The Illusory Precision of TTV Masses: Hidden Solutions Behind Kepler-9's Tight Mass Ratio

TL;DR

TTV data for Kepler-9 planets can be explained by many degenerate mass solutions, highlighting the limitations of current sampling methods and the importance of considering multiple solutions in planetary mass estimation.

Contribution

Developed an efficient mode-first search algorithm to identify multiple TTV solutions and demonstrated the degeneracy in planetary mass estimates for Kepler-9.

Findings

TTV solutions for Kepler-9 span broad mass ranges.

Mass ratio between planets follows a linear relationship in degenerate solutions.

Global convergence in TTV posterior sampling is fundamentally unachievable with Markovian algorithms.

Abstract

Transit timing variations (TTV) are considered a tool for constraining the masses of transiting planets in the absence of radial-velocity data. Although theoretical studies have long revealed that TTV mass determinations intrinsically suffer from degeneracies, existing analyses of TTV data typically report a single-mode solution under a model with a specified number of planets. This is because fitting TTV curves in the high-dimensional solution space of TTV posterior is extremely challenging; even locating a single solution requires substantial computational resources. We developed an efficient mode-first searching algorithm that can locate multiple solutions in a single MCMC run. We applied this algorithm to Kepler-9 b and c, which have the highest-quality TTV data. We found that the observed TTV can be reproduced by many combinations of planetary masses spanning a broad range, rather than the previously assumed precise determination. The mass of Kepler-9 b can range from 31.6 to 47.1 $M_{\oplus}$, while that of Kepler-9 c can range from 21.8 to 32.3 $M_{\oplus}$, and even more broadly under looser constraints. These degenerate solutions follow a linear relationship under a tight mass ratio between the two planets, consistent with previous theoretical predictions. Furthermore, we demonstrate that achieving a globally converged posterior distribution for Kepler-9's TTV is impossible using a sampling algorithm that preserves the Markovian property. This underscores the need for caution when interpreting results from sampling algorithms that lack mathematical guarantees of global convergence.