5. THE CARTWRIGHT & WHITWORTH Q PARAMETER

To study hierarchical structure in a different way, Cartwright & Whitworth (2004) introduced a parameter, Q. This is the ratio of the average separation in a minimum spanning tree to the average 2-point separation. For example, suppose there are 5 stars clustered together in one region with a typical separation of 1 unit, and another 5 stars clustered together in another region with a typical separation of 1 unit, and these two regions are separated by 10 units. Then the minimum spanning tree has 4 separations of 1 unit in each region and 1 separation of 10 units (for the two closest stars among those two regions), for an average of (8 × 1 + 1 × 10) / (8 + 1) = 2 units length. The average separation for all possible pairs is counted as follows: there are 5 stars with separation from another star equal to about 1 in each region, so that means 5 stars taken 2 at a time in each region, or 10 pairs with a separation of 1 in each region, or 20 pairs with this separation total, plus each star in one group has a separation of 10 units from each star in the other group, which is 5 × 5 separations of 10 units. The average is (20 × 1 + 25 × 10) / (20 + 25) = 6. The ratio of these is Q = 2/6 = 0.33. Smaller Q means more subclumping because for multiple subgroups, the mean 2-point separation has a lot of distances equal to the overall size of the region, so the denominator of Q is large, but the minimum spanning tree has only a few distances comparable to the overall size of the system, one for each subgroup, and then the numerator in Q is small.

Bastian et al. (2009) looked at the correlated properties of stars in the LMC, using a compilation from Zaritsky et al. (2004). There were about 2000 sources in each of several age ranges on the color-magnitude diagram. Bastian et al. determined the zero-points and slopes of the two point correlation function for each different age. They found that younger regions have higher correlation slopes and greater correlation amplitudes, which means more hierarchical substructure. Most of this substructure is erased by 175 Myr. They also evaluated the Cartwright & Whitworth (2004) Q parameter and found a systematic decrease in Q with decreasing age, meaning more substructure for younger stars. Gieles et al. (2008) did the same kind of correlation and Q analysis for stars in the Small Magellanic Cloud, and found the same general result.