Metrics for Optimizing Searches for Orbital Precession and Tidal Decay via Transit- and Occultation-Timing

TL;DR

The study tackles the problem of distinguishing apsidal precession from tidal decay in short-period exoplanets using transit- and occultation-timing data. It develops a Bayesian Information Criterion (BIC)–based framework and derives analytic and semi-analytic expressions for $\Delta{\rm BIC}$ under different ephemeris models, enabling forecasting of when future timing data will favor precession or decay and how occultation data improve discrimination. Applying these methods to systems including WASP-12 b, HAT-P-37 b, and WASP-19 b, the authors show how occultation timings can decisively sharpen conclusions and how data quality and outliers influence robustness. The work also highlights the practical value of citizen-science transit observations in extending baselines and provides concrete metrics to plan and assess long-term timing campaigns.

Abstract

Short-period exoplanets may exhibit orbital precession driven by several different processes, including tidal interactions with their host stars and secular interactions with additional planets. This motion manifests as periodic shifts in the timing between transits which may be detectable via high-precision and long-baseline transit- and occultation-timing measurements. Detecting precession and attributing it to a particular process may constrain the tidal responses of planets and point to the presence of otherwise undetected perturbers. However, over relatively short timescales, orbital decay driven by the same tidal interactions can induce transit-timing signals similar to the precession signal, and distinguishing between the two processes requires robust assessment of the model statistics. In this context, occultation observations can help distinguish the two signals, but determining the precision and scheduling of observations sufficient to meaningfully contribute can be complicated. In this study, we expand on earlier work focused on searches for tidal decay to map out simple metrics that facilitate detection of precession and how to distinguish it from tidal decay. We discuss properties for a short-period exoplanet system that can maximize the likelihood for detecting such signals and prospects for contributions from citizen-science observations.

Figures (10)

• Figure 1: Examples of precessionally driven transit-timing departures, $\Delta t$, from a strictly periodic Keplerian orbit. (a) Precession driven by tidally induced distortion of a short-period gas giant with a sidereal period $P_{\rm s}$, an assumed mass $M_{\rm p} = 1$ Jupiter mass (${\rm M_{Jup}}$), an orbital eccentricity $e = 0.003$, and a range of Love numbers $k_2$. The Love number depends on the planet's internal structure and tidal response -- see 2009ApJ...698.1778R for additional details. (b) Precession for the smaller, inner planet driven by secular interactions in a two-planet, WASP-47-like system consisting of a $9$-Earth mass planet with a sidereal period $P_{\rm s}$ interior to a co-planar, $1\,{\rm M_{Jup}}$ planet with a semi-major axis $a = 0.05\,{\rm AU}$ and orbital eccentricity $e = 0.0028$, all around a star with a mass equal to the Sun's. Here, we have neglected tidal damping, which would act over much longer timescales than shown. The size of $\Delta t$ is insensitive to the interior planet's orbital eccentricity, as long as it is not identically zero.
• Figure 2: (a) The blue dots represent synthetic transit-timing data based on (but not actually using) data reported in 2020ApJ...888L...5Y. For this synthetic dataset, we took the orbital epochs of transit observations of WASP-12 b and generated a synthetic precession ephemeris according to Equation \ref{['eqn:transit_precession_ephemeris']}. Into that dataset, we injected Gaussian noise with a timing scatter equal to the average timing scatter for the data reported in 2020ApJ...888L...5Y. The precession ephemeris parameters used were $T_0 =2456305.45488\,{\rm BJD}$, $P_{\rm s} = 1.091419633\,{\rm days}$, $e = 0.00310$, $d\omega/dE = 0.000984\,{\rm rad\ orb^{-1}}$, and $\omega_0 = 2.62\,{\rm rad}$. The orange line shows the best-fit precession model for the whole dataset. (b) Evolution of $\Delta {\rm BIC}$ comparing a linear and a precession model fit to a precession ephemeris. $\Delta {\rm BIC} > 0$ favors the precession over the linear model. The solid, blue line shows the direct numeric estimate of $\Delta {\rm BIC}$, while the dashed, orange line shows implementation of the quasi-analytic estimate, Equation \ref{['eqn:Delta_BIC_lin_fit_vs_prec_fit_to_prec_ephem']}. The shaded blue and orange regions show the range of $\Delta {\rm BIC}$ values expected from random variations.
• Figure 3: (a) The blue dots represent a synthetic transit-timing data based on the same approach as used in Figure \ref{['fig:Testing_Linear_Fit_to_Precession_Ephemeris']}. The symbols have the same meaning as in that figure. (b) Evolution of $\Delta {\rm BIC}$ comparing a quadratic and a precession model, both fit to a precession ephemeris. $\Delta {\rm BIC} > 0$ favors the precession over the quadratic model. The solid blue line shows the direct numeric estimate of $\Delta {\rm BIC}$, while the dashed orange line shows implementation of the quasi-analytic estimate, Equation \ref{['eqn:Delta_BIC_quad_to_prec']}. The shaded blue and orange regions show the range of $\Delta {\rm BIC}$ values expected from random variations.
• Figure 4: (a) The blue dots show the real transit timing data from 2020ApJ...888L...5Y for WASP-12 b. The orange line shows a precession model fit to those data. (b) Evolution of $\Delta {\rm BIC}$ comparing a precession and a quadratic model, both fit to the real data in panel (a); $\Delta {\rm BIC}>0$ indicates a preference for a quadratic model. On the basis of these transit-timing, occultation timing, as well as radial velocity data, 2020ApJ...888L...5Y concluded that the transit-timing variations seen for WASP-12 b result from tidal decay, so we consider the example shown here to represent a quadratic ephemeris. No closed-form quasi-analytic expression is available to estimate $\Delta {\rm BIC}$ for fitting a precession model to a quadratic ephemeris, so panel (b) only includes what we have called in previous figures the "Numeric" estimate, along with the estimated range of variation (shaded blue).
• Figure 5: Exploring the impact of occultation timing data on model-fitting. (a) The blue dots show the transit-timing data from Yee et al. (2020) for WASP-12 b, while the black diamonds show the occultation-timing data. The solid, blue line shows a quadratic (tidal decay) model fit to both datasets, while the solid, orange line shows a transit-timing precession model fit to both datasets. The dashed, orange line shows the model occultation times for the same precession parameters. Because the occultation times have much larger error bars, the best-fit model tolerates a larger disagreement between the observed and the modeled occultation times. (b) The evolution of $\Delta{\rm BIC}$ for comparing the quadratic and the precession model both fit to the mounting data. The blue line shows the the evolution considering only the transit data (the blue shaded region shows the expected range), while the orange line shows the evolution considering both the transit and occultation data (again, the orange shaded region shows the range). $\Delta{\rm BIC} > 0$ favors tidal decay over precession. (c) Evolution of the eccentricity metric (Equation \ref{['eqn:eccentricity_metric']}). The blue line shows the direct, numeric result, while the orange line shows the approximate quasi-analytic approach (Equation \ref{['eqn:anal_eccentricity']}).