Instability of a stalled accretion shock: evidence for the advective-acoustic cycle

We analyze the linear stability of a stalled accretion shock in a perfect gas with a parametrized cooling function L ~ rho^{beta-alpha} P^alpha. The instability is dominated by the l=1 mode if the shock radius exceeds 2-3 times the accretor radius, depending on the parameters of the cooling function. The growth rate and oscillation period are comparable to those observed in the numerical simulations of Blondin & Mezzacappa (2006). The instability mechanism is analyzed by separately measuring the efficiencies of the purely acoustic cycle and the advective-acoustic cycle. These efficiencies are estimated directly from the eigenspectrum, and also through a WKB analysis in the high frequency limit. Both methods prove that the advective-acoustic cycle is unstable, and that the purely acoustic cycle is stable. Extrapolating these results to low frequency leads us to interpret the dominant mode as an advective-acoustic instability, different from the purely acoustic interpretation of Blondin & Mezzacappa (2006). A simplified characterization of the instability is proposed, based on an advective-acoustic cycle between the shock and the radius r_nabla where the velocity gradients of the stationary flow are strongest. The importance of the coupling region in this mechanism calls for a better understanding of the conditions for an efficient advective-acoustic coupling in a decelerated, nonadiabatic flow, in order to extend these results to core-collapse supernovae.Comment: 29 pages, 18 figures, to appear in ApJ (1 new Section, 2 new Figures

Global Anisotropies in Supernova Explosions and Pulsar Recoil

We show by two-dimensional and first three-dimensional simulations of neutrino-driven supernova explosions that low (l=1,2) modes can dominate the flow pattern in the convective postshock region on timescales of hundreds of milliseconds after core bounce. This can lead to large global anisotropy of the supernova explosion and pulsar kicks in excess of 500 km/s.Comment: 3 pages, 2 figures, contribution to Procs. 12th Workshop on Nuclear Astrophysics, Ringberg Castle, March 22-27, 200

Nucleosynthesis-relevant conditions in neutrino-driven supernova outflows: I. Spherically symmetric hydrodynamic simulations

We investigate the behavior and consequences of the reverse shock that terminates the supersonic expansion of the baryonic wind which is driven by neutrino heating off the surface of (non-magnetized) new-born neutron stars in supernova cores. To this end we perform long-time hydrodynamic simulations in spherical symmetry. In agreement with previous relativistic wind studies, we find that the neutrino-driven outflow accelerates to supersonic velocities and in case of a compact, about 1.4 solar mass (gravitational mass) neutron star with a radius of about 10 km, the wind reaches entropies of about 100 k_B per nucleon. The wind, however, is strongly influenced by the environment of the supernova core. It is decelerated and shock-heated abruptly by a termination shock that forms when the supersonic outflow collides with the slower preceding supernova ejecta. The radial position of this reverse shock varies with time and depends on the strength of the neutrino wind and the different conditions in progenitor stars with different masses and structure. Its basic properties and behavior can be understood by simple analytic considerations. We demonstrate that the entropy of matter going through the reverse shock can increase to a multiple of the asymptotic wind value. Seconds after the onset of the explosion it therefore can exceed 400 k_B per nucleon. The temperature of the shocked wind has typically dropped to about or less than 10^9 K, and density and temperature in the shock-decelerated matter continue to decrease only very slowly. Such conditions might strongly affect the important phases of supernova nucleosynthesis in a time and progenitor dependent way. (abridged

Multidimensional supernova simulations with approximative neutrino transport. II. Convection and the advective-acoustic cycle in the supernova core

