  The presence of hot gaseous coronae around present-day massive spiral
galaxies is a fundamental prediction of galaxy formation models. However, our
observational knowledge remains scarce, since to date only four gaseous coronae
were detected around spirals with massive stellar bodies
($\gtrsim2\times10^{11} \ \rm{M_{\odot}}$). To explore the hot coronae around
lower mass spiral galaxies, we utilized Chandra X-ray observations of a sample
of eight normal spiral galaxies with stellar masses of $(0.7-2.0)\times10^{11}
\ \rm{M_{\odot}}$. Although statistically significant diffuse X-ray emission is
not detected beyond the optical radii ($\sim20$ kpc) of the galaxies, we derive
$3\sigma$ limits on the characteristics of the coronae. These limits,
complemented with previous detections of NGC 1961 and NGC 6753, are used to
probe the Illustris Simulation. The observed $3\sigma$ upper limits on the
X-ray luminosities and gas masses exceed or are at the upper end of the model
predictions. For NGC 1961 and NGC 6753 the observed gas temperatures, metal
abundances, and electron density profiles broadly agree with those predicted by
Illustris. These results hint that the physics modules of Illustris are broadly
consistent with the observed properties of hot coronae around spiral galaxies.
However, a shortcoming of Illustris is that massive black holes, mostly
residing in giant ellipticals, give rise to powerful radio-mode AGN feedback,
which results in under luminous coronae for ellipticals.
