Superfluid Neutrons in the Core of the Neutron Star in Cassiopeia A

# Superfluid Neutrons in the Core of the Neutron Star in Cassiopeia A

Dany Page
Instituto de Astronomía, Universidad Nacional Autónoma de México,
Mexico, DF 04510, Mexico
E-mail: page@astro.unam.mx
Speaker.
Madappa Prakash
Department of Physics and Astronomy, Ohio University,
Athens, OH 45701-2979, USA
E-mail: prakash@harsha.phy.ohiou.edu
James M. Lattimer
Department of Physics and Astronomy, State University of New York at Stony Brook,
Stony Brook, NY 11794-3800, USA
E-mail: lattimer@mail.astro.sunysb.edu
Andrew W. Steiner
Institute for Nuclear Theory, University of Washington,
Seattle, WA 98195, USA
E-mail: steiner3@uw.edu
###### Abstract

The supernova remnant Cassiopeia A contains the youngest known neutron star which is also the first one for which real time cooling has ever been observed. In order to explain the rapid cooling of this neutron star, we first present the fundamental properties of neutron stars that control their thermal evolution with emphasis on the neutrino emission processes and neutron/proton superfluidity/superconductivity. Equipped with these results, we present a scenario in which the observed cooling of the neutron star in Cassiopeia A is interpreted as being due to the recent onset of neutron superfluidity in the core of the star. The manner in which the earlier occurrence of proton superconductivity determines the observed rapidity of this neutron star’s cooling is highlighted. This is the first direct evidence that superfluidity and superconductivity occur at supranuclear densities within neutron stars.

Superfluid Neutrons in the Core of the Neutron Star in Cassiopeia A

Andrew W. Steiner

Institute for Nuclear Theory, University of Washington,

Seattle, WA 98195, USA

E-mail: steiner3@uw.edu

\abstract@cs

XXXIV edition of the Brazilian Workshop on Nuclear Physics, 5-10 June 2011 Foz de Iguaçu, Parana state, Brasil

## 1 Introduction: "neutron stars" do exist

Neutron stars contain the densest form of cold matter observable in the Universe. Larger energy densities are transiently reached in relativistic heavy ion collisions, but the resulting matter is extremely "hot". Black holes contain a much denser form of matter, but their interior is not observable. Two simple arguments can convince us that such stars, very small and very dense, do exist. First, consider the fastest known radio pulsar, "Ter5ad" (PSR J1748-2446ad) [21], and posit that the observed period of its pulses, ms, is its rotational period. Using causality, that is, imposing that the velocity at its equator is smaller than the speed of light , one then obtains

 vequator=ΩR=2πRP

This value of 65 km for the radius is only a strict upper limit; detailed theoretical models indicate radii on the order of 10 km. Secondly, assuming that the star is bound by gravity, we can require that the gravitational acceleration at the equator is larger than the centrifugal acceleration and obtain

 agr=GMR2>acf=Ω2R=4π2RP2orMR3>4π2GP2⇒¯¯¯ρ=M43πR3>8×1013gcm−3. (1.0)

Obviously, Newtonian gravity is not accurate in this case, but we can nevertheless conclude that the central density of these stars is comparable, or likely larger, than the nuclear density g cm. Theoretical models show that densities up to are possibly reachable [26]. In short, a "neutron star" is a gigantic, and compressed, nucleus of the size of a city.

In what follows, we outline the basic properties of neutron stars relevant for describing their thermal evolution with emphasis on the neutrino emission processes and neutron/proton superfluidity/superconductivity. This allows us to present simple analytical models of the cooling of a neutron star in order to gain physical insight. The results of these analytical models are complemented by those of numerical simulations in which full general relativity and the state of the art microphysics are employed. Finally, an interpretation of the observed rapid cooling of the neutron star in Cassiopeia A as due to the recent onset of neutron superfluidity in it core is proffered. The role of proton superconductivity in determining the rapidity of Cas A’s cooling is addressed.

