Nonradial superfluid modes in oscillating neutron stars
aa r X i v : . [ a s t r o - ph . S R ] J a n Mon. Not. R. Astron. Soc. , 1–6 (2011) Printed 2 September 2018 (MN L A TEX style file v2.2)
Nonradial superfluid modes in oscillating neutron stars
A. I. Chugunov ⋆ , M. E. Gusakov † Ioffe Physical-Technical Institute of the Russian Academy of Sciences, Politekhnicheskaya 26, 194021 Saint-Petersburg, Russia
Accepted 2011 xxxx. Received 2011 xxxx; in original form 2011 xxxx
ABSTRACT
For the first time nonradial oscillations of superfluid nonrotating stars are self-consistently studied at finite stellar temperatures. We apply a realistic equation ofstate and realistic density dependent model of critical temperature of neutron andproton superfluidity. In particular, we discuss three-layer configurations of a star withno neutron superfluidity at the centre and in the outer region of the core but with su-perfluid intermediate region. We show, that oscillation spectra contain a set of modeswhose frequencies can be very sensitive to temperature variations. Fast temporal evo-lution of the pulsation spectrum in the course of neutron star cooling is also analysed.
Key words: stars: neutron – stars: oscillations – stars: interiors.
Studying the pulsations of neutron stars (NSs) is very im-portant and actively developing area of research, since com-parison of pulsation theory with observations can potentiallygive valuable information about the properties of superdensematter (Andersson et al. 2011). Yet, it is an extremely dif-ficult theoretical problem even for normal (nonsuperfluid)stars. Superfluidity of baryons additionally complicates thetheory because in superfluid (SFL) matter one should dealwith several independent velocity fields. As a consequence,the hydrodynamics describing pulsations of SFL NSs is muchmore complicated in comparison to ordinary (normal) hy-drodynamics (e.g., Gusakov 2007).In this letter we report on a substantial progress inmodeling and understanding the nonradial oscillations ofnonrotating SFL NSs in full general relativity. It shouldbe noted that the nonradial oscillations of such starshave been intensively studied in the literature startingfrom the seminal paper by Lindblom & Mendell (1994).In particular, nonradial oscillations of general relativisticNSs were considered by Comer, Langlois & Lin (1999);Andersson, Comer, & Langlois (2002); Yoshida & Lee(2003); Lin, Andersson & Comer (2008). Because of thecomplexity of the problem, most of these papers used asimplified microphysics input, i.e., toy-model equations ofstate and simplified models of baryon superfluidity (see,however, Lin et al. 2008; Haskell, Andersson & Passamonti2009; Haskell & Andersson 2010). Moreover, in all thesestudies SFL matter was treated as being at zero temper-ature, an assumption that is unjustified at not too lowstellar temperatures and may lead to a quantitatively ⋆ [email protected]ffe.ru † [email protected]ffe.ru incorrect oscillation spectra (Gusakov & Andersson 2006;Kantor & Gusakov 2011; this letter).Here we improve on this by considering nonradial os-cillations of SFL neutron stars at finite temperatures. Wefollow an approach of Gusakov & Kantor (2011) (hereafterGK11) which allows us to analyse oscillations of general rel-ativistic NSs employing realistic equation of state, densitydependent profiles of nucleon critical temperatures and fullyrelativistic finite-temperature SFL hydrodynamics.As it was first found by Lindblom & Mendell (1994),oscillation spectrum of a SFL NS consists of two distinctclasses of modes, the so called normal and superfluidmodes. The frequencies of normal modes almost coin-cide with the oscillation frequencies of a normal starand hence are independent of temperature. The spec-trum of these modes is therefore very well studiedin the literature (see, e.g., Thorne & Campolattaro1967; McDermott, Van Horn & Hansen 1988;Benhar, Ferrari & Gualtieri 2004). On the contrary,SFL modes can be very temperature-dependent. Using theapproach of GK11 