The Sphere of Influence of Sagittarius A*: Where the Black Hole Takes Over the Cluster

Every black hole embedded in a star cluster rules a definite territory, the sphere within which its gravity, not the cluster's, sets the motions. This paper computes that territory for Sagittarius A* two ways. From the velocity dispersion of the surrounding cluster, about 105 kilometres per second, the influence radius is 1.68 parsecs, 42 arcseconds on the sky, some 334 times the semi-major axis of S2's orbit, with an orbital period at its edge of 98084 years. From the mass side, the Bahcall-Wolf cusp normalised to a million solar masses inside a parsec accumulates twice the black hole's mass only at 5.6 parsecs, showing the two definitions answer different questions and bracket the transition zone between hole-dominated and cluster-dominated dynamics. Across the plausible dispersion range of 75 to 150 kilometres per second the influence radius runs from 3.3 to 0.82 parsecs if the tabulated values are read in order. The crossing time at the influence radius is 15611 years. Computed from published dispersion and cusp values.