PoS(HTRS 2011)034 http://pos.sissa.it/ † ∗ [email protected] Speaker. I acknowledge support from the European Union via a Marie Curie fellowship. In this paper I will discuss how X-rayemission observations mechanisms can be and used interior to constrain dynamics thewill of gravitational briefly wave superfluid review the neutron multifluid stars formalism for inin modelling accreting particular superfluid on systems. neutron the stars r-mode I and instability. thencomposed I focus exclusively will of show neutrons, that protons the (non "minimal"with superconducting) scenario, and i.e. the electrons a observed is neutron temperatures not star consistent andsignificantly spins more rigid of than the expected. An Lowhyperons alternative Mass is or X-ray that that the Binaries, neutron it unless star may coretube the may be interactions. crust be in contain is Finally a I typegravitational will II wave observations discuss superconducting could how state, help future distinguish giving between X-ray rise the missions to models. such strong as vortex/flux LOFT or IXO and ∗ † Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. c

Fast X-ray timing and spectroscopy atFebruary extreme 7-11, count 2011 rates Champéry, Switzerland University of Amsterdam E-mail: Brynmor Haskell Signatures of superfluid neutron-star dynamics PoS(HTRS 2011)034 its i x (2.2) (2.1) (2.3) v is the i j g ]), a General . 2 j x v Brynmor Haskell i ∇ its mass and j x π x m +  Φ x m , + 2 x !# v x j xy m w 1 2 xy − α x y µ ∑ its chemical potential, 

i x 2 ∇ µ 2 x − n j x v + x n  j x x i m D " i j + g x i x π Γ = j x encodes the so-called “entrainment” effect, which leads to a non- v x i  π ] we can write the equations of motion as coupled Euler equations xy j ) = 2 i x α ∇ v x + ]). Furthermore several microphysical inputs are required for the models, n ( 3 x i i ]. In fact NS superfluidity is thought not only to be at the heart of several π ∇ t 1 ∂ + x = n x t i f ∂ K, which will happen very shortly after its birth. This clearly affects the dynamics 9 is the number density of the species, 10 X n ≈ Clearly a multi-disciplinary effort is required to tackle this problem, with contributions coming Let us first of all present the Newtonian multifluid formalism that will be used to discuss Naturally such rich and complex physics does not come without its challenges. Modelling a Neutron stars (NSs) represent one of the best laboratories to test our understanding of matter The momentum of each species (or “fluid”) is: where velocity and the parameter dissipative coupling between the different components. I use covariant notation , and and continuity equations for each species x (where x can indicate neutrons, protons etc...) : from the nuclear physics, neutron star modellingshow and astrophysical how communities. inputs In from this X-ray paper observations Iconstrain can will the be used, physics together of with NS (future)instability. interiors. GW observations, In to particular I will discuss the GW emission2. driven r-mode The multifluid model superfluid NSs. Following [ Relativistic formulation, including the effects ofplete dissipation (for and a vortex review dynamics, see is [ such still as not com- the equation ofuncertain at state, supranuclear transport densities. coefficients, superfluid gaps etc., all of which are highly phenomena, such as radio pulsar glitches,wave but (GW) also emission to mechanisms play and a inin significant determining accreting role the X-ray in response binaries. several of gravitational the star to an external torque superfluid/superconducting NS requires us totons, follow possibly hyperons several or massive deconfined components quarks) weakly (neutrons,Although coupled the pro- by superfluid Newtonian and framework dissipative effects. for such a task has been developed (see e.g. [ of the system, asprotons superfluid neutrons are will likely rotate to byquantised form forming flux a tubes[ a quantised type array II of superconductor, vortices, thus while carrying the stellar magnetic field in at high densities. With coresthe that low exceed the temperature, nuclear high saturation densityparticle density accelerators sector they and of can heavy ion allow the colliders. us QCD Furthermore,NS to phase at interiors, probe the diagram, matter extreme is densities which that believed cannot characterise tobelow be be in studied a in superfluid/superconducting state once the star has cooled Signatures of superfluid neutron-star dynamics 1. Introduction PoS(HTRS 2011)034 ]), the must be 14 ]) and the , ]) and has GW 7 10 11 τ , , 6 9 Brynmor Haskell ]), which is well below the obeys the Poisson equation 13 Φ is set by the fastest process at , V 12 τ . Usually this is illustrated in terms NS with a 10 km radius, assuming an V

