Investigating the vertical distribution of the disk as a function of radial action

As heating processes can broaden the distributions of radial actions and the vertical distributions of the Galactic disks, we investigate the vertical distribution of the Galactic disks as a function of radial action based on Apache Point Observatory Galactic Evolution Experiment(APOGEE) and Gaia data in order to deepen our understanding of the formation and heating history of the Galactic disks. We find that the distributions of radial action for both the thin and thick disks can be approximately described by pseudo-isothermal distributions, which give a statistical measurement for the temperature of the disk as indicated by the mean radial action of the star sample. Estimations of the scale heights in different radial action ranges for these pseudo-isothermal distributions of the disks seem to show fixed relationships between radial action $J_R$ and scale height $h$. We describe these relationships with a two-parameter function of $h = \sqrt{J_R /a} + b$, where $a$ and $b$ are free parameters. When testing with a three-parameter function of $h=\sqrt[\alpha]{J_R/a}+b$, we find that this two-parameter function describes the thin disk well, but we note the function should be used with care for the thick disk. When comparing the best-fit relationships between the inner and outer disk for both of the thin and thick disks, we find that the relationships are nearly the same for the thin disks but are different for the thick disks. The inner thick disk shows a nearly flattened relationship, while the outer thick disk presents a gradually increasing relationship. This work highlights an alternative way to unveil the heating history of the disks by investigating the relationship between scale height and radial action, as these relationships encode the formation and heating history of the Galactic disks.