Debades Bandyopadhyay

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R-modes of neutron stars with exotic matter

• Debades Bandyopadhyay
• Saha Institute of Nuclear Physics
• Kolkata, India
• With
• Debarati Chatterjee (SINP)

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Outline of the talk
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Pulsation modes of neutron stars

• Large number of families of pulsation modes
• Modes are classified according to restoring
  forces acting on the fluid motion
• Important modes among them are,
• f-mode associated with global oscillation of the
  fluid
• g-mdoe due to buoyancy and p-mode due to pressure
  gradient
• w-mode associated with the spacetime
• Finally, the inertial r-mode.

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R-modes

• R-modes derive its name from (R)ossby waves
• Rossby waves are inertial waves
• Inertial waves are possible in rotating fluids
  and propagate through the bulk of the fluid
• The Coriolis force is the restoring force in this
  case
• Responsible for regulating sipns of rapidly
  rotating neutron stars/ accreting pulsars in
  LMXBs
• Possible sources of gravitational radiation

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Gravitational Radiation Reaction driven
instability

• For rapidly rotating and oscillating neutron
  stars, a mode that moves backward relative to
  the corotating frame appears as a forward
  moving mode relative to the inertial observer
• The prograde mode in the inertial frame has
  positive angular momentum whereas that of the
  retrograde mode in the corotating frame is
  negative
• Gravitational radiation removes positive angular
  momentum from the retrograde mode making its
  angular momentum increasingly negative and leads
  to the Chandrasekhar-Friedman-Schutz (CFS)
  instability

CreditYoshida Rezzolla
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Growth vs Damping

• Bulk viscosity arises because the pressure and
  density variations associated with the mode
  oscillation drive the fluid away from chemical
  equilibrium. It estimates the energy dissipated
  from the fluid motion as weak interaction tries
  to re-establish equilibrium
• Viscosity tends to counteract the growth of the
  GW instability
• Viscosity would stabilize any mode whose growth
  time is longer than the viscous damping time
• There must exist a critical angular velocity ?c
  above which the perturbation will grow, and below
  which it will be damped by viscosity
• If ? gt ?c , the rate of radiation of angular
  momentum in gravity waves will rapidly slow the
  star, till it reaches ?c and can rotate stably

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Structure of a neutron star

• Atmosphere (atoms) n ? 10 4 g/cm3
• Outer crust ( free electrons, lattice of
  nuclei ) 10 4 - 4 x 1011 g/cm3
• Inner crust ( lattice of nuclei with free
  electrons and neutrons)
• Outer core (atomic particle fluid)
• Inner core ( exotic subatomic particles? ) n ?
  10 14 g/cm3

Credit D. Page
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Damping of r-modes
P.B. Jones, PRD 64 (2001) 084003
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Coefficient of Bulk Viscosity ?

• Ref Landau and Lifshitz,
  Fluid Mehanics,2nd ed. ( Oxford,1999)
• Lindblom ,Owen and
  Morsink , Phys. Rev. D 65, 063006
• p -?p - ? ? . v
  ...(1)
• Perturbation of particle no.density, ? n n
  n 0
• Time dependence of the perturbation e -
  i? t
• - ? ?. ? v - i ? ? ? n /n
  ...(2)
• Let fluid variable x characterize the
  process that produced BV
• ? t x v .? x - ( x - ?x ) / ?
  (3)
• where ? is the relaxation timescale
  for the process
• As x oscillates about the background
  equilibrium value x0 ,
• (? t v .? ) ( x - x 0 ) - i ? ( x
  - x 0 ) (4)
• ? x x 0 ( x - ?x ) / i ? ? (?x x 0
  ) / ( 1 - i ? ? ) (5)

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• As the particle no. density changes, the
  fluid variable?x
• changes as
• ?x x 0 d?x (?n n 0 ) d?x ?
  n (6)
• d n
  d n
• by definition , ?n n
• ?p - p 0 ( ? p ) ( ? p ) . d ?x
  ...(7)
• ? n x ? x
  n d n
• p - p 0 ( ? p ) 1
  ( ? p ) . d ?x ...(8)
• ? n x (1 - i ?
  ? ) ? x n d n
• p -?p i ? ? ( ? p )
  . d?x ? n ...(9)
• (1 - i ? ? ) ?
  x n d n
• ? - n ? ( ? p
  ) d?x ...(10)
• ( 1- i ? ? ) ?
  x n d n

