quasi-normal modes of a relativistic superﬂuid neutron...

Leonardo Gualtieri

Sapienza Università di Roma

Quasi-normal modes

of a relativistic superfluid neutron star

M.E. Gusakov, E.M. Kantor, A.I. Chugunov, L. G., MNRAS 428, 1518 (2013).L. G., M.E. Gusakov, E.M. Kantor, in preparation

Leonardo Gualtieri Problemi Attuali di Fisica Teorica (XX Edizione) Vietri sul Mare, 11-16 April, 2014

Neutron stars are among the main expected sources for GW detectors.They would provide us unvaluable information on the behaviour of matter at

extreme conditions, occurring mainly in their inner core(conditions of density, temperature, pressure, gravitational field, magnetic field).

Our theoretical understanding of the behaviour of nuclear matter is limited, because we can not reproduce such extreme conditions in the lab,and because we do not fully understand the hadronic interaction.

A very promising way to get information on the NS interior consists in looking to their oscillations.

When a NS is excited by an external perturbation (for instance a glitch, interaction with an orbital companion - such as in binary coalescence - or as

a result of the collapse giving birth to the neutron star), it can be set into non-radial oscillations, emitting GWs at the characteristic

frequencies of its quasi-normal modes (QNMs). Such oscillations are damped, due to gravitational wave emission,

therefore they have a complex frequency: ω = σ + ι / τ

NSs and their oscillations


Detection of the GW emission from a NS in non-radial oscillations will allow us to measure the frequencies (σ/2π) and damping times (τ) of its QNMs

which carry the imprint of the equilibrium and non-equilibrium properties of matter in the NS core. This strategy has been called gravitational wave

asteroseismology (N. Andersson & K. Kokkotas, ’98, O. Benhar, V. Ferrari, L.G. ’04, ’07)

NSs and their oscillations

In order to extract useful information from a future GW detection,theoretical modeling of the NS QNMs is needed, in order to find the relation

between the NS properties and the QNM frequencies and damping times.We need to model - in a general relativistic framework - NS oscillations,taking into account all relevant features of the matter composing the NS.

However, present computations of NS QNMs do not take into accounta crucial feature of NS matter: superfluidity in the NS core(or include it under strong simplifying assumptions, i.e. T=0)

How does superfluidity affect the QNM spectrum of NSs?


NSs are superfluid!

In recent years, theoretical calculations provided evidence thatneutrons and protons in NS cores should be superfluid/superconducting

(see e.g. Yakovlev et al. ’99, Lombardo & Schultze ’01, and references therein).

These results have recently been supported by strong observational evidenceRecent observations of the NS in Cassiopeia A (the youngest in our galaxy)

showed for the first time the cooling of an isolated NS.An analysis of the cooling curve (Shternin et al. ’11, Page et al. ’11)

showed that it can be explained by assuming superfluidity in the NS core.


Dissipation in relativistic superfluid NSs 1527

Figure 1. Left-hand panel: nucleon critical temperatures Tck (k = n, p) versus density ! for model 1. Right-hand panel: redshifted critical temperatures T !ck

versus radial coordinate r (in units of R) for model 1.

Figure 2. The same as in Fig. 1 but for model 2.

On the contrary, in the model 1 only two-layer configurations arepossible.

The entrainment matrix Yik is calculated for the superfluiditymodels 1 and 2 in a way similar to how it was done in Kantor &Gusakov (2011).

When analysing viscous dissipation in oscillating NSs we al-low for the damping due to shear and bulk viscosities. For the shearviscosity coefficient " we take the electron shear viscosity "e, calcu-lated in Shternin & Yakovlev (2008). We neglect the nucleon shearviscosity because (i) it is poorly known even for non-superfluidmatter and (ii) it appears to be less than the electron shear viscosityin the core at T " Tcp (Shternin & Yakovlev 2008).

The bulk viscosity coefficients are calculated as described byGusakov (2007), Gusakov & Kantor (2008) and Kantor & Gusakov

(2011). Since the direct URCA process is closed for our stellarmodel with M = 1.4 M#, the main contributor to the bulk viscosityis the modified URCA process.

7.2 Oscillations of a non-superfluid star

As follows from Section 6.2, before considering oscillations of asuperfluid NS one should study those of a normal (non-superfluid)star of the same mass. To this aim, we have determined the eigenfre-quencies and eigenfunctions of the radial and non-radial oscillationmodes for a non-superfluid NS of mass M = 1.4 M# and equationof state APR (see Section 7.1). We have solved the equations de-scribing radial and non-radial perturbations of a non-rotating starin general relativity. These equations are derived by expanding the

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We shall consider two models, compatible with theory and observations:Dissipation in relativistic superfluid NSs 1527

Figure 1. Left-hand panel: nucleon critical temperatures Tck (k = n, p) versus density ! for model 1. Right-hand panel: redshifted critical temperatures T !ck

versus radial coordinate r (in units of R) for model 1.

Figure 2. The same as in Fig. 1 but for model 2.

On the contrary, in the model 1 only two-layer configurations arepossible.

The entrainment matrix Yik is calculated for the superfluiditymodels 1 and 2 in a way similar to how it was done in Kantor &Gusakov (2011).

When analysing viscous dissipation in oscillating NSs we al-low for the damping due to shear and bulk viscosities. For the shearviscosity coefficient " we take the electron shear viscosity "e, calcu-lated in Shternin & Yakovlev (2008). We neglect the nucleon shearviscosity because (i) it is poorly known even for non-superfluidmatter and (ii) it appears to be less than the electron shear viscosityin the core at T " Tcp (Shternin & Yakovlev 2008).

The bulk viscosity coefficients are calculated as described byGusakov (2007), Gusakov & Kantor (2008) and Kantor & Gusakov

(2011). Since the direct URCA process is closed for our stellarmodel with M = 1.4 M#, the main contributor to the bulk viscosityis the modified URCA process.

7.2 Oscillations of a non-superfluid star

As follows from Section 6.2, before considering oscillations of asuperfluid NS one should study those of a normal (non-superfluid)star of the same mass. To this aim, we have determined the eigenfre-quencies and eigenfunctions of the radial and non-radial oscillationmodes for a non-superfluid NS of mass M = 1.4 M# and equationof state APR (see Section 7.1). We have solved the equations de-scribing radial and non-radial perturbations of a non-rotating starin general relativity. These equations are derived by expanding the

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A: 2 layers B: 2 or 3 layers

QNMs of a compact star: the standard (non-SFL) approachMany sets of general relativistic equations have been derived in the years

(Thorne & Campolattaro ’67; Lindblom & Detweiler ’85; Chandrasekhar & Ferrari ’90).We follow the notation of Lindblom & Detweiler (LD).

