  The relation between X-ray luminosity (L_X) and ambient gas temperature (T)
among massive galactic systems is an important cornerstone of both
observational cosmology and galaxy-evolution modeling. In the most massive
galaxy clusters, the relation is determined primarily by cosmological structure
formation. In less massive systems, it primarily reflects the feedback response
to radiative cooling of circumgalactic gas. Here we present a simple but
powerful model for the L_X-T relation as a function of physical aperture R
within which those measurements are made. The model is based on the
precipitation framework for AGN feedback and assumes that the circumgalactic
medium is precipitation-regulated at small radii and limited by cosmological
structure formation at large radii. We compare this model with many different
data sets and show that it successfully reproduces the slope and upper envelope
of the L_X-T-R relation over the temperature range from ~0.2 keV through >10
keV. Our findings strongly suggest that the feedback mechanisms responsible for
regulating star formation in individual massive galaxies have much in common
with the precipitation-triggered feedback that appears to regulate
galaxy-cluster cores.
