  Wavelength dependent photodesorption rates have been determined using
synchrotron radiation, for condensed pure and mixed methanol ice in the 7 -- 14
eV range. The VUV photodesorption of intact methanol molecules from pure
methanol ices is found to be of the order of 10$^{-5}$ molecules/photon, that
is two orders of magnitude below what is generally used in astrochemical
models. This rate gets even lower ($<$ 10$^{-6}$ molecules/photon) when the
methanol is mixed with CO molecules in the ices. This is consistent with a
picture in which photodissociation and recombination processes are at the
origin of intact methanol desorption from pure CH$_3$OH ices. Such low rates
are explained by the fact that the overall photodesorption process is dominated
by the desorption of the photofragments CO, CH$_3$, OH, H$_2$CO and
CH$_3$O/CH$_2$OH, whose photodesorption rates are given in this study. Our
results suggest that the role of the photodesorption as a mechanism to explain
the observed gas phase abundances of methanol in cold media is probably
overestimated. Nevertheless, the photodesorption of radicals from methanol-rich
ices may stand at the origin of the gas phase presence of radicals such as
CH$_3$O, therefore opening new gas phase chemical routes for the formation of
complex molecules.
