Superfluid instability of r-modes in ‘‘differentially rotating’’ neutron stars

N. Andersson,1 K. Glampedakis,2,3 and M. Hogg1

1School of Mathematics, University of Southampton, Southampton SO17 1BJ, United Kingdom2Departamento de Fisica, Universidad de Murcia, E-30100 Murcia, Spain

3Theoretical Astrophysics, University of Tubingen, Auf der Morgenstelle 10, Tubingen, D-72076, Germany(Received 13 August 2012; published 18 March 2013)

Superfluid hydrodynamics affects the spin evolution of mature neutron stars and may be key to

explaining timing irregularities such as pulsar glitches. However, most models for this phenomenon

exclude the global instability required to trigger the event. In this paper we discuss a mechanism that may

fill this gap. We establish that small scale inertial r-modes become unstable in a superfluid neutron star that

exhibits a rotational lag expected to build up due to vortex pinning as the star spins down. Somewhat

counterintuitively, this instability arises due to the (under normal circumstances dissipative) vortex-

mediated mutual friction. We explore the nature of the superfluid instability for a simple incompressible

model allowing for entrainment coupling between the two fluid components. Our results recover a

previously discussed dynamical instability in systems where the two components are strongly coupled. In

addition, we demonstrate for the first time that the system is secularly unstable (with a growth time that

scales with the mutual friction) throughout much of parameter space. Interestingly, large scale r-modes are

also affected by this new aspect of the instability. We analyze the damping effect of shear viscosity, which

should be particularly efficient at small scales, arguing that it will not be sufficient to completely suppress

the instability in astrophysical systems.

DOI: 10.1103/PhysRevD.87.063007 PACS numbers: 97.60.Jd, 04.30.Db, 95.30.Qd

I. INTRODUCTION

Since the first observation of a glitch in the spin-down ofthe Vela pulsar in 1969 [1,2], there have been a largenumber of similar events observed in other pulsars [3,4].Several competing, in some cases complimentary, mecha-nisms have been suggested as explanation for these occur-rences [5–9]. The two most widely discussed classes ofmodels relate either to changes in the star’s elastic crust orthe dynamics of the superfluid neutrons that are presentboth in the core and the inner crust. The first set of modelsinvolves fractures in the crust, leading to changes in themoment of inertia of the star [10]. The second set involvesthe unpinning of vortices associated with a superfluidcomponent from the star’s inner crust [11] (although it isworth noting recent evidence that this model may needrevision [12,13]). The unpinning event is attributed to thebuild up of ‘‘differential rotation’’ between the two com-ponents (elastic crust and interpenetrating superfluid). It isthe second of these two mechanisms with which we con-cern ourselves in this paper.

We consider a promising mechanism for triggering theobserved events; a superfluid instability present in systemswith ‘‘differential rotation,’’ like the lag that plays a centralrole in all vortex-based glitch models. The basic mecha-nism has already been discussed in [14], where it wasargued that the instability may trigger large-scale vortexunpinning leading to the observed events. The first part ofthe argument was a demonstration that r-modes might bedriven unstable by the difference in rotation of the chargedparticles (protons and electrons) and the superfluid

neutrons in the star. In particular, it was shown in [14]that such unstable modes exist for a strongly coupledsystem. The second part of the argument consisted of ademonstration that viscosity would not suppress this insta-bility completely, but would allow growing modes to existfor a range of wavelengths once a critical rotational lag wasreached. The third part of the argument placed the mecha-nism in an astrophysical context by comparing the predic-tions of the model to observational data.In this paper we consider this new r-mode instability in

more detail. We extend the analysis of [14] beyond thestrong-coupling limit, and show that dynamically unstablemodes (with a growth time similar to the star’s rotationrate) exist for a wide range of parameter values. In addi-tion, we discover that the r-modes also suffer a secularinstability (with growth time proportional to the mutualfriction) throughout much of parameter space. This aspectis new. It is particularly interesting as it affects also theglobal scale r-modes, while the dynamical instability isrestricted to small scale modes. As in the previous work,we limit ourselves to discussion of the nonrelativistic case.Again following the approach of [14], we introduce shearviscosity and demonstrate that the instability is not com-pletely suppressed by the inclusion of the associateddamping.The main purpose of this investigation is to establish the

robustness of the superfluid instability, highlight the keyingredients that lead to its presence and set the stage formore detailed studies of this mechanism. The instabilitythat we consider may be generic (belonging to the class oftwo-stream instabilities that are known to operate in a wide

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variety of physical systems [15]), but at this point it is notclear how it is affected by other aspects of neutron starphysics. In particular, future work needs to extend theanalysis to account for the crust elasticity and the presenceof the star’s magnetic field.

II. SETTING THE STAGE

We consider the conditions that are expected to prevailin the outer core of a mature neutron star, where a neutronsuperfluid is thought to coexist with a proton superconduc-tor. A key qualitative aspect of this system is that it allowsthe two ‘‘fluids’’ to flow ‘‘independently,’’ leading todynamics that is well represented by a two-fluid model.In order to keep the analysis manageable, we assume thatboth fluid components are charge neutral. This allows us toneglect (in this first proof-of-principle analysis) thedynamics of the star’s magnetic field. This is obviously asevere simplification, but to analyze the full problemwould be difficult at this point. The key aspect that weadd to previous studies in this problem area is the relativerotation between the two components expected to build upas the observed component spins down due to brakingassociated with the exterior magnetic field. We are inter-ested in the global oscillations of a star that exhibits thiskind of ‘‘differential’’ rotation. Despite this being a keyaspect of the generally accepted ‘‘explanation’’ for ob-served pulsar glitches, there have not yet been any detailedstudies of the dynamics of such a system. In this sense, thepresent work provides an important step towards realism.

The main question that motivates us stems from thediscovery that two-stream instabilities are generic in thetwo-fluid model [16]. Given this, it is relevant to askwhether the rotational lag that builds up in a matureneutron star may lead to such instabilities and, if so,what the observational consequences may be. Conversely,given that we do not yet know what the mechanism thattriggers the observed glitches may be, but that it ought to besome kind of ‘‘global’’ instability, it is interesting to askwhether the two-stream instability may be relevant in thiscontext.

