Large scale star formation in Auriga region

The four wave-bands of $WISE$ are useful to detect mid-IR emission from cold circumstellar disk/envelope material in YSOs. This makes $WISE$ all-sky survey as a readily available tool to identify and classify YSOs, in a similar way to what can be done with $Spitzer$ (Allen et al., 2004; Gutermuth et al., 2008; Gutermuth et al., 2009, and others) but over the entire sky. We use the AllWISE Source Catalog to search for YSOs adopting the criteria given by Koenig & Leisawitz (2014). We refer Figure 3 of Koenig & Leisawitz (2014) to summarize the entire scheme. This method includes the selection of candidate contaminants (AGN, AGB stars and star forming galaxies). Out of 40788 $WISE$ source meeting photometric quality flags, 22238 sources were excluded from the data file as contaminants on the basis of selection criteria of Koenig & Leisawitz (2014). In the present study we followed and applied the photometric quality criteria for different $WISE$ bands as given in Koenig & Leisawitz (2014).

The $(J-H)/(H-K)$ NIR TCD is also a useful tool to identify pre-main sequence (PMS) objects. It is possible that some of the candidate YSOs might have not been detected on the basis of $WISE$ data, hence we have also used 2MASS $JHK$ three band data to identify additional YSOs. Fig. 6 (bottom right-hand panel) displays the NIR TCD for all the stars which have not been detected in the WISE survey or do not meet the photometric quality criterion (cf. Section 3.1.1) in the region studied. The 2MASS magnitudes and colours have been converted into the California Institute of Technology (CIT) system⁶⁶6http://www.astro.caltech.edu/$\sim$jmc/2mass/v3/transformations/. The solid and long dashed lines in Fig. 6 (bottom left-hand panel) represent unreddened main sequence (MS) and giant branch loci (Bessell & Brett, 1988), respectively. The dotted line indicates the intrinsic loci of CTTSs (Meyer et al., 1997). The parallel dashed lines are the reddening vectors drawn from the tip (spectral type M4) of the giant branch (‘left reddening line’), from the base (spectral type A0) of the MS branch (‘middle reddening line’) and from the tip of the intrinsic CTTSs line (‘right reddening line’). The extinction ratios $A_{J}/A_{V}$ = 0.265, $A_{H}/A_{V}$ = 0.155 and $A_{K}/A_{V}$ = 0.090 have been adopted from Cohen et al. (1981). The sources lying in the ‘F’ region could be either field stars (MS stars, giants), Class iii or Class ii sources with small NIR excesses. The sources lying in the ‘T’ region are considered to be mostly classical T-Tauri stars (CTTSs, i.e., Class ii objects). The sources lying in the ‘P’ region - redward of the right reddening line - are most likely Class i objects (protostellar-like objects; Ojha et al., 2004b). In this scheme we consider only those sources as YSOs that lie at a location above the intrinsic loci of CTTSs with margins larger than the errors in their colours. This classification criterion yields 21 and 204 probable Class i and Class ii YSOs, respectively.

In the present study we have not attempted to identify diskless YSOs. The Class iii sources may possibly have a different spatial distribution in the Auriga region as compared to the Class i and  ii YSOs and inclusion of these sources may have impact on the parameters described in the ensuing sections. It is worthwhile to mention that disk fraction estimates in young clusters having age $\leq 3$ Myr varies from 60-70% (M $\lesssim$ 2 M_⊙) and 35 - 40% (M$>$2 M_⊙) (Ribas et al., 2015), whereas the disk half-life estimates varies from 1.3 Myr to 3.5 Myr (Richert et al., 2018, and references therein). Since the missing YSO mass due to the incompleteness of our YSO search criteria as well as to unidentified Class iii sources may play a role in the analysis to be carried out in the ensuing sections, we will assume that 50% of the total YSO population is missed in the present YSO sample.