The perturbed spacetime metric is expanded in tensor spherical harmonics, in the frequency domain:

ds2 = −eν�1 + r�H�m

0 Y�meiωt�dt2 + eλ

�1 + r�H�m

2 Y�meiωt�dr2 − 2iωr�+1H�m

1 Y�meiωtdtdr

+r2�1− r�K�mY�meiωt

�(dϑ2 + sin2 θdϕ2)

2

quasi-stationary configurations, which are essentially the en-tropy profile (which will appear to be the most important inthis respect) and the lepton composition.The article is organized as follows. In Sec. II we review

the computation of the quasi-normal oscillation modes of aPNS. In Sec. III we provide details regarding the calculationof the EOS of hot nuclear matter, and the construction of ide-alized PNS profiles during the stellar evolution. In Sec. IV wepresent and interpret our numerical results, and draw conclu-sions in Sec. V.

• dire della EOS e del fatto che sono circa 20 s dopo ilbounce.

• dire che trascuriamo la rotazione

• dire che calcoliamo i QNM

II. THE QUASI-NORMALMODES OF NEUTRON STARS

A. Equations of motion ???

In order to find frequencies and damping times of the quasi-normal modes (QNMs), we solve the equations describingnon-radial perturbations of a (spherically symmetric) star ingeneral relativity, using the numerical approach described in[8], which we briefly summarize.The perturbed spacetime metric is expanded in tensor

spherical harmonics, as (we use geometrized units, assumingc= G= 1)

ds2 =!e!1+ r!H!m

0 Y!mei t"dt2

+e!1+ r!H!m

2 Y!mei t"dr2

!2i r!+1H!m1 Y!mei tdtdr

+r2!1! r!K!mY!mei t

"(d 2+ sin2 d 2) , (1)

where is the frequency, Y!m( , ) are the scalar spheri-cal harmonics, and H !m

i (r), K!m(r) describe the metric per-turbations with polar parity, i.e., transforming as (!1) ! un-der parity transformations. In this paper we do not considerperturbations with axial parity, which transform as (!1) !+1.The functions (r), (r) describe the spherically symmetricstellar background, and are found by solving the Tolman-Oppenheimer-Volkov equations. The four-velocity of the fluidis

u! = u!0 + u! = (e! /2,0,0,0)+ i e! /2(0, r, , ) ,(2)

where ! is the Lagrangian displacement of the fluid, and isexpanded in vector spherical harmonics as

r(t,r, , ) = e /2r!!1W !m(r)Y!m( , )ei t ,

(t,r, , ) =!r!V !m(r) Y!m( , )ei t ,

(t,r, , ) =!r!V !m(r) Y!m( , )ei t . (3)

The fluid is also characterized by its pressure and energydensity

p(r)+ p(t,r, , ) = p(r)+ r! p!m(r)Y!m( , )ei t ,

(r)+ (t,r, , ) = (r)+ r! !m(r)Y!m( , )ei t . (4)

We denote with the Eulerian perturbations, and with theLagrangian perturbations, so that for instance the Lagrangianpressure perturbation is

p= p+ rpr, (5)

p!m = p!m+e! /2

rW !m p

r. (6)

Einstein’s equations, linearized in the perturbations, yielda system of ordinary differential equations for the perturbedfunctions. Different equivalent sets of equations have beenderived in the literature, using different gauge choices or dif-ferent combinations of the relevant equations [9–12]. We usethe formulation of Lindblom and Detweiler [10, 11], con-sisting of a system of four first-order differential equations(hereafter, the LD equations) in the perturbation functions{H!m

1 ,K!m,W !m,X !m}, where

X !m =!e /2 p!m , (7)

and algebraic relations give the perturbation functions{H!m

0 ,H!m2 ,V !m} in terms of the others. To close the system of

equations we also need the EOS, relating the energy densityand the pressure p; and this will be discussed in Section III.Can/Should we be more explicit on the LD eqs. ?

B. Computation of the quasi-normal modes

A QNM is a solution of the LD equations, which is reg-ular at the center and continuous on the surface, and whichbehaves as a pure outgoing wave at infinity. Since in generalrelativity a non-radial oscillation is associated to gravitationalwave emission, the frequencies of such solutions are necessar-ily complex:

= +iGW

, (8)

where is the pulsation frequency, and GW is the dampingtime of the mode due to gravitational wave emission. If themode is unstable, its imaginary part is negative and ! GW isthe growth time of the instability.The procedure to find the QNMs is then the following [8]:

(i) We choose a value of l and a (complex) value of (sincethe background is spherical, the equations do not depend onthe index m); (ii) We integrate the LD equations, imposingregular boundary conditions at the center and p = 0 at thestellar surface; (iii) Imposing continuity at the stellar surface,we obtain the metric perturbations outside the star; (iv) In vac-uum, the perturbation equations reduce to a simple, second-order differential equation (the Zerilli equation), which we in-tegrate up to infinity; (v) We check if our solution satisfies

2

quasi-stationary configurations, which are essentially the en-tropy profile (which will appear to be the most important inthis respect) and the lepton composition.The article is organized as follows. In Sec. II we review

the computation of the quasi-normal oscillation modes of aPNS. In Sec. III we provide details regarding the calculationof the EOS of hot nuclear matter, and the construction of ide-alized PNS profiles during the stellar evolution. In Sec. IV wepresent and interpret our numerical results, and draw conclu-sions in Sec. V.

• dire della EOS e del fatto che sono circa 20 s dopo ilbounce.

