8300 pc [@2013IAUS..289...29G] . Then we transformed back into the cluster-centric Galactic Cartesian system $X_P,Y_P,Z$, and finally $Z_P$ = $Z -Z_c$. Note that this transformation is not necessary for the way we determine over-densities, it only puts Praesepe into the centre of the volume we consider.

A general method to find over-densities {#ITT}
---------------------------------------

In Paper I we cut a small rectangle in the tangential velocity plane where we suspected the signature of the tails, and then looked for over-densities in 3-D space. Here we follow a different, more general approach: we search simultaneously for over-densities in 5-D space. The main purpose is to separate the over-densities such as clusters or moving groups (signal) from the local Galactic background (noise). In other words, for each star in the sub-sample of 78,163 stars we want to decide, if it belongs to the general Galactic field, or to an over-density significantly above the general background. We make the decision by counting neighbours of this star in a 5-D neighbourhood and compare this with the expected background counts where the background is modelled by a Poisson distribution. We make the following neighbourhood definition: a star $j$ with 5-D coordinates $(X_{P_{j}},Y_{P_{j}},Z_{P_{j}},\Delta V_{\parallel_{j}},\Delta V_{\bot_{j}})$ is 