# Presentation: 119259

*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

**Notes:**
First I will briefly outline the possible exotic forms of matter that could appear in the neutron star interior. Then I will discuss the field theoretical model and equation of state that describes such a system. I would also show the spin evolution of rotating neutron stars due to r-mode instability. For given non-leptonic weak interactions, i will compare the effect of presence of hyperons and antikaon condensates on bulk viscosity in 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.......

-              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*
*     *

**Notes:**
The frequency of r-modes in corotating frame is given by this generic equation, where l,m denote spherical harmonics. 

## 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/cm**3*
*Outer crust** ( free electrons,  lattice         of nuclei ) 10 **4** - 4 *x* 10**11 **g/cm**3*
*Inner crust** ( lattice of nuclei with free electrons and neutrons)*
*Outer core** (atomic   particle fluid)*
*Inner core**   ( exotic  subatomic particles? ) n **** 10 **14** g/cm**3*

- Credit: D. Page

## Damping of r-modes

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

**Notes:**
However, there are viscous forces inside the star which tend to compensate the instability. Viscosity arises because the pressure and density variations associated with the mode oscillation drive the fluid away from chemical equilibrium. 
  At low temperatures, damping is dominated by shear viscosity. At high temperatures, the damping is due to bulk viscosity, either from the Urca process or from nonleptonic processes involving strange particles such as hyperons and kaons.
Viscosity would stabilize any mode whose growth time is longer than the viscous damping time.

##  Equation of State

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

**Notes:**
We consider a first order phase transition from the hadronic to the kaon-condensed phase. We adopt a relativistic feld theoretical model to describe the beta equilibrated and charge neutral hadronic matter. The constituents of matter are the octet baryons, electrons and muons. In this model, baryon-baryon interaction is mediated by the exchange of scalar sigma, vector omega and the isovector rho meson. Psi_B is the baryon spinor, m_B is the vacuum baryon mass, tau_B the isospin operator, omega_mu nu and rho_mu nu are the field strength tensors. U_sigma is the scalar self interaction introduced to get correct compressibilty of matter. We have also incorporated hyperon-hyperon interaction through this [L_YY] term, which is assumed to be mediated through the exchange of the strange scalar and vector mesons.  

## Composition of hyperon matter

## Coefficient of Bulk Viscosity 

*               ****  **=  - *_*    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**  =   - *_* n*_*b **2   *_**__* p*_*    *_*d*__**__*x*_*                       *
*                                       **p     **** n**n**   d n**b**  *
* *
*                   ** **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*

**Notes:**
The expression for the coefficient of bulk viscosity is taken from the above references. Here, p and n denote the pressure and number density,  is the frequency of the perturbation,  is the relaxation time for the non-leptonic process, x is the fluid variable that characterizes the process (here taken to be the fraction of baryons comprised of neutrons), and $\bar x$ represents x in the thermodynamic state. We can define the infinite freq (fast) adia index and zero freq (slow) adia index. Then the above formula can be re-expressed in terms of the difference between adiabatic indeces to arrive at the final expr for the coeff of bulk viscosity.

* *
*    **We consider the non-leptonic reaction,  **n + p **** p + **** *
*     **x**n** = n**n** / n**B **:  fraction of baryons comprised of neutrons*
*           **( **** **t ** + ** **v .**** ) x**n **= - (  x**n** -**** x**n**  ) / **** = - **** **n ** / n**B *
*     **where **** **n** ** is the production rate of neutrons / volume,*
*     **which is proportional to the chemical potential imbalance*
*                      **** **= **** - ***
*      **The relaxation time is given by*
*          *
*                       *_*1*_*    =  *_*  *__**_**_* *_*  *_*    *__**__*  *__*.*_
*                      ****           ****     **n**B** ****x**n*
*    *
*           **where   **** x**n** **= x**n  **-****  x**n*
*     **The reaction rate ****  may be calculated using *
*                      **4*
*   **** **= *_*  1     *_*  * *    *_*d*_* 3*_***p***_*i*M2 *** **(3)**( ****p****1**+ ****p****2** - ****p****3** - ****p****4** )F(****i**) **** (****1**+****2** -****3**-****4 **)*
*          **4096****8**    i=1      ** ****i*