## 2 Neutron Stars: "pure neutron stars" do not exist

A "pure neutron star", as originally conceived by Baade & Zwicky [3] and Oppenheimer & Volkoff [31] cannot really exist. Neutrons in a ball should decay into protons through

 n→p+e−+¯¯¯νe.\specialhtml:\specialhtml: (2.0)

This decay is possible for free neutrons as . However, given the densities expected within the neutron star interior, the relevant quantities are not the masses, but instead the chemical potentials ( denoting the species) of the participants. The matter is degenerate as typical Fermi energies are on the order of tens to hundreds of MeV’s, whereas the temperature drops below a few MeV within seconds after the birth of the neutron star in a core collapse supernova [10]. Starting with a ball of almost degenerate neutrons, the decay of Eq. (2) will generate a Fermi sea of protons, electrons and anti-neutrinos. The interaction mean free paths of anti-neutrinos (and neutrinos) far exceed the size of the star and these can be assumed to leave the star. Thus, the decay will be possible until

 μn=μp+μe,\specialhtml:\specialhtml: (2.0)

where we have neglected . Under this condition, the inverse reaction

 p+e−→n+νe\specialhtml:\specialhtml: (2.0)

also becomes energetically possible. Hence, under equilibrium conditions, which a neutron star will reach within a few tens of seconds after its birth, reactions such as Eq. (2) and (2) will adjust the chemical composition of matter to that characteristic of (or chemical) -equilibrium determined by Eq. (2). A neutron star is not born as a "ball of neutrons" which may decay according to Eq. (2), but from the collapse of the iron core of a massive star. It is thus born with an excess of protons so that the reaction Eq. (2) initially dominates over Eq. (2) in order to reduce the proton fraction and it is only when the neutrinos escape that the final cold -equilibrium configuration, Eq. (2), is reached.

Notice that once MeV, muons will appear, and be stably present with . The condition for the appearance of muons is fulfilled when the density is slightly above . Thus, in all processes we describe below, there will always be the possibility to replace electrons by muons when the density is large enough to allow for their presence.

Let us consider simple expressions for the chemical potentials 111Relativistic expressions for and also exist, but are omitted here in the interest of simplicity.:

 μn=mn+pF(n)22mn+Vn and μp=mp+pF(p)22mp+Vp (2.0) μe=√m2ec4+pF(e)2c2=pF(e)c and μμ=√m2μc4+pF(μ)2c2, (2.0)

where is the Fermi momentum of species , and and are the mean-field energies of and . For the leptons, and are negligibly small, and we have considered that electrons, but not necessarily muons, are ultra-relativistic. With a knowledge of and , the two -equilibrium relations Eq. (2) and can be solved. With four ’s and two equations, a unique solution is obtained by imposing charge neutrality

 np=ne+nμ\specialhtml:\specialhtml: (2.0)

and fixing the baryon density

 nB=np+np.\specialhtml:\specialhtml: (2.0)

With the ’s and ’s known, one can calculate any thermodynamic potential, in particular the pressure and energy density . Varying the value of gives us the equation of state (EOS): . Given an EOS, an integration of the Tolman-Oppenheimer-Volkoff (TOV) equations of hydrostatic equilibrium provides us with a well defined model of a neutron star.

The potentials and in Eq. (2) turn out to be rapidly growing functions of density, and one can anticipate that eventually reactions such as

 p+e−→Λ+¯¯¯νeor/andn+e−→Σ−+¯¯¯νe\specialhtml:\specialhtml: (2.0)

may produce hyperons. Hyperons can appear, and be stable, once the corresponding -equilibrium conditions are satisfied, i.e., or/and . At the threshold, where or , one can expect that and and thus and . Since and are larger than the nucleon mass by only about 200 MeV these hyperons222The is less favored as its -equilibrium condition is . Heavier baryon are even less favored, but cannot a priori be excluded. are good candidates 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 form meson condensates. At even larger densities, the ground state of matter is likely to be one of deconfined quarks. All these possibilities depend crucially on the strong interactions terms, and . As we will not employ them in this chapter, we refer, e.g., to [38] for more details and entries to the original literature. Figure 1 illustrates our present understanding (or misunderstanding) of the interior of a neutron star, with a black 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 ’s and ’s as implied by charge neutrality and constrained by -equilibrium, are considered, the EOS can be calculated with much more confidence than in the presence of "exotic" forms of matter. For illustrative puposes, we will employ the EOS of Akmal, Pandharipande & Ravenhall [1] ("APR" hereafter) in presenting our results.

Although there is no evidence that any observed "neutron star" or pulsar might actually instead be a pure quark star, theory allows this possibility. Such a star would be nearly completely composed of a mixture of up, down and strange quarks, and would differ from a neutron star in that it would be self-bound rather than held together by gravity.

