We have investigated the linear X-ray filament, G359.944-0.052, previously identified as a likely X-ray counterpart of a parsec-scale jet from the Galactic Center super-massive black hole (SMBH), Sagittarius A* (Sgr A*), using a total of ~5.6 Ms ultra-deep Chandra observations taken from September 1999 to July 2017. This unprecedented dataset enables us to examine flux and spectral variations that might be related to intrinsic properties of the weakly accreting SMBH. We find no flux or spectral variation in G359.944-0.052 after the G2 periapsis passage around early 2014, however, a moderate flux increase of ~2 sigma significance might be associated with the periapsis passage of G1 in early 2001. The filament exhibits an unusually hard spectrum (photon-index < 1) in its portion closest to Sgr A* (i.e., near-side) and a significant spectral softening in the more distant portion, which can be interpreted as synchrotron cooling of the relativistic electrons moving along the jet path. In particular, the hard spectrum of the near-side suggests a piling up of quasi-monoenergetic electrons caused by rapid radiative cooling. The spectral and temporal properties of G359.944-0.052 strengthen the case of it being the X-ray counterpart of a jet launched by Sgr A*.
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