Numerical relativity simulations of thick accretion disks around - - PowerPoint PPT Presentation

Introduction and astrophysical motivation Initial models and diagnostic tools Results

Numerical relativity simulations of thick accretion disks around tilted Kerr black holes

Vassilios Mewes Collaborators: Toni Font Pedro Montero Nikolaos Stergioulas Filippo Galeazzi

Universitat de València vassilios.mewes@uv.es

April 15, 2015

Introduction and astrophysical motivation Initial models and diagnostic tools Results

Outline

1

Introduction and astrophysical motivation

2

Initial models and diagnostic tools

3

Results

Introduction and astrophysical motivation Initial models and diagnostic tools Results

Thick accretion disks in the universe

Thick accretion disks: believed to be formed in NS-NS and NS-BH mergers (a mechanism for sGRB), as well as in the CC of massive stars (a mechanism for lGRB). Rezzolla et al (2010) have shown that massive, thick disks form in simulations of unequal mass NS-NS mergers. to explain sGRB as energy released from accreted material coming from a thick disk, it must survive long enough > stability.

Introduction and astrophysical motivation Initial models and diagnostic tools Results

Instabilities of disks

Papalouizou-Pringle Instability (PPI) (Papaloizou and Pringle (1984)): axisymmetry in the disk is broken and m planetary structures emerge, where m is the dominant mode. Runaway Instability (RI) (Abramowicz et al (1983)): initially stable disk is being accreted almost completely in a few dynamical time-scales onto the central object. Aim to understand under which conditions (and if) these instabilities develop, and investigate the effect the BH tilt has on these instabilities.

Introduction and astrophysical motivation Initial models and diagnostic tools Results

Tilted Disks: Motivation and previous work

Pioneering work in this field by Fragile et al (2005,2006) who have analysed tilted disks in the Cowling (fixed background spacetime) approximation. No reason to expect that SBH is aligned with the orbital plane of the NS-BH merger. Perform simulations with spacetime evolution to investigate effects of BH tilt to BH+torus evolution. Computationally cheaper (due to symmetries) to consider SBH and Ldisk aligned. Test effect of Spin magnitude and Spin direction on evolution of disk and search for imprint on GW.

Introduction and astrophysical motivation Initial models and diagnostic tools Results

Model and initial data

Self-gravitating, massive tori around non-rotating stellar mass BH, Stergioulas (2011). Starting from an AJS disk (Polish doughnut), the field equations of the QI spacetime and the hydrostatic equilibrium equations are solved iteratively until an equilibrium solution is found. Tilted simulations: Kerr BH in improved QI coordinates (Liu et al 2009) Tilt BH by rotating the coordinate system by an angle θ about the x-axis

Introduction and astrophysical motivation Initial models and diagnostic tools Results

Simulation software

Simulations were performed using the publicly available Einstein Toolkit (www.einsteintoolkit.org). We solve the 3D Einstein equations: Gµν = 8 π T µν in the so-called BSSN formulation (MacLachlan thorn). Solve the relativistic hydrodynamic equations in conservative form (Valencia formulation) for a perfect fluid, using High Resolution Shock Capturing schemes, coupling the hydro evolution to the spacetime via the stress-energy tensor (GRHydro thorn). Mesh: Carpet Code, providing AMR.

Introduction and astrophysical motivation Initial models and diagnostic tools Results

Disk Models

Model ρmax [G=c=M⊙=1] l Mtorus/MBH forb [Hz] D2 1.05e-05 3.75, const. 4.4e-02 1360 C1B 5.91e-05 3.67, const. 1.6e-01 1300 NC1 1.69e-05 3.04, non-const. 1.1e-01 843

Introduction and astrophysical motivation Initial models and diagnostic tools Results

Analysis of Twist and Tilt

We analyse the response of the disk to the tilted BH by two quantities: The twist: σ(r) = ∠(SBH × Sxy−90, P(JDisk(r), SBH)), where P(a, b) = a − a · b |b|2 b , (1) is the projection of vector a onto the plane with normal b. and tilt: ν(r) = ∠(SBH, JDisk(r)) The disk is said to become twisted (warped), if σ(r) (ν(r)) vary with r

Introduction and astrophysical motivation Initial models and diagnostic tools Results

Analysis of Twist and Tilt II

SBH JDisk(r)

ν(r)

SBH × Sxy−90 P(JDisk(r),SBH)

σ(r)

Sxy Sxy−90

Introduction and astrophysical motivation Initial models and diagnostic tools Results

