  The second Gaia data release (Gaia-DR2) contains, beyond the astrometry,
three-band photometry for 1.38 billion sources. We have used these three broad
bands to infer stellar effective temperatures, Teff, for all sources brighter
than G=17 mag with Teff in the range 3000-10 000 K (161 million sources). Using
in addition the parallaxes, we infer the line-of-sight extinction, A_G, and the
reddening, E[BP-RP], for 88 million sources. Together with a bolometric
correction we derive luminosity and radius for 77 million sources. These
quantities as well as their estimated uncertainties are part of Gaia-DR2. Here
we describe the procedures by which these quantities were obtained, including
the underlying assumptions, comparison with literature estimates, and the
limitations of our results. Typical accuracies are of order 324 K (Teff), 0.46
mag (A_G), 0.23 mag (E[BP-RP]), 15% (luminosity), and 10% (radius). Being based
on only a small number of observable quantities and limited training data, our
results are necessarily subject to some extreme assumptions that can lead to
strong systematics in some cases (not included in the aforementioned accuracy
estimates). One aspect is the non-negativity contraint of our estimates, in
particular extinction. Yet in several regions of parameter space our results
show very good performance, for example for red clump stars and solar
analogues. Large uncertainties render the extinctions less useful at the
individual star level, but they show good performance for ensemble estimates.
We identify regimes in which our parameters should and should not be used and
we define a "clean" sample. Despite the limitations, this is the largest
catalogue of uniformly-inferred stellar parameters to date. More precise and
detailed astrophysical parameters based on the full BP/RP spectrophotometry are
planned as part of the third Gaia data release.
