SStellar Superfluids
Dany Page , James M. Lattimer , Madappa Prakash , and Andrew W. Steiner Instituto de Astronom´ıa, Universidad Nacional Aut´onoma de M´exico,Mexico, DF 04510, Mexico Department of Physics and Astronomy, State University of New York at Stony Brook,Stony Brook, NY 11794-3800, USA Department of Physics and Astronomy, Ohio University,Athens, OH 45701-2979, USA Institute for Nuclear Theory, University of Washington,Seattle, WA 98195, USA
Abstract
Neutron stars provide a fertile environment for exploring superfluidity underextreme conditions. It is not surprising that Cooper pairing occurs in densematter since nucleon pairing is observed in nuclei as energy differences be-tween even-even and odd-even nuclei. Since superfluids and superconductorsin neutron stars profoundly affect neutrino emissivities and specific heats, theirpresence can be observed in the thermal evolution of neutron stars. An ever-growing number of cooling neutron stars, now amounting to 13 thermal sources,and several additional objects from which upper limits to temperatures can beascertained, can now be used to discriminate among theoretical scenarios andeven to dramatically restrict properties of nucleon pairing at high densities. Inaddition, observations of pulsars, including their spin-downs and glitch histories,additionally support the conjecture that superfluidity and superconductivity areubiquitous within, and important to our understanding of, neutron stars.
In this contribution, we describe the roles of neutron superfluidity and protonsuperconductivity in the astrophysical setting of neutron stars, drawing uponlessons learned from similar phenomena occuring in laboratory nuclei. We willfocus on both the thermal evolution (i.e., cooling) as well as the dynamicalevolution (i.e., spin-down) of neutron stars. In the former, pairing dramaticallyaffects neutrino emission processes and the specific heat of dense matter. Inthe latter, pairing may be responsible for the observed anomalous values of thebraking index and the glitch phenomenon. We will also briefly describe thepossibility of pairing in the presence of hyperons and deconfined quarks.In fermionic systems, superfluidity and superconductivity occur due to thepairing of neutral and charged fermions, respectively. The
Cooper Theorem [1]1 a r X i v : . [ a s t r o - ph . H E ] M a r tates that, in a system of degenerate fermions the Fermi surface is unstabledue to the formation of “pairs” if there is an attractive interaction in some spin-angular momentum channel between the two interacting particles. The essenceof the BCS theory [2] is that as a result of this instability there is a collectivereorganization of particles at energies around the Fermi energy and the appear-ance of a gap in the quasi-particle spectrum. This reorganization manifests itselfin the formation of “Cooper pairs”. At high-enough temperature, the energygap disappears and the system reverts to its normal state.To begin, in Sec. 2, we motivate the existence of pairing in neutron stars byexamining the pairing phenomenon in laboratory nuclei. We then summarizethe relevant properties of neutron stars, including their interior compositionsand properties of their crusts in Sec. 3. We describe in Sec. 4 how pairing indense matter is achieved and, in Sec. 5, we present a brief description of ex-pectations of pairing in deconfined quark matter that may be present in theinner core of the most massive neutron stars. Initially, neutron stars cool pri-marily due to neutrino emission in their interiors before surface photon coolingtakes over later in their lives. In Sec. 6, we summarize the various neutrinoprocesses that can occur in dense matter. Theoretically, these processes canproceed either very rapidly (enhanced neutrino emission) or relatively slowly.We also describe an important secondary process that greatly influences the in-terpretation of observations – neutrino pair emission from the pair breaking andformation (PBF) of Cooper pairs. This process is triggered as the ambient tem-perature, decreasing because of cooling, approaches the critical temperatures forsuperfluidity and superconductivity. The discussion in this section shows howthe existence of superfluidity or superconductivity dramatically influences neu-trino emissions, leading to both the quenching of enhanced neutrino emissionand bursts of neutrino emission due to the PBF processes.We present in Sec. 7 simplified analytical models of neutron star cooling inorder to gain physical insight. These analytical models are complemented by de-tailed numerical simulations which include general relativity and state-of-the-artmicrophysics, such as the dense matter equation of state, thermal conductivi-ties, neutrino emissivities, and specific heats. We also summarize in this sectionthe abundant observational data, consisting of estimates of surface temperaturesand ages, which collectively describe the thermal evolution of neutron stars. Wewill show in Sec. 8 that the bulk of this data supports the so-called “MinimalCooling Paradigm”, which supposes that no drastically enhanced neutrino emis-sion processes occur, or if they do, they are quickly quenched by superfluidityor superconductivity. Nevertheless, a few sources suggesting enhanced coolingare observed, and we discuss the implications.In Sec. 9, we describe an outstanding recent development in which the firstreal-time cooling of any isolated neutron star – the young neutron star in thesupernova remnant of Cassiopeia A – is observed. The observed rate of coolingis more than 10 times faster than expected, unless both neutron superfluidityand proton superconductivity are present in the star’s core. These observationsprovide the first direct evidence for superfluidity or superconductivity in theinterior of a neutron star, and can be verified by continued observations of the2eutron star in Cassiopeia A.Section 10 is devoted to other observations of neutron stars and their dynam-ical evolution that may also indicate the presence of superfluidity. One concernsthe deceleration observed in the spin-down of pulsars, which could be due tosuperfluidity in their cores, while the other is related to sporadic spin jumps,commonly known as glitches, thought to stem from superfluidity in neutron starcrusts.Summarizing discussions and conclusions are contained in Sec. 11. Soon after the development of the BCS theory, Bohr, Mottelson & Pines [3]pointed out that excitation energies of nuclei exhibit a gap, as shown in theleft panel of Fig. 1. A nucleon in even-even nuclei, whether neutron or proton,clearly requires a minimum energy for excitation. This energy was interpretedas being the binding energy of a Cooper pair which must break to produce anexcitation. In contrast, odd-even nuclei do not show such a gap, and this is dueto the fact that they have one unpaired nucleon which can be easily excited.The right panel of Fig. 1 shows that pairing also manifests itself in the bindingenergies of nuclei, even-even nuclei being slightly more bound than odd-even orodd-odd nuclei .
150 170 190 210 230 2500.10.51.0E (MeV) Mass number Aodd−even/even−odd nucleieven−even nuclei G A P
50 100 150 200 250Mass number A8.68.07.4E/A (MeV)
Figure 1: Left panel: Lowest excitation levels of nuclei (adapted from [3]). Rightpanel: Binding energy per nucleon for the most beta-stable isobars (adaptedfrom [4]).The systematics of neutron pairing energies in nuclei, defined through∆
Z,N = ±
12 ( E Z,N +1 − E Z,N + E Z,N − ) , (1) Notice that, as a result of pairing, the only stable odd-odd nuclei are H( Z = 1 , N = 1), Li(3,3), B(5,5), and N(7,7). All heavier odd-odd nuclei are beta-unstable and decay intoan even-even nuclei. ! ( ) " ( M e V ) Neutron number N Odd N -0.5 0 0.5 1 1.5 2 2.5 3 20 40 60 80 100 120 140 ! ( ) e ( M e V ) Neutron number N Even N
Figure 2: Upper panel: Odd-N pairing energies. Lower panel: Even-N pairingenergies. Figure adapted from [5].where E Z,N is the binding energy for charge Z and neutron number N , and+( − ) refers to odd-N and even-N nuclei, are shown in Fig. 2. A few key factsto note are:1. Pairing energies range from about 3 to 0.5 MeV, decreasing in magnitudewith increasing neutron numbers; their behavior with the mass number A = N + Z is well fit by [5]∆ N,Z = 24 /A + 0 . ± . , for N odd , (2)∆ N,Z = 41 /A + 0 . ± . , for N even . (3)2. Dips (peaks) occur adjacent to (at) the sequence of magic numbers N = 14 , , ,
82 and 126 for N odd (even).Systematics of proton pairing energies for odd- and even- Z as a function ofproton number Z (see Fig. 2 and Table 1 of [5]) show similar qualitative be-4avior, but the magic number effects are less pronounced. If the pairing energywas to be extrapolated to infinite matter using Eq. (2) and A → ∞ , the gapwould vanish. As gaps in infinite matter are predicted to be finite, a saturationpenomenon is at play. Extended matter in He, albeit for different reasons, alsoexhibits finite gaps.In addition to the excitation spectra and binding energies of nuclei, the pair-ing phenomenon plays important roles in the dynamical properties of nuclei suchas their rotational inertia and the large amplitude collective motion encounteredin fissioning nuclei. Tunnelling effects in spontaneously fissioning nuclei receivean enhancement factor 2∆ /G , amounting to an order of magnitude or more ( G is a typical pairing interaction matrix element between neighboring mean-fieldconfigurations).Pairing effects are also evident in nuclear reactions. For example, thermalneutrons of energy only ∼ .
025 eV are needed to cause the fission of
U(which results in the even-even compound nucleus
U), whereas fast neutronsof higher energy ∼ U (the compoundnucleus in this case is the even-odd nucleus
U). What is interesting is that thisphenomenon was appreciated well before the BCS theory was formulated, andit lies at the root of building nuclear reactors and purifying naturally-occurringuranium to contain more of the 235 isotope than the 238 isotope.Besides BCS pairing, the pairing energies shown in Fig. 2 receive contribu-tions from other sources since nuclear sizes are much smaller than the coherencelength of the pairing field. The odd-even staggering is caused by a combina-tion of effects including the pair-wise filling of the orbitals, diagonal matrixelements of the two-body interaction, three-nucleon interactions, the bunchingof single particle levels near the Fermi energy, and the softness of nuclei withrespect to quadrupolar deformations. The global description of the pairing phe-nomenon in nuclei is based on the Hartree-Fock-Bogoliubov approximation andrecent accounts may be found in Refs. [5, 6, 7]. The basic cause for pairing innuclei is, however, easy to identify. The nuclear interaction between identicalnucleons is strongly attractive in the spin S = 0 channel, the di-neutron beingnearly bound. The even stronger attraction between neutrons and protons inthe spin S = 1 channel produces bound deuterons, but its effects are mitigatedin heavy nuclei due to the imbalance of neutrons and protons and attendantmany-body effects. In any case, with proton and neutron pairing energies onthe order of an MeV, nuclei represent the highest temperature superconductorsand superfluid objects in the laboratory. It is interesting that gaps of similarorder-of-magnitude are expected for nucleon pairing in neutron stars. Given that nucleon pairing is important in nuclei, we should expect that pairingwill also occur within neutron stars, as was originally pointed out by Migdal in1959 [8]. Although matter within neutron stars may be heated to more than10 K during birth, and may remain warmer than 10 K for hundreds of thou-5ands of years, the nucleons are generally extremely degenerate. Furthermore,given the high ambient densities, the critical temperatures for pairing to occurare large, ∼ − K. The onset of pairing is expected to take place in someparts of a neutron star’s interior within minutes to thousands of years afterbirth, and is expected to lead to alteration of several important properties ofmatter. While pairing will not affect the pressure-density relation significantly and, therefore, the overall structure of neutron stars, the specific heat of densematter and the emissivity of neutrinos are dramatically influenced. Both emis-sivites and specific heats are altered at and below the critical temperature, andwhen the temperature falls well below the pairing critical temperature, bothvanish exponentially. This has important consequences for the thermal evolu-tion of neutron stars that will be described in several subsequent sections of thischapter.Superfluidity can also be important in the dynamical evolution of neutronstars. It has long been suspected that the so-called “glitch” phenomenon ob-served in pulsars is due to the existence of superfluids within neutron star crustsand perhaps their outer cores. Superfluidity within neutron stars might also sig-nificantly contribute to the anomalous braking indices, which are related to theobserved long-term deceleration of the spin-down of pulsars. These phenomenawill be discussed in Sec. 10.Neutron stars contain the densest form of cold matter observable in theUniverse, in excess of several times the central densities of nuclei (which is oftenreferred to as the nuclear saturation density ρ nuc (cid:39) . × g cm − ). Notethat ρ nuc corresponds to the density where cold matter with a proton fraction x p = 1 / M (cid:12) are thought to producebriefly existing proto-neutron stars before they collapse further into black holes[10]. Nevertheless, the vast majority of gravitational-collapse supernovae, dueto the preponderance of lower-mass stars, will produce stable neutron stars.Two simple arguments can convince us that neutron stars, very small andvery dense, can exist. First, consider the fastest known radio pulsar, Terzan5 ad (AKA PSR J1748-2446ad) [11], and posit that the observed period of itspulses, P = 1 .
39 ms, is its rotational period. (Pulses produced by binaries oroscillations of neutron stars are ruled out because nearly all pulsars are observedto be slowing down, while orbital and vibrational frequencies increase as energyis lost.) Using causality , that is, imposing that the rotation velocity at its At asymptotically-high densities where deconfined quark matter is thought to exist, pair-ing gaps could be of order 100 MeV, in which case the EOS is moderately affected by thepairing phenomenon. c , one obtains v eq = Ω R = 2 πRP < c ⇒ R < cP π = 65 km . (4)This value of 65 km for the radius R is but a strict upper limit; detailed theo-retical models and observations indicate radii on the order of 12 km. Secondly,assuming that the star is bound by gravity , we can require that the gravitationalacceleration g eq at the equator is larger than the centrifugal acceleration a eq and obtain g eq = GMR > a eq = Ω R = 4 π RP or MR > π GP ⇒ ρ = M πR > × g cm − . (5)Obviously, Newtonian gravity is not accurate in this case, but we can neverthe-less conclude that the central density of these stars is comparable to, or likelylarger than, ρ nuc . Theoretical models show that densities up to 10 ρ nuc [12]are possibly reachable. In short, a neutron star is a gigantic, and compressed,nucleus the size of a city. A “pure neutron star”, as originally conceived by Baade & Zwicky [9] andOppenheimer & Volkoff [13], cannot really exist. Neutrons in a ball shoulddecay into protons through n → p + e − + ν e . (6)This decay is possible for free neutrons since m n > m p + m e , where the massesdenote rest masses. However, given the large densities expected within theneutron star interior, the relevant quantities are not the masses, but instead thechemical potentials µ i ( i denoting the species) of the participants. The matter isdegenerate as typical Fermi energies are on the order of 10 −
100 MeV, whereasthe temperature drops below a few MeV within seconds after the birth of theneutron star [14]. Starting with a ball of nearly degenerate neutrons, the decayof Eq. (6) will generate a degenerate sea of protons, electrons and anti-neutrinos.The interaction mean free paths of anti-neutrinos (and neutrinos) far exceed thesize of the star. In a neutron star, these can be assumed to immediately vacatethe star, implying that µ ν = 0. Thus, this reaction will ultimately result in the β -equilibrium condition µ n = µ p + µ e , (7)which is equivalent to the energy minimization condition ∂ε/∂x p = 0, where ε is the energy density and the proton fraction is x p . At finite, but smalltemperatures, the inverse reaction p + e − → n + ν e (8)7lso occurs since not all particles are in their lowest energy states at all times.A neutron star, however, is not born from the collapse of a “ball of neutrons”,but rather from the collapse of the iron core of a massive star. At densitiestypical of pre-collapse configurations, ρ ∼ g cm − and x p > ∼ .
