Using the Boltzmann-radiation-hydrodynamics code, which solves the Boltzmann equation for neutrino transport, we present the results of the simulations with the nuclear equations of state (EOSs) of Lattimer and Swesty (LS) and Furusawa and Shen (FS). We extend the simulation time of the LS model and conduct thorough investigations, though our previous paper briefly reported some of the results. Only the LS model shows the shock revival. This seems to originate from the nuclear composition: the different nuclear composition results in the different energy loss by photodissociation and hence the different strength of the prompt convection and the later neutrino-driven convection. The protoneutron star seen in the FS model is more compact than that in the LS model because the existence of multinuclear species softens the EOS. For the behavior of neutrinos, we examined the flux and the Eddington tensor of neutrinos. In the optically thick region, the diffusion of neutrinos and the dragging by the motion of matter determine the flux. In the optically thin region, the free-streaming determines it. The Eddington tensor is compared with that obtained from the M1-closure relation. The M1-closure scheme overestimates the contribution from the velocity-dependent terms in the semitransparent region.

Neutrinos are believed to have a key role in the explosion mechanism of core-collapse supernovae as they carry most of the energy released by the gravitational collapse of a massive star. If their flavor is converted fast inside the neutrino sphere, the supernova explosion may be influenced. This paper is reporting the results of the extended work of our previous paper. We perform a thorough survey of the electron lepton number (ELN) crossing in one of our self-consistent, realistic Boltzmann simulations in two spatial dimensions under axisymmetry for the existence of the crossings between νe and νe angular distributions, or the ELN crossing. We report for the first time the positive detections deep inside the core of the massive star in the vicinity of neutrino sphere at r≈16-21 km. We find that low values of the electron fraction Ye produced by convective motions together with the appearance of light elements are critically important to give rise to the ELN crossing by enhancing the chemical potential difference between proton and neutron, and hence by mitigating the Fermi-degeneracy of νe. Since the region of positive detection are sustained and, in fact, expanding with time, it may have an impact on the explosion of core-collapse supernovae, observational neutrino astronomy, and nucleosynthesis of heavy nuclei.

Neutrinos are densely populated deep inside the core of massive stars after their gravitational collapse to produce supernova explosions and form compact stars such as neutron stars and black holes. It has been considered that they may change their flavor identities through so-called fast-pairwise conversions induced by mutual forward scatterings. If that is really the case, the dynamics of supernova explosion will be influenced, since the conversion may occur near the neutrino sphere, from which neutrinos are effectively emitted. In this paper, we conduct a pilot study of such possibilities based on the results of fully self-consistent, realistic simulations of a core-collapse supernova explosion in two spatial dimensions under axisymmetry. As we solved the Boltzmann equations for neutrino transfer in the simulation not as a postprocess but in real time, the angular distributions of neutrinos in momentum space for all points in the core at all times are available, a distinct feature of our simulations. We employ some of these distributions extracted at a few selected points and times from the numerical data and apply linear analysis to assess the possibility of the conversion. We focus on the vicinity of the neutrino sphere, where different species of neutrinos move in different directions and have different angular distributions as a result. This is a pilot study for a more thorough survey that will follow soon. We find no positive sign of conversion unfortunately at least for the spatial points and times we studied in this particular model. We hence investigate rather in detail the condition for the conversion by modifying the neutrino distributions rather arbitrarily by hand.

With the Boltzmann-radiation-hydrodynamics code, which we have developed to solve numerically the Boltzmann equations for neutrino transfer, the Newtonian hydrodynamics equations, and the Newtonian self-gravity simultaneously and consistently, we simulate the collapse of a rotating core of the progenitor with a zero-age- main-sequence mass of 11.2 M and a shellular rotation of at the center. We pay particular attention in this paper to the neutrino distribution in phase space, which is affected by the rotation. By solving the Boltzmann equations directly, we can assess the rotation-induced distortion of the angular distribution in momentum space, which gives rise to the rotational component of the neutrino flux. We compare the Eddington tensors calculated both from the raw data and from the M1-closure approximation. We demonstrate that the Eddington tensor is determined by complicated interplays of the fluid velocity and the neutrino interactions and that the M1-closure, which assumes that the Eddington factor is determined by the flux factor, fails to fully capture this aspect, especially in the vicinity of the shock. We find that the error in the Eddington factor reaches ∼20% in our simulation. This is due not to the resolution but to the different dependence of the Eddington and flux factors on the angular profile of the neutrino distribution function, and hence modification to the closure relation is needed.

