Decoupling of superfluid and normal modes in pulsating neutron stars

Mikhail E. Gusakov1 and Elena M. Kantor1,2

1Ioffe Physical Technical Institute, Politekhnicheskaya 26, 194021 St. Petersburg, Russia2St. Petersburg State Polytechnical University, Polytekhnicheskaya 29, 195251 St. Petersburg, Russia

(Received 16 July 2010; published 22 April 2011)

We show that equations governing pulsations of superfluid neutron stars can be split into two sets of

weakly coupled equations, one describing the superfluid modes and another one, the normal modes. The

coupling parameter s is small, jsj � 0:01–0:05, for realistic equations of state. Already an approximation

s ¼ 0 is sufficient to calculate the pulsation spectrum within the accuracy of a few percent. Our results

indicate, in particular, that emission of gravitational waves from superfluid pulsation modes is suppressed

in comparison to that from normal modes. The proposed approach allows to drastically simplify modeling

of pulsations of superfluid neutron stars.

DOI: 10.1103/PhysRevD.83.081304 PACS numbers: 97.60.Jd, 47.37.+q, 47.75.+f, 97.10.Sj

I. INTRODUCTION

The pulsations of neutron stars (NSs) can be excitedeither due to internal instabilities or owing to externalperturbations. Currently the detectors that will be able,according to preliminary estimates, to register gravitationalwaves from pulsating NSs, are under development [1]. Forthe correct interpretation of future observations, it is nec-essary to have a well-developed theory of NS pulsations.The formulation of such a theory is complicated by the factthat at a temperature T & 108–1010 K, baryons in theinternal layers of NSs become superfluid. Thus, to modelpulsations one has to employ superfluid hydrodynamics,which is much more complicated than the ordinary one,describing ‘‘normal’’ (nonsuperfluid) matter.

For the first time, the global pulsations of superfluid NSswere analyzed by Lindblom and Mendell in 1994 [2].Considering a simple model of a Newtonian star, theynumerically found two distinct classes of pulsation modes:(i) normal modes which practically coincide with the cor-responding modes of a normal star; and, (ii) superfluidmodes in which the matter pulsates in such a way thatthe mass current density approximately vanishes. The sub-sequent numerical studies of various pulsation modes (theliterature is vast; see, e.g., Refs. [3,4] and referencestherein) confirmed the result of Ref. [2], though generalexplanation of this result has not yet been proposed [5]. Inthis work, we give such an explanation. In addition, wepresent an approximate scheme which allows to greatlysimplify calculations of pulsating superfluid NSs. In whatfollows, the speed of light c ¼ 1.

II. SUPERFLUID HYDRODYNAMICS

For simplicity, we consider NS cores composed of neu-trons (n), protons (p), and electrons (e). We also assumethat protons are normal while neutrons are superfluid insome region of a NS core. As demonstrated in Ref. [8],possible admixture of other particle species (e.g., muons)and proton superfluidity do not affect our principal results.

Finally, we first consider a nonrotating NS. Effects ofrotation will be incorporated later in the text.It is well known that in superfluid matter, several inde-

pendent motions with different velocities may coexistwithout dissipation [9]. In our case the system is fullydescribed by two four-vectors, u� and w

�ðnÞ. The vector

u� is the velocity of electrons and protons as well asnormal neutrons; the vector w

�ðnÞ arises from additional

degrees of freedom associated with neutron superfluidity.In the nonrelativistic limit the spatial components of thefour-vector w

�ðnÞ are related to superfluid velocities Vsn of

the Landau-Khalatnikov theory [9] by the equality wðnÞ ¼mnðVsn � uÞ, wheremn is the neutron mass; u is the spatialcomponent of the normal four-velocity u�. The electronj�ðeÞ, proton j�ðpÞ, and neutron j�ðnÞ current densities are

expressed through the vectors u� and w�ðnÞ as [10,11]:

j�ðeÞ ¼ neu

�, j�ðpÞ ¼ npu

�, and j�ðnÞ ¼ nnu

� þ Ynnw�ðnÞ.