4. Duringcollapse, the reaction Eq. (8) therefore initially dominates over Eq. (6) in orderto reduce the proton fraction. As the density increases, however, the neutrinomean free paths become smaller than the collapsing core’s size, so ν e temporarilybecome trapped within the core. In this case, µ ν e >
0, altering β -equilibriumand permits x p to remain relatively large. Only after neutrinos are able todiffuse away, a time of approximately 10 seconds [14], will the final β -equilibriumcondition Eq. (7) be achieved.Eq. (7) allows one to determine the composition of cold dense nucleonicmatter. Near ρ nuc , a rough approximation to the difference of nucleon chemicalpotentials is µ n − µ p (cid:39) S v (cid:112) ρ/ρ nuc (1 − x p ) , (9)where S v (cid:39)
31 MeV is the bulk nuclear symmetry energy parameter. In denseneutron star matter, electrons are relativistic and degenerate, so charge neu-trality implies equality between electron and proton number densities and µ e = ¯ hc (cid:0) π n B x p (cid:1) / , (10)where n B = n n + n p = ρ/m B is the baryon density and m B is the baryon mass.Therefore, Eq. (7) implies x p = (cid:18) S v ¯ hc (cid:19) m B π ρ nuc (cid:18) ρρ nuc (cid:19) / (1 − x p ) . (11)This is a cubic equation for x p ( ρ ), and at the density ρ nuc the solution is x p, nuc (cid:39) . x p increases roughly with √ ρ near ρ nuc .Notice that once µ e > m µ (cid:39)
105 MeV, muons will appear and be stablypresent with the conditions of beta equilibrium µ µ = µ e (12)and charge neutrality n p = n µ + n e . (13)The condition for the appearance of muons is fulfilled when the density is slightlyabove ρ nuc . However, even though n p and n e are no longer equal, the trend that x p slowly increases with density is not altered by the presence of muons. Fur-thermore, in all processes we describe below, there will always be the possibilityto replace electrons by muons when the density is large enough for them toappear.At all but the highest densities, nucleons can be regarded as non-relativisticin neutron star matter, but muons can be either relativistic or non-relativistic8 eutron vortex AtmosphereEnvelopeCrustOuter coreInner core
Nuclei in a lattice +Neutron superfluidNeutron vortex proton superconductorMagnetic flux tubeNeutron superfluidhomogeneous
Core: Crust: nuclei+neutronsuperfluidmatter S p a gh e tti L a s a gn ac h ee s e S w i ss CA B
Figure 3: Schematic illustration of the structure of a neutron star; figure takenfrom [15]. The outermost layers of a neutron star, the atmophere, envelope,and crust are described in Sec. 3.2. Superfluidity in the crust is schematicallyrepresented in inset “A”, and a diagram of the pasta phases in the crust is shownin inset “B”. The core is separated into the outer core, which has the structuregiven in inset “C”, and the inner core whose nature is currently unknown.9epending on their abundance. For nucleons, the generalization of the simpleapproximation Eq. (9) is : µ n = m n + p F n m n + V n , µ p = m p + p F p m p + V p . (14)For muons, µ µ = (cid:113) m µ c + p F µ c , (15)where p F i = ¯ hk F i is the Fermi momentum of species i , and V n and V p are themean-field energies of n and p . The Fermi momenta are related to the particledensities by k F i = 3 π n i . For the leptons, V e and V µ are negligibly small.With a knowledge of V n and V p , the two β -equilibrium relations Eqs. (7) and(12) can be solved. With four chemical potentials and two equations, a uniquesolution is obtained by imposing charge neutrality, Eq. (12), and fixing n B .With the particle densities and chemical potentials known, one can calculateany thermodynamic potential, in particular the pressure P and energy density (cid:15) = ρc . Varying the value of n B gives us the equation of state (EOS): P ( ρ ).Given an EOS, an integration of the Tolman-Oppenheimer-Volkoff (TOV, [13])equations of hydrostatic equilibrium provides us with a well defined model of aneutron star.The potentials V n and V p in Eq. (14) turn out to be rapidly growing functionsof density, and one can anticipate that eventually reactions such as p + e − → Λ + ν e and / or n + e − → Σ − + ν e (16)may produce hyperons. Hyperons can appear once the corresponding β -equi-librium conditions are satisfied, i.e., µ n = µ Λ or/and µ n + µ e = µ Σ − . At thethreshold, where p F Λ = 0 or p F Σ − = 0, one can expect that | V Λ | (cid:28) m Λ and | V Σ − | (cid:28) m Σ − and thus µ Λ (cid:39) m Λ and µ Σ − (cid:39) m Σ − . Since m Λ and m Σ − arelarger than the nucleon mass by only about 200 MeV these hyperons are goodcandidates for an “exotic” form of matter in neutron stars. Along similar lines,the lightest mesons, pions and/or kaons, may also appear stably, and can formmeson condensates. At even larger densities, the ground state of matter islikely to be one of deconfined quarks. All these possibilities depend cruciallyon the strong interactions terms, V n and V p . Figure 3 illustrates our presentunderstanding (or misunderstanding) of the interior of a neutron star, with ablack question mark “ ? ” in its densest part. The outer part of the star, its crust ,is described briefly in the following subsection.When only nucleons, plus leptons as implied by charge neutrality and con-strained by β -equilibrium, are considered, the EOS can be calculated with muchmore confidence than in the presence of “exotic” forms of matter. For illustra-tive puposes, we will generally employ the EOS of Akmal, Pandharipande &Ravenhall [16] (“APR” hereafter) in presenting our results. Relativistic expressions for µ n and µ p also exist, but are omitted here in the interest ofsimplicity. The Σ + is less favored as its β -equilibrium condition is µ Σ + = µ p = µ n − µ e . Heavierbaryon are even less favored, but cannot a priori be excluded. In the outer part of the star, where ρ < ∼ ρ nuc /
2, a homogeneous liquid of nucle-ons is mechanically unstable (known as the spinodal instability). Stability is,however, restored by the formation of nuclei, or nuclear clusters. This region,called the crust , has a thickness of < ∼ atmosphere , but there is the possibility of havinga solid surface, condensed by a sufficiently-strong magnetic field [17]. A fewmeters below the surface, ions are completely pressure-ionized (the radius ofthe first Bohr orbital is larger than the inter-nuclear distance when ρ > ∼ gcm − ). Matter then consists of a gas/liquid of nuclei immersed in a quantumliquid of electrons. When ρ ≈ g cm − , µ e is of the order of 1 MeV and theelectrons become relativistic. Here, and at higher densities, Coulomb correctionsare negligible – electrons form an almost perfect Fermi gas. However, Coulombcorrections to the ions are not negligible. From a gaseous state at the surface,ions will progressively go through a liquid state (sometimes called the ocean )and finally crystallize, at densities between 10 up to ∼ g cm − dependingon the temperature (within the range of temperatures for which neutron starsare thermally detectable). These low-density layers are commonly referred toas the envelope .With growing ρ , and the accompanying growth of µ e , it is energeticallyfavorable to absorb electrons into neutrons and, hence, nuclei become progres-sively neutron-rich. When ρ ∼ × g cm − (the exact value depends on theassumed chemical composition), one achieves the neutron drip point at whichthe neutron density is so much larger than that of the protons that some neu-trons become unbound (i.e., µ n > innercrust . In most of this inner crust, because of the long-range attractive natureof the nucleon-nucleon interaction, the dripped neutrons are predicted to forma superfluid (in a spin-singlet, zero orbital angular momentum, state S ). Allneutron stars we observe as pulsars are rotating. While a superfluid cannotundergo rigid body rotation, it can simulate it by forming an array of vortices(in the cores of which superfluidity is destroyed). (See, e.g., [18].) The resultingstructure is illustrated in inset A of Fig. 3.At not too-high densities, nucleons are correlated at short distances by stronginteraction and anti-correlated at larger distances by Coulomb repulsion be-11ween the nuclei, the former producing spherical nuclei and the latter resultingin the crystallization of the matter. As ρ approaches ∼ . ρ nuc , the shapesof nuclei can undergo drastic changes: the nuclear attraction and Coulomb re-pulsion length-scales become comparable and the system is “frustrated”. Fromspherical shapes, as the density is increased, nuclei are expected to deform, be-come elongated into 2D structures (“spaghetti”), and then form 1D structures(“lasagna”), always with denser nuclear matter surrounded by the dilute neu-tron gas/superfluid which occupies an increasing portion of the volume. Whenthe phases achieve approximately equal volume fractions, the geometry can in-vert, with dripped neutrons confined into 2D (“anti-spaghetti” or “ziti”) andfinally 3D (“swiss cheese”) bubbles. The homogeneous phase, i.e. the core ofthe star, is reached when ρ (cid:39) . − . ρ nuc . This “pasta” regime is illustrated ininset B of Fig. 3 and is thought to resemble a liquid crystal [19]. A compilationof the most recent progress on neutron star crust physics can be found in thebook [20]. Expectations from measured phase shifts
As a two-particle bound state, the Cooper pair can appear in many spin-orbitalangular momentum states (see the left panel of Fig. 4). In terrestrial super-conducting metals, the Cooper pairs are generally in the S channel, i.e., spin-singlets with L = 0 orbital angular momentum, whereas in liquid He theyare in spin-triplet states. What can we expect in a neutron star? In the rightpanel of Fig. 4, we adapt a figure from one of the first works to study neutronpairing in the neutron star core [21] showing laboratory-measured phase-shiftsfrom nucleon-nucleon scattering. A positive phase-shift implies an attractiveinteraction. From this figure, one can expect that nucleons could pair in a spin-singlet state, S , at low densities, whereas a spin-triplet, P , pairing shouldoccur at high densities. We emphasize that this is only a presumption (phaseshifts reflect free-space interaction) as medium effects can strongly affect particleinteractions. The energy gap
In a normal Fermi system at T = 0, all particles are in states with energies (cid:15) ≤ (cid:15) F . When T >
0, states with energies (cid:15) > ∼ (cid:15) F can be occupied (left panelof Fig. 5) resulting in a smearing of the particle distribution around (cid:15) F in arange ∼ k B T . It is precisely this smooth smearing of energies around (cid:15) F whichproduces the linear T dependence of c v , Sec. 7.1, and the T or T dependenceof the neutrino emissivities, Sec. 6.5.In a superfluid/superconducting Fermi system at T = 0, all particles are instates with energies (cid:15) ≤ (cid:15) F (actually, (cid:15) ≤ (cid:15) F − ∆). For nonzero temperatures12 pin−triplet pairsS = 1Spin−singlet pairsL > 0L = 0 S = 0 L > 0L = 0 −20 o −30 o E (MeV) labN−N
E (MeV) F (10 g cm ) ρ
14 −3
Phase shift (in degrees) o o o o −10 o
100 200 300 400 500 60025 501 2 754 1006 8125 10 1215030 P S P D G P S Figure 4: Left panel: Possible spin-angular momentum combinations forCooper-pairs. Right panel: Phase shifts for N-N scattering as a function ofthe laboratory energy (middle axis) or the neutron Fermi energy and densityfor a neutron star interior (lower axis). Adapted from [21].that permit the presence of Cooper pairs (and hence a gap ∆( T )), states withenergy (cid:15) ≥ (cid:15) F + ∆ can be populated. However, in contrast to the smooth fillingof levels above (cid:15) F in the case of a normal Fermi liquid, the presence of the2∆( T ) gap in the spectrum implies that the occupation probability is stronglysuppressed by a Boltzmann-like factor ∼ exp[ − T ) /k B T ]. As a result, boththe specific heat of paired particles and the neutrino emissivity of all processesin which they participate are strongly reduced. The phase transition
The transition to the superfluid/superconducting state through pairing `a laBCS is usually a second order phase transition and the gap ∆( T ) is its orderparameter (see central panel of Fig. 6). Explicitly, ∆( T ) = 0 when T > T c ,the critical temperature, and, when T drops below T c , ∆( T ) grows rapidly butcontinuously, with a discontinuity in its slope at T = T c . There is no latent heatbut a discontinuity in specific heat. (Examples: superfluid ↔ normal fluid; fer-romagnetic ↔ paramagnetic.) In the BCS theory, which remains approximatelyvalid for nucleons, the relationship between the zero temperature gap and T c is∆( T = 0) (cid:39) . k B T c . (17)In a first order phase transition there is a discontinuous change of ∆( T ) at T c and the transition occurs entirely at T c (see left panel of Fig. 6). There is a latentheat due to the entropy difference between the two states. (Examples: solid ↔ liquid; liquid ↔ gas below the critical point.) In a smooth state transition there13 ε Normal Fermi Liquid Superfluid Fermions ε F k ε ∆ε (k)k F (k) ε k F ε F Figure 5: Comparison of quasi-particle spectra, (cid:15) ( k ), for a normal and a super-fluid Fermi liquid. The reorganization of particles at (cid:15) ∼ (cid:15) F into Cooper pairsresults in the development a gap 2∆ in the spectrum so that no particle canhave an energy between (cid:15) F − ∆ and (cid:15) F + ∆. (T) ∆ (T) ∆ T ∆ T c T c (T) ∆∆ T ∆ TFirst Order Phase Transition Second Order Phase Transition Smooth State Transition
Figure 6: Temperature evolution of the state of a system parametrized by an“order” parameter, ∆( T ).is a continuous change of ∆( T ) with no critical temperature (see the right panelof Fig. 6). (Examples: liquid ↔ gas above the critical point; atomic gas ↔ plasma.) A simple example
A simple model can illustrate the difficulty in calculating pairing gaps. Con-sider a dilute Fermi gas with a weak, attractive, interaction potential U . Theinteraction is then entirely described by the corresponding scattering length , a ,which is negative for an attractive potential. In this case, the model has a singledimensionless parameter, | a | k F , and the dilute gas corresponds to | a | k F (cid:28) U (which is calledthe BCS approximation ), the gap equation at T = 0 can be solved analytically, The scattering length a is related to U by a = ( m/ π ¯ h ) U with U k = (cid:82) d r exp( i k · r ) U ( r ). weak-coupling BCS-approximation gap:∆( k F ) | a | k F → −−−−−→ ∆ BCS ( k F ) = 8 e (cid:18) ¯ h k F M (cid:19) exp (cid:20) − π | a | k F (cid:21) . (18)This result is bad news: the gap depends exponentially on the pairing potential U . The Cooper pairs have a size of the order of ξ ∼ ¯ hv F / ∆ (the coherencelength ) and thus ξk F ∝ exp[+ π/ | a | k F ] (cid:29) | a | k F →
0. These particles will react and can screen, orun-screen, the interaction. Including this medium polarization on the pairing iscalled beyond BCS , and in the weak coupling limit its effect can be calculatedanalytically [22], giving∆( k F ) | a | k F → −−−−−→ ∆ GMB ( k F ) = 1(4e) / ∆ BCS ( k F ) (cid:39) . BCS ( k F ) (19)So, screening by the medium can reduce the gap by more than a factor two,even in an extremely dilute system. A significant amount of work has been devoted to the calculation of pairinggaps in the neutron star environment: see, e.g., [23, 24, 25] or A. Schwenk’scontribution to this volume for reviews. Below we first briefly describe theGorkov formalism [26, 27, 23] that will allow us to set up the stage for thepresentation of representative results for nucleon gaps. Specifically, we willaddress the neutron S and P-F and the proton S and P-F gaps andbriefly mention hyperon gaps. The effects of pairing on the thermal evolutionof neutron stars are described in Sec. 4.3. Calculations of pairing gaps in quarkmatter will be described in Sec. 5. General formalism
Several significant effects of pairing are due to the change in the quasi-particlespectrum that is obtained from the poles of the propagator G . In Fig. 7, Eq. (A),we show the definition of G , the two point Green’s function with one particle inand one particle out, and its Dyson equation which relates the free propagatorand the self energy Σ in the case of a normal system. The resulting quasi-particlespectrum is then ω = (cid:15) ( k , ω ) = ¯ h k m + Σ( k , ω ) − (cid:15) F (cid:39) ¯ h k F m ∗ ( k − k F ) . (20)(A) iG (1 ,
2) = (cid:104) T { ψ (1) ψ † (2) }(cid:105) = = + Σ Figure 7: The normal state propagator and its Dyson equation.15n obtaining the right-most result above, we assumed the system to be isotropicand the spectrum is evaluated for k (cid:39) k F with the effective mass m ∗ definedthrough ∂(cid:15) ( k, ω = 0) ∂k (cid:12)(cid:12)(cid:12)(cid:12) k = k F = ¯ h k F m ∗ . (21)The resulting spectrum (cid:15) F + ω is depicted in the left panel of Fig. 5.In the presence of a pairing instability, and the concomitant development ofa condensate, an anomalous propagator F and its adjoint F † can be defined,see Eq. (B) and Eq. (B’) in Fig. 8, with their corresponding Gorkov equationsthat replace the Dyson equation. In addition to the self energy Σ, the Gorkovequations feature an anomalous self energy , or the gap function ∆. The prop-agator F violates particle number conservation as it propagates a hole into aparticle, and vanishes in the absence of a condensate in a normal system. Thegap function is a 2 × k , ω ) = (cid:18) ∆ ↑↑ ( k , ω ) ∆ ↑↓ ( k , ω )∆ ↓↑ ( k , ω ) ∆ ↓↓ ( k , ω ) (cid:19) . (22)In the case the ground state is assumed to be time-reversal invariant, ˆ∆( k , ω )(denoted ˆ∆ for short below) has a unitary structure satisfyingˆ∆ ˆ∆ † = ˆ∆ † ˆ∆ = ∆ ˆ1 (23)where ˆ1 is a 2 × = ∆ ( k , ω ) = det ˆ∆( k , ω ) . (24)The quantity ∆ above will appear as the energy gap in the quasi-particle spec-trum.The normal propagator G is also modified, as depicted in Eq. (C) in Fig. 8.Solving the Gorkov equations gives G = ( ω + (cid:15) ) /D with D ( k , ω ) = ω − (cid:15) ( k , ω ) − ∆( k , ω ) (25)and its modified poles yield a quasi-particle spectrum with two branches: ω = ± (cid:112) (cid:15) ( k , ω ) + ∆( k , ω ) . (26)The resulting spectrum (cid:15) F + ω is depicted in the right panel of Fig. 5. Note that, G and Σ are also 2 × F gives F = ˆ∆ † /D . Finally, Σ and ∆are defined by Eq. (D) and (E) in Fig. 8 from a kernel K . Equation (D) is thegap equation and reads∆ αβ ( k ) = i (cid:90) d k (cid:48) (2 π ) (cid:88) α (cid:48) ,β (cid:48) (cid:104) kα, − kβ | K | k (cid:48) α (cid:48) , − k (cid:48) β (cid:48) (cid:105) ∆ α (cid:48) β (cid:48) ( k (cid:48) ) D ( k (cid:48) ) , (27)where k = ( k , ω ), k (cid:48) = ( k (cid:48) , ω (cid:48) ) and α , β , ... denote spin indices.16B) iF (1 ,
2) = (cid:104) T { ψ (1) ψ (2) }(cid:105) = = Σ + * ∆ (B’) iF † (1 ,
2) = (cid:104) T { ψ † (1) ψ † (2) }(cid:105) = = Σ + ∆ (C) iG (1 ,
2) = (cid:104) T { ψ (1) ψ † (2) }(cid:105) = = + Σ + ∆ (D) Σ = K (E) ∆ = K Figure 8: The Gorkov equationsSolving the full set of equations in Fig. 8 requires many approximations, no-tably in the choice of the kernel K . In particular, different approximate kernels K are used in (D) and (E) as it appears in the particle-hole channel in (D),whereas in (E) it is in the particle-particle channel. In the BCS approxima-tion , the self energy is calculated in the normal phase (e.g., with a Br¨uckner-Hartree-Fock or BHF scheme) with its energy dependence being neglected, i.e.Σ( k , ω ) → Σ( k ), and the kernel for ∆ is simply taken as the bare interaction.The latter can be a two-body force (2BF) from a potential adjusted to labora-tory N-N scattering data or that derived from an effective interaction such asa Skyrme force. Recently, the effect of three-body forces (3BF), absent in thelaboratory N-N scattering experiment, has been considered. Inclusion of 3BFis necessary to reproduce the nuclear saturation density; they are, in the bulk,repulsive and their importance grows with increasing density. Even at the 2BFlevel, a severe problem is encountered: to date, none of the N-N interactionmodels reproduce the measured phase shifts for E lab larger than 300 MeV inthe channels needed (particularly for P ) for the conditions prevailing in thecore of a neutron star. The source of this problem is easy to identify; beyond290 MeV (the threshold for pion production), inelastic channels begin to becomedominant.Models beyond BCS have proceeded in two directions. In the first approach,in the gap equation, the bare interaction is supplemented by the inclusion ofshort-range correlations. In a further step, long-range correlations to account formedium polarization are added. As illustrated above with the weak-coupling re-sult of Eq. (19), polarizations effects can be significant. In the second approach,which goes beyond the BHF level, the self energy Σ is calculated by includingits energy dependence. Calculations of Σ in the paired phase, see Eq. (D) inFig. 8, do not yet exist and are necessary. In the following, we will neglect theenergy dependence of the gap, i.e., write it as ˆ∆( k ) instead of ˆ∆( k , ω ). Pairing in single spin-angular momentum channels
In pairing calculations, the potential and the gap function are usually expandedin partial waves so as to focus on specific spin-angular momentum channels, λ =17 s, j ). At low k F n , or k F p , it is theoretically predicted that the preferred channelis λ = (0 ,
0) in S -wave, i.e., the spin-singlet S . At large Fermi momenta, the S interaction becomes repulsive and the preferred channel is λ = (1 ,
2) in P and F waves (the mixing being due to the tensor interaction [28]) , i.e., the spin-triplet P-F . In the S channel, which has also been called the “A” phase, thegap is spherically symmetric and can be written asA phase ( S ) : ˆ∆ (0 , ( k ) = (cid:18) k ) − ∆( k ) 0 (cid:19) [∆( k F ) = energy gap] (28)In the P-F channel, ˆ∆ λ has contributions from all possible orbital angularmomenta l and their m j components, i.e., ˆ∆ λ = (cid:80) l,m j ∆ m j l λ ( k ) ˆ G m j l λ (ˆ k ), wherethe ˆ G m j l λ (ˆ k ) are 2 × λ which is thus not spherically symmetric. Microscopic calculations restricted tothe P channel [29, 30] indicate that the largest component of ˆ∆ λ correspondsto the m j = 0 sub-channel or, possibly, the m j = ± k F ) is given by [29]B phase ( P , m j = 0) : ∆ ( k F ) = 12 (cid:2) ∆ λ ( k F ) (cid:3) θ k π (29)C phase ( P , m j = ±
2) : ∆ ( k F ) = (cid:2) ∆ λ ( k F ) (cid:3) θ k π , (30)where θ k is the angle between k F and the arbitrary quantization axis. Noticethat in the B phase, the gap is nodeless whereas in the C phase it vanishes onthe equator, θ k = π/ Temperature dependence of ∆ and T c The preceding discussion was restricted to the zero temperature case. It is natu-rally extended to finite temperature whence the gap becomes ˆ∆( k ; T ). However,effects of thermal excitations are important only for values of k (cid:39) k F . We willoften omit either of the arguments k F or T when not necessary, but they arealways implied (as is its ω dependence). Notice that microscopic calculationsare often limited to the T = 0 case only.The relationship between the critical temperature T c for the phase transitionand the energy gap ∆( k F ) is approximately given by the usual result k B T c ≈ .
57 ∆( k F ; T = 0) (31)for all three phases A, B, and C [29, 30], where ∆( k F ; T ) is obtained by angleaveraging of ∆ ( k F , T ) over the Fermi surface (cid:2) ∆( k F ; T ) (cid:3) ≡ (cid:90) (cid:90) d Ω4 π ∆ ( k F ; T ) . (32)Obviously, ∆( k F ; T ) = ∆( k F ; T ) for an isotropic S gap. The temperaturedependence of the energy gap ∆( k F , T ) for S pairing and of the angle averaged∆( k F ; T ) for the P pairing in the m j = 0 case are shown in Fig. 9.18 .00.80.60.40.200 0.2 0.4 0.6 0.8 P S ( T ) / ( T = ) ∆ ∆ T/T c Figure 9: Temperature dependence of the energy gap ∆( k F , T ) for S pairingand of the angle averaged ∆( k F ; T ) for the P pairing in the m j = 0 case.Continuous lines show the commonly used analytical fits of [31] whereas thedots are from the calculations of [30]. P pairing in the m j = 2 case results invalues very close to the m j = 0 case. The isotropic S neutron gap In Fig. 10, we show sets of predicted T c for the neutron S pairing in uniformpure neutron matter. The two dotted lines marked “BCS” and “GMB” showthe simple analytical results of Eq. (18) and Eq. (19), respectively, with a = − . T c = 0 .
