X-ray scaling relations of early-type galaxies
  X-ray luminosity, temperature, gas mass, total mass, and their scaling
relations are derived for 94 early-type galaxies using archival $Chandra$ X-ray
Observatory observations. Consistent with earlier studies, the scaling
relations, $L_X \propto T^{4.5\pm0.2}$, $M \propto T^{2.4\pm0.2}$, and $L_X
\propto M^{2.8\pm0.3}$, are significantly steeper than expected from self
similarity. This steepening indicates that their atmospheres are heated above
the level expected from gravitational infall alone. Energetic feedback from
nuclear black holes and supernova explosions are likely heating agents. The
tight $L_X - T$ correlation for low-luminosities systems (i.e., below 10$^{40}$
erg/s) are at variance with hydrodynamical simulations which generally predict
higher temperatures for low luminosity galaxies. We also investigate the
relationship between total mass and pressure, $Y_X = M_g \times T$, finding $M
\propto Y_{X}^{0.45\pm0.04}$. We explore the gas mass to total mass fraction in
early-type galaxies and find a range of $0.1-1.0\%$. We find no correlation
between the gas-to-total mass fraction with temperature or total mass. Higher
stellar velocity dispersions and higher metallicities are found in hotter,
brighter, and more massive atmospheres. X-ray core radii derived from
$\beta$-model fitting are used to characterize the degree of core and cuspiness
of hot atmospheres.
