\begin{table}%T6
%\centering
\small
  \caption{A model for the instrument bias of the \hipp{} catalogue, based on the analysis of \cite{lindgren1997}. The binary systems satisfying the above constraints are resolved with \hipp{} in our models. For the comparison between the observations and the simulated observations, we consider two sets of \hipp{} binaries: the visual binaries and the astrometric binaries. No orbital motion is detected for the (C) binaries; these are visually resolved and therefore technically visual binaries. The \hipp{} astrometric binaries contain the targets with (X), (O), (G), and optionally (S) entries. No difference between the latter categories is made for the comparison with the astrometric binaries. Binary systems that do not satisfy the constraints listed in this table remain undetected in our simulated observations for \hipp{}. We do not model the (V)-binaries (variability-induced movers; VIMs) and (S)-binaries (suspected non-single stars). Note that in our model we {\em overpredict} the number of binaries in categories (X), (O), and (S), as not all binaries with the properties above are detected by \hipp{} as such. \label{table: true_hipparcosbiases} }
  \begin{tabular}{lllcl}
    \hline\hline
    Constraints on $\rho$ and $\Delta H_{\rm p}$ & Period constraints & Solution & Symbol & Elements \\
    \hline
    $2 \leq \langle \rho \rangle \leq 100$~mas  or $\Delta H_{\rm p} > 4$   & $P \leq 0.1$~year      & Stochastic  & (X) & no \\
    $2 \leq \langle \rho \rangle \leq 100$~mas  or $\Delta H_{\rm p} > 4$   & $0.1 < P \leq 10$~year & Orbital     & (O) & yes \\
    $2 \leq \rho \leq 100$~mas  or $\Delta H_{\rm p} > 4$                   & $5 < P \leq 30$~year   & Acceleration& (G) & no \\
    $0.1 \leq \rho \leq 100$~arcsec and $\Delta H_{\rm p} \leq 4$              & $P > 30$~year       & Resolved    & (C) & no \\
    \multicolumn{2}{l}{Not modeled}                                                            & Suspected   & (S) & no \\
    \multicolumn{2}{l}{Not modeled}                                                            & VIM         & (V) & no \\
    \hline
  \end{tabular}
\end{table}