By 2D hydrodynamic simulations including a detailed equation of state and neutrino transport, we investigate the interplay between different non-radial hydrodynamic instabilities that play a role during the postbounce accretion phase of collapsing stellar cores. The convective mode of instability, which is driven by negative entropy gradients caused by neutrino heating or by time variations of the shock strength, can be identified clearly by the development of typical Rayleigh-Taylor mushrooms. However, in cases where the gas in the postshock region is rapidly advected towards the gain radius, the growth of such a buoyancy instability can be suppressed. In such a situation the shocked flow nevertheless can develop non-radial asymmetry with an oscillatory growth of the amplitude. This phenomenon has been termed ``standing accretion shock instability'' (SASI). It is shown here that the SASI oscillations can trigger convective instability and like the latter they lead to an increase of the average shock radius and of the mass in the gain layer. Both hydrodynamic instabilities in combination stretch the advection time of matter through the neutrino-heating layer and thus enhance the neutrino energy deposition in support of the neutrino-driven explosion mechanism. A rapidly contracting and more compact nascent NS turns out to be favorable for explosions, because the accretion luminosity and neutrino heating are larger and the growth rate of the SASI is higher. Moreover, we show that the oscillation period of the SASI and a variety of other features in our simulations agree with estimates for the advective-acoustic cycle (AAC), in which perturbations are carried by the accretion flow from the shock to the neutron star and pressure waves close an amplifying global feedback loop. (abridged)Comment: 23 pages, 20 figures; revised version with extended Sect.5, accepted by Astronomy & Astrophysics; high-resolution images can be obtained upon reques

Supernova Asymmetries and Pulsar Kicks -- Views on Controversial Issues

Two- and three-dimensional simulations demonstrate that hydrodynamic instabilities can lead to low-mode (l=1,2) asymmetries of the fluid flow in the neutrino-heated layer behind the supernova shock. This provides a natural explanation for aspherical mass ejection and for pulsar recoil velocities even in excess of 1000 km/s. We propose that the bimodality of the pulsar velocity distribution might be a consequence of a dominant l=1 mode in case of the fast component, while higher-mode anisotropy characterizes the postshock flow and SN ejecta during the birth of the slow neutron stars. We argue that the observed large asymmetries of supernovae and the measured high velocities of young pulsars therefore do not imply rapid rotation of the iron core of the progenitor star, nor do they require strong magnetic fields to play a crucial role in the explosion. Anisotropic neutrino emission from accretion contributes to the neutron star acceleration on a minor level, and pulsar kicks do not make a good case for non-standard neutrino physics in the nascent neutron star.Comment: 10 pages, 5 figures, full resolution figures available on request or from Preprint P-MPA1651e on MPA web page. In: The Fate of the Most Massive Stars, Proc. Eta Carinae Science Symposium (Jackson Hole, May 2004); revision discusses new Cas A observation

Neutrino signatures of supernova shock and reverse shock propagation

A few seconds after bounce in a core-collapse supernova, the shock wave passes the density region corresponding to resonant neutrino oscillations with the ``atmospheric'' neutrino mass difference. The transient violation of the adiabaticity condition manifests itself in an observable modulation of the neutrino signal from a future galactic supernova. In addition to the shock wave propagation effects that were previously studied, a reverse shock forms when the supersonically expanding neutrino-driven wind collides with the slower earlier supernova ejecta. This implies that for some period the neutrinos pass two subsequent density discontinuities, giving rise to a ``double dip'' feature in the average neutrino energy as a function of time. We study this effect both analytically and numerically and find that it allows one to trace the positions of the forward and reverse shocks. We show that the energy dependent neutrino conversion probabilities allow one to detect oscillations even if the energy spectra of different neutrino flavors are the same as long as the fluxes differ. These features are observable in the \bar\nu_e signal for an inverted and in the \nu_e signal for a normal neutrino mass hierarchy, provided the 13-mixing angle is ``large'' (sin^2\theta_{13}\gg 10^{-5}).Comment: 23 pages, 27 eps figures (high resolution plots are available on request), JCAP style; v2: figure 8 extended, matches published versio

Verband tussen die fluoriedinhoud van drinkwater en die voorkoms van tandfluorose in geselekteerde gebiede in Suid-Afrika : 'n medies-geografiese studie

