Gravitational Wave Signatures of Magnetohydrodynamically-Driven Core-Collapse Supernova Explosions
aa r X i v : . [ a s t r o - ph . H E ] A ug Gravitational Wave Signatures ofMagnetohydrodynamically-Driven Core-Collapse SupernovaExplosions
Tomoya Takiwaki and Kei Kotake , Center for Computational Astrophysics, National Astronomical Observatory of Japan,2-21-1, Osawa, Mitaka, Tokyo, 181-8588, Japan Division of Theoretical Astronomy, National Astronomical Observatory of Japan, 2-21-1,Osawa, Mitaka, Tokyo, 181-8588, Japan [email protected],[email protected]
ABSTRACT
By performing a series of two-dimensional, special relativistic magnetohydro-dynamic (MHD) simulations, we study signatures of gravitational waves (GWs)in the magnetohydrodynamically-driven core-collapse supernovae. In order toextract the gravitational waveforms, we present a stress formula including con-tributions both from magnetic fields and special relativistic corrections.By changing the precollapse magnetic fields and initial angular momentumdistributions parametrically, we compute twelve models. As for the microphysics,a realistic equation of state is employed and the neutrino cooling is taken intoaccount via a multiflavor neutrino leakage scheme. With these computations, wefind that the total GW amplitudes show a monotonic increase after bounce formodels with a strong precollapse magnetic field (10 G) also with a rapid rotationimposed. We show that this trend stems both from the kinetic contribution ofMHD outflows with large radial velocities and also from the magnetic contribu-tion dominated by the toroidal magnetic fields that predominantly trigger MHDexplosions. For models with weaker initial magnetic fields, the total GW ampli-tudes after bounce stay almost zero, because the contribution from the magneticfields cancels with the one from the hydrodynamic counterpart. These featurescan be clearly understood with a careful analysis on the explosion dynamics.We point out that the GW signals with the increasing trend, possibly visible tothe next-generation detectors for a Galactic supernova, would be associated withMHD explosions with the explosion energies exceeding 10 erg. Subject headings: supernovae: collapse — gravitational waves — neutrinos —hydrodynamics 2 –
1. Introduction
Successful detection of neutrinos from SN1987A paved the way for
Neutrino Astronomy (Hirata et al. 1987; Bionta et al. 1987), alternative to conventional astronomy by electromag-netic waves. Core-collapse supernovae are now expected to be opening yet another astron-omy,
Gravitational-Wave Astronomy . Currently long-baseline laser interferometers LIGO(Abbott et al. 2005), VIRGO , GEO600 , TAMA300 (Ando & the TAMA Collaboration 2005),and AIGO with their international network of the observatories, are beginning to take dataat sensitivities where astrophysical events are predicted (see, e.g., Hough et al. (2005) for arecent review). For these detectors, core-collapse supernovae have been proposed as one ofthe most plausible sources of gravitational waves (GWs) (see, e.g., Kotake et al. (2006); Ott(2009) for recent reviews).Although the explosion mechanism of core-collapse supernovae has not been completelyclarified yet, current multi-dimensional simulations based on refined numerical models showseveral promising scenarios. Among the candidates is the neutrino heating mechanismaided by convection and standing accretion shock instability (SASI) (e.g., Marek & Janka(2009); Bruenn et al. (2010); Scheck et al. (2004, 2008); Suwa et al. (2009)), the acousticmechanism (Burrows et al. 2006, 2007b), and the magnetohydrodynamic (MHD) mecha-nism (e.g., Ardeljan et al. (2000); Kotake et al. (2004a, 2005); Obergaulinger et al. (2006b);Burrows et al. (2007a); Takiwaki et al. (2009) and references therein). For the former twoto be the case, the explosion geometry is expected to be unipolar and bipolar, and for theMHD mechanism to be bipolar.Since the GW signatures imprint a live information of the asphericity at the moment ofexplosion, they are expected to provide us an important hint to solve the supernova mecha-nism. So far, most of the theoretical predictions of GWs have focused on the bounce signalsin the context of rotational core-collapse (e.g., M¨onchmeyer et al. (1991); Zwerger & Mueller(1997); Kotake et al. (2003b); Shibata & Sekiguchi (2004); Ott et al. (2004, 2007a,b); Dimmelmeier et al.(2002a, 2007, 2008); Scheidegger et al. (2008)). For the bounce signals having a strong andcharacteristic signature, the iron core must rotate enough rapidly. The waveforms are cate-gorized into the following three types, namely types I, II, and III. Type II and III waveformsare shown less likely to appear than type I, because a combination of general relativity (GR)and electron capture near core bounce suppresses multiple bounce in the type II waveforms http://geo600.aei.mpg.de/ ∼
1% ofmassive star population (e.g., Woosley & Bloom (2006)). However this can be really thecase for progenitors of rapidly rotating metal-poor stars, which experience the so-calledchemically homogeneous evolution (Woosley & Heger 2006; Yoon & Langer 2005). The highangular momentum of the core as well as a strong precollapse magnetic field is preconditionedfor the MHD mechanism, because the MHD mechanism relies on the extraction of rotationalfree energy of the collapsing core via magnetic fields. The energetic MHD explosions arereceiving great attention recently as a possible relevance to magnetars and collapsars (e.g.,Harikae et al. (2009, 2010) for collective references), which are presumably linked to theformation of long-duration gamma-ray bursts (GRBs) (e.g., Meszaros (2006)).Among the previous studies mentioned above, only a small portion of papers has beenspent on determining the GW signals in the MHD mechanism (Yamada & Sawai 2004;Kotake et al. 2004b; Obergaulinger et al. 2006b; Cerd´a-Dur´an et al. 2007; Shibata et al. 2006;Scheidegger et al. 2010). This may be because the MHD effects on the dynamics as well astheir influence over the GW signals can be visible only for cores with precollapse mag-netic fields over B & G (Obergaulinger et al. 2006b; Kotake et al. 2004b). Consid-ering that the typical magnetic-field strength of GRB progenitors is at most ∼ − G (Woosley & Heger 2006), this is already an extreme situation. Interestingly in a moreextremely case of B ∼ G, a secularly growing feature in the waveforms was ob-served (Obergaulinger et al. 2006a; Shibata et al. 2006; Scheidegger et al. 2010). MoreoverObergaulinger et al. (2006a) called a waveform as type IV in which quasi-periodic large-scale oscillations of GWs near bounce are replaced by higher frequency irregular oscillations.Some of these MHD simulations follow adiabatic core-collapse, in which a polytropic EOSis employed to mimic supernova microphysics. At this level of approximation, the bounceshock generally does not stall and a prompt explosion occurs within a few ten milliseconds 4 –after bounce. Therefore a main focus in these previous studies has been rather limited tothe early postbounce phase ( . several 10 ms). However, for models with weaker precol-lapse magnetic fields akin to the current GRB progenitors, the prompt shocks stall firstlyin the core like a conventional supernova model with more sophisticated neutrino treatment(e.g., Burrows et al. (2007a)). In such a case, the onset of MHD explosions, depending onthe initial rotation rates, can be delayed till ∼
