Unveiling two expanding stellar groups formed through violent relaxation in The Lagoon Nebula Cluster.

During the formation of a stellar cluster, the potential well becomes deeper as the cloud collapses. Since the varying potential performs work on the stars, the energy of individual particles is not conserved (Lynden-Bell, 1967). Furthermore, the particle can gain or lose energy depending on the orbital phase and the timing of the potential variation. The net effect is that particles are scattered in energy, and as long as the potential keeps varying in time, this random scattering will persist. This process is called violent relaxation, and its timescale is of a couple of dynamical timescales (Binney & Tremaine, 2008).

One can envisage two possible behaviors for the variation of the gravitational potential during violent relaxation. On the one hand, if the stars produce the potential, the violent relaxation is self-limiting: the more relaxed the particles become, the fewer fluctuations are available for driving relaxation. In contrast, if the potential is dominated by an external agent, such as the collapsing molecular cloud during the formation of the cluster, or the expanding molecular cloud during cloud disruption, violent relaxation will continue as long as the potential keeps fluctuating.

A key ingredient is that violent relaxation produces equal velocity dispersion for all particles (Lynden-Bell, 1967). The reason for this effect is that the mass of the stars is utterly negligible compared to the mass represented by the potential fluctuations. It is important to note that, although this process does not produce energy equipartition between particles (they all have the same velocity dispersion), the whole system does it. Indeed, at the end of a collapsing stage, either gas (e.g., Vázquez-Semadeni et al., 2007) or particle systems (e.g., Noriega-Mendoza & Aguilar, 2018) will tend globally to virialization, with the total gravitational energy being twice the total kinetic energy of the system (see also Ballesteros-Paredes et al., 2020).

Another relaxation mechanism relevant in few-body systems is collisional relaxation. When particles undergo strong encounters such that their trajectories are substantially deflected, energy is exchanged between them. Although the specific outcome depends on the details of the encounter, the interchange of energy between particles will tend to statistically produce energy equipartition between them, meaning that more massive particles will end up with less velocity than low-mass particles.

Nonetheless, gravitational systems have negative heat capacity (Antonov, 1985; Lynden-Bell & Wood, 1968), and thus, the statistical tendency of collisions to produce energy equipartition produces the opposite effect in gravitational systems, i.e., as the system loses energy, it becomes hotter dynamically²²2As a consequence, the entropy of a gravitational system is not bound, and thus, an isolated system can never reach equilibrium (see, e.g., Binney & Tremaine, 2008, §4.10.1).. In order to understand this effect, one can imagine a system of only two types of particles: heavy and light, initially thoroughly mixed and in equilibrium with their own potential. As collisions occur, heavy particles lose velocity. Consequently, lacking support, they fall inward to a tighter new equilibrium configuration with larger velocities. The net effect is that mass and velocity segregation occurs as particles try to get energy equipartition via collisions. Heavy particles become more concentrated with larger velocity dispersions than light ones ³³3A point worth noticing is that spatial segregation should be present for collisions to be responsible for the heavier particles having a larger velocity dispersion. If the latter is observed without the former, collisions are not responsible for the dynamical heating.. In an extreme version of this effect, the massive stars become so concentrated that they evolve on their own collisional timescale and become dynamically detached from the rest of the cluster. This is the Spitzer instability (Spitzer, 1969).

Numerical simulations have consistently found that stellar clusters produce the dynamical heating of massive stars. Then, it is considered that the Spitzer instability typically does occur (e.g., Trenti & van der Marel, 2013; Spera et al., 2016; Bianchini et al., 2016; Parker & Wright, 2016), with the last authors arguing that it may occur within a few Myrs for young stellar clusters.

Theoretically, the difference between the Spitzer instability and the regular mass segregation is a matter of timescales. Without the Spitzer instability, mass segregation grows on the global collisional timescale. With the Spitzer instability, the heavier stars detach dynamically from the rest of the system and collapse at an accelerated pace, that of their own collisional timescale. Observationally, it will be difficult to disentangle whether a cluster has experienced the Spitzer (1969) instability, or just collisional relaxation. Being essentially the same process, they both share the same features, though the latter is more extreme than the former. In what follows, thus, we will just talk about collisional relaxation.

