NAOJ 14th. May (2016)
25 min. including discussion

Rotational equilibria byvariational Lagrange schemewith realistic EOS
Chiba Institute of Technology
Nobutoshi Yasutake
MNRAS Letter (2015) 446, 56
MNRAS (2016) submitted
NY, K.Fujisawa, S.Yamada

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“MOTIVATION”

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・Supernovae mechanism
mass → compactness

dynamical simulation(3D)of star formation

But how does it appear ?

?
Ugliano et al. (2012)APJ, 757, 69

evolution(〜1D)

Most of general toolkits for stellar evolution
calculations are based on Heney method since 1964.
16年5月14日土曜日Machida et al. (2009)APJ, 685, 690

IMPORTANCE OF STELLAR EVOLUTION

Rotational law ?
numerical simulation

astroseismology(Sun)NY, Hashimoto, Eriguchi, PTP(2005)
Differential rotation

Uniform rotation？

Schou et al.(1998)

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The other diﬃculties ofdeformed stellar evolution① Barotropic EOS, P(ρ) → Realistic (baroclinic) EOS, P(ρ, Yi, T)
Uryu&Eriguchi (1994,1995), Roxburgh (2006), Espinoza-lala&Rietord(2007),Fujisawa(2015)② Trajectory of mass(element).
③ Convection.
④ Mass ejection.
etc. ...

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The other diﬃculties ofdeformed stellar evolution
This work① Barotropic EOS, P(ρ) → Realistic (baroclinic) EOS, P(ρ, Yi, T)
Uryu&Eriguchi (1994,1995), Roxburgh (2006), Espinoza-lala&Rietord(2007),Fujisawa(2015)② Trajectory of mass(element).
③ Convection.
④ Mass ejection.
etc. ...

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dynamical simulation(not this study)

“OUR NEW METHOD”

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“DEFORMED STAR/PASTA DUALITY”
Deformed stars(non-spherical stars)

Hydro-static equilibrium

Pasta structures which are
non-uniform structures inphase transitions; neutron drip,quark-hadron phase transition,etc.)

Chemical equilibriumex) quark-hadron phase transition
repulsion = pressure and rotation

repulsion = Coulomb interaction

attraction = gravitation

attraction = surface tension

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If you give each node the physical quantities; position, and
StepIV
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).
6
1

8

4
7

2

935

each node: x, y, m, j, s, Ye, Yn, Yp, YHe ..... → dv, ρ, P, T, u, ...
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StepV
Find the most optimalarrangement by changing
the positions of nodes
finding the minimum totalenergy.

CautionNow, 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.
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“APPLICATIONAL RESULT”

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A simple example

initial guess

resultexact solution

3e+10

2.5e+102e+10

1.5e+10

1e+105e+09

0
0

5e+09

1e+10

1.5e+102e+10

2.5e+103e+10Appropriate initial guesses go to exact solutions.

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uniqueness of solution

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A part of local minimums comes fromdeformation of triangles
failuresuccess
3e+10

3e+10

2.5e+102.5e+10

2e+10

2e+10

1.5e+101.5e+10

1e+10

1e+10

5e+09

5e+09

0

0
0

5e+09

1e+10

1.5e+102e+10

2.5e+10

3e+10

5e+09

1e+10

1.5e+102e+10

2.5e+10-8e-06

total energy-8e-06

3e+10

exact

exact-8.05e-06-8.05e-06-8.1e-06-8.1e-06-8.15e-06-8.15e-06-8.2e-06-8.2e-06-8.25e-06-8.25e-06-8.3e-06
0

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050

100150200250-8.3e-06

smoothing

050

100

150200250300350400

TRAP OF LOCAL ENERGY MINIMUM(WITHOUT SMOOTHING)

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MATHEMATICAL/NUMERICAL PROBLEMS
Lagrangian perturbation fails because of the guarge freedom.

Hour-glass problem

iso potential surface→

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COMPARISON WITH OTHER METHODSdensity
3

Contours of density and K(=P/ρ^Γ)

14012033228060
140200Z [105 km]

100

2

Z [105 km]

Z [105 km]

Fujisawa method

Ours

Hachisu method

Ours

10012305

X [10 km]03

12angular velocity

310.0012

20.00160.001410.00120.001100.001012

3500.0008
1235

X [10 km]X [10 km]

P=P(ρ)
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0.001
02

X [10 km]0.0018
Z [105 km]

0.001310.0020.00150.001405

X [10 km]0.0016203

5angular velocity
Z [105 km]

1

P=P(ρ, Yi, T)

3Stellar structures without H¢iland criteria

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

→Bjerkness theorem.
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RIGID TYPE ROTATION(LEFT) & SHELLULAR ROTATION(RIGHT)
density120l

3

YFY scheme

100

28060

1400.0005

10.00045

norotation

20.0004
0.00035

10.00032000012030X [105 km]

density
31008060

14000125X [10 km]3density
3YFY scheme

angular velocity
120l

20.0004
0.00035

1280601400.0003
00.00025
0125X [10 km]

331000.00045

200

X [10 km]

0.00051352

YFY scheme

25Z [10 km]

120l

Z [105 km]

YFY scheme

1angular velocity

5

Z [10 km]30.00025

Z [105 km]

YFY scheme

Z [105 km]

Z [105 km]

3angular velocity

YFY scheme

2120000125

X [10 km]

300125X [10 km]【KEY POINT】
The distribution of angular momentum depends on the evolution.
→ We can check it !!
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0.00040.000380.000360.000340.00032
0.00030.000280.00026

13

ON GOING PROJECTS
weak solution(this talk)=0

weak solution

exact solution

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SUMMARY
We 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.

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