56 ∆( k F ). Formally, these results are only valid when | a | k F (cid:28)
1, i.e., k F (cid:28) . − . The curve “SCLBL”, from [32], illustratesthe results of a numerical solution of the gap equation, using the Argone V N-N potential, within the BCS approximation. The results merge with the“BCS” curve in the weak coupling limit k F →
0. Also shown are results fromcalculations that take into account more sophisticated medium effects, includingmedium polarization (with different schemes): “CCDK”, from [33], employeda variational method within the correlated basis functions scheme, “WAP”,from [34], employed an extension of the induced interaction scheme, whereas“SFB”, from [35], went beyond [34] with renormalization group methods. Inline with the simple GMB result of Eq. (19), these model calculations show thatpolarization has a screening effect that quenches the gap, by a factor ∼
3. Thesethree calculations yield some agreement, particularly in the predicted maximumvalue of T c , but with a non-negligible difference in the density dependence. Theother two curves show more recent results: “GIPSF”, from [36], utilizes theauxiliary-field diffusion Monte Carlo technique while “GC”, from [37], stemsfrom a Quantum Monte Carlo calculation. These last two models result in gapsthat are intermediate between the previous models and the BCS approximation;moreover, they converge toward the GMB value when k F → CS C o r e C r u s t GMBGC SCLBL 2.52.01.51.00.50k [fm ] 1.5
F −1 T [ K ]c ∆ [ M e V ] Figure 10: Some theoretical predictions of T c and ∆, vs neutron k F , for theneutron S gap in uniform pure neutron matter. The value of k F correspondingto the transition from the crust to the core is indicated. See text for description.Transcribed to the neutron star context, the range of Fermi momenta forwhich these neutron S gaps are non vanishing corresponds mostly to thedripped neutrons in the inner crust. The presence of nuclei, or nuclear clus-ters in the pasta phase, may modify the sizes of these gaps from their values inuniform matter. The coherence length ξ of the dripped neutrons is larger thanthe sizes of nuclei, leading to proximity effects. This issue has received someattention, see, e.g., [38, 39, 40, 41], and position dependent gaps, from insideto outside of nuclei, have been calculated. However, in most of the crust ξ issmaller than the internuclear distance, and the size of the gap far outside thenuclei is close to its value in uniform matter. The isotropic S proton gap The magnitudes of proton S gaps are similar to those of neutrons, but with theimportant difference that, in the neutron star context in which beta equilibriumprevails, protons are immersed within the neutron liquid, and constitute onlya small fraction of the total baryon number (3 to 20% in the density rangewhere they are expected to be superconducting). Proton-neutron correlationscause the effective mass of the proton to be smaller than that of the neutron,a simple effect that reduces the size of the proton S gap compared to that ofthe neutron.Several theoretical predictions of T c for the proton S gap are shown inFig. 11: “CCY” from [42], “T” from [43], and “AO” from [44] that are amongthe first historical calculations, whereas “BCLL” from [30], “CCDK” from [33],and “EEHO” from [45] are more recent results. All of these calculations wereperformed within the BCS approximation and very few works have gone beyond20 [ K ] c BS F −1 C r u s t − C o r e k [fm ] BS [ M e V ] ∆ Figure 11: Some theoretical predictions of T c and ∆, vs proton k F , for theproton S gap in β -equilibrium uniform neutron-proton matter. The value of k F corresponding to the transition from the crust to the core is indicated: valueson the right of this line correspond to the neutron star core but values on theleft are not realized since protons in the crust are confined within nuclei whichare finite size systems while this figure presents results for infinite matter. Onthe top margin are marked the values of the proton k F at the center of a 1 . .
4, 1 .
8, and 2 . M (cid:12) star built with the APR EOS [16]. See text for description.BCS for the proton S gap. Among the latter, we show results from [46]:these authors used either only two body forces in the interaction kernel, curve“BS ”, or two body forces supplemented by the inclusion of three body forces,curve “BS ” which shows that three body forces are repulsive in the S channel. These “BS” results also include effects of medium polarization. Recallthat for the S pairing of neutrons in pure neutron matter, polarization has ascreening effect and quenches the gap. However, in neutron star matter, wherethe medium consists mostly of neutrons, the strong np -correlations result inmedium polarization inducing anti-screening [47] for the S pairing of protons. The anisotropic P-F neutron (and proton) gap The S neutron gap vanishes at densities close to the crust-core transitionand the dominant pairing for neutrons in the core occurs in the mixed P-F channel. Uncertainties in the actual size and the range of density in whichthis gap persists are, however, considerable. As previously mentioned, a majorsource of uncertainty is the fact that even the best models of the N-N interaction in vacuum fail to reproduce the measured phase shift in the P channel [48].Also significant are the effects of the medium on the kernel and 3BF, even at21he level of the BCS approximation. It was found in [49] that 3BF at the Fermisurface are strongly attractive in the P-F channel in spite of being repulsive inthe bulk. Moreover, due to medium polarization a long-wavelength tensor forceappears that is not present in the interaction in vacuum and results in a strongsuppression of the gap [50]. SF 3bfHGRR NijICDB AV C r u s t C o r e NijII a bca2 C r u s t C o r e Figure 12: Left panel: some theoretical predictions of T c for the neutron P-F gap in uniform pure neutron and β -equilibrium matter. See text for description.Right panel: some phenomenological models of T c for the neutron P-F gapused in neutron star cooling simulations. Models “a”, “b”, and “c” are from[51] and [52], model “a2” from [53]. On the top margin are marked the valuesof k F n at the center of a 1 .
0, 1 .
4, 1 .
8, and 2 . M (cid:12) star built with the APR EOS[16].Figure 12 shows examples of theoretical predictions of T c for the neutron P-F gap. The three dotted lines show some of the first published models:“HGRR” from [54], “T” from [55] and “AO” from [29]. The four continuouslines show results of models from [48] calculated using the Nijmegen II (“NijII”),Nijmegen I (“NijI”), CD-Bonn (“CDB”), and Argonne V (“AV18”) potentials(displayed values are taken from the middle panel of Figure 4 of [48]). Theresults of these four models start to diverge at k F n above 1.8 fm − and illustratethe failure of all four N-N interactions models to fit the P laboratory phase-shifts above E lab (cid:39)
300 MeV. All of these calculations were performed for pureneutron matter using the BCS approximation.In the case of the S gap, medium polarization is known to result in screen-ing and to reduce the size of the gap. In the case of a P gap, polarizationwith central forces is expected to result in anti-screening and to increase thesize of the gap. However, Schwenk & Friman [50] showed that spin-dependentnon-central forces do the opposite and strongly screen the coupling in the P channel, resulting in a T c lower than 10 K: this “SF” value is indicated in the22gure by an arrow.The two dashed lines in Fig. 12 present results from [56] where the “2BF”model only considers 2-body forces (from the Argonne V ) while the “3BF”model includes a meson exchange model 3-body force: the result is a growing P-F gap which shows no tendency to saturate at high density. This work,for β -equilibrium matter, moreover emphasized the importance of the protoncomponent.Other delicate issues are the effect of the proton contaminant and the likelydevelopment of a π condensate which also strongly affects the size of theneutron (and proton) gap(s).In summary, the size and extent in density of the neutron P-F gap in theneutron star core are poorly known. Given these large uncertainties in the sizeof the neutron P-F gap (about three orders of magnitude) and the fact thatneutrino emissivity is suppressed by an exponential Boltzmann-like factor, thisgap is often considered as a free parameter in neutron star cooling models. Theextreme sensitivity of the cooling history on the size of this gap can be utilizedto one’s advantage by inverting the problem, as it may allows us to measure it byfitting theoretical models to observational data [57]. The right panel of Fig. 12presents the phenomenological neutron P-F gaps used in cooling calculationsin a later section.In the case of protons, their P-F gaps have generally been overlooked due totheir small effective masses, and considered to be likely negligible [58]. However,in view of the strong enhancing effect of the 3BF on the neutron P-F gap, thisissue has been reconsidered in [49] where it was shown that the proton P-F gap can be sizable. Hyperon gaps and nucleon gaps in hyperonic matter
Many calculations of dense matter indicate that strangeness-bearing hyperonswill be present in neutron star matter once the neutron chemical potential ex-ceeds the rest masses of hyperons [59] (see Sec. 3.1). In the likely presence ofhyperons (denoted by Y) arise the issues of, first, the effect of their presence onthe nucleon gaps, and second, the possibility of hyperon pairing. Nucleon gapsin the presence of hyperons have been studied in [49] and [60]: depending onthe N-N, N-Y, and Y-Y interaction models employed, the nucleon gaps may beeither enhanced or reduced by the presence of hyperons.Since the suggestion in [61] and the first detailed work of [62], hyperon gapshave attracted some attention. All uncertainties present in the nucleon caseimmediately translate to the hyperon case. An additional problem is that verylittle is known about hyperon-hyperon interactions , which is generally guessed In the presence of a charged π − condensate a new Urca neutrino emission pathway isopen, see Table 1. The development of a neutral π condensate has, however, little effect onneutrino emission. Some experimental information is available from hypernuclei [63, 64] and hadronicatoms [65], but the data do not yet uniquely determine the hyperon-nucleon or hyperon- S channel. In both cases of Λ’s andΣ − ’s, the estimated gap sizes are similar to those of nucleons. We refer thereader to [67, 68, 69] and references therein for details.At very high densities, the presence of deconfined quark matter is also likely.Quarks are expected to pair and form a color superconductor. This subject hasdeveloped into a field of its own, and we dedicate Sec. 5 for a brief account andrefer the reader to other articles in this monograph. The occurrence of pairing leads to three important effects of relevance to neutronstar cooling:A) Alteration, and possible strong suppression when T (cid:28) T c , of the specificheat c V of the paired component.B) Reduction, and possible strong suppression when T (cid:28) T c , of the emissivity (cid:15) ν of the neutrino processes the paired component is involved in.C) Triggering of the “Cooper pair breaking and formation” (PBF) with con-comitant neutrino pair emission which is very efficient in the case of spin-triplet pairing.These effects are direct consequences of the development of the energy gap ∆( k )and the resulting two branches in the quasi-particle spectrum, Eq. (26). Thegap severely limits the available phase space when T (cid:28) T c and the spectrum isusually treated in the effective mass approximation with an angle averaged gap,see Eq. (32), (cid:15) ( k ) = ± (cid:113) [¯ hv F ( k − k F )] + ∆( k F , T ) , (33)where v F ≡ ¯ hk F /m ∗ .In cooling calculations, these effects are introduced through “control func-tions”: c V −→ R c c V and (cid:15) Xν −→ R X (cid:15) Xν . (34)There is a large family of such control functions for the various types of pair-ing and the numerous neutrino processes “X”. For nodeless gaps, the R ’s areBoltzmann-like factors ∼ exp[ − D ( T ) /k B T ] and result in a strong suppressionwhen T (cid:28) T c , whereas for gaps with nodes the suppression is much milder.Regarding the specific heat, there is a sudden increase, by a factor ∼ . T = T c , followed by a reduction at lower T . Examples of such control functionsare shown in Fig. 13. hyperon interaction. Future work in lattice QCD [66] may prove fruitful, but current resultsare limited to unphysically large pion masses. S P S P C on t r o l f un c ti on R ν T/T c T/T c p n Specific Heat C on t r o l f un c ti on R Neutrino Emission C S P Figure 13: Control functions for neutrino emission from the modified Urca pro-cess (as, e.g., n + n → n + n + νν ) (left panel) and the specific heat (right panel),in the presence of S pairing and P in the B phase (see Eq. (29)), from theanalytical fits of [31] and [70].The effect C, neutrino emission from the formation and breaking of Cooperpairs [71, 72], can be interpreted as an inter-band transition (as, e.g., n → n + νν ) where a neutron/proton quasiparticle from the upper (+) branch ofthe spectrum of Eq. (33) falls into a hole in the lower ( − ) branch. Such areaction is kinematically forbidden by the excitation spectrum of the normalphase, Eq. (20), but becomes possible in the presence of an energy-gap, Eq. (33).This process is described in more detail in Sec. 6.6. The resulting emissivitycan be significantly larger than that of the modified Urca process (as, e.g., n + n → n + n + νν ) in the case of spin-triplet pairing. The central densities of neutron stars can exceed the nuclear density ρ nuc ∼ . × g cm − by significant amounts. At sufficiently high densities, a de-scription of neutron star interiors in terms of nucleons becomes untenable andsub-nucleonic degrees of freedom, namely quarks, must be invoked. Interac-tions between quarks is fundamentally grounded in Quantum Chromodynamics(QCD), the theory of strong nuclear interactions. The theory has a gauge sym-metry based on the Lie group SU(3), and the associated charge is referred toas “color”. QCD is asymptotically free: interactions between quarks mediatedby gluons become weak at short distances, or equivalently, high densities. Atlow densities, strong interactions “confine” quarks into neutrons and protonswhich are color neutral. Asymptotic freedom guarantees that, at some largedensity, the ground state of zero-temperature matter will consist of nearly-free,“deconfined” quarks [73].QCD has been amply tested by experiments at high energies where asymp-totic freedom has been confirmed [74]. Lattice-gauge calculations of hadron25asses, and of baryon-free matter at finite temperature, have made enormousstrides in recent years [75]. However, first-principle QCD calculations for fi-nite baryon density have been stymied due to the fermion-sign problem in lat-tice gauge calcuations. While many guesses are available, perturbation theoryis unable to accurately predict the density at which the deconfinement phasetransition occurs. However, it is possible that the phase transition occurs ata density lower than the central density of some (or even all) neutron stars.In that case, at least some neutron stars will contain deconfined quark matter.Such objects are referred to as hybrid quark stars. In their cores, up ( u ), down( d ), and strange ( s ) quarks are the principal degrees of freedom, the other threequark flavors (charm ( c ), bottom ( b ), and top ( t )) being excluded because oftheir large masses. It is theoretically possible that the energy of zero-pressurestrange quark matter has a lower energy than Fe [76, 77, 78], in which case ahybrid star would be metastable or unstable, and nucleonic matter would spon-taneously convert into strange quark matter, creating pure quark stars that areself bound [79, 80]. There is no experimental or observational evidence for purequark stars, however, and we do not consider them further in this contribution. Cooper pairing between quarks was first investigated in the late 1970s [81, 82].As all quarks are charged, pairing between quarks is often referred to as colorsuperconductivity because the paired phase breaks the SU(3) gauge symmetryof QCD. First estimates of pairing gaps in quark matter were of order 1 MeV. Inthis case, neutron stars with deconfined quark matter cool very rapidly throughthe quark direct Urca processes (see Table 1). Because some older neutron starsare observed to be relatively warm, this naturally implies that not all neutronstars can contain quark matter.This situation changed drastically with the discovery [83, 84] that colorsuperconductivity implies gaps as large as 100 MeV (see a recent review in[85]). These works suggested two possibilities: either the so-called “color-flavor-locked” (CFL) phase in which all nine combinations of flavor (up, down, strange)and color (red, green, blue) participate in pairing, or the “2SC” phase whereonly four of the nine combinations pair (corresponding to up and down quarkswhich are either red or green).
There are several formalisms which have been applied to describe color super-conducting quark matter. High-density effective theories (HDET) were firstdeveloped in 1990s [86, 87] and then developed further for color superconduc-tivity in [88, 89]. To construct an effective theory, one begins with the QCDLagrangian, rewrites it in terms of a 1 /µ expansion ( µ being the chemical poten-tial), and then integrates out hard gluons and fermionic modes correspondingto the Dirac sea. 26nother commonly used alternative consists of using Nambu–Jona-Lasinio(NJL) [90] models. The original NJL model was a theory of strong interactionsbefore the advent of QCD. The four-fermion interaction of the NJL Lagrangianbears close resemblance to that in the BCS theory of superconductivity andgives rise to analogous effects. Originally framed in terms of nucleon fields,quartic interactions serve to give the nucleon its mass through a self-energygenerated by the formation of a condensate. Modern versions involve quarkfields that develop a “quark condensate” which is then related to the mass ofconstituent quarks [91]. NJL models for color superconductivity presume thatgluonic degrees of freedom have been integrated out resulting in point-like cou-plings between quarks. For a review of the NJL model applied to dense quarkmatter see [92]. Confinement is sometimes implemented by the addition ofPolyakov loop terms giving rise to “PNJL” models. HDET and NJL methodsgive qualitatively similar results, but the NJL Lagrangian is a bit more trans-parent, so we describe some of its details here. A chiral SU(3) Lagrangian withsuperconducting quarks (adapted from [93]) is L = ¯ q iα (cid:0) i∂ µ γ µ δ ij δ αβ − m ij δ αβ − µ ij,αβ γ (cid:1) q jβ + G S (cid:88) a =0 (cid:104)(cid:0) ¯ qλ af q (cid:1) + (cid:0) ¯ qiγ λ af q (cid:1) (cid:105) + G ∆ (cid:88) k (cid:88) γ (cid:0) ¯ q iα (cid:15) ijk (cid:15) αβγ q Cjβ (cid:1) (cid:0) ¯ q Ci (cid:48) α (cid:48) (cid:15) i (cid:48) j (cid:48) k (cid:48) (cid:15) α (cid:48) β (cid:48) γ (cid:48) q j (cid:48) β (cid:48) (cid:1) + G ∆ (cid:88) k (cid:88) γ (cid:0) ¯ q iα iγ (cid:15) ijk (cid:15) αβγ q Cjβ (cid:1) (cid:0) ¯ q Ci (cid:48) α (cid:48) iγ (cid:15) i (cid:48) j (cid:48) k (cid:48) (cid:15) α (cid:48) β (cid:48) γ (cid:48) q j (cid:48) β (cid:48) (cid:1) , (35)where Roman indices are for flavor and greek indices are for color, except for a which enumerates the SU(3) matrices, m ij is the quark mass matrix, µ ij,αβ is the chemical potential matrix, q is the quark field, q C = C ¯ q T , and (cid:15) is theLevi-Civita tensor. The first term is the Dirac Lagrangian which describes freerelativistic massive quarks at finite density. The second term is a combination offour quark fields which model non-superfluid quark-quark interactions and obeysthe SU (3) L × SU (3) R chiral symmetry present in QCD. The third and fourthterms, which give rise to color superconductivity, are the chirally symmetricanalog of the second term in the quark-quark channel. QCD breaks U A (1)symmetry, and so these four-fermion interactions can be supplanted by six-fermion interactions in order to do the same [94].The first step in obtaining the thermodynamic potential in the mean-fieldapproximation is to replace the quark bilinears ¯ q i q i and ¯ q iα iγ (cid:15) ijk (cid:15) αβγ q Cjβ withtheir ground-state expectation values. The former is the quark condensate as-sociated with the breaking of chiral symmetry and the latter gives rise to thesuperconducting gap, ∆ kγ . Having made this replacement, the non-constantterms in the Lagrangian take the form ¯ qM q , where M is a matrix represent-ing the inverse propagator. This matrix can be diagonalized in the standardway to obtain the individual quark dispersion relations and the thermodynamic27otential Ω = − G S (cid:88) i = u,d,s (cid:104) ¯ q i q i (cid:105) − (cid:88) k (cid:88) γ (cid:12)(cid:12) ∆ kγ (cid:12)(cid:12) G ∆ − (cid:90) d p (2 π ) (cid:88) i (cid:20) λ i T ln(1 + e − λ i /T ) (cid:21) , (36)where λ i gives the energy eigenvalues and i runs over 3 flavors, 3 colors, andthe Dirac indices (36 total). In this formulation, there is a manifest parallelismbetween the quark condensates, (cid:104) ¯ qq (cid:105) , and the superconducting gaps. The mini-mum of the thermodynamic potential with respect to the superconducting gapgives the gap equation, and the minimum of the thermodynamic potential withrespect to the quark condensates gives the “mass gap” equation, i.e. the equa-tion which controls the dependence of the dynamically generated quark masses.The energy eigenvalues λ i cannot be computed analytically at all densities,except in two limiting situations. At low densities, where chiral symmetry isspontaneously broken, the gaps are zero. In this case, the quark dispersionrelations are (cid:112) p + m ∗ i ± µ , where m ∗ i = m i − G S (cid:104) ¯ q i q i (cid:105) are effective masses.The corresponding quark condensates are given by [95] (cid:104) ¯ q i q i (cid:105) = − π (cid:90) Λ p Fi p dp m ∗ i (cid:112) p + m ∗ i . (37)At high densities, the gaps are larger than the quark masses, hence the latterdo not play significant roles and can be ignored. With progressively increasingdensity color-flavor locking becomes increasingly perfect, hence the name theCFL phase. In this phase, we can assume flavor symmetry and 8 of 9 quarks(3 colors times 3 flavors) have the dispersion relation (cid:112) ( p − µ ) + ∆ , whilethe remaining quark has the dispersion relation (cid:112) ( p − µ ) + 4∆ . In general,these properties are coupled so that both the masses and the gaps appear in thedispersion relations in a nontrivial fashion.Results of calculations based on the above model are displayed in Fig. 14using Eq. (35). The dynamically generated quark masses are larger than thecurrent quark masses at low density where chiral symmetry is spontaneouslybroken. In the CFL phase, the superconducting gaps form among all threecombinations of unlike flavors, up-down, up-strange, and down-strange. In the2SC phase, the only pairing is between up and down quarks. This model exhibitsa first-order phase transition between the gapped and ungapped phases, sothe gaps do not continuously go to zero at low densities. The decrease in thegaps as a function of increasing density or large values of the quark chemicalpotential µ is an artifact of the ultraviolet cutoff (a necessity imposed by thenonrenormalizable Lagrangian). HDET models show that the gaps increasewith increasing µ . 28igure 14: Down and strange dynamical quark masses (left panel) and super-conducting gaps (right panel) as a function of density from [93]. The abscissashows the quark chemical potential µ . The thin (bold) curves show values whencolor neutrality is not (is) enforced. Different results are obtained in the CFLand 2SC phases as shown. There are many different possible pairing configurations in addition to the CFLand 2SC phases described above, including gapless phases [96], and color-spinlocked pairing. Color superfluids also admit a new set of Goldstone bosonsassociated with flavor rotations of the pairing condensate which have a similargroup structure to the pseudoscalar Goldstone bosons in QCD ( π, K , etc.).These bosons can condense [97], forming a new phase of superconducting quarkmatter. The most common is the “CFL-K” phase which contains CFL quarkswith a K meson condensate. All of these phases have their own associatedexotic neutrino emissivities, including their own associated quark PBF neutrinocooling processes. A caricature phase diagram is shown in Fig. 15.For densities near the deconfinement phase transition, the ground state ofthe quark superfluid may be similar to the Larkin-Ovchinnikov-Fulde-Ferrell [99,100] (LOFF) pairing observed in condensed matter systems. LOFF pairing oc-29 Hadrons
Nuclear
Plasma
Matter µ EE exotics LOFF nuclearsuperfluid
Figure 15: A schematic model QCD phase diagram, adapted from [98].curs when two species participating in a pairing interaction have different Fermimomenta thus creating Cooper pairs with nonzero momentum. This pairingconfiguration breaks translational symmetry and encourages the formation acrystal. This creates a novel mechanism for pulsar glitches: the superfluid vor-tices pin to the crystalline part of the quark phase and during a glitch event thevortices move outwards by unpinning and repinning to the lattice [101]. Thismechanism has not been either verified or ruled out by the data.
