ARDENT: Dynamical Detection Limit Refinement

• The paper introduces ARDENT, an algorithm that integrates data-driven detection with N-body stability tests to refine exoplanet detection limits.
• It employs analytical criteria and fast N-body simulations to systematically identify and exclude dynamically unstable orbital configurations.
• ARDENT enhances RV survey completeness by yielding more accurate planet occurrence rates and guiding targeted follow-up observations.

The Algorithm for the Refinement of Detection Limits via N-body Stability Threshold (ARDENT) systematically advances the assessment of planetary system architectures by computing not only the sensitivity of radial velocity (RV) data to unseen planets, but also enforcing stringent dynamical constraints derived from N-body stability criteria. By integrating data-driven completeness with analytic and fast numerical stability evaluations, ARDENT yields detection limits that reflect both observational capacity and the physical impossibility of certain planetary orbits, leading to a more rigorous and informative completeness landscape for exoplanet surveys.

1. Data-driven Detection Limits in Radial Velocity Surveys

Detection limits in RV planet searches are traditionally established via injection–recovery experiments. Synthetic planetary signals, spanning a grid of test masses (or RV semi-amplitudes, $K$) and orbital periods ($P$), are sequentially inserted into the observed (model-subtracted) RV time series. A recovery algorithm—typically involving the generalized Lomb–Scargle periodogram or similar detection metrics—quantifies the fraction of detectable planets above given mass and period thresholds. Conventionally, the 95% detection sensitivity curve ($\mathcal{P}_d \approx 1$) in the $P$–$M$ plane demarcates where additional planets can be robustly excluded based solely on data completeness.

Crucially, this approach ignores the possibility that a hypothetical planet might render the system dynamically unstable, regardless of its observational detectability. Such omissions lead to inflated estimates of system capacity and obscure genuine correlations between system architecture and occurrence rates (Stalport et al., 16 Sep 2025).

2. Dynamical Constraints: Stability Probability and its Role

ARDENT introduces a dynamical probability $\mathcal{P}_s$, quantifying the fraction of allowed orbits that remain dynamically viable in the presence of additional planets. The total probability of missing a planet at $(P, M)$ is then

$$\mathcal{P}_{\text{miss}} = (1 - \mathcal{P}_d) \times \mathcal{P}_s.$$

Thus, only those orbits which are both undetectable and dynamically allowed contribute to the possibility of a missed detection. If $\mathcal{P}_s = 0$—for example, due to instability—the region is excised from the completeness diagram, regardless of data-driven sensitivity.

To evaluate $\mathcal{P}_s$, ARDENT systematically explores the period–mass grid below the $\mathcal{P}_d=1$ curve. For each configuration, the planet mass is incremented until crossing a threshold where orbital stability is lost. The corresponding mass boundary yields the "dynamically refined" detection limit.

3. Analytical and Numerical Stability Criteria

ARDENT employs a hierarchical two-tiered stability analysis.

A. Analytical Constraints

1. Orbit Crossing: For adjacent planet pairs $(i, o)$, instability is immediately flagged if their orbits satisfy

$a_i (1 + e_i) \geq a_o (1 - e_o),$

unless protected by a mean-motion resonance (MMR). This criterion rapidly eliminates obviously unstable orbital architectures.

1. Angular Momentum Deficit (AMD) Framework: The Angular Momentum Deficit (AMD) quantifies orbital excitation:

$$\text{AMD} = \sum_{k} \Lambda_k [1 - \sqrt{1 - e_k^2} \cos i_k],$$

with $\Lambda_k = M_k \sqrt{G M_* a_k}$. System stability is evaluated using critical values $C_c^H$ (Hill regime) and $C_c^{\text{MMR}}$ (near resonances). The system is stable if

$$\beta_H = (\text{AMD}/\Lambda_2)/C_c^H < 1 \quad \text{(Hill regime)},$$

or unstable if a similar inequality is violated in the MMR regime. These tests efficiently triage most planet pairs.

