tokens: they are not noticeably affected by the motions of individual stars within binary systems (because PMs provide a measure of the motions of stars integrated over a baseline and not an instantaneous measure of the velocity as RVs do). In the text that follows, we denote the velocity dispersions in RA and Dec. as σ α and σ δ , though technically the velocity dispersions of the PMs in RA and Dec. should be written as and . Fig. 5 shows the vector point diagram of PMs for the majority of Cyg OB2 members. The PMs of non-members of Cyg OB2 are not shown, but these have a similar distribution, implying that Cyg OB2 members cannot be kinematically separated from the field population. A number of outliers with very large relative PMs are evident in this diagram and are discussed in more detail in Section 3.5 . These outliers cause the velocity dispersion (Section 3.1 ) to come out very differently for a Gaussian dispersion compared to that calculated from an outlier-resistant method such as the interquartile range (IQR), and the use of a forward modelling approach (Section 3.2 ) reinforces this. Figure 5. Vector point diagram for members of Cyg OB2 with PMs (note that PMs are not placed on an absolute reference system but are only calculated relative to each other) with the spectroscopically known OB stars shown as red dots. The left-hand panel shows 869 of the 873 stars with PMs (four stars with very high PMs are not shown), highlighting the presence of a number of runaways or possible contaminants with high PMs. The right-hand panel (inset shown in the left-hand panel with a blue box) shows a close-up of the centre of the velocity space, with 822 of the 873 stars shown. 3.1 Calculating the velocity dispersions Fig. 6 shows the binned velocity distribution (over the range covered by the majority of stars), which has a broadly Gaussian distribution with low-amplitude, broad wings. The simplest estimate of the velocity dispersion is the standard deviation, which for our sample of 873 stars gives mas yr −1 and mas yr −1 (uncertainties calculated from a Monte Carlo simulation in which the individual PMs are varied by their estimated uncertainties). These velocity dispersions are considerably larger than the observed distribution because they are being enlarged by a number of outliers with large velocity offsets. There are 23 (17) stars with |μ| > 10 mas yr −1 in the RA (Dec.) dimension, and 1 (2) of these have |μ| > 100 mas yr −1 . These outliers could be removed by a process of ‘sigma clipping’, but this is a procedure fraught with difficulties of subjectivity and irreproducibility. Figure 6. PM velocity distributions along both axes (black histogram) showing the interquartile range velocity dispersions (green dashed lines) and the two-component Gaussian mixture model for the velocity dispersion (red dashed lines). The best-fitting velocity dispersions are listed in each case, which for the Gaussian mixture model is the dominant Cyg OB2 component.
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