text:
\section{Introduction}\label{section1}

The molecular cloud complex in the southern constellation of  Chamaeleon hosts one of the richest populations of T~Tauri stars in the Solar neighbourhood. It consists of three molecular clouds with angular sizes of a few degrees: Chamaeleon~I (Cha~I), Chamaeleon~II (Cha~II), and Chamaeleon~III (Cha~III). Star formation activity is restricted to Cha~I and Cha~II as no young stars have been detected in Cha~III \citep[see][]{Luhman2008_review}. The cloud morphology has been investigated in previous studies from star counts and extinction maps \citep[see e.g.][]{Gregorio-Hetem1988,Cambresy1997,Mizuno2001,Dobashi2005} revealing moderate extinction levels in Chamaeleon as compared to other nearby star-forming regions such as for example Ophiuchus \citep[][]{Cambresy1999}. The modest extinction, proximity, compact structure and isolated location with respect to other young stellar groups led to Chamaeleon being one of the most targeted regions over the past decades for the study of low-mass star formation.

The first young stellar objects associated to the Chamaeleon clouds were mostly classical T~Tauri stars (CTTSs) identified from objective prism surveys based on their strong H$\alpha$ emission \citep{Henize1973,Schwartz1977,Hartigan1993}. Deeper H$\alpha$ surveys conducted in the years that followed added a number of very low-mass members and brown dwarfs to the region \citep{Comeron1999,Comeron2000,Neuhauser1999}. \textit{ROSAT X-ray} pointed observations led to the discovery of a significant number of weak-emission-line T~Tauri stars (WTTSs) in this region \citep{Feigelson1993,Alcala1995,Alcala2000} and the first brown dwarf detected in \textit{X-}rays \citep{Neuhauser1998}. Additional members were later discovered from data collected with the \textit{XMM-Newton}  \citep{Stelzer2004,Robrade2007} and \textit{Chandra X-ray} observatories \citep{Feigelson2004}. Several optical and infrared surveys followed by ground-based spectroscopic observations expanded the census of Chamaeleon stars and confirmed the most likely members associated to the molecular clouds \citep[see e.g.][]{Hughes1992,Prusti1992,Cambresy1998,Persi2000,Vuong2001,Carpenter2002,Barrado2004,Comeron2004,LopezMarti2004,LopezMarti2005,Luhman2004a,Luhman2004b,Allers2007,Luhman2007}. The most recent census of the stellar population of Cha~I is given by \citet{Esplin2017} and contains 250~stars. Analogously, the list of 63~stars studied by \citet{Alcala2008} and \citet{Spezzi2008} represents the most complete sample of stars in Cha~II to date.

The distance to Chamaeleon has undergone extensive revision in recent decades. Early studies reported distance estimates to Cha~I in the range from 115 to 215~pc \citep{Grasdalen1975,Rydgren1980,Hyland1982,Schwartz1992}, and suggested that the distance to Cha~II could be as large as 400~pc \citep{Fitzgerald1976,Graham1988}. \citet{Franco1991} investigated the interstellar extinction of field stars projected towards the Chameleon clouds and estimated distances of 140~pc and 158$\pm$40~pc to Cha~I and Cha~II, respectively. A subsequent study conducted by \citet{Hughes1992} based on the same technique derived the distance of 200$\pm$20~pc to Cha~II. In the following years, \citet{Whittet1997} derived the more robust distance estimates of 160$\pm$15~pc and 178$\pm$18~pc for Cha~I and Cha~II, respectively, based on multiple distance indicators. \citet{Bertout1999}  computed the distance of 168$^{+14}_{-12}$~pc to Cha~I from the trigonometric parallaxes delivered by the \textit{Hipparcos} satellite \citep{Hipparcos} for a few stars in this region. A major contribution to the effort in constraining the distance to the Chamaeleon clouds was made by \citet{Voirin2018}, where the authors combined the extinction distribution of field stars projected towards the clouds with the parallaxes delivered by the first data release of the \textit{Gaia} space mission \citep[Gaia-DR1,][]{GaiaDR1} to estimate the distances of $179^{+11+11}_{-10-10}$~pc, $181^{+6+11}_{-5-10}$~pc, $199^{+8+12}_{-7-11}$~pc to Cha~I, Cha~II and Cha~III, respectively. This study put Cha~I about 20~pc further away from previous estimates and returned the first distance determination to the Cha~III molecular cloud. However, the systematic uncertainties of 0.3~mas in the Gaia-DR1 parallaxes \citep{Lindegren2016} largely dominated the distance uncertainties obtained in that study and called for a revision of the results. More recently, \citet{Roccatagliata2018} used the parallaxes from the second data release of the \textit{Gaia} space mission \citep[Gaia-DR2,][]{GaiaDR2} to revisit the distance to Cha~I. These latter authors reported distances of $192.7^{+0.4}_{-0.4}$~pc and $186.5^{+0.7}_{-0.7}$~pc to the northern and southern subgroups of stars in this cloud, respectively. The improved precision level in the distance determination is  related to the more precise parallaxes, but also to the fact that the systematic errors of the Gaia-DR2 catalogue \citep[see e.g.][]{Lindegren2018,Luri2018} were not modelled in that solution.

Kinematic studies have proven to be fundamental in distinguishing between the different subgroups of the Chamaeleon region and to searching for new cluster members. For example, \citet{LopezMarti2013a} found evidence that the stars in Cha~I and Cha~II have different proper motions, and that they also differ from the adjacent $\epsilon$~Cha and $\eta$~Cha associations that populate the same region of the sky. In a subsequent study, \citet{LopezMarti2013b} identified new kinematic members in Cha~I and Cha~II located in the outskirts of the molecular clouds and argued that this dispersed population could be larger, but more accurate data would be required to confirm this hypothesis. Indeed, the scarcity of trigonometric parallaxes and radial velocity (RV) information for most stars in Chamaeleon has been the main limitation to studying the kinematic properties of this region. This situation has dramatically changed with the advent of the Gaia-DR2 catalogue combined with the spectroscopic observations conducted by the \textit{Gaia}-ESO Survey \citep{Gilmore2012} in the Chamaeleon region \citep[see e.g.][]{Sacco2017}. Altogether, this puts us in a timely position to investigate the 3D structure and 3D space motion of the Chamaeleon star-forming complex with unprecedented precision as discussed throughout this study.

This paper is one in a series conducted in the context of the Dynamical Analysis of Nearby Clusters project \citep[DANCe,][]{Bouy2013}. Here, we investigate the census of stars, and the structure and kinematic properties of the Chamaeleon star-forming region in light of Gaia-DR2 data. The paper is organised as follows. In Section~\ref{section2} we compile the lists of stars in Cha~I and Cha~II published in the literature, and perform a new membership analysis to confirm the historical members associated with these clouds and discover new ones. In Section~\ref{section3} we revisit the distance, spatial velocity, and age of the Chamaeleon subgroups based on our new sample of cluster members selected in this study. Finally, we summarise our results and conclusions in Section~\ref{section4}.

\section{Membership analysis}\label{section2}

Our strategy to assess membership is based on the methodology previously developed by our team \citep{Sarro2014,Olivares2019}. In this section, we describe the main steps of our membership analysis applied to the Chamaeleon star-forming region and we refer the reader to the original papers for more details on the performance and implementation of our classifier.

\subsection{Field and cluster models}\label{section2.1}

The representation space (i.e., set of observables) that we use in the membership analysis includes the astrometric and photometric features of the stars provided in the Gaia-DR2 catalogue. We do not include the blue photometry $G_{BP}$ in the analysis because of the calibration problems in this band as reported in the literature \citep[see e.g.][]{Apellaniz2018} which can affect our selection of cluster members particularly in the faint end. We therefore restrict the membership analysis to the space of observables defined by $\mu_{\alpha}\cos\delta$, $\mu_{\delta}$, $\varpi$, $G_{RP}$ and $G-G_{RP}$. We then downloaded the Gaia-DR2 catalogue in the region of the sky defined by $295^{\circ}\leq l \leq 305^{\circ}$ and $-20^{\circ}\leq b \leq -10^{\circ}$ in Galactic coordinates which encompasses the three molecular clouds of the complex (Cha~I, Cha~II, and Cha~III). This field includes 4\,433\,409~sources and 3\,904\,492 of them have complete data in our representation space.

\begin{figure*}[!h]
\begin{center}
\includegraphics[width=0.33\textwidth]{pmra_pmdec.pdf}
\includegraphics[width=0.33\textwidth]{pmra_parallax.pdf}
\includegraphics[width=0.33\textwidth]{pmdec_parallax.pdf}
\caption{Proper motions and parallaxes of the stars in Cha~I and Cha~II in the samples of \citet{Esplin2017} and \citet{Alcala2008}, respectively. Open symbols denote potential outliers in these samples based on Gaia-DR2 data identified from the MCD estimator (see Sect.~\ref{section2}).
\label{fig_pmra_pmdec_init_list}
}
\end{center}
\end{figure*}

The field population was modelled using Gaussian Mixture Models (GMMs). We tested the field model with a random sample of 10$^{6}$ sources using GMMs with 20, 40, 60, 80, 100, 120, 140 and 160 components, and we chose the model with 80 components, which returns the smallest Bayesian information criteria (BIC) value. The field model was computed only once at the beginning while the cluster model was built iteratively during the process based on an initial list of members (see below). The cluster model uses the inferred parameters from the initial list of members to define the cluster locus in the space of the astrometric features using GMMs and defines the cluster sequence in the photometric space as a principal curve with a spread at any point along the curve given by a multivariate Gaussian (both the principal curve and its spread are initialised with a fit to the initial list of members). The method then computes membership probabilities for all sources in the field using the fraction of sources in each category (member vs. non-member) obtained in the previous iteration to estimate the marginal class probabilities. The sources are classified into members and non-members based on a probability threshold $p_{in}$ predefined by the user. The list of stars that results from this process is used as input for the next iteration and this procedure is repeated until convergence. The solution is said to converge when the list of cluster members remains fixed after successive iterations.

