Context. There are a number of methods that identify stellar sub-structure in star forming regions, but these do not quantify the degree of association of individual stars – something which is required if we are to better understand the mechanisms and physical processes that dictate structure. Aims. We present the new novel statistical clustering tool “INDICATE” which assesses and quantifies the degree of spatial clustering of each object in a dataset, discuss its applications as a tracer of morphological stellar features in star forming regions, and to look for these features in the Carina Nebula (NGC 3372). Methods. We employ a nearest neighbour approach to quantitatively compare the spatial distribution in the local neighbourhood of an object with that expected in an evenly spaced uniform (i.e. definitively non-clustered) field. Each object is assigned a clustering index (“I”) value, which is a quantitative measure of its clustering tendency. We have calibrated our tool against random distributions to aid interpretation and identification of significant I values. Results. Using INDICATE we successfully recover known stellar structure of the Carina Nebula, including the young Trumpler 14-16, Treasure Chest and Bochum 11 clusters. Four sub-clusters contain no, or very few, stars with a degree of association above random which suggests these sub-clusters may be fluctuations in the field rather than real clusters. In addition we find: (1) Stars in the NW and SE regions have significantly different clustering tendencies, which is reflective of differences in the apparent star formation activity in these regions. Further study is required to ascertain the physical origin of the difference; (2) The different clustering properties between the NW and SE regions are also seen for OB stars and are even more pronounced; (3) There are no signatures of classical mass segregation present in the SE region – massive stars here are not spatially concentrated together above random; (4) Stellar concentrations are more frequent around massive stars than typical for the general population, particularly in the Tr14 cluster; (5) There is a relation between the concentration of OB stars and the concentration of (lower mass) stars around OB stars in the centrally concentrated Tr14 and Tr15, but no such relation exists in Tr16. We conclude this is due to the highly sub-structured nature of Tr16. Conclusions. INDICATE is a powerful new tool employing a novel approach to quantify the clustering tendencies of individual objects in a dataset within a user-defined parameter space. As such it can be used in a wide array of data analysis applications. In this paper we have discussed and demonstrated its application to trace morphological features of young massive clusters.

This paper presents the sharpest near-IR images of the massive cluster R 136 to date, based on the extreme adaptive optics of the SPHERE focal instrument implemented on the ESO Very Large Telescope and operated in its IRDIS imaging mode. The crowded stellar population in the core of the R 136 starburst compact cluster remains still to be characterized in terms of individual luminosities, age, mass and multiplicity. SPHERE/VLT and its high contrast imaging possibilities open new windows to make progress on these questions. Stacking-up a few hundreds of short exposures in J and Ks spectral bands over a field of view (FoV) of 10.9 × 12.3 centered on the R 136a1 stellar component, enabled us to carry a refined photometric analysis of the core of R 136. We detected 1110 and 1059 sources in J and Ks images respectively with 818 common sources. Thanks to better angular resolution and dynamic range, we found that more than 62.6% (16.5%) of the stars, detected both in J and Ks data, have neighbours closer than 0.2 (0.1 ). The closest stars are resolved down to the full width at half maximum (FWHM) of the point spread function (PSF) measured by Starfinder. Among resolved and/or detected sources R 136a1 and R 136c have optical companions and R 136a3 is resolved as two stars (PSF fitting) separated by 59 ± 2 mas. This new companion of R 136a3 presents a correlation coefficient of 86% in J and 75% in Ks. The new set of detected sources were used to re-assess the age and extinction of R 136 based on 54 spectroscopically stars that have been recently studied with HST slit-spectroscopy (Crowther et al. 2016, MNRAS, 458, 624) of the core of this cluster. Over 90% of these 54 sources identified visual companions (closer than 0.2 ). We found the most probable age and extinction for these sources are 1.8 +1.2 −0.8 Myr, A J = (0.45 ± 0.5) mag and A K = (0.2 ± 0.5) mag within the photometric and spectroscopic error-bars. Additionally, using PARSEC evolutionary isochrones and tracks, we estimated the stellar mass range for each detected source (common in J and K data) and plotted the generalized histogram of mass (MF with error-bars). Using SPHERE data, we have gone one step further and partially resolved and studied the initial mass function covering mass range of (3-300) M at the age of 1 and 1.5 Myr. The density in the core of R 136 (0.1-1.4 pc) is estimated and extrapolated in 3D and larger radii (up to 6 pc). We show that the stars in the core are still unresolved due to crowding, and the results we obtained are upper limits. Higher angular resolution is mandatory to overcome these difficulties.

Context. Better understanding of star formation in clusters with high-mass stars requires rigorous dynamical and spatial analyses of star-forming regions. Aims. We seek to demonstrate that “INDICATE” is a powerful spatial analysis tool which when combined with kinematic data from Gaia DR2 can be used to probe star formation history in a robust way. Methods. We compared the dynamic and spatial distributions of young stellar objects (YSOs) at various evolutionary stages in NGC 2264 using Gaia DR2 proper motion data and INDICATE. Results. The dynamic and spatial behaviours of YSOs at different evolutionary stages are distinct. Dynamically, Class II YSOs predominately have non-random trajectories that are consistent with known substructures, whereas Class III YSOs have random trajectories with no clear expansion or contraction patterns. Spatially, there is a correlation between the evolutionary stage and source concentration: 69.4% of Class 0/I, 27.9% of Class II, and 7.7% of Class III objects are found to be clustered. The proportion of YSOs clustered with objects of the same class also follows this trend. Class 0/I objects are both found to be more tightly clustered with the general populous/objects of the same class than Class IIs and IIIs by a factor of 1.2/4.1 and 1.9/6.6, respectively. An exception to these findings is within 0.05° of S Mon where Class III objects mimic the behaviours of Class II sources across the wider cluster region. Our results suggest (i) current YSOs distributions are a result of dynamical evolution, (ii) prolonged star formation has been occurring sequentially, and (iii) stellar feedback from S Mon is causing YSOs to appear as more evolved sources. Conclusions. Designed to provide a quantitative measure of clustering behaviours, INDICATE is a powerful tool with which to perform rigorous spatial analyses. Our findings are consistent with what is known about NGC 2264, effectively demonstrating that when combined with kinematic data from Gaia DR2 INDICATE can be used to study the star formation history of a cluster in a robust way.