The vertical motion history of disk stars throughout the Galaxy

It has long been known that the vertical motions of Galactic disk stars increase with stellar age, commonly interpreted as vertical heating through orbit scattering. Here we map the vertical actions of disk stars as a function of age ($\tau\le$ 8 Gyrs) and across a large range of Galactocentric radii, $\overline{R}_{\rm GC}$, drawing on APOGEE and Gaia data. We fit $\widehat{J_z}(\overline{R}_{\rm GC},\tau )$ as a combination of the vertical action at birth, $\widehat{J_{z,0}}$, and subsequent heating $\widehat{\Delta J_z}_{\, {\rm 1Gyr}}(\overline{R}_{\rm GC})$ that scales as $\tau^{\gamma(\overline{R}_{\rm GC})}$. The inferred birth temperature, $\widehat{J_{z,0}}(\overline{R}_{\rm GC})$ is 1 kpc km/s for 3 kpc < $\overline{R}_{\rm GC}$ < 10 kpc, consistent with the ISM velocity dispersion; but it rapidly rises outward, to 8 kpc km/s for $\overline{R}_{\rm GC}$ = 14 kpc, likely reflecting the stars' birth in a warped or flared gas disk. We find the heating rate $\widehat{\Delta J_z}_{\, {\rm 1Gyr}}$ to be modest and nearly constant across all radii, 1.6 kpc km/s Gyr$^{-1}$. The stellar age dependence $\gamma$ gently grows with Galactocentric radius, from $\gamma \simeq$ 1 for $\overline{R}_{\rm GC}\lesssim R_\odot$ to $\gamma \simeq$ 1.3 at $\overline{R}_{\rm GC}$ =14 kpc. The observed $J_z - \tau$ relation at all radii is considerably steeper ($\gamma\gtrsim$ 1) than the time dependence theoretically expected from orbit scattering, $J_z\propto t^{0.5}$. We illustrate how this conundrum can be resolved if we also account for the fact that at earlier epochs the scatterers were more common, and the restoring force from the stellar disk surface mass density was low. Our analysis may re-instate gradual orbital scattering as a plausible and viable mechanism to explain the age-dependent vertical motions of disk stars.