Simulations of core-collapse supernovae in spatial axisymmetry with full Boltzmann neutrino transport
Hiroki Nagakura, Wakana Iwakami, Shun Furusawa, Hirotada Okawa, Akira Harada, Kohsuke Sumiyoshi, Shoichi Yamada, Hideo Matsufuru, Akira Imakura
aa r X i v : . [ a s t r o - ph . H E ] F e b Draft version February 14, 2018
Typeset using L A TEX twocolumn style in AASTeX61
SIMULATIONS OF CORE-COLLAPSE SUPERNOVAE IN SPATIAL AXISYMMETRY WITH FULLBOLTZMANN NEUTRINO TRANSPORT
Hiroki Nagakura, Wakana Iwakami,
2, 3
Shun Furusawa, Hirotada Okawa,
Akira Harada, Kohsuke Sumiyoshi, Shoichi Yamada,
Hideo Matsufuru, and Akira Imakura TAPIR, Walter Burke Institute for Theoretical Physics, Mailcode 350-17, California Institute of Technology, Pasadena, CA 91125, USA Yukawa Institute for Theoretical Physics, Kyoto University, Oiwake-cho, Kitashirakawa, Sakyo-ku, Kyoto, 606-8502, Japan Advanced Research Institute for Science & Engineering, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan Interdisciplinary Theoretical Science (iTHES) Research Group, RIKEN, Wako, Saitama 351-0198, Japan Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan Numazu College of Technology, Ooka 3600, Numazu, Shizuoka 410-8501, Japan Department of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan High Energy Accelerator Research Organization, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan University of Tsukuba, 1-1-1, Tennodai Tsukuba, Ibaraki 305-8577, Japan
ABSTRACTWe present the first results of our spatially axisymmetric core-collapse supernova simulations with full Boltzmannneutrino transport, which amount to a time-dependent 5-dimensional (2 in space and 3 in momentum space) problemin fact. Special relativistic effects are fully taken into account with a two-energy-grid technique. We performed twosimulations for a progenitor of 11 . M ⊙ , employing different nuclear equations-of-state (EOS’s): Lattimer and Swesty’sEOS with the incompressibility of K = 220MeV (LS EOS) and Furusawa’s EOS based on the relativistic mean fieldtheory with the TM1 parameter set (FS EOS). In the LS EOS the shock wave reaches ∼ ∼ rθ -component reaches ∼
10% of the dominant rr-componentand, more importantly, it dictates the evolution of lateral neutrino fluxes, dominating over the θθ -component, in thesemi-transparent region. These data will be useful to further test and possibly improve the prescriptions used in theapproximate methods. Keywords: supernovae: general—neutrinos—hydrodynamics
Corresponding author: Hiroki [email protected]
Nagakura et al. INTRODUCTIONThe theoretical study of the explosion mechanismof core-collapse supernovae (CCSNe) has heavily re-lied on numerical simulations. This is mainly becausenearby CCSNe are rare (van den Bergh & Tammann1991; Cappellaro et al. 1993; Tammann et al. 1994;Reed 2005; Diehl et al. 2006; Maoz & Badenes 2010;Li et al. 2011) and, in fact, SN1987A is the only oneclose enough to extract some useful information on whathappened deep inside the massive star from, amongother things, the detection of neutrinos (Bionta et al.1987; Hirata et al. 1987). Since the CCSNe are intrin-sically multi-scale, multi-physics and multi-dimensional(multi-D) phenomena, their mechanism can be ad-dressed only with detailed numerical simulations.Unfortunately, even the most advanced multi-D sim-ulations of CCSNe employed approximations one wayor another in their numerical treatment of neutrinotransport (Marek & Janka 2009; M¨uller et al. 2012;Bruenn et al. 2013; Takiwaki et al. 2014; Bruenn et al.2016; Dolence et al. 2015; Lentz et al. 2015; Melson et al.2015; Kuroda et al. 2016; Skinner et al. 2016; O’Connor & Couch2015; Pan et al. 2016; Just et al. 2015; Summa et al.2016; Roberts et al. 2016; Andresen et al. 2017; Burrows et al.2016). Most of them somehow integrated out the angu-lar degrees of freedom in momentum space or neglectednon-radial fluxes in neutrino transport. Ott et al. (2008)is the only exception, in which they conducted time-dependent 5-dimensional simulations in spatial axisym-metry. However, they ignored relativistic correctionscompletely, dropping all fluid-velocity-dependent terms,which are crucial for qualitatively correct descriptionsof the angular distribution of neutrinos in momentumspace (see e.g., Buras et al. (2006); Lentz et al. (2012)).The best way to calibrate all these approximate meth-ods should be to compare them with simulations thatsolve full Boltzmann equations, retaining the angulardegree of freedom, for neutrino transport. Under ax-isymmetry in space, this is possible now indeed and wehave achieved such simulations with the K computer inJapan, one of the currently available best supercomput-ers with ∼ − + ++. Unless otherwise stated, we work in units with c = G = 1, where c is the speed of light and G is thegravitational constant. METHODS AND MODELSWe solve numerically the equations of neutrino-radiation hydrodynamics. We apply the so-calleddiscrete-ordinate (DO) method to the Boltzmann equa-tions for neutrino transport, taking fully into accountspecial relativistic effects by virtue of a two-energy-grid technique (Nagakura et al. 2014). It has alreadyincorporated general relativistic capabilities as well, apart of which is utilized to track the proper motion ofPNS (Nagakura et al. 2017). The hydrodynamics andself-gravity are still Newtonian: the so-called centralscheme of second-order accuracy in both space and timeis employed for the former and the Poisson equation issolved for the latter.It should be noted that our treatment of neutrinotransport is essentially different from other approximatemethods such as the M1 