they can be decoupled from the normalmodes and studied separately. Since radial SFL modes havealready been thoroughly analysed in Kantor & Gusakov(2011), here we focus on the nonradial SFL modes. In whatfollows, the speed of light c = 1. In this section we briefly discuss the equation describingSFL oscillation modes [Eq. (5)]. The detailed derivation ofthis equation can be found in GK11. We consider a NS withnucleonic core and assume that both neutrons (n) and pro-tons (p) can be superfluid. Following GK11, we introducethe baryon current density j µ (b) = j µ (n) + j µ (p) , where c (cid:13) A. I. Chugunov, M. E. Gusakov j µ ( i ) = n i u µ + Y ik w µ ( k ) (1)is the current density for particles i = n or p (e.g.,Kantor & Gusakov 2011). Here and below the summationis assumed over the repeated nucleon index k = n, p. InEq. (1) n i is the number density; u µ is the four-velocityof “normal” liquid component; w µ ( k ) is the four-vector thatcharacterizes motion of superfluid neutron ( k = n) or proton( k = p) component with respect to normal matter. Finally,the symmetric temperature-dependent matrix Y ik (= Y ki ) isa relativistic analogue of the SFL entrainment matrix. Sinceelectrons (e) are normal, their current density is j µ (e) = n e u µ ,where n e is the electron number density. The quasineutralityimplies n e = n p (for simplicity, we ignore possible admixtureof muons in the core).In this letter we study small-amplitude (linear) oscil-lations of a nonrotating star being initially in hydrostaticequilibrium. Hence, for the unperturbed star one has u µ =(e − ν/ , , ,
0) and w µ (n) = w µ (p) = 0, while the metric isd S = − e ν d t + e λ d r + r dΩ , where r and t are the ra-dial and time coordinates, respectively; ν ( r ) and λ ( r ) arethe metric functions; and Ω is the solid angle in a sphericalframe with the origin at the stellar centre. For definiteness,we assume that the unperturbed matter in the stellar corewas in beta-equilibrium, δµ ≡ µ n − µ p − µ e = 0, where µ d is the chemical potential for particles d = n, p, e. Wealso restrict ourselves to oscillations with vanishing electri-cal current, j µ (p) − j µ (e) = 0. The latter condition couples theSFL degrees of freedom, w µ (p) = − ( Y pn /Y pp ) w µ (n) . (2)As it was shown in GK11 the interaction betweenthe SFL and normal oscillation modes is controlled bythe coupling parameter s , which is given by s =( n e ∂P/∂n e ) / ( n b ∂P/∂n b ) where P ( n b , n e ) is the pressureand n b = n n + n p is the baryon number density. This pa-rameter is small for a wide set of realistic equations of state, | s | . .
05 (see fig. 1 in GK11), so that the approximationof completely decoupled SFL and normal modes ( s = 0) isalready sufficient to calculate the pulsation spectrum withinan accuracy of a few per cent. In the s = 0 approximationthe quantities j µ (b) , P , and the metric g µν remain unper-turbed for SFL oscillation modes. This opens up a possi-bility to formulate an equation describing SFL modes thatdepends only on SFL degrees of freedom, i.e., on w µ ( k ) . Thisequation follows from the energy-momentum conservationand potentiality condition for motion of SFL neutrons. Fora nonrotating star it takes the form (see GK11) ıω (cid:0) µ n Y n k w ( k ) j − n b w (n) j (cid:1) = n e ∂ j ( δµ ∞ ) , (3)where δµ ∞ ≡ δµ e ν/ and the disbalance δµ ( n b , n e ) equals δµ = − ı e ν/ B n e ω " Y n k n e n b ∂n e ∂x µ w µ ( k ) + (cid:18) Y n k n b w µ ( k ) (cid:19) ; µ . (4)In Eqs. (3) and (4) j = 1, 2, or 3 is the space index, ∂ j ≡ ∂/ ( ∂x j ), B ≡ ∂δµ ( n b , n e ) /∂n e , and all perturbations areassumed to be ∝ exp( ıωt ). Because Eq. (3) is linear, onecan generally present δµ ∞ as: δµ ∞ ( r, Ω) = δµ l ( r ) Y lm (Ω),where Y lm (Ω) is the spherical harmonic [notice that δµ l ( r )does not depend on the index m , see Eq. (5)]. Combining then Eqs. (2), (3), and (4) one obtains the following equation0 = δµ ′′ l + (cid:18) h ′ h − λ ′ r (cid:19) δµ ′ l − e λ (cid:20) l ( l + 1) r + e − ν/ ω h B (cid:21) δµ l . (5)Here prime means derivative with respect to r and h = e ν/ n / ( µ n n b y ), where y = n b Y pp / [ µ n ( Y nn Y pp − Y )] − T onlythrough the parameter y = y ( T /T cn , T /T cp ), where T c i ( r )is the profile of critical temperatures for particles i = n orp. The boundary conditions to Eq. (5) are following. If neu-trons in the stellar centre are superfluid, the regularity ofthe solution requires δµ l ∝ r l at r →
