On the Impossibility of Aggregating Individual Cloud Accretion Rates
By Joshua Peek
The rate of accretion onto the Milky Way (MW) by an individual cloud (or satellite) can be written as dM/dt = M v / R, where M is the cloud mass, v the cloud velocity toward the MW, and R is the distance from the cloud to the MW. Over the time t = R / v a mass M will accrete. This rate can be evaluated for many accreting objects; in the MW about a dozen high-velocity clouds (HVCs) are currently accreting onto the Galactic disk. This said, one cannot sum these accretion rates to determine the MW total rate of growth. For instance, in a scenario with all clouds having mass M, fixed accretion velocity v, and linearly increasing distances R, 2R, 3R, 4R... we would expect a mass M to accrete every R / v, providing a fixed accretion rate of M v / R. Yet by summing the accretion rates of each cloud we find M v / R times the harmonic series 1 + 1/2 + 1/3 + 1/4... This results in an infinite accretion rate. In practice, for currently known HVC distances, the summed accretion rate overestimates by 50% to 100%.
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