Stability of an Ultra-Relativistic Blast Wave in an External Medium with a Steep Power-Law Density Profile

We examine the stability of self-similar solutions for an accelerating relativistic blast wave which is generated by a point explosion in an external medium with a steep radial density profile of a power-law index > 4.134. These accelerating solutions apply, for example, to the breakout of a gamma-ray burst outflow from the boundary of a massive star, as assumed in the popular collapsar model. We show that short wavelength perturbations may grow but only by a modest factor <~ 10.Comment: 12 pages, 3 figures, submitted to Physical Review

The Non-Relativistic Evolution of GRBs 980703 and 970508: Beaming-Independent Calorimetry

We use the Sedov-Taylor self-similar solution to model the radio emission from the gamma-ray bursts (GRBs) 980703 and 970508, when the blastwave has decelerated to non-relativistic velocities. This approach allows us to infer the energy independent of jet collimation. We find that for GRB 980703 the kinetic energy at the time of the transition to non-relativistic evolution, t_NR ~ 40 d, is E_ST ~ (1-6)e51 erg. For GRB 970508 we find E_ST ~ 3e51 erg at t_NR ~ 100 d, nearly an order of magnitude higher than the energy derived in Frail, Waxman and Kulkarni (2000). This is due primarily to revised cosmological parameters and partly to the maximum likelihood fit we use here. Taking into account radiative losses prior to t_NR, the inferred energies agree well with those derived from the early, relativistic evolution of the afterglow. Thus, the analysis presented here provides a robust, geometry-independent confirmation that the energy scale of cosmological GRBs is about 5e51 erg, and additionally shows that the central engine in these two bursts did not produce a significant amount of energy in mildly relativistic ejecta at late time. Furthermore, a comparison to the prompt energy release reveals a wide dispersion in the gamma-ray efficiency, strengthening our growing understanding that E_gamma is a not a reliable proxy for the total energy.Comment: Submitted to ApJ; 13 pages, 6 figures, 1 table; high-resolution figures can be found at: http://www.astro.caltech.edu/~ejb/NR

Navigation in Curved Space-Time

A covariant and invariant theory of navigation in curved space-time with respect to electromagnetic beacons is written in terms of J. L. Synge's two-point invariant world function. Explicit equations are given for navigation in space-time in the vicinity of the Earth in Schwarzschild coordinates and in rotating coordinates. The restricted problem of determining an observer's coordinate time when their spatial position is known is also considered

Confinement of supernova explosions in a collapsing cloud

We analyze the confining effect of cloud collapse on an expanding supernova shockfront. We solve the differential equation for the forces on the shockfront due to ram pressure, supernova energy, and gravity. We find that the expansion of the shockfront is slowed and in fact reversed by the collapsing cloud. Including radiative losses and a potential time lag between supernova explosion and cloud collapse shows that the expansion is reversed at smaller distances as compared to the non-radiative case. We also consider the case of multiple supernova explosions at the center of a collapsing cloud. For instance, if we scale our self-similar solution to a single supernova of energy 10^51 ergs occurring when a cloud of initial density 10^2 H/cm^3 has collapsed by 50%, we find that the shockfront is confined to ~15 pc in ~1 Myrs. Our calculations are pertinent to the observed unusually compact non-thermal radio emission in blue compact dwarf galaxies (BCDs). More generally, we demonstrate the potential of a collapsing cloud to confine supernovae, thereby explaining how dwarf galaxies would exist beyond their first generation of star formation.Comment: 3 pages, 4 figure

