2013A&A...550A..44S

The Gaia catalogue will contain observations and physical parameters of a vast number of objects, including ultra-cool dwarf stars, which we define here as stars with a temperature below 2500K. We aimed to assess the accuracy of the Gaia T_eff and log(g) estimates as derived with current models and observations. We assessed the validity of several inference techniques for deriving the physical parameters of ultra-cool dwarf stars: Gaussian processes, support vector machines, k-nearest neighbours, kernel partial least squares and Bayesian estimation. In addition, we tested the potential benefits of data compression for improving robustness and speed. We used synthetic spectra derived from ultra-cool dwarf models to construct (train) the regression models. We derived the intrinsic uncertainties of the best inference models and assessed their validity by comparing the estimated parameters with the values derived in the bibliography for a sample of ultra-cool dwarf stars observed from the ground. We estimated the total number of ultra-cool dwarfs per spectral subtype, and obtained values that can be summarised (in orders of magnitude) as 400000 objects in the M5-L0 range, 600 objects between L0 and L5, 30 objects between L5 and T0, and 10 objects between T0 and T8. A bright ultra-cool dwarf (with T_eff=2500K and log(g)=3.5) will be detected by Gaia out to approximately 220pc, while for T_eff=1500K (spectral type L5) and the same surface gravity, this maximum distance reduces to 10-20pc. We found the cross-validation RMSE prediction error to be 10K for regression models based on the k-nearest neighbours and 62K for Gaussian process models in the faintest limit (Gaia magnitude G=20). However, these values correspond to the evaluation of the regression models with independent test sets of synthetic spectra of the same model families as used in the training phase (internal errors). For the k-nearest neighbours model, this seems an overly optimistic error estimate due to the use of a dense grid of examples in the training set, together with a relatively high signal-to-noise ratio for the end-of-mission data. The RMSE of the prediction deduced from ground-based spectra of ultra-cool dwarfs simulated at the Gaia spectral range and resolution, and for a Gaia magnitude G=20 is 213K and 266K for the models based on k-nearest neighbours and Gaussian process regression, respectively. These are total errors in the sense that they include the internal and external errors, with the latter caused by the inability of the synthetic spectral models (used for the construction of the regression models) to exactly reproduce the observed spectra, and by the large uncertainties in the current calibrations of spectral types and effective temperatures. We found maximum-likelihood methods (minimum χ², k-nearest neighbours, and Bayesian estimation with flat priors) to be biased in the L0-T0 range in that they systematically assign a temperature around 1700K. Finally, the likelihood landscape is significantly multimodal in spectra with realistic noise.