Analysis of membership probability in nearby young moving groups with Gaia DR2

LACEwING uses the spatial and kinematic information of an object such as Right Ascension ($\alpha$), Declination ($\delta$), distance, proper motion in Right Ascension ($\mu_{\alpha}$), proper motion in Declination ($\mu_{\delta}$), and radial velocity ($\gamma$) to check whether the object is a member of any of the known NYMGs. Inside the LACEwING framework each NYMG is considered as two triaxial ellipsoids having base values of U, V, W and X, Y, Z. For the members of each moving group, LACEwING predicts the observable values such as distance, $\mu_{\alpha}$ and $\mu_{\delta}$ at the $\alpha$ and $\delta$ of the input star. These predicted values are compared with the measured quantities. The working of LACEwING mainly comprises of three steps. In the first step, LACEwING predicts the properties such as proper motion, radial velocity and distances corresponding to $\alpha$ and $\delta$. To generate the spread of U, V, W space velocities of each NYMG, 100,000 Monte-Carlo iterations are made within the triaxial ellipsoids to find out the actual membership probability. The U, V, W velocities are transformed into proper motion components, $\mu_{\alpha}$, $\mu_{\delta}$ and $\gamma$ by inverted matrices given in Johnson & Soderblom (1987) to simulate a star for each moving group at a standard distance of 10 parsecs for the same $\alpha$ and $\delta$ of the target. By making use of this predicted values and the available kinematic data of the star, four goodness-of-fit matrices such as proper motion, distance, radial velocity and position will be produced. The number of matrices can vary upto four based on the availability of observables of input star. The quality of match of the star with one of the moving group is represented by each of these matrices. In the second step, these matrices are combined in quadrature and the goodness-of-fit values are converted into membership probabilities. Also LACEwING accounts all the 7 possible combinations of $\alpha,\delta$, $\mu_{\alpha}$, distance, $\mu_{\delta}$ and $\gamma$ each having its own goodness-of-fit values. These probabilities are quantified by making use of a Solar Neighborhood simulation which is ran through the LACEwING algorithm. Hence, even if one of the parameter is missing, the other combinations will be used to estimate the membership probability.

Further analysis of membership probability is carried out on this simulated star (Riedel et al., 2017). A small variation in $\mu_{\alpha}$, $\mu_{\delta}$ can reflect in the distance, U, V, W and X, Y, Z values. This can easily be evaluated by comparing new estimations from the Gaia DR2 data with those listed in CSNYMGS.

Vector point diagram is one of the basic tools that can be used to distinguish moving group members from field stars (e.g., Hoogerwerf & Aguilar, 1999). Figure 1 represents the vector point diagram of stars in Octans moving group, constructed using proper motion information from Gaia DR2. Also shown are the proper motion values of the members listed in CSNYMGS. It is evident from the figure that the proper motion values from the Gaia DR2 shows less error when compared to the values listed in CSNYMGS. This explains the accuracy of Gaia DR2 proper motion values over previous estimates.

Since our moving group stars are close to the Sun, they are spread over a large area in the sky. Hence, it is difficult to identify their association by means of stellar density, as in the case of star clusters. The best way to group these stars is based on their common space motion (Torres et al., 2006). The inclusion and elimination of members from a moving group rely on the space velocity values, for which we need accurate parallax, proper motion and radial velocity measurements (Zuckerman & Song, 2004). Gaia DR2 currently provides the best possible proper motion and radial velocity values, from which we can calculate the space velocity (U, V, W) values. We selected $\mu_{\alpha}$, $\mu_{\delta}$ and $\gamma$ from Gaia DR2 database for our sample of 890 stars. Since 367 stars were having Gaia radial velocity measurement, we have used this sub-sample for this particular analysis. The U, V, W space velocities are calculated using the matrices from Johnson & Soderblom (1987), in which right-handed convention is used. This means that U, V, W values are considered positive in the direction of Galactic center, Galactic rotation and the North Galactic Pole (NGP), respectively. Along with the Right Ascension ($\alpha$) and Declination ($\delta$) of the objects, the matrices make use of the following quantities and their uncertainties, i.e., the parallax in arcsec ($\pi\pm\sigma_{\pi}$), the radial velocity in $km~{}s^{-1}$ ($\gamma\pm\sigma_{\gamma}$), the proper motion in right ascension in $arcsec~{}yr^{-1}$ ($\mu_{\alpha}\pm\sigma_{\mu_{\alpha}}$) and the proper motion in declination in $arcsec~{}yr^{-1}$ ($\mu_{\delta}\pm\sigma_{\mu_{\delta}}$). The matrices, as given in Johnson & Soderblom (1987), are reproduced below.