0. If the outer bound-ary of the SFL-region coincides with the crust-core interface(where r = R cc ), then the SFL current should not penetratethe crust. This condition implies δµ ′ l ( R cc ) = 0. Finally, if T is so high, that the SFL-region does not spread all overthe core, then its boundaries are determined by the condi-tion T = T cn ( r ) (see Sec. 3 for more details). In that case,the regularity of the solution at such boundaries requires δµ ′ l = e λ − ν/ ω δµ l / ( h ′ B ).Thus, we reduce the problem of calculation of SFLmodes to solving simple second-order differential Eq. (5).For any fixed multipolarity l the solution to Eq. (5) consistsof a set of eigenfrequencies ω ln and eigenfunctions δµ ln ( r )which differ by the number of radial nodes n = 0, 1, 2, . . . Prior to studying the SFL oscillations using Eq. (5), onehas to specify equation of state (including profiles of T cn and T cp ) and construct a hydrostatic model of an un-perturbed star. In addition, one has to specify the pro-file of internal stellar temperature. High thermal conduc-tivity leads to a rapid equilibration of T in the NS core(see, e.g., Gnedin, Yakovlev & Potekhin 2001). As a re-sult, the red-shifted internal temperature T ∞ = T e ν/ becomes almost constant. Moreover, as it was shown byGusakov & Andersson (2006), for the SFL-region to be inhydrostatic and beta-equilibrium it must be in thermal equi-librium. Thus, in what follows we assume that T ∞ = constin the SFL-region.In the present letter we employ the equation of statesuggested by Akmal, Pandharipande & Ravenhall (1998)(APR). The coupling parameter s for such an equation ofstate is small, | s | ∼ .
02. The density profiles of T cn and T cp that we use here are shown in the left panel of Fig. 1. Theydo not contradict to results of microscopic calculations (see,e.g., Lombardo & Schulze 2001) and are similar to the nu-cleon pairing models used to explain observations of the cool-ing NS in Cas A supernova remnant (Shternin et al. 2011).For definiteness, all calculations are performed for a star ofthe mass 1 . M ⊙ and circumferential radius R = 12 . T ∞ , but not T , is constant in the SFL-region, it isconvenient to introduce the red-shifted nucleon critical tem-peratures T ∞ c k ≡ e ν/ T c k ( k = n, p) to analyse how the sizeof SFL-region changes with T ∞ . The functions T ∞ cn ( r ) and c (cid:13) , 1–6 uperfluid modes in oscillating neutron stars T c k [ K ] ρ [g cm − ] T cn , P – triplet T c p , S – s i n g l e t c r u s t c o r e r/R t ≈
380 yearsAPR M = 1 . M ⊙ , R = 12 . T ∞ c n , P – t r i p l e t T ∞ cp , S – singlet c r u s t c o r e Figure 1. (color online) Left panel: Nucleon critical temperatures T c k versus density ρ ( k = n, p). Right panel: Red-shifted criticaltemperatures T ∞ c k versus radial coordinate r . T ∞ cp ( r ) are shown on the right panel of Fig. 1. The red-shiftedproton critical temperature is high, T ∞ cp ( r ) ∼ × K, sothat superfluid protons occupy the entire core almost im-mediately after the NS birth. The function T ∞ cn ( r ) has amaximum T ∞ cn ≈ . × K at r = r maxcn ≈ . R . Near thestellar centre the density varies slowly with r which resultsin a weak dependence of T ∞ cn on the radial coordinate. As thestar cools down to T ∞ . T ∞ cn , the SFL-region is formed, ini-tially, as a narrow spherical layer. Upon subsequent coolingthe layer becomes wider and, for example, at T ∞ = 4 × Kit is shown by the hatched region in the figure. As the tem-perature decreases further, the SFL-region extends to thecrust and, eventually, at T ∞ = T ∞ cn (0) ≈ × K it pene-trates the stellar centre.