100 ms after bounce (Burrows et al. 2007a;Takiwaki et al. 2009). There remains a room to study GW signatures in such a case, whichwe hope to study in this work.In this study, we choose to take precollapse magnetic fields less than 10 G based on a re-cent GRB-oriented progenitor models. By this choice, it generally takes much longer time af-ter bounce than the adiabatic MHD models to amplify magnetic fields enough strong to over-whelm the ram pressure of the accreting matter, leading to the magnetohydrodynamically-driven (MHD, in short) explosions. Even if the speed of jets in MHD explosions is only mildlyrelativistic, Newtonian simulations are not numerically stable because the Alfv´en velocity( ∝ B/ √ ρ ) could exceed the speed of light unphysically especially when the strongly magne-tized jets (i.e., large B ) propagate to a stellar envelope with decreasing density ( ρ ). To followa long-term postbounce evolution numerically stably, we perform special relativistic MHD(SRMHD) simulations (Takiwaki et al. 2009), in which a realistic EOS is employed and theneutrino cooling is taken into account via a multiflavor neutrino leakage scheme. Note inour previous study of GWs in magneto-rotational core-collapse (Kotake et al. 2004b) thatwe were unable to study properties of the GWs long in the postbounce phase because theemployed Newtonian simulations quite often crashed especially in the case of strong MHD ex-plosions. To include GR effects in this study, we follow a prescription in Obergaulinger et al.(2006a) which is reported to capture basic features of full GR simulations quite well. Bychanging precollapse magnetic fields as well as initial angular momentum distributions para-metrically, we compute twelve models. By doing so, we hope to study the properties of GWsin MHD explosions systematically and also address their detectability.The paper opens up with descriptions of the initial models and numerical methodsemployed in this work (section 2). Formalism for calculating the gravitational waveforms inSRMHD is summarized in section 3. The main results are given in Section 4. We summarizeour results and discuss their implications in Section 5. 5 –
2. Models and Numerical Methods2.1. Initial Models
We make precollapse models by taking the profiles of density, internal energy, andelectron fraction distribution from a rotating presupernova model of E25 in Heger & Langer(2000). This model has mass of 25 M ⊙ at the zero-age main sequence, however loses thehydrogen envelope and becomes a Wolf-Rayet (WR) star of 5.45 M ⊙ before core-collapse.Our computational domain involves the whole iron-core of 1 . M ⊙ . Note that this model issuggested as a candidate progenitor of long-duration GRBs because type Ib/c core-collapsesupernovae originated from WR stars have a observational association with long-durationGRBs (e.g., Woosley & Bloom (2006)).Since little is known about the spatial distributions of rotation and magnetic fields inevolved massive stars, we add the following profiles in a parametric manner to the non-rotating core mentioned above. For the rotation profile, we assume a cylindrical rotationof Ω( X, Z ) = Ω X X + X Z Z + Z , (1)where Ω is angular velocity and X and Z denotes distance from the rotational axis andthe equatorial plane, respectively. The parameter X represents the degree of differentialrotation, which we choose to change in the following three ways, 100km (strongly differ-ential rotation), 500km, (moderately differential rotation), and 2000km (uniform rotation),respectively. The parameter Z is fixed to 1000km.Regarding the precollapse magnetic field, we assume that the magnetic field is nearlyuniform and parallel to the rotational axis in the core and dipolar outside. This can bemodeled by the following effective vector potential, A r = A θ = 0 , (2) A φ = B r r + r r sin θ, (3)where A r,θ,φ is vector potential in the r, θ, φ direction, respectively, r is radius, r is radiusof the core, and B is a model constant (see Takiwaki et al. (2004) for detail). In this study, r is set to 2000 km which is approximately the size of the precollapse iron core.By changing initial angular momentum, degree of differential rotation, and the strengthof magnetic fields, we compute twelve models. The model parameters are shown in Table 1. 6 –The models are named after this combination, with the first letters, B12, B11 representingstrength of the initial magnetic field, the second letters, X , X , X
20 indicating the degreeof differential rotation ( X = 100 , , β = 0 . , β . Here β represents ratio of the rotational energy to theabsolute value of the gravitational energy prior to core-collapse. The original progenitorof model E25 in Heger & Langer (2000) has a uniform rotation profile in the iron coreand the initial β parameter is ∼ β . β . ∼ − G and β ∼ . β (%)0.1% 1% X (km)100km 500km 2000km 100km 500km 2000km B : 10 G B11X1 β β β β β β G B12X1 β β β β β β β represents ratio of initial rotational energy to the absolute value of theinitial gravitational energy. From left to right in the table, Ω in unit of rad/s (equation (1))is 24, 2.8, 0.95, 76, 8.9, and 3.0, respectively. Note that X and B is defined in equation (1)and (3), respectively. Numerical results in this work are calculated by the SRMHD code developed in Takiwaki et al.(2009). In the following, we first mention the importance of SR and then briefly summarizethe numerical schemes.The Alfv´en velocity of MHD jets propagating into the outer layer of the iron core canbe estimated as v A ∼ cm / s ( B/ G)( ρ/ g / cm ) − / , where ρ and B are the typicaldensity and the magnetic field near along the rotational axis. It can be readily inferred thatthe Alfv´en velocity could exceed the speed of light unphysically in Newtonian simulations. 7 –SR corrections are also helpful to capture correctly the dynamics of infalling material in thevicinity of the protoneutron star, because their free-fall velocities and rotational velocitiesbecome close to the speed of light. Such conditions are quite ubiquitous in MHD explosions.Even if the propagation speeds of the jets are only mildly relativistic, we have learned that(at least) SR treatments are quite important for keeping the stable numerical calculationsin good accuracy over a long-term postbounce evolution (e.g., Harikae et al. (2009)).The MHD part of our code is based on the formalism of De Villiers et al. (2003). Thestate of the relativistic fluid element at each point in the space time is described by itsdensity, ρ ; specific energy, e ; velocity, v i ; and pressure, p . And the magnetic field in thelaboratory frame is described by the four-vector √ πb µ = ∗ F µν U ν , where ∗ F µν is the dual ofthe electro-magnetic field strength tensor and U ν is the four-velocity. The basic equations ofthe SRMHD code are written as, ∂D∂t + 1 √ γ ∂ i √ γDv i = 0 (4) ∂E∂t + 1 √ γ ∂ i √ γEv i = − p ∂W∂t − p √ γ ∂ i √ γW v i − L ν (5) ∂ ( S i − b t b i ) ∂t + 1 √ γ ∂ j √ γ (cid:0) S i v j − b i b j (cid:1) = − (cid:0) ρh ( W v k ) − ( b k ) (cid:1) ∂ i γ kk − (cid:16) ρhW − b t (cid:17) ∂ i Φ tot − ∂ i (cid:18) p + | b | (cid:19) (6) ∂ ( W b i − W b t v i ) ∂t + ∂ j (cid:0) W v j b i − W v i b j (cid:1) = 0 (7) ∂ k ∂ k Φ = 4 π h ρh ( W + ( v k ) ) + 2 p + | b | ! − (cid:0) ( b ) + ( b k ) (cid:1) i (8)where W = √ − v k v k , D = ρW , E = eW and S i = ρhW v i are the Lorentz boost factor,auxiliary variables correspond to density, energy, and momentum, respectively. All of themare defined in the laboratory frame. Equations (4,5,6) represents the mass, energy, andmomentum conservation, respectively. In equation (6), note that the relativistic enthalpy, h = (1 + e/ρ + p/ρ + | b | /ρ ) includes the magnetic energy. Equation (7) is induction equationin SR. In solving the equation, the method of characteristics is implemented to propagateaccurately all modes of MHD waves (see Takiwaki et al. (2009) for more detail). Equation 8 –(8) is Poisson equation for the gravitational potential of Φ, which is solved by the modifiedincomplete Cholesky conjugate gradient method. For an approximate treatment of generalrelativistic (GR) gravity, Φ tot in equation (6) includes a GR correction to Φ as in Buras et al.