  We present the results of 1.3 mm observations of the Crab Nebula, performed
with the MPIfR bolometer arrays at the IRAM 30-m telescope. The maps obtained,
of unprecedented quality at these wavelengths, allow a direct comparison with
high-resolution radio maps. Although the spatial structure of the Crab Nebula
does not change much from radio to millimetre wavelengths, we have detected
significant spatial variations of the spectral index between 20 cm and 1.3 mm.
The main effect is a spectral flattening in the inner region, which can be
hardly explained just in terms of the evolution of a single population of
synchrotron emitting electrons. We propose instead that this is the result of
the emergence of a second synchrotron component, that we have tried to extract
from the data. Shape and size of this component resemble those of the Crab
Nebula in X rays. However, while the more compact structure of the Crab Nebula
in X rays is commonly regarded as an effect of synchrotron downgrading, it
cannot be explained why a similar structure is present also at mm wavelengths,
where the electron lifetimes far exceed the nebular age. Our data, combined
with published upper limits on spatial variations of the radio spectral index,
also imply a low-energy cutoff for the distribution of electrons responsible
for this additional synchrotron component. Although no model has been developed
so far to explain the details of this component, one may verify that the total
number of the electrons responsible for it is in agreement with what predicted
by the classical pulsar-wind models, which otherwise are known to fail in
accounting for the number of radio emitting electrons. We have also detected a
spectral steepening at mm wavelengths in some elongated regions, whose
positions match those of radio synchrotron filaments.