Text in AfrikaansDie tydruimtelike variasie in die fluoriedinhoud vannatuurlike drinkwater en in die voorkoms van tandfluorose, die verband tussen hierdie veranderlikes asook hulle kovariasie is op 'n hoe resolusievlak in geselekteerde endemiese gebiede ondersoek. Die drie studiegebiede verskil grootliks van mekaar wat sommige fisies- en menslik-geografiese kenmerke betref, maar ondergrondse water was oral die belangrikste bron van drinkwater. Ioonspesifieke analise en laboratoriumtegnieke is gebruik om die fluoriedinhoud van die drinkwater uit 517 bronne te bepaal. Die tande van 3 068 kinders is klinies en fotografies ondersoek. Die erkende Tooth Fluorosis Index en die indeks van Dean is gebruik om die voorkoms van fluorose te beskryf. Grafiese tegnieke, beskrywende statistieke en nieparametriese analise van variansietoetse is gebruik om die voorkoms en tydruimtelike variasiepatrone van die sleutelveranderlikes in elke studiegebied, asook die verskillende ruimtelike eenhede binne die studiegebiede, te beskryf. Die verband tussen die sleutelveranderlikes asook hul intra- en interareale kovariasie is ontleed. Daar is gevind dat die fluoriedinhoud van die ondergrondse water in die Pilanesberg- en Hammanskraal-studiegebied relatief hoog tot hoog is en die in die Vrystaat-studiegebied relatief laag. Ruimtelike veranderlikheid met groot verskille oor klein afstande kenmerk die fluoriedinhoud van ondergrondse water ongeag die gesteentes waaruit dit onttrek word. Middelwaardes bied ontoereikende beskrywings van die fluoriedgehalte van die drinkwater. Die skep van nuwe drinkwaterbronne veroorsaak langtermynvariasie in die fluoriedinhoud van die beskikbare drinkwater. Fluoroseprevalensie in die studiegebiede bet van 62% tot 87% gewissel, met meer ernstige aantasting in die Pilanesberg- en Hammanskraal-gebied as in die Suid-Vrystaat. Beduidende intra-areale ruimtelike variasie is 'n wesenlike kenmerk van die voorkoms van fluorose in al drie studiegebiede. In alle gevalle het die variasiepatroon in die fluoriedinhoud van die drinkwater die in die voorkoms van fluorose slegs gedeeltelik verklaar. Sporadiese en/of periodieke kortstondige blootstelling aan hoe fluoriedkonsentrasies lei tot emstige aantasting ten spyte van die gereelde gebruik van water met 'n laefluoriedinhoud. Ligte fluorose ontwikkel geredelik in assosiasie met <0,5 mgF-/l en matige fluorose in assossiasie met 0,5-0,7 mgF-/l, die optimum konsentrasieinterval vir fluoridasie in Suid-Afrika.The spatiotemporal variation in the fluoride content of natural drinking water and the occurrence of dental fluorosis, the relation between these variables as well as their covariation were investigated at a high resolution level in selected endemic areas. Groundwater was the most important source of drinking water in all three study areas, but they differed markedly in respect of some physical and human geographical characteristics. Ion specific analysis and laboratory techniques were used to determine the fluoride content of the water from 517 sources. The teeth of 3 068 children were examined clinically and photographically. The fluorosis was scored according to the Tooth Fluorosis Index and Dean's classification. Graphical techniques, descriptive statistics and nonparametric analysis of variance were used to describe the occurrence and variation patterns of the key variables in the different spatial units is each study area. The relation between the variables as well as their intra and interareal covariation were analysed. The fluoride content of the groundwater in the Pilanesberg and Hammanskraal areas was relatively high to high; in the southern Free State it was relatively low. Spatial variability and significant differences over small distances typify the fluoride content of the groundwater, irrespective of the aquifer. Central statistics inadequately describe the fluoride quality of the natural drinking water. The development of new water sources causes long term variation in the fluoride content of the available drinking water. Fluorosis prevalence varied form 62% to 87%, with more severe fluorosis in Pilanesberg and Hammanskraal than in the southern Free State. Significant intra-areal spatial variation is an attribute of fluorosis in all three study areas. In all cases the variation pattern in the fluoride content of the drinking water partly explained the spatial pattern in the occurrence of fluorosis. Sporadic and/or periodic brief exposure to high fluoride concentrations leads to severe fluorosis despite regul~ usage of water with a low fluoride content. Mild fluorosis readily develops in association with <0,5 mgF-1~ and medium fluorosis in association with 0,5-0,7 mgF-/l, the optimum concentration interval for fluoridation in South Africa.GeographyD. Phil. (Geografie

