On the black hole - torus systems

On the black hole - torus systems

Andrzej Odrzywołek

Dept. of General Relativity & Astrophysics, IF UJ

29 Nov. 2018, Thu, 17:15

Work with : Patryk Mach, M. Piróg, W. Kulczycki, E. Malec, Z. Karkowski, W. Dyba

GR simulation data provided by: Roberto De Pietri, Alessandra FEO, Universit`a degli Studi di PARMA Magnetised disks: Jose Font, S. Gimeno-Soler

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Ring, circle, torus, toroid, polish donut

1 circle (1D) (Piotrowski paradox)

2 ring (2D) (orbital megastructure)

3 disk (2D) (Saturn’s ring, Galaxy)

4 toroid (3D)

(self-gravitating with central mass)

Circle and disk as limiting cases of toroids If the mass of the toroid m Ñ 0 we get:

flat disk for Keplerian rotation

circle or mass gap (no solution) otherwise

A. Odrzywołek Seminarium NKF

Ring, circle, torus, toroid, polish donut

1 circle (1D) (Piotrowski paradox)

2 ring (2D) (orbital megastructure)

3 disk (2D) (Saturn’s ring, Galaxy)

4 toroid (3D)

(self-gravitating with central mass)

Circle and disk as limiting cases of toroids If the mass of the toroid m Ñ 0 we get:

flat disk for Keplerian rotation

circle or mass gap (no solution) otherwise

A. Odrzywołek Seminarium NKF

Ring, circle, torus, toroid, polish donut

1 circle (1D) (Piotrowski paradox)

2 ring (2D) (orbital megastructure)

3 disk (2D) (Saturn’s ring, Galaxy)

4 toroid (3D)

(self-gravitating with central mass)

Circle and disk as limiting cases of toroids If the mass of the toroid m Ñ 0 we get:

flat disk for Keplerian rotation

circle or mass gap (no solution) otherwise

A. Odrzywołek Seminarium NKF

Ring, circle, torus, toroid, polish donut

1 circle (1D) (Piotrowski paradox)

2 ring (2D) (orbital megastructure)

3 disk (2D) (Saturn’s ring, Galaxy)

4 toroid (3D)

(self-gravitating with central mass)

Circle and disk as limiting cases of toroids If the mass of the toroid m Ñ 0 we get:

flat disk for Keplerian rotation

circle or mass gap (no solution) otherwise

A. Odrzywołek Seminarium NKF

Ring, circle, torus, toroid, polish donut

1 circle (1D) (Piotrowski paradox)

2 ring (2D) (orbital megastructure)

3 disk (2D) (Saturn’s ring, Galaxy)

4 toroid (3D)

(self-gravitating with central mass)

Circle and disk as limiting cases of toroids If the mass of the toroid m Ñ 0 we get:

flat disk for Keplerian rotation

circle or mass gap (no solution) otherwise

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Stationary configurations

Configuration of interest is: „Kerr” black hole + massive, stationary, axisymmetric torus.

Object is fully specified by:

1 Black Hole and torus masses & angular momenta

2 Equation of State

3 last but not least: rotation law

4 magnetic field

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Numerical implementation

Surface capturing vs surface tracking We must handle two essential surfaces:

1 black hole horizon

2 toroid edge

Any of them could be tracked or captured on numerical grid.

A. Odrzywołek Seminarium NKF

Numerical implementation

Surface capturing vs surface tracking We must handle two essential surfaces:

1 black hole horizon

2 toroid edge

Any of them could be tracked or captured on numerical grid.

A. Odrzywołek Seminarium NKF

Numerical implementation

Surface capturing vs surface tracking We must handle two essential surfaces:

1 black hole horizon

2 toroid edge

Any of them could be tracked or captured on numerical grid.

A. Odrzywołek Seminarium NKF

Numerical implementation

Surface capturing vs surface tracking We must handle two essential surfaces:

1 black hole horizon

2 toroid edge

Any of them could be tracked or captured on numerical grid.

A. Odrzywołek Seminarium NKF

Numerical implementation

Surface capturing vs surface tracking We must handle two essential surfaces:

1 black hole horizon

2 toroid edge

Any of them could be tracked or captured on numerical grid.