Figure 2 presents normalized eigenfrequencies ω ln (in unitsof ˜ ω = c/R ≈ . × s − ) versus internal temperature T ∞ for SFL oscillations of multipolarity l = 0, 1, 2, and 3.For each l we plot a set of oscillation modes that differ bythe number of radial nodes n = 0 (solid lines), n = 1 (dots), n = 2 (long dashes), n = 3 (long-short dashes), and n = 4(dashes). One sees that the higher the n the larger the ω ln .By the hatches in the bottom panel of the figure we showthe SFL-region; as expected, the size of this region dependson T ∞ . The grey-shaded area corresponds to the crust and aregion in the core where all neutrons are unpaired. A similarshaded area on the four upper panels shows temperatures T ∞ > T ∞ cn for which all neutron matter in the core is normal.In the latter case there are no SFL modes in NS.Before further discussing spectra in Fig. 2 it is con-venient to describe briefly Fig. 3 that presents eigenfunc-tions δµ ln ( r ) normalized to unity in the maximum. The solidlines correspond to radial oscillation modes ( l = 0), dottedand dashed lines describe dipole ( l = 1) and quadrupole( l = 2) modes, respectively. Each column in the figure con-tains four panels which are plotted for the following temper-atures (from bottom to the top): T ∞ = 10 , 2 × , 3 × ,and 4 × K. For any of these temperatures we have fivepanels in a row, which correspond to (from left to right) n = 0, 1, 2, 3, and 4 radial nodes of δµ ln ( r ). The stellarregions where neutrons are normal, are shaded in Fig. 3.At T ∞ . T ∞ cn the size of SFL-region rapidly increases r / R T ∞ [K] SUPERFLUID ω n / ˜ ω
123 4 T ∞ c n ( ) ≈ × K T ∞ c n ≈ . × K l = 0 ω n / ˜ ω l = 1 T ∞ c n ( ) ≈ × K T ∞ c n ≈ . × K ω n / ˜ ω l = 2 T ∞ c n ( ) ≈ × K T ∞ c n ≈ . × K ω n / ˜ ω T ∞ c n ( ) ≈ × K T ∞ c n ≈ . × K l = 3 Figure 2. (color online) Eigenfrequencies ω ln versus T ∞ for mul-tipolarities l = 0, 1, 2, and 3. For each l a set of curves is plottedwith n = 0, 1, 2, 3 or 4. At T ∞ T cn (0) ≈ × K (see thevertical short-dashed line) superfluidity occupies the stellar cen-tre. The bottom panel shows variation of SFL-region with T ∞ .For more details see the text. as the star cools down, whereas the eigenfrequencies are al-most temperature-independent for T ∞ & × K (exceptfor the modes with n = 0, see Fig. 2 and discussion below).This is so because of compensation of two opposite tenden-cies: (i) expansion of the SFL-region and (ii) increasing ofthe local speed of SFL sound v sf = e − ν/ √− h B with de-creasing T ∞ (see Gusakov & Andersson 2006 for details on v sf ). The tendency (i) leads to decreasing while (ii) leads togrowing of the eigenfrequencies ω ln .The qualitative behaviour of ω ln and δµ ln ( r ) at 3 × K . T ∞ . T ∞ cn follows from simple arguments. The pa-rameter h is small there, so that it is sufficient to retain onlythe terms ∝ h − in Eq. (5). As a result, for T ∞ & × K δµ ln ( r ) will be almost independent of l (see Fig. 3). Fur-thermore, at T ∞ & × K the function h can be ap-proximated as h ≈ A (cid:2) ( T ∞ cn − T ∞ ) − B ( r − r maxcn ) (cid:3) , where A and B are some constants depending on a NS model.Using this approximation, one can analytically solve Eq.(5) and find that eigenfunctions δµ ln ( r ) are proportionalto the Legendre polynomials P n while the eigenfrequencies ω ln ∝ p n ( n + 1) and are independent of T ∞ (see Figs. 2and 3 for T ∞ & × K). For the modes with n = 0 c (cid:13) , 1–6 A. I. Chugunov, M. E. Gusakov -1.00.00.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0-1.00.0-1.00.0-1.00.01.0 T ∞ = 1 . ×
108 K δ µ l n r/R r/R r/R r/R r/R T ∞ = 2 . ×
108 K δ µ l n T ∞ = 3 . ×
108 K δ µ l n T ∞ = 4 . ×
