The Orbital Eccentricity--Radius Distribution for Warm, Single Planets in TESS

Abstract: We characterize the radius-dependent eccentricity distribution of 347 warm (P = 8-200 days) systems with only one transiting planetary candidate identified during Sectors 1-69 of the TESS mission. Using the ``photoeccentric effect'' in a hierarchical Bayesian framework, we first model the population using discrete planetary size bins (sub-Neptunes, sub-Saturns, and Jovians). We then develop a continuous mixture model with weights governed by a logistic sigmoid function of radius. We find that the warm-single population is best described by two components: a dominant low-eccentricity mode ( <e_low> = 0.070-0.068+0.026) and a secondary dynamically excited mode (<e_high> = 0.616-0.075+0.091). The fraction of planets belonging to this high-eccentricity component increases strongly with planet radius, characterized by a transition at a break radius of R_br = 9.8-1.1+1.4 R_e. This trend places warm sub-Saturns predominantly on the same low-eccentricity track as sub-Neptunes. In contrast, warm Jovians (8--16 R_e) are frequently eccentric, with 59+-13% of the population in the high eccentricity mode. We detect this bimodality at >4sigma, providing statistically significant evidence that warm gas giants are sculpted by two distinct pathways, or a single mechanism with subsequent eccentricity excitation. Finally, we identify a non-negligible tail of highly eccentric sub-Neptunes (1--4 R_e), which comprise 14.9-6.5+5.1% of the population, consistent with excitation by non-transiting external companions.

• The paper combines the photoeccentric effect with hierarchical Bayesian modeling of 347 warm, single-transiting TESS planets to measure their radius-dependent eccentricity distribution.
• The analysis identifies low- and high-eccentricity modes with means near 0.07 and 0.62, while the excited component grows sharply above a break radius of 9.8 Earth radii.
• Warm sub-Saturns resemble sub-Neptunes dynamically, whereas about 59% of warm Jovians belong to the excited population, motivating follow-up searches for unseen companions and improved selection modeling.

Fairnington et al. present the first homogeneous, all-sky characterization of the orbital eccentricity distribution of warm, single-transiting planets from TESS, spanning the full planet-radius range from sub-Neptunes to Jovians (2602.20015). The analysis combines the photoeccentric effect with hierarchical Bayesian modeling (HBM) and introduces a radius-continuous mixture model in which the eccentricity distribution is a two-component Beta mixture whose weights vary with planet size via a logistic sigmoid. The central result is that the warm-single population resolves into a low-eccentricity mode ($\mu_{\rm low}=0.070^{+0.068}_{-0.068}$... reported as $0.070_{-0.068}$) and a dynamically excited mode ($\mu_{\rm high}=0.616_{-0.075}$), with the excited component's share increasing sharply with radius and crossing at a break radius of $R_{\rm br}=9.8_{-1.1}\,R_\oplus$.

Sample construction

The starting catalog is the 6881 TOI candidates through Sectors 1–69. Cuts include $T_{\rm mag}<12$ (needed to resolve ingress/egress and break the $b$–$\rho_\star$ degeneracy), Gaia RUWE $<2$ to suppress false positives, orbital periods of 8–200 days (below 8 days tidal circularization becomes significant), single-transiting status, and removal of FP/FA/APC dispositions, leaving 620 targets. An additional SNR $>15$ requirement (implemented via a theoretical noise floor and an empirical one-hour noise metric) removes 182 candidates. Later, unresolved period aliases (17) and grazing or highly degenerate transits with $b\ge0.9$ in over half the posterior (47) are removed, yielding a final sample of 347 planets used in the hierarchical analysis. The authors note that removing period-aliased targets, while necessary for homogeneity, may exclude interesting long-period systems confirmed by ground-based follow-up (e.g., TOI-2134c).