scheme that are commonly em-ployed in the currently most elaborate supernova sim-ulations and are based on the truncated moment for-malism one way or another. It is combined with theray-by-ray approximation in some applications (see e.g.,M¨uller et al. (2012)). In the moment formalism theBoltzmann equation is angle-integrated in momentumspace to obtain an infinite number of equations for an-gular moments, which are then truncated at some or-der somehow (see Sec. 4 for more details). Such ap-proximations reduce the computational cost drastically.On the other hand, they inevitably introduce the so-called closure relation among lo-order moments, whichare the artificial prescriptions to make the truncatedequations self-contained. Although the validity of thoseprescriptions has been assessed for spherically symmet-ric cases in the literature (Richers et al. 2017), it re-mains to be demonstrated in multi-dimensional and,
ASTEX Axisymmetric Boltzmann simulations of CCSNe
200 400 600 r a d i u s ( k m ) (a) L ( e r g / s ) E m ( M e V ) time after bounce (ms) (b) LE LS ν e LS ν e - LS ν x FS ν e FS ν e - FS ν x η M h ( - M ⊙ ) LS ( η )FS ( η )LS (M h )FS (M h ) (c) T a d v / T h ea t χ time after bounce (ms) (d) T adv /T heat χ LSFS
Figure 1. (a) Shock radii as functions of time. The color-shaded regions show the ranges of the shock radii, red for the LS EOSand blue for the FS EOS. The solid lines are the angle-average values. For comparison, the corresponding results in sphericalsymmetry are displayed with dashed lines. (b) Time evolutions of the angle-integrated luminosities ( L , solid lines) and theangle-averaged mean energies ( E m , dashed lines) for different species of neutrinos. Both of them are measured at r = 500km.(c) Neutrino-heating efficiency (solid lines) and total mass in the gain region (dashed lines). The heating efficiency is definedas the ratio of the energy deposition rate in the gain region to the sum of the neutrino luminosities of ν e and ¯ ν e (d) The ratioof the advection to heating timescales ( T adv /T heat , with solid lines) and the χ parameter (dashed lines). The dotted black linerepresents T adv /T heat = 1 and χ = 3 for reference. Nagakura et al.
0 25 50 75 100time after bounce (ms) 50 100 150 200 250 r a d i u s ( k m ) LS: |V θ |LS: |V θ |
0 25 50 75 100time after bounce (ms) 50 100 150 200 250 r a d i u s ( k m ) FS: |V θ | V a n i s o / c FS: |V θ | V a n i s o / c
0 5 10 15 20time after bounce (ms) 20 40 60 80 100 r a d i u s ( k m ) LS: ω B V a n i s o / c LS: ω B V a n i s o / c
0 5 10 15 20time after bounce (ms) 20 40 60 80 100 r a d i u s ( k m ) LS: ω B FS: ω B ω B ( m s - ) LS: ω B FS: ω B ω B ( m s - ) Figure 2.
Color contours showing time evolutions of the radial profile of angle-averaged lateral velocities ( | v θ | ) until 100msafter bounce (top) and of Brunt-V¨ais¨al¨a frequencies in the very early post-bounce phase up to 20ms (bottom). Left and rightpanels present LS- and FS EOS models, respectively. The solid line indicates the minimum shock radius in each panel. Notethat a positive (negative) sign is assigned to imaginary (real) Brunt-V¨ais¨al¨a frequencies in this figure for convenience. ASTEX Axisymmetric Boltzmann simulations of CCSNe r, θ ) covering 0 ≤ r ≤ ◦ ≤ θ ≤ ◦ in the meridian section. Wedeploy 384( r ) × θ ) grid points. Momentum spaceis also discretized non-uniformally with 20 energy meshpoints covering 0 ≤ ε ≤ θ ) ×
6( ¯ φ ) angu-lar grid points over the entire solid angle. The polar andazimuthal angles (¯ θ, ¯ φ ) are locally measured from the ra-dial direction. Three neutrino species are distinguished:electron-type neutrinos ν e , electron-type anti-neutrinos¯ ν e and all the others collectively denoted by ν x .We pick up a non-rotating progenitor model of11.2 M ⊙ from Woosley et al. (2002). We employ twonuclear EOS’s: Lattimer & Swesty’s EOS with the in-compressibility of K = 220MeV (Lattimer & Swesty1991) and Furusawa’s EOS derived from H. Shen’srelativistic mean-field EOS with the TM1 parameterset (Furusawa et al. 2011, 2013); the former is softerthan the latter (see Sumiyoshi et al. (2004)). In the fol-lowing, they are referred to as the ”LS” and ”FS” EOS’s,respectively . The choice of EOS’s is simply based onthe fact that most of previous simulations employedone of these EOS’s. We are currently running similarsimulations, but with another EOS: Togashi’s nuclearEOS based on the variational method with realisticnuclear potentials (Togashi & Takano 2013) extendedby Furusawa et al. (2017) to sub-nuclear densities; ittakes into account the full ensemble of heavy nuclei innuclear-statistical equilibrium (NSE). The results willbe reported elsewhere (Nagakura et al. 2018). Neutrino-matter interactions are based on those given by Bruenn(1985) but we have implemented the up-to-date electroncapture rates for heavy nuclei (Juodagalvis et al. 2010;Langanke & Mart´ınez-Pinedo 2000; Langanke et al.2003); they are calculated based on the abundance ofheavy nuclei obtained in the FS EOS; the same ratesare employed in the LS EOS model just for simplicity;note also that the LS EOS employs a single-nucleusapproximation and the detailed information on the pop-ulation of various nuclei is unavailable. In the cur-rent simulations we incorporated the non-isoenergeticscatterings on electrons and positrons as well as thebremsstrahlung in nucleon collisions. We refer readers The maximum gravitational masses at zero temperature andnon-rotating neutron stars are 2 . M ⊙ for LS EOS and 2 . M ⊙ for FS EOS, respectively. LS - entropy1612840
500 km
LS - |V| FS - entropyFS - |V|3 x 10 (cm/s) Figure 3.
