Modeling the Solar System as an Observed Multi-Transit System I: Characterization Limits from Analytic Timing Variations

TL;DR

This study investigates how transit timing variations from two transiting terrestrial bodies, Venus and Earth–Moon, can reveal masses and orbits in a solar-system-like system seen as an observed multi-transit system. Using a coplanar analytic TTV framework (TTVFaster) and DE440 CalcEph ephemerides, the authors simulate 15–30 year observing baselines with varying timing noise to test 2-, 3-, and 4-planet configurations and search for Mars and Jupiter analogs. They find that a 3-planet model is generally preferred and that Jupiter-like perturbations are detectable under realistic timing precision, while Mars remains difficult to characterize unless timing precision is unrealistically high; Mars’ presence, however, can improve Jupiter’s parameter retrieval. The work provides mass-precision scaling relations and discusses implications for future high-precision transit surveys and complementary RV observations, highlighting the feasibility and limits of detecting Solar-System–like architectures with TTVs.

Abstract

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 $σ$ 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.

Figures (16)

• Figure 1: Top-down view of solar system orbits computed from JPL ephemerides over 15 yrs; the x-y axes represent the projected coordinates of the international celestial reference frame Charlot2020 used by the JPL ephemeris. We indicate a projection of the direction near which the mid-transits of Venus (triangles) and EM (circles) can be seen.
• Figure 2: Schematic of the simulated multi-transit system (i), along with the configurations considered in this work (ii--iv). We treat the Earth + Moon as planet c, with a location provided by the JPL ephemerides of the EMB.
• Figure 3: Likelihood as a function of mass-ratio and period of a third perturbing planet. Each curve corresponds to a value in ten grid points of the mass-ratio, where $\gamma = log_{10}(m_p/M_\odot)$. The log-likelihood was computed over 200 logarithmic steps in $P = [1.5,22]$ years, while optimizing over the orbital phase.
• Figure 4: Model comparison for simulated observations of Venus (planet b) and Earth + Moon (planet c): $\mathcal{H}_{PP}$ in dashed blue, $\mathcal{H}_{PPP}$ in solid orange, $\mathcal{H}_{PPP}$ in dot-dashed green. Right: histograms showing the distribution of scatter for each planet. The dashed black line corresponds to a Gaussian with the same mean and variance as the injected noise. The red curve is a Normal distribution with zero mean and a standard deviation of 0.5 minutes.
• Figure 5: Simulated TTVs of 2 transiting terrestrial planets given in Table \ref{['tab:sims']} -- Venus/b (top) and Earth + Moon/c (bottom) -- with a best-fit linear ephemeris removed, colored by source. Left panels: The grey curve is the sum of all the perturbations, calculated from the median values of the posterior sampling of the best-fit 3-planet model. Right panels: Same as the left, but here the solid grey line was created from the median posteriors in our best-fit 4-planet model.