Problematic Keplerian rotation

Problematic Keplerian rotation

Andrzej Odrzywołek

Zakład Teorii Względności i Astrofizyki

11.12.2019, 10:15, D-2-02

The simple research plan

1 extract data from numerical GR simulation of binary neutron star merger

2 calculate rotation law j pΩq inside apparently stationary, low-mass toroid emerging after Kerr black hole formation

3 compare with analytical formula derived assuming circular orbits in Boyer-Lindquist or other „standard” coordinates (so-called Relativistic Keplerian Rotation Law )

4 fill slots (or whatever your motivation is) Any questions? Only one:

What took you so long? (18 months so far . . . )

1. BNS simulation and data extraction

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

1 XY and XZ slices of full 3D data

2 400 evolution snapshots covering from late NS-NS inspiral to toroid stabilization

3 Carpet-HDF5 BSSN fixed mesh-refinemet (6 levels) files for:

slicing: α (1+log), β^X, β^Y, β^Z (Γ-driver ?) [ADM-base]

space metric: gXX, gXY, gYY, gXZ, gZZ, gYZ [ADM-base]

moving fluid: VX, VY, VZ, ρ [HYDRO-base]

4 G “ c “ 1 Md [CU] units - Cactus Units

T-XYZ coordinates?

Assuming above, we could simply transform to „polar coords” on XY plane:

X “ R cos ϕ, Y “ R sin ϕ where R is „radius” and ϕ „angle”.

Angular velocity and momentum

Lorentz factor:

W “ 1

a1 ´ g_ijVⁱV^j, U^t” U⁰ “ W α

4-velocity:

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

Ω and j :

Ω “ U^ϕ

U^t, j “ U_ϕU^t.

Data was painfully extracted using ScatterPlot and built-in Carpet-HDF5 reader in LLNL Visit.

Relativistic Kepler’s law

1 Newton (3rd Kepler law):

1

j “ pΩ{pGmq²q¹³

2 Schwarzschild:

1

j “ pΩ{pGmq²q¹³ ´ 3Ω{c²

3 Kerr:

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

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

Expected j pΩq

Expected j pΩq

Expected j pΩq

Raw j pΩq data from simulation

Raw j pΩq data from simulation

Raw j pΩq data from simulation

Initial stage of apparent „success”

1 above analysis was done „instantly” by A.O. & P.M in June 2018

2 initial impression:

during inspiral: Newtonian rotation law from orbiting neutron stars visible

during merger: completely random movements

after collapse to the black hole: instant formation of U-shaped j pΩq curve, as expected for low mass (6.2 ˆ 10^´5M_dq toroid

3 TODO: find fitting parameters m, a of the formula

It does not match . . .

More detailed analysis revealed irreducible differences between j pΩq from numerical simulation and analytical formula.

1 Q: is the most dense part of toroid is relevant (ρ cut) ? A: yes, but at least ρ ă 0.5ρ_max must be rejected.

2 Q: is only some part of toroid relevant (above/below ISCO/minimum, R cut)?

A: unclear selection criteria, reduces j pΩq to slope/single point

3 Q: meridional circulation?

A: no, boundary condition V^Z “ 0 at equator prevent it

4 Q: Is toroid non-stationary (vortex/shock/spiral arm - like)?

A: Looks stable for many orbits, but slowly loose mass.

However, V^r{V^ϕ ! 1.

5 Q: other rotation law (κ ‰ 3)?

A: Yes, near horizon κ » 2.

6 Q: poor resolution/quality of GR simulation?

A: Better model from L. Rezzola behaves similarly.

7 Q: is this data processing issue?

A: no, cross checked VisIt/SimTools, results identical.

More questions

1 outcome is likely Kerr metric, but in what kind of coordinates?

2 is Kepler law j pΩq invariant (geometric object with transformation law)?

3 P.M. shows in „cylindrical” coords, both j and Ω must be corrected by a factor:

c g_tt g_t¹_t¹

where g_tt is in coords where analytical form j pΩq has been derived, and g_t¹_t¹ from numerics.

4 matching is improved, but still not satisfactory

5 is rotation of initial neutron stars relevant?

6 is coordinate system determined in simulation and how?

7 does it „remember” initial distortion caused by inspiraling neutron stars?

8 how 4 different definitions of angular momentum are related to each other? which one is the best for data analysis?

Kozłowski, Jaroszyński & Abramowicz 1978

Kozłowski, Jaroszyński & Abramowicz 1978

Kozłowski, Jaroszyński & Abramowicz 1978

Kozłowski, Jaroszyński & Abramowicz 1978

Example for Schwarzschild metric

4 newtonowsko równoważne definicje j

1

j “ R_c²Ω “

?mR

2

j “ U_ϕU^t “

?mR 1

b 1 ´^3m_R

3

j “ Uϕ “

?mR 1

1 ´^3m_R

4

l “ ´Uϕ

U_t “

?mR 1

1 ´^2m_R

How to use numerically evolved coords?

In numerics both metric and coords are evolved!