We consider the linear perturbations of a system thatexhibits a rotational lag. Because of this setup, it is naturalto work in the inertial frame (rather than choosing one ofthe rotating frames). It is also natural, given the complexnature of the perturbed velocity fields etcetera to carry outthe analysis in a coordinate basis. This means that vectorslike the velocity are expressed in terms of their compo-nents, vi (say), and a distinction is made between covariantand contravariant objects, with the former following fromthe latter as vi ¼ gijv

j where gij is the flat three-

dimensional metric. This description of the problem isalso advantageous since it involves the use of the covariantderivative ri associated with the given metric, whichautomatically encodes the scale factors associated with

the curvilinear coordinates ðr; �; ’Þ that are appropriatefor the problem.We consider a system with two interpenetrating fluids

labeled n and p (from now on), which are assumed to rotaterigidly in such a way that

vix ¼ �xe

i’; x ¼ n; p (1)

where ei’ represents the azimuthal basis vector (and ei�later the polar one). The angular velocities�x are assumedto be constant, but we allow for differential rotation, i.e.�i

n � �ip. To keep the analysis tractable, we assume that

the two rotation axes coincide, and use

�ip ¼ �i and �i

n ¼ ð1þ�Þ�i: (2)

Making contact with pulsar observations, �i would be theobserved (angular) rotation frequency (i.e. that of the crust)while �i

n represents the rotation of the unseen (interior)superfluid component. As the superfluid is expected to lagbehind as the crust spins down, one would typically expect� to be small (of the order of 10�4 at the time of a glitch[3]) and positive.Key to understanding the dynamics of glitching pulsars

is the appreciation that the neutrons may form Cooperpairs, and hence act as a superfluid. The upshot of this isthat bulk rotation can only be achieved by the formation ofan array of quantized vortices. As discussed in, forexample, [17] the quantization condition is imposed onthe momentum that is conjugate to the velocity vi

n.This momentum is given by

pni ¼ mpðvn

i þ "nwpni Þ (3)

where mp is the nucleon mass (we ignore the small differ-

ence between the bare neutron and proton masses), "nrepresents the entrainment between neutrons and protons(more of which later) and the relative velocity is

wiyx ¼ vi

y � vix; y � x: (4)

Given this, and the relevant quantization condition (see[17] for discussion), the local vortex density (per unit area)nv follows from

nv�i¼�ijkrj½vn

kþ"nwpnk Þ�¼2½�i

nþ"nð�ip��i

nÞ�¼2½1þð1�"nÞ���i (5)

where �i is the vector aligned with the vortex array withmagnitude � ¼ h=2mp (h is Planck’s constant). Here we

have introduced yet another simplifying assumption; wehave taken the fluids to be incompressible which impliesthat the entrainment coefficient may (at least in the small�case that we are focusing on) be taken to be constant. It isalso worth noting that �n"n ¼ �p"p where �x ¼ mpnx are

the respective mass densities.Let us now consider the linear perturbations (repre-

sented by �) of this kind of configuration. Assuming thatthe flow is incompressible also at this level (incidentally

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not a bad approximation for the inertial flows that we willconsider) we have, first of all,

ri�vix ¼ 0: (6)

Meanwhile, the perturbed momenta

�pxi ¼ �vx

i þ "x�wyxi (7)

are governed by the Euler equations

Exi ¼ i!�px

i þ �vjxrjp

xi þ vj

xrj�pxi

þ "xð�wyxj riv

jx þ wyx

j ri�vjxÞ þ ri��x

¼ �fxi (8)

where

�x ¼ �þ ~�x (9)

combines the gravitational potential � and the chemicalpotential ~�x ¼ �x=mp for each fluid. We have assumed

that the time dependence is harmonic, / exp ði!tÞ, as weare interested in the oscillation modes of the system. Later,we will make extensive use of the equations that govern theperturbed vorticity. Specifically, we will work with thequantity

W ix ¼ �ijkrjEx

k: (10)

The right-hand side of (8) can be used to account for any‘‘external’’ forces acting on the fluids. It can also be used toincorporate interactions between them. The main suchinteraction force is the mutual friction that arises due tothe presence of the quantized vortices [18,19]. The unper-turbed force takes the form [20]

fxi ¼�n

�x

B0nv�ijk�jwkxy þ �n

�x

Bnv�ijk�j�klm�lw

xym (11)

where B and B0 are coefficients to be determined frommicrophysics. In the standard scenario these coefficientscan be expressed in terms of a single resistivity R asso-ciated with scattering off of the vortices, such that [20]

B ¼ R1þR2

(12)

and

B 0 ¼ R2

1þR2: (13)

The weak- and strong-coupling limits discussed later cor-respond to, respectively, R ! 0 and R ! 1. It is worthnoting that the system can build up a differential rotationlag only as long as some additional force (like vortexpinning) prevents the mutual friction from acting. Thepresence of such a force is implicitly assumed in thefollowing discussion.

Perturbing (11), we get

�fxi ¼�n

�x

B0½�ðnv�jÞ�ijkwkxy þ nv�

j�ijk�wkxy�

þ �n

�x

B�ijk�klm½�ðnv�lÞ�jw

xym

þ nv�lðwxym ��j þ �j�wxy

m Þ�: (14)

To evaluate this we need

�ðnv�iÞ ¼ �ijkrj�pnk (15)

and

��i ¼ 1

nv�ð�i

j � �i�j�jlmrl�pnm (16)

where �i is a unit vector aligned with �i, together with,cf. (5),

nv� ¼ 2½1þ ð1� "nÞ���: (17)

In the perturbation equations discussed in the next section,we will not work with the momentum equations (8) di-rectly. Rather, we use two combinations that represent thetotal (perturbed) momentum and the difference. It is wellestablished that these combinations isolate the two dy-namical degrees of freedom in an ‘‘uncoupled’’ two-fluidsystem [21]. Hence, this decomposition is often used instudies of oscillating superfluid neutron stars [22]. Thismeans that we need

�n�fni þ �p�f

pi ¼ 0 (18)

and

�fpi � �fni ¼ � 1

xp�fni (19)

where we have introduced the proton fraction xp ¼�p=ð�n þ �pÞ. Only the second degree of freedom is

explicitly affected by the mutual friction.Finally, we want to account for the presence of shear

viscosity (in the superfluid case mainly due to electron-electron scattering). In the incompressible case, this meansthat we add a force to (the right-hand side of) the protonequation of form

�fisv ¼ �eer2�vip (20)

where �ee is the kinematic viscosity coefficient. We ignorebulk viscosity for two reasons. First of all, glitching pulsarsare cold enough that shear viscosity should be the domi-nant damping mechanism. Secondly, the particular class offluid motion that we consider is not efficiently damped bybulk viscosity [23].

