Probing exotic matter in neutron stars from the study of R-modes - PowerPoint

							Probing exotic matter in neutron stars from the study of R-modes

Debarati Chatterjee
Saha Institute of Nuclear Physics, Kolkata
Supervisor: Prof. Debades Bandyopadhyay

Neutron Stars
• Compact, massive objects • Remnants of supernova explosions (Type II) • Mass: ~ 1-2 Μsolar

• Radius: ~ 10 Km • Density: ~1015 g/cm3

Structure of a neutron star

Possible phases in the neutron star core
• Hyperons • Bose-Einstein condensates of pions and kaons • Quarks

NS as sources of GW
NS Oscillations
GW

r ,r
EoS

Instabilities in neutron stars
• g-modes: the main restoring force is buoyancy • p-modes: main restoring force is the pressure • f-modes: has an intermediate character of pand g-mode • w-modes: pure space time modes ( only in GR ) • r-modes (inertial modes): main restoring force is the Coriolis force

Why r-modes?
• Newborn neutron stars: r-mode could govern the spin evolution of nascent neutron stars Lead to gravitational waves, detectable to Virgo Cluster with Advanced LIGO • Accreting neutron stars: persistent periodic GW emission expected Several/yr detectable by advanced LIGO Could halt the accretion induced spin up of NS in LMXBs

Inertial frame 

CFS Instability
( Chandrasekhar 1970, Friedman and Schutz 1978 )

Inertial frame 

Corotating frame

for r-modes, l=m=2

Gravitational radiation reaction driven instability
• A backward mode in corotating frame appears as forward moving mode in inertial frame • GR removes positive angular momentum from a mode whose angular momentum is negative ( moves backward relative to the fluid ) • Angular momentum of the perturbation becomes increasingly negative  GR drives the mode unstable

Here comes theTorero: Viscosity!
Shear viscosity • In a normal fluid neutron star, n-n scattering can give rise to shear viscosity
Bulk viscosity In neutron stars, r-modes can perturb the pressure and densities of the fluid  equilibrium is lost Reactions between the components tend to drive the system back to equilibrium with a delay, that depends on the characteristic timescale of the interaction Bulk viscosity is generated by processes having time scale comparable to the period of the perturbation

•  • •

Critical Angular velocity
• The rotating frame energy E E = ½   | v |
2

d3 x

= ½  ² ² R

-2l+2

  r 2l+2 dr
0

R

Lindblom , Owen and Morsink, Phys Rev Lett. 80 (1998) 4843

• imaginary part of the frequency of the r-mode 1 = - 1 r 2E

[

dE ] = - 1 dt  GR

+ 1

v

• Mode stable when r > 0 , unstable when r < 0 ( BV >  GR ) • Critical angular velocity c : 1 = 0

r

Growth timescale GR
GR
1 = - 1 2E

[ dE ]
dt
GR

R

1 = - 32 G  2l+2 (l -1) 2l (l +2)2l+2 0  (r) r 2l+2 dr GR c 2l+3 [ 2l+3!!]2 ( l+1)

Shear viscosity damping timescale SV

•

SV

1

= - 1 [ dE ] 2E dt SV
R R
0

SV

1 =

(l-1)(2l+1)   r ²l dr (  r ²l+2 dr ) - 1
0

Leptonic Bulk Viscosity
• Direct Urca process:
requires proton conc > 11%

• Modified Urca process:

• Coefficient of Modified Urca Bulk Viscosity:

Damping timescale B(h)
• Time derivative of corotating frame energy due to BV is

[ dE ] = - 4 0 Re . v² r ² dr
dt
BV

R

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

Damping of r-modes
Viscosity
9

T < 10 K

Shear Viscosity

T > 10 K

9

Bulk Viscosity

n+n+e

-(Urca n + p +  Processes)

Leptonic

Nonleptonic

n+nn+p+e+

e

n + p  p + np+K-

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

 p – p = -   .  v  v ~ e it
from particle conservation equation, - .v =-in/n (  t + v . )  x = - i   x (  t + v . ) x = - ( x -  x ) / 

(1)

(2)

(3) (4)

 x = x , 1- i  

x = dx
dn

 n

(5)

 p=
p =

p [(  n )x + (  p )n. x

dx dn

]

n
dx dn

(6)