(2006). We employ a realistic equation of state based on the relativistic mean field theory(Shen et al. 1998).We approximate the neutrino cooling by a neutrino multiflavor leakage scheme (Epstein & Pethick1981; Rosswog & Liebend¨orfer 2003), in which three neutrino flavors: electron neutrino( ν e ),electron antineutrino(¯ ν e ), and heavy-lepton neutrinos( ν µ , ¯ ν µ , ν τ , ¯ ν τ , collectively referredto as ν X ), are taken into account. The implemented neutrino reactions are electron cap-ture on proton and free nuclei; positron capture on neutron; photo-, pair, plasma processes(Fuller et al. 1985; Takahashi et al. 1978; Itoh et al. 1989, 1990). A transport equation forthe lepton fractions (namely Y e − Y e + , Y ν e , Y ¯ ν e and Y ν X , are solved in an operator-splittingmanner (see equation (7) in Takiwaki et al. (2009) for more detail). L ν in equation (5)represents the neutrino cooling rate summed over all the reactions, which can be also esti-mated by the leakage scheme (see Epstein & Pethick (1981); Rosswog & Liebend¨orfer (2003);Kotake et al. (2003a) for more detail).In our two dimensional simulations, the spherical coordinates are employed with 300( r ) × θ ) grid points to cover the computational domain assuming axial and equatorial sym-metry. The radial grid is nonuniform, extending from 0 to 4000km with finer grid near thecenter. The finest grid for the radial direction is set to 1km. The polar grid uniformlycovers from θ = 0 to π/
2. The finest grid for the polar direction is 25 m. The numericaltests and convergence with this choice of the numerical grid points are given in section 5 ofTakiwaki et al. (2009).To measure the strength of explosion, we define the explosion energy as follows, E exp = Z D dV e local = Z D dV ( e kin + e int + e mag + e grav ) , (9)here e local is the sum of e kin , e int , e mag and e grav with being kinetic, internal, magnetic, andgravitational energy, respectively defined as e kin = ρW ( W − , (10) e int = eW + p (cid:0) W − (cid:1) , (11) e mag = | b | (cid:18) − W (cid:19) − b W , (12) e grav = − ρhW Φ , (13)and D in equation (9) represents the domain where the local energy ( e local ) is positive, 9 –indicating that the matter is gravitationally unbound. The explosion energy is evaluatedwhen the MHD jets pass through 1000 km along the polar direction.
3. Formulae for Gravitational Waves in SRMHD
To extract the gravitational waveforms in MHD explosions, we present the stress formulain SRMHD for later convenience. As shown below, this can be done straightforwardly byextending the Newtonian MHD formulation presented in Kotake et al. (2004b).From the Einstein equation, one obtains the following formula as a primary expressionfor the leading part of the gravitational quadrupole field emitted by the motion of a fluid inthe post-Newtonian approximation (e.g., M¨onchmeyer et al. (1991); Finn & Evans (1990);Blanchet et al. (1990)), h T Tij ( X , t ) = 4 Gc R P ijkl ( N ) Z d x T kl , (14)where G and c are the gravitational constant and the velocity of light, respectively, T kl is the energy momentum tensor of the source, R ≡ ( δ ij X i X j ) / = | X | is the distancebetween the observer and the source. P ijkl , with N = X /R denotes the transverse-traceless(TT) projection operator onto the plane orthogonal to the outgoing wave direction N (e.g.,M¨onchmeyer et al. (1991)) . T ij consists of the three parts, namely of perfect fluid, electromagnetic field, and gravi-tational potential as follows, T ij = T ij (hyd) + T ij (mag) + T ij (grav) . (15)The first term which we refer as the hydrodynamic part is explicitly written as, T ij (hyd) = ρ ∗ W v i v j + pδ ij , (16)where ρ ∗ is effective density defined as, ρ ∗ = ρ + e + p + | b | c . (17)The second term in equation (15) represents the contribution from magnetic fields as, T ij (mag) = − b i b j . (18)And the last term in equation (15) is the contribution from the gravitational potential, T ij (grav) = 14 πG (cid:18) Φ ,i Φ ,j − δ ij Φ ,m Φ ,m (cid:19) (19) 10 –where Φ corresponds to the self-gravity in equation (8).In our axisymmetric case, there remains only one non-vanishing quadrupole term in themetric perturbation, namely ℓ = 2 , m = 0 in terms of the pure-spin tensor harmonics, as h T Tij ( X , t ) ℓ =2 ,m =0 = 1 R A E (cid:18) t − Rc (cid:19) T E , ij ( θ, φ ) , (20)where T E , ij ( θ, φ ) is T E , ij ( θ, φ ) = 18 r π sin θ, (21)(e.g., Thorne (1980)). The projection operator in equation (14) acts on T ij as, P ijkl T kl = 2 T zz − T xx − T yy . (22)Transforming equation (14) to the spherical coordinates, and expressing b i and v i in termsof unit vectors in the r , θ , φ direction, we obtain for A E220 the expression, A E220 = A E220 (hyd) + A E220 (mag) + A E220 (grav) , (23)where A E220 (hyd) = Gc π / √ Z dµ Z ∞ r drf E220 (hyd) , (24) f E220 (hyd) = ρ ∗ W ( v r (3 µ −
1) + v θ (2 − µ ) − v φ − v r v θ µ p − µ ); (25) A E220 (grav) = Gc π / √ Z dµ Z ∞ r drf E220 (grav) , (26) f E220 (grav) = " ρh ( W + ( v k /c ) ) + 2 c p + | b | ! − c (cid:0) ( b ) + ( b k ) (cid:1) × h − r∂ r Φ(3 µ −
1) + 3 ∂ θ Φ µ p − µ i ; (27)and A E220 (mag) = − Gc π / √ Z dµ Z ∞ r drf E220 (mag) , (28) f E220 (mag) = [ b r (3 µ −
1) + b θ (2 − µ ) − b φ − b r b θ µ p − µ ]; (29)where µ = cos θ . For later convenience, we write the total GW amplitude as, h TT = h TT(hyd) + h TT(mag) + h T T (grav) , (30) 11 –where the quantities of the right hand are defined by combining equations (20) and (23) withequations (25), (27), and (29). By dropping O ( v/c ) terms, the above formulae reduce to theconventional Newtonian stress formula (e.g., M¨onchmeyer et al. (1991)). In the followingcomputations, the observer is assumed to be located in the equatorial plane ( θ = π/ R = 10 kpc). 12 –
4. Results
The gravitational waveforms obtained in this work can be categorized into two, whichwe call increasing type or cancellation type just for convenience. Note that the latter typedoes not mean a new waveform as will be explained later in this section. Regarding theformer one, such a waveform was presented in previous literature (Obergaulinger et al. 2006a;Shibata et al. 2006; Scheidegger et al. 2010), however their properties have not been clearlyunderstood yet. In section 4.1, we first overview their characters, which are peculiar in thecase of MHD explosions. In section 4.2, we move on to analyze their properties by carefullycomparing each contribution in equation (30) to the total GW amplitudes. Then in section4.3, we perform the spectra analysis and discuss their detectability.