Here nl is the number density of particles l ¼ n, p, or e.The expression for j�ðnÞ consists of two terms reflecting the

fact that both normal and superfluid liquid componentscontribute to neutron current density. The coefficient Ynn

has been calculated in Ref. [12]; it is a relativistic analogueof superfluid density of neutrons �sn. In the nonrelativisticlimit Ynn ¼ �sn=m

2n. This coefficient depends on T and

increases steadily from 0 for normal matter (whenT � Tcn, where Tcn is the neutron critical temperature)to nn=�n for entirely superfluid matter (T ¼ 0). Hereand below, �l is the chemical potential for particlesl ¼ n, p, or e.To proceed further, we assume that: (i) the quasineutral-

ity condition holds both for equilibrium and pulsatingmatter, np ¼ ne; and, (ii) an unperturbed NS is in beta-

equilibrium, i.e. the disbalance of chemical potentials�� � �n ��p ��e ¼ 0. It is convenient then to formu-

late the system of nondissipative hydrodynamic equationsusing the baryon number density nb � nn þ np and �� as

independent variables [11]. The system consists of (1) thecontinuity equations for baryons and electrons

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j�ðbÞ;� ¼ 0; j�ðeÞ;� ¼ 0; (1)

where the baryon current density is j�ðbÞ � j

�ðnÞ þ j

�ðpÞ ¼

nbu� þ Ynnw

�ðnÞ; (2) Einstein equations

R�� � 1

2g��R ¼ 8�GT�� (2)

with the energy-momentum tensor T�� ¼ ðPþ "Þu�u� þPg�� þ Ynnðw�

ðnÞw�ðnÞ þ�nw

�ðnÞu

� þ�nw�ðnÞu

�Þ; (3) the

potentiality condition for superfluid motion of neutrons

@�½wðnÞ� þ�nu�� � @�½wðnÞ� þ�nu�� ¼ 0; (3)

and, (4) the second law of thermodynamics

d"¼�ndnb���dneþTdSþYnndðw�ðnÞwðnÞ�Þ=2: (4)

In Eqs. (2)–(4), R��, R, and g�� are Ricci tensor, scalarcurvature, and metric tensor, respectively; @� � @=@x�; G

is the gravitation constant; P � �"þ�nnb � ��ne þTS is the pressure; " and S are the energy and entropy

densities, respectively. All the thermodynamic quantitiesare defined in the comoving frame in which u� ¼ð1; 0; 0; 0Þ. This imposes an additional constraint on w

�ðnÞ

[11], u�w�ðnÞ ¼ 0. The solution to Eqs. (1)–(4) in the su-

perfluid region should be matched with that in the residual(normal) region of a star. Equations, describing pulsationsof normal matter can be obtained from the system (1)–(4) ifone puts Ynn ¼ 0 and ignores the condition (3).

III. LINEAR APPROXIMATION

In this work, we assume that in the unperturbed NSwðnÞ ¼ 0, i.e., velocities of normal and superfluid compo-

nents coincide. For a nonrotating NS, this simply meansthat both components are at rest. Then it follows from theconstraint u�w

�ðnÞ ¼ 0 that w�

ðnÞ vanishes in equilibrium,

w�ðnÞ ¼ 0 (because u0 � 0). As a consequence, the terms in

Eq. (4) and in the expression for T��, underlined with oneline, are quadratically small. Similarly, the terms whichdepend on T and underlined twice, are small in the stronglydegenerate matter of NSs and can be omitted [11]. Becauseof the same reasons one can consider, e.g., the quantities Pand �� as functions of only nb and ne (and neglect theirdependence on scalars w�

ðnÞwðnÞ� and T).