III. THE AXIAL PERTURBATION EQUATIONS

Despite the various simplifying assumptions, the generalperturbation problem is challenging. It is well known that,

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even in the slow-rotation limit where the star remainsspherical, there exists a class of inertial modes [24] whichrequire the coupling of many (spherical harmonics) multi-poles for their description. We are, however, not going toattempt to solve the general problem. Instead we will ask avery specific question. How does the presence of the rota-tional lag affect the r-modes of the system? This question isrelevant for a number of reasons, perhaps the most impor-tant being related to the fact that the large scale r-modesmay be driven unstable by the emission of gravitationalradiation [23]. From a practical point of view, it is naturalto focus on the r-modes since they are associated withparticularly simple velocity fields. The hope would bethat the corresponding problem remains tractable evenwhen we add the rotational lag.

The r-modes are a subclass of inertial modes (restoredby the Coriolis force) that are purely axial/toroidal toleading order. This means that the perturbed velocitiestake the form

�vix ¼ � im

r2 sin �Ul

xYml e

i� þ

1

r2 sin�Ul

x@�Yml e

i’ (21)

where Yml ð�; ’Þ are the standard spherical harmonics,

UlxðrÞ are the mode amplitudes, and the symmetry of the

problem is such that the m multipoles (proportional toeim’) decouple. In the single (barotropic) fluid case, r-mode solutions exist for each individual l ¼ m � 1. Themain focus in previous work has been on the quadrupolemode (l ¼ 2) since it is associated with the fastest growinggravitational-wave instability [23]. In this work we willend up studying a large range of scales, including smallscale modes with l the order of 100. In a neutron star, suchperturbations would be relatively local, corresponding to atypical length scale of 100 m.

As hinted at in the previous section, we prefer to workwith slightly different perturbation variables. Experiencefrom other problems involving the two-fluid model [22]suggests that it is advantageous to work with the totalperturbed momentum flux defined as

�Ul ¼ �nUln þ �pU

lp (22)

with � ¼ �n þ �p. We take the second variable to be given

by the velocity difference

ul ¼ Ulp �Ul

n: (23)

In order to ‘‘simplify’’ the final perturbation equations it isuseful to express the frequency in the ‘‘rotating frame,’’i.e. work with � defined by

!þm� ¼ ��: (24)

It is also convenient to introduce

L ¼ lðlþ 1Þ; (25)

Z ¼ 1� xp � "p; (26)

and

�Z ¼ xpZ

1� xp: (27)

Finally, we use (as in [22]) the scaled mutual frictioncoefficients

�B 0 ¼ B0=xp and �B ¼ B=xp: (28)

The perturbation equations can now be obtained byinserting the expected velocity field in the equations fromthe previous section. After some manipulations this leadsto the following set of the equations to be solved: From�nW r

n þ �pW rp, we first of all get

½L�� 2m�UlYml ¼ �m�ðL� 2Þ½Ul � �Zul�Ym

l : (29)

(Here, and in the following, summation over l � m isimplied.) The advantage of working with this combinationthat represents to total vorticity, is that there are no mutualfriction terms in the equation. Such terms are associatedwith the relative flow. To see this we consider the combi-nation W r

p �W rn leading to

�L� �Z

xp� 2mð1� �B0Þ � 2i �BðL�m2Þ

�ulYm

l

¼ �m�

�½ðL� 4Þxp þ 2� �Zxp

� 2ð1� xpÞ�ulYm

l þm�

�L� 2 �Z

xp

�UlYm

l þm�L �B0ð �Zul �UlÞYml

� 2m �B0�ð �Zþ 1� xpÞulYml þ i �B�L½ �Zðr@rul � ulÞ � r@rU

l þUl�Yml þ 2i �B�ðL�m2Þð �Zþ 1� xpÞulYm

l

þ i �BL�½r@rUl � 3Ul � �Zðr@rul � 3ulÞ�cos 2�Yml þ i �B�½2r@rUl � LUl � �Zð2r@rul � LulÞ� cos � sin �@�Ym

l :

(30)

We will also use the radial components of the Euler equations. From the combination �nErn þ �pEr

p we find

½xpr@r��lp þ ð1� xpÞr@r��l

n�Yml � 2�Ul sin �@�Y

ml ¼ ���ð1� xpÞ½ðxp � �ZÞr@rul þ 2 �Zul � 2Ul� sin �@�Ym

l ; (31)

while Erp � Er

n leads to

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ðr@r��lp � r@r��

lnÞYm

l � 2�ð1� �B0Þul sin�@�Yml � 2im �B�ul cos�Ym

l

¼���

��1� �Z

xp

�r@rU

l þ 2 �Z

xpUl þ ð1� 2xpÞ

��1� �Z

xp

�r@ru

l þ 2 �Z

xpul�� 2ð1� xpÞul

�sin�@�Y

ml

þ �B0��½r@rUl � �Zr@rul � 2ð1� xp þ �ZÞul� sin�@�Ym

l þ im �B��½r@rUl � �Zr@rul þ 2ð1� xp þ �ZÞul� cos�Ym

l :

(32)

Finally, it is convenient to work with a ‘‘divergence equation’’ [22] that follows from the combination

sin�@�½sin �ð�nE�n þ �pE�

pÞ� þ @’½�nE’n þ �pE

’p � ! �L½xp��l

p þ ð1� xpÞ��ln�Ym

l þ 2�Ul½L cos �Yml þ sin �@�Y

ml �

¼ ���ð1� xpÞf2Ul½L cos�Yml þ sin �@�Y

ml �

� Lxpul½2 cos �Ym

l þ sin�@�Yml � þ ðL� 2Þ �Zul sin �@�Ym

l g: (33)

