Decoupling of superfluid and normal modes in pulsating neutron stars

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by  M. E. Gusakov

Decoupling of superfluid and normal modes in pulsating neutron stars
Mikhail E. Gusakov1 and Elena M. Kantor1,2
1Ioffe Physical Technical Institute, Polytekhnicheskaya 26, 194021 St.-Petersburg, Russia
2St.-Petersburg State Polytechnical University, Polytekhnicheskaya 29, 195251 St.-Petersburg, Russia
We show that equations governing pulsations of superfluid neutron stars (NSs) 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, |s|∼0.01−0.05, for realistic equations of state
(EOSs). Already an approximation s = 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.
2 The proposed approach allows to drastically simplify modeling of pulsations of superfluid NSs.
1
0 PACSnumbers: 97.60.Jd,47.75.+f,97.10.Sj,47.37.+q
2

n
Introduction. — The pulsations of NSs can be ex- described by two four-vectors, uµ and wµ . The vec-
a (n)
J cited either due to internal instabilities or owing to ex- tor uµ is the velocity of electrons and protons as well as
8 ternal perturbations. Currently the detectors that will “normal”neutrons;thevectorwµ arisesfromadditional
(n)
1 be able, according to preliminary estimates, to register degrees of freedom associated with neutron superfluid-
gravitational waves from pulsating NSs, are under de-
ity. In the nonrelativistic limit the spatial components
R] velopment [1]. For the correct interpretation of future of the four-vector wµ are related to superfluid veloc-
observationsit isnecessaryto haveawelldevelopedthe- (n)
S ory of NS pulsations. The formulation of such a the- ities VVVsn of the Landau-Khalatnikov theory [7] by the
h. ory is complicated by the fact that at a temperature equality www(n) = mn(VVVsn−uuu), where mn is the neutron
mass; uuu is the spatial component of the “normal” four-
-p Tbec<∼om10e8s−up1e0r1fl0uKid,.baTrhyuosn,stionmthoedienlteprunlsaaltliaoynesrsonofeNhSass velocity uµ. The electron j(µe), proton j(µp), and neutron
o jµ current densities are expressed through the vectors
r toemploysuperfluidhydrodynamicswhichismuchmore (n)
st complicated than the ordinary one, describing “normal” uµ and w(µn) as [8, 9]: j(µe) = neuµ, j(µp) = npuµ, and
a (nonsuperfluid) matter. jµ =n uµ+Y wµ . Here n is the number density of
[ (n) n nn (n) l
For the first time the global pulsations of superfluid particles l = n, p, or e. The expression for jµ consists
(n)
2 NSswereanalyzedbyLindblomandMendellin1994[2]. oftwotermsreflectingthe factthatbothnormalandsu-
v
ConsideringasimplemodelofaNewtonianstartheynu- perfluidliquidcomponentscontributetoneutroncurrent
2
merically found two distinct classes of pulsation modes: density. The coefficient Y has been calculated in Ref.
5 nn
7 (i)normalmodeswhichpracticallycoincidewiththecor- [10]; it is a relativistic analogue of superfluid density of
2 responding modes of a normal star; and, (ii) superfluid neutronsρsn. Inthe nonrelativisticlimit Ynn =ρsn/m2n.
. modes in which the matter pulsates in such a way that ThiscoefficientdependsonT andincreasessteadilyfrom
7
0 the mass current density approximately vanishes. The 0fornormalmatter(whenT Tcn,whereTcnistheneu-
≥
0 subsequent numericalstudies of various pulsation modes tron critical temperature) to nn/µn for entirely super-
1 (theliteratureisvast;see,e.g.,Refs.[3,4]andreferences fluid matter (T =0). Here and below, µ is the chemical
l
: therein) confirmed the result of Ref. [2] though general potential for particles l =n, p, or e.
