Equation Of State and back bending phenomenon in rotating neutron stars

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Equation Of State and back bending phenomenon in
rotating neutron stars
M. Bejger E. Gourgoulhon P. Haensel L. Zdunik
1st Astro-PF Workshop CAMK, 14 October
2004 Compact Stars structure, dynamics, and
gravitational waves
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Plan

1. Historical remarksGlendenning 1997 Spyrou,
   Stergioulas 2002
2. Back bending phenomenon for Neutron Stars (with
   hyperons)new approach MB Req dependence at
   fixed frequency J(f) at fixed baryon mass
3. Polytropic EOS and back bending Phase transition
   to quark phase through mixed phase
4. The role of instability

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• rapidly rotating pulsar spins down (looses
  angular momentum J)
• central density increases with time
• the density of the transition to the mixed
  quark-hadron phase is reached
• the radius of the star and the moment of inertia
  significantly decreases
• the increase of central density of the star is
  more important than decrease of angular
  momentum JI O dJI dOO dI
  dO dJ /I - O dI /I gt 0
• the epoch of SPIN-UP BY ANGULAR MOMENTUM
  LOSS

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Braking index -singularity
Energy loss equation
Spin up
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Consequences of back-bending

• the braking index has very large value
• the isolated pulsar may be observed to be
  spinning up

Signature of the transition to the mixed phase
with quarks
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Re-invistigate the deconfinement phase-transition
of spinning-down PSR

• fully relativistic, rapidly rotating models
  
  (vs. Slow-rotation approximation)
• analytic expression for quark phase (vs
  interpolation of tabulated EOS)
• high accuracy of the code and EOS extremely
  important

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For normal pulsar the quark core appears without
back-bending behaviour
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Braking index no singular behaviour
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Back-bending for NS with hyperons

• 2-D multidomain LORENE code based on spectral
  methods
• softening of the EOS due to the appereance of
  hypeons

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MB Req at fixed frequency

• Analysis of the BB problem in the baryon mass MB
  equatorial radius Req plane
• MB is constant during the evolution of solitary
  pulsars
• at fixed frequency the frequency are directly
  connected to the back-bending definition
• numerical reasons frequency is basic input
  parameter in the numerical calculations of
  rotating star (with central density ?c)
• no need to calculate the evolution of the star
  with fixed MB
• numerical procedure input - (f, ?c) , output
  (M,MB ,J,R)

Discussion based on MB (?c) fconst or MB (R)
fconst
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Softening of the EOS due to the core with
hyperons
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Signature of BB minimum of MB at fixed frequency
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Back-bending and MB (x) fconst
xReq x ?c x Pc
The softening of the EOS due to the
hyperonization leads to the flattening of the MB
(x) fconst curves. Back bending - between two
frequencies defined by the existence of the point
x of vanishing first and second derivative (point
of inflexion).
This condition does not depend on the choice of x.
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The onset of back-bending
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Interesting points

• minimum frequency for BB
• maximum frequency for deceleration after BB
• acceleration from Keplerian configuration

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Importance of angular momentum
Why to use angular momentum J instead of moment
of inertia I ?

• J is well defined quantity in GR describing the
  instantaneous state of rotating star
• the evolution of rotating star can be easily
  calculated under some assumptions about the
  change of J

magnetic braking n3GW emission n5

• the moment of inertia defined as J/O does not
  describe the response of the star to the
  change of J or O (rather dJ/dO)
• J enters the stability condition of rotating
  stars with respect to axially symmetric
  perturbations

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Instability
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ANGULAR MOMENTUM vs MOMENT OF INERTIA
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Angular momentum vs rotational frequency
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Angular momentum vs rotational frequency
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Importance of the accuracy
2 domains inthe interior of the star The
boundary have to be adjusted to the point of the
discontinuity of properties of EOS
The innermost zone boundary not adjusted to the
surface of hyperonthreshold except for f920 Hz
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Conclusions for NS with hyperons

• the presence of hyperons neutron-star cores can
  strongly affect the spin evolution of solitary NS
  (isolated pulsar)
• epochs with back-bending for normal rotating NS
  were found for two of four EOS
• for these models pulsar looses half of its
  initial angular momentum without changing much
  its rotation period

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Mixed Phase analytical EOS

• ??lt?1 nuclear matter - polytrope
• ??1lt ? lt?2 mixed phase polytrope
• ??gt?2 quark matter linear EOS

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Mixed Phase analytical EOS
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Mixed Phase analytical EOS
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Mixed Stable
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Mixed Stable
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Mixed Unstable
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Mixed Unstable
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Mixed Marginally Stable
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Mixed Marginally Stable
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Rotation and stability
If nonrotating stars are stable (ie. softening of
the EOS does not result in unstable branch) then
for any value of total angular momentum J (fixed)
MB increases. If MB (x) J0 has local maximum
and minimum (unstable region) than for any value
of total angular momentum J (fixed) such region
exists. In most cases rotation neither
stabilizes nor destabilizes configurations with
phase transitions.
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Onset of instability test of the code

• Test of the code (GR effects)
• Test of the thermodynamic consistency of the
  equation of state

First law of thermodynamics

• Total angular momentum J
• Gravitational mass M
• Baryon mass MB

The extrema of two of these quantities at third
fixed at the same point
Cusps in Figures
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Mixed unstable M(J)
MB const
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Mixed unstable M(MB)
J const