Physics of Neutron Star Crusts
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Abstract
The physics of neutron star crusts is vast, involving many different research fields, from nuclear and condensed matter physics to general relativity. This review summarizes the progress, which has been achieved over the last few years, in modeling neutron star crusts, both at the microscopic and macroscopic levels. The confrontation of these theoretical models with observations is also briefly discussed.
A List of Notations
- A
mass number
- A
electromagnetic potential vector
- a_{0}
Bohr radius
- B, B
magnetic field
- c
speed of light
- c_{s}
speed of sound
- Δ(k)
wave number dependent pairing gap
- Δ_{F}
pairing gap at the Fermi level Δ(k_{ F })
- d_{n}
interneutron spacing
- d_{υ}
intervortex spacing
- E{A, Z}
energy of a nucleus of mass number A and charge number Z
- e
proton electric charge
- η
shear viscosity
- ϵ
single particle energy
- ϵ_{F}
Fermi energy
- ε
energy density
- \({\mathcal F}\)
force per unit length
- f
force density
- f_{μ}
4-force density
- G
gravitational constant
- γ
adiabatic index
- ħ
Dirac constant
- h
Planck constant
- I
moment of inertia
- J
angular momentum
- k
wave vector
- k
wave number
- k_{B}
Boltzman constant
- k_{F}
Fermi wavenumber
- κ
thermal conductivity (Section 9)
- κ, κ
circulation of a vortex (Section 8)
- Λ
Lagrangian density (Section 10)
- μ
shear modulus
- μ_{e}
electron chemical potential
- μ_{n}
neutron chemical potential
- μ_{p}
proton chemical potential
- M
neutron star gravitational mass
- M_{⊙}
solar mass
- m
mass
- \(m_c^\ast\)
dynamic effective mass of confined nucleons in neutron star crust
- m_{e}
electron mass
- \(m_e^\ast\)
electron (relativistic) effective mass in dense matter
- \(m_{\rm{f}}^\ast\)
dynamic effective mass of the free neutrons in neutron star crust
- m_{n}
neutron mass
- \(m_n^\ast\)
neutron effective mass in dense matter
- m_{p}
proton mass
- m_{u}
atomic mass unit
- n
number density of particles
- n_{b}
number density of baryons
- n_{c}
number density of confined nucleons in neutron star crust
- n_{e}
number density of electrons
- n_{f}
number density of free neutrons in neutron star crust
- n_{n}
number density of neutrons
- n_{N}
number density of nuclei (ions)
- n_{p}
number density of protons
- n_{υ}
surface density of vortex lines
- Ω
angular velocity
- π_{μ}
momentum 4-covector
- P
pressure
- p, p
momentum
- p_{F}
Fermi momentum
- p_{Fe}
electron Fermi momentum
- φ
single particle wave function
- ρ
mass density
- ρ_{ND}
neutron drip density
- r_{g}
Schwarzschild radius
- r_{υ}
radius of a superfluid vortex core
- R
neutron star circumferential radius
- R_{cell}
radius of a Wigner-Seitz sphere
- σ
surface thermodynamic potential (Section 3)
- σ
electric conductivity (Section 9)
- σ
Poisson coefficient (Section 7)
- σ_{ij}
elastic stress tensor
- σ_{max}
breaking strain
- σ_{s}
surface tension
- T
temperature
- T_{c}
critical temperature
- T_{F}
Fermi temperature
- T_{Fe}
electron Fermi temperature
- T_{m}
melting temperature
- T_{pe}
electron plasma temperature
- T_{pi}
ion plasma temperature
- \(T_{\mathcal S}^\infty\)
effective surface temperature as seen by a distant observer
- \(T_\nu ^\mu\)
stress-energy tensor
- ν_{cell}
volume of the Wigner-Seitz cell
- ν_{N}
volume of the proton cluster
- υ_{F}
Fermi velocity
- υ_{Fe}
electron Fermi velocity
- υ
velocity
- υ_{υ}
vortex velocity
- υ_{f}
velocity of neutron superfluid in neutron star crust
- υ_{c}
velocity of confined nucleons in neutron star crust
- ω
volume fraction occupied by nuclear clusters
- ξ
coherence length
- x_{r}
relativity parameter
- Z
charge (proton) number
- ζ
bulk viscosity
B List of Abbreviations
- bcc
body-centered cubic
- BCS
Bardeen-Cooper-Schrieffer
- BE
Boltzman equation
- BEC
Bose-Einstein condensate
- EoS
equation of state
- LMXB
low-mass X-ray binary
- QED
quantum electrodynamics
- QPO
Quasi Periodic Oscillation
- SGR
Soft Gamma Repeater
- SXT
Soft X-ray Transient
- W-S
Wigner-Seitz
1 Introduction
Constructing models of neutron stars requires knowledge of the physics of matter with a density significantly exceeding the density of atomic nuclei. The simplest picture of the atomic nucleus is a drop of highly incompressible nuclear matter. Analysis of nuclear masses tells us that nuclear matter at saturation (i.e. at the minimum of the energy per nucleon) has the density ρ_{0} = 2.8 × 10^{14} g cm^{−3}, often called normal nuclear density. It corresponds to n_{0} = 0.16 nucleons per fermi cubed. The density in the cores of massive neutron stars is expected to be as large as ∼ 5 − 10ρ_{0} and in spite of decades of observations of neutron stars and intense theoretical studies, the structure of the matter in neutron star cores and in particular its equation of state remain the well-kept secret of neutron stars (for a recent review, see the book by Haensel, Potekhin and Yakovlev [184]). The physics of matter with ρ ∼ 5 − 10ρ_{0} is a huge challenge to theorists, with observations of neutron stars being crucial for selecting a correct dense-matter model. Up to now, progress has been slow and based overwhelmingly on scant observation [184].
The outer layer of neutron stars with density ρ < ρ_{0} — the neutron star crust — which is the subject of the present review, represents very different theoretical challenges and observational opportunities. The elementary constituents of the matter are neutrons, protons, and electrons — like in the atomic matter around us. The density is “subnuclear”, so that the methods developed and successfully applied in the last decades to terrestrial nuclear physics can be applied to neutron star crusts. Of course, the physical conditions are extreme and far from terrestrial ones. The compression of matter by gravity crushes atoms and forces, through electron captures, the neutronization of the matter. This effect of huge pressure was already predicted in the 1930s (Sterne [390], Hund [202, 203]). At densities ρ ≳ 4 × 10^{11} g cm^{−3} a fraction of the neutrons is unbound and forms a gas around the nuclei. For a density approaching 10^{14} g cm^{−3}, some 90% of nucleons are neutrons while nuclei are represented by proton clusters with a small neutron fraction. How far we are taken from terrestrial nuclei with a moderate neutron excess! Finally, somewhat above 10^{14} g cm^{−3} nuclei can no longer exist — they coalesce into a uniform plasma of nearly-pure neutron matter, with a few percent admixture of protons and electrons: we reach the bottom of the neutron star crust.
The crust contains only a small percentage of a neutron star’s mass, but it is crucial for many astrophysical phenomena involving neutron stars. It contains matter at subnuclear density, and therefore there is no excuse for the theoretical physicists, at least in principle: the interactions are known, and many-body theory techniques are available. Neutron star crusts are wonderful cosmic laboratories in which the full power of theoretical physics can be demonstrated and hopefully confronted with neutron star observations.
To construct neutron star crust models we have to employ atomic and plasma physics, as well as the theory of condensed matter, the physics of matter in strong magnetic fields, the theory of nuclear structure, nuclear reactions, the nuclear many-body problem, superfluidity, physical kinetics, hydrodynamics, the physics of liquid crystals, and the theory of elasticity. Theories have to be applied under extreme physical conditions, very far from the domains where they were originally developed and tested. Therefore, caution is a must!
The ground state of the crust (one of the possible “crusts”) is reviewed in Section 3. We also discuss there the uncertainties concerning the densest bottom layers of the crust and we mention possible deviations from the ground state. As we describe in Section 4, a crust formed via accretion is expected to be very different from that formed during the aftermath of a supernova explosion. We show how it can be a site for nuclear reactions. We study its thermal structure during accretion, and briefly review the phenomenon of X-ray bursts. We quantitatively analyze the phenomenon of deep crustal heating.
To construct a neutron star model one needs the equation of state (EoS) of the crust, reviewed in Section 5. We consider separately the ground-state crust, and the accreted crust. For the sake of comparison, we also describe another EoS of matter at subnuclear density — that relevant for the collapsing type II supernova core.