B. Fast N-body Integrations

When analytic results are inconclusive, ARDENT executes short-timescale simulations using the WHFast symplectic integrator (via REBOUND). Each system is evolved over $10^4$ orbits of the outermost planet; the drift

$$\Delta a = \underset{i}{\max} \ |a_{2,i} - a_{1,i}|/a_{0,i}$$

(for $a_{k,i}$ the median semimajor axis over two integration epochs) is compared against a calibrated threshold (default $\Delta a_\text{max} \approx 0.025\%$). Exceedance indicates unresolved instabilities (e.g., secular chaos or resonance overlap) even when close encounters do not occur in the integration window (Stalport et al., 16 Sep 2025).

4. Refinement of RV Survey Completeness and Exoplanet Architectures

By imposing $\mathcal{P}_s$ in addition to $\mathcal{P}_d$, ARDENT excludes not just undetected planets but orbits that cannot exist stably. This restricts planet parameter spaces to those consistent with both the data and dynamical feasibility. For planetary system occurrence statistics, this approach yields:

• More accurate planet occurrence rates (especially for assessing correlations or anti-correlations, such as those between cold giants and inner super-Earths/sub-Neptunes).
• Guidance for follow-up strategies, refining period/mass intervals where additional planets may plausibly reside.
• A framework for contextualizing Solar System architecture with respect to exoplanetary system diversity.

5. Illustrative Application: The TOI-1736 System

In TOI-1736, comprising a 7-day sub-Neptune and a 570-day eccentric cold giant, ARDENT’s analysis of SOPHIE RV data proceeds as follows (Stalport et al., 16 Sep 2025):

• The standard 95% detection limit is computed via injection–recovery in the residual RVs.
• For each trial period and mass below this limit, the package applies the orbit crossing/AMD analytic tests, supplementing with N-body integrations where warranted.
• The resulting dynamical detection limit demonstrates that orbits beyond 150 days for an additional planet are destabilized by the gravitational influence of the cold giant; i.e., $\mathcal{P}_s = 0$ for $P > 150$ d.

The exclusion of long-period orbits substantially improves the "completeness" estimate—the region of parameter space where RV data genuinely rule out planets—by leveraging the underlying dynamical constraints of the system.

6. Algorithmic Integration and Usage

ARDENT is implemented as an open-source Python package and is designed to be broadly applicable across RV-detected systems. Its integration of both analytical and rapid numerical stability tests ensures computational efficiency while maintaining rigor. The algorithmic core—sequentially invoking orbit crossing, AMD, and drift-based N-body criteria—can be represented in the following logical sequence (Editor’s term):

for period in period_grid: for mass in mass_grid: # 1. Data-driven detection: is planet detectable? if not detectable(period, mass): # 2. Dynamical tests: if not orbit_crossing(planets, [period, mass]): if not AMD_instability(planets, [period, mass]): if not Nbody_drift_unstable(planets, [period, mass]): record_viable_orbit(period, mass)

Here, detectable, orbit_crossing, AMD_instability, and Nbody_drift_unstable represent clauses corresponding to the hierarchy outlined above.

7. Scientific Implications and Broader Significance

ARDENT fundamentally enables the distinction between observationally permitted and physically possible planetary system configurations. By enforcing the requirement: $$\mathcal{P}_{\text{miss}} = (1-\mathcal{P}_d)\,\mathcal{P}_s,$$ exoplanet demographic analyses can, for the first time, systematically exclude both undetectable and dynamically implausible planetary companions—greatly enhancing scientific inference on architecture, evolution, and planet formation.

In empirical applications, such as for TOI-1736, this yields sharpened constraints: additional planets cannot exist at long orbital periods due to secular and resonant destabilization. The resulting completeness maps thus reflect true astrophysical limitations, not merely observational ones, providing a model for rigorous completeness assessment in all RV exoplanet surveys (Stalport et al., 16 Sep 2025).

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ARDENT’s synthesis of data-driven and dynamical analyses sets a new standard for detection-limit refinement, ensuring that completeness claims are grounded in both experiment and the fundamental physics of gravitational systems.