The massive black hole
By following the motions of individual stars we have measured the mass associated with radio source Sgr A* - we see stars on Keplerian ellipses surrounding an object of 4 million times the mass of the Sun. The physics at play is extremely simple: It is Newton's law of gravity. The precision of these measurements is stunning - see the orbit of the star S2 (Gillessen et al. 2009, Gillessen et al. 2013)
S2 is the best example of the orbits, and constraints the mass most. Overall, we can measure the mass with a precision of around 1%. Furthermore, we can locate the mass and show that its position agrees to better than 1mas with where the radio source Sgr A* is located (Plewa et al. 2015). This measurement uses a few SiO maser stars, which are visible both in the infrared, and at radio wavelengths.
Since we can measure both the proper motion of S2 on the sky as well as its radial velocity along the line of sight, our measurements allow us to calculate both the mass of the black hole as well as the distance R0 to the galactic center. The GRAVITY instrument has allowed us to measure these two quantities with unprecedented precision and accuracy. By combining the precise astrometry from GRAVITY with the spectral measurements of SINFONI, we determined the distance to the galactic center to be 8178±13 stat ±22 sys (Gravity Collaboration et al. 2019). The statistical error is dominated by the uncertainty in measuring the radial velocity, while the systematic error stems from the GRAVITY instrument.