Analogously, we consider the ‘‘difference’’ equation

sin �@�½sin �ðE�p � E�

nÞ� þ @’½E’p � E’

n �! Lð��n � ��pÞYm

l þ 2�ð1� �B0Þul½L cos �Yml þ sin�@�Y

ml � þ 2im� �Bul½2 cos�Ym

l þ sin �@�Yml �

¼ ��LðUl � xpulÞ½2 cos�Ym

l þ sin �@�Yml � � ��

�ðL� 2Þ �Zxp

ðUl þ ulÞ � ðL� 2Þð1� xp þ 2 �ZÞul�sin �@�Y

ml

þ�� �B0f�LUl þ ½2ðZþ xpÞ � ð1� xpÞ�Lulg½2 cos �Yml þ sin�@�Y

ml �

��� �B0ðL� 2ÞðZþ xpÞul sin �@�Yml � 2im� �B�ðxp þ ZÞul½2 cos �Ym

l þ sin�@�Yml �

� im� �B�½2r@rUl þ LUl þ ð1� 2xp � ZÞð2r@rul þ LulÞ� cos�Yml : (34)

Next, we separate the l multipoles by means of thestandard recurrence relations

cos�Yml ¼ Qlþ1Y

mlþ1 þQlY

ml�1 (35)

and

sin �@�Yml ¼ lQlþ1Y

mlþ1 � ðlþ 1ÞQlY

ml�1 (36)

where

Q2l ¼

ðl�mÞðlþmÞð2l� 1Þð2lþ 1Þ : (37)

These relations lead to

cos 2�Yml ¼ ðQ2

lþ1 þQ2l ÞYm

l þQlþ1Qlþ2Ymlþ2

þQlQl�1Yml�2 (38)

and

cos� sin�@�Yml ¼ ½lQ2

lþ1 � ðlþ 1ÞQ2l �Ym

l

þ lQlþ1Qlþ2Ymlþ2 � ðlþ 1ÞQlQl�1Y

ml�2:

(39)

It should be quite clear at this point that the problem weconsider is rather complex, even for purely axial modes.One must also be careful, because it is not clear from theoutset that modes of this particular character exist (as, inprinciple, rotation would couple the axial/toroidal degree

of freedom to the polar/spheroidal one [24]). However, inthe particular case that we are considering the problemsimplifies in an almost miraculous fashion. We do notexpect this level of simplification in a more general situ-ation, leaving the problem exceedingly difficult.

IV. THE UNSTABLE R-MODES

The r-modes are very special members of the generalclass of inertial modes because their eigenfunctions‘‘truncate’’ at l ¼ m (at least in the standard single-fluidsetting) making their eigenfunctions particularly simple(generic inertial modes involve coupling a number ofl � m multipoles [24]). Inspired by the fact that thisremains true also for corotating superfluids [22], it makessense to ask whether it may be the case when a rotationallag is present as well. Somewhat to our surprise, it turnsout that such simple r-mode solutions do, indeed, exist.Moreover, we find that these modes may become unstableonce the vortex-mediated mutual friction is accounted for.Assuming that the perturbed velocity fields take the

same form as in the corotating case, i.e.

Ulx ¼

�Axr

mþ1 for l ¼ m

0 for l � m(40)

and noting that Ql¼m ¼ 0, the equations from the previoussection collapse to two scalar relations for the amplitudes.

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Expressing these in terms of Um and um, we have

½ðmþ 1Þ�� 2þ �ð1� xpÞðm� 1Þðmþ 2Þ�Um

� ð1� xp � "pÞ�xpðm� 1Þðmþ 2Þum ¼ 0 (41)

and

� ½ðm� 1Þðmþ 2Þ þ 2 �"�mðmþ 1Þð �B0 þ i �BÞ��Um

þ fð1� �"Þðmþ 1Þ�� 2ð1� �B0 þ i �BÞþ�xpðm� 1Þðmþ 2Þ � �"�f½mðmþ 1Þ � 4�xp þ 2g�mðmþ 1Þ�xpð1� �"Þð �B0 þ i �BÞþ 2ð1� "nÞð �B0 � i �BÞ�gum ¼ 0 (42)

where we have used �" ¼ "n=xp ¼ "p=ð1� xpÞ. In the

present case, where all the coefficients are taken to beconstant, it is easy to see that the problem reduces to aquadratic for �. The general solution to this quadratic is,however, rather messy and not particularly instructive. Inorder to understand the nature of the solutions, it is better tofocus on simplified cases.

A. Dynamical instability

Let us first consider the case of vanishing entrainment.In this case we have

ðmþ 1Þ2�2 þ P1�þ P0 ¼ 0 (43)

with

P0 ¼ 2ðm� 1Þðmþ 2Þð �B0 � i �BÞ�2

þ 4½1� ð1þ xpÞð �B0 � i �BÞ�þ 2�½ðm� 1Þðmþ 2Þðxp �B0 � 1Þþ ð �B0 � i �BÞðm2 þm� 4Þ þ ixp

�Bðm2 þmþ 2Þ�(44)

and

P1 ¼ ðmþ 1Þf�4þ 2ð1þ xpÞð �B0 � i �BÞþ�½2ð �B0 � i �BÞ þ ðm� 1Þðmþ 2Þð1� xp

�B0Þ� ixp

�Bðm2 þmþ 2Þ�g: (45)

The explicit solutions are, of course, still not transparent.However, if we consider the limit of strong mutual frictioncoupling (R ! 1), such that B � 0 and B0 � 1, then weobtain the roots

� ¼ � 1

ðmþ 1Þxp ½1� xp þ ��D1=2� (46)

with

D ¼ ð1þ xpÞ2 þ 2�f1þ xp½3�mðmþ 1Þ�g: (47)

These solutions highlight one of the main new results inthis paper (and [14]): For m � 1 we have

D � ð1þ xpÞ2 � 2xpm2�; (48)

showing that we have unstable r-modes (Im�< 0) for

m * mc; where mc ¼1þ xpffiffiffiffiffiffiffiffiffiffiffi2xp�

q : (49)

Expressed in terms of the (observed) rotation periodP ¼ 2=� of the system, the growth time scale for theseunstable modes is

grow � mP

2

�xp

1þ xp

��m2

m2c

� 1

��1=2: (50)

For m � mc (in practice, m * 2mc), the growth rate iswell approximated by

grow � P

2

�xp

2�

�1=2 ) grow � 3

�xp

0:05

�1=2

�10�4

�

�1=2

P:

(51)

With the same scaling, the critical multipole beyond whichthe instability is present is

mc � 300

�xp0:05

��1=2�

�

10�4

��1=2: (52)