The appearance of quark degrees of freedom often lowers the pressure at highdensities, yielding neutron stars with smaller radii and smaller maximum massescompared to those in which quarks are absent. This is not guaranteed, however,as quark-quark interactions are sufficiently uncertain that quark matter can benearly indistinguishable from matter consisting entirely of neutrons, protons,electrons and muons [102]. The presence (or absence) of quark matter will bedifficult to determine from observations of neutron star structure alone. Theneutron star mass-radius relation is dependent on the pressure of matter at agiven energy density, but is insensitive to the nature of the particular degreesof freedom which provide that pressure. For this reason, it is natural to look toneutron star cooling to discern the composition of a neutron star’s core. Thistopic will be considered in Sec. 7.7. 30
Neutrino Emission Processes
The thermal evolution of neutron stars with ages < ∼ yrs is driven by neutrinoemission. We will here briefly describe the dominant processes; the interestedreader can find a detailed description in [103] and an alternative point of view in[104]. Table 1 presents a short list of neutrino processes with estimates of theiremissivities. Most noticeable is the clear distinction between processes involving5 degenerate fermions with a T dependence, which are labeled as “slow”, andthose with only 3 degenerate fermions with a T dependence, which are severalorders of magnitude more efficient and labeled as “fast”. The difference in the T dependence is important and is simply related to phase space arguments whichare outlined in Sec. 6.5. The last subsection, § The simplest neutrino emitting processes are Eq. (6) and Eq. (8) (see also Ta-ble 1), which collectively are generally referred to as the nucleon direct Urca (“DU” or “DUrca”) cycle. By the condition of β -equilibrium, both reactionsnaturally satisfy energy conservation, but momentum conservation is more del-icate. Due to the high degree of degeneracy, all participating particles havemomenta p ( i ) equal (within a small T (cid:28) T F correction) to their Fermi mo-menta p F ( i ). As p F ( i ) ∝ n / i and n p ∼ n e (cid:28) n n in neutron star matter,momentum conservation is not a priori guaranteed. In the absence of muonsand hence with n p = n e , the “triangle rule” for momentum conservation requiresthat the proton concentration x p > / (cid:39) ρ ∼ ρ nuc we have x p (cid:39) ρ nuc , the conditionis stronger and one needs x p larger than about 15% [105]. The proton fraction x p grows with density (see Eq. (11)), its growth being directly determined bythe growth of the nuclear symmetry energy, so that the critical proton fractionfor the DU process is likely reached at some supra-nuclear density [105]. For theAPR EOS [16] that we will frequently use, the corresponding critical neutronstar mass for the onset of the nucleon DU process is 1 . (cid:12) , but other EOSscan predict smaller critical densities and masses.It should also be noted that the direct Urca process, and for that matter,all the processes discussed in this section, can involve other leptons. Thus, for ρ > ∼ ρ nuc , where µ − appear, one also has n −→ p + µ − + ¯ ν µ , p + µ − −→ n + ν µ . (38)31ame Process Emissivity Efficiencyerg cm − s − Modified Urca(neutron branch) (cid:40) n + n (cid:48) → p + n (cid:48) + e − + ¯ ν e p + n (cid:48) + e − → n + n (cid:48) + ν e ∼ × RT SlowModified Urca(proton branch) (cid:40) n + p (cid:48) → p + p (cid:48) + e − + ¯ ν e p + p (cid:48) + e − → n + p (cid:48) + ν e ∼ RT SlowBremsstrahlung n + n (cid:48) → n + n (cid:48) + ν + ¯ νn + p → n + p + ν + ¯ νp + p (cid:48) → p + p (cid:48) + ν + ¯ ν ∼ RT SlowCooper pair (cid:40) n + n → [ nn ] + ν + ¯ νp + p → [ pp ] + ν + ¯ ν ∼ × RT ∼ × RT MediumDirect Urca(nucleons) (cid:40) n → p + e − + ¯ ν e p + e − → n + ν e ∼ RT FastDirect Urca(Λ hyperons) (cid:40) Λ → p + e − + ¯ ν e p + e − → Λ + ν e ∼ RT FastDirect Urca(Σ − hyperons) (cid:40) Σ − → n + e − + ¯ ν e n + e − → Σ − + ν e ∼ RT FastDirect Urca(no-nucleon) (cid:40)
Λ + e − → Σ − + ν e Σ − → Λ + e − + ¯ ν e ∼ × RT FastDirect Urca( π − condensate) (cid:40) n + < π − > → n + e − + ¯ ν e n + e − → n + < π − > + ν e ∼ RT FastDirect Urca( K − condensate) (cid:40) n + < K − > → n + e − + ¯ ν e n + e − → n + < K − > + ν e ∼ RT FastDirect Urca cycle( u − d quarks) (cid:40) d → u + e − + ¯ ν e u + e − → d + ν e ∼ RT FastDirect Urca cycle( u − s quarks) (cid:40) s → u + e − + ¯ ν e u + e − → s + ν e ∼ RT FastTable 1: A sample of neutrino emission processes. T is temperature T in unitsof 10 K and the R ’s are control factors to include the suppressing effects ofpairing (see Sec. 4.3). 32 .2 The Modified Urca Process At densities below the threshold density for the nucleon DU process, wherethe DU process is forbidden at low temperatures, a variant of this process,the modified Urca (“MU” or “MUrca”) process (see Table 1) can operate, asadvantage is taken of a neighboring nucleon in the medium [106] to conservemomentum. As it involves the participation of five degenerate particles, theMU process is much less efficient than the DU process. Unlike the nucleon DUprocess, which requires sufficient amount of protons, both branches of the MUprocess operate at any density when neutrons and protons are present.
Related to the MU processes is another class of processes, bremsstrahlung, madepossible through neutral currents [107]. These differ from MU processes in thateach reaction results in the production of a ν ¯ ν pair, and the pair can have anyneutrino flavor. Bremsstrahlung reactions are less efficient, by about 2 orders ofmagnitude, than the MU processes, but may make important contributions inthe case that the MU process is suppressed by pairing of neutrons or protons.Bremsstrahlung involving electron-ion scattering is also an important source ofneutrino emission in neutron star crusts: e − + ( A, Z ) −→ e − + ( A, Z ) + ν + ν , (39)where (A,Z) designates the participating ion. In the presence of hyperons, DU processes which are obvious generalizations ofthe nucleon-only process, can also occur [59] and several are displayed in Table 1.When they appear, the Λ’s initially have a density much smaller than that ofthe neutron and hence a smaller Fermi momentum. Consequently, momentumconservation in the Λ DU cycle is easily satisfied, requiring a Λ concentration x Λ ∼ − DU process is also kinematically forbidden, whereas no-nucleon DU pro-cesses, of which one example in shown in Table 1, are possible. This particularno-nucleon DU process requires relatively low Λ and Σ − threshold concentra-tions. Other examples involving Σ − , Σ , Σ + , Ξ − and Ξ hyperons are given in[59].In deconfined quark matter, DU processes involving all three flavors arepossible, as indicated in Table 1. Rates for these processes have been calculatedby Iwamoto [108].Although not shown in Table 1, hyperons or quarks could also be involved inMU-like and bremsstrahlung processes (a quark MU process would involve anadditional quark in the entrance and the exit channels, for example), but withgreatly reduced rates compared to their corresponding DU processes and a T We turn now to briefly describe how the specific temperature dependence ofthe neutrino processes described above emerges. Consider first the simple caseof the neutron β -decay. The weak interaction is described by the Hamiltonian H I = ( G F / √ B µ L µ , where G F is Fermi’s constant, and L µ = ψ e γ µ (1 − γ ) ψ ν and B µ = ψ p γ µ ( C V − C A γ ) ψ n are the lepton and baryon weak currents,respectively. In the non relativistic approximation, one has B = cos θ c Ψ † p Ψ n and B i = − cos θ c g A Ψ † p σ i Ψ n where θ c is the Cabibbo angle and g A the axial-vector coupling. Fermi’s Golden rule gives us for the neutron decay rate W i → f = (cid:90)(cid:90)(cid:90) d p ν (2 π ) d p e (2 π ) d p p (2 π ) (2 π ) δ ( P f − P i ) · | M fi | , (40)i.e., a sum of (2 π ) δ ( P f − P i ) ·| M fi | over the phase space of all final states f =( (cid:126)p ν , (cid:126)p e , (cid:126)p p ). The integration gives the well known result W β = G F cos θ c (1 +3 g A ) m e c w β / (2 π ), where w β ∼ τ n (cid:39)
15 minutes. Alternatively, ameasurement of τ n determines G F (modulo cos θ c and w β ).The emissivity (cid:15) DU of the DU process (the Feynman diagram for this processis shown in Fig. 16) can be obtained by the same method, leading to (cid:15) DU = (cid:90)(cid:90)(cid:90)(cid:90) d p ν (2 π ) d p e (2 π ) d p p (2 π ) d p n (2 π ) (1 − f e )(1 − f p ) f n (2 π ) δ ( P f − P i ) | M fi | E ν (41)with an extra factor E ν for the neutrino energy and the phase space sum nowincludes the initial n . The f i terms, f i being the Fermi-Dirac distribution forparticle i at temperature T , take into account: (1) the probability to have a n inthe initial state, f n , and (2) the probabilities to have available states for the final e and p , denoted by (1 − f e ) and (1 − f p ), respectively. We do not introduce aPauli blocking factor (1 − f ν ) for the anti-neutrino as it is assumed to be able tofreely leave the star (i.e., f ν = 0). When performing the phase space integrals,each degenerate fermion gives a factor T , as particles are restricted to be withina shell of thickness k B T of their respective Fermi surfaces. The anti-neutrinophase space gives a factor T . The factors E ν is ∼ T and the delta function δ ( P f − P i ) gives a factor T − from δ ( E f − E i ). Altogether, we find that (cid:15) DU ∝ T · T · T · T · T · (1) · T = T , (42)34here the (1) factor emphasizes that the squared matrix element | M fi | isT-independent. An explicit expression for the neutrino emissivity for the DUprocess can be found in [105]. e ! _ ! _ e ! _ n npnp e e ! nModified Urca BremsstrahlungDirect Urca n n n nnn Figure 16: Feynman diagrams for the indicated neutrino emitting processes.Figure 16 shows a Feynman diagram for the MU process. There are twomore such diagrams in which the weak interaction vertex is attached to one ofthe two incoming legs. In this case, the T -power counting gives (cid:15) MU ∝ T · T · T · T · T · T · T · (1) · T = T . (43)In this case, the | M fi | involves two strong interaction vertices, connected by thewavy line in Fig. 16, which is momentum independent and hence T -independent.The numerical difference ∼ − T between the MU and the DU rates, seeTable 1, comes mostly from the extra phase space limitation ∝ T from thetwo extra nucleons: as a dimensionless factor it is ( k B T /E F ) (cid:39) − T with E F (cid:39)
100 MeV and k B K (cid:39) . (cid:39) µ e which does not introduce anyextra T -dependence as we are working in the case E F ( e ) > ∼
100 MeV (cid:29) T .Reference [106] contains the expression from which neutrino emissivity from theMU process can be calculated.Turning to the n − n bremsstrahlung process, one diagram is shown in Fig. 16and there are three more diagrams with the weak interaction vertices attachedto the other three external lines. The T -power counting now gives (cid:15) Br ∝ T · T · T · T · T · T · T · (cid:18) T (cid:19) · T = T (44)with two T factors for the neutrino pair. The factor ( T − ) arises from thematrix element as the intermediate neutron is almost on-shell, with an energydeficit ∼ T , and its propagator gives us a T − dependence for M fi . A workingexpression for the bremsstrahlung process can be found in [106].35 .6 The Cooper Pair Neutrino Process The formation of the fermonic pair condensate also triggers a new neutrinoemission process [71, 111, 112] which has been termed the “pair breaking andformation”, or PBF, process [113]. Whenever any two fermions form a Cooperpair, the binding energy can be emitted as a ν − ν pair. Under the rightconditions, this PBF process can be the dominant cooling agent in the evolutionof a neutron star [114]. Such efficiency is due to the fact that the pairingphase transition is second order in nature. During the cooling of the star, thephase transition starts when the temperature T reaches T c when pairs begin toform, but thermal agitation will constantly induce the breaking of pairs withsubsequent re-formation and possible neutrino pair emission.The emissivity of the PBF process (see the left panel of Fig. 17 for a Feynmandiagram) can be written as (cid:15) PBF = (cid:90)(cid:90)(cid:90)(cid:90) d p ν (2 π ) d p ν (2 π ) d p (2 π ) d p (cid:48) (2 π ) f ( E p ) f ( E p (cid:48) ) · (2 π ) δ ( P f − P i ) | M fi | · E ν (45)Under degenerate conditions, the expression above can be reduced to read as (cid:15) PBF = 12 G F m ∗ f p F,f π ¯ h c ( k B T ) a f,j R j [∆ j ( T ) /T ]= 3 . × ergcm s × ˜ m f ˜ p F,f T a f,j R j [∆ j ( T ) /T ] (46)for a fermion f in a pairing state j = S or P . The coefficients a f,j dependon the type of fermion and on the vector and axial couplings C V and C A (see,e.g., [52]). The control functions R j are plotted in the right panel of Fig. 17.These functions encapsulate the effect that the PBF process turns on when T reaches T c and practically turns off at T < ∼ . T c when there is not enoughthermal energy to break pairs.The PBF process has had an interesting history. It was first discovered byFlowers, Ruderman, and Sutherland [71] and, independently, by Voskresenskyand Senatorov [111]. It was, however, overlooked for 20 years, until implementedin a cooling calculation in [113] and its importance emphasized in [114]. Then in2006, Leinson and Perez [115] showed that the previous computations of the PBFemissivity were incompatible with vector current (e.g. baryon number) conser-vation. Neutrino pair-production is mediated by the weak interaction, which canbe decomposed in the traditional manner into vector and axial-vector parts. Inpure neutron matter, the vector part of the PBF emissivity is suppressed be-cause of vector current conservation by a factor of order ( p F /m ∗ c ) = ( v F /c ) .This is equivalent to the simple observation that a one-component system ofcharges does not have a time-varying dipole moment [116]. The axial part ofthe PBF process is, however, unimpeded and dominates the emissivity.36 B F P ν _ ν n n 1.0 T/T C on t r o l f un c ti on R S0.5 c ∆ Figure 17: Left panel: Feynman diagram for νν emission from the formationof a nn Cooper pair (pair breaking and formation, PBF, process). Right panel:control functions R PBF for the PBF process.