Figure 1 shows examples of the two categories, which we call as the increasing (leftpanels) or cancellation type (right panels), respectively. In the increasing type, the totalwave amplitudes (red line) have a monotonically increase trend after bounce ( t − t b = 0in the figures). While in the cancellation type (right panels), the total amplitudes afterbounce stay almost zero. This is because the contribution from the magnetic fields (blue line,equation (29)) cancels with the one from the sum of the hydrodynamic and gravitational parts(green line, equations (25,27)). Regardless of the difference in the two types, it is commonthat the magnetic contribution (blue line) increases almost monotonically with time. Notsurprisingly, the bounce GW signals (( t − t b .
20 ms) are categorized into the so-called typeI or II waveforms. Note here that the MHD simulations are terminated at around 100 msafter bounce for all the computed models. This is simply because the GW amplitudes in amore later phase decrease because the MHD shock comes out of the computational domainand the enclosed mass in the domain becomes smaller.Table 2 depicts a classification of the computed models, in which ”C” and ”I” indicatesthe cancellation and increasing type, respectively. I ∗ in the table indicates the mixture of thetwo types, which we call as intermediate type. The table shows that the bifurcation of thetwo types is predominantly determined by the precollapse magnetic fields, so that the modelswith stronger magnetic fields ( B G) are basically classified to the increasing type. Formodels colored by orange, which are slow rotator with uniform rotation in our models, thefield amplification works less efficiently than for models with stronger differential rotation(such as X = 100 ,
500 km) (see also Cerd´a-Dur´an et al. (2008)). This suppresses the increasein the wave amplitudes due to magnetic fields, which gives rise to the intermediate statebetween the two types. 13 – -20-15-10-5 0 5 10 15 20 25 30 35 0 20 40 60 80 100 h [ - a t k p c ] t-t b [ms] B12X1 β h [ - a t k p c ] t-t b [ms] B11X1 β h [ - a t k p c ] t-t b [ms] B12X5 β h [ - a t k p c ] t-t b [ms] B11X5 β h [ - a t k p c ] t-t b [ms] B12X5 β h [ - a t k p c ] t-t b [ms] B11X5 β Fig. 1.— Gravitational waveforms with the increasing (left) or the cancellation trend (right)(see text for more detail). At the right bottom in each panel, the model names are givensuch as B12X1 β . ∼
30 ms (see models colored by red in Table 3)and another is launched rather later after bounce ( &
30 ms) (models colored by green inTable 3). In this sense, our computed models could be roughly categorized into two, namelypromptly MHD explosion (colored by red in Table 3) or delayed MHD explosion (colored bygreen in Table 3), respectively. This simply reflects that it takes longer time for the weaklymagnetized models to amplify the magnetic pressure behind the stalled shock enough strongto overwhelm the ram pressure of accreting matter. By comparing Table 2 to 3, the twocharacters in the waveforms have a rough correlation with the difference of the explosiondynamics.Table 4 shows a summary regarding the explosion energy (e.g., equation (9)). ComparingTable 2 to 4, the explosion energy for the increasing type (models colored by yellow in Table2 and 4) is higher compared to the cancellation type (models colored by light blue). Giventhe rotation rate of β = 0 . erg (models colored by yellow).Having summarized the waveform classification together with the explosion dynamics,we move on to look more in detail what makes the difference between the two types in thenext section. β rot (%)0 .
1% 1% X (km)100km 500km 2000km 100km 500km 2000km B (Gauss) 10 G C C I* C C C10 G I I I* I I I
Table 2: Same as Table 1 but for the classification of the computed models. ”C” and ”I”indicates the cancellation and increasing type, respectively, while I ∗ indicates the mixture ofthe two types, which we refer to as intermediate type. 15 – β rot (%)0 .
1% 1% X (km)100km 500km 2000km 100km 500km 2000km B (Gauss) 10 G 38ms 57ms 91ms 18ms 87ms 75ms10 G 7ms 18ms 28ms 3ms 17ms 19ms1 β rot (%)0 .
1% 1% X (km)100km 500km 2000km 100km 500km 2000km B (Gauss) 10 G 40ms 58ms 92ms 25ms 95ms 81ms10 G 10ms 20ms 30ms 8ms 20ms 22ms
Table 3: Same as Table 1 but for the interval measured from the stall of the bounce shockto the MHD-driven revival of the stalled shock (top panel) and the one measured from corebounce (bottom panel). The computed models are classified whether the launch of the MHDjets occurs relatively promptly after bounce (models colored by red, with the intervals beingshorter than ∼
30 ms typically) or rather later (models colored by green), which we referto as promptly or delayed MHD explosion for convenience in this work (see text for moredetail). Note that these timescales are estimated just by looking at the velocity evolutionsalong the polar axis. β rot (%)0 .
