NUCLEAR IMPURITIES IN THE SUPERFLUID CRUST OF NEUTRON STARS: QUANTUMCALCULATION AND OBSERVABLE EFFECTS ON THE COOLING

P. M. Pizzochero,1,2

F. Barranco,3E. Vigezzi,

2and R. A. Broglia

1,2,4

Received 2001 July 31; accepted 2001 December 19

ABSTRACT

We perform a quantum mechanical calculation of the neutron-specific heat in the Wigner-Seitz cells thatcharacterize the inner crust of a neutron star, where a Coulomb lattice of neutron-rich nuclei is permeated bya sea of unbound superfluid neutrons. We then evaluate the observable thermal properties associated withsuch a microscopic structure and compare them to the standard values obtained in a uniform density modelof the crust superfluidity with no nuclear impurities. We first argue that the bottleneck for the diffusion ofheat is represented by the outermost layers of the inner crust. We then show that the crust-cooling timesobtained for the nonuniform system are significantly different from those obtained in the standard uniformmodel. We find that the sign and magnitude of this difference depend sensitively on the local matter tempera-ture and the pairing potential. We conclude that in order to give predictive power to observations of the ther-mal emission from neutron stars, it is crucial to include in any realistic cooling code the effect of nuclearimpurities on the neutron superfluid.

Subject headings: dense matter — diffusion — stars: interiors — stars: neutron

1. INTRODUCTION

The observation of thermal emission from the surface ofa neutron star has the potential to provide strong con-straints on its inner structure. In the existing literature,detailed cooling calculations have been compared to the fewestimates available for the surface temperatures of neutronstars and used to draw some conclusions about the presenceof superfluidity or the existence of exotic matter in theircores (see, e.g., Brown et al. 1988; Page & Baron 1990; VanRiper 1991; Pizzochero 1991; Page & Applegate 1992).More recently, Lattimer et al. (1994) have argued that thelikely occurrence of rapid cooling processes in the denseinterior of neutron stars and the observation of the subse-quent precipitous drop in the surface temperature can effec-tively constrain the mass-radius relation for these compactobjects, thus providing invaluable information about theunderlying equation of state (EOS) of dense matter. Thepotential of this approach is closely related to the growingdatabase of observed neutron star surface temperatures.

It is clear, however, that any quantitative predictionabout the core structure based on observations of the sur-face will require a realistic and accurate description of theintermediate regions of the star, namely, the crust. In partic-ular, the presence of neutron superfluidity in the inner crust,where a gas of unbound neutrons permeates a Coulomb lat-tice of neutron-rich nuclei, will affect the thermal propertiesof this region and hence its thermal response to the insidecooling. This, in turn, will influence the cooling time, whichis the time interval between the formation of the star and thedrop in surface temperature due to the rapid cooling of thecore.

Some time ago it was suggested that considering the pres-ence of the lattice of nuclei in the inner crust could signifi-cantly modify the specific heat corresponding to theunbound neutrons alone, affecting the observable thermalproperties of the neutron star (Broglia et al. 1994; De Blasioet al. 1996). In particular, the larger neutron density corre-sponding to the atomic nucleus was shown to yield locallysmaller pairing gaps and hence altogether larger specificheats. As a consequence, the diffusion times along the wholecrust obtained for the nonuniform system, consisting offinite nuclei embedded in a sea of neutrons (nonuniformneutron matter), turned out to be about a factor of 2 longerthan those coming from the standard assumption of uni-form neutron matter without nuclear impurities. These cal-culations, however, were based on a semiclassical model,the local or Thomas-Fermi (TF) approximation, to deter-mine the spatial behavior of the neutron pairing gap insidethe Wigner-Seitz cells of the lattice, each containing onenucleus immersed in the neutron gas. By neglecting the non-locality of the Cooper pairs and the associated proximityeffects, the TF approach is expected to overestimate the spa-tial dependence of the gap, thus producing specific heatsthat are too large. In the meantime, a quantum treatment ofneutron superfluidity in the nonuniform inner crust matterhas been developed based on the Hartree-Fock-Bogoliubovequations (Barranco et al. 1996, 1998). Indeed, the specificheat from the quantum calculation turned out to be smallerthan that from the TF approximation, at least at the partic-ular density (� ¼ 3� 1013 g cm�3) where it was evaluated,quite deep inside the inner crust. In turn, these results raiseagain the original question of whether the presence of thenuclear lattice actually affects the observed cooling times ascompared to uniform matter and, in the positive case, inwhich direction and magnitude.

In the present paper we answer this question, at least forthe nuclear interactions we have studied. Namely, after per-forming a quantum mechanical calculation of the neutron-specific heat throughout the inner crust of a neutron star,we evaluate the crust-cooling times when influenced by thepresence of nuclear impurities and compare them to the val-

1 Dipartimento di Fisica, Universita degli Studi di Milano, Via Celoria16, 20133Milan, Italy.

2 INFN, Sezione diMilano, Via Celoria 16, 20133Milan, Italy.3 Departamento de Fisica Aplicada, Escuela Superior de Ingenieros,

Universidad de Sevilla, Seville, Spain.4 Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100

Copenhagen Ø, Denmark.

The Astrophysical Journal, 569:381–394, 2002 April 10

# 2002. The American Astronomical Society. All rights reserved. Printed in U.S.A.

381

ues obtained in the standard uniform density model of thecrust superfluidity.

We first solve the Hartree-Fock-Bogoliubov equationsfor the pairing gap in 10 different Wigner-Seitz cells, withsets of parameters corresponding to different densities alongthe inner crust (Negele & Vautherin 1973), which extendsfrom neutron drip density �d up to the density �c, wherenuclei merge into a uniform nuclear sea of neutrons andprotons (�d ¼ 4� 1011 g cm�3 and �c � 0:6�0, with� ¼ 2:4� 104 g cm�3 the standard nuclear matter density;see Lorenz, Ravenhall, & Pethick 1993). This calculationrequires choosing the neutron pairing interaction. On theone hand, calculations of the energy gap based on realisticmicroscopic nucleon-nucleon interactions (see, e.g., Parisand Argonne interactions) have been performed, whichhave the problem of not considering the induced interactionproduced by the surrounding medium of nucleons. Muchwork has been done concerning this induced interaction inneutron matter (see, e.g., Chen et al. 1993; Wambach, Ains-worth, & Pines 1993; Schulze et al. 1996), and usually itseffect is found to reduce the original pairing gaps by a sub-stantial amount. However, the issue is still far from beingsettled. On the other hand, effective potentials (see, e.g.,Gogny and Skyrme interactions) have also been used, butthese have been developed for ordinary finite nuclei, and itis not clear how they work for bulk neutron-rich matter.The values of the gaps with effective interactions are gener-ally larger than those with realistic potentials (Baldo et al.1990, 1991). Since the nature of the pairing interaction isnot the issue we want to address in this paper, in what fol-lows we have considered two situations: an effective poten-tial (Gogny) and a realistic interaction (Argonne).

We then calculate the specific heat of the superfluid neu-trons in the Wigner-Seitz cells as a function of temperatureand compare them to the values obtained for uniform neu-tron matter at the same average density, i.e., for cells withthe same number of nucleon as before but without a nucleusin the center. The main results of the quantum calculationsare (1) unlike the TF approximation, the nonuniform spe-cific heats can be larger or smaller than the uniform ones,the latter case occurring for the lowest densities along theinner crust, and (2) altogether the nonuniform specific heatsare significantly different than the uniform ones, particularlyat low temperatures (T � 0:1 MeV). This should have aneffect on the thermal response of the crust as a whole andhence on the observable properties of the neutron star.