Temperature dependence of the PBF neutrino emissivity
The temperature dependence of the PBF process (left panel of Fig. 17) can beascertained from Eq. (45) according to the following T -power counting: (cid:15) PBF ∝ T · T · T · · T · R (∆ /T ) · T = T R (∆ /T ) , (47)where the two T and the first T factors arise from the phase space integrationsof the neutrino pair and the first participating nucleon, respectively. The fac-tor 1 results from the phase space integration of the second nucleon. As thereare only two degenerate fermions in this process (in contrast to the Urca andbremsstrahlung processes that involve 3, 4, or 5 degenerate fermions), the mo-menta of the neutrino pair and the first nucleon are chosen the momentum ofthe second nucleon is fixed by the three-momentum conserving delta function.Thus, this second nucleon does not introduce any T dependence. The T − de-pendence arises from the energy conserving delta function. The last T factor isfrom the neutrino pair’s energy, whereas the T and ∆ dependence of the matrixelement of the reaction are included in the function R (∆ /T ), which is shown inthe right panel of Fig. 17.An alternative way of looking at the PBF process is simply as an interbandtransition of a nucleon [117]. Considering the particle spectrum in a paired state(the right panel of Fig. 5), the lower branch (with (cid:15) < (cid:15) F − ∆) corresponds topaired particles whereas the upper branch to excited ones, i.e., the “brokenpair” leaves a hole in the lower branch. A transition of a particle from theupper branch to a hole in the lower branch corresponds to the formation of aCooper pair. 37 ominance of triplet-pairing In the non-relativistic limit for nucleons, the leading contribution from the axial-vector part is proportional to ( v F /c ) . To this order, the control function R ( z =∆ /T ) receives a contribution from the axial-vector part which can be expressedas [117] R ( z ) = c A π (cid:90) d Ω (cid:90) ∞ z dx ( e z + 1) I , (48)where x = v F ( p − p F ) /T and the quantity I = I xx + I yy + I zz with I ik = (cid:88) ηη (cid:48) (cid:104) B | ˆΨ † σ i ˆΨ | A (cid:105)(cid:104) B | ˆΨ † σ k ˆΨ | A (cid:105) ∗ , (49)where ˆΨ is the second-quantized non-relativistic spinor wave function of thenucleons in superfluid matter (see [117] for its detailed structure in terms ofthe Bogoliubov transformation matrix elements U ση ( p ) and V ση ( p )), | A (cid:105) is theinitial state of the system and | B (cid:105) its final state. The total spin states of thepair η , η (cid:48) and σ each take on values ± σ i ’s are the Pauli spin matrices.The energy of a paring quasiparticle is given by E = (cid:113) (cid:15) + ∆ p , where (cid:15) = v F ( p − p F ). For singlet-state pairing, the momentum-dependent gap ∆ p isindependent of p , so that the occupation probabilities u p and v p associatedwith the matrix elements U ση ( p ) and V ση ( p ) depend only on p = | p | . In thecase of singlet pairing, the Bogoliubov matrix elements satisfy the symmetryproperties V αβ ( − p ) = V αβ ( p ) and V αβ ( p ) = − V βα ( p ), so that the diagonalelements of this 2 × v p and − v p , respectively. These symmetry properties, together with the tracelessproperty of the Pauli matrices, ensure that the quantity I = 0. Thus, to order( v F /c ) , the axial-vector part does not contribute in the spin-singlet channelrendering the triplet pairing channel, which does not vanish, to be the solecontribution to the PBF process. Time history of the PBF process
The Cooper pair neutrino process operates at different times in a neutron star’scooling history according to the time during which the local temperature isnearly equal to the critical temperature of any superfluid gap. In neutron starsconsisting of neutrons, protons, and electrons, there are three relevant super-fluid gaps: singlet neutron superfluidity, singlet proton superfluidity, and tripletneutron superfluidity at high densities when the singlet channel of the neutron-neutron interaction becomes repulsive above the saturation density. In neutronstars which contain exotic matter in their interiors, each additional superfluidfermion potentially opens up new Cooper pair cooling processes. If neutron starscontain deconfined quark matter in their cores, then pairing between quark fla-vors creates new Cooper pair neutrino processes which involve pairing betweenunlike fermions [118]. 38
Cooling of Neutron Stars
The study of neutron star cooling is a Sherlock Holmes investigation, followingthe tracks of energy. At its birth, some 300 B (1 Bethe = 10 ergs) of grav-itational energy are converted largely into thermal energy. About 98% of it isemitted in neutrinos during the first minute, the proto-neutron star phase, 1%is transferred to the supernova ejecta (with 1% of this 1% powering the lightshow), and the remainder is left in thermal energy of the new-born neutron star ,i.e., the star produced during the proto-neutron star phase. Following the tracksof energy, the subsequent evolution of the neutron star can, in a simplified way,be described by an energy balance equation dE th dt = C V dTdt = − L ν − L γ + H , (50)where E th is the star’s total thermal energy, C V its specific heat, and L γ and L ν its photon and neutrino luminosities, respectively. The term H , for “heat-ing”, represents possible dissipative processes, such as friction from differentialrotation or magnetic field decay. In this simplified equation it is assumed thatthe star’s interior is isothermal with temperature T , a state reached within afew decades after birth in the core-collapse supernova (see Sec. 7.4). A more de-tailed study would include general relativistic effects and consider a local energybalance equation for each layer in the star, instead of the global one of Eq. (50),complemented by a heat transport equation, in order to follow the evolutionof the temperature profile in the stellar interior (see, e.g., [51] and referencestherein).After the proto-neutron star phase, matter is highly degenerate within mostof the star, except the outermost, lowest density, layers. As a consequence,the gross structure of the star does not evolve with time and is determined,once and for all, by solving the Tolman-Oppenheimer-Volkoff equations [13]of hydrostatic equilibrium. An equation of state is required to not only solvethese equations, which determine the mass and radius of the star, but alsoto evaluate the internal chemical composition of each species of nucleus andparticle as well as their effective masses, chemical potentials, specific heats, etc.A complete cooling model requires, moreover, inclusion of neutrino and surfacephoton emissions as well as a description of the pairing properties of matter,i.e., the pairing gaps for each fermonic species, together with their respectivedensity dependences.Within the isothermal approximation of Eq. (50), the three major ingre-dients needed for the study are C V , L γ , and L ν . Neutrino emission processeswere described in Sec. 6 and the specific heat and photon emission are briefly de-scribed below. We continue this section by describing simple analytical solutionsof Eq. (50) and displaying the results of representative numerical simulations ofthe complete set of general relativistic evolutionary equations.39 eutronsprotons m uon s electrons Figure 18: Cumulative specific heats of e, µ , p, and n as a function of stellarvolume within the core of a 1.4 M (cid:12) star built using the APR EOS at T = 10 K. Nucleons are assumed to be unpaired which implies c v ∝ T . No hyperons orquarks are permitted by the EOS. This figure is adapted from [51]. The dominant contributions to the specific heat C v come from the core, whichmakes up more than 90% of the total volume and 98% of the mass. Its con-stituents are quantum liquids of leptons, baryons, mesons, and, possibly, decon-fined quarks at the highest densities. Hence, one has C V = (cid:88) i C V ,i with C V ,i = (cid:90)(cid:90)(cid:90) c v ,i d v , (51)where c v ,i is the specific heat per unit volume of constituent i ( i = e, µ , n, p,hyperons, quarks), but those of meson condensates is usually neglected. Fornormal (i.e., unpaired) degenerate fermions, the standard Fermi liquid result[119] c v i = N (0) π k B T with N (0) = m ∗ i p F i π ¯ h (52)can be used, where m ∗ is the fermion’s effective mass. In Fig. 18, the variouscontributions to C V are illustrated.When baryons, and quarks, become paired, as briefly described in Sec. 4.3,their contribution to C V is strongly suppressed at temperatures T (cid:28) T c ( T c being the corresponding critical temperature). Extensive baryon, and quark,pairing can thus significantly reduce C V , but by no more than a factor of or-der ten because leptons do not pair. The crustal specific heat is, in principle,dominated by neutrons in the inner crust but, as these are certainly extensivelypaired, only the nuclear lattice and electrons contribute in practice.40 .2 Photon Thermal Luminosity and the Envelope The photon thermal luminosity L γ is commonly expressed through the effectivetemperature T e defined by L γ = 4 πR σ SB T e , (53)where σ SB is the Stefan-Boltzmann constant. Thermal photons from the neu-tron star surface are effectively emitted at the photosphere , which is usually inan atmosphere, but could be located on the solid surface if a very strong mag-netic field exists [17]. The atmosphere, which is only a few centimeters thick,contains a temperature gradient; T e gives an estimate of its average tempera-ture. The opacity in the atmosphere receives a strong contribution from free-freescattering that has a strong ( ∼ E − ) energy dependence. As a result photons ofincreasing energy escape from deeper and hotter layers and the emitted thermalspectrum shows an excess of emission in its Wien’s tail compared to a black-body of the same temperature T = T e [120]. In the presence of heavy elements,“metals” in astronomical parlance, which may be not fully ionized, absorptionlines increase the bound-free opacity contributions, and push the Wien’s tail ofthe observable spectrum closer to the blackbody one. The presence of a strongmagnetic field also alters the opacity in such way as to mimic a blackbody withthe same chemical composition and T e . There were great expectations that,with the improved spectral capabilities of Chandra and
Newton observatories,many absorption lines would be observed and allow the determination of thegravitational redshifts and chemical composition of isolated neutron star sur-faces. This expectation has, unfortunately, not been fulfilled; only in a very fewcases have lines been detected, and their interpretation is controversial.Observationally, L γ and T e are red-shifted and Eq. (53) is rewritten as L ∞ γ = 4 πR ∞ σ SB ( T ∞ e ) , (54)where L ∞ γ = e φ L γ , T ∞ e = e φ T e , and R ∞ = e − φ R . Here e − φ = 1 + z , with z being the redshift, and e φ is the g coefficient of the Schwarzschild metric,i.e., e φ ≡ (cid:114) − GMRc . (55)Notice that R ∞ has the physical interpretation of being the star’s radius corre-sponding to its circumference divided by 2 π , and would be the radius one wouldmeasure trigonometrically, if that were possible [121]. L ∞ γ and T ∞ e are the observational quantitities that are compared with the-oretical cooling models. In principle, both are independently observable: T ∞ e is deduced from a fit of the observed spectrum while L ∞ γ is deduced from theobserved total flux , and knowledge of the distance D , via L γ = 4 πD F . Thestar’s distance can be deduced either from the radio signal dispersion measure,if it is a radio pulsar, or from the distance of an associated supernova remnant, The flux must, however, be corrected for interstellar absorption.
41f any. Then Eq. (53) or (54) provides a consistency check: the inferred radius R should be of the order of 10 - 15 km. Given the lack of determination of theatmospheric composition from spectral lines, this consistency check is generallythe only criterion to decide on the reliability of a T e measurement from an at-mosphere model spectral fit (besides the obvious requirement that the modelmust give a good fit to the data, i.e. a χ (cid:39) enve-lope . Encompassing a density range from ρ b at its bottom (typically ρ b = 10 g cm − ) up to ρ e at the photosphere ( ρ e < ∼ − ), and a temperature rangefrom T b to T e , the envelope is about one hundred meters thick. Due to thehigh thermal conductivity of degenerate matter, stars older than a few decadeshave an almost uniform internal temperature, except within the envelope whichacts as a thermal blanket insulating the hot interior from the colder surface. Asimple relationship between T b and T e can be formulated [122]: T e (cid:39) K × (cid:18) T b K (cid:19) . α (56)with α (cid:28)
1. The precise T e − T b relationship depends on the chemical com-position of the envelope. The presence of light elements like H, He, C, or O,which have large thermal conductivities, leads to a larger T e for the same T b relative to the case of a heavy element, such as iron, envelope. Light elementsare not expected to survive densities larger than ∼ g cm − due to pycnonu-clear reactions. Thus, the maximum possible mass in light elements amounts to∆ M Light (cid:39) − M (cid:12) , which is enough to raise T e by a factor of two. Magneticfields also alter the T e − T b relationship, but to a lesser extent (see, e.g., [123]for more details) unless they are super-strong as in the case of magnetars, i.e. B s ∼ G. As the essential ingredients entering Eq. (50) can all be approximated by power-law functions, one can obtain simple and illustrative analytical solutions (seealso [124]). We adopt the notation C V = C T , L ν = N T , and L γ = S T α , (57)where T ≡ T / (10 K) refers to the isothermal temperature T b in the star’sinterior. As written, L ν considers slow neutrino emission involving five degener-ate fermions from the modified Urca and the similar bremsstrahlung processes,summarized in Table 1. The photon luminosity L γ is obtained from Eq. (53)using the simple expression in Eq. (56). We will ignore redshift. Typical pa-rameter values are C (cid:39) erg K − , N (cid:39) erg s − and S (cid:39) ergs − (see Table 3 in [123] for more details). In young stars, neutrinos dominate42he energy losses (in the so-called neutrino cooling era ), and photons take overafter about 10 years (the photon cooling era ). Neutrino cooling era:
In this case L γ can be neglected in Eq. (50), so that t = 10 C N (cid:32) T − T , (cid:33) → T = (cid:16) τ MU t (cid:17) / (when T (cid:28) T ) (58)with a MU cooling timescale τ MU = 19 C / N ∼ T (cid:28) T , T being the initial temperature at time t = 0).The observed slope of the cooling track during the asymptotic stage is (cid:12)(cid:12)(cid:12)(cid:12) d ln T e d ln t (cid:12)(cid:12)(cid:12)(cid:12) (cid:39)
112 + α , (59)using the core-envelope relation Eq. (56). This slope is not sensitive to thecore-envelope relation. Photon cooling era:
In this era, L ν is negligible compared to L γ . Since | α | (cid:28)
1, one finds t = t + τ γ [ln T − ln T , ] → T = T , e ( t − t ) /τ γ , (60)where T is T at time t = t and τ γ = 10 C /S ∼ × years, the photoncooling timescale. The observed slope of the cooling track is, when T (cid:28) T , (cid:12)(cid:12)(cid:12)(cid:12) d ln T e d ln t (cid:12)(cid:12)(cid:12)(cid:12) (cid:39) t τ γ . (61)This slope becomes steeper with time. Numerical simulations of a cooling neutron star use an evolution code in whichthe energy balance and energy transport equations, in their fully general rel-ativistic forms, are solved, usually assuming spherical symmetry and with anumerical radial grid of several hundred zones . Compared to the analytic so-lutions discussed in Sec. 7.3, detailed conductivities, neutrino emissivities, andpossible pairing are taken into account.A set of cooling curves that illustrate the difference between cooling drivenby the modified Urca and the direct Urca processes is presented in Fig. 19.Cooling curves of eight different stars of increasing mass are shown, using anequation of state model from [125], which allows the DU process at densitiesabove 1 . × g cm − , i.e., above a critical neutron star mass of 1.35 M (cid:12) .Notice that the equation of state used has parameters specifically adjusted toobtain a critical mass of 1.35 M (cid:12) which falls within the expected range ofisolated neutron star masses; other equations of state can result in very different Such a code,
NSCool , is available at: CLog Age (yrs) D L og T ( K ) e CBA6.66.25.85.45.00 1 2 3 4 5 D 6 7
Figure 19: Cooling curves illustrating the difference between slow cooling drivenby the modified Urca process, for masses below 1 . M (cid:12) , and fast cooling fromthe direct Urca process in more massive stars [57].critical masses. The difference arising from slow and fast neutrino processes isclear. The various cooling phases, A to D are discussed below.To further illustrate some of the possible cooling behaviors of neutron stars,and the effects of pairing, we show in Fig. 20 simulations based on a 1 . M (cid:12) star built with the APR EOS [16] and a heavy-element envelope. The “slowcooling” models include, in the core, the slow neutrino processes of Table 1and the PBF process only. For the “fast cooling” models, a fast process withemissivity (cid:15) ( q ) F ast = 10 q · T erg cm − s − , with q = 25, 26, and 27, was added at ρ > ρ nuc . These q values simulate neutrino emission from a kaon condensate,a pion condensate, or a direct Urca, respectively. These models, being basedon the same EOS, are not self-consistent, but they have the advantage thatthe only differences among them is the presence or absence of the fast coolingprocess with (cid:15) ( q ) F ast and the presence or absence of nucleon pairing. The modelswith pairing include the neutron S gap “SFB” of Fig. 10 in the inner crust,the S proton gap model “T” of Fig. 11, and the phenomenological neutron P-F gap “b” of the right panel of Fig. 12.The distinctive phases of evolution are labelled “A”, “B”, “C”, and “D” onthe cooling curves in Fig. 19 and above the cooling curves in the left panel ofFig. 20. Phases A and B are determined by the evolution of the crust while Cand D reflect the evolution of the core. We describe these four phases in moredetail according to Fig. 20: 44 hase A: The effective surface temperature T e here is determined by the evo-lution of the outer crust only. At such early stages, the temperature profile inthe outer crust is independent of the rest of the star and, as a result, all modelshave the same T e . Phase B:
The age of the star during this phase is similar to the thermal relax-ation timescale of the crust, the heat flow in which controls the evolution of T e .The evolution of the temperature profile for the fast cooling model with q = 26in the absence of pairing (marked as “Normal” in the left panel of Fig. 20) isdepicted in the right panel of this figure 20. Very early in the evolution, a cold“pit” develops in the core where fast neutrino emission is occurring. During thefirst 30 years, heat flows from the outer core into this pit until the core becomesisothermal. Afterwards, heat from the crust rapidly flows into the cold core andthe surface temperature T e drops rapidly. Well before 300 years, during phaseC, the stellar interior becomes isothermal and it is only within the shallow en-velope, not shown in this figure to preserve clarity, that a temperature gradientis still present.Notice that models of with pairing have shorter crust relaxation times dueto the strong reduction of the neutron specific heat in the inner crust by S neutron superfluidity there. Phase C:
This is the “neutrino cooling phase” in which the star’s evolutionis driven by neutrino emission from the core: L ν (cid:29) L γ . The difference be-tween “slow” and “fast” neutrino emission, with or without core superfluidity,is clearly seen.Quite noticeable is the effect of pairing-induced suppression of the neutrinoemissivity in the fast cooling models. Once T drops below T c , which happensonly a few seconds or minutes after the star’s birth, neutrino cooling is quenched.Which fast cooling process occurs is much less important in the presence ofpairing than in its absence. It takes half a minute, if q = 27, or half an hour, if q = 25, for the “pit’s” temperature to fall below T c , which does not matter whenlooking at the star thousands of years later. The evolution is more dependenton T c than on q .In the case of the slow cooling models, the effects of pairing are more sub-tle than that in the fast cooling models, if one ignores the artificial case withthe PBF processed turned off. The burst of neutrino emission occurring when T (cid:39) T c , from the PBF process, induces an additional, if temporary, rapid cool-ing episode. The impact of the PBF process, however, depends on the size of theneutron P-F gap. If the gap is large enough, the PBF cooling occurs duringstage B and is hidden; if the gap is small enough, the PBF cooling occurs duringstage D and is again hidden. Only intermediate size gaps reveal the presence ofthe PBF process.
The effect of this gap on the evolution is considered in moredetail in Sec. 8. The effect of the proton S gap is more subtle still, and itseffects are considered in detail in Sec. 9.45 A C D " P it " C o r e C r u s t N e u t r on d r i p O u t e r bound a r y Slow coolingFast cooling q=27
Paired ( " PBF " t u r n e d o ff ) PairedNormal q=26q=25
Normal
Figure 20: Left panel: cooling curves for various illustrative cooling scenarios.Right panel: temperature profile evolution for the fast cooling model with q =26. The numbers on the curves give the age of the star, in years. See text fordescription. Figure taken from [126].
