The VMC Survey XXVII. Young Stellar Structures in the LMC$'$s Bar Star-Forming Complex

Star formation is a hierarchical process, forming young stellar structures of star clusters, associations, and complexes over a wide scale range. The star-forming complex in the bar region of the Large Magellanic Cloud is investigated with upper main-sequence stars observed by the VISTA Survey of the Magellanic Clouds. The upper main-sequence stars exhibit highly non-uniform distributions. Young stellar structures inside the complex are identified from the stellar density map as density enhancements of different significance levels. We find that these structures are hierarchically organized such that larger, lower-density structures contain one or several smaller, higher-density ones. They follow power-law size and mass distributions as well as a lognormal surface density distribution. All these results support a scenario of hierarchical star formation regulated by turbulence. The temporal evolution of young stellar structures is explored by using subsamples of upper main-sequence stars with different magnitude and age ranges. While the youngest subsample, with a median age of log($\tau$/yr)~=~7.2, contains most substructure, progressively older ones are less and less substructured. The oldest subsample, with a median age of log($\tau$/yr)~=~8.0, is almost indistinguishable from a uniform distribution on spatial scales of 30--300~pc, suggesting that the young stellar structures are completely dispersed on a timescale of $\sim$100~Myr. These results are consistent with the characteristics of the 30~Doradus complex and the entire Large Magellanic Cloud, suggesting no significant environmental effects. We further point out that the fractal dimension may be method-dependent for stellar samples with significant age spreads.


INTRODUCTION
It has been suggested that star formation is a hierarchical process, forming young stellar structures over a wide range of scales Elmegreen et al. 2000;Elmegreen 2011;Gusev 2014;Gouliermis et al. 2015Gouliermis et al. , 2017Sun et al. 2017). These structures include, for increasing size and decreasing density, star clusters, associations, and complexes. Studies of local star-forming regions (e.g. Gomez et al. 1993;Larson 1995;Simon 1997;Kraus & Hillenbrand 2008) and nearby galaxies (e.g. Elmegreen & Elmegreen 2001;Elmegreen et al. 2006;Gouliermis et al. 2010Gouliermis et al. , 2015Gouliermis et al. , 2017 suggest that the young stellar structures display a high degree of substructuring and fractal properties, which may be inherited from the natal gas from which they form. After birth, the young stellar structures evolve rapidly toward uniform distributions be-fore they are completely dispersed (Gieles et al. 2008;Bastian et al. 2009;Gouliermis et al. 2015). Beyond this simple picture, however, more exploration is needed to fully understand their properties, formation, evolution, and especially the roles played by related physical processes (e.g. gravity, turbulence, galactic dynamics, etc.). Stellar complexes are important nurseries of new stars. They usually have kiloparsec scales and contain smaller young stellar structures such as associations and aggregates, which themselves are subclustered into compact star clusters (Efremov 1995). Star formation in stellar complexes is not only influenced by global galactic properties, e.g. bars and spiral arms (Binney & Merrifield 1998), but also regulated by local processes, e.g. gravity, turbulence, magnetic field, and stellar feedback (Mac Low & Klessen 2004). On the other hand, the Large and Small Magellanic Clouds (LMC and SMC) are close neighbors of the Milky Way at distances of 50 and 60 kpc, respectively (de Grijs et al. 2014;de Grijs & Bono 2015). The LMC is the prototype of barred Magellanic spiral galaxies. It has a flat stellar disk, a single-looping spiral arm, a stellar bar which is off-centered from the galaxy's dynamical center, and a large star-forming complex at the northwestern end of the bar (Wilcots 2009). The SMC, however, is a late-type dwarf galaxy, with an elongated, cigarshaped structure seen edge-on (D'Onghia & Fox 2016). They show signatures of interactions with one another as well as with the Milky Way's gravitational potential and halo gas (de Boer et al. 1998;D'Onghia & Fox 2016;Belokurov et al. 2017;Subramanian et al. 2017). Both Clouds exhibit active past and ongoing star formation Oliveira 2009;Rubele et al. 2012Rubele et al. , 2015. As a result, stellar complexes in the Magellanic Clouds provide unique laboratories for understanding young stellar structures. Bastian et al. (2009) studied stellar structures in the LMC. Based on young massive (OB) stars, they found that their identified stellar groups have no characteristic length scale and exhibit a power-law luminosity function with index −2. Using stellar subsamples of different ages, they showed that while stars are born with a high degree of substructuring, older subsamples are progressively less clumpy, reaching a uniform distribution at ∼175 Myr. Bonatto & Bica (2010) investigated the spatial correlation of star clusters and "non-clusters" (which are basically nebula complexes and stellar associations) in the Magellanic Clouds. Using two-point correlation functions (TPCFs), they found that young star clusters present a high degree of spatial correlation with themselves and with non-clusters, which does not occur for old star clusters. They also noticed that star clusters (young and old) and non-clusters all have power-law size distributions but with different slopes. In general, these results are in agreement with the scenario referred to above.
Both studies focused on the global population of young stellar structures across the entire LMC (and SMC). Thus, they do not reveal whether there is any environmental dependence in the properties, formation, and evolution of the young stellar structures. It is possible to explore this issue by studying individual stellar complexes in the LMC. In Sun et al. (2017), we reported the hierarchical patterns of the young stellar structures in the 30 Doradus-N158-N159-N160 complex (30 Dor complex hereafter) in the LMC. The structures were identified based on upper main-sequence (upper-MS) stars observed by the VISTA Survey of the Magellanic Clouds (VMC; Cioni et al. 2011). The results suggest that the 30 Dor complex is highly substructured in a scale-free manner, supporting the scenario of hierarchical star formation from a turbulent interstellar medium (ISM). The derived projected fractal dimension, D 2 = 1.6 ± 0.3, is consistent with those of Galactic star-forming regions and NGC 346 in the SMC. Thus, no significant environmental dependence was discovered in the fractal dimension of the 30 Dor complex.
In this paper, we carry out a similar study of the stellar complex at the northwestern end of the LMC bar, which we shall refer to as the bar complex. A major part of this complex is covered by VMC tile LMC 6 4, which is used for our analysis in this work ( Fig. 1; see Section 2 for the definition of a tile). The bar complex is an important component of the LMC. As an active star-forming nursery, it is abundant in molecular gas, H II regions, young stellar objects, and star clusters (e.g. Fukui et al. 2008;Carlson et al. 2012;Piatti et al. 2015). The previous studies of Bastian et al. (2009) and Bonatto & Bica (2010), however, have used small samples or subsamples, which contain several hundreds or thousands of objects. As a result, they do not provide sufficient spatial sampling to resolve the detailed inner structures of the bar complex. On the other hand, perturbations from the bar (Gardiner et al. 1998; but see Harris & Zaritsky 2009 for a discussion against a dynamical bar) may have an influence on the bar complex; compared with the galaxy outskirts, it is less affected by external tidal forces (e.g. Fujimoto & Noguchi 1990;Bekki & Chiba 2007;D'Onghia & Fox 2016). It still remains unexplored whether these environmental processes cause any difference in the properties, formation, and evolution of the young stellar structures in the bar complex.
The goal of this paper is to understand the properties, formation, and evolution of young stellar structures in the bar complex. Using wide color and magnitude cuts, we construct a sample of more than 2.5 × 10 4 upper-MS stars younger than ∼1 Gyr, an order of magnitude more than the previous samples mentioned above. The sample provides sufficient spatial sampling, allowing us to identify young stellar structures on scales 10 pc. We then analyze the properties of these structures and discuss the effects of physical processes associated with their formation. We also use TPCFs to study the degree of substructuring in upper-MS subsamples of different magnitude and age ranges, which will help demonstrate any evolution of the young stellar structures. We discuss the environmental effects through comparisons with other regions or with the entire LMC.
This paper is organized as follows. Section 2 describes the data used in this work. Our selection of upper-MS stars and their spatial distributions are outlined in Section 3. Young stellar structures are identified and analyzed based on the full sample in Section 4, while in Section 5 we use subsamples of upper-MS stars to explore their temporal evolution. Finally, we complete this paper with a summary and conclusions.