We then turn to study this issue. To this end, we must firstrecall that two possible cooling scenarios have been consid-ered in the literature. In the ‘‘ standard cooling ’’ picture, thecore of the star (or anyway a large fraction of it) cools byemission of neutrinos through the relatively slow modifiedUrca process. The observable surface temperature decreasesgradually after a plateau-like phase of a few hundred yearscaused by the thermal inertia of the crust. Several studies(e.g., Page & Baron 1990; Van Riper 1991; Page & Apple-gate 1992) have shown that variations of the crust-specificheat yield significant differences in the surface temperatureonly after cooling by photon emission dominates over thatfrom neutrino emission, that is, for old neutron stars (olderthan �104–105 yr). In the ‘‘ rapid cooling ’’ scenario instead,the core cools mostly by any of several fast processes thathave been proposed in the last decade as a likely alternativeto the modified Urca emission. In this case, the surface tem-perature shows a precipitous drop after a plateau phase of a

few decades, the time necessary for the thermal signal fromthe core to cross the crust and reach the surface. The specificheat of the crust is now expected to influence the duration ofthe plateau phase by changing the thermal conductivity ofthe crust itself, and therefore it will affect the behavior ofyoung neutron stars.

In this paper we choose to concentrate on the rapid cool-ing scenario for different reasons: on the one hand, severalrecent studies seem to favor the occurrence of exotic coolingprocesses under the conditions found in neutron star cores(Lattimer et al. 1991; Prakash et al. 1992; Akmal, Pandhari-pande, & Ravenhall 1998; Banik & Bandyopadhyay 2000;Voskresensky et al. 2000). On the other hand, as alreadypointed out, the determination of the cooling time (durationof the plateau phase) for a young neutron star has the poten-tial of setting constraints on the EOS of dense matter, aresult that would be of paramount importance for severalfields of physics (Lattimer et al. 1994). Finally, while therelationship between crust-specific heat and cooling of oldneutron stars has been extensively studied in the existing lit-erature, the same is not true for the case of young neutronstars undergoing rapid cooling. The present issue will thenbe to determine whether given changes in the neutron-spe-cific heat in the crust will affect the surface cooling times sig-nificantly above observational uncertainties.

We therefore estimate the time for heat diffusion alongthe inner crust following an initial rapid cooling of the core.A very simple analysis of the thermal diffusivity shows thatfor both the realistic and the effective pairing interactions,the bottleneck for this process is represented by the moreexternal regions of the inner crust, i.e., �d < � < 7�d . Inturn, these regions are quickly cooled down to temperaturesaround T � 0:1MeV by early neutrino processes (Itoh et al.1996). Consequently, the cooling times will be particularlysensitive to the low-temperature properties of the low-den-sity regions of the inner crust. As a consequence of thebehavior of the total specific heat, to which both neutronsand electrons contribute, we find that the cooling times cal-culated for nonuniform matter are significantly differentthan those obtained for uniform matter. The direction andmagnitude of the difference depend in a sensitive way on thevalue of the temperature in the region �d < � < 7�d as wellas on the nucleon-nucleon interaction used to calculate theneutron pairing. Altogether, the presence of the nuclear lat-tice can change the diffusion times across the inner crust byamounts that are comparable to the cooling times them-selves, namely, several years. Therefore, the presence ofnuclear impurities in the inner crust is expected to have sig-nificant observable consequences on the cooling of youngneutron stars of magnitude well above observational uncer-tainties. Incidentally, we will argue that a similar situationalso occurs for slowly cooling old neutron stars, whose sur-face temperature will be influenced by the inclusion of thenuclear lattice.

We shall thus conclude that as already suggested in thelocal (TF) approximation study of the inner crust structure,the quantum treatment of neutron pairing also indicatesthat the effects of the nuclear lattice in the sea of superfluidneutrons must be taken into account in any realistic calcula-tion of the thermal evolution of a neutron star. Only in thisway will the associated results have predictive power, thusproviding a solid framework to interpret future observa-tions of these objects. At the same time, more work isrequired on the nuclear structure side in order to pin down

382 PIZZOCHERO ET AL. Vol. 569

the detailed nature of the pairing interaction acting betweennucleons.

2. SPECIFIC HEAT OF SUPERFLUID NEUTRONS INTHE INNER CRUST

In what follows we shall calculate the pairing gap of theneutrons in the inner crust of a neutron star making use ofdifferent approximations and discuss some of the conse-quences this quantity has on the specific heat of the system.The reader interested only in the astrophysical implicationsof our calculations can skip directly to x 2.3.

2.1. The Uniform System

As a first approximation in the study of the superfluidityproperties of the system, one can neglect the inhomogeneityof the crust and calculate the pairing gap as a function ofdensity, as if it was a piece of uniform neutron or nuclearmatter. These calculations have been performed by severalauthors (e.g., Pethick & Ravenhall 1995b and referencestherein). We shall only consider correlations in the 1S0 chan-nel, as suited to the densities typical of the inner crust ofneutron stars.

The pairing gap Dunif then depends only on the absolutevalue of the linear momentum k of the nucleons, and at agiven temperature T it is determined by the equation (see,e.g., Fetter &Walecka 1971; Tinkham 1996)

DunifðkÞ ¼ �Z

d3k0

ð2�Þ3Dunifðk0Þ2Ek0

vðjk� k0jÞ 1� 2nk0 ðTÞ½ � ; ð1Þ

where vðjk� k0jÞ is the Fourier transform of the two-bodyinteraction vðr12Þ in the 1S0 channel, which is supposed todepend only on the relative distance r12. The quasi particleenergy Ek is given by Ek ¼ ½ð�k � �FÞ2 þ D2

unifðkÞ�1=2, where

�F is the Fermi energy, �k denotes the energy of a single-par-ticle state of momentum k, and nkðTÞ ¼ 1= eEk=T þ 1ð Þ is theoccupation number of the quasiparticle Ek at temperatureT. For a translationally invariant system, the single-particlestates are plane waves, and their energy is given in the effec-tive mass approximation by �k ¼ �h2k2=2m�.

For simplicity, in our calculations we shall not introduceany effective mass so that the nucleon mass will be assumedto be equal to its bare mass, m� ¼ 1. This should be a goodapproximation for the cells in the outermost layers of theinner crust; these cells will turn out to be the most importantfor the thermal properties of the star. Detailed studies ofneutron matter (Takatsuka 1984; Elgarøy et al. 1996; Lom-bardo 1999) find that at densities corresponding to those ofthe inner part of the crust m� � 0:9, so that we shall some-what overestimate the gaps in that region but in a similarway for the uniform and the inhomogeneous system.