Non-spherical core collapse supernovae III. Evolution towards homology and dependence on the numerical resolution

(abridged) We study the hydrodynamic evolution of a non-spherical core-collapse supernova in two spatial dimensions. We find that our model displays a strong tendency to expand toward the pole. We demonstrate that this expansion is a physical property of the low-mode, SASI instability. The SASI leaves behind a large lateral velocity gradient in the post shock layer which affects the evolution for minutes and hours later. This results in a prolate deformation of the ejecta and a fast advection of Ni-rich material from moderate latitudes to the polar regions. This effect might actually be responsible for the global asymmetry of the nickel lines in SN 1987A. The simulations demonstrate that significant radial and lateral motions in the post-shock region, produced by convective overturn and the SASI during the early explosion phase, contribute to the evolution for minutes and hours after shock revival. They lead to both later clump formation, and a significant prolate deformation of the ejecta which are observed even as late as one week after the explosion. As pointed out recently by Kjaer et al., such an ejecta morphology is in good agreement with the observational data of SN 1987A. Systematic future studies are needed to investigate how the SASI-induced late-time lateral expansion depends on the dominant mode of the SASI, and to which extent it is affected by the dimensionality of the simulations. The impact on and importance of the SASI for the distribution of iron group nuclei and the morphology of the young SNR argues for future three-dimensional explosion and post-explosion studies on singularity-free grids that cover the entire sphere. Given the results of our 2D resolution study, present 3D simulations must be regarded as underresolved, and their conclusions must be verified by a proper numerical convergence analysis in three dimensions.Comment: 16 pages, 20 figures, accepted for publication in Astronomy & Astrophysic

Equation-of-State Dependent Features in Shock-Oscillation Modulated Neutrino and Gravitational-Wave Signals from Supernovae

We present 2D hydrodynamic simulations of the long-time accretion phase of a 15 solar mass star after core bounce and before the launch of a supernova explosion. Our simulations are performed with the Prometheus-Vertex code, employing multi-flavor, energy-dependent neutrino transport and an effective relativistic gravitational potential. Testing the influence of a stiff and a soft equation of state for hot neutron star matter, we find that the non-radial mass motions in the supernova core due to the standing accretion shock instability (SASI) and convection impose a time variability on the neutrino and gravitational-wave signals. These variations have larger amplitudes as well as higher frequencies in the case of a more compact nascent neutron star. After the prompt shock-breakout burst of electron neutrinos, a more compact accreting remnant radiates neutrinos with higher luminosities and larger mean energies. The observable neutrino emission in the direction of SASI shock oscillations exhibits a modulation of several 10% in the luminosities and ~1 MeV in the mean energies with most power at typical SASI frequencies of 20-100 Hz. At times later than 50-100 ms after bounce the gravitational-wave amplitude is dominated by the growing low-frequency (<200 Hz) signal associated with anisotropic neutrino emission. A high-frequency wave signal is caused by nonradial gas flows in the outer neutron star layers, which are stirred by anisotropic accretion from the SASI and convective regions. The gravitational-wave power then peaks at about 300-800 Hz with distinctively higher spectral frequencies originating from the more compact and more rapidly contracting neutron star. The detectability of the SASI effects in the neutrino and gravitational-wave signals is briefly discussed. (abridged)Comment: 21 pages, 11 figures, 45 eps files; revised version including discussion of signal detectability; accepted by Astronomy & Astrophysics; high-resolution images can be obtained upon reques

Integrability and chaos: the classical uncertainty

In recent years there has been a considerable increase in the publishing of textbooks and monographs covering what was formerly known as random or irregular deterministic motion, now named by the more fashionable term of deterministic chaos. There is still substantial interest in a matter that is included in many graduate and even undergraduate courses on classical mechanics. Based on the Hamiltonian formalism, the main objective of this article is to provide, from the physicist's point of view, an overall and intuitive review of this broad subject (with some emphasis on the KAM theorem and the stability of planetary motions) which may be useful to both students and instructors.Comment: 24 pages, 10 figure