Section 6 is devoted to the stellar-structure aspects of neutron star crusts. We start with the simplest case of a spherically-symmetric static neutron star and derive approximate formulae for crust mass and moment of inertia. Then we study the deformation of the crust in a rotating star. Finally, we consider the effects of magnetic fields on the crust structure. Apart from isotropic stress resulting from pressure, a solid crust can support an elastic strain. Elastic properties are reviewed in Section 7. A separate subsection is devoted to the elastic parameters of the so-called “pasta” layers, which behave like liquid crystals. The inner crust is permeated by a neutron superfluid. Various aspects of crustal superfluidity are reviewed in Section 8. After a brief introduction to superconductivity and its relevance for neutron star crusts, we start with the static properties of neutron superfluidity, considering first a uniform neutron gas, and then discussing the effects of the presence of the nuclear crystal lattice. In the following, we consider superfluid hydrodynamics. We stress those points, which have been raised only recently. We consider also the important problem of the critical velocity above which superfluid flow breaks down. The interplay of superfluid flow and vortices is reviewed. The section ends with a discussion of entrainment effects. Transport phenomena are reviewed in Section 9. We present calculational methods and results for electrical and thermal conductivity, and shear viscosity of neutron star crusts. Differences between accreted and ground-state crusts, and the potential role of impurities are illustrated by examples. Finally, we discuss the very important effects of magnetic fields on transport parameters. In Section 10, we review macroscopic models of the crust, and we describe in particular a two-fluid model, which takes into account the stratification of crust layers, as well as the presence of a neutron superfluid. We show how entrainment effects between the superfluid and the charged components can be included using the variational approach developed by Brandon Carter [70]. Section 11 is devoted to a description of the wealth of neutrino emission processes associated with crusts. We limit ourselves to the basic mechanisms, which, according to existing calculations, are the most important ones at subsequent stages of neutron-star cooling.
The confrontation of theory with observations is presented in Section 12. Neutron stars are born very hot, and we briefly describe in Section 12.1 the present status of the theory of hot dense matter at subnuclear densities; this layer of the proto-neutron star will eventually become the neutron star crust. The crust is crucial for neutron star cooling, as observed by a distant observer. Namely, the crust separates the neutron star core from its surface, where the observed X-ray radiation is produced. The relation between crust physics and observations of cooling neutron stars is studied in Section 12.2. In Section 12.3 we briefly consider possible r-processes associated with the ejection and subsequent decompression of the neutron star crusts. Pulsar glitches are thought to originate in neutron star crust and glitch models are confronted with observations of glitches in pulsar timing data in Section 12.4. The asteroseismology of neutron stars from their gravitational wave radiation is discussed in Section 12.5. Due to its elasticity, the solid crust can support mountains and shear (torsional) oscillations, both associated with gravitational wave emission. The crust-core interface can be crucial for the damping of r-modes, which, if unstable, could be a promising source of gravitational waves from rotating neutron stars. Observations of oscillations in the giant flares from Soft Gamma Repeaters are confronted with models of torsional oscillations of crusts in Section 12.6. As discussed in Section 12.7, the modeling of phenomena associated with low-mass X-ray binaries (LMXB) requires a rather detailed knowledge of the physics of neutron star crusts. New phenomena discovered in the last decade (and some very recently) necessitate realistic physics of accreted neutron star crusts, including deep crustal heating and the correct degree of purity. All aspects of accreted crusts, relevant for soft X-ray transients, X-ray superbursts, and persistent X-ray transients, are discussed in this section.
2 Plasma Parameters
2.1 No magnetic field
In this section we introduce several parameters that will be used throughout this review. We follow the notations of the book by Haensel, Potekhin and Yakovlev [184].
We consider a one-component plasma model of neutron star crusts, assuming a single species of nuclei at a given density ρ. We restrict ourselves to matter composed of atomic nuclei immersed in a nearly ideal and uniform, strongly degenerate electron gas of number density n_{ e }. This model is valid at ρ > 10^{5} g cm^{−3}. A neutron gas is also present at densities greater than neutron drip density ρ_{ND} ≈ 4 × 10^{11} g cm^{−3}.
2.2 Effects of magnetic fields
Critical temperature (in K) below which a condensed phase exists at P = 0, for several magnetic field strength B_{12} = B/10^{12} G and matter composition. From [287].
B_{12} = | 10 | 100 | 1000 |
---|---|---|---|
56_{Fe} | 7 × 10^{5} | 3 × 10^{6} | 2 × 10^{7} |
12_{C} | 3 × 10^{5} | 3 × 10^{6} | 2 × 10^{7} |
^{4} _{He} | 3 × 10^{5} | 2 × 10^{6} | 9 × 10^{7} |
The magnetic field can strongly modify transport properties (Section 9.5) and neutrino emission (Section 11.7). Its effect on the equation of state is significant only if it is strongly quantizing (see Section 6.4).
3 The Ground State Structure of Neutron Star Crusts
According to the cold catalyzed matter hypothesis, the matter inside cold non-accreting neutron stars is assumed to be in complete thermodynamic equilibrium with respect to all interactions at zero temperature and is therefore supposed to be in its ground state with the lowest possible energy. The validity of this assumption is discussed in Section 3.4.
The ground state of a neutron star crust is obtained by minimizing the total energy density ε_{tot} for a given baryon density n_{b} under the assumption of β-equilibrium and electric charge neutrality. For simplicity, the crust is assumed to be formed of a perfect crystal with a single nuclear species at lattice sites (see Jog & Smith [221] and references therein for the possibility of heteronuclear compounds).
3.1 Structure of the outer crust
An exact calculation of the lattice energy for cubic lattices yields similar expressions except for the factor 9/10, which is replaced by 0.89593 and 0.89588 for body-centered and face-centered cubic-lattices, respectively (we exclude simple cubic lattices since they are generally unstable; note that polonium is the only known element on Earth with such a crystal structure under normal conditions [258]). This shows that the equilibrium structure of the crust is expected to be a body-centered cubic lattice, since this gives the smallest lattice energy. Other lattice types, such as hexagonal closed packed for instance, might be realized in neutron star crusts. Nevertheless, the study of Kohanoff and Hansen [242] suggests that such noncubic lattices may occur only at small densities, meaning that r_{ e } ∼ a_{0}, while in the crust r_{ e } ≪ a_{0}, where r_{ e } ≡ (3/4πn_{ e })^{1/3} and a_{0} = ħ^{2}/m_{ e }e^{2} is the Bohr radius. Equation (23) shows that the lattice energy is negative and therefore reduces the total Coulomb energy. The lattice contribution to the total energy density is small but large enough to affect the equilibrium structure of the crust by favoring large nuclei. Corrections due to electron-exchange interactions, electron polarization and quantum zero point motion of the ions are discussed in the book by Haensel, Potekhin and Yakovlev [184].
The main physical input is the energy E{A, Z}, which has been experimentally measured for more than 2000 known nuclei [25]. Nevertheless, this quantity has not been measured yet for the very neutron rich nuclei that could be present in the dense layers of the crust and has therefore to be calculated. The most accurate theoretical microscopic nuclear mass tables, using self-consistent mean field methods, have been calculated by the Brussels group and are available on line [211].
Sequence of nuclei in the ground state of the outer crust of neutron star calculated by Rüster et al. [357] using experimental nuclear data (upper part), and the theoretical mass table of the Skyrme model BSk8 (lower part).