1% 1% X (km)100km 500km 2000km 100km 500km 2000km B (Gauss) 10 G 8 × × × × × × G 2 . × × × × × . × Table 4: Same as Table 1 but for the explosion energy (defined in equation (9)). Comparingwith Table 2 to 4, the explosion energies for the models with the increasing trend generallyexceed 10 erg (models colored by yellow). 16 – By taking model B12X1 β β (right-half, the ratio of matter to magnetic pressure) at 100 ms after bounce. It can be seen thatthe outgoing jets indicated by the velocity fields (arrows in the left-half) are driven by themagnetic pressure behind the shock (see bluish region (i.e., low plasma β ) in the right-halfpanel).The right panel of Figure 2 shows contributions to the total GW amplitudes (equa-tion (27)), in which the left-hand-side panels are for the sum of the hydrodynamic andgravitational part, namely log (cid:16) ± h f E220 (hyd) + f E220 (grav) i(cid:17) (left top(+)/bottom( − )(equations(25,27)), and the right-hand-side panels are for the magnetic part, namely log (cid:16) ± f E220 (mag) (cid:17) (right top(+)/bottom( − )) (e.g., equation (29)). By comparing the top two panels in Fig-ure 2, it can be seen that the positive contribution is overlapped with the regions wherethe MHD outflows exist. The major positive contribution is from the kinetic term of theMHD outflows with large radial velocities (e.g., + ρ ∗ W v r in equation (25)). The magneticpart also contributes to the positive trend (see top right-half in the right panel (labeled bymag(+))). This comes from the toroidal magnetic fields (e.g., + b φ in equation (29)), whichdominantly contribute to drive MHD explosions. The magnetic contribution was alreadymentioned in Kotake et al. (2004b). This study furthermore adds that the kinetic energy ofMHD outflows more importantly contributes to the positive trend.Figure 3 shows a normalized cumulative contribution of each term in A E220 , which isestimated by the volume integral of A E220 within a given sphere enclosed by certain radius. Itcan be seen that the contribution of the hydrodynamic and gravity parts (indicated by ”hyd& grav”) is prominent for radius outside ∼ Now we proceed to focus on the cancellation-type waveform by taking model B11X1 β β (right-half). Note that the jet head of MHD outflows is located at ∼
400 km along thepole which sticks out of the plots in Figure 5 (compare the difference in scales of Figure 2).In fact, the bluish regions (low β ) around the pole have a lower density because materialthere has been already blown up due to the passage of MHD-driven shocks. The middlepanel of Figure 5 shows the term-by-term contribution to A E220 as in the right panel of Figure2. From the right-half panel, it is shown that the magnetic contribution dominantly makesa positive contribution (labeled by mag(+)), which is also the case of the increasing type asmentioned in the previous section. Note that the negative contribution from magnetic fields(bluish region in the bottom-right-half of the middle panel) comes from the regions, whereare off-axis from the propagation of the MHD jets. In these regions, the poloidal componentsof magnetic fields are stronger than the toroidal ones, which makes the negative contributionmainly through − [ b r (3 µ −
1) + b θ (2 − µ )] in equation (29).Looking at the sum of the hydrodynamic and gravitational part (left-half in the middlepanel), a large negative contribution comes from regions near in the rotational axis (coloredby red, bottom-left-half). The bottom panel of Figure 5 further shows a contribution fromthe hydrodynamic (left-half) and gravitational part (right-half), separately. Regarding thehydrodynamic part, the negative contribution is highest in the vicinity of the equatorialplane which closely coincides with the oblately deformed protoneutron star (colored by red,the bottom-left-half in the bottom panel). This is because the negative contribution comesfrom the centrifugal forces (e.g., the term related to the rotational energy, − ρv φ in equation(25)). For the gravity part (right-half in the bottom panel), a big negative contribution comesfrom regions in the vicinity of the rotational axis (bottom-right-half). This comes from theterm of − r∂ r Φ(3 µ −
1) in equation (25), which is the radial gradient of the gravitationalpotential. Remembering that ∂ r Φ > µ = cos θ which is a directional cosine measured from therotational axis. As a result, the gravity part makes a negatively contribution in the vicinityof the rotational axis ( µ ∼ This may be the reason why the stress formula has been often employed in supernova researches so far.
18 –Fig. 2.— Left panel shows distributions of entropy [ k B /baryon] (left) and logarithm ofplasma β (right) for model B12X1 β . c ). Right panel shows the sum of the hydrodynamic and gravitational parts (indicatedby “hyd and grav” in the left-hand side) and the magnetic part (indicated by ”mag” in theright-hand side), respectively. The top and bottom panels represent the positive and negativecontribution (indicated by (+) or (-)) to A E220 , respectively (see text for more detail). Theside length of each plot is 4000(km)x8000(km). 19 – -0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1 10 100 1000 C on t r i bu t i on t o G W A m p li t ude Radius [km] M25B12X1 β Fig. 3.— Normalized cumulative contribution of each term in A E220 as a function of radius formodel B12X1 β A E220 within sphere of a givenradius.Fig. 4.— Gravitational waveform extracted either by the stress formula (indicated by QPM,red line) or by the first-moment of momentum-density formalism (FDM, green line) for modelB12X1 β . ∼
10 km in Figure 6).Outside it, the magnetic part becomes almost comparable to the sum of the hydrodynamicand gravity part, which makes the cancellation type shown in the right panels of Figure 1. 21 –Fig. 5.— The top panel shows distributions of logarithm of density [g / cm ] (left-half) andlogarithm of β (right-half) for model B11X1 β -18-16-14-12-10-8-6-4-2 0 2 4 1 10 100 1000 C on t r i bu t i on t o G W A m p li t ude Radius [km] M25B11X1 β Fig. 6.— Same as Figure 3 but for model B11X1 β
1. 23 –
Now we move on to perform a spectral analysis (Figure 7). Both for the two types (leftpanel (increasing-type), right panel (cancellation-type)), the peak amplitudes in the spectraare around 1 kHz, which comes from the GWs near at bounce. It can be seen that thespectra for lower frequency domains (below ∼
100 Hz) are much larger for the increasingtype (left panels) compared to the cancellation type (right panels). This reflects a slowertemporal variation of the secular drift inherent to the increase-type waveforms (e.g., Figure1). As a measure to characterize the dominance in the lower frequency domains, we define˜ h low , which represents average amplitudes below 100 Hz (see Table 5). Although the peakamplitudes, ˜ h peak , in the spectra have no clear correlation with the two types, we pointout that the final GW amplitudes (the first column) and the ˜ h low (the third column) aremuch larger for the increasing type (colored by yellow) compared to the cancellation type(colored by light blue). In Figure 7, the peak amplitudes near 1 kHz are, irrespective ofthe two types, marginally within the detection limits of the currently running detector ofthe first LIGO and the detection seems more feasible for the next-generation detectors suchas LCGT and the advanced LIGO for a Galactic supernova. It is true that the GWs inthe low frequency domains mentioned above are relatively difficult to detect due to seismicnoises, but a recently proposed future space interferometers like Fabry-Perot type DECIGO isdesigned to be sensitive in the frequency regimes (Kawamura et al. 2006; Kudoh et al. 2006).The sensitivity curve of the detector is plotted with the black line in Figure 7. Our resultssuggest that these low-frequency signals, if observed, could be one important messenger ofthe increase-type waveforms that are likely to be associated with MHD explosions exceeding10 erg.
