[[spi:math]] (2) From (4), we note that the beam density falls off as z-2 in the beam propagation direction but more importantly the beam is confined to a transverse region [[spi:math]]. Since z and v0 are given for our case, the constant [[spi:math]] determines the cross-sectional area of the beam. In charge-exchange considerations of Section V.A, certain complex drift path averages will be needed to determine ring current particle lifetimes. Consequently, (4) may be too complex for such purposes; instead, an averaging approach can be used to estimate the neutral jet characteristics here. Assuming the exhaust to expand away from the beam axis at an average speed w ~ u/10 [[spi:math]] 0.4 km/sec, the neutral exhaust cloud would occupy an approximately conical volume [[spi:math]] at any time t > 0. The density distribution N inside this volume depends on the details of the expansion as descibed above. For our purposes here an average estimate is N ~ Ft/V. This would imply an average density of 1013/(4t2) atoms/cm3, where t is measured in 10 normalized (to the exhaust molecules propellant flow rate F) source function is where F (molecules/sec) is [[spi:math]]. The exact solution of (1) for S given by (2) can be obtained as [[spi:math]] (3) The density N is given by [[spi:math]] . (4)
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