  Comet 8P/Tuttle is a Nearly Isotropic Comet (NIC), whose physical properties
are poorly known and could be different from those of Ecliptic Comets (EC)
owing to their different origin. Two independent observations have shown that
8P has a bilobate nucleus. Our goal is to determine the physical properties of
the nucleus (size, shape, thermal inertia, albedo) and coma (water and dust) of
8P/Tuttle. We observed the inner coma of 8P with the infrared spectrograph
(IRS) and the infrared camera (MIPS) of the Spitzer Space Telescope (SST). We
obtained one spectrum (5-40 $\mu$m) on 2 November 2007 and a set of 19 images
at 24 $\mu$m on 22-23 June 2008 sampling the nucleus rotational period. The
data were interpreted using thermal models for the nucleus and the dust coma,
and considering 2 possible shape models of the nucleus derived from
respectively Hubble Space Telescope visible and Arecibo radar observations. We
favor a nucleus shape model composed of 2 contact spheres with respective radii
of 2.7+/-0.1 km and 1.1+/-0.1 km and a pole orientation with RA=285+/-12 deg
and DEC=+20+/-5 deg. The nucleus has a thermal inertia in the range 0-100
J/K/m^2/s^0.5 and a R-band geometric albedo of 0.042+/-0.008. The water
production rate amounts to 1.1+/-0.2x10^28~molecules/s at 1.6 AU from the Sun
pre-perihelion, which corresponds to an active fraction of 9%. At the same
distance, the $\epsilon f \rho$ quantity amounts to 310+/-34 cm at 1.6~AU, and
reaches 325+/-36 cm at 2.2~AU post-perihelion. The dust grain temperature is
estimated to 258+/-10 K, which is 37 K larger than the thermal equilibrium
temperature at 1.6 AU. This indicates that the dust grains contributing to the
thermal infrared flux have a typical size of 10 $\mu$m. The dust spectrum
exhibits broad emissions around 10 $\mu$m (1.5-sigma confidence level) and 18
$\mu$m (5-sigma confidence level) that we attribute to amorphous pyroxene.
