We describe recent progress in the numerical study of the structure of
rapidly rotating superfluid neutron star models in full general relativity. The
superfluid neutron star is described by a model of two interpenetrating and
interacting fluids, one representing the superfluid neutrons and the second
consisting of the remaining charged particles (protons, electrons, muons). We
consider general stationary configurations where the two fluids can have
different rotation rates around a common rotation axis. The previously
discovered existence of configurations with one fluid in a prolate shape is
confirmed.Comment: 5 pages, 2 figures. Conference proceedings for the 26th Spanish
  Relativity Meeting (ERE 2002), Menorca, Spain, 22-24 Sept. 200

Comer, G. L.

Novak, J.

Prix, R.

English

arXiv

We describe recent progress in the numerical study of the structure of rapidly rotating superfluid neutron star models in full general relativity. The superfluid neutron star is described by a model of two interpenetrating and interacting fluids, one representing the superfluid neutrons and the second consisting of the remaining charged particles (protons, electrons, muons). We consider general stationary configurations where the two fluids can have different rotation rates around a common rotation axis. The previously discovered existence of configurations with one fluid in a prolate shape is confirmed

Comer, G.

MPG.PuRe

arXiv:gr-qc/0211105v1  29 Nov 2002Stationary structure ofrelativistic superfluid neutron starsR. Prix1, J. Novak2 and G.L. Comer31University of Southampton, UK2Observatoire de Paris-Meudon, France3Saint Louis University, USA.We describe recent progress in the numerical study of the structure of rapidlyrotating superfluid neutron star models in full general relativity. The superfluidneutron star is described by a model of two interpenetrating and interactingfluids, one representing the superfluid neutrons and the second consisting ofthe remaining charged particles (protons, electrons, muons). We consider gen-eral stationary configurations where the two fluids can have different rotationrates around a common rotation axis. The previously discovered existence ofconfigurations with one fluid in a prolate shape is confirmed.1 IntroductionMost current studies of neutron star oscillations are based on a simple perfectfluid model for a neutron star, which neglects (besides other things like the magneticfield, crust etc.) the crucial importance of superfluidity. The presence of substantialamounts of superfluid matter in neutron stars is backed by a number of theoreti-cal calculations of the state of matter at these extreme densities (e.g. see [1, 2]),and by the qualitative success of superfluid models to accommodate observed fea-tures of glitches and their relaxation (e.g. see [3] and references therein). For boththe study of oscillation modes of superfluid neutron stars, and the modelling of theparticular glitch instability, it is important to know the (quasi-) stationary initialstate of the unperturbed neutron star. The principal difference with ordinary-fluidmodels consists of the fact that in a superfluid model there are two fluids (i.e. thesuperfluid neutrons and all the rest), that can flow independently. As a result of theslow spindown process of the neutron star, the neutron superfluid will lag behind thecharged constituents, due to its complete absence of viscosity. The generic configu-ration therefore corresponds to a (quasi-) stationary two-fluid model, with the two(strongly coupled!) fluids rotating around the same axis, but with different rotationrates. This model has been studied before in both the Newtonian ([4, 5]) and therelativistic framework [6], using the slow-rotating approximation and a simple “poly-tropic” two-fluid equation of state. Here we describe a fully relativistic numericalapproach to solve the problem for arbitrary rotation rates.12 R. Prix, J. Novak and G.L. Comer2 The two-fluid modelWe briefly present the notation and formalism used to describe the two-fluidmodel for superfluid neutron stars, a more detailed discussion can be found in [7, 8].We consider two fluids, representing the neutrons (denoted by the label n) and theremaining charged components (denoted