Snapshots of entropy per baryon (upper) andfluid-speed (lower) at t = 200ms. Left and right panels arefor the LS- and FS EOS, respectively. to Nagakura et al. (2014, 2017); Sumiyoshi & Yamada(2012) for more details of our code.We start the simulations in spherical symmetry andswitch them to axisymmetric computations at ∼ .
1% in the radial ve-locities at 30 ≤ r ≤ t = 300ms after bounce. DYNAMICSAs displayed in Fig. 1(a), the shock wave producedat core bounce expands rather gradually with time forthe LS EOS and its maximum radius reaches ∼ t = 300ms. For the FS EOS, on the other hand,the shock wave stalls at r ∼ t ∼ t ∼ r ∼ t ∼ ∼ Nagakura et al. ferences have already appeared in the post-shock flowsby this time in fact.In the top two panels of Fig. 2, we compare the angle-averaged amplitudes of lateral velocity for the two mod-els. The more reddish the color is, the stronger thelateral motions are. It is apparent that they become ap-preciable initially at t ∼ r ∼ r ∼ − t ∼ L ) andmean energies ( E m , defined as the ratio of energy densityto number density) are almost identical between the twocases (Fig. 1(b)). It should be noted, however, that theneutrino-heating efficiency is different, being higher forthe LS EOS (see solid lines in Fig. 1(c)). This is mainlybecause the total baryon mass in the gain region, whereheating dominates over cooling and the net heating oc-curs, is consistently larger for the LS EOS than for theFS EOS (dashed lines in the same panel). This in turnseems to be a consequence of the turbulent motions thatare more vigorous for the LS EOS as we mentioned inthe previous paragraphs.Figure 3 compares the entropy and velocity distribu-tions between the two models at t = 200ms. Their post-shock morphologies are quite similar to each other andonly the scales are different. In fact, the convection is dominant over SASI in most of the post-bounce phasefor both models (see the χ parameter (Foglizzo et al.2006; Iwakami et al. 2014) in Fig. 1(d)). In the samepanel, we also show the ratio of the advection timescale( T adv = M g / ˙ M with M g and ˙ M denoting the mass in thegain region and the mass accretion rate, respectively) tothe heating timescale ( T heat = | E tot | / ˙ Q ν with E tot and˙ Q ν being the total energy and the heating rate in thegain region, respectively) as solid lines. One can seethat it is consistently larger for the LS EOS than for theFS EOS, meaning that the former has more favorableconditions for shock revival than the latter.The decline of this ratio near the end of the simula-tion for the LS EOS in spite of a continuous growth ofthe maximum shock radius is an artifact originated fromour choice of the minimum shock radius in the evalua-tion of the ratio. As displayed in Fig. 1(a), the minimumshock radius is still decreasing with time at the end ofthe simulation. Then the volume of gain region is under-estimated and, as a result, T heat is overestimated. Thefact that the ratio occasionally exceeds unity but stillyields no shock revival for the FS EOS indicates thatthe criterion is not a rigorous condition, which is un-derstood also from the uncertainty in its definition justmentioned. We do not intend to discuss the applicabil-ity of the diagnostics any further in this paper but westill think it is useful in judging, albeit roughly, whichmodel is closer to shock revival. ν -DISTRIBUTIONS IN MOMENTUM SPACENext we turn our attention to novel features of theneutrino distributions in momentum space. We findin our calculations significant non-axisymmetry with re-spect to the radial direction in the neutrino angular dis-tributions. It is produced by lateral inhomogeneities inmatter, which are in turn generated by hydrodynamicalinstabilities. The asymmetry hence appears inevitablyin multi-D simulations.Figure 4 shows as an example the angular distribu-tions of ν e with an energy of ε = 11 . r = 23km (red surface) whilethey become forward peaked (green and blue surfaces)as the radius increases, a fact that is well known. Whatis really new here is that they are non-axisymmetric withrespect to the radial direction, which is more apparentin Fig. 5, in which the isotropic contributions are sub-tracted from the original distributions and the resultant ASTEX Axisymmetric Boltzmann simulations of CCSNe e r - e φ -e θ - Figure 4.
Angular distributions of ν e in momentum spaceat t pb = 15ms for the LS EOS. Different colors correspond todifferent radial positions (red: r = 23km, green: r = 39km,blue: r = 49km) along the radial ray with the zenith angleof θ = 8 π/
15. The neutrino energy is ε = 11 . ones are re-normalized by their maximum values. Notethat the feature is robust, occurring irrespective of neu-trino energies or species.It should be mentioned, however, that the non-axisymmetric angular distributions obtained in the cur-rent simulations still have a symmetry with respect tothe azimuthal angle ( ¯ φ ) in momentum space. This isdue to the fact that these are non-rotating models andthere is a mirror symmetry with respect to the planespanned by ¯ e r and ¯ e θ in momentum space in the ab-sence of rotation. Once rotation is taken into account,the symmetry is lost even in (spatial) axisymmetry.This is the reason why we do not assume this symmetryin our code. In 3D simulations, no symmetry remainsin the angular distribution in momentum space. Itscharacterization is an interesting subject of spatially 3Dsupernova simulations with multi-angle neutrino trans-port, which are currently being undertaken and will bereported elsewhere later.The multi-angle treatment of neutrino transport inour simulations enables us to evaluate the so-calledEddington tensor ( k ij ), which characterizes these non- axisymmetric angular distributions more quantitatively.The Eddington tensor is obtained from the neutrino dis-tribution function ( f ) as follows: