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. We determine stellar age estimates from combined studies of complementary surveys including GALAH, LAMOST, APOGEE, and the NASA Kepler and K2 missions, and obtain parallax and proper motion from {\it Gaia} DR2. We find that stellar samples from these surveys, 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 dispersion of stars in the solar neighborhood depends on orbital angular momentum and metallicity. The dispersion has a power-law dependence on age with exponents of 0.446$\pm 0.008$ and 0.264$\pm 0.005$ for vertical and radial components respectively, and the power law is valid even for the oldest stars. The apparent break in the power law for older stars, as seen in previous studies, is due to the anti-correlation of angular momentum with age. The dispersion decreases with increasing angular momentum until we reach the Sun's orbital angular momentum, after which the dispersion increases, which implies flaring in the outer disc. For any given stellar age and angular momentum, the dispersion increases with decreasing metallicity, suggesting that the dispersion increases with birth radius. For any given age, angular momentum, and metallicity, the dispersion also increases linearly with height above the midplane of the Galaxy, indicating that the disc stellar populations are not strictly isothermal. The same relation that works in the solar neighborhood also works for stars with Galactocentric radius $R$ ranging from $3<R/{\rm kpc}<20$. Finally, stars with high-[$\alpha$/Fe] abundance follow the same relations as stars with low-[$\alpha$/Fe] abundance.