5. Summary and Discussion
By performing a series of two-dimensional SRMHD simulations, we studied signaturesof GWs in the MHD-driven core-collapse supernovae. In order to extract the gravitationalwaveforms, we presented a stress formula including contributions both from magnetic fieldsand special relativistic corrections. By changing the precollapse magnetic fields and initialangular momentum distributions parametrically, we computed twelve models. As for themicrophysics, a realistic equation of state was employed and the neutrino cooling was takeninto account via a multiflavor neutrino leakage scheme. With these computations, we foundthat the total GW amplitudes show a monotonic increase after bounce for models with astrong precollapse magnetic field (10 G) also with a rapid rotation imposed. We pointed out 24 – h c ha r Frequency [Hz]B12X1 β h c ha r Frequency [Hz]B11X1 β h c ha r Frequency [Hz]B12X5 β h c ha r Frequency [Hz]B11X5 β Fig. 7.— Gravitational-wave spectrum for representative models with the expected detectionlimits of the first LIGO (Abbott et al. 2005), the advanced LIGO (Weinstein 2002), Large-scale Cryogenic Gravitational wave Telescope (LCGT) (Kuroda & the LCGT Collaboration2006), and Fabry-Perot type DECIGO (Kawamura et al. 2006; Kudoh et al. 2006). It isnoted that h char is the characteristic gravitational wave strain defined in Flanagan & Hughes(1998). The supernova is assumed to be located at the distance of 10 kpc. 25 – Model h [10 − ] ˜ h peak [10 − ] ˜ h low [10 − ]B11X1 β β β β β β β β β β β β Table 5: Summary of GW amplitudes for all the models. The colors used for clarify are thesame as Table 2. The first column represents the GW amplitudes when we terminated thesimulation (100 ms after bounce). The second column, ˜ h peak , is the peak GW amplitudes inthe spectra. The third column, ˜ h low , is the average amplitudes below 100 Hz. The supernovais assumed to be located at the distance of 10 kpc.that this trend stems both from the kinetic contribution of MHD outflows with large radialvelocities and also from the magnetic contribution dominated by the toroidal magnetic fieldsthat predominantly trigger MHD explosions. For models with weaker initial magnetic fields,the total GW amplitudes after bounce stay almost zero, because the contribution from themagnetic fields cancels with the one from the hydrodynamic counterpart. These featurescan be clearly understood with a careful analysis on the explosion dynamics. It was pointedout that the GW signals with an increasing trend, possibly visible to the next-generationdetectors for a Galactic supernova, would be associated with MHD explosions exceeding 10 erg. Although the presented simulations have utilized the leakage scheme to approximate thedeleptonization, it would be more accurate (especially before bounce) to employ a formuladeveloped by Liebend¨orfer (2005), which was designed to fit 1D Boltzmann results. Figure 8shows snapshots at around 33 ms after bounce for model B12X20 β . Y e prescription (right panel) is employed, respectively. Note that“G15” is taken in our simulation until bounce among the parameter sets in Liebend¨orfer(2005) and is switched to the leakage scheme at the postbounce phase. As is shown, the The inner-core mass at bounce for the employed parameter set is 0.6 M ⊙ for our non-rotating 25 M ⊙ progenitor. This value is higher than that obtained in GR simulations (0.45-0.55 M ⊙ ) in Dimmelmeier et al.(2008). This may be because the pseudo GR potential employed in this work underestimates the GR gravity,which could potentially lead to a large inner-core mass.
26 – −300 −150 068101214
33 ms −300 −150 0
Density
Entropy
Leakage
Fig. 8.— Snapshots at 33 ms after bounce for model B12X20 β . Y e prescription (right panel) is employed, respectively. In eachpanel, density (logarithmic, left-half) and entropy (right-half) distributions are shown. Theside length of each plot is 600x600(km). P r e ss u r e a t po l e [ e r g / c m ] Radius [km] B12X20 β P r e ss u r e a t equa t o r [ e r g / c m ] Radius [km] B12X20 β R ad i a l v e l o c i t y [ x c m / s ] Radius [km] B12X20 β P r e ss u r e a t po l e [ e r g / c m ] Radius [km] B12X20 β P r e ss u r e a t equa t o r [ e r g / c m ] Radius [km] B12X20 β R ad i a l v e l o c i t y [ x c m / s ] Radius [km] B12X20 β Fig. 9.— Left and middle panels show magnetic pressure (red line) vs. ram pressure (blueline) for model B12X20 β . Y e prescription (top panels)or the leakage scheme (bottom panels) along the polar axis (left panel) or the equatorialplane (middle panel). Matter pressure is shown by green line as a reference. The rightpanels show velocity profiles along the pole near after the stall of the bounce shock (redlines). Both in the two different deleptonization schemes, the MHD-driven explosions areindeed obtained. 27 – -250-200-150-100-50 0 50-2 -1 0 1 2 3 4 5 h [ - a t k p c ] t-t b [ms] B12X1 β Fig. 10.— Gravitational waveforms for model B12X1 β t − t b = -2 ms with t b being the epoch of bounce),correspond to models with different angular grid points (30 (green), 60(red), 120(blue)) whilefixing the radial grid points to be 300. The bottom three lines are set to start from -100in the GW amplitudes (just for convenience), and they correspond to models with differentradial grid points (250 (pink), 300 (orange), 600 (brown)) while fixing the lateral grid pointsto be 60. Note that the fiducial set employed in this work is 300(r)x60( θ ).shock revival also occurs for the model with the Y e prescription (right panel). Figure 9depicts the magnetic pressure (red line) vs. ram pressure (blue line) along the polar axis(left panel) or the equatorial plane (middle) near the rebirth of the stalled shock for themodel with the Y e prescription (top panels) or the leakage scheme (bottom panels). For theequator, the magnetic pressure is much less than the ram pressure (middle panel), while themagnetic pressure amplified by the field wrapping along the pole becomes as high as theram pressure of the infalling material at the shock front, leading to the MHD-driven shockformation (see right panels). Regardless of the two different deleptoniaztion schemes, theseimportant features associated with the MHD explosions are shown to be quite similar.Now we mention a comparison between the obtained results and relevant MHD simula-tions. Model R4E1CF in Scheidegger et al. (2010) whose precollapse rotational parameter is β = 0 . G, is close to our model B12X20 β .
1. From their Figure 23, the jet propagates to ∼
300 km along the rotational axis at around ∼
18 ms after bounce. In our counterpart model,the MHD-driven shock revives after around 30 ms after bounce, and it reaches to 300 km ataround ∼
10 ms, which is equivalent to ∼
40 ms after bounce. Considering that our model( β = 0 . β = 0 . β = 0 . G, isclose to our model B12X20 β .
