A planet-host ratio relation to synthesize microlensing and transiting exoplanet demography from Roman

Technical Summary: A Planet–Host Ratio Relation to Synthesize Microlensing and Transiting Exoplanet Demography from Roman

Problem Statement
The upcoming NASA Nancy Grace Roman Space Telescope (Roman) is expected to detect approximately 1,400 cold exoplanets via gravitational microlensing and up to 200,000 hot, transiting planets across Galactic distances. A significant challenge in synthesizing demographic results from these two distinct detection methods lies in their differing sensitivities to planet bulk properties. Microlensing directly yields the planet–host mass ratio ($q$), while transits yield the transit depth ($\delta$, the square of the planet–host radius ratio). Traditional approaches to linking these samples rely on a planet mass–radius relation (MRR). However, MRRs suffer from significant scatter, are derived from small datasets (only $\sim$20% of known exoplanets have both mass and radius measurements), and often require assumptions about composition that may not hold for the full Roman sample. Consequently, a direct relation between the observables that are always measured ($q$ and $\delta$) is needed to enable consistent demographic forward-modelling without relying on ill-constrained mass or radius measurements for the majority of detections.

Methodology
The authors propose and test a Planet–Host Ratio Relation (PHRR) to directly couple microlensing and transit observables. The study utilizes a dataset of 908 confirmed exoplanets from the NASA Exoplanet Archive, selected through rigorous cuts to ensure data precision and physical plausibility. Selection criteria included:

• Adherence to the IAU definition of an exoplanet ($q < 2/(25 + \sqrt{621})$ and $M_p < 13 M_J$).
• Exclusion of planets with "super-iron" densities (inconsistent with pure iron equations of state).
• Removal of data points with large uncertainties in $\log q$, $\log \delta$, or host effective temperature ($T_*$), specifically requiring $\sigma_{\log X} \le 0.2$.
• Exclusion of planets orbiting large stars ($R_* > 2.5 R_\odot$) identified as outliers likely subject to Malmquist bias, and iterative sigma clipping to remove remaining statistical outliers.

The authors modeled the relationship $\delta(q, T_*, P)$ using orthogonal distance regression (ODR) to account for errors in both independent and dependent variables. They tested several candidate models, including:

1. Broken power laws in $q$ with breaks ($q_{br}$) that are either universal or dependent on $T_*$.
2. Power-law indices ($n$) that are either constant or vary with $T_*$.
3. Quadratic terms in $q$ to account for non-linear behavior.
4. Inclusion of orbital period ($P$) as a power-law term.

Model performance was evaluated using the Bayesian Information Criterion (BIC) to balance goodness-of-fit against model complexity.

Key Results

• Continuity of the PHRR: Unlike the MRR, which exhibits disjoint regimes for rocky and ice giant systems, the relationship between $\delta$ and $q$ is continuous across all planet scales.
• Optimal Model Structure: The BIC favors a model where transit depth scales as a two-stage power law in $q$, with a break at a characteristic mass ratio $q_{br}$. Both the break location and the power-law slopes are found to be power-law functions of the host effective temperature ($T_*$).
• Orbital Period Dependence: Including orbital period ($P$) significantly improves the model (lowering BIC scores by $\sim$6,000–10,000). The favored model indicates a weak inverse dependence on period ($\delta \propto P^{-0.10}$), suggesting planets at larger separations are slightly smaller, potentially due to atmospheric bloating effects in closer-in planets.
• Precision: The favored PHRR achieves a relatively uniform 50% relative precision in predicting $\delta$ across all $q$.
• Outliers: Approximately 5% of the initial sample showed strong over-prediction of transit depth. These were primarily associated with large host stars ($R_* > 2.5 R_\odot$), consistent with Malmquist bias, and were excluded from the final analysis.
• Comparison to Occurrence: The characteristic break mass ratio ($q_{br}$) in the PHRR shows a trend of decreasing with increasing $T_*$, similar to findings in planet occurrence studies, but the absolute values differ significantly from previous occurrence break measurements (e.g., Suzuki et al. 2016; Pascucci et al. 2018). No definitive link is established between the PHRR break and the occurrence break.

Significance and Claims
The paper claims that the PHRR offers a superior alternative to the MRR for synthesizing Roman's microlensing and transit datasets. The primary advantage is that the PHRR relies on parameters ($q$ and $\delta$) that are directly measured for all Roman detections, whereas direct mass or radius measurements will be available for only a fraction of the sample. By using a PHRR, the entire Roman exoplanet sample can be utilized in joint demographic studies.

The authors emphasize that their derived relation is purely empirical, reflecting a combination of intrinsic planet–host physics and observational selection biases inherent in the current transit and radial velocity-dominated sample. They caution that the specific PHRR for Roman may differ due to Roman's unique sensitivity to cold planets via microlensing and its ability to directly characterize host stars. However, the framework allows for the self-calibration of the PHRR using Roman's own sub-samples with secure host mass or radius measurements, enabling a convergent joint demographic analysis of the full exoplanet architecture.