While the characteristic timescale for violent relaxation is of the order of 1-2 free-fall timescales

with $\rho$ the mean density of the system, and $G$ the gravitational constant, the characteristic timescale for collisional relaxation is the collisional timescale,

where $N$ is the number of particles, and $t_{\rm cross}\sim R/\sigma_{\upsilon}$ the crossing time of the system, given by its size $R$ and velocity dispersion $\sigma_{\upsilon}$ (Binney & Tremaine, 2008) . In general terms, the violent relaxation timescale (1-2 free-fall timescales) is substantially smaller than the collisional relaxation timescale. For instance, the free-fall time of a molecular cloud core at a density of 10⁴ cm^-3 is of the order of $\tau_{\rm ff}\sim$1 Myr. In contrast, the collisional timescale of a young stellar cluster with $\sim$1000 particles (as the Lagoon or the Orion Nebula clusters) and a crossing time of $\sim 1-3$ Myr is of the order of 14-50 Myr. However, it is worth pointing out that numerical simulations by Parker et al. (2016) of young clusters have shown dynamical heating and ejection of massive stars in $\sim 5-10$ Myr.

From this perspective, one can expect that young clusters, with ages $\lesssim$5 Myr, should exhibit signs of violent relaxation (constant velocity dispersion) preferentially, while older clusters may show dynamical heating of their more massive stars.

To understand the origin of the apparent large velocity dispersion for the most massive stars in the LNC, we define Sample 1A as the cross-match between the original 819 stars from Wright et al. (2019) with the Gaia DR3 catalog. We found 817 stars in the original sample have Gaia DR3 counterparts within 1″. However, 23 stars were rejected because they lacked parallaxes or had negative values in Gaia DR3. Thus, Sample 1A includes 794 stars.

We constrain the sample further by keeping those stars from Sample 1A with good astrometric quality, i.e., those stars with RUWE$<$1.4 (Lindegren et al., 2021) and parallax errors better than 20%. It is important to mention that the RUWE parameter is recommended to evaluate the quality of the astrometric solution. Sources with large RUWE are problematic for the astrometric solution, likely to be non-single stars. This step reduced the sample from 794 to 547.

We defined Sample 1B by drawing from the 547 stars in the previous list, those stars for which proper motions are within three times their corresponding mean absolute deviations (MAD) from the median value in right ascension (pmra) and declination (pmdec). As a result, Sample 1B includes 502 stars. A cautionary point has to be made here: since we are interested in understanding whether the LNC has undergone some degree of dynamical relaxation, eliminating stars with large proper motions may skew our results towards not finding it. In appendix A we show that this is not the case, since the rejected stars are not the most massive, and are not particularly concentrated towards the center of the cluster.

In Fig. 1 we show the distribution of distances of Sample 1B based on the parallaxes corrected by the zero point bias (solid line histogram). As a reference, we also include the distance distribution previous to this correction (dashed line histogram). Statistically speaking, the distribution of distances to the stars in the LNC is substantially smaller than the typical value adopted of 1.33 kpc in previous studies (e.g., Kuhn et al., 2019; Wright et al., 2019; Wright & Parker, 2019). These differences can affect the derivation of stellar parameters as stellar luminosities and stellar masses.

In Fig. 2, we show the map of the LNC for Samples 1A (green) and 1B (pink). The proper motions of each sample are denoted with arrows and are plotted in the frame of reference of the whole cluster. For purposes that will be clear later, we also denote with a star symbol those stars with masses larger than 20 $M_{\odot}$, according to Wright & Parker (2019).

We defined Sample 2 as follows: Since we are interested in an accurate estimation of the masses of the stars, from Sample 1A we select those stars from Wright et al. (2019) for which effective temperatures have been estimated. We additionally included four stars with effective temperatures estimated by Prisinzano et al. (2019), who used similar spectroscopic data and method to Wright et al. (2019). We also applied the same filters as in the case of Sample 1B. While this sample has only 40% of the stars in Sample 1B (286 stars), it still gives us good statistics that may allow to confirm or discard the results found with the masses tabulated in Wright & Parker (2019).

