Sedov Solution Research Articles

About regular expansion of a two-phase ball

The work is devoted to the research of an expansion into a vacuum of a ball, which is filled with a two-phase fluid. In assumptions that the dynamics passes in a regular mode, the velocities of the phases are linear functions by space coordinate and the first phase spreads into the void faster than the second phase there is obtained a solution of equations of two-phase fluid dynamics, which describes an expansion of a ball into a vacuum. This solution generalizes the known Sedov solution, which determines an expansion of a gas cloud into a vacuum, to the case of a ball filled with a gas suspension. In the last part of the work there are derived asymptotic formulas determining a relationship between the velocities of the phases and the radii of two balls.

Reverberation of pulsar wind nebulae – II. Anatomy of the ‘thin-shell’ evolution

ABSTRACT During its early evolution, a pulsar wind nebula (PWN) sweeps the inner part of the supernova ejecta and forms a thin massive shell. Later on, when the shell has been reached by the reverse shock of the supernova remnant, the evolution becomes more complex, in most cases reverting the expansion into a compression: this later phase is called ‘reverberation’. Computations done so far to understand this phase have been mostly performed in the thin-shell approximation, where the evolution of the PWN radius is assimilated to that of the swept-up shell under the effect of both the inner pressure from the PWN, and the outer pressure from the supernova remnant. Despite the thin-shell approach seems rather justifiable, its implementations have so far been inaccurate, and its correctness, never tested. The outer pressure was naively assumed to be scaled according to the Sedov solution (or a constant fraction of it) along the entire evolution. The thin-shell assumption itself fails along the process, being the shell no longer thin in comparison with the size of the PWN. Here, through a combination of numerical models, dimensional arguments, and analytical approximations, we present a detailed analysis of the interaction of the PWN with the supernova remnant. We provide a new analytical approximation of the outer pressure, beyond the Sedov solution, and a revised ‘thin-shell’ able to reproduce results from numerical simulations. Finally, we compute the efficiency by which the PWN is compressed during reverberation over a wide population of sources.

Effect of volumetric expansion on shock-induced ignition of H2–NO2/N2O4 mixtures

The competition between chemical energy release rate and volumetric expansion related to shock wave’s dynamics is of primary importance for a number of situations relevant to explosion safety. While studies have been performed on this topic over the years, they have been limited to mixtures with monotonous energy release profile. In the present study, the ignition of H2–NO2/N2O4 mixtures, which exhibit a single-step or a two-step energy release rate profile depending on the equivalence ratio, has been investigated under volumetric expansion conditions. The rate of expansion has been calculated using the Taylor–Sedov solution and accounted for using 0-D numerical simulations with time-dependent specific volume. The results were analyzed in terms of a Damkohler number defined as the ratio of the expansion to ignition times. For mixtures with non-monotonous energy release rate profiles, two critical Damkohler numbers can be identified for individual steps of energy release. It was also shown that the fluid element behind a shock propagating at the Chapman–Jouguet velocity is most likely to ignite. The thermo-chemical dynamics have been analyzed about the critical conditions using energy release rate per reaction, rate of production and sensitivity analyses.

Shock-accelerated cosmic rays and streaming instability in the adaptive mesh refinement code Ramses

Cosmic rays (CRs) are thought to play a dynamically important role in several key aspects of galaxy evolution, including the structure of the interstellar medium, the formation of galactic winds, and the non-thermal pressure support of halos. We introduce a numerical model solving for the CR streaming instability and acceleration of CRs at shocks with a fluid approach in the adaptive mesh refinement code RAMSES. CR streaming is solved with a diffusion approach and its anisotropic nature is naturally captured. We introduce a shock finder for the RAMSES code that automatically detects shock discontinuities in the flow. Shocks are the loci for CR injection, and their efficiency of CR acceleration is made dependent on the upstream magnetic obliquity according to the diffuse shock acceleration mechanism. We show that the shock finder accurately captures shock locations and estimates the shock Mach number for several problems. The obliquity-dependent injection of CRs in the Sedov solution leads to situations where the supernova bubble exhibits large polar caps (homogeneous background magnetic field), or a patchy structure of the CR distribution (inhomogeneous background magnetic field). Finally, we combine both accelerated CRs with streaming in a simple turbulent interstellar medium box, and show that the presence of CRs significantly modifies the structure of the gas.