In textbook coordinates:

Ω “ U^ϕ U^t In arbitrary coordinates:

Ω “ ηµU^µ ηνξ^ν ´ ξµU^µ ηνη^ν ξ_µU^µ ξ_νη^ν´ ηµU^µ ξ_νξ^ν

where η is (asymptotically) timelike, ξ spacelike Killing vector.

Example: Schwarzschild metric in skew coords

¨

˚

˚

˚

˚

˚

˝

2M

R ´ 1 0 0 0

0 ^R

3´2Mp^y2`z2q

R²pR´2Mq

2Mxy R²pR´2Mq

2Mxz R²pR´2Mq

0 _R2pR´2Mq^2Mxy

R³´2Mp^x2`z2q

R²pR´2Mq

2Myz R²pR´2Mq

0 _R2pR´2Mq^2Mxz

2Myz R²pR´2Mq

R³´2Mp^x2`y2q

R²pR´2Mq

˛

‹

‹

‹

‹

‹

‚

With linear transformation of xy -coords:

ˆ x¹ y¹

˙

“

ˆ λ11 λ12

λ₂₁ λ₂₂

˙ ˆ x y

˙

Kepler law j pΩq after skew transform of Schwarzschild

metric.

Conclusions: What to do now?

it looks like Kozłowski, Jaroszyński & Abramowicz provided valid procedure for extraction of j pΩq in general coords is this the only valid procedure?

how to obtain (or at least verify) Killing vectors from numerical metric?

any volunteers to verify KJA (1978) formulas for Kerr metric in skew coords by direct calculation? (not orthogonal Killing vectors, non-zero shifts)

is above applicable also for heavy toroids (non-Kerr spacetime)?

is above related to j pΩq misfit problem at all?

deeper understanding of dynamical slicing side-effects in numerical GR is mandatory to proper interpretation of the results (so far we were happy as long as it did not crash)

Conclusions: What to do now?

it looks like Kozłowski, Jaroszyński & Abramowicz provided valid procedure for extraction of j pΩq in general coords is this the only valid procedure?

how to obtain (or at least verify) Killing vectors from numerical metric?

any volunteers to verify KJA (1978) formulas for Kerr metric in skew coords by direct calculation? (not orthogonal Killing vectors, non-zero shifts)

is above applicable also for heavy toroids (non-Kerr spacetime)?

is above related to j pΩq misfit problem at all?

deeper understanding of dynamical slicing side-effects in numerical GR is mandatory to proper interpretation of the results (so far we were happy as long as it did not crash)

Conclusions: What to do now?

it looks like Kozłowski, Jaroszyński & Abramowicz provided valid procedure for extraction of j pΩq in general coords is this the only valid procedure?

how to obtain (or at least verify) Killing vectors from numerical metric?

any volunteers to verify KJA (1978) formulas for Kerr metric in skew coords by direct calculation? (not orthogonal Killing vectors, non-zero shifts)

is above applicable also for heavy toroids (non-Kerr spacetime)?

is above related to j pΩq misfit problem at all?

deeper understanding of dynamical slicing side-effects in numerical GR is mandatory to proper interpretation of the results (so far we were happy as long as it did not crash)

Conclusions: What to do now?

it looks like Kozłowski, Jaroszyński & Abramowicz provided valid procedure for extraction of j pΩq in general coords is this the only valid procedure?

how to obtain (or at least verify) Killing vectors from numerical metric?

any volunteers to verify KJA (1978) formulas for Kerr metric in skew coords by direct calculation? (not orthogonal Killing vectors, non-zero shifts)

is above applicable also for heavy toroids (non-Kerr spacetime)?

is above related to j pΩq misfit problem at all?

deeper understanding of dynamical slicing side-effects in numerical GR is mandatory to proper interpretation of the results (so far we were happy as long as it did not crash)

Conclusions: What to do now?

it looks like Kozłowski, Jaroszyński & Abramowicz provided valid procedure for extraction of j pΩq in general coords is this the only valid procedure?

how to obtain (or at least verify) Killing vectors from numerical metric?

any volunteers to verify KJA (1978) formulas for Kerr metric in skew coords by direct calculation? (not orthogonal Killing vectors, non-zero shifts)

is above applicable also for heavy toroids (non-Kerr spacetime)?

is above related to j pΩq misfit problem at all?

deeper understanding of dynamical slicing side-effects in numerical GR is mandatory to proper interpretation of the results (so far we were happy as long as it did not crash)

Conclusions: What to do now?

it looks like Kozłowski, Jaroszyński & Abramowicz provided valid procedure for extraction of j pΩq in general coords is this the only valid procedure?

how to obtain (or at least verify) Killing vectors from numerical metric?

any volunteers to verify KJA (1978) formulas for Kerr metric in skew coords by direct calculation? (not orthogonal Killing vectors, non-zero shifts)

is above applicable also for heavy toroids (non-Kerr spacetime)?

is above related to j pΩq misfit problem at all?

deeper understanding of dynamical slicing side-effects in numerical GR is mandatory to proper interpretation of the results (so far we were happy as long as it did not crash)

Conclusions: What to do now?