DATA
Data used in this work are from the VMC survey (Cioni et al. 2011), which is carried out with the Visible and Infrared Survey Telescope for Astronomy (VISTA; Sutherland et al. 2015). The VMC survey is a multiepoch, uniform, and homogeneous photometric survey of the Magellanic System (LMC, SMC, Bridge, and Stream). It uses the near-infrared Y , J, and K s bands, and the typical spatial resolution is ∼1 ′′ or better. Because there are gaps between its 16 detectors, a sequence of six offsets is needed to observe a contiguous area of sky; the combined image is then referred to as a tile. Each tile covers an area of ∼1.5 deg 2 and is designed to be observed at three epochs in the Y and J bands and at 12 epochs in the K s band, corresponding to total exposure times of 2400 s, 2400 s, and 7500 s, respectively. The saturation limits are usually Y = 12.9 mag, J = 12.7 mag, and K s = 11.4 mag; typical 5σ magnitude limits are density map of all stars with V < 20 mag from the Magellanic Cloud Photometric Survey (MCPS; Zaritsky et al. 2004). We obtain the map by simple star counts in bins of 1 ′ ×1 ′ . The nearly horizontal black line is caused by slight gaps between scans in their observations. The LMC bar corresponds to the northeast-southwest elongated structure with prominently high densities in this map. Blue points: MCPS stars with V < 14.5 mag and B − V < 0.5 mag, which are young and massive stars and trace recent star formation (Bastian et al. 2009). At the northwestern end of the bar, there is a concentration of such bright and blue stars, corresponding to the bar complex. Its approximate extent is indicated by the white ellipse. Note the other end of the bar contains significantly fewer bright and blue stars. The white rectangle shows the extent of VMC tile LMC 6 4, which covers the major part of the bar complex. The map is centered R.A.(J2000) = 05 h 18 m 48 s , Dec.(J2000) = −68 o 42 ′ 00 ′′ . Y = 21.9 mag, J = 22.0 mag, and K s = 21.5 mag for stacked observations combining all epochs. Note, however, that these limits may vary with source crowding and sky conditions. The VMC survey is still ongoing, and is expected to cover 170 deg 2 on completion within its ∼9 years of observations. The star-forming complex analyzed in this work is based on tile LMC 6 4. We retrieved the data of this tile as part of VMC Data Release 3 from the VISTA Science Archive (VSA). The VSA and the VISTA data flow pipeline are described by Cross et al. (2012) and Irwin et al. (2004), respectively. We use point-spread function (PSF) photometry obtained with PSF-homogenized, stacked images from different epochs (Rubele et al. 2015, their Appendix A); the photometric errors and local completeness have also been calculated with artificial star tests (Rubele et al. 2012(Rubele et al. , 2015. The "top" half of detector #16 of the VISTA infrared camera (VIRCAM) has worse signal-to-noise ratios, since its pixel-to-pixel quantum efficiency varies on short timescales, leading to inaccurate flat-fields. This affects the southwestern corner of the analyzed tile, but it will be shown that the bar complex does not overlap with this region (Section 3.2). Thus we do not attempt to deal with this effect. Figure 2 shows the (J − K s , K s ) color-magnitude diagram (CMD) of stars in tile LMC 6 4. The MS is clearly visible at colors −0.3 < (J − K s ) < 0.2 mag. The red, populous branch is the red-giant branch (RGB), overlapping with the red clump (RC) at (J − K s ) = 0.5 mag and K s = 17.0 mag. Slightly bluer than the RGB, red supergiants (RSG) can be seen at K s < 13.0 mag. The vertical strip at (J − K s ) = 0.35 mag arises from foreground Galactic stars, and sources redder than (J − K s ) = 1.0 mag are essentially background galaxies. In addition, an overdensity of stars can also be found around (J − K s ) = 0.1 mag and 13.5 < K s < 14.5 mag, most of which are primarily blue-loop stars.

