Studies of self-gravitating tori around black holes and of self-gravitating rings Pedro Montero Max Planck Institute for Astrophysics Garching (Germany) Collaborators: Jose Antonio Font (U. Valencia) Masaru Shibata (Tokyo U.) David Petroff (U. Jena) Wolfgang Kastaun (U. Tuebingen) Garching 16-02-2009
Outline of the talk Introduction and motivation The Nada code: review Simulations of self-gravitating tori Oscillations of self-gravitating rings References: PM, Font & Shibata, PRD (2008)
Introduction Self-gravitating tori around black holes are common end-products in several astrophysical scenarios: May form when a NS merges with another compact object, either a NS or a BH. Numerical simulations (Shibata et al, Baiotti et al. 2008) show that most of the material disappears beyond the event horizon in a few ms, and a thick accreting disk or tori with mass of about 10% of total mass may be formed. Such tori are dense (close to nuclear matter density) and hot, with maximum temperature of about 10 11 K Material from the disk will then accrete onto the black hole, releasing energy, and at these high density and temperature neutrinos play a very important role in cooling the accretion disk.
Introduction Magnetic field are important for the formation process and evolution Another formation mechanism is the gravitational collapse of the rotating core of massive stars, i.e. collapsar scenario: As a result of the collapse a BH of about 3 solar masses is formed and an accretion tori which is fed by infalling stellar material develops. These type of astrophysical objects are very interesting because most current models for the central engine of GRBs involve an accretion torus orbiting around a central black hole Another motivation is that these objects may also be a promising new source of GWs Zanotti, Rezzolla, and Font, MNRAS, 341, 832 (2003) Zanotti, Font, Rezolla, and Montero, MNRAS, 356, 1371 (2005) Montero, Zanotti, Font, and Rezzolla, MNRAS, 378, 1101 (2007) Nagar, Font, Zanotti, and de Pietri, Phys. Rev. D, 75, 044016 (2007)
Introduction Using post-newtonian approximation. Similar results have been obtained for tori with a non-constant distribution of specific ang. mom. and orbiting a Kerr BH. Also for models with a toroidal magnetic field configuration.
Motivation: runaway instability Found by Abramowicz, Calvani and Nobili (1983): in a BH+thick disk system the gas flows in an effective (gravitational+centrifugal) potential whose structure is similar to that of a close binary. The torus filling Roche-lobe has a cusp-like inner edge at the Lagrange point L 1 where mass transfer driven by the radial pressure gradient is possible. These systems may be subjected to a runaway instability: due to accretion from the disk the BH mass and spin increase and the gravitational field changes. Two evolutions are feasible: 1. Cusp moves inwards toward the BH, mass transfer slows down. Stable. 2. Cusp moves deeper inside the disc material, mass transfer speeds up. Unstable.
Motivation: runaway instability 1. Cusp moves deeper inside the disc material, mass transfer speeds up. Unstable. 2. Cusp moves inwards toward the BH, mass transfer slows down. Stable.
Motivation: runaway instability Font and Daigne, MNRAS, 334, 383 (2002)
Motivation: runaway instability Previous studies of the runaway: Font and Daigne, MNRAS, 334, 383 (2002)
Motivation: runaway instability Constant specific ang.mom. Not Cowling (unstable) Cowling (stable) Non-const. ang.mom suppress of instability Numerical hydrodynamical simulations in GR of non-self-gravitating discs show that constant angular momentum discs are unstable while those with power-law distributions of the specific angular momentum (increasing outwards) are stable. Font and Daigne, MNRAS, 334, 383 (2002) Font and Daigne, MNRAS, 349, 841 (2004)
Motivation: runaway instability Time evolution of the rest-mass accretion rate for different values of the radial velocity perturbation and for a marginally stable initial model Zanotti,Rezzolla, Font MNRAS(2003)
Motivation: runaway instability Constant angular momentum discs are unstable Power-law distributions of the specific angular momentum are stable. 1. Rotation of the central BH has a stabilizing effect 2. Non-constant distribution of specific angular momentum has also stabilizing effect 3. Self-gravity of the torus tends to favour the instability
The 2D code: Nada 2D axisymmetric code to solve the coupled system of Einstein equations and GRHD equations: BSSN formulation of Einstein eqs: 4th(6th) order finite differencing Cartoon method: axisymmetry using a Cartesian grid Puncture approach for BH treatment: Puncture BH ID and moving puncture gauge GRHD eqs in conservation form: HRSC schemes: Roe and HLLE solvers Slope-limited TVD and PPM reconstructions Time integration using MoL: 4th-order RK scheme OpenMP
GRHD Equations GRHD eqs written as a first-order flux-conservative system: which allows for the use of the HRSC schemes Ref: Shibata (2003)
Formulation of Einstein Equations 3+1 decomposition of Einstein equations BSSN formulation: Shibata & Nakamura (1995), Baumgarte & Shapiro(1999)
Puncture approach Moving puncture: (" t # $ i " i )% = #2%K " 2 t # i = 3 " 4 t$ i %&" t # i 1. φ-method: the original BSSN variable is evolved 2. χ-method: introduces a new variable 3. W-method: w " e #2$ " # e $4% (Campanelli et al 2006; Baker et al 2006, Marronetti et al 2007)
Code tests performed Vacuum tests: Single BH evolutions with geodesic slicing and also with the moving puncture gauge conditions Hydrodynamic tests: 1D shock tube test, planar shock reflection test 2D Spherical shock reflection test Einstein+Hydro tests: Spherical relativistic stars in equilibrium Rotating relativistic stars Gravitational collapse of spherical stars and subsequent evolution of the BH formed without excision and relying only on gauge conditions, i.e. no dissipation added (see also Baiotti&Rezzolla, PRD 2006)
Initial data for BH+self-gravitating torus We use the initial data recently computed by Shibata (2006) in the puncture framework. Assuming the 3+1 formalism line element written in quasi-isotropic form: Configurations are computed assuming a perfect fluid stress-energy tensor and polytropic EOS (Γ=4/3). u t and u ϕ non-zero fluid velocity components We discuss a model with constant specific ang.mom. around Schwarzschild BH. Torus-to-BH mass ratio is M t /M BH =0.1 r in =7.1M, r out =14.0M, r max =9.8M orbital period at the centre of the torus t orb =223M, M is the ADM mass of the system.