Phase D:
At late times, L ν has decreased significantly due to its strong T dependence and photon emission, L γ , which is less T -dependent, now drives theevolution. This is reflected by the larger slopes of the cooling curves. Duringthis “photon cooling era”, models with pairing cool faster due to the reductionof the specific heat from superfluidity. The effect of a very strong magnetic field
In this contribution, we have neglected effects of the presence of a magneticfield. A strong magnetic field can alter the cooling of a neutron star in twoways. In the envelope and the crust, heat is transported by electrons. A surfacemagnetic field of strength ∼ G, a typical value for the majority of pulsars,is sufficient to induce anisotropy in the thermal conductivity of the enveloperesulting in a non-uniform surface temperature [121] that manifests itself as amodulation of the thermal flux with the pulsar’s rotational period. As mentionedin Sec. 7.2, this effect only slightly alters the T b − T e relationship of Eq. (56)[127]. However, a strong magnetic field deep in the crust [128] will have largereffects. For example, a > ∼ G toroidal field within the crust can act as anefficient insulator, rendering most of the star’s surface very cold, but having twohot spots on the symmetry axis of the torus [129, 130]. This results in peculiarcooling trajectories [131]. The second effect of a magnetic field is that a slowlydecaying field can act as a source of energy (i.e., the “H” term in Eq. (50))46hich can keep old neutron stars warm. If the magnetic energy reservoir is largeenough, and small scale magnetic structures exist that can decay rapidly ,aided by the Hall drift, the thermal evolution is significantly altered [132]. It is useful at this juncture to compare these general behaviors with observations.We summarize the observational information relevant to neutron star cooling inFig. 21. The data is separated into three subsets of stars. In the upper panel ofFig. 21 are presented 13 stars for which a thermal spectrum, in the soft X-rayband (0.1 - 3.0 keV) is clearly detected. In the lower panel, we show data from 4pulsars, labeled (A) through (D), that are seen in the X-ray band with a power-law spectrum, but whose detected emission is of magnetospheric origin. Sincethe surface thermal emission from these 4 stars is undetected, being covered bythe magnetospheric emission, only upper limits on their effective temperaturescould be inferred. Finally, the lower panel of this figure contains 6 upper limitsresulting from the absence of detection of any emission from compact objectsin 6 gravitational collapse supernova remnants, labeled (a) through (f), from asearch by Kaplan and collaborators [133]. Since no compact object has beendetected in these supernova remnants, some of them may contain an isolatedblack hole, but that is unlikely to be the case for all 6 of them. The observationsand estimates of ages and temperatures, or upper limits, are detailed in [51] andupdated in [52].Estimates of surface temperatures (redshifted to infinity) require atmosphericmodeling in which three factors are involved: (1) the composition of the atmo-sphere (i.e., H, He or heavy element); (2) the column density of X-ray absorbingmaterial (mostly H) between the star and the Earth; and (3) the surface grav-itational redshift. The column density is important because the bulk of theemitted flux is absorbed before reaching the Earth. The redshift does not af-fect blackbody models, but can influence heavy-element atmosphere models. Inmost cases, this quantity is not optimized, but set to the canonical value for M = 1 . M (cid:12) and R = 10 km stars.Narrow spectral lines are not observed in any of these sources, so their atmo-spheric compositions are unknown. However, some information can be deducedfrom the shape of the spectral distribution as heavy-element atmospheres closelyresemble blackbodies. Fitting the flux and temperature of a source to a modelyields the neutron star radius, if the distance is known. In some cases, clothinga star with a light-element atmosphere results in a predicted radius much largerthan the canonical value, and one can infer the presence of a heavy-elementatmosphere. Chang & Bildsten [134] have noted from such radius argumentsthat there exists a trend for stars younger than 10 years to be better fit bylight-element atmospheres and stars older than 10 years to be better fit byheavy-element atmospheres. The possible evolution of stars leading to this trendis discussed further in [51]. The Ohmic decay time, e.g., is ∝ l , l being the length-scale of the field structure. ) PSR B1706−444) PSR 1E1207−522) PSR J1119−61277) PSR B0538+28175) PSR B0833−451) CXO J232327.8+584842 (in Cas A)(in Vela)
3) PSR J0822−4300 (in Puppis A)
11) PSR B0633+1748 ("Geminga")
10) PSR B1055−529) PSR B0656+1412) RX J1856−375413) RX J0720.4−31258) PSR B2334+61 101000100101 Age [yrs] T [ K ]e f) (? in G084.2+0.8)e) (? in G074.0−8.5)c) (? in G127.1+0.5)b) (? in G093.3+6.9)a) (? in G315.4−2.3) d) (? in G065.3+5.7) (in CTA 1)D) PSR J0007.0+7303 (in Crab) A) PSR B0531+21B) PSR J1124−5916 (in 3C58)
C) PSR J0205+644910 T [ K ]e Figure 21: The present data set of cooling neutron stars. See text for presenta-tion. 48efore embarking on the presentation of detailed cooling scenarios and com-paring them with data, it is worthwhile to discuss one general feature here.The four oldest stars, numbered 10 to 13 in Fig. 21, will appear to be muchwarmer than most theoretical predictions. These stars are prime candidates forthe occurence of some late “heating”, i.e. the “ H ” term in Eq. (50). Models canbe easily adapted to incorporate such heating (which has nothing to do withneutrino emission or pairing) that may become important at these times [123].We will therefore concentrate on younger objects in the comparison with data. The thermal evolution of a neutron star undergoing fast neutrino emission wasillustrated in Fig. 20 which showed that such a star, after the initial crustrelaxation phase, has a very low T e unless the neutrino emission is suppressedby pairing. In the presence of a gap that quenches fast neutrino emission, theresulting T e is more sensitive to the size of the quenching gap than to the specificfast neutrino process in action.In Fig. 22, we compare one specific scenario with the observational datadescribed in the previous Sec. 7.5. This scenario employs the same EOS as in themodels of Fig. 19 and is more realistic than what was described in Sec. 7.4 insofaras it takes into account uncertainties concerning the chemical composition ofthe envelope. If the neutron star has an envelope containing the maximumpossible amount of light elements, ( ∼ − M (cid:12) ), its T e is raised by a factorof two compared to the case when its envelope contains only heavy elements.Results for these two extreme scenarios of envelope composition are separatelyshown in the two panels of Fig. 20. In each panel, the first noticeable feature isthe mass dependence of results when the mass exceeds 1 . M (cid:12) and the directUrca process is activated. This is partially due to the increasing size of the fastneutrino emitting “pit” as the star’s mass increases, but more dominantly dueto the decrease in the neutron P-F gap with increasing density. More massiveneutron stars, at least in this model, will have the direct Urca process acting intheir inner cores for a longer time until their central temperatures drop belowthe corresponding central values of T c . Notice that in a model where the P-F gap keeps growing with density as, in the “3bf” model of Fig. 12, this massdependence essentially disappears.When comparing the cooling trajectories of Fig. 22 with the displayed datawe find that, with the exception of some of the oldest stars, all observed tem-peratures can be reasonably well fit. This result implies that different neutronstars have different masses, and some stars require an envelope containing lightelements. This conclusion is common to all fast cooling scenarios that involveeither the simple direct Urca process with nucleons or any direct Urca processwith hyperons or quarks, as well as scenarios with charged mesons condensates.The successful recipe consists of having some fast neutrino process allowed inmassive enough neutron stars. Then, in order to prevent intermediate massstars from cooling too much, it is necessary to have a gap, or several gaps, largeenough so that all fast neutrino processes are quenched at some point.49 ba fedC DAB 46 7 9 1012 1321 113 85 10 T [ K ]e edcba fDA CB 132 983 456 71 1011 1210 T [ K ]e Figure 22: Comparision of a cooling scenario with data. The dense matter EOSused to build the stars is a PAL [125] version [57] that has only nucleons andleptons, and allows for the direct Urca process to occur in stars of mass largerthan M cr = 1 . M (cid:12) . Cooling trajectories are shown for seven different massesand two different envelope chemical compositions. The neutron P-F gap frommodel “b” of Fig. 12 is implemented while the neutron and proton S gaps arefrom models “SFB” of Fig. 10 and “T” from Fig. 11, respectively.50 .7 Can Quark Matter Be Detected from Cooling Obser-vations? The previous subsection presented one specific fast cooling scenario, while Sec.7.4 only described very general trends of fast cooling scenario with pairing. Itis interesting, however, to briefly consider if observations could determine notonly whether or not fast cooling has occurred but also which fast cooling mech-anism operates. One of the first studies of hybrid stars with superconductingquarks [135] showed, in fact, that it will be difficult to establish that neutronstars contain deconfined quark matter based solely on the cooling data of iso-lated neutron stars. When quark superconductivity is present, for example inthe 2SC phase, neutrino emission is suppressed by a factor of e − ∆ /T for eachflavor and color quark which is paired. Cooling in the CFL phase is special be-cause of the Goldstone bosons, which provide new cooling processes, and otherpairing configurations also have their own cooling behaviors. Quark gaps thatare typically larger than baryonic gaps therefore imply that hybrid star coolingis driven by other degrees of freedom present at lower densities. On the otherhand, small quark gaps, or the absence of quark matter entirely, imply similarcooling behaviors that are easily reproduced by models with varying masses.The above conclusions are demonstrated in Fig. 23, in different ways inthe two panels, using two pairs of EOSs. In the left panel, cooling curveswith a fixed neutron star mass and employing two EOSs, one containing onlynucleons (np) and the other with nucleons with quark matter (npQ), are shown.These simulations utilize a range of nucleon and quark pairing gaps. In theright panel, cooling curves of neutron stars of various masses and employingtwo EOSs, one with nucleons and hyperons (npH) and the other with quarkmatter added (npHQ), are compared. This second set of simulations utilizesfixed nucleon/hyperon/quark pairing gaps.We refer the reader to [135] for details, but the figure illustrates that, givenour poor knowledge of the size and density extent of the various pairing gaps,nearly indistinguishable cooling curves can be obtained with very different densematter models, i.e with or without hyperons and/or with or without quarkmatter. Complementary studies, as, e.g., in [136, 137, 138, 139], confirm thisconclusion. We are led, by the discussions in Sec. 7.6 and Sec. 7.7, to an intriguing result:
Any equation of state that permits fast neutrino emission is compatible with thepresently available cooling data on isolated neutron stars IF pairing is presentwith gap(s) of the appropriate size(s) . This result, unfortunately, is ambigu-ous: it means that theoretical uncertainties in the description of dense matter The ultimate theory of neutron star matter will yield the equation of state and the super-fluid gaps. However, at present, this is far from being the case since the gaps, as described inSec. 4, are very sensitive to many intricate Fermi surface processes to which the equation ofstate, which is a bulk property, is quite insensitive. A,B,C,D] Log Age [yrs] 76.56.05.55.00 1 2 3 4 5 6 8 [Mc] e [c][b] L og T [ K ] [a][z] [A][B,C,D][Z] e L og T [ K ] Masses of npHQ matter stars:
Figure 23: Left panel: cooling of a 1 . M (cid:12) neutron star composed only ofnucleons (continuous lines) or nucleons and quark matter (dashed and dottedcurves), with various assumptions on the pairing gaps. Right panel: mass depen-dent cooling histories for two dense matter scenarios: “npH” matter containsnucleons and hyperons; “npHQ” matter also includes deconfined quarks. Inthese two scenarios, pairing gaps are assumed fixed. Figures from [135].make it difficult to determine the composition of neutron star cores from isolatedneutron star cooling observations alone.One could, however, take an optimistic approach and reverse the argument.As the cooling of neutron stars is very sensitive to the presence, and the sizes, ofpairing gaps, we can apply this fact to attempt to measure the gap(s) [57]. Thisis in line with the idea of using neutron stars as extra-terrestrial laboratories tostudy dense matter. However, as massive neutron stars may contain hyperons,meson condensates or deconfined quarks, the question arises: which gap wouldwe be measuring [135] ? This situation, as demonstrated in the maximally com-plicated models of Fig. 23, may appear to render further studies of neutron starcooling a waste of time.However, our discussion so far has assumed the presence of “fast” cooling.Is fast cooling, in fact, actually necessary? This question motivates the minimal cooling scenario, to which we turn in Sec. 8, which considers only models inwhich enhanced cooling processes do not occur.
The comparison in the previous section of theoretical cooling models with avail-able data suggests the presence of extensive pairing in the cores of neutron stars.However, the plethora of possible scenarios makes it difficult to go beyond thisgeneric conclusion. And, in spite of the fact that many theoretical models of52ense matter predict the presence of some form of “exotica”, one should stillask the question “do we need them?” and, if yes, how strong is the evidence forthem? To address these questions, the minimal cooling paradigm was developedin [51, 52]. In this scenario, all possible fast neutrino emission processes (fromdirect Urca processes involving baryons or quarks, and from any form of exot-ica) are excluded a priori . Superfluid effects along with the PBF process are,however, included in the slow neutrino emission processes, such as the modifiedUrca processes involving neutrons and protons. This is a very restrictive sce-nario that minimizes the number of degrees of freedom, but fully incorporatesuncertainties associated with the equation of state, envelope composition and itsmass, etc. A detailed presentation of this scenario can be found in [51, 52] and avariant of it was developed by the Ioffe group in [140, 141]. Extensive studies ofthese two groups have pinned down two major sources of uncertainties: (i) ourpresent lack of knowledge on the chemical composition of the envelope, and (ii)the size and extent of the neutron P-F gap. It turns out that, in the absence ofany “fast process”, neutrino emissivity resulting from the PBF process involving P-F Cooper pairs is a major factor in the thermal evolution of a neutron starand its effect is strongly dependent on the size of the neutron P-F gap. Wedescribe in some detail effects of the PBF process in the next subsection beforeturning to a comparison of the predictions of the minimal cooling paradigm withavailable data. The PBF process is distinctive in the sense that, in any layer in the star’sinterior, it turns on when the temperature T reaches the corresponding T c ofthe ambient density. Then the pairing phase transition is triggered and while T is not too much lower than T c , there is constant formation and breakingof Cooper pairs induced by thermal excitation. However, when T falls below ∼ . T c , there is not enough thermal energy to break pairs and the processshuts off (see its control function displayed in the right panel of Fig. 17). Asdescribed in Sec. 6.6 the neutrino emissivity of the PBF process is significantonly in the case of anisotropic pairing, as in the case of a P-F gap.The schematic Fig. 24 illustrates the effect of the pairing phase transitionon the total neutrino luminosity of the star, L ν , depicted in the upper part ofeach panel as a function of the core temperature T . The long-dashed curveslabelled “MUrca” show the modified Urca luminosity, L MU ν ∝ T in the absenceof neutron P-F pairing while the short-dashed lines labelled “PBF” show themaximum possible PBF luminosity, L PBF ν ∝ T , i.e. from Eq. (46) with thecontrol function R j set to 1. The lower part of each panel shows a neutron P-F T c versus radius curve (displayed with its T axis horizontal to coincidewith the upper part’s T axis). Assuming the temperature T to be nearly uniformin space in the entire core, the core T profile is just a vertical straight line thatmoves from the right to the left as the star cools down. Each panel considers adifferent type of T c curve and in its upper part, the thick continuous line showsthe actual evolution of the total L ν in the presence of pairing. As the star cools,53nd while T > T max cn , the total L ν is dominated by the MUrca process with smallcontributions from other much less efficient processes. When T reaches T max cn ,there is a thick layer in which the phase transition starts and where neutrinoemission from the PBF process is triggered which increases L ν . As T decreases,but is still larger then T min cn , there will be one or two layers in the star where T is only slightly lower than the corresponding value of T c in that layer(s) andwhere the PBF process is efficiently acting. (At these times, neutrino emissionis strongly suppressed in the layers where T is much smaller than T c .) Finally,when T drops well below T min cn , neutrino emission by the dominant processes issuppressed everywhere in the core and L ν drops rapidly to reflect the much lessefficient neutrino emissivity from the crust. The higher the value of T max cn theearlier in the star’s history will the enhanced cooling from the PBF process betriggered, while the lower the value of T min cn the longer this enhanced cooling willlast, as illustrated by the difference between the left and right panels of Fig. 24.This description will be pursued with a simple analytical model in Sec. 9.1 and9.2. T r a d i u s TT r a d i u s T L T cnmax ν L T cnmin T cnmax T cnmin P B F M U r c a P B F ν M U r c a Figure 24: Schematic illustration of the PBF process induced neutrino luminos-ity as controlled by the shape of the T c curve. See text for a detailed description.From the above discussion, we learn that the effect of the PBF processdepends strongly on the size and density dependence of the neutron P-F gap.To explore the possible impact of these two physical ingredients, we show inFig. 25 two sets of cooling trajectories resulting from different assumptions.For contrast, both panels show a curve with no pairing, i.e., the same as theslow cooling model marked “Normal” in the left panel of Fig. 20. The variouscooling curves are labelled by T max cn , the maximum value of T c reached in thestellar core. In the left panel, large values of T max cn corresponding to the three54henomenological gaps “a”, “b”, and “c” of the right panel of Fig. 12, areemployed. All three gaps lead to the same result during the photon coolingera ( t > yrs), but case “a” strongly differs from the other two cases duringthe neutrino cooling era. All three models have large enough gaps and T max cn that the enhanced cooling from the PBF process started well before the end ofthe crust relaxation phase. The gaps and T min cn of models “b” and “c” are solarge that they induce a very strong suppression of L ν at early times and hencehigh values of T e during the neutrino cooling phase. In contrast, the gap “a”has a moderate T max cn and also a very small T min cn which guarantees that at ages10 − yrs there is alway a significant layer going through the pairing phasetransition in which L PBF ν is large which results in a colder star.As small gaps have the strongest effect in inducing some enhanced cooling ofthe neutron star, we explore in the right panel of Fig. 25 the effect of increasinglysmall neutron P-F gaps by scaling down the model “a” gap by a factor 0.6,0.4, and 0.2, respectively. A new feature now emerges: with decreasing valuesof T max cn , the pairing phase transition is initiated at progressively later stages.This feature is signaled by cooling trajectories (dotted curves) departing fromthe T max cn = 0 trajectory. This late onset of pairing then manifests itself as asudden rapid cooling of the star (due to the sudden increase of L ν from L MU ν to L PBF ν at the moment when T reaches T max cn ) and interestingly within the agerange for which we have many observations available.Armed with these considerations, we turn now to compare the predictionsof the minimal cooling paradigm with data.
