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Characterization of the known T type dwarfs towards the Sigma Orionis cluster

(Abridged) A total of three T type candidates (SOri70, SOri73, and SOriJ0538-0213) lying in the line of sight towards Sigma Orionis were characterized by means of near-infrared photometric, astrometric, and spectroscopic studies. H-band methane images were collected for all three sources and an additional sample of 15 field T type dwarfs using LIRIS/WHT. J-band spectra of resolution of ~500 were obtained for SOriJ0538-0213 with ISAAC/VLT, and JH spectra of resolution of ~50 acquired with WFC3/HST were employed for the spectroscopic classification of SOri70 and 73. Proper motions with a typical uncertainty of +/-3 mas/yr and a time interval of ~7-9 yr were derived. Using the LIRIS observations of the field T dwarfs, we calibrated this imager for T spectral typing via methane photometry. The three SOri objects were spectroscopically classified as T4.5+/-0.5 (SOri73), T5+/-0.5 (SOriJ0538-0213), and T7$^{+0.5}_{-1.0}$ (SOri70). The similarity between the observed JH spectra and the methane colors and the data of field ultra-cool dwarfs of related classifications suggests that SOri70, 73, and SOriJ053804.65-021352.5 do not deviate significantly in surface gravity in relation to the field. Additionally, the detection of KI at ~1.25 microns in SOriJ0538-0213 points to a high-gravity atmosphere. Only the K-band reddish nature of SOri70 may be consistent with a low gravity atmosphere. The proper motions of SOri70 and 73 are measurable and are larger than that of the cluster by >3.5 sigma. The proper motion of SOriJ0538-0213 is consistent with a null displacement. These observations suggest that none of the three T dwarfs are likely Sigma Orionis members, and that either planetary-mass objects with masses below ~4 MJup may not exist free-floating in the cluster or they may lie at fainter near-infrared magnitudes than those of the targets (this is H>20.6 mag), thus remaining unidentified to date.