We report on proper-motion measurements of water masers in the massive-star forming region W 51A and analyses of the 3-D kinematics of the masers in three maser clusters of W 51A (W 51 North, Main, and South). In W 51 North, we found a clear expanding flow having an expansion velocity of ~ 70 \,km \,s⁻¹, which indicates deceleration. The originating point of the flow coincides within 0 .1 with a silicon-monoxide maser source near the H II region W 51d. In W 51 Main, no systematic motion was found over the whole velocity range (- 58 \,km \,s⁻¹ ≤ VLSR < 158 \,km \,s⁻¹), although a stream motion was previously reported in a limited range of the Doppler velocity (54 \,km \,s⁻¹ ≤ VLSR < 68 \,km \,s⁻¹). Multiple driving sources of outflows are thought to explain the kinematics of W 51 Main. In W 51 South, an expansion motion like a bipolar flow was marginally visible. Analyses based on diagonalization of the variance–covariance matrix of maser velocity vectors demonstrate that the maser kinematics in W 51 North and Main are significantly tri-axially asymmetric. We estimated the distance to W 51 North to be 6.1 ± 1.3 \,kpc based on a model-fitting method adopting a radially expanding flow.
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