we first define the sec-ond angular moment M µν as M µν ( ε ) ≡ ε Z f ( ε, Ω m ) p µ p ν d Ω m , (1)where p µ is the four-momentum of neutrino and ε andΩ m are the corresponding energy and solid angle mea-sured in the fluid-rest frame; then the Eddington tensor k ij is given as k ij ( ε ) ≡ P ij ( ε ) E ( ε ) , (2)where P ij and E are defined from M µν as P ij ( ε ) ≡ γ iµ γ jν M µν ( ε ) , (3) E ( ε ) ≡ n µ n ν M µν ( ε ) , (4)with n µ and γ iµ (= δ iµ + n i n µ ) being the unit vector or-thogonal to a hypersurface of constant coordinate timeand the projection tensor onto this hypersurface, respec-tively.We pay particular attention here to one of the off-diagonal components of the Eddington tensor, k rθ ,which are zero in spherical symmetry in space, i.e., theyare a measure of genuinely multi-dimensional trans-fer. The left panel in Fig. 6(a) shows k rθ for ν e withthe mean energy at each point. As expected, it isalmost zero inside the PNS, where matter is opaqueenough to make the neutrino distribution isotropic. Itbecomes non-zero outside the PNS, however, and in-creases with radius in accord with the appearance ofthe non-axisymmetric structures in the neutrino angu-lar distribution (see Fig. 4). In fact, the k rθ correspondsto the mode with ℓ = 2 , m = 1 in the spherical harmon-ics expansion of the distribution function.The right panel in Fig. 6(a) compares k rθ obtainedfrom our simulation with that which is evaluated ac-cording to the M1 prescription: the Eddington tensor inthe M1 prescription ( k ij M1 ) is obtained by replacing P ij in Eq. (3) with P ij M1 ( ε ) = 3 ζ ( ε ) − P ij thin ( ε ) + 3(1 − ζ ( ε ))2 P ij thick ( ε ) , (5)where ζ is referred to as the variable Eddington factor,which we set as ζ ( ε ) = 3 + 4 ¯ F ( ε ) p − F ( ε ) . (6)In this expression, ¯ F denotes the so-called flux factor,which is the energy-flux normalized with the energy den-sity in the fluid-rest frame. The flux factor that we use Nagakura et al. (a) e r - e φ -e θ - θ = π /3, φ = 2 π /3 - - (b) e r -e φ -e θ - θ = π /3, φ = 4 π /3 - - Figure 5.
Similar to Fig. 4 but the deviations from spherical symmetry emphasized and viewed from different angles: (a)¯ θ = π/ φ = 2 π/ θ = π/ φ = 4 π/
3. In each panel, the minimum is subtracted isotropically from the originalangular distribution and the resultant distribution is normalized so that the maximum value should be always identical. Theblue surface corresponds to the one with the same color in Fig. 4 while the purple surface shows another subtracted surface atthe same radius but at a different zenith angle, θ = 17 π/ in this paper is measured in the fluid-rest frame (seeShibata et al. (2011) for another option);¯ F ( ε ) = h µν H µ ( ε ) H ν ( ε ) J ( ε ) ! / , (7)where J and H µ can be expressed in terms of M µν as J ( ε ) = u µ u ν M µν ( ε ) ,H µ ( ε ) = − h µα u β M αβ ( ε ) , (8)with u µ and h µν (= δ µν + u µ u ν ) being the fluid fourvelocity and the projection tensor onto the fluid-restframe, respectively. The optically thick and thin lim-its of P ij are denoted by P ij thick and P ij thin (Just et al. 2015; Shibata et al. 2011; O’Connor & Couch 2015;Kuroda et al. 2016), which are written as P ij thick ( ε ) = J ( ε ) γ ij + 4 V i V j H i ( ε ) V j + V i H j ( ε ) ,P ij thin ( ε ) = E ( ε ) F i ( ε ) F j ( ε ) F ( ε ) , (9)where V i denotes the three dimensional vector of fluidvelocity. F i can be expressed in terms of M µν as F i ( ε ) = − γ iµ n ν M µν ( ε ) . (10)As clearly seen in this panel, the values of k rθ are sub-stantially different between the two cases. We find that ASTEX Axisymmetric Boltzmann simulations of CCSNe (a) 300 km100 km k r θ BZ k r θ BZ - k r θ M1 k r θν e k r θν e - (H θ /J) ν e (H θ /J) ν e - Figure 6. (a) The ( rθ ) component of the Eddington tensor ( k rθ ) for ν e in the northern hemisphere obtained in our simulationfor the FS EOS (left) and its deviation from the M1 prescription (right). The values of k rθ are evaluated at the mean neutrinoenergy at each point. (b) k rθ for ν e (left) and ¯ ν e (right) on a smaller spatial scale of 100km. The neutrino energy is fixedto 8 . H θ /J with H and J being the energy-flux and energydensities measured in the fluid-rest frame, respectively. The time is t = 190ms in all cases. Nagakura et al. such discrepancies in k rθ are rather generic, being insen-sitive to the choice of the prescription for the Eddingtonfactor (see Just et al. (2015) for various options). Theyare also systematic in the sense that the increase in thenumber of grid points in the M1 prescription does notreduce the difference. This is in contrast to our ap-proach, in which the accuracy is simply improved withthe resolution.Moreover, we find in k rθ an intriguing correlation/anti-correlation between ν e and ¯ ν e . The two panels ofFig. 6(b) compare k rθ for ν e and ¯ ν e with the sameenergy of ε = 8 . . > . k rθ roughly coincideswith that of the lateral neutrino flux, which is shown inFig. 6(c). In fact, it is apparent that the lateral flux isoriented in the opposite directions for ν e and ¯ ν e . This isin turn due to the Fermi-degeneracy of ν e at r . ν e and ¯ ν e . Since neutrinos flow from high to low ν number density regions in the diffusion regime, thefluxes of ν e and ¯ ν e should be naturally anti-correlatedas a result of the opposite trend in the number densitiesof ν e and ¯ ν e . We do not know for the moment how thisanti-correlation in the fluxes is transferred to that in k rθ . It will be necessary to analyze more in detail theequations of motion for higher moments including k rθ .Importantly, the anti-correlation is then carried tolarger radii by the radial flux and remains non-vanishingeven at r ∼ ν e is no longer degenerate. Onthe other hand, at even larger radii, where matter is op-tically thin to neutrinos, k rθ is correlated with the locallateral velocity of