Catania 20th. June (2015)30 min. including discussion

Pasta / Deformed-StarsDuality

Chiba Institute of TechnologyNobutoshi Yasutake

MNRAS Letter (2015) 446, 56NY, K.Fujisawa, S.Yamada

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“MOTIVATION”

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Thanks to collaborations & discussion...

EXOCT 2007

→Nov.2011-Mar.2012 PD INFN DI CATANIA15年6月20日土曜日

EOS & Mass-Radius relationA.W.Steiner, J.M.Lattimer, E.F.BrownApJ 722 (2010) 33

0

0.5

1

1.5

2

2.5

6 8 10 12 14 16 18

M [M

S]

R [km]

.

m40m80

BOB

DS+BHF(BOB+TBF) with pasta [NY, Huan, Maruyama, Tatsumi in prep.]　

EPNJL+χPT[Sasaki, NY, Khono, Kouno, YahiroarXiv:1307.0681]

BHF(TBF by Pomeron )[Yamamoto, Furumoto, NY, Rijken, PRC(2014)]

nlNJL+BHF(BOB+TBF) with pasta [NY, Lastowiecki, Sanjin, Blaschke, Maruyama, Tatsumi, PRC2014]

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EOS & Mass-Radius relationA.W.Steiner, J.M.Lattimer, E.F.BrownApJ 722 (2010) 33

0

0.5

1

1.5

2

2.5

6 8 10 12 14 16 18

M [M

S]

R [km]

.

m40m80

BOB

DS+BHF(BOB+TBF) with pasta [NY, Huan, Maruyama, Tatsumi in prep.]　

EPNJL+χPT[Sasaki, NY, Khono, Kouno, YahiroarXiv:1307.0681]

BHF(TBF by Pomeron )[Yamamoto, Furumoto, NY, Rijken, PRC(2014)]

nlNJL+BHF(BOB+TBF) with pasta [NY, Lastowiecki, Sanjin, Blaschke, Maruyama, Tatsumi, PRC2014]

They look similar.

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cooling rate (neutrino)＋

heating rate (magnetic field)

heatcapacity

flux(thermal conductivity)

thermal diffusion eq.

Cooling…→many physical properties

COOLING OF NEUTRON STARS

2D: 10^5×10^5 matrix inversion

1D: 300×300 matrix inversion

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Observation, Yujin, et al.　(2009) PASJ

Low T (keV)

high T(keV)

Temperature distributionNY, Kotake, Kutsuna, Shigeyama (2014) PASJ

Our resultsNY, Kotake, Kutsuna,Shigeyama (2014) PASJ

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c.f. See the reference;(2010) Physics Today

PNSs(R～50km)

T～30MeV

multi-dimensional evolution

K.Kotake, K.Sumiyoshi, S.Yamada,T.Takiwaki, T.Kuroda, Y.Suwa,H.Nagakura (2012) PTEP

?

Others >> C. Jacobi, R. Dedekind, P.L. Dirichlet, B. Riemann...

NSs(R～10km)

A big problem in astrophysicsThere is no scheme of time evolution for ``deformed stars” in (quasi-)equilibrium !!

Also in star formations, Be-stars, proto-planets, etc.

dynamical simulation(3D) evolution (1D)

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Uniform rotation Differential rotation

Without assumtions of rotational law,no one can get rotating structures.

？

Rotating structures of neutron stars in fully GR formulationNY, Hashimoto, Eriguchi, PTP(2005)

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Basic equations for star evolution(1D)Laglange coodinateEular coordinate

hydro-static equilibrium　

mass conservation

radiation

energy conservation

equation of state

There is no scheme of time evolution for ``deformed stars” in (quasi-)equilibrium !!

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Basic equations for star evolution(1D)Laglange coodinateEular coordinate

hydro-static equilibrium　

mass conservation

radiation

energy conservation

equation of state

There is no scheme of time evolution for ``deformed stars” in (quasi-)equilibrium !!

even15年6月20日土曜日

“HOW TO MAKE ?”