The YSOs can also be characterized from their SED. The SED fitting provides evolutionary stages and physical parameters such as mass, age, disk mass, disk accretion rate and photospheric temperature of YSOs and hence is an ideal tool to study their evolutionary status. We constructed the SEDs of the YSOs using the grid models and Python version of SED fitting tools⁷⁷7https://sedfitter.readthedocs.io/en/stable/ of Robitaille et al. (2006, 2007)⁸⁸8WISE fluxes were acquired from Dr. Thomas Robitaille through private communication. The models were computed by using a Monte-Carlo based 20000 2-D radiation transfer calculations from Whitney et al. (2003a, b, 2004) and by adopting several combinations of a central star, a disk, an infalling envelope, and a bipolar cavity in a reasonably large parameter space and with 10 viewing angles (inclinations).

The SEDs were constructed by using the multiwavelength data (i.e. optical to MIR) with the condition that a minimum of 5 data points should be available. While fitting the models to the data we assumed the extinction and the distance as free parameters. Considering the errors associated with the distance estimates available in the literature, the range in distance estimate was assumed to vary between 2.0 kpc to 2.4 kpc. Since the extinction in the region is variable (cf. Jose et al., 2008; Pandey et al., 2013), we used a range for $A_{V}$ of 1.6 to 30 mag. We further set photometric uncertainties of 10% for optical and 20% for both NIR and MIR. These values are adopted instead of the formal errors in the catalog in order to fit without any possible bias in underestimating the flux uncertainties. In Fig. 8, we show example SEDs of Class i and Class ii sources, where the solid black curves represent the best-fit and the gray curves are the subsequent well-fits satisfying our requirements for good fit discussed below.

Since the SED models are highly degenerate, the best-fit model is unlikely to give an unique solution and the estimated physical parameters of the YSOs tabulated in Table 7 are the weighted mean with standard deviation of the physical parameters obtained from the models that satisfy $\chi^{2}-\chi^{2}_{min}\leq 2N_{data}$, where $\chi^{2}_{min}$ is the goodness-of-fit parameter for the best-fit model and $N_{data}$ is the number of input data points.

We have identified 710 YSOs in the Auriga Bubble region (cf. Table 5) and derived the physical parameters of 489 YSOs from the SED fitting analysis (cf. Table 7). These parameters were used in further analysis. Histograms of the age and mass of these YSOs are shown in Fig. 9. The distribution of the ages estimated on the basis of SEDs indicates that $\sim$76% (370/489) of the sources have ages $\leq$ 3.5 Myr. The masses of the YSOs are found to range between 0.75 to 9 M_⊙, and a majority ($\sim$86%) of them are in the range of 1.0 to 3.5 M_⊙. The $A_{V}$ distribution shows a long tail indicating its large spread from $A_{V}$=1 - 27 mag, which is consistent with the nebulous nature of this region. The average age, mass and extinction ($A_{V}$) for this sample of YSOs are $2.5\pm 1.7$ Myr, $2.4\pm 1.1$ M_⊙ and $7.1\pm 4.1$ mag, respectively.

The evolutionary classes of the identified 710 YSOs given in the Table 5 reveal that $\sim$25% (175 out of 710) sources are Class i YSOs. The comparatively high percentage of Class i YSOs indicates the youth of this region.

Details on the environment and distribution of YSOs can be used to probe the star formation scenario in the region. In Fig. 10 (top left-hand panel), the H i contours by Furst et al. (1990) (black contours) and the ¹²CO contour map (cyan contours) from Dame et al. (2001) along with the distribution of the YSOs are overlaid on the $WISE$ 12 $\mu$m image. The $WISE$ 12 $\mu$m image covers the prominent PAH features at 11.3 $\mu$m, which is indicative of star formation activity (see e.g. Peeters et al., 2004). This figure reveals that ionized as well as molecular gas distribution is well correlated with that of YSOs. The distribution of ionized gas, molecular gas and YSOs indicates a ring-like structure spread over an area of a few degrees. The distribution of YSOs also reveals that a majority of Class i sources belong generally to this ring-like structure, whereas the comparatively older population, i.e. Class ii objects, are rather randomly distributed throughout the region. As stated already we term this structure as Auriga Bubble1. Furthermore there is another, very well defined distribution of Class i and Class ii objects towards the north-west of the H ii complex G173+1.5, forming another bubble feature, which we call as Auriga Bubble2. Its nature will be discussed in an ensuing study.

