Photodynamical modeling of TOI-4504 reveals its deeply resonant state and similarity to GJ 876

Abstract

The K-dwarf TOI-4504 hosts two giant planets in 2:1 mean-motion resonance, with orbital periods of 41.3 days (planet d) and 82.8 days (planet c). They exhibit among the largest known absolute transit-timing variations, with respective peak-to-node amplitudes up to 5 and 3 days. Newer TESS data show that the previously non-transiting planet d has now precessed into transiting, and we derive updated system parameters with significant discrepancies with the discovery paper. The revised parameters place planets d and c deep in the resonance and close to or in the fully-relaxed limit-cycle state, with the resonant and secular modes interfering nonlinearly to induce non-zero relaxed free eccentricities which precess at the same rate as the forced eccentricities and the longitude of conjunctions, in turn enabling precise measurement of the full eccentricities and apsidal angles. We discuss the predictions of linear theory and how it can be used to understand the true state of the system revealed by N-body integrations, and more generally why it is that the posteriors of systems more compact than 2:1 tend to suffer from significant eccentricity degeneracy. We show that the extraordinary dynamical states of the giant pairs orbiting TOI-4504 and the M-dwarf GJ 876 are remarkably similar, in spite of the significant difference in their host-star masses, and discuss the implications for damping timescales during the relatively gentle formation process of Type II migration.

Figures (12)

• Figure 1: Photodynamical modeling of the transits of planets d (left) and c (right). The dots represent the observations. The black line shows the oversampled transit model. Each panel is centered at the MAP transit timing and labeled with the epoch number and the sector for TESS observations. For planet c, the dashed vertical lines delimit a transit duration of 8 hours, close to the maximum value during the observations.
• Figure 2: Posterior TTV predictions of planets d (blue) and c (orange) computed relative to a linear ephemeris (Table \ref{['table:results']}). A thousand random draws from the posterior distribution were used to estimate the median TTV values and their uncertainties (68.3% confidence interval). The upper panel shows the posterior TTV values and compares them with the individual transit-time determinations (Table \ref{['table:transit_times']}, error bars). In the lower panel, the posterior median transit-timing value was subtracted to emphasize the uncertainty in the distribution and facilitate a clearer comparison with the individually determined transit times.
• Figure 3: Posterior of the transit duration of planets d (blue) and c (orange). We used a thousand random draws from the posterior distribution to estimate the median transit duration and its uncertainty (68.3% CI). A duration of zero correspond with the planet not transiting.
• Figure 4: Orbital projection of the MAP model over the time-span of the superperiod for planet b (green), d (blue), and c (orange). The origin is the system barycenter, and the orbits are projected in the X-Z plane (system top view, movement is clockwise, the positive Z-axis points towards the observer). One orbit near $t_{\mathrm{ref}}$ is shown in black. The black circles mark the position of the star (size to scale) and the planets (enlarged by a factor of 10) at $t_{\mathrm{ref}}$.
• Figure 5: Stellar‑crossing paths of planets d and c as seen by the observer for the MAP model. The stellar disc is shown in gray, and the MAP planet radii are indicated by blue (planet d) and orange (planet c) disks. The upper panel displays the TESS light curve (gray dots) together with the MAP transit model, with the model color‑coded by time. The same color scale is used in the main panel for the crossing paths. The TESS sector is annotated in red.