by c) respectively. Superfluidity allows theneutrons to flow independently of the remaining constituents, without any directfrictional force between the neutrons and the second fluid. Another implicationof superfluidity is that the rotating neutron fluid will be threaded by a lattice ofmicroscopic quantized vortices, but their effect on the macroscopic scale is neglectedhere, so we treat the neutrons simply as an inviscid fluid. The kinematics of the twofluids is described by the respective rest-frame particle number densities nn and nc,together with the four-velocities un and uc. The corresponding particle four-currentsare therefore nn = nnun and nc = ncuc, and we assume strict conservation of bothfluids (i.e. we exclude the possibility of “transfusion” [8] via β-reactions betweenprotons and neutrons), which means∇ · nn = 0 , and ∇ · nc = 0 . (1)The dynamics is governed by the total energy density E(nn, nc, ∆2), i.e. “equation ofstate”, where ∆ represents the relative velocity between the two fluids, and is definedas∆2 ≡ 1 −(nnncnn · nc)2, (2)corresponding to a gamma factor Γ∆ of the relative motion, which isΓ∆ = −un · uc =(1 − ∆2)−1/2. (3)The energy scalar E determines the first law of thermodynamics, namelydE = µn dnn + µc dnc + α d∆2 , (4)which defines the chemical potentials µn and µc of the two fluids, and the “entrain-ment function” α, which measures the dependence of the energy density on the rela-tive velocity. The equations of motion for the two interacting fluids are derived froma “convective” variational principle [9, 8] based on the hydrodynamic LagrangianLH = −E . This defines the canonical momenta pn and pc via the total differentialδLH = pn · δnn + pc · δnc. These canonical momenta are found explicitly aspn = µnun +2αnnΓ3∆(uc − Γ∆un) , (5)pc = µcuc +2αncΓ3∆(un − Γ∆uc) , (6)Stationary structure of relativistic superfluid neutron stars 3which shows that the effect of the entrainment α is to make the momenta deviatefrom their respective four-velocity directions. This effect has first been discussed (indifferent terms) by Andreev and Bashkin in the context of mixtures of superfluid 4Heand 3He [10], and is also equivalent to a description in terms of “effective masses”[5]. In the absence of mutual friction and pinning forces between the two fluids, theequations of motion have a very simple form, despite the fact that two fluids arecoupled via the equation of state, namelyun · dpn = 0 , and uc · dpc = 0 , (7)where d denotes the exterior derivative, i.e. (dp)αβ = ∇αpβ −∇βpα. The corre-sponding two-fluid energy-momentum tensor T is obtained from the variational prin-ciple asT = nn ⊗ pn + nc ⊗ pc + Ψg , (8)where g is the metric tensor, and Ψ is the generalized pressure, defined asΨ + E = nnµn + ncµc . (9)3 Symmetries and integrals of motionWe consider spacetimes that are stationary, axisymmetric and asymptoticallyflat. Stationarity and axisymmetry are associated with two (commuting) Killing vec-tor fields t and ϕ respectively. We choose adapted polar-type coordinates {t, r, θ, ϕ},which are such that t = ∂/∂t and ϕ = ∂/∂ϕ. We assume the two fluids are in purelycircular motion around the z-axis (i.e. we exclude convective meridional currents),so we writeun = γn (t + Ωnϕ) , and uc = γc (t + Ωcϕ) , (10)where Ωn and Ωc are the respective rotation rates of the two fluids. The most generalmetric (in quasi-isotropic maximal slicing gauge) [11] can be written asds2 = −(N2 − NϕNϕ) dt2 − 2Nϕ dϕdt + A2(dr2 + r2dθ2)+ B2r2 sin2θ dϕ2 , (11)where N , Nϕ, A and B are functions of (r, θ), and Nϕ ≡ gϕϕNϕ. With the additionalassumption of uniform rotations (∇Ω = 0), the equations of motion (7) simplifygreatly and can be shown to reduce to two first integrals of motion [6], namelyµn = γn constn , and µc = γc constc . (12)4 Results and discussionUnder the assumptions of the previous section, the Einstein equations reduceto four coupled elliptic equations for the four unknown functions N, Nϕ, A and4 R. Prix, J. Novak and G.L. ComerB, with the source terms given by the energy-momentum tensor (8). These canbe solved very accurately using an iteration scheme involving the first integrals ofmotion (12). The numerical scheme is based on a multi-domain pseudo-spectralmethod, which is described in more