Spectroscopic Bolometric Corrections and Empirical Zero-point Constants of \textit{Gaia} Magnitudes, $G$, $G_{\rm BP}$, and $G_{\rm RP}$, from \textit{Gaia} XP Spectra

Abstract: The International Astronomical Union 2015 Resolution B2 (IAU2015GARB2) has resolved the long-standing problem of zero-point constants for the absolute and apparent bolometric magnitude scales and opened a new window in fundamental astrophysics. The empirical zero-point constants of the bolometric corrections, $C_2(ξ)$, and the absolute/apparent magnitudes, $C_ξ/c_ξ$, for the {\it Gaia} passbands were obtained from 88 {\it Gaia} XP spectra, and absolute bolometric/filtered magnitudes. The individual zero-point constants $\langle C_{\rm 2}\rangle$ of the bolometric corrections ($BC_ξ$) for each star revealed weighted averages of $\langle C_{\rm 2}(G)\rangle=0.8677\pm0.0109$ mag, $$\langle C_{\rm 2}(G_{\rm BP})\rangle=1.0449\pm0.0116$$ mag, and $$\langle C_{\rm 2}(G_{\rm RP})\rangle=2.0510\pm0.0087$$ mag. Furthermore, $C_{\rm Bol}=71.197425...$ mag and $c_{\rm Bol} =-18.997351...$mag announced by IAU2015GARB2, and using the definition of $C_{\rm 2}=C_{\rm Bol}-C_ξ=c_{\rm Bol}-c_ξ$, where the subscript $2$ indicate the wavelength ranges of two in which one is for bolometric and the other for one of the three filters, the zero-point constants of magnitudes for {\it Gaia} filters as $C_{\rm G}=70.1525\pm0.0109$ mag and $c_{\rm G}=-19.8651\pm0.0105$ mag, $C_{\rm G_{\rm BP}}=70.1525\pm0.0116$ mag and $c_{\rm G_{\rm BP}}=-20.0423\pm0.0116$ mag, and $C_{\rm G_{\rm RP}}=69.1464\pm0.0087$ mag and $c_{\rm G_{\rm RP}}=-21.0484\pm0.0087$ mag, if $L_ξ$ and $f_ξ$ are in SI units in case no extinctions. Lastly, spectroscopic $BC$s for {\it Gaia} magnitudes of 88 stars and the spectroscopic $BC-T_{\rm eff}$ relation for each {\it Gaia} filter are presented.