The present lecture describes similar results which are derived from the laws of nebular distribution as formulated with the alternative scales of distance. The preliminary reconnaissance had indicated that the large-scale distribution is at least roughly uniform. Guided by this information accurate surveys were planned for the purpose of formulating precisely the law of distribution. Five such surveys have now been reported, made with three large reflectors, by two observers working independently. One survey was made by Dr. Mayall with the 36-inch reflector, at the Lick Observatory, the others were made at the Mount Wilson Observatory, two with the 66-inch reflector and two with the 100-inch. The results are internally consistent, and the diversity of the investigations offers some assurance against the presence of hidden systematic errors - that nightmare of the cautious observer - which might vitiate the conclusions.
Each survey, although it required many months or some years for completion, is summarized by a single symbol N_{m}, which represents the average number of nebulae per unit area, brighter than a specified limit of apparent faintness. It is unnecessary to describe the simple but laborious methods by which the nebular counts were reduced to a standard, homogeneous system, and the limits of apparent faintness determined with considerable accuracy. It is sufficient to say that, in their final forms, the surveys represent about 100,000 nebulae, and that the distances corresponding to the limits of the surveys range from about 150 to 400 million light-years. The surveys, it will be noticed, extended far back into time as well as far out into space.
Analysis of the surveys consists in the comparison of numbers of nebulae with the volumes of space they occupy. If the large-scale distribution is uniform, the numbers will evidently be proportional to the volumes. If the proportionality does not emerge from the data, we must conclude that the observable region is not homogeneous. The discrepancies must then be examined in order to find whether they represent random variations within the sample, or departures from uniformity which vary systematically with distance. Systematic departures would be important because the trend could be pushed out beyond the limits of the telescope to furnish positive information concerning the universe at large. On the other hand, homogeneity, or random variations, would suggest that our sample is a small, insignificant portion of the universe.
Now let us follow this programme of investigation. Each survey indicates the number of nebulae in the sky that are brighter than a certain limit of apparent faintness (corrected for local obscuration within the, stellar system). Apparent faintness can be transformed into distances on either of two possible scales. Consequently, each survey furnishes a certain number of nebulae distributed through either of two, spheres with specified radii and specified volumes.
When the surveys are compared, the crude observations, uncorrected for red-shifts - the World Picture, to use Professor Milne's happy phrase - indicate that the number of nebulae increase less rapidly than the volumes of space through which they are distributed; the nebulae appear to thin out with distance. However, the measured luminosities must be corrected for energy-effects, regardless of the interpretation of red-shifts. The corrections will reduce the estimated distances and volumes, and, consequently, will tend to compensate the apparent thinning out.
m | d / | log N |
20.34 | 0.231 | 3.162 |
19.53 | 0.158 | 2.686 |
19.03 | 0.125 | 2.342 |
18.68^{*} | 0.106 | 2.161 |
18.11 | 0.086 | 1.886 |
m is limiting magnitude actually observed, corrected for local obscuration and for energy-effects; d / is the red-shift corresponding to the limiting magnitude m; log N is the logarithm of the average number of nebulae per square degree brighter than magnitude m. ^{*} The fourth survey was made by N. U. Mayall using the 36-inch reflector at the Lick Observatory. The five surveys are discussed in Contributions of the Mount Wilson Observatory, No. 557; Astrophysical Journal, 84, 517, Dec. 1936. |
The energy-corrections lead to the first of the two scales of distance, namely, that which follows from the assumption that red-shifts are not velocity-shifts. On this scale the law of red-shifts is strictly linear over an observed range of about 250 million light-years, and, presumably, will continue to be approximately linear over a considerably greater range. Therefore, the red-shifts at the limits of the various surveys are readily ascertained, and uncertainties are restricted to the deepest survey alone. The energy-corrections, as previously explained, are simple functions of the red-shifts.