v
i explanation of this result has not yet been proposed [5]. To proceed further, we assume that: (i) the quasineu-
X Inthisworkwegivesuchanexplanation. Inaddition,we trality condition holds both for equilibrium and pulsat-
r present an approximate scheme which allows to greatly ing matter, n = n ; and, (ii) an unperturbed NS is
a p e
simplifycalculationsofpulsatingsuperfluidNSs. Inwhat in beta-equilibrium, i.e. the disbalance of chemical po-
follows the speed of light c=1. tentials δµ µ µ µ = 0. It is convenient then
n p e
≡ − −
Superfluid hydrodynamics. — For simplicity, we to formulate the system of nondissipative hydrodynamic
considerNScorescomposedofneutrons(n),protons(p), equationsusingthe baryonnumberdensitynb nn+np
≡
and electrons (e). We also assume that protons are nor- andδµasindependentvariables[9]. Thesystemconsists
mal while neutrons are superfluid in some region of a of (1) the continuity equations for baryonsand electrons
NS core. As demonstratedin Ref. [6] possible admixture
of other particle species (e.g., muons) and proton super- jµ =0, jµ =0, (1)
(b);µ (e);µ
fluidity do not affect our principal results. Finally, we
first consider a nonrotating NS. Effects of rotation will where the baryon current density is jµ jµ +jµ =
be incorporated later in the text. n uµ+Y wµ ; (2) Einstein equation(sb) ≡ (n) (p)
b nn (n)
It is well known that in superfluid matter, several in-
dependent motions with different velocities may coexist 1
without dissipation [7]. In our case the system is fully Rµν gµνR= 8πG Tµν (2)
− 2 −
2
withtheenergy-momentumtensorTµν =(P+ε)uµuν+ whereweassumedthatwµ dependsontimetaswµ
Pgµν +Ynn(w(µn)w(νn) +µnw(µn)uν +µnw(νn)uµ); (3) the exp(iωt). In Eq. (6) all th(ne)quantities except for δµ(na)n∼d
potentiality condition for superfluid motion of neutrons wµ are taken in equilibrium. Near the equilibrium, the
(n)
function δµ(n ,n ) can be expanded in the Taylor series
∂ w +µ u ∂ w +µ u =0, (3) b e
ν(cid:2) (n)µ n µ(cid:3)− µ(cid:2) (n)ν n ν(cid:3) and presented, in the linear approximation, as
and (4) the second law of thermodynamics
δµ=n (∂δµ/∂n )(zD +D ). (7)
e e 1 2
dε=µ dn δµdn +T dS+Y d wα w /2. (4)
n b− e nn (cid:16) (n) (n)α(cid:17) Here z [n ∂δµ/∂n ]/[n ∂δµ/∂n ]; D δn /n ; D
b b e e 1 b b 2
≡ ≡ ≡
In Eqs. (2)–(4), Rµν, R, and gµν are Ricci tensor, scalar δne/ne. Thesymbolδinfrontofsomequantitydenotesa
curvature, and metric tensor, respectively; ∂ ∂/∂xµ; deviationofthisquantityfromitsequilibriumvalue. The
µ
G is the gravitationconstant; P ε+µnnb ≡δµne+ dimensionlessfunctionsD1andD2canbefoundfromEq.
T S is the pressure; ε and S a≡re−the energy−and en- (1) and depend on w(µn), uµ, and gµν. Thus, generally,
tropy densities, respectively. All the thermodynamic Eq.(6)isnotindependentandshouldbesolvedtogether
quantities are defined in the comoving frame in which with Einsteinequations (2). In the linear approximation
uµ = (1,0,0,0). This imposes an additional constraint Eq. (2) can be written in the following symbolic form:
on w(µn) [9], uµw(µn) =0. The solution to Eqs. (1)–(4) in δ(Rµν−1/2 gµνR)=−8πG δTµν. The left-handsideof
the superfluidregionshouldbe matchedwiththatinthe this equation contains only perturbations of metric. To
residual (normal) region of a star. Equations, describ- writeouttheright-handsideitisconvenienttointroduce
ing pulsations of normal matter can be obtained from new independent variables, the four-velocity of baryons
the system (1)–(4) if one puts Ynn = 0 and ignores the Uµ ≡j(µb)/nb =uµ+Ynnw(µn)/nb andWµ ≡Ynnw(µn)/nb,
condition (3). insteadofuµandwµ . Notethatforanunperturbedstar
(n)
Linear Approximation. — In this work we assume Uµ = uµ and Wµ = 0 (since in equilibrium wµ = 0).
