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PoS(MPCS2015)012 http://pos.sissa.it/ , M. Oertel, D. Chatterjee and A. Sourie ∗ [email protected] Speaker. Realistic global neutron star models requirerecall numerical the approach. general-relativistic framework Here, for wehow the briefly global description observable motivate quantities and of can rotating be computed compactresults unambiguously. stars Then, on we and the present show some building recent oflowing more two detailed directions. physical First, models, weglobal including describe equilibrium the realistic and inclusion microphysics, equation of fol- of magnetic state,an field and attempt and to finally its model we effects glitch discuss on phenomena. superfluid both, (two-fluid) models in ∗ Copyright owned by the author(s) under the terms of the Creative Commons c Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). The Modern Physics of Compact Stars30 2015 September 2015 - 3 OctoberYerevan, Armenia 2015 J. Novak LUTH, Observatoire de Paris, PSL ResearchUniversité University, CNRS, Paris Diderot, Sorbonne Paris Cité, 5E-mail: place Jules Janssen, 92195 Meudon Numerical models of isolated rotating compact stars PoS(MPCS2015)012 ) r ( p (1.1) (1.3) J. Novak a different static equi- 5 for a black . 0 ; (1.2) i.e. = 2 , defined as -equilibrium. The ϕ a priori C β d θ 2 , sin ) r + ( 2 θ Gp d π 4 compactness ratio 2 r + 2 + c 2 ) r 2 , dr ( r 2 for our Sun. 1 6 − Rc Gm − / ) . 10 2 r 1 ν ( GM is the so-called 2 − dr ∼ d rc R = : )] Gm C ) r r 2 r C ( ( 2 p , − the speed of light. The highest value is ) rc 1 r 2 Gm c ( c ) + 2 r ε ( 2 + − and radius ε r 2 [ T; and special relativity, with rotation velocities approaching that 1 . This figure also shows the influence of rotation, by displaying π M 1 dt − 4 13 2 c ], coupled to an equation of state (EoS). The line elements, giving the = = = ) 2 r ( , ν ν profiles. TOV equations are then a system of coupled ordinary differential 1 2 dr dr dr dp d dm ) e r − ( are the two gravitational potentials. Matter is described by the pressure ε 2 for compact stars, and ) = . r 0 2 ( ν ' ds C and ) r the gravitational constant and ( m G Each model can be parameterized by a value of the central of pressure (or, equivalently, den- First, in order to motivate the use of general relativity (GR), we here give a qualitative differ- The building of compact stars models requires inputs from many different fields of physics, The simplest model in GR is obtained when assuming spherical symmetry, with mass-radius diagrams for rotating sequences at 700 Hz, which is about the fastest known pulsar. GR metric reads: sity) and varying this value,played one gets as a a sequence curve of in equilibrium the configurations, mass-radius which can plane be for dis- a given EoS. Each EoS gives hole, while librium, together with the factstructure that of matter such is a madeVolkoff compact (TOV) of system star a [ can cold be perfect described fluid, by at the well-known Tolman – Oppenheimer – with and energy density equations in terms of the coordinate radius mass-radius relation, see Fig. ence between neutron star models inof Newtonian a theory maximal and general mass. relativisticing ones: Indeed, central the if density existence one as solves a forlarge parameter values. spherically in On symmetric the Newtonian configurations contrary, theory with ifreaches of one increas- a gravity, uses mass instead maximal GR, can value there increase andself-gravitating is to object above a of arbitrary which critical mass density it for decreases. which the A mass measure of the influence of GR on a of light for rapidlymakes rotating it stars. often necessary to Thus, use numerical the simulations combination for of the setting equations of arising a global from theoretical these model. fields with overall conditions that cancold, highly hardly asymmetric nuclear be matter; tested general