The transformation matrix is given by,

and the coordinate matrix is defined as,

The space velocity components are given by,

where B = T$\cdot$A, and k = 4.74057, the equivalent in km $s^{-1}$ of one astronomical unit in one tropical year. The associated error is estimated by,

where, $c_{ij}=b_{ij}^{2}$ for all i and j.

As an example, the U, V, W values calculated for stars in the ABDor moving group are given in Table 1. Also shown are the values listed in CSNYMGS. The average percentage error associated with the previous U, V, W estimates are 14$\%$, 4.8$\%$ and 17.2$\%$, respectively. But the average percentage error obtained from our estimate using Gaia DR2 reduced to 6.4$\%$, 1.9$\%$ and 5.1$\%$, respectively. Since the error associated with the estimates reduced significantly it is clear that the U, V, W space velocities of stars estimated from Gaia DR2 are accurate than previous estimates.

Based on the kinematic analysis of moving group members, Zuckerman & Song (2004) suggested that U, V, W velocities of almost all the young stars are within 10 km s^-1 of the average U, V, W value of Eggen’s Local Association (Jeffries, 1995). It is then generalized that, for young stars U, V, W velocities fall in the “good box” with U, V, W dimensions in the range 0 – -15 km s^-1, -10 – -34 km s^-1, and +3 – -20 km s^-1, respectively (Zuckerman & Song, 2004). We checked whether our sample of moving group candidates fall inside the “good box” while using Gaia DR2 data. As an example, we have defined the “good box” for the moving group Octans in Figure 2. It is evident from the figure that the U, V, W velocities of the moving group members calculated from the Gaia DR2 data spans out of the “good box”. Hence, the “good box” criterion is not completely satisfied by the catalogued members in the Octans moving group. This suggests the need to re-estimate the membership of moving group candidates using Gaia DR2 data.

We have analyzed the membership probability of candidates in the moving groups with LACEwING algorithm using Gaia DR2 data. LACEwING is used to calculate the membership probability of a star in a particular moving group (Riedel et al., 2017). Since LACEwING is capable of determining the membership probability even with the incomplete kinematic data, by making use of the goodness-of-metrices and the simulation of solar neighbourhood as discussed in section 3.1, we were able to reanalyse the membership probability of the 890 stars. During the analysis we found the membership of some stars changed from one moving group to another. As an example, we explain the case of Tuc-Hor moving group, which hosts maximum number of stars. Although Horologium moving group (Torres et al., 2000) and Tucana moving group (Zuckerman et al., 2001) were discovered separately, Zuckerman (2001) confirmed that both of them contain similar stars and hence was classified as Tucana-Horologium (Tuc-Hor) moving group. Our analysis suggested that most of the members in Tuc-Hor has a definite probability to belong to either Columba or $\beta$ pic moving groups. This may be due to the overlap in the properties of these moving groups. One important aspect we saw during the analysis is that a few stars which were identified as moving group members now belong to open clusters. The stars RX J1005.3-7749 and RX J0915.5-7609 are now confirmed as $\eta Cha$ members whereas 2MASS J08082487+4347557, HD 221539, 2MASS J02501167-0151295, HIP 15247A, HIP 15247B, HD 22213, 2MASS J03445673-1145126, 2MASS J04174743-2129191 and 2MASS J04365738-1613065 are re-assigned to Hyades open cluster. This can be seen as a direct consequence due to the change in the input data used for the present analysis.

From the analysis of membership probability using Gaia DR2 data, we found 611 stars remaining as members of any of the thirteen moving groups. It may be noted that from Riedel et al. (2017) we have obtained 890 candidates as moving group members having Gaia DR2 data. Hence, from our analysis by making use of accurate Gaia DR2 data, we found that 279 stars do not belong to any of the known moving groups. The new estimates of distance, age and space velocity values of moving group members from the available Gaia DR2 is listed in Table 2. So, for the present study, we will be using a sample of 611 member stars, belonging to 13 moving groups.