We present the first results of our spatially axisymmetric core-collapse supernova simulations with full Boltzmann neutrino transport, which amount to a time-dependent five-dimensional (two in space and three in momentum space) problem. Special relativistic effects are fully taken into account with a two-energy-grid technique. We performed two simulations for a progenitor of 11.2 M⊙, employing different nuclear equations of state (EOSs): Lattimer and Swesty's EOS with the incompressibility of K =220 MeV (LS EOS) and Furusawa's EOS based on the relativistic mean field theory with the TM1 parameter set (FS EOS). In the LS EOS, the shock wave reaches ∼700 km at 300 ms after bounce and is still expanding, whereas in the FS EOS it stalled at ∼200 km and has started to recede by the same time. This seems to be due to more vigorous turbulent motions in the former during the entire postbounce phase, which leads to higher neutrino-heating efficiency in the neutrino-driven convection. We also look into the neutrino distributions in momentum space, which is the advantage of the Boltzmann transport over other approximate methods. We find nonaxisymmetric angular distributions with respect to the local radial direction, which also generate off-diagonal components of the Eddington tensor. We find that the rθ component reaches ∼10% of the dominant rr component and, more importantly, it dictates the evolution of lateral neutrino fluxes, dominating over the θθ component, in the semitransparent region. These data will be useful to further test and possibly improve the prescriptions used in the approximate methods.

We present a newly developed moving-mesh technique for the multi-dimensional Boltzmann-Hydro code for the simulation of core-collapse supernovae (CCSNe). What makes this technique different from others is the fact that it treats not only hydrodynamics but also neutrino transfer in the language of the 3 + 1 formalism of general relativity (GR), making use of the shift vector to specify the time evolution of the coordinate system. This means that the transport part of our code is essentially general relativistic, although in this paper it is applied only to the moving curvilinear coordinates in the flat Minknowski spacetime, since the gravity part is still Newtonian. The numerical aspect of the implementation is also described in detail. Employing the axisymmetric two-dimensional version of the code, we conduct two test computations: oscillations and runaways of proto-neutron star (PNS). We show that our new method works fine, tracking the motions of PNS correctly. We believe that this is a major advancement toward the realistic simulation of CCSNe.

We investigate the criterion for the acoustic mechanism to work successfully in core-collapse supernovae. The acoustic mechanism is an alternative to the neutrino-heating mechanism. It was proposed by Burrows et al., who claimed that acoustic waves emitted by g-mode oscillations in proto-neutron stars (PNS) energize a stalled shock wave and eventually induce an explosion. Previous works mainly studied to which extent the g-modes are excited in the PNS. In this paper, on the other hand, we investigate how strong the acoustic wave needs to be if it were to revive a stalled shock wave. By adding the acoustic power as a new axis, we draw a critical surface, which is an extension of the critical curve commonly employed in the context of neutrino heating. We perform both 1D and 2D parametrized simulations, in which we inject acoustic waves from the inner boundary. In order to quantify the power of acoustic waves, we use the extended Myers theory to take neutrino reactions into proper account. We find for the 1D simulations that rather large acoustic powers are required to relaunch the shock wave, since the additional heating provided by the secondary shocks developed from acoustic waves is partially canceled by the neutrino cooling that is also enhanced. In 2D, the required acoustic powers are consistent with those of Burrows et al. Our results seem to imply, however, that it is the sum of neutrino heating and acoustic powers that matters for shock revival.

