Quadrupole Correction to Wave extraction for uniformly rotating

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Motivation

• Differetially rotating configurations
• a) proto-neutron stars, b) in a component of
  a binary system due to
• accretion, c) Remants of Hypermassive star
  merger.
• Non-axisymmetric instabilities ?T / W
• - secular instabilities driven by
  gravitational radiation (CFS) or
• viscosity ?s? 0.14 (polar modes).
• - dynamical instabilities i) classical bar
  mode instabilities
• ?d 0.25-0.27
• ii) low T / W instabilities (m1,2) for ?
  0.01-0.09.
• Possible interpretation shear
  instabilities of inviscid fluid due to
• the presence of corotation modes (Watts
  et al. 2005 ).
• Asteroseismology to infer the rotational rates
  and the differential law
• from the splitting of non-axisymmetric modes.

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Motivation

• QNMs in GR for rotating stars
• In perturbation theory problems for the BC at
  infinitity.
• In Numerical Relativity, is a 3D problem,
  then high performance
• computers and large simulation time are
  required.
• Aims of our project
• 1) Determine the spectral properties of
  non-axisymmetric adiabatic
• oscillations both for barotropic and
  non-barotropic relativistic stars.
• 2) Study the low T / W instability and its
  connection with corotation
• modes.
• Assumptions
• 1) Slow rotation approximation. It can
  describe quite accurately also
• fast uniformly rotating NS (Hartle
  1967)
• 2) Cowling approximation we focus on the
  fluid modes and neglect
• all the metric perturbations.

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Comparison with RNS code
B1 model Polytropic model M / R 0.146 T
1.719 ms A 12 km ?e???e????k? 0.161

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Fluid Perturbation Equations
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Dependence on the number of couplings
- Perturbative Eqs. contain some
intrinsecally 2nd order corrections in ?,
due to the axial variable u3lm - We neglect
these 2nd order corrections in order to be
consistent with a first order slowly
rotation approximation.
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Corotation band

• Rotational parameters
• in order to have
• corotation modes.

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Summary

• We have studied the spectral properties of
  differentially and slowly rotating stars.
• In the Cowling approximation, the rotational
  splitting of the non-axisymmetric modes has been
  studied with respect to the compactness and the
  degree of differential rotation.
• For a sequence of relativistic polytropic models,
  we have determined the necessary rotational
  configurations in order to have corotation modes.
• We are currently studying the inertial modes and
  g-modes of barotropic and non-barotropic stars.
• We are currently working on the full 1-order
  slow-rotation approximation problem, i.e. without
  Cowling approximation.