108 K δ µ l n n = 0 n = 1 n = 2 n = 3 n = 4 Figure 3. (color online) Eigenfunctions δµ ln versus r for T ∞ = 10 , 2 × , 3 × , and 4 × K (5 panels in a row for each temperature).The columns of panels are for n = 0 ,
1, 2, 3, and 4 radial nodes. Solid, dotted, and dashed lines correspond to multipolarities l = 0, 1,and 2, respectively. For more details see the text. -3 -2 -1 -2 -1 0 1 21.01.52.02.53.0 r / R t [years] SUPERFLUID ν n [ k H z ] t r / R t − t [years] SUPERFLUID ν n [ k H z ] r / R t − t [years] SUPERFLUID ν n [ k H z ] Figure 4. (color online) Upper panels: The oscillation frequencies ν n of quadrupole modes with n = 0, . . . ,4 versus stellar age t .Vertical dashed lines show the stellar age t at which neutron superfluidity first appears in the stellar centre. Bottom panels: Variationof SFL-region with t . For more details see the text and captions to Figs. 2 and 3. this estimate gives ω l = 0. As follows from Fig. 2, in thatcase the eigenfrequencies are indeed small but nonzero. Toimprove the estimate for ω l one should take into accountthe term depending on l ( l + 1) /r in Eq. (5). Because at T ∞ & × K δµ l ( r ) is almost constant (see top-leftpanel in Fig. 3), one can put δµ ′ l ≈ δµ ′′ l ≈
0. Us-ing then Eq. (5), one obtains the following estimate for ω l : ω l ∝ p l ( l + 1) v sf /r maxcn ∝ p l ( l + 1) p − T ∞ / T ∞ cn . Thisformula indicates that ω l depends only on T ∞ / T ∞ cn (but noton the size of SFL-region) and vanishes at T ∞ = T ∞ cn (seeFig. 2). It remains to note that for l = 0 Eq. (5) has a staticsolution δµ = δµ ∞ = const and ω = 0, which describesa star in hydrostatic and diffusive equilibrium (but not nec-essarily in beta-equilibrium, see Gusakov & Andersson 2006for more details). It should also be stressed that approximateformulas for eigenfrequencies found above is not a special feature of our microphysical model. Rather, it is inherent in all NSs for which the maximum of T ∞ cn ( r ) lies between thecentre and the crust-core boundary.At T ∞ slightly exceeding T ∞ cn (0) the size of SFL-regionrapidly expands to the stellar centre as the star cools down.The reason for that is weak dependence of T ∞ cn on r at r . . R (see the right panel of Fig. 1). Since T ∞ ≈ T ∞ cn for r . . R , v sf is close to 0 there. This results in a noticeabledecrease of ω ln with n > T ∞ cn (0) (see Fig. 2). Inparticular, at T ∞ = 2 × K the corresponding oscillationmodes are all localized in the vicinity of the stellar centre,where v sf is small (see Fig. 3). During subsequent cooling[for T ∞ . T ∞ cn (0)] the entire core is already occupied by theneutron superfluidity and increasing of v sf , especially in thecentral regions of the star, leads to growing of ω ln . When T ∞ drops below ∼ × K the eigenfrequencies ω ln and c (cid:13) , 1–6 uperfluid modes in oscillating neutron stars eigenfunctions δµ ln ( r ) become almost independent of T ∞ ,since h (and v sf ) approach their asymptotes at T ∞ = 0.Let us now discuss the temporal evolution of oscilla-tion spectra during the star cooling. To simulate coolingwe apply a slightly updated version of the code discussedby Gusakov et al. (2005). The onset of neutron superflu-idity is accompanied by an abrupt acceleration of coolingdue to the Cooper pairing neutrino emission process (seeShternin et al. 2011 for recent observational evidences ofsuch cooling). As is demonstrated in Fig. 4, this processleads to a rapid evolution of oscillation spectrum. Upperpanels in Fig. 4 show the eigenfrequencies ν n ≡ ω n / (2 π ) ofquadrupole modes ( n = 0 , . . . ,
4) versus NS age t . The lowerpanels illustrate change of the SFL-region with t . On theleft upper panel the functions ν n ( t ) are plotted for the timeinterval from 200 to 2 × years. After the onset of neu-tron superfluidity (at t ≈
330 years) the oscillation spectrumrapidly changes and approaches its zero-temperature asymp-tote only at t & years. At the star age t ≈