Sample Selection
In the right-hand panel of Fig. 2, PARSEC isochrones (version 1.2S; Bressan et al. 2012) of metallicity [Fe/H] = −0.3 dex (which is typical for massive LMC stars; Rolleston et al. 2002) and ages log(τ /yr) = 7.0, 8.0, and 9.0 are overplotted. Offsets of 0.026 mag in J and 0.003 mag in K s are subtracted from the isochrones to convert model magnitudes from the Vega system to the VISTA system (for details, see Rubele et al. 2012Rubele et al. , 2015. The isochrones are shifted by the LMC's distance modulus of (m − M ) 0 = 18.49 mag (Pietrzyński et al. 2013;de Grijs et al. 2014) and then reddened by an extinction of A V = 0.6 mag. This extinction is found by matching the isochrone of log(τ /yr) = 7.0 (whose MS extends the full magnitude range in the CMD) to the bluest edge of the MS at K s < 18 mag. We apply the extinction coefficients A J /A V = 0.283 and A Ks /A V = 0.114, computed from the Cardelli et al. (1989) extinction curve with R V = 3.1 (Girardi et al. 2008).
Stars in the upper-MS are relatively young given that higher-mass stars have shorter MS lifetimes than their lower-mass counterparts. Indeed, stars in the upper-MS brighter than K s = 18.0 mag should be primarily younger than 1 Gyr, as indicated by the theoretical isochrones; thus, we select the upper-MS stars brighter than K s = 18.0 mag for the analysis in the following sections. We use a color window, −0.3 < (J −K s ) < 0.2 mag, to distinguish the upper-MS stars from the RGB and RC stars. The interstellar extinction in this region shows significant spatial variations, leading to the broad width of the upper-MS. Very high extinctions may shift stars out of the color window. However, such cases should be rare, since the adopted color window has a wide range in (J − K s ) 1 . An upper magnitude limit of K s = 14.5 mag is applied to avoid the blue-loop stars. While there may be some contamination of blue-loop stars fainter than K s = 14.5 mag, their number is small compared with the upper-MS stars; because of the rapid evolutionary phase, they will not spend much time there. The selection of upper-MS stars is indicated by the box in Fig. 2 (right-hand panel), and the final upper-MS sample contains 25,232 stars in total.
It has long been known that young stellar structures are transient structures, except for bound star clusters on small scales, which may survive for a significant period . They are usually younger than several hundred million years (Efremov 1995;Gieles et al. 2008;Bastian et al. 2009). Thus, the upper-MS sample includes many stars which are older than the typical age of young stellar structures (see also Section 5.1). Despite this potential contamination, the young stellar structures can still be revealed with the upper-MS sample. It will be shown that old stars ( 100 Myr) in the sample are almost uniformly distributed (Section 5); as a result, the surface density enhancements in the stellar distribution, which are identified as young stellar structures (Section 4.1), are dominated only by young stars. On the other hand, although it is possible to reduce the contamination by adopting a brighter upper magnitude limit, this will lead to a very small sample with poor spatial sampling insufficient to resolve the small-scale young stellar structures. Moreover, the upper-MS sample containing both young and old stars allows us to investigate their temporal evolution (Section 5). Thus, we use this upper-MS sample for the following analysis. Figure 3 shows the spatial distribution of the selected upper-MS stars. Simply for clarity, we show stars brighter and fainter than K s = 16.5 mag in two panels separately. All 2827 stars brighter than K s = 16.5 mag are displayed in the left-hand panel, while the same number of stars are randomly selected and displayed in the right-hand panel from the 22,405 stars fainter than K s = 16.5 mag. We do not show all the stars fainter than K s = 16.5 mag to avoid symbols crowding in the figure.

Spatial Distributions
From the left-hand panel, it is apparent that the distribution of the upper-MS stars brighter than K s = 16.5 mag is not uniform but highly clumpy and substructured.
Compared with a dispersed stellar distribution across the tile, many of the stars reside in groups with high surface densities. These groups correspond well with eight Lucke & Hodge (LH;1970) associations, the positions and extents of which are labeled in the figure. Some other groups not cataloged by Lucke & Hodge (1970) are also visible; for instance, the stellar group to the northwest of LH 31 corresponds primarily to the young, populous star cluster NGC 1850 (Vallenari et al. 1994). Another large concentration of upper-MS stars can also be found between LH 39, LH 41, and LH 42.
The right-hand panel shows the spatial distribution of stars fainter than K s = 16.5 mag, which is less clumpy than that of their brighter counterparts. Some of the LH associations, e.g., LH 33 and LH 39, become less prominent and can be barely seen in the stellar distribution. Considering that the fainter stars have an older average age than the brighter ones, their different distributions may suggest the presence of an evolutionary effect. This will be discussed in detail in Section 5. Figure 3 also shows the IRAC 8.0 µm emission map from the Spitzer legacy program "Surveying the Agents of Galaxy Evolution" (SAGE; Meixner et al. 2006). The 8.0 µm emission comes mainly from hot dust, which is heated by young stars and re-radiated at infrared wavelengths. It can be seen that some of the stellar groups are associated with dust emission, for instance, NGC 1850, LH 31, LH 33, LH 35, LH 41, and LH 42. In contrast, LH 27, LH 30, and LH 39 are not associated with dust emission; and moreover, the large stellar concentration between LH 39, LH 41, and LH 42 lies in a void of dust emission, with the surrounding dust emission exhibiting a half-circular boundary. It is possible that these stars have dispersed the ISM through their radiation, stellar The same number of stars are randomly selected and displayed in the right-hand panel from the 22,403 stars with 16.5 < Ks < 18.0 mag. This is simply for clarity and for the ease of comparison between the two panels. The background colorscale is the Spitzer/IRAC 8.0 µm dust emission map from SAGE. The color bar is in units of MJy sr −1 ; contours of 3 MJy sr −1 are also overplotted. Eight LH associations are labeled, with the cyan circles showing their approximate extents; the circles' radii are calculated based on the associations' geometric means of the major and minor axes given by Lucke & Hodge (1970). If an LH association is associated with a Henize (1956) nebula, the nebula name is also labeled in the bracket after the LH designation. To the northwest of LH31 is the young, populous star cluster winds, and/or supernovae. Henize (1956) has cataloged Hα-emitting nebulae in the Magellanic Clouds, and this region is very abundant in such nebulae, which are primarily H II regions, wind-driven shells, supernova remnants, etc., or complexes of different types (e.g. Laval et al. 1992;Ambrocio-Cruz et al. 1998; see also Carlson et al. 2012). Some of these nebulae are associated with the abovementioned stellar groups, e.g. N103, N105, N113, N119, and N120, which are indicated in the figure. An extended nebula, N117, is associated with the stellar concentration between LH 39, LH 41, and LH 42. In addition, a number of Henize nebulae with small sizes are also located in this region (see Fig. 6 of Henize 1956). The presence of Hα-emitting nebulae suggests active feedback from the young stars to the ISM.

