Statistical deprojection of galaxy pairs

Aims. The purpose of the present paper is to provide methods of statistical analysis of the physical properties of galaxy pairs. We perform this study to apply it later to catalogs of isolated pairs of galaxies, especially two new catalogs we recently constructed that contain approximate to 1000 and approximate to 13 000 pairs, respectively. We are particularly interested by the dynamics of those pairs, including the determination of their masses. Methods. We could not compute the dynamical parameters directly since the necessary data are incomplete. Indeed, we only have at our disposal one component of the intervelocity between the members, namely along the line of sight, and two components of their interdistance, i.e., the projection on the sky-plane. Moreover, we know only one point of each galaxy orbit. Hence we need statistical methods to find the probability distribution of 3D interdistances and 3D intervelocities from their projections; we designed those methods under the term deprojection. Results. We proceed in two steps to determine and use the deprojection methods. First we derive the probability distributions expected for the various relevant projected quantities, namely intervelocity v(z), interdistance r(p), their ratio, and the product r(p)v(z)(2), which is involved in mass determination. In a second step, we propose various methods of deprojection of those parameters based on the previous analysis. We start from a histogram of the projected data and we apply inversion formulae to obtain the deprojected distributions; lastly, we test the methods by numerical simulations, which also allow us to determine the uncertainties involved.