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

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

Data - 2 Remillard et al. (2002) XTE J1550-564 Strohmayer (2001) GRO J1655-40 Two kHz QPO peaks in Black Hole systems, showing a 3:2 correspondence. (Abramowicz & Kluzniak 2001).

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

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)

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

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

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

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)

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

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

Global variables - 1 Rubio-Herrera & Lee 2005

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

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

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.

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

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

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