THE FINITE UNIVERSE OF GENERAL RELATIVITY
The mean density of space can be used to determine the dimensions of the finite but boundless universe of general relativity. De Sitter[27] made the calculations some years ago, but used values for the density, 10–26 and greater, which are of an entirely different order from that indicated by the present investigations. As a consequence, the various dimensions, both for spherical and for elliptical space, were small as compared with the range of existing instruments.
For the present purpose, the simplified equations which Einstein has derived for a spherically curved space can be used.[28] When R, V, M, and ρ represent the radius of curvature, volume, mass, and density, and k and c are the gravitational constant and the velocity of light,
| (14) |
| (15) |
| (16) |
Substituting the value found for ρ, 1.5×10–31, the dimensions become
| R = 8.5×1028 cm = 2.7×1010 parsecs, | (17) |
| V = 1.1×1088 cm = 3.5×1032 cubic parsecs, | (18) |
| M = 1.8×1057 grams = 9×1022 ☉. | (19) |
The mass corresponds to 3.5×1015 normal nebulae.
The distance to which the 100-inch reflector should detect the normal nebula was found to be of the order of 4.4×1075 parsecs, or about 1⁄600 the radius of curvature. Unusually bright nebulae, such as M 31, could be photographed at several times this distance, and with reasonable increases in the speed of plates and size of telescopes it may become possible to observe an appreciable fraction of the Einstein universe.
Mount Wilson Observatory
September 1926
[1] Contributions from the Mount Wilson Observatory, No. 324.
[2] These are the two Magellanic Clouds, M 31, and M 33.
[3] Bailey, Harvard Annals, 60, 1908.
[4] Hardcastle, Monthly Notices, 74, 699, 1914.
[5] This estimate by Seares is based on a revision of Fath’s counts of nebulae in Selected Areas (Mt. Wilson Contr., No. 297; Astrophysical Journal, 62, 168, 1925).
[6] “A General Study of Diffuse Galactic Nebulae,” Mt. Wilson Contr., No. 241; Astrophysical Journal, 56, 162, 1922.
[7] The classification was presented in the form of a memorandum to the Commission on Nebulae of the International Astronomical Union in 1923. Copies of the memorandum were distributed by the chairman to all members of the Commission. The classification was discussed at the Cambridge meeting in 1925, and has been published in an account of the meeting by Mrs. Roberts in L’Astronomie, 40, 169, 1926. Further consideration of the matter was left to a subcommittee, with a resolution that the adopted system should be as purely descriptive as possible, and free from any terms suggesting order of physical development (Transactions of the I.A.U., 2, 1925). Mrs. Roberts’ report also indicates the preference of the Commission for the term “extra-galactic” in place of the original, and then necessarily non-committal, “non-galactic.”
Meanwhile K. Lundmark, who was present at the Cambridge meeting and has since been appointed a member of the Commission, has recently published (Arkiv für Matematik, Astronomi och Fysik, Band 19B, No. 8, 1926) a classification, which, except for nomenclature, is practically identical with that submitted by me. Dr. Lundmark makes no acknowledgments or references to the discussions of the Commission other than those for the use of the term “galactic.”
[8] Problems of Cosmogony and Stellar Dynamics, 1919.
[9] N.G.C. 4486 (M 87) may be an exception. On the best photographs made with the 100-inch reflector, numerous exceedingly faint images, apparently of stars, are found around the periphery. It was among these that Belanowsky’s nova of 1919 appeared. The observations are described in Publications of the Astronomical Society of the Pacific, 35, 261, 1923.
[10] “Early” and “late,” in spite of their temporal connotations, appear to be the most convenient adjectives available for describing relative positions in the sequence. This sequence of structural forms is an observed phenomenon. As will be shown later in the discussion, it exhibits a smooth progression in nuclear luminosity, surface brightness, degree of flattening, major diameters, resolution, and complexity. An antithetical pair of adjectives denoting relative positions in the sequence is desirable for many reasons, but none of the progressive characteristics are well adapted for the purpose. Terms which apply to series in general are available, however, and of these “early” and “late” are the most suitable. They can be assumed to express a progression from simple to complex forms.
