Unveiling the Sagittarius Dwarf Spheroidal Galaxy Core in Gaia DR3: A Red Clump Distance Precise to 2%

The Sagittarius dwarf spheroidal galaxy provides us with the unique opportunity to study an ongoing Galactic cannibalistic event between the Milky Way and a satellite dwarf galaxy. Understanding this event crucially requires memberships and high-precision metallicities. Here, we present the first major membership star catalog of the Sagittarius dwarf core ($\approx$140,000 sources) and Messier 54 ($\approx$2000 sources) with positions, proper motions, and parallaxes from $Gaia$ DR3, supplemented with metallicities from APOGEE DR17. We initially isolate and separate the Sagittarius dwarf core and Messier 54 spatially from prior literature positions. Using evolutionary sub-samples separated within a color-magnitude diagram, we analyze the substructures of the Sagittarius core and infer its positional relationship with Messier 54 within 5D phase space. A sample of Milky Way stars from a similar galactic latitude were used to identify contaminants and separate member stars from the cores of Sgr dSph and Messier 54 using a Gaussian Mixture Model. We present the derived proper motion, parallaxes, and metallicities for these evolutionary sub-samples while demonstrating the precision of our sample using red clump standard candles. We find a distance modulus for the Sagittarius core and Messier 54 of $(m-M)_{0}=16.958^{+0.044}_{-0.044}$ mag and $(m-M)_{0}=16.94^{+0.047}_{-0.056}$ mag, corresponding to a heliocentric distance of $d=24.635^{+0.49}_{-0.49}$ kpc and $d=24.452^{+0.537}_{-0.602}$ kpc respectively. With red clump distance analysis, our results imply there is no separation between the Sagittarius core and Messier 54 for the evolved stellar populations. Finally, we describe the metallicity distributions of the evolved stars within these two systems, finding evidence in favor of the infall scenario.