480 yearssuperfluidity penetrate the stellar centre [this correspondsto T ∞ = T ∞ cn (0)]. The period of time | t − t | ≪ t is accom-panied by a very fast evolution of the spectrum. In moredetail the evolution is shown in the central panels whichspan an interval of approximately 600 years of NS cooling.The time t in these panels is counted from t . The rightpanels in Fig. 4 show the only four-year episode of NS life inthe vicinity of t . One sees that even for such a small periodof time the frequencies of some oscillation modes increaseby more than 10%, whereas the frequency of the mode withfour radial nodes changes noticeably on a time-scale of a fewmonths (!). In this letter we study, for the first time, nonradial oscilla-tions of SFL NSs at finite temperatures. Use of an approachdeveloped in GK11 enable us to solve this problem in fullgeneral relativity, employing the realistic equation of state(APR) and realistic, density-dependent profiles of nucleoncritical temperatures (Fig. 1). It is shown that equations de-scribing SFL modes can generally be reduced to the simplesecond-order differential Eq. (5). The eigenfrequency spec-trum (Fig. 2) and eigenfunctions (Fig. 3) for this equationare carefully analysed. It is demonstrated that dependence ofeigenfrequencies ω ln on T ∞ is determined by two competingeffects: (i) decreasing of ω ln with expanding SFL-region and(ii) growing of ω ln with increasing of the local speed of SFLsound v sf . These results agree with the conclusions made byKantor and Gusakov (2011) for a radially oscillating NS.In addition, we examine the evolution of oscillationspectrum in the course of NS cooling (Fig. 4). We find thatacceleration of NS cooling soon after the onset of tripletneutron superfluidity leads to a very fast modification ofthe spectrum – the eigenfrequencies can vary dramaticallyon a time-scale of months.In the present work we completely ignore various dissi-pative effects in pulsating SFL NSs. Meanwhile, our resultsprovide a strong basis to study these effects using the per-turbative scheme suggested in GK11. In particular, to de-termine the damping time τ of some normal or SFL modeone could proceed in the following steps: (i) find the vec- tors u µ and w µ (n) assuming s = 0 and neglecting dissipation(see GK11 and Sec. 2); (ii) use them to calculate the dis-sipative terms entering the SFL hydrodynamics (and hencethe rate of change ˙ E of the oscillation energy E , see, e.g.,Kantor & Gusakov 2011); (iii) calculate τ as τ = − E/ (2 ˙ E ).The other important problem concerns gravitational ra-diation from pulsating SFL NSs. The radiation from normalmodes can be accurately calculated already in the s = 0limit (and will be the same as that for a nonsuperfluid NS).In turn, SFL modes are not coupled with the metric in the s = 0 limit, so that to calculate gravitational radiation fromthem one must use the first-order perturbation theory in s .Interestingly, this problem is equivalent to finding gravita-tional radiation from a normal NS experiencing oscillationsunder an action of a small external force ∝ s . An intensityof such radiation will be reduced by a factor of s ∼ − in comparison to normal modes with the same E . Detailedstudies of dissipation in SFL NSs are a very important task,e.g., for calculation of the instability windows of r-modes;we plan to address it in the near future. ACKNOWLEDGMENTS
The authors are grateful to E.M. Kantor, D.G. Yakovlev,and A.V. Brillante for useful comments. This work waspartially supported by RFBR (grant 11-02-00253-a), byRF president program (grants NSh-3769.2010.2 and MK-5857.2010.2), and by the Dynasty foundation.
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