ρ_{max} [g cm^{−3}] | Element | Z | N | R_{cell} [fm] |
---|---|---|---|---|
8.02 × 10^{6} | ^{56}Fe | 26 | 30 | 1404.05 |
2.71 × 10^{8} | ^{62}Ni | 28 | 34 | 449.48 |
1.33 × 10^{9} | ^{64}Ni | 28 | 36 | 266.97 |
1.50 × 10^{9} | ^{66}Ni | 28 | 38 | 259.26 |
3.09 × 10^{9} | ^{86}Kr | 36 | 50 | 222.66 |
1.06 × 10^{10} | ^{84}Se | 34 | 50 | 146.56 |
2.79 × 10^{10} | ^{82}Ge | 32 | 50 | 105.23 |
6.07 × 10^{10} | ^{80}Zn | 30 | 50 | 80.58 |
8.46 × 10^{10} | ^{82}Zn | 30 | 52 | 72.77 |
9.67 × 10^{10} | ^{128}Pd | 46 | 82 | 80.77 |
1.47 × 10^{11} | ^{126}Ru | 44 | 82 | 69.81 |
2.11 × 10^{11} | ^{124}Mo | 42 | 82 | 61.71 |
2.89 × 10^{11} | ^{122}Zr | 40 | 82 | 55.22 |
3.97 × 10^{11} | ^{120}Sr | 38 | 82 | 49.37 |
4.27 × 10^{11} | 118_{Kr} | 36 | 82 | 47.92 |
3.2 Structure of the inner crust
The inner crust of a neutron star is a unique system, which is not accessible in the laboratory due to the presence of this neutron gas. In the following we shall thus refer to the “nuclei” in the inner crust as “clusters” in order to emphasize these peculiarities. The description of the crust beyond neutron drip therefore relies on theoretical models only. Many-body calculations starting from the realistic nucleon-nucleon interaction are out of reach at present due to the presence of spatial inhomogeneities of nuclear matter. Even in the simpler case of homogeneous nuclear matter, these calculations are complicated by the fact that nucleons are strongly interacting via two-body, as well as three-body, forces, which contain about twenty different operators. As a result, the inner crust of a neutron star has been studied with phenomenological models. Most of the calculations carried out in the inner crust rely on purely classical (compressible liquid drop) and semi-classical models (Thomas-Fermi approximation and its extensions). The state-of-the-art calculations performed so far are based on self-consistent mean field methods, which have been very successful in predicting the properties of heavy laboratory nuclei.
3.2.1 Liquid drop models
Equation (49) shows that the equilibrium composition of the cluster is a result of the competition between Coulomb effects, which favor small clusters, and surface effects, which favor large clusters. This also shows that the lattice energy is very important for determining the equilibrium shape of the cluster, especially at the bottom of the crust, where the size of the cluster is of the same order as the lattice spacing. Even at the neutron drip, the lattice energy reduces the total Coulomb energy by about 15%.
The liquid drop model is very instructive for understanding the contribution of different physical effects to the structure of the crust. However this model is purely classical and consequently neglects quantum effects. Besides, the assumption of clusters with a sharp cut surface is questionable, especially in the high density layers where the nuclei are very neutron rich.
3.2.2 Semi-classical models
The idea for obtaining the energy functional is to assume that the matter is locally homogeneous: this is known as the Thomas-Fermi or local density approximation. This approximation is valid when the characteristic length scales of the density variations are much larger than the corresponding interparticle spacings. The Thomas-Fermi approximation can be improved by including density gradients in the energy functional.
As discussed in Section 3.1, the electron density is almost constant so that the local density approximation is very good with the electron energy functional given by Equation (22).
3.2.3 Quantum calculations
Quantum calculations of the structure of the inner crust were pioneered by Negele & Vautherin [303]. These types of calculations have been improved only recently by Baldo and collaborators [29, 30]. Following the Wigner-Seitz approximation [422], the inner crust is decomposed into independent spheres, each of them centered at a nuclear cluster, whose radius is defined by Equation (16), as illustrated in Figure 5. The determination of the equilibrium structure of the crust thus reduces to calculating the composition of one of the spheres. Each sphere can be seen as an exotic “nucleus”. The methods developed in nuclear physics for treating isolated nuclei can then be directly applied.
Equations (57) reduce to ordinary differential equations by expanding a wave function on the basis of the total angular momentum. Apart from the nuclear central and spin-orbit potentials, the protons also feel a Coulomb potential. In the Hartree-Fock approximation, the Coulomb potential is the sum of a direct part eϕ(r), where ϕ(r) is the electrostatic potential, which obeys Poisson’s Equation (54), and an exchange part, which is nonlocal in general. Negele & Vautherin adopted the Slater approximation for the Coulomb exchange, which leads to a local proton Coulomb potential. As a remark, the expression of the Coulomb exchange potential used nowadays was actually suggested by Kohn & Sham [243]. It is smaller by a factor 3/2 compared to that initially proposed by Slater [380] before the formulation of the density functional theory. It is obtained by taking the derivative of Equation (55) with respect to the proton density n_{ p }(r). Since the clusters in the crust are expected to have a very diffuse surface and a thick neutron skin (see Section 3.2.2), the spin-orbit coupling term for the neutrons (which is proportional to the gradient of the neutron density) was neglected.
Equations (57) have to be solved self-consistently. For a given number N of neutrons and Z of protons and some initial guess of the effective masses and potentials, the equations are solved for the wave functions of N neutrons and Z protons, which correspond to the lowest energies \(\epsilon_\alpha ^{(q)}\) (These wave functions are then used to recalculate the effective masses and potentials. The process is iterated until the convergence is achieved.
Sequence of nuclear clusters in the ground state of the inner crust calculated by Negele & Vautherin [303]. Here N is the total number of neutrons in the Wigner-Seitz sphere (i.e., it is a sum of the number of neutrons bound in nuclei and of those forming a neutron gas, per nucleus). Isotopes are labelled with the total number of nucleons in the Wigner-Seitz sphere.
ρ [g cm^{−3}] | Element | z | N | R_{cell} [fm] |
---|---|---|---|---|
4.67 × 10^{11} | ^{180}Zr | 40 | 140 | 53.60 |
6.69 × 10^{11} | ^{200}Zr | 40 | 160 | 49.24 |
1.00 × 10^{12} | ^{250}Zr | 40 | 210 | 46.33 |
1.47 × 10^{12} | ^{320}Zr | 40 | 280 | 44.30 |
2.66 × 10^{12} | ^{500}Zr | 40 | 460 | 42.16 |
6.24 × 10^{12} | ^{950}Sn | 50 | 900 | 39.32 |
9.65 × 10^{12} | ^{100}Sn | 50 | 1050 | 35.70 |
1.49 × 10^{13} | ^{1350}Sn | 50 | 1300 | 33.07 |
3.41 × 10^{13} | ^{1800}Sn | 50 | 1750 | 27.61 |
7.94 × 10^{13} | ^{1500}Zr | 40 | 1460 | 19.61 |
1.32 × 10^{14} | ^{982}Ge | 32 | 950 | 14.38 |
Sequence of nuclear clusters in the ground state of the inner crust calculated by Baldo et al. [31, 32] including pairing correlations (their P2 model). The boundary conditions are the same as those of Negele and Vautherin [303]. Similarly, N is the total number of neutrons in the Wigner-Seitz sphere. The isotopes are labelled with the total number of nucleons in the Wigner-Seitz sphere, as in Table 3.
ρ [g cm^{−3}] | Element | Z | N | R_{cell} [fm] |
---|---|---|---|---|
4.52 × 10^{11} | ^{212}Te | 52 | 160 | 57.19 |
1.53 × 10^{12} | ^{562}Xe | 54 | 508 | 52.79 |
3.62 × 10^{12} | ^{830}Sn | 50 | 780 | 45.09 |
7.06 × 10^{12} | ^{1020}Pd | 46 | 974 | 38.64 |
1.22 × 10^{13} | ^{1529}Ba | 56 | 1473 | 36.85 |
1.94 × 10^{13} | ^{1351}Pd | 46 | 1305 | 30.31 |
2.89 × 10^{13} | ^{1269}Zr | 40 | 1229 | 25.97 |
4.12 × 10^{13} | ^{636}Cr | 20 | 616 | 18.34 |
5.65 × 10^{13} | ^{642}Ca | 20 | 622 | 16.56 |
7.52 × 10^{13} | ^{642}Ca | 20 | 622 | 15.05 |
9.76 × 10^{13} | ^{633}Ca | 20 | 613 | 13.73 |
3.2.4 Going further: nuclear band theory
The unbound neutrons in the inner crust of a neutron star are closely analogous to the “free” electrons in an ordinary (i.e. under terrestrial conditions) metal^{2}. Assuming that the ground state of cold dense matter below saturation density possesses the symmetry of a perfect crystal, which is usually taken for granted, it is therefore natural to apply the band theory of solids to neutron star crusts (see Carter, Chamel & Haensel [78] for the application to the pasta phases and Chamel [90, 91] for the application to the general case of 3D crystal structures).
Equations (63) need to be solved inside only one such cell. Indeed once the wave function in one cell is known, the wave function in any other cell can be deduced from the Floquet-Bloch theorem (61). This theorem also determines the boundary conditions to be imposed at the cell boundary.