We estimated extinctions ($A_{V}$), luminosities, masses, and ages of Sample 2 using the code, MassAge (Hernández et al. in prep). This code uses as input the effective temperature, the parallaxes, the Gaia-DR3 (Gp, Rp, Bp), and 2MASS (J and H) photometry. In brief, $A_{V}$ is estimated by minimizing the differences between the observed colors corrected by extinction and the theoretical colors obtained from the evolutionary models. We used two evolutionary models in this work, PARSEC (Marigo et al., 2017) and MIST (Dotter, 2016). Luminosities are derived using the extinction-corrected J magnitude, the theoretical bolometric correction, and the distances estimated as the inverse of the parallax. Finally, stellar masses and ages are obtained by comparing the location of each source on the HR diagram and theoretical evolutionary models. We obtain the mass and age corresponding to specific effective temperature and luminosity values using a linear interpolation method on the theoretical grid.

In Fig. 3 we show the distribution of extinctions towards Sample 2. As can be seen, the extinctions in the sample are statistically larger than the value assumed by Wright & Parker (2019, dashed vertical line). Moreover, a range of extinctions is expected in star-forming regions; thus, assuming a unique value of extinction for a very young stellar population could produce erroneous results.

To show the differences between these samples, in Fig. 4, we show the vector-point diagram, i.e., the proper motion in right ascension vs. proper motion in declination, for the stars in Sample 1A (green) and 1B (pink). In addition, we show with star symbols the six most massive stars in Sample 1A, i.e., those stars with masses larger than 20 $M_{\odot}$, according to Wright & Parker (2019). Finally, the black dots denote the stars entering Sample 2, which is basically a subset of Sample 1B, with the addition of those 4 stars from Prisinzano et al. (2019). The physics of this plot is discussed in §5. Here we just point out that the original sample has many stars which proper motions are substantially more scattered than 3$\sigma$.

When looking at kinematical features, one of the first ones are whether a cluster is expanding or not. While it was clear that the LNC is expanding, it was yet to be clear to what extent. Gaia DR2 studies of the LNC have shown contradictory results. On the one hand, Kuhn et al. (2019) reported a moderate expansion. However, although Wright et al. (2019) confirmed the expansion along the declination, they found no signs of expansion in the right ascension. It is interesting to notice, nonetheless, that visual inspection of Fig. 9 of Wright et al. (2019) hinted at some degree of expansion along the right ascension, which their fit missed, likely because of lower-quality data of Gaia DR2 compared to Gaia DR3. In what follows, we will call “Primary” group to the group having most of the stars, and “Secondary” to the smaller group.

Aimed to disentangle to which degree the LNC is expanding, in Figure 5 we show the (ra, pmra) and (dec, pmdec) diagrams of our Sample 1B, which are the equivalent of those in Fig. 9 of Wright et al. (2019), but for our 3$\sigma$ cleaned Gaia DR3 data (see §3). From this figure, it is clear that the LNC, which has been identified as a well-defined young cluster in the plane of the sky (Walker, 1956; Adams et al., 1983), is actually composed of two expanding groups that are well-mixed in ra, dec, and pmdec, but clearly separated in the pmra as measured by Gaia DR3.

In order to find out which stars belong to each group, it is necessary to apply a clustering method. There are many possibilities to do so, and each one will not necessarily provide the same outcome since the result depends on the method’s assumptions. Given the evident velocity gradients in Fig. 5, we decided to apply the K-means algorithm to the (ra, pmra) plane in order to assign the points to one or the other velocity gradients. The details of our procedure are described in appendix B.

The two groups found with our method are composed of 385 and 110 stars, respectively. Fig. 6 shows the (ra, pmra) and (dec, pmdec) diagrams of the LNC, with the Primary and Secondary clusters denoted by orange and blue dots, respectively. The (ra, dec) map with proper motions is shown in Fig. 7, where the proper motions of each star are shown in the frame of reference of the group they belong to. The big stars denote the position of the centroid of each group, while the corresponding arrows denote their proper motions in the frame of reference of the LNC. The proper motions difference between the centroids of both groups ($\Delta$pmra$\sim$0.728 mas yr^-1 and $\Delta$pmdec$\sim$0.106 mas yr^-1 correspond to a relative velocity between the groups of $\sim$3 km s^-1 at the median distance of the LNC (d = 1231 pc, see Fig. 1).