relativity,itational on with field stars Earth. possessing (only a These a very strong black includemagnetic grav- field hole nuclear up can physics, to possess with about a 10 stronger one); electromagnetism, with an intense Compact star numerical models 1. Introduction PoS(MPCS2015)012 ] ] et 6 7 J. Novak ]. These (1986) [ 12 et al. ; and the two-fluid 3 ], showing excellent 13 radii are given. (1998) [ eq @700 Hz @700 Hz R @ 0 Hz circ circ circ eq pol et al. SLy4 R SLy4 R SLy4 R and equatorial library, based on the approach by Bonazzola pol 3 R code by Stergioulas and Friedmann (1995) [ Neutron star radius [km] LORENE rns star. Therefore, one must be careful when defining radii for compact from the ] incorporated realistic EoSs for rapidly rotating neutron star models. 10 12 14 16

8 M ] were the ﬁrst to include magnetic ﬁeld, thus solving coupled Einstein- ]. Here, we shall give some details about rotating relativistic star models 4 9 . 15 1 2

rostar 2.5 1.5 0.5 Neutron star mass [Solar masses] [Solar mass star Neutron (1990) [ ], using the so-called Lewis-Papapetrou coordinates. A year later, Wilson [ ], 5 ], able to describe strongly deformed stars and XNS, handling magnetic field too, (1995) [ 10 ], the widely used 14 (2014) [ et al. 11 Mass radius relations for static and rotating compact stars, using the EoS SLy4 by Douchin & ]. For the rotating case, stars are rotating at 700 Hz, which is about the fastest rotating pulsar ], called slow rotation approximation. The starting point is the solution of the TOV et al. 3 , before showing two lines of improvement of the models: the inclusion of magnetic field, 4 et al. 2 (1989) [ (2002) [ The TOV system has been devised in the 30’s, and is quite easy to solve numerically. It is (1993) [ in Sec. extended the model to theand case Lattimer of differentially rotating stars. Then, Friedmann al. by Pili with Einstein-Maxwell equations and a magnetic field dependent EoS in Sec. Bocquet Maxwell equations. Furthermore,relativistic there star now structure, exist some severalet numerical of al. codes them solving being the publicly rotating available: the KEH solver by Komatsu observed. In the rotating case, both curves for polar a very interesting approach torotating compact test star rapidly models an now have EoSto a compute or long such some history, models. too new The and first idea there(1968) and in exist oldest [ one several microphysics. is easy-to-use the tools However, perturbative approachsystem by upon Hartle and which Thorne a rotatingsmall. solution The is first given numerical in model powersMaschio of of (1971) rotating the [ relativistic angular stars velocity, has supposed been to obtained be by Bonazolla and Figure 1: Haensel [ three numerical approaches have been comparedagreement. by More Nozawa recently, new codes have emerged,et in al. particular, the very accurate one by Ansorg Compact star numerical models The difference between the radius measured atreach the up pole to or 1 at km the for equatorstars: a is 1 if noticeable, rotation too, is as not it taken can into account, important errors can be made. PoS(MPCS2015)012 )- (2.1) (2.2) (2.3) (2.5) (2.6) (2.7) (2.8) 2.2 (2.9) (2.4) is the (2.10) p J. Novak − turn into a ) p ) Ω β + + ε 2 ( , ν ) 2 ( 2 . Γ ) . ∂ν ∞ 4 ( dt = ]: ) ∂ν ∂ ω − ν → E 11 2 − r − ) , − 2 ∂ ∂θ ) N ϕ β / d ∂ω 3 θ ) ( ( ( ∂ω ω ( θ θ 1 tan 2 2 , θ and takes the form (quasi-isotropic 2 − 2 r 2 ) and µν sin sin 2 Ω N θ θ ∂ω ∂ 2 ( 2 + B sin , N 4 r pg r r θ 2 2 2 2 2 2 . ( ln 2 2 r sin B + B 2 ∂ ∂ r b θ sin 3 ∂θ ∂θ = ν B : + ∂ ∂θ u 2 2 − 1 1 r r Br µ sin β a 2 ] + , u U ) + 2 ∂ ) θ ∂θ 4 + + θ ) 2 = A p d 2 p U 2 r r the Lorentz factor. Note that sources of Eqs. ( Bpr for the matter, which implies the absence of merid- 1 U 2 ) r ∂ ∂ U ) 2 1 ln ∂ ∂ r + p 2 + sin p / r r 1 2 + 1 ε ) in spherical symmetry. + E 2 = + NA : ( r + r − 2 b + + 2 ) π E ∂ ∂ 1.3 α E 2 = ( − 2 2 dr B , 2 2 r