1. From their Figure 10, the MHD jet propagates to ∼ Y e formula (Liebend¨orfer 2005).As mentioned above, the dynamics is rather close to our corresponding model. Among thecomputed models in Burrows et al. (2007a), model M15B12DP2A1H which has a precollapseangular velocity of π rad/s (the rotational parameter should be close to β = 0 . G, is close to our model B12X20 β .
1. The intervalbefore the launch of the MHD shock for their model is 80 ms after bounce (e.g., their Table1) is much later than our model (28 ms after bounce). This may be due to the larger initialangular momentum ( β = 0 . β = 0 .
9% with a differential rotation imposed (the radial cut off is 500 km) and the initialpoloidal magnetic field is 10 G, is closer to our model B12X1 β .
1. The MHD jet reachesto 500 km at around 7 ms after bounce, which is also the case of our counterpart model. Asdiscussed above, our results are compatible to the ones obtained in the relevant foregoingresults.The major limitation of this study is the assumption of axisymmetry. Recently it wasreported in three-dimensional (3D) MHD core-collapse simulations (Scheidegger et al. 2008,2010) that the fast growth of the spiral SASI hinders the efficient amplification of the toroidalfields, which could suppress the formation of jets rather easily realized in 2D simulations. Asa sequel of this study, we plan to investigate the 3D effects in SRMHD. Regarding a resolutiondependence of our results, Figure 10 indicates that our standard resolution is adequate tofollow the evolution of the computed models. However, it is not sufficient at all to capturethe magneto-rotational instability (MRI, e.g., Balbus & Hawley (1998)). At least 10 − REFERENCES
Abbott et al., M. 2005, Phys. Rev. D, 72, 122004Ando, M. & the TAMA Collaboration. 2005, Classical and Quantum Gravity, 22, 881Ardeljan, N. V., Bisnovatyi-Kogan, G. S., & Moiseenko, S. G. 2000, Astron. Astrophys., 355,1181Aso, Y., M´arka, Z., Finley, C., Dwyer, J., Kotake, K., & M´arka, S. 2008, Classical andQuantum Gravity, 25, 114039 30 –Balbus, S. A. & Hawley, J. F. 1998, Reviews of Modern Physics, 70, 1Bionta, R. M., Blewitt, G., Bratton, C. B., Caspere, D., & Ciocio, A. 1987, Physical ReviewLetters, 58, 1494Blanchet, L., Damour, T., & Schaefer, G. 1990, MNRAS, 242, 289Bruenn, S. W., Mezzacappa, A., Hix, W. R., Blondin, J. M., Marronetti, P., Messer, O. E. B.,Dirk, C. J., & Yoshida, S. 2010, ArXiv e-printsBuras, R., Rampp, M., Janka, H.-T., & Kifonidis, K. 2006, A&A, 447, 1049Burrows, A., Dessart, L., Livne, E., Ott, C. D., & Murphy, J. 2007a, Astrophys. J., 664, 416Burrows, A. & Hayes, J. 1996, Physical Review Letters, 76, 352Burrows, A., Livne, E., Dessart, L., Ott, C. D., & Murphy, J. 2006, Astrophys. J., 640, 878—. 2007b, Astrophys. J., 655, 416Cerd´a-Dur´an, P., Font, J. A., Ant´on, L., & M¨uller, E. 2008, Astron. Astrophys., 492, 937Cerd´a-Dur´an, P., Font, J. A., & Dimmelmeier, H. 2007, A&A, 474, 169De Villiers, J.-P., Hawley, J. F., & Krolik, J. H. 2003, Astrophys. J., 599, 1238Dimmelmeier, H., Font, J. A., & M¨uller, E. 2002a, Astron. Astrophys., 393, 523Dimmelmeier, H., Font, J. A., & M¨uller, E. 2002b, A&A, 393, 523Dimmelmeier, H., Ott, C. D., Janka, H.-T., Marek, A., & M¨uller, E. 2007, Physical ReviewLetters, 98, 251101Dimmelmeier, H., Ott, C. D., Marek, A., & Janka, H. 2008, Phys. Rev. D, 78, 064056Epstein, R. I. & Pethick, C. J. 1981, Astrophys. J., 243, 1003Finn, L. S. & Evans, C. R. 1990, Astrophys. J., 351, 588Flanagan, ´E. ´E. & Hughes, S. A. 1998, Phys. Rev. D, 57, 4566Fryer, C. L. 2004, Astrophys. J. Lett., 601, L175Fuller, G. M., Fowler, W. A., & Newman, M. J. 1985, Astrophys. J., 293, 1Harikae, S., Kotake, K., & Takiwaki, T. 2010, Astrophys. J., 713, 304 31 –Harikae, S., Takiwaki, T., & Kotake, K. 2009, Astrophys. J., 704, 354Heger, A. & Langer, N. 2000, Astrophys. J., 544, 1016Hirata, K., Kajita, T., Koshiba, M., Nakahata, M., & Oyama, Y. 1987, Physical ReviewLetters, 58, 1490Hough, J., Rowan, S., & Sathyaprakash, B. S. 2005, Journal of Physics B Atomic MolecularPhysics, 38, 497Itoh, N., Adachi, T., Nakagawa, M., Kohyama, Y., & Munakata, H. 1989, Astrophys. J.,339, 354—. 1990, Astrophys. J., 360, 741Janka, H. & Moenchmeyer, R. 1989, A&A, 209, L5Kawagoe, S., Takiwaki, T., & Kotake, K. 2009, Journal of Cosmology and Astro-ParticlePhysics, 9, 