Identification and Dendrograms
In the previous section, we have shown that the upper-MS stars exhibit large numbers of overdensities in the bar complex. In this paper, these overdensities are all referred to as young stellar structures, regardless of their mass or length scales, whether gravitationally bound or unbound. In order to quantitively analyze the young stellar structures in the bar complex, we need to identify them in a systematic way; to do this, we adopt the same method as detailed in Gouliermis et al. (2015Gouliermis et al. ( , 2017. Firstly, we construct a surface density map of the upper-MS stars (Fig. 4, left-hand panel) through kernel density estimation (KDE). This is done by convolving the map of the upper-MS stars with a Gaussian kernel. The choice of an optimal kernel width is best decided through experimentation (Gouliermis et al. 2017). The kernel width specifies the resolution of the KDE map, but small kernels lead to significant noise as well. Testing various kernel sizes shows that a standard-deviation width of 10 pc offers a good balance between resolution and noise. With this width it is possible to resolve structures of sizes comparable to or larger than 10 pc. We can achieve this resolution because the upper-MS sample provides sufficient spatial sampling based on its ∼2.5 × 10 4 stars. Smaller samples will unnecessarily lead to poorer resolutions. The resultant KDE map has a median value of 0.011 stars pc −2 , a mean value of 0.015 stars pc −2 , and a standard deviation of 0.016 stars pc −2 .
We then identify young stellar structures as stellar surface overdensities above the mean value, of different sig- nificance levels from 1σ to 10σ, in steps of 1σ. To avoid spurious detections, we require that the iso-density contour of each structure should enclose at least N min = 5 upper-MS stars. As a result, 52, 23, 20, 13, 6, 3, 2, 2, 1, and 1 structures are identified at the 10 levels of increasing significance, i.e., 123 structures in total. The catalog of identified structures, along with their physical parameters (see Sections 4.2-4.4), is given in Table 1. The choice of N min is arbitrary, since larger values of N min may miss out small real structures while smaller values lead to more spurious identifications. Changing N min does not affect the conclusions in this section but may be important, and will be discussed, in the context of parameter statistics of the young stellar structures (Secions 4.2-4.4).
The KDE map shows that many of the young stellar structures exhibit very irregular morphologies; and moreover, the young stellar structures at different significance levels are organized in a hierarchical way. This is especially obvious for Structures A and B, the two large-sized young stellar structures at 1σ significance level located in the southeast and northwest of the field, respectively. Structures A and B contain five and three substructures at 2σ level, respectively; the substructures are smaller in size than their parent structures, and going up to even higher significance levels they may vanish, survive but contract in size, or fragment into several even smaller sub-substructures. These are illustrated with dendro-grams -structure trees showing the "parent-child" relations of young stellar structures found at various significance levels (Fig. 4,. This hierarchical subclustering of young stars has also been reported for a number of star-forming regions and galaxies (Gouliermis et al. 2010(Gouliermis et al. , 2015Kirk & Myers 2011;Gusev 2014;Sun et al. 2017), and is an indicator of hierarchical star formation over a range of length scales (Efremov 1995;Elmegreen et al. 2000;Elmegreen 2011).

Size Distribution
The young stellar structures span a wide range of sizes, from the largest Structure A, to the smallest ones, which are comparable to the kernel width. The size of each young stellar structure, R, is estimated based on the radius of a circle which has the same area as that covered by the iso-density contour of the structure. The results for all structures are listed in Table 1. Figure 5 shows the cumulative size distribution. The distribution is approximately a single power law for larger sizes but shows significant flattening at smaller sizes. We note that there are two main factors affecting the completeness of young stellar structures (Section 4.1). On the one hand, the KDE map, from which they have been identified, has resolution of 10 pc; thus, structures smaller than this size will probably be smeared out by the convolution. On the other hand, each structure has been required to contain at least N min = 5 stars; as a result, small struc-  Note.
-Column 1: ID number for each young stellar structure; Column 2: the significance level; Columns 3: the right ascension of the center of each young stellar structure, defined as α = (α min + αmax)/2, where α min and αmax are the minimum and maximum right ascension of the iso-density contour of each structure, respectively; Column 4: same as Column 3 but for the declination; Columns 5-7: parameters of size, mass (expressed in N * ), and surface density (see Sections 4.2-4.4). Only the first 15 records are shown for example. The complete catalog is available online. tures may not contain enough upper-MS stars at the less populated high-mass end of the stellar initial mass function (IMF). To assess the latter effect, we carried out an experiment similar to that of Sun et al. (2017), i.e. by changing N min to 10 and 15 and repeating the structure identification process. The resulting structure size distributions below 10 pc do indeed change with different values of N min ; beyond 10 pc, however, the distributions remain unaffected (not shown). Considering all this, the young stellar structures are complete beyond R = 10 pc. We fit a single power-law function to the size distribution above this value, which suggests a power-law slope of α(R) = −1.5 ± 0.1.
The sizes of the substructures inside a fractal follow where D is the fractal dimension (Mandelbrot 1983;Elmegreen & Falgarone 1996). Thus, the size distribution of the young stellar structures is consistent with a (projected) fractal dimension of D 2 = 1.5 ± 0.1. The subscript '2' indicates that the young stellar structures have been identified and analyzed based on twodimensional projections. It is not easy to obtain the three-dimensional (volume) fractal dimension, D 3 . A relation, D 3 = D 2 + 1, has been proposed by Beech (1992); however, this relation applies only when the perimeterarea dimension of the object's projection is the same as that of a slice (Elmegreen & Scalo 2004). The relation between D 2 and D 3 has also been investigated based on simulations (e.g. Sánchez et al. 2005;Gouliermis et al. 2014).
There is a similarity between the ISM and the young stellar structures identified here. First, similar to the young stellar structures, the ISM also displays irregular morphologies and contains large amounts of substructures (clouds, clumps, cores, and filaments, etc.) which are hierarchically organized (Rosolowsky et al. 2008). Second, the ISM substructures also follow a power-law size distribution, which indicates a scale-free behavior (e.g. Elmegreen & Falgarone 1996). The third aspect of their similarity comes from the fractal dimension. The projected fractal dimension of the ISM has been investigated based on the perimeter-area relation of its projected boundaries. Typical values are close to D 2 = 1.4-1.5 (e.g. Beech 1987;Scalo 1990;Falgarone et al. 1991;Vogelaar & Wakker 1994;Lee 2004;Lee et al. 2016; although smaller values have also been reported by e.g. Dickman et al. 1990;Hetem & Lepine 1993), which are consistent with the fractal dimension as derived for the young stellar structures. Using power-spectrum analysis, Stanimirovic et al. (1999Stanimirovic et al. ( , 2000 reported D 2 = 1.4 or 1.5 for the ISM in the SMC, also close to that of the young stellar structures. On the other hand, it is possible to measure the volume fractal dimension of the ISM, since clouds along the line of sight can be distinguished by their velocities. For instance, Elmegreen & Falgarone (1996) reported D 3 = 2.3 ± 0.3 based on the size distribution for a number of Galactic molecular clouds, and Roman-Duval et al. (2010) found D 3 = 2.36 ± 0.04 using the mass-size relation. If the relation D 3 = D 2 + 1 holds for the ISM, these results would not be far from the fractal dimension of the young stellar structures. Unfortunately, there is no reported measurement of the fractal dimension of the ISM in the LMC bar region. However, it has been suggested that, despite a few exceptions, the fractal dimension is invariant from cloud to cloud, regardless of their nature as star-forming or quiescent, whether gravitationally bound or unbound (e.g. Williams et al. 2000).
A fractal dimension of D 3 = 2.4 is consistent with laboratory results of numerical turbulent flows (Sreenivasan 1991;Elmegreen & Scalo 2004;Federrath et al. 2009). As a result, turbulence has been argued to play a major role in creating the hierarchical structures in the ISM. In addition, agglomeration with fragmentation (Carlberg & Pudritz 1990) and self-gravity (de Vega et al. 1996) may also contribute (see also Elmegreen et al. 2000). The similarity discussed in the previous paragraph suggests that the young stellar structures may have inherited the irregular morphologies, hi- erarchy, size distribution, and fractal dimension from the ISM substructures where they form. After birth, the young stellar structures are expected to evolve toward uniform distributions before they are finally dispersed (Section 5; see also Gieles et al. 2008;Bastian et al. 2009;Gouliermis et al. 2015). As a result, D 2 is expected to increase with age toward an ultimate value of 2. For the bar complex, the small value of D 2 = 1.5 ± 0.1 suggests insignificant evolutionary effects in the structures' size distribution. The reason for this will be further discussed in Section 5.3.
A power-law size distribution has also been found for the 30 Dor complex ) with a projected fractal dimension of 1.6 ± 0.3, which is in agreement with the value found here for the bar complex, within errors. It has been proposed that star formation in the LMC is influenced by the perturbation of the offcenter bar (Gardiner et al. 1998) or by interactions with the SMC (e.g. Fujimoto & Noguchi 1990;Bekki & Chiba 2007); specifically, star formation in the 30 Dor complex may be induced by the bow shock as the LMC moves through the Milky Way's halo (de Boer et al. 1998). The 30 Dor and bar complexes are located in very different galactic environments; however, no significant difference in D 2 is found, considering the measurement uncertainties. Stars in other Galactic or SMC starforming regions have D 2 = 1.4-1.5 (e.g. Larson 1995;Simon 1997;Gouliermis et al. 2014), also close to that of the bar complex. On the other hand, galaxy-wide young stellar distributions exhibit a larger range of D 2 from ∼1.0 to 1.8, which may reflect different clustering properties, evolutionary effects, or a method dependence (Elmegreen & Elmegreen 2001;Elmegreen et al. 2006Elmegreen et al. , 2014Gouliermis et al. 2015). We refer the reader to Sun et al. (2017, their Section 6) for a more detailed discussion of comparisons of D 2 .