• dire che trascuriamo la rotazione

• dire che calcoliamo i QNM

II. THE QUASI-NORMALMODES OF NEUTRON STARS

A. Equations of motion ???

In order to find frequencies and damping times of the quasi-normal modes (QNMs), we solve the equations describingnon-radial perturbations of a (spherically symmetric) star ingeneral relativity, using the numerical approach described in[8], which we briefly summarize.The perturbed spacetime metric is expanded in tensor

spherical harmonics, as (we use geometrized units, assumingc= G= 1)

ds2 =!e!1+ r!H!m

0 Y!mei t"dt2

+e!1+ r!H!m

2 Y!mei t"dr2

!2i r!+1H!m1 Y!mei tdtdr

+r2!1! r!K!mY!mei t

"(d 2+ sin2 d 2) , (1)

where is the frequency, Y!m( , ) are the scalar spheri-cal harmonics, and H !m

i (r), K!m(r) describe the metric per-turbations with polar parity, i.e., transforming as (!1) ! un-der parity transformations. In this paper we do not considerperturbations with axial parity, which transform as (!1) !+1.The functions (r), (r) describe the spherically symmetricstellar background, and are found by solving the Tolman-Oppenheimer-Volkov equations. The four-velocity of the fluidis

u! = u!0 + u! = (e! /2,0,0,0)+ i e! /2(0, r, , ) ,(2)

where ! is the Lagrangian displacement of the fluid, and isexpanded in vector spherical harmonics as

r(t,r, , ) = e /2r!!1W !m(r)Y!m( , )ei t ,

(t,r, , ) =!r!V !m(r) Y!m( , )ei t ,

(t,r, , ) =!r!V !m(r) Y!m( , )ei t . (3)

The fluid is also characterized by its pressure and energydensity

p(r)+ p(t,r, , ) = p(r)+ r! p!m(r)Y!m( , )ei t ,

(r)+ (t,r, , ) = (r)+ r! !m(r)Y!m( , )ei t . (4)

We denote with the Eulerian perturbations, and with theLagrangian perturbations, so that for instance the Lagrangianpressure perturbation is

p= p+ rpr, (5)

p!m = p!m+e! /2

rW !m p

r. (6)

Einstein’s equations, linearized in the perturbations, yielda system of ordinary differential equations for the perturbedfunctions. Different equivalent sets of equations have beenderived in the literature, using different gauge choices or dif-ferent combinations of the relevant equations [9–12]. We usethe formulation of Lindblom and Detweiler [10, 11], con-sisting of a system of four first-order differential equations(hereafter, the LD equations) in the perturbation functions{H!m

1 ,K!m,W !m,X !m}, where

X !m =!e /2 p!m , (7)

and algebraic relations give the perturbation functions{H!m

0 ,H!m2 ,V !m} in terms of the others. To close the system of

equations we also need the EOS, relating the energy densityand the pressure p; and this will be discussed in Section III.Can/Should we be more explicit on the LD eqs. ?

B. Computation of the quasi-normal modes

A QNM is a solution of the LD equations, which is reg-ular at the center and continuous on the surface, and whichbehaves as a pure outgoing wave at infinity. Since in generalrelativity a non-radial oscillation is associated to gravitationalwave emission, the frequencies of such solutions are necessar-ily complex:

= +iGW

, (8)

where is the pulsation frequency, and GW is the dampingtime of the mode due to gravitational wave emission. If themode is unstable, its imaginary part is negative and ! GW isthe growth time of the instability.The procedure to find the QNMs is then the following [8]:

(i) We choose a value of l and a (complex) value of (sincethe background is spherical, the equations do not depend onthe index m); (ii) We integrate the LD equations, imposingregular boundary conditions at the center and p = 0 at thestellar surface; (iii) Imposing continuity at the stellar surface,we obtain the metric perturbations outside the star; (iv) In vac-uum, the perturbation equations reduce to a simple, second-order differential equation (the Zerilli equation), which we in-tegrate up to infinity; (v) We check if our solution satisfies

∆p = −e−ν/2rlY lmeiωtX lm ∆� =�+ p

γp∆p

Einstein’s equations, linearized in the metric perturbations, yield a 4th-order system of ODEs inside the star in

and a single 2nd-order wave equation (the Zerilli equation) in vacuum.They are solved assuming regularity at the center, continuity at the surface

(together with Δp=0 as r=R), outgoing wave boundary conditions at infinity. Solutions only exist for a discrete set of frequencies, which are complex:

ω = 2πν + i/τ : the quasi-normal modes of the star.

H lm1 , Klm, X lm, W lm


Perturbations of superfluid NSs

In recent years many works have studied perturbations of superfluid (SFL) NSs(see e.g. Lindblom & Mendell ’94, Comer et al. ’99, Andersson & Comer ’01, Yoshida & Lee ’03,

Gusakov & Andersson ’06, Lin et al. ’08, Haskell et al. ’09, Andersson et al. ’09, Haskell & Andersson ’10).

Previous studies found that SFL creates a new class of modes, the superfluid modes, and changes frequencies and damping times of the others.

Our work is based on the approach introduced in Gusakov & Kantor ’11,(see also Kantor & Gusakov ‘11, Chugunov & Gusakov ’11, Gusakov et al. ’13)

in which for the first time NS oscillations are studied taking into account bothgeneral relativity and finite temperature SFL hydrodynamics

(but see also the recently proposed approach of Andersson et al. ’12)

In presence of SFL, independent motions can coexist without dissipation. We have “normal” n,p,e, with 4-velocity uμ, and SFL n,p, with vμs(n) , vμs(p).

Defining wμ(n/p)=μ(n/p)(vμs(n/p)-uμ) , the conserved currents are: jμ(n)=nn uμ+Ynk wμ(k)

jμ(e)=ne uμ

and since at the frequencies of NS oscillations ne=np , jμ(p)= jμ(e) .Ynk : entrainment matrix


Perturbations of superfluid NSs

The SFL degrees of freedom are described by a new perturbation quantity:δμ=μn-μp-μe=e-ν/2rlYlmeiωtδμlm

The perturbation equations are then the LD equations, expressed in terms of the quantities (obtained from the harmonic expansion of δUμ(b)) + an equation for δμlm , and with some modifications which couple

the “modified” LD perturbations ( ) with δμlm .

H lm1 (b), Klm

(b) , X lm(b) , W lm

(b)

H lm1 (b), Klm


(b)

Let us consider linear perturbations of a spherically symmetric background.It is possible to show that introducing the quantities

Xμ=(Ynk wμ(k))/nb , and Uμ(b)= uμ+Xμ , and the NS perturbations can be formally described in terms of Uμ(b):

δTμν written in terms of δUμ(b) , has the same form as in the non-SFL case

10 M. E. Gusakov et al.

(i) Continuity equations for baryons (8) and electrons (9), that can be written in terms of the baryon and electron number

density perturbations, !nb and !ne, as

!nb =i

" e!!/2

!

#j(nb)Uj(b) + nb U

µ(b) ;µ

"

, (83)

!ne = !ne (norm) + !ne (sfl), (84)

where j is the spatial index and we defined

!ne (norm) ! i

" e!!/2

!

#j(ne)Uj(b) + ne U

µ(b) ;µ

"

, (85)

!ne (sfl) ! " i

" e!!/2

!

