Properties of a variational equation of state in core-collapse supernovae

H. Togashi (RIKEN)

PACIFIC2018 @ Kiroro, Hokkaido, February 14, 2017

1：Introduction

2：Supernova EOS with realistic nuclear forces

3：Application to astrophysical objects

4：Summary

Outline

Collaborators: M. Takano, K. Nakazato, Y. Takehara, S. Yamamuro, K. Sumiyoshi, H. Suzuki

common feature of models that include the appearance of ‘exotic’hadronic matter such as hyperons4,5 or kaon condensates3 at densitiesof a few times the nuclear saturation density (ns), for example modelsGS1 and GM3 in Fig. 3. Almost all such EOSs are ruled out by ourresults. Our mass measurement does not rule out condensed quarkmatter as a component of the neutron star interior6,21, but it stronglyconstrains quark matter model parameters12. For the range of allowedEOS lines presented in Fig. 3, typical values for the physical parametersof J1614-2230 are a central baryon density of between 2ns and 5ns and aradius of between 11 and 15 km, which is only 2–3 times theSchwarzschild radius for a 1.97M[ star. It has been proposed thatthe Tolman VII EOS-independent analytic solution of Einstein’sequations marks an upper limit on the ultimate density of observablecold matter22. If this argument is correct, it follows that our mass mea-surement sets an upper limit on this maximum density of(3.74 6 0.15) 3 1015 g cm23, or ,10ns.

Evolutionary models resulting in companion masses .0.4M[ gen-erally predict that the neutron star accretes only a few hundredths of asolar mass of material, and result in a mildly recycled pulsar23, that isone with a spin period .8 ms. A few models resulting in orbital para-meters similar to those of J1614-223023,24 predict that the neutron starcould accrete up to 0.2M[, which is still significantly less than the>0.6M[ needed to bring a neutron star formed at 1.4M[ up to theobserved mass of J1614-2230. A possible explanation is that someneutron stars are formed massive (,1.9M[). Alternatively, the trans-fer of mass from the companion may be more efficient than currentmodels predict. This suggests that systems with shorter initial orbitalperiods and lower companion masses—those that produce the vastmajority of the fully recycled millisecond pulsar population23—mayexperience even greater amounts of mass transfer. In either case, ourmass measurement for J1614-2230 suggests that many other milli-second pulsars may also have masses much greater than 1.4M[.

Received 7 July; accepted 1 September 2010.

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Supplementary Information is linked to the online version of the paper atwww.nature.com/nature.

Acknowledgements P.B.D. is a Jansky Fellow of the National Radio AstronomyObservatory. J.W.T.H. is a Veni Fellow of The Netherlands Organisation for ScientificResearch. We thankJ. Lattimer for providing the EOSdataplotted inFig. 3, and P. Freire,F. Ozel and D. Psaltis for discussions. The National Radio Astronomy Observatory is afacility of the US National Science Foundation, operated under cooperative agreementby Associated Universities, Inc.

Author Contributions All authors contributed to collecting data, discussed the resultsand edited the manuscript. In addition, P.B.D. developed the MCMC code, reduced andanalysed data, and wrote the manuscript. T.P. wrote the observing proposal andcreated Fig. 3. J.W.T.H. originally discovered the pulsar. M.S.E.R. initiated the survey thatfound the pulsar. S.M.R. initiated the high-precision timing proposal.

Author Information Reprints and permissions information is available atwww.nature.com/reprints. The authors declare no competing financial interests.Readers are welcome to comment on the online version of this article atwww.nature.com/nature. Correspondence and requests for materials should beaddressed to P.B.D. (pdemores@nrao.edu).

0.07 8 9 10 11

Radius (km)12 13 14 15

0.5

1.0

1.5

2.0

AP4

J1903+0327

J1909-3744

systemsDouble neutron sDouble neutron star sysy

J1614-2230

AP3

ENG

MPA1

GM3

GS1

PAL6

FSUSQM3

SQM1

PAL1

MS0

MS2

MS1

2.5 GR

Causality

Rotation

P < ∞

Mas

s (M

()

Figure 3 | Neutron star mass–radius diagram. The plot shows non-rotatingmass versus physical radius for several typical EOSs27: blue, nucleons; pink,nucleons plus exotic matter; green, strange quark matter. The horizontal bandsshow the observational constraint from our J1614-2230 mass measurement of(1.97 6 0.04)M[, similar measurements for two other millisecond pulsars8,28

and the range of observed masses for double neutron star binaries2. Any EOSline that does not intersect the J1614-2230 band is ruled out by thismeasurement. In particular, most EOS curves involving exotic matter, such askaon condensates or hyperons, tend to predict maximum masses well below2.0M[ and are therefore ruled out. Including the effect of neutron star rotationincreases the maximum possible mass for each EOS. For a 3.15-ms spin period,this is a =2% correction29 and does not significantly alter our conclusions. Thegrey regions show parameter space that is ruled out by other theoretical orobservational constraints2. GR, general relativity; P, spin period.

LETTER RESEARCH

2 8 O C T O B E R 2 0 1 0 | V O L 4 6 7 | N A T U R E | 1 0 8 3

Macmillan Publishers Limited. All rights reserved©2010

P. B. Demorest et al., NATURE 467 (2010)

Mass-radius relation of cold neutron stars Phase diagram of cold nuclear matter 2/20

1. IntroductionEquation of state (EOS) for infinite nuclear matter

plays important roles for astrophysical studies.

