Angular momentum variation of the Milky Way thick disk: The dependence of chemical abundance and the evidence on inside-out formation scenario

We investigate the angular momentum of mono-abundance populations (MAPs) of the Milky Way thick disk by using a sample of 26,076 giant stars taken from APOGEE DR17 and Gaia EDR3. The vertical and perpendicular angular momentum components, $L_Z$ and $L_P$, of MAPs in narrow bins have significant variations across the [$\alpha$/M]-[M/H] plane. $L_Z$ and $L_P$ systematically change with [M/H] and [$\alpha$/M] and can be alternatively quantified by the chemical gradients: $d[\mathrm{M}\mathrm{/}\mathrm{H}]\mathrm{/}d{L}_Z=1.2\times {10}^{-3}$,dex,kpc${}^{-1}$,km${}^{-1}$,s, $d[\mathrm{M}\mathrm{/}\mathrm{H}]\mathrm{/}d{L}_P=-5.0\times {10}^{-3}$,dec,kpc${}^{-1}$,km${}^{-1}$,s, and $d[\alpha \mathrm{/}\mathrm{M}]\mathrm{/}d{L}_Z=-3.0\times {10}^{-4}$,dex,kpc${}^{-1}$,km${}^{-1}$,s, $d[\alpha \mathrm{/}\mathrm{M}]\mathrm{/}d{L}_P=1.2\times {10}^{-3}$,dec,kpc${}^{-1}$,km${}^{-1}$,s. These correlations can also be explained as the chemical-dependence of the spatial distribution shape of MAPs. We also exhibit the corresponding age dependence of angular momentum components. Under the assumption that the guiding radius ($R_g$) is proportional to $L_Z$, it provides direct observational evidence of the inside-out structure formation scenario of the thick disk, with $d{R}_g\mathrm{/}dAge=-1.9$,kpc,Gyr${}^{-1}$. The progressive changes in the disk thickness can be explained by the upside-down formation or/and the consequent kinematical heating.