[Paper Review] Sterile and Fertile Planetary Systems - Statistical Analysis of Multi-Planet Systems in Kepler's data

Amir Weissbein, Elad Steinberg+1 more|arXiv (Cornell University)|Mar 27, 2012

Stellar, planetary, and galactic studies3 citations

TL;DR

This study analyzes Kepler multi-planet systems to test whether planet occurrence is an independent process, finding that a Poisson model with planar systems fails to explain the observed excess of systems with four or more transiting planets. The key result is that additional correlations—either mutual planetary formation enhancement or variable planetary system fertility—are required, with radial distribution dependencies in multi-planet systems indicating non-uniform formation efficiency across stars.

ABSTRACT

The Kepler mission has discovered a large number of planetary systems. We analyze the implications of the discovered single/multi-exoplanet systems from Kepler's data. As done in previous works, we test a simple model in which the intrinsic occurrence of planet is an independent process, and with equal probability around all planet producing stars. This leads to a Poisson distribution for the intrinsic number of planets around each host. However, the possibility of zero/low mutual inclination is taken into account, creating a correlation between detecting different planets in a given stellar system, leading to a non Poisson distribution for the number of transiting planets per system. Comparing the model's predictions with the observations made by Kepler, we find that the correlation produced by planarity is insufficient and a higher correlation is needed; either the formation of one planet in the system enhances the likelihood of other planets to form, and/or that some stars are considerably more fertile than others. Kepler's data presents evidences that both correlations might play a part, in particular a significant dependency in the radial distribution of planets in multi-planet systems is shown. Followup observations on Kepler planet's hosts can help pinpoint the physical nature of this correlation.

Motivation & Objective

• To test whether the observed distribution of transiting planets in Kepler systems can be explained by an independent planet occurrence process.
• To evaluate whether planarity alone—implying correlated transits due to low mutual inclinations—accounts for the observed excess of high-multiplicity systems.
• To determine whether additional correlations, such as mutual planetary formation enhancement or variable stellar fertility, are necessary to explain the data.
• To investigate the radial distribution of planets in multi-planet systems as a potential indicator of underlying correlations.
• To assess the robustness of the model to assumptions about data completeness, particularly the choice of maximum orbital radius for completeness.

Proposed method

• Models the intrinsic number of planets per star using a Poisson distribution with parameter λ, representing planet-forming efficiency.
• Incorporates planarity by assuming low mutual inclinations, which induces correlation between transits of multiple planets in a system.
• Derives the predicted distribution P(m) of systems with m transiting planets under the planar independent model, dependent only on λ.
• Compares model predictions to Kepler data using the observed multiplicity distribution (e.g., 27 systems with four transiting planets).
• Introduces a second parameter C to account for a fraction of stars being 'fertile' (planet-producing) and others 'sterile', to test if variable stellar fertility explains the data.
• Performs statistical comparison using likelihood ratios to assess model fit, and evaluates sensitivity to the assumed completeness limit r_max.

Experimental results

Research questions

• RQ1 Does the observed distribution of transiting planets in Kepler systems follow a Poisson distribution under the assumption of independent planet occurrence?
• RQ2 Can the observed excess of high-multiplicity systems (e.g., four or more transiting planets) be explained by the correlation induced by planar orbital configurations alone?
• RQ3 Is there evidence for additional correlations in planet occurrence, such as mutual planetary formation enhancement or variable stellar fertility?
• RQ4 Do the radial separations between planets in Kepler systems deviate from expectations under an independent model, suggesting underlying dependencies?
• RQ5 How robust are the conclusions to assumptions about data completeness, particularly the choice of maximum orbital radius r_max?

Key findings

• The observed multiplicity distribution in Kepler data shows a significantly longer tail than predicted by a Poisson distribution, with 27 systems having four transiting planets versus only 2.4 expected.
• The planar independent model, while producing a longer tail than Poisson, still fails to explain the observed data, as the predicted number of high-multiplicity systems remains too low.
• The inclusion of a fertility parameter C (fraction of fertile stars) does not fully reconcile the model with observations, indicating that additional correlations are required.
• Analysis of radial separation ratios in Kepler planet pairs shows a significant deviation from synthetic populations under the independent model, suggesting mutual planetary influence or formation correlation.
• The data suggests that planet-forming efficiency (λ) is not constant across stars, with metallicity-based models offering a plausible but not definitive explanation for the observed deviations.
• The model’s conclusions are robust to changes in the completeness limit r_max, as likelihood ratios remain low for r_max > 45R*, indicating that the results are not sensitive to this choice.