#j(ne)Xj + ne X

µ;µ

"

. (86)

(ii) Einstein equations, which can schematically be presented as

!(Rµ! " 1/2 gµ! R) = 8$G !T µ! , (87)

where the perturbation !T µ! of the energy-momentum tensor (11) can be expressed in terms of the perturbations of baryon

four-velocity !Uµ(b), metric !gµ! , pressure !P and energy density !% as

!T µ! = (!P + !%)Uµ(b)U

!(b) + (P + %)

!

Uµ(b) !U

!(b) + U!

(b) !Uµ(b)

"

+ !P gµ! + P !gµ! . (88)

In Eq. (87) Rµ! and R are the Ricci tensor and scalar curvature, respectively; G is the gravitation constant.(iii) ‘Superfluid’ equation, that can be derived from Eqs. (10) and (13) of Sec. 2 (here we present only the spatial

components j of this equation) 6

i" (µn Ynk w(k)j " nb w(n)j) = ne #j(e!/2 !µ). (89)

Expressing the vectors wj(i) through Xj in this equation [see Eqs. (3) and (4)], and introducing the redshifted imbalance of

chemical potentials !µ" ! e!/2 !µ, one can rewrite Eq. (89) as

Xj =ine

µnnb " y#j(!µ

"), (90)

where y is defined by Eq. (68). Notice, that this equation dictates the most general form of the superfluid Lagrangiandisplacement &j(sfl), that was already obtained in Eq. (52) from the symmetry arguments.

Eqs. (83)–(90) should be supplemented with the expressions for the perturbations !P , !µ, and !%. To derive them, let us

notice that any thermodynamic quantity (e.g., P ) in the superfluid matter can be presented as a function of nb, ne, T , andw(i)µw

µ(k) (see, e.g., Gusakov 2007). In strongly degenerate matter the dependence of P , !µ, and % on T can be neglected (see,

e.g., Reisenegger 1995; Gusakov et al. 2005), while the scalars w(i)µwµ(k) are quadratically small in a slightly perturbed star

[see Sec. 3.2]. Thus, P = P (nb, ne), !µ = !µ(nb, ne), and % = %(nb, ne). Expanding these functions into Taylor series near

the equilibrium, one obtains

!P = nb#P#nb

#

!nb

nb+ s̃

!ne (norm)

ne+ s

!ne (sfl)

ne

$

, (91)

!µ = ne#!µ#ne

#

z!nb

nb+

!ne (norm)

ne+

!ne (sfl)

ne

$

, (92)

!% = µn !nb. (93)

where we made use of Eq. (84), and introduced dimensionless coupling parameter s and the quantities s̃ and z,

s ! ne

nb

(#P/#ne)(#P/#nb)

, (94)

s̃ ! ne

nb

(#P/#ne)(#P/#nb)

, (95)

z ! nb

ne

(#!µ/#nb)(#!µ/#ne)

. (96)

Notice that the variable s̃ is equal to s here. The reason for discriminating between s̃ and s is purely technical: To solve

oscillation equations (see Secs. 6 and 7) it turns out to be convenient to develop a perturbation theory in (small) parameters, at the same time treating the terms depending on s̃ in a non-perturbative way (see Sec. 6.2, and, in particular, footnote 9

6 It is worth to make a number of comments on Eq. (89): (i) In Gusakov & Kantor (2011) this equation was derived under the assumptionthat the only superfluid species are neutrons (that is Ypi = 0). A generalization of this equation to the case of possible proton superfluidityis presented in Chugunov & Gusakov (2011) [see their Eq. (3)]; (ii) In both papers, Gusakov & Kantor (2011); Chugunov & Gusakov(2011), this equation is written with the same mistake. In particular, in Chugunov & Gusakov (2011) one should write ne !j(e!/2 "µ)instead of ne e!/2 !j("µ) in the right-hand side of Eq. (3).

jµ(b) = jµ(n) + jµ(p) = nbUµ(b) , jµ(e) = ne(U(b) −Xµ) ,


Perturbations of superfluid NSs: decoupled case

In Gusakov et al., ’13 we have considered the decoupling limit

in which the parameter which couples

the superfluid an non-superfluid degrees of freedom, was neglected: s=0. This approximation is justified to some extent, since s ~ 0.01-0.05.

1524 M. E. Gusakov et al.

1!bulk

= " 2

4Emech(0)

! R

0r2(l+1) e#/2

"#$2 %1 +

#$3n %2

$2dr,

(77)

1!shear

= " 2

2Emech(0)

! R

0& r2(l!1) e#/2

%32

('1)2

+2l(l + 1) ('2)2 + l(l + 1)&

12

l(l + 1) ! 1'

V 2(

dr, (78)

where

%1(r) = K + 12

H2 ! 1r

e!#/2&

dW

dr+ 1

r(l + 1) W

'! l(l + 1)

V

r2,

(79)

%2(r) = !1r

e!#/2&

d(nb Wsfl)dr

+ 1r

(l + 1) nb Wsfl

'

!l(l + 1)nb Vsfl

r2, (80)

'1(r) = r2

3

%2r

e!#/2&

dW

dr+ (l ! 2)

W

r

'

+ K ! H2 ! l(l + 1)V

r2

(, (81)

'2(r) = r

2

&dV

dr+ (l ! 2)

V

r! e#/2 W

r

'e!#/2. (82)

As for the mechanical energy (71), to obtain from these formu-las ! bulk and ! shear for a non-superfluid star, one has to put Wsfl =Vsfl = 0. In that case our equations (77) and (78) should coincidewith the corresponding formulas (5) and (6) of Cutler et al. (1990).Unfortunately, direct comparison of these formulas reveals that our! bulk and ! shear appear to be two times larger. Using, as tests ex-amples, damping of (i) NS radial oscillations, (ii) p modes in theNS envelopes and (iii) sound waves in the non-superfluid matterof NSs, we checked that our results reproduce those of Gusakovet al. (2005), Chugunov & Yakovlev (2005) and Kantor & Gusakov(2009), obtained in a quite a different way.

5 O S C I L L AT I O N E QUAT I O N S

In order to calculate ! bulk and ! shear one has to determine the os-cillation eigenfrequencies " and eigenfunctions H0, H1, H2, K, Wb,Vb, Wsfl and Vsfl. To do that one needs to formulate oscillation equa-tions. Since the dissipation is weak, when deriving the oscillationequations one can neglect the dissipative terms in the superfluidhydrodynamics of Section 2 and put !µ( = 0 and !n = 0.

As it was shown in Gusakov & Kantor (2011), equations describ-ing small linear oscillations of an NS include the following.