The presence of this critical value and the associatedemergence of unstable modes is illustrated in Fig. 1.We see that the instability sets in as two r-modes mergeat a critical scale represented by mc. This is the character-istic behavior of a dynamical instability in a nondissipativesystem; two real-valued modes merge to form a complexconjugate pair of solutions along a one parameter sequenceof models [25].Accounting for the entrainment obviously complicates

the analysis. However, the result remains transparent in thestrong-coupling limit. Again setting B ¼ 0 and B0 ¼ 1,we find the roots

� ¼ �

ðmþ 1Þxp ½�ð1þ "nÞ þ ð1� "nÞðxp � �Þ �D1=2�(53)

with

� ¼ ð1� "n � "pÞ�1 (54)

and

D ¼�1þ xp

�

�2 þ 2ð1� "nÞ

�1� ðm2 þm� 3Þ xp

�

��

þ ð1� "nÞ2�1� 2xp

�ðmþ 2Þðm� 1Þ

��2: (55)

For m � 1 and � � 1 this is approximately

D � ð�þ xpÞ2�2

�1�m2

m2c

�; where m2

c ¼ð�þ xpÞ2

2xp��ð1� "nÞ :(56)

N. ANDERSSON, K. GLAMPEDAKIS, AND M. HOGG PHYSICAL REVIEW D 87, 063007 (2013)

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We see that in this case there are unstable modes withgrowth time:

grow � mP

2

�xp

�þ xp

��m2

m2c

� 1

��1=2: (57)

For m � mc this reduces to

grow � P

2

�xp

2�

�1=2

�1=2? (58)

where we have introduced

�? ¼ 1� "n � "p1� "n

� m�p

mp

: (59)

Given that the effective proton mass, m�p, in a neutron star

core is expected to be in the range m�p=mp � 0:5–0:9 [26]

we conclude that entrainment has a minor impact on thegrowth time scale. An illustration of this result is providedin Fig. 1.In summary, these results show that for the typical

magnitude of rotational lag inferred from radio pulsarglitches [3], we would have unstable modes with a char-acteristic horizontal length scale of 10’s to 100’s of meters.Smaller scale r-modes would be unstable, with a growthtime as fast as a few rotation periods. As the unstablemodes grow extremely fast compared to the evolutionarytime scale (spin-down, cooling, etc.) of the system it seemsreasonable to expect that they may affect any real systemthat develops the required lag, �. It is worth noting that thepredicted growth time is much shorter than the currentobservational constraint for the rise of pulsar glitches(10’s of seconds [27]), unless the star is slowly rotating.Hence, the instability could grow fast enough to serve astrigger for the observed events.

B. Secular instability

Having established the existence of an instability in thestrong-coupling limit, let us consider the problem for less‘‘extreme’’ parameters. It is, of course, straightforward tosolve (43) for given parameter values. A sample of resultsobtained by considering the problem for fixed R, leadingto the mode frequencies depending on m, are provided inFigs. 2–4. These graphs show the behavior of the r-modesolutions as R decreases from 100 to 2, i.e. as we move

10 100m

-0.4

-0.2

0

0.2

0.4

εp = 0

εp = 0.6

FIG. 1. Real (solid lines) and imaginary parts (dashed) of theroots to the dispersion relation (the r-modes) in the strong-coupling limit, R ¼ 103, for two illustrative cases, when"p ¼ 0 (thin lines) and "p ¼ 0:6 (thick lines). The rotational

lag is fixed to � ¼ 5 10�4. In each case, the presence of acritical value for the azimuthal index mc, beyond which themodes are unstable (the roots are complex) is apparent. Theresults also demonstrate how the entrainment affects the range ofthe instability by shifting mc. Meanwhile, the growth rate of theinstability (the magnitude of the imaginary part in the unstableregime) is not affected much.

50 100 150 200m

-0.5

-0.25

0

0.25

0.5

Re

σ

50 100 150 200m

-10

-5

0

log

| Im

σ|

mc

mc

FIG. 2. Real (left panel) and imaginary parts (right panel) of the r-modes forR ¼ 100, xp ¼ 0:1 and a rotational lag � ¼ 5 10�4.Beyond a critical value for the azimuthal index,mc, the modes are dynamically unstable. The onset of this instability is associated with(near) merger of the real part of the two frequencies. The absence of sign change of the imaginary part of the unstable branch indicatesthe presence of a secular instability for smaller values of m.

SUPERFLUID INSTABILITY OF r-MODES IN . . . PHYSICAL REVIEW D 87, 063007 (2013)

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away from the strong-coupling regime. The results are veryinteresting. First of all, we see that the general trend of a‘‘dynamical’’ instability (associated with modes with amarkedly larger imaginary part to their frequency) settingin near the critical value mc remains down to R ¼ 10 orso. For smaller values, e.g. R ¼ 2 as in Fig. 4, there is nolonger a clear change in the imaginary part near mc.We also see that, away from the extreme strong-couplinglimit, the dissipative aspect of the mutual friction becomesimportant. In particular, it leads to the dynamical instabil-ity no longer being associated with exact mode mergers.Rather, the instability sets in at ‘‘near misses’’ in thecomplex frequency plane. This is probably what shouldbe expected. Another aspect of the results was notexpected. Considering the imaginary parts of the roots inmore detail (the right-hand panels of Figs. 2–4) we see thatneither imaginary part changes sign in the displayed inter-val (as we show log jIm�j a sign change would show up asa sharp singularity). This demonstrates the most interestingnew result in this paper. As we know that one of the modesis unstable beyond mc we must conclude from Figs. 2–4that this mode is, in fact, unstable for all lower values of mas well. Of course, for smaller m the growth time of theunstable modes is much longer. Later we will demonstrate

that it is linked to the mutual friction parameters, makingthis a secular instability (plausibly related to the Donnelly-Glaberson vortex instability in laboratory superfluids[28–30]). Particularly interesting may be the fact thatthis instability is not restricted to the small scaler-modes. It is also active for the large scale modes. Thisis interesting since these modes, especially the m ¼ 2r-mode, are also secularly unstable due to gravitational-wave emission. Basically, our results show that the mutualfriction may not provide damping of these modes. Rather,it could provide an additional driving mechanism for theinstability. It may also be, given the strong scaling of thegravitational radiation reaction with the modes oscillationfrequency (essentially the star’s spin rate), that the mutualfriction driven instability dominates for slowly rotatingsystems. We will return to this question later.Let us first see if we can find approximate solutions that

demonstrate the behavior seen in the numerical results.Returning to (41) and (42) we see that the two degreesof freedom (represented by Um and um) only couple atorder �. Since we are interested in mode solutions forsmall�, this suggests that the solutions are either predomi-nantly comoving or countermoving, depending on whetherUm dominates over um, or vice versa. Considering first the

50 100 150 200m

-0.5

-0.25

0

0.25

0.5

Re

σ

50 100 150 200m

-10

-5

0

log

| Im

σ|

mcm

c

FIG. 3. Same as Fig. 2 but for R ¼ 10.