To quantify the extinction in the region and to characterize the structure of the molecular gas associated with various SFRs in the Auriga Bubble, we derived $A_{K}$ extinction maps using the $(H-K)$ colours of field stars (cf. Gutermuth et al., 2011). To produce the extinction map we excluded the candidate YSOs (cf. Section 3.2) and probable contaminating sources (AGNs, AGB stars and star forming galaxies) using the procedure by Koenig & Leisawitz (2014) from the sample stars. Similar approaches were used by other studies also (e.g., Allen et al., 2008; Gutermuth et al., 2009; Jose et al., 2013; Jose et al., 2016; Sharma et al., 2016, and references therein). Mean values of $A_{K}$ were derived by using the nearest neighbor (NN) method as described in detail by Gutermuth et al. (2005) and Gutermuth et al. (2009). Briefly, the mean value of $(H-K)$ colours of five nearest stars at each position in a grid of 30 arcsec was calculated for the entire Auriga region ($\sim 6^{\circ}\times 6^{\circ}$). The sources deviating above 3$\sigma$ were excluded to calculate the final mean colours of each grid. The reddening law $A_{K}$ = 1.82 $\times$ ($(H-K)_{obs}-(H-K)_{int}$) by Flaherty et al. (2007) was used to convert the $(H-K)$ colours into $A_{K}$, where $(H-K)_{int}$ = 0.2 was assumed as an average intrinsic colours of the field stars (see. Allen et al., 2008; Gutermuth et al., 2009). To eliminate the foreground contribution in generating the extinction map we used only those stars which have $A_{K}>$ 0.15$\times$D, where D is the distance in kpc (Indebetouw et al., 2005). The extinction map is sensitive down to $A_{K}\sim$2.8 mag (=$A_{V}\sim$30 mag). However, the derived $A_{K}$ values are to be considered as lower limits, because the sources in the region with higher extinction might have not been detected in the present sample. The extinction map, smoothened to a resolution of 18 arcmin, is shown in blue colour in Fig. 10 (top right-hand panel). It is interesting to note that the extinction map resembles the general distribution of the molecular gas as outlined by the ¹²CO emission map shown in Fig. 10 (top left-hand panel). It shows a concentration of molecular clouds towards Sh2 - 231 - 235. A comparison of the stellar density distribution and the morphology of the molecular material can provide a clue to the history of star formation in the region. The surface density maps of YSOs were generated by using the NN method as described by Gutermuth et al. (2005). We used the radial distance that contains 5 nearest YSOs to compute the local surface density in a grid size of 30 arcsec. The grid size identical to that of the extinction map was used to compare the stellar density and the gas column density. The density distribution of YSOs is shown in red colour in Fig. 10 (top right-hand panel). The distribution of the YSOs and molecular material shows a nice correspondence. Gutermuth et al. (2011) also found similar trends in eight nearby molecular clouds. Gutermuth et al. (2008) have shown that the sources in each of the Class i and Class ii evolutionary stages have very different spatial distributions relative to that of the dense gas in their natal cloud. We have also compared the extinction map with the positions of YSOs of different evolutionary stages and found that the Class i sources are located towards the places with higher extinction. These properties agree well with previous findings in several star-forming regions such as W5 (Koenig et al., 2008; Deharveng et al., 2012), i.e., younger Class i sources are more clustered and closely associated with the densest molecular clouds in which they were born presumably, while the Class ii sources are scattered probably by drifting away from their birthplaces.