τ M ]). ) and then estimating the damping timescale 19 2.2 3 ]) or one could have a mode (and the r-mode is the best ) and ( 18 , . The damping timescale 2.1 ]). The gravitational potential V reprensents the sum of all external forces, in particular the 5 17 τ , x i 4 f = , ] to explain the apparent cutoff in the spin-distribution of these 2 GW 11 τ , while ]. j , 8 x i D , and the system is closed by supplying an equation of state. x ρ x ]) or the magnetic field ([ ∑ 16 Left panel: The r-mode instability window for a 1.4 G , π 15 4 LMXBs were first suggested as sources of GWs more than thirty years ago ([ We shall mainly be interested in the r-mode instability. An r-mode is a toroidal mode of Let us concentrate on the latter mechanism and examine the conditions under which the insta- 1 polytrope for the equation of state and a neutron, proton and electron core. Right panel: A typical = = n thermal instability cycle. The starthe spins shear up into due the to unstable thedown region, below mode after the until which instability it the curve, will thermal and heat the runaway up cycle is rapidly starts halted from again. by neutrino emission.three-dimensional At flat-space this metric. point The it spins encoded dissipative effects, in such the as tensor bulksuperfluid and mutual shear friction force viscosity, ([ are idea was revived by Bildsten [ 3. Gravitational waves from LMXBs and the r-mode instability systems (and of the millisecond radio pulsars) at around 700 Hz ([ candidate) being driven unstable by GW emission ([ oscillation of the NS, for which thein restoring this force is context the as Coriolis force. it It has is particularly been interesting shown to be generically unstable to GW emission ([ bility can occur. The main constraint is that timescale on which GWs drive the mode, ∆Φ been suggested to playthe a following role sections in the analysis thelinearised of spin version the evolution of r-mode of the instability Low will equationsdue Mass to be in various carried X-ray mechanisms, ( out such Binaries as by hyperon (LMXBs).approach, integrating bulk as viscosity the outlined or In in mutual friction, [ following an integral Keplerian breakup limit. The main emissionessentially mechanism two: that could either be a at quadrupolar workcore in ([ “mountain” these is systems present, are sustained by the crust ([ shorter than the timescale on which viscosity acts to damp it, Figure 1: Signatures of superfluid neutron-star dynamics of the critical curve on which PoS(HTRS 2011)034 (4.1) ) shows 2 ” labels the var- Brynmor Haskell i the gravitational ]) and thus assume g 22 ] (such as that due to the 4 42 . 2  ]). The right panel of ( K 27 9 T 10 1 ] would be on the instability window. . , where the subscript ” i 1 τ 29 18 i ∑ 10 Km.  = = 2 V − 1 R τ ]. For such a model the r-mode will however un- 4 20 g cm s ], strong mutual friction [ and a 14 28 ). The duty cycle for such a process is very small, with

10 1 M ]), with respect to the critical curve for the "minimal" NS  4 . 13 ]: = 1 , 4 23 =  26 , M K ]. The star will heat up rapidly from the shear due to the mode until 6 e f f 25 T , 21 10 G and shows no indication of additional GW torques.  24 8 ) I show the inferred temperatures and spins of the LMXBs for which we have 2 is the observed temperature, obtained from spectral fitting, and ) I show the typical instability window for a “minimal” neutron star model. I assume a 1 e f f T In figure ( This clearly indicates the need for additional physics in our model (as do theoretical calcula- It is clear from the discussion in the preceding section that in order to understand a system’s acceleration at the NS(when surface. most of We the can thermalfrom then emission the disc) is use to thought X-ray obtain to an observationsstar. come estimate In of of directly particular the LMXBs we from core shall the in temperature take NS by quiescence assuming surface a rather mass than and radius for the what the effect of hyperonvortex/flux bulk tube viscosity interaction [ in the core) or a rigid crust [ tions of NS temperatures in the presence of an unstable r-mode [ that the core temperature isconsidering simply accreting that systems at we the shalland use base interior the of temperature relation the given between in ’heat [ the blanketing’ effective envelope. surface temperature As we are Where model. It is clearappear that, to even be with well the inside’catch’ inherent many the uncertainties systems instability in in window. the the unstablea Given