For each wave vector k, there exists only a discrete set of single particle energies \(\epsilon_\alpha ^{(q)}({{k}})\), labeled by the principal quantum number α, for which the boundary conditions (61) are fulfilled. The energy spectrum is thus formed of “bands”, each of them being a continuous (but in general not analytic) function of the wave vector k (bands are labelled by increasing values of energy, so that \(\epsilon_\alpha ^{(q)}({{k}}) \le \epsilon_\beta ^{(q)}({{k}})\). The band index α is associated with the rotational symmetry of the nuclear clusters around each lattice site, while the wave vector k accounts for the translational symmetry. Both local and global symmetries are therefore properly taken into account. Let us remark that the band theory includes uniform matter as a limiting case of an “empty” crystal.
The (nonlinear) three-dimensional partial differential Equations (63) are numerically very difficult to solve (see Chamel [90, 91] for a review of some numerical methods that are applicable to neutron star crusts). Since the work of Negele & Vautherin [303], the usual approach has been to apply the Wigner-Seitz approximation [422]. The complicated Wigner-Seitz cell (shown in Figure 13) is replaced by a sphere of equal volume. It is also assumed that the clusters are spherical so that Equations (63) reduce to ordinary differential Equations (57). The Wigner-Seitz approximation has been used to predict the structure of the crust, the pairing properties, the thermal effects, and the low-lying energy-excitation spectrum of the clusters [303, 55, 360, 237, 413, 31, 294].
However, the Wigner-Seitz approximation overestimates the importance of neutron shell effects, as can be clearly seen in Figure 15. The energy spectrum is discrete in the Wigner-Seitz approximation (due to the neglect of the k-dependence of the states), while it is continuous in the full band theory. The spurious shell effects depend on a particular choice of boundary conditions, which are not unique. Indeed as pointed out by Bonche & Vautherin [54], two types of boundary conditions are physically plausible yielding a more-or-less constant neutron density outside the cluster: either the wave function or its radial derivative vanishes at the cell edge, depending on its parity. Less physical boundary conditions have also been applied, like the vanishing of the wave functions. Whichever boundary conditions are adopted, they lead to unphysical spatial fluctuations of the neutron density, as discussed in detail by Chamel et al. [96]. Negele & Vautherin [303] average the neutron density in the vicinity of the cell edge in order to remove these fluctuations, but it is not clear whether this ad hoc procedure did remove all the spurious contributions to the total energy. As shown in Figure 15, shell energy gaps are on the order of ∆∊ ∼ 100 keV, at ρ ≃ 7 × 10^{11} g cm^{−3}. Since these gaps scale approximately like \(\Delta \epsilon\infty {\hbar ^2}/(2{m_n}R_{cell}^2)\) (where m_{ n } is the neutron mass), they increase with density ρ and eventually become comparable to the total energy difference between neighboring configurations. As a consequence, the predicted equilibrium structure of the crust becomes very sensitive to the choice of boundary conditions in the bottom layers [30, 96]. One way of eliminating the boundary condition problem without carrying out full band structure calculations, is to perform semi-classical calculations including only proton shell effects with the Strutinsky method, as discussed by Onsi et al. [311].
In recent calculations [278, 304, 168] the Wigner-Seitz cell has been replaced by a cube with periodic boundary conditions instead of Bloch boundary conditions (61). Although such calculations allow for possible deformations of the nuclear clusters, the lattice periodicity is still not properly taken into account, since such boundary conditions are associated with only one kind of solutions with k = 0. Besides the Wigner-Seitz cell is only cubic for a simple cubic lattice and it is very unlikely that the equilibrium structure of the crust is of this type (the structure of the crust is expected to be a body centered cubic lattice as discussed in Section 3.1). Let us also remember that a simple cubic lattice is unstable. It is, therefore, not clear whether these calculations, which require much more computational time than those carried out in the spherical approximation, are more realistic. This point should be clarified in future work by a detailed comparison with full band theory. Let us also mention that recently Bürvenich et al. [64] have considered axially-deformed spheroidal W-S cells to account for deformations of the nuclear clusters.
Whereas the Wigner-Seitz approximation is reasonable at not too high densities for determining the equilibrium crust structure, full band theory is indispensable for studying transport properties (which involve obviously translational symmetry and, hence, the k-dependence of the states). Carter, Chamel & Haensel [78] using this novel approach have shown that the unbound neutrons move in the crust as if they had an effective mass much larger than the bare mass (see Sections 8.3.6 and 8.3.7). This dynamic effective neutron mass has been calculated by Carter, Chamel & Haensel [78] in the pasta phases of rod and slab-like clusters (discussed in Section 3.3) and by Chamel [90, 91] in the general case of spherical clusters. By taking consistently into account both nuclear clusters, which form a solid lattice, and the neutron liquid, band theory provides a unified scheme for studying the structure and properties of neutron star crusts.
3.3 Pastas
These pasta phases have been studied by various nuclear models, from liquid drop models to semiclassical models, quantum molecular dynamic simulations and Hartree-Fock calculations (for the current status of this issue, see, for instance, [417]). These models differ in numerical values of the densities at which the various phases occur, but they all predict the same sequence of configurations shown in Figure 17 (see also the discussion in Section 5.4 of [326]). Some models [274, 97, 124, 282] predict that the spherical clusters remain energetically favored throughout the whole inner crust. Generalizing the Bohr-Wheeler condition to nonspherical nuclei, Iida et al. [206, 207] showed that the rod-like and slab-like clusters are stable against fission and proton clustering, suggesting that the crust layers containing pasta phases may be larger than that predicted by the equilibrium conditions. It has also been suggested that the pinning of neutron superfluid vortices in neutron star crusts might trigger the formation of rod-like clusters [293]. Nevertheless, the nuclear pastas may be destroyed by thermal fluctuations [419, 420]. Quite remarkably, Watanabe and collaborators [417] performed quantum molecular dynamic simulations and observed the formation of rod-like and slab-like nuclei by cooling down hot uniform nuclear matter without any assumption of the nuclear shape. They also found the appearance of intermediate sponge-like structures, which might be identified with the ordered, bicontinuous, double-diamond geometry observed in block copolymers [284]. Those various phase transitions leading to the pasta structures in neutron star crusts are also relevant at higher densities in neutron star cores, where kaonic or quark pastas could exist [281].
The pasta phases cover a small range of densities near the crust-core interface with ρ ∼ 10^{14} g cm^{−3}. Nevertheless, by filling the densest layers of the crust, they may represent a sizable fraction of the crustal mass [274] and thus may have important astrophysical consequences. For instance, the existence of nuclear pastas in hot dense matter below saturation density affects the neutrino opacity [201, 384], which is an important ingredient for understanding the gravitational core collapse of massive stars in supernova events and the formation of neutron stars (see Section 12.1). The dynamics of neutron superfluid vortices, which is thought to underlie pulsar glitches (see Section 12.4), is likely to be affected by the pasta phase. Besides, the presence of non-spherical clusters in the bottom layers of the crust influences the subsequent cooling of the star, hence the thermal X-ray emission by allowing direct Urca processes [274, 179] (see Section 11) and enhancing the heat capacity [112, 113, 135]. The elastic properties of the nuclear pastas can be calculated using the theory of liquid crystals [325, 419, 420] (see Section 7.2). The pasta phase could thus affect the elastic deformations of neutron stars, oscillations, precession and crustquakes.
3.4 Impurities and defects
There are many reasons why the real crust of neutron stars can be imperfect. In particular, apart from a dominating (A, Z) nuclide at a given density ρ, it can contain an admixture of different nuclei (“impurities”). The initial temperature at birth exceeds 10^{10} K. At such a high T, thermodynamic equilibrium is characterized by a statistical distribution of A and Z. With decreasing T, the A, Z peak becomes narrower [55, 63]. After crystallization at T_{m}, the composition is basically frozen. Therefore, the composition at T < T_{m} reflects the situation at T ∼ T_{m}, which can differ from that in the absolute ground state at T = 0. For example, between the neighboring shells, with nuclides (A_{1}, Z_{1}) and (A_{2}, Z_{2}), respectively, one might expect a transition layer composed of a binary mixture of the two nuclides [108, 109]. Another way of forming impurities is via thermal fluctuations of Z and N_{cell}, which, according to Jones [224, 225], might be quite significant at ρ ≳ 10^{12} g cm^{−3} and T ≳ 10^{9} K.
