Debades BandyopadhyaySaha Institute of Nuclear Physics

Kolkata, India

With

Debarati Chatterjee (SINP)

Bulk viscosity and r-modes of neutron stars

Neutron Stars

Outline of the talk

R-mode instability Bulk Viscosity

Non-leptonic

Weak interaction

Spin EvolutionExotic Matter in NS

Core

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 fluidg-mdoe due to buoyancy and p-mode due to pressure

gradientw-mode associated with the spacetime

Finally, the inertial r-mode…….

Pulsation modes of neutron stars

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

• 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• 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 c and can rotate stably

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

Possible forms of exotic matter• Hyperons• Bose-Einstein condensates of

pions and kaons• Quarks

Credit: D. Page

Damping of r-modes

Viscosity

Shear Viscosity

Bulk Viscosity

Leptonic (modified Urca Processes)

Nonleptonic

T < 109 K

T > 109 K

n + p p +n p + K -

n + n + e - n + p + e

n + n n + p + e +e

P.B. Jones, PRD 64 (2001) 084003D. Chatterjee and DB, PRD 75 (2007) 123006

Equation of State

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

Composition of hyperon matter

Coefficient of Bulk Viscosity

= - n ( p ) dx ( 1- i ) x n d n

infinite frequency (“fast”) adiabatic index = n ( p ) p n x

zero frequency (“slow”) adiabatic index

0 = [( p ) + ( p ) . dx ] n x x n d n - 0 = - nb

2 p dx p nn d nb

Re = p ( - 0 ) 1 + ( )2

where = 2m rot

l(l+1) for l=m=2 r-modes

Landau and Lifshitz, Fluid Mehanics,2nd ed. ( Oxford,1999)

Lindblom and Owen, Phys. Rev. D 65, 063006

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

3piM2

(3)( p1+ p2 - p3 - p4 )F(i) (1+2 -3-4 )

40968 i=1 i

where

M2 = 4 GF 2 sin2 2 c [ 2 mn mp

2 m (1- g np2

) (1- gp2)

- mn mp p2 . p4 (1 - g np2

) (1+ gp2)

- mp m p1 . p3 (1 + g np2

) (1 - gp2)

+ p1 . p2 p3 . p4 {(1 + g np2

) (1 + gp2) + 4 gnp gp }

+ p1 . p4 p2 . p3 {(1 + g np2

) (1 + gp2) - 4 gnp gp }]

After performing the energy and angular integrals,

= 1 <M2 > p4 (kT)2 192 3

where <M2 > is the angle-averaged value of M2

1 = ( kT )2 p < M2 > 192 3 nB xn

Hyperon bulk viscosity coefficient

Modified Urca Bulk viscosity Bulk viscosity coefficient due to modified Urca process of nucleons: B(u) = 6 x 10 25 c

2 T 6 r – 2 Lindblom , Owen and Morsink , Phys. Rev. Lett. 80 (1998) 4843

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 . v² r ² dr dt BV 0

The angle averaged expansion squared is determined numerically . v² = 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

• 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 Velocity

n + p p+

L. Lindblom and B. J. Owen,Phys. Rev. D 65 (2002) 063006M. Nayyar and B. J. Owen,Phys. Rev. D 73 (2006) 084001D. Chatterjee and D. B.,Phys. Rev. D 74 (2006) 023003

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

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

We consider the process n p + K -

The relaxation time is given by 1 = . nn

K

The reaction rate may be calculated using

= 1 d

3p1 d

3p2 d

3p3 M2

(3)( p1- p2 - p3 ) F(i) (1-2 -3 )

8 (2)5 1 2 3

where

M2 = 2 [( n p - pFn pFp + mn* mp

* ) A2 + ( n p - pFn pFp – mn

* mp* ) B2 ]

A = -1.62 x 10 -7 , B = -7.1 x10 -7

After performing the energy and angular integrals,

= 1 M2 pFn 2

16 3 K -

where is the production rate of neutrons / volume

proportional to the chemical potential imbalance = nK - p

K - K -

Composition of Bose condensed matter

Bulk viscosity profile

n p + K -

n + p p+

D. Chatterjee and D. B., Phys. Rev. D 74 (2006) 023003D. Chatterjee and D. B.,Phys. Rev. D 75 (2007) 123006

Critical Angular Velocity

n p + K -

D. Chatterjee and D. B.,Phys. Rev. D 75 (2007) 123006

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

Composition of condensed matter

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