(i) Continuity equations for baryons (8) and electrons (9) thatcan be written in terms of the baryon and electron number densityperturbations, )nb and )ne, as

)nb = i* e!(/2

"!j (nb) U

j(b) + nb U

µ(b) ;µ

$, (83)

)ne = )ne (norm) + )ne (sfl), (84)

where j is the spatial index and we defined

)ne (norm) " i* e!(/2

"!j (ne) U

j(b) + ne U

µ(b) ;µ

$, (85)

)ne (sfl) " ! i* e!(/2

)!j (ne) Xj + ne Xµ

;µ

*. (86)

(ii) Einstein equations, which can schematically be presented as

)(Rµ( ! 1/2 gµ( R) = 8"G )T µ(, (87)

where the perturbation )Tµ( of the energy–momentum tensor (11)can be expressed in terms of the perturbations of baryon four-velocity )U

µ(b), metric )gµ( , pressure )P and energy density )+ as

)T µ( = ()P + )+) Uµ(b)U

((b) + (P + +)

)U

µ(b) )U

((b) + U (

(b) )Uµ(b)

*

+ )P gµ( + P )gµ( . (88)

In equation (87) Rµ( and R are the Ricci tensor and scalar curvature,respectively, and G is the gravitation constant.

(iii) ‘Superfluid’ equation that can be derived from equations (10)and (13) of Section 2 (here we present only the spatial componentsj of this equation)6

i * (µn Ynk w(k)j ! nb w(n)j ) = ne !j (e(/2 )µ). (89)

Expressing the vectors wj(i) through Xj in this equation (see equations

3 and 4), and introducing the redshifted imbalance of chemicalpotentials )µ# " e(/2 )µ, one can rewrite equation (89) as

Xj = i ne

µnnb * y!j ()µ#), (90)

where y is defined by equation (68). Note that this equation dictatesthe most general form of the superfluid Lagrangian displacement$

j(sfl) that was already obtained in equation (52) from the symmetry

arguments.

Equations (83)–(90) should be supplemented with the expres-sions for the perturbations )P, )µ and )+. To derive them, let usnote that any thermodynamic quantity (e.g. P) in the superfluid mat-ter can be presented as a function of nb, ne, T and w(i) µw

µ(k) (see e.g.

Gusakov 2007). In strongly degenerate matter the dependence of P,)µ and + on T can be neglected (see e.g. Reisenegger 1995; Gusakovet al. 2005), while the scalars w(i) µw

µ(k) are quadratically small in a

slightly perturbed star (see Section 3.2). Thus, P = P(nb, ne), )µ =)µ(nb, ne) and + = +(nb, ne). Expanding these functions into Taylorseries near the equilibrium, one obtains

)P = nb!P

!nb

&)nb

nb+ s̃

)ne (norm)

ne+ s

)ne (sfl)

ne

', (91)

)µ = ne!)µ

!ne

&z

)nb

nb+ )ne (norm)

ne+ )ne (sfl)

ne

', (92)

)+ = µn )nb, (93)

where we made use of equation (84), and introduced dimensionlesscoupling parameter s and the quantities s̃ and z,

s " ne

nb

(!P/!ne)(!P/!nb)

, (94)

6 It is worth to make a number of comments on equation (89): (i) in Gusakov& Kantor (2011) this equation was derived under the assumption that theonly superfluid species are neutrons (i.e. Ypi = 0). A generalization ofthis equation to the case of possible proton superfluidity is presented in(Chugunov & Gusakov 2011, see their equation 3). (ii) In both papers(Chugunov & Gusakov 2011; Gusakov & Kantor 2011), this equation iswritten with the same mistake. In particular, in Chugunov & Gusakov (2011)one should write ne !j (e(/2 )µ) instead of ne e(/2 !j ()µ) in the right-handside of equation (3).

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In this limit the LD equations are decoupled from the equation for δμ, and have the same form as in the non-SFL case. Therefore the “normal”modes are not modified in this limit, but there are also SFL modes, in

which the non-SFL perturbationsvanish, and are solutions of

the δμ equation.

Dissipation in relativistic superfluid NSs 1533

Figure 8. Damping times ! b+s versus T! for various oscillation modes for model 2 of nucleon superfluidity. On each panel we plot one normal mode (shownby solid line; its multipolarity and name is indicated) and first 15 superfluid modes (dashed lines). Dotted lines show ! b+s(T!) for normal modes calculatedusing normal-fluid hydrodynamics (see the text for more details). In the shaded area all neutrons are normal and superfluid modes do not exist.

Figure 9. Eigenfrequencies " (upper panels) and damping times ! b+s (lower panels) versus T! for quadrupole (l = 2) oscillation modes in an increasinglylarger scale. The normal p1 mode is shown by solid lines. Lower left-hand panel coincides with Fig. 8(e). Other lower panels are zoomed in versions ofFig. 8(e). Notations are the same as in Figs 6 and 8.

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The viscous damping time changes (expecially in radial

and p-modes) by a factor ~1-5 far from“resonances”

( i.e., T such that νSFL~νnon-SFL )and sharply decrease

at “resonances”Leonardo Gualtieri Problemi Attuali di Fisica Teorica (XX Edizione) Vietri sul Mare, 11-16 April, 2014

Perturbations of superfluid NSs: coupled case We have then derived the full coupled form of the extended LD

equations describing non-radial perturbations of a SFL NS.We have the LD equations in the variables

with a modification which introduces a coupling with the SFL variable δμ; this is due to the fact that

Finally, there is the equation for δμ, coupled to the others.

H lm1 (b), Klm


(b)

∆plm = (1− γ2)γp

�+ p∆�lm + γ3nbe

ν/2δµlm


We considered model A (two layers) and B (two or three layers),APR2 EoS, M=1.4 Msun9

107

108

109

1010

1014 1015107

108

109

1010

0.0 0.2 0.4 0.6 0.8 1.0

Tck

[K]

! [g cm!3]

Tcn, 3P2 – triplet

Tcp,1 S0

– singlet

crust

core

T" ck

[K]

r/R

T! = 4 # 108 K

APRM = 1.4M", R = 12.2 km

T!cn

, 3P2 – triplet

T!cp , 1S0 – singlet

crust

core

FIG. 2: (color online) Same as in Fig. 1, for model B.