Various two-body interactions have been proposed. First,there are ‘‘ bare ’’ interactions, like, for example, theArgonne v14 (Wiringa, Smith, & Ainsworth 1984) and v18(Wiringa, Smith, & Ainsworth 1995) Paris (Lacombe et al.1980) or Bonn (Machleidt, Holinde, & Elster 1987) poten-tials, which reproduce the experimental nucleon-nucleonphase shifts in the 1S0 channel. These interactions lead togaps of similar magnitude in uniformmatter (see, e.g., Elga-røy & Hjorth-Jensen 1998). The maximum value for thepairing gap at the Fermi energy in neutron matter,DF � DunifðkFÞ, is obtained for kF � 0:9 fm�1, with a typicalvalue (using m� ¼ 1) of DF � 3 MeV. Other groups have

adopted ‘‘ effective ’’ interactions of zero or finite range thatare able to reproduce rather successfully the pairing proper-ties in finite nuclei; an example is the Gogny D1 interaction(Decharge & Gogny 1980), which for neutron matter, leadsto DF � 3.8 MeV, again for kF � 0.9 fm �1 and m� ¼ 1 (orabout 3.2MeV withm� � 0:9). The main difference betweenthese two classes of interactions lies in the strong repulsivecore present in the v14 potential, which is needed in order toreproduce the negative phase shifts observed in nucleon-nucleon scattering at nucleon energies larger than 250MeV.The repulsive core, which scatters pairs to states of very highenergies with respect to �F, is strongly felt at densities on theorder of the saturation density. This is the reason why thepairing gap associated with v14 vanishes at values of kF thatare smaller than those calculated using the Gogny (effective)interaction.

The dependence of vðr12Þ on r12 is shown in Figure 1 for abare interaction (the Argonne potential) and for an effectiveinteraction (the Gogny D1 potential). The gaps DF, calcu-lated at the Fermi energy in neutron matter with the sameinteractions, are shown in Figure 2a as a function of theFermi momentum kF. The pairing gap calculated as a func-tion of the relative momentum k at a fixed value of kF(kF ¼ 1 fm�1) is shown in Figure 2b. The change of sign inDunif(k) at large momenta is caused by the repulsive core(Baldo et al. 1990). To obtain convergence of the resultingpairing gaps, the calculations have been performed integrat-ing the gap equation (1) up to klim ¼ 8 fm�1 in the case ofthe Argonne potential and up to klim ¼ 4 fm �1 in the case ofthe Gogny interaction.

In the following discussion, we shall compare the pairinggaps obtained in the uniform system, as described above,with those obtained in a more accurate calculation, takingexplicitly into account in eachWigner-Seitz cell the presenceof the nucleus immersed in the neutron sea. The propertiesof nuclei in this regime have been studied by several authors(e.g., Pethick & Ravenhall 1995c and references therein).We will study densities up to about 1� 1014 g cm�3; in thisrange, detailed calculations find that the favored shape of

0 1 2 3 4r12 (fm)

−150

−100

−50

0

50

100

150

200

v (M

eV)

Fig. 1.—Dependence of the two-body interaction v on the relative dis-tance r12 for the Gogny and for the Argonne potentials, as indicated by thesolid and dashed curves, respectively; in the case of Argonne, the repulsivecore raises up to about 2500MeV.

No. 1, 2002 NUCLEAR IMPURITIES IN NEUTRON STARS 383

the Wigner-Seitz cell is spherical (Lorenz, Ravenhall, &Pethick 1993; Oyamatsu 1993).

One of the most detailed studies is the Hartree-Fock cal-culation of Negele & Vautherin (1973), who determined thenumbers of protons and neutrons that are energeticallyfavored as well as the mean field acting on the nucleons, tak-ing shell effects into account as a function of density in thecrust. We shall now study the superfluidity in the inner cruststarting from their results. Our calculations will be self-con-sistent only in the pairing channel: in particular, we shallcalculate the pairing gap of the system using interactionsdifferent than those employed by Negele & Vautherin. Weremark that they neglected pairing in their calculations, esti-mating that its effect on the overall structure of the crust issmall.

2.2. Nonuniform System

Following the work of Negele & Vautherin (1973), wehave subdivided the inner crust into a number of zones thatcorrespond to different values of the matter (i.e., baryon)density � (see Table 1). Corresponding to each zone, there is

a typical value for the radius RWS of the Wigner-Seitz cell,which was deduced from the baryon density and the numberof neutrons and protons (N and Z) in each cell as given byNegele & Vautherin (1973). In each cell,Nbound neutrons arebound to the nucleus placed at the center of the cell, whilethe remaining neutrons occupy orbitals at positive energiesand their wave functions extend throughout the cell.

The calculations to be carried out to obtain the pairinggap, taking into account the inhomogeneity of this system,are considerably more involved than for the uniform case.This is because the system is not any more translationallyinvariant. Consequently, the single-particle wave functionsare not plane waves anymore, the gap becoming a functionof position as well as momentum. However, as it will be dis-cussed in x 2.3, we have found that in the outer layers of thecrust it is possible to account for the presence of the nucleussimply by readjusting the Fermi energy of the system. Thefollowing considerations will therefore be applied only tothe four inner zones,Nzone ¼ 1 4.

For a system with spherical symmetry, such as the one weshall consider here, single-particle states are characterizedby the quantum numbers nljm, denoting the number ofnodes of the radial wave function, the orbital angularmomentum, the total angular momentum, and its projec-tion on the z-axis, respectively.

The radial wave functions �nlj(r) in a given cell wereobtained by solving the Schrodinger equation with a meanfield potential V(R) acting on the neutrons. We have para-meterized V(R) as a Saxon-Woods potential, correspondingto

VðRÞ ¼ � V0

1þ eðR�RN Þ=a0: ð2Þ

The values of the depthV0 of the radius RN and of the diffu-sivity parameter a0 for the four inner cells have been fittedso as to reproduce the density profiles shown in Figure 4 ofNegele & Vautherin (1973). These parameters are listed inTable 1. The single-particle levels associated with the poten-tials of each zone have been obtained by solving the Schro-dinger equation in a box of radius equal to the Wigner-Seitzcell radius RWS. The spin-orbit potential has been neglected,as was also done by Negele & Vautherin.

We treat each cell independently of the others and requirethat the single-particle wave functions vanish at the cellboundary. We calculate both bound and continuum single-particle levels lying at high positive energies (up to about800 MeV), as required for the solution of the pairing gapequationmaking use of interactions having a strongly repul-sive core (see the discussion below).

In the standard Bardeen-Cooper-Schrieffer approxima-tion, pairs of particles coupled to zero angular momentumare constructed by coupling wave functions having the samenumber of nodes. A more general solution of the pairingHamiltonian, which is the one we shall use below, makesuse of a generalized Bogoliubov quasiparticle transforma-tion, considering also Cooper pairs built out of states havingdifferent numbers of nodes n and n0 (Dobaczewski, Flocard,& Treiner 1984; Dobaczewski et al. 1996; Barranco et al.1996, 1998).

The associated state-dependent pairing gap Dnn0lj is inde-pendent of the magnetic quantum numberm and is obtainedby solving a system of coupled equations, diagonal in thequantum numbers l and j. They can be written at T ¼ 0 (see,

Fig. 2.—(a) Pairing gap at the Fermi energy shown as a function of theFermi momentum kF for the Gogny potential (solid line) and the Argonnepotential (dashed line). (b) Dependence of the gap on the relative momen-tum k for a fixed value of the Fermi momentum, kF ¼ 1 fm�1, shown forthe Gogny potential (solid line) and for the Argonne potential (dashed line).