Stellar densities and transit modeling

Independent true stellar densities $0.070_{-0.068}$0 are derived with astroARIADNE SED fitting mapped onto MIST isochrones, using Gaia DR3 parallax priors, Bayesian model averaging over five atmosphere grids, and empirical $0.070_{-0.068}$1–$0.070_{-0.068}$2 and mass–radius relations for $0.070_{-0.068}$3 K hosts. Transit light curves (SPOC and QLP SAP, shortest available cadence) are fit with PyMC/NUTS using quadratically limb-darkened models, fitting the "pseudo" density $0.070_{-0.068}$4 under a circular-orbit assumption in log space to mitigate the known $0.070_{-0.068}$5–$0.070_{-0.068}$6 covariance bias. TTV systems have individual transit times freed. The photoeccentric relation $0.070_{-0.068}$7 then yields joint $0.070_{-0.068}$8–$0.070_{-0.068}$9 posteriors per planet, sampled in $\mu_{\rm high}=0.616_{-0.075}$0 and $\mu_{\rm high}=0.616_{-0.075}$1. A KS test against a uniform distribution flags 13 unconstrained systems ($\mu_{\rm high}=0.616_{-0.075}$2), assigned $\mu_{\rm high}=0.616_{-0.075}$3; dropping them does not change the conclusions. For planets with published RV eccentricities, the photometric posteriors agree within uncertainties, with the notable exception of HIP 97166b, where the RV solution ($\mu_{\rm high}=0.616_{-0.075}$4) differs from the photoeccentric mode ($\mu_{\rm high}=0.616_{-0.075}$5), attributed to a high impact parameter ($\mu_{\rm high}=0.616_{-0.075}$6) mimicking a shortened duration — a concrete illustration of the method's main systematic.

Population-level eccentricity distribution

A single Beta distribution fit to the full sample gives a population mean of $\mu_{\rm high}=0.616_{-0.075}$7, but the two-component Beta mixture reveals the underlying structure: a dominant low-$\mu_{\rm high}=0.616_{-0.075}$8 component with weight $\mu_{\rm high}=0.616_{-0.075}$9 and mean $R_{\rm br}=9.8_{-1.1}\,R_\oplus$0, and a high-$R_{\rm br}=9.8_{-1.1}\,R_\oplus$1 component with $R_{\rm br}=9.8_{-1.1}\,R_\oplus$2 and $R_{\rm br}=9.8_{-1.1}\,R_\oplus$3. The likelihood includes a geometric transit-probability factor proportional to $R_{\rm br}=9.8_{-1.1}\,R_\oplus$4, conditioning on the transiting event; the authors are explicit that this corrects viewing-geometry bias but not survey selection functions, so the result is an "observed transiting" rather than intrinsic distribution. They also do not compute Bayesian evidences between the single-Beta and mixture models, using the mixture as a physically motivated description rather than a formally selected model class — a caveat worth noting when interpreting the "bimodality at $R_{\rm br}=9.8_{-1.1}\,R_\oplus$5" claim, which refers to the credibility that the high-$R_{\rm br}=9.8_{-1.1}\,R_\oplus$6 weight exceeds zero, not a model comparison.

In archetype radius classes (single-Beta fits), sub-Neptunes (1–4 $R_{\rm br}=9.8_{-1.1}\,R_\oplus$7, $R_{\rm br}=9.8_{-1.1}\,R_\oplus$8) and sub-Saturns (4–8 $R_{\rm br}=9.8_{-1.1}\,R_\oplus$9, $T_{\rm mag}<12$0) have means $T_{\rm mag}<12$1 and $T_{\rm mag}<12$2 respectively, while Jovians (8–16 $T_{\rm mag}<12$3, $T_{\rm mag}<12$4) reach $T_{\rm mag}<12$5, with a mixture high-$T_{\rm mag}<12$6 weight of $T_{\rm mag}<12$7 for giants.

The eccentricity–radius relation

The paper's methodological contribution is a three-stage hierarchical model treating each planet's radius as a latent variable and the low-$T_{\rm mag}<12$8 mixture fraction as a sigmoid function of $T_{\rm mag}<12$9. The inferred hyperparameters are $b$0, $b$1, $b$2, $b$3, $b$4, and $b$5.