The reader can find a more detailed presentation and entries to the key literature in [38].

### The Neutron Star Crust

In the outer part of the star, where , a homogeneous liquid of nucleons is mechanically unstable (spinodal instablity). Stability is, however, restored by the formation of nuclei, or nuclear clusters. At the surface, defined as the layer where , we expect the presence of an atmosphere, but there is the possibility of having a solid surface condensed by a sufficiently strong magnetic field [25]. A few meters below the surface, ions are totally ionized by the increasing density (the radius of the first Bohr orbital will be larger than the inter-nuclear distance when g cm). Matter then consists of a gas/liquid of nuclei immersed in a quantum liquid of electrons. When g cm, is of the order of 1 MeV and the electrons become relativistic. From here on, the Coulomb correction to the electron gas EOS is negligible and electrons form an almost perfect Fermi gas. However, the Coulomb correction to the ion EOS is not negligible. From a gaseous state at the surface, ions will progressively go through a liquid state and finally crystallize, at densities between up to g cm depending on the temperature (within the range of temperatures for which the neutron star is thermally detectable). With growing , and the accompanying growth of , it is energetically favorable to absorb electrons into nuclei and, hence, the nuclear species expected to be present have a neutron fraction strongly growing with . When g cm (the exact value depending on the assumed chemical composition), one reaches the neutron drip point at which the neutron density is so much larger than that of the proton that not all neutrons are bound to nuclei. Matter then consists of a crystal of nuclei immersed in a perfect Fermi gas of electrons and a quantum liquid of dripped neutrons. This region is usually called the inner crust. In most of this inner crust, the dripped neutrons are predicted to form a superfluid (in a spin-singlet, zero orbital angular momentum, state S). All neutron stars we observe are rotating; a superfluid cannot undergo rigid body rotation, but it can simulate it by forming an array of vortices (in the core of which superfluidity is destroyed). (See, e.g., [48].) The resulting structure is illustrated in the inset A of Fig. 1.

At not too high densities, nucleons are correlated at short distances by the strong interaction and anti-correlated at larger distances by the Coulomb repulsion between the nuclei, the former producing spherical nuclei and the latter resulting in the crystallization of the matter. As approaches the shape of the "nuclei" can undergo drastic changes: the nuclear attraction and Coulomb repulsion length-scales become comparable and the system is "frustrated". From spherical shapes, nuclei are expected to become elongated ("spaghettis"), form 2D structures ("lasagnas"), always surrounded by the neutron gas/superfluid which occupies an increasingly small portion of the volume. Then the geometry is inverted, with the dripped neutrons confined into 2D, then 1D ("anti-spaghettis") and finally 0D ("swiss cheese") bubbles. The homogeneous phase, i.e. the core of the star, is reached when 333 correspond to the density of symmetric nuclear matter, i.e. with a proton fraction %, and zero pressure, whereas in a neutron star crust at one has %.. This "pasta" regime is illustrated in the inset B of Fig. 1 and is thought to resemble a liquid crystal [40].

## 3 Neutrino Emission Processes

The thermal evolution of neutron stars with ages yrs is driven by neutrino emission. Here we describe the dominant processes. The simplest neutrino emitting processes, Eq. (2) and Eq. (2), are

 {n⟶p+e−+¯¯¯νep+e−⟶n+νe,\specialhtml:\specialhtml: (3.0)

and are generally referred to as the direct Urca ("DU") cycle.

By the condition of -equilibrium the cycle naturally satisfies energy conservation, but momentum conservation is much more delicate. Due to the high degeneracy, all participating particles have momenta equal (within a small correction) to their Fermi momenta . As and , momentum conservation is not a priori guaranteed. It is easy to see that, in the absence of muons and hence with , the "triangle rule" for momentum conservation requires that %, whereas at we have %. In the presence of muons, which appear just above , the condition is stronger and one needs larger than about 15% [27]. The proton fraction grows with density, its growth being directly determined by the growth of the nuclear symmetry energy, so that the critical proton fraction for the DU process is likely reached at some supra-nuclear density [27]. For the EOS of APR, the corresponding critical neutron star mass for the allowance of the DU process is , but other EOS’s predict smaller values.