Fixed spacetime evolution of BH+self-gravitating torus Δx=Δz=0.05M, N x xn z =600x600, 0 x,z 30M Spacetime: 4th-order finite differencing; Hydro: Roe solver, MC rec. Isocontours of the logarithm of the rest-mass density at t=0 (left) and after 5 orbits (torb=223m 1.1ms if M=M ) Stationarity of the torus preserved for long period of time (no sign of numerical instabilities). t=0 t=1000m=5orb
Fixed spacetime evolution of BH+self-gravitating torus rest-mass error<0.1% Now adding a radial perturbation central rest-mass density error<1% o 1 /f~1.4 angular momentum error<0.1%
Dynamical spacetime evolution of BH+self-gravitating torus Δx=Δz=0.05M, N x xn z =600x600, 0 x,z 30M Spacetime: 4th-order finite differencing; Hydro: Roe solver, MC rec. rest-mass density isocontours at t=0 and t=2.7 orbits overlap very well Spacetime evolution very dynamical till t=30m due to initial adjustment of the gauge which produces a small pulse of the metric funcs. that propagates outwards
Dynamical spacetime evolution of BH+self-gravitating torus rest-mass Time evolution of the log of rest-mass density profile (Movie1, total time evolution: t=1200m) central rest-mass density angular momentum
Dynamical spacetime evolution of BH+self-gravitating torus Model M t /M BH R in R out Ang.Mom. t prb (j=kω -n ) M1 0.1 7M 14M const 223M (n=0) RM1 0.1 4M 9M const 147M RM2 RM3 0.5 0.5 4M 4M 20M 20M Const n=0 Non-const n=1/8 198M 229M Marginally stable models RM4 0.1 4M 20M Non-const 245M
Dynamical spacetime evolution of BH+self-gravitating torus Model RM3 Δx=Δz=0.05M, N x xn z =600x600, 0 x,z 30M Spacetime: 4th-order finite differencing; Hydro: HLLE solver, PPM rec. M T /M BH =0.504 non-constant specific angular momentum 1orbital period Total restmass ~0.2% Cusp:4M Outer edge 20M, (Movie2, total time: 1300M)
Dynamical spacetime evolution of BH+self-gravitating torus Same model RM3, introducing a perturbation in the x-component of the 3-velocity (η 0.01) 1orbital period Total restmass Cusp:4M Movie3, total time: 1300M
Dynamical spacetime evolution of BH+self-gravitating torus Current status of simulations Model M t /M BH R in R out Ang.Mom. t orb t run RM1 0.1 4M 9M const 147M 500M stable RM3 0.5 4M 20M Non-const 229M 1000M stable RM4 0.1 4M 40M Non-const 245M 700M stable
Evolution of rings in Cowling approximation As a first step, the ring considered is quite Newtonian. Total rest-mass=0.015. The coordinate inner radius is at x=0.75 and the outer one at x=1. The ring is almost circular in cross section (the vertical extent is roughly 0.23). Δx=Δz=0.00375, N x xn z =100x100, 0 x,z 1.5, orbit=56
Evolution of rings in Cowling approximation central rest-mass density error<1%
Evolution of rings in Cowling approximation 27.6 KHz 63.9 KHz 40.4 KHz 50.4 KHz
Evolution of rings in Cowling approximation Relativistic ring: total rest-mass=1.259; orbital period=26; central lapse=0.2 Nada code (KHz) 11.239 14.510 16.314 17.407 19.901 Pizza code (KHz) 11.002 14.546 16.456 17.471 19.420
Conclusions We are currently using the Nada-code to investigate the dynamics of self-gravitating tori and rings, among other astrophysical scenarios. Simulations of self-gravitating torus in equilibrium around BHs performed: robustness assessed. So far, accreting tori seem stable against the runaway instability for several dynamical timescales. It seems so far that self-gravity may not be enough to destabilize tori with possibly non-constant distribution of specific angular momentum in just a few dynamical timescales. But still longer simulations are needed, as well as other initial perturbations to enhance the accretion of matter for concluding results. We are also investigating the oscillations of rings in the Cowling approximation.