ALBERT

Notes: Abstract When compact objects or black holes move through a fluid medium, or when turbulent plasma and magnetic fields so conspire, a gas flow is set up which closely resembles the flow of water down a plug-hole (Section 1). A similar hypothesis, but in reverse, was suggested by Jeans in 1928, and would nowadays be referred to as the white hole concept. The dynamics of the flow (Section 2) lead to expressions for the rotational velocity of the fluid far away from (2.1) and near to (2.2) the origin of the vorticity. Rotation curves derived from the model (Section 3) are closely akin to actual galactic rotation curves, but observational data on the latter are not precise enough to permit a delineation to be made between (i) flow around a singularity and (ii) flow around a non-singular sink or source. The other acceptable model, that of (iii) a spreading line vortex, is ruled out by comparison with astrophysical observations (Section 4). The basic analysis for all the models shows that the old problem of the winding-up of spiral arms can be avoided, since the galactic flow system is in a steady state. Section 5 identifies Jeans' speculation as being a hypothesis compatible with singular vortex flow and so with observation, but perhaps not with the usual interpretation of general relativity metrics, even though the requisite dual space does complete the topology in a mathematically satsifying manner. Section 6 concludes that the predictions of the hypothesis of vortex flow agree with the shape, dynamics and structure of galaxies.

Notes: Abstract Clusters of galaxies are approximated by the Schwarzschild interior solution (with non-zero cosmological constant) embedded in a Robertson/Walker background. The conditions that the two metrics join up smoothly and that the cluster be stable imply either (a)k=−1 with Λ lying in the range −1.1×10−27≲Λ≲1.5×10−36 (s−2), or (b)k=0. Also, superclustering on a scale larger than 0.5 Mpc is unacceptable unless Λ〈0.

Notes: Abstract Following the re-expression of the metric for a hierarchical cosmology in Section 1 (Introduction), the metric curvature is considered in Section 2 in showing that the hierarchy models are not, in general, related to the Robertson/Walker models even though the metrics can be made to appear superficially similar. The classes of models examined are quasi-Robertson/Walker models with Λ=0 (Section 3), quasi-Robertson/Walker models with Λ≠0 (Section 4), constant curvature (k(t)=constant) models (Section 5) and zero-curvature (k(t)=0) models (Section 6). The first group have ϱ0αt −3,S(t)αt 2 in a suitable limit; the second group have solutions expressible as incomplete elliptic integrals of the first and third kind; the next group (k(t)=const.) includes an oscillating model withϱ 0 (t)α[sin (t/2T)]−2,S(t)α[sin (t/2T)]2/3 and a model withk(t)αΛ(ϱ 0αt −2), and a model withk(t)αΛ −β 2/3(ϱ 0αe βt , β〈0); the last group comprises three models, one having Λ = 0(ϱ 0αt −2), one having Λ=−1/3T 2 where 2πT is the bounce period of the (oscillating) model, and an analogue of the de Sitter Universe with $$\varrho _0 (t)\alpha e^{ - \sqrt {3\Lambda t} } $$ and $$S(t)\alpha e^{ - \sqrt {(\Lambda /3)t} } $$ . Section 7 contains our conclusions.

Notes: Abstract An approximate metric is found which represents a sphere of matter embedded in a background of dust. The use of this metric in conjunction with the Friedmann equations gives values of Λ for the three possible values ofk as Λ≃+6×10−36 (k=+1), Λ≃+3×10−35 (k=0), Λ≈+10−36 (k=−1). These values depend on data regarding clusters of galaxies, and are probably accurate to within an order of magnitude given the correctness of the assumptions on which their derivation rests.

Notes: Abstract Attention is given to the radiation of microwaves by charged dust in space. Presently-used particle distributions do not restrict the presence in space of large numbers of small (r〈10−6 cm) silicate grains, but it is shown that such densities (ϱ≈10−25–10−26 g cm−3) of small grains would produce a microwave background with an energy density of the same order of magnitude as the energy density of the (presumed) cosmological 3 K background. Limits set by the isotropy of the latter are: ϱ(HI clouds)≲10−26, ϱ(Galactic plane)≲10−30, ϱ(Halo)≲10−32, ϱ(Local Group)≲10−34 g cm−3. These limits imply that either there is a cutoff in particle distributions atr≃10−6 cm, or that the density of silicate grains in space has been generally overestimated, or that cosmic rays have broken up a lot of grains so that they now form a population of grains of very small size (≃10−7 cm) which are difficult to detect by conventional methods. One way to look for the latter population is by studying expected distortions of the 3 K spectrum to the short wavelength side of the portion hitherto observed (grains may have a size distribution able to give an approximate black-body curve for radiation from larger grains of 10−6 cm size), and by testing the effective energy density of the 3 K field in other galaxies.

Notes: Abstract Recent computer simulations by Haggerty (1974) and Haggerty and Janin (1974) on the gravitational clumping of bodies into clusters and superclusters in Newtonian cosmology have given an approximate value of Θ=1.9 for the thinning factor (defined for two systems of sizesR 1,R 2 and densitiesϱ 1,ϱ 2 byϱ 1/ϱ 2 = (R 2/R 1)θ), close to the observed value of Θ=1.7. To get an almost exact value of Θ in a general relativistic hierarchy, algebraic conditions on the metric tensor are employed: the result is Θ=2, with the hierarchy characterized by a dimensionless constantη 1 (≋Gϱb 2/c2) of valueη 1 ⋍ 2 x 10-7 (ϱ=density,b=characteristic dimension, of any system at any level of the hierarchy). A condition on the rotation of systems is also found for objects of sufficiently high angular momentum.