matter due to relativistic aberration.Note that this positive correlation at large distances isless remarkable than the anti-correlation in the vicinityof PNS (see the equatorial region in Fig. 6(b)), since theangular distribution is no longer determined locally andthe correlation is somewhat smeared out.As will be discussed in Sec. 6, the appropriate treat-ment of k rθ is related with the accurate calculationof the neutrino flux, in particular its lateral compo-nent (see Eqs. (11) and (12)). It is true that thesecorrelation/anti-correlation look rather minor but theymay play an important role through the lateral fluxesof neutrinos. In fact they clearly indicate the intricacyof neutrino transport in non-spherically dynamical set-tings. It will be interesting to see how well the M1scheme can reproduce these features and to conceivepossible improvements of its prescription. ANGULAR RESOLUTION IN MOMENTUMSPACEThis study is the first ever attempt to perform spa-tially 2D supernova simulations with multi-angle andmulti-energy neutrino transport, taking into account allspecial relativistic effects completely. It is a legitimateconcern, however, that the current simulations may nothave a sufficient numerical resolution especially in mo-mentum space (Richers et al. 2017). In this section wehence discuss this resolution issue, focusing on the an-gular resolution in momentum space.For that purpose we perform a new high-resolutionsimulation for the early post-bounce phase, whereas forthe discussion of the late post-bounce phase we employthe results of our previous analyses (Richers et al. 2017)of time-independent solutions of the Boltzmann equa-tions for neutrinos in given matter distributions; closecomparisons were made with the data obtained withMonte Carlo Simulations (Richers et al. 2015). Notethat although the use of the time-independent solutionsfor the fixed matter distributions enabled us to conductrigorous comparisons, its applicability may be limitedto the late post-bounce phase, where the time scale ofvariations in the background is indeed long. For the ear-lier phase, however, we need to consider time-dependentsolutions. We hence run a higher-resolution simulation,in which the time evolutions of both neutrino and mat-ter distributions are computed for only 15ms from thebounce with the LS EOS. We compare the results so ob-tained with the original ones to see to what extent theangular resolution could affect the outcome. Note, how-ever, that the comparisons are not so clear-cut as in theprevious paper, since the matter dynamics in this phaseis chaotic and small perturbations induced by the changein the angular resolution modify not only the neutrinodistributions but also the matter configurations in thebackground substantially.Richers et al. (2017) demonstrated that our Boltz-mann solver tends to underestimate the forward peakin the angular distributions of neutrinos in momentumspace at large radii if the number of the angular meshpoints is not large enough. This is actually just as ex-pected and was indeed pointed out by Yamada et al.(1999) in their 1D study. As a matter of fact, neutri-nos are moving almost radially at large distances fromthe neutrino sphere no matter what happens to them atsmall radii and if the angular spread becomes smallerthan the smallest width of the angular bin employed inthe Boltzmann solver, it is no longer resolved.Such properties of our Boltzmann solver should havesome implications for the success or failure of explo-sion in our simulations, since the underestimation of the
ASTEX Axisymmetric Boltzmann simulations of CCSNe S / n B ( k B ) NormalHigh 0.1 0.2 0.3 0.4 0.5 Y e | V | / c R (km) 0.01 0.02 0.03 0.04 0.05 0 30 60 90 120 150 | V θ | / c R (km)
Figure 7.
Angle-averaged radial profiles of fluid quantities. Upper left: entropy per baryon. Upper right: electron fraction.Bottom left: fluid speed. Bottom right: absolute values of lateral velocity. The red line shows the result of the normal-resolutionwhile the blue lines correspond to the high-resolution simulation. The time is t = 15ms post-bounce. forward peak in the angular distribution in momentumspace leads in turn to the overestimation of the localnumber density of neutrinos and, as a result, the over-estimation of neutrino heating in the gain region. Onthe other hand, Richers et al. (2017) also found that thefinite energy resolution tends to underestimate the neu-trino heating. We then surmise from these results thatthe volume-integrated net energy deposition in the gainregion is probably underestimated in the current simu-lations by a few percent.For the study of the resolution dependence in the earlypost-bounce phase, we conduct a high-resolution simu-lation for a short period as mentioned earlier. This timethe matter distribution is not fixed but calculated justas in the ordinary run. We deploy 14(¯ θ ) ×
10( ¯ φ ) an-gular grid points over the entire solid angle while spaceand energy grids are unchanged from the normal run.In Fig. 7, we compare the radial profiles of some angle-averaged quantities at 15ms after bounce between the models with the normal and high angular resolutions.As can be seen in this figure, the prompt shock waveis a bit faster and reaches a larger radius in the high-resolution model than in the normal-resolution model(upper left panel); in association with this, the delep-tonization behind the shock is slightly stronger in theformer around 20 ≤ r ≤ Nagakura et al. F NormalHigh θ = π /4| 0 0.2 0.4 0.6 0.8 1 F θ = π /4| θ = 3 π /4| 0.3 0.4 0.5 0.6 0.7 k rr Normal (BZ)High (BZ)Normal (M1)High (M1) θ = π /4| θ = 3 π /4| 0.3 0.4 0.5 0.6 0.7 k rr θ = π /4| θ = 3 π /4|-2-1.5-1-0.5 0 0.5 1 1.5 2 0 30 60 90 120 150 k r θ ( - ) R (km) θ = π /4| θ = 3 π /4| -2-1.5-1-0.5 0 0.5 1 1.5 2 0 30 60 90 120 150 k r θ ( - ) R (km) θ = π /4| θ = 3 π /4| Figure 8.
