the covariance matrix of the $\mu_{\alpha*}, \mu_{\delta}$, and $\varpi$.

For further analysis we did not use the full sphere with $\varpi \leq$ 2.4 mas, but constrained the volume around Praesepe by a cut in the z-plane as $|{Z - Z_c}| \leq 50$ pc where $Z_c$ is given in Eq. \[COPO\]. This gives a volume of 4.867$\times 10^7~\rm{pc}^3$. We also introduced a restriction in the tangential velocity plane by requiring $|\Delta V_{\parallel}|\leq $7 km s$^{-1}$ and $|\Delta V_{\bot}| \leq$7 km s$^{-1}$, a total area of 196 ${(\rm{km~s}^{-1})}^2$. These cuts are ample compared to the predicted extent of the model tails, and they reduced our selection to a sub-sample containing 78,163 stars which populate this search volume.

As we are interested in Praesepe and its possible tails, we also transform the spacial coordinates $X,Y,Z$ to a coordinate system $X_P,Y_P,Z_P$ (the subscript $P$ refers to Praesepe) with origin at the centre of the Praesepe cluster, where the $X_P$-axis points to the Galactic centre, $Y_P$-axis to the direction of Galactic rotation and the $Z_P$-axis to the Galactic North Pole. To achieve this we first went from the solar system bary-centric Galactic Cartesian coordinates $X,Y,Z$ to the Galactocentric cylindrical coordinate system $R,\theta,Z$ with $R_\odot$ = 