We examine the relation between presupernova stellar structure and the distribution of ejecta in core-collapse supernovae of Types Ib, Ic, and II, under the approximations of adiabatic, spherically symmetric flow. We develop a simple yet accurate analytical formula for the velocity of the initial forward shock that traverses the stellar envelope. For material that does not later experience a strong reverse shock, the entropy deposited by this forward shock persists into the final, freely expanding state. We demonstrate that the final density distribution can be approximated with simple models for the final pressure distribution, in a way that matches the results of simulations. Our results indicate that the distribution of density and radiation pressure in a star's ejecta depends on whether the outer envelope is radiative or convective, and if convective, on the composition structure of the star. Our models are most accurate for the high-velocity ejecta cast away from the periphery of a star. For stellar structures that limit to a common form in this region, the resulting ejecta limit to a common distribution at high velocities because the blast wave forgets its history as it approaches the stellar surface. We present formulae for the final density distribution of this material as a function of mass, for both radiative and efficiently convective envelopes. These formulae limit to the well-known planar, self-similar solutions for mass shells approaching the stellar surface. However, the assumption of adiabatic flow breaks down for shells of low optical depth, so this planar limit need not be attained. The event of shock emergence, which limits adiabatic flow, also produces a soft X-ray burst of radiation. Formulae are given for the observable properties of this burst and their dependence on the parameters of the explosion. Motivated by the relativistic expansion recently inferred by Kulkarni et al. for the synchrotron shell around SN 1998bw, we estimate the criterion for relativistic mass ejection and the rest mass of relativistic ejecta. We base our models for the entire ejecta distribution on the high-velocity solution, on our shock-velocity formula, and on realistic radiation pressure distributions. We also present simpler, but less flexible, analytical approximations for ejecta distributions. We survey the ejecta of the polytropic hydrogen envelopes of red supergiants. Our models will be useful for studies of the light curves and circumstellar or interstellar interactions of core-collapse supernovae, and of the birth of pulsar nebulae in their ejecta.

The expulsion of stellar envelopes in core-collapse supernovae

We study the growth of the explosion energy after shock revival in neutrino-driven explosions in two and three dimensions (2D/3D) using multi-group neutrino hydrodynamics simulations of an $11.2 M_\odot$ star. The 3D model shows a faster and steadier growth of the explosion energy and already shows signs of subsiding accretion after one second. By contrast, the growth of the explosion energy in 2D is unsteady, and accretion lasts for several seconds as confirmed by additional long-time simulations of stars of similar masses. Appreciable explosion energies can still be reached, albeit at the expense of rather high neutron star masses. In 2D, the binding energy at the gain radius is larger because the strong excitation of downward-propagating g-modes removes energy from the freshly accreted material in the downflows. Consequently, the mass outflow rate is considerably lower in 2D than in 3D. This is only partially compensated by additional heating by outward-propagating acoustic waves in 2D. Moreover, the mass outflow rate in 2D is reduced because much of the neutrino energy deposition occurs in downflows or bubbles confined by secondary shocks without driving outflows. Episodic constriction of outflows and vertical mixing of colder shocked material and hot, neutrino-heated ejecta due to Rayleigh-Taylor instability further hamper the growth of the explosion energy in 2D. Further simulations will be necessary to determine whether these effects are generic over a wider range of supernova progenitors.

The dynamics of neutrino-driven supernova explosions after shock revival in 2D and 3D

Using the all-sky ROSAT soft X-ray and 408 MHz radio continuum data, we show that the North Polar Spur (NPS) and its western and southern counterspurs draw a giant dumbbell shape necked at the Galactic plane. We interpret these features as due to a shock front originating from a starburst 15 million years ago with a total energy of the order of ~1056 ergs or 105 Type II supernovae. We simulate all-sky distributions of radio continuum and soft X-ray intensities based on the bipolar hypershell Galactic center starburst model. The simulations can well reproduce the radio NPS and related spurs, as well as radio spurs in the tangential directions of spiral arms. Simulated X-ray maps in the 0.25, 0.75, and 1.5 keV bands reproduce the ROSAT X-ray NPS, its western and southern counterspurs, and the absorption layer along the Galactic plane. We propose to use the ROSAT all-sky maps to probe the physics of gas in the halo-intergalactic interface, and to directly date and measure the energy of a recent Galactic center starburst.

http://iopscience.iop.org/article/10.1086/309297/pdf

Bipolar hypershell galactic center starburst model: further evidence from rosat data and new radio and x-ray simulations

The Magellan images showed presence of radar-dark shadowlike disks surrounding small impact craters on the Venusian surface, which are also found without craters. This paper examines the hypothesis of Phillips et al. (1991) that the dark shadows formed where shock waves from airbursting impactors met the ground, pulverizing decimeter-scale surface features. A simple analytic model is used successfully to simulate the catastrophic disruption and the deceleration of impactors in a thick atmosphere to reproduce observed Venusian cratering statistics and generate radar-dark disks by the impact of atmospheric shock waves on the surface.