At densities below the threshold density for the DU process, a variant of this process, the modified Urca (MU) process

 {n+n′⟶p+n′+e−+¯¯¯νep+n′+e−⟶n+n′+νe\specialhtml:\specialhtml: (3.0)

can operate as advantage is taken of neighboring particles in the medium [17]. A bystander neutron can take or give the needed momentum for momentum conservation. The processes in Eq. (3) show the neutron branch of the MU process and in the proton branch is replaced by a proton . As it involves the participation of five degenerate particles, the MU process is much less efficient than the DU process. Unlike the DU processes which require sufficient amount of protons, both branches of the MU process operate at any density when neutrons and protons are present.

In the presence of hyperons, DU and MU processes which are obvious generalizations of the nucleon-only process can also occur [42]. When they appear, the ’s have a density much smaller than that of the neutron and hence a smaller Fermi momentum. Consequently, momentum conservation in the DU cycle and is easily satisfied, requiring an of only %. Notice that if the nucleon DU process is kinematically forbidden, the hyperon DU process with is also kinematically forbidden, whereas the no-nucleon DU process together with may be possible and require very low threshold fractions and .

In deconfined quark matter, DU processes such as and are possible [23]. Reactions in which the quark is replaced by an quark can also occur in the case quarks are present.

In the presence of a meson condensate copious neutrino emission in processes such as

 {n+e−⟶n+π−+νen+π−⟶n+e−+¯¯¯νe\specialhtml:\specialhtml: (3.0)

and

 {n+e−⟶n+K−+νen+K−⟶n+e−+¯¯¯νe\specialhtml:\specialhtml: (3.0)

occur [29, 9]. As the meson condensate is a macroscopic object there is no restriction arising from momentum conservation in these processes.

Finally, another class of processes, bremsstrahlung, is possible through neutral currents [16]. Reactions such as

 n+n′⟶n+n′+νe+¯¯¯νe (3.0) p+p′⟶p+p′+νe+¯¯¯νe (3.0) n+p′⟶n+p′+νe+¯¯¯νe (3.0)

are less efficient, by about 2 orders of magnitude, than the MU processes, but may make some contribution in the case that the MU process is suppressed by pairing of neutrons or protons as we will see in §5.

In Table 1 we list these processes with order of magnitude estimates of their neutrino emissivities. Most noticeable is the clear distinction between processes involving 5 degenerate fermions with a dependence, which are labeled as "slow", and those with only 3 degenerate fermions with a dependence, which are several orders of magnitude more efficient and labeled as "fast". The difference in the dependence is important and simply related to phase space arguments which are outlined below. The Cooper pair process [15, 51] will be described in §5.

The reader can find a detailed description of neutrino emission processes in [54] and an alternative point of view in [50].

### Temperature dependence of neutrino emission

We turn now to briefly describe how the specific temperature dependence of the neutrino processes described above emerges. Consider first the simple case of the neutron decay. The weak interaction is described by the Hamiltonian , where is Fermi’s constant, and and are the lepton and baryon currents, respectively. In the non relativistic approximation, one has and where is the Cabibbo angle and the axial-vector coupling. Fermi’s Golden rule gives us for the neutron decay rate

 Wi→f=∫∫∫d3pν(2π)3d3pe(2π)3d3pp(2π)3(2π)4δ4(Pf−Pi)⋅|Mfi|2, (3.0)

i.e., a sum of over the phase space of all final states . The integration gives the well known result: , where takes into account small Coulomb corrections. This gives us the neutron mean life, min., or, measuring , a measurement of (modulo and ).

The emissivity of the DU process (the Feynman diagram for this process is shown in Fig. 2) can be obtained by the same method as above leading to

 ϵDU=∫∫∫∫d3p¯¯¯ν(2π)3d3pe(2π)3d3pp(2π)3d3pn(2π)3(1−fe)(1−fp)fn⋅(2π)4δ4(Pf−Pi)|Mfi|2⋅Eν\specialhtml:\specialhtml: (3.0)

with an extra factor for the neutrino energy and the phase space sum now includes the initial . The terms, being the Fermi-Dirac distribution for particle at temperature , take into account: (1) the probability to have a in the initial state, , and (2) the probabilities to have available states for the final and , denoted by and , respectively. We do not introduce a Pauli blocking factor for the anti-neutrino as it is assumed to be able to freely leave the star (i.e., ). When performing the phase space integrals, each degenerate fermion gives us a factor , as particles are restricted to be within a shell of thickness of their respective Fermi surfaces. The anti-neutrino phase space gives a factor . The factors and the delta function gives a factor (it cancels one of the ’s from the degenerate fermions). Altogether, we find that

 ϵDU∝T3⋅T⋅T⋅T⋅1T⋅(1)2⋅T=T6,\specialhtml:\specialhtml: (3.0)

where the factor emphasizes that the squared matrix element is T-independent. An explicit expression for the neutrino emissivity for the DU process can be found in [27] .

