The Deviation of the Broad-line Region Size Between Reverberation Mapping and Spectroastrometry

The combination of the linear size from reverberation mapping (RM) and the angular distance of the broad line region (BLR) from spectroastrometry (SA) in active galactic nuclei (AGNs) can be used to measure the Hubble constant $H_0$. Recently, Wang et al. (2020) successfully employed this approach and estimated $H_0$ from 3C 273. However, there may be a systematic deviation between the response-weighted radius (RM measurement) and luminosity-weighted radius (SA measurement), especially when different broad lines are adopted for size indicators (e.g., \hb\ for RM and \pa\ for SA). Here we evaluate the size deviations measured by six pairs of hydrogen lines (e.g., \hb, \ha\ and \pa) via the locally optimally emitting cloud (LOC) models of BLR. We find that the radius ratios $K$(=$R_{\rm SA}$/$R_{\rm RM}$) of the same line deviated systematically from 1 (0.85-0.88) with dispersions between 0.063-0.083. Surprisingly, the $K$ values from the \pa(SA)/\hb(RM) and \ha(SA)/\hb(RM) pairs not only are closest to 1 but also have considerably smaller uncertainty. Considering the current infrared interferometry technology, the \pa(SA)/\hb(RM) pair is the ideal choice for the low redshift objects in the SARM project. In the future, the \ha(SA)/\hb(RM) pair could be used for the high redshift luminous quasars. These theoretical estimations of the SA/RM radius pave the way for the future SARM measurements to further constrain the standard cosmological model.