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%.