The final step of the membership analysis consists in evaluating the performance of our classifier. We generate synthetic data based on the cluster and field properties inferred from the previous steps, and measure the quality of the classifier to define an optimum probability threshold $p_{opt}$ as described in Sect.~4.2.7 of \citet{Olivares2019}. The sources in the catalogue are finally re-classified as members ($p\geq p_{opt}$) and non-members.

The samples of 250 stars from \citet{Esplin2017} and 63~stars from \citet{Alcala2008} represent the most complete censuses of the stellar population in Cha~I and Cha~II, respectively, known to date. We found Gaia-DR2 astrometry for 194 and 48~stars, respectively, where we note the existence of a few stars with discrepant proper motion and parallax in these samples as shown in Figure~\ref{fig_pmra_pmdec_init_list}. We computed robust distances based on the covariance matrix obtained from the minimum covariance determinant \citep[MCD,][]{Rousseeuw1999} estimator and removed potential outliers from these samples as described in Section~2.1 of \citet{Galli2020}. This step reduces the lists of stars (with Gaia-DR2 data) in Cha~I and Cha~II to 161 and 36~stars, respectively. We use these clean samples of stars as the initial list of members in our membership analysis. The stars in Cha~I and Cha~II exhibit distinct properties as discussed throughout this paper (see also Figure~\ref{fig_pmra_pmdec_init_list}) despite them being part of the same molecular cloud complex. We therefore decided to run two independent membership analyses (one for each cluster) using the same catalogue of Gaia-DR2 sources downloaded for this sky region, representation space, and strategy to select the most likely cluster members, as described above, but using different input lists of stars that are specific for each case.

In Table~\ref{tab_comp_pin} we compare the solutions obtained for Cha~I and Cha~II using different values of the user predefined probability threshold $p_{in}$. We compute the true positive rate (TPR, i.e. the fraction of synthetic cluster members recovered by our methodology) and contamination rate (CR, i.e. the fraction of synthetic field stars identified by our model as cluster members) of the classifier to better evaluate our results obtained with different $p_{in}$ values. These numbers were obtained from synthetic data sets sampled from the inferred model for each cluster, and so they represent only rough estimates of these indicators computed in the absence of the true distributions.

\subsection{Projection effects}\label{section2.2}

Our membership analysis conducted over the relatively large field that encompasses the Chamaeleon molecular clouds (as defined in Sect.~\ref{section2.1}) identifies a few more dispersed field sources as cluster members independent of the adopted $p_{in}$ threshold. In particular, we note the existence of one source (namely Gaia DR2 5789232155389250304) that is closer to the Cha~II molecular clouds, but exhibits proper motion and parallax that are consistent with membership in Cha~I. The RV of this source published in the literature yields a spatial velocity ($U=-11.0\pm1.5$~km/s, $V=-21.6\pm2.0$~km/s, $W=-1.8\pm0.6$~km/s) that is not consistent with the space motion of Cha~I stars (see Sect.~\ref{section3.3}). This is due to projection effects that render proper motion and parallax consistent with membership in Cha~I at the spatial location of this source despite the different space motion. We proceed as follows to minimise the existence of potential contaminants in our solution due to projection effects.

\begin{figure*}[!h]
\begin{center}
\includegraphics[width=0.49\textwidth]{Cha1_case1_final.pdf}
\includegraphics[width=0.49\textwidth]{Cha2_case1_final.pdf}
\includegraphics[width=0.49\textwidth]{Cha1_case2_final.pdf}
\includegraphics[width=0.49\textwidth]{Cha2_case2_final.pdf}
\caption{Projection effects on the stellar proper motion of Chamaeleon stars. Upper panels show the proper motion distribution of synthetic stars randomly distributed over the fields used in our membership analysis moving with the same space motion of Cha~I and Cha~II stars. Lower panels illustrate the expected proper motion distribution of synthetic $\epsilon$~Cha and $\eta$~Cha located in the same fields. We generated a total of 1\,000 sources in our simulations. We compare the proper motion of the synthetic stars with the observed values for the candidate members previously identified in the literature and members identified in this study (in this case we use the solution with $p_{in}=0.9$, see Table~\ref{tab_comp_pin}). The mean, dispersion and correlation coefficient obtained from each proper motion distribution are indicated in the panels. The solid lines denote the fitted proper motion distributions and the crosses mark the mean proper motion in each case.
\label{fig_projection_effects}
}
\end{center}
\end{figure*}

First, we conduct an independent membership analysis following the same methodology described before but in two smaller regions centred around the Cha~I and Cha~II molecular clouds. Our results obtained in these regions are labelled with the term `central region' which we use hereafter to distinguish from the membership analysis conducted over the entire Chamaeleon complex. The size of each field is defined based on the position of the candidate members previously identified in the literature with available Gaia-DR2 data that constitute our input list for the membership analysis (see Sect.~\ref{section2.1}). Thus, the field for the membership analysis in Cha~I covers the sky region defined by  $295.9^{\circ}\leq l \leq 297.8^{\circ}$, $-16.1^{\circ}\leq b \leq -13.1^{\circ}$ in Galactic coordinates and includes 196\, 330 sources. The field centred around Cha~II is defined by $303.2^{\circ}\leq l \leq 304.2^{\circ}$, $-15.1^{\circ}\leq b \leq -13.6^{\circ}$ and includes 56\,100 sources.

Second, we generated synthetic cluster members located in the fields surveyed by our membership analysis from the velocity and distance distribution of the Cha~I and Cha~II candidate members previously identified in the literature. The 3D positions and spatial velocities of the stars were transformed into the observable proper motions in the corresponding sky regions that we use to discuss the importance of projection effects in our analysis. As illustrated in Figure~\ref{fig_projection_effects} the location and scatter of the proper motion distribution for synthetic stars, candidate members from the literature, and members identified in this study are consistent between themselves. The orientation of the proper motion distribution generated in our simulations is different because we allow the synthetic cluster members to be randomly distributed in the fields covered by our membership analysis. This effect is more apparent in the case of Cha~I. Therefore, our methodology employed for the membership analysis could be missing some cluster members if we assume that the stars are equally probable to be found at any sky position of the field. However, what we observe in practice is that the true members are mostly projected towards the molecular clouds (and in their immediate vicinity) which explains the different orientation observed for the simulated proper motions. The proper motion distribution of the members identified in this study is in good agreement with the results given in the literature. This confirms that our results for the membership analysis in the central regions are not affected by projection effects.

Third, we performed additional simulations to investigate the existence of contaminants from other young stellar groups due to projection effects. As described in Sect.~\ref{section2.1}, our membership analysis uses not only the astrometric features of the stars, but also their photometry to select cluster members with similar ages. The only two young stellar groups in the Chamaeleon complex that have similar ages to the stellar populations in the molecular clouds are the $\epsilon$~Cha and $\eta$~Cha associations. We generated synthetic stars (as described above) using the distance and space motion of these stellar groups given in the literature \citep[see e.g.][]{Murphy2013} in the fields covered by our membership analysis of Cha~I and Cha~II. Figure~\ref{fig_projection_effects} shows that the proper motions of the synthetic $\epsilon$~Cha and $\eta$~Cha stars are significantly different from the observed proper motion distribution of Cha~I and Cha~II cluster members. We therefore conclude that our sample of members in the central regions is not contaminated by the other young stellar groups of the Chamaeleon complex due to projection effects.

\subsection{Final list of cluster members}\label{section2.3}

The high TPRs and low CRs for all solutions given in Table~\ref{tab_comp_pin} confirm the robustness of our results. In particular, we note that our results for the membership analysis in the central regions have higher TPRs and lower CRs as compared to the ones obtained over the whole Chamaeleon complex. We see little variation in sample size in the two cases and confirm that running the membership analysis over extended regions of the Chamaeleon complex will not allow us to detect more cluster members (but will lead to the inclusion of more contaminants due to projection effects). On the contrary, performing the membership analysis over smaller regions allowed us to recover a few members that have been overlooked in the first analysis. We therefore decided to report the solutions obtained with $p_{in}=0.9$ in the central regions as our final lists of clusters members in Cha~I (188~stars) and Cha~II (41~stars). The cluster members in each population are listed in Tables~\ref{tab_members_Cha1} and \ref{tab_members_Cha2}. We provide in Tables~\ref{tab_prob_Cha1} and \ref{tab_prob_Cha2} the membership probabilities obtained with different $p_{in}$ values for all sources in the field for the analyses conducted in Cha~I and Cha~II, respectively, to allow the reader to choose a different probability threshold that is more suited to their study.

