EMPEROR I. Exoplanet MCMC parallel tempering for RV orbit retrieval

TL;DR

EMPEROR advances exoplanet RV analysis by integrating Dynamic Nested Sampling and Adaptive Parallel Tempering MCMC to robustly explore multi-modal posteriors under diverse noise models such as Gaussian Processes and Moving Averages. The framework supports automated, Bayesian model comparison using evidence and information criteria, and its modular block structure enables flexible inclusion of dynamical stability priors and multiple parameterisations. Benchmark results across 51 Pegasi, HD 55693, and Barnard's star demonstrate improved sampling efficiency, effective disentanglement of stellar activity from planetary signals, and precise Earth-mass planet constraints in favorable cases. The open-source nature and automated, publish-ready outputs position EMPEROR as a powerful tool for RV exoplanet detection and characterization, with potential extensions to photometry and astrometry for population-level studies.

Abstract

We present EMPEROR, an open-source Python framework designed for efficient exoplanet detection and characterisation with radial velocities (RV). EMPEROR integrates Dynamic Nested Sampling (DNS) and Adaptive Parallel Tempering (APT) Markov Chain Monte Carlo (MCMC), supporting multiple noise models such as Gaussian Processes (GPs) and Moving Averages (MA). The framework enables systematic model comparison using statistical metrics, including Bayesian evidence ($\ln{\mathcal{Z}}$) and Bayesian Information Criterion (BIC), while providing automated, publish-ready visualisations. EMPEROR is evaluated across three distinct systems to assess its capabilities in different detection scenarios. Sampling performance, model selection, and the search for Earth-mass planets are evaluated in data for 51 Pegasi, HD 55693 and Barnard's Star (GJ 699). For 51 Pegasi, APT achieves an effective sampling increase over DNS by a factor 3.76, while retrieving tighter parameter estimates. For HD 55693 the stellar rotation $P_{\text{rot}}=29.72^{+0.01}_{-0.02}$ and magnetic cycle $P_{\text{mag}}=2557.0^{+70.1}_{-36.7}$ are recovered, while demonstrating the sensitivity of $\ln{\mathcal{Z}}$ to prior selection. For Barnard's star, several noise models are compared, and the confirmed planet parameters are successfully retrieved with all of them. The best model shows a period of 3.1536$\pm$0.0003~d, minimum mass of 0.38$\pm$0.03 M$_{\rm{\oplus}}$, and semi-major axis of 0.02315$\pm$0.00039~AU. Purely statistical inference might be insufficient on its own for robust exoplanet detection. Effective methodologies must integrate domain knowledge, heuristic criteria, and multi-faceted model comparisons. The versatility of EMPEROR in handling diverse noise structures, its systematic model selection, and its improved performance make it a valuable tool for RV exoplanetary studies.

Figures (12)

• Figure 1: Three distinct steps of the code, shown in red: pre-processing, where user inputs and data are read; run compilation, where the code is executed; and post-processing, where statistical measures are calculated. The purple blocks with dashed arrows represent the optional step of repeating the loop. Blue highlights the main stages of each step, and available alternatives are displayed in green.
• Figure 2: 51 Peg Keplerian best fit model with reddemcee. Top: LICK RVs phase-folded to the period with the best-fit model (black line). Bottom: Residuals. Right: Histograms of the observations and residuals.
• Figure 3: 51 Peg corner plot of Keplerian parameters. Period, semi-amplitude, and eccentricity are well-defined narrow Gaussians. The angular parameters, $M_0$ and $\omega$, appear as wide Gaussians, highly correlated with each other, which is the case for $e=0$.
• Figure 4: HD 55693 periodogram for TERRA1 data. Descending, RVs, $S$-Index, FWHM, BIS, and window function. FAP lines included for 10%, 1%, and 0.1%, in dashed red, dotted purple, and dotted blue, respectively. Circle markers show the five periods with the greatest power, coloured by FAP region. The orange coloured region corresponds to $P_{\mathrm{rot}}=27.4\pm3.2$ and the green one to $P_{\mathrm{mag}}=2403^{266}_{-218}$.
• Figure 5: HD 55693 correlogram for TERRA1 data. Shows the Pearson Correlation coefficient ($\rho$) between RVs, $S$-Index, FWHM, and BIS. Diagonal displays the samples distribution. RV presents significant correlation with $S$-Index $\rho$=0.8 and BIS $\rho$=0.56. $S$-Index with BIS have $\rho$=0.76, suggesting they describe the same physical phenomena.