50 100 150 200m

-0.5

-0.25

0

0.25

0.5

Re

σ

50 100 150 200m

-10

-5

0

log

| Im

σ|

mc

mc

FIG. 4. Same as Fig. 2 but for R ¼ 2.

N. ANDERSSON, K. GLAMPEDAKIS, AND M. HOGG PHYSICAL REVIEW D 87, 063007 (2013)

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comoving case, we assume that Um � 0 and um � 0,which leads to

� ¼ 1

mþ 1½2��ð1� xpÞðm� 1Þðmþ 2Þ� �0: (60)

At this level, the frequency is real and the modes are stable.Meanwhile, in the countermoving case we have modes

with frequency

�¼ 1

ð1� �"Þðmþ1Þf2ð1��B0 þ i �BÞ��xpðm�1Þðmþ2Þ

þ �"�f½mðmþ1Þ�4�xpþ2gþmðmþ1Þ�xpð1� �"Þð �B0 þ i �BÞ�2ð1�"nÞð �B0 � i �BÞ�g: (61)

From this we see that the imaginary part of the frequency isproportional to

2þmðmþ 1Þ�xpð1� �"Þ þ 2ð1� "nÞ�;which suggests that the countermoving modes tend to bestable, at least in the nonentrainment case. When the en-trainment is accounted for, these modes may become un-stable. In the short length scale limit, when m2xp� � 1

(corresponding to m � 103 or so for typical parameters),an instability is present as long as "n > xp, which is not a

particularly severe criterion.Let us now consider the corotating modes at the next

level of approximation. We have already seen from (60)that, to leading order in �, these modes are marginallystable, with frequency approximated by �0. Yet, we knowfrom the numerical results that an instability should bepresent for some range of R, certainly R � 2. Hence,we need to estimate the solution at the next order in �. Todo this, we take

� ¼ �0 þ �2�2 (62)

which leads to

�2 ¼ ðm� 1Þðmþ 2Þ2ðmþ 1Þ

xpð1� xpÞð �B0 � i �BÞ ½ðm� 1Þðmþ 2Þ

�mðmþ 1Þð �B0 þ i �BÞ�: (63)

We are (primarily) interested in the imaginary part of thefrequency. Since we know that an instability exists in thestrong-coupling limit we note that this limit corresponds to�B0 � 1=xp and hence we have (to order �B)

Im�2 ¼ �ðm� 1Þðmþ 2Þ2ðmþ 1Þ xpð1� xpÞ

½mðmþ 1Þð2� xpÞ þ 2xp� �B: (64)

We learn that these modes are unstable for all values of �.

We also see that the growth rate scales as 1= �B making thisa secular instability. The estimate (64) establishes thepresence of the instability for a wide range of parameters,and provides more detailed insight into the dependence onthe parameters.It is obviously relevant to establish what the critical

value of R may be. We can get an idea of this by workingout where the imaginary part of (63) changes sign. Thisleads to

B 0c ¼ R2

c

1þR2c

¼ ðm� 1Þðmþ 2Þ2mðmþ 1Þ xp: (65)

ForR � Rc the system should be stable. It is easy to showthat this is the case by considering the weak-coupling limit.

Then we have �B0 � �B2 so the dominant behavior will be

Im�2 ¼ ðm� 1Þ2ðmþ 2Þ22ðmþ 1Þ xpð1� xpÞ 1�B : (66)

As expected, the modes are always stable in this limit. Thisis, of course, as expected.Returning to the numerical results, the typical behavior

for Rc <R< 1 is similar to that shown in Fig. 5 (whichcorresponds to R ¼ 0:5). That is, for a given value of Rthe instability is present for a range of multipoles, up to a

50 100 150 200m

-3

-2

-1

0

1

2

Re

σ

50 100 150 200m

-10

-5

0

log

| Im

σ|

50 100 150 200-0.002

00.0020.0040.0060.008

FIG. 5. Same as Fig. 2 but forR ¼ 0:5. The inset in the right panel shows the detailed behavior of Im� as function ofm. The dashedcurve in the right-hand panel corresponds to the approximate solution (64), which is an accurate representation for low multipoles.

SUPERFLUID INSTABILITY OF r-MODES IN . . . PHYSICAL REVIEW D 87, 063007 (2013)

063007-9

critical value where the modes become stable. AboveR ¼ 1 all modes are unstable.

We get a complementary view of the new instability by

fixing the multipole m and varying R. This leads to the

results shown in Figs. 6 and 7, for m ¼ 2 and m ¼ 100,respectively. These results demonstrate that the instability

exists throughout the R � 1 regime, and that it extends

into the ‘‘weak-coupling’’ regime as well. This is yet

another demonstration of the generic nature of the super-

fluid r-mode instability. It is notable that the approximation

(64) is excellent for low values of m, see the right-hand

panel of Fig. 6, but deteriorates as m increases. This is not

surprising, since for large values of m one cannot simply

neglect higher order terms in � as these may be multiplied

by factors of m. In essence, the comoving and counter-

moving degrees of freedom are no longer neatly decoupled

on shorter (angular) scales. Nevertheless, the approxima-

tion serves its purpose by illustrating the behavior in an

important part of parameter space.