Many ground-based NIR surveys of molecular clouds (e.g., Lada et al., 1991; Strom et al., 1993; Carpenter et al., 2000; Porras et al., 2003; Lada & Lada, 2003) have shown that molecular clouds host both dense ‘clustered’ and diffuse ‘distributed’ population. As discussed earlier the Auriga Bubble region contains several star forming subregions, hence a clustered distribution of YSOs is expected. Gutermuth et al. (2009) have used an empirical method based on the minimum spanning tree (MST) technique to isolate groupings (sub-clusters) from the more diffuse distribution of YSOs in nebulous regions. This method effectively isolates sub-structures without any type of smoothening (see e.g., Cartwright & Whitworth, 2004; Schmeja & Klessen, 2006; Bastian et al., 2007; Bastian et al., 2009; Gutermuth et al., 2009). The sub-groups detected in this way have no biases regarding the shapes of the distribution and preserve the underlying geometry of the distribution (Gutermuth et al., 2009). In Fig. 10 (bottom left-hand panel) we plot the derived MSTs for the YSOs in the region. In order to isolate the sub-structures, we adopted a surface density threshold expressed by a critical branch length. With the help of an adopted MST branch length threshold we can identify local surface density enhancements. To do that, we used a similar approach to that suggested by Gutermuth et al. (2009). In Fig. 11, we plot the cumulative distribution for MST branch lengths, which shows a three-segment curve; a steep segment at short lengths, a transition segment at the intermediate lengths, and a shallow-sloped segment at long lengths. The majority of the sources are found in the steep segment, where the lengths are small (i.e., sub-cluster). Therefore, to isolate sub-cluster regions in the Auriga Bubble, we fitted two straight lines to the shallow and steep segments of the cumulative distribution function (CDF) and extended them to connect together. The intersection is adopted as the MST critical branch length, as shown in Fig. 11 (see also, Gutermuth et al., 2009). The sub-clusters of the SFRs were then isolated from the lower density distribution by clipping the MST branches longer than the critical length described above. Similarly, we defined the extended area for the SFR by selecting the point where the curved transition segment meets the shallow-sloped segment at longer spacings. This range represents the extended region of star formation or the area where YSOs might have moved away from the sub-clusters due to dynamical evolution, and we have named this region as the active region. The black dots connected with black lines and the blue dots connected with blue lines in Fig. 10 (bottom left-hand panel) are the branches smaller than the critical length for the sub-clusters and the active region, respectively. We have also plotted the convex hulls (cf. Gutermuth et al., 2009) for the active region in Fig. 10 (bottom left-hand panel) with solid purple lines. The physical details of the sub-groups (sub-clusters) and the active regions are given in Tables 8, 9 and 10. In total, we have identified 9 active regions and 26 probable sub-clusters having at-least 5 YSO members in the Auriga Bubble region. Although sub-clusters with a small number of members have been reported in previous studies, e.g., Gutermuth et al. (2009, N=10), Cartwright & Whitworth (2004, N=7-10) and Sharma et al. (2016, N=10), however it is worthwhile to mention that smaller numbers of YSOs in some of the probable sub-clusters may introduce relatively large errors in the derived physical parameters discussed in the ensuing sub-sections. The median value of the critical branch lengths for the sub-clusters and the active regions are 2 pc and 9 pc, respectively. In Fig. 10 (bottom right-hand panel) we can see a correlation between the identified active regions with the extinction contours and the YSO locations in the Auriga Bubble.

In many SFRs YSOs are observed to have both diffuse and clustered spatial distribution. For example, Koenig et al. (2008) have analyzed the clustering properties across the W5 region and found 40-70% of the YSOs belong to groups with $\geq$10 members and the remaining were described as scattered populations. Using the AllWISE database, Fischer et al. (2016) identified 479 YSOs in a $10\times 10$ degree² region centered on the Canis Major star-forming region. Their YSO sample contains 144 and 335 Class i and Class ii YSOs, respectively. On the basis of the MST of the YSO distribution, Fischer et al. (2016) concluded that there were 16 groups with more than four members. Of the 479 YSOs, 53% are in such groups. Gutermuth et al. (2009) presented a uniform MIR imaging and photometric survey of 36 nearby young clusters and groups using Spitzer IRAC and MIPS. They found 39 clusters/sub-clusters with 10 or more YSO members. Of the 2548 YSOs identified, 1573 (62%) are members of one of these clusters/sub-clusters. Although the sub-clusters in the Auriga Bubble region have sizes of the order of a few parsecs, however, some of the member stars might have moved away in the last few Myr due to dynamical and environmental effects (Miao et al., 2006). Weidner et al. (2011) used N-body calculations to study the numbers and properties of escaping stars from young embedded star clusters during the first 5 Myr of their existence prior to the removal of gas from the system. They found that these clusters can lose substantial amounts (up to 20%) of stars within 5 Myr. In the present sample, the YSOs formed in the sub-clusters having a mean velocity of $\sim$2 km s^-1 (cf. Weidner et al., 2011) can travel a distance of $\sim$2-6 pc in 1-3 Myr of their formation. Therefore, we expect that the effect of escaping members from the sub-clusters/active regions must be insignificant. We have estimated that the fraction of the scattered YSO population (the YSOs outside sub-clusters, but in the active regions) is about $\sim$37% of the total YSOs in the whole active regions. Similar numbers ($\sim$40%) in the case of 8 bright rimmed clouds (Sharma et al., 2016) and 5 embedded clusters (Chavarría et al., 2014) have also been reported in previous studies. The explanation for the scattered populations may include: escape of sub-clusters/cluster members due to their dynamical interaction and isolated star formation (for details, Elmegreen & Efremov, 1997; Lada & Lada, 2003; Koenig et al., 2008; Evans et al., 2009; Elmegreen et al., 2014; Chavarría et al., 2014; Panwar et al., 2019).