The real composition of neutron star crusts can also differ from the ground state due to the fallback of material from the envelope ejected during the supernova explosion and due to the accretion of matter. In particular, an accreted crust is a site of X-ray bursts. The ashes of unstable thermonuclear burning at accretion rates \({10^{- 8}}{M_ \odot}{\rm{y}^{- 1}} \gtrsim \dot M \gtrsim {10^{- 9}}{M_ \odot}{\rm{y}^{- 1}}\) could be a mixture of A ≃ 60 − 100 nuclei and could therefore be relatively “impure” (heterogeneous and possibly amorphous) [365, 364]. If the initial ashes are a mixture of many nuclides, further compression under the weight of accreted matter can keep the heterogeneity. If the crust is weakly impure but rather amorphous, its thermal and electrical conductivities in the solid phase would be orders of magnitude lower than in the perfect crystal as discussed in Section 9. This would have dramatic consequences as far as the rate of the thermal relaxation of the crust is concerned (Section 12.7.3).
4 Accreting Neutron Star Crusts
4.1 Accreting neutron stars in low-mass X-ray binaries
A hydrogen atom falling on a neutron star surface from infinity releases ∼ 200 MeV of gravitational binding energy. Therefore, accretion onto a neutron star releases ∼ 200 MeV per accreted nucleon. Most of this energy is radiated in X-rays, so that the total X-ray luminosity of an accreting neutron star can be estimated as LX ∼ (Ṁ/10^{−10} M_{⊙}/y) 10^{36} erg s^{−1}. Space X-ray observations of accreting neutron stars were at the origin of X-ray astronomy [159]. Accreted matter is usually hydrogen rich. It forms the outer envelope of a neutron star, which contains a hydrogen burning shell, with an energy release of about 5 MeV/nucleon in a stable burning. Helium ashes from hydrogen burning accumulate in the helium layer, which ignites under specific density-temperature conditions. For some range of accretion rate, helium burning is unstable, so that its ignition triggers a thermonuclear flash, burning within seconds all the envelope into nuclear ashes composed of nuclides of the iron group and beyond it; the energy release in the flash is less than 5 MeV/nucleon. These flashes are observed as X-ray bursts, with luminosity rising in a second to about 10^{38} ergs^{−1} (≈ Eddington limit for neutron stars, L_{Edd}), and then typically decaying in a few tens of seconds^{3}.
Multiplying the burst luminosity by its duration we get an estimate of the total burst energy ∼ 10^{39}−10^{40} erg. The X-ray bursts are quasiperiodic, with typical recurrence time ∼ hours-days. Since their discovery in 1975 [177], about seventy X-ray bursters have been found. Many bursters are of transient character, and form a group of soft X-ray transients (SXTs), with typical active periods of days — weeks, separated by periods of quiescence of several months — years long. During quiescent periods, there is very little or no accretion, while during much shorter periods of activity there is an abundant accretion, due probably to disc flow instability. Some SXTs, with active periods of years separated by decades of quiescence, are called persistent SXTs. In 2000, a special rare type of X-ray superbursts was discovered. Superbursts last for a few to twelve hours, with recurrence times of several years. The total energy radiated in a superburst is ∼ 10^{42} erg. Superbursts are explained by the unstable burning of carbon in deep layers of the outer crust.
In all cases, ignition of the thermonuclear flash takes place in the neutron star crust, and is sensitive to the crust structure and to the physical conditions within it. This aspect will be discussed in Section 4.4. An accreted crust has a different structure and composition than the ground state one, as discussed in Section 4.2. It has, therefore, a different equation of state than the ground-state crust (see Section 5.2). Moreover, it is a reservoir of nuclear energy, which is released in the process of deep crustal heating, accompanying accretion, reviewed in Section 4.3. Observations of SXTs in quiescence prove the presence of deep crustal heating (Section 12.7.2). Cooling of the neutron star surface in quiescence after long periods of accretion (years — decades) in persistent SXTs also allows one to test physical properties of the accreted crust (Section 12.7.3).
4.2 Nuclear processes and formation of accreted crusts
In what follows we will use a simple model of the accreted crust formation, based on the one-component plasma approximation at T = 0 [185, 187]. The (initial) X-burst ashes are approximated by a one-component plasma with (A_{i}, Z_{i}) nuclei.
Two different compositions of X-ray burst ashes at ≲ 10^{8} g cm^{−3}, A_{i}, Z_{i}, were assumed. In the first case, A_{i} = 56, Z_{i} = 26, which is a “standard composition”. In the second scenario A_{i} = 106, to imitate nuclear ashes obtained by Schatz et al. [364]. The value of Z_{i} = 46 stems then from the condition of beta equilibrium at ρ = 10^{8} g cm^{−3}. As we see in Figure 21, after the pycnonuclear fusion region is reached, both curves converge (as explained in Haensel & Zdunik [187], this results from A_{i} and Z_{i} in two scenarios).
4.3 Deep crustal heating
A neutron star crust that is not in full thermodynamic equilibrium constitutes a reservoir of energy, which can then be released during the star’s evolution. The formation and structure of nonequilibrium neutron star crusts has been considered by many authors [410, 50, 362, 186, 187, 188, 178]. Such a crust can be produced by accretion onto a neutron star in compact LMXB, where the original crust built of a catalyzed matter (see Section 3) is replaced by a crust with a composition strongly deviating from that of nuclear equilibrium. However, building up the accreted crust takes time. The outer crust (Section 3.1), containing ∼ 10^{−5} M_{⊙}, is replaced by the accreted crust in (10^{4}/Ṁ_{−9}) y. To replace the whole crust of mass ∼ 10^{−2} M_{⊙} by accreted matter requires (10^{7}/Ṁ_{−9}) y. After that time has passed, the entire “old crust” is pushed down through the crust-core interface, and is molten into the liquid core. The time (10^{7}/Ṁ_{−9}) y may seem huge. However, LMXBs can live for ∼ 10^{9} y, so that a fully accreted crust on a neutron star is a realistic possibility.
Heating due to nonequilibrium nuclear processes in the outer and inner crust of an accreting neutron star (deep crustal heating) was calculated, using different scenarios and models [186, 187, 188]. The effect of crustal heating on the thermal structure of the interior of an accreting neutron star can be seen in Figures 25 and 26. In what follows, we will describe the most recent calculations of crustal heating by Haensel & Zdunik [188]. In spite of the model’s simplicity (one-component plasma, T = 0 approximation), the heating in the accreted outer crust obtained by Haensel & Zdunik [188] agrees nicely with extensive calculations carried out by Gupta et al. [178]. The latter authors considered a multicomponent plasma, a reaction network of many nuclides, and included the contribution from the nuclear excited states. They found that electron captures in the outer crust proceed mostly via the excited states of the daughter nuclei, which then de-excite, the excitation energy heating the matter; this strongly reduces neutrino losses, accompanying nonequilibrium electron captures. The total deep crustal heating obtained by Haensel & Zdunik [188] is equal to Q_{tot} = 1.5 and 1.9 MeV per accreted nucleon for A_{i} = 106 and A_{i} = 56, respectively.
4.4 Thermal structure of accreted crusts and X-ray bursts
5 Equation of State
In this section we discuss the Equation of State (EoS) of the neutron star crust. Three different cases will be considered: cold catalyzed matter in Section 5.1, accreted crust matter in Section 5.2 assuming the formation scenario described in Section 4, and hot dense matter in supernova cores in Section 5.4.
5.1 Ground state crust
On the contrary, the inner crust nuclei cannot be studied in a laboratory because their properties are influenced by the gas of dripped neutrons, as reviewed in Section 3.2. This means that only theoretical models can be used there and consequently the EoS after neutron drip is much more uncertain than in the outer layers. The neutron gas contributes more and more to the total pressure with increasing density. Therefore, the problem of correct modeling of the EoS of a pure neutron gas at subnuclear densities becomes important. The true EoS of cold catalyzed matter stems from a true nucleon Hamiltonian, expected to describe nucleon interactions at ρ ≲ ρ_{0}, where ρ_{0} is the nuclear saturation density. To make the solution of the many-body problem feasible, the task is reduced to finding an effective nucleon Hamiltonian, which would enable one to calculate reliably both the properties of laboratory nuclei and the EoS of cold catalyzed matter for 10^{11} g cm^{−3} ≲ ρ ≲ ρ_{0}. The task also includes the calculation of the crust-core transition. We will illustrate the general results with two examples of the EoS of the inner crust, calculated in the compressible-liquid-drop model (see Section 3.2.1) using the effective nucleon-nucleon interactions FPS (Friedman-Panharipande-Skyrme [320]) and SLy (Skyrme-Lyon [86, 88, 87]).