Converging shock flows for a Mie-Grüneisen equation of state

Previous work has shown that the one-dimensional (1D) inviscid compressible flow (Euler) equations admit a wide variety of scale-invariant solutions (including the famous Noh, Sedov, and Guderley shock solutions) when the included equation of state (EOS) closure model assumes a certain scale-invariant form. However, this scale-invariant EOS class does not include even simple models used for shock compression of crystalline solids, including many broadly applicable representations of Mie-Grüneisen EOS. Intuitively, this incompatibility naturally arises from the presence of multiple dimensional scales in the Mie-Grüneisen EOS, which are otherwise absent from scale-invariant models that feature only dimensionless parameters (such as the adiabatic index in the ideal gas EOS). The current work extends previous efforts intended to rectify this inconsistency, by using a scale-invariant EOS model to approximate a Mie-Grüneisen EOS form. To this end, the adiabatic bulk modulus for the Mie-Grüneisen EOS is constructed, and its key features are used to motivate the selection of a scale-invariant approximation form. The remaining surrogate model parameters are selected through enforcement of the Rankine-Hugoniot jump conditions for an infinitely strong shock in a Mie-Grüneisen material. Finally, the approximate EOS is used in conjunction with the 1D inviscid Euler equations to calculate a semi-analytical Guderley-like imploding shock solution in a metal sphere and to determine if and when the solution may be valid for the underlying Mie-Grüneisen EOS.

Off-center blast in a shocked medium

When multiple blasts occur at different times, the situation arises in which a blast wave is propagating into a medium that has already been shocked. Determining the evolution in the shape of the second shock is not trivial, as it is propagating into air that is not only non-uniform, but also non-stationary. To accomplish this task, we employ the method of Kompaneets to determine the shape of a shock in a non-uniform media. We also draw from the work of Korycansky (Astrophys J 398:184–189. https://doi.org/10.1086/171847 , 1992) on an off-center explosion in a medium with radially varying density. Extending this to treat non-stationary flow, and making use of approximations to the Sedov solution for the point blast problem, we are able to determine an analytic expression for the evolving shape of the second shock. In particular, we consider the case of a shock in air at standard ambient temperature and pressure, with the second shock occurring shortly after the original blast wave reaches it, as in a sympathetic detonation.

The Sedov Blast Wave as a Radial Piston Verification Test

The Sedov blast wave is of great utility as a verification problem for hydrodynamic methods. The typical implementation uses an energized cell of finite dimensions to represent the energy point source. This approximation can be avoided by directly finding the effects of the energy source as a boundary condition (BC). The proposed method transforms the Sedov problem into an outward moving radial piston problem with a time-varying velocity. A portion of the mesh adjacent to the origin is removed and the boundaries of this hole are forced with the velocities from the Sedov solution. This verification test is implemented on two types of meshes, and convergence is shown. The results from the typical initial condition (IC) method and the new BC method are compared.

Beam-Riding Flight of a Laser Propulsion Vehicle Using Actively Controlled Pulse

This paper discusses active laser control using a genetic algorithm and a mirror-actuating system to achieve kilometer-order in-air flight using a laser propulsion vehicle while riding a beam. A 10 kg vehicle reaction driven by a strong shock wave is examined using our flight simulator to analyze interaction with unsteady blast wave propagation based on coupling calculation between hydrodynamic simulation of the shock wave propagation and orbital simulation of the vehicle flight motion, and the generated impulses are characterized by the spherically symmetric Sedov solution. Beam-riding flight with an initial offset of 5 mm is successfully sustained by controlling angular offset, while vehicle acceleration is kept low for safe launch to the target altitude. A system delay of laser control is introduced into the flight simulator, and beam-riding flight is maintained for a 20 ms system delay using delay correction following prediction by six-degree-of-freedom equations of motion. This study also examines robustness of the flying technique for wind perturbation, and an active control scheme that can ensure stable flight with a wind of up to . The stability of flight control is assessed when there is a positioning error of laser incident light, and flight is stably sustained by keeping the angular offset small for the positioning error with a standard deviation 1 cm. This paper examines the lower limit of repetition frequency of multiple pulses for stable flight, and found that a repetition frequency exceeding 15 Hz should be selected in order to maintain posture stability. Stable flight is not maintained for the combination of larger system delay and positioning error because translational velocity and angular offset simultaneously increase. However, the vehicle can fly to kilometer-order altitude while maintaining posture stability if the translational and angular impulses are adjusted. Flight performance can be improved by adjusting impulse direction and size using our flight simulator as an impulse design tool.