Measuring BH spin direction in 3D Cartesian grids

One of the ways to measure spin direction (Campanelli et al, 2006) is by using flat-space rotational Killing vectors ξx = (0, −z, y) ξy = (z, 0, −x) ξz = (−y, x, 0) (2) in the angular momentum integral of the isolated and dynamical horizon formalism (Ashtekar and Krishnan): Si = 1 8π

• S
• ξa

i RbKab

• dS,

(3)

Introduction and astrophysical motivation Initial models and diagnostic tools Results

Measuring BH spin direction in 3D Cartesian grids II

One can derive the angular momentum integral ( 3) from the Weinberg pseudotensor (Weinberg, 1972) when the pseudotensor is expressed in Gaussian coordinates (α = 1, βi = 0), (Mewes et al, submitted to PRD). Both integrals are equal to the Komar angular momentum integral, when the Komar integral is written in a foliation adapted to the axisymmetry of the problem. Pseudotensors are problematic, because they are not coordinate independent quantities, however, by using Gaussian coordinates, we restore coordinate freedom in the Weinberg pseudotensor.

Introduction and astrophysical motivation Initial models and diagnostic tools Results

Results

PPI universality and BH movement Bardeen-Petterson effect and tilt evolution

Introduction and astrophysical motivation Initial models and diagnostic tools Results

Results

PPI universality and BH movement Bardeen-Petterson effect and tilt evolution

Introduction and astrophysical motivation Initial models and diagnostic tools Results

PPI universality and BH movement

The models that develop the PPI do so irrespective of initial tilt angle and BH spin magnitude. The over-density lump (planet) that develops causes the BH to start moving in a spiral. For tilted models, the spiral plane is tilted and causes a mild kick in the vertical direction when the PPI saturates.

Introduction and astrophysical motivation Initial models and diagnostic tools Results

Model ρmax [G=c=M⊙=1] l Mtorus/MBH forb [Hz] D2 1.05e-05 3.75, const. 4.4e-02 1360 C1B 5.91e-05 3.67, const. 1.6e-01 1300 NC1 1.69e-05 3.04, non-const. 1.1e-01 843

Introduction and astrophysical motivation Initial models and diagnostic tools Results

PPI development in untilted C1B, xy-plane

Introduction and astrophysical motivation Initial models and diagnostic tools Results

Model ρmax [G=c=M⊙=1] l Mtorus/MBH forb [Hz] D2 1.05e-05 3.75, const. 4.4e-02 1360 C1B 5.91e-05 3.67, const. 1.6e-01 1300 NC1 1.69e-05 3.04, non-const. 1.1e-01 843

Introduction and astrophysical motivation Initial models and diagnostic tools Results

No PPI development in untilted model D2

Introduction and astrophysical motivation Initial models and diagnostic tools Results

Tilted disk D2 isovolume animation

Introduction and astrophysical motivation Initial models and diagnostic tools Results

PPI in tilted disks, a=0.1

β0 = 5◦ β0 = 15◦ β0 = 30◦ C1B NC1 D2

Introduction and astrophysical motivation Initial models and diagnostic tools Results

PPI in tilted disks, a=0.1

β0 = 5◦ β0 = 15◦ β0 = 30◦ C1B

Introduction and astrophysical motivation Initial models and diagnostic tools Results

PPI in tilted disks, a=0.1

β0 = 5◦ β0 = 15◦ β0 = 30◦ NC1

Introduction and astrophysical motivation Initial models and diagnostic tools Results

PPI in tilted disks, a=0.1

β0 = 5◦ β0 = 15◦ β0 = 30◦ D2

Introduction and astrophysical motivation Initial models and diagnostic tools Results

M=1 mode evolution in C1B and NC1

Dm=

• ρ e−imφ d3x

5 10 15 20 t/torb 10−6 10−5 10−4 10−3 10−2 10−1 100 D1/max(D1) C1Ba00 C1Ba00 C1Ba01b05 C1Ba01b15 C1Ba01b30 C1Ba03b05 C1Ba03b15 C1Ba03b30 5 10 15 20 t/torb 10−6 10−5 10−4 10−3 10−2 10−1 100 D1/max(D1) NC1a00 NC1a00 NC1a01b05 NC1a01b15 NC1a01b30 NC1a03b05 NC1a03b15 NC1a03b30

Introduction and astrophysical motivation Initial models and diagnostic tools Results

Mode evolution in C1Ba01b30 and NC1a01b30

Dm=

• ρ e−imφ d3x

5 10 15 20 t/torb 10−6 10−5 10−4 10−3 10−2 10−1 100 Dm/max(D1) ρBH m = 1 m = 2 m = 3 m = 4 5 10 15 20 t/torb 10−6 10−5 10−4 10−3 10−2 10−1 100 Dm/max(D1) ρBH m = 1 m = 2 m = 3 m = 4