detail in [12], and which we have appropriatelyextended to the two-fluid model considered here. In principle the problem can besolved for any equation of state (EOS), but currently only a polytropic toy-modelEOS is implemented. This, however, is sufficient to show the qualitative features ofa superfluid neutron star models as opposed to a single fluid model. The EOS usedhere is a “generalized polytrope” of the formE = (mnnn + mcnc)c2 +12κnnγ1n +12κcnγ2c + κncnγ3n nγ4c + βnγ5n nγ6c ∆2 . (13)We note that the two fluids are coupled via a “symmetry energy”-type term propor-tional to κnc and an “entrainment” term proportional to β∆2.The numerical two-fluid code is currently in the test phase, internal consistencychecks like the relativistic viriel (cf. [11]) are satisfied to about 10−5 at a convergenceof 10−6. However, further tests are necessary, in particular comparisons with previousresults in the slow-rotation approximation, both relativistic [6] and Newtonian [4, 5].The latter, in particular, allowed an analytic solution for a subclass of the EOS(13). Furthermore, the existence of prolate solutions for the slower rotating fluid wasfound analytically in [5], and we have been able to reproduce these solutions withour numerical code (see Fig. 1). In Figure 1 we see the influence of the interaction(characterized by κnc and β) between the two fluids on the stationary configurationof a neutron stars model with roughly M ≈ 1.4 M⊙ and R ≈ 11 km. The EOSparameters used are γ1 = γ2 = 2, and γ3 = γ4 = γ5 = γ6 = 1, and κn = κc = 0.06.In this configuration, one fluid is rotating at Ωn = 2000 π s−1, while the secondfluid rotates much slower at only Ωc = 200 π s−1. The left-hand graph shows thesolution for two fluids which are only gravitationally coupled (i.e. κnc = β = 0),while in the right-hand graph there is additional coupling via the “symmetry”-termκnc and entrainment β. The effect is to push the faster fluid further out and to“squeeze” the slower fluid, in this case to the point of actually making it prolate.Both the uncoupled and the coupled configuration are characterized by the samecentral chemical potentials µn(0) = µc(0) = 0.3 mc2, which is why their resultingtotal masses and radii differ by about 10 %.Apart from further quantitative tests, more work is necessary in order to allowfor more realistic equations of state, and to include the possibility of vortex-mediatedpinning forces between the two fluids.Acknowledgements. RP acknowledges support from the EU Programme ’Im-proving the Human Research Potential and the Socio-Economic Knowledge Base’(Research Training Network Contract HPRN-CT-2000-00137). GLC acknowledgespartial support from NSF grant PHYS-0140138.Stationary structure of relativistic superfluid neutron stars 5Figure 1: Two-fluid configurations with Ωn = 2000 π s−1 and Ωc = 200 π s−1. Thethick lines represent the surfaces of the two fluids respectively, while the dashed linesare the isosurfaces of “gravitational potential” lnN . In the left-hand configuration,the two fluids are locally non-interacting, i.e. κnc = 0 and β = 0, while in theright-hand configuration we have chosen κnc = 0.02 and β = 0.12.References[1] M. Baldo, J. Cugnon, A. Lejeune, and U. Lombardo. Nucl. Phys. A 536, 349(1992).[2] O. Sjöberg. Nucl. Phys. A 265, 511 (1976).[3] B. Link and R.I. Epstein. ApJ 457, 844 (1996).[4] R. Prix. A&A 352, 623 (1999).[5] R. Prix, G.L. Comer, and N. Andersson. A&A 381, 178 (2002).[6] N. Andersson and G.L. Comer. Class. Quant. Grav. 18, 969 (2001).[7] B. Carter and D. Langlois. Nucl. Phys. B 531, 478 (1998).[8] D. Langlois, D.M. Sedrakian, and B. Carter. MNRAS 297, 1189 (1998).[9] B. Carter. In Relativistic Fluid Dynamics (Noto, 1987), eds. A. Anile andM. Choquet-Bruhat, (Springer, Heidelberg, 1989), pp. 1–64.[10] A.F. Andreev and E.P. Bashkin. JETP 42, 164 (1975).[11] S. Bonazzola, E. Gourgoulhon, M. Salgado, and J.A. Marck. A&A 278, 421(1993).[12] S. Bonazzola, E. Gourgoulhon and J.A. Marck. Phys. Rev. D 58, 104020 (1998).

Stationary structure of relativistic superfluid neutron stars

Prix, Reinhard

Novak, Jerome

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Stationary structure of relativistic superfluid neutron stars

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