r r a 16 3 U NA ∂ ∂ + ( ∂ 2 ∂ ∂ ∂ µν ∆ + ( N p π − circularity A p T [ ln 1 = = = = 2 3 ] = 16 + : : : : A θ = 2 − + : 2 3 3 b π = ( ˜ ∆ ∆ ∆ dt ∂ 8 E are respectively the fluid 3-velocity and energy density, both as mea- ν sin 2 Γ ( a r 2 E N ∂ ) = ) A 1 − θ π ) = 4 − and α = sin 2 r + = ]): U NB ds ω ν ), 11 ν ( [( ( -direction, [ 3 3 2 2 ϕ ˜ 2.5 ∆ ∆ ∆ ∆ e.g. )-( 2.2 The matter source terms come from the model of perfect fluid for the energy-momentum ten- As we plan to describe rotating stars in GR, spacetime is assumed to possess some symmetries, In this gauge, the Einstein equation for a star rigidly rotating at the frequency ) contain non-linear metric terms of non-compact support. So, contrary to spherical symmetry 2.5 there is no matching to any knownand vacuum boundary solution conditions at the for surface the of metric the system star can (no only Birkhoff theorem), be imposedsor: at with the additional requirement of with the following notations ( ional currents and no mixed toroidal/poloidalfrom magnetic the field. one Note used that for this the gauge TOV is system quite ( different sured by the locally non-rotating observer: which are described in GRexist by two the Killing existence vector fields: of a Killingclosed timelike orbits, vector for one fields. axisymmetry. to Under account Thus, such for conditions, wesuch stationarity, it suppose and that is that a possible the to spacelike there choose metric one, adapted with coordinates, depends only on two coordinates 2. Stationary and axisymmetric approach gauge) (see system of four coupled non-linear elliptic partial differential equations [ In Eqs. ( Compact star numerical models approach, to describe superfluid phenomena such as pulsar glitches in Sec. velocity in the PoS(MPCS2015)012 T 8 ]. , 4 10 11 ) over (2.11) (2.12) 1–10 s J. Novak → 2.1 ∼ ]), 4 3 P (1993) [ , or in a supercon- 3 et al. ], Beig and Schmidt is the mass felt by a 17 defined by Eq. ( ds in Bonazzola e.g. , gravitational mass . ) ]). They are observed as anomalous X-ray H its total energy density. This simple matter p 2 const ( 20 c ε ]), ε + B , = ) n ε 16 SLy4 by Douchin and Haensel 2001 [ Γ H 5 ( ln p ln e.g. − , and to compute it from the asymptotic behavior of the = ν ], . . . ). H + 19 H its pressure and p 2012 [ is computed integrating the line element et al. T. Such a magnetic field should have an influence on the global structure . With the magnetar model, their dipolar magnetic field is estimated at the ˙ P ]: 11 (Duncan and Thompson 1992 [ . However, it is known that very highly magnetized neutron stars exist, which 1 total angular momentum 11 . Formulas and more details can be found –10 ω 10 ], Gundlach is just the total number of baryons in the star, multiplied by the baryon mean mass. G 18 4 magnetars is the fluid 4-velocity, µ the baryon density. As in the TOV case, the system is closed by the description of micro- u 2003 [ superfluidity can be taken into account, under the two-fluid approachelectromagnetic as field shown in in a Sec. perfect conductor with magnetization, see Sec. a fluid crust with EoS-based model (as an elastic crust, with a solid-type model (Carter and Quintana 1972 [ ductor (Bonazzola and Gourgoulhon 1996 [ B circumferential radius 1 T = 10 The equilibrium of this perfect fluid can be described in the stationary, axisymmetric and Compact stars are known to possess strong magnetic field from pulsar observations. Indeed, In order to extract from the numerical results observationally relevant information, it is nec- n 1 • • • • metric potential with the log-enthalpy a star circumference at thepossible equator to or define passing a through the poles. In our case of axisymmetry, it is pulsars (AXP) or soft gamma-ray repeaters (SGR), they show a low rotation period circular case, by the conservationintegral of of the motion energy-momentum [ tensor, which turns into a simple first their magnetic