Sharma, SanjibHayden, MichaelBland-Hawthorn, JossStello, DennisBuder, SvenZinn, Joel C.Kallinger, ThomasAsplund, MartinDe Silva, Gayandhi M.D'Orazi, ValentinaFreeman, KennethKos, JanezLewis, Geraint F.Jane, LinLind, KarinMartell, SarahSimpson, Jeffrey D.Wittenmyer, Rob A.Zucker, Daniel B.Zwitter, TomazChen, BoquanCotar, KlemenEsdaile, JamesHon, MarcHorner, JonathanHuber, DanielKafle, Prajwal R.Khanna, ShouryaTing, Yuan-SenNataf, David M.Nordlander, ThomasSaadon, Mohd Hafiz MohdTepper-Garcia, ThorTinney, C. G.Traven, GregorWatson, FredWright, DuncanWyse, Rosemary F. G.2023-01-192023-01-190035-8711http://hdl.handle.net/1885/283846We 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 Lz, 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 ± 0.007 and 0.251 ± 0.006 for σz and σR, respectively, and the power law is valid even for the oldest stars. For the solar neighbourhood stars, the apparent break in the power law for older stars, as seen in previous studies, is due to the anticorrelation of Lz with age. The dispersions decrease with increasing Lz until we reach the Sun's orbital angular momentum, after which σz increases (implying flaring in the outer disc) while σR flattens. For a given age, 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 neighbourhood also work for stars between 3 < R/kpc < 20. Finally, the high-[α/Fe] stars follow the same relations as the low-[α/Fe] stars.SS is funded by a Senior Fellowship (University of Sydney), an ASTRO-3D Research Fellowship and JBH’s Laureate Fellowship from the Australian Research Council (ARC). JBH’s research team is supported by an ARC Laureate Fellowship (FL140100278) and funds from ASTRO-3D. MJH is supported by an ASTRO-3D 4- year Research Fellowship. DS is the recipient of an ARC Future Fellowship (project number FT1400147). SB and KL acknowledge funds from the Alexander von Humboldt Foundation in the framework of the Sofja Kovalevskaja Award endowed by the Federal Ministry of Education and Research. KL acknowledges funds from the Swedish Research Council (Grant 2015-00415 3) and Marie Sklodowska Curie Actions (Cofund Project INCA 600398). JK, KC, and TZ acknowledges financial support from the Slovenian Research Agency (research core funding No. P1-0188). DMN was supported by the Allan C. and Dorothy H. Davis Fellowship. JZ acknowledges support from NASA grants 80NSSC18K0391 and NNX17AJ40G. DH acknowledges support from the Alfred P. Sloan Foundation and the National Aeronautics and Space Administration (80NSSC19K0108). The GALAH Survey is supported by the ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project CE170100013. This work has made use of data acquired through the Australian Astronomical Observatory under programmes GALAH, TESS-HERMES and K2-HERMES. We acknowledge the traditional owners of the land on which the AAT stands, the Gamilaraay people, and pay our respects to elders past and present.application/pdfen-AU© 2021 The Author(s) Published by Oxford University Press on behalf of Royal Astronomical SocietyGalaxy: discGalaxy: evolutionGalaxy: formationGalaxy: kinematics and dynamics202110.1093/mnras/stab10862021-11-28