Neutron Stars: Stiffness (EOS at 0 MeV) ⇔ Self-gravity

K. Sumiyoshi @ NUFRA 2011

Nuclear EOS and Core-Collapse SupernovaeNuclear EOS at finite temperature is one of the crucial ingredients

Scenario of the Core-Collapse Supernovae (SNe) for the numerical simulations of Core-Collapse Supernovae.

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K. Sumiyoshi @ NUFRA 2011

Nuclear EOS and Core-Collapse SupernovaeNuclear EOS at finite temperature is one of the crucial ingredients

Scenario of the Core-Collapse Supernovae (SNe) for the numerical simulations of Core-Collapse Supernovae.

-  The stiffness of high-density nuclear matter-  Species of nuclides in hot matter Nuclear EOS

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Phase diagram of nuclear matter [based on HT et al., NPA 961 (2017) 78]

Uniform

Non-uniform

- SN-EOS should provide thermodynamic quantities in the wide ranges.

•  Temperature T : 0 ≤ T ≤ 30 MeV

•  Density ρ : 105.1 ≤ ρB ≤ 1015.0g/cm3

•  Proton fraction Yp : 0 ≤ Yp ≤ 0.50

Nuclear EOS for supernova simulations

- SN matter contains uniform and non-uniform phases.

5/20

Current status of SN-EOS

We aim to construct a new SN-EOS with the variational method starting from bare nuclear forces.

There is no SN-EOSs based on the microscopic many-body theory.

Skyrme-type effective interaction

Relativistic Mean Field Theory

(M. Oertel et al., Rev. Mod. Phys. 89 (2017) 015007)

6/20

1: Cluster variational method with AV18 + UIX potentials

2: Thomas-Fermi calculation for non-uniform matter

Uniform

Non-uniform

Our procedure to construct a new EOS table

7/20

Nuclear Hamiltonian

AV18 two-body nuclear potential UIX three-body nuclear potential

Jastrow wave function ΦF: Fermi-gas wave function

(Phys. Lett. 87B(1979) 11, PRC 75(2007) 035802)

fij: Correlation function

•  The prescription by Schmidt and Pandharipande is employed to obtain the free energy at finite temperature.

•  The expectation value of the Hamiltonian is calculated in the two-body cluster approximation.

2. Supernova EOS with realistic nuclear forces

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n0[fm-3]

0.16

E0[MeV] Esym[MeV] K [MeV]

-16.1 245 30.0 APR : A. Akmal, V. R. Pandharipande, D. G. Ravenhall,

PRC 58 (1998) 1804

Our EOS : HT and M. Takano, NPA 902 (2013) 53

FHNC : A. Mukherjee, PRC 79(2009) 045811

Nuclear EOS for uniform matter

9/20

Bulk energy Gradient energy

Coulomb energy

Free energy of a Wigner-Seitz cell

Free energy density of uniform matter: f = fN + fα

We use the Thomas-Fermi method by Shen et al. (PTP 100 (1998) 1013, APJS 197(2011) 20)

Particle number density distributions Protons and neutrons (i = p, n)

Alpha-particles

Nuclear EOS for non-uniform matter

10/20

Phase Diagram of Nuclear Matter

HT et al., NPA 961 (2017) 78 11/20

EOS for Hot Nuclear Matter

Free energy Entropy

HT et al., NPA 961 (2017) 78

Particle fraction

12/20

http://www.np.phys.waseda.ac.jp/EOS/!

Home Page of Variational EOS Table

(HT et al., NPA961 (2017) 78)

13/20

J1614-2230: Nature 467 (2010) 1081 J0348+0432: Science 340 (2013) 1233232

Shaded region: Astrophys. J. 722 (2010) 33

Mass-Radius relation of neutron stars3. Application to astrophysical objects

GW170817: PRL 119 (2017) 161101arXiv:1711.02644

14/20

SN simulation numerical code: K. Sumiyoshi, et al., NPA 730 (2004) 227

1D neutrino-radiation hydrodynamics simulations Progenitor: Woosley Weaver 1995, 15M◉ Astrophys. J. Suppl. 101 (1995) 181

Radial trajectories of mass elements

Application to Core-Collapse Supernovae

15/20

Comparison of Results (Collapse Phase)

The numbers (1)–(5) :the times when the central density reaches 1010, 1011, 1012, 1013, 1014 g/cm3

Heavy Nuclei in Supernova Matter

→　The density derivative coefficient of the symmetry energy L (Our EOS: L = 35 MeV Shen EOS: L = 111 MeV)

HT et al., NPA 961 (2017) 78 17/20

Energies of uniform nuclear matter at 0MeV

4.7 × 1014 g/cm3 (Variational)3.6 × 1014 g/cm3 (Shen EOS)

Comparison of Results (Postbounce Phase)

18/20

Stiffness of the nuclear EOS

Variational EOS is softer than the Shen EOS.

The density profiles at tpb = 0 ms The adiabatic indices at tpb = 0 ms

Adiabatic index:

→ Our EOS is advantageous for supernova explosion!? 19/20

SummaryNuclear EOS for supernova simulations is constructed

with realistic nuclear forces (AV18 + UIX).

Our SN-EOS is available at http://www.np.phys.waseda.ac.jp/EOS/!

Uniform nuclear matter : Cluster variational method

Non-uniform nuclear matter : Thomas-Fermi approximation

•  Our EOS is softer than Shen EOS in 1D supernova simulation.

•  Multi-dimensional supernova simulations •  Application to other astrophysical simulations •  Hyperon mixing in high-density matter

Future Plans

•  Neutron star structure is consistent with observational data.

20/20