V. ASTROPHYSICAL CONTEXT: ACCOUNTINGFOR SHEAR VISCOSITY

As suggested in [14], the superfluid instability discussedin the previous section could be relevant for astrophysicalneutron stars, in particular glitching pulsars. A‘‘minimum’’ requirement for this to be the case is thatthe instability grows fast enough to overcome the dissipa-tive action of viscosity in neutron star matter.A mature neutron star core is sufficiently cold to contain

both superfluid neutrons and protons (recent evidence sug-gests that this is the case for a core temperature T & 5–9108 K [31,32]). Under these conditions the fluid motion isprimarily damped by vortex mutual friction (which we havealready accounted for, and which drives the instability weare considering) and shear viscosity due to electron-electroncollisions [33]. Drawing on the considerable amount ofwork that has been done on the gravitational-wave drivenr-mode instability [23], we can estimate the shear viscositydamping time scale using an energy-integral approach

0.01 0.1 1 10 100R

-8

-6

-4

-2

0

Re

σ

0.01 0.1 1 10 100R

-10

-5

0

log

| Im

σ|

0.005 0.01 0.015 0.020.663

0.664

0.665

0.666

0.6670.5 1 1.5 2

-1e-07

0

1e-07

2e-07

FIG. 6. Real (left panel) and imaginary parts (right panel) of the r-modes for m ¼ 2, xp ¼ 0:1 and a rotational lag � ¼ 5 10�4.Unstable modes are present beyond a critical value of R, see inset in the right panel. We compare the obtained imaginary part for themodes that become unstable to the estimate (64) (dashed curve in the right-hand panel). In this case, this is clearly a very goodapproximation. The inset in the left panel shows that the real parts of the mode frequencies cross at a low value of R, seeminglyunrelated to the onset of instability.

R

-0.15

-0.1

-0.05

0

0.05

Re

σ

0.01 0.1 1 10 100 0.01 0.1 1 10 100R

-7

-6

-5

-4

-3

-2

-1

0

log

| Im

σ|

0 0.5 1

-0.005

0

0.005

FIG. 7. The same as Fig. 6 but for m ¼ 100. In this case, (64) no longer provides a good approximation to the imaginary part.

N. ANDERSSON, K. GLAMPEDAKIS, AND M. HOGG PHYSICAL REVIEW D 87, 063007 (2013)

063007-10

sv ¼ 2Emode

_Esv

(67)

where Emode is the mode energy and _Esv is the shear vis-cosity damping rate. The damping time scale (67) can beeasily calculated using the two-fluid r-mode results of theprevious sections. However, as we are only interested in arough estimate we use the result for ordinary single-fluidr-modes in a uniform density star [34] to approximate sv.This leads to

sv ¼ 3

4

M

�eeR

1

ð2mþ 3Þðm� 1Þ

� 1:3 106

m2

�0:05

xp

�3=2 R7=2

6

M1=21:4

T28 s (68)

where R6 ¼ R=106 cm and M1:4 ¼ M=1:4M representthe radius and mass of the star, respectively. The relevantshear viscosity coefficient (�ee ¼ �p�ee) has been taken to

be [33]

�ee ¼ 1:5 1019�xp0:05

�3=2

�3=214 T�2

8 g cm�1 s�1 (69)

where �14 ¼ �=1014 g cm�3 and T8 ¼ T=108 K.We are now in a position where we can compare the

growth time scale, grow, to that due to shear-viscosity

damping, sv. Such a comparison is provided in Fig. 8for modes exhibiting the dynamical instability (in thestrong-coupling regime) and typical neutron-star parame-ters. The time scales are shown as functions of the multi-pole m, and we provide results for two choices of theentrainment parameter "p, cf. Fig. 1. The damping rate

sv is shown for two representative core temperaturesT ¼ 107 K and 5 107 K representing mature neutronstars. Dynamically unstable superfluid r-modes exist forthe range of m above the grow curve but below the svcurve. The results show that the grow profile levels off

(as a function ofm) form * mc and that the instability canovercome the viscous damping for a range of scales. Wealso see that the entrainment has a small effect on theasymptotic behavior of grow but can significantly affect

the critical multipole mc. Finally, it is apparent that therange of unstable modes decreases as the star cools. This isan interesting observation since glitches are only seen inrelatively young pulsars. The results in Fig. 8 indicate thatshear viscosity would prevent the instability from devel-oping soon after the star has cooled below 107 K. It isworth keeping in mind that the core temperature mayremain above 108 K for (at least) the first 105 years of apulsar’s life [35].

The predicted spin-lag for the dynamical instability toset in also makes a connection with pulsar glitches seemplausible. Balancing the mode growth and the viscousdamping, i.e., setting grow ¼ sv, we find the critical

spin-lag �c above which the instability is active.Combining (51) and (68) and setting m ¼ mc we obtain

�c � 3:3 10�5

�xp0:05

�2=3

�P

1 s

�2=3

�m�

p

mp

��1=3T�4=38 : (70)

This result was first derived in [14]. As discussed in thatpaper, the predicted critical lag compares well with theavailable data for large pulsar glitches. This could be anindication that large glitches are indeed triggered by thelarge m r-mode instability.The results for the secular instability for smaller values

of m are quite similar. In Fig. 9 we show results both form ¼ 2 and m ¼ 100. In each case we see that unstablemodes will be present above a certain temperature.Keeping in mind that the critical temperature for coresuperfluidity is expected to be below 109 K [31,32], whilea typical glitching pulsar like the Vela is expected to havecore temperature just above 108 K, these results make itseem plausible that the secular instability can becomeactive in these systems. Whether astrophysical systemswith R in the required range for the instability existis not clear at this point. Most work has focused onsmaller values, in which case the system would notexhibit the instability we have discussed here, but therehave been suggestions of larger values [36]. It is alsoworth noting that a stronger mutual friction may help

100 500 1000m

10-2

10-1

100

101

102

103

Tim

esca

les

(s)

τgrow

τsv

εp = 0

εp = 0.6

T = 5x107 K

T = 107 K

FIG. 8 (color online). The dynamical r-mode instabilitygrowth and viscous damping time scales grow and sv [from

Eqs. (57) and (68), respectively] as functions of the multipole mfor typical pulsar parameters: P ¼ 0:1 s, xp ¼ 0:1 and strong

drag R ¼ 104. The stellar mass and radius are fixed at thecanonical values M1:4 ¼ R6 ¼ 1. The shear viscosity dampingrate sv is shown for two core temperatures: T ¼ 107 K andT ¼ 5 107 K. The spin-lag � is fixed to 5 10�4 and weshow results for two values of the entrainment parameter, "p ¼ 0

and "p ¼ 0:6. The grey region indicates the range of unstable

r-modes for T ¼ 107 K.

SUPERFLUID INSTABILITY OF r-MODES IN . . . PHYSICAL REVIEW D 87, 063007 (2013)

063007-11

reconcile the discrepancy between our theoretical under-standing of the r-mode instability with observed astrophys-ical systems [37].