We consider an isothermal temperature profile, i.e., weassume that the redshifted temperature T! = e!/2T isuniform over the core of the star. We assume the tripletpairing of neutrons and singlet pairing of protons in theNS core. The neutron superfluidity in the stellar crustis ignored; this assumption should not a!ect noticeablyglobal oscillations of NSs.Following Ref. [39], we consider two models of nucleon

superfluidity, which we denote by ’A’ and ’B’, as repre-sentatives of a two-layer and a three-layer structure forthe superfluid phase, respectively.In model A the redshifted proton critical temperature

is constant over the core, T!cp ! Tcp e!/2 = 5"109 K; the

redshifted neutron critical temperature T!cn ! Tcn e!/2

increases with the density ! and reaches the maximumvalue T!

cnmax = 6 " 108 K at the stellar centre (r=0).A similar model of neutron superfluidity (with the max-imum of T!

cn (!) at the stellar centre) has been recentlyconsidered in Ref. [56] and agrees with the results of somemicroscopic calculations [57].In model B both critical temperatures T!

cn and T!cp are

density-dependent, and, depending on the value of thetemperature, the superfluid phase can have two or threelayers. A similar model of neutron superfluidity has beenrecently used to explain observations of the cooling NSin Cassiopea A supernova remnant [16, 17], and agreeswith the results of microscopic calculations (see, e.g., Ref.[11, 12]).Models A and B are shown in Figs. 1 and 2. These

figures coincide with, respectively, Figs. 1 and 2 of Ref.[39]. The functions Tci(!) are shown in the left panelsof both figures; the right panels show the dependenceT!ci (r) (i = n and p). As the redshifted temperature T!

decreases, the size of the superfluid region [given by thecondition T < Tcn(r), or, equivalently, T! < T!

cn (r)] in-creases or remains una!ected. For illustration, we shadedin Figs. 1 and 2 the superfluid region corresponding toT! = 4" 108 K. One can see that in model B there canbe three-layer configurations of a star with no neutronsuperfluidity in the centre and in the outer region butwith superfluid intermediate region, or, for lower temper-atures, two-layer configurations. In contrast, in model Aonly two-layer configurations are possible.

1 1034

2 1036

4 1038

8 1040

0 2 4 6 8 10 12

h(r)

r (km)

T=6 107 KT=1.5 108 K

T=3 108 K

1 1034

2 1036

4 1038

8 1040

0 2 4 6 8 10 12

h(r)

r (km)

T=3.16 107 KT=1.9 108 K

T=2 108 KT=3 108 K

FIG. 3: (color online) Profile of h(r) for model A (upperpanel) and model B (lower panel).

This can also be seen looking at the profiles of thefunction h(r) defined in Eq. (64), which vanishes in thenon-superfluid region, and is non-vanishing in the super-fluid region. In Fig. 3 we show h(r) for models A, B andfor di!erent values of the temperature. We can see thatmodel A yields two-layer configurations, while in modelB we have two-layer configurations for T ! 2 " 108 K,and three-layer configurations for T " 2 " 108 K. ForT # 6 " 108 K (model A) or T " 5 " 108 K (model B),the superfluid region disappear.We remark that the entrainment matrix Yik depends

on the critical temperature profiles T!ci (!), and on the

value of the stellar temperature T! (which is assumedto be constant over the core) as well. We have computedYik for the models A and B following the same procedureas in Refs. [39, 49].

V. RESULTS

Here we describe the results of our numerical integra-tions of the perturbative equations derived in Sec. III E,to find the QNMs of superfluid NSs.

A. General structure of the QNM spectrum

As first noted by Epstein [22] and Lindblom andMendell [23], when a superfluid phase is present, thereare two classes of QNMs. The first class is formed by“normal” (or “ordinary”) modes, which correspond (withsmall deviations) to modes of a non-superfluid NS. These

9

107

108

109

1010

1014 1015107

108

109

1010

0.0 0.2 0.4 0.6 0.8 1.0

Tck

[K]

! [g cm!3]

Tcn, 3P2 – triplet

Tcp,1 S0

– singlet

crust

core

T" ck

[K]

r/R

T! = 4 # 108 K

APRM = 1.4M", R = 12.2 km

T!cn

, 3P2 – triplet

T!cp , 1S0 – singlet

crust

core

FIG. 2: (color online) Same as in Fig. 1, for model B.

We consider an isothermal temperature profile, i.e., weassume that the redshifted temperature T! = e!/2T isuniform over the core of the star. We assume the tripletpairing of neutrons and singlet pairing of protons in theNS core. The neutron superfluidity in the stellar crustis ignored; this assumption should not a!ect noticeablyglobal oscillations of NSs.Following Ref. [39], we consider two models of nucleon

superfluidity, which we denote by ’A’ and ’B’, as repre-sentatives of a two-layer and a three-layer structure forthe superfluid phase, respectively.In model A the redshifted proton critical temperature

is constant over the core, T!cp ! Tcp e!/2 = 5"109 K; the

redshifted neutron critical temperature T!cn ! Tcn e!/2

increases with the density ! and reaches the maximumvalue T!

cnmax = 6 " 108 K at the stellar centre (r=0).A similar model of neutron superfluidity (with the max-imum of T!

cn (!) at the stellar centre) has been recentlyconsidered in Ref. [56] and agrees with the results of somemicroscopic calculations [57].In model B both critical temperatures T!

cn and T!cp are

density-dependent, and, depending on the value of thetemperature, the superfluid phase can have two or threelayers. A similar model of neutron superfluidity has beenrecently used to explain observations of the cooling NSin Cassiopea A supernova remnant [16, 17], and agreeswith the results of microscopic calculations (see, e.g., Ref.[11, 12]).Models A and B are shown in Figs. 1 and 2. These

figures coincide with, respectively, Figs. 1 and 2 of Ref.[39]. The functions Tci(!) are shown in the left panelsof both figures; the right panels show the dependenceT!ci (r) (i = n and p). As the redshifted temperature T!

decreases, the size of the superfluid region [given by thecondition T < Tcn(r), or, equivalently, T! < T!

cn (r)] in-creases or remains una!ected. For illustration, we shadedin Figs. 1 and 2 the superfluid region corresponding toT! = 4" 108 K. One can see that in model B there canbe three-layer configurations of a star with no neutronsuperfluidity in the centre and in the outer region butwith superfluid intermediate region, or, for lower temper-atures, two-layer configurations. In contrast, in model Aonly two-layer configurations are possible.

1 1034

2 1036

4 1038

8 1040

0 2 4 6 8 10 12h(

r)r (km)

T=6 107 KT=1.5 108 K

T=3 108 K

1 1034

2 1036

4 1038

8 1040

0 2 4 6 8 10 12

h(r)

r (km)

T=3.16 107 KT=1.9 108 K

T=2 108 KT=3 108 K

FIG. 3: (color online) Profile of h(r) for model A (upperpanel) and model B (lower panel).