Known SFRs show a wide range of sizes, morphologies and star numbers (cf. Gutermuth et al., 2008; Gutermuth et al., 2009, 2011; Chavarría et al., 2014). We use cluster’s convex hull radii ($R_{H}$) and aspect ratios to investigate their morphology (see Table 9 and Fig. 12). Here we would like to point out that the present procedure is applied to the sample which contains only Class i and Class ii YSOs and does not include Class iii YSOs. The similar approach has been adopted in previous studies also (Gutermuth et al., 2009; Sharma et al., 2016). However, as discussed in Section 3.2.1, the contribution of diskless YSOs (i.e. Class iii sources) in star-forming regions having age 2-3 Myr may be about 50% of the total YSO population. This missing population may play a significant role in estimating various parameters discussed in the ensuing sections. Hence we also estimate parameters by assuming the contribution of Class iii sources as 50% of the total YSO population.

A majority of the sub-clusters identified in the present sample show an elongated morphology with the median value of the aspect ratios around 1.6. The median number of YSOs in the sub-clusters and the active regions are 9 and 38, respectively (cf. Table 10). The median MST branch length for these sub-clusters is found to be $\sim$0.5 pc. The total sum of YSOs in the active regions is 546, out of which 345 (63%) falls in the sub-clusters. The YSOs in the Auriga Bubble have mean surface densities mostly between 0.3 and 7.5 pc^-2 (see Table 8 and Fig. 13). The median values for the surface densities for the sub-clusters and the active region come out to be around 1.35 pc^-2 and 0.1 pc^-2, respectively. The peak surface densities vary between $\sim$ 0.5 - 18 pc^-2 with a median value of 4 pc^-2 for our sample (cf. Table 8 and Fig. 13). As in the case of low-mass embedded clusters studied by Chavarría et al. (2014), we found a weak proportionality between the peak surface density and the number of cluster members, suggesting that the clusters are better characterized by their peak YSO surface density.

The spatial distribution of YSOs in a SFR can be investigated with the help of the structural $Q$ parameter (Cartwright & Whitworth, 2004; Schmeja & Klessen, 2006). It is used to measure the level of hierarchical versus radial distributions of a set of points, and it is defined by the ratio of the MST-normalized mean branch length to the normalized mean separation between points (cf. Chavarría et al., 2014, for details). If the normal values are used, the $Q$ parameter becomes independent on the cluster size (Schmeja & Klessen, 2006). A group of points distributed radially will have a high $Q$ value ($Q$ $>$ 0.8), while clusters with a more fractal distribution will have a low $Q$ value ($Q$ $<$ 0.8) (Cartwright & Whitworth, 2004). We find that the groups of the YSOs in the present study (including only Class i and ii sources) have median $Q$ values less than 0.8 (0.71 in sub-clusters and 0.52 in active regions, cf. Tables 9 and 10), indicating a more fractal distribution. Chavarría et al. (2014) have found a weak trend in the distribution of $Q$ values per number of members, suggesting a higher occurrence of sub-clusters merging in the most massive cluster, which reduces the $Q$ value. A similar trend can be noticed in Fig. 14 (left-hand panel).