An accepted precedent for this usage is found in the series of stellar spectral types. There also the progression is assumed to be from the simple to the complex, and in view of the great convenience of the terms “early” and “late,” the temporal connotations, after a full consideration of their possible consequences, have been deliberately disregarded.
[11] Publications of the Lick Observatory, 13, 12, 1918.
[12] Hβ is brighter than N2. Patches with similar spectra are often found in the arms of late-type spirals—N.G.C. 253, M 33, M 101. The typical planetary spectrum, where Hβ is fainter than N2, is found in the rare cases of apparently stellar nuclei of spirals; for instance, in N.G.C. 1068, 4051, and 4151. Here also the emission spectra are localized and do not extend over the nebulae.
[13] Monthly Notices. 74, 699, 1914.
[14] Annalen der Wiener Sternwarte, 20, 1907.
[15] Astronomische Nachrichten, 214, 425, 1921.
[16] Publications of the Lick Observatory, 13, 1918.
[17] Since C is constant for all nebulae in a given class, the linear relation between Δ log d and C for the different classes is something more than a mere geometrical relation arising from the observed equality of the mean mT in the various classes.
[18] This is apparent even among the observed classes. Referring to [formula (3)], mT + 5 log b will be constant in so far as Ce + 5 log (1 – e) is constant. The following table indicates that the latter term is approximately constant throughout the sequence of elliptical nebulae. The values of Ce were read from the smooth curve in [Fig. 6].
| e | Ce | 5 log (1 – e) | C0 | Res. |
|---|---|---|---|---|
| 0 | 10.30 | 0.0 | 10.30 | –0.14 |
| 1 | 10.65 | – .23 | 10.42 | – .02 |
| 2 | 11.00 | .48 | 10.52 | + .10 |
| 3 | 11.35 | 0.78 | 10.57 | + .13 |
| 4 | 11.70 | 1.11 | 10.59 | + .15 |
| 5 | 12.05 | 1.50 | 10.55 | + .11 |
| 6 | 12.40 | 1.99 | 10.41 | – .03 |
| 7 | 12.75 | –2.62 | 10.13 | –0.31 |
| Mean | 10.44 | 0.12 |
[19] Mt. Wilson Contr., No. 191; Astrophysical Journal, 52, 162, 1920.
[20] Pease, Mt. Wilson Comm., No. 51; Proceedings of the National Academy of Sciences, 4, 21, 1918.
[21] Pease, Mt. Wilson Comm., No. 32: ibid., 2, 517, 1916.
[22] Astrophysical Journal, 55, 406, 1922.
[23] Monthly Notices, 82, 133, 1922.
[24] Astronomical Journal, 28, 75, 1914.
[25] Mt. Wilson Contr., No. 297; Astrophysical Journal, 62, 168, 1925.
[26] The latest and most reliable results bearing on the distribution of faint (hence apparently distant) nebulae are found in Seares’s revision and discussion of the counts made by Fath on plates of the Selected Areas with the 60-inch reflector. When the influence of the cluster in Virgo is eliminated the density appears to be roughly uniform for all latitudes greater than about 25°.
[27] Monthly Notices, 78, 3, 1917.
[28] Haas, Introduction to Theoretical Physics, 2, 373, 1925.
Transcriber’s Notes
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https://hdl.handle.net/2027/uc1.b3805680 (see pg 353). - The outline “CLASSIFICATION OF NEBULAE” starting on [page 3] lists N.G.C. 2117 as a type E7 nebula. The version of this paper printed in Contributions from the Mount Wilson Observatory No. 324 lists N.G.C 3115. Otherwise the papers are essentially identical.