### 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.