The analysis of space velocity plays a key role in the membership determination of stars in a moving group. This is further important in deciding the “good box” criterion for moving groups (Zuckerman & Song, 2004). Gaia DR2 provides homogeneity in the measurement of parameters such as parallax, proper motion and radial velocity, which is one of the fundamental requirement for estimating $U$, $V$, $W$ space velocity values. We found that among the moving group candidates listed in CSNYMGS, 250 have parallax, proper motion and radial velocity values in the Gaia DR2, which can be used for estimating space velocity. It may be noted that the number of stars for this analysis got reduced since only 250 stars have radial velocity values from Gaia DR2.

We estimated the U, V, W values of 250 stars using the method described in previous section (Sec. 3.1). The U, V, W estimates from this study shows considerably lesser error with a mean value of around 0.54 km s^-1, 0.76 km s^-1 and 0.58 km s^-1, respectively, which is better than previous reported values (Montes et al., 2001; Malo et al., 2013; Maldonado et al., 2010; Shkolnik et al., 2012; Makarov, 2007). For example, the average error estimate in the space velocity values for the re-analyzed members of AB Dor moving group is around 0.32 km s^-1 whereas those compiled in CSNYMGS has an average error of 1 km s^-1. This is because the input values for space velocity estimation were taken from different resources in previous studies. For example, Malo et al. (2013) compiled parallax, proper motion and radial velocity values from various resources such as van Leeuwen (2007), Koen et al. (2010), Messina et al. (2010) and Bailey et al. (2012). The advantage of present space velocity estimation is due to the homogeneous dataset of Gaia DR2, from which all the input data is taken. In addition, the input values such as proper motion and radial velocity are more accurate than previous missions/surveys.

The velocity ellipses between the different space velocity components ($U$ and $V$, $U$ and $W$, and $V$ and $W$) are generated for all the moving groups and is shown in Figure 3. The left panel shows the ellipse generated using a sample of 367 stars from CSNYMGS having $U,V,W$ velocity estimates from Gaia DR2. The right panel represents the space velocities of the new sample of 250 members whose membership is confirmed from this study. We have included the “good box” criterion in the figure with the dimensions 0 – -15 km s^-1, -10 – -34 km s^-1, and +3 – -20 km s^-1 for $U$, $V$ and $W$ respectively. It is seen from Figure 3 that the velocity ellipses of all the moving groups are well constrained within the “good box”. Hence, from the analysis of space velocities, we found that 117 stars (i.e., 367 - 250) do not belong to any of the known moving groups. We had shown in previous section that 279 stars are identified as non-members from the analysis of proper motion in LACEwING framework. Among this sample, the space velocity analysis has reconfirmed the non-membership of 117 stars. In the Gaia era, there is a need to re-estimate the “good box” criterion. Since, LACEwING makes use of “good box” defined in Zuckerman & Song (2004) for estimating the membership probability, we are also considering the same criterion for the present work. Distribution of space velocity estimates obtained for the confirmed moving group stars suggests a more wider “good box” compared to that proposed by Zuckerman & Song (2004). However, as a future work, we plan to re-define the “good box” criteria by making use of Gaia DR 3 data.

Stars belonging to nearby young moving groups are quite near ($<$ 100 pc) and hence the protoplanetary disc in them can be studied in detail using imaging techniques such as interferometry (e.g., Rebull et al., 2008; Zuckerman et al., 2011; Rodriguez et al., 2015; Binks & Jeffries, 2017). The proximity of these stars makes them the best targets for direct detection of low-mass planetary companions (e.g., Chauvin et al., 2004, 2005; Lagrange et al., 2010; Bowler et al., 2013, 2017; Dupuy et al., 2018). For studies related to protoplanetary disc and planetary mass companions, it is important to accurately estimate the age of the host star.