\begin{table*}[!h]
\renewcommand\thetable{1}
\centering
\caption{Comparison of membership results in Cha~I and Cha~II using different values for the probability threshold $p_{in}$.
\label{tab_comp_pin}}
\begin{tabular}{ccccccccc}
\hline\hline
&\multicolumn{4}{c}{Cha~I (large field)}&\multicolumn{4}{c}{Cha~II (large field)}\\
\hline\hline
$p_{in}$&$p_{opt}$&Members&TPR&CR&$p_{opt}$&Members&TPR&CR\\
\hline
0.5&0.80&189&$0.964\pm0.010$&$0.040\pm0.017$&0.74&48&$0.978\pm0.018$&$0.039\pm0.004$\\
0.6&0.90&183&$0.947\pm0.005$&$0.024\pm0.011$&0.78&47&$0.982\pm0.013$&$0.030\pm0.005$\\
0.7&0.80&189&$0.962\pm0.014$&$0.052\pm0.024$&0.59&47&$0.977\pm0.006$&$0.030\pm0.016$\\
0.8&0.84&184&$0.972\pm0.006$&$0.037\pm0.016$&0.83&40&$0.970\pm0.023$&$0.030\pm0.005$\\
0.9&0.81&187&$0.972\pm0.020$&$0.034\pm0.011$&0.81&38&$0.983\pm0.010$&$0.021\pm0.011$\\
\hline\hline
&\multicolumn{4}{c}{Cha~I (central region)}&\multicolumn{4}{c}{Cha~II (central region)}\\
\hline\hline
$p_{in}$&$p_{opt}$&Members&TPR&CR&$p_{opt}$&Members&TPR&CR\\
\hline
0.5&0.80&192&$0.995\pm0.005$&$0.021\pm0.013$&0.74&43&$0.980\pm0.017$&$0.022\pm0.003$\\
0.6&0.84&191&$0.996\pm0.003$&$0.014\pm0.008$&0.77&41&$0.988\pm0.006$&$0.012\pm0.003$\\
0.7&0.69&192&$0.998\pm0.003$&$0.012\pm0.003$&0.76&41&$0.985\pm0.013$&$0.008\pm0.008$\\
0.8&0.86&190&$0.993\pm0.008$&$0.007\pm0.012$&0.78&41&$0.983\pm0.016$&$0.008\pm0.003$\\
0.9&0.90&188&$0.991\pm0.008$&$0.002\pm0.003$&0.79&41&$0.985\pm0.010$&$0.017\pm0.011$\\
\hline\hline
\end{tabular}
\tablefoot{We provide the optimum probability threshold, the number of cluster members, the true positive rate (TPR) and contamination rate (CR) obtained for each solution and cluster. We present the results of our membership analysis performed over a large field that encompasses the entire Chamaeleon complex (see Sect.~\ref{section2.1}) and the central regions around the Cha~I and Cha~II molecular clouds (see Sect.~\ref{section2.2}).}
\end{table*}

Figure~\ref{fig_location} confirms the existence of a more dispersed population of cluster members in the immediate vicinity of the Cha~I molecular cloud that is known from pre-\textit{Gaia} studies \citep[see e.g.][]{LopezMarti2013b}. Figures~\ref{fig_pm_plx_cha1} and \ref{fig_pm_plx_cha2} show the distribution of proper motions and parallaxes of the Cha~I and Cha~II members identified in our analysis. As expected, the cluster members in the outskirts of the distributions and with the largest uncertainties have the lowest membership probabilities. The colour-magnitude diagrams of Cha~I and Cha~II in the chosen photometric space are shown in Figure~\ref{fig_cmd}, and the empirical isochrones are given in Tables~\ref{tab_isochrone_Cha1} and \ref{tab_isochrone_Cha2}. Our samples of cluster members cover the magnitude range from about $G=6$ to $G=20$~mag and $G=12$ to $G=18$~mag in Cha~I and Cha~II, respectively, and they represent the most complete censuses of members in the Chamaeleon clouds (with available astrometry in the Gaia-DR2 catalogue) to date.

\begin{figure*}[!h]
\begin{center}
\includegraphics[width=1.0\textwidth]{fig_extmap_large_final_sample_small.png}
\caption{Location of the Chamaeleon stars overlaid on the extinction map of \citet{Dobashi2005} in Galactic coordinates. Open and filled symbols indicate the position of the stars from the literature \citep{Alcala2008,Esplin2017} and cluster members identified in this study, respectively. The blue and red colours denote the stars in the Cha~I and Cha~II molecular clouds, respectively.
\label{fig_location}
}
\end{center}
\end{figure*}

\begin{figure*}[!h]
\begin{center}
\includegraphics[width=0.33\textwidth]{pmra_pmdec_Cha1.pdf}
\includegraphics[width=0.33\textwidth]{pmra_parallax_Cha1.pdf}
\includegraphics[width=0.33\textwidth]{pmdec_parallax_Cha1.pdf}
\caption{Proper motions and parallaxes of the 188 stars in Cha~I identified in our membership analysis. The stars are colour-coded based on their membership probabilities which are scaled from zero to one. Triangles indicate the stars with RUWE $\geq$ 1.4 (see Sect.~\ref{section3.1}).
\label{fig_pm_plx_cha1}
}
\end{center}
\end{figure*}

\begin{figure*}[!h]
\begin{center}
\includegraphics[width=0.33\textwidth]{pmra_pmdec_Cha2.pdf}
\includegraphics[width=0.33\textwidth]{pmra_parallax_Cha2.pdf}
\includegraphics[width=0.33\textwidth]{pmdec_parallax_Cha2.pdf}
\caption{Proper motions and parallaxes of the 41 stars in Cha~II identified in our membership analysis. The stars are colour-coded based on their membership probabilities which are scaled from zero to one. Triangles indicate the stars with RUWE $\geq$ 1.4 (see Sect.~\ref{section3.1}).
\label{fig_pm_plx_cha2}
}
\end{center}
\end{figure*}

\begin{figure*}[!h]
\begin{center}
\includegraphics[width=0.45\textwidth]{CMD_Cha1.pdf}
\includegraphics[width=0.45\textwidth]{CMD_Cha2.pdf}
\caption{Colour-magnitude diagram of the cluster members identified in our membership analysis in Cha~I \textit{(left panel)} and Cha~II \textit{(right panel)}. The black solid line indicates the empirical isochrone derived for each cluster (see Tables~\ref{tab_isochrone_Cha1} and \ref{tab_isochrone_Cha2}). The stars are colour-coded based on their membership probabilities which are scaled from zero to one. Triangles indicate the stars with RUWE $\geq$ 1.4 (see Sect.~\ref{section3.1}). The arrow indicates the extinction
vector of $A_{V}=1$~mag that we converted to the Gaia bands based on the relative extinction values computed by \citet{Wang2019}.
\label{fig_cmd}
}
\end{center}
\end{figure*}

In Figure~\ref{fig_venn} we compare the samples of cluster members identified in this paper with the lists of Cha~I and Cha~II stars (with available Gaia-DR2 data) published in other studies. We note that most members in the Cha~I sample (i.e., 169 stars) were also included in the study conducted by \citet{Esplin2017}. However, 25~sources presented in that study could not be recovered by our membership analysis. We verified that they have proper motions and parallaxes that are either not consistent with membership in Cha~I (see e.g. Figure~\ref{fig_pmra_pmdec_init_list}) or result from poor astrometric solutions. For example, if we use the re-normalised unit weight error (RUWE) criterion\footnote{see technical note \href{https://www.cosmos.esa.int/web/gaia/dr2-known-issues}{GAIA-C3-TN-LU-LL-124-01} for more details} to filter the literature sources with poor astrometry (see Sect.~\ref{section3.1}) we note that 17~stars among the rejected sources fail to pass this selection criterion (i.e. RUWE $<$ 1.4). These 17 sources can be considered at this stage as potential candidate members and future data releases of the Gaia space mission will allow us to confirm (or exclude) membership with more precise data. Three stars among the other eight rejected candidate members from the literature lie outside the region surveyed by our analysis. Only one rejected star from the literature among the remaining five sources (namely, Gaia DR2 5201125062389375872) has proper motions and parallaxes consistent with membership in Cha~I. However, it is located below the empirical isochrone of the cluster which explains why it is rejected by our methodology. The same conclusion holds if we try to correct its photometry by the effect of interstellar extinction using the value of $A_{V}\simeq2.5$~mag that is estimated from the \citet{Dobashi2005} maps at the location of the source.

\begin{figure*}[!h]
\begin{center}
\includegraphics[width=0.49\textwidth]{VennDiagram_Cha1.pdf}
\includegraphics[width=0.49\textwidth]{VennDiagram_Cha2.pdf}
\caption{Venn diagram comparing the number of members  (with Gaia-DR2 data) identified in previous studies of the literature \citep{Alcala2008,Esplin2017} and the samples of cluster members derived in this paper from our membership analysis.
\label{fig_venn}
}
\end{center}
\end{figure*}

Similar arguments apply to the analysis in Cha~II. We confirmed 34~members from \citet{Alcala2008} and rejected 14~stars from that study. We note that among the rejected members there are eight sources that are more likely to be background objects ($\varpi<1$~mas) unrelated to the Cha~II population (see Figure~\ref{fig_pmra_pmdec_init_list}). Four stars have poor astrometric solutions based on the RUWE criterion which could possibly explain the discrepant proper motions and parallaxes. The remaining two stars among the other rejected sources, namely Gaia~DR2~5788884464898536960 and Gaia~DR2~5788200298087320320, have proper motions and parallaxes consistent with membership in Cha~II, but they lie below the empirical isochrone defined by the other cluster members. We tentatively corrected their photometry from the interstellar extinction estimated at their position from the \citet{Dobashi2005} extinction maps, but this correction does not place them on the empirical isochrone of the cluster.

The membership analysis conducted in this study allowed us to identify 19 and 7 new members of Cha~I and Cha~II, respectively. This represents an increase of about 11\% and 21\% in the stellar population of Cha~I and Cha~II with respect to the number of confirmed cluster members from previous studies.

\section{Discussion}\label{section3}

\subsection{Refining the sample of cluster members}\label{section3.1}

Let us now refine our sample of cluster members by filtering the stars with the best data available before deriving the overall kinematic properties of the Chamaeleon subgroups. To do so, we use the RUWE criterion as one possible indicator of the goodness of fit of the Gaia-DR2 astrometric solution. The value of 1.4 is often used in the literature to distinguish between the sources with reliable (i.e., RUWE $<$ 1.4) and poor astrometric solutions in the Gaia-DR2 catalogue, but some studies adopt different thresholds \citep[see e.g.][]{Luhman2020}. In this paper we adopt the RUWE threshold of 1.4 to select the sources with good astrometric solutions for consistency with the methodology adopted in other papers of this series \citep[see e.g.][]{Galli2020,Galli2020b}. This step reduces the samples of members to 160 and 31 stars in Cha~I and Cha~II, respectively. The rejected stars exhibit mostly the largest uncertainties in proper motions and parallaxes as shown in Figures~\ref{fig_pm_plx_cha1} and \ref{fig_pm_plx_cha2}. Many of them have been identified as binaries or multiple systems in the literature (see also Figure~\ref{fig_cmd}) which explains the poor astrometry in the Gaia-DR2 catalogue. Future data releases of the Gaia space mission will deliver an improved astrometry for such systems.