temperatures, region measured the and, several spin furthermore, short systems up one duty in of cyclefield the outburst of systems, we and approximately IGR spin do 10 00291, down not has in expect quiescence, to which is consistent with a magnetic estimates of both (from [ behaviour with respect to the r-mode instability weis need not to estimate a the straightforward NS’s task, core temperature. asmap This the the stellar observed interior surface cannot emission be toand probed a crust directly, core of but temperature. rather the To we star do needcrust are this to is nearly we high, isothermal shall as assume (which indicated that is by the very recent core nearly cooling the observations of case X-ray if transients the [ conductivity of the ious dissipative processes that are atIn work, figure i.e. ( shear viscosity, bulk viscosity, mutual friction etc.. the system spending less than athe mode, few in percent the of unstable the region time,estimates where depending of it NS on is spins the emitting and saturation GWs. temperatures amplitude Let compare of us to thus this examine picture. how observational 4. Neutron star core temperatures the thermal runaway is halted bythe neutrino instability emission, curve, after as which shown the in star figure rapidly ( spins down below Signatures of superfluid neutron-star dynamics a given spin rate and temperature and is such that core of neutrons, non-superconducting protons andis electrons taken and to the be damping due atmainly to high due bulk temperature to viscosity shear from at modified thedergo URCA crust/core a reactions interface thermal [ while instability at [ low temperature it is PoS(HTRS 2011)034 01 for weak . 0 ], even though = Brynmor Haskell 32 ]. R 33 ] with K, the thermal runaway 4 8 ] taken to be 0.5, of hyperon bulk 29 K and 10 7 , 673 (1969) ]. Such a scenario would have clear obser- 31 224 , 30 5 ]. 34 1 and of strong mutual friction from [ , 5505 (2006) = 23 ξ ] with the parameter 28 The left panel shows the estimated core temperatures of the LMXBs with respect to the "minimal" We have shown how current X-ray observations can be used to constrain the physics of the [2] N. Andersson, G. L. Comer, CQG [1] G. Baym, C. J. Pethick, D. Pines, Nature (London) r-mode instability window, andconstrain thus the models of it the is however complex necessaryspins. to physics Timing obtain of of further these observations systems NS of is NSand interiors. also temperatures IXO crucial, and would a In be task for ideally order which suited.is the to likely This proposed to truly is X-ray be especially missions very LOFT important challengingfrom to as detect, the the and background GW detailed noise signature modelling of of is the such thus detectors events crucial [ to extract the signal could be halted; resulting in thea star rate reaching that an balances equilibrium the along spinup the accretion curve torque and [ emitting GWs at recent assessments of the spinthat evolution there of is a this need system, for together extra with GW spin XTE down 1814, torques do to explain not this suggest behaviour5. [ Conclusions References vational consequences on the timing asbeen there would suggestions be that an the additional spin spin down of torque. SAX There J1808 have may be constant during outburst [ superfluidity. All these additional mechanism could make theobtain observed further systems observational stable constraints and to it distinguish isample, thus between the imperative the instability to models. curve rises Furthermore with if, temperature for between ex- 10 Figure 2: instability curve. Clearly manypanel systems shows would the effect be of inparticular different the physical I unstable mechanisms show the which region, effect could which of be is a at unexpected. rigid work crust, to The with stabilise the right the slippage r-mode. factor In of [ Signatures of superfluid neutron-star dynamics viscosity from [ PoS(HTRS 2011)034 Brynmor Haskell , 584 (2009) 692 , 231101 (2007) 99 , 1409 (2011) , 381 (2001) 10 412 , 1424 (2007) , 839 (2008) 660 , 101101 (2011) , 103009 (2009) 389 , 1224 (2002) , 307 (1999) 79 , 1464 (2009) 107 , 415 (1996) 337 516 , 902 (2000) , 162 (2006) 397 323 319 368 6 , 044040 (2006) 74 , 501 (1978) , 714 (1998) , 1897 (2010) , 84001 (2006) 184 , 1020 (2009) 502 73 408 , 1044 (2005) , L14 (2011) 698 , 345 (1984) 623 , 211101 (2005) 738 , 42 (2003) 278 95 424 , 084025 (2002) , 708 (1998) , 2673 (2009) 66 , L89 (1998) , 909 (2010) 502 702 , 328 (1999) 501 , (2007), 1. URL (cited on 15/04/2011): http://www.livingreviews.org/lrr-2007-1 722 517 10 R. 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