Now we make use of the energy-momentum conserva-tion law T��

;� ¼ 0 which can be derived from Eq. (2).Composing a vanishing combination T��

;� þ u�u�T��;�

and subtracting from it Eq. (3) multiplied by nbu�, one

gets, with the help of Eq. (4) and the expression for P,

� ne@���� u�u�ne@���� ne��u�ðu�Þ;�

þ ðg�� þ u�u�Þu�X�;� þ X�u�;� þ X�u

�;�

� nbu�½@�wðnÞ� � @�wðnÞ�� ¼ 0: (5)

Here X� � �nYnnw�ðnÞ. The obtained ‘‘superfluid’’ equa-

tion is very attractive because each term in it dependseither on �� or w

�ðnÞ. Both these quantities are small in a

slightly perturbed matter (and vanish in equilibrium). Thismeans that in the linear approximation Eq. (5) does notdepend explicitly on the perturbations of the metric g��

and the four-velocity u�. Thus, one can replace g�� and u�

in Eq. (5) by their equilibrium values. For a nonrotatingNS, the spatial components of Eq. (5) can be rewritten in aremarkably simple form (j ¼ 1, 2, 3)

i!ð�nYnn � nbÞwðnÞj ¼ ffiffiffiffiffiffiffiffiffiffiffi�g00p

ne@jð��Þ; (6)

where we assumed that w�ðnÞ depends on time t as w�

ðnÞ �expði!tÞ. In Eq. (6), all the quantities except for ��and w

�ðnÞ are taken in equilibrium. Near the equilibrium,

the function ��ðnb; neÞ can be expanded in the Taylorseries and presented, in the linear approximation, as

�� ¼ neð@��=@neÞðzD1 þD2Þ: (7)

Here z � ½nb@��=@nb�=½ne@��=@ne�; D1 � �nb=nb;D2 � �ne=ne. The symbol � in front of some quantitydenotes a deviation of this quantity from its equilibriumvalue. The dimensionless functions D1 and D2 can befound from Eq. (1) and depend on w

�ðnÞ, u

�, and g��.

Thus, generally, Eq. (6) is not independent and should besolved together with Einstein Eqs. (2). In the linearapproximation, Eq. (2) can be written in the followingsymbolic form: �ðR�� � 1=2g��RÞ ¼ 8�G�T��. Theleft-hand side of this equation contains only perturbationsof metric. To write out the right-hand side, it is convenientto introduce new independent variables, the four-velocityof baryons U� � j�ðbÞ=nb ¼ u� þ Ynnw

�ðnÞ=nb and W� �

Ynnw�ðnÞ=nb, instead of u� and w�

ðnÞ. Note that for an un-

perturbed star U� ¼ u� andW� ¼ 0 (since in equilibriumw�

ðnÞ ¼ 0). The same is also true for a pulsating NS if

the matter is nonsuperfluid (because then Ynn ¼ 0).Employing the new variables, an expression for �T�� takesthe form

�T�� ¼ ð�Pþ �"ÞU�U� þ ðPþ "ÞðU��U� þU��U�Þþ �Pg�� þ P�g��: (8)

Here the quantities P, ", U� and g�� are taken at equilib-rium. As follows from Eq. (4), the variation �" equals�" ¼ �n�nb and depends on �U� and �g�� (and isindependent of �W�). The variation �P can be expandedin analogy with Eq. (7),

�P ¼ nbð@P=@nbÞðD1 þ sD2Þ; (9)

where the function D1 depends on �U� and �g��, and D2

depends on the difference (�U� � �W�) and �g��

[see the definitions for D1 and D2, and Eq. (1)].The parameter s, hereafter referred to as the ‘‘couplingparameter,’’ is given by s � ðne@P=@neÞ=ðnb@P=@nbÞ.

