We review a range of statistical methods for analysing the structures of star clusters, and derive a new measure Q, which both quantifies and distinguishes between a (relatively smooth) large-scale radial density gradient and multiscale (fractal) subclustering.The distribution of separations p(s) is considered, and the normalized correlation lengths (i.e. the mean separation between stars, divided by the overall radius of the cluster) is shown to be a robust indicator of the extent to which a smooth cluster is centrally concentrated. For spherical clusters having volume-density n ∝ r −α (with α between 0 and 2)s decreases monotonically with α, from ∼0.8 to ∼0.6. Sinces reflects all star positions, it implicitly incorporates edge effects. However, for fractal star clusters (with fractal dimension D between 1.5 and 3)s decreases monotonically with D (from ∼0.8 to ∼0.6). Hences, on its own, can quantify, but cannot distinguish between, a smooth large-scale radial density gradient and multiscale (fractal) subclustering.The minimal spanning tree (MST) is then considered, and it is shown that the normalized mean edge lengthm [i.e. the mean length of the branches of the tree, divided by (N total A) 1/2 /(N total − 1), where A is the area of the cluster and N total is the number of stars] can also quantify, but again cannot on its own distinguish between, a smooth large-scale radial density gradient and multiscale (fractal) subclustering.However, the combination Q =m/s does both quantify and distinguish between a smooth large-scale radial density gradient and multiscale (fractal) subclustering. IC348 has Q = 0.98 and ρ Ophiuchus has Q = 0.85, implying that both are centrally concentrated clusters with, respectively, α 2.2 ± 0.2 and α 1.2 ± 0.3. Chamaeleon and IC2391 have Q = 0.67 and 0.66, respectively, implying mild substructure with a notional fractal dimension D 2.25 ± 0.25. Taurus has even more substructure, with Q = 0.45 implying D 1.55 ± 0.25. If the binaries in Taurus are treated as single systems, Q increases to 0.58 and D increases to 1.9 ± 0.2.

Measurements of trace gases in planetary atmospheres help us explore chemical conditions different to those on Earth. Our nearest neighbour, Venus, has cloud decks that are temperate but hyperacidic. Here we report the apparent presence of phosphine (PH 3) gas in Venus's atmosphere, where any phosphorus should be in oxidized forms. Single-line millimetre-waveband spectral detections (quality up to ~15σ) from the JCMT and ALMA telescopes have no other plausible identification. Atmospheric PH 3 at ~20 ppb abundance is inferred. The presence of PH 3 is unexplained after exhaustive study of steady-state chemistry and photochemical pathways, with no currently known abiotic production routes in Venus's atmosphere, clouds, surface and subsurface, or from lightning, volcanic or meteoritic delivery. PH 3 could originate from unknown photochemistry or geochemistry, or, by analogy with biological production of PH 3 on Earth, from the presence of life. Other PH 3 spectral features should be sought, while in situ cloud and surface sampling could examine sources of this gas.

The statistical descriptor is a robust and useful tool for distinguishing and quantifying the degree of radial or multiscale clustering in objects such as open clusters. is calculated as m/s, where is the mean edge length of the minimum spanning tree and is the mean distance between cluster members, or correlation length. is obtained using only two‐dimensional position data. Here, we investigate the performance of in three dimensions, both when true three‐dimensional data are available and when the radial velocity of cluster components is used as a proxy for position: this is known as 2dv space. True three‐dimensional data offer an improvement in the resolution of and as diagnostic indicators of clustering, a scatter plot of versus proving to be a particularly clear method of interpreting the information. Results are not satisfactory when 2dv information is used, as the data from cluster types which are clearly distinguishable using 2d information alone become overlapping and confused when 2dv information is used. We therefore recommend that the 2d method is used, unless true 3d positions of cluster members are available. The use of the versus plot is particularly recommended, as adding extra discrimination between cluster types, compared with that achieved using alone.

The extent to which the projected distribution of stars in a cluster is due to a large-scale radial gradient, and the extent to which it is due to fractal sub-structure, can be quantified -- statistically -- using the measure ${\cal Q} = \bar{m}/\bar{s}$. Here $\bar{m}$ is the normalized mean edge length of its minimum spanning tree (i.e. the shortest network of edges connecting all stars in the cluster) and $\bar{s}$ is the correlation length (i.e. the normalized mean separation between all pairs of stars). We show how ${\cal Q}$ can be indirectly applied to grey-scale images by decomposing the image into a distribution of points from which $\bar{m}$ and $\bar{s}$ can be calculated. This provides a powerful technique for comparing the distribution of dense gas in a molecular cloud with the distribution of the stars that condense out of it. We illustrate the application of this technique by comparing ${\cal Q}$ values from simulated clouds and star clusters.Comment: Accepted 2010 October 27. Received 2010 October 25; in original form 2010 September 13 The paper contains 7 figures and 2 table