that in the unperturbed NS www = 0, i.e. velocities of (n)
(n) The same is also true for a pulsating NS if the matter
normalandsuperfluidcomponentscoincide. Foranonro-
is nonsuperfluid (because then Y =0). Employing the
tatingNSthissimplymeansthatbothcomponentsareat nn
new variables an expression for δTµν takes the form
rest. Thenitfollowsfromtheconstraintu wµ =0that
µ (n)
wµ vanishes in equilibrium, wµ =0 (because u0 =0). δTµν =(δP +δε)UµUν
(n) (n) 6
Asaconsequence,thetermsinEq.(4)andintheexpres- +(P +ε)(UµδUν +UνδUµ)+δPgµν +P δgµν.(8)
sion for Tµν, underlined with one line, are quadratically
small. Similarly, the terms which depend on T and un- Here the quantities P, ε, Uµ and gµν are taken at equi-
derlinedtwice, are smallinthe stronglydegenerate mat- librium. As follows from Eq. (4) the variation δε equals
ter of NSs and can be omitted [9]. Because of the same δε = µ δn and depends on δUµ and δgµν (and is inde-
n b
reasonsonecanconsider,e.g.,thequantitiesP andδµas pendent of δWµ). The variation δP can be expanded in
functionsofonlyn andn (andneglecttheirdependence analogy with Eq. (7),
b e
on scalars wµ w and T).
(n) (n)µ
Now we make use of the energy-momentum conser- δP =nb(∂P/∂nb)(D1+sD2), (9)
vation law Tµν = 0 which can be derived from Eq. (2).
Composinga;νvanishingcombinationTµν+uµu Tαν and wherethefunctionD1 dependsonδUµ andδgµν,andD2
subtractingfromitEq.(3)multiplied;bνynbuµ,νon;αegets, dependsonthedifference(δUµ−δWµ)andδgµν [seethe
definitions for D and D , and Eq. (1)]. The parameter
with the help of Eq. (4) and the expression for P, 1 2
s, hereafter referred to as the “coupling parameter”, is
−ne∂µδµ−uµuνne∂νδµ−neδµ uν(uµ);ν given by s≡(ne∂P/∂ne)/(nb∂P/∂nb).
+(g +u u )uαXν +Xνu +X uν Superfluid and normal modes. — If s = 0 then
µν µ ν ;α µ;ν µ ;ν
Eq.(8)forδTµν doesnotdependonδWµandhasexactly
n uν ∂ w ∂ w =0. (5)
− b (cid:2) ν (n)µ− µ (n)ν(cid:3) the same form as in the absence of superfluidity. In that
HereXµ µ Y wµ . Theobtained“superfluid”equa- case the Einstein equations (and boundary conditions to
≡ n nn (n) them)coincide,inthelinearapproximation,withthecor-
tion is very attractive because each term in it depends
either on δµ or wµ . Both these quantities are small in responding equations for normal matter. They can be
(n) solved separately from the “superfluid” Eq. (6) so that
a slightly perturbed matter (and vanish in equilibrium).
the solution(the spectrum ofeigenfrequencies ω and the
ThismeansthatinthelinearapproximationEq.(5)does
eigenfunctions δUµ) will be indistinguishable from that
not depend explicitly on the perturbations of the metric
for a nonsuperfluid star.
g and the four-velocity uµ. Thus, one can replace g
µν µν Let us now focus on the following question. Assume
anduµ inEq.(5)bytheir equilibriumvalues. Foranon-
that s still vanishes. Is it possible for a NS to oscil-
rotating NS the spatial components of Eq. (5) can be
late on a frequency which is not an eigenfrequency of a
rewritten in a remarkably simple form (j =1, 2, 3)
normal star? Suppose that it is indeed the case. Then
iω(µ Y n )w =n ∂ (√ g δµ), (6) the linearized Einstein equations will be satisfied only if
n nn b (n)j e j 00
− −
3
In the consideration above we supposed that s = 0.