ESCAPE FRACTION OF IONIZING PHOTONS DURING REIONIZATION: EFFECTS DUE TO SUPERNOVA FEEDBACK AND RUNAWAY OB STARS

The fraction of hydrogen ionizing photons escaping from galaxies into the intergalactic medium is a critical ingredient in the theory of reionization. We use two zoomed-in, high-resolution (4 pc), cosmological radiation hydrodynamic simulations with adaptive mesh refinement to investigate the impact of two physical mechanisms (supernova feedback and runaway OB stars) on the escape fraction (f_esc) at the epoch of reionization (z>7). We implement a new, physically motivated supernova feedback model that can approximate the Sedov solutions at all (from the free expansion to snowplow) stages. We find that there is a significant time delay of about ten million years between the peak of star formation and that of escape fraction, due to the time required for the build-up and subsequent destruction of the star-forming cloud by supernova feedback. Consequently, the photon number-weighted mean escape fraction for dwarf galaxies in halos of mass 10^8-10^10.5 Msun is found to be <fesc>~11%, although instantaneous values of f_esc>20% are common when star formation is strongly modulated by the supernova explosions. We find that the inclusion of runaway OB stars increases the mean escape fraction by 22% to <fesc>~14%. As supernovae resulting from runaway OB stars tend to occur in less dense environments, the feedback effect is enhanced and star formation is further suppressed in halos with Mvir>10^9 Msun in the simulation with runaway OB stars compared with the model without them. While both our models produce enough ionizing photons to maintain a fully ionized universe at z>7 as observed, a still higher amount of ionizing photons at z>9 appears necessary to accommodate the high observed electron optical depth inferred from cosmic microwave background observations.

The impact of supernova-driven winds on stream-fed protogalaxies

Supernova (SN) driven winds are widely thought to be very influential in the high-redshift Universe, shaping the properties of the circumgalactic medium, enriching the intergalactic medium with metals and driving the evolution of low-mass galaxies. However, it is not yet fully understood how SN-driven winds interact with their surroundings in a cosmological context, nor is it clear whether they are able to significantly impact the evolution of low-mass galaxies from which they originate by altering the amount of the cold material these accrete from the cosmic web. Indeed, due to the strong constraints on resolution imposed by limited computational power, all cosmological hydrodynamic simulations to date resort to implementing more or less physically well motivated and complex subgrid models to trigger galactic winds. To explore this issue, we implement a standard Taylor–Sedov type solution, widely used in the community to depict the combined action of many SN explosions, in a cosmological resimulation of a low-mass galaxy at z≥ 9 from the ‘nut’ suite. However, in contrast with previous work, we achieve a resolution high enough to capture individual SN remnants in the Taylor–Sedov phase, for which the Taylor–Sedov solution actually provides an accurate description of the expansion. We report the development of a high-velocity, far-reaching galactic wind produced by the combined action of SNe in the main galaxy and its satellites, which are located in the same or a neighbouring dark matter halo. Despite this, we find that (i) this wind carries out very little mass (the measured outflow is of the order of a tenth of the inflow/star formation rate); and (ii) the cold gas inflow rate remains essentially unchanged from the run without SN feedback. Moreover, there are epochs during which star formation is enhanced in the feedback run relative to its radiative-cooling-only counterpart. We attribute this ‘positive’ feedback to the metal enrichment that is present only in the former. We conclude that at very high redshift, efficient SN feedback can drive large-scale galactic winds but does not prevent massive cold gas inflow from fuelling galaxies, resulting in long-lived episodes of intense star formation.