Figure Fig. 2 shows a Feynman diagram for the MU process. There are two more such diagrams in which the weak interaction vertex is attached to one of the two incoming legs. In this case, the -power counting gives

 ϵMU∝T3⋅T⋅T⋅T⋅T⋅T⋅1T⋅(1)2⋅T=T8,\specialhtml:\specialhtml: (3.0)

where the now involves a strong interaction vertex, the wavy line in Fig. 2, but is still -independent. Notice that in the MU case, the internal neutron is off-shell by an amount which does not introduce any extra -dependence as we are working in the case MeV . Reference [17] contains the expression from which neutrino emissivity from the MU process can be calculated. Considering finally the bremsstrahlung process. One diagram is shown in Fig. 2 and there are three more diagrams with the weak interaction vertices attached to the other three external lines. The -power counting now gives

 ϵBr∝T3⋅T3⋅T⋅T⋅T⋅T⋅1T⋅(1T)2⋅T=T8 (3.0)

with two factors for the neutrino pair and a from the matrix element: the intermediate neutron is almost on-shell, with an energy deficit , and its propagator gives us a dependence for . A working expression for the bremsstrahlung process can be found in [17].

## 4 Neutron Star Cooling

The basic features of the cooling of a neutron star are best illustrated by considering the energy balance equation for the star in its Newtonian formulation444Numerical results to be shown later include full general relativistic effects.:

 dEthdt=CvdTdt=−Lν−Lγ+H,\specialhtml:\specialhtml: (4.0)

where is the thermal energy content of the star, its internal temperature, and its total specific heat. The two energy sinks are neutrino luminosity , described in §3, and the surface photon luminosity , discussed in §4.2. The source term would include heating mechanisms as, e.g., magnetic field decay, which we will not consider here.

### 4.1 Specific Heat

The dominant contributions to the specific heat come from the core, constituting more than 90% of the total volume, whose constituents are quantum liquids of leptons, baryons, mesons, and possibly deconfined quarks at the highest densities. Hence one has

 CV=∑iCV,iwithCV,i=∫∫∫cv,idv(i=e,μ,n,p,hyperons,quarks,...),\specialhtml:\specialhtml: (4.0)

where is the specific heat, per unit volume, of component . For normal (i.e., unpaired) degenerate fermions, the standard Fermi liquid result [6]

 cvi=N(0)π23k2BTwithN(0)=m∗ipFiπ2ℏ3\specialhtml:\specialhtml: (4.0)

can be used, where is the fermion’s effective mass. In Fig. 3, the various contributions to are illustrated. When baryons, and quarks, become paired, as briefly described in §5, their contribution to is strongly suppressed at temperatures ( being the corresponding critical temperature). Extensive baryon, and quark, pairing can thus significantly reduce , but by at most a factor of order ten as the leptons do not pair. The crustal contribution is in principle dominated by neutrons in the inner crust but, as these are certainly extensively paired, practically only the nuclear lattice and electrons contribute.

### 4.2 Photon emission and the neutron star envelope

Thermal photons from the neutron star surface are emitted at the photosphere, which is usually in an atmosphere but could be a solid surface in the presence of a very strong magnetic field. The atmosphere, which is only a few centimeters thick, presents a temperature gradient. Consequently, photons of increasing energy escape from deeper and hotter layers. It is customary to define an effective temperature, , so that the total surface photon luminosity, by analogy with the blackbody emission, is written as

 Lγ=4πR2σSBT4e,\specialhtml:\specialhtml: (4.0)

where is the Stefan-Boltzmann constant. Observationally, and are red-shifted and Eq. (4.2) is rewritten as

 Lγ∞=4πR2∞σSBT4e∞\specialhtml:\specialhtml: (4.0)

where , , and . Here , with being the red-shift, and is the coefficient of the Schwarzschild metric, i.e.,

 eϕ≡√1−2GMRc2.\specialhtml:\specialhtml: (4.0)

Notice that has the physical interpretation of being the star’s radius that an observer would measure trigonometrically, if this were possible.