The flux factor (top), and the rr (middle) and rθ (bottom) components of the Eddington tensor for electron-typeneutrinos. The left column presents the radial profiles along the radial ray with θ = π/
4, while the right one displays the samequantities but for θ = 3 π/
4. The colors of lines and the time of the snap shot ( t = 15ms post-bounce) are the same as in Fig. 7. ASTEX Axisymmetric Boltzmann simulations of CCSNe θ = π/ θ = 3 π/ ν e distributionat 15ms after bounce. The neutrino energy is set to theaverage value at each point. In the top panels, the fluxfactors ( ¯ F ) defined in Eq. (7) are shown. One immedi-ately recognizes that it is systematically smaller for thehigh-resolution case in the post-shock region. This isnot directly related with the angular resolution, though.Instead it is simply because the shock radius is largerin the high-resolution run and, as a result, the flux fac-tor increases more slowly from the optically thick limit( ¯ F = 0) to the thin limit ( ¯ F = 1). On the other hand,the flux factor is always smaller for the normal case thanfor the high-resolution case at large radii. This is a di-rect resolution effect, i.e., the low-resolution simulationfails to reproduce the forward peak in the angular dis-tribution at large radii.The rr components of the Eddington tensor, k rr , areshown in the middle panels of Fig. 8. It is observedthat they also increase a bit more slowly initially in thehigh-resolution run. This is again a mere consequenceof the larger shock radius in that case. In these pan-els, we also display as additional dotted lines the samecomponents of the Eddington tensor that are obtainedwith the M1 prescription. Except in the inner opticallythick region, they are always slightly greater than thoseobtained with the Boltzmann code for both resolutions.Considering the result in Richers et al. (2017) that low-resolution computations with the Boltzmann solver tendto underestimate k rr , one may think that the results ofthe M1 prescription is closer to the true values. It shouldbe noted, however, that the differences found here in k rr between the Boltzmann and M1 results are larger thanthose obtained in Richers et al. (2017) (see Fig.17 intheir paper). This may imply that the M1 prescriptionhas its own problem in reproducing k rr for highly-timedependent and highly-inhomogeneous matter distribu-tions considered here. This issue will be further studiedin our forthcoming paper. It is incidentally pointed outthat the M1 prescription needs the flux factor to ob-tain the Eddington tensor (see Eqs. (5) and (6)). Inthe present comparison it is provided by the Boltzmannsolver although it should be calculated on its own inthe actual simulations with the M1 approximation. Itis hence desirable to make comparisons, employing theresults of such M1 simulations, which is another subjectworth further investigations. The bottom panels in Fig. 8 are again the Eddingtontensors but for the rθ component k rθ this time. It shouldbe noted first that k rθ is very sensitive to the mattermotion in the background. As a result, their profiles arequite different between the normal and high-resolutionsimulations and it is rather difficult to discuss the con-vergence in the current dynamical setting. Nevertheless,it is evident that the Boltzmann and M1 results are sub-stantially different from each other even qualitatively inthe semi-transparent region although they agree in boththe optically thin and thick limits irrespective of reso-lutions. This is indeed consistent with the findings byRichers et al. (2017), who also came to the same conclu-sion that the difference in k rθ between the Boltzmanntransport with multi-angles and the M1 prescription inthe semi-transparent regime is intrinsic and never re-duced by increasing resolution. As will be demonstratedin Sec. 6, inaccurate k rθ may give a ∼
10% level of errorsin the neutrino luminosity and, more importantly, willlead to qualitatively wrong lateral fluxes of neutrinos inthe semi-transparent region.In Fig. 9, we compare the angular distributions in mo-mentum space obtained with the two resolution. Notethat, the isotropic contributions are subtracted as pre-viously in these pictures so that the anisotropies couldbe better recognized. In panel (a), the purple surface isidentical to the one presented in Fig. 5, while the blacksurface is the high-resolution counterpart. In Fig. 9(b),we change the viewing angle to facilitate readers’ un-derstanding of the non-axisymmetric features. As men-tioned above, since the matter distributions in the back-ground are different between the two cases, the neutrinoangular distributions differ qualitatively. It is impor-tant, however, that the degree of asymmetry is evenmore prominent in the high-resolution simulation. Thisis again consistent with the finding in Richers et al.(2017) that k rθ tends to be underestimated in low-angular resolution simulations (see the right panel ofFig.15 in their paper). POSSIBLE IMPLICATIONS OF OFF-DIAGONALCOMPONENTS ON SUPERNOVA DYNAMICSThe existence of the non-axisymmetric features in theangular distributions of neutrinos and the appearanceof the non-vanishing off-diagonal components of the Ed-dington tensor as a result are the main novel findings inthis paper. The legitimate question then is how signifi-cant they are for supernova dynamics. In order to fullyaddress this issue, it is required to run additional simula-tions with some approximate neutrino transport schemesuch as the ray-by-ray and/or M1 methods, which eithercompletely ignore or employ a makeshift prescription for4
Nagakura et al. (a) e r - e φ -e θ - θ = π /3, φ = 2 π /3 - - (b) e r -e φ -e θ - θ = π /3, φ = 4 π /3 - - Figure 9.