VI. FINAL REMARKS

We have provided a more detailed analysis of the super-fluid r-modes in a simple uniform density model with(solid body) differential rotation between the twocomponents. As already discussed in [14], the results ofthis analysis are interesting, both conceptually andfrom the point of view of astrophysical applications.It is obviously interesting that the vortex-mediated mutualfriction may lead to the presence of an instability.However, this is perhaps not surprising given the wealthof results relating to instabilities triggering superfluid tur-bulence [38,39] and previous results in the context ofpulsar precession [40,41], but the present analysis providesthe first demonstration of this kind of instability for globalmode oscillations. We are also breaking new ground byconsidering perturbations for background configurationswith differential rotation. It is notable that, as soon as weallow for this extra degree of freedom the problem be-comes richer.

A particularly interesting aspect of the results is thepresence of both secular and dynamical instability behav-ior in (more or less) distinct parts of parameter space. Thebehavior that we have unveiled is summarized in the phaseplane in Fig. 10 that shows the unstable region in the m-Rplane. As far as we are aware, the present analysis repre-sents the first detailed study of a problem that has secularlyunstable modes entering a regime where they becomedynamically unstable. The associated behavior is, inmany ways, predictable. For example, instead of havingmodes becoming dynamically unstable as real-valued fre-quency pairs merge and form complex conjugate pairs, wenow have dynamical instability behavior associated with

‘‘near misses’’ in the complex frequency plane. Our resultssuggest that dynamical instability only results providedthat the damping of the modes is not too large (in ourcaseR � 10 or so). To improve our understanding further,we need to consider the necessary and sufficient criteria forthe superfluid instability to operate. This is a challengingproblem but it seems likely that one could make progressby adding the perturbed mutual friction force to the resultsin [42].

0.1 1 10R

105

106

107

108

Tim

esca

le (

s)

5x108K

108K

2x108K

unstable region

0.1 1 10R

1

102

104

Tim

esca

le (

s)

108K

107K

unstable region

FIG. 9. The secular r-mode instability growth and viscous damping time scales grow and sv [obtained from the numerical solution to(43) and (68), respectively] as functions of R for two multipoles: m ¼ 2 (left panel) and m ¼ 100 (right panel). The other parametersare the same as in Fig. 8: P ¼ 0:1 s, xp ¼ 0:1 and M1:4 ¼ R6 ¼ 1. The spin-lag � is fixed to 5 10�4. The shear viscosity damping

rate sv is shown for various temperatures as indicated in each panel (dashed horizontal lines). The r-mode growth time is short enoughfor an instability to be present for a range of R for temperatures characteristic of young neutron stars.

1

10

102

103

104

101

102

103

104

105

106

m

0

0.2

0.4

0.6

0.8

1 R

stable

secular

dynamical instability

instability

mc

Rc

FIG. 10. Summary of the parameter space R vs m indicatingthe different regions of instability for xp ¼ 0:1 and � ¼ 510�4. Dashed curves show the critical multipole mc where thebehavior changes from secular to dynamical instability on thestrong-coupling regime (upper panel), and the estimated criticaldragRc from (65) where the secular instability sets in for low mmultipoles (lower panel).

N. ANDERSSON, K. GLAMPEDAKIS, AND M. HOGG PHYSICAL REVIEW D 87, 063007 (2013)

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An interesting analogous problem concerns the l¼m¼2bar mode in rotating neutron stars. This mode is known tobecome secularly unstable due to the emission of gravita-tional waves well before (along a sequence with increasingdegree of differential rotation) it becomes dynamicallyunstable, see [25] for a review and [43] for recent work.The behavior of that instability may be quite similar to thatof the superfluid problem we discuss here. If this is thecase, then a key question concerns whether there is alwaysa dramatic change in gravitational-wave emission ratebefore and after the critical parameter for onset of thedynamical instability is reached. In the bar-mode problemthere is also another class of instability, commonly referredto as the low T=W instability [44]. We have no evidence foran analogous instability in the present problem.

Even though we do not discuss possible astrophysicalrepercussions in any detail, it is clear that the potentiallink with the unresolved problem of radio pulsar glitchesprovides strong motivation for further work on this problem.Of course, we cannot at this point really tell how relevant thenew r-mode instability is in this context. In order to act in anastrophysical system, the instability must be robust enoughto remain once we account for the star’s magnetic field andthe elastic crust. These features are likely to affect theinstability considerably, but more work is required if wewant to quantify what the effects may be. The two-streaminstability mechanism is sufficiently generic that it would beremarkable if it would cease to operate in more complexsettings, but it could be that the instability threshold movesout of reach for a real system. At this point, we cannot say.There is, however, fresh evidence supporting the existenceof short wavelength superfluid instabilities in the presenceof a magnetic field and/or an elastic crust, see [45,46]. Theexact relation between these instabilities and our r-modeinstability is still unclear, apart from the fact that both

require vortex mutual friction to operate. The connectionbetween the global-mode instability we discuss here and theshort-range vortex instability investigated in [45,46] need tobe clarified by future work. It is also important to establishwhether these instabilities may lead to the onset of super-fluid turbulence, which seems plausible. If this is the case,then one would need to explore the connection with numeri-cal simulations of the kind pioneered in [39]. As the super-fluid instability acts on short scales, there may also be aconnection with the kind of local trigger discussed in [47].Once these issues are better understood, we may be able tostart piecing together a coherent picture of the glitch phe-nomenon. Another issue, which may require numericalsimulations, relates to the nonlinear behavior of the two-stream instability. The effect that the instability may have onan astrophysical system may depend on the unstabler-modes growing to a large amplitude. In the case of thegravitational-wave driven instability of the r-modes, it isknown that the saturation is at a low level due to coupling toa sea of other inertial modes, see [48,49]. Whether the samemechanism operates in a superfluid system is not yet clear.It is a key issue for future work, but unfortunately one thatinvolves severe technical challenges. Finally, the secularinstability behavior that we unveil is obviously also worthyof further attention. In particular, since it is relevant also forlarge scale modes, including the lowest multipoles that arethe most important from the gravitational-wave point ofview. The problem is clearly extremely interesting andwell worth returning to in the future.

ACKNOWLEDGMENTS

N.A. is supported by STFC in the United Kingdom.K.G. is supported by the Ramon y Cajal Programme inSpain.

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