Mass Distribution and the Mass-Size Relation
The number of upper-MS stars in a structure, N * , provides an approximate representation of the structure mass, assuming that all structures have a similar age and that the stellar IMF is fully sampled. To obtain this quantity, we first correct the photometric incom- pleteness by assigning weights to the stars, defined as w = 1.0/min[f J , f Ks ], where f J and f Ks are the local completeness in the J and K s bands, respectively (Section 2). We then calculate N * with in which the subscript i runs for all stars enclosed by the structure's iso-density contour, Σ bg = 0.015 stars pc −2 is the mean surface density of the KDE map, and A is the area of the iso-density contour (see Section 4.1 and Fig. 4). We use the last term to estimate the number of background stars in chance alignment with the structure. Table 1 lists N * for all the structures. Figure 6 shows the cumulative distribution of N * , which for brevity we shall refer to as the mass distribution. Beyond N * = 100 stars, the mass distribution is well described by a single power law; at N * < 100 stars, however, the mass distribution shows a deficiency of structures with respect to the extrapolation from the higher-mass end. Similarly to the size distribution, this is also caused by the incompleteness of young stellar structures. This is supported by Fig. 7, which shows a strong correlation of mass and size (with a Pearson correlation coefficient of 0.90). It is apparent that structures more massive than N * = 100 stars are all larger than 10 pc, while structures below this mass may be larger or smaller than 10 pc. Recall that the young stellar structures are complete beyond this size, the mass-size relation suggests that the completeness threshold lies at N * = 100 stars in the mass distribution.
Beyond N * = 100 stars, the cumulative mass distribution has a power-law slope of α(N * ) = −1.0 ± 0.1. This translates into a differential mass function of the form n(M )dM ∝ M −β dM , with β = 2.0 ± 0.1. A mass function slope of 2 is predicted by the scenario of hierarchical star formation (Fleck 1996;Elmegreen 2008). This is in also agreement with studies of young stellar structures in NGC 628 (Elmegreen et al. 2006), M33 (Bastian et al. 2007), the LMC (Bastian et al. 2009), and the SMC (Oey, King, & Parker 2004). The same slope is also found for young star clusters in a wide variety of environments, e.g. the solar neighborhood (Battinelli et al. 1994), the Magellanic Clouds (Hunter et al. 2003; As mentioned in the first paragraph of this subsection, N * is proportional to mass if the young stellar structures have similar ages and if the stellar IMF is fully sampled. Age differences or stochastic sampling of the IMF may lead to uncertainties in the slope of the mass function. Note, however, that the slope is derived for structures with N * ≥ 100 stars. As a result, the stochastic sampling effect is very small. On the other hand, we have assessed the K s -band luminosity functions (LFs) of the upper-MS stars which are located inside the iso-density contours of these young stellar structures (not shown). There are 34 structures with N * > 100 stars, and 28 of them have LFs with indistinguishable shapes. Thus, their age differences are not very significant. Only 7 out of the 34 structures have LFs with significantly different shapes, suggesting possible age differences. These structures have N * ranging from 103 to 358 stars; thus, they may only slightly affect the first few bins above N * = 100 in Fig. 6. Considering all this, we suggest that possible age differences do not influence the derived mass function slope.
Returning to the mass-size relation (Fig. 7), the mass and size of young stellar structures follow a powerlaw correlation of N * ∝ R κ , in which κ = 1.5 ± 0.1. Power-law mass-size relations with fractional slopes are also reported for young stellar structures in e.g. M33 (Bastian et al. 2007), NGC 6503 (Gouliermis et al. 2015), and NGC 1566 (Gouliermis et al. 2017). This power-law relation is expected from a fractal distribution of upper-MS stars (Elmegreen & Falgarone 1996) 2 . Moreover, this power-law slope is another definition of the fractal dimension (Mandelbrot 1983), which is consistent with that from the structure size distribution (Section 4.2). Figure 8 shows the cumulative distribution of the structures' surface densities. The surface density (values given in Table 1) is calculated simply by the number of upper-MS stars divided by the contour area, i.e. Σ = N * /A. As previously mentioned, the completeness of young stellar structures is affected by the resolution of the KDE map and the requirement of N min = 5 stars for each structure. The relation between surface density and size is shown in Fig. 9, in which the limits of the two constraits are also plotted. It is immediately obvious that the requirement of N min = 5 stars rejects low-surface density structures at the small-size end. Beyond R = 10 pc, however, this requirement does not affect the completeness (which reaffirms our conclusion in Section 4.2). On the other hand, the surface density does not show any apparent correlation with size. As a result, we expect that the surface density distribution for structures with R > 10 pc should be a good representation of the distribution of the underlying young stellar structure population. Thus in summary, we can use structures with R > 10 pc to investigate the distribution of the surface density. The result is shown as the lower data points in Fig. 8.