The explosion mechanism of core-collapse supernovae (CCSNe) has not been fully understood yet, but multidimensional fluid instabilities such as standing accretion shock instability and convection are now believed to be crucial for shock revival. Another multidimensional effect that has been recently argued is the asymmetric structures in progenitors, which are induced by violent convections in silicon/oxygen layers that occur before the onset of collapse, as revealed by recent numerical simulations of the last stage of massive star evolutions. Furthermore, it has been also demonstrated numerically that accretions of such nonspherical envelopes could facilitate shock revival. These two multidimensional effects may hence hold a key to successful explosions. In this paper, we performed a linear stability analysis of the standing accretion shock in CCSNe, taking into account nonspherical, unsteady accretion flows onto the shock to clarify the possible links between the two effects. We found that such preshock perturbations can excite the fluid instabilities efficiently and hence help the shock revive in CCSNe.

The volume penalization (VP) method for compressible flows is investigated as a tool of direct numerical simulation of aeroacoustic sound in problems where not only acoustic pressure but also hydrodynamic pressure depends on time and position. First, it is shown that the method proposed by Liu and Vasilyev (2007) [30] is not Galilean invariant. It is corrected to satisfy Galilean invariance. Next, numerical accuracy of the corrected VP method is investigated in problems of simple geometry which can be simulated also by a standard method on a body-fitted coordinate system: sound generation in (i) flow past a fixed square/circular cylinder, (ii) flow past an oscillating square/circular cylinder, and (iii) flow past two square cylinders. The results confirm that the corrected VP method gives reasonably accurate results for sound pressure which is much smaller than hydrodynamic pressure within 5% error. Finally, the corrected method is applied to two examples of complex geometry, which cannot be simulated by standard methods using body-fitted coordinate systems without considerable difficulty: sound generation in (i) flow past an oscillating cylinder and a fixed cylinder behind it and (ii) flow past a bundle of cylinders. The results show that the present method is in principle applicable to aeroacoustic problems in any complex geometry including practical engineering ones.

A new approach for reducing error of the volume penalization method is proposed. The mask function is modified by shifting the interface between solid and fluid by √νη toward the fluid region, where ν and η are the viscosity and the permeability, respectively. The shift length √νη is derived from the analytical solution of the one-dimensional diffusion equation with a penalization term. The effect of the error reduction is verified numerically for the one-dimensional diffusion equation, Burgers' equation, and the two-dimensional Navier-Stokes equations. The results show that the numerical error is reduced except in the vicinity of the interface showing overall second-order accuracy, while it converges to a non-zero constant value as the number of grid points increases for the original mask function. However, the new approach is effectivewhen the grid resolution is sufficiently high so that the boundary layer, whose width is proportional to √νη, is resolved. Hence, the approach should be used when an appropriate combination of ν and η is chosen with a given numerical grid.

The effect of rotation on the explosion of core-collapse supernovae is investigated systematically in three-dimensional simulations. In order to obtain the critical conditions for explosion as a function of mass accretion rate, neutrino luminosity, and specific angular momentum, rigidly rotating matter was injected from the outer boundary with an angular momentum, which is increased every 500 ms. It is found that there is a critical value of the specific angular momentum, above which the standing shock wave revives, for a given combination of mass accretion rate and neutrino luminosity, i.e., an explosion can occur by rotation even if the neutrino luminosity is lower than the critical value for a given mass accretion rate in non-rotational models. The coupling of rotation and hydrodynamical instabilities plays an important role in characterizing the dynamics of shock revival for the range of specific angular momentum that are supposed to be realistic. Contrary to expectations from past studies, the most rapidly expanding direction of the shock wave is not aligned with the rotation axis. Being perpendicular to the rotation axis on average, it can be oriented in various directions. Its dispersion is small when the spiral mode of the standing accretion shock instability (SASI) governs the dynamics, while it is large when neutrino-driven convection is dominant. As a result of the comparison between two-dimensional and three-dimensional rotational models, it is found that m ≠ 0 modes of neutrino-driven convection or SASI are important for shock revival around the critical surface.

