Rotation law emerging from GR numerical
simulation of merging neutron stars
Andrzej Odrzywołek
Dept. of General Relativity & Astrophysics, IF UJ
25 Sept 2018, Tue, 19:00
Stationary configurations
Configuration of interest is a Kerr black hole + massive, stationary, axisymmetric torus (see also talks by P. Mach, E. Malec, W.
Kulczycki, M. Piróg).
Object is fully specified by:
1 Black Hole and torus masses & angular momenta
2 Equation of State
last but not least: rotation law
General remarks on rotating fluids
so-called rotation law connecting angular velocity Ω “ uϕ{ut or angular momentum j “ uϕut with „distance” from rotation axis is a free function j pΩq, e.g. j “ const, j “ Ωδ, . . .
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 (m “ 1) play a special role in astrophysics
1 Newton (3rd Kepler law):
1 j 9 Ω1{3
2 Schwarzschild:
1
j 9 Ω1{3´ 3Ω
3 and finally Kerr:
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` 3w43Ω23p1 ´ aΩq43q ı
, w “ f pmq 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 )
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, gXX, gXY, gYY, βX, βY, VX, VY, ρ
Neutron star mergers (kilonova)
Neutron star mergers (kilonova)
Neutron star mergers (kilonova)
Neutron star mergers (kilonova)
Neutron star mergers (kilonova)
Angular velocity and momentum in isotropic coordinates
Lorentz factor:
W “ 1
b
1 ´ gxxVx2´ gyyVy2´ 2gxyVxVy Transversal velocity component:
Vϕ“ ´y pgxxVx` gxyVyq ` x pgyyVy` gxyVxq Angular velocity:
Ω “ x pαVy´ βyq ´ y pαVx´ βxq x2` y2
Angular momentum:
W2V
Expected j pΩq
Raw j pΩq data from simulation
Raw j pΩq data from simulation
Raw j pΩq data from simulation
Raw j pΩq data fits
Fit j pΩq after cut
10´9 ă ρ ă ρmax “ 2.76 ˆ 10´9
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 » 1.50
5 above result is surprising but not forbidden
6 more investigation required
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