We searched the CDS databases \citep{Wenger2000} for RV information of the stars in our sample. Our search for RVs in the literature is based on \citet{Dubath1996}, \citet{Covino1997}, \citet{Joergens2001}, \citet{Torres2006}, \citet{James2006}, \citet{Gontcharov2006}, \citet{Guenther2007}, \citet{Nguyen2012}, \citet{Biazzo2012}, \citet{Sacco2017}, and the Gaia-DR2 catalogue. We found RV measurements for 82 and 19~stars in our samples of cluster members in Cha~I and Cha~II, respectively. Some stars have multiple RV measurements in the literature (collected from different studies) and in such cases we decided to use the most precise value in our analysis.

We applied the interquartile range (IQR) criterion to identify the stars with discrepant RVs in the sample. This method rejects outliers that lie over $1.5\times$IQR below the first quartile or above the third quartile. Doing so, we found four stars in Cha~I with discrepant RVs as illustrated in Figure~\ref{fig_RV}. We discarded these RV measurements from the analysis presented below, but we still retain the stars in the sample of cluster members. Their proper motions and parallaxes are consistent with membership in Cha~I and the corresponding RVs are more likely to be affected by other reasons (e.g. undetected binaries). This reduces the sample of sources in Cha~I with available RV information to 78~stars. We do not discard any RV measurement in the Cha~II sample based on the IQR criterion.

Our search for published RV data for Chamaeleon stars in the literature allowed us to retrieve this information for about 49\% and 61\% of the stars in Cha~I and Cha~II, respectively. The main source of RV data in Cha~I is the study conducted by \citet{Sacco2017} as part of the Gaia-ESO survey, and we therefore refer the reader to that paper for further details on these measurements. In the case of the Cha~II molecular cloud, the main source of RV data is the study conducted by \citet{Biazzo2012}. The mean precision of the RVs in our sample is 0.5~km/s and 4.7~km/s in Cha~I and Cha~II, respectively. In the following, we use the precise Gaia-DR2 astrometry combined with published RV data to investigate the distance and spatial velocity of Chamaeleon stars.

\begin{figure*}[!h]
\begin{center}
\includegraphics[width=0.49\textwidth]{RV_Cha1.pdf}
\includegraphics[width=0.49\textwidth]{RV_Cha2.pdf}
\caption{Kernel density estimate of the RV distribution for the stars in Cha~I \textit{(left panel)} and Cha~II \textit{(right panel)}. We used the mean precision of the RVs in each sample (see text in Sect.~\ref{section3.1}) as kernel bandwidth. The tick marks in the horizontal axis mark the RVs of individual stars. The star with the most discrepant RV in the Cha~I sample, namely Gaia DR2 5201347408553455616 \citep[$V_{r}=164.11\pm1.61$~km/s, ][]{Sacco2017}, is not shown to improve the visibility of the plot.
\label{fig_RV}
}
\end{center}
\end{figure*}

\subsection{Proper motions and parallaxes of the subgroups}\label{section3.2}

The stellar population of Cha~I is composed of two subclusters that have been historically separated based on their position in the sky using the declination of $\delta=-77^{\circ}$ as midpoint to separate the northern and southern subgroups \citep[see e.g.][]{Luhman2007}. However, a recent study conducted by \citet{Roccatagliata2018} shows that the subclusters overlap and that they cannot be strictly divided by their sky positions.

In this paper, we apply the Partitioning Around Medoids \citep[PAM,][]{PAM} clustering algorithm to separate the two subclusters in Cha~I. The PAM algorithm is an unsupervised machine-learning method for partitioning a $N$-dimensional dataset into $k$ groups (i.e. clusters) that works similarly to the $k$-Means algorithm \citep{kMeans}, but uses medoids to represent the centers of the clusters. We perform the clustering in the 5D space of astrometric observables $(\alpha,\delta,\mu_{\alpha}\cos\delta,\mu_{\delta},\varpi)$ of the stars using the \textit{pam} routine from the \texttt{cluster} package implemented in R programming language. We use the Average Silhouette \citep{ROUSSEEUW198753} and Gap Statistic \citep{GapStatistic} methods to confirm that the optimal number of clusters for our dataset is indeed $k=2$, which we use as input for the clustering analysis. The clustering analysis with PAM divides the sample into 76~stars and 84~stars for Cha~I (north) and Cha~I (south), respectively. Figure~\ref{fig_clustering} confirms that the subclusters in Cha~I overlap in position, proper motion, and parallax in good agreement with the findings reported by \citet{Roccatagliata2018}.

In Table~\ref{tab_subgroups} we compare the proper motions and parallaxes of the Chamaeleon subgroups in our sample of members. It is apparent from this comparison that the stellar populations in Cha~I and Cha~II exhibit distinct kinematic properties (as already anticipated in Sect.~\ref{section2}). However, we also note a mean difference of about 1~mas/yr in the proper motions of the stars in the two subgroups of Cha~I despite the overlap of the subgroups shown in Figure~\ref{fig_clustering}. \citet{Roccatagliata2018} performed a two-sample Kolmogorov-Smirnov test and concluded that the proper motion distribution of the subclusters is not drawn from the same parent distribution, but the authors do not discuss whether the observed difference is due to projection effects or a different space motion. In Section~\ref{section3.3} we compute the spatial velocity of the stars and investigate whether the two subclusters in Cha~I are indeed kinematically distinct.

\begin{figure*}[!h]
\begin{center}
\includegraphics[width=0.49\textwidth]{pam_ra_dec_k2.pdf}
\includegraphics[width=0.49\textwidth]{pam_pmra_pmdec_k2.pdf}
\includegraphics[width=0.49\textwidth]{pam_pmra_parallax_k2.pdf}
\includegraphics[width=0.49\textwidth]{pam_pmdec_parallax_k2.pdf}
\caption{Clustering results obtained for Cha~I with the PAM algorithm in the 5D space of position, proper motion, and parallax.
\label{fig_clustering}
}
\end{center}
\end{figure*}

\begin{table*}[!h]
\centering
\scriptsize{
\caption{Proper motions and parallaxes of the Chamaeleon subgroups in our sample of cluster members.
\label{tab_subgroups}}
\begin{tabular}{lccccccccccc}
\hline\hline
Sample&$N_{init}$&$N_{RUWE}$&\multicolumn{3}{c}{$\mu_{\alpha}\cos\delta$}&\multicolumn{3}{c}{$\mu_{\delta}$}&\multicolumn{3}{c}{$\varpi$}\\
&&&\multicolumn{3}{c}{(mas/yr)}&\multicolumn{3}{c}{(mas/yr)}&\multicolumn{3}{c}{(mas)}\\
\hline\hline
&&&Mean$\pm$SEM&Median&SD&Mean$\pm$SEM&Median&SD&Mean$\pm$SEM&Median&SD\\
\hline

Cha I (north) & 87 & 76 & $ -21.903 \pm 0.063 $& -21.953 & 0.548 & $ -0.054 \pm 0.099 $& -0.023 & 0.863 & $ 5.178 \pm 0.016 $& 5.175 & 0.138 \\
Cha I (south) & 101 & 84 & $ -23.053 \pm 0.069 $& -23.048 & 0.629 & $ 1.127 \pm 0.118 $& 1.022 & 1.083 & $ 5.314 \pm 0.023 $& 5.322 & 0.211 \\
\hline
Cha I & 188 & 160 & $ -22.507 \pm 0.065 $& -22.403 & 0.824 & $ 0.566 \pm 0.091 $& 0.405 & 1.146 & $ 5.250 \pm 0.015 $& 5.235 & 0.192 \\
Cha II & 41 & 31 & $ -20.207 \pm 0.170 $& -19.954 & 0.945 & $ -7.635 \pm 0.129 $& -7.491 & 0.717 & $ 5.037 \pm 0.021 $& 5.061 & 0.115 \\

\hline
\end{tabular}
\tablefoot{We provide for each subgroup the initial number of stars, final number of stars after the RUWE filtering (see Sect.~\ref{section3.1}), mean, standard error of the mean (SEM), median and standard deviation (SD) of proper motions and parallaxes.}
}
\end{table*}

\subsection{Distance and spatial velocity of Chamaeleon stars}\label{section3.3}

In the following, we convert the stellar proper motions and parallaxes into distances and 2D velocities using Bayesian inference. This analysis uses the \textit{Kalkayotl} package\footnote{The code is available at \href{https://github.com/olivares-j/Kalkayotl}{https://github.com/olivares-j/Kalkayotl}.} \citep{kalkayotl,Olivares2020} in python programming language, which implements a number of priors for the distance. We took advantage of this code to investigate two priors for the distance that are based on purely statistical probability density distributions (hereafter, statistical priors), and another two priors that are based on astrophysical assumptions (hereafter, astrophysical priors). The statistical priors considered in this paper include the Uniform and Gaussian distributions. The two astrophysical priors that we use are based on the surface brightness profile of the Large Magellanic Cloud derived from star counts by \citet{Elson1987} and the King's profile distribution observed in globular clusters \citep{King1962}. More details on the implementation and performance of each prior are given in the \textit{Kalkayotl} paper \citep{Olivares2020}. The prior that we used in the case of the angular velocity (i.e. 2D tangential velocity) is a beta function following the online tutorials available in the \textit{Gaia} archive \citep{Luri2018}. This procedure takes as input the astrometric observables and covariance matrix of the stars as given by the Gaia-DR2 catalogue.