A. Odrzywołek Seminarium NKF

Numerical implementation

Surface capturing vs surface tracking We must handle two essential surfaces:

1 black hole horizon

2 toroid edge

Any of them could be tracked or captured on numerical grid.

A. Odrzywołek Seminarium NKF

Numerical implementation

Surface capturing vs surface tracking We must handle two essential surfaces:

1 black hole horizon

2 toroid edge

Any of them could be tracked or captured on numerical grid.

A. Odrzywołek Seminarium NKF

Numerical implementation

Surface capturing vs surface tracking We must handle two essential surfaces:

1 black hole horizon

2 toroid edge

Any of them could be tracked or captured on numerical grid.

A. Odrzywołek Seminarium NKF

Numerical implementation

Surface capturing vs surface tracking We must handle two essential surfaces:

1 black hole horizon

2 toroid edge

Any of them could be tracked or captured on numerical grid.

A. Odrzywołek Seminarium NKF

Numerical implementation

Surface capturing vs surface tracking We must handle two essential surfaces:

1 black hole horizon

2 toroid edge

Any of them could be tracked or captured on numerical grid.

A. Odrzywołek Seminarium NKF

1 Nishida, Eriguchi, Lanza (1992): both (?) surfaces captured

2 Ansorg & Petroff (2005):

both surfaces tracked

3 Shibata (2008): horizon tracked, toroid captured

A. Odrzywołek Seminarium NKF

1 Nishida, Eriguchi, Lanza (1992): both (?) surfaces captured

2 Ansorg & Petroff (2005):

both surfaces tracked

3 Shibata (2008): horizon tracked, toroid captured

A. Odrzywołek Seminarium NKF

1 Nishida, Eriguchi, Lanza (1992): both (?) surfaces captured

2 Ansorg & Petroff (2005):

both surfaces tracked

3 Shibata (2008): horizon tracked, toroid captured

0 5 10 15 20

r sin θ 0

5 10 15 20

r cos θ

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Motivation for re-implementation

1 arbitrary rotation laws (incl. Keplerian)

2 magnetic fields

3 study full parameter space: uniqueness, bifurcation, topology, existence . . .

4 cross-check of results obtained with different codes

5 post-newtonian & newtonian comparison

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General remarks on rotating fluids

so-called rotation law connecting angular velocity Ω “ u^ϕ{u^t or angular momentum j “ uϕu^t with „distance” from rotation axis is a free function j pΩq, e.g. j “ const, j “ Ω^δ, . . .

above reflect freedom of how are you stirring sugar in a glass of tea (neglecting viscosity & meridional currents)

Keplerian rotation law of a test particle around point mass/black hole play a special role in astrophysics

1 Newton (3rd Kepler law):

1

j 9 pΩ{pGmq²q¹³

2 Schwarzschild:

1

j 9 pΩ{pGmq²q¹³ ´ 3Ω{c²

3 and finally Kerr:

j 9 `a ` ξ³˘ `a<sup>2</sup>´ 2aξ ` ξ⁴˘

ξ³p2a ` ξ³´ 3ξq , ξ “ p1{Ω ´ aq^1{3, pm “ 1q

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Conjecture of „attractor” in rotating GR systems

Postulate: all generic/realistic GR disks should rotate according to

„Keplerian” rotation law:

j pΩq “ ´1 2

d d Ωln

”

1 ´ pa²Ω²` 3m²³Ω²³p1 ´ aΩq⁴³q ı

, where m, a are now free parameters unrelated to those of central Kerr black hole.

How to verify above statement?

Compare computed toroid structure with:

astronomical observations

toroid emerging in GR simulation of NS-NS merger (binary neutron star merger, kilonova, e.g. GW170817 )

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Numerical GR data used for comparison

Data used later is from article: Roberto De Pietri, Alessandra Feo, Francesco Maione, Frank L¨offler, Modeling equal and unequal mass binary neutron star mergers using public codes, Physical Review D, Volume 93, Issue 6, id.064047.