The same picture as in Fig. 5 but for two different angular resolutions. The purple wired frame is identical to thesame purple one in Fig. 5. The black one is a high-resolution counterpart. these non-axisymmetric features, for the same progeni-tor, resolution, EOS and input physics and make a de-tailed comparison, which is certainly beyond the scope ofthis paper. Instead, in this section, we compare differentcomponents of the Eddington tensor quantitatively anddiscuss how the off-diagonal components might becomeimportant.Note first that the equations for both the zeroth andfirst moments of the angular distribution include in prin-ciple all components of Eddington tensor (see, e.g., Eqs.(3.37) and (3.38) in Shibata et al. (2011)). It should bealso pointed out that reaction rates of some neutrino-matter interactions such as non-isoenergetic scatteringsand pair processes depend on higher-order moments in-cluding the Eddington tensor. The neglect of them mayhave some implications for CCSNe dynamics. Althoughthis is an interesting issue and is in fact on our to-do- list, in the following, we will limit our discussion to theadvection part of the neutrino transport.The principal part of the equations for the first angu-lar moment or the flux can be approximately written as(see also Eq. (3.38) in Shibata et al. (2011)) ∂ t ( F r ) ∼ − ∂ r ( Ek rr ) − r ∂ θ ( Ek rθ ) , (11) ∂ t ( F θ ) ∼ − ∂ r ( Ek rθ ) − r ∂ θ ( Ek θθ ) , (12)where we ignore collision terms and assume that thespacetime is flat and the background matter is axisym-metric and non-rotating. The off-diagonal componentof Eddington tensor k rθ appears in the second and firstterms on the right hand side of Eqs. (11) and (12), re-spectively. Note that it does not show up in the principalpart of the zeroth-order equation for the energy density. ASTEX Axisymmetric Boltzmann simulations of CCSNe R a t i o NormalHigh θ = π /4For F r θ = 3 π /4For F r θ = 3 π /4For F θ θ = 3 π /4For F θ Figure 10.
The radial profiles of the absolute ratios of ∂ θ ( Ek rθ ) /r to ∂ r ( Ek rr ) (upper panels) and ∂ r ( Ek rθ ) to ∂ θ ( Ek θθ ) /r (lower panels). These quantities measure the relative importance of the terms on the right hand side of Eqs. (11) and (12) forthe r- and θ - components of neutrino flux. The left and right panels show profiles along the radial rays with θ = π/ π/ t = 15ms postbounce. In Fig. 10, we display radial profiles of the absolutevalues of the ratios of ∂ θ ( Ek rθ ) /r to ∂ r ( Ek rr ) (upperpanels) and ∂ r ( Ek rθ ) to ∂ θ ( Ek θθ ) /r (lower panels) ontwo radial rays with θ = π/ π/ k rr with k rθ being at most 10%.This is certainly not a large value but still may not beignored, since, as Burrows et al. (2016) claims, an ac-cumulation of seemingly minor effects may turn out tobe crucially important. On the other hand, k rθ playsmore important roles in the equation for the lateral com-ponent of neutrino flux as demonstrated in the bottompanels. In fact, the ratio of radial gradient of Ek rθ to the lateral gradient of Ek rθ /r exceeds unity in some post-shock regions. This is also the case for the result of thehigh-resolution simulation although the radial profilesthemselves are quite different from those in the normal-resolution run, which is a consequence of the fact thatmatter distributions in the background become differentbetween the two cases.As discussed in sections 4 and 5, the M1 prescriptionis not very successful in reproducing k rθ in the semi-transparent region particularly in non-spherical settings.Although there is no artificially preferred direction in theM1 transport unlike in the ray-by-ray approximation,the lateral neutrino flux may be still inaccurate. It ismisleading to argue that the Eddington tensor is repro-duced again very well in the transparent regime with itsoff-diagonal component becoming negligible comparedwith the dominant k rr . This is because the errors in the6 Nagakura et al. semi-transparent region will not be confined there andspread to transparent region in time. The errors in theflux will lead to those in the Eddington tensor throughthe closure relation, which will again contribute to errorsin the flux. This may eventually affect CCSNe dynam-ics. The quantitative assessment of this effect requiresdetailed comparisons in collaboration with other groupsand is much beyond the scope of this first report of ournew simulations.It is finally mentioned that the above analysis is basedon the result of the early post-bounce phase, in whichthe semi-transparent region is highly dynamical owing tothe prompt convection, and the k rθ effect may be muchsmaller in the later phase. The errors in early times havesome influences on the evolution in later times in prin-ciple, though. It should be also added that convectionsin the proto-neutron star and other hydrodynamical in-stabilities such as SASI and convections in the heatingregion occur more often than not even in the late phase.It is repeated that the quantitative assessments are cer-tainly in order and will be studied in subsequent papers. COMPARISON WITH PREVIOUS WORKSIn this section, we attempt to make a comparisonof our results with other CCSNe simulations. Thesame progenitor model has been employed by many au-thors so far (M¨uller et al. 2012; Takiwaki et al. 2014;Summa et al. 2016). It is mentioned first that our re-sults are qualitatively in line with them in that softerEOS’s are advantageous for shock revival. It shouldbe pointed out, however, that there are some stud-ies, in which softer EOS including the LS EOS havesmaller shock radii initially than the stiffer ones (see,e.g., Fischer et al. (2014)), in apparent contradictionwith our results.According to Fischer et al. (2014), the difference inthe shock trajectory originates mainly not from the stiff-ness of EOS but from the treatment of electron captureson heavy nuclei: representative heavy nuclei tend to besmaller in the softer LS EOS than in the stiffer STOSEOS, which is essentially the same as our FS EOS exceptfor the single-nucleus approximation in the former, re-sulting in the greater deleptonization in the LS EOS dur-ing the collapse phase; this in turn leads to the smallerinner core and hence the weaker prompt shock wave forthe LS EOS. It should be recalled, however, that theelectron capture rates employed in our simulation withthe LS EOS are the same as those for the simulation withthe FS EOS. As a result, the effects just mentioned arenot taken into account in our current simulations andthe shock trajectories reflect the difference in the stiff-ness of EOS’s alone. The treatment of nuclear weak interactions consistentwith the EOS employed is important to compute CC-SNe dynamics accurately. We stress that the currentapproximate treatment is meant just for simplicity inmodels with EOS’s that employ the single-nucleus ap-proximation. We believe that multi-nucleus