We compared the distances derived from different priors and confirmed that they do not depend on the choice of the prior. We therefore decided to report the distances derived from the uniform prior as our final results, because this is the most simple prior. We then used the resulting distances to compute the $XYZ$ position of the stars in a reference system that has its origin at the Sun where $X$ points to the Galactic centre, $Y$ points in the direction of Galactic rotation, and $Z$ points to the Galactic north pole. We converted the 2D tangential velocities and RVs of the stars into the $UVW$ components of the Galactic velocity (in the same reference system described before) using the transformation outlined by \citet{Johnson1987}. The distance and spatial velocity of the stars are given in Tables~\ref{tab_members_Cha1} and \ref{tab_members_Cha2}.

\begin{table*}[!h]
\centering
\scriptsize{
\caption{Distance and spatial velocity of the Chamaeleon subgroups in our sample of cluster members.
\label{tab_distance_velocity}}
\begin{tabular}{lcccccccccccc}
\hline\hline
Sample&$N_{d}$&$N_{UVW}$&$d$&\multicolumn{3}{c}{$U$}&\multicolumn{3}{c}{$V$}&\multicolumn{3}{c}{$W$}\\
&&&(pc)&\multicolumn{3}{c}{(km/s)}&\multicolumn{3}{c}{(km/s)}&\multicolumn{3}{c}{(km/s)}\\
\hline\hline
&&&&Mean&Median&SD&Mean&Median&SD&Mean&Median&SD\\
\hline
Cha I (north) & 76 & 39 & $ 191.4 _{ -0.8 }^{+ 0.8 }$ &$ -10.6 \pm 0.4 $& -10.6 & 0.7 & $ -19.3 \pm 0.6 $& -19.4 & 1.0 & $ -11.5 \pm 0.2 $& -11.5 & 0.6 \\
Cha I (south) & 84 & 39 & $ 186.7 _{ -1.0 }^{+ 1.0 }$ &$ -11.5 \pm 0.6 $& -11.6 & 0.9 & $ -19.7 \pm 0.9 $& -19.8 & 1.2 & $ -10.9 \pm 0.3 $& -10.8 & 0.8 \\
\hline
Cha I (all stars) & 160 & 78 & $ 189.4 _{ -0.7 }^{+ 0.8 }$ &$ -11.0 \pm 0.5 $& -11.0 & 0.9 & $ -19.5 \pm 0.8 $& -19.6 & 1.1 & $ -11.2 \pm 0.3 $& -11.1 & 0.8 \\
Cha II (all stars) & 31 & 19 & $ 197.5 _{ -0.9 }^{+ 1.0 }$ &$ -11.0 \pm 2.9 $& -11.2 & 1.7 & $ -18.1 \pm 4.2 $& -17.8 & 2.7 & $ -8.5 \pm 1.4 $& -8.6 & 1.1 \\

\hline\hline
\end{tabular}
\tablefoot{We provide for each subgroup the number of stars used to compute the distance (after the RUWE filtering) and spatial velocity, Bayesian distance, mean, median and standard deviation (SD) of the $UVW$ velocity components. The uncertainties in the mean velocities are computed as explained in the text of Sect.~\ref{section3.3}.}
}
\end{table*}

Table~\ref{tab_distance_velocity} lists the distance of the Chamaeleon subgroups in our sample. We confirm with our methodology that the two subgroups of Cha~I (north and south) are located at different distances within the reported uncertainties as previously reported by \citet{Roccatagliata2018}. In addition, we measure the distance variation along the line of sight of $\Delta d=8.1^{+1.3}_{-1.1}$~pc between Cha~I and Cha~II, and find that they are separated by $23.1\pm0.3$~pc in the space of 3D positions.

The distances derived in this paper based on Gaia-DR2 data are more precise than the results of $179^{+11 +11}_{-10 -10}$~pc and $181^{+6 +11}_{-5 -10}$~pc obtained by \citet{Voirin2018} for Cha~I and Cha~II, respectively (the error bars reported in those solutions refer to random and systematic uncertainties in this order). The latter made use of the parallaxes from the Tycho-Gaia Astrometric Solution catalogue \citep[TGAS,][]{Lindegren2016} which were affected by systematic errors of about 0.3~mas, which explains the much larger uncertainties in the distances. Our results are more precise than the distances of $192\pm6$~pc and $198\pm6$ provided by \citet{Dzib2018} for Cha~I and Cha~II, respectively, using different samples of stars. Our distance estimates are also consistent with the results of $192.7^{+0.4}_{-0.4}$~pc and $186.5^{+0.7}_{-0.7}$~pc derived by \citet{Roccatagliata2018} for the northern and southern subclusters of Cha~I, respectively. However, in this case we note that our uncertainties are larger by a factor of almost two. As explained in Sect.~2 of \citet{Roccatagliata2018} the authors did not include the systematic errors on the Gaia-DR2 parallaxes in their analysis and derived the distance from the inverse of the parallax. In this study, we derived the distances from Bayesian inference using the \textit{Kalkayotl} code \citep{Olivares2020} which is designed to deal with some characteristics of the Gaia-DR2 catalogue including the parallax zero-point correction and spatial correlations which are modelled with the covariance function of \citet{Vasiliev2019}. The existence of systematic errors in the Gaia-DR2 catalogue does not compromise the valuable astrometry delivered by the Gaia satellite, but needs to be considered to avoid underestimating uncertainties.

Before computing the distance and spatial velocity of the stars we corrected the Gaia-DR2 parallaxes by the zero-point shift of -0.030~mas and added the systematic errors of 0.1~mas/yr in quadrature to the formal uncertainties on proper motions \citep[see][for more details]{Lindegren2018}. It is nevertheless important to mention that other zero-point corrections for the Gaia-DR2 parallaxes exist in the literature. They range from $-0.031\pm0.011$~mas \citep{Graczyk2019} to $-0.082\pm0.033$~mas \citep{Stassun2018}. The lowest value is consistent with the zero-point correction provided by the Gaia collaboration \citep{Lindegren2018}, which was adopted throughout this study. We compare in Table~\ref{tab_distance_zeropoint} the minimum and maximum distances of the Chamaeleon subgroups when we consider different zero-point corrections applied to the Gaia-DR2 parallaxes. The largest zero-point correction available in the literature  \citep[$-0.082\pm0.033$~mas,][]{Stassun2018} puts the Chamaeleon subgroups about 2~pc closer to the Sun, but the resulting distances are still consistent with the results obtained in Table~\ref{tab_distance_velocity} within the reported uncertainties. We therefore conclude that our results are not sensitive to the adopted zero-point correction for the parallaxes.

\begin{table}[!h]
\centering
\footnotesize{
\caption{Distance of the Chamaeleon subgroups using different zero-point corrections for the Gaia-DR2 parallaxes.
\label{tab_distance_zeropoint}}
\begin{tabular}{lccc}
\hline\hline
Sample&$N_{d}$&\multicolumn{2}{c}{distance}\\
&&\multicolumn{2}{c}{(pc)}\\
\hline\hline
&&\citet{Stassun2018}&no correction\\
\hline
Cha~I (north)&76&$189.6^{+0.8}_{-0.8}$&$192.5^{+0.8}_{-0.7}$\\
Cha~I (south)&84&$184.9^{+0.9}_{-0.9}$&$187.8^{+0.9}_{-0.9}$\\
\hline
Cha~I &160&$187.5^{+0.8}_{-0.8}$&$190.5^{+0.8}_{-0.8}$\\
Cha~II &31&$195.5^{+0.9}_{-0.9}$&$198.6^{+1.0}_{-0.9}$\\
\hline\hline
\end{tabular}
\tablefoot{We provide for each subgroup the number of stars and the Bayesian distances derived from the largest zero-point shift reported in the literature \citep[$-0.082\pm0.033$~mas,][]{Stassun2018} and without any zero-point correction applied to the Gaia-DR2 parallaxes.}
}
\end{table}

Although the mean space motion of stellar groups provides useful information, the non-trivial question that arises in our discussion is whether the Chamaeleon subgroups discussed throughout this study are kinematically distinct. In Section~\ref{section3.2} we report a mean difference of about 1~mas/yr in the proper motions of the two subclusters in Cha~I which translates into a relative tangential velocity of the order of 1~km/s. \citet{Sacco2017} also reported a mean difference of about 1~km/s between the RVs of the northern and southern subgroups of Cha~I. This relatively small difference in velocity can result from other effects such as for example intrinsic velocity dispersion, undetected binaries, and correlated noise. For example, the transformation of  parallaxes, proper motions, and RVs to 3D velocities can result in correlated errors between the velocity components even in the absence of correlations between the astrometric observables \citep[see e.g.][]{Brown1997,Perryman1998}. We followed the procedure outlined in Sect.~7.2 of \citet{Perryman1998} to investigate this effect in our results. The mean motion of the Chamaeleon subgroups is obtained by averaging the measured velocities and the uncertainty in the mean is computed from the mean of the covariance matrices of the individual stars. The resulting spatial velocities of the Chamaeleon subgroups are given in Table~\ref{tab_distance_velocity}. Thus, the relative motion between the two subclusters of Cha~I (in the sense `north' minus `south') is $(\Delta U,\Delta V, \Delta W)=(0.9,0.4,-0.6)\pm(0.7,1.1,0.4)$~km/s. Similarly, the relative motion between Cha~I and Cha~II (in the sense `Cha~II' minus `Cha~I') is $(\Delta U,\Delta V, \Delta W)=(0.0,1.4,2.7)\pm(2.9,4.3,1.4)$~km/s. We therefore conclude that the reported differences in the velocity of the subgroups are not significant at the 3$\sigma$ level when we take into account the covariances in our analysis.

