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

A simple toy model of the advective-acoustic instability I. Perturbative approach

Some general properties of the advective-acoustic instability are described and understood using a toy model which is simple enough to allow for analytical estimates of the eigenfrequencies. The essential ingredients of this model, in the unperturbed regime, are a stationary shock and a subsonic region of deceleration. For the sake of analytical simplicity, the 2D unperturbed flow is parallel and the deceleration is produced adiabatically by an external potential. The instability mechanism is determined unambiguously as the consequence of a cycle between advected and acoustic perturbations. The purely acoustic cycle, considered alone, is proven to be stable in this flow. Its contribution to the instability can be either constructive or destructive. A frequency cut-off is associated to the advection time through the region of deceleration. This cut-off frequency explains why the instability favours eigenmodes with a low frequency and a large horizontal wavelength. The relation between the instability occurring in this highly simplified toy model and the properties of SASI observed in the numerical simulations of stellar core-collapse is discussed. This simple set up is proposed as a benchmark test to evaluate the accuracy, in the linear regime, of numerical simulations involving this instability. We illustrate such benchmark simulations in a companion paper.Comment: 14 pages, 10 figures, ApJ in pres

Entropic-acoustic instability of shocked Bondi accretion I. What does perturbed Bondi accretion sound like ?

In the radial flow of gas into a black hole (i.e. Bondi accretion), the infall of any entropy or vorticity perturbation produces acoustic waves propagating outward. The dependence of this acoustic flux on the shape of the perturbation is investigated in detail. This is the key process in the mechanism of the entropic-acoustic instability proposed by Foglizzo & Tagger (2000) to explain the instability of Bondi-Hoyle-Lyttleton accretion. These acoustic waves create new entropy and vorticity perturbations when they reach the shock, thus closing the entropic-acoustic cycle. With an adiabatic index 1<gamma<=5/3, the linearized equations describing the perturbations of the Bondi flow are studied analytically and solved numerically. The fundamental frequency of this problem is the cut-off frequency of acoustic refraction, below which ingoing acoustic waves are refracted out. This cut-off is significantly smaller than the Keplerian frequency at the sonic radius and depends on the latitudinal number l of the perturbations. When advected adiabatically inward, entropy and vorticity perturbations trigger acoustic waves propagating outward, with an efficiency which is highest for non radial perturbations l=1. The outgoing acoustic flux produced by the advection of vorticity perturbations is always moderate and peaks at rather low frequency. By contrast, the acoustic flux produced by an entropy wave is highest close to the refraction cut-off. It can be very large if gamma is close to 5/3. These results suggest that the shocked Bondi flow with gamma=5/3 is strongly unstable with respect to the entropic-acoustic mechanism.Comment: 14 pages, 11 figures, accepted for publication in A&

Criteria for Core-Collapse Supernova Explosions by the Neutrino Mechanism

We investigate the criteria for successful core-collapse supernova explosions by the neutrino mechanism. We find that a critical-luminosity/mass-accretion-rate condition distinguishes non-exploding from exploding models in hydrodynamic one-dimensional (1D) and two-dimensional (2D) simulations. We present 95 such simulations that parametrically explore the dependence on neutrino luminosity, mass accretion rate, resolution, and dimensionality. While radial oscillations mediate the transition between 1D accretion (non-exploding) and exploding simulations, the non-radial standing accretion shock instability characterizes 2D simulations. We find that it is useful to compare the average dwell time of matter in the gain region with the corresponding heating timescale, but that tracking the residence time distribution function of tracer particles better describes the complex flows in multi-dimensional simulations. Integral quantities such as the net heating rate, heating efficiency, and mass in the gain region decrease with time in non-exploding models, but for 2D exploding models, increase before, during, and after explosion. At the onset of explosion in 2D, the heating efficiency is $\sim$2% to $\sim$5% and the mass in the gain region is $\sim$0.005 M_{\sun} to $\sim$0.01 M_{\sun}. Importantly, we find that the critical luminosity for explosions in 2D is $\sim$70% of the critical luminosity required in 1D. This result is not sensitive to resolution or whether the 2D computational domain is a quadrant or the full 180$^{\circ}$. We suggest that the relaxation of the explosion condition in going from 1D to 2D (and to, perhaps, 3D) is of a general character and is not limited by the parametric nature of this study.Comment: 32 pages in emulateapj, including 17 figures, accepted for publication in ApJ, included changes suggested by the refere

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

Stability of the Accretion Flows with Stalled Shocks in Core-Collapse Supernovae

Bearing in mind the application to the theory of core-collapse supernovae, we performed a global linear analysis on the stability of spherically symmetric accretion flows through a standing shock wave onto a proto neutron star. As unperturbed flows, we adopted the spherically symmetric steady solutions to the Euler equations obtained with realistic equation of state and formulae for neutrino reaction rates taken into account. Then we solved the equations for linear perturbations numerically, and obtained the eigen frequencies and eigen functions. We found (1) the flows are stable for all modes if the neutrino luminosity is lower than $\sim 1\times 10^{52}$ ergs/s for $\dot{M}=1.0M_{\odot}/{\rm s}$. (2) For larger luminosities, the non-radial instabilities are induced, probably via the advection-acoustic cycles. Interestingly, the modes with $\ell=2$ and 3 become unstable at first for relatively low neutrino luminosities, e.g. $\gtrsim 2-3\times 10^{52}$ ergs/s for the same accretion rate, whereas the $\ell=1$ mode is the most unstable for higher luminosities, $\sim 3-7\times 10^{52}$ ergs/s. These are all oscillatory modes. (3) For still larger luminosities, $\sim 7\times 10^{52}$ ergs/s for $\dot{M}=1.0M_{\odot}/{\rm s}$, non-oscillatory modes, both radial and non-radial, become unstable. These non-radial modes were identified as convection. We confirmed the results obtained by numerical simulations that the instabilities induced by the advection-acoustic cycles are more important than the convection for lower neutrino luminosities.Comment: 46 pages, 19 figures, Accepted by Ap