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IV. SUPERFLUID AND NORMAL MODES

If s ¼ 0, then Eq. (8) for �T�� does not depend on �W�

and has exactly the same form as in the absence of super-fluidity. In that case, the Einstein equations (and boundaryconditions to them) coincide, in the linear approximation,with the corresponding equations for normal matter. Theycan be solved separately from the superfluid Eq. (6) so thatthe solution (the spectrum of eigenfrequencies ! and theeigenfunctions �U�) will be indistinguishable from thatfor a nonsuperfluid star.

Let us now focus on the following question. Assume thats still vanishes. Is it possible for a NS to oscillate on afrequency which is not an eigenfrequency of a normal star?Suppose that it is indeed the case. Then the linearizedEinstein equations will be satisfied only if �U� ¼ 0 and�g�� ¼ 0. As follows from Eq. (1), in this case �nb ¼ 0(i.e.,D1 ¼ 0), whileD2 depends only on �W

� (or, in otherwords, onW�, sinceW� ¼ 0 in equilibrium). In particular,for a nonrotating NS

D2 ¼ ½ð@jne=neÞWj þWj;j�=ði!U0Þ: (10)

Here j ¼ 1, 2, and 3; all the quantities, except for Wj,are taken in equilibrium; when calculating the covariantderivative one should use the unperturbed metric.Equations (7) and (10) allow to formulate Eq. (6) purelyin terms of Wj. A boundary condition to this equation,W? ¼ 0, also depends only on Wj and can be obtainedfrom the requirement that the baryon current density j

�ðbÞ is

continuous through the normal-superfluid interface (W? isthe component of a vector Wj, perpendicular to the inter-face). Thus, Eq. (6) is self-contained and can be solvedindependently of Eq. (2). Its solution (eigenfrequenciesand eigenfunctions Wj) describes superfluid modes whichwere first considered in Ref. [2] and do not have ananalogue for a normal star. To our best knowledge, thestriking properties of such modes have not been discussedfor a realistic model of a general relativistic NS at finite T.First of all, the superfluid pulsation modes do not perturbmetric (�g�� ¼ 0) and hence cannot emit gravitationalwaves. In addition, because for these modes �U� ¼ 0 and�nb ¼ 0, the variations of j

�ðbÞ and P vanish, �j

�ðbÞ ¼ 0

and �P ¼ 0 [see Eqs. (1) and (9)]. As a consequence,pulsations are localized entirely in the superfluid regionof a star. In particular, they do not go to the NS surface.

In the consideration above, we supposed that s ¼ 0. Yet,it is clear that superfluid and normal modes should remainapproximately decoupled also at small but finite s. Asfollows from Fig. 1, s is indeed small for realistic equationsof state (EOSs) and changes, on the average, from�0:01 to�0:05 [13]. Taking into account that the parameter z inEq. (7) is z��1 for the same EOSs, it is easy to show thatfor normal modes D1 * D2 [then the second term inEq. (9) is much smaller than the first one], while forsuperfluid modes D1 � sD2 [then the first term in Eq. (7)

is much smaller than the second one]. Generally, the exactsolution of linearized pulsation Eqs. (2) and (6) can bepresented as a series in parameter s [8]. However, since s isvery small, the approximation of noninteracting Eqs. (2)and (6) considered above (hereafter ‘‘zero approxima-tion’’) is already sufficient to calculate the pulsationspectrum within the accuracy �s (i.e., a few percent).

V. EXAMPLE: RADIAL PULSATIONS

Let us illustrate the obtained results with an example of aradially pulsating NS with the mass M ¼ 1:4M�. We con-sider a simple NS model which was analyzed in detail inRef. [10]. In that paper, it was assumed that the redshiftedcritical temperature of neutrons T1

cn is constant throughoutthe stellar core, T1

cn ¼ 6� 108 K. The results of approxi-mate calculation of pulsation spectrum are illustrated inFig. 2(a). The spectrum is calculated in zero approximationin s. In the figure, the pulsation frequency ! (in units of!0 � c=RNS, where RNS ¼ 12:17 km is the circumferen-tial radius of a star) is plotted as a function of internalredshifted stellar temperature T1 for 3 normal (solid lines)and 6 superfluid (dashes) pulsation modes. At T1 > T1

cn,only the normal modes (I, II, and III) survive since then thestar is nonsuperfluid. For comparison, in Fig. 2(b), wepresent the exact solution to the system of linearizedEqs. (2) and (6). The first 6 modes are shown by alternatesolid and dashed lines. The spectrum was not plotted inthe shaded region. All other notations are the same as in