In a detailed cooling calculation, the time evolution of the whole temperature profile in the star is followed. However, the uppermost layers have a thermal time-scale much shorter than the interior of the star and are practically always in a steady state. It is hence common to treat these layers separately as an envelope. Encompassing a density range from at its bottom (typically g cm) up to that at the photosphere, and a temperature range from to , the envelope is about one hundred meters thick. Due to the high thermal conductivity of degenerate matter, stars older than a few decades have an almost uniform internal temperature except within the envelope which acts as a thermal blanket insulating the hot interior from the colder surface. A simple relationship between and can be formulated as in [19]:

 Te≃106(Tb108K)0.5+α\specialhtml:\specialhtml: (4.0)

with . The precise relationship depends on the chemical composition of the envelope. The presence of light elements, resulting in larger thermal conductivities, implies a larger for the same compared to the case of a heavy element envelope. Magnetic fields also alter this relationship (see, e.g., [34] for more details).

### 4.3 Some simple analytical solutions

As the essential ingredients entering Eq. (4) can all be approximated by power-law functions, one can obtain simple and illustrative analytical solutions. Let us write

 CV=C9⋅T9,Lν=N9⋅T89andLγ=S9⋅T2+4α9,\specialhtml:\specialhtml: (4.0)

where . As written, considers slow neutrino emission involving five degenerate fermions from the modified Urca and the similar bremsstrahlung processes, summarized in Table 1. The photon luminosity is obtained from Eq. (4.2) using the simple expression in Eq. (4.2). Typical values are erg K, erg s and erg s (see Table 3 in [34] for more details). In young stars neutrinos dominate the energy losses (in the so-called neutrino cooling era) and photons take over after about yrs (in the photon cooling era).

Neutrino cooling era: In this case, we can neglect in Eq. (4) and find

 t=109C96N9(1T69−1T609)⟶T=109K(τMU/t)1/6(whenT≪T0)\specialhtml:\specialhtml: (4.0)

with a MU cooling time-scale yr when the star reaches the asymptotic solution (, being the initial temperature at time ).

Photon cooling era: In this era, becomes negligible compared to so that we obtain

 t=t1+C94αS9109+36α(1T4α9−1T4α19)⟶T∝t−14α(whenT≪T1)\specialhtml:\specialhtml: (4.0)

where is at time . Notice here that the slope, , of the asymptotic solution is strongly affected by any small change in the envelope structure, Eq. (4.2).

### 4.4 Some numerical solutions

Numerical simulations of a cooling neutron star use an evolution code in which the energy balance and energy transport equations in their fully general relativistic forms are solved, usually assuming spherical symmetry and with a numerical radial grid of several hundred zones 555Such a code, NSCool, is available at: http://www.astroscu.unam.mx/neutrones/NSCool/. A set of cooling curves that illustrate the difference between cooling driven by the modified Urca and the direct Urca processes is presented in Fig. 4. Cooling curves of eight different stars of increasing mass are shown, using an equation of state model, from [41], which allows the DU process at densities above g cm, i.e., above a critical neutron star mass of 1.35 . Notice that the equation of state used is a parametric one and its parameters were specifically adjusted to obtain a critical mass of 1.35 which falls within the expected range of isolated neutron star masses; other equations of state can result in very different critical masses. The difference arising from slow and fast neutrino processes is clear.

Four successive cooling stages are marked in the figure. The neutrino cooling era is marked stage 3 and the photon cooling era is marked stage 4. As the figure shows , and since , the slopes of the cooling curves in this figure are rescaled from the values of Eq. (4.3) and (4.3), i.e., they become and , respectively. The simple analytical model of the previous section considered a single temperature in the stellar interior. For a young star, whose age is smaller that its thermal relaxation time, the interior temperature has a very complicated radial profile. In particular, the core cools much faster than the crust, due to its much stronger neutrino emission. During stage 1 in Fig. 4, the surface temperature is controlled by processes occurring in the outer layers of the crust and is totally independent of the temperature deeper in the star. (One can appreciate that the more massive stars have a larger at that time; this is mostly due to the fact that their blanketing envelopes are thinner than in low mass stars.) The rapid drop in occurring during stage 2 corresponds to the thermal relaxation of the crust; the star’s age becomes comparable to the heat transport time-scale from the crust to the core and the crustal heat flows into the core. After stage 2, the stellar interior is essentially isothermal with a strong temperature gradient present only in the envelope and the simple analytical solutions presented above apply.