33Kawamura, S., Nakamura, T., Ando, M., Seto, N., Tsubono, K., Numata, K., Takahashi, R.,Nagano, S., Ishikawa, T., Musha, M., Ueda, K., Sato, T., Hosokawa, M., Agatsuma,K., Akutsu, T., Aoyanagi, K., Arai, K., Araya, A., Asada, H., Aso, Y., Chiba, T.,Ebisuzaki, T., Eriguchi, Y., Fujimoto, M., Fukushima, M., Futamase, T., Ganzu, K.,Harada, T., Hashimoto, T., Hayama, K., Hikida, W., Himemoto, Y., Hirabayashi, H.,Hiramatsu, T., Ichiki, K., Ikegami, T., Inoue, K. T., Ioka, K., Ishidoshiro, K., Itoh, Y.,Kamagasako, S., Kanda, N., Kawashima, N., Kirihara, H., Kiuchi, K., Kobayashi, S.,Kohri, K., Kojima, Y., Kokeyama, K., Kozai, Y., Kudoh, H., Kunimori, H., Kuroda,K., Maeda, K., Matsuhara, H., Mino, Y., Miyakawa, O., Miyoki, S., Mizusawa, H.,Morisawa, T., Mukohyama, S., Naito, I., Nakagawa, N., Nakamura, K., Nakano, H.,Nakao, K., Nishizawa, A., Niwa, Y., Nozawa, C., Ohashi, M., Ohishi, N., Ohkawa,M., Okutomi, A., Oohara, K., Sago, N., Saijo, M., Sakagami, M., Sakata, S., Sasaki,M., Sato, S., Shibata, M., Shinkai, H., Somiya, K., Sotani, H., Sugiyama, N., Tagoshi,H., Takahashi, T., Takahashi, H., Takahashi, R., Takano, T., Tanaka, T., Taniguchi,K., Taruya, A., Tashiro, H., Tokunari, M., Tsujikawa, S., Tsunesada, Y., Yamamoto,K., Yamazaki, T., Yokoyama, J., Yoo, C., Yoshida, S., & Yoshino, T. 2006, Classicaland Quantum Gravity, 23, 125Kotake, K., Iwakami, W., Ohnishi, N., & Yamada, S. 2009a, Astrophys. J., 704, 951—. 2009b, Astrophys. J. Lett., 697, L133 32 –Kotake, K., Iwakami-Nakano, W., & Ohnishi, N. 2011, ApJ, 736, 124Kotake, K., Ohnishi, N., & Yamada, S. 2007, Astrophys. J., 655, 406Kotake, K., Sato, K., & Takahashi, K. 2006, Reports of Progress in Physics, 69, 971Kotake, K., Sawai, H., Yamada, S., & Sato, K. 2004a, Astrophys. J., 608, 391Kotake, K., Yamada, S., & Sato, K. 2003a, Astrophys. J., 595, 304—. 2003b, Phys. Rev. D, 68, 044023—. 2005, ApJ, 618, 474Kotake, K., Yamada, S., Sato, K., Sumiyoshi, K., Ono, H., & Suzuki, H. 2004b, Phys. Rev. D,69, 124004Kudoh, H., Taruya, A., Hiramatsu, T., & Himemoto, Y. 2006, Phys. Rev. D, 73, 064006Kuroda, K. & the LCGT Collaboration. 2006, Classical and Quantum Gravity, 23, 215Leonor, I., Cadonati, L., Coccia, E., D’Antonio, S., Di Credico, A., Fafone, V., Frey, R.,Fulgione, W., Katsavounidis, E., Ott, C. D., Pagliaroli, G., Scholberg, K., Thrane,E., & Vissani, F. 2010, Classical and Quantum Gravity, 27, 084019Liebend¨orfer, M. 2005, ApJ, 633, 1042Marek, A. & Janka, H.-T. 2009, Astrophys. J., 694, 664Marek, A., Janka, H.-T., & M¨uller, E. 2009, Astron. Astrophys., 496, 475Meszaros, P. 2006, Reports of Progress in Physics, 69, 2259M¨onchmeyer, R., Schaefer, G., Mueller, E., & Kates, R. E. 1991, Astron. Astrophys., 246,417M¨uler, E. & Janka, H.-T. 1997, Astron. Astrophys., 317, 140M¨uller, E., Rampp, M., Buras, R., Janka, H.-T., & Shoemaker, D. H. 2004, Astrophys. J.,603, 221Murphy, J. W., Ott, C. D., & Burrows, A. 2009, ApJ, 707, 1173Obergaulinger, M., Aloy, M. A., Dimmelmeier, H., & M¨uller, E. 2006a, Astron. Astrophys.,457, 209 33 –Obergaulinger, M., Aloy, M. A., & M¨uller, E. 2006b, Astron. Astrophys., 450, 1107Obergaulinger, M., Cerd´a-Dur´an, P., M¨uller, E., & Aloy, M. A. 2009, A&A, 498, 241Ott, C. D. 2009, Classical and Quantum Gravity, 26, 063001Ott, C. D., Burrows, A., Dessart, L., & Livne, E. 2006, Physical Review Letters, 96, 201102—. 2008, ApJ, 685, 1069Ott, C. D., Burrows, A., Livne, E., & Walder, R. 2004, Astrophys. J., 600, 834Ott, C. D., Dimmelmeier, H., Marek, A., Janka, H., Hawke, I., Zink, B., & Schnetter, E.2007a, Physical Review Letters, 98, 261101Ott, C. D., Dimmelmeier, H., Marek, A., Janka, H., Zink, B., Hawke, I., & Schnetter, E.2007b, Classical and Quantum Gravity, 24, 139Rosswog, S. & Liebend¨orfer, M. 2003, MNRAS, 342, 673Scheck, L., Janka, H., Foglizzo, T., & Kifonidis, K. 2008, A&A, 477, 931Scheck, L., Plewa, T., Janka, H.-T., Kifonidis, K., & M¨uller, E. 2004, Physical ReviewLetters, 92, 011103Scheidegger, S., Fischer, T., Whitehouse, S. C., & Liebend¨orfer, M. 2008, Astron. Astrophys.,490, 231Scheidegger, S., K¨appeli, R., Whitehouse, S. C., Fischer, T., & Liebend¨orfer, M. 2010, A&A,514, A51+Shen, H., Toki, H., Oyamatsu, K., & Sumiyoshi, K. 1998, Nuclear Physics A, 637, 435Shibata, M., Liu, Y. T., Shapiro, S. L., & Stephens, B. C. 2006, Phys. Rev. D, 74, 104026Shibata, M. & Sekiguchi, Y.-I. 2004, Phys. Rev. D, 69, 084024Suwa, Y., Kotake, K., Takiwaki, T., Whitehouse, S. C., Liebendoerfer, M., & Sato, K. 2009,ArXiv e-printsTakahashi, K., El Eid, M. F., & Hillebrandt, W. 1978, Astron. Astrophys., 67, 185Takiwaki, T., Kotake, K., Nagataki, S., & Sato, K. 2004, Astrophys. J., 616, 1086Takiwaki, T., Kotake, K., & Sato, K. 2009, Astrophys. J., 691, 1360 34 –Thorne, K. S. 1980, Reviews of Modern Physics, 52, 299van Elewyck, V., Ando, S., Aso, Y., Baret, B., Barsuglia, M., Bartos, I., Chassande-Mottin,E., di Palma, I., Dwyer, J., Finley, C., Kei, K., Kouchner, A., Marka, S., Marka,Z., Rollins, J., Ott, C. D., Pradier, T., & Searle, A. 2009, International Journal ofModern Physics D, 18, 1655Walder, R., Burrows, A., Ott, C. D., Livne, E., Lichtenstadt, I., & Jarrah, M. 2005, ApJ,626, 317Weinstein, A. 2002, Classical and Quantum Gravity, 19, 1575Woosley, S. E. & Bloom, J. S. 2006, ARA&A, 44, 507Woosley, S. E. & Heger, A. 2006, Astrophys. J., 637, 914Yamada, S. & Sawai, H. 2004, Astrophys. J., 608, 907Yoon, S.-C. & Langer, N. 2005, Astron. Astrophys., 443, 643Zwerger, T. & Mueller, E. 1997, Astron. Astrophys., 320, 209