field can be estimated by the accelerated dipole model to be in the range 10 and scopic properties of the fluid through the EoS particle orbiting around the star;mass in (from our stationarity case property), it canbaryon or be mass as equivalently the computed ADM either massThe as (asymptotic the Komar flatness of spacetime). The 3. Magnetic field for most of pulsars are called and high deceleration essary in GR to define gauge independent quantities. The of the compact star. surface up to 10 Compact star numerical models where model can be extended in the following directions: PoS(MPCS2015)012 , are 23 and, , ]). (3.3) (3.5) (3.6) (3.4) (3.1) (3.2) ψ b 22 29 , J. Novak , 21 αβ 2015 [ F αβ and magnetization F et al. ) µν 4 b g , , H + L ( , . Moreover, only poloidal ν ε . α 2 µν στ F the existence of parallel and g ] is thus quite similar to that µν F . ν F µα + 29 ) [ ∇ x F µν e.g. ψ µ ) F στ ), the equilibrium equation for mat- µ ∇ 0 F 0 1 0 µ 1 D µ . et al. 3.3 σ µ 4 ν x µ α γ 2 F − b − β + + ) − u x ν , energy density τν ( ) , the magnetic field in the fluid rest frame: D free M b f ree σ µ ψ ]. In these studies, very simple EoSs have been µ j , µ µ τ ) b j γ αβ µν 0 H F ( 6 28 m ε ( µ ¯ , νµ 0 ψ p coordinates, as in Sec. ( + x + µ F 1 2 x ) 27 µ τµ , γ θ − 1 = − + , µ M r 1 ν 26 ( µν ν τ α D f µν , : F T F = )( M µ 25 µν x µα ( , F 2 1 ∇ σ µ ¯ 9 F ψ F 0 + = 1 − µ µ ). In particular, the model solves coupled Einstein-Maxwell equa- ∇ µν µν = − ], but with the equations containing new terms depending on the mag- 3.4 T 9 : p g [ = µ x L ∇ + µν ν T u et al. µ u ) p + defined in Eq. ( ε x decoupled from inert mass movement are sources of Maxwell equations, with additional the magnetization tensor in presence of = ( ) in presence of electromagnetic field reads: as functions of both enthalpy and magnetic field. i µν free σ ) j M b µν From the expression of the energy-momentum tensor ( This question has been addressed mainly in two ways: either focusing on microphysics or on The starting point is a microscopic Lagrangian where the density functions of fermions The model, which details are presented in Chatterjee 2.11 , T ]. On the other hand, global models have been computed in axisymmetry, including Lorentz force h H ( with The microphysics calculations take into account the magnetic field in the fluid rest frame netization tions with all fields depending only on coupled to an electromagnetic field presented in Bocquet magnetic field is considered, and therents magnetic moment is aligned withterms the from rotation magnetization axis. Free cur- within our model thex EoS provides pressure ter ( and consistent magnetic pressure [ 24 used, mostly polytropic, which didpresent a not numerical take model into that combines account both the approaches magnetic (see also field. Chatterjee Hereafter, we shall global structure. The first approachand consists uses in rather looking simple at globalis the model, not effect with compatible of the with magnetic TOV the system, fieldassumptions fact assuming on on that spherical the the symmetry, there EoS which structure are of no magnetic the monopoles. energy-momentum tensor However, using ( additional Compact star numerical models perpendicular pressures) several authors have studied magnetic-field dependent EoSs [ From this one gets the microscopic energy-momentum tensor: which, after a thermal average, givesthe the global model: macroscopic energy-momentum tensor, which is used in PoS(MPCS2015)012 x et (3.8) (3.7) ). This J. Novak Bocquet 2.10 e.g. . 0 ]). This occurs when = and the magnetic field at , which means that the i 25 3