EOS’s areindispensable for the quantitative study of the nuclearweak interactions mentioned above. Such a study in-deed under way (Nagakura et al. 2018) with the multi-nucleus extension by Furusawa et al. (2017) of Togashi’sEOS (Togashi & Takano 2013), which is based on thevariational method for realistic nulcear potentials.It is also important to point out that the shock ex-pansion in our model looks less energetic than those inother simulations with the same progenitor model (seee.g., Takiwaki et al. (2014)). It is difficult to pin downthe cause of the discrepancy, since there are many dif-ferences in input physics as well as numerical methodsfor hydrodynamics and neutrino transport, but the ray-by-ray approximation employed for neutrino transportin their simulations may be one of the main causes ofthe difference. In fact, Skinner et al. (2016) pointed outthat the ray-by-ray approximation tends to artificiallyfacilitate explosion in 2D, enhancing sloshing motions inaxisymmetry. A similar concern was also expressed bySumiyoshi et al. (2015), in which they showed that theasymmetry in the neutrino heating tends to be overesti-mated in the ray-by-ray approximation. More detailedcomparisons in collaborations with other groups are re-quired to substantiate the claim, though. SUMMARY AND DISCUSSIONWe have presented the first report of spatially axisym-metric CCSNe simulations with the full Boltzmann neu-trino transport. We have found both similarities and dif-ferences between the two models with two different nu-clear EOS’s. On the one hand, the neutrino luminositiesand mean energies as well as the post-shock morpholo-gies except the scale are very similar between the two.This seems to be a consequence of the cancellation of thestronger bounce that would be expected in the softer LSEOS by the greater electron captures that produced thesmaller inner core in the LS EOS model. On the otherhand, the neutrino-heating efficiency and the mass inthe gain region are consistently higher for the LS EOS.This seems to be due to more vigorous turbulent mo-tions in the post-shock flow for the LS EOS than for theFS EOS, the fact which results in the greater expansionof the shock wave: it has reached ∼ ASTEX Axisymmetric Boltzmann simulations of CCSNe k rθ , on neutrino trans-port have been also discussed quantitatively: it plays anon-negligible role for the time evolutions of neutrinofluxes; it may give a ∼
10% level of contribution to theneutrino luminosity and, more importantly, can be adominant factor for the time evolution of lateral flux inthe semi-transparent region.We have found an interesting correlation/anti-correlationin k rθ between ν e and ¯ ν e depending on the radius. It isrelated with the lateral fluxes of these neutrinos. It willbe interesting to see how well the M1 approximationfares in reproducing these features and hence the lateralfluxes. The close comparison between our Boltzmannsolver and other approximate methods possibly in col-laboration with other groups will be indispensable toassess critically and quantitatively the significance ofthe findings in this paper for the CCSNe dynamics. Itwill also enable us to calibrate and possibly improvethe prescriptions, which should be given by hand in ap-proximate transport schemes. This is indeed importantpractically, since our method is very costly in terms ofrequired numerical resources.We have made an attempt to compare our results withthose obtained by other groups for the same progenitormodel. We have found that the general trend that softerEOS’s are favorable for shock revival is also true of oursimulations. On the other hand, the continuous shockexpansion observed for the softer LS EOS looks less en-ergetic than that found by others. Although this seemsto be consistent with the finding by Skinner et al. (2016)that the ray-by-ray approximation in spatial axisymme-try may artificially enhance shock revival, more detailedcomparisons are certainly necessary to draw some con-clusions.There are also certainly many other issues remainingto be addressed. The top priority is to make detailedcomparisons with other approximate methods to assess the importance of multi-angle treatments for supernovadynamics by possibly collaborating with other groups.We will also proceed to explore other progenitors withdifferent masses. The EOS dependence should be fur-ther clarified. Rotation is another concern, since theangular distribution in momentum space is then qual-itatively changed: e.g., the principal axis will not bealigned with the radial direction in general and anotheroff-diagonal component, k rφ , will no longer be vanish-ing. We are currently implementing general relativity inour code to investigate its influences, which are expectedto be non-negligible. The angular distributions for dif-ferent species of neutrinos we obtained in this study arevaluable in their own right for e.g. the analysis of col-lective oscillations of neutrino flavors (Duan et al. 2010;Mirizzi 2013; Capozzi et al. 2017; Izaguirre et al. 2017),which feed on the differences in the angular distributionsamong different neutrino species. They are currentlybeing investigated and the results will be reported else-where.H.N. acknowledges to C. D. Ott, S. Richers, L.Roberts, D. Radice, M. Shibata, Y. Sekiguchi, K. Ki-uchi and T. Takiwaki for valuable comments and dis-cussions. The numerical computations were performedon K computer, at AICS, FX10 at Information Tech-nology Center of Tokyo University, SR16000 at YITPof Kyoto University, and SR16000 and Blue Gene/Qat KEK under the support of its Large Scale Simula-tion Program (14/15-17, 15/16-08, 16/17-11), ResearchCenter for Nuclear Physics (RCNP) at Osaka University,the XC30 and the general common use computer sys-tem at the Center for the Computational Astrophysics,CfCA, the National Astronomical Observatory of Japan.Large-scale storage of numerical data is supported byJLDG constructed over SINET4 of NII. H.N and S.Fwere supported in part by JSPS Postdoctoral Fellow-ships for Research Abroad No. 27-348 and 28-472 andH.N was partially supported at Caltech through NSFaward No. TCAN AST-1333520. This work was sup-ported by Grant-in-Aid for the Scientific Research fromthe Ministry of Education, Culture, Sports, Scienceand Technology (MEXT), Japan (15K05093, 24103006,24105008, 24740165, 24244036, 25870099, 26104006,16H03986, 17H06357, 17H06365) and HPCI StrategicProgram of Japanese MEXT and K-computer at theRIKEN and Post-K project (Project ID: hp 140211,150225, 160071, 160211, 170230, 170031, 170304).REFERENCES Andresen, H., M¨uller, B., M¨uller, E., & Janka, H.-T. 2017,MNRAS, 468, 2032 Bionta, R. M., Blewitt, G., Bratton, C. B., Casper, D., &Ciocio, A. 1987, Physical Review Letters, 58, 1494 Nagakura et al.
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ASTEX Axisymmetric Boltzmann simulations of CCSNe19