Technically, we used:

1 equatorial plane „XY” slices of full 3D data

2 all 400 timesteps covering from late NS-NS inspiral to toroid stabilization

3 Carpet-HDF5 BSSN fixed mesh-refinemet files for α, g_XX, g_XY, g_YY, g_XZ, g_ZZ, g_YZ, β^X, β^Y, V_X, V_Y, V_Z, ρ

4 G “ c “ 1 Md units

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Neutron star mergers (kilonova)

Binary NS params:

M₁ “ M2 “ 1.6 Md

BH formed instantly:

M‚ “ 2.83 Md, a » 0.44 Toroid mass:

MT “ 6.2 ˆ 10^´5 Md

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Neutron star mergers (kilonova)

Binary NS params:

M₁ “ M2 “ 1.6 Md

BH formed instantly:

M‚ “ 2.83 Md, a » 0.44 Toroid mass:

MT “ 6.2 ˆ 10^´5 Md

A. Odrzywołek Seminarium NKF

Neutron star mergers (kilonova)

Binary NS params:

M₁ “ M2 “ 1.6 Md

BH formed instantly:

M‚ “ 2.83 Md, a » 0.44 Toroid mass:

MT “ 6.2 ˆ 10^´5 Md

A. Odrzywołek Seminarium NKF

Neutron star mergers (kilonova)

Binary NS params:

M₁ “ M2 “ 1.6 Md

BH formed instantly:

M‚ “ 2.83 Md, a » 0.44 Toroid mass:

MT “ 6.2 ˆ 10^´5 Md

A. Odrzywołek Seminarium NKF

Neutron star mergers (kilonova)

Binary NS params:

M₁ “ M2 “ 1.6 Md

BH formed instantly:

M‚ “ 2.83 Md, a » 0.44 Toroid mass:

MT “ 6.2 ˆ 10^´5 Md

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Angular velocity and momentum in isotropic coordinates

Lorentz factor:

W “ 1

a1 ´ gijVⁱV^j, U^t” U⁰ “ W α

Transversal velocity components:

U_ϕ“ W p´Y VX`X VYq, U<sup>ϕ</sup> “ WX pV<sup>Y</sup> ´ β<sup>Y</sup>{αq ´ Y pV<sup>X</sup> ´ β<sup>X</sup>{αq X<sup>2</sup>` Y²

Angular velocity Ω and angular momentum j : Ω “ U^ϕ

U^t, j “ UϕU^t.

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Expected j pΩq

Conclusion: „stabilized” toroid expected between jmin and ISCO.

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Expected j pΩq

Conclusion: „stabilized” toroid expected between j_min and ISCO.

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Expected j pΩq

Conclusion: „stabilized” toroid expected between j_min and ISCO.

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Raw j pΩq data from simulation

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Raw j pΩq data from simulation

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Raw j pΩq data from simulation

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Raw j pΩq data fit

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Fit j pΩq after cut

1

2ρmaxă ρ ă ρmax “ 2.76 ˆ 10^´9

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Conclusions

1 data extracted from GR simulation

2 j pΩq curve formed quickly after black hole formation

3 naive raw data fit seems poorly related to Mach-Malec formula

4 fit weighted with matter density gives rotation law with Kerr parameter a » 1

5 above result is surprising but not forbidden

6 reason for fit failure still unclear

7 are we dealing with accretion or excretion disk ?

8 accretion/excretion disk probably located between ISCO and j_min

9 dj {d Ω changes sign - stability criteria affected?

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References

1 Self-gravitating axially symmetric disks in general-relativistic rotation, Janusz Karkowski, Wojciech Kulczycki, Patryk Mach, Edward Malec, Andrzej Odrzywołek, and Michał Piróg Phys. Rev. D 97, 104017 – Published 15 May 2018

2 General-relativistic rotation: Self-gravitating fluid tori in motion around black holes, Janusz Karkowski, Wojciech Kulczycki, Patryk Mach, Edward Malec, Andrzej Odrzywołek, and Michał Piróg Phys. Rev. D 97, 104034 – Published 21 May 2018

3 Modeling equal and unequal mass binary neutron star mergers using public codes Roberto De Pietri, Alessandra Feo,

Francesco Maione, and Frank L¨offler Phys. Rev. D 93, 064047 – Published 21 March 2016

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