In this study, we conduct three-dimensional hydrodynamic simulations systematically to investigate the flow patterns behind the accretion shock waves that are commonly formed in the post-bounce phase of core-collapse supernovae. Adding small perturbations to spherically symmetric, steady, shocked accretion flows, we compute the subsequent evolutions to find what flow pattern emerges as a consequence of hydrodynamical instabilities such as convection and standing accretion shock instability for different neutrino luminosities and mass accretion rates. Depending on these two controlling parameters, various flow patterns are indeed realized. We classify them into three basic patterns and two intermediate ones; the former includes sloshing motion (SL), spiral motion (SP), and multiple buoyant bubble formation (BB); the latter consists of spiral motion with buoyant-bubble formation (SPB) and spiral motion with pulsationally changing rotational velocities (SPP). Although the post-shock flow is highly chaotic, there is a clear trend in the pattern realization. The sloshing and spiral motions tend to be dominant for high accretion rates and low neutrino luminosities, and multiple buoyant bubbles prevail for low accretion rates and high neutrino luminosities. It is interesting that the dominant pattern is not always identical between the semi-nonlinear and nonlinear phases near the critical luminosity; the intermediate cases are realized in the latter case. Running several simulations with different random perturbations, we confirm that the realization of flow pattern is robust in most cases.

Core-collapse supernovae are dramatic explosions marking the catastrophic end of massive stars. The only means to get direct information about the supernova engine is from observations of neutrinos emitted by the forming neutron star, and through gravitational waves which are produced when the hydrodynamic flow or the neutrino flux is not perfectly spherically symmetric. The multidimensionality of the supernova engine, which breaks the sphericity of the central core such as convection, rotation, magnetic fields, and hydrodynamic instabilities of the supernova shock, is attracting great attention as the most important ingredient to understand the long-veiled explosion mechanism. Based on our recent work, we summarize properties of gravitational waves, neutrinos, and explosive nucleosynthesis obtained in a series of our multidimensional hydrodynamic simulations and discuss how the mystery of the central engines can be unraveled by deciphering these multimessengers produced under the thick veils of massive stars. © 2012 Kei Kotake et al.

By performing three-dimensional (3D) simulations that demonstrate the neutrino-driven core-collapse supernovae aided by the standing accretion shock instability (SASI), we study how the spiral modes of the SASI can impact the properties of the gravitational-wave (GW) emission. To see the effects of rotation in the nonlinear postbounce phase, we give a uniform rotation on the flow advecting from the outer boundary of the iron core, the specific angular momentum of which is assumed to agree with recent stellar evolution models. We compute fifteen 3D models in which the initial angular momentum and the input neutrino luminosities from the protoneutron star are changed in a systematic manner. By performing a ray-tracing analysis, we accurately estimate the GW amplitudes generated by anisotropic neutrino emission. Our results show that the gravitational waveforms from neutrinos in models that include rotation exhibit a common feature; otherwise, they vary much more stochastically in the absence of rotation. The breaking of the stochasticity stems from the excess of the neutrino emission parallel to the spin axis. This is because the compression of matter is more enhanced in the vicinity of the equatorial plane due to the growth of the spiral SASI modes, leading to the formation of the spiral flows circulating around the spin axis with higher temperatures. We point out that recently proposed future space interferometers like Fabry-Perot-type DECIGO would permit the detection of these signals for a Galactic supernova. © 2011. The American Astronomical Society. All rights reserved.

The study of standing accretion shock instability (SASI) in core-collapse supernova cores has been done with three-dimensional (3D) computer simulations. Rotations with various perturbations were introduced from outer boundary of an initial steady accreting flow. We found that one or two armed spiral accreting flow onto the proto-neutron star (PNS) is formed inside the shock wave depending on perturbations. The linear growth of spiral modes are clearly diagnosed by the mode analysis of the shock surface, and the lower m modes grow quickly in the linear regime. © 2008 Springer Science+Business Media B.V.

