In the dark clouds of interstellar gas where stars are born, matter gathers into dense knots — future cradles of stars. Astronomers measure distances between these knots from sky photographs, but a photograph is flat while the cloud is volumetric. Converting from plane to volume is not as simple as it seems.
Usually, one assumes that the flat distance between neighboring knots should be multiplied by 4/π ≈ 1.27 to get the volumetric distance. But the authors showed that projection not only shortens distances but also changes who is considered a neighbor. Knots that appear close on the sky may be far apart in space, and vice versa. And if the telescope sees blurry, nearby knots merge into one, making distances seem larger.
The scientists ran computer experiments with models of clouds of various shapes and sizes. It turned out that the correction depends on how many knots are visible and how clearly the telescope distinguishes them. For small or blurry samples, the true volumetric distance is only 20–40% larger than the flat one. For large and sharp samples, it is about twice as large.
Now astronomers can more accurately convert distances between stellar nurseries. This is important for understanding how a cloud fragments and why stars form at certain distances from each other. Errors in such conversion are typically 30–40%, but that is better than a blind guess.
as written by the authors arXiv:2601.20928
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 $$4/\pi\simeq1.27$$ 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]
In a typical molecular cloud, there can be from dozens to hundreds of dense knots where stars might ignite, and the distances between them are comparable to the size of the Solar System multiplied thousands of times.