Effect of Earth's Oblateness on Black Hole Imaging Through Earth-Space and Space-Space VLBI

Earth-based Very Long Baseline Interferometry (VLBI) has made rapid advances in imaging black holes. However, due to the limitations imposed on terrestrial VLBI by the Earth's finite size and turbulent atmosphere, it is imperative to have a space-based component in future VLBI missions. Herein, this paper investigates the effect of Earth's oblateness, also known as the $J_{2}$ effect, on orbiters in Earth-Space and Space-Space VLBI. The paper provides an extensive discussion on how the $J_{2}$ effect can directly impact orbit selection for black hole observations and how through informed choices of orbital parameters, the effect can be used to the mission's advantage, a fact that has not been addressed in existing space-VLBI investigations. We provide a comprehensive study of how the orbital parameters of several current space VLBI proposals will vary specifically due to the $J_{2}$ effect. For black hole accretion flow targets of interest, we have demonstrated how the $J_{2}$ effect leads to modest increase in shorter baseline coverage, filling gaps in the $(u,v)$ plane. Subsequently, we construct a simple analytical formalism that allows isolation of the impact of the $J_{2}$ effect on the $(u,v)$ plane without requiring computationally intensive orbit propagation simulations. By directly constructing $(u,v)$ coverage using the $J_{2}$ affected and invariant equations of motion, we obtain distinct coverage patterns for M87* and SgrA* that show extremely dense coverage on short baselines as well as long term orbital stability on longer baselines.