Oscillations in Accretion Discs around Black Holes Bárbara Trovão Ferreira & Gordon Ogilvie DAMTP, University of Cambridge

Problems 1. Turbulent viscosity  damping No reflection boundaries 2. Linear oscillations (small amplitude) need an excitation mechanism to be detected no standing waves No resonant cavity

Disc Oscillations Equilibrium state: Linear perturbations: e.g. Kato 2001 Lubow & Pringle 1993 Equilibrium state: Ignore accretion - assume timescale of oscillations much faster then timescale for accretion WKB - radial variation of perturbed quantities much faster than variation of equilibrium quantities Local separation of variables: the function of z can actually also be a slowly varying function of r as well - anyway, locally is function of z only Get set of ODEs in r Linear perturbations: Further separation of variables (approx):  System of 1st order ODEs in for

Dispersion Relation Far from resonances: DR  f / 2D modes  p modes 1.WKB anstaz 2.w hat - frequency in fluid frame, kappa - epicyclic frequency (particle in orbity with ang. Velocity Omega - kick in radial direction - particle oscillates with epicyclic frequency), Omega_z - same but for vertical kick 3.More general DR - Lubow & Pringle 1993.  f / 2D modes  p modes  r modes Let (simpler 3D mode)

Keplerian Disc p r freq2 [in units of (c3/10GM)2] radius [in units of GM/c2]

Relativistic Disc Using particle orbit, relativistic expressions for characteristic frequencies (Kato 1990) p freq2 [in units of (c3/10GM)2] Kato & Fukue 1980 Okazaki, Kato, Fukue 1987 r radius [in units of GM/c2]

Numerical Calculation of Trapped r mode eigenvector matrix ≠ identity depending on b. c. eigenvalue Li, Goodman, Narayan - potential analogy - harmonic oscillator potential and wavefunctions

Deformed Disc Kato 2004, 2007 Goal: excite the r-mode; Idea: r-mode + warp => f-mode +warp => r-mode => r-mode grows w_r+-w_w=w_i, m_r+-m_w=m_i, n_r+-n_w=n_i

Wave Coupling Goal: excite the r-mode; Idea: r-mode + warp => f-mode +warp => r-mode => r-mode grows w_r+-w_w=w_i, m_r+-m_w=m_i, n_r+-n_w=n_i

dissipation term () to be included in equations for interm. mode Energy Exchanges Co-rotation resonance: CR interm. mode r mode Warp - frequency zero - energy approximately zero - not relevant in energy exchanges E<0 - Wave passage through a medium decreases its energy Have to include damping term in f-mode equations warp/eccentricity energy ≈ zero - interm. & r mode exchanges: damping of negative energy wave  draws positive energy from disc rotation  available to excite r mode (who replenishes intermediate mode negative energy) dissipation term () to be included in equations for interm. mode

Numerical Calculations of Growth Rate complicated matrix includes coupling terms & interm. mode dissipation eigenvalue After separating variables we get a linear (on the unknows) system of ODEs that can be solved as a generalized eigenvalue problem. eigenvector - r and interm. modes quantities evaluated at discrete points matrix ≠ identity depending on b. c.

Growth of Oscillations - Results Ferreira & Ogilvie 2008

Conclusions In a relativistic disc, inertial modes can, in principle, avoid turbulent effects by being trapped in a small region; These modes can be excited via the coupling mechanism described here, provided that: Negative energy (intermediate) mode dissipates in order to remove rotational energy from the disc; Global deformations reach the inner disc region with non-negligible amplitude; Measure black hole spin.

The End