QPOs and Nonlinear Pendulums

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

• Quasi (nearly) Periodic Oscillations
• Oceanography
• Electronic circuits
• Neural systems
• ASTROPHYSICS

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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
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High frequency (kHz) QPOs

• In Black Holes
• stable
• In Neutron Stars
• the peaks move-much more rich
  phenomenology (state, spin, Q)

32although not always
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Low Mass X-ray Binaries
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ACCRETION DISKS
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Shakura Sunyaev model (1973)

• Thin disks
• No radial pressure gradient
• llk, keplerian angular momentum
• neglect higher order terms

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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 ClintockRemillard 2004 )

TEST GR in STRONG fields!
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HFQPOs and ULXs
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Models (many!)

• Relativistic precession model (StellaVietri
  1998)
• Beat-frequency models (LambMiller
  2003,SchnittmanBertschinger 2004)
• Resonance models (KluzniakAbramowicz 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

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Our group, lead by Marek Abramowicz and Wlodek
Kluzniak
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Analogy with the Mathieu equation
Kluzniak Abramowicz (2000)
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Transition curves
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HFQPOs and nonlinear pendulums

• wr lt wq ------? n3
• Subharmonics signature of
• nonlinear resonance
• The frequency are corrected
• Not exact rational ratio

Bursa 2004
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The effective potential
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Epicyclic Eigenfrequencies

Schwarzschild
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TOY MODEL - Perturbed geodesics

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

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Geodesics
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Perturbed geodesics
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Analytical results
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The Method of Poincarè-Lindstedt
It is a technique for calculating periodic
solutions
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Transition curves
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Weakly nonlinear coupled oscillators
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SCO X-1 numericsanalytics

• Risonanza 23

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the Bursa line
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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)

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

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What we know

• The perturbation of the geodesics leads to two
  nonlinear coupled harmonic oscillators
• The strongest instability occurs when the ratio
  is 32 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 32
• the observed frequencies are close, but not
  equal to the eigenfrequencies

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What we do not knowEXCITATION MECHANISM

• Direct forcing (NS)
• Stochastic forcing (BH)
• Disk instabilities (Mami Machidas simulations
  PP instability)

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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)

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NEUTRON STARS

• ?? ?spin or ?spin/2 (NO!MendezBelloni 2007)
• The asymptotic expansion is not valid for large
  amplitudes

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Perturbed geodesics (numericstheory)
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Black Holes

• 32, GRS 1915150 (53)..
• on those small numbers (4) the model worksTB

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Turbulence? (Vio et al 2005)
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The right turbulence feeds the resonance
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BH vs NS

• Different excitation
• Different modulation (GR/boundary layer)
• Is it a different phenomenon?!?

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What we do not know - DAMPING

• Stochastic damping ?!?
• What controls the coherence Q ?/???
• Didier Barret et al. - QlowgtQhigh
• 
  - Qlow increases with ?
• and
  drops

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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 -gt 1/M
  ?!?
• - g-modes (but do not
  survive)
• which modes are special?

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Conclusions

• There is strong evidence that HFQPOs
• arise from nonlinear resonance in accretion
  disks in GR (KA 2000)
• BUT
• some ingredients are missing

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Successive approximations

• a
• b

(e)
(e2)
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Secular and nearly secular terms
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Tuning parameter
Small-divisor terms are converted into secular
terms
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Stability
Linearization
Linearizzazione