## 5 Pairing and its Effects

Pairing, which induces superfluidity in the case of degenerate neutral fermions and superconductivity for charged fermions, is expected to occur between neutrons/protons in the interior of neutron stars [30]. The Cooper Theorem [12] states that, in a system of degenerate fermions the Fermi surface is unstable, at , due to the formation of Cooper pairs if there is an attractive interaction in some spin-angular momentum channel. The essence of the BCS theory [5] is that as a result of this instability there is a collective reorganization of the particles at energies around the Fermi energy and the appearance of a gap in the quasi-particle spectrum (see Fig. 5) which is the binding energy of a Cooper pair. At high enough temperature the gap vanishes and the system is in the normal state. The transition to the superfluid/superconducting state is a second order phase transition and the gap is its order parameter (see Fig. 6). Explicitly, when and, when drops below , grows rapidly but continuously, with a discontinuity in its slope at . This is in sharp contrast with a first order phase transition, in which the transition occurs entirely at (see left panel of Fig. 6) and will be of paramount importance for our purpose. In the BCS theory, which remains approximately valid for nucleons, the relationship between the gap and is

 Δ(T=0)≃1.75kBTc.\specialhtml:\specialhtml: (5.0)

In a normal Fermi system at , all particles are in states with energies . When , states with energies can be smoothly occupied (left panel of Fig. 5) resulting in a smearing of the particle distribution around in a range . It is precisely this smooth smearing of energies around which produces the linear dependence of , § 4.1, and the or dependence of the neutrino emissivities, § 3.

In a superfluid/superconducting Fermi system at , all particles are in states with energies (actually, ). When , states with energy (actually, ) can be populated. However, in contrast to the smooth filling of levels above in the case of a normal Fermi liquid, the presence of the gap in the spectrum implies that the occupation probability is strongly suppressed by a Boltzmann factor . Consequently, all physical properties/processes depending on thermally excited particles, such as the specific heat and the neutrino emission processes described in § 3, are strongly suppressed when . In practice, for numerical simulations of neutron star cooling, these suppressions are implemented through "control functions" such that

 cv→cPairedv = RccNormalv (5.0) ϵν→ϵPairedν = RνϵNormalν. (5.0)

There is a large family of such functions, for each process and they moreover depend on how many of the participating particles are paired and on the specific kind of pairing; a few examples are displayed in Fig. 7 [54].

### 5.1 Theoretical expectations on pairing gaps

Soon after the development of the BCS theory, Bohr, Mottelson & Pine [8] pointed out that excitation energies of nuclei show a gap, as shown in the left panel of Fig. 8. Even-even nuclei clearly require a finite minimal energy for excitation. This energy was interpreted as being the binding energy of the Cooper pair which must break to produce an excitation. In contrast, odd nuclei do not show such a gap, and this is due to the fact that they have one nucleon, neutron or proton, which is not paired and can be easily excited. The right panel of Fig. 8 shows that pairing also manifests itself in the binding energies, even-even nuclei being slightly more bound than odd nuclei666Notice that, as a result of pairing, the only stable odd-odd nuclei are H(1,1), Li(3,3), B(5,5), and N(7,7). All heavier odd-odd nuclei are beta-unstable and decay into an even-even nucleus..

As a two-particle bound state, the Cooper pair can appear in many spin-orbital angular momentum states (see left panel of Fig. 9). In terrestrial superconducting metals, the Cooper pairs are generally in the S channel, i.e., spin-singlets with orbital angular momentum, whereas in liquid He they are in spin-triplet states. What can we expect in a neutron star ? In the right panel of Fig. 9, we adapt a figure from one of the first works to study neutron pairing in the neutron star core [47] showing laboratory measured phase-shifts from N-N scattering. A positive phase-shift implies an attractive interaction. 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 should appear at higher densities. We emphasize that this is only a presumption as medium effects can strongly affect particle interactions.