b x − , too. Thus the term ∂ M ∂ b 10 b 2 . 0 x . 2001 [ µ x − free et al. , ρ i j b x ρ i ∂ ∂ F p b − ∂ ∂ + ], the last term containing the magnetization Γ i i x x 29 H 7 ln ∂ ∂ ∂ ∂ p H − ∂ ∂ i x (2015) [ = depends on the magnetic field ∂ν ∂ i p p x + ∂ ∂ i p x et al. ∂ ∂ p shows the magnetic field resulting from our simulation for a non- 1 + 2 ε ], which consists in massless three-flavor MIT bag model (with a bag ), together with a NJL-type pairing interaction. As expected, we observe ) 3 30 p + ε ( T. Thick line indicates the star’s surface. 13 10 is the part of the energy-momentum tensor associated with a perfect fluid ( ]). In particular, this means that there should be no assumption about the existence of × ). It means that equilibrium depends only on enthalpy gradient, together with gravita- 9 Magnetic field for a non-rotating star. The gravitational mass is 2 2 . αβ f 3.7 T As an illustration, Fig. As an application, we have implemented the EoS for quark matter in MCFL phase by Noronha 1995 [ rotating star, with adeformation magnetic is field-dependent due to EoS the andfigure Lorentz shows magnetization force, the taken which limits into “pinches” oftoo our the account. strongly code, star distorted too, along The in as the a it magnetic torus-like cannot axes. shape give (see This accurate also results Cardall when the star becomes constant of 60 MeV/fm the first term giving the usualin gradient Eq. of ( enthalpy and the second canceling out with the last term the pole 8 de Haas van Alphen oscillations,overall but effect the on maximal compact star magnetization structure should be small. and Shovkovy (2008) [ Figure 2: expression can be made more explicit: al. tional, centrifugal and Lorentz forces, as for the case without magnetization (Cf. parallel or perpendicular pressure. Compact star numerical models where However, as shown by Chatterjee cancels when one writes that pressure PoS(MPCS2015)012 0), or 6= J. Novak x electromagnetic shows the relative via 3 ]. As a consequence, 31 200 ), within the case we have considered. 3 150 − . p ) 2 : with the loss of energy due to pulsar emis- Ω 4 Am & 30 n 8 Ω 100 Magnetic moment (10 50 are sketched in Fig. p Ω remains almost constant. At a critical lag between both velocities, the EoS(B), no M EoS(B), M n Ω and the pulsar observed frequency. The second fluid is composed of superfluid , linked with strong interaction between nucleons, and a dissipative one, n Ω 0 Ω , linked with pinning and unpinning of superfluid vortices to/from the neutron -4 -4 -4 -4 0.0 = 10 10 10 10 × × × × p

8.0 6.0 4.0 2.0 G G

(no EoS(B),no M) EoS(B),no (no /M M

∆ Ω

max max max Relative influence on the maximal mass, as a function of magnetic moment of the star. entrainment decreases, while p mutual friction Ω Figure 3: The evolution of In this section, we do not consider magnetic field, but superfluid properties of neutron stars. interactions on very shortangular timescales. velocity This fluidneutrons shall with a be slightly higher generically angular called velocity “protons”, with an the magnetic pressure at the center becomes equivalent to the fluid one. Fig. The observations of pulsar glitches, orevidence sudden spin-up, for with the long relaxation existence time-scales of have given a superfluid component in neutron stars [ difference in the computation of thethe maximal magnetic mass moment, of when a taking non-rotatingEoS into compact or star account none of as the them. a magnetization, From function theof this of the figure, magnetic magnetic one field-dependent field can in say the that EoS theThis are influence may very of small be magnetization (less (if due than 10 towould the give very higher specific numbers. EoS However, we ithere have is are chosen to much higher and, be than noted perhaps, the that some strongest the hadronic-type observed very EoS magnetars. high magnetic moments4. used Two-fluid models there should be several dynamicallytwo distinct fluids: components the inside first neutron one is stars. composed of We here all charged consider particles, locked together sion, two components lock together, neutrons givingare part spun of up their (the angular glitch). momentumThere to are After