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Hyperons

• Hyperons produced at the cost of nucleons
• n p ? p ? K0 , n n ? n ? - K
• Chemical equilibrium through weak processes
• p e- ? ? ? e , ? e- ? ? - ? e
• General condition for ?-equilibrium
• ? i bi ?
  n - qi ? e

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Hadronic Phase
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Composition of hyperon matter

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EOS including hyperons
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Coefficient of Bulk Viscosity ?
Landau and Lifshitz, Fluid
Mehanics,2nd ed. ( Oxford,1999)
Lindblom and Owen, Phys. Rev. D 65, 063006

• ? - n ? ( ? p
  ) d?x
• ( 1- i ? ? ) ?
  x n d n
• infinite frequency (fast) adiabatic
  index
• ?? n ( ? p )
  
• p ? n x
  
• zero frequency (slow) adiabatic index
• ? 0 ( ? p ) ( ? p ) .
  d?x
• ? n x ?
  x n d n
• ?? - ? 0 - nb 2 ? p
  d?x
• p ? nn
  d nb
• Re ? p ( ?? - ?0 ) ?
  
• 1 (? ?
  )2

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• We consider the non-leptonic reaction, n p
  ? p ?
• xn nn / nB fraction of baryons
  comprised of neutrons
• ( ? t v .? ) xn - ( xn -? xn )
  / ? - ? n / nB
• where ? n is the production rate of
  neutrons / volume,
• which is proportional to the chemical
  potential imbalance
• ?? ? - ??
• The relaxation time is given by
• 1 ?? ?? .
• ? ?? nB ?xn
• where ? xn xn -? xn
• The reaction rate ? may be calculated using
  
• 4

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• where
• ?M??2 4 GF 2 sin2 2? c 2 mn mp2 m? (1-
  g np2 ) (1- gp?2)
• - mn
  mp p2 . p4 (1 - g np2 ) (1 gp?2)
• - mp
  m? p1 . p3 (1 g np2 ) (1 - gp?2)
• p1 . p2 p3 . p4 (1 g
  np2 ) (1 gp?2) 4 gnp gp?
• p1 . p4 p2 . p3 (1 g np2
  ) (1 gp?2) - 4 gnp gp?
• After performing the energy and angular
  integrals,
• ? 1 lt?M?2 gt p4 (kT)2 ? ?
• 192? 3
• where lt?M?2 gt is the angle-averaged value of
  ?M?2
• 1 ( kT )2 p? lt
  ?M??2 gt ??
• ? 192? 3
  nB ?xn

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Temperature dependence of relaxation time
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Hyperon bulk viscosity coefficient
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r-mode damping time ?B(h)

• The rotating frame energy E for r-modes is
• R
• E ½ ? 2 ? 2 1 ? ? r 2 dr
• R2 0
  
• Lindblom , Owen and Morsink, Phys Rev
  Lett. 80 (1998) 4843
• Time derivative of corotating frame energy due
  to BV is
• 
  R
• dE - 4 ? ? Re? ???.? v?²? r ² dr
• dt BV 0
• The angle averaged expansion squared is
  determined numerically
• ???.? v?²? ?² ? ² ( r )6 1 0.86
  ( r )2 ( ? ² )2
• 690 R
  R ? G?
• Lindblom , Mendell and Owen, Phys Rev D
  60 (1999) 064006
• The time scale ?BV on which bulk
  viscosity damps the mode is
• 1 - 1 dE
• ?BV 2E dt BV