This can also be seen looking at the profiles of thefunction h(r) defined in Eq. (64), which vanishes in thenon-superfluid region, and is non-vanishing in the super-fluid region. In Fig. 3 we show h(r) for models A, B andfor di!erent values of the temperature. We can see thatmodel A yields two-layer configurations, while in modelB we have two-layer configurations for T ! 2 " 108 K,and three-layer configurations for T " 2 " 108 K. ForT # 6 " 108 K (model A) or T " 5 " 108 K (model B),the superfluid region disappear.We remark that the entrainment matrix Yik depends

on the critical temperature profiles T!ci (!), and on the

value of the stellar temperature T! (which is assumedto be constant over the core) as well. We have computedYik for the models A and B following the same procedureas in Refs. [39, 49].

V. RESULTS

Here we describe the results of our numerical integra-tions of the perturbative equations derived in Sec. III E,to find the QNMs of superfluid NSs.

A. General structure of the QNM spectrum

As first noted by Epstein [22] and Lindblom andMendell [23], when a superfluid phase is present, thereare two classes of QNMs. The first class is formed by“normal” (or “ordinary”) modes, which correspond (withsmall deviations) to modes of a non-superfluid NS. These


Perturbations of superfluid NSs: coupled case 11

1000

2000

3000

4000

5000

6000

5 1084 1083 1082 1081080

(Hz)

T (K)

Model A

1810

1860

1.3 108 1.4 108

0

1000

2000

3000

4000

5000

6000

5 1084 1083 1082 1081080

(Hz)

T (K)

Model B

1600

2000

2400

1.2 108 1.8 108 2.4 108

FIG. 4. (color online) Frequencies of the first l = 2 modes asfunctions of the temperature for model A (upper panel) andmodel B (lower panel). The thinner lines show the superfluidmodes in the decoupling limit. In the inset we show a detailof avoided crossing.

smaller than the real part, numerical errors make it di!-cult to compute the damping times with good accuracy;this problem seems to be more severe for temperatures! 5! 107 K, and for damping times " 103 " 104 s. Still,we think that the values shown in Fig. 5 provide a reli-able estimate at least of the order of magnitude of thedamping times, and of their dependence on the temper-ature.We can see that at the crossing temperatures Ti, the

curves of the damping times do cross, and the modeschange their nature from normal to superfluid and viceversa. Fig. 5 also shows that, far from the crossing tem-peratures Ti, the superfluid modes have damping times" 102 " 103 s, much larger than those of the normalmodes. Conversely, at temperatures close to the Ti, thedamping times of the superfluid modes sharply decrease,becoming comparable with those of the correspondingnormal modes. This behaviour is due to the fact thatat T # Ti, the normal and superfluid degrees of freedombecome significantly coupled.The viscous damping time of superfluid modes !b+s,

which has been computed in [36] in the decoupled limit,shows the same qualitative behaviour. Comparing our re-sults with those of [36], we find that the viscous dampingtimes of superfluid modes are significantly larger thantheir gravitational damping times. However, this com-

10-1

100

101

102

103

104

105

106

1 108 2 108 3 108

GW

(s)

T (K)

Model A

10-1

100

101

102

103

1.0 108 1.2 108 1.4 108 1.6 108

GW

(s)

T (K)

Model B

FIG. 5. (color online) Damping times for the lowest frequencymodes shown in Fig. (4) for model A (upper panel) and modelB (lower panel).

parison has been only done for the lowest lying QNMs,because we have been able to compute !GW for thesemodes only. We note that, as the order of the mode in-creases, the gravitational damping time increases, whilethe viscous damping time decreases [36], therefore it isreasonable to expect that !GW becomes larger than !b+s

for high-order modes.QNMs with shorter damping times are more e!cient

in emitting gravitational waves. Indeed, the gravitationalwave flux can be estimated as LGW $ 2Emech/!GW [19],where Emech is the mechanical pulsation energy stored inthe mode, introduced in Sec. III C. Therefore, for genericvalues of the temperature the superfluid modes are notgood sources of GWs, because their damping times arelarge; but, at temperatures close to the crossing temper-atures Ti, their damping times become comparable tothose of the normal modes, and they can become muchmore e!cient in emitting GWs. We can expect, then,that during the cooling of a NS, when it reaches one ofthe crossing temperatures, a new QNM - in principle de-tectable from gravitational wave observers - can appearin the spectrum.

3. Eigenfunctions

[LG: Here I expanded the discussion on the separation ofdegrees of freedom. See also the new Section, IIIC]In Fig. 6 we show the velocity eigenfunctions for the

f -mode (upper panel), the p1-mode (middle panel) andthe sf0-mode (lower panel), for model A at T = 6 · 107

11

1000

2000

3000

4000

5000

6000

5 1084 1083 1082 1081080

(Hz)

T (K)

Model A

1810

1860

1.3 108 1.4 108

0

1000

2000

3000

4000

5000

6000

5 1084 1083 1082 1081080

(Hz)

T (K)

Model B

1600

2000

2400

1.2 108 1.8 108 2.4 108

FIG. 4. (color online) Frequencies of the first l = 2 modes asfunctions of the temperature for model A (upper panel) andmodel B (lower panel). The thinner lines show the superfluidmodes in the decoupling limit. In the inset we show a detailof avoided crossing.

smaller than the real part, numerical errors make it di!-cult to compute the damping times with good accuracy;this problem seems to be more severe for temperatures! 5! 107 K, and for damping times " 103 " 104 s. Still,we think that the values shown in Fig. 5 provide a reli-able estimate at least of the order of magnitude of thedamping times, and of their dependence on the temper-ature.We can see that at the crossing temperatures Ti, the

curves of the damping times do cross, and the modeschange their nature from normal to superfluid and viceversa. Fig. 5 also shows that, far from the crossing tem-peratures Ti, the superfluid modes have damping times" 102 " 103 s, much larger than those of the normalmodes. Conversely, at temperatures close to the Ti, thedamping times of the superfluid modes sharply decrease,becoming comparable with those of the correspondingnormal modes. This behaviour is due to the fact thatat T # Ti, the normal and superfluid degrees of freedombecome significantly coupled.The viscous damping time of superfluid modes !b+s,

which has been computed in [36] in the decoupled limit,shows the same qualitative behaviour. Comparing our re-sults with those of [36], we find that the viscous dampingtimes of superfluid modes are significantly larger thantheir gravitational damping times. However, this com-

10-1

100

101

102

103

104

105

106

1 108 2 108 3 108

GW

(s)

T (K)

Model A

10-1

100

101

102

103

1.0 108 1.2 108 1.4 108 1.6 108

GW

(s)