A simple model can illustrate the difficulty in calculating pairing gaps. Consider a dilute Fermi gas with a weak, attractive, interaction potential . The interaction is then entirely described by the corresponding scattering length, , 777The scattering length is related to by with . which is negative for an attractive potential. In this case, the model has a single dimensionless parameter, , and the dilute gas corresponds to . Assuming the pairing interaction is just the bare interaction (which is, improperly, called the BCS approximation), the gap equation at can be solved analytically, giving the weak-coupling BCS-approximation gap:

 Δ(kF)⟶ΔBCS(kF)=8e2(ℏ2k2F2M)exp[−π2|a|kF]when|a|kF→0.\specialhtml:\specialhtml: (5.0)

This result is bad news: the gap depends exponentially on the pairing potential . The Cooper pairs have a size of the order of (the coherence length) and thus in the weak coupling limit. There appear to be many other particles within the pair’s coherence length. These particles will react, and can screen or un-screen, the interaction. Including this medium polarization on the pairing is called beyond BCS, and in the weak coupling limit its effect has been calculated analyticaly [18], giving

 Δ(kF)⟶ΔGMB(kF)=1(4e)1/3ΔBCS(kF)≃0.45ΔBCS(kF)when|a|kF→0.\specialhtml:\specialhtml: (5.0)

So, screening by the medium reduces the gap by more than a factor two, even in an extremely dilute system.

Pairing correlations in nuclei are part of everyday nuclear physics, and a significant amount of work has also been devoted to the neutron star environment (see, e.g., [13] and [28] for reviews). We show in Fig. 10 three sets of predicted for the neutron star interior. At low density, corresponding to the neutron star crust in the regime of dripped neutrons, the expectation of a neutron S superfluid is amply confirmed by the models. This regime was also illustrated in the inset A of Fig. 1. At higher densities, corresponding to the neutron star core, the situation is much more ambiguous. Due to their low concentrations, protons have small Fermi momenta in the core and are expected to form a S superconductor. There is, however, a significant uncertainty in the size of their gap, with predicted values of ranging from K up to K, and a larger uncertainty in the range of Fermi momenta in which is non-zero, which translates into an uncertainty of a factor of more than 3 on the density range covered by the superconductor. In the "pessimistic" case protons would be superconducting only in the outer part of the core, whereas in the "optimistic" case the whole core may be superconducting.

For neutrons in the neutron star core, there is no agreement between the many published models about either the maximum value of or on the density range in which pairing is significant. Notice that, due to the tensor interaction, pairing is expected to be in the P-F channel. However, even the best models of the N-N interaction in vacuum fail to reproduce the measured phase shift in the P channel [4]. Due to medium polarization a long-wavelength tensor force appears that is not present in the in vacuum interaction and results in a strong suppression of the gap [45]. Recently, the effect of three-body forces (TBF), absent in the laboratory N-N scattering experiment, has been considered. TBF are necessary to reproduce the nuclear saturation density; they are, in the bulk, repulsive and their importance grows with increasing density. However, it was found in [57, 58] that, at the Fermi surface, they are strongly attractive in the P-F channel and result in very large neutron P-F gaps. Other delicate issues are the effect of the proton contaminant and the likely development of a condensate888In the presence of a charged condensate a new Urca neutrino emission pathway is open, see Table 1 and Eq (3). The development of a neutral condensate has, however, little effect on neutrino emission. which also strongly affects the size of the neutron (and proton) gap(s). In short, the size and extent in density of the neutron P-F gap in the neutron star core is poorly known.

### 5.2 The Cooper pair neutrino process

The formation of the fermonic pair condensate also triggers a new neutrino emission process, which has been termed as the "pair breaking and formation", or PBF, process [44]. When an ff pair (f = n, p, or any fermion undergoing pair condensation) forms, its binding energy can be emitted as a pair. Under the right conditions, this PBF process can be the dominant cooling agent in the evolution of a neutron star [32]. Such efficiency is due to the fact that the pairing phase transition is second order in nature. During the cooling of the star, the phase transition starts when the temperature reaches when pairs begin to form, but thermal agitation will constantly induce the breaking of pairs with subsequent re-formation and possible neutrino pair emission.

The emissivity of the PBF process can be written as

 ϵPBF=12G2Fm∗fpF,f15π5ℏ10c6(kBT)7af,jRj[Δj(T)/T] =3.51×1021