An improved method to experimentally determine temperature and pressure behind laser-induced shock waves at low Mach numbers

Laser–matter interactions are frequently studied by measuring the propagation of shock waves caused by the rapid laser-induced material removal. An improved method for calculating the thermo-fluid parameters behind shock waves is introduced in this work. Shock waves in ambient air, induced by pulsed Nd : YAG laser ablation of aluminium films, are measured using a shadowgraph apparatus. Normal shock solutions are applied to experimental data for shock wave positions and used to calculate pressure, temperature, and velocity behind the shock wave. Non-dimensionalizing the pressure and temperature with respect to the ambient values, the dimensionless pressure and temperature are estimated to be as high as 90 and 16, respectively, at a time of 10 ns after the ablation pulse for a laser fluence of F = 14.5 J cm−2. The results of the normal shock solution and the Taylor–Sedov similarity solution are compared to show that the Taylor–Sedov solution under-predicts pressure when the Mach number of the shock wave is small. At a fluence of 3.1 J cm−2, the shock wave Mach number is less than 3, and the Taylor–Sedov solution under-predicts the non-dimensional pressure by as much as 45%.

Comptonization of the cosmic microwave background by high energy particles residing in AGN cocoons

X-ray cavities and extended radio sources (`cocoons') surrounding active galactic nuclei (AGN) have been detected by the Chandra X-ray mission and radio interferometers. A joint analysis of X-ray and radio maps suggests that pressure values of non-thermal radio-emitting particles derived from the radio maps are not sufficient to inflate the X-ray cavities. We propose using the Sunyaev-Zel'dovich (SZ) effect, whose intensity strongly depends on the pressure, to find the hitherto undetected, dynamically-dominant component in the radio cocoons. We demonstrate that the spectral function at a frequency of 217 GHz has an absolute maximum at a temperature higher than $10^9$ K, therefore the measurement of the SZ effect at this frequency is a powerful tool for potentially revealing the dynamically-dominant component inside AGN jet-driven radio cocoons. A new method is proposed for excluding the contribution from the low energy, non-relativistic electrons to the SZ effect by means of observations at two frequencies. We show how one may correct for a possible contribution from the kinematic SZ effect. The intensity maps of the SZ effect are calculated for the self-similar Sedov solution, and application of a predicted ring-like structure on the SZ map at a frequency of 217 GHz is proposed to determine the energy released during the active jet stage. The SZ intensity map for an AGN cocoon in a distant elliptical is calculated using a 2-D numerical simulation and including relativistic corrections to the SZ effect. We show the intensity spectrum of the SZ effect is flat at high frequencies if gas temperature is as high as $k_\mathrm{b} T_{\mathrm{e}}=500$ keV.

Supernova Blast Waves in Low‐Density Hot Media: A Mechanism for Spatially Distributed Heating

Most supernovae are expected to explode in low-density hot media, particularly in galactic bulges and elliptical galaxies. The remnants of such supernovae, although difficult to detect individually, can be profoundly important in heating the media on large scales. We characterize the evolution of this kind of supernova remnant, based on analytical approximations and hydrodynamic simulations. We generalize the standard Sedov solution to account for both temperature and density effects of the ambient media. Although cooling can be neglected, the expansion of such a remnant deviates quickly from the standard Sedov solution and asymptotically approaches the ambient sound speed as the swept-up thermal energy becomes important. The relatively steady and fast expansion of the remnants over large volumes provides an ideal mechanism for spatially distributed heating, which may help to alleviate the overcooling problem of hot gas in groups and clusters of galaxies, as well as in galaxies themselves. The simulations were performed with the FLASH code.

Water impact of an asymmetric floating wedge

The flow generated during the early stage of the impact of an asymmetric wedge is here investigated, with the help of a conformal-mapping technique. The wedge is assumed to be initially at rest and floating on a still free surface of an inviscid and incompressible liquid. The study is aimed at understanding the main features of the flow that develops in the vicinity of the apex of the wedge. Due to the inviscid-fluid assumption, a velocity (and pressure) singularity takes place at the apex of asymmetric wedges, which is here removed by reintroducing viscous effects, in terms of the shedding of a wake from the apex. In the present numerical approach, the wake is modelled through point vortices, circulations of which are provided via a suitable Kutta condition. Attention being focused at the early stage of the impact, the free surface is kept during the penetration of the body. This allows the use of an asymmetric extension of Sedov's solution to describe the flow field generated by the wedge entry. Changes induced by the vortical flow on the velocity field and on the pressure distribution on the wedge are discussed.