In order to compute the expansion rates of each cluster and their dynamical times and to evaluate each group’s projected morphology, we computed the positions and velocities of the stars, assuming they all have the median distance to the LNC. We associated their $x$ direction to the right ascension and the $y$ direction to the declination. The linear regressions in the space phase ($x$, $v_{x}$) gave us expanding rates of 0.68 km/s/pc and 0.26 km/s/pc for the Primary and Secondary groups, with Pearson correlation coefficients of $r=$ 0.62 and 0.55 respectively. In the ($y$, $v_{y}$) space phase, the corresponding expanding rates are 0.69 km/s/pc and 0.68 km/s/pc, with Pearson correlation coefficients of $r=$ 0.57 and 0.55. From these values, one can notice that the dynamical times for the Primary group are consistent in both directions: $\tau_{\rm dyn}\sim$1.46 and 1.44 Myr in the ra and dec directions, respectively. Instead, the dynamical timescales for the Secondary group appear less consistent: $\tau_{\rm dyn}\sim$3.89 and 1.47 Myr for the ra and dec, respectively. All these dynamical ages, however, are consistent with the most ($\sim 90\%$) of the cluster’s stars having ages smaller than 10 Myr, with a mean of $2-3$ Myr.

This bipolar nature of the Secondary group is consistent with the values in the diagonal of the covariance matrices, which are $var(\bf{x})=10.5$, $var(\bf{y})=2.8$. These values show that the Secondary group is strongly elongated in the $x$ (ra) direction, while the Primary group is substantially less elongated. Its values of the covariance matrix along the diagonal are $var(\bf{x})=3$, $var(\bf{y})=1.8$.

The fact that the LNC comprises two expanding groups has probably passed unadvertised for several reasons. First of all, when plotting the proper motions of the whole cluster, it is not clear that there are two groups, neither in the Sample by Wright & Parker (2019, Sample 1A, green points in Fig. 2) nor in Sample 1B (pink points). Indeed, although the Secondary group is located at the western edge of the main group, it still shares similar positions with the Primary group. In addition, the lack of accurate proper motions even with Gaia DR2, has made difficult the separation. Finally, while the Primary group is denser at its center, the Secondary group is not strongly centrally concentrated, making it difficult its identification as a group. Instead, it has some degree of internal structure, since it is composed of several overdensities that have been reported by Kuhn et al. (2014, see their Fig. 2b), with the north-western group being one of the more prominent overdensities.

As commented before, the masses estimated by Wright et al. (2019) are based on assuming a single distance and single extinction. This assumption may give inaccurate mass estimations, potentially changing the main result, namely, that massive stars have larger velocity dispersion. In order to verify that this is not the case, we used our Sample 2, which is based on the crossmatch between Sample 1B and those stars that have reliable estimations of the effective temperature, and thus, that may allow us to estimate more reliably the masses of the stars.

In Fig. 8 we show the masses estimated with MassAge⁶⁶6The new masses for 286 stars are reported in an electronic Table. and compare them to the masses reported by Wright et al. (2019). In panels (a) and (b), we compare the masses estimated by Wright et al. (2019, $x$ axis) with the masses estimated with MassAge-mist and MassAge-parsec, respectively. In panel (c), we compare the differences between our two estimations. The solid lines in each panel are the identity.

There are two points to notice from this figure. On the one hand, the masses estimated by Wright et al. (2019) are statistically larger than those estimated with MassAge. On the other hand, differences between the MIST and PARSEC models of early stellar evolution do not appear to give statistically different results.

The fact that the masses estimated previously are larger than ours must be because the distance assumed to the stars is statistically larger than the distances from Gaia DR3 corrected by the zero-point bias (see Fig. 1 and discussion in §3), and thus, the inferred luminosities of the stars will also be larger.

It is interesting to investigate whether mass segregation exists in the LNC, especially because although spatial mass segregation does not necessarily imply collisional relaxation, the lack of the former necessarily discards the latter (see §2.2).

Aimed to quantify the degree of mass segregation in the LNC, we performed KS tests to assess the statistical significance of the difference between the spatial distribution of massive and low-mass stars, using projected distances from the center of mass at the mean distance of the group to determine their spatial position. Since the mass $m_{\mathrm{lim}}$ above which we will distinguish massive from low-mass stars is rather arbitrary, we computed their spatial distributions and applied KS to check differences using different values of $m_{\mathrm{lim}}$ in the range 0.4 $M_{\odot}$$\leq m_{\mathrm{lim}}\leq$ 8 $M_{\odot}$.