PALIII240
SLy4 Yet, it is clear that superfluid and normal modes should
remain approximately decoupled also at small but finite
PALIII180 s. As follows from Fig. 1, s is indeed small for realistic
EOSs and changes, on the average,from 0.01 to 0.05
− −
[18]. Takingintoaccountthattheparameterz inEq.(7)
APR PA
is z 1 for the same EOSs, it is easy to show that for
∼−
s normalmodesD >D [then the secondtermin Eq.(9)
1 2
PALI240 is much smaller th∼an the first one], while for superfluid
modes D sD [then the first term in Eq. (7) is much
PALI180 1 ∼ 2
smaller than the second one]. Generally, the exact solu-
LNS
tion of linearized pulsation equations (2) and (6) can be
presented as a series in parameter s [6]. However, since
PALII180 PALII240 sisverysmall,the approximationofnoninteractingEqs.
(2) and (6) considered above (hereafter “zero approxi-
mation”) is already sufficient to calculate the pulsation
logρ (gcm−3) spectrum within the accuracy s (i.e., a few percent).
∼
Example: Radial pulsations. — Let us illustrate
theobtainedresultswithanexampleofaradiallypulsat-
FIG.1: ThecouplingparametersversusdensityρforEOSs
PAL [11], PA [12], APR [13], SLy4 [14], and LNS [15]. EOSs ingNSwiththemassM =1.4M⊙. Weconsiderasimple
NSmodelwhichwasanalyzedindetailinRef.[8]. Inthat
of the PAL family differ by the symmetry energy (models I,
II, or III) and by the value of the compression modulus, 180 paperitwasassumedthattheredshiftedcriticaltemper-
or 240 MeV. The models with the compression modulus 120 ature of neutrons T∞ is constant throughout the stellar
cn
MeV are not plotted since they give too small maximum NS core,T∞ =6 108K.Theresultsofapproximatecalcula-
cn ×
masses that seem to contradict observations [16]. Note that tionofpulsationspectrumareillustratedinFig.2a. The
therecentlymeasured massM =(1.97±0.04)M⊙ [17]of the spectrumiscalculatedinzeroapproximationins. Inthe
millisecondpulsarPSRJ1614-2230 furtherrulesoutPAEOS figure,thepulsationfrequencyω(inunitsofω c/R ,
0 NS
and all PAL EOSs except for PAL I240 and PAL II240. ≡
where R =12.17 km is the circumferential radius of a
NS
δUµ = 0 and δgµν = 0. As follows from Eq. (1), in star)isplottedasafunctionofinternalredshiftedstellar
this case δn = 0 (i.e. D = 0), while D depends only temperature T∞ for 3 normal (solid lines) and 6 super-
b 1 2
on δWµ (or, in other words, on Wµ, since Wµ = 0 in fluid (dashes) pulsation modes. At T∞ > Tc∞n only the
equilibrium). In particular, for a nonrotating NS normal modes (I, II, and III) survive since then the star
is nonsuperfluid. For comparison, in Fig. 2b we present
D =[(∂ n /n )Wj +Wj]/(iωU0). (10) the exact solution to the system of linearized equations
2 j e e ;j
(2) and (6). The first 6 modes are shown by alternate
Here j = 1, 2, and 3; all the quantities, except for Wj, solid and dashed lines. The spectrum was not plotted in
are taken in equilibrium; when calculating the covariant the shaded region. All other notations are the same as
derivative one should use the unperturbed metric. Eqs. in Fig. 2a. It is easy to see that the structure of both
(7) and (10) allow to formulate Eq. (6) purely in terms spectra is very similar. However, there is one principal
of Wj. A boundary conditionto this equation,WWW⊥ =0, difference. Instead of crossings of superfluid and normal
also depends only on Wj and can be obtained from the modes in Fig. 2a, we have avoided crossings in Fig. 2b.