In the case of the SLy EoS, the crust-liquid core transition takes place as a very weak first-order phase transition, with a relative density jump on the order of one percent. Notice that, for this model, spherical nuclei persist to the very bottom of the crust [126]. As seen from Figure 30, the crust-core transition is accompanied by a noticeable stiffening of the EoS. For the FPS EoS the situation is different. Namely, the crust-core transition takes place through a sequence of phase transitions with changes of nuclear shapes as discussed in Section 3.3. These phase transitions make the crust-core transition smoother than in the SLy case, with a gradual increase of stiffness (see Figure 35). While the presence of exotic nuclear shapes is expected to have dramatic consequences for the transport, neutrino emission, and elastic properties of neutron star matter, their effect on the EoS is rather small.
5.2 Accreted crust
A model of the EoS of accreted crusts was calculated by Haensel & Zdunik [185]. They used the compressible liquid drop model (see Section 3.2.1) with a “single nucleus” scenario.
The difference between the cold catalyzed and accreted matter EoSs decreases for large density. Both curves are very close to each other for ρ > 10^{13} g cm^{−3}. This is because for such a high density the pressure is mainly produced by the neutron gas and is not sensitive to the detailed composition of the nuclear clusters. In view of this, one can use the EoS of the catalyzed matter for calculating the hydrostatic equilibrium of the high-density (ρ > 10^{13} g cm^{−3}) internal layer of the accreted crust.
5.3 Effect of magnetic fields on the EoS
Typical values of the surface magnetic field of radio pulsars are B ∼ 10^{12} g. For magnetars, surface magnetic fields can be as high as ∼ 10^{15} G. Effects of such fields on the EoS and structure of the crust are briefly reviewed in Section 6.4. A detailed study of the effect of B on neutron star envelopes can be found in Chapter 4 of [184].
5.4 Supernova core at subnuclear density
The outer layers of the supernova core, which after a successful explosion will become the envelope of a proto-neutron star, display a similar range of densities ρ ≲ 10^{14} g cm^{−3} and are governed by the same nuclear Hamiltonian as the neutron star crust. This is why we include it in the present review.
The striking differences between the adiabatic index of supernova matter, γ_{SN}(ρ), and that for cold catalyzed matter in neutron stars, γ_{SN}(ρ), deserves additional explanation. In the core collapse, compression of the matter becomes adiabatic as soon as ρ≳ 10^{11} g cm^{−3}, so that the entropy per nucleon s = const. Simultaneously, due to neutrino trapping, the electron-lepton fraction is frozen, Y_{ l } = const. The condition s = const. ≈ 1k_{B} blocks evaporation of nucleons from the nuclei; the motion of nucleons have to remain ordered. Therefore, the fraction of free nucleons stays small and they do not contribute significantly to the pressure, which is supplied by the electrons, until the density reaches 10^{14} g cm^{−3}.
At ρ ≳ 10^{14} g cm^{−3}, nuclei coalesce forming uniform nuclear matter. Thus, there are two density regimes for γ_{SN}(ρ). For ρ ≳ 10^{14} g cm^{−3}, pressure is supplied by the electrons, while nucleons are confined to the nuclei, so that γ_{SN} ≃ 4/3 ≈ 1.3. Then, for ρ ≳ 10^{14} g cm^{−3} nuclei coalesce into uniform nuclear matter, and the supernova matter stiffens violently, with the adiabatic index jumping by a factor of about two, to γ_{SN} ≈ 2 −3. This stiffening is actually responsible for the bounce of infalling matter. An additional factor stabilizing nuclei at ρ ≲ 10^{14} g cm^{−3} in spite of a high T > 10^{10} K, is a large lepton fraction, Y_{ l } ≈ 0.4, enforcing a relatively large proton fraction, \(Y_p^{{\rm{SN}}} \approx 0.3\), to be compared with \(Y_p^{{\rm{NS}}} \approx 0.05\) for neutron stars.
Finally, for supernova matter we notice the absence of a neutron-drip softening, so well pronounced in γ_{SN}, Figure 31. This is because neutron gas is present in supernova matter also at ρ < 10^{11} g cm^{−3}, and the increase of the free neutron fraction at higher density is prevented by strong neutron binding in the nuclei (large \(Y_p^{{\rm{SN}}}\)), and low s ≈ 1k_{B}.
6 Crust in Global Neutron Star Structure
6.1 Spherical nonrotating neutron stars
Let us limit ourselves to the static case t = const. Fixing r, θ = π/2, and then integrating ds over ϕ from zero to 2π, we find that the proper length of the equator of the star, i.e., its circumference, as measured by a local observer, is equal to 2πr. This is why r is called the circumferential radius. Notice that Equation (90) implies that the infinitesimal proper radial distance (corresponding to the infinitesimal difference of radial coordinates dr) is given by dℓ = e^{λ}dr.
Let us consider the differential Equations (92) and (93), which determine the global structure of a neutron star. They are integrated from the star center, r = 0, with the boundary conditions ρ(0) = ρ_{ c } [P(0) = P(ρ_{c})] and m(0) = 0. It is clear from Equation (92), that pressure is strictly decreasing with increasing r. The integration is continued until P = 0, which corresponds to the surface of the star, with radial coordinate r = R, usually called the star radius.
6.2 Approximate formulae
For astrophysically relevant neutron star masses M > M_{ ⊙ }, the gravitational mass of the crust M_{ cr } = M − m(r_{cc}), where r_{cc} is the radial coordinate of the crust-core interface, constitutes less than 3% of M. Moreover, for realistic EoSs and M > M_{⊙}, the difference ΔR = R − r_{cc} does not exceed 15% of R. Clearly, M_{cr}/M ≪ 1 and an approximation in which the terms \(\mathcal{O}({M_{{\rm{cr}}}}/M)\) are neglected is usually sufficiently precise. Neglecting the terms \({\mathcal O}(\Delta R/R)\) gives a less accurate but still useful approximation. In what follows we will use the above “light and thin crust approximation” to obtain useful approximate expressions for the crustal parameters.
6.3 Crust in rotating neutron stars
The crustal baryon mass (not to be confused with the gravitational mass) M_{b,cr}(Ω) of a neutron star rotating at angular frequency Ω, is larger than the crustal baryon mass M_{b,cr}(0) of the static star (with the same total baryon mass). The baryon mass (also called the rest mass) of a star is equal to the number A of baryons it contains times an assumed baryon mass m_{b}. One may take m_{b} = m_{ n } or m_{b} = m_{u}. We take M_{b,cr} = Am_{u}. For Ω not too close to Ω_{ms}, we have M_{b,cr}(Ω) − M_{b,cr}(0) ∝ Ω^{2}. Due to the radiation of electromagnetic waves and particles, a pulsar spins down, so that \(\dot \Omega < 0\). Consequently, the baryon mass of the pulsar crust decreases in time, \({\dot M_{{\rm{b,cr}}}}\infty \dot \Omega \Omega < 0\). Nucleons pass from the crust to the liquid core, releasing some heat. As shown in Figure 40, the crust is decompressed near the equator and compressed near the pole. These deformations trigger various nuclear reactions involving electrons, neutrons, and nuclei. These reactions tend to drive the deformed crust towards its equilibrium shape and release heat, which influences the cooling of a spinning down pulsar [205]. Additional heating results from crust cracking when local shear strain exceeds the maximal one.
6.4 Effects of magnetic fields on the crust structure
The structure of the outer layers of the crust (neutron star “envelope”) can be affected by the presence of a magnetic field. Effects of magnetic fields on the EoS were briefly mentioned in Section 2.2. Here we consider examples showing effects of B on the crust structure.
7 Elastic Properties
A solid crust can sustain an elastic strain up to a critical level, the breaking strain. Neutron stars are relativistic objects, and therefore a relativistic theory of elastic media in a curved spacetime should be used to describe elastic effects in neutron star structures and dynamics. Such a theory of elasticity has been developed by Carter & Quintana [82], who applied it to rotating neutron stars in [83, 84] (see also Beig [44] and references therein). Recently, Carter and collaborators have extended this theory to include the effects of the magnetic field [73], as well as the presence of the neutron superfluid, which permeates the inner crust [72, 85]. For the time being, for the sake of simplicity, we ignore magnetic fields and free neutrons. However, in Section 7.2 the effect of free neutrons on the elastic moduli of the pasta phases is included, within the compressible liquid drop model. Since relativistic effects are not very large in the crust, we shall restrict ourselves to the Newtonian approximation (see, e.g., [249]).
The thermodynamic equilibrium of an element of neutron-star crust corresponds to equilibrium positions of nuclei, which will be denoted by a set of vectors {r}, which are associated with the lattice sites. Neutron star evolution, driven by spin-down, accretion of matter or some external forces, like tidal forces produced by a close massive body, or internal electromagnetic strains associated with strong magnetic fields, may lead to deformation of this crust element as compared to the equilibrium state.