Conventionally, the age of a star is estimated from their location in the Color-Magnitude Diagram (CMD), which is overplotted with isochrones generated from stellar models (Ortega et al., 2002, 2004; Song et al., 2003). Age can also be estimated by other means such as Lithium abundance, rotation rate and stellar activity (Zuckerman & Song, 2004; da Silva et al., 2009). Age estimates based on Lithium Depletion Boundary (LDB) technique was proposed as one of the most reliable age estimation methods since it is less model dependent compared to other age estimation techniques (Soderblom et al., 2014; Mathew et al., 2017). However, LDB technique provide reliable age estimates for stars in which lithium burning happens in the early phase of their lifetime. This restricts the spectral range over which this technique can be used. The dynamical age of a moving group can also be estimated from convergent point analysis (e.g. Barenfeld et al., 2013). Bell et al. (2015) employed semi-empirical pre-main sequence models with $\tau^{2}$ minimization to estimate the age of moving groups. Most of these studies estimate the common age of the moving group, considering the fact that all stars which belong to moving group are bona fide members. It is to be noted that there are issues related to the model dependency of age estimates using isochrone fitting (Bell et al., 2012, 2013, 2014) when factors of stellar variability and binarity are taken into account. But the high quality photometry and distance data available from Gaia DR2 enable us to create better CMD. We have estimated the individual age of the moving group members from Gaia CMD.

The science data of Gaia DR2 consists of broad-band photometry in the unfiltered G band and integrated $G_{BP}$ and $G_{RP}$ magnitudes (Gaia Collaboration et al., 2016). This will equip us with a complete data set for constructing CMD purely based on Gaia DR2 data (Gaia Collaboration et al., 2018b). It may be noted that the advantage of this CMD analysis is due to the homogeneous use of stellar parameters from a single mission/database. By making use of distances estimated from the Gaia parallax, absolute G magnitude ($M_{G}$) is calculated. Based on this analysis we have estimated the age of 505 stars in our sample. Nearly 106 stars do not have age estimates since majority of them were placed either below the low mass limit of the isochrones or below the main sequence in the CMD (Pecaut & Mamajek, 2013). Also, it is to be noted that we are not considering the BP/RP excess factor in the sample selection. These factors may account for the non-estimation of ages for the 17$\%$ of our sample.

The stars belonging to a particular moving group is included in the CMD where isochrones belonging to ‘Modules for Experiments in Stellar Astrophysics (MESA) Isochrones and Stellar Tracks’ (MIST) are plotted. MIST provides stellar evolutionary models for different ages, masses and metallicities (Choi et al., 2016; Dotter, 2016). Isochrones using Gaia DR2 data are also included in the latest update of MIST. We have overplotted a grid of MIST isochrones over the CMD and found the closet two isochrones and took the average of the two as the age. The Gaia CMD ($M_{G}$ vs $G_{BP}$–$G_{RP}$) of all the members in the AB Dor moving group is shown in Figure 4. Although most of the stars are found to be in the age range 30–50 Myr, a few lower age candidates are also visible. Rather than a single age, a spread in age is visible for all the moving groups. A comparison between the present age estimates and those mentioned in literature is given in Table 3. For most of the moving groups our age estimates agree with those identified from previous studies. However, we found moving groups such as Carina-near to be younger. This may be due to different age estimation methods employed in previous studies.

The new age estimates of all the 505 stars, belonging to thirteen moving groups are listed in Table 2. As an example, we discuss the age estimation of 123 members belonging to AB Dor moving group. A comparison of our estimates with those listed in CSNYMGS is shown in Figure 5. It is seen that for $\sim$73% of stars, our new age estimates are considerably near or in the range predicted by previous studies. However, large variations occur for three stars, whose literature values are adopted from Maldonado et al. (2010). Instead, if we adopt the age of these stars from other works, it matches with our estimates. For example, in the case of G112-035, age mentioned in (Maldonado et al., 2010) is 2200 Myr whereas that listed in Lafrenière et al. (2007) is 150–300 Myr. This suggests that the estimation of the age of a star is heavily dependent on the techniques and assumptions adopted for estimating the age (e.g, Bell et al., 2012, 2013). Since we used Gaia CMD for the age analysis, the uncertainties in age estimation due to the use of data from diverse resources is mitigated. Hence, the present age estimates are more reliable and can be used for future work. The stellar parameters of moving groups estimated from this work is listed in Table 4.