Small‐Scale Structure of the SN 1006 Shock withChandraObservations

The northeast shell of SN 1006 is the most probable acceleration site of high-energy electrons (up to ~100 TeV) with the Fermi acceleration mechanism at the shock front. We resolved nonthermal filaments from thermal emission in the shell with the excellent spatial resolution of Chandra. The thermal component is extended over ~100'' (about 1 pc at 1.8 kpc distance) in width, consistent with the shock width derived from the Sedov solution. The spectrum is fitted with a thin thermal plasma of kT = 0.24 keV in nonequilibrium ionization, typical for a young supernova remnant. The nonthermal filaments are likely thin sheets with scale widths of ~4'' (0.04 pc) and ~20'' (0.2 pc) upstream and downstream, respectively. The spectra of the filaments are fitted with a power-law function of index 2.1-2.3, with no significant variation from position to position. In a standard diffusive shock acceleration model, the extremely small scale length in the upstream region requires the magnetic field nearly perpendicular to the shock normal. The injection efficiency (η) from thermal to nonthermal electrons around the shock front is estimated to be ~1 × 10-3 under the assumption that the magnetic field in the upstream region is 10 μG. In the filaments, the energy densities of the magnetic field and nonthermal electrons are similar to each other, and both are slightly smaller than that of thermal electrons. These results suggest that the acceleration occurs in more compact regions with larger efficiency than suggested by previous studies.

Pulsar wind nebulae in supernova remnants

A spherically symmetric model is presented for the interaction of a pulsar wind with the associated supernova remnant. This results in a pulsar wind nebula whose evolution is coupled to the evolution of the surrounding supernova remnant. This evolution can be divided in three stages. The rst stage is characterised by a supersonic expansion of the pulsar wind nebula into the freely expanding ejecta of the progenitor star. In the next stage the pulsar wind nebula is not steady; the pulsar wind nebula oscillates between contraction and expansion due to interaction with the reverse shock of the supernova remnant: reverberations which propagate forward and backward in the remnant. After the reverberations of the reverse shock have almost completely vanished and the supernova remnant has relaxed to a Sedov solution, the expansion of the pulsar wind nebula proceeds subsonically. In this paper we present results from hydrodynamical simulations of a pulsar wind nebula through all these stages in its evolution. The simulations were carried out with the Versatile Advection Code.

A Diffuse X-Ray Source, AX J1843.80352: Association with the Radio Complex G28.6-0.1 and Identification of a New Supernova Remnant

Abstract ASCA discovered an extended source in the constellation Scutum. The X-ray morphology is an elliptical shape elongated from north to south with a mean diameter of about $ {10{}^{\mathrm {\prime }}}$. The image center is located at $ \mathrm{RA}_{2000}$ = $ {18{}^{\mathrm {h}}43{}^{\mathrm {m}}53{}^{\mathrm {s}}}$, $ \mathrm{DEC}_{2000}$ = $ - {03{}^{\mathrm {\circ }}52{}^{\mathrm {\prime }}55{}^{\mathrm {\prime \prime }}}$ (hereafter, AX J1843.8–0352). The north and south rims of AX J1843.8–0352 are associated with non-thermal radio sources C and F of the G28.6–0.1 complex (Helfand et al. 1989, AAA $ 49.125.029$). The X-ray spectrum was fitted with a model of either a thin thermal plasma in non-equilibrium ionization of a temperature 5.4keV or a power-law of photon index 2.1. The absorption column is $ (2.4$-$ 4.0)\times 10^{22}\mathrm{cm}^{-2}$, which probably places this source in the Scutum arm. With a reasonable distance assumption of about 7kpc, we estimate the mean diameter and X-ray luminosity to be $ \sim 20 \,\mathrm{pc}$ and $ \sim 3 \times 10^{34}\,\mathrm{erg} \,\mathrm{s}^{-1}$, respectively. Although a Sedov solution for a thin thermal plasma model gives parameters of a young shell-like SNR, no strong emission lines are found with the metal abundances being $ \sim$ solar. Thus, a more likely scenario for both the radio and X-ray spectra and the morphologies is that AX J1843.8–0352 is a shell-like SNR which predominantly emits synchrotron X-rays.