We study the effects of rotation on standing accretion shock instability (SASI) by performing three-dimensional hydrodynamics simulations. Taking into account a realistic equation of state and neutrino heating/cooling, we prepare a spherically symmetric and steady accretion flow through a standing shock wave onto a proto-neutron star (PNS). When the SASI enters the nonlinear phase, we impose uniform rotation on the flow advecting from the outer boundary of the iron core, whose specific angular momentum is assumed to agree with recent stellar evolution models. Using spherical harmonics in space and Fourier decompositions in time, we perform mode analysis of the nonspherical deformed shock wave to observe rotational effects on the SASI in the nonlinear phase. We find that rotation imposed on the axisymmetric flowdoes not make any spiralmodes and hardly affects sloshingmodes, except for steady l = 2,m = 0 modes. In contrast, rotation imposed on the nonaxisymmetric flow increases the amplitude of spiral modes so that some spiral flows accreting on the PNS are more clearly formed inside the shock wave than without rotation. The amplitudes of spiral modes increase significantly with rotation in the progressive direction.

We study the properties of gravitational waves (GWs) based on three-dimensional (3D) simulations, which demonstrate neutrino-driven explosions aided by standing accretion shock instability (SASI). Pushed by evidence supporting slow rotation prior to core collapse, we focus on the asphericities in neutrino emissions and matter motions outside the protoneutron star. By performing a ray-tracing calculation in 3D, we estimate accurately the gravitational waveforms from anisotropic neutrino emissions. In contrast to the previous work assuming axisymmetry, we find that the gravitational waveforms vary much more stochastically because the explosion anisotropies depend sensitively on the growth of SASI which develops chaotically in all directions. Our results show that the GW spectrum has its peak near 100 Hz, reflecting SASI-induced matter overturns of O(10) ms. We point out that the detection of such signals, possibly visible to the LIGO-class detectors for a Galactic supernova, could be an important probe into the long-veiled explosion mechanism. © 2009. The American Astronomical Society. All rights reserved.

We propose a ray-tracing method to estimate gravitational waves (GWs) generated by anisotropic neutrino emission in supernova cores. To calculate the gravitational waveforms, we derive analytic formulae in a useful form, which are applicable also for three-dimensional computations. Pushed by evidence of slow rotation prior to core-collapse, we focus on asphericities in neutrino emission and matter motions outside the protoneutron star. Based on the two-dimensional models, which mimic standing accretion shock instability (SASI)-aided neutrino heating explosions, we compute the neutrino anisotropies via the ray-tracing method in a post-processing manner and calculate the resulting waveforms. For simplicity, neutrino absorption and emission by free nucleons, dominant processes outside the protoneutron stars, are only taken into account, while the neutrino scattering and the velocity-dependent terms in the transport equations are neglected. With these computations, it is found that the waveforms exhibit more variety in contrast to the ones previously estimated by the ray-by-ray analysis. In addition to a positively growing feature, which was predicted to determine the total wave amplitudes predominantly, the waveforms are shown to exhibit large negative growth for some epochs during the growth of SASI. These features are found to stem from the excess of neutrino emission in lateral directions, which can be precisely captured by the ray-tracing calculation. Reflecting the nature of SASI which grows chaotically with time, there is little systematic dependence of the input neutrino luminosities on the maximum wave amplitudes. Due to the negative contributions and the neutrino absorptions appropriately taken into account by the ray-tracing method, the wave amplitudes become more than one order of magnitude smaller than the previous estimation, thus making their detections very hard for a Galactic source. On the other hand, it is pointed out that the GW spectrum from matter motions have its peak near 100 Hz, reflecting the SASI-induced matter overturns of O(10) ms. Such a feature could be characteristic for the SASI-induced supernova explosions. The proposed ray-tracing method will be useful for the GW prediction in the first generation of three-dimensional core-collapse supernova simulations that do not solve the angle-dependent neutrino transport equations as part of the numerical evolution. © 2009. The American Astronomical Society. All rights reserved.