Density Distribution and the Density-Size Relation
We further fit the data with which is the cumulative form of the lognormal function where p 0 is a normalization constant, p 1 the natural logarithm of the mean value, and p 2 the dispersion in efoldings, respectively. The solid line in Fig. 8 is the best-fitting result, which provides a reasonable description of the data. Consistent with this work, lognormal surface density distributions have also been reported by Bressert et al. (2010) for the young stellar objects in the solar neighborhood, and recently by Gouliermis et al. (2017) for the young stellar structures in the granddesign galaxy NGC 1566. Lognormal distributions of volume and/or column densities are also found for the ISM, either in hydrodynamical simulations of turbulent, isothermal gas (e.g. Klessen 2000;Federrath et al. 2010;Konstandin et al. 2012) or in observations of molecular clouds (e.g. Lombardi et al. 2010;Schneider et al. 2012). The origin of lognormal density distributions can be understood in a purely statistical way in the context of hierarchical star formation with turbulence (Vázquez-Semadeni 1994). In an ISM regulated by turbulence, a substructure with average density Σ can be considered as hierarchically produced by an n-step sequence of density fluctuations, each occurring within a lower-density substructure whose average density is generated via fluctuation from the previous step. We denote this hierarchical sequence by {Σ 0 , Σ 1 = ǫ 1 Σ 0 , Σ 2 = ǫ 2 Σ 1 , ..., Σ n = ǫ n Σ n−1 }, where ǫ 1 , ǫ 2 , ..., ǫ n would follow the same probability distribution function when pressure and self-gravity are unimportant, since in that case the hydrodynamic equations become self-similar and invariant to rescaling in density (see the discussion of Vázquez-Semadeni 1994, their Section 2). As a result, Σ is proportional to the product of this large number of identically-distributed random variables, and would follow a lognormal distribution according to the central limit theorem if the fluctuations are independent (see also Federrath et al. 2010, their Section 3.3). Furthermore, it is a natural expectation that the young stellar structures may have inherited the lognormal density distribution from their parental ISM substructures. Indeed, Gutermuth et al. (2011) report a correlation between the surface densities of young stellar objects and gas in eight nearby molecular clouds. The above theoretical considerations may be violated when the star-forming ISM is subject to strong self-gravity, shocks, rarefaction waves, etc., leading to non-Gaussian deviations from lognormal density distributions (Klessen 2000;Federrath et al. 2010;Girichidis et al. 2014). The young stellar structures, however, do not demonstrate such deviations to any statistical significance.

The Upper-MS Subsamples
The upper-MS sample contains stars of various ages, allowing us to investigate the temporal evolution of young stellar structures. To do this, we divide the full upper-MS sample into four subsamples (a), (b), (c), and (d), by applying three cuts in the K s -band magnitude at K s = 16.0, 17.0, and 17.5 mag, respectively. The numbers of stars in the subsamples are given in Table 2. While the brightest subsample (a) consists of only the youngest stars, continuously fainter subsamples contain stars of wider age ranges. We estimate the stellar ages by constructing a model upper-MS sample which fits the luminosity function (LF) of the observed sample (Fig. 10, left-hand panel). In deriving the observed LF, we take into account photometric incompleteness by assigning weights to stars in the same way as in Section 4.3.
The observed sample is assumed to be a linear combination of "stellar partial models" (SPMs) -model representations of stellar populations covering small age intervals (Harris & Zaritsky 2001;Kerber et al. 2009;Rubele et al. 2012Rubele et al. , 2015. Each SPM is simulated with PARSEC isochrones, the Chabrier (2001) lognormal IMF, and a 30% binary fraction. This binary fraction is typical for LMC star clusters (Elson et al. 1998;Li et al. 2013), and the simulated binaries are non-interacting systems with primary/secondary mass ratios evenly distributed from 0.7 to 1.0 (below these mass ratios, the secondary does not affect the photometry of the system significantly). The SPMs are reddened with an extinction value of A V = 0.6 mag as derived in Section 3.1; we do not consider spatial variations of the extinction since the effect is small for K s -band magnitudes. Neither do we consider the photometric errors, typical values of which (σ Ks = 0.02 mag at K s = 18 mag) are much smaller than the LF's bin size (0.1 mag). The upper-MS stars of each SPM are selected based on the same criteria as have been applied to the data in Section 3.1. We use 21 SPMs, each with an age interval of 0.1 dex in log(τ /yr) and central ages log(τ /yr) = 7.0 to 9.0 in steps of 0.1 dex; and for simplicity, we assume that the 21 SPMs have a single metallicity of [Fe/H] = −0.3 dex, which is typical for massive MS stars in the LMC (see e.g. Rolleston et al. 2002). Note that although the isochrone of a population of log(τ /yr) = 9.0 does not cross the selection box for upper-MS stars (Fig. 2), its binary systems may still be bright enough to satisfy the selection criteria; thus, we choose log(τ /yr) = 9.0 as the oldest age bin.
A linear combination of the SPMs corresponds to a certain star-forming history (SFH), which can be characterized by the star-forming rate (SFR) as a function of look-back time. Adopting different SFHs will produce model upper-MS samples with different LFs. Via tests with manually adjusted SFHs, we find that a SFH of the form SFR ∝ exp (log(τ /yr) − 8.5) 2 0.75 (5) (as shown in the middle panel of Fig. 10), provides a reasonable fit to the observed LF. The only large deviation exists in the three brightest magnitude bins; however, in the tests we have not considered the fitting performance here because there may be an increasing contribution from blue-loop stars (Section 3.1). Adopting this SFH we are able to model the observed upper-MS sample and estimate the stellar age distributions in the full sample and subsamples, which are shown in the right-hand panel of Fig. 10. The full sample covers a wide age range from log(τ /yr) = 7.0 to 9.0, with median and mean values of both 7.7. As expected, the subsamples, from the brightest to the faintest, cover progressively larger age ranges; their median and mean logarithmic stellar ages are given in Table 2. Figure 11 shows the spatial distributions of upper-MS stars in the subsamples. It is apparent that subsample (a) exhibits the most subclustered distribution. Subsample (b) also displays a non-uniform distribution, but the stellar distribution is more dispersed and less subclustered. By contrast, subsamples (c) and (d) have rather smooth stellar distributions. Visible in all four subsamples, the northeastern and southwestern areas always contain fewer stars than the central region; this is due to the large-scale density gradient of the bar struc- a The fractal dimension is derived as D 2 = η + 2, where η is the slope of the TPCF in the range of 30 < θ < 300 pc. ture (see also Fig. 1 for the location of the bar with respect to the analyzed tile).

