Moving Cluster Parallax

 

 

 

 

 

 

 

 

 

The actual procedure goes as follows:
       (1)  One identifies the cluster members in a field by measuring proper motions (μis) and selecting all stars moving toward a common
             point (the "convergent point").
       (2)  vRi and λi are measured for each selected star.  vSi = vRi sec λi is calculated for each.  The vSis determined in this manner
             serve as a further discriminant for removing non-cluster members.  Generally |vSi - v| £ 1 km/s.
       (3)  v = Σ vSi / n, where n is the remaining number of stars in the cluster, is determined.
       (4)  vTi = v sin λi is determined for each cluster member.
       (5)  ri = vTi / (4.74 μi) is determined for each cluster member.

In principle this method should be applicable to all clusters.  In practice it is not for two reasons:
       (1)  For distant clusters the μis are rather small and uncertain and therefore the location of the convergent point and identification of
             cluster members becomes difficult.
       (2)  The probability of finding a cluster with a particular value of λ is greatest at λ = 90° (such a value is clearly unacceptable because
             of the singularity in sec λ) and least at λ = 0° and 180°.  Furthermore, because of projection effects, clusters with suitably small
             values of λ are much less likely to have sufficiently large values of μ than are clusters with unacceptably large values.  If we
             assume that, in order to be amenable to the moving cluster method, a cluster must have μ ³ μ0, where μ0 is the smallest μ such
             that the method is applicable, then the probability that λ falls within a range λ to  λ + dλ can be shown to be equal to
             P(λ) dλ = 8 sin4λ dλ / 3π.  Hence 1.17% of suitable cluster fall within the favorable λ ranges 0°-30° and 150°-180°, 24.14% fall
             within the moderately favorable ranges 30°-60° and 120°-150°, and 74.68% fall within the unfavorable 60°-120° range.