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Critical Angular Velocity

• imaginary part of the frequency of the r-mode
• 1 - 1 1
  1
• ?r ?GR ?BV
  ?B(u)
• where ?GR timescale over which GR
  drives mode unstable
• 
  R
• 1 131072 ? ? 6 ?0 ? (r) r
  6 dr
• ?GR 164025
• ?B(u) Bulk viscosity timescale due to
  Modified Urca
• process of
  nucleons
• Mode stable when ?r gt 0 , unstable when ?r lt 0
• Critical angular velocity ?c 1 0
• 
  ?r
• Above ?c the perturbation will grow, below ?c it
  is damped by viscosity
• If ? gt ?c , the rate of radiation of angular
  momentum in gravity waves will rapidly slow the
  star, till it reaches ?c and can rotate stably

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Evolutionary sequences
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Critical Angular Velocity
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Bose-Einstein condensates

• Processes responsible for p-wave pion condensate/
  s-wave kaon condensate in compact stars
• n ? p ? - n ? p K -
• e - ? ? - ?e e - ? K - ?e
• Threshold condition for Bose
• condensation of mesons
• For K - ??K - ? K - ? e
• For ? - ??? - ? e
• S Banik , D. Bandyopadhyay, Phys Rev C64 (2001)
  055805
• S Banik , D. Bandyopadhyay, Phys Rev C66 (2002)
  065801

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Equation of State

• J.Schaffner and I.N.Mishustin, PRC
  53,1416 (1996)

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N.K. Glendenning and J. Schaffner-Bielich, PRL
81(1998) PRC 60 (1999)S. Banik and D.
Bandyopadhyay, PRC 64, 055805 (2001)
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• We consider the process n ? p K -
• M ?up ( p2 ) A
  ?5 B un( p1 ) ?
• Determination of A and B using SU(3)
• weak nonleptonic decay of octet hyperons (? ,?
  ,? ) can be described by processes
• S B ? B P
• ?Y 1, ?I ½ ? spurion
  transforms like ?6
• J.Schaffner-Bielich, R.
  Matiello and H. Sorge, nucl-th/9908043
• General SU(3) symmetry and CP
  invariance results in Lagrangian
• L D Tr?B B P, ?6 F Tr?B P, ?6
  B
• G Tr?B P ?5 B ?6 H Tr?B ?6
  ?5 B P J Tr?B P, ?6 ?5 B
• ? D 4.72 ,F -1.62 (in units of
  10-7 ) for A amplitudes and
• G 40.0 ,H47.8 , J
  -7.1 (in units of 10-7 ) for B amplitudes
• gives good agreement with
  experimental data

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• We consider the process n ? p K -
• The relaxation time is given by
• 1 ? ?? .
• ? ?? ? nnK
• The reaction rate ? may be
  calculated using
• ? 1 ? d 3p1 d 3p2 d 3p3 ?M?2 ?
  (3)( p1- p2 - p3 ) F(?i) ? (?1-?2 -?3 )
• 8 (2?)5 ?1 ?2 ?3
• where
• ?M?2 2 (? n ? p - pFn pFp mn mp )
  ?A?2 (? n ? p - pFn pFp mn mp ) ?B?2
• A -1.62 x 10 -7 , B
  -7.1 x10 -7
• After performing the energy and angular
  integrals,

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Composition of Bose condensed matter
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EOS with antikaon condensate
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Difference in adiabatic indices

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Relaxation time
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Bulk viscosity profile

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Modified Urca Bulk viscosity

• Bulk viscosity coefficient due to modified Urca
  process of nucleons
• ?B(u) 6 x 10 25 ? c2 T
  6 ?r 2
• Lindblom , Owen and
  Morsink , Phys. Rev. Lett. 80 (1998) 4843

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Critical Angular Velocity
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Hyperon bulk viscosity in superfluid matter

• Significant suppression of hyperon bulk viscosity
  due to neutron, proton or hyperon superfluidity
• In this situation, hyperon bulk viscosity may not
  be able to damp the r-mode
• The hyperon bulk viscosity due to the process
• n p ? p? in kaon condensed matter and its
  role on r-modes

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Composition of condensed matter
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Conclusions

• The bulk viscosity coefficient due to the weak
  process involving antikaon condensate is several
  orders of magnitude smaller than the hyperon bulk
  viscosity
• The bulk coefficient in the former case may not
  damp the r-mode instability
• Hyperon bulk viscosity is suppressed in a Bose
  condensate