For simplicity, we will neglect thermal contributions to thermodynamic quantities and restrict ourselves to the T = 0 approximation. Deformation of a crust element with respect to the equilibrium configuration implies a displacement of nuclei into their new positions r′ = r + u, where u = u(r) is the displacement vector. In the continuum limit, valid for macroscopic phenomena, both r and u are treated as continuous fields. Nonzero u is associated with elastic strain (i.e., forces which tend to return the matter element to the equilibrium state of minimum energy density ε_{0}), and with the deformation energy density ε_{def} = ε(u) − ε_{0}^{4}.
The elastic contribution to the stress tensor \(\Pi _{ik}^{{\rm{elast}}} \equiv {\sigma _{ik}}\).
7.1 Isotropic solid (polycrystal)
Monte Carlo calculations of the effective shear modulus of a polycrystalline bcc Coulomb solid were performed by Ogata & Ichimaru [309]. The deformation energy, resulting from the application of a specific strain u_{ ik }, was evaluated through Monte Carlo sampling.
Let us remember that the formulae given above hold for the outer crust, where the size of the nuclei is very small compared to the lattice spacing and P ≃ P_{ e }. For the inner crust these formulae are only approximate.
7.2 Nuclear pasta
Some theories of dense matter predict the existence of “nuclear pasta” — rods, plates, tubes, bubbles — in the bottom layer of the crust with ρ ≳ 10^{14} g cm^{−3} (see Section 3.3). In what follows we will concentrate on rods (spaghetti) and plates (lasagna). They are expected to fill most of the bottom crust layers. The matter phases containing rods and plates have properties intermediate between solids and liquids. The displacement of an element of matter parallel to the plane containing rods or plates is not opposed by restoring forces: this lack of a shear strain is typical for a liquid. On the contrary, an elastic strain opposes any bending of planes or rods: this is a property of a solid. Being intermediate between solids and liquids, these kinds of matter are usually called mesomorphic phases, or liquid crystals (see, e.g., [249]).
In the case of the rod phase, also called the columnar phase [115], the number of elastic moduli is larger. They describe the increase in energy density due to compression, dilatation, transverse shearing, and bending of the rod lattice. Elastic moduli were calculated within the liquid drop model by Pethick & Potekhin [325] and by Watanabe, Iida & Sato [419, 420].
At the microscopic scale (fermis), the elastic properties of the nuclear pastas are very different from those of a body-centered-cubic crystal of spherical nuclei. Nevertheless, the effects of pasta phases on the elastic properties of neutron star crusts may not be so dramatic at large scales (let’s say meters). Indeed these nuclear pastas are necessarily of finite extent since one and two-dimensional long-range crystalline orders cannot exist in infinite systems (see, for instance, [157] and references therein). How the nuclear pastas arrange themselves remains to be studied, but it is likely that the resulting configurations look more-or-less isotropic at macroscopic scales.
8 Superfluidity and Superconductivity
Except for a brief period after their birth, neutron stars are expected to contain various super-fluid and superconducting phases [363, 116, 29, 367]. In this section, after a brief discussion of superconductivity and its possible occurrence in neutron star crusts, we will review our current theoretical understanding of the static and dynamic properties of neutron superfluid in the inner crust of neutron stars. For a general introduction and a recent overview on superfluidity and superconductivity, see, for instance, the book by Annett [22].
8.1 Superconductivity in neutron star crusts
We can, thus, firmly conclude that electrons in neutron star crusts (and, a fortiori, in neutron star cores) are not superconducting. Nevertheless, superconductivity in the crust is not completely ruled out. Indeed, at the crust-core interface some protons could be free in the “pasta” mantle (Section 3.3), and could be superconducting due to pairing via strong nuclear interactions with a critical temperature far higher than that of electron superconductivity. Microscopic calculations in uniform nuclear matter predict transition temperatures on the order of T_{ cp } ∼ 10^{9}−10^{10} K, which are much larger than typical temperatures in mature neutron stars. Some properties of superconductors are discussed in Sections 8.3.3 and 8.3.4.
8.2 Static properties of neutron superfluidity
Soon after its formulation, the Bardeen-Cooper-Schrieffer (BCS) theory of electron superconductivity [36] was successfully applied to nuclei by Bohr, Mottelson and Pines [51] and Belyaev [46]. In a paper devoted to the moment of inertia of nuclei, Migdal [291] speculated about the possibility that superfluidity could occur in the “neutron core” of stars (an idea which was raised by Gamow and Landau in 1937 as a possible source of stellar energy; see, for instance, [184]). The superfluidity inside neutron stars was first studied by Ginzburg and Kirzhnits in 1964 [161, 162]. Soon after, Wolf [424] showed that the free neutrons in the crust are very likely to be superfluid. It is quite remarkable that the possibility of superfluidity inside neutron stars was raised before the discovery of pulsars by Jocelyn Bell and Anthony Hewish in 1967. Later, this prediction seemed to be confirmed by the observation of the long relaxation time, on the order of months, following the first glitch in the Vela pulsar [41]. The neutron superfluid in the crust is believed to play a key role in the glitch mechanism itself. Pulsar glitches are still considered to be the strongest observational evidence of superfluidity in neutron stars (see Section 12.4).
At the heart of BCS theory is the existence of an attractive interaction needed for pair formation. In conventional superconductors, this pairing interaction is indirect and weak. In the nuclear case the occurrence of superfluidity is a much less subtle phenomenon since the bare strong interaction between nucleons is naturally attractive at not too small distances in many JLS channels (J-total angular momentum, L-orbital angular momentum, S-spin of nucleon pair). Apart from a proton superconductor similar to conventional electron superconductors, two different kinds of neutron superfluids are expected to be found in the interior of a neutron star (for a review, see, for instance, [363, 271, 116, 29, 367]). In the crust and in the outer core, the neutrons are expected to form an isotropic superfluid like helium-4, while in denser regions they are expected to form a more exotic kind of (anisotropic) superfluid with each member of a pair having parallel spins, as in superfluid helium-3. Neutron-proton pairs could also exist in principle; however, their formation is not strongly favored in the asymmetric nuclear matter of neutron stars.
8.2.1 Neutron pairing gap in uniform neutron matter at zero temperature
The gap Equations (128) and (129) solved for the bare interaction with the free single particle energy spectrum, Equation (131), represent the simplest possible approximation to the pairing problem. A more consistent approach from the point of view of the many-body theory, is to calculate the single particle energies in the Hartree-Fock approximation (after regularizing the hard core of the bare nucleon-nucleon interaction). The next step is to “dress” the pairing interaction by medium polarization effects. Calculations have been carried out with phenomenological nucleon-nucleon interactions such as the Gogny force [117, 140], that are constructed so as to reproduce some properties of finite nuclei and nuclear matter. Another approach is to derive this effective interaction from a bare nucleon-nucleon potential (two-body and/or three-body forces) using many-body techniques. Still the gap equations of form (128) neglect important many-body aspects.
Parameters for the analytic formula Equation (134) of a few representative ^{1}S_{0} pairing gaps in pure neutron matter: BCS-BCS pairing gap shown in Figure 45, Brueckner — pairing gap of Cao et al. [69] based on diagrammatic calculations (shown in Figure 46) and RG — pairing gap of Schwenk et al. [366] based on the Renormalization Group approach (shown in Figure 46). Δ_{0} is given in MeV. k_{1}, k_{2}, k_{3} and kmax are given in fm^{−1}.
model | Δ_{0} | k _{1} | k _{2} | k _{3} | k _{max} |
---|---|---|---|---|---|
BCS | 910.603 | 1.38297 | 1.57068 | 0.905237 | 1.57 |
Brueckner | 11.4222 | 0.556092 | 1.38236 | 0.327517 | 1.37 |
RG | 16.5709 | 1.13084 | 1.47001 | 0.582515 | 1.5 |
8.2.2 Critical temperature for neutron superfluidity
Zero temperature pairing gaps on the order of 1 MeV are therefore associated with critical temperatures of the order 10^{10} K, considerably larger than typical temperatures inside neutron stars except for the very early stage of their formation. The existence of a neutron superfluid in the inner crust of a neutron star is therefore well established theoretically. Nevertheless the density dependence of the critical temperature predicted by different microscopic calculations differ considerably due to different approximations of the many-body problem. An interesting issue concerns the cooling of neutron stars and the crystallization of the crust: do the neutrons condense into a superfluid phase before the formation of the crust or after?
