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The circumstellar environment of the AGB star L$_{\rm 2}$ Puppis was observedwith ALMA in cycle 3, with a resolution of $15 \times 18 \rm\ mas$. Themolecular emission shows a differentially rotating disk, inclined to a nearlyedge-on position. In the first paper in this series (paper I) the molecularemission was analysed to accurately deduce the motion of the gas in theequatorial regions of the disk. In this work we model the optically thick$^{\rm 12}$CO $J=$3$-$2 and the optically thin $^{\rm 13}$CO $J=$3$-$2rotational transition to constrain the physical conditions in the disk. Torealise this effort we make use of the 3D NLTE radiative transfer code {\ttLIME}. The temperature structure and velocity structure show a high degree ofcomplexity, both radially and vertically. The radial H$_{\rm 2}$ densityprofile in the disk plane is characterised by a power law with a slope of$-3.1$. We find a $^{\rm 12}$CO over $^{\rm 13}$CO abundance ratio of 10 insidethe disk. Finally, estimations of the angular momentum in the disk surpass theexpected available angular momentum of the star, strongly supporting theindirect detection of a compact binary companion reported in paper I. Weestimate the mass of the companion to be around 1 Jupiter mass.

ALMA observations of the nearby AGB star L$_{\rm 2}$ Puppis. II. Gas disk properties derived from $^{\rm 12}$CO and $^{\rm13}$CO $J=$3$-$2 emission