In addition, we also note from Figure~\ref{fig_UVW} that the space motion of most members in the two subclusters of Cha~I is consistent within $1\sigma$ of the observed velocity dispersion. The observed scatter of the stars in the velocity space results from both measurement errors and the intrinsic velocity dispersion of the cluster. The values listed in Table~\ref{tab_distance_velocity} for the one-dimensional velocity dispersion of Cha~I in each velocity component are in good agreement with the velocity dispersion of $1.10\pm0.15$~km/s reported by \citet{Sacco2017} based on the RV of the stars. The median uncertainties in the UVW velocity components of Cha~I stars are 0.6, 0.5, and 0.4~km/s, suggesting that the internal velocity dispersion is resolved. On the other hand, the median uncertainties in the three velocity components of Cha~II stars are 3.4, 4.7, and 1.8~km/s, implying that the internal velocity dispersion is not resolved. Thus, the observed scatter of Cha~II stars in the space of velocities is most probably due to measurement errors and can be explained by the large uncertainties in the RV of the stars (see Sect.~\ref{section3.1}) which propagate to the 3D spatial velocities.

We now compare the space motion of the stellar populations associated with the molecular clouds with the other two young stellar groups located in the Chamaeleon star-forming region. The $\epsilon$~Cha and $\eta$~Cha associations are located in the same sky region of the molecular clouds, but they constitute a foreground population of young stars at a distance of about 100~pc \citep{Gagne2018b}. We cross-matched our samples of Cha~I and Cha~II stars identified in this study with the lists of $\epsilon$~Cha and $\eta$~Cha given in the literature \citep[see e.g.][]{Gagne2018a,Gagne2018c}, but we found no sources in common. The relative motion of Cha~I with respect to the space motion of $\epsilon$~Cha and $\eta$~Cha derived by \citet{Murphy2013} is $(\Delta U,\Delta V,\Delta W)=(-0.1,0.9,-1.3)\pm(0.9,1.5,1.4)$~km/s and $(\Delta U,\Delta V,\Delta W)=(-0.8,1.2,0.0)\pm(0.5,0.8,0.3)$~km/s, respectively (in the sense `Cha~I' minus `$\epsilon$~Cha' or `$\eta$~Cha'). Similarly, the relative motion of Cha~II with respect to $\epsilon$~Cha and $\eta$~Cha is $(\Delta U,\Delta V,\Delta W)=(-0.1,2.3,1.4)\pm(3.0,4.4,2.0)$~km/s and  $(\Delta U,\Delta V,\Delta W)=(-0.8,2.6,2.7)\pm(2.9,4.2,1.4)$~km/s, respectively. Therefore, we conclude that the space motion of $\epsilon$~Cha and $\eta$~Cha is consistent with the space motion of the stars in the Cha~I  and Cha~II molecular clouds within 3$\sigma$ of the reported uncertainties in the spatial velocities. The common space motion and similar age of the stars in these stellar groups \citep[see e.g.][]{Luhman2007,Gagne2018b} suggest that they may have formed from the same parent cloud, but further investigation study is warranted to confirm this hypothesis.

\begin{figure*}[!h]
\begin{center}
\includegraphics[width=1.0\textwidth]{Cha1_UVW_corr.pdf}
\includegraphics[width=1.0\textwidth]{Cha1_Cha2_UVW_corr.pdf}
\caption{Distribution of the 3D spatial velocity of Chamaeleon stars. The upper panels compare the velocity of Cha~I (north) and Cha~I (south), and the lower panels compare the velocity distribution of Cha~I and Cha~II. Filled and open symbols denote single stars and known binaries (or multiple systems), respectively. The contours indicate the  68.3\%, 95.4\%, and 99.9\% confidence levels computed from the covariance matrix of the ensemble of data points illustrated in each panel.
\label{fig_UVW}
}
\end{center}
\end{figure*}

\subsection{Hertzsprung-Russell diagram and relative ages of the subgroups}\label{section3.4}

In this section we use the distances derived from Gaia-DR2 parallaxes (see Sect.~\ref{section3.3}) and spectroscopic data collected from previous studies to construct the most accurate Hertzsprung-Russell diagram (HRD) of the Chamaeleon star-forming region.

We proceed as follows to place the stars in the HRD. First, we compile the spectral types and extinctions for individual stars  determined from past studies thanks to the numerous spectroscopic surveys performed in this region. We found spectral types and extinctions for 144~stars (out of 160~stars) in Cha~I and 28~stars (out of 31~stars) in Cha~II which are given by \citet{Esplin2017} and \citet{Spezzi2008}, respectively. Second, we compute the photospheric luminosities from the $J$-band given by the 2MASS catalogue \citep{Cutri2003} which is available for all sources with measured spectral type in our sample. We corrected the apparent magnitudes for the extinction values given by \citet{Esplin2017} and \citet{Spezzi2008} using the the extinction relations for various bands given by \citet{Cieza2005} to convert them to the adopted band when necessary. We used the bolometric corrections and effective temperatures for the adopted spectral types given in Table~6 of \citet{Pecaut2013} which is specific for pre-main sequence stars. For a few sources in our sample with spectral types earlier than F0 and later than M5 which are not included in this table we adopt the bolometric corrections and effective temperatures for dwarf stars given in Table~5 of that same study as an approximation. We assumed an uncertainty of half a subclass in the spectral types compiled from the literature which results in an uncertainty of about 20 to 120~K on the effective temperatures for most stars. The resulting stellar luminosities are presented in Table~\ref{tab_HRD}.

\begin{figure*}[!h]
\begin{center}
\includegraphics[width=0.49\textwidth]{HRD_BHAC15.pdf}
\includegraphics[width=0.49\textwidth]{HRD_SDF00.pdf}
\caption{HRD of the Chamaeleon star-forming region compared to the grid of isochrones and tracks from pre-main sequence stars models. The green solid and black dashed lines denote the isochrones and tracks for each model, respectively, with the ages (in Myr) and masses (in M$_{\odot}$) indicated in the panels. The different colours and symbols represent the Chamaeleon subgroups discussed throughout this paper. The most massive star in our sample, namely Gaia DR2 5201128124701636864, is not shown here to improve the visibility of the low-mass stars that largely dominate our sample.
\label{fig_HRD}
}
\end{center}
\end{figure*}

Figure~\ref{fig_HRD} shows the HRD with the pre-main sequence star models of \citet{BHAC15} and \citet{Siess2000}. The two grids of models combined together cover the entire mass domain of our sample which ranges from about 0.02 to 3~M$_{\odot}$. HD~97048 (Gaia~DR2~5201128124701636864) in Cha~I is the most massive cluster member in our sample and a A0V Herbig Ae/Be star \citep[see e.g.][]{Chen2016}. Although age determination at these early stages of stellar evolution is rather uncertain, the HRD analysis suggests that the stars in our sample are mostly younger than 5~Myr. In particular, we note that most sources in the HRD are distributed along the 1~Myr isochrone. Some of them are probably binaries or high-order multiple systems that will require further investigation in future studies. However, it is interesting to note that the Chamaeleon stars appear to be younger as compared to previous studies in light of our new analysis based on the stellar distances derived from the Gaia-DR2 parallaxes. For example, the bolometric luminosities derived in this paper are systematically higher than the values obtained by \citet{Luhman2007} who derived the ages of 3-4~Myr and 5-6~Myr for the northern and southern subclusters of Cha~I, respectively. The latter study adopted a distance modulus of 6.05 which defines a (common) distance of 162~pc for all stars in the region (i.e., about 30~pc closer to the Sun than the value reported in Table~\ref{tab_distance_velocity}) and leads to overestimated ages in the HRD. A similar argument also applies to the results obtained by \citet{Spezzi2008} in Cha~II who adopted the distance of 178$\pm$18~pc derived by \citet{Whittet1997} to this cloud and reported ages of 3-4~Myr for most sources in their sample. We computed the isochronal age of the stars in our sample by interpolating between the isochrones of the \citet{BHAC15} and
\citet{Siess2000} models (see Table~\ref{tab_HRD}). This analysis is restricted to the stars covered by these grids of models (see Figure~\ref{fig_HRD}). The median age of the Cha~I and Cha~II subgroups is 1-2~Myr (see Table~\ref{tab_YSO} and discussion below) which suggests that they are indeed younger than previously thought.

Let us now compare the relative ages of the Chamaeleon subgroups based on the fraction of disc-bearing stars in each sample. \citet{Luhman2008} classified the spectral energy distribution (SED) of 122~stars in our sample of 160~cluster members in Cha~I based on the spectral index $\alpha$ \citep{Lada1987} computed from infrared photometry. We compared these results with the ones derived from the classification scheme developed by \citet{Koenig2014} based on infrared colours from the AllWISE catalogue \citep{WISE}. The latter method was originally designed to identify Class~I and Class~II sources which exhibit strong infrared excess emission as illustrated in Figure~\ref{fig_Koenig2014}. We note that many sources in our sample fall into the region between $W2-W3<1.0$ and $W1-W2<0.5$ where Class~III and asymptotic giant branch (AGB) stars reside \citep[see e.g.][]{Koenig2014}. However, given the very young ages of the stars in our sample (as derived from the HRD) it seems unlikely that these sources are AGB stars and we therefore classify them as Class~III stars. When we compare the SED classification derived by \citet{Luhman2008} with the methodology proposed by \citet{Koenig2014} we find a perfect match for all sources in common between the two methods. By combining the two methodologies we were able to classify the SED of 142~stars in Cha~I. We proceeded in a similar manner for the Cha~II sample. We compiled the SED classification for 28~stars from the study of \citet{Alcala2008}, and derived the SED subclass for one additional star based on the method of \citet{Koenig2014}.

