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4. Number Counts and Distributions

I could be bounded in a nutshell and count
myself a king of infinite space.
Shakespeare, Hamlet

If, like Hamlet, you count yourself king of
an infinite space, I do not challenge your
sovereignty. I only invite attention to certain
disquieting rumours which have arisen
as to the state of Your Majesty's Nutshell.

One dominant stroke transformed thousands of years of-increasingly refined speculation on the structure of our Universe into fact. Hubble (1925a,b,c, 1926, 1929a,b) clinched the extragalactic nature of the "white nebulae" by discovering their Cepheid variable stars. This vastly expanded the known distance scale.

Cepheids are unusually bright stars that pulsate with regular periods ranging from about 10 to 30 days. (The first one was found in the constellation Cepheus in the Milky Way.) Their crucial property is the relation between a Cepheid's period and its peak intrinsic luminosity, recognized in 1908 (Leavitt, 1912; Hertzsprung, 1913). Brighter Cepheids have longer periods. From the observed periods of Cepheids in nebulae, Hubble could obtain their intrinsic luminosity and thus find the distance from their observed apparent luminosity. The main uncertainty was in calibrating their period-luminosity relation from the independently known distances of Cepheids in our Galaxy. Early calibrations turned out to be wrong, mainly because there are different types of Cepheids, which give somewhat different period-luminosity relations. (Occam's Razor fails again.) These later corrections were substantial (a factor of about two) but not qualitatively significant (Baade, 1963).

Different types of Cepheids illustrate the caution needed when applying Occam's Razor to astronomy. It failed at least three times for Herschel (standard candle luminosities for stars, all nebulae essentially the same objects, Uranus originally thought more likely to be an ordinary common comet rather than a new unexpected planet), once for Hubble (all Cepheids the same) and would be destined to mislead on many later occasions. The Universe is far more various and heterodox than we are apt to imagine. Moreover, Occam's Razor, naively applied, is just an empty slogan. He said (see Russell, 1945) "It is in vain to do with more what can be done with fewer." This means that scientific explanations should not invoke more hypotheses than necessary. As logic it is very sensible, but it ignores the empirical question of whether fewer hypotheses will in fact do. Nor does it address the problem that some hypotheses are stronger than others and their implications may only partially overlap.

Controversy over whether nebulae were outside our Galaxy hinged on whether new hypotheses were necessary. Ultimately the answer is empirical. Before Hubble found Cepheids, evidence pointed both ways. There were Slipher's (1913, 1915) spectra showing Doppler shifts in the nebular absorption lines, which could be interpreted as radial velocities much larger than any in our Galaxy if no new physical hypothesis for the Doppler shift was necessary. In opposition, van Maanen claimed to observe an angular rotation of spiral nebulae. If these nebulae were at extragalactic distances, their outer arms would be moving faster than the speed of light. Many astronomers were convinced by this evidence, which turned out to be spurious (see Hetherington, 1988, for a detailed account). However, the discoveries by Ritchey and Curtis (see Shapley, 1917) of novae in the nebulae seemed to favor the extragalactic interpretation, provided the novae had the same absolute (intrinsic) brightness as those in our own Galaxy, i.e., provided no new hypothesis was necessary. But they did not. Some were supernovae. If these extra-bright novae in the nebulae were iden- tified with ordinary novae in our Galaxy, the nebulae would seem closer. (Shapley suggested that ordinary novae were stars running into gas clouds.) A third aspect of this controversy swirled around whether individual stars had been resolved in the outer parts of the M31 (Andromeda) and M33 nebulae. They had, but the images looked soft; so perhaps they were not stars after all but something else (Shapley, 1919; Lundmark, 1921). Was a new hypothesis necessary here? The confused situation was summarized by Shapley (1919) and Curtis (1919) as a prelude to their famous quasi-debate at the National Academy of Sciences in 1920 (see Shapley, 1921; Curtis, 1921). Curtis correctly concluded that the nebulae were extragalactic, but he failed to convince many astronomers. Shapley gave apparently better arguments for the wrong conclusion. Öpik (1922) argued that the Andromeda nebula was extragalactic, based on a relation between its observed rotational velocity and its absolute magnitude. So the scene had been set for Hubble's discovery.

Once the nebulae were clearly known to be cosmological, the question of their distribution quickened and took on new life.

Hubble, this time, was first off the mark. By 1926 he realized that the Mt. Wilson photographic plates taken earlier were unsuitable for galaxy counting. Their telescopes, emulsions, exposure times, development, zenith distances, atmospheric seeing, and positions on the sky were too varied and irregular. So he began a major program using the 60" and 100" telescopes to provide a large and uniform survey for galaxies brighter than about 20th magnitude. Five years later he had preliminary results based on 900 plates with 20,000 galaxies (Hubble, 1931). The "zone of avoidance" found by the Herschels - a band of sky along the Milky Way where interstellar obscuration reduces the apparent number of galaxies - was delineated more precisely than ever before. Its width varies irregularly, usually from 10 to 40 degrees as a function of galactic longitude. In the direction of our galactic center there is partial obscuration to latitudes of ± 40°. but much less toward our galactic anticenter. Outside the zone of avoidance the distribution appeared roughly uniform with occasional large clusters. Correlations of number counts with exposure times supported the view that the galaxies were uniformly distributed in depth, and Hubble calculated a mean density rhobar0 of about 10-30 g/cm-3 for luminous matter in space. Through a fortuitous approximate cancellation of errors in the absolute magnitude and estimated average mass of a galaxy, he came close to the modern value for rhobar0.

