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QPOs and Nonlinear Pendulums

This presentation provides an in-depth exploration of Quasi-Periodic Oscillations (QPOs) and their relation to nonlinear pendulum dynamics. Special emphasis is placed on high-frequency QPOs observed in black holes and neutron stars, detailing their robust relations to general relativity. It discusses various models explaining QPOs, including relativistic precession and resonance models, underscoring the prominence of a 3:2 frequency ratio. Additionally, the implications of these findings in astrophysical contexts, such as accretion disks and their stability, are analyzed, offering insights into the complexities of stellar phenomena.

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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

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