Two-Point Correlation Functions
In the previous sections we have identified young stellar structures as surface overdensities from the KDE map (Section 4.1). As will be discussed in Section 5.3, however, structures identified in this way have a fractal dimension biased toward that of young stars. By contrast, the TPCF is a more suitable tool to quantify the distribution of stars of all ages in a sample. The TPCF, w(θ), is defined as the number of excess pairs of objects with a given separation, θ, over the expected number for a reference distribution (Peebles 1980). Here, we recast the TPCF as 1 + w(θ), which is proportional to the more intuitive quantity of mean surface density of companions (Larson 1995;Kraus & Hillenbrand 2008). To save computing time and avoid edge effects, we evaluate the TPCFs via a Monte Carlo method. For each subsample, 1200 stars are randomly selected and the separation distribution of all their possible pairs, N p (θ), is calculated; on the other hand, another 1200 artificial stars, which are randomly distributed across the tile, are generated as reference, and the separation distribution of all their possible pairs, N r (θ), is derived. The TPCF from such a simulation is defined as 1 + w(θ) = N p (θ)/N r (θ), and we repeat this process 1000 times; the final TPCF is obtained by averaging the results of all simulations, and their standard deviation is taken as the uncertainty in the TPCF.
The TPCFs of the four subsamples are shown in Fig. 12. Beyond θ = 300 pc, all four subsamples show a steep drop in 1 + w(θ), falling below unity at large separations. Below θ = 300 pc, subsamples (a) and (b) display broken power-law TPCFs, with steeper slopes at θ < 30 pc compared with shallower slopes for 30 < θ < 300 pc; in contrast, the TPCFs of subsamples (c) and (d) exhibit no statistically significant deviations from single power laws over the entire range of θ < 300 pc. The slopes are given in Table 2 and also indicated in the Fig 12. The TPCFs in different separation regimes are discussed in detail in the following sections.

The large-separation regime: effect of density gradient
In calculating the TPCFs, we have used a uniform distribution as reference. As previously mentioned, however, the bar causes a density gradient across the tile (Fig. 11). The upper-MS stars are concentrated in a southeast-northwest locus, and there are fewer stars in the southwestern and northeastern regions. Thus, there are more large-separation stellar pairs in the reference than in the upper-MS subsamples. As a result, the TPCFs drop sharply beyond θ = 300 pc and fall below unity at even larger separations.
On the other hand, the density gradient occurs on a large scale comparable to the tile size, and is not significant on small scales. Thus, the small-scale clustering properties are still preserved in the TPCFs. For instance, the slopes of the TPCFs at θ < 300 pc, which may relate to the fractal dimensions and/or star clusters (see the next sections), should not be affected by the bar.

The small-separation regime: effect of star clusters
We were curious about the break at θ = 30 pc in the TPCFs of subsamples (a) and (b). To explore its origin, we show in Fig. 13 the spatial distributions of all possible pairs in the two subsamples with separations θ < 30 pc. In addition to some dispersed, low-density distributions, the stellar pairs are concentrated in one or several compact areas which are spatially associated with known star clusters. For both subsamples, the most significant concentration of stellar pairs is located in the northwest and associated with the populous star cluster NGC 1850. For subsample (a), there are a few additional compact concentrations, whose associated star clusters have been labeled in the figure. Thus, it is reasonable to speculate that the break at θ = 30 pc is caused by the star clusters, whose member stars are so densely distributed, leading to an enhanced number of stellar pairs with θ < 30 pc than that expected from single power laws. The break is not seen in either subsample (c) or (d), possibly because the numbers of non-cluster stars are so much larger than of the cluster stars; as a result, this effect is no longer significant.
Star clusters are subject to strong self-gravity and rapid dynamical evolutions. They are usually nonfractal, centrally concentrated objects, and reflect a different clustering mode from the fractal, hierarchical com- ponent (see e.g. Gouliermis et al. 2014). Strictly speaking, star clusters can also be regarded as the high-density end of the continuous hierarchy of young stellar structures; moreover, young star clusters may in turn contain subclusters (Gutermuth et al. 2005;Schmeja et al. 2008;Sánchez & Alfaro 2009). However, sophisticated simulations are needed to distinguish star clusters from the hierarchical component (e.g. Gouliermis et al. 2014). On the other hand, stochastic sampling effects become increasingly important on small scales. Thus, we do not further explore the TPCFs at θ < 30 pc.

