  Radio observations of some Asymptotic Giant Branch (AGB) star envelopes show
the excess emission at frequencies below 100 GHz which cannot be explained by
thermal dust emission (hereafter anomalous microwave emission-AME). Moreover,
AGB envelopes are a common place where gas molecules condense to form
nanoparticles (e.g., polycyclic aromatic hydrocarbons) and large grains. In
this paper, we will study whether electric dipole emission from rapidly
spinning nanoparticles can reproduce the AME observed toward AGB stars. To
properly model the size distribution of nanoparticles in the AGB envelope, we
take into account both the increase of nanoparticles due to rotational
disruption of large grains spun-up by radiative torques and the decrease of
smallest nanoparticles due to rotational disruption driven by stochastic gas
collisions. We then perform detailed modeling of microwave emission from
rapidly spinning nanoparticles from both C-rich and O-rich AGB envelopes using
the grain size distribution constrained by rotational disruption. We find that
spinning dust emission is dominant over thermal dust emission at frequencies
below 100 GHz. We attempt to fit the observational data of AME using our
spinning dust model and demonstrate that spinning dust can reproduce the
observed AME in six AGB stars. Finally, we discuss that microwave emission from
spinning dust in AGB envelopes could be observed with high-resolution upcoming
radio telescopes such as ngVLA and ALMA Band 1. This would be a major leap for
understanding AGB envelopes, formation, evolution, and internal structures of
dust. Observations would help to distinguish the carrier of AME from comparing
C-rich to O-rich stars, because PAHs are formed in C-rich AGB stars while
silicates are formed in O-rich stars.
