Abstract
We used the spectroscopic and astrometric data provided from the GALactic Archaeology with HERMES (GALAH) Data Release (DR2) and Gaia DR2, respectively, for a large sample of stars to investigate the behaviour of the [
$\alpha$
/Fe] abundances via two procedures, that is, kinematically and spectroscopically. With the kinematical procedure, we investigated the distribution of the [
$\alpha$
/Fe] abundances into the high-/low-probability thin disc, and high-/low-probability thick-disc populations in terms of total space velocity, [Fe/H] abundance, and age. The high-probability thin-disc stars dominate in all sub-intervals of [
$\alpha$
/Fe], including the rich ones: [
$\alpha$
/Fe]
$\,>\,0.3$
dex, where the high-probability thick-disc stars are expected to dominate. This result can be explained by the limiting apparent magnitude of the GALAH DR2 (
$V \lt 14$
mag) and intermediate galactic latitude of the star sample. Stars in the four populations share equivalent [
$\alpha$
/Fe] and [Fe/H] abundances, total space velocities, and ages. Hence, none of these parameters can be used alone for separation of a sample of stars into different populations. High-probability thin-disc stars with abundance
$-1.3 \lt {\rm[Fe/H]}\leq -0.5$
dex and age
$9 \lt \tau\leq13$
Gyr are assumed to have different birth places relative to the metal-rich and younger ones. With the spectroscopic procedure, we separated the sample stars into
$\alpha$
-rich and
$\alpha$
-poor categories by means of their ages as well as their [
$\alpha$
/Fe] and [Fe/H] abundances. Stars older than 8 Gyr are richer in [
$\alpha$
/Fe] than the younger ones. We could estimate the abundance [
$\alpha$
/Fe] = 0.14 dex as the boundary separating the
$\alpha$
-rich and
$\alpha$
-poor sub-samples in the [
$\alpha$
/Fe]
$\,\times\,$
[Fe/H] plane.
Variation formulas for principal functions, II: Applications to variation for harmonic spans