384 PIZZOCHERO ET AL. Vol. 569

e.g., Ring & Schuck 1980) as

ð�nlj � �FÞUqnlj þ

Xn0

Dnn0ljVqn0lj ¼ Eq

ljUqnlj ; ð3Þ

�ð�nlj � �FÞVqnlj þ

Xn0

Dnn0ljUqn0lj ¼ Eq

ljVqnlj : ð4Þ

where the index q labels the different quasiparticle states and�nlj is the energy of the single-particle state nlj. The eigenval-ues E

qlj of the coupled equations written above are the ener-

gies of the quasiparticle states, while the eigenvectors Uqnlj

andVqnlj are the occupation amplitudes on the single-particle

states.Equations (3) and (4) are solved for an initial guess of the

pairing matrix elements Dnn0lj. Then a new value of Dnn0lj isobtained making use of the relation

Dnn0lj ¼ �X

n1n01l1j1

Xq

Uqn1l1j1

Vqn01l1j1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2j1 þ 1

2j þ 1

s

� nlj; n0lj; J ¼ 0jvðr12Þjn1l1j1; n01l1j1; J ¼ 0h i : ð5Þ

Equations (3)–(5), together with the conditionXnljq

ð2j þ 1ÞðVqnljÞ

2 ¼ N ; ð6Þ

are then iterated until convergence is achieved. We note thatduring the iteration process, the radial wave functions �nlj

are kept fixed.From the resulting occupation amplitudes one can con-

struct the pairing density, given by (Barranco et al. 1998)

�ðr1; r2Þ ¼Xnn0lj

ð2j þ 1Þ8�

Xq

UqnljV

qn0lj

!

� �nljðr1Þ�n0ljðr2ÞPlðcos �21Þ ; ð7Þ

as well as the pairing field

Dðr1; r2Þ ¼ �vðr12Þ�ðr1; r2Þ ; ð8Þ

where h21 is the angle between r1 and r2. It is also useful todefine the Fourier transform of this quantity, namely,

Dðk;RÞ ¼Z

d3r12Dðr1; r2Þe�ik x r12 : ð9Þ

Here R denotes the center of mass of the pair and r12 the rel-ative distance. The quantities Dnn0lj can also be expressed asthe projection of the pairing field D(r1, r 2) over the S ¼ 0component of the wave function of two particles coupled toJ ¼ 0,

�nn0ljðr1; r2Þ ¼1

4��nljðr1Þ�n0ljðr2ÞPlðcos �21Þ : ð10Þ

In fact,

Dnn0lj ¼Z

d3r1d3r2Dðr1; r2Þ�nn0ljðr1; r2Þ ; ð11Þ

or equivalently,

Dnn0lj ¼1

ð2�Þ3

Zd3kd3RDðk;RÞ�nn0ljðk;RÞ ; ð12Þ

where the Fourier transform �nn0lj(k, R) of the two-particlewave function�nn0lj(r 1, r2),

�nn0ljðk; RÞ ¼Z

d3r12�nn0ljðr1; r2Þe�ik�r12 ; ð13Þ

was introduced. Note that for the special case n ¼ n0, it coin-cides with the Wigner transform of the density matrix asso-ciated with the single-particle wave function �nlj.

In the absence of the nucleus at the center of the Wigner-Seitz cell, equations (3)–(5) reduce to equation (1) for uni-form nuclear matter, at least in the limit of a large cell (Bar-ranco et al. 1998). In fact, it has been checked that, keepingthe Fermi energy �F fixed, the gaps calculated within theWigner cell using the Gogny force with equations (3)–(5)coincide with those calculated in neutronmatter using equa-tion (1), aside from small fluctuations connected with thediscreteness of the Wigner cell single-particle levels. Thepresence of the finite nucleus at the center of the Wigner-

TABLE 1

Properties of Wigner-Seitz Cells along the Inner Crust

Nzone

�

(g cm�3)

RWS

(fm) N Nbound

EF

(MeV)

EF(unif)

(MeV)

V0

(MeV)

RN

(fm)

a0(fm)

10....... 4.7 � 1011 54 140 110 0.2 0.7 . . . . . . . . .9......... 6.7 � 1011 49 160 110 0.4 0.9 . . . . . . . . .

8......... 1.0 � 1012 46 210 110 0.8 1.2 . . . . . . . . .

7......... 1.5 � 1012 44 280 110 1.3 1.7 . . . . . . . . .6......... 2.7 � 1012 42 460 110 2.2 2.6 . . . . . . . . .

5......... 6.2 � 1012 39 900 110 4.1 4.6 . . . . . . . . .

4......... 9.7 � 1012 36 1050 110 4.8 5.3 40 7 0.8

3......... 3.3 � 1013 28 1750 110 13.5 14.4 31 7.5 0.9

2......... 7.8 � 1013 20 1460 70 24 25.2 19 7 1.1

1......... 1.3 � 1014 14 950 40 32 33 13.5 5.3 1.25

Note.—The physical parameters of the 10 zones in the inner crust analyzed in this paper are reported asderived from the results of Negele & Vautherin 1973. For each zone, we list the neutron density �, the radiusof the cell RWS, the total number of neutrons in the cell N, the number of neutrons bound in the nucleusNbound, the Fermi energyEF, and the Fermi energyEF(unif) obtained if there was no nucleus at center of thecells so that no neutron was bound. For the zones 1–4, we also give the parameters of the Saxon-Woodspotential used to simulate the density profile obtained by Negele and Vautherin: the depth of the potentialis denoted byV0, its radius byRN, and its diffusivity by a0.

No. 1, 2002 NUCLEAR IMPURITIES IN NEUTRON STARS 385

Seitz cell (see Fig. 3) modifies equations (3)–(5) in severalrespects, as compared to the uniform situation:

1. The energies and the matrix elements of the bound lev-els now enter equations (3)–(5).2. The wave functions of the unbound levels are affected

by the Saxon-Woods potential.3. The Fermi energy decreases, as compared to the uni-

form case, in order to ensure the same number of neutronswithin the cell since Nbound neutrons are now found in thebound single-particle levels.

In order to disentangle these effects at least partially, westart by fixing the Fermi energy at �F ¼ 13:5 MeV (kF � 0:8fm�1), a value for which the gap depends only weakly on kF(see top panel of Fig. 2), so that effect 3 is not important inthis case. We then show in Figure 4 the diagonal state-dependent pairing gaps Dnnlj, calculated with the Gognyinteraction with and without the nuclear potential for� ¼ 3:3� 1013 g cm�3 (Nzone ¼ 3; see Table 1). While theoverall dependence of the gaps of the unbound levels on the

single-particle energy is quite similar for both sets of results,following quite closely the behavior of the pairing gap inneutron matter (see bottom panel of Fig. 2b), the presenceof the nucleus in theWigner-Seitz cell leads to a reduction ofthe gap of about 10% close to the Fermi energy.

In Figure 5a we show the Fourier transform (see eq. [9])of the pairing field D(k, R) obtained from equations (3)–(5).For comparison, we display in Figure 5b the result of a semi-classical calculation in which the gap at a point (k, R) istaken from the gap in neutron matter Dunif(k), calculatedat a Fermi energy equal to the local Fermi energy�FðRÞ ¼ �F � VðRÞ. It is seen that the semiclassical calcula-tion reproduces the overall behavior of the quantum solu-tion. The reduction of the gap inside the nucleus is due tothe fact that the density, and hence the local Fermi momen-tum, are higher inside the nucleus than in the surroundingneutron gas (see Figs. 2a and 3). However, the difference ofthe gap in the two regions is smoothed out in the quantumcalculation because of quantum fluctuations (for moredetails, see Barranco et al. 1998). Through the semiclassicalapproximation, it is also possible to gain insight into thebehavior of the state-dependent gap Dnnlj shown in Figure 4.

