  The Crab nebula originated from a core-collapse supernova (SN) explosion
observed in 1054 A.D. When viewed as a supernova remnant (SNR), it has an
anomalously low observed ejecta mass and kinetic energy for an Fe-core collapse
SN. Intensive searches were made for a massive shell that solves this
discrepancy, but none has been detected. An alternative idea is that the SN1054
is an electron-capture (EC) explosion with a lower explosion energy by an order
of magnitude than Fe-core collapse SNe. In the X-rays, imaging searches were
performed for the plasma emission from the shell in the Crab outskirts to set a
stringent upper limit to the X-ray emitting mass. However, the extreme
brightness of the source hampers access to its vicinity. We thus employed
spectroscopic technique using the X-ray micro-calorimeter onboard the Hitomi
satellite. By exploiting its superb energy resolution, we set an upper limit
for emission or absorption features from yet undetected thermal plasma in the
2-12 keV range. We also re-evaluated the existing Chandra and XMM-Newton data.
By assembling these results, a new upper limit was obtained for the X-ray
plasma mass of <~ 1Mo for a wide range of assumed shell radius, size, and
plasma temperature both in and out of the collisional equilibrium. To compare
with the observation, we further performed hydrodynamic simulations of the Crab
SNR for two SN models (Fe-core versus EC) under two SN environments (uniform
ISM versus progenitor wind). We found that the observed mass limit can be
compatible with both SN models if the SN environment has a low density of <~
0.03 cm-3 (Fe core) or <~ 0.1 cm-3 (EC) for the uniform density, or a
progenitor wind density somewhat less than that provided by a mass loss rate of
10-5 Mo yr-1 at 20 km s-1 for the wind environment.