protons, two some which sorts time, of theone, coupling proton called between fluid both relaxes fluids andcalled start occurring slowing in down this again. scenario: a non-dissipative Compact star numerical models PoS(MPCS2015)012 ) p n ], or , (4.4) (4.1) (4.3) (4.2) et al. n n 33 ( entrain- J. Novak time . We have built two such δ and ρ , , 2 ω , ∆ . αβ , σ d g µ µ p p Ψ α n n + pp + np meson). Both were adapted to a two-fluid . p p ) β δ 2 , but with a non-realistic EoS. p K K 2 dn ∆ , α p + + p p µ µ µ n being respectively the 4-currents of neutrons and n 9 µ n (with , n n + p α + n δ n as a function of neutron and proton densities n pn nn n β µ . Equivalently, using the first law of thermodynamics ( p E dn ∆ K K α Ψ n and n µ n = = α n n = = p µ µ n p p E αβ d T p n ]) that momenta are linear combinations of currents, using the meson) and DDH energy density ]. 32 δ e.g. 34 , the corresponding conjugate 4-momenta. Indeed, without entrainment 4- 2005 [

p α :

p angular velocities angular Schematic evolution of the two-fluid angular velocities during glitch phenomena. XY (2009) [ et al. and K n α ], using a similar approach to that of Sec. is the generalized pressure, et al. p Prix 32 Figure 4: Ψ We have developed a new EoS based on microphysics. In the case of two fluids and entrain- To start with, we defined the energy-momentum tensor in the two-fluid case: e.g. The underlying microscopic model is based of relativisticinteractions mean are field modeled theory, by where the nucleon-nucleon exchange of effective mesons system coupled by entrainment following previous approaches by Comer and Joynt (2003) [ where protons, momenta are proportional to 4-velocities and(see therefore, to 4-currents, too. However, here we have the EoS can provide generalized pressurespeed: as a function of both chemical potentials and the relative EoSs: DDH (without the ment matrix and the relative speed between both fluids Gusakov ment the EoS is giving star crust. Here, we show preliminaryfigurations results of from rotating the superfluid numerical neutron computations stars, ofWe considering equilibrium apply con- two-fluid then EoSs these based configurations on to microphysics. a simple numerical model for pulsar glitches. Compact star numerical models Numerical models with two fluids(2005) in [ rotating compact stars in GR were developed by Prix PoS(MPCS2015)012 ]), ), is 40 s (4.6) (4.7) 32 (4.5) et al. . r J. Novak 2.11 τ 2005 [ ], the evolution et al. 38 is the macroscopic ] and Sidery . As expected, the n ]. On the other hand, 5 is the volume element 39 ϖ 36 ). For the case here, we , ϕ n d are given by the evolution Γ (see Prix 4.6 ) (2010) [ θ t 1998 [ ( in Fig. p 6 . dr d Ω − has been constrained to Σ θ 3 r in Eq. ( et al. d τ et al. LORENE and sin 2 ⊥ B ) 2 h t . n r ( τ Br n , we can model the glitch rise by a series ϖ / 2 r t n Ω , . τ A − n n e = int int Γ and Lorentz factor ∝ Γ Γ Σ we neglect during the rise time the energy loss n ) Z 3 t n ) − d ( 5 s, which is compatible with observations. Note n . 10 p i.e. 8 Ω = = + Ω ∼ p n − : ˙ ˙ − (see also Langlois J J ) r p ) p τ t J Ω ( ( , ( n n . There are very few observational constraints concerning the J Ω B c ( = ∆Ω ). In particular, the rising time 4 int & , one finds ], and about a few hours for the Crab pulsar [ Γ p 5 35 Ω ]. As this is much shorter than − n is a geometric coefficient, 37 Ω , but some better resolved observations of pulsar glitches should give us more 2 ⊥ 6 h = − mutual friction moment 10 ∆Ω = B is the ] for details): int the mutual friction parameter. Γ 38 As preliminary results, we show a sequence of equilibrium models, starting from an initial We have presented here several models of rotating compact stars. There is a need for such In the two-fluid model, glitches start when the difference in angular velocities is greater than Following the study in Newtonian framework by Sidery B superfluid vorticity, however that this rising time is quite sensitive to the value of relative difference in angular velocities (glitch amplitude) of 10 2010 [ In this equation, beside the neutron fluid density and angular velocity of protons increases,transferred whereas from that the of fluid neutron ofvelocity. decreases. Typical neutrons rise time to Angular can momentum that be is of deduced from protons. a fit Both of fluids these curves tend by to an exponential a function: common angular In the case displayed in Fig. have taken information about this mostly unknown variable. 