With so many galaxies in his homogeneous sample, Hubble could calculate their distribution function f(N), the probability for finding N galaxies in a given size area on the sky. He looked only at large areas, with more than about 25 galaxies, and in this regime found that f(N) had a lognormal distribution. The distribution function would become an important but long neglected clue to understanding the dynamical clustering of galaxies; we shall return to it often in this book. Hubble had competition. Harlow Shapley, who did not get along well with him, had previously left the Mt. Wilson Observatory staff and staffed his own galaxy distribution program at Harvard. This was based on an extensive survey of about 100,000 galaxies brighter than 18.2m taken with a 24-inch refractor at the Harvard Boyden Station. The photographic plates, some dating from the beginning of the century, were a much more varied lot than Hubble's, but Shapley (1933) thought he could remove the effects of different emulsions, exposure times, seeing, etc. by comparing the limiting magnitudes of stars on each plate to obtain a normalized galaxy count. While agreeing on the zone of avoidance, he found galaxies outside this zone much less homogeneous than Hubble had claimed. According to Shapley (1931)". . . the irregularities in apparent distribution are real and indicate actual groupings of external galaxies. The irregularities are obviously too pronounced to be attributed to chance; they are rather a demonstration of evolutionary tendencies in the metagalactic system." Soon Hubble (1 936a,b) would agree, but echoes of this controversy would be heard throughout astronomy on larger and larger length scales until the late 1960s.

Meanwhile, under Hubble's instigation, the Lick Observatory was making a third great survey of the galaxy distribution. Mayall (1934) combined 184 plates of his own taken from 1923 to 1933 with 56 taken by Keeler and Perrine around the turn of the century and 249 by Curtis around 1917, all on the Crossley 36.5-inch reflector. He collected about 15,000 galaxies and concluded, like Hubble, that they were consistent with a uniform spatial distribution and had a lognormal f(N). The same year Hubble (1934) published the detailed analysis of his complete number counts of about 44,000 galaxies brighter than 20th magnitude on 1,288 plates covering 2% of the three fourths of the sky north of -30° declination. He concluded: "There are as yet no indications of a super-system of nebulae analogous to the system of stars. Hence, for the first time, the region now observable with existing telescopes may possibly be a fair sample of the universe as a whole." He was, of course, aware of the large groups with hundreds of galaxies, but thought them exceptional, containing only about 1% of all galaxies in his sample:

When groups are included, the clustering tendency becomes more appreciable and significant, especially when considered in connection with the systematic advance in average type from the dense clusters to the random nebula.

On the grand scale, however, the tendency to cluster averages out. The counts with large reflectors conform rather closely with the theory of sampling for a homogeneous population. Statistically uniform distribution of nebulae appears to be a general characteristic of the observable region as a whole.

Hubble was confused. He evidently failed to distinguish between random (Poisson) clustering, in which there were no correlations, and a statistically uniform distribution, which, nevertheless, could be correlated. In the latter case, the correlations appear on average to be the same when viewed from any point in the system. This confusion increased when Shapley (1934a,b) found that the ratios of galaxy numbers in the northern to southern parts of his Harvard survey were large for bright galaxies but nearly unity for faint (m gtapprox 17.6) galaxies. He concluded that this implied a large-scale gradient in the "metagalactic system" - a supersystem of galaxies. It contradicted Hubble's view, but only out to scales dominated by 18th magnitude galaxies. So perhaps the difference in their views was only a matter of scale? Shapley's results might be thought a prelude to modem superclustering of galaxies, but the plates later proved too heterogeneous to give accurate conclusions.

Bok (1934) attempted to clarify the situation. He simply and perceptively emphasized how very different f(N) was from a random Poisson or normal (for large average N) distribution on all available scales. He used the Shapley-Ames catalog for galaxies brighter than 13th magnitude, the Harvard survey, which went to 18.2 magnitude, and then Hubble's own survey. Figure 4.1 shows Bok's graph for the Hubble survey. Of course, this is nothing more than Hubble's own lognormal f(N) distribution, but it definitely shows we are not dealing with "random nebulae" even at the faint magnitudes and large scales that Hubble thought represented a "fair sample."

Figure 4.1

Figure 4.1. The unnormalized frequency distribution of Hubble's galaxy counts compared to a random Gaussian distribution, after Bok (1934). The inset shows a recent fit of Bok's points to the distribution of Equation 27.24.

