Global and Local Three-dimensional Studies of The Residual Vector Field from 2MASS and Hipparcos-2 Catalog

[EN] The Gaia mission will provide a six-parameter solution for millions of stars, including a tridimensional map of our Galaxy. The estimation of distances has been made for the Tycho-Gaia Astrometric Solution (TGAS), while to contrast the proper motions it is interesting to consider positions from the different Gaia Data Release with older ones given in ground-based massive catalogs. This process has been followed to build, for example, the PMA catalog using the 2MASS. Our aim is to improve the positions of this catalog (although the process is applicable to any other). The first stage, presented here, consists of carrying out a three-dimensional study using vector spherical harmonics (VSH) development of the systematisms in position for the stars common with Hipparcos-2; we take into account the distances, magnitudes, and spectral types. To this aim, we use linear polynomial regression of first order that fits vector fields and the derivatives of their components. We verify that the coefficients of the developments of first order have different behavior according to the characteristics of stars and distances. To deepen the study, we focus on the conservative component of the field, applying the Helmholtz theorem. Each potential function is obtained solving a Poisson equation on the sphere, after finding the divergence of the corresponding vector field. Both vector and potential fields present patterns, at certain points, that depend on the three considered parameters (distance, magnitude, and spectral type); their sources and shrinks correspond to maxima and minima. In this sense, we observe that these critical points are also critical points of the surface that represents the VT magnitude of Tycho-2, which makes sense because this catalog was used in the reduction of 2MASS positions. Finally, we selected some stars near the critical points of the vector fields and apply the adjustments obtained in the previous sections. 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