We studied the standing accretion shock instability (SASI) for a core-collapse supernova explosion. SASI induces a nonspherically symmetric motion of a standing spherical shock wave. In order to investigate the growth of SASI, we solved the three-dimensional compressible Euler equations using ZEUS-MP/2 code based on the finite-difference method with a staggered mesh of spherical polar geometry. Although the von Neumann and Richtmyer artificial viscosity is used in ZEUS-MP/2 code to capture shock waves, we propose utilizing a tensor artificial viscosity in order to overcome the numerical instability regarded as the carbuncle phenomenon. This numerical instability emerges around the grid polar axis and precludes mode analysis of SASI.

Standing accretion shock instability (SASI) is one of the candidates to solve the mystery of why we cannot reproduce the explosion with the present core-collapse supernova models. We have studied this phenomenon with including neutrino heating and realistic EOS and found that SASI may enhance neutrino heating. Although g-mode of proto-neutron star may enhance the SASI growth, the simulations just including the pressure perturbation as a mimic of g-mode induced sound wave reveal no significant effect on the shock dynamics. Moreover, we discuss the required conditions toward the possible laboratory experiment of SASI.

We have studied nonaxisymmetric standing accretion shock instabilities, or SASI, using three-dimensional (3D) hydrodynamical simulations. This is an extension of our previous study of axisymmetric SASI. We have prepared a spherically symmetric and steady accretion flow through a standing shock wave onto a proto-neutron star, taking into account a realistic equation of state and neutrino heating and cooling. This unperturbed model is meant to represent approximately the typical postbounce phase of core-collapse supernovae. We then added a small perturbation (∼1%) to the radial velocity and computed the ensuing evolutions. Both axisymmetric and nonaxisymmetric perturbations have been imposed. We have applied mode analysis to the nonspherical deformation of the shock surface, using spherical harmonics. We have found that (1) the growth rates of SASI are degenerate with respect to the azimuthal index m of the spherical harmonics Ylm, just as expected for a spherically symmetric background; (2) nonlinear mode couplings produce only m = 0 modes for axisymmetric perturbations, whereas m ≠= 0 modes are also generated in the nonaxisymmetric cases, according to the selection rule for quadratic couplings; (3) the nonlinear saturation level of each mode is lower in general for 3D than for 2D, because a larger number of modes contribute to turbulence in 3D; (4) low-l modes are dominant in the nonlinear phase; (5) equipartition is nearly established among different m modes in the nonlinear phase; (6) spectra with respect to l obey power laws with a slope slightly steeper for 3D; and (7) although these features are common to the models with and without a shock revival at the end of the simulation, the dominance of low-l modes is more remarkable in the models with a shock revival. © 2008. The American Astronomical Society. All rights reserved.

The sound generated by two square cylinders placed in a side-by-side arrangement in a uniform flow at low Mach numbers is studied by direct solution of the two-dimensional unsteady compressible Navier-Stokes equations. Special attention is paid to the effect of the spacing between the two cylinders on the generation mechanism of the sound. Results show that sound pressure fields as well as flow fields show different features depending on the spacing; six different wake patterns (nonsynchronized, antiphase and in-phase synchronized, flip-flopping, single bluff-body, and steady patterns) are observed and, depending on the wake pattern, sound pressure fields also show different features. Two types of bifurcation phenomenon are observed where two different wake patterns exist for the same spacing and either of the two patterns appears depending on an initial condition; different sound pressure fields appear in response to the wake patterns. Results also show that Curle's dipole solution does not predict well the direct numerical simulation results in the antiphase-synchronized pattern at a small spacing and the flip-flopping pattern, but that it gives a good approximation in the nonsynchronized, in-phase-synchronized, and single-bluff-body patterns. © 2006 American Institute of Physics.