Authors: A. Moya1, F. Zuccarino2, William Chaplin1, Guy R. Davies1

Empirical relations interconnecting different stellar characteristics are well known since early 1900, as the well-known Mass-Luminosity relation or the HR-Diagram.

Almost every time a technique or physical knowledge improved the stellar characterization, these relations were revised. With the recent increasing in the number of accurate determined eclipsing binaries, the irruption of asteroseismology as a very precise tool for studying the stellar structure and evolution, the publication of the first catalogues of stars characterized using interferometry, and the development of computational facilities for the massive data analysis, we have been able to carry out a revision of empirical relations for the estimation of the stellar mass and radius. Gaia’s data releases are also an opportunity for these relations since we will have, for the first time, accurate luminosities for millions of isolated stars.

In this work, we have gathered a total of 93 stars with very precise mass, radius, effective temperature, luminosity, gravity, density, and metallicity. We have searched for any accurate and precise linear relation estimating the stellar mass and/or radius as a function of any of the other variables. For doing so, we have used an error-in-variables algorithm to ensure the fair treatment of the variable uncertainties. We have studied hundreds of relations and selected the best ones. We present a total of 38 new or revised relations, those with a R2 statistic (don’t confuse with the stellar radius R) larger than 0.85, and an accuracy and precision better than 10% for almost all the cases. They cover barely all the possible combinations of observables, almost ensuring that, whatever list of observables available, there is at least one relation for estimating the stellar mass and radius. In addition, we present an empirical relation for the stellar density as a function of its surface gravity.

We have also taken the advantage of the very comprehensive and precise data sampling we have gathered for applying machine learning techniques with the objective of obtaining the best stellar masses and radii estimations possible. For doing so, we have trained a Random Forest model using a 70% of the Main-Sequence stars of our sampling. The reaming 30% has been used as testing group.

In Figs. 1 and 2 we show a comparison of the radius and mass estimations, respectively, obtained using this Random Forest model and the real testing values, where “real” means the values obtained using asteroseismology, eclipsing binaries and/or interferometry. The Standard deviation of the residuals is around 0.07 in both cases, offering a remarkable precision in the estimation of both stellar characteristics using only standard observables.

Fig. 1: Predicted stellar radius using the Random Forest model vs. the real one obtained using asteroseismology, eclipsing binaries or interferometry, for the stars in the test subsample

Fig. 2: Predicted stellar mass using the Random Forest model vs. the real one obtained using asteroseismology, eclipsing binaries or interferometry, for the stars in the test subsample.

The paper can be found in arXiv

1 School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK
2 Universidad Internacional de Valencia (VIU), E-46021 Valencia, Spain