\begin{figure}[!h]
\begin{center}
\includegraphics[width=0.49\textwidth]{W2-W3_x_W1-W2_final_Cha1_Cha2.pdf}
\caption{Colour-colour diagram constructed from infrared colours for the Chamaeleon stars identified in our membership analysis. This is one of the diagrams used in the classification scheme proposed by \citet{Koenig2014}.
\label{fig_Koenig2014}
}
\end{center}
\end{figure}

\begin{table}[!h]
\centering
\scriptsize{
\caption{Fraction of disc-bearing stars and age estimates of the Chamaeleon subgroups.
\label{tab_YSO}}
\begin{tabular}{lccrrcccc}
\hline\hline
&&&&\multicolumn{2}{c}{BHAC15}&\multicolumn{2}{c}{SDF00}\\
&&&&\multicolumn{2}{c}{models}&\multicolumn{2}{c}{models}\\
\hline
Sample&$N_{\star}$&Class~II&Class~III&$N_{\star}$&Age&$N_{\star}$&Age \\
&&&&&(Myr)&&(Myr)\\
\hline\hline

Cha~I (north)&70&33 (47\%)&37 (53\%)&33&1.9&49&2.4\\
Cha~I (south)&72&30 (42\%)&42 (58\%)&30&1.4&53&2.4\\
\hline
Cha~I&142&63 (44\%)&79 (56\%)&63&1.7&102&2.4\\
Cha~II&29&22 (76\%)&7 (24\%)&15&1.7&23&2.3\\
\hline\hline
\end{tabular}
\tablefoot{We provide the number of stars and relative fraction of the SED subclasses (in parenthesis), number of stars and median age inferred from \citet[][BHAC15]{BHAC15} and \citet[][SDF00]{Siess2000} models. }
}
\end{table}

In Table~\ref{tab_YSO} we compare the fractions of SED classes and age estimates in the Chamaeleon subgroups. This comparison confirms that the two subclusters in Cha~I exhibit approximately the same number of Class~II and Class~III stars as shown previously by \citet{Luhman2008}. However, our analysis suggests that the fraction of disc-bearing stars is these subgroups is somewhat lower than the results reported in that study which is also related to the different samples of stars used in each study. On the other hand, when we compare Cha~I and Cha~II we see a different situation. The fraction of disc-bearing stars in Cha~II is almost twice as large as in Cha~I, although the two populations appear to have similar ages as inferred from the HRD. This suggests that the survival time of circumstellar discs in the Cha~I subgroup is somewhat shorter. Similarly, we do not detect significant age differences between the northern and southern subclusters of Cha~I. The two SED classes overlap in the 3D space of stellar positions, but the most dispersed cluster members are mostly Class~III stars as shown in Figure~\ref{fig_XYZ}. We find that the two subgroups of Cha~I define two parallel filamentary structures with lengths of about 10~pc in the $Y$ direction (see middle panel of Figure~\ref{fig_XYZ}). We investigated the 3D position of Chamaeleon stars using the distances derived from the different priors discussed in Sect.~\ref{section3.3} and confirmed that the existence of this structure is not an artifact caused by our choice of prior. The median uncertainties in the XYZ positions of Cha~I stars are 1.1, 2.1 and 0.6~pc. The larger uncertainties in Y can be explained by the higher correlation of the distance with Y ($\rho_{Y}=-0.95$) as compared to the X and Z coordinates ($\rho_{X}=0.73$ and $\rho_{Z}=-0.09$). Thus, the filamentary structure observed for Cha~I is more likely to be explained by the large uncertainties in Y.

\begin{figure*}[!h]
\begin{center}
\includegraphics[width=0.33\textwidth]{XY.pdf}
\includegraphics[width=0.33\textwidth]{YZ.pdf}
\includegraphics[width=0.33\textwidth]{XZ.pdf}
\caption{Spatial distribution of Chamaeleon stars. The different colours denote the subgroups of stars in our sample. Filled and open symbols indicate Class~II and Class~III stars, respectively.
\label{fig_XYZ}
}
\end{center}
\end{figure*}

We now compare the results obtained in this paper with the ones derived by our team for the Corona-Australis \citep{Galli2020} and Lupus \citep{Galli2020b} star-forming regions using the same methodology applied in this paper. The two age indicators used in this study (isochronal ages and fraction of disc-bearing stars) suggest that the Chamaeleon stars are younger than the stellar population in the Corona-Australis star-forming region. On the other hand, the Chamaeleon subgroups appear to be coeval with the Lupus association and we do not detect important age differences among the various subgroups in these two star-forming regions.

\section{Conclusions}\label{section4}

In this paper we revise the census of stars with available astrometry in the Gaia-DR2 catalogue that are associated to the molecular clouds of the Chamaeleon star-forming region. We applied a probabilistic method to infer membership probabilities and our  analysis allowed us to identify 188~members and 41~members in Cha~I and Cha~II, respectively. We confirm most of the historical members from the literature (with available Gaia-DR2 astrometry) and increase the samples of cluster members by 11\% and 21\% in Cha~I and Cha~II, respectively.

We combined the Gaia-DR2 astrometry with ancillary RV data from previous studies to investigate the 3D spatial distribution and 3D space motion of Chamaeleon stars. We confirm that Cha~I and Cha~II are located at different distances and are separated by about 23~pc in the 3D space of positions. The two subclusters of Cha~I (north and south) are also located at different distances and the observed difference in their proper motion distributions is more likely to be due to projection effects. Our analysis shows that Cha~I (north) and Cha~I (south) have consistent space motions within the reported uncertainties. The HRD analysis reveals that the stars in our sample are mostly younger than 5~Myr and cover the mass range from 0.02 to 3~$M_{\odot}$. The median age of the stars is about 1-2~Myr. We detect no significant age differences between Cha~I and Cha~II, but show that these stellar populations exhibit different fractions of disc-bearing stars. When we compare the results obtained in the present study with those of previous studies conducted by our team in other star-forming regions, we conclude that the Chamaeleon subgroups appear to be coeval with the Lupus association and are younger than the stellar population in the Corona-Australis star-forming region.

In this study we restricted our analysis of the stellar population of the Chamaeleon clouds to the sources with available data in the Gaia-DR2 catalogue. Our team is measuring precise proper motions of faint sources (beyond the sensitivity limit of the \textit{Gaia} satellite) as part of the DANCe project and we will soon be able to expand the current census of Chamaeleon stars based on ancillary data and spectroscopic observations to accurately derive the initial mass function of this stellar group.

\begin{acknowledgements}
We thank the referee for constructive criticism that improved the manuscript. This research has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 682903, P.I. H. Bouy), and from the French State in the framework of the ``Investments for the future" Program, IdEx Bordeaux, reference ANR-10-IDEX-03-02. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. This work has made use of data from the European Space Agency (ESA) mission {\it Gaia} (\url{https://www.cosmos.esa.int/gaia}), processed by the {\it Gaia} Data Processing and Analysis Consortium (DPAC, \url{https://www.cosmos.esa.int/web/gaia/dpac/consortium}). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the {\it Gaia} Multilateral Agreement. This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration.
\end{acknowledgements}

\bibliographystyle{aa}

redpajama_set_name: RedPajamaArXiv


text:
Enlighten Series
Escape Chronicles
Deities Series
Reverse Harem Books
Slayers & Protectors Series
Element Keeper Series
Book Release! A Mirrors Beginning: John Brooks by K.G. Stutts
High school drop out, former addict, now fights for intergalactic peace.
Life can be rough for some. For one man, life was insurmountable. One dead end after another lead him in and out of jail more times than he ventured to count. One such occasion changed everything.
Charlie Westlake is a man on a mission to bring together one of the top teams the universe has ever seen. On this trip, he will bring an empty man into the fold. Will he make it?
John Brooks enters the ISC in this, a prequel to Mirror Image.
Five random facts about the series:
1. Before Jackson joined the ISC, he was married and had a little girl. You'll be able to read his backstory next year.
2. Maddie has a file on her computer of ideas for future products, including one for genetic enhancements. It becomes the Genesis Project. More on that when the Amethyst Chronicles: Winter's Kiss launches in November.
3. I started writing Mirror Shattered immediately following Mirror Image.
4. This is book two in a five part series, including a prequel for Mack.
5. Short stories for Jackson and the Brooks brothers will follow next year.
Five Facts about the ISC:
1. The Intergalactic Security Commission encompasses 115 planets and has a total of 40 bases spread out over the galaxy.
2. The ISC isn't formally part of the military. It acts as a separate entity.
3. Even though it isn't involved in the military, the ISC is mostly made up of scientists and service personnel.
4. On the outside, the ISC appears like a normal city building. The building uses self-generating shields and holographic projectors to hide the roof top hanger. The more secure levels are underground.
5. Before the start of Mirror Shattered, Charlie Westlake has been the only Commander.
Five personal facts while writing this story:
1. Several scenes in this book made me cry but there is one in particular that I cried while writing.
2. Like a lot of sci-fi fans, I've always been fascinated with time travel. When I began planning this series, I knew the second book would involve going to the past. The results were anything behind I had ever dreamed.
3. I subscribe to the Joss Whedon school of philosophy: "Make it dark, make it grim, make it tough, but then, for the love of God, tell a joke." That describes the latter half of Mirror Shattered.
4. I pose the question if you could go back in time to affect the future, not knowing if what you do is for the better or not, would you do it? That question kept me up for two nights.
5. The changes both Mack and Maddie go through in this book inspired me and will forever change the course of the series.
Link for Mirror Image: http://www.amazon.com/dp/B00JWZZ20W
http://www.barnesandnoble.com/w/mirror-image-kg-stutts/1119689102?ean=9781633100022
Bio of Author, K.G Stutts:
Growing up in Texas and later South Carolina as the youngest in a house full of science-fiction fans, K.G. Stutts had her natural curiosity and imagination nurtured since birth by family movie nights where they would watch Star Wars, Indiana Jones and even timeless Disney favorites.
A prolific writer of sci-fi, romance and mystery, K.G. draws much of her inspiration from those amazing works that gave her an appreciation for telling rich, compelling, character-driven stories for all audiences.
Keep up with her on Facebook at http://www.facebook.com/KGStutts or follow her on Twitter at http://twitter.com/KGStutts. She also is on Tumblr at kgstutts.tumblr.com. She lives in North Carolina with her husband, Brad and works for a customer service call center full time. She's a lover of Star Wars, Star Trek (original, TNG, and Voyager), Stargate SG-1 and Atlantis, Muppets, Garfield, Disney, Indiana Jones and is a big football, wrestling, and hockey fan.