According to equation (12), the value of Dnnlj is the aver-age of D(k, R) over the Wigner-Seitz cell, weighted with theassociated single-particle density, since �nnlj(k, R) is just theWigner phase-space density associated with the single-par-ticle state nlj. If one approximates this quantity with the cor-responding semiclassical local momentum expression

�nnljðk; RÞ ¼ �2nnljðRÞ� k � knljðRÞ

� �; ð14Þ

where �h2k2nljðRÞ=2mN � �nlj � VðRÞ, one obtains from equa-tion (12) the simple semiclassical estimate

Dnnlj ¼Z

d3R�2nljðRÞD knljðRÞ; R

� �: ð15Þ

In other words, the value of the state-dependent gap associ-ated with a given single-particle state is the average ofD[k(R), R] over the Wigner-Seitz cell, weighted with the

0 5 10 15 20 25R (fm)

−40

−30

−20

−10

0

10

20

V (

MeV

)

0 5 10 15 20 25R (fm)

0

0.05

0.1

0.15

n (f

m−

3 )

Fig. 3.—Top: Saxon-Woods potential V(R) used for the Wigner-Seitzcell Nzone ¼ 3 shown by the solid line, while the dashed line represents theFermi energy (�F ¼ 13:5 MeV; see Table 1). Bottom: Resulting neutrondensity displayed as a function of the position of the cell; it vanishes at thecell boundary.

−30 −20 −10 0 10 20 30E s.p. (MeV)

2

3

4

5

∆ nn (

MeV

)

Fig. 4.—Diagonal elements of the state-dependent pairing gap Dnnlj cal-culated using the Gogny interaction as a function of the single particle ener-gies, with (squares) and without (circles) the nucleus at the center of theWigner-Seitz cell for Nzone ¼ 3. Also shown are the corresponding gapsaveraged over an interval of 5MeV (solid and dashed lines).

386 PIZZOCHERO ET AL. Vol. 569

associated single-particle density. For states close to the(positive) Fermi energy, the density is distributed rather uni-formly throughout the Wigner-Seitz cell, and since insidethe nucleus the local momentum k(R) is higher than outside(see Fig. 3), and consequently the gap is lower, the associ-ated values of Dnnlj must be smaller than those associatedwith a uniform system. The importance of this effect willdepend on the portion of the volume of the Wigner celloccupied by the nucleus or, more exactly, on its overlap withthe single-particle density of the states around the Fermienergy, which becomes progressively smaller the larger thecell becomes, that is, moving toward the surface of the neu-tron star. In fact, we have found that for densities smallerthan about 1013 g cm�3, or for Nzone 4, the main effect ofthe nucleus is the reduction of the Fermi energy because ofthe fact that a fraction of the neutrons is bound in theSaxon-Woods potential (effect 3 above). The effect of thesebound neutrons on quantities calculated close to the Fermienergy is only marginal so that in the low-density region wecan reproduce the effects of the nucleus with a simple uni-form calculation involving only the free neutrons.

2.3. Specific Heat

In what follows, we shall discuss some of the effects thatthe change in the pairing gap observed in going from theuniform to the nonuniform system has on the specific heatof the Wigner-Seitz cell. In this section we consider only theneutron contribution. In x 3 we shall also take into accountthe electronic contribution to the specific heat.

A 10% reduction of the gap close to the Fermi energy,such as the one found for Nzone ¼ 3, can lead to importanteffects on the specific heat of the cell. The specific heat CV isdefined as

CV ¼ 1

V

@E

@T; ð16Þ

where V is the volume of the Wigner-Seitz cell. Within theframework of the mean field theory used above, the energyof the system is simply obtained summing over the quasipar-ticle states q,

E ¼Xq

nqEq ; ð17Þ

the quantity nq ¼ ð1þ eEq=TÞ�1 being the occupation num-ber of the quasiparticle state q at temperature T. The specificheat for a superfluid system, at temperatures T muchsmaller than DF, is well approximated by the expression(Landau &Lifshitz 1980)

CðsfÞV ¼ C

ðnormÞV

ffiffiffi2

p

�3=2

DF

T

� �5=2

e�DF=T ; ð18Þ

where CðnormÞV is the specific heat of the system in absence of

superfluidity. From equation (18) it can be seen that areduction of the gap close to the Fermi energy leads to anexponential increase of the specific heat. The specific heatcalculated with and without the nucleus is shown in Figure 6as a function of temperature for Nzone ¼ 2. It is seen that ata temperature of 0.1 MeV, the presence of the nucleuscauses an increase of about 1 order of magnitude.

As was commented on at the end of the previous subsec-tion, it turns out from our numerical calculations with the

0 0.2 0.4 0.6T (MeV)

10−28

10−24

10−20

10−16

10−12

Cv(

MeV

K−

1 fm−

3 )

0.0 0.2 0.4 0.6T

0.1

1.0

10.0

Rat

io

Fig. 6.—Neutron-specific heat of the Wigner-Seitz cell Nzone ¼ 2 as afunction of temperature, calculated using the Gogny interaction with (solidcurve) and without (dashed curve) the nucleus at the center of the cell. In theinset the ratio of the two values is shown.

2 4 6 8 10 12

R (fm)

0

2

4

6

∆(k

,R)

(MeV

) TF

(b)

∆(kF(R))

Fig. 5.—Neutron pairing gap D(k, R) as a function of R in the Wigner-Seitz cell Nzone ¼ 3, calculated using the Gogny interaction using (a) thequantum theory (Hartree-Fock-Bogoliubov) or (b) a semiclassical approxi-mation (local TF approximation). The gap becomes essentially constant forR 12 fm. The various curves refer to different values of k in steps of 0.5fm�1 in the interval k ¼ 0:25 (top curve) to k ¼ 2:25 fm�1 (bottom curve). In(b) we also show the value of the gap calculated at the local Fermi momen-tum kF(R) (dashed curve).

No. 1, 2002 NUCLEAR IMPURITIES IN NEUTRON STARS 387

Gogny interaction that it is important to deal with thedetailed effects of the nucleus on the specific heat only fordensities on the order of, or larger than, 1013 g cm�3, that is,for Wigner-Seitz cells smaller than about 30 fm (Nzone 4;see Table 1). For larger cells, we can instead use the resultsof a uniformmatter calculation in which the Fermi energy isdetermined by the number of free neutrons in the cell; this isobtained by subtracting the number of neutrons bound inthe nucleus from the total number of neutrons in the cell,Nfree ¼ N �Nbound. The Fermi momentum kF is thenobtained from the density of free neutrons Nfree=V accord-ing to the Fermi gas expression

Nfree

V¼ k3F

3�2: ð19Þ

The results obtained in this way will be compared to thoseobtained in the absence of the nucleus. In this case, all theneutrons are free so that the Fermi momentum is larger,

being obtained from the equation

N

V¼ k3F

3�2: ð20Þ

The decrease in Fermi energy associated with the presenceof the nucleus leads to a decrease of the gap because for� < 1013 g cm�3, DF is an increasing function of �F; as a con-sequence, the presence of the nucleus tends to also increasethe specific heat in the low-density region. However, this istrue only as long as the system remains superfluid (as can beseen in Fig. 8, where we show the behavior of the specificheat in the crust, both in the presence and in the absence ofthe nucleus, at three temperatures, T ¼ 0:1, 0.2, and 0.3MeV). At very low densities the critical temperature Tc

becomes so small that the system ceases to be superfluid: atT ¼ 0:1 this happens for Nzone 9 and at T ¼ 0:3 MeV forNzone 7. For temperatures above Tc the specific heat inthe presence of the nucleus is smaller than in the uniformcase. This is illustrated in more detail in Figure 7, where weshow the specific heat as a function of temperature calcu-lated with the Gogny interaction for Nzone ¼ 10, which liesat the outer boundary of the crust. In this case, only 140neutrons occupy a very large cell with a radius of 54 fm (seeTable 1). Now, if we consider the system as a piece of uni-form neutron matter, the Fermi energy has the value�F ¼ 0:7 MeV, while if we consider that 110 of 140 neutronsare bound in the nucleus, �F is equal to only 0.2 MeV (seeTable 1). In the first case, the system ceases to be superfluidat the critical temperature Tc � 0:18 MeV, while in the sec-ond case Tc � 0:03MeV.

