Dynamics of thick discs around black holes
Relativistic tori (namely geometrically thick discs) orbiting around black holes are expected to form in a number of different scenarios, such as after the gravitational collapse of the core of a rotating massive star or after a neutron star binary merger. Numerical simulations of these scenarios, both in the Newtonian and in the relativistic regime, have shown that, under certain conditions, a massive disc may be produced. This disc surrounds a newly formed black hole, has large rest-mass density and highly super-Eddington mass fluxes, and its angular momentum obeys a sub Keplerian distribution. There are several reasons for considering these objects astrophysically interesting
1) An accretion torus around a black hole is probably the progenitor source of short duration (&rt; 2s) gamma ray bursts (GRBs).
2) An energetic long duration( > 2s) GRB accompanied by a Type Ib/Ic supernova may be produced as the result of a thick accretion disc process.
3) Even in the absence of other instabilities triggered by non-axisymmetric disturbances, thick discs around black holes may be dynamically unstable, due to the progressive penetration of the cusp of the equilibrium configuration into the disc when some accretion of matter is induced.
THE RUN-AWAY INSTABILITY
Consider a marginally stable disc, therefore filling its Roche-lobe with the cusp representing the Lagrangian point. Everywhere in the torus the inertial forces (gravitational, centrifugal and pressure gradients) compose to produce a zero acceleration; at the cusp this composition occurs only at on a circular ring.
Any perturbation of this configuration will produce a small mass overflow and results into a change of the position of the cusp.
There are two possible evolutions:
a) the cusp moves towards the black hole -> the new configuration is STABLE
b) the cusp moves away from the black hole -> new matter will be outside of the Lagrangian point ? the new configuration is UNSTABLE
Click on image to see animation of the rest mast density
Numerical simulations show that, if suitably perturbed, thick discs become unstable to the accretion of matter in a dynamical space-time (i.e. when the increase of the black hole mass is properly taken into account). The instability makes the disc swallow onto the black hole in few orbital periods.
For small perturbations, the exponential growth of the mass accretion is anticipated by a periodic oscillation.
PERIODIC DYNAMIC BEHAVIOR
When the accreted mass is negligible with respect to the mass of the black hole, the spacetime can be safely considered fixed, and the response of the disc to external perturbations is studied both through numerical simulations and through a linear perturbative analysis aimed at computing the eigenfunctions and eigenfrequencies of an equilibrium configuration around a black hole.
The central density of the disc is a good tracer to monitor the periodic oscillation induced by the perturbation.
Click on image to see animation of the rest mast density
LINEAR PERTURBATION ANALYSIS
In accretion discs, a first restoring force to external perturbations is given by the centrifugal force, which is responsible for the appearance of the so called inertial waves, tightly related to the orbital motion of the disc and hence to epicyclic oscillations. A second restoring force is offered by pressure gradients, and the oscillations that are produced in this way are related to p modes and have close connections with the propagation of sound waves in the perturbed fluid. A third restoring force is the gravitational field in the direction orthogonal to the orbital plane. If a portion of the disc is perturbed in the vertical direction, in fact, the vertical component of the gravitational field will produce a harmonic oscillation across the equatorial plane with oscillation frequency equal to the orbital frequency. These oscillations are related to corrugation waves.
Under the following assumptions:
1) Stationarity and axisymmetry of the equilibrium model
2) Non-self gravitating disc
3) Barotropic equation of state
4) Non Keplerian flow
5) Fixed Schwarzschild or Kerr spacetime
it is possible to solve the perturbed relativistic Euler equations in order to find the eigenfunctions of the perturbations and the corresponding eigenfrequencies.
Eigenfunctions of the pressure perturbation for the fundamental mode f and the first three overtones in a model with a constant distribution of the specific angular momentum.
The comparison between the results of the non linear numerical simulations and those coming from the linear perturbation analysis indicate that the fundamental mode f and the first overtones in the numerical simulations do represent the first p-modes of the system and that these are in a ratio o1/f very close to 3:2, with deviations that can be as large as 15%. On the other hand, the computed spectra show overtones at integer multiples, in particular at 2f, plus additional modes which are linear combinations of f and o1 and that are all the results of non linear coupling effects.
Comparison of the eigenfrequencies of the fundamental mode as computed from the numerical simulations and from the linear perturbative analysis.
Power spectrum of the L2 norm of the rest mass density obtained from the numerical simulations. It shows the presence of a fundamental mode f, two genuine overtones o1 and o2, plus a large number of modes given by the non-linear coupling between f, o1 and o2.
The overtones can be selectively excited by choosing as initial perturbations the eigenfunctions of the velocity and of the rest mass density calculated through the linear perturbative analysis.
GRAVITATIONAL WAVE EMISSION
Among traditional isolated sources of gravitational waves, such as supernova explosion, black hole formation, binary mergers, we should actually consider also high density oscillating thick discs. Indeed, when the simple Newtonian Quadrupole formula is used to compute the gravitational wave emission from oscillating thick discs, the numbers obtained are very promising and usually well above the sensitivity curve of the laser interferometric detectors like LIGO, VIRGO, GEO.
In particular, the gravitational radiation emitted by these sources is comparable to or larger than that expected during the gravitational collapse of a stellar iron core, with a rate of detectable events that could also be larger given the variety of physical scenarios leading to the formation of a massive torus orbiting a black hole. Overall, the strength of the gravitational waves emitted and their periodicity are such that signal-to-noise ratios ~O(1)- O(10) can be reached for sources at 20 000 or 10 kpc, respectively, making these new sources of gravitational waves detectable and potentially important.
The comparison with present and future interferometric detectors show that thick discs are effectively very promising sources of gravitational waves.
Essential bibliography:
Abramowicz M., Calvani M., Nobili L., Nat., 302, 597, 1983
Font J., Daigne F., MNRAS, 334, 383, 2002
Rezzolla L., Yoshida S., Zanotti O., MNRAS, 344, 978, 2003
Zanotti O., Rezzolla L., Font A., MNRAS, 341, 832, 2003
Zanotti O., Font J., Rezzolla L., Montero P., MNRAS, 356, 1371, 2005
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