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“DEFORMED STAR/PASTA DUALITY”Pasta structures which are non-uniform structures in phase transitions; neutron drip, quark-hadron phase transition, etc.)

Deformed stars (non-spherical stars)

Chemical equilibrium

repulsion = Coulomb interaction

Hydro-static equilibrium

attraction = surface tension

repulsion = pressure and rotation

attraction = gravitation

ex) quark-hadron phase transition

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“NODES & EDGES”

Step 1 Put some nodes, numbering them randomly.

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Step IIConnect the nodes with edges making triangles as you want.

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5

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StepIII Make the matrix which shows the relationship between the nodes; if they have an edge between two nodes, put “1”, otherwise put “0” in the matrix. Then, you will get a beautiful symmetric matrix, which is known as an “adjacency matrix” in the graph theory.

1 2 3 4 5 6 7 8 90 1 1 1 0 1 0 0 01 0 1 1 1 0 0 0 01 1 0 0 1 0 0 0 01 1 0 0 1 1 1 0 00 1 1 1 0 0 1 0 11 0 0 1 0 0 1 1 00 0 0 1 1 1 0 1 10 0 0 0 0 1 1 0 10 0 0 0 1 0 1 1 0

123456789

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StepIV If you give each node the physical quantities; position, and Lagrange values(mass, angular momentum, entropy, fractions ...), you can get Eulerian values (partial volume, density, ...) considering with the relation between neighboring nodes (and/or the adjacency matrix).

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5each node: x, y, m, j, s, Ye, Yn, Yp, YHe ..... → dv, ρ, P, T, u, ...

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Caution

Now, you may think that this is about two dimension. Of course, if you give (xi,yi) as the positions on nodes, it becomes a two-dimensional topic. But, if you give (xi,yi, zi, ...), it becomes the multi-dimensional.

StepV

Find the most optimal arrangement by changing the positions of nodes finding the minimum total energy.

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“APPLICATIONAL RESULT”

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A simple example

exact solution

Appropriate initial guesses go to exact solutions.

initial guess

0

5e+09

1e+10

1.5e+10

2e+10

2.5e+10

3e+10

0 5e+09 1e+10 1.5e+10 2e+10 2.5e+10 3e+10

result

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We succeeded to get hydro-static equilibriums for stars with realistic (baroclinic) EOS in Lagrange coordinate. The distribution of specific angular moment j is not cylindrical as shown in the right panel.

→Bjerkness theorem.

Stellar structures without H¢iland criteria

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COMPARISON WITH OTHER METHODSNY, K.Fujisawa,S.Yamada, (2015) in prep

Hachisu method

0 1 2 3X [105 km]

0

1

2

3

Z [1

05 km

]

0.001

0.0011

0.0012

0.0013

0.0014

0.0015

0.0016

0 1 2 3X [105 km]

0

1

2

3

Z [1

05 km

]

0 20 40 60 80 100 120 140density

angular velocity

Fujisawa methodOurs Ours

0

1

2

3

0 1 2 3

Z [1

05 km

]

X [105 km]

Contours of density and K(=P/ρ^Γ)

angular velocity

0

1

2

3

0 1 2 3

Z [1

05 km

]

X [105 km]

0 1 2 3X [105 km]

0

1

2

3

Z [1

05 km

]

0.0008

0.001

0.0012

0.0014

0.0016

0.0018

0.002

P=P(ρ) P=P(ρ, Yi, T)15年6月20日土曜日

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We are here.　↓

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SUMMARYWe construct a new formulation to calculate structures of deformed stars with rotation.

This technique has the similarity with calculations of pasta structures.

Our method is applicable for all types of deformed stars.

Future works・Next step is the time evolution of the deformed stars.・Triangulations in 3D is easy, and we have already prepared.→But, it needs a new and special formulation for３D.

-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1-1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1-1

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inside

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