The SDSS-V Black Hole Mapper Reverberation Mapping Project: Multi-Line Dynamical Modeling of a Highly Variable Active Galactic Nucleus with Decade-long Light Curves

We present dynamical modeling of the broad-line region (BLR) for the highly variable AGN SDSS J141041.25+531849.0 ($z = 0.359$) using photometric and spectroscopic monitoring data from the Sloan Digital Sky Survey Reverberation Mapping project and the SDSS-V Black Hole Mapper program, spanning from early 2013 to early 2023. We model the geometry and kinematics of the BLR in the H$\beta$, H$\alpha$, and MgII, emission lines for three different time periods to measure the potential change of structure within the BLR across time and line species. We consistently find a moderately edge-on $(i_{\rm full-state} = 53.29^{\circ} \,{}^{+7.29}_{-6.55})$ thick-disk $(\theta_{\rm opn, \; full-state} = 54.86^{\circ} \,{}^{+5.83}_{-4.74})$ geometry for all BLRs, with a joint estimate for the mass of the supermassive black hole (SMBH) for each of three time periods, yielding $\log_{10}(M_{\rm BH} / M_{\odot}) = 7.66^{+0.12}_{-0.13}$ when using the full dataset. The inferred individual virial factor $f$ $\sim 1$ is significantly smaller than the average factor for a local sample of dynamically modeled AGNs. There is strong evidence for non-virial motion, with over $80\%$ of clouds on inflowing/outflowing orbits. We analyze the change in model parameters across emission lines, finding the radii of BLRs for the emission lines are consistent with the following relative sizes $R_{\rm H\beta} \lesssim R_{\rm MgII } \lesssim R_{\rm H\alpha}$. Comparing results across time, we find $R_{\rm low-state} \lesssim R_{\rm high-state}$, with the change in BLR size for H$\beta$, being more significant than for the other two lines. The data also reveal complex, time-evolving, and potentially transient dynamics of the BLR gas over decade-long timescales, encouraging for future dynamical modeling of fine-scale BLR kinematics.