Introduction and astrophysical motivation Initial models and diagnostic tools Results

BH xy-movement in C1Ba01b30 and NC1a01b30

−6 −4 −2 2 4 x ×10−1 −6 −4 −2 2 4 6 y ×10−1

PPI saturation

−3 −2 −1 1 2 3 4 5 x ×10−1 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2 y

PPI saturation

Introduction and astrophysical motivation Initial models and diagnostic tools Results

Full BH trajectory and radiated linear momentum in z-direction for C1Ba01b30

2 4 6 8 10 12 14 16 (t − rdet)/torb −6 −5 −4 −3 −2 −1 1 2 ˙ Pz ×10−20 C1Ba01b05 C1Ba01b15 C1Ba01b30

Introduction and astrophysical motivation Initial models and diagnostic tools Results

Results

universality and BH movement Bardeen-Petterson effect and tilt evolution

Introduction and astrophysical motivation Initial models and diagnostic tools Results

Bardeen-Petterson effect

BP-effect causes alignment of BH spin and disk angular momentum (ν(r) = 0) in the inner regions of the disk. Generally only expected for thin disks. We observe that during the growth of the PPI the BH spin and total disk angular momentum vector tend to align. There are strong oscillations in the tilt for models NC1 for the inner disk region due to the persistent m = 1 structure.

Introduction and astrophysical motivation Initial models and diagnostic tools Results

Global disk tilt evolution for C1B and NC1

5 10 15 20 t/torb 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 νDisk [radians] C1Ba01b05 C1Ba01b15 C1Ba01b30 C1Ba03b05 C1Ba03b15 C1Ba03b30 5 10 15 20 t/torb 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 νDisk [radians] NC1a01b05 NC1a01b15 NC1a01b30 NC1a03b05 NC1a03b15 NC1a03b30

Introduction and astrophysical motivation Initial models and diagnostic tools Results

BH nutation rate about the z-axis for C1B and NC1

5 10 15 20 t/torb −0.10 −0.05 0.00 0.05 ˙ νBH [radians/torb] C1Ba01b05 C1Ba01b15 C1Ba01b30 C1Ba03b05 C1Ba03b15 C1Ba03b30 5 10 15 20 t/torb −0.030 −0.025 −0.020 −0.015 −0.010 −0.005 0.000 0.005 0.010 0.015 ˙ νBH [radians/torb] NC1a01b05 NC1a01b15 NC1a01b30 NC1a03b05 NC1a03b15 NC1a03b30

Introduction and astrophysical motivation Initial models and diagnostic tools Results

T-r diagrams of complete tilt evolution C1B

β0 = 5◦ β0 = 15◦ β0 = 30◦ a01

5 10 15 20 r 5 10 15 t/torb 0.000 0.015 0.030 0.045 0.060 0.075 0.090 0.105 0.120 0.135 0.150 5 10 15 20 r 5 10 15 20 t/torb 0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.40 5 10 15 20 r 5 10 15 t/torb 0.00 0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64

a03

5 10 15 20 r 5 10 15 20 t/torb 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.250 5 10 15 20 r 5 10 15 20 t/torb 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 5 10 15 20 r 5 10 15 t/torb 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Introduction and astrophysical motivation Initial models and diagnostic tools Results

T-r diagrams of complete tilt evolution NC1

β0 = 5◦ β0 = 15◦ β0 = 30◦ a01

5 10 15 20 25 30 r 5 10 15 20 t/torb 0.000 0.015 0.030 0.045 0.060 0.075 0.090 0.105 0.120 0.135 5 10 15 20 25 30 r 5 10 15 20 t/torb 0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 5 10 15 20 25 30 r 5 10 15 20 t/torb 0.00 0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64

a03

5 10 15 20 25 30 r 5 10 15 20 t/torb 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 5 10 15 20 25 30 r 5 10 15 20 t/torb 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 5 10 15 20 25 30 r 5 10 15 20 t/torb 0.00 0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64 0.72 0.80

Introduction and astrophysical motivation Initial models and diagnostic tools Results