5. Conclusion models, where rotation isof taken into compact account, stars. in particular Moreover, if when one looking is at discussing the or magnetic studying field radii in such stars, it is absolutely of the order of 0.1 ms [ rising of the frequency (see Fig. some threshold Compact star numerical models the hydrodynamic time scale, which is the typical time scale for theof equilibrium equilibrium equation configurations ( taken from thewhich numerical has library been updated in orderEoSs to cannot be be able computed to within use the theare rotating realistic then star EoSs equilibrium read described code, here and they above. interpolated are As bytotal saved these the as baryon tables, code. number which The and series angularby of momentum pulsar equilibrium ( emission). models are taken at constant equations for both angular velocities as functions of time – 2 min for the Vela pulsar [ of each specie’s angular momentum Here, PoS(MPCS2015)012 (1939) J. Novak LORENE 55 -equilibrium β (green). The star is 60 Phys. Rev. ) f , 0 p ( Ω p n p crust thermal properties. Ω = f Ω Ω / n 50 )) Ω e.g. 0 ( p Ω − ) 40 t ( p Ω ( 30 time (s) .

11 (red) and M ) 0 ( 20 p Ω / )) 0 ( p 10 o o Ω p n − Ω Ω ) t ( n Ω 0 ( 0 -6 -7 -7 -7 -7 -6

0.0.10 1.0.10 8.0.10 6.0.10 4.0.10 2.0.10 1.2.10

/ / Static Solutions of Einstein’s Field Equations for Spheres of Fluid Ω ∆Ω p Numerical integration of the two fluid angular velocities relative differences as functions of time. 364. The rotating compact star models shown here can be improved in many important ways. First, [1] R.C Tolman, our glitch model needs toconcerning the be EoS. improved Next, and rotating star acrust. models parametric in This study GR is still must quite lack be of importantfeatures. a undertaken, because good many in In implementation observational particular of particular, neutron a glitch star solid crust. models properties depend Therefore, include on more pinning detailed crust and models unpinning are of expected superfluid here, as vortices well in as the library, we have studied effect ofmagnetization magnetic (derivative of field pressure onto with thelatter respect EoS effect to of is magnetic compact very field) stars, small, on asthe because the well there equilibrium global as is equation structure. that a for of The cancellationneutron the of stars, star. terms which depending are Another on very magnetization interesting studywe to in are is model able dealing the to glitch with recover phenomenonobservational two-fluid the in constraints. configurations glitch pulsars. rise of In time, particular, which seems reasonable, having in mind the very few Temperature effects have numerous observational(proto)-neutron stars. consequences, Thus, for rotating instance modelsEoSs including for should temperature the be effects built, cooling of too. out of of References compulsory to go beyond sphericallypublicly symmetric available models. to There study are rotatingtested now or and several magnetic numerical are stars codes now in fast, GR. running These on codes a have simple been laptop. extensively With such codes, taken from the rotating at 400 Hz, with a total baryon mass of 1.7 Figure 5: The plotted quantities are Compact star numerical models PoS(MPCS2015)012 , , 439 , , (1989) ]. 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