The calculations discussed above were performed withthe D1 Gogny interaction. We have also performed similarcalculations with the Argonne v14 interaction. As in the caseof the Gogny interaction, we have calculated the specificheat in the presence of the nucleus for the cells Nzone ¼ 1 4,while the results for larger cells are obtained from a calcula-tion in neutron matter with the appropriate Fermi energy(see eq. [19]). The overall trends resemble those obtainedwith the Gogny interaction, as can be seen from Figures 9–

0 0.2 0.4 0.6T (MeV)

10−16

10−15

10−14

10−13

Cv

(MeV

K−

1 fm−

3 )

0 0.2 0.4 0.6T

0.1

1.0

10.0

Rat

io

Fig. 7.—Same as Fig. 6, but forNzone ¼ 10

−25 −15 −5 5 15 25Es.p.(MeV)

0

1

2

3

4

5

∆ nn (

MeV

)

Fig. 9.—State-dependent pairing gap Dnnlj, calculated making use of theArgonne interaction with (solid line) and without (dashed line) the nucleusin the cellNzone ¼ 3, averaged over an interval of 5MeV as a function of thesingle-particle energies.

1 2 3 4 5 6 7 8 9 10Nzone

10−26

10−23

10−20

10−17

10−14

CV (

MeV

K−

1 fm−

3 )

Fig. 8.—Neutron-specific heats in the various zones, calculated atT ¼ 0:1 (lower curves), 0.2, and 0.3 (upper curves) MeV, using the Gognyinteraction with (solid curves) and without (dashed curves) the nucleus at thecenter of the associated Wigner-Seitz cells. The corresponding Fermi ener-gies are given in Table 1.

388 PIZZOCHERO ET AL. Vol. 569

11, reflecting the similar density dependence of the pairinggap in neutron matter. However, the strong repulsive corepresent in the Argonne interaction suppresses the gap atlarge Fermi energies, and this causes the large differencebetween the gaps inside and outside the nucleus seen in thesemiclassical calculation (top panel of Fig. 10). This differ-ence, however, is much reduced in the quantum calculation(bottom panel of Fig. 10) showing the greater importance ofthe proximity effects with respect to the Gogny interaction,as can be appreciated by comparing Figure 10 with Figure5. The dependence of the specific heats on the position in thecrust, shown in Figure 11, is similar to that displayed by theGogny force in Figure 8.

3. HEAT DIFFUSION ALONG THE INNER CRUST

We now estimate the time for heat diffusion along theinner crust following an initial rapid cooling of the core. Inthe previous section we showed how the presence of nuclearimpurities in the inner crust significantly modifies the pair-ing properties of the superfluid neutrons and hence the neu-tron-specific heat. This is not enough, however, in order todecide whether this effect can have observational conse- quences and hence if it is worth considering any further. On

the one hand, the total specific heat receives a substantialcontribution from the relativistic degenerate electron gas,which is not affected by the details of neutron superfluidityand may therefore mask the effects obtained in x 2. On theother hand, even in the case that the total specific heat ismodified in some regions of the inner crust by the presenceof the nuclear lattice, it is not clear whether this will influ-ence significantly the thermal evolution since these regionsmay have large thermal conductivity and thus contributevery little to the total diffusion time.

To address the first issue, we calculate the total specificheat of matter CV at the different densities considered in x 2.The contributions to CV from the Coulomb lattice and fromthe protons are negligible when compared to those fromthe neutrons and the electrons. We thus haveCV ¼ CV ; n þ CV ; e, where the neutron contribution is thatobtained before for the Wigner-Seitz cells, while the contri-bution of the relativistic degenerate electrons is calculatedwith the standard formula (see, e.g., Landau & Lifshitz1969). We then plot the ratio of total specific heat betweenthe nonuniform and uniform cases. In Figure 12 we give theresults obtained for the two pairing interactions at (a)T ¼ 80 keV, (b) T ¼ 100 keV, and (c) T ¼ 120 keV. We willexplain later why this range of temperature values is of par-ticular astrophysical interest. From the figures we see thatfor both the free nucleon-nucleon potential and the effectiveinteraction the ratio of specific heats is different from unityin the region �d < � < 7�d , while it is practically equal to 1in the denser parts of the inner crust. This is because at suchdensities the electron contribution to CV is dominant. Forthe zones Nzone ¼ 6, 7, 8 (that is, 2:5�d < � < 7�d), the ratiois significantly larger than 1, its value being larger for lowertemperatures and in correspondence to the Argonne inter-action. On the contrary, the ratio is smaller than 1 for thefirst point in the crust (Nzone ¼ 10). The zone Nzone ¼ 9 is aborderline case; i.e., the ratio is larger or smaller than 1depending on the temperature and the pairing interaction.We thus conclude that the presence of nuclear impuritiesindeed has a sizable effect on the total specific heat in themore external parts of the inner crust.

1 2 3 4 5 6 7 8 9 10Nzone

10−24

10−21

10−18

10−15

10−12

CV (

MeV

K−

1 fm−

3 )

Fig. 11.—Neutron-specific heats calculated at T ¼ 0:1 (lower curves),0.2, and 0.3 (upper curves) MeV in the various zones making use of theArgonne potential in the uniform case (dashed lines) or considering thepresence of the nucleus at the center of theWigner-Seitz cell (solid lines).

2 4 6 8 10 12R (fm)

−2

0

2

4

∆(k,

R)

(MeV

)

HFB

2 4 6 8 10 12R (fm)

−2

0

2

4

∆(k,

R)

(MeV

)