10 3 1 0Model gap: c b a
T (10 K) = cn T (10 K) =
Figure 25: Neutron star cooling trajectories varying the magnitude of the neu-tron P-F gap. See text for description. Fig. 26 shows ranges of predicted thermal evolutions of a canonical neutron starof 1 . M (cid:12) built with the EOS of Akmal & Pandharipande [16]. The densitydependence of the symmetry energy in this EOS precludes the occurrence of55 ea fb c DBA C1 2 3 46 78 9 101112 135 max (T = 10 K) HEE LEE LEE HEE Neutron P−F gap: "a" cn cde fbaA C DB 101 2 3 46 78 9 1112 135 Neutron P−F gap: "b"
LEE (T = 3x10 K) cnmax 92
LEE HEEHEE Figure 26: Comparison of two minimal cooling scenarios with observationaldata. The neutron P-F gaps employed are shown in Fig. 12. See text fordescription. Adapted from the results of [52]56he direct Urca process in a 1 . M (cid:12) neutron star. Furthermore, this EOS doesnot permit hyperons, kaon condensates, pions or deconfined quark matter, allof which could lead to enhanced cooling. It was shown in [51] that all EOS’scompatible with the restrictions imposed by the minimal cooling scenario yieldalmost identical predictions. Moreover, within the range of neutron P-F pair-ing gaps explored, for which T max cn is reached at a density that is smaller thanthe central density of any neutron star (see right panel of Fig. 12), the neutronstar mass of any model also has very little effect .The various possible assumptions about the neutron S gap have only asmall effect on the early crust thermal relaxation phase, phase (B) of Fig. 20.Changes in the proton S gap produce significantly differing effects, but dom-inant effects are due to changes in the neutron P-F gaps. The full range ofpossibilities is (almost) covered by considering the two neutron P-F gaps inmodels “a” and “b” of Fig. 12 and is depicted in the two panels of Fig. 26. Thevarious curves in the grey shaded areas show the uncertainty in the predictionsdue to lack of knowledge of the envelope compositions, which can range frompure heavy elements (“HEE”) to pure light elements (“LEE”). In agreementwith the results of Fig. 25 and the discussion of the previous subsection, the largegap “b” results in warmer stars than the smaller gap “a” during the neutrinocooling era.Comparing model tracks with observations, we can conclude, following [51]and [52], that if the neutron P-F gap is of small size ( T max cn ∼ K) as that ofmodel “a”, all neutron stars with detected thermal emission are compatible withthe minimal cooling scenario with the exception of the oldest objects (labelled10, 12, and 13 in Fig. 26, see the discussion at the end of Sec. 7.5). Theseolder stars are candidates for the presence of internal heating, i.e., the “H”term in Eq. (50) [123], a possibility that can easily be incorporated within theminimal cooling scenario. In addition, it is found that young neutron starsmust have heterogeneous envelope compositions: some must have light-elementcompositions and some must have heavy-element compositions, as noted above.The updated comparison by [52] more precisely quantifies the required sizeof this gap. In the notations used in Fig. 24, maximal compatibility of theminimal scenario requires the neutron P-F gap to satisfy T max cn > ∼ × Kand T min cn < ∼ × K. The constraint on T max cn , which determines when the PBFprocess will be triggered, derives from the necessity of having a PBF enhanced L ν already acting in the youngest observed stars at ages < ∼ yrs. On theother side, the constraint on the low T min cn , which determines when core neutrinoemission will be strongly suppressed, including the PBF process, is obtainedby the requirement that L ν should not be strongly suppressed before the starreaches an age of a few times 10 yrs. This constraint assures compatibility withthe low T e of objects 5 and 6. Considering the smooth k F , and thus density,dependence of gaps, the constraint on T min cn likely prevents T max cn from being too The Ioffe group, in their version of minimal cooling [140, 141], assumed neutron P-F gaps that are small at ρ nuc and grow rapidly at high densities. Consequently, T max cn is reachedin the center of the star and stars of increasing mass have increasing values of T max cn resultingin a mass dependence of the cooling evolution. T max cn (cid:29) K usually also have T min cn (cid:29) K.In the case the neutron P-F gap is as large as our model “b”, one reachesthe opposite conclusion that minimal cooling cannot explain about half of theyoung isolated stars, implying the occurence of some fast neutrino process. Thesame conclusion is also reached in the case this gap is very small. As seen inthe left panel of Fig. 25, a vanishing gap leads to cooling trajectories very closeto those obtained using large gaps during the neutrino cooling era.Only one object, the pulsar in the supernova remnant CTA 1 (object “D” inthe figure) for which only an upper limit on T e is available, stands out as beingsignificantly below all predictions of minimal cooling.Finally, irrespective of the magnitude of the neutron P-F gap, if any ofthe six upper limits marked “a” to “f” in Fig. 26 are, in fact, neutron stars,they can only be explained by enhanced cooling. As it is unlikely that allof these remnants contain black holes, since the predicted neutron star/blackhole abundance ratio from gravitational collapse supernovae is not small, theseobjects provide additional evidence in favor of enhanced cooling in some neutronstars . As the nuclear symmetry energy likely increases continuously withdensity, larger-mass stars, which have larger central densities, likely have largerproton fractions and a greater probability of enhanced cooling. The dichotomybetween minimal cooling and enhanced cooling might simply be due to a criticalneutron star mass above which the direct Urca process can operate before beingquenched by superfluidity. The study of the minimal cooling scenario allows progress beyond the conclu-sions reached in Sec. 7.8, but we are still faced with a clear dichotomy:(1) In the case the neutron P-F gap is small in size (satisfying T min cn < ∼ × K and T max cn > ∼ × K), neutron stars undergoing fast neutrinocooling must be relatively rare as among the dozen of known young isolatedneutron stars, only one candidate, the pulsar in the supernova remnant CTA1,exists. This candidate is augmented by possible neutron stars represented byone or more of the(green) upper limits in Fig. 26. Nevertheless, the total numberof fast cooling candidates remains small.(2) In the case the neutron P-F gap is either larger, or smaller than in (1),a much larger fraction of neutron stars appear to undergo fast neutrino cooling.Both conclusions are extremely interesting, but more information is neededto choose the (hopefully) correct one. Such information may be at hand and ispresented in the next section. These conclusions also have implications in termsof neutron star masses. Neutron star mass measurements are only available Further evidence for enhanced cooling is provided by two neutron stars, SAX J1808.4-3658 and 1H 1905+00, in transiently accreting low-mass X-ray binaries. In contrast to thethe six, yet to be detected, candidates “a” to “f”, these two stars are known to be neutronstars from their characteristics during the accretion phases, but their thermal emission is notdetectable after accretion stops, implying extremely fast neutrino emission occurring in theircores [142, 143]. M cr for fast neutrino cooling would appear to be much largerthan the average mass, perhaps in the range 1 . − . M (cid:12) . Alternatively, ifconclusion (2) above is correct, M cr ∼ . M (cid:12) , around the average mass.Figure 27: The initial mass function of neutron stars as predicted by stellarevolution models. The continuous line shows results from [145] and the dottedline is adapted from [10]. Figure from [15]. The Cassiopeia A (Cas A) supernova remnant (SNR) was discovered in radio in1947 and is the second brightest radio source in the sky (after the Sun). It hassince then been observed at almost all wavelengths. Very likely, the supernovawas observed by the first
Astronomer Royal , John Flamsteed [146] who, onAugust 16, 1680, when describing the stars in the Cassiopeia constellation,listed the star “3 Cassiopeia” at a position almost coincident with the supernovaremnant. This star had never been reported previously, and was never to beseen again, until August 1999 when the first light observation of
Chandra founda point source in the center of the remnant (see Fig. 28).The distance to the SNR is 3 . +0 . − . kpc [147], and the direct observation,by the Hubble Space Telescope , of the remnant expansion implies a birth inthe second half of the 17 th century [148] and supports Flamsteed’s observation.These observations give a present age of 333 yrs for the neutron star in Cas A.The optical spectrum of the supernova has been observed through its light echofrom scattering of the original light by a cloud of interstellar dust and shows59igure 28: The Cassiopeia supernova remnant in X-rays: first light of Chandra ,August 1999. (The neutron star is highlighted by the authors.) Image fromc (cid:13)
NASA/CXC/SAO.the supernova was of type IIb [149]. The progenitor was thus a red supergiantthat had lost most of its hydrogen envelope, with an estimated zero age mainsequence (ZAMS) mass of 16 to 20 M (cid:12) [150, 151, 152] or even up to 25 M (cid:12) in the case of a binary system [153]. This implies a relatively massive neutronstar, i.e. likely > ∼ . M (cid:12) [153]. The large amount of circumstellar materialassociated with mass loss from its massive progenitor could have diminishedits visibility from Earth and could explain why it wasn’t as bright as the twoRenaissance supernovae, Kepler’s SN 1572 and Tycho’s SN 1604.The soft X-ray spectrum of the point source in the center of the SNR inCas A is thermal, but its interpretation has been challenging [154]. With aknown distance, a measurement of the temperature implies a measurement ofthe star’s radius, but spectral fits with a blackbody or a H atmosphere modelresulted in an estimated radius of 0.5 and 2 km, respectively. This suggestsa hot spot, but that should lead to spin-induced variations in the X-ray flux,which are not observed. It was only in 2009 that a successful model was found: anon-magnetized C atmosphere , which implies a stellar radius between 8 to 18km [155]. Models with heavier elements, or a blackbody, produced significantlypoorer fits. With the C model, and analyzing 5 observations of the SNR, Heinke& Ho [156] found that the inferred neutron star surface temperature had droppedby 4% from 2000 to 2009, from 2 .
12 to 2 . × K, and the observed flux haddecreased by 21%. The neutron star in Cas A, the youngest known neutronstar, is the first one whose cooling has been observed in real time! There is, to date, no evidence for the presence of a significant magnetic field in the CasA neutron star. .1 Superfluid Neutrons in the Core of Cas A The
Chandra observations of Cas A, together with its known distance, implythat the photon luminosity of the neutron star is L γ (cid:39) erg s − . (62)With a measured T ∞ e (cid:39) × K [155], we deduce an internal T (cid:39) × Kfrom Eq. (56). The star’s total specific heat is thus C V (cid:39) × erg K − (fromFig. 18 or Eq. (57)). The observed ∆ T ∞ e /T ∞ e (cid:39)
4% [156] gives a change of coretemperature ∆
T /T (cid:39)
8% over a ten years period since T ∼ ( T ∞ e ) . Assumingthe observed cooling corresponds to the global cooling of the neutron star, itsthermal energy loss is˙ E th = C V ˙ T (cid:39) (4 × erg K − ) × (0 . − ) (cid:39) × erg s − , (63)which is 3–4 orders of magnitude larger than L γ ! For a young neutron star,neutrinos are the prime candidates to induce such a large energy loss.The cooling rate of this neutron star is so large that it must be a transitoryevent, which was initiated only recently. Otherwise it would have cooled so muchit would probably now be unobservable. Something critical occurred recentlywithin this star! “Something critical” for a cooling neutron star points toward acritical temperature, and a phase transition is a good candidate. The previoussection highlighted that a phase of accelerated cooling occurs when the neutron P-F pairing phase transition is triggered. With T max cn (cid:39) × K, a transitorycooling episode can occur at an age (cid:39)
300 yrs as shown in the right panel ofFig. 25 and in Fig. 29. C T /10 K = e T ( K ) Figure 29: Similar to the right panel of Fig. 25, but showing the observed ageand temperature of the Cas A neutron star (the star) and its consistency with T C (cid:39) × K for the magnitude of the neutron P-F gap. Taken from [53].The interpretation that the observed rapid cooling of the neutron star inCas A was triggered by the recent onset of the neutron P-F superfluid phase61ransition and consequent neutrino emission from the formation and breakingof Cooper pairs in the neutron superfluid was recently proposed in [53] and,independently, in [157]. A simple analytical model
The simple analytical solution of Eq. (58) gives some insight into the observedbehavior. When
T > T C ≡ T max cn , but (cid:28) T , the star follows the asymptotic“MU trajectory”, T = ( τ MU /t ) / , and when T reaches T C , at time t = t C ,the neutrino luminosity suddenly increases (see Fig. 24). Despite the compli-cated T dependence of (cid:15) PBF , Eq. (46), the resulting luminosity, once integratedover the entire core (also, aided by the bell shape of the T c ( k F ) curve), is wellapproximated by a T power law in the T regime in which some thick shell ofneutrons is going through the phase transition. If we write L PBF ν = f · L MU ν = f N T , (64)with f ∼
10, the solution of Eq. (58), replacing t by t C and T by T C , gives T = T C [1 + f ( t − t C ) /t C ] / → T = (cid:18) τ MU f t (cid:19) / ( T (cid:28) T C ) . (65)Thus, at late times, the asymptotic “PBF trajectory” is a factor f − / (cid:39) . T = ( τ MU /f t ) / , is τ TR = t C /f . This behavior is shown schematically in the left panel of Fig. 30.Since the initiation of the rapid transit, when the neutron star left the MUtrajectory, must have occurred recently, i.e. t C (cid:39) (0 . − . ×
333 yrs, we obtain T C = T max cn = 10 K ( τ MU /t C ) / ∼ × K . (66)This is the first important result from this simple analytical model: given theobservation of rapid cooling, the inferred value of T C depends only on the knownage, t = 333 yrs, of this neutron star and the value of τ MU ∼ / How “rapid” is the observed “rapid cooling” of the Cas A neutron star can bequantified by the observed slope s obs = (cid:12)(cid:12)(cid:12)(cid:12) d ln T ∞ e d ln t (cid:12)(cid:12)(cid:12)(cid:12) (cid:39) . s MU = (cid:12)(cid:12)(cid:12)(cid:12) d ln T ∞ e d ln t (cid:12)(cid:12)(cid:12)(cid:12) = d ln T ∞ e d ln T × (cid:12)(cid:12)(cid:12)(cid:12) d ln Td ln t (cid:12)(cid:12)(cid:12)(cid:12) (cid:39) × (cid:39) . . (68)62 B F t r a j ec t o r y C t CC T Log t S M U t r a j ec t o r y M U t r a j ec t o r y L og T Log t M U t r a j ec t o r y P B F t r a j ec t o r y C T’ t’ C C L og T Figure 30: Schematic cooling trajectories (heavy curves) showing the effectof superconductivity. Left panel: Without superconductivity, T initially followsthe modified Urca (MU) trajectory, T = ( τ MU /t ) / until T reaches the neutroncritical temperature T C at time t C . The pair breaking and formation (PBF)process turns on and the neutrino luminosity L ν abruptly increases by a factor f . Thereafter, T rapidly transits, on a time scale τ TR = t C /f toward the PBFtrajectory, T = ( τ MU /f t ) / . Empirically, this transition takes a time < ∼ t C .Right panel: When protons are superconducting, the initial evolution followsa superconducting-suppressed modified Urca (SMU) path. For the transit tostart at a time t (cid:48) C ≈ t C the trajectory requires T (cid:48) C > T C . The early transit hasa shorter time scale, τ (cid:48) TR = t (cid:48) C /f (cid:48) with f (cid:48) (cid:29) f , and a significantly larger slope.The late time evolution converges to that of the left panel. The left (right) panelcorresponds to models in Fig. 29 (Fig. 31 and 32). From [53].In contrast, the “transit trajectory” of Eq. (65) gives a much larger slope s TR = 12 × f (cid:18) TT C (cid:19) tt C , (69)which has a maximum value f /
12. Nevertheless, to match the observed slope s obs , f (cid:29)
10 is required.The value of f considered above, f = L PBF ν /L MU ν (cid:39)
10, arose from a freelyacting modified Urca neutrino emission. However, despite the many theoreticaluncertainties discussed in Sec. 4, there seems little doubt that proton supercon-ductivity exists around a few times ρ nuc , as illustrated in Fig. 11. Moreover,expected values of T max cp are somewhat larger than 10 K, implying that protonswere likely already superconducting in some part of the core of the Cas A neu-tron star when neutron anisotropic superfluidity set in. If such was the case, theprevious neutrino luminosity of this star was due to a proton-pairing-suppressedmodified Urca process with L SMU ν < L MU ν . This implies a much higher relativeefficiency of the neutron PBF process, i.e., f (cid:48) = L PBF ν /L SMU ν (cid:29) f ∼
10. Theresulting transit from a “SMU trajectory” to the “PBF trajectory” is depictedin the right panel of Fig. 30 and exhibits a transit slope enhanced by a factor63 T = K T = C C T = . K Figure 31: A fit to observations of the neutron star in Cas A assuming recentonset of neutron P-F superfluidity and PBF cooling. The 1 . M (cid:12) model shownassumes the APR EOS [16] with a C atmosphere [155]. With model “CCDK”of Fig. 11 ( T max cp (cid:29) K), proton are superconducting from early times. Theneutron P-F gap is model “a2” of Fig. 12 with T C = T max cn = 5 . × K. This neutron superfluid phase transition triggered the PBF process thatresults in a sudden cooling of the neutron star. Observations [156, 157] suggest | d ln T ∞ e /d ln t | (cid:39) .
4, shown in the inset. Two dotted curves with T C = 0 and1 × K, respectively, illustrate the sensitivity to T C . Figure adapted from[53]. f (cid:48) /f .With the above considerations, a very good fit to the observations can beobtained, as shown in Fig. 31, implying a maximum neutron P-F pairing T C = 5 . × K along with superconducting protons with a larger T max cp . Verysimilar results were independently obtained by [157] . This observation ofthe cooling of the youngest known neutron star is unique and its interpretationpotentially imposes very strong constraints on the physics of ultra-dense matter.The requirement that protons became superconducting before the onset ofneutron superfluidity places a constraint not only on the proton S pairing butalso on the neutron star mass. If the neutron star mass is not too large, mod-els show that proton superconductivity does not extend to very high densities. Presumably, at the time of the onset of proton superconductivity, another PBF episodehad occurred, but, as the PBF process for singlet pairing is much less efficient than for tripletpairing, and also because protons are much less abundant than neutrons, it did not result ina significant cooling of the star. Moreover, this cooling occurred on timescales much smallerthan the crustal thermal timescale. These authors, however, assumed that proton superconductivity extends to very highdensities, with T cp > ∼ (2 − × K in the whole neutron star core. This results in a verystrong suppression of L ν prior to the onset of neutron P-F superfluidity, and a neutronstar much warmer than in the model of [53], from which a larger T max cn (cid:39) (6 − × K isdeduced. However, strong proton superconductivity at high densities is, at present time, notsupported by the microscopic models presented in Fig. 11.
C 9
Figure 32: Dependence of the slope s = | d ln T ∞ e /d ln t | of the cooling curveon the star mass at t = 330 years: s = 1 . , .
9, and 0.5 for M = 1 . , .
6, and1 . M (cid:12) , respectively (from [53]).The neutron superfluid explanation for the rapid cooling of the neutron starin Cas A fits well within the minimal cooling scenario [51, 52]. As described inSec. 8.2, maximal compatibility of the minimal cooling scenario [52] with datarequired the neutron P-F gap to have T max cn > ∼ × . This lower limit on T max cn was deduced for compatibility with the measured T e of the youngest neutronstars of age ∼ yrs The upper limit on T min cn was deduced for compatibilitywith the oldest middle-aged stars as, e.g., the Vela pulsar. The compatibilityof the neutron P-F gap inferred from the cooling data of the neutron star inCas A with observations of other isolated neutron stars is confimed in Fig. 33.The only marked difference between Fig. 33 and the left panel of Fig. 26 is theoccurrence of the rapid cooling phase at ages (cid:39)
300 yrs to ∼
500 yrs, due to thereduced value of T max cn in the former, 0 . × K compared to 1 . × K. Alternative explanations for the observed rapid cooling of Cas A have beenproposed. One could consider observed cooling of this star to be due to a sig-nificantly longer thermal relaxation timescale in the crust or core than assumedin [53, 157]. In such a case, the estimate of Eq. (63), which assumes the starto be isothermal, becomes invalid. In all models shown in this chapter, thecrust thermal relaxation occurs on a timescale of a few decades. However, ifthe crust thermal conductivity is, in fact, significantly smaller, it is conceivablethat the observed rapid cooling corresonds to the thermal relaxation of the crust(see, e.g., [158]). Such a low crust thermal conductivity is, however, in conflictwith the observed crust relaxation time in transiently accreting neutron stars[159, 160] and is based on the assumption that the crust is in an amorphous65
BA C 51 2 3 46 78 9 101112 138 (T = 5.5x10 K)
Neutron P−F gap: "a2"
Figure 33: Comparison of the cooling scenario of Fig. 31 with data for isolatedneutron stars. As in Fig. 26, the various lines show the effect of varying theamount of light elements in the envelope (from [126]).solid state instead of a crystalline one, a possibility that is not supported bymicroscopic studies [161, 162].Similarly, the core thermal relaxation time may be much larger than usuallyconsidered. For example, Blaschke et al. [163] have proposed that the inner coreof the star cools rapidly and that it also takes a few hundreds years for the starto become isothermal. The latter time is when the rapid decrease of T ∞ e wouldbe observed. This scenario requires that the core thermal conductivity be lowerthan usually considered, by a factor 4 or larger, and also requires that neutrons do not form a superfluid until the star is much colder. This scenario, basedon the “Medium-Modified Urca” neutrino emission process [72, 104], is alsocompatible with the cooling data, but only if the suppression of core conductivityis adjusted to fit the observed cooling of the neutron star in Cas A. More work isrequired to confront these alternative possibilities with other facets of neutronstar phenomenology.Finally, there are important systematic uncertainties related to the observa-tions which may affect our ability to interpret the cooling of this neutron star.Among these uncertainties are: the incorrect idenfication of two simultaneousphotons as a single photon of larger energy, detector calibration issues, and con-tributions from material in the line of sight between the neutron star and theobserving satellite. Recent analyses of these uncertainties cannot conclusivelyconfirm that cooling is present, but cannot unambiguosly rule out such coolingeither [164, 165]. 66 Pulsars are rotating neutron stars whose spin rates are generally observed tobe decreasing. The regularity of pulsars is outstanding; as timekeepers theyrival atomic clocks. Although the pulses are remarkably regular, the time be-tween pulses slowly but predictably increases. Their spin-downs are attributedto magnetic dipole radiation [166, 167] – the conversion of rotational energyinto electromagnetic energy – and the pulsar wind from ejection of the mag-netospheric plasma [168]. That the observed evolution of pulsar spins exhibitsevidence for both core and crustal superfluidity is the subject of this section.
