Requirements for Joint Orbital Characterization of Cold Giants and Habitable Worlds with Habitable Worlds Observatory

We determine optimal requirements for the joint detection of habitable-zone planets and cold giant planets with the Habitable Worlds Observatory (HWO). Analysis of 164 nearby stars shows that a coronagraph outer working angle of 1440 milliarcseconds (mas) is necessary to achieve 80%-90% visibility of cold giants. Approximately 40 precursor radial velocity (RV) measurements with 1 m s⁻¹ precision are required to adequately constrain orbital parameters before HWO observations. We demonstrate that six to eight astrometric measurements and 40 RV measurements distributed across the mission timeline, compared to RV constraints alone and to astrometry constraints alone, significantly improve orbital parameter precision, enabling direct determination of orbital inclination with uncertainties of <3°. For habitable-zone planet characterization, four to five epochs provide moderate confidence, while high-confidence (95%) confirmation requires eight of more observations. These specifications are essential for the comprehensive characterization of planetary system architectures and understanding the potential habitability of terrestrial exoplanets.