Testing mass-radius relations using Gaia distances as constraints in Machine Learning algorithms

Abstract

White dwarfs (WDs) are the endpoint of evolution for over 90% of the stars in our Galaxy. They serve as essential instruments for studying various phenomena. For example, 25 −50% are metal polluted, however the timescales that elements heavier than He remain in their surface are ex tremely short (i.e., they are self-cleaning). The contamination can be explained by recent-or current- accretion of planetary remnants onto the surface of the star, which means we can study the fate of planetary systems by analysing the chemical abundances of WDs. Not only this, but because they no longer produce energy through nuclear fusion, they cool gradually in well known sequences, which means we can determine their age, and with this, the age of the Milky way. Both require an accurate determination of their main atmospheric parameters: effec tive temperature (Teff) and surface gravity (log(g)). Mass-Radius (M-R) relations are necessary to determine the evolution and cooling of WDs, which in turn makes them necessary when trying to determine the atmospheric parameters mentioned above; they are determined by the degenerate nature of the star. The two main M-R relations are Montreal and La Plata. There have been efforts in constraining these M-R relations observationally, but the conclusions are not straightforward, because of the dispersion of the objects (they are small, therefore faint and difficult to observe) and the dependence of the M-R relations in multiple parameters. Fortunately, ongoing and upcoming surveys are providing more observations of WDs, which should help with increasing the size of the sample. The problem now is how we analyse the data. Current techniques to fit a model atmosphere to a WD are time consuming and very computationally expensive, because they require numerically solving the differential equations of the stellar structure for each parameter combination, until we find the best fit by minimizing a metric like χ2. To confront this problem, we implement Machine Learning models to predict the atmospheric parameters of 262 hydrogen WDs, using ultraviolet spectroscopy from the Hubble Space Telescope. We use two decision tree-based ensemble algorithms: Random Forest (RF) and Extreme Gradient Boosting (XGBoost). We train the algorithms with synthetic atmosphere models. In order to compare the M-R relations mentioned above, we take two precautions: (1) In order to have model independent evaluation metrics, we introduce the distance as a third variable to predict, since we can compare this result to the Gaia distances, (2) We create two training sets, with synthetic fluxes created with the same Teff, log(g), and distance, but that represent WDs with radii according to each M-R relation. We find that the best performing model on observational data is XGBoost for both M-R relations. With the model we find a distance Mean Absolute Error of ∼ 12.5 pc on average for the synthetic models and of ∼ 19.5 pc for the observational data. We find a mean difference (± st. deviation) between Montreal and La Plata of ∼145±738 K for Teff, ∼ −0.019±0.21 [dex] for log(g), and ∼−0.46±15.72 pc for the distance. We also confirm that there is a problematic zone in the M-R relations, a fact that was already proposed in the literature, since our highest distance errors and the worst fits to the observational spectra are located at Teff≲ 25000 K and log(g)≲ 8.0 [dex]. We compare our resulting parameters with previous works that use the traditional fitting techniques, and while we find that our errors are significantly higher, we obtain an accurate mass distribution with a mean mass of 0.60 ± 0.18M⊙ for Montreal and 0.62 ± 0.22M⊙ for La Plata. We find that our results are affected by observational bias, and further analyses should be conducted on volume limited samples, with objects with distances < 50 pc. Given the error we obtain with our ML model, we believe it is still preferable to fit spectroscopic data with the conventional techniques in samples < 1000 objects. However, our algorithm still has room for improvement, and future work involves analysing the effects of feature sampling (that is, the flux points that serve as input), size of training set (to improve the resolution of the parameter space), and upgrade the computational capabilities, to test if it is possible to improve the performance of the model and obtain lower errors that are comparable to the results of using the current available techniques.