T (K)

Model B

FIG. 5. (color online) Damping times for the lowest frequencymodes shown in Fig. (4) for model A (upper panel) and modelB (lower panel).

parison has been only done for the lowest lying QNMs,because we have been able to compute !GW for thesemodes only. We note that, as the order of the mode in-creases, the gravitational damping time increases, whilethe viscous damping time decreases [36], therefore it isreasonable to expect that !GW becomes larger than !b+s

for high-order modes.QNMs with shorter damping times are more e!cient

in emitting gravitational waves. Indeed, the gravitationalwave flux can be estimated as LGW $ 2Emech/!GW [19],where Emech is the mechanical pulsation energy stored inthe mode, introduced in Sec. III C. Therefore, for genericvalues of the temperature the superfluid modes are notgood sources of GWs, because their damping times arelarge; but, at temperatures close to the crossing temper-atures Ti, their damping times become comparable tothose of the normal modes, and they can become muchmore e!cient in emitting GWs. We can expect, then,that during the cooling of a NS, when it reaches one ofthe crossing temperatures, a new QNM - in principle de-tectable from gravitational wave observers - can appearin the spectrum.

3. Eigenfunctions

[LG: Here I expanded the discussion on the separation ofdegrees of freedom. See also the new Section, IIIC]In Fig. 6 we show the velocity eigenfunctions for the

f -mode (upper panel), the p1-mode (middle panel) andthe sf0-mode (lower panel), for model A at T = 6 · 107


Perturbations of superfluid NSs: coupled case

Velocity eigenfunctions

f-mode

p1-mode

sfl0-mode

12

-8

-6

-4

-2

0

2

4

0 2 4 6 8 10 12 14r (km)

W(b)W(sfl)

V(b)V(sfl)

0

-0.05

0

0.05

0.1

0 2 4 6 8 10 12 14r (km)

W(b)W(sfl)

V(b)V(sfl)

0

-0.2

-0.1

0

0.1

0.2

0 2 4 6 8 10 12 14r (km)

W(b)W(sfl)

V(b)V(sfl)

0

FIG. 6. (color online) Velocity eigenfunctions for the f -mode(upper panel), the p1-mode (middle panel) and the sf0-mode(lower panel), for model A at T = 6 · 107 K.

K. We show the (l = 2) quantities W lm(b) , V

lm(b) , obtained

expanding the radial and angular components, respec-tively, of the perturbation !Uµ

(b) in spherical harmonics

(27), and the quantities W lm(sfl), V

lm(sfl) obtaind expanding

in the same way !Xµ (31). As discussed in Sec. III C,the functions W lm, V lm obtained by the expansion of!uµ (23) are given by

W lm = W lm(b) !W lm

(sfl)

V lm = V lm(b) ! V lm

(sfl) . (70)

We can see that for the fundamental modeW lm(sfl) " W lm

(b) ,

V lm(sfl) " V lm

(b) ; for the first pressure mode W lm(b) # W lm

(sfl),

V lm(b) # V lm

(sfl); for the first superfluid mode W lm(sfl) $ W lm

(b) ,

V lm(sfl) $ V lm

(b) . This supports the interpretation (see [36]

and the discussion in Sec. III C) of W lm(b) , V

lm(b) as describ-

ing non-superfluid degrees of freedom and W lm(sfl), V

lm(sfl) as

describing superfluid degrees of freedom.Fig. 6 also shows that the radial velocity, W , has no

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14

!

r (km)

sfl0 Coupledsfl0 Decoupled

sfl1 Coupledsfl0 Decoupled

FIG. 7. (color online) Chemical potential imbalance eigen-function for the l = 2 sf0-mode, for model A at T = 6 · 107K.

Model T108 K

f p1 sf0

0.6 3.8! 10!5 6.0! 10!3 1.5! 102

A 1.5 4.6! 10!4 1.6! 10!3 1.7! 102

3.0 1.2! 10!4 1.6! 10!4 0.7! 102

0.6 2.1! 10!5 3.0! 10!2 0.9! 102

B 1.5 4.5! 10!3 1.6! 10!3 0.8! 102

3.0 6.1! 10!6 1.5! 10!4 0.2! 102

TABLE II. Ratio Emech (sf)/Emech (b) for the f -mode, the p1-mode and the sf0-mode, for di!erent values of temperature,for the models A, B.

nodes inside the star in the case of the f -mode, while ithas one node in the case of the p1-mode and in the caseof the sf0 mode, as mentioned in Sec. VA.The eigenfunction of !µ, for the l = 2 sf0 mode, model

A and T = 6 ·107 K, is shown in Fig. 7 in the coupled anddecoupled case. [LG: I think that either we say somethingabout this eigenfunction, or we remove it, i.e., figure 7 andthis sentence.]

4. Pulsation energy

We have computed the mechanical pulsation energyEmech stored in the QNMs,

Emech = Emech (sfl) + Emech (b) (71)

using Eqs. (29), (30). In Table II we show the ratioEmech (sfl)/Emech (b) for the f -mode, the p1-mode and thesf0-mode, for models A and B, at three values of tem-perature. We can see that when the star oscillates in anon-superfluid mode, Emech (sfl) " Emech (b), i.e., mostof the energy is stored in non-superfluid degrees of free-dom (this is more evident for the f -mode than for thep1-mode), while when it oscillates in a superfluid mode,Emech (sfl) $ Emech (b), i.e., most of the energy is storedin superfluid degrees of freedom.

W(sfl) � W(b)

V(sfl) � V(b)

W(sfl) ∼ W(b)

V(sfl) ∼ V(b)

W(sfl) � W(b)

V(sfl) � V(b)

Conclusions:

Perturbations of a superfluid, spherically symmetric star can be formulated in a way formally equivalent to one-fluid hydrodynamics. In this framework, we have derived a set of equations which extends the Lindblom-Detweiler equations describing perturbations of non-SFL NSs

Our approach, in a general relativistic framework, includes the effect of finite temperature in the quantities associated to SFL (entrainment matrix, etc.) This is crucial to determine the domain of the SFL region.

We computed frequencies and (gravitational) damping times of the f and p1 modes, and of the first SFL modes.

As the temperature of the star increases, the SFL region shrinks. At some values of T, the frequencies of non-SFL and SFL modes are close; at these values, the frequencies show avoided crossings, and damping times of SFL modes (generally >> than those of the non-SFL modes) sharply decrease. SFL modes, then, may become detectable at those temperatures.

It is now possible to include SFL in gravitational wave asteroseismology.


quasi-normal modes of a relativistic superﬂuid neutron...

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