Mathematical methods of constructing new models of continua

Mathematical methods of constructing new models of continu

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

Discrete Self-Similarity in Type-II Strong Explosions

We present new solutions to the strong explosion problem in a non-power law density profile. The unperturbed self-similar solutions discovered by Waxman & Shvarts describe strong Newtonian shocks propagating into a cold gas with a density profile falling off as $r^{-\omega}$, where $\omega>3$ (Type-II solutions). The perturbations we consider are spherically symmetric and log-periodic with respect to the radius. While the unperturbed solutions are continuously self-similar, the log-periodicity of the density perturbations leads to a discrete self-similarity of the perturbations, i.e. the solution repeats itself up to a scaling at discrete time intervals. We discuss these solutions and verify them against numerical integrations of the time dependent hydrodynamic equations. Finally we show that this method can be generalized to treat any small, spherically symmetric density perturbation by employing Fourier decomposition

An exact self-similar solution for an expanding ball of radiation

We give an exact solution of the $5D$ Einstein equations which in 4D can be interpreted as a spherically symmetric dissipative distribution of matter, with heat flux, whose effective density and pressure are nonstatic, nonuniform, and satisfy the equation of state of radiation. The matter satisfies the usual energy and thermodynamic conditions. The energy density and temperature are related by the Stefan-Boltzmann law. The solution admits a homothetic Killing vector in $5D$, which induces the existence of self-similar symmetry in 4D, where the line element as well as the dimensionless matter quantities are invariant under a simple "scaling" group.Comment: New version expanded and improved. To appear in Int. J. Mod. Phys.

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

An all-optical event horizon in an optical analogue of a Laval nozzle

Exploiting the fact that light propagation in defocusing nonlinear media can mimic the transonic flow of an equivalent fluid, we demonstrate experimentally the formation of an all-optical event horizon in a waveguide structure akin to a hydrodynamic Laval nozzle. The analogue event horizon, which forms at the nozzle throat is suggested as a novel platform for analogous gravity experiments

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

Emergence of a filamentary structure in the fireball from GRB spectra

It is shown that the concept of a fireball with a definite filamentary structure naturally emerges from the analysis of the spectra of Gamma-Ray Bursts (GRBs). These results, made possible by the recently obtained analytic expressions of the equitemporal surfaces in the GRB afterglow, depend crucially on the single parameter R describing the effective area of the fireball emitting the X- and gamma ray radiation. The X- and gamma ray components of the afterglow radiation are shown to have a thermal spectrum in the co-moving frame of the fireball and originate from a stable shock front described self-consistently by the Rankine-Hugoniot equations. Precise predictions are presented on a correlations between spectral changes and intensity variations in the prompt radiation verifiable, e.g., by the Swift and future missions. The highly variable optical and radio emission depends instead on the parameters of the surrounding medium. The GRB 991216 is used as a prototype for this model.Comment: 9 pages, 3 figures, to appear on International Journal of Modern Physics

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

The stability of decelerating shocks revisited

We present a new method for analyzing the global stability of the Sedov-von Neumann-Taylor self-similar solutions, describing the asymptotic behavior of spherical decelerating shock waves, expanding into ideal gas with density \propto r^{-\omega}. Our method allows to overcome the difficulties associated with the non-physical divergences of the solutions at the origin. We show that while the growth rates of global modes derived by previous analyses are accurate in the large wave number (small wavelength) limit, they do not correctly describe the small wave number behavior for small values of the adiabatic index \gamma. Our method furthermore allows to analyze the stability properties of the flow at early times, when the flow deviates significantly from the asymptotic self-similar behavior. We find that at this stage the perturbation growth rates are larger than those obtained for unstable asymptotic solutions at similar [\gamma,\omega]. Our results reduce the discrepancy that exists between theoretical predictions and experimental results.Comment: 10 pages, 9 figures. Accepted to ApJ; Expanded discussion of boundary condition