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Companies are being extremely focused on hiring and retaining great talent but it takes more to drive high performance. Another key requirement is talent management and development. The leaders who align their company strategy with their talent management and development strategy are most likely to win the war against their competitors and achieve growth.
Its true it takes a lot of time and effort to implement effective talent management and development practices. But what are the strategies to create a high performing company using your talent?
Aligning employee goals to company vision and strategy. Every company should follow a top-down approach to communicate the strategy to the teams. Every manager should be able to access the objectives of other teams to allow for cross-team collaboration as well as faster goal completion. Goal alignment will enhance talent management and improve performance.
Create internal talent pools. In many occasions companies where unable to pivot due to lack of a talent pool with a specific skill-set. A company should not rely on external talent pools but it should focus in building a strong internal talent pool. The creation of an internal talent pool will allow for specific skill-set development and easier to plan talent succession. Also, a company will be able to fill leadership roles internally rather than hiring externally someone which might not have the required skill-set. A great company example that has strong internal talent pools is General Electric. Internal talent pools are part of GE's culture for the past 50 years.
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Talent management and development strategies are essential to build a high performing company with a workforce that successfully executes company objectives.


text:
Q: Split html body in two parts Is it possible to split the body tag in my web site into two parts. For example having the first 50% (in height) in the background-color in white and the rest 50% in black?
Does anyone know how I can do that?
Thanks

A: If it's enough to work with CSS3, then the following do the trick:
body {
  background: #000 url(white.png) 0 100% repeat-x;
  -webkit-background-size: 0 50%;
  -moz-background-size: 0 50%;
  -o-background-size: 0 50%;
  background-size: 0 50%;
}

Or you need a (vertically) big enough image, then:
body {
  background: #000 url(white.png) 0 50% repeat-x;
}

A: You can make two DIVs for the two colors and place the content in a third div over them.

A: Not sure what you mean to 'split' the body but here is how you could do 50% 50% background

redpajama_set_name: RedPajamaStackExchange


Blue with black base and grey/grey-brown interior trim. This large scale model has all the usual Bburago features: opening rear engine cover and front compartment, opening 'scissor' doors and working steering. With 'glazed' lights, and detailed engine, dashboard and suspension. Chrome 'alloy' wheels with black tyres.
Generally in near mint condition, but one of the front indicator lenses has broken off and is missing.


text:
Q: Coefficient matrix for an implicit scheme for heat equation I have a problem where I have 1D filament that is heated in one end (x=0) at a temperature T=210C (this is the first non-homogenous BC), and the other end (x=L) we have two boundary conditions: T(x=L)=25C and there is no flux (Neumann BC) at this point where dT(x=L)/dx=0.
I am using the implicit Euler scheme in time and central difference in space to solve the !D heat equation and model this system. However, looking at the solution I can see that the coefficient matrix of this scheme is given by:

Which has been confusing me a lot! Could you please shed some light on why the matrix looks like this. Specifically speaking, why A(0,0) = A(N,N) = 1 instead of -2, and why A(0,1)=0, and lastly why A(N-1)=A(N-2)=-1!
Thanks in advance.

A: The first thing to note is that the finite difference scheme
$$
\Bigl[\frac{\Delta x^2}{\alpha\Delta t}+2\Bigr]T_i^n-[1]T_{i+1}^n-[1]T_{i-1}^n=\Bigl[\frac{\Delta x^2}{\alpha\Delta t}\Bigr]T_i^{n-1}
$$
applied to all interior points $i=2 \cdots N-1$ can be represented as
$$a_iT_i^n + b_iT_{i+1}^n + c_iT_{i-1}^n=d_i$$
where $$a_i = \Bigl[\frac{\Delta x^2}{\alpha\Delta t}+2\Bigr]$$
$$b_i = c_i =-1$$
and $$d_i = \Bigl[\frac{\Delta x^2}{\alpha\Delta t}\Bigr]T_i^{n-1}$$
which represents a tri-diagonal matrix, so that there is no need for the storage of a full matrix.
The boundary conditions are implemented as
$$
a_1 = 1,\hspace{0.5 in}b_1=0,\hspace{0.5 in}c_1=0,\hspace{0.5 in}d_1=T_0
$$
and using a simple backward finite-difference for the Neuman condition at $x=L$, ($i=N$), we have
$$
a_N = 1,\hspace{0.5 in}b_N=0,\hspace{0.5 in}c_N=-1,\hspace{0.5 in}d_N=0
$$
So you start with the initial condition $T_i^0 = T(x)$ at all nodes and then solve the tri-diagonal system at each time step, and then update the temperatures until the desired time interval has been simulated.
The tri-diagonal system can be very efficiently solved with the Tri-Diagonal Matrix Algorithm (TDMA or Thomas) Algorithm which is described at
https://en.wikipedia.org/wiki/Tridiagonal_matrix_algorithm
I wrote the section in the wiki page above that also has the code for solving a TDMA so feel free to ask me if anything isn't clear.
Hope this helps. Note that my indexing runs from $1$ to $N$, but can just as easily be implemented from $0$ to $N$.


text:
FA: Replica trophy presented to current holders
Director of external affairs makes presentation to Manchester United
Current FA Cup holders Manchester United took receipt of their record equalling twelfth replica trophy ahead of their third round meeting with Reading earlier this month, and it fell to United's Director of External Affairs John Nixon – current chairman of the FA Cup committee – to hand over the three-quarter sized memento of the achievement.
Club legend Sir Bobby Charlton and Old Trafford's Executive Vice President Edward Woodward accepted the trophy ahead of a home victory which the Premier League giants hope will set them on the way to more success in the competition this year.
Speaking about the presentation this week, Mr Nixon explained: "The FA Cup Committee is always chaired by someone from the either the Premier League or the EFL, on a year-by-year rotation.
"It was the EFL's turn to take up the position this season and it was an honour for me to be asked to become chairman of the committee, on their behalf, back in August. The role of the committee is to oversee and run the competition, all the way from the preliminary qualification rounds through to the final.
"For example, if any issues such as floodlight failures or use of unregistered substitutes arise we deal with them and make decisions based on the evidence presented. People talk about the magic of the FA Cup and it's something I've seen first hand at every level now, having taken up this position.
"Every club just wants to progress to the next round and stay involved for as long as they can. Financially it can be quite lucrative for some of the smaller teams in the earlier rounds, so there's that to factor in as well.
"On top of governance of the competition itself we are also responsible for the organisation of the draws, from the first qualifying round to the semi-finals, which includes looking at who does them and where they're done, in conjunction with BT Sport and the BBC."
"One of the pleasant duties which came up with the role of Chairman was to present a replica of the FA Cup to the current holders, Manchester United," he continued. "Every winner gets one of these each season, and it's generally a presentation which takes place on or around the date of their first game in the following season's competition. That meant a home game against Reading was an ideal time for it to take place.
"It was interesting when I went along to Old Trafford to hear the amount of discussion about the fact they are now joint record holders with Arsenal, having won the competition 12 times. There's a determination there for them to go on and make the record their own this season.
"They made a great start by winning the game, but I know they were all really pleased to see Jaap Stam [manager of Reading FC] again because of the service he had given to the club."
"I have to say it was an honour to make the presentation on behalf of the FA, but I always had one eye on what was happening at Colchester," he confirmed. "I watched as the scores came through on my phone and I caught the last bit of the game on Radio Cumbria as I travelled back up north. It was disappointing not to be at the Colchester game in the first place and even more disappointing that it wasn't a positive result.
"However, having now been to a club as big as Manchester United on an FA Cup match day I can say with complete confidence that the competition holds the same level of excitement and expectation as it does for clubs at every other level.
"Inevitably we will see squad rotation, because they're involved in a number of competitions, but they still very much want to win it and they put out a team which they think will comfortably see them through each round. To be fair, it was a very strong side, with Wayne Rooney equalling a club record, so there was no doubt they were taking it very seriously indeed."
"Being chairman of the committee means I will end up going to the FA Cup final this season," he concluded. "It's a fascinating insight into how one of the most respected competitions in the world operates and it's good to be involved in many of the decision making processes.
"Hopefully we'll see another exciting competition, with plenty of twists and turns, and it'll be interesting to see if anyone can stop either Manchester United or Arsenal from taking their win tally to thirteen."
CLUB: Cumberland Community Stand undergoes facelift
The Cumberland Community Stand [A Stand] is undergoing a facelift over the coming weeks.
SKY BET POPPY AUCTION: Signed 2018/19 Poppy Shirts up for grabs
Signed shirts from 58 of the clubs within the Sky Bet EFL are going under the hammer in aid of The Royal British Legion's Poppy Appeal.
CLUB: Farewell to former-manager Ian MacFarlane
Carlisle United are deeply saddened to learn, this week, of the passing of former-manager Ian MacFarlane at the age of 86.
INTERVIEW: We have two clubs trying to take him
Another apparent subject of summer interest has been striker Hallam Hope, who spoke via his agent just after the season had ended to indicate that he would like to play at a higher level, should the...


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The idea of wearing mascara any color other than black may make you cringe, but if you choose the right shade it really makes your eyes pop. Use our guide.
The idea of wearing mascara any color other than black or brown may give you flashbacks to high school, and understandably so. But if you choose the right shade—that is, nothing sparkly or neon—it can really make your eyes pop, says Blair Patterson, director of global makeup artistry for Estée Lauder.
For a subtle, eye-catching look, Patterson suggests using the tinted formula up against your lash line, then covering the rest of the lashes with your usual mascara.
Looking for even more ways to enhance your eyes? Check out the The Golden Rules for Wide, Bright Eyes and The Pretty Way to Do Bright Eye Makeup.