation can be discussed in terms of the particle injection from the pulsar in two different directions : the equatorial plane containing the torus and the symmetry axis along the jet .
In the equatorial plane , spectra within the torus are the hardest , with photon index \alpha \approx 1.9 , and are almost independent of the surface brightness .
At the periphery of the torus , the spectrum gradually softens in the outer , lower surface brightness regions , up to \alpha \approx 3.0 .
This indicates that synchrotron losses become significant to X-ray emitting particles at the outer boundary of the torus .
We discuss the nature of the torus , incorporating information from observations at other wavelengths .
Spectral variations are also seen within the southern jet .
The core of the jet is the hardest with \alpha \approx 2.0 , and the outer sheath surrounding the core becomes softer with \alpha up to 2.5 at the outermost part .
Based on the similarity between the spectra of the jet core and the torus , we suggest that the electron spectra of the particles injected from the pulsar are also similar in these two different directions .
The brightness ratio between the near and far sides of the torus can be explained by Doppler boosting and relativistic aberration ; however , the observed ratio can not be derived from the standard weakly magnetized pulsar wind model .
We also found a site where an optical filament comprised of supernova ejecta is absorbing the soft X-ray emission ( < 2 keV ) .