T-r diagrams of complete tilt evolution D2

β0 = 5◦ β0 = 15◦ β0 = 30◦ a01

5 10 15 20 25 30 r 5 10 15 20 t/torb 0.000 0.015 0.030 0.045 0.060 0.075 0.090 0.105 0.120 5 10 15 20 25 30 r 5 10 15 20 t/torb 0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 5 10 15 20 25 30 r 5 10 15 20 t/torb 0.00 0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64

a03

5 10 15 20 25 30 r 5 10 15 20 t/torb 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200 0.225 0.250 5 10 15 20 25 30 r 5 10 15 20 t/torb 0.00 0.06 0.12 0.18 0.24 0.30 0.36 0.42 0.48 0.54 5 10 15 20 25 30 r 5 10 15 20 t/torb 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Introduction and astrophysical motivation Initial models and diagnostic tools Results

Conclusions

The PPI seems to be a universal feature of the models that develop it: we have seen that it grows for all initial tilt angles and spin magnitudes investigated. The PPI causes a realignment between BH and global disk angular momentum during it’s growth. For models NC1, the persistent m = 1 structure in the disk causes a rapid

• scillation of the tilt angle ν(r) in the inner region of the

disk.

Introduction and astrophysical motivation Initial models and diagnostic tools Results

The End

Thank you for your attention! This work is supported by the Spanish Ministerio de Educación y Ciencia in the project Computational Relativistic Astrophysics (AYA2010-21097-c03-01).

Introduction and astrophysical motivation Initial models and diagnostic tools Results

Other codes and their results

Until very recently, full 3D GR simulations of self-gravitating tori around BH were not possible. Kiuchi et al (2011), Montero et al (2010) (in 2D) and Korobkin et al (2011,2012) performed simulations of self-gravitating thick tori with different codes. We use a fourth code and different setup in our current work to redo some of the simulations of Korobkin et al (2011/2012) with similar initial data.

Introduction and astrophysical motivation Initial models and diagnostic tools Results

Convergence

p =

1 log(f) log

• Hlow

2 −Hmed 2

• Hmed

2

−Hhigh

2

• 0.0

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 t/torb 10−7 10−6 H2 ∆x = 0.02 ∆x = 0.04 ∆x = 0.08 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 t/torb 0.5 1.0 1.5 2.0 2.5 3.0 3.5 convergence order p H2

Introduction and astrophysical motivation Initial models and diagnostic tools Results

Where does the non-axisymmetric perturbation in the torus causing the PPI come from?

Spherically symmetric (Schwarzschild BH) or axisymmetric (Kerr BH) gauge pulse travelling through the torus initially. Cartesian grid causes small delays or advances in the perturbation > it is not axisymmetric any more. In Spherical codes, PPI has to be seeded manually.

Introduction and astrophysical motivation Initial models and diagnostic tools Results

evolution of ρmax for C1B and NC1

5 10 15 20 t/torb 0.5 1.0 1.5 2.0 ρmax/ρmax(0) C1Ba00 C1Ba01b05 C1Ba01b15 C1Ba01b30 C1Ba03b05 C1Ba03b15 C1Ba03b30 5 10 15 20 t/torb 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 ρmax/ρmax(0) NC1a00 NC1a01b05 NC1a01b15 NC1a01b30 NC1a03b05 NC1a03b15 NC1a03b30

Introduction and astrophysical motivation Initial models and diagnostic tools Results

Accretion rate for C1B and NC1

5 10 15 20 t/torb 1013 1014 1015 1016 1017 1018 ˙ M/ ˙ MEdd C1Ba00 C1Ba01b05 C1Ba01b15 C1Ba01b30 C1Ba03b05 C1Ba03b15 C1Ba03b30 5 10 15 20 t/torb 1012 1013 1014 1015 1016 1017 ˙ M/ ˙ MEdd NC1a00 NC1a01b05 NC1a01b15 NC1a01b30 NC1a03b05 NC1a03b15 NC1a03b30

Introduction and astrophysical motivation Initial models and diagnostic tools Results

l = m = 2 mode of the real part of the Weyl scalar Ψ4 for C1B and NC1

2 4 6 8 10 12 14 16 (t − rdet)/torb −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 rΨ2,2

4

×10−5 C1Ba01b05 C1Ba01b15 C1Ba01b30 2 4 6 8 10 12 14 16 18 (t − rdet)/torb −1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 rΨ2,2

4

×10−6 NC1a01b05 NC1a01b15 NC1a01b30

Introduction and astrophysical motivation Initial models and diagnostic tools Results

Angular momentum transport for C1Ba00 and NC1a00

5 10 15 20 25 30 35 40 r 5 10 15 20 t/torb 0.00 0.15 0.30 0.45 0.60 0.75 0.90 5 10 15 20 25 30 35 40 r 5 10 15 20 t/torb 0.00 0.15 0.30 0.45 0.60 0.75 0.90