Velocity Dispersion ${\sigma }_{\mathrm{a}\mathrm{p}\mathrm{e}\mathrm{r}}$ Aperture Corrections as a Function of Galaxy Properties from Integral-field Stellar Kinematics of 10,000 MaNGA Galaxies

The second moment of the stellar velocity within the effective radius, denoted by ${\sigma }_{\mathrm{e}}^2$, is a crucial quantity in galaxy studies as it provides insight into galaxy properties and their mass distributions. However, large spectroscopic surveys typically do not measure ${\sigma }_{\mathrm{e}}$ directly, instead providing ${\sigma }_{\mathrm{a}\mathrm{p}\mathrm{e}\mathrm{r}}$, the second moment of the stellar velocity within a fixed fiber aperture. In this paper, we derive an empirical aperture correction formula, given by ${\sigma }_{\mathrm{a}\mathrm{p}\mathrm{e}\mathrm{r}}\mathrm{/}{\sigma }_{\mathrm{e}}=(R_{\mathrm{a}\mathrm{p}\mathrm{e}\mathrm{r}}\mathrm{/}{R}_{\mathrm{e}}{)}^{\alpha }$, using spatially resolved stellar kinematics extracted from approximately 10,000 Sloan Digital Sky Survey-Mapping Nearby Galaxies at Apache Point Observatory (SDSS-MaNGA) integral field unit observations. Our analysis reveals a strong dependence of $\alpha$ on the $r$-band absolute magnitude $M_{\mathrm{r}}$, $g-i$ color, and Sersic index $n_{\mathrm{S}\mathrm{e}\mathrm{r}}$, where $\alpha$ values are lower for brighter, redder galaxies with higher Sersic indices. Our results demonstrate that the aperture correction derived from previous literature on early-type galaxies cannot be applied to predict the aperture corrections for galaxies with intermediate Sersic indices. We provide a lookup table of $\alpha$ values for different galaxy types, with parameters in the ranges of $-18>M_{\mathrm{r}}>-24$, $0.4<g-i<1.6$, and $0<n_{\mathrm{S}\mathrm{e}\mathrm{r}}<8$. A Python script is provided to obtain the correction factors from the lookup table.