We study the sub-Keplerian rotation and dust content of the circumstellar material around the asymptotic giant branch (AGB) star L$_2$ Puppis. We find that the thermal pressure gradient alone cannot explain the observed rotation profile. We find that there is a family of possible dust populations for which radiation pressure can drive the observed sub-Keplerian rotation. This set of solutions is further constrained by the spectral energy distribution (SED) of the system, and we find that a dust-to-gas mass ratio of $sim10^{-3}$ and a maximum grain size that decreases radially outwards can satisfy both the rotation curve and SED. These dust populations are dynamically tightly coupled to the gas azimuthally. However grains larger than $sim0.5mu$m are driven outward radially by radiation pressure at velocities $sim5$kms$^{-1}$, which implies a dust replenishment rate of $sim3times10^{-9}$M$_odot$yr$^{-1}$. This replenishment rate is consistent with observational estimates to within uncertainties. Coupling between the radial motion of the dust and gas is weak and hence the gas does not share in this rapid outward motion. Overall we conclude that radiation pressure is a capable and necessary mechanism to explain the observed rotation profile of L$_2$ Pup, and offers other additional constraints on the dust properties.
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