Fundamental relations for the velocity dispersion of stars in the Milky
  Way
  We explore the fundamental relations governing the radial and vertical
velocity dispersions of stars in the Milky Way, from combined studies of
complementary surveys including GALAH, LAMOST, APOGEE, the NASA $Kepler$ and K2
missions, and $Gaia$ DR2. We find that different stellar samples, even though
they target different tracer populations and employ a variety of age estimation
techniques, follow the same set of fundamental relations. We provide the
clearest evidence to date that, in addition to the well-known dependence on
stellar age, the velocity dispersions of stars depend on orbital angular
momentum $L_z$, metallicity and height above the plane $|z|$, and are well
described by a multiplicatively separable functional form. The dispersions have
a power-law dependence on age with exponents of 0.441$\pm 0.007$ and 0.251$\pm
0.006$ for $\sigma_z$ and $\sigma_R$ respectively, and the power law is valid
even for the oldest stars. For the solar neighborhood stars, the apparent break
in the power law for older stars, as seen in previous studies, is due to the
anti-correlation of $L_z$ with age. The dispersions decrease with increasing
$L_z$ until we reach the Sun's orbital angular momentum, after which $\sigma_z$
increases (implying flaring in the outer disc) while $\sigma_R$ flattens. The
dispersions increase with decreasing metallicity, suggesting that the
dispersions increase with birth radius. The dispersions also increase linearly
with $|z|$. The same set of relations that work in the solar neighborhood also
work for stars between $3<R/{\rm kpc}<20$. Finally, the high-[$\alpha$/Fe]
stars follow the same relations as the low-[$\alpha$/Fe] stars.
