Fractal dimension in star formation regions

Historically, interest in geometry has been stimulated by its applications to nature. The ellipse assumed importance as the shape of planetary orbits, as did the sphere as the shape of the Earth. Geometry of ellipse and sphere can be applied to these physical situations. A similar situation pertains to fractals. Recent physics literature shows a variety of natural objects that are described as fractals, cloud boundaries, topographical surfaces, coastlines, turbulence in fluids, and so on. None of these are actual fractals, their fractal features disappear if they are viewed at sufficiently small scales. Nevertheless, over certain ranges of scale they appear very much like fractals, and at such scales may usefully be regarded as such.
Several works suggest a fractal structure at different scales of the universe such as galaxy clusters or HII regions (Pietronero, 1987; Elmegreen & Elmegreen, 2001; Sanchez & Alfaro, 2010). Fractal objects share a symmetry called scale invariance, so they are invariant under a transformation which change a small part of a picture for a bigger one. We can characterize fractal systems by calculating their fractal dimension (FD).
In Sec. 2, we introduce the data of the three star formation regions that have been studied. Then in Sec. 3, the box-counting dimension is explained as a way to estimate the FD for a data set, and the computed FD is given for each region. Finally, we present the conclusions and the importance of a good algorithm for FD estimation (see Sec. 4).

Gaia is a mission to chart a three-dimensional map of our Galaxy, (Gaia Collaboration et al., 2016). It serves, among other things, to study the composition, formation and evolution of the Galaxy. Gaia provides radial velocity and position measurements with the necessary precision to produce data that will allow the study of more than one billion stars in our Galaxy and in the entire Local Group.
The Data Release 2 of the Gaia mission (Gaia Collaboration et al., 2018) was used to obtain the Cartesian coordinates of stars for three star formation regions RCW 38, Orion Nebula and M16, which distances are directly taken from Bailer-Jones et al. (2018). With this, a spatial graph of the stars can be achieved and the box-counting dimension computed (see Sec. 3). For RCW 38 we consider a distance of 1700$\leavevmode\nobreak\ \rm{pc}$ with an error of $\pm 300\leavevmode\nobreak\ \rm{pc}$, and a cube of $300\times 300\leavevmode\nobreak\ \rm{pc}$, for M16 a distance of 1750$\leavevmode\nobreak\ \rm{pc}$ with an error of $\pm 300\leavevmode\nobreak\ \rm{pc}$, so we consider a cube of $300\times 300\leavevmode\nobreak\ \rm{pc}$, and for Orion Nebula a distance of $400$$\leavevmode\nobreak\ \rm{pc}$ with an error of $\pm 100\leavevmode\nobreak\ \rm{pc}$, so consider a cube of cube of $100\times 100\leavevmode\nobreak\ \rm{pc}$.

A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly is not equal to the euclidean dimension. The Hausdorff-Besicovitch dimension is not practical to compute so alternative definitions are used. The most common and used definition is the Box-Counting dimension (Feder, 2013). Let be $F$ a subset $F\in\mathbf{R}^{n}$, with $N_{\delta}(F)$ the number of $\delta-mesh$ cubes that intersect $F$, the boxcounting dimension ($dim_{b}$) is:

Note that $dim_{b}F$ is the slope of $\log N_{\delta}(F)$ vs. $-\log\delta$. We developed the algorithm to calculate this dimension in 2D and 3D in R  (R Core Team, 2019), avoiding both boundary and small data set problems. In order to test the algorithm, we compute the fractal dimension of a known fractal. The used fractal was the Pascal triangle module 3 for which we obtained a $FD=1.623$ while the real FD is $1.6309$ (Fig. 1).
The definition of the Hausdorff dimension of a set of particles requires the diameter of the covering sets to vanish. In our case of study, we have a characteristic smallest length scale, this is where the fractal structure of HII regions disappears. We associate this value with $\delta_{min}$ which considering the three regions gives us an average of $44.33\leavevmode\nobreak\ \rm{pc}$. To find the $\delta_{min}$ we analyze the $\log_{10}N_{\delta}(F)$ vs. $-\log_{10}\delta$ plot for all the stars and analyze when we lose the linear behavior.
If we assume unchanged density, we can consider the fractal dimension as the mass dimension, this is based on the idea of how a system mass scales, so:

Where $\delta_{max}$ is the size of the studied domain. If we assumed that matter with constant density is distributed over the fractal, then the mass of the fractal enclosed in a volume of a characteristic size R satisfies the power law of Eq. 2, with $dim_{b}\notin\mathbb{Z}$. Whereas for a regular $3D$ Eucliadian object the dimension mass scales: $M(R)\sim R^{3}$. For more details see Tarasov (2011).

Through our box-counting algorithm, we were able to obtain the FD of three star formation regions in the Milky Way using Gaia data. The obtained FDs are: $FD_{M16}=2.468$, $FD_{OrionNebula}=2.126$ and $FD_{RCW38}=2.435$ (see Tab. LABEL:ajustes2 and Fig. 2). These values are in concordance with FD values ($FD\sim 2.3$) obtained by Elmegreen & Elmegreen (2001) for stars formation regions in different galaxies.

We found that the three star formation regions have a fractal dimension near $2.3$. This is important to support that the mass-size relationship can result from the fractal nature of this type of regions. Note that, these results are independent of the star formation distance, as was previously argued by Elmegreen & Falgarone (2009).
The innovative aspect of this work is that the FD is calculated for the first time using Gaia data in the Milky Way, without using projections as necessary for two dimensions, achieving values similar to those obtained by Elmegreen & Elmegreen (2001). Gaia release can be used to further studied the fractal properties of the Milky Way.