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Strongly lensed quasars can provide measurements of the Hubble constant($H_{0}$) independent of any other methods. One of the key ingredients isexquisite high-resolution imaging data, such as Hubble Space Telescope (HST)imaging and adaptive-optics (AO) imaging from ground-based telescopes, whichprovide strong constraints on the mass distribution of the lensing galaxy. Inthis work, we expand on the previous analysis of three time-delay lenses withAO imaging (RXJ1131-1231, HE0435-1223, and PG1115+080), and perform a jointanalysis of J0924+0219 by using AO imaging from the Keck Telescope, obtained aspart of the SHARP (Strong lensing at High Angular Resolution Program) AOeffort, with HST imaging to constrain the mass distribution of the lensinggalaxy. Under the assumption of a flat $\Lambda$CDM model with fixed$\Omega_{\rm m}=0.3$, we show that by marginalizing over two different kinds ofmass models (power-law and composite models) and their transformed massprofiles via a mass-sheet transformation, we obtain $\Delta t_{\rmBA}h\hat{\sigma}_{v}^{-2}=6.89\substack{+0.8\\-0.7}$ days, $\Delta t_{\rmCA}h\hat{\sigma}_{v}^{-2}=10.7\substack{+1.6\\-1.2}$ days, and $\Delta t_{\rmDA}h\hat{\sigma}_{v}^{-2}=7.70\substack{+1.0\\-0.9}$ days, where$h=H_{0}/100~\rm km\,s^{-1}\,Mpc^{-1}$ is the dimensionless Hubble constant and$\hat{\sigma}_{v}=\sigma^{\rm ob}_{v}/(280~\rm km\,s^{-1})$ is the scaleddimensionless velocity dispersion. Future measurements of time delays with 10%uncertainty and velocity dispersion with 5% uncertainty would yield a $H_0$constraint of $\sim15$% precision.

SHARP VIII: J0924+0219 lens mass distribution and time-delay prediction through adaptive-optics imaging