The mean $A_{K}$ values for the identified sub-clusters have been found in the range of 0.6 and 1.3 mag, with a median value of 0.9 mag (cf. Table 10 and Fig. 15). A weak correlation (Spearman’s correlation coefficient ‘r’ = 0.6 with 95% confidence interval of 0.3 to 0.8) between the peak $A_{K}$ and the number of cluster members can be noticed in Fig. 15. The median $A_{K}$ value for the active regions is 0.6 mag, which is lower than the sub-cluster value, naturally indicating the YSO distribution of higher density towards the molecular clouds of higher density. The observed correlation indicates that active regions/ sub-clusters having higher number of YSOs have higher peak $A_{K}$ value. Fig 16 (left panel) suggests that higher number of YSOs in active regions/ sub-cluster are associated with the higher mass cloud. We presume that massive clouds have higher peak $A_{K}$ value.

The extinction maps generated in §4.2.1 have been used to estimate the molecular mass associated with the identified sub-clusters/active regions. The $A_{V}$ value (corrected for the foreground extinction) in each grid of the 30 arcsec was converted to $H_{2}$ column density by using the relation given by Dickman (1978) and Cardelli et al. (1989), i.e., $\rm N(H_{2})=1.25\times 10^{21}\times A_{V}~{}cm^{-2}mag^{-1}$. The $H_{2}$ column density was integrated over the convex hull of each region and multiplied by the $H_{2}$ molecule mass to get the cloud mass. The extinction law, $A_{K}/A_{V}=0.09$ (Cohen et al., 1981) has been used to convert $A_{K}$ values to $A_{V}$. The foreground contributions have been corrected for by using the relation: $A_{K_{foreground}}$ =0.15$\times$D (Indebetouw et al., 2005, D is distance in kpc). The properties of the molecular clouds associated with the sub-clusters and active regions are listed in Table 9. The molecular gas associated with the sub-clusters in the present sample shows a wide range in the mass distribution ($\sim$5.8 to 7621 M_⊙), with a median value around $\sim$146 M_⊙. SED analysis would have allowed us to estimate the mass of 489 YSOs (cf. Section 3.3.1), while for other 221 YSOs available data-points are not enough to properly constrain stellar parameters. Hence, the total mass of all the candidate YSOs in the sub-clusters/ active regions has been estimated by multiplying the total number of the YSOs (i.e., 710) with the average SED mass (2.4 M_⊙ as discussed in Section 4.1).

An analysis of YSO spacings in sub-clusters of 36 star-forming clusters by Gutermuth et al. (2009) suggested that Jeans fragmentation is a starting point for understanding the primordial structure in SFRs. We estimated the minimum radius required for the gravitational collapse of a homogeneous isothermal sphere (Jeans length ‘$\lambda_{J}$’) in order to investigate the fragmentation scale by using the formula given in Chavarría et al. (2014). The Jeans length $\lambda_{J}$ estimated for the sub-clusters in the current study has values between 0.8 - 3.3 pc for the sample having Class i and Class ii sources as well the sample having assumed missing mass of Class iii. The contribution of missing mass of Class iii sources has been accounted by assuming a disk fraction of 50%. The resultant number of missed stars has been multiplied with an average SED mass (2.4 M_⊙, as discussed in Section 4.1) to get contribution of unidentified Class iii sources. The estimated value of Jeans length ‘$\lambda_{J}$’ are given in Tables 9 and 10. The median values of ‘$\lambda_{J}$’ are estimated as $\sim$2 pc for both of the two samples as discussed. We also compared the $\lambda_{J}$ and the mean separation ‘$S_{YSO}$’ between the cluster members (Fig. 14, right-hand panel) and found that the ratio $\lambda_{J}/S_{YSO}$ has an average value of $3.4\pm 0.9$ and $3.2\pm 0.8$ for the two samples, respectively. Chavarría et al. (2014) reported the ratio for their sample of embedded clusters as $4.3\pm 1.5$. The present results agree with a non-thermal driven fragmentation since it takes place at scales smaller than the Jeans length (Chavarría et al., 2014).