Airburst origin of dark shadows on Venus

Using the all-sky ROSAT soft X-ray and 408-MHz radio continuum data, we show that the North Polar Spur and its western and southern counter-spurs draw a giant dumbbell-shape necked at the galactic plane. We interpret these features as due to a shock front originating from a starburst 15 million years ago with a total energy of the order of $\sim 10^{56}$ ergs or $10^5$ type II supernovae. We simulate all-sky distributions of radio continuum and soft X-ray intensities based on the bipolar-hyper-shell galactic center starburst model. The simulations can well reproduce the radio NPS and related spurs, as well as radio spurs in the tangential directions of spiral arms. Simulated X-ray maps in 0.25, 0.75 and 1.5 keV bands reproduce the ROSAT X-ray NPS, its western and southern counter-spurs, and the absorption layer along the galactic plane. We propose to use the ROSAT all-sky maps to probe the physics of gas in the halo-intergalactic interface, and to directly date and measure the energy of a recent Galactic Center starburst.

Bipolar-Hyper-Shell Galactic Center Statrburst Model: Further Evidence from ROSAT Data and New Radio and X-ray Simulations

The problem considered is that of a strong shock propagating from a point energy source into a cold atmosphere whose density varies exponentially with altitude. An explicit analytic solution is obtained by taking the flow field as ‘locally radial’ and using an integral method with an energy constraint. A scaling law is given which eliminates the parametric dependence of the solution on the explosion energy, scale height, and atmospheric density at the point of the explosion. The scaling law also transforms the infinity of solutions for various polar angles into two distinct solutions which show that all motions of the ascending portion of the shock may be scaled from the vertically upward behaviour and all motions of the descending portion of the shock may be scaled from the vertically downward behaviour. The limit in the lateral direction of both of the fundamental solutions corresponds to the case of the uniform density atmosphere. The results for the uniform density atmosphere show remarkable agreement with the exact Taylor—Sedov results. Comparison with finite difference calculations of Troutman & Davis for the vertically upward and downward directions shows excellent agreement with respect to the prediction of shock propagation velocity, position, and the flow variables behind the shock. A scaling law for the time, shock velocity, and pressure for different values of the adiabatic exponent γ is proposed which correlates the results of the present analysis for different values of γ over the entire range of shock positions where the analysis applies. The solution shows that, contrary to the result obtained by Kompaneets, there is no theoretical limit as to how far downward a strong shock may propagate. The far field behaviour of the shock wave in the upward and downward directions is found to be of the same form as the self-similar asymptotic solutions obtained by Raizer for a plane shock. It is shown by relaxing the energy constraint in the vertically downward direction that the asymptotic result obtained agrees closely with that obtained by Raizer. The energy constraint, however, is the appropriate one for all but the far field behaviour. The far field limit of the present solution in the upward direction is found to compare favourably with the approximate asymptotic calculations of Hayes for an ascending curved shock. The empirical concept of ‘modified Sachs scaling’ for calculating the overpressure is considered and shown within this model to have a justification in the downward direction but a limited range of applicability in the upward direction.

A point explosion in a cold exponential atmosphere

The self‐similar one‐dimensional flow behind a plane shock propagating upward into an exponentially decreasing atmosphere is considered. The flow is taken to be isothermal in view of the large radiation mean free paths associated with high altitudes and the intense radiation heat transfer accompanying the high temperatures characteristic of an accelerating shock wave. The equations of motion are formulated in Lagrangian coordinates and are integrated exactly for all values of the shock density ratio. Solutions are presented for the cases where the boundary conditions at the shock correspond to a Hugoniot shock and to a Chapman‐Jouguet shock. A significant result of the analysis is that in both of these cases the shock propagates much faster than for the case of adiabatic flow.

Self-Similar Strong Shocks with Radiation in a Decreasing Exponential Atmosphere

: The problem considered is that of a strong shock propagating from a point energy source into a cold exponential atmosphere with radiative heat transfer in the flow behind the shock. The radiation mean free path is taken to be small compared to the shock radius so that the shock may be treated as discontinuous and the radiative heat flux represented by the Rosseland diffusion approximation. The solution obtained is an approximate one based upon the 'local radiality' assumption and integral method previously utilized by the authors for the case of adiabatic flow. A radiation parameter is developed which provides an index as to when the effects of radiation may be neglected and the flow taken to be adiabatic. The physical interpretation of this parameter is that of the ratio of a characteristic radiation energy flux to a characteristic kinetic energy flux. It is shown that when the radiation mean free path varies as a power of the temperature (T to the nth power), where n = - 17/6, the infinity of solutions for various polar angles can be transformed into two distinct solutions thereby essentially eliminating the parametric dependence on the polar angle and the atmospheric scale height. (Author)

Dallas D. Laumbach

Papers

A point explosion in a cold exponential atmosphere

Self-Similar Strong Shocks with Radiation in a Decreasing Exponential Atmosphere

A point explosion in a cold exponential atmosphere. Part 2. Radiating flow