The intermediate-separation regime: temporal evolution
In the range of 30 < θ < 300 pc, the TPCFs of all four subsamples agree well with single power laws of the form 1 + w(θ) ∝ θ η . The exponent is related to the projected fractal dimension as D 2 = η + 2 (Larson 1995;Kraus & Hillenbrand 2008), whose values are given in Table 2. Smaller η and D 2 , corresponding to a steeper TPCF, suggest larger amounts of substructures, while larger values of η and D 2 , on the other hand, indicate a more uniform distribution. Fig. 14 shows the values of η as a function of the median stellar age of the four subsamples. From subsample (a) to (d), η evolves steadily from −0.25 to −0.05, suggesting that they contain continuously fewer stellar structures as they become older. Subsample (d) has η very close to zero and is almost indistinguishable from a random distribution in this separation regime 3 . Its median age, log(τ /yr) = 8.0 or 100 Myr, provides a rough estimate of the timescale over which the young stellar structures disperse.
Specifically, Bastian et al. (2009) report a timescale of 175 Myr, or log(τ /yr) = 8.2, for losing all substructures in the stellar distributions over the entire LMC. They note that this timescale is comparable to the dynamical crossing time of stars in this galaxy, suggesting that the dominant driver of the structures' evolution is general galactic dynamics. Dissolving star clusters may lead to the erasure of substructures; ejected stars from star clusters, e.g. runaway stars, may also account for the temporal evolution of the TPCFs (see e.g. Pellerin et al. 2007). If these were the dominant mechanisms for the substructure dispersal, one would expect that the distribution of stars evolves more rapidly than that of star clusters, since in this case stars would have an additional expansion or ejection velocity component. However, Bastian et al. (2009) showed that the evolution of star clusters and stars is largely similar. Thus, although these effects may take place, they are not the dominant cause of the temporal evolution.  (b) (left-and right-hand panels, respectively). A stellar pair's position is represented by its center. The maps are convolved with a Gaussian kernel which is 10 pc in the standard deviation. The two maps have maximum surface densities of 0.78 pc −2 and 3.49 pc −2 , respectively. The blue contours correspond to low-density levels of 1%, 5%, and 10% of the maximum, while the red contours correspond to high-density levels of 30%, 60%, and 90% of the maximum. The star clusters associated with the red, high-density contours are labeled in the figure  It still remains unanswered whether the structure dispersion timescale may vary with location in the galaxy, as the young stellar structures may reside in different environments and undergo different physical processes. For instance, the bar complex is expected to be influenced by bar perturbations, if any (Gardiner et al. 1998; see also the discussion against a dynamical bar in Section 5.1.1 of Harris & Zaritsky 2009 Bastian et al. 2009) and for the bar complex only (100 Myr).
However, we note that the dispersion timescale is subject to significant uncertainties. On the one hand, the derived median ages may vary with the adopted SFH and metallicity of the SPMs (Section 5.1). We redid the calculation in Section 5.1 by changing the metallicity to [Fe/H] = −0.5 or 0.0 dex (the SFH changes accordingly to fit the observed LF). The derived median ages of the subsamples change by at most 0.1 dex, suggesting that the effect of this change is unimportant. On the other hand, the largest uncertainty comes from the significant spread of stellar ages within a subsample. In particular, while the brighter subsamples consist of only young stars, the fainter ones have larger age spreads because they contain both young and old stars. The interquartile range of log(τ /yr) offers an estimate of the age spread. For instance, the interquartile range of subsample (d) is as large as [7.4 dex, 8.4 dex]. This is a very large spread, even exceeding the difference between the median ages of the subsamples. In other words, the subsamples are not single-aged stellar populations, and the TPCFs reflect the mixed distribution of stars of different ages. Thus, it is not easy to obtain an accurate measurement of the structure dispersion timescale with upper-MS stars. Considering this fact, the timescale of 100 Myr as derived here is not inconsistent with the value from Bastian et al. (2009), which is 175 Myr for the whole LMC. In this section we have demonstrated that young stellar structures evolve rapidly after birth and are finally dispersed on a timescale of ∼100 Myr. We have also shown that the upper-MS stars cover a wide age range, with a significant fraction older than 100 Myr. Still, a small fractal dimension of D 2 = 1.5 ± 0.1 was previously derived from the structures' size distribution and masssize relation. This is very close to that expected for the ISM and newly formed stars (∼1.4; see Section 4.2), and seems contradictory to the presence of old stars in the upper-MS sample. We recall, however, that the young stellar structures in Section 4 were identified as surface overdensities from the KDE map. Given that younger populations are more substructured, the overdensities are dominated by the young stars in the upper-MS sample. Slightly older stars make minor contributions, and very old stars lead to density enhancements only through their statistical fluctuations. Thus, derivation of small fractal dimension is expected.

Fractal Dimension from Mixed Populations
The fractal dimension can also be derived from the TPCF (e.g. Section 5.2), to which, however, both young and old stars contribute importantly. With an increasing number of old stars, the relative number of smallseparation stellar pairs decreases, leading to shallower TPCFs and larger fractal dimensions. Thus, even with the same sample of stars, the fractal dimension from the TPCF is often higher than that from surface overdensities, as long as the sample has a significant age spread.
To verify this, we calculate the TPCF for the full upper-MS sample (Fig. 15) in the same way as for the subsamples. The full-sample TPCF is also well described by a single power law below θ = 300 pc, the slope of which is η = −0.09 ± 0.01. This suggests a fractal dimension of D 2 = 1.91 ± 0.01, significantly larger than that from the size distribution or mass-size relation. Actually, this value is very close to the TPCF-derived fractal dimensions of subsamples (c) and (d), since these two subsamples make up the majority of stars in the full sample (Table 2).
From this perspective, we can also understand the fractal dimension of subsample (a) derived with the TPCF (1.75 ± 0.01, Table 2). Although this subsample has a young median age, it also contains many old stars up to ∼30 Myr (Figs. 10 and 14). Within this timescale, the young stellar structures have already undergone significant evolution. Thus, its fractal dimension from the TPCF is significatly higher than ∼1.4, which is expected for the ISM and newly formed stars.

SUMMARY AND CONCLUSIONS
In this paper, the star-forming complex at the northwestern end of the LMC bar region (the bar complex) has been investigated with the VMC survey. The analysis is based on ∼2.5 × 10 4 upper-MS stars selected from the CMD, which is an order of magnitude larger than the samples used in previous studies. As a result, we have been able to trace young stellar structures down to scales of ∼10 pc.
The upper-MS stars have highly non-uniform spatial distributions. Young stellar structures in the complex are identified as stellar surface overdensities from the KDE map at different significance levels. The structures are organized in a hierarchical way such that larger, lowerlevel structures host one or several smaller, higher-level ones inside. This "parent-child" relation is further illustrated with dendrograms for two typical young stellar structures along with their substructures.
Size, mass, and surface density are also calculated for the young stellar structures. In the range not affected by incompleteness, the size distribution can be well described by a single power law, which is consistent with a projected fractal dimension D 2 = 1.5 ± 0.1. The structures follow a power-law mass function of the form n(M )dM ∝ M −β dM , with β = 2.0 ± 0.1. Their surface densities, on the other hand, agree well with a lognormal distribution. These results support the scenario of hierarchical star formation regulated by turbulence, in which newly-formed stars follow the gas distribution. The effects of other physical processes, e.g. gravity, strong shocks, rarefaction waves, and environmental influences, are not obviously visible from our results.
We further divide the full upper-MS sample into subsamples (a-d) with different magnitude ranges; the subsamples have increasing average ages, as confirmed by detailed LF modeling. The stellar distributions of the subsamples are quantitively studied based on the TPCFs, where the presence of the bar and star clusters is also apparent beyond 300 pc and below 30 pc, respectively. In the spatial range of 30-300 pc, the TPCFs are well described by single power laws, the slopes of which increase continuously from subsamples (a) to (d). Over this separation range, the youngest subsample (a) contains most substructures, while the oldest subsample (d) is almost indistinguishable from a uniform distribution. This suggests rapid temporal evolution so that the young stellar structures are completely dispersed on a timescale of ∼100 Myr. Considering the uncertainties, however, this timescale is not significantly different from that previously reported for the entire LMC. Thus, the environmental influence from the bar, if any, is not revealed from the data to any statistical significance.
Considering this evolutionary effect, we further point out that the fractal dimension may be method-dependent even using the same sample, as long as the stellar sample have a significant age spread. D 2 obtained with surface overdensities is biased toward that of young stars. On the other hand, D 2 from the TPCF is also sensitive to the old stars. As a result, D 2 will often appear lower if it is derived with the former method.
We thank the anonymous referee for the helpful sugges-