FIG. 1. The coupling parameter s versus density � for EOSsPAL [15], PA [16], APR [17], SLy4 [18], and LNS [19]. EOSs ofthe PAL family differ by the symmetry energy (models I, II,or III) and by the value of the compression modulus, 180 or240 MeV. The models with the compression modulus 120 MeVare not plotted since they give too small maximum NS massesthat seem to contradict observations [20]. Note that the recentlymeasured mass M ¼ ð1:97� 0:04ÞM� [21] of the millisecondpulsar PSR J1614-2230 further rules out PA EOS and all PALEOSs except for PAL I240 and PAL II240.

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Fig. 2(a). It is easy to see that the structure of both spectrais very similar. However, there is one principal difference.Instead of crossings of superfluid and normal modes inFig. 2(a), we have avoided crossings in Fig. 2(b). At thesepoints, the superfluid mode becomes normal and viceversa. Such avoided crossings are not described in approxi-mate treatment (Fig. 2(a)) because when frequencies ofsuperfluid and normal modes are close to each other,Eqs. (2) and (6) become strongly interacting and cannotbe considered as independent. For comparison, we plotboth spectra in Fig. 2(c). The exact solution is shown bysolid lines, dashes correspond to the approximate solution.Other notations are the same as in Figs. 2(a) and 2(b). Onaverage, the approximate solution differs from the exactone by �1:5–2%. For normal modes, the difference be-comes smaller with increasing of T. In this case the numberof superfluid neutrons decreases (Ynn ! 0), consequently,

Wj � YnnwjðnÞ=nb ! 0 and zero approximation works bet-

ter and better.

VI. TAKING INTO ACCOUNT ROTATION

Rotation leads to formation of Feynman-Onsagervortices inside NSs with the interspacing distance�10�2–10�4 cm. The hydrodynamic equations averagedover the volume containing large amount of vortices for-mally have the same form as in their absence [14] (if weneglect the small contribution of vortices to the internalenergy density of matter). The only exception is thepotentiality condition (3) that should be replacedby u�f@�½wðnÞ� þ�nu�� � @�½wðnÞ� þ�nu��g ¼O��W

�,

where the tensor O�� is specified in Ref. [8] and is respon-

sible for the interaction between the normal and superfluidcomponent. It can be found from the requirement that theentropy does not decrease. Because of this condition, thenew term nbO��W

� appears in the right-hand side of

Eq. (5). Since this term depends on a small quantity W�,all our reasoning about decoupling of superfluid and nor-mal modes remain valid for rotating NSs as well.

VII. COMPARISON WITH PREVIOUS WORKS

For comparison, we choose two papers, Refs. [4,6], sinceat first sight it is not clearwhether our results complement orcontradict the conclusions drawn in these references.The authors of Ref. [6] considered a model of

Newtonian star at T ¼ 0. They demonstrated that super-fluid modes decouple from the normal modes only for anidealized case of nonstratified NSs, for which ne=nb ¼const throughout the stellar core.This result does not contradict ours because one can

show that the neutron-star matter is nonstratified only if@Pðnb; neÞ=@ne ¼ 0 (that is s ¼ 0). As follows from ouranalysis, in the latter case superfluid and normal modes areindeed strictly decoupled.The second conclusion made in Ref. [6] is based on

the observation that for most of the neutron-star models,the stellar matter is stratified. Using this observation, theauthors of Ref. [6] argued that generally there should be noclear distinction between the superfluid and normal modes,or, in other words, equations describing superfluid- andnormal-type pulsations are strongly interacting.This conclusion is not correct because, as we demon-