The results of this procedure are plotted in Fig. 9, where we show the KS-test’s $p-$value ($y-$axis) between the spatial distributions of stars with mass $M>m_{\mathrm{lim}}$ and $M<m_{\mathrm{lim}}$, as a function of $m_{\mathrm{lim}}$ (lower $x-$axis). The upper $x$ axis, furthermore, denotes the percentage of stars with masses below $m_{\mathrm{lim}}$ for a set of 5 values of $m_{\mathrm{lim}}$.

The horizontal red line indicates $p-$value = 0.05, below which we would reject the hypothesis that the two data sets come from the same intrinsic distribution function. As it can be seen, with $p>0.05$ for most values of $m_{\mathrm{lim}}$, there is insufficient evidence to suggest that massive and low-mass stars have intrinsically different spatial distributions, which would indicate spatial mass segregation.

To further argue that the LNC and their substructures have not undergone dynamical relaxation, in Fig. 10, we show the cumulative distributions of massive (solid lines) and low-mass (dotted lines) stars for the whole Sample 1B (left panel), the Primary Group (middle panel), and the Secondary Group (right panel). The distributions shown correspond to $m_{\mathrm{lim}}=$ 0.8, 0.8, and 4.8  $M_{\odot}$ for the left, middle, and right panels, respectively. These are the values for the worst-case scenarios denoted by a red arrow in Fig.9, i.e., by those cases where the $p$-value is minimum ($p\lesssim$ 0.05). In Sample 1B and the Primary Group (left and middle panels), the cumulative functions show minimal differences between the distributions of high-mass and low-mass stars, and thus, it is clear that there is no larger concentration of massive stars towards the center of their respective groups, compared to the low-mass stars. As for the Secondary Group (right panel), judging from this figure, it appears to be a larger concentration of massive stars towards the group’s center.

To further understand these results, in Fig. 11, we show the histograms of the high- (open histogram) and low-mass stars (gray histogram) corresponding to the cases shown in Fig. 10. As it can be seen, the whole Sample 1B and the Primary Group show no clear differences between spatial distributions of the high- and low-mass stars, consistent with the discussion above. As for the Secondary Group, we notice that, although Fig. 10c shows signs of segregation, there are only 8 high-mass stars, and thus, one cannot draw conclusions with such poor statistics.

As a summary from the previous discussion, we conclude that there are no signs that the whole group, or its subgroups, are mass-segregated for the following reasons: (a) Only in a few cases the $p$-values of the KS test are below 0.05, indicating that, typically, it cannot be said that the low- and high-mass stars are drawn from different distributions. (b) Even in the case of the Secondary Group, which has a $p$-value slightly above 0.05 and its cumulative histogram of the high-mass stars is more centrally concentrated than that of the low-mass stars, the group has too few high-mass stars to draw conclusions. Thus, we can argue that, in general, there are no signs of dynamical relaxation in the LNC.

Since we want to understand whether the massive stars in the LNC are undergoing dynamical heating as the result of collisional relaxation, we first want to verify whether the dynamical heating reported previously with Gaia DR2 data remains in Gaia DR3. Therefore, in Fig. 12 a, b, we show⁷⁷7It is worth recalling that these plots are not histograms, i.e., these are not number per mass bin, but velocity dispersion-mass plots. We show them as histograms because in order to compute a velocity dispersion, it is necessary to take a mass-bin., using the same masses and mass bins as Wright & Parker (2019), the velocity dispersion per mass bin for the stars in (a) Sample 1A and (b) Sample 1B. In contrast, Panel (c) shows the velocity dispersion of Sample 1B with equally spaced mass bins.

We note that the velocity dispersion as a function of the mass bin is substantially flatter than the one reported by Wright & Parker (2019) in all three panels. A striking feature at first glance is that the second most massive bin has a substantial drop between panel (a) and (b). However, the reason for this discrepancy is just a poor-statistics effect. Panel (a) has ten stars in that bin, while panel(b) has eight, making unreliable the statistics provided with such low numbers. This is precisely the reason for including panel (c): to avoid low-number statistics by plotting Sample 1 B with the same number of stars in each mass bin. From this panel, it is clear that the velocity dispersion does not increase as a function of the mass bin of the stars. Instead, it looks substantially flat.

