Ph.D. Thesis of Glyn George:
The Alignment and Clustering of Quasars

This thesis was examined in 1983 September and the degree of Ph.D. was conferred by the University of Wales on 1983 December 14.   The summary and a core result follow.


Recently [1980-83] there has been considerable interest in the question of how many excellent alignments of triplets of quasars are expected on a field on the sky.   The answer is provided, (in the case where the quasars are distributed uniformly at random), as a more precise expression than has been available hitherto.   A correction factor for a square boundary is also derived.

It is shown that the expectation is very sensitive to clustering.   Various tests are described and used on particular fields of quasars to assess the level of clustering present, if any, (Chapters 3, 4 and 5).   A discussion of various selection effects which might cause a random distribution in space to appear clustered on a telescope plate is given in Chapter 3.

Approximate significance levels for the number of alignments in a field are derived, and a correction for clustering is suggested and used on the test fields, (Chapter 6).   Some evidence for clustering, (which can be explained by selection effects), is found for some fields.

After due correction for clustering, however, no convincing evidence for “physical” alignments of quasars is found in any of the test fields.

The Expected Number of Alignments

In section 5.2 of the thesis, (pages 128-138), the derivation of the expected number of chance alignments is presented.

[Illustration of three aligned points] Three points are considered to be aligned if the middle point lies in a box of half-width p with the other two points at the two ends of the box and if the length of the box is at most d.

If N points are placed randomly on a square of side L, then the number of alignments n occurring by chance will, on average, be (thesis equation 5.2.5, page 130)
        E[nP] = (2pi/3)(p*d^3)/(L^4)*N*(N-1)*(N-2)
This expression is valid for “periodic” boundaries, (that is, for squares whose opposite sides are identified with each other, so that points close to one edge of the square are considered to be neighbours of points near the opposite edge).

When boundary effects are taken into account, the expectation decreases to (thesis equation 5.2.22, page 137)
        E[nF] = E[nP] * (1 - (3/pi)(d/L) + (3/5)(4/pi-1)(d/L)^2)
This accounts for alignments “lost over the edges” of the square.

For the typical quasar data available in 1983, the parameter values were
        p = 30"   (30 arc seconds)
        d = 1°
        L = 5°
        N = various values, from 23 to 270.
With these values, the expectations become
        E[nP] = pi*N*(N-1)*(N-2)/112500
        E[nF ] = 0.8156×E[nP ]     —     almost 20% of all alignments are “lost over the edges”.

A derivation of these results may be found at this separate page.

The expression E[nP ] can be generalised (from work just subsequent to this thesis) to the case of alignments (chains) of k points (k > 2):
        E[nP, k] = {pi*N*(N-1)/k}*{(N-2)C(k-2)} *
 (2p/L)^(k-2) * (d/L)^k

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Created 2003 07 31 and most recently modified 2003 08 03 by Dr. G.H. George.