 [(  np )x + 1 i   (  p ) .n 1x

]

n

(7)

 p – p = i   1- i  
 = n ( 1- i   )

 (  xp ) n
dx dn

dx  n dn

(8)

p (  x )n

(9)

infinite frequency (“fast”) adiabatic index  = n ( p) p n x zero frequency (“slow”) adiabatic index  0 = [(  p ) + (  p ) . dx ] n x x n dn

 =

p (  - 0 )  ( 1- i   )

(10)

Theoretical Model

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

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

Parameters of the theory
Nucleon-meson couplings from properties of symmetric nuclear matter :
• • • • • Binding energy B/A= -16.3 MeV Isospin asymmetry energy coefficient , asym = 32.5 MeV. Saturation density n0 = 0.153 fm -3 Compressibility K = 240 MeV Effective nucleon mass m*/m = 0.78
Compressibility: K = 9 dp dn
1 [ 2 (/n) ] t=0 2  t2

Isospin asymmetry energy coefficient : asym =
where t = ( nn –np ) / n

Hyperon-meson coupling constants
The vector meson-kaon couplings : ½ g = ½ g = g = ⅓ gN ½ g  = g  = gN ; g  = 0 2 g = 2 g  = g  = - 22 gN 3 scalar meson ( )-hyperon coupling constants UY ( n0 ) = - gY  + gY 0

From Hypernuclear data : U ( n0 ) = - 30 MeV, U ( n0 ) = + 30 MeV, U ( n0 ) = - 18 MeV
N N N

* - hyperon couplings from Double-hypernuclear data:
 U (

n0 ) =

 U (

n0 ) =

 U (

n0 ) = 2

 U (

n0 ) = - 40 MeV

Kaon-meson coupling constants
vector meson-kaon couplings determined using quark model and isospin counting rule gK = ⅓ gN and gK = gN scalar meson-kaon coupling constants obtained from U K ( n0 ) = - gK  - gK 0 where U K ( n0 ) from -100 MeV to -180 MeV
strange scalar and vector meson couplings with (anti)kaons determined from the decay of f0 and SU(6) symmetry relations respectively g* K = 2.65 and  2 gK = 6.04

K=240 MeV, m*/m = 0.78, U K = -120 MeV

Equations of state

I : 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

= 1

40968

 d
i=1

4

3



piM2 i

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

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

Relaxation time

Coefficient of bulk viscosity

Temperature dependence of hyperon bulk viscosity

Modified Urca Bulk Viscosity
Bulk viscosity coefficient due to modified Urca process of nucleons:

B(u) = 6  10 25  2 T 6 r – 2
Lindblom , Owen and Morsink , Phys Rev Lett. 80 (1998) 4843

Instability Window

II :

We consider the non-leptonic reaction, n  p + K (  t + v . ) nnK = - ( nnK - nnK ) /  = -  n

-

where  n is the production rate of neutrons / volume, proportional to the chemical potential imbalance  = nK - pK - K The relaxation time is given by 1 = 





 .  nn K

The reaction rate  may be calculated using 3 3 3 (3) = 1  d p1 d p2 d p3< M2 >  ( p1- p2 - p3 ) F(i)  (1-2 -3 ) 8 (2)5 1 2 3 where <MK2 > = 2 [( n  p - pFn pFp + mn mp ) A2 + [( n  p - pFn pFp - mn mp ) B2 ]
After performing the energy and angular integrals, K = 1 <MK2 > pFn 2   16  3  K -

Relaxation time

Coefficient of bulk viscosity

Bulk viscosity profile

n + p  p +

np+K-

D.C. and D. Bandyopadhyay, Phys. Rev. D 74 (2006) 023003

Critical Angular Velocity

n + p  p +

np+K-

Conclusions
• Bulk viscosity due to the nonleptonic weak interactions involving hyperons ( n + p  p + ) is sufficient to damp r-mode instability effectively • Bulk viscosity due to the nonleptonic process involving antikaon condensate ( n  p + K - ) is not an effective mechanism for damping the r-mode oscillations

Outlook…

• To study the presence of antikaon condensates on the bulk viscosity due to the non-leptonic process ( n + p  p + ) involving hyperons and its effect on r-mode instability • To probe neutron star matter containing antikaon condensates using w-mode instability.

Ciao!