requirement that the baryon current density jµ is con- Atthesepointsthesuperfluidmodebecomesnormaland
(b)
tinuous through the normal-superfluid interface (WWW⊥ is vice versa. Such avoided crossings are not described in
the component of a vector Wj, perpendicular to the in- approximate treatment (Fig. 2a) because when frequen-
terface). Thus,Eq.(6)isself-containedandcanbesolved cies of superfluid and normal modes are close to each
independently of Eq. (2). Its solution (eigenfrequencies other, Eqs. (2) and (6) become strongly interacting and
andeigenfunctionsWj)describessuperfluidmodeswhich cannot be considered as independent. For comparison,
were first considered in Ref. [2] and do not have an ana- we plot both spectra in Fig. 2c. The exact solution is
logueforanormalstar. Toourbestknowledge,thestrik- shown by solid lines, dashes correspond to the approxi-
ing propertiesof suchmodes havenotbeen discussedfor mate solution. Other notations are the same as in Figs.
a realistic model of a general relativistic NS at finite T. 2a and 2b. On average,the approximate solution differs
Firstofall,thesuperfluidpulsationmodesdonotperturb from the exact one by 1.5 2%. For normal modes
∼ −
metric (δgµν = 0) and hence cannot emit gravitational the difference becomes smaller with increasing of T. In
waves. Inaddition,becauseforthesemodesδUµ =0and this case the number of “superfluid” neutrons decreases
δnb = 0, the variations of j(µb) and P vanish, δj(µb) = 0 (Ynn → 0), consequently, Wj ≡ Ynnw(jn)/nb → 0 and
and δP = 0 [see Eqs. (1) and (9)]. As a consequence, zero approximation works better and better.
pulsations are localized entirely in the superfluid region Taking into account rotation. —Rotationleadsto
ofastar. Inparticular,theydonotgototheNSsurface. formation of Feynman-Onsager vortices inside NSs with
4
III III III
K K K
8 8 8
0 0 0
1 II 1 II 1 II
× × ×
0
ω
6 6 6
/
= = =
ω
∞n ∞n ∞n
Tc I Tc I Tc I
(a) (b) (c)
T∞ (K) T∞ (K) T∞ (K)
8 8 8
FIG. 2: Frequency ω in units of ω0 versus T8∞ ≡ T∞/(108 K) for various pulsation modes. (a) approximate spectrum; (b)
exact spectrum; (c) approximate (dashed lines) and exact (solid lines) spectra. For more details see thetext.
the interspacingdistance 10−2 10−4 cm. The hydro- s can be small even for strongly stratified NSs (and is
∼ −
dynamic equations averaged over the volume containing indeed small for realistic EOSs).
large amount of vortices formally have the same form Now let us discuss the results of Ref. [4]. This paper
as in their absence [19] (if we neglect the small con- analyzed gravitational radiation from superfluid nonro-
tribution of vortices to the internal energy density of tatingNSsatT =0intheframeofthegeneralrelativity.
matter). The only exception is the potentiality condi- It was argued that superfluid modes must radiate gravi-
tion(3)thatshouldbereplacedbyuν ∂µ[w(n)ν+µnuν] tationalwavesinpracticallyallsituations,withintensity
∂ν[w(n)µ +µnuµ] = OµνWν, whe{re the tensor Oµν of radiation comparable to that from the normal modes
− }
is specified in Ref. [6] and is responsible for the inter- (unless an EOS has a very specific form satisfying Eq.
action between the normal and superfluid component. (74) of Ref. [4]).
It can be found from the requirement that the entropy
When modeling the neutron-star pulsations the au-
does not decrease. Because of this condition the new
thors of Ref. [4] used toy-model EOSs that give com-
term n O Wν appears in the right-hand side of Eq.
b µν pletely unrealistic values for the coupling parameter s.
