  It is theoretically expected that a supermassive black hole (SMBH) in the
centre of a typical nearby galaxy disrupts a Solar-type star every ~ 10^5
years, resulting in a bright flare lasting for months. Sgr A*, the resident
SMBH of the Milky Way, produces (by comparison) tiny flares that last only
hours but occur daily. Here we explore the possibility that these flares could
be produced by disruption of smaller bodies - asteroids. We show that asteroids
passing within an AU of Sgr A* could be split into smaller fragments which then
vaporise by bodily friction with the tenuous quiescent gas accretion flow onto
Sgr A*. The ensuing shocks and plasma instabilities may create a transient
population of very hot electrons invoked in several currently popular models
for Sgr A* flares, thus producing the required spectra. We estimate that
asteroids larger than ~ 10 km in size are needed to power the observed flares,
with the maximum possible luminosity of the order 10^39 erg s^-1. Assuming that
the asteroid population per parent star in the central parsec of the Milky Way
is not too dissimilar from that around stars in the Solar neighbourhood, we
estimate the asteroid disruption rates, and the distribution of the expected
luminosities, finding a reasonable agreement with the observations. We also
note that planets may be tidally disrupted by Sgr A* as well, also very
infrequently. We speculate that one such disruption may explain the putative
increase in Sgr A* luminosity ~ 300 yr ago.
