Deformed stellar structures with rotation Tohoku 14th. Nov. (2016) - - PowerPoint PPT Presentation

Tohoku 14th. Nov. (2016) 12 min. talk+3 min

Deformed stellar structures with rotation

Chiba Institute of Technology Nobutoshi Yasutake MNRAS Letter (2015) 446, 56, MNRAS(2016) 463, 3705 NY, K.Fujisawa, S.Yamada + H.Okawa

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“MOTIVATION”

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PNSs(R～50km)

T～30MeV Yl～0.3

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

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NSs(R～10km)

A big problem in astrophysics There is no scheme of time evolution for ``deformed stars” in quasi-equilibrium !!

Supernovae dynamical simulation(3D) evolution (1D) T～0 MeV Yν～0

Machida, Matsumoto, Inutsuka, (2008) ApJ

Star formations dynamical simulation(3D) Heyney method (1964)

Massive Stars, Planets, ... etc

evolution (1D) Heyney method (1964) in GR

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Proto-stars

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Uniform rotation Differential rotation

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

→No one knows rotation inside stars. ？

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

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P Pdot diagram

Dipole radiation = Loss rate of rotational energy Magnetic filed B = 3.2 x 1019 (P ∂tP)1/2 G Characteristic age τ = P /(2∂tP) s Observations with the ansatz

⇩

We are looking at rotation !!

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“WEAK SOLUTION”

=0 MNRAS Letter (2015) 446, 56, MNRAS(2016) 463, 3705 NY, K.Fujisawa, S.Yamada

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Mass coordinate

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

Variational Principle IMPORTANT POINTS

①Free boundary condition. ②Realistic EOS (baroclinic). ③100% conservative scheme. ④ However, numerical error is large close to the boundary with low mass element.

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YFY scheme

• 1

2 3 X [105 km] 1 2 3 Z [105 km] 0.00025 0.0003 0.00035 0.0004 0.00045 0.0005 YFY scheme

• 1

2 3 X [105 km] 1 2 3 Z [105 km] 20 40 60 80 100 120 YFY scheme

• 1

2 3 X [105 km] 1 2 3 Z [105 km] 20 40 60 80 100 120 YFY scheme

• 1

2 3 X [105 km] 1 2 3 Z [105 km] 0.00025 0.0003 0.00035 0.0004 0.00045 0.0005

YFY scheme

• 1

2 3 X [105 km] 1 2 3 Z [105 km] 20 40 60 80 100 120

YFY scheme

• 1

2 3 X [105 km] 1 2 3 Z [105 km] 0.00026 0.00028 0.0003 0.00032 0.00034 0.00036 0.00038 0.0004

We can trace stellar evolutions with ``entropy (⬇)”, ``angular momentum (➡)”. →If arbitral distributions of mass, entropy, angular momenta, and fraction, we can obtain hydrostatic equilibria in a conservative way.

APPLICATIONAL RESULTS

uniform-like sellular-type

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uniqueness of solution

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“STRONG SOLUTION”

MNRAS(2017) in prep. NY, K.Fujisawa, H.Okawa, S.Yamada

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PROBLEMS

①Lagrangian perturbation fails because of the gauge freedom. (Freedman&Shutz 1978) ②Hour-glass problem iso potential surface→

HOW TO SOLVE ?

f

dxdy = | J | dXdY

f- -1

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x1+A1 x1+ B1 F =0

.. .

x2+A2 x2+ B2 F =0

.. . : F =

H.Okawa, K.Fujisawa, R.Hirai, Y.Yamamoto, NY, H.Nagakura (in prep.) 【tangent vector space on S1】 Find a fixed point⇔ Find a solution for 1st order differential eq. (SOR, Newton method etc.) →Sometime, there is no fixed point. 【 tangent vector space on S2】 Find a fixed point⇔ Find a solution for 2nd order differential eq. (W4, MD calculation etc.) → There must be fixed points. Naive Newton-Raphon methods does not work for multi-dimensional hydrostatic equilibria in Lagrange coordinate [Ref. Friedman & Schutz 1978]. We, then, introduce a new method, named as ``W4 method”, which is based on the fixed point theorem.

A new method based on a fixed point theorem

Here, F=F(x1,x2,...y1,y2,...).

In Lagrange scheme, the variables are ``coordinates” of nodes, which have mass, entropy, and angular momenta.

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20 40 60 80 100 120 140 160 180 200 0.5 1 1.5 2 2.5 3 3.5 [g/cc] r [105 km] x1.2 x0.8 20 40 60 80 100 120 140 160 180 200 0.5 1 1.5 2 2.5 3 3.5 [g/cc] r [105 km] x1.2 x0.8

A extended initial-model and a shrinked initial-model provide the same result. residual = |∇P/ρ+Φ|/(|∇P/ρ|+|Φ|) のmax < 1.e-5 convergence condition viral constant

“STRONG SOLUTION (ID TEST)”

MNRAS (in prep) N=30 → VC= 6.3e-2 N=60 → VC= 3.3e-2 N=100 → VC= 2.0e-2 High resolution ``does” !! High resolution ``does not” improve numerical accuracy, since finite volume does not contribute to total energy.

Weak solution Strong solution

Initial models Final result

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SUMMARY

We introduce two new methods to obtain baroclinic equilibria in multi-dimension, for weak solutions and strong solutions. Our weak-solution method can trace mass element for the entropy and the angular-momentum evolution, but the local balance eq. may be broken. weak solution→strong solution Our strong-solution method is successful for 1D test-calculation. Our scheme will be a break-through not only for studies on main-sequence stars but also on planets, compact stars, etc.

Future work

radiation, convection, mass loss, observation, GR, 3D, magnetic filed....

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