-      *where*
-    M**2   = *4 G**F *2* sin*2* 2**** **c* [ * 2 m**n** m**p*2 *m**** *(*1- g** np*2 ) (*1- g**p***2)
-                                            - *m**n** m**p  ****p******2  ******. p******4**** *(*1 - g** np*2 ) (*1+ g**p***2)
-                                            - *m**p **m****  ****p******1  ******. p******3**** *(*1 + g** np*2 ) (*1 -  g**p***2)
-                   + ***p******1  ******. p******2  *** ***p******3  ******. p******4**** *{(*1 + g** np*2 ) (*1 +  g**p***2) + 4 *g**np **g**p**** *}
-                  + ***p******1  ******. p******4  *** ***p******2 ******. p******3**** *{(*1 + g** np*2 ) (*1 +  g**p***2) - 4 *g**np **g**p**** *}]
-    *After performing the energy and angular integrals,*
*              **** **= *_*  1     *_* <*M2  *> ****p******4****** ****(kT)**2** **** ***
*                       **192* 3*   *
*  **where <*M2  *> is the angle-averaged value of *M2
-                        _*1*_*    =    *_*( kT )*_*2**  p**** **< *M**2 *>*_*  *__**__* *_
*               ****           **192* 3*   **                         n**B** ****x**n*

## 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*

**Notes:**
The coefficient of bulk viscosity due to modified Urca process of nucleons decreases with the decrease in temperature. Hence it cannot efficiently damp GW instability at low temperatures.

##  r-mode damping time B(h) 

*   **The **rotating frame energy E** for r-modes is  *
*                                                **R*
*                **E = 1⁄2 **** **2** **** **2*_* 1  *_*    ****   **** r **2** dr*
*                                       **R**2      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** *
*    *

**Notes:**
Taking the perturbation to be of the form e-it , the timescale BV  on which bulk viscosity damps the mode is given by [the last equation].

## 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

**Notes:**
At higher temperatures, GW instability is damped by bulk viscosity due to modified Urca process of nucleons. But at low temperatures, bulk viscosity due to nonleptonic weak interaction of antikaons cannot damp r-mode instability. Thus, for a rotating neutron star, if antikaon condensed matter appears in its core, there could be copious emission of GW. 

##        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*

**Notes:**
We have already seen that Bose-Einstein condensates of mesons could appear in the neutron star core. S-wave condensation sets in when the effective energy of the meson in the medium equals its chemical potential, which is in turn equal to electron chemical potential.

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

**Notes:**
The pure antikaon condensate is composed of octet baryons, leptons and antikaons. In this phase, baryons are embedded in the Bose-Einstein condensate. The antikaon baryon interaction is treated on the same footing as the baryon-baryon interaction.

**Notes:**
The mixed phase of pure hadronic and antikaon condensed matter is described by Gibbs phase equilibrium rules, which states that the pressure and chemical potentials are equal across the phase boundary. The conditions of global charge neutrality and baryon number conservation are imposed through the relations given.  is the volume fraction of the K^ - condensed phase.

* **We consider the  process  **n **** p  + K ** -*
*          *
*The relaxation time is given by*
*          *
*                       *_*1*_*    =  *_*  *__**__* *_*  *_*    *__**__*  .*_
*                      ****           ****      **** **n**n**K *
*    *
*              **The reaction rate ****  may be calculated using *
*     **** **= *_*  1     *_*   ** *_*d*_* 3*_***p***_*1 *_*d*_* 3*_***p***_*2 *_*d*_* 3*_***p***_*3** *M2  *** **(3)**( ****p****1**- ****p****2** - ****p****3** ) F(****i**) **** (****1**-****2** -****3 **)*
*          **8 (2****)**5    **  ****1         ****2     ****3 *
*    **where*
-   M2   = *2 * [(*** **n** **** **p **-  p**Fn * *p**Fp **+ m**n***** m**p ***** *) A2  + (*** **n** **** **p **-  p**Fn * *p**Fp **– m**n*** **m**p*** *)  B2 ]
-                      A =*  -1.62 *x* 10 **-7** ,*  B =*  ** -7.1 *x*10** **-7*
*    **After performing the energy and angular integrals,*
*              **** **= *_*     1                *_M2  * ****p******Fn ****2**  **** **** *
*                          **16 * 3*  **** **K **- *
-       *where **** ** ** is the production rate of neutrons / volume*
*proportional to the chemical potential imbalance   **** = ****n**K** - ****p**K** - ****K **-*

**Notes:**
Let the neutron frac in the antikaon condensed phase relax towards its equilibrium value with the relaxation time  . We must note here that the reaction rate for this process K is independent of temperature.

## Composition of Bose condensed matter

## Bulk viscosity profile 

*                         *

**Notes:**
For this calculation, we note that the factor omega tau <<1 over the range of baryon densities considered and is neglected in the bulk viscosity coefficient. The coefficient of bulk viscosity due to the nonleptonic process is independent of temperature. 
The hyperons in the neutron star interior can be taken to be in a non-superfluid state. Since a condensate is a superconducting state, it suppresses bulk viscosity. 

##       Critical Angular Velocity

**Notes:**
At higher temperatures, GW instability is damped by bulk viscosity due to modified Urca process of nucleons. But at low temperatures, bulk viscosity due to nonleptonic weak interaction of antikaons cannot damp r-mode instability. Thus, for a rotating neutron star, if antikaon condensed matter appears in its core, there could be copious emission of GW. 

## 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 *