Modeling the Solar System I: Characterization Limits from Analytic Timing Variations

Planetary systems with multiple transiting planets are beneficial for understanding planet occurrence rates and system architectures. Although we have yet to find a solar system analogue, future surveys may detect multiple terrestrial planets transiting a Sun-like star. In this work, we simulate transit timing observations of our system based on the actual orbital motions of Venus and the Earth+Moon (EM) -- influenced by the other solar system objects -- and retrieve the system's dynamical parameters for varying noise levels and observing durations. Using an approximate coplanar N-body model for transit-time variations, we consider test configurations with 2, 3, and 4 planets. For various observing baselines, we can robustly retrieve the masses and orbits of Venus and EM; detect Jupiter at high significance (for < 90-second timing error and baseline $\leq$ 15 yrs); and detect Mars at 5 $\sigma$ confidence (with < 20-second timing error and baseline $\geq$ 27 yrs) using TTVFaster. We also find that the 3-planet model is generally preferred, and provide equations to estimate the mass precision of Venus/Earth/Jupiter-analogues. The addition of Mars -- which is near a 2:1 mean-motion resonance with Earth -- improves our retrieval of Jupiter's parameters, suggesting that unseen terrestrials could interfere in the characterization of multi-planetary systems. Our findings are comparable to theoretical limits based upon stellar variability and may eventually be possible.