(5). Since thistermdepends onasmallquantityWµ, all
Inparticular,we foundthattheir mostrealisticmodelII
ourreasoningaboutdecouplingofsuperfluidandnormal
gives s 0.1 at the center and s= at the superfluid-
modes remain valid for rotating NSs as well. ∼ ∞
normal interface. Moreover, because their EOSs are ar-
Comparison with previous works. — For com- tificial,theywereforcedtorelaxanassumptionofchem-
parisonwe choosetwo papers,Refs. [20]and [4], since at icalequilibriumin the core. As itis demonstratedin the
first sight it is not clear whether our results complement presentpaper,thelatterassumptionisveryimportantfor
or contradict the conclusions drawn in these references. the decoupling of modes and cannot be ignored. Thus,
The authors of Ref. [20] considereda model of Newto- it is not surprising that our results disagree with the re-
nian star at T = 0. They demonstrated that superfluid sultsofRef.[4];whensisnotsmall,superfluidmodescan
modes decouple from the normal modes only for an ide- be aseffectiveinradiatinggravitationalwavesasnormal
alizedcase ofnonstratifiedNSs, for whichne/nb =const modes.
throughout the stellar core.
In the end, it is worth mentioning one more result of
This result does not contradict ours because one can Ref. [4]. In that paper it is claimed that any (nonradial)
showthattheneutron-starmatterisnonstratifiedonlyif pulsation mode must emit gravitational waves unless an
∂P(nb,ne)/∂ne =0 (that is s=0). As follows from our EOSsatisfies some specific criterion[their Eq. (74)]. We
analysis, in the latter case superfluid and normal modes checkedthatthis criterionisnotequivalentanddoesnot
are indeed strictly decoupled. follow from our criterion s = 0, which is a necessary
The second conclusion made in Ref. [20] is based on conditionfordecouplingofsuperfluidmodesfrommetric.
the observation that for most of the neutron-star mod- Conclusion. — Summarizing, equations describing
els the stellar matter is stratified. Using this observa- pulsations of superfluid NSs can be split into two sys-
tion the authors of Ref. [20] argued that generally there tems of weakly coupled equations. The coupling pa-
shouldbenocleardistinctionbetweenthesuperfluidand rameter s of these systems is small for realistic EOSs,
normal modes, or, in other words, equations describing s 0.01 0.05. One system of equations describes
superfluid- and normal-type pulsations are strongly in- n|o|rm∼almod−es, another one – superfluid modes. Already
teracting. zero approximation in parameter s (when the systems
This conclusion is not correct because, as we demon- are fully decoupled) is sufficient to calculate the pulsa-
strated earlier in this work, the real coupling parameter tionspectrum with an accuracyof a few percent. In this
5
approximationthe normal modes coincide with ordinary [2, 3] and suggest simple perturbative (in parameter s)
modes of nonsuperfluid NS, while superfluid modes do scheme which drastically simplifies the problem of cal-
notperturbmetric,pressure,baryoncurrentdensityand culation of the pulsation spectrum for superfluid NSs.
are localized in superfluid region of a star. Note that Thepresentedapproachallowstoeasilytakeintoaccount
an emission of gravitational waves by superfluid modes realistic EOSs, dissipation, various composition of mat-
is possible only in the next (first) order of perturbation ter, temperature effects, baryon superfluidity, density-
theoryins. Thus,itshouldbesuppressedincomparison dependent profiles of critical temperatures, and rotation
to gravitational radiation from the normal modes. ofNSs. Inmoredetailthese issueswillbe discussedelse-
Our finding that superfluid modes do not appear at where [6].
the NS surface and do not emit gravitational waves in Acknowledgements. — We thank D.P. Barsukov,
the s=0 limit indicate that these modes should be very A.I. Chugunov, and D.G. Yakovlev for valuable com-
difficulttoobserveatsmallbutfinites. Thismeansthat ments. This research was supported by the Dynasty
observationalpropertiesofapulsatingsuperfluidstarand Foundation, Ministry of Education and Science of Rus-
a normal star of the same mass should be very similar, sianFederation(ContractNo. 11.G34.31.0001withSPb-
so that it will be very hard to discriminate one from the SPU and leading scientist G.G. Pavlov), RFBR (Grant
other. No. 11-02-00253-a), and by FASI (Grant No. NSh-
The obtained results explain numerical calculations 3769.2010.2).
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