Dimensional analysis gives us a simple estimate of the pulsar energy losses.Energy and angular momentum are irreversibly lost when either the magneticfield or the plasma reaches the light cylinder where co-rotation with the pulsarimplies a speed equal to c and whose radius is thus R lc = c/ Ω (cid:39) P/ P = 2 π/ Ω the spin period. The magnetic field atthe light cylinder is B lc = ( R/R lc ) B s assuming a dipolar field with a strength B s at the stellar surface. Just writing that an energy density B lc / π is lost atthe speed of light c from a sphere of area 4 πR lc , one obtains˙ E P SR (cid:39) c ( R B s ) Ω . (70)Energy loss by in vacuum magneto-dipolar radiation gives the same result withjust an extra factor (2 /
3) sin α , α being the angle between the rotationnalaxis and the dipolar moment. An aligned rotator, i.e. with α = 0, will notspin-down from magneto-dipolar radiation but rather by plasma ejection [168].Numerical and consistent calculations of the energy loss from the ejected magne-tospheric plasma have been possible only recently ([169] and [170] for a review)and the result is that the total ˙ E P SR is given by Eq. (70) with an extra factor (cid:39) (1 + sin α ). Considering the rotational energy of a uniformly rotating sphere E rot = I Ω /
2, where I is the star’s moment of inertia, the pulsar’s spin-downis determined by equating ˙ E rot with − ˙ E P SR , giving˙Ω = − KI Ω (71)with K (cid:39) R B s /c , assuming that I and B s remain constant. This also providesan estimate of B s , probably reliable within a factor of a few, B s (cid:39) (cid:115) − c ˙ΩΩ IR (cid:39) . × ( P ˙ P ) / G (cid:39) . × P .
01 s (cid:18) τ c (cid:19) / G , (72)where we assumed M (cid:39) . (cid:12) and R (cid:39)
12 km. It is traditional to also deducean observable characteristic pulsar age τ c = − Ω2 ˙Ω (73)67nd an observable braking index n ≡ Ω ¨Ω˙Ω (74)In the case of the magnetic dipole model and assuming both I and B s areconstant, n = 3.The ATFN catalogue [171] lists 1759 pulsars (as of December 2012) withmeasured values of ˙ P . The measurement of a second derivative ¨Ω, allowingthe determination of n , requires a very accurate long term and smooth fit of˙Ω and is much more difficult due in part to timing noise and to glitches. Todate, there are only 11 published values of n [172], all with values less than3 and ranging from − . .
91 (see Table 10.2). Anomalous values n < S p and P-F n components (further evidence for this stems fromobservations of the cooling of the neutron star in the Cassiopeia A supernovaremnant, which is discussed in Sec. 9). Consider the evolution of the spin of a pulsar if we allow changes in the surfacefield B s and the moment of inertia I . Assume the star has both normal matterand superfluid matter with moments of inertia I n and I s , respectively. Althoughthe onset of superfluidity is unlikely to significantly change the total momentof inertia, the portion of the star that is superfluid may be considered to spinfrictionlessly and its spin rate Ω s to remain constant. The observed spin rate ofthe star is that of the star’s surface, which is composed of normal matter. Wetherefore have I = I n + I s , ˙ I = 0 , Ω n = Ω , ˙Ω s = 0 . (75)With these two components, the rate of change of angular momentum in themagnetic dipole model is ddt [ I n Ω n + I s Ω s ] = I n ˙Ω + ˙ I n (Ω − Ω s ) = − K Ω . (76)Taking a time derivative of the above, one finds n = 3 + 4 τ c (cid:34) ˙ I n I n − ˙ B s B s (cid:35) + τ c (cid:18) Ω s Ω − (cid:19) ˙ I n I n (cid:34) τ c (cid:32) ¨ I n ˙ I n − B s B s (cid:33)(cid:35) . (77)Since Ω s − Ω (cid:28) Ω, we can drop the last term in Eq. (77). Thus, the growthof either the surface magnetic field and/or the superfluid component leads to n <
3. In fact, the growth of a core superfluid component could lead to field On-line version:
P τ c B s n A τ c A s kyr 10 G 10 − /d %B0531+21 (Crab) 0.0331 1.24 3.78 2.5J0537-6910 (N157B) 0.0161 4.93 0.925 -1.5 2.40 0.9B0540-69 (0540-693) 0.0505 1.67 4.98 2.1J0631+1036 0.2878 43.6 5.55 0.48 1.5B0833-45 (Vela) 0.0893 11.3 3.38 1.7 1.91 1.6J1119-6127 (G292.2-0.5) 0.4080 1.61 41 2.9B1338-62 (G308.8-0.1) 0.1933 12.1 7.08 1.31 1.2B1509-58 (G320.4-1.2) 0.1513 1.56 15.4 2.8J1734-3333 1.1693 8.13 52.2 0.9B1737-30 0.6069 20.6 17 0.79 1.2J1747-2958 0.0988 25.5 2.49 < . P , characteristic age τ c [Eq. (73)] and dipole surfacefield strength B s [Eq. (72)] rounded from the ATFN catalogue [171]. Values ofthe braking index n [Eq. (74)] rounded from [172]. Values of A [Eq. (79)] takenfrom [182].ejection from the core and an increase in the surface field strength. Note thatthere has been a long history of study of the opposite possibility of decay ofpulsar magnetic fields. However, statistical studies of the pulsar population[175] no longer support this idea and the present consensus is that magneticfield decay is significant only in the case of super-strong fields as in magnetarsor after a long phase of accretion in a binary system.Nevertheless, observed values n < n can bereproduced [179] without invoking superfluidity. Alternatively, magnetosphericcurrents may deform the field into a non-dipolar geometry [180, 181] that resultsin n < n (cid:28) n < I n < B s >
0, and non-dipolar fields. 69igure 34: Schematic illustration of a Vela pulsar glitch. The observed pulsarspin frequency ν has been corrected by the average accumulated spindown since1969.0 ( ν o = ˙ ν ( t − . ν in spin frequency. The frequencyafterwards relaxes roughly exponentially on a timescale τ to a new spindownline (dotted) which has a higher average frequency (1 − Q )∆ ν than the pre-glitchspindown line ν = ν (dashed). The second indication of superfluidity in neutron stars stems from the fact thatmany pulsars exhibit sporadic spin jumps, or glitches (see Fig. 34). The JodrellBank glitch catalogue lists 420 glitches (as of December 2012) [183]. Theseare thought to represent angular momentum transfer between the crust and aliquid, possibly superfluid, interior [184, 185, 186]. As the star’s crust spins downunder the influence of magnetic torque, differential rotation develops betweenthe crust (and whatever other parts of the star are tightly coupled to it) and adifferent portion of the interior containing a superfluid (recall in the above that˙Ω s = 0), but possibly a superfluid component different from those in the core.The now more rapidly rotating (superfluid) component then acts as an angularmomentum reservoir which exerts a spin-up torque on the crust as a result ofan instability. The Vela pulsar, one of the most active glitching pulsars, glitchesabout every 3 years, with a fractional change in the rotation rate averaging apart in a million [187, 188], as shown in Fig. 35.The stocastic nature of glitches implies that they represent a self-regulatinginstability for which the star prepares over a waiting interval. The amount ofangular momentum transferred is observed to increase linearly with time, as On-line version: J0573-6910 B0833-45
Figure 35: The accumulated relative angular velocity 10 − (cid:80) i ∆Ω i / Ω as a func-tion of time in days for the X-ray pulsar J0537-6910 and for Vela (B0833-45).∆Ω i is the change in angular velocity of a single glitch. Linear fits are indi-cated, with respective slopes [ Eq. (79)] A = 2 . × − d − and 1 . × − d − . (From [182])shown in Fig. 35 for Vela and for the X-ray pulsar J0537-6910. Assume that themoment of inertia of the superfluid component responsible for glitches is I g andthat of the crust and whatever parts of the star are tightly coupled to it is I c .In an individual glitch i , an amount of angular momentum I c ∆Ω i is transferredto the crust, where ∆Ω i is the observed jump in angular velocity. Over theinterval t i between the last glitch and the current glitch, the star spins down bythe amount ∆Ω = − ˙Ω t i and the total angular momentum differentially storedin the superfluid component is I g ∆Ω. These two angular momenta are equal,and over a total observed time t obs = (cid:80) i t i , we have I g > ∼ τ c A I c , (78)where the observed quantity A is related to the accumulated jumps, A = (cid:32)(cid:88) i ∆Ω i / Ω (cid:33) t − obs . (79)In the case of the Vela pulsar, the magnitude of A implies that I g /I c = 2 τ c A > ∼ . S type superfluid)is I sc and satisfies I sc (cid:39) . I, (80)where I is the star’s total moment of inertia, glitches can be naturally explainedby the inner-crust superfluid since I g < ∼ I sc . Although the magnitudes of in-dividual glitches varies somewhat, the maximum-sized glitches have stable andlimited sizes. This would be difficult to explain if glitches originated from the71nner core’s superfluid component, whose associated moment of inertia is a muchlarger fraction of the star’s total.Recently it has been shown that most of the neutron mass of the drippedneutrons in the inner crust is entrained by Bragg scattering with the nuclearlattice, effectively increasing the neutron mass by factors of 4–5 [190, 184]. Withthis entrainment, Eq. (78) becomes, for Vela, I g > ∼ . I c , (81)precluding an inner crust superfluid explanation for I g if I c (cid:39) I . Entrainmentresults in most of the neutron fluid spinning down with the crust, and theunentrained conduction neutrons cannot accumulate angular momentum at ahigh-enough rate to produce the largest observed glitches [191].B. Link [191], however, has maintained that typically observed glitch behav-ior is almost certainly a crustal feature, or perhaps due to some small regionof the core where vortex pinning is not occurring. In other words, it is likelythat I c < I , and perhaps substantially so, as if the outer core decouples fromthe inner core over timescales ranging from weeks to years. In fact, this is theobserved timescale τ of the post-glitch relaxation (see Fig. 34), which may sim-ply represent the dynamical recovery of the outer core. For older pulsars, whichare cooler, the relaxation timescale is long, of the order of years, and this couldexplain the nearly step-like behavior of many pulsar glitches. As a consequence,for example, if I c (cid:39) I/
2, Eq. (81) and (80) regain consistency. The upshot isthat spin glitches could then originate in either the inner crust or the outercore. The observed limited maximum magnitude of glitches retains its naturalexplanation in terms of the small and regular sizes of neutron star crusts asopposed to the wider variations in sizes of core superfluid regions.It should be noted that the above is at odds with the results of Sidery andAlpar [192], who obtain relaxation times of only about 2 days. This discrepancyis the subject of ongoing discussions.
11 Discussion and Conclusions
In this chapter, the influence of pairing, leading to neutron superfluidity andproton (and quark) superconductivity, on key observables of neutron stars is de-scribed. The observables include aspects of their thermal evolution, composedof surface temperatures, cooling rates, and ages, and dynamical evolution, com-prised of pulsar spin-down characteristics and glitch information. The majoreffects of pairing on the thermal evolution of an isolated neutron star are thequenching of neutrino emissions and the reduction of specific heat of the pairedfermions, be they nucleons, hyperons or deconfined quarks, in their cores. How-ever, the onset of pairing also triggers short episodes of increased neutrino emis-sion through pair breaking and formation (PBF) processes when the ambienttemperature falls below the superfluid critical temperature. The PBF emissionof neutrino pairs through weak interactions of strongly interacting particles isunique to dense neutron-star matter, as a similar phenomenon does not occur72n nuclei for S paired fermions as it is forbidden on the basis of symmetry. Themajor effects of pairing on dynamical evolution include the reduction of the so-called braking index, a measure of spin deceleration, of pulsars, and the possibletriggering of glitches from weak coupling of superfluid vortices to neutron starcrusts.The sizes and density dependences of superfluid and superconducting gapsplay a crucial role. Pairing is observed in nuclei as an energy difference betweeneven-even and odd-even nuclei with a typical magnitude, ranging from 0.5 to 3MeV, that decreases with atomic number. The basic cause of pairing in nucleiis due to the attractive interaction between neutrons in the spin S = 0 channel.Gaps of similar magnitude for nucleon pairing in neutron stars are expected.Both spin-singlet S (at lower densities) and spin-triplet P (at higher den-sities) configurations appear possible from scattering phase shifts arguments.Pairing appears as a gap ∆ in the single particle energy spectrum, leading toa strong suppression ( ∼ e − /T ) of both specific heat and neutrino emissivi-ties at low temperatures. In the simple BCS approximation, however, the gapdepends exponentialy on the pairing potential. Hence, uncertainties associatedwith in-medium effects of strong-interactions at high density have prevented aconsensus about the sizes and density dependences of gaps from being reached.In addition, the nature and abundance of possible candidates for pairing (nu-cleons and strangeness-bearing hyperons, quarks, Bose condensates) are alsouncertain.In the neutron star crust, the density of unbound neutrons is high enoughthat S pairing is expected to occur. Most theoretical models suggest that theassociated gap disappears at neutron densities higher than that of the core-crustinterface, so it is confined to the crust. The latest calculations indicate a max-imum gap magnitude of about 1 . × K. Since proton-neutron correlationsreduce the effective mass of the proton below that of the neutron, the size ofthe proton S gap is smaller than that of the neutron. “Unbound” protonsexist only at densities greater than ρ nuc where nuclei disappear, so proton su-perconductivity in the spin-singlet state is expected to exist from the core-crustboundary to deep into the core once temperatures fall below a few times 10 K. The S neutron gap vanishes close to the core-crust interface and the dom-inant pairing for neutrons in the core occurs in the anisotropic P-F channel.Uncertainties in the size and density range for this gap are larger than for S gaps, however, with maximum magnitudes ranging from a few times 10 K toa few times 10 K.The greatest influence of pairing will be on the thermal evolution of neutronstars. The occurrence of pairing leads to three important effects for neutronstar cooling: alteration and eventual suppression of nucleon specific heats, sup-pression of neutrino emissivities, and triggering of PBF neutrino emission fortemperatures just below the critical temperature. Presently, 13 isolated neu-tron stars with thermal spectra in the soft X-ray band have been identified. Inaddition, there are four pulsars with detected X-rays but with only upper lim-its to thermal emission. Finally, there are six gravitational-collapse supernovaremnants that might contain neutron stars (if not, then black holes), but no73etected thermal emission as yet. Atmospheric modeling of the thermal sourcesyields estimates of surface temperatures, and together with age estimates, al-lows these stars to be compared to theoretical cooling models. Most observedsources are younger than a million years, during which time they cool primarilythrough neutrino emission.Neutrino cooling can be either very fast (i.e., enhanced ), or relatively slow.Enhanced cooling occurs by way of the direct Urca process on nucleons, hyper-ons, Bose condensates, or deconfined quark matter. It is allowed when energyand momentum can be conserved with 3 or fewer degenerate fermions involved.If additional “bystander” nucleons are required to conserve momentum, neu-trino emission is suppressed by about a factor of a million. If pairing is notpresent and enhanced neutrino emission does not occur, it is found that severalobserved neutron stars are too cold to match cooling models. Therefore, animportant first conclusion is that either pairing or enhanced neutrino emissionmust occur in some neutron stars.
On the other hand, if pairing is not presentand enhanced neutrino emission does occur, all observed neutron stars are toohot to match cooling models. Thus, a second conclusion is that in the presenceof enhanced cooling, superfluidity and/or superconductivity must occur.
How-ever, as there are many combinations of pairing gap sizes and extents, neutronstar masses and envelope compositions, and enhanced cooling candidates thatcan match observations, it is not possible to determine either gap properties orthe specific enhanced cooling reactions involved. Therefore, for example, neu-tron star cooling cannot lead to an unambiguous detection of hyperons, Bosecondensates or deconfined quark matter in the interior of neutron stars at thistime.A final scenario, known as the minimal cooling paradigm , assumes enhancedneutrino emission does not occur, but allows pairing. In this case, all observedstars, with the possible excption of the pulsar in the supernova remnant CTA 1,for which only an upper temperature limit is available, and the six undetectedneutron stars in gravitational-collapse supernova remnants, none of which iscertain to be a neutron star at all, can be fit by theoretical cooling models.However, fitting observations requires two constraints on neutron stars: • The neutron P-F gap must have a maximum critical temperature T max cn larger than ∼ . × K, but unlikely larger than 1 . − × K; and • some, but not all, neutron stars must have envelopes composed of lightelements (H/He/C) and some, but not all, must have envelopes composedof heavy elements.The assumed mass of a neutron star has only a minor influence on the resultsof the minimal cooling scenario. The source CTA 1 and some of the undetectedneutron stars in gravitational-collapse supernova remnants might then be candi-dates for enhanced neutrino emission via the direct Urca process. One obviousway in which this could occur is that, since there is a density threshold for theonset of the direct Urca process, only neutron stars above a critical mass couldparticipate. From observed and theoretically-predicted neutron star mass dis-74ributions, this critical mass is estimated to be in the range 1 . − . M (cid:12) . Thisresult is itself supported by recent experimental and observational restrictionson how fast the symmetry energy of nuclear matter can increase with density[144], which suggest that the nucleon direct Urca threshold density is muchgreater than ρ nuc .The recently detected rapid cooling of the neutron star in Cas A provides anunprecedented opportunity to overcome the dilemma of deciding whether “exot-ica” are needed, or if the minimal cooling paradigm is sufficient, to account forobservational cooling data. The rapidity of the stellar cooling in Cas A remnantpoints to both the onset of core neutron superfluidity within the last few decadesand the prior existence of core proton superconductivity with a larger criticaltemperature. The PBF process, with its time-dependent burst of neutrinos atthe critical temperature, is central to the success of model calculations fittingboth Cas A cooling observations and the entire body of data for other observedisolated neutron stars. Furthermore, the Cas A cooling observations can be fitonly if T max cn (cid:39) − × K, precisely in the range independently found forthe minimal cooling scenario based on other isolated neutron star cooling data.Sherlock Holmes would not have deemed this a coincidence. “Enhanced” pro-cesses appear to be ruled out in the case of Cas A, as its star would presentlybe too cold. Further observations of the cooling of the neutron star in Cas Acan confirm these conclusions.Final indications of the existence of pairing in neutron stars is afforded by ob-servations of the deceleration of the spin frequency and glitches of pulsars. Thestandard paradigm to explain the pulsar mechanism is based on the magneticrotating dipole model, for which it can be shown that the spin-down parameter n = − Ω ¨Ω / ˙Ω = 3. All non-accreting pulsars for which n has been determinedhave n <
3, which can be understood either if the surface dipolar magnetic fieldis increasing in strength or if there is a growing superluid or superconductingcomponent within the neutron star’s core. An increasing surface field could,in fact, be due to the expulsion of magnetic flux from the core by supercon-ductors. The leading model for pulsar glitches is that a more-rapidly rotating,and essentially frictionless, superfluid in the crust sporadically transfers angularmomentum to the more slowly-rotating crust and the parts of the star stronglycoupled to it.In conclusion, there is abundant evidence that the ultimate “high-temper-ature” superfluid (or superconductor) exists in nearly every neutron star. Un-mistakably, superfluidity and superconductivity exhibited by the constituentsof neutron stars provide avenues by which the observed cooling behaviors ofthese stars can be explained. Theoretical predictions of pairing gaps being asyet uncertain, astronomical data are pointing to a path forward in their deter-minations. A remarkable recent development is that, despite the nearly three-order-of-magnitude theoretical uncertainty in the size of the neutron P-F gap,and the factor of 3 uncertainty in the size of the proton S gap, observations ofthe neutron star in Cas A now appear to restrict their magnitudes to remark-ably small ranges. It will be interesting to see if these considerable restrictionstranslate into a more complete understanding of the pairing interaction between75ucleons and provide insights into other aspects of condensed matter and nuclearphysics. Continued X-ray observations of this star and other isolated sources,and, hopefully, the discovery of additional cooling neutron stars, will furtherenhance these efforts. Acknowledgments
DP acknowledges support by grants from UNAM-DGAPA,
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