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QPOs and Nonlinear Pendulums

QPOs and Nonlinear Pendulums

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QPOs and Nonlinear Pendulums

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  1. QPOs and Nonlinear Pendulums Weizmann Institute - January 13th 2008 by Paola Rebusco

  2. QPOs= Quasi (=nearly) Periodic Oscillations • Oceanography • Electronic circuits • Neural systems • ASTROPHYSICS

  3. QPOs • Lorentzian Peaks • X-ray power spectra • Highly Coherent • 0.1-1200 Hz • Show up alone OR in pairs OR more … Both NS and BH sources Van der Klis, 2000

  4. High frequency (kHz) QPOs M. van der Klis 2000 T. Strohmayer 2001 • In Black Holes: stable • In Neutron Stars: the peaks “move”-much more rich phenomenology (state, spin, Q) 3:2…although not always

  5. Low Mass X-ray Binaries

  6. ACCRETION DISKS

  7. Shakura & Sunyaev model (1973) Thin disks: • No radial pressure gradient • l=lk, : keplerian angular momentum • neglect higher order terms

  8. HFQPOs and General Relativity • High frequency (kHz) QPOs lie in the range of ORBITAL FREQUENCIES of geodesics just few Schwarzschild radii outside the central source • The frequencies scale with 1/M (Mc Clintock&Remillard 2004 ) TEST GR in STRONG fields!

  9. HFQPOs and ULXs

  10. Models (many!): • Relativistic precession model (Stella&Vietri 1998) • Beat-frequency models (Lamb&Miller 2003,Schnittman&Bertschinger 2004) • Resonance models(Kluzniak&Abramowicz 2000) • Discosesmeic models (Wagoner et al 2001) • Non-axisymmetric trapped modes (S. Kato 2001,2007) • Hydrodynamical oscillations model (Rezzolla et al 2003) NONE WORKS 100%

  11. Our group, lead by Marek Abramowicz and Wlodek Kluzniak

  12. Analogy with the Mathieu equation Kluzniak &Abramowicz (2000)

  13. Transition curves

  14. HFQPOs and nonlinear pendulums • wr < wq------ n=3 • Subharmonics: signature of nonlinear resonance • The frequency are corrected • Not exact rational ratio Bursa 2004

  15. The effective potential

  16. Epicyclic Eigenfrequencies Schwarzschild

  17. TOY MODEL - Perturbed geodesics • Geodesics Equations • Taylor series to the III order • Isotropic non-geodesic term (Abramowicz et al 2003, Rebusco 2004, Horak 2005)

  18. Geodesics

  19. Perturbed geodesics

  20. n:m Plane symmetry n:2p p=1 n=3 Analytical results

  21. The Method of Poincarè-Lindstedt It is a technique for calculating periodic solutions

  22. Transition curves

  23. Weakly nonlinear coupled oscillators

  24. SCO X-1 numerics&analytics • Risonanza 2:3

  25. the Bursa line

  26. Successive approximations • a Multiple Scales “ Imagine that we have a watch and attempt to observe the behaviour of the solution using the different scales of the watch (s, m, h) ”

  27. To know that we know what we know, and to know that we do not knowwhat we do not know, that is true knowledge. Nicolaus Copernicus

  28. What we know • The perturbation of the geodesics leads to two nonlinear coupled harmonic oscillators • The strongest instability occurs when the ratio is 3:2 : this is a direct consequence of the symmetry • In this case the asymptotic solution shows two peaks in correspondence of a ratio between the frequencies near to 3:2 • the observed frequencies are close, but not equal to the eigenfrequencies

  29. What we do not knowEXCITATION MECHANISM • Direct forcing (NS) • Stochastic forcing (BH) • Disk instabilities (Mami Machida’s simulations: P&P instability)

  30. SIMULATIONS • Slender tori slightly out of equilibrium do not produce QPOs (only transient - Mami Machida) • They do if the torus is kicked HOWEVER it is difficult to recognize the modes (William Lee,Omer Blaes, Chris Fragile, Mami Machida, Eva Sramkova)

  31. NEUTRON STARS •  = spin or spin/2 (NO!Mendez&Belloni 2007) • The asymptotic expansion is not valid for large amplitudes

  32. Perturbed geodesics (numerics&theory)

  33. Black Holes • 3:2, [ GRS 1915+150 (5:3)..] “on those small numbers” (4) “the model works”TB

  34. Turbulence? (Vio et al 2005)

  35. The “right” turbulence feeds the resonance

  36. BH vs NS • Different excitation • Different modulation (GR/boundary layer) Is it a different phenomenon?!?

  37. What we do not know - DAMPING • Stochastic damping ?!? • What controls the coherence Q= /? Didier Barret et al. - Qlow>Qhigh - Qlow increases with and drops

  38. EPI or NOT EPI?!? • EPI: - the advantage is that they survive to damping. Pressure and gravity coupling? NO!!!(slender torus-Jiri Horak)… • NOT EPI: - the eigenfreq. depend on thermodynamics -> 1/M ?!? - g-modes (but do not survive) which modes are “special”?

  39. Conclusions There is strong evidence that HFQPOs arise from nonlinear resonance in accretion disks in GR (K&A 2000) BUT some ingredients are missing

  40. Successive approximations • a • b (e) (e2)

  41. Secular and nearly secular terms

  42. Tuning parameter Small-divisor terms are converted into secular terms

  43. Stability Linearization Linearizzazione