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Debades Bandyopadhyay Saha Institute of Nuclear Physics Kolkata, India With Debarati Chatterjee (SINP). Bulk viscosity and r-modes of neutron stars. Neutron Stars. Spin Evolution. Exotic Matter in NS Core. Non-leptonic Weak interaction. R-mode instability. Bulk Viscosity.

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debades bandyopadhyay saha institute of nuclear physics kolkata india with debarati chatterjee sinp
Debades Bandyopadhyay

Saha Institute of Nuclear Physics

Kolkata, India

With

Debarati Chatterjee (SINP)

Bulk viscosity and r-modes of neutron stars

outline of the talk

Neutron Stars

Spin Evolution

Exotic Matter in NS Core

Non-leptonic

Weak interaction

R-mode instability

Bulk Viscosity

Outline of the talk
slide3

Pulsation modes of neutron stars

1. Large number of families of pulsation modes

2. Modes are classified according to restoring forces acting on the fluid motion

3. 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…….

r modes
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
gravitational radiation reaction driven instability
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

Credit:Yoshida & Rezzolla

growth vs damping
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  > c , the rate of radiation of angular momentum in gravity waves will rapidly slow the star, till it reaches cand can rotate stably
structure of a neutron star

Possible forms of exotic matter

  • Hyperons
  • Bose-Einstein condensates of pions and kaons
  • Quarks
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

slide8

Viscosity

T < 109 K

Shear Viscosity

T > 109 K

Bulk Viscosity

Leptonic

(modified Urca Processes)

Nonleptonic

n + n + e - n + p + e

n + n  n + p + e +e

n + p  p +

n  p + K -

Damping of r-modes

P.B. Jones, PRD 64 (2001) 084003

D. Chatterjee and DB, PRD 75 (2007) 123006

coefficient of bulk viscosity
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  pdx

p  nn d nb

Re = p (  - 0 ) 

1 + ( )2

where = 2m rot

l(l+1)

for l=m=2 r-modes

slide13
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

 = 1  d 3piM2  (3)( p1+ p2 - p3 - p4 )F(i)  (1+2 -3-4 )

40968 i=1 i

slide14
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 <M2 > p4(kT)2  

192 3

where <M2 > is the angle-averaged value of M2

1 = ( kT )2 p < M2 > 

 192 3 nB xn

modified urca bulk viscosity
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

r mode damping time b h
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

critical angular velocity
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 > 0 , unstable when r < 0
  • Critical angular velocityc :1 = 0

r

  • Above c the perturbation will grow, below c it is damped by viscosity
  • If  > c , the rate of radiation of angular momentum in gravity waves will rapidly slow the star, till it reaches c and can rotate stably
critical angular velocity1

n + p  p+

L. Lindblom and B. J. Owen,

Phys. Rev. D 65 (2002) 063006

M. Nayyar and B. J. Owen,

Phys. Rev. D 73 (2006) 084001

D. Chatterjee and D. B.,

Phys. Rev. D 74 (2006) 023003

Critical Angular Velocity
bose einstein condensates
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:

ForK - K - = K -=e

For-- = e

S Banik , D. Bandyopadhyay, Phys Rev C64 (2001) 055805

S Banik , D. Bandyopadhyay, Phys Rev C66 (2002) 065801

slide21
N.K. Glendenning and J. Schaffner-Bielich, PRL 81(1998)& PRC 60 (1999)S. Banik and D. Bandyopadhyay, PRC 64, 055805 (2001)
slide23
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,

 = 1 M2 pFn 2

16 3K -

where  is the production rate of neutrons / volume

proportional to the chemical potential imbalance  = nK - pK - K -

bulk viscosity profile

n + p  p+

n  p + K -

D. Chatterjee and D. B.,

Phys. Rev. D 74 (2006) 023003

D. Chatterjee and D. B.,

Phys. Rev. D 75 (2007) 123006

Bulk viscosity profile
critical angular velocity2

n  p + K -

D. Chatterjee and D. B.,

Phys. Rev. D 75 (2007) 123006

Critical Angular Velocity
hyperon bulk viscosity in superfluid matter
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

conclusions
Conclusions
  • The bulk viscosity coefficient due to the weak process involving antikaon condensate is several orders of magnitude smaller than the hyperon bulk viscosity
  • Hyperon bulk viscosity is suppressed in a Bose condensate
  • Hyperon bulk viscosity in (non )superfluid medium may damp the r-mode instability in neutron stars