strated earlier in this work, the real coupling parameter scan be small even for strongly stratified NSs (and is indeedsmall for realistic EOSs).Now let us discuss the results of Ref. [4]. This paper

analyzed gravitational radiation from superfluid nonrotat-ing NSs at T ¼ 0 in the frame of the general relativity. Itwas argued that superfluid modes must radiate gravita-tional waves in practically all situations, with intensity ofradiation comparable to that from the normal modes (un-less an EOS has a very specific form satisfying Eq. (74) ofRef. [4]).When modeling the neutron-star pulsations, the authors

of Ref. [4] used toy-model EOSs that give completelyunrealistic values for the coupling parameter s. In particu-lar, we found that their most realistic model II gives s� 0:1at the center and s ¼ 1 at the superfluid-normal interface.Moreover, because their EOSs are artificial, they were

FIG. 2 (color online). Frequency ! in units of !0 versus T18 � T1=ð108 KÞ for various pulsation modes. (a) approximate spectrum;

(b) exact spectrum; (c) approximate (dashed lines) and exact (solid lines) spectra. For more details see the text.

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forced to relax an assumption of chemical equilibrium inthe core. As it is demonstrated in the present paper, thelatter assumption is very important for the decoupling ofmodes and cannot be ignored. Thus, it is not surprising thatour results disagree with the results of Ref. [4]; when s isnot small, superfluid modes can be as effective in radiatinggravitational waves as normal modes.

In the end, it is worth mentioning one more result ofRef. [4]. In that paper, it is claimed that any (nonradial)pulsation mode must emit gravitational waves unlessan EOS satisfies some specific criterion [their Eq. (74)].We checked that this criterion is not equivalent and doesnot follow from our criterion s ¼ 0, which is a necessarycondition for decoupling of superfluid modes from metric.

VIII. CONCLUSION

Summarizing, equations describing pulsations of super-fluid NSs can be split into two systems of weakly coupledequations. The coupling parameter s of these systems issmall for realistic EOSs, jsj � 0:01–0:05. One system ofequations describes normal modes, another one, superfluidmodes. Already, zero approximation in parameter s (whenthe systems are fully decoupled) is sufficient to calculatethe pulsation spectrum with an accuracy of a few percent.In this approximation, the normal modes coincide withordinary modes of nonsuperfluid NS, while superfluidmodes do not perturb metric, pressure, baryon currentdensity and are localized in superfluid region of a star.Note that an emission of gravitational waves by superfluid

modes is possible only in the next (first) order of perturba-tion theory in s. Thus, it should be suppressed in compari-son to gravitational radiation from the normal modes.Our finding that superfluidmodes do not appear at the NS

surface and do not emit gravitational waves in thes ¼ 0 limit indicate that these modes should be very diffi-cult to observe at small but finite s. This means that obser-vational properties of a pulsating superfluid star and anormal star of the same mass should be very similar, sothat it will be very hard to discriminate one from the other.The obtained results explain numerical calculations [2,3],

and suggest a simple perturbative (in parameter s) schemewhich drastically simplifies the problemof calculation of thepulsation spectrum for superfluid NSs. The presented ap-proach allows to easily take into account realistic EOSs,dissipation, various composition of matter, temperatureeffects, baryon superfluidity, density-dependent profiles ofcritical temperatures, and rotation of NSs. In more detail,these issues will be discussed elsewhere.

ACKNOWLEDGMENTS

We thank D. P. Barsukov, A. I. Chugunov, and D.G.Yakovlev for valuable comments. This research wassupported by the Dynasty Foundation, Ministry ofEducation and Science of Russian Federation (ContractNo. 11.G34.31.0001 with SPbSPU and leading scientistG.G. Pavlov), RFBR (Grant No. 11-02-00253-a), and byFASI (Grant No. NSh-3769.2010.2).

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