it looks like Kozłowski, Jaroszyński & Abramowicz provided valid procedure for extraction of j pΩq in general coords is this the only valid procedure?

how to obtain (or at least verify) Killing vectors from numerical metric?

any volunteers to verify KJA (1978) formulas for Kerr metric in skew coords by direct calculation? (not orthogonal Killing vectors, non-zero shifts)

is above applicable also for heavy toroids (non-Kerr spacetime)?

is above related to j pΩq misfit problem at all?

deeper understanding of dynamical slicing side-effects in numerical GR is mandatory to proper interpretation of the results (so far we were happy as long as it did not crash)

EXTRA SLIDES

Zderzenie/zlanie się gwiazd neutronowych: gęstość

Zderzenie/zlanie się gwiazd neutronowych: gęstość

Zderzenie/zlanie się gwiazd neutronowych: gęstość

Zderzenie/zlanie się gwiazd neutronowych: gęstość

Zderzenie/zlanie się gwiazd neutronowych: gęstość

Zderzenie/zlanie się gwiazd neutronowych: gęstość

Zderzenie/zlanie się gwiazd neutronowych: gęstość

Zderzenie/zlanie się gwiazd neutronowych: gęstość

Zderzenie/zlanie się gwiazd neutronowych: gęstość

Zderzenie/zlanie się gwiazd neutronowych: gęstość

Zderzenie/zlanie się gwiazd neutronowych: gęstość

Zderzenie/zlanie się gwiazd neutronowych: gęstość

Zderzenie/zlanie się gwiazd neutronowych: gęstość

Zderzenie/zlanie się gwiazd neutronowych: gęstość

Zderzenie/zlanie się gwiazd neutronowych: gęstość

Zderzenie/zlanie się gwiazd neutronowych: gęstość

Zderzenie/zlanie się gwiazd neutronowych: gęstość

Zderzenie/zlanie się gwiazd neutronowych: gęstość

Zderzenie/zlanie się gwiazd neutronowych: gęstość

Zderzenie/zlanie się gwiazd neutronowych: α

Zderzenie/zlanie się gwiazd neutronowych: β

Zderzenia gwiazd neutronowych (kilonova)

Masy gwiazd neutronowych:

M₁ “ M2 “ 1.6 Md

Czarna dziura:

M_‚ “ 2.83 Md, a » 0.44 Masa toroidu:

MT “

6.2 ˆ 10^´5 M_d

Zderzenia gwiazd neutronowych (kilonova)

Masy gwiazd neutronowych:

M₁ “ M2 “ 1.6 Md

Czarna dziura:

M_‚ “ 2.83 Md, a » 0.44 Masa toroidu:

MT “

6.2 ˆ 10^´5 M_d

Zderzenia gwiazd neutronowych (kilonova)

Masy gwiazd neutronowych:

M₁ “ M2 “ 1.6 Md

Czarna dziura:

M_‚ “ 2.83 Md, a » 0.44 Masa toroidu:

MT “

6.2 ˆ 10^´5 M_d

Zderzenia gwiazd neutronowych (kilonova)

Masy gwiazd neutronowych:

M₁ “ M2 “ 1.6 Md

Czarna dziura:

M_‚ “ 2.83 Md, a » 0.44 Masa toroidu:

MT “

6.2 ˆ 10^´5 M_d

Zderzenia gwiazd neutronowych (kilonova)

Masy gwiazd neutronowych:

M₁ “ M2 “ 1.6 Md

Czarna dziura:

M_‚ “ 2.83 Md, a » 0.44 Masa toroidu:

MT “

6.2 ˆ 10^´5 M_d

Zderzenia gwiazd neutronowych (kilonova)

Masy gwiazd neutronowych:

M₁ “ M2 “ 1.6 Md

Czarna dziura:

M_‚ “ 2.83 Md, a » 0.44 Masa toroidu:

MT “

6.2 ˆ 10^´5 M_d

Zderzenia gwiazd neutronowych (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

Zderzenia gwiazd neutronowych (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

Zderzenia gwiazd neutronowych (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

Zderzenia gwiazd neutronowych (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

Zderzenia gwiazd neutronowych (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

Initial conditions

w teorii newtona układ podwójny jest „stacjonarny”

w OTW z powodu emisji fal grawitacyjnych parametry orbity zmieniają się

powyższy efekt można „skasować” napromieniowując układ falami grawitacyjnymi

Synchronizacja spin-orbita

1 typowo w astrofizyce ciasne układy podwójne są zsynchronizowane

2 przykłady: Ziemia-Księżyc (synchroniczny obrót), Pluton-Charon (pełna synchronizacja)

3 maksymalna częstość rotacji gwiazd neutronowych do 1 kHZ

4 maksymalna częstość orbitalna 2 kHz

GW 170817 & GRB 170817A

1 częstość wzrasta od 32 Hz do 2048 Hz w ciągu 60 s

2 3000 pełnych orbit

GW 170817 & GRB 170817A

1 częstość wzrasta od 32 Hz do 2048 Hz w ciągu 60 s

Density/radial cuts

Meridional circulation

Non-stationary flow