It is not clear how Hubble reacted to this. But the following year he changed his view somewhat, without referring to Bok. In the Silliman Lectures, later published as The Realm of the Nebulae, Hubble (1936a) strongly emphasizes (for the first time?) the difference between a Gaussian normal distribution for N, the number of galaxies per sample, and such a distribution with log N as its variable. He seems to regard the transformation from N to log N as a well-known mathematical device to remove the asymmetry of the curve, an asymmetiy "presumably associated with the tendency of nebulae to cluster." Then he continues with a statement that might have been Newton's reply to a more modern Bentley:

It is clear that the groups and clusters are not superimposed on a random (statistically uniform) distribution of isolated nebulae, but that the relation is organic. Condensations in the general field may have produced the clusters, or the evaporation of clusters may have populated the general field. Equations describing the observed distribution can doubtless be formulated on either assumption, and, when solved, should contribute significantly to speculations on nebular evolution.

Finding the equations Hubble suggested would take nearly half a century; we will meet them in Sections 15.2 and 27.2.

Still, Hubble continued to think there should be some large scale on which clustering is random, even if he had not yet discovered it. "Samples which, on the average, are large compared with a single cluster, should tend to conform with the theory of random sampling. The frequency-distributions of N per sample should approximate normal error curves. Smaller samples should give unsymmetrical frequency distributions of N, with an excess of thinly populated fields." This appears essentially correct, but again its discovery was to take half a century (see Section 15.4 and Chapter 33).

In a subsequent and rather strange paper, mainly trying to show that galaxy redshifts are not velocity shifts and the Universe is not expanding significantly, Hubble (1936b) also elaborated his view of the distribution function:

While the large-scale distribution appears to be essentially uniform, the small-scale distribution is very appreciably influenced by the well-known tendency toward clustering. [Here, again, the idea of a uniformly clustered distribution had not occurred to Hubble. He did not distinguish between a uniformly correlated and a uniformly uncorrelated state, only between a uniform state and an irregular state.] The phenomena might be roughly represented by an originally uniform distribution from which the nebulae have tended to gather around various points until now they are found in all stages from random scattering, through groups of various sizes, up to the occasional great clusters. 10 The tendency appears to operate on a relatively modest scale - at least no clusters are known with more than a few hundred members; hence irregularities tend to average out among samples that are large compared to a single cluster. 11 In small samples irregularities are conspicuous, and, as an empirical fact, they lead to a random distribution of log N rather than of N.

The two footnotes in this paragraph are significant. The second one states "11 Samples in the regions of great clusters were rejected in the present surveys; and, for this reason, irregularities tend to average out for samples that are large compared to single groups of nebulae." Thus Hubble had confused the "uniformity" implied by finding the same average values of log N and its dispersion in different regions of space with the much stricter requirement that the distribution be Poisson (i.e., "random" in his terminology). It is a confusion between a description in terms of "clusters" and a description in terms of "clustering"; we shall consider this problem further in Chapter 6.

The earlier footnote refers to dynamics in the spirit of Herschel, but it presents an opposite view from Newton's: "10 This representation is purely formal and has no genetic implications. For purposes of speculation, the reverse development seems preferable, namely, that nebulae were formed in great clusters whose gradual evaporation has populated the general field." Like Herschel, Hubble imagined the observed distribution might be formed either by clustering or by fragmentation and dispersal.

Genetic implications? Did Hubble harken back to von Humboldt here (see Chapter 3), or had he something else in mind? The latter possibility arose in a conversation with Allan Sandage (Cambridge, England, August 1986) where we speculated on what originally inspired Hubble to fit a lognormal curve to his data. Hubble's statement (above) that the relation between clusters and isolated nebulae is "organic" might be a clue. At Cal Tech he was a sociable man about campus with many friends in other departments. A chance meeting with a microbiologist or bacteriologist might have led to a casual suggestion that Hubble try fitting his strange-looking skew distribution data with a lognormal function. Counting the sizes of bacterial colonies growing on agar in petri dishes was fashionable then, and their distribution was often lognormal. So perhaps Hubble tried it, and to his amazement it worked. He did not test it on smaller scales, where it would have failed, nor did he seem to think much more about the physics behind it.

Subsequent analyses confirmed the nonrandom distribution of galaxies. Mowbray (1938) essentially repeated Bok's work for different samples, and Shapley (1934a,b) continued to defend his metagalaxy. Katz and Mulders (1942). following Zwicky's suggestion, used a somewhat different technique to show that the galaxies brighter than 12.7 magnitude in the Shapley-Ames Catalog were also nonrandom. They divided the catalog into thirty-six strips of 100 width in galactic longitude, took different groupings of these strips, computed the dispersions of the galaxy number counts among these different groups, and found that this dispersion was significantly greater than a Gaussian distribution would have given.

Between 1943 and 1950. the effects of World War II intervened and very little was published on galaxy clustering, or on astronomy in general. What few extragalactic observations were made mostly concerned individual galaxies. But the discoveries of the decade 1925-1935 had fundamentally changed our picture of infinite space, and the theoreticians began to stir.

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