Self-similar cosmologies in 5D: spatially flat anisotropic models

In the context of theories of Kaluza-Klein type, with a large extra dimension, we study self-similar cosmological models in 5D that are homogeneous, anisotropic and spatially flat. The "ladder" to go between the physics in 5D and 4D is provided by Campbell-Maagard's embedding theorems. We show that the 5-dimensional field equations $R_{AB} = 0$ determine the form of the similarity variable. There are three different possibilities: homothetic, conformal and "wave-like" solutions in 5D. We derive the most general homothetic and conformal solutions to the 5D field equations. They require the extra dimension to be spacelike, and are given in terms of one arbitrary function of the similarity variable and three parameters. The Riemann tensor in 5D is not zero, except in the isotropic limit, which corresponds to the case where the parameters are equal to each other. The solutions can be used as 5D embeddings for a great variety of 4D homogeneous cosmological models, with and without matter, including the Kasner universe. Since the extra dimension is spacelike, the 5D solutions are invariant under the exchange of spatial coordinates. Therefore they also embed a family of spatially {\it inhomogeneous} models in 4D. We show that these models can be interpreted as vacuum solutions in braneworld theory. Our work (I) generalizes the 5D embeddings used for the FLRW models; (II) shows that anisotropic cosmologies are, in general, curved in 5D, in contrast with FLRW models which can always be embedded in a 5D Riemann-flat (Minkowski) manifold; (III) reveals that anisotropic cosmologies can be curved and devoid of matter, both in 5D and 4D, even when the metric in 5D explicitly depends on the extra coordinate, which is quite different from the isotropic case.Comment: Typos corrected. Minor editorial changes and additions in the Introduction and Summary section

Self-similar imploding relativistic shock waves

Self-similar solutions to the problem of a strong imploding relativistic shock wave are calculated. These solutions represent the relativistic generalisation of the Newtonian Gouderley-Landau-Stanyukovich problem of a strong imploding spherical shock wave converging to a centre. The solutions are found assuming that the pre-shocked flow has a uniform density, and are accurate for sufficiently large times after the formation of the shock wave.Comment: 22 pages, 4 figures. Minor corrections and a discussion of the singular C_ characteristic added. Accepted for publication in Physics of Fluid

Egological study of the phenomenon of subjective space

The article presents a phenomenological analysis of the space of transcendental subjectivity. The relevance of this study is to form an idea of the internal spatial structures of the pure I. The purpose of this study is to identify the complex nature of self-consciousness. To achieve this goal, the egological analysis was based on a mereological approach, which prepares the consideration of the space of pure consciousness from the point of view of the whole, composed of parts. The application of spatial categories to the pure Self is based on the expansion of the proximity space. The results obtained indicate the need to create an egology of space in order to study the nature of spatial vision. The proposed research focuses not on the variety of available spaces, but on the process in which their formation takes place in subjective spatial vision. The influence of the subjective factor on the understanding of space is demonstrated by the example of mathematics. The mathematical notion of “proximity space” is subjected to a phenomenological extension, which reveals the phenomenon of spatiality of the pure I, or the inner space of the subject, which reflects the hierarchy of preferences or evaluations. The primary spatial structure of pure consciousness is revealed, which is the division of pure I into two parts. The splitting of the Self into two constituent parts and its representation in the form of a spatial model with its top and bottom is funded by a deeper phenomenon — subjective spatiality. This structure can serve as a foundation for the subsequent correlative study of objective and subjective factors influencing the formation of various egological structures. Based on the analysis, a conclusion is made about the essential relationship of the pure I and the inner space. The importance of this conclusion is supported by an analysis of Husserl’s works, in which he touches upon the issues of the spatiality of transcendental subjectivity. The scientific novelty of the proposed work is that the distinction between absolute and relative intimacy is introduced into egological research. Only within the framework of relative intimacy is it possible to observe and systematically investigate the pure I. In the absolute position it remains anonymous because of its extreme intimacy

Self-similar solutions for relativistic shocks emerging from stars with polytropic envelopes

We consider a strong ultrarelativistic shock moving through a star whose envelope has a polytrope-like density profile. When the shock is close to the star's outer boundary, its behavior follows the self-similar solution given by Sari (2005) for implosions in planar geometry. Here we outline this solution and find the asymptotic solution as the shock reaches the star's edge. We then show that the motion after the shock breaks out of the star is described by a self-similar solution remarkably like the solution for the motion inside the star. In particular, the characteristic Lorentz factor, pressure, and density vary with time according to the same power laws both before and after the shock breaks out of the star. After emergence from the star, however, the self-similar solution's characteristic position corresponds to a point behind the leading edge of the flow rather than at the shock front, and the relevant range of values for the similarity variable changes. Our numerical integrations agree well with the analytic results both before and after the shock reaches the star's edge.Comment: 18 pages, 5 figures, submitted to Ap