1. Torus Oscillations in Accretion Disks and kHz QPOs
William Lee
Instituto de Astronomía,
Universidad Nacional Autónoma de México
In collaboration with Marek Abramowicz (Chalmers), Wlodek Kluzniak (CAMK), Eduardo Rubio-Herrera (UNAM)
MIT, Oct. 2006

2. Data - 1
Sco X-1
Twin peaks…
Van der Klis et al 1997

3. Data - 2
Remillard et al. (2002)
XTE J
Strohmayer (2001)
GRO J
Two kHz QPO peaks in Black Hole systems, showing a 3:2 correspondence.
(Abramowicz & Kluzniak 2001).

4. Data - 3 SAXJ1808 … one or two peaks. Source n(spin) Dn
Wijnands et al 2003

5. Oscillations-What do we model?
Accretion disk = finite torus
Slender = thin (L,H << r) or thick (L,H~r)
No self-gravity (m<<M)
Torus = axisymmetric
Really a fluid (not a wire): ideal gas eos, P( r )
Strong gravity = pseudo-potential (central pit, ISCO)

6. How do we model it - 1?
Lagrangian hydrodynamics, 2D, cylindrical symmetry (SPH)
Initial conditions in hydrostatic equilibrium
Initial conditions with constant or power law
distributions for specific angular momentum.
Potential F = GM[1 - exp(rms/r)]/rms (Kluzniak & Lee 2002), similar to that of Paczynski & Wiita

7. How do we model it - 2?
Meridional section (density) of a slender torus
Blaes 1985, Blaes et al. 2006
Sramkova et al. 2006
Lee, Abramowicz & Kluzniak 2004

8. Characteristic frequencies
M=1.4Mo
Newtonian GR
n=k=z
n=z
Frequency (Hz)
k<n
r/rg
r/rg

9. What do we look for?
Variability in “local” quantities, e.g., the position of the circle of maximum density or pressure.
Variability in “global” quantitites, e.g., the total internal energy of the torus.
Impact upon X-ray luminosity? (Bursa et al. 2004, Schnittman 2005)

10. Applying a perturbation
Radial perturbation with a fixed repetition frequency.
Lee, Abramowicz & Kluzniak 2004

11. Non-linear response
Coupling between radial and vertical modes, due to pressure.
Strong response when np and Dn in 1:1: 3:2 or 2:1 ratio (but see Sramkova et al. 2004).
Lee, Abramowicz & Kluzniak 2004

12. Global variables - 1
Rubio-Herrera & Lee 2005

13. Global variables - 2 Power in total internal energy
A localized perturbation may excite global modes
(see Rezzolla et al. 2003).
Rubio-Herrera & Lee 2005

14. Global variables - 3 nr nz nb ~ 3 nz /2 z/rg ||Uint||
Breathing and epicyclic modes for slender tori show a ~3:2 correspondence (Blaes et al. 2006).
What happens for thick tori?
r/rg
Frequency (Hz)
Lee 2006, in prep

15. The end
The excitation of internal fluid modes at frequencies other than the epicyclic frequencies is possible in an accretion disk.
What is the perturbation? NS spin? Turbulence?
How does this impact X-ray emission? (Bursa et al. 2004, Schnittman 2005)
Torus vs. full disk? Mode leakage and accretion? (Blaes 1987, Fragile 2005)
Modes in thick disks may be more difficult to excite efficiently.

16. Radially drifting tori - 1
Angular momentum is removed on a timescale
Tl >> Torb (e.g., viscosity).
The perturbation is applied continuously, from rout to rin.
Tests with varying MNS, np.
Lee 2005

17. Radially drifting tori - 2
Power in radial and vertical motions
Lee, 2005

18. Radially drifting tori - 3
Power in vertical motions
Range of excited frequencies, from zmin to zmax.
Secondary peak with variable power, at about z+np/2.
Lee 2005