Notes: Abstract Attention is given to four reasons for believing that the upper limit on the rotation of the Universe ω set by isotropy of the 3K background may not be appropriate to the local system because of its hierarchical structure. In particular, recent work of Rubinet al. (1973) on the anisotropy of Hubble's parameter (H) as determined by certain galaxies is examined. The anisotropy inH is a 1st order harmonic effect, inconsistent with an origin in an acceleration of the expansion of the Universe (U α;4≠0), but explicable as being due to a large peculiar velocity of the Local Group. This compromises limits set on ω by isotropy of the 3K field, as does the realization that only weak limits can be set if the last-scattering surface (z *) is notz *→∞ but is at smallz * (as expected in a hierarchy). In a rotating Universe, the 3-spaces of constant density cannot be orthogonal on the world lines of matter: a number test of Gödell based on this is generalized and applied (after consideration of Galactic obscuration) to the local Universe, by taking data on clusters of galaxies from the Abell and Zwicky catalogues. Data from the former give only a marginally significant result for the component ω1 of ω in one direction, but a bootstrap argument is applied which takes significance over from Abell's data (considered as a class of galaxies) to Zwicky's data (taken as a class of clusters), giving a statistically significant result on the hypothesis that clusters are the fundamental units of the Universe: it seems likely that ω1r≃(const)r-n with 0≲n≲1 over the interval 500–1000 Mpc (H=60 km s−1 Mpc−1) with a total rotation of ω〈2ω1, and ω1 = 1.2 (+0.25) x 10-18 s-1 evaluated on data out to 103 Mpc. Strictly, the quoted value of the rotation only applies to a region of space that in some sense has an isotropic limit: if the actual hierarchy has a large density-dependence away from a local origin (i.e., large thinning factor), then the numerical value of the rotation is smaller than the quoted value but still finite and significant.

Notes: Abstract Data on a statistic derived from the angular covariance function show that (contrary to the claim of Peebles that galaxies are distributed continuously with no distinct scales), superclusters and the maximum size of clusters are probably defined at scales of 15 and 2.0h −1 Mpc. This suggests some stepped-density profile like the idealized models of de Vaucouleurs and Wertz: consideration is therefore given to a semi-continuous hierarchy in which there are galaxies outside clusters, clusters outside superclusters etc. Theories of the origin of clustering by gravitational clumping and the escape of galaxies from clusters suggests the hypothesis that the average mass (m g) of galaxies outside clusters is smaller than that of those inside (=fractionf of the total), a hypothesis supported by results on the continuity of the angular and spatial covariance functions. In a semi-continuous hierarchy, the overall packing fractionf e and the fraction (1-f) of galaxies outside clusters both appear to increase as the distancer from a local origin increases, because a line-of-sight to greater depths intersects systems of the hierarchy of continually greater size (R i). If the hypothesis is valid thatm g inside clusters is slightly larger thanm g outside, the apparent effect is to makem g systematically distance-dependent from a local origin with $$m_g = m_g (r)\alpha r^{ - \eta _1 } $$ and η1≃0.3. No direct data on galaxy masses exist to refute such a small trend, but since the absolute magnitudes of galaxies are known to be correlated (very weakly) with their masses, a semi-continuous hierarchy has a location-dependent luminosity function, ϕ(M). Within uncertainties as to the steepness of ϕ(M) at the bright end, the model is consistent with optical number counts to a limiting photographic magnitudem pg (isotropic slope,q′=0.6; semicontinuous modelq′=0.64; observation,q′=0.67±0.03, standard error.) this removes the discrepancy between the determinations by de Vaucouleurs and Sandageet al. of the thinning factor ⊝(≃1.7). Predictions of the semi-continuous model are made which are at present observationally feasible to carry out. In particular, it is predicted thatq′(20〈|M|〈22)/q′(14〈|M|〈19)≃2(±0.2).

Notes: Abstract An exact solution of Einstein's field equations is given representing an inhomogeneous sphere of matter in static isothermal equilibrium with a density profile of the form ϱ ∝R −2. A perturbation analysis shows that initial small density inhomogeneities grow self-similarly in a non-extreme configuration for which general relativistic effects are important. The perturbed configuration develops asymptotically like the expanding solution to Einstein's equations considered previously by Henriksen and Wesson, and the present solution is therefore identified as the initial stage of that evolving model. As a cosmology, the solutions in tandem provide an expanding cosmology that did not begin as a Big Bang, but evolved from an inhomogeneous ‘primeval atom’, tending at late epochs to a model with homogeneous density. The new static solution has applications also in other branches of astrophysics.

Notes: Abstract The response of material to a rotating magnetic dipole, considered as primeaval, the axis of which liesin the galactic plane of a model galaxy, is examined. In the three cases of (2.1) gas gradient dominant; (2.2) magnetic pressure and gravity dominant; and (2.3) gas pressure, magnetic pressure and gravity dominant with viscosity neglected, the flow pattern is found to be always characterised by two streamers of high-velocity matter emerging in the plane of the galaxy. The accompanying density distribution suggests a ready analogy with spiral galaxies, especially of SBc and SBb type; the main implication of the hypothesis, however, is that galactic dipoles will inevitably set up density perturbations of a form suitable for the generation of spiral arms via the mechanism of density waves.