  Recent far-infrared (IR) observations of supernova remnants (SNRs) have
revealed significantly large amounts of newly-condensed dust in their ejecta,
comparable to the total mass of available refractory elements. The dust masses
derived from these observations assume that all the grains of a given species
radiate at the same temperature, regardless of the dust heating mechanism or
grain radius. In this paper, we derive the dust mass in the ejecta of the Crab
Nebula, using a physical model for the heating and radiation from the dust. We
adopt a power-law distribution of grain sizes and two different dust
compositions (silicates and amorphous carbon), and calculate the heating rate
of each dust grain by the radiation from the pulsar wind nebula (PWN). We find
that the grains attain a continuous range of temperatures, depending on their
size and composition. The total mass derived from the best-fit models to the
observed IR spectrum is 0.019-0.13 solar masses, depending on the assumed grain
composition. We find that the power-law size distribution of dust grains is
characterized by a power-law index of 3.5-4.0 and a maximum grain size larger
than 0.1 microns. The grain sizes and composition are consistent with what is
expected for dust grains formed in a Type IIP SN. Our derived dust mass is at
least a factor of two less than the mass reported in previous studies of the
Crab Nebula that assumed more simplified two-temperature models. The results of
this study show that a physical model resulting in a realistic distribution of
dust temperatures can constrain the dust properties and affect the derived dust
masses. Our study may also have important implications for deriving grain
properties and mass estimates in other SNRs and for the ultimate question of
whether SNe are major sources of dust in the Galactic interstellar medium (ISM)
and in external galaxies.