RX J050736−6847.8: A Large Supernova Remnant around an X-Ray Binary in the Large Magellanic Cloud

RX J050736-6847.8 in the Large Magellanic Cloud (LMC) is an intriguing X-ray source consisting of a large ring (diameter ~ 150 pc) of diffuse emission and a central compact source. It is projected in the vicinity of the superbubble N103 around the star cluster NGC 1850. RX J050736-6847.8's ring of diffuse X-ray emission, offset from the superbubble N103, is not bounded by any optical shell structure, while RX J050736-6847.8's central compact X-ray source is projected within the cluster HS122. We have analyzed archival ROSAT observations of RX J050736-6847.8 to determine the physical properties of its X-ray–emitting gas. The X-ray luminosity of the diffuse X-ray emission is 5–6 × 1035 ergs s-1 in the 0.1–2.4 keV band, within the range for supernova remnants (SNRs). Assuming a shell geometry with a fractional shell thickness (ΔR/R) of 0.05–0.2, we find the density of the hot gas to be 0.05–0.09 cm-3 and a hot gas mass of ~820 M⊙. The physical properties of this shell of hot gas are consistent with Sedov's solution for a ~5 × 104 yr old SNR in a low-density (~0.015 cm-3) medium formed by a supernova of an explosion energy of 3 × 1051 ergs. The density is so low that no detectable optical emission is expected. Therefore, we suggest that this ring of X-ray emission originates in a SNR, the largest known in the LMC. The large size, low density, and the regular X-ray morphology suggest that this SNR is located in the halo of the LMC; the small absorption column density further suggests that this SNR is on the near side of the LMC halo. The central compact source of RX J050736-6847.8 is probably an X-ray binary in the cluster HS122. The relationship between the compact source and the diffuse ring is unknown.

Modeling W44 as a Supernova Remnant in a Density Gradient with a Partially Formed Dense Shell and Thermal Conduction in the Hot Interior. I. The Analytical Model

We show that many observations of W44, a supernova remnant in the Galactic plane at a distance of about 2500 pc, are remarkably consistent with the simplest realistic model. The model remnant is evolving in a smooth ambient medium of fairly high density, about 6 cm-3 on average, with a substantial density gradient. At the observed time it has an age of about 20,000 yr, consistent with the age of the associated pulsar, and a radius of 11-13 pc. Over most of the outer surface, radiative cooling has become important in the postshock gas; on the denser end there has been sufficient compression of the cooled gas to develop a very thin dense half-shell of about 450 M☉, supported against further compression by nonthermal pressure. The half-shell has an expansion velocity of about 150 km s-1 and is bounded on the outer surface by a radiative shock with that speed. The deep interior of the remnant has a substantial and fairly uniform pressure, as expected from even highly idealized adiabatic models; our model, however, is never adiabatic. Thermal conduction, while the remnant is young and hot, reduces the need for expansion cooling and prevents formation of the intensely vacuous cavity characteristic of adiabatic evolution. It radically alters the interior structure from what one might expect from familiarity with the Sedov solution. At the time of observation, the temperature in the center is about 6 × 106 K, the density about 1 cm-3. The temperature decreases gradually away from the center, while the density rises. Farther out, where cooling is becoming important, the pressure drops precipitously, and the temperature in the denser gas there is quite low. We provide several analytic tools for the assembly of models of this type. We review the early evolution and shell formation analyses and their generalizations to evolution in a density gradient. We also calculate the density and temperature that should be present in the hot interior of a remnant with thermal conduction. We supply the van der Laan mechanism in a particularly useful form for the calculation of radio continuum from radiative remnants. Finally, we estimate the optical emission that should be present from fluorescence of UV light, emitted by the forming shell and the radiative shock and absorbed in the cold shell and the ambient medium, and the associated 63 μm [O I] emission. Both are in agreement with the intensity and spatial structures found in recent observations. Neither requires interaction with a dense molecular cloud for its generation. We calculate the gamma rays that should be emitted by cosmic-ray electrons and ions in the shell, interacting with the cold material, and find each capable of generating about 25% of the flux reported by EGRET for the vicinity.

A generic dynamical model of gamma-ray burst remnants

The conventional generic model is considered to explain the dynamics of gamma-ray burst remnants very well, no matter whether they are adiabatic or highly radiative. However, we find that, for adiabatic expansion, the model cannot reproduce the Sedov solution in the non-relativistic phase, and thus it needs to be revised. In this paper a new differential equation is derived. The generic model based on this equation is shown to be correct for both radiative and adiabatic fireballs, and in both ultrarelativistic and non-relativistic phases.