Lada et al. (2010) have reported that the number of YSOs in a cluster are linearly related to the dense cloud mass M_0.8 (the mass above a column density equivalent to $A_{K}\sim$ 0.8 mag) with a slope equal to unity. Recently Chavarría et al. (2014) have also found a similar relation for a sample of embedded clusters. This suggests that the star formation rates depend linearly on the mass of the dense cloud (Lada et al., 2010). We also estimated the M_0.8 (cf. Table 9), accounting only for the Class i and ii objects, and found that the number of YSOs in the sub-cluster or active region is linearly correlated with the dense cloud mass with a value of Spearman’s correlation coefficient ‘r’ = 0.8 with 95% confidence interval of 0.6 to 0.9 (cf. Fig. 16, left-hand panel).

The right-hand panel of Fig. 16 shows the hull radius of sub-clusters/ active regions as a function of the total YSO mass, which manifests that the hull radius is linearly correlated (r=0.8) to the total YSO mass. The radius versus mass relation gives a clue of whether the cluster will be bound or unbound. The radius limit of a group moving in the Galactic tidal field is defined as the distance from the center of the group at which the attraction of a given star from the cluster is balanced by the tidal force of external masses (Kholopov, 1969). The limiting radius $r_{lim}$ for a group that is moving in elliptical orbit around the Galactic center is given by the relation (King, 1962).

where $R_{p}$ is the perigalactic distance of the group, $M_{*}$ is the mass of the group and $M_{G}$ is the mass of the Galaxy. Assuming $R_{p}$ of the Sun as 8.5 kpc and $M_{G}\sim 2\times 10^{11}M_{\odot}$, the limiting radius $r_{lim}$ as a function of mass of the cluster is shown as the continuous curve in Fig. 16 (right-hand panel). This figure indicates that the radii of all the sub-clusters are below the limiting radius, which suggests that all the sub-clusters will be bound systems, whereas all the active regions may be unbound systems.

The star formation efficiency (SFE), defined as the percentage of the gas mass converted into stars, is an important parameter to determine whether a cluster will be a bound or unbound system etc. Several studies have been carried out, suggesting that the formation of a bound system requires a SFE of $\geq$$\sim$ 50% when the gas dispersal is quick or $\geq$$\sim$ 30% when the gas dispersal time is $\sim$3 Myr (cf. Elmegreen, 1983; Lada et al., 1984). Lada et al. (1984) have also concluded that in the case of slow dispersal a lower SFE of $\sim$ 15% may also produce a bound system.

The observed surface density of the YSOs in the sub-clusters and active regions provides an opportunity to study how this quantity is related to the observed SFE and other properties of the associated molecular cloud. Recent works indicate that SFE increases with the stellar density; e.g., Evans et al. (2009) reported that YSO clusters of higher surface density have higher SFE (30%) than their lower density surroundings (3%-6%). Similarly Koenig et al. (2008) also found SFEs of $>$10%-17% for high surface density clusters, whereas in lower density regions the SFEs are found to be $\sim$ 3%. We have calculated the SFE by using the cloud mass derived from $A_{K}$ inside the cluster convex hull areas and the number of YSOs found in the same areas (see also Koenig et al., 2008). The total mass of the stellar content was estimated as discussed in Section 4.2.3. It is found that for the sub-cluster regions the SFEs varies between 5% to 20% with an average of $\sim 10.2\pm 1.2$%. These SFE estimates must be considered as lower limits in these regions as we are considering stellar contents composed of Class i and II sources. Inclusion of Class iii sources will increase the SFE values.

In the case of embedded clusters Chavarría et al. (2014) have obtained SFE as 3-45% with an average 20%. The SFE distribution as a function of the number of the cluster members and the mean surface density of each of our regions is shown in Fig. 17. The SFE seems to be anti-correlated with the number of the cluster members (Spearman’s correlation coefficient ‘r’ =-0.7 with 95% confidence interval of -0.5 to -0.8) in the sense that the regions associated with high SFEs have smaller numbers of closely packed stars. The right-hand panel of Fig. 17 also indicates a tight correlation between the SFE and mean surface density (Spearman’s correlation coefficient ‘r’ =0.9 with 95% confidence interval of 0.8 to 1.0) in the sense that the regions having higher YSO densities exhibit higher SFEs. However, Fig. 17 left-hand panel indicates that the regions associated with high SFEs have smaller numbers of stars. The probable explanation for the regions having a smaller number of stars but a higher SFE could be that these regions are closely packed as revealed in Fig. 13.