TF

Fig. 10.—Same as Fig. 5, but for the Argonne interaction

No. 1, 2002 NUCLEAR IMPURITIES IN NEUTRON STARS 389

We now turn to the second issue and point out that theparameter controlling the diffusion of heat is neither thethermal conductivity nor the total specific heat alone, but itis rather their ratio, the thermal diffusivity D ¼ �=CV ,which appears in the heat equation (see, e.g., Landau & Lif-shitz 1959). For the case of the inner crust, the thermal con-ductivity � is mostly due to electrons, and thus it is notaffected by the superfluid properties of the neutrons. Wetake for � the values reported in Lattimer at al. (1994),which in turn were deduced from the detailed analysis ofItoh and collaborators (Itoh et al. 1983, 1984; Mitake, Ichi-maru, & Itoh 1984), and using the results of x 2, we calculatethe thermal diffusivity for both nonuniform and uniforminner crust matter. This is plotted for the two pairing inter-actions used in the present paper at T ¼ 0:1MeV in Figures13 and 14. From the figures we see that for both interac-tions, the diffusivity is about 2 orders of magnitude smallerin the region �d < � < 7�d than in the denser parts of theinner crust for both nonuniform (solid line) and uniform(dashed line) neutron matter. Similar results are obtainedfor temperatures T ¼ 80 and 120 keV. We thus conclude

that the bottleneck for the diffusion of heat from the core tothe surface is represented by the low-density regions of theinner crust. Since these are exactly the regions where theeffects of nuclear impurities are important (as is also evidentfrom Figs. 13 and 14), we see that in principle we can expectobservable consequences from the presence of the nuclearlattice. Before turning to evaluate the magnitude of sucheffects, however, we justify an important point. As we justdiscussed, the cooling time will depend mostly on the low-density parts of the inner crust since there the thermal signalwill diffuse very slowly. From numerical simulations (see,e.g., Van Riper 1991), these parts are found to cool veryquickly to temperatures around T � 0:1 MeV because ofearly neutrino processes (mostly plasma neutrino emission,as can be seen in the exhaustive treatment by Itoh et al.1996). This explains why we have considered as physicallyrelevant to our analysis only the temperature range aroundT � 0:1MeV.

In order to calculate the diffusion time along the innercrust, one should solve the complete heat equation with theproper temperature and density profiles of the matter

Fig. 12a Fig. 12b

Fig. 12c

Fig. 12.—Ratio of total specific heat between the nonuniform case with nuclear impurities and the standard case of uniform neutron matter. The tempera-ture of the inner crust is (a) T ¼ 80 keV, (b) T ¼ 100 keV, and (c) T ¼ 120 keV. The solid line refers to the effective Gogny interaction and the dashed line tothe realistic Argonne potential. The symbols indicate the zones defined in Table 1.

390 PIZZOCHERO ET AL. Vol. 569

encountered by the heat signal or, more correctly, by thecooling wave that propagates across the crust from the coldcore out to the surface. Several sophisticated computercodes have been developed to that goal, but here we wouldbe satisfied with a simple yet reasonable estimate that ena-bles us to assess whether the effect of nuclear impurities isactually relevant to the observations of neutron stars or not.To achieve this, we first notice that for the case under study,the spherical geometry is well approximated by a planargeometry since the crust is thin (�1 km across) compared tothe cold reservoir represented by the core (�10 km inradius). The problem can thus be described as one-dimen-sional diffusion of heat across a crustal ‘‘ slab.’’ Now, for auniform medium with constant thermal diffusivity D, verygeneral dimensional arguments show that the diffusion timeacross a layer of matter must be proportional to the squareof its thickness; i.e., one has the relation x2 ¼ �Dt (Landau& Lifshitz 1959). Incidentally, a similar conclusion can bereached by random walk arguments (Reif 1965). The factor� depends on the boundary conditions of the problem. For

a semi-infinite slab in contact with a heat reservoir, one findsthe exact value � ¼ 4 since the heat equation admits for thepropagating temperature front an explicit solution in termsof the error function. For a finite slab with one end in con-tact with a heat reservoir and the other end insulated, oneevaluates the time for temperature equilibration and obtainsthe approximate value � ’ �2=4, which follows from takingthe first term (with lowest eigenvalue) in the eigenfunctionexpansion of the solution to the heat equation (Landau &Lifshitz 1959; Haberman 1983). The former case maydescribe the process of thermal diffusion along the innercrust, with the outer crust representing the ‘‘ rest ’’ of thesemi-infinite medium; the latter case may describe the ther-malization time of the whole crust, the low-emitting surfacerepresenting the (quasi) insulated end.

From the previous discussion we conclude that the diffu-sion time tdiff across a crustal shell of thickness Rshell of thecooling wave from the core will satisfy a similar expression.For a homogeneous medium with constant D, we can thuswrite R2

shell ¼ �Dtdiff . The appropriate value of � can bedetermined by solving the heat equation, and it will alsodepend on the specific definition of tdiff: propagation time ofa temperature front, thermalization time of the shell, timewhen the cooling curve has the maximum negative slope(the tw of Lattimer et al. 1994), etc. If the medium is nothomogeneous, as in the case of the crust where both the den-sity and the temperature vary with the radial coordinate, wenaturally generalize the previous relation as

tdiff ¼ �

Z Rshell

0

1

DðRÞ dR2 ; ð21Þ

where R ¼ 0 corresponds to the core-crust interface, i.e., tothe density � ¼ �c, and the integration proceeds outwardalong the crust in the direction of the propagating coolingwave. The thermal diffusivity depends on the positionthrough density and temperature, that is, DðRÞ ¼D �ðRÞ; TðRÞ½ �, where the dependence of the density and thetemperature on the radius comes from solving the Tolman-Oppenheimer-Volkov equations for hydrostatic equili-brium coupled with the EOS of matter appropriate for thedifferent regions that form the neutron star (core, inner andouter crust, surface envelope).

Inverting the � ¼ �ðRÞ and T ¼ TðRÞ relations and tak-ing the density as the integration variable, equation (21) canbe rewritten as

tdiff ¼ �1

D

� ��shell

R2shell ; ð22Þ

where the averaged value of 1=D along the shell, from �cdown to the density �shell corresponding to R ¼ Rshell, isdefined as

1

D

� ��shell

� 1

R2shell

Z �shell

�c

1

D �; Tð�Þ½ � dR2ð�Þ : ð23Þ

We notice that equation (23) represents a harmonic mean;namely, we use the average of 1=D rather than the inverse ofthe average of D, in analogy with the conductivity of resis-tors in series. This makes sense if one thinks of the contribu-tion to the total diffusion time of zones with either verysmall or very large diffusivity. In particular, it shows thatthe low-diffusivity external zones are indeed a bottleneck for

Fig. 13.—Thermal diffusivity for inner crust matter atT ¼ 0:1MeVwiththe effective Gogny interaction for the pairing. The solid line represents thecase of nonuniform neutron matter with nuclear impurities, while thedashed line is the standard uniform neutron matter. As before, the symbolsrepresent the zones where the calculation has actually been performed.

Fig. 14.—Thermal diffusivity at T ¼ 0:1 MeV as in Fig. 13, but with theArgonne interaction.

No. 1, 2002 NUCLEAR IMPURITIES IN NEUTRON STARS 391

the transfer of heat, the average value of 1=D being deter-mined mostly by them.

The average value ð1=DÞ�shell will only depend on the low-density boundary �shell of the shell and not on the particular(and so far still unknown) nuclear EOS that describes thedense core. This is because for different choices of thenuclear EOS, the thickness DRAB between given densities �Aand �B in the crust scales with a function of the total massand radius of the neutron star (Pethick & Ravenhall 1995a),and this function cancels out in the ratio of equation (23).Therefore, equation (22) predicts that the diffusion timeacross a given density range in the crust will simply be pro-portional to the square of its thickness and will not dependon the total mass and radius of the star. This is consistentwith the numerical results of Lattimer et al. (1994), whichshow that, whatever the nuclear EOS used, the dependenceof the cooling time tw on the total thickness of the crustRcrust

is well described by the relation tw / Rncrust with

1:7 n 1:8, the deviations from n ¼ 2 coming from gen-eral relativistic corrections.

