Projection effects in star-forming regions: I. Nearest-neighbour statistics and observational biases

TL;DR

This work quantifies how projecting a three-dimensional fragmentation pattern of dense cores into two dimensions biases nearest-neighbour statistics. Using spherical and fractal toy models, it shows that simple geometric deprojection ($\frac{4}{\pi}$) is insufficient because projection rewires neighbour connections and finite resolution blends close pairs, altering inferred fragmentation scales. The authors derive an empirical correction surface $\mathcal{C}(N,\mathrm{SDR})$ with $\mathcal{C}_\infty = 1.94$, $S_0 = 21.8$, and $\beta = 0.173$ that links measured 2D NN spacings to intrinsic 3D values, valid across regimes from unresolved to well-resolved, and provide the corespacing3d package for practical use. The calibration reveals morphology-driven uncertainties of order 30–40% and has important implications for interpreting fragmentation scales in both observations (e.g., ASHES) and simulations, urging careful treatment of projection and resolution effects in star-forming regions.

Abstract

Stars form as molecular clouds fragment into networks of dense cores, filaments, and subclusters. The characteristic spacing of these cores is a key observable imprint of fragmentation physics and is commonly measured using nearest-neighbour (NN) statistics. However, NN separations are derived from projected two-dimensional (2D) positions, while fragmentation occurs in three dimensions (3D). Using spherical and fractal toy models, we show that the standard geometric deprojection factor of $4/π\simeq1.27$ is inadequate because projection not only foreshortens separations but also rewires the NN network, while finite angular resolution merges close neighbours and inflates apparent spacings. We quantify these competing biases with Monte Carlo experiments spanning a wide range of morphologies, sample sizes, and effective resolutions. From these we derive an empirical correction factor that depends on both sample size and resolution: for small ($N\lesssim10$) or poorly resolved samples ($\lesssim$10 resolution elements across the field), intrinsic NN spacings exceed projected values by only 20 to 40%, whereas for well-sampled ($N\gtrsim100$), well-resolved data ($\gtrsim$30-50 resolution elements), true 3D separations are typically larger by a factor of $\sim$2. This calibration enables observers to convert measured 2D NN spacings into corresponding 3D estimates, with typical morphology-driven uncertainties of order 30 to 40%, and we demonstrate how it alters inferred fragmentation scales in observed and simulated core populations. [abridged]

Figures (12)

• Figure 1: Example of a 1.3 mm continuum image from the ASHES survey Morii2023, overlaid with the NN graph (orange lines). The background colour scale and contours show the ALMA 12 m plus 7 m continuum emission, with contour levels at $3\times2^{\,n}\sigma$ ($n = 0, 1, 2, \dots$), where $\sigma = 9.5\times10^{-5}$ Jy beam$^{-1}$ is the rms noise level. Detected core positions are marked by orange circles, and the scale bar in the lower left corresponds to 0.1 pc at the assumed distance of 5.5 kpc.
• Figure 2: Nearest--neighbour comparisons for $N=200$ points drawn from a uniform spherical distribution within $R=1.0$. Top: NN graph constructed in 3D, where the blue connections are the 3D NNs (left); 2D projection, where the orange connections are the 2D NNs (Centre); and the corresponding distributions of NN edge lengths (right) highlighting the overall shortening in projection. Bottom: Detailed breakdown. The left panel repeats the 2D overlay of 3D and 2D edges, while the right panels show the distributions of $\ell_{\rm 3D}$, $\ell_{\rm 2D}$ for shared edges and the ratios $\ell_{\rm 3D}/\ell_{\rm 2D}$. Vertical dashed lines mark the medians, and KDEs are overplotted on the histograms. Together these panels illustrate how projection both compresses lengths and reassigns neighbours, with only the very shortest pairs surviving unchanged.
• Figure 3: Nearest-neighbour comparisons for uniformly sampled point distributions. Top: Example realisation with $N = 10$ points drawn within a sphere of radius of $R = 1.0$. The NN graph constructed in 3D (left), its 2D projection obtained by dropping the line-of-sight coordinate (centre), and the corresponding edge-length distributions (right) illustrate how projection systematically shortens apparent separations and modifies the network connectivity even in a simple isotropic configuration. Bottom: Results from the Monte Carlo ensembles ($10^3$ realisations each) for $N = 5$, 10, 20, and 50 showing stacked kernel density estimates of 3D (blue) and projected 2D (orange) NN edge lengths. The distributions converge at a stable median ratio as $N$ increases, while stochastic fluctuations dominate at small values of $N$.
• Figure 4: Illustration of the impact of a finite SDR on NN statistics using a uniform distribution of $N = 200$ points within a sphere of radius $R = 1.0$. Left: Intrinsic 3D NN graph. Centre: Two-dimensional projection after applying a beam--blending step that merges points closer than one beam width, corresponding here to a SDR of $\mathrm{SDR} = \mathrm{FoV}/\mathrm{FWHM}_{\rm beam} = 10$. Circles mark the original 2D positions (open) and the resulting beam--blended centroids (filled). Right: Distributions of NN edge lengths in 3D and 2D after blending. In this example, $135$ of the $200$ projected cores (67.5%) are merged into $65$ effective groups, erasing all topological correspondence between the intrinsic and projected networks ($J = 0.00$, overlap fraction = 0). The typical 3D and 2D NN lengths become nearly equal ($\langle \ell_{\mathrm{3D}}\rangle / \langle \ell_{\mathrm{2D}}\rangle \simeq 1.0$), as beam blending suppresses the shortest intrinsic separations that normally produce the geometric compression factor ($4/\pi \simeq 1.27$). This example illustrates that limited spatial resolution can strongly distort the apparent connectivity and scale distribution of dense cores even in an intrinsically uniform configuration.
• Figure 5: Nearest--neighbour graphs for a fractal distribution with $N=200$ points, fractal dimension $D=1.6$, and sub-division $n_{\rm div}=3$, within $R=1.0$. Each row shows the NN network in 3D (left), the projected NN network in 2D (centre), and the corresponding NN edge--length distributions (right). The three rows illustrate different viewing geometries: no rotation (top), rotation about the $y$--axis ($\beta=90^\circ$; middle), and rotation about the $z$--axis ($\gamma=90^\circ$; bottom). Projection systematically shortens the apparent NN separations and rewires connectivity, but the degree of overlap and compression depends on the line of sight, reflecting the anisotropic and clumpy structure of the fractal distribution.