Class	$b$6	$b$7	Mixture mean $b$8
Sub-Neptunes (1–4 $b$9)	0.851 ± 0.051	0.149 ± 0.065	0.155 ± 0.037
Sub-Saturns (4–8 $\rho_\star$0)	0.837 ± 0.055	0.163 ± 0.077	0.163 ± 0.037
Jovians (8–16 $\rho_\star$1)	0.41 ± 0.13	0.59 ± 0.13	0.381 ± 0.037

Two implications follow directly. First, the break radius lies above the sub-Saturn regime, so warm sub-Saturns predominantly share the low-$\rho_\star$2 track of sub-Neptunes; the authors argue this disfavors a naive "failed gas giant" picture in which sub-Saturns would inherit the dynamical signatures of giant-planet scattering. Second, roughly $\rho_\star$3 of warm Jovians occupy the excited component, consistent with Dong et al.'s warm-Jupiter bimodality (~55/45 split) and detected at $\rho_\star$4. Randomly dropping 30% of the sample and refitting both the discrete binned and continuous models reproduces the trend within 1$\rho_\star$5, and removing F-type hosts ($\rho_\star$6 K) preserves the small-planet result while suggesting possible stellar-type dependence in giant-planet eccentricities — an effect the sample size does not allow them to establish.

Dynamical interpretation and eccentric small planets

For warm giants, the high-$\rho_\star$7 fraction is consistent with either high-eccentricity migration channels (planet–planet scattering, Kozai–Lidov, secular chaos, or coplanar high-$\rho_\star$8 migration, the latter favored by the observed spin–orbit alignment of eccentric warm Jupiters) or in-situ scattering after disk dispersal. The paper notes that the Safronov-number scaling $\rho_\star$9 permits eccentricities of order unity for a typical warm Jupiter, so in-situ scattering alone can supply the observed eccentricities; the high-$<2$0 component mean also appears roughly constant with radius ($<2$1–0.72 across classes), which the authors flag as potentially unexpected.

The non-zero high-$<2$2 fraction among sub-Neptunes ($<2$3) motivates a target list of ten systems with $<2$4 and eccentricity posterior mode $<2$5, including the confirmed Kepler-1656b (inferred $<2$6, consistent with the RV value $<2$7). The favored interpretation is excitation by non-transiting external companions, testable through RV trends and TTVs, though the authors caution that false-positive probability is likely elevated in this subset.

Relation to prior work

The results unify earlier TESS analyses (warm Jupiters; warm sub-Saturns with a ~15% high-$<2$8 sub-population) and extend them down to the sub-Neptune regime within one framework. Relative to the Kepler-based radius trend of Gilbert et al., who found a transition near $<2$9 with selection-function corrections, the larger TESS break radius is attributed to differences in period coverage, sample composition, and the fact that the TESS analysis characterizes the observed transiting population, in which eccentric orbits carry enhanced transit probability. Kepler studies including shorter-period planets (where circularization operates) may also infer lower eccentricity scales than this strictly warm sample.

Limitations and open questions

Several limitations are conceded explicitly. The analysis does not model TESS survey completeness or vetting selection as functions of period, radius, and host properties, so the reported distributions are observed rather than intrinsic; injection–recovery forward modeling would be required for the latter. The sample mixes confirmed planets and candidates, so residual false positives cannot be excluded. Unresolved binaries and blended photometry can bias SED-based stellar densities and hence photoeccentric constraints, motivating high-resolution imaging and spectroscopic vetting of the sample. The mixture model is not formally selected against alternatives via evidence, and the stellar-type dependence of giant-planet eccentricities remains unresolved.

Conclusion

Using the photoeccentric effect and hierarchical Bayesian inference on 347 warm, single-transiting TESS planets, this work establishes that the warm-single eccentricity distribution is a radius-dependent mixture of a near-circular component and a dynamically excited component with mean $>15$0, with the transition occurring near $>15$1. Warm sub-Saturns are dynamically more akin to sub-Neptunes than to giants, while a majority of warm Jovians show signatures of excitation. The identified population of candidate eccentric sub-Neptunes provides a concrete follow-up program, and extending the framework to warm multi-transiting systems — and to intrinsic distributions via survey-selection forward modeling — remains the immediate open step.