From the previous discussion it follows that the averageð1=DÞ�shell can be calculated along any density profile for thecrust, irrespective of the particular nuclear model chosen todescribe the neutron star core. We have taken the profile� ¼ �ðRÞ reported in Lorenz et al. (1993) and approximatedthe integral of equation (23) by a sum over the regions ana-lyzed in x 2. As for the temperature at which to evaluate thethermal diffusivity, we have taken a flat temperature profileacross the whole inner crust, with values T � 0:1 MeV typi-cal of its more external regions. Actually, detailed computersimulations yield as initial temperature conditions of theinner crust, before the passage of the cooling wave, a shal-low gradient going from T ’ 0:1MeV at the lower densitiesto T ’ 0:2 0:3 MeV at the core-crust interface (Page 1989;Van Riper 1991). However, as already discussed, the highertemperature inner zones contribute little to the diffusiontime so that one expects the zero-gradient approximation tobe quite satisfactory. Indeed, we checked that taking a tem-perature gradient changes the average value ð1=DÞ�shell by nomore than some percent.

We can now present our results for heat diffusion in non-uniform and uniform neutron matter. In Figures 15 and 16we report the propagation time of the cooling wave alongthe inner crust for the two pairing interactions; namely, weplot tdiff as a function of �shell when this varies from �c downto �d. In order to obtain the characteristic timescales, wehave set typical values � ¼ 4 and Rshell ¼ 1 km in equation(22). The plots correspond to the flat temperature profileT ¼ 100 keV, but similar qualitative results are obtained forT ¼ 80 and 120 keV. As expected from our discussion aboutthe thermal diffusivity, the figures show that the propaga-tion of the thermal signal is rapid in the denser parts of theinner crust, while it drastically slows down in its more exter-nal parts. For both the Gogny and Argonne interactions, itis precisely in these parts that the influence of the nuclearimpurities becomes relevant, changing the value of the diffu-sion time. Again, from the previous discussion about thethermal diffusivity, it follows that in the region2:5�d < � < 7�d the signal will be slower in the nonuniformcase than in the uniform one, while in the outermost layersof the inner crust the behavior will depend on the tempera-ture and the pairing interaction. For instance, Figure 15shows that the total diffusion time across the inner crust isaltogether not affected by the nuclear lattice when one takes

the Gogny interaction and a matter temperature T ¼ 0:1MeV. On the contrary, using the Argonne potential at thesame temperature, we see from Figure 16 that the time ittakes the thermal signal to cross a 1 km long inner crust is�11 yr if one assumes uniform matter, while the effect ofnuclear impurities raises this value to�16 yr (�tdiff ¼ 45%).

In order to be more detailed in our conclusions, in Table2 we report the results for ð1=DÞ�d , namely, the averagevalue of 1=D over the whole inner crust from �c to �d. Thisquantity is the one affected by the presence of nuclearimpurities, while it does not depend on the EOS of densematter, i.e., on the thickness of the crust or the mass andradius of the neutron star. When multiplied by the propervalue of � and the square of the inner crust thickness as inequation (22), it yields directly the diffusion time across it.In the table we give the values corresponding to the twopairing interactions and to a range of physically relevanttemperatures, i.e., T ¼ 80, 90, 100, 110, and 120 keV. Inaddition to the values of ð1=DÞ�d for nonuniform and uni-

Fig. 16.—Diffusion time along the inner crust as in Fig. 15, but with theArgonne interaction.

Fig. 15.—Diffusion time along the inner crust for nonuniform (solid line)and uniform (dashed line) neutron matter, with the effective Gogny interac-tion for the pairing. Eq. (23) was evaluated with a flat temperature profileT ¼ 0:1MeV, while we set � ¼ 4 andRshell ¼ 1 km in eq. (22).

392 PIZZOCHERO ET AL. Vol. 569

form neutron matter, we give their percentage difference(referred to the uniform case). Since the multiplicative fac-tors of equation (22) cancel out, this will also represent thepercentage difference in diffusion times

�tdiff ¼tdiffðNUÞ � tdiffðUÞ

tdiffðUÞ ; ð24Þ

which is the physical quantity we are interested in. FromTable 2 we can draw some important general conclusions:(1) The effects of the nuclear impurities on the diffusiontimes are altogether quite significant and therefore cannotbe neglected. In particular, with typical cooling times on theorder of 10 yr (see, e.g., Van Riper 1991; Lattimer et al.1994), our calculations can yield time differences betweennonuniform and uniform cases up to several years, whichare well above observational uncertainties and comparablein magnitude to the cooling times themselves. (2) As typicalfor pairing-related phenomena, the results are quite sensi-tive to the local temperature. The general trend is that thepercentage difference in diffusion time �tdiff between nonuni-form and uniformmatter is a decreasing function of temper-ature. Obviously, a detailed cooling calculation that goesbeyond our simple estimate is required in order to evaluatethe actual diffusion times in realistic models. (3) The direc-tion and magnitude of the effect due to the nuclear impur-ities depend on the pairing interaction used. The Gognyinteraction yields values for �tdiff that go from positive tonegative with temperature increasing in the range80 keV < T < 120 keV. The Argonne potential, instead,yields values for �tdiff that are mostly positive in the sametemperature range and that are significantly larger than

their Gogny counterparts. Therefore, the need of furtherstudies that pin down the correct nucleon-nucleon (renor-malized) interaction needed to calculate the pairing proper-ties of the inner crust of neutron stars also appears evidentfrom the astrophysical point of view.

Before concluding this section, we point out thatalthough so far we have concentrated on young neutronstars undergoing rapid cooling of the core, the results weobtained in this paper are expected to also have significantconsequences for old neutron stars cooling in the standardscenario. Indeed, after �104–105 yr the star reaches an iso-thermal state with temperatures on the order of �0.01 MeVand cools mostly by surface photon emission, its surfacetemperature being quite sensitive to the thermal propertiesof the crust (see, e.g., Page & Baron 1990; Van Riper 1991;Page & Applegate 1992). Therefore, since the total specificheat for low-density nonuniform crust matter is altogethersignificantly larger than that for uniform matter, particu-larly at low temperatures (see Fig. 12), one expects to obtainhigher surface temperatures when the presence of nuclearimpurities is taken into account.

4. CONCLUSION

In this article we have given a state-of-the-art descriptionof the superfluid properties of the inner crust of neutronstars. In particular, we have performed a consistent quan-tum calculation that goes beyond the approximations usedso far, and we have studied the observational consequencesof the results obtained. The main conclusion we can draw isthat the presence of a nuclear lattice immersed in the sea ofneutrons significantly modifies their thermal properties andhence the observable cooling behavior of both young andold neutron stars. In turn this implies that no detailed cool-ing calculation can get away without including these effectsunless it loses predictive power. Unfortunately, our resultsalso raise a difficulty in the completion of any program inwhich observations can, in principle, be used to constrainphysical models of neutron stars. Indeed, the sensitivity ofthe cooling times (and of the thermal properties in general)on the pairing interaction used makes it impossible to drawany firm quantitative conclusion as long as the correctnucleon-nucleon interaction in bulk neutron-rich matter isnot established. In particular, a complete calculation ofpairing should go beyond the mean field level followed inthe calculation of x 2 and should include the induced inter-action, which takes into account the effects of polarizationin the medium (see, e.g., Pines & Alpar 1992). We are cur-rently investigating this issue and will analyze its impact onneutron star cooling in a successive paper. We also expectthat the quantum calculation for the inhomogeneous systempresented in this paper will prove instrumental for a realisticcalculation of the vortex pinning in the inner crust.

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TABLE 2

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