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## QPOs and Nonlinear Pendulums

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**QPOs and Nonlinear Pendulums**Weizmann Institute - January 13th 2008 by Paola Rebusco**QPOs=**Quasi (=nearly) Periodic Oscillations • Oceanography • Electronic circuits • Neural systems • ASTROPHYSICS**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**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**Shakura & Sunyaev model (1973)**Thin disks: • No radial pressure gradient • l=lk, : keplerian angular momentum • neglect higher order terms**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!**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%**Analogy with the Mathieu equation**Kluzniak &Abramowicz (2000)**HFQPOs and nonlinear pendulums**• wr < wq------ n=3 • Subharmonics: signature of nonlinear resonance • The frequency are corrected • Not exact rational ratio Bursa 2004**Epicyclic Eigenfrequencies**Schwarzschild**TOY MODEL - Perturbed geodesics**• Geodesics Equations • Taylor series to the III order • Isotropic non-geodesic term (Abramowicz et al 2003, Rebusco 2004, Horak 2005)**n:m**Plane symmetry n:2p p=1 n=3 Analytical results**The Method of Poincarè-Lindstedt**It is a technique for calculating periodic solutions**SCO X-1 numerics&analytics**• Risonanza 2:3**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) ”**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**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**What we do not knowEXCITATION MECHANISM**• Direct forcing (NS) • Stochastic forcing (BH) • Disk instabilities (Mami Machida’s simulations: P&P instability)**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)**NEUTRON STARS**• = spin or spin/2 (NO!Mendez&Belloni 2007) • The asymptotic expansion is not valid for large amplitudes**Black Holes**• 3:2, [ GRS 1915+150 (5:3)..] “on those small numbers” (4) “the model works”TB**BH vs NS**• Different excitation • Different modulation (GR/boundary layer) Is it a different phenomenon?!?**What we do not know - DAMPING**• Stochastic damping ?!? • What controls the coherence Q= /? Didier Barret et al. - Qlow>Qhigh - Qlow increases with and drops**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”?**Conclusions**There is strong evidence that HFQPOs arise from nonlinear resonance in accretion disks in GR (K&A 2000) BUT some ingredients are missing**Successive approximations**• a • b (e) (e2)**Tuning parameter**Small-divisor terms are converted into secular terms**Stability**Linearization Linearizzazione