In order to furthermore understand the large velocity dispersion of the massive stars in the original sample by Wright & Parker (2019), we recall Fig. 4, where we showed the vector-point diagram, i.e., the proper motion in right ascension vs. proper motion in declination, for the stars in Sample 1A (green) and 1B (yellow). As commented before, the green points do not pass the tests given by astrometry quality described in §3.1 and have velocities substantially different than those of the whole cluster. In addition, we show with star symbols the six most massive stars in Sample 1A, i.e., those stars with masses larger than 20 $M_{\odot}$, according to Wright & Parker (2019). From this plot, it is clear that one of these massive stars has substantially larger proper motions compared to the characteristic proper motion of the LNC. This star is mostly responsible for the large velocity dispersion of the high-mass bin in Wright & Parker (2019).

At first glance, one may be tempted to argue that our result is flawed from origin: our $3\sigma$ cleaning process eliminates precisely the massive star that could have been ejected by collisional relaxation. However, in order to show that it is unlikely that the velocity of this star is the result of collisional relaxation in the LNC, we notice that this star has a proper motion almost perpendicular to its radius towards the cluster’s center. This is seen in Fig. 2a, where the massive star rejected from Sample 1A to Sample 1B is the green star symbol at RA$=$271.06 deg, DEC$=-24.18$ deg. If the large velocity of this star were due to collisional relaxation, it should be located either at the center of the cluster or anywhere else, but with its proper motions radially aligned, as a consequence of its ejection from the center. Since its velocity is almost tangential, it is unlikely that it has been expelled from the center of the cluster due to any sort of dynamical heating (see §2.2).

Furthermore, we argue that it is unlikely that this star belongs to the cluster. The tangential proper motions of this star correspond to a velocity of $\sim$9 km/sec at the median distance of the LNC. It is located at a distance of $\sim$6 pc from the cluster’s center. In order to be bound, the mass of the LNC had to be of the order of 2$\times$$10^{5}$ $M_{\odot}$, an unusual mass for an open cluster. This mass is ten times the estimated virial mass of the cluster and 50-200 times larger than the estimations by Prisinzano et al. (2019, 1000 $M_{\odot}$), Wright et al. (2019, 2500 $M_{\odot}$) Kuhn et al. (2015, 4000 $M_{\odot}$).

From the previous discussion, it seems unlikely that this star belongs to the LNC. An intriguing possibility, however, is whether it has undergone substantial gravitational acceleration from the parental cloud. Indeed, numerical simulations show that as the newborn stars in stellar clusters expel their parental gas, the gravitational potential decreases, accelerating the stars in different directions (Geen et al., 2018; Zamora-Avilés et al., 2019). This process is called gravitational feedback (Zamora-Avilés et al., 2019). A further numerical and observational investigation is necessary, however, in order to estimate whether this effect has a relevant impact on the dynamics of stars in star-forming regions.

The analysis derived from Fig. 12 was obtained using the masses reported by Wright & Parker (2019). It is still necessary, however, to verify whether the masses computed with MassAge show evidence of dynamical heating. For this purpose, in Fig. 13, we first plot the velocity dispersion as a function of the mass bin for Sample 2. Panel (a) shows masses computed withMassAge-parsec while panel (b) shows masses computed with MassAge-mist. Since, in this calculation, we do not have the same mass ranges as Wright & Parker (2019), we opt to show bins with the same number of stars. This allows us to have similar statistics in all bins. As can be seen, there is no evidence of the dynamical heating of the massive stars.

Similarly, we computed the velocity dispersion per mass bin for the two smaller clusters. The corresponding velocity dispersion-mass plots are shown in Fig. 14, using the masses from Wright & Parker (2019). Panel (a) corresponds to the Main group, while Panel (b) corresponds to the Secondary group. From this figure, we do not see any sign of dynamical heating in each one of the individual groups.

The situation is different, however, if we use the masses inferred with MassAge. Indeed, as can be seen in Fig. 15, although the Primary group does not exhibit massive stars having larger velocity dispersion than low-mass stars (upper panel), the massive stars in the Secondary group (lower panel) do it.

The cause for massive stars’ larger velocity dispersion than low-mass stars in the Secondary group is unclear. In dynamical relaxation, one would expect massive stars to not only have larger velocity dispersion but also to be preferentially concentrated towards the center of the group. However, in §4.3 we found that massive stars are not preferentially concentrated towards the group’s center. Thus, it is hard to argue in favor of dynamical relaxation being the cause of the larger velocity dispersion of massive stars in the Secondary group.