For the BCS and Brueckner calculations of the pairing gap, in the density range of ∼ 10^{12} − 10^{14} g cm^{−3}, the neutrons may become superfluid before the matter crystallizes into a solid crust. As discussed in Section 8.3.2, as a result of the rotation of the star, the neutron superfluid would be threaded by an array of quantized vortices. These vortices might affect the crystallization of the crust by favoring nuclear clusters along the vortex lines, as suggested by Mochizuki et al. [293]. On the contrary, the calculations of Schwenk et al. [366] indicate that, at any density, the solid crust would form before the neutrons become superfluid. Recently, it has also been shown, by taking into account the effects of the inhomogeneities on the neutron superfluid, that in the shallow layers of the inner crust, the neutrons might remain in the normal phase even long after the formation of the crust, when the temperature has dropped below 10^{9} K [294].
8.2.3 Pairing gap in neutron star crusts
In the denser layers of the crust, the coherence length is smaller than the mean inter-neutron spacing, suggesting that the neutron superfluid is a Bose-Einstein condensate of strongly-bound neutron pairs, while in the shallower layers of the inner crust the neutron superfluid is in a BCS regime of overlapping loosely-bound pairs. Quite remarkably, for screened pairing gaps like those of Schwenk et al. [366], the coherence length is larger than the mean inter-neutron spacing in the entire inner crust, so that in this case, at any depth, neutron superfluid is in the BCS regime.
8.3 Superfluid hydrodynamics
8.3.1 Superflow and critical velocity
The roton local minimum has also been interpreted as a characteristic feature of density fluctuations marking the onset of crystallization [200, 334, 307]. According to Nozières [307], rotons are “ghosts of Bragg spots”. Landau’s theory has been very successful in explaining the observed properties of superfluid helium-4 from the postulated energy spectrum of quasiparticles.
This expression can be derived more rigorously from the microscopic BCS theory [35]. It shows that a system of fermions is superfluid (i.e. the critical velocity is not zero) whenever the interactions are attractive, so that the formation of pairs becomes possible. It is also interesting to note that the BCS spectrum can be interpreted in terms of rotons. Indeed, expanding Equation (149) around the minimum leads, to lowest order, to an expression similar to Equation (146). In this case, p_{0} is obtained by solving ϵ(p) = μ. The other parameters are given by Δ_{r} = Δ(p_{0}) and \({\tilde \rho _{\rm{G}}}\) where v_{0} = dϵ/dp is the group velocity evaluated at p_{0}.
8.3.2 Rotating superfluid and vortices
8.3.3 Type II superconductors and magnetic flux tubes
8.3.4 Superfluid vortices and magnetic flux tubes in neutron stars
8.3.5 Dynamics of superfluid vortices
- A viscous drag force (not to be confused with entrainment, which is a nondissipative effect; see Section 8.3.6) opposes relative motion between a vortex line and the crust, inducing dissipation. At sufficiently small relative velocities, the force per unit length of the vortex line can be written aswhere \(\mathcal{J} = 0\) is a positive resistivity coefficient, which is determined by the interactions of the neutron vortex line with the nuclear lattice and the electron gas. The pinning of the vortex line to the crust is the limit of very strong drag entailing that υ_{ υ } = υ_{ c }.$${{\mathcal F}_{\rm{d}}} = - {\mathcal R}({v_\upsilon} - {v_{\rm{c}}})\,,$$(169)
- Relative motion of a vortex line with respect to bulk superfluid (caused by drag or pinning) gives rise to a Magnus or lift force (analog to the Lorentz force), given bywhere ρ_{f} is the mass density of the free superfluid neutrons and κ is a vector oriented along the superfluid angular velocity and whose norm is given by h/2m_{ n } (see Carter & Chamel [75] for the generalization to multi-fluid systems).$${{\mathcal F}_{\rm{m}}} = {\rho _{\rm{f}}}\kappa \times ({v_{\rm{f}}} - {v_\upsilon})\,,$$(170)
- A tension force resists the bending of the vortex line and is given bywhere u is the two-dimensional displacement vector of the vortex line directed along the z-axis. \(\mathcal{R}\) is a rigidity coefficient of order$${{\mathcal F}_{\rm{t}}} = - {\rho _{\rm{f}}}\kappa {{\mathcal C}_{\rm{t}}}{{{\partial ^2}u} \over {\partial {z^2}}}\,,$$(171)where κ = h/2m_{ n }, d_{ υ } is the intervortex spacing and r_{ υ } the size of the vortex core [383].$${{\mathcal C}_{\rm{t}}}\sim{\kappa \over {4\pi}}\ln {{{d_\upsilon}} \over {{r_\upsilon}}}\,,$$(172)
All forces considered above are given per unit length of the vortex line. Let us remark that even in the fastest millisecond pulsars, the intervortex spacing (assuming a regular array) of order d_{ υ } ∼ 10^{−3} − 10^{−4} cm is much larger than the size of the vortex core r_{ υ } ∼ 10 − 100 fermis. Consequently the vortex-vortex interactions can be neglected.
Different dissipative mechanisms giving rise to a mutual friction force have been invoked: scattering of electrons/lattice vibrations (phonons)/impurities/lattice defects by thermally excited neutrons in vortex cores [141, 189, 222], electron scattering off the electric field around a vortex line [48], and coupling between phonons and vortex line oscillations (Kelvin modes) [139, 223]. In the weak coupling limit \({n_\upsilon} \sim 1/\pi d_\upsilon ^2\), the vortices co-rotate with the bulk superfluid (Helmholtz theorem), while in the opposite limit \(\mathcal{B} \to 0\), they are “pinned” to the crust. In between these two limits, in a frame co-rotating with the crust, the vortices move radially outward at angle atan(ℛ) with respect to the azimuthal direction. The radial component of the vortex velocity reaches a maximum at ℛ = 1.
Vortex pinning plays a central role in theories of pulsar glitches. The strength of the interaction between a small segment of the vortex line and a nucleus remains a controversial issue [5, 138, 333, 134, 120, 121, 122, 27]. The actual “pinning” of the vortex line (i.e., υ_{ υ } = υ_{ c }) depends not only on the vortex-nucleus interaction, but also on the structure of the crust, on the rigidity of lines and on the vortex dynamics. For instance, assuming that the crust is a polycrystal, a rigid vortex line would not pin to the crust simply because the line cannot bend in order to pass through the nuclei, independent of the strength of the vortex-nucleus interaction! Recent observations of long-period precession in PSR 1828−11 [387], PSR B1642−03 [371] and RX J0720.4−3125 [180] suggest that, at least in those neutron stars, the neutron vortices cannot be pinned to the crust and must be very weakly dragged [372, 266].
8.3.6 Superfluid hydrodynamics and entrainment
One of the striking consequences of superfluidity is the allowance for several distinct dynamic components. In 1938, Tisza [404] introduced a two-fluid model in order to explain the properties of the newly discovered superfluid phase of liquid helium-4, which behaves either like a fluid with no viscosity in some experiments or like a classical fluid in other experiments. Guided by the Fritz London’s idea that superfluidity is intimately related to Bose-Einstein condensation (which is now widely accepted), Tisza proposed that liquid helium is a mixture of two components, a superfluid component, which has no viscosity, and a normal component, which is viscous and conducts heat, thus, carrying all the entropy of the liquid. These two fluids are allowed to flow with different velocities. This model was subsequently developed by Landau[247, 246] and justified on a microscopic basis by several authors, especially Feynman [144]. Quite surprisingly, Landau never mentioned Bose-Einstein condensation in his work on superfluidity. According to Pitaevskii (as recently cited by Balibar [33]), Landau might have reasoned that superfluidity and superconductivity were similar phenomena (which is indeed true), incorrectly concluding that they could not depend on the Bose or Fermi statistics (see also the discussion by Feynman in Section 11.2 of his book [144]).
The confusion between velocity and momentum is very misleading and makes generalizations of the two-fluid model to multi-fluid systems (like the interior of neutron stars) unnecessarily difficult. Following the approach of Carter (see Section 10), the two-fluid model can be reformulated in terms of the real velocity υ of the helium atoms instead of the superfluid “velocity” υ_{ S }. The normal fluid with velocity υ_{ N } is then associated with the flow of entropy and the corresponding number density is given by the entropy density. At low temperatures, heat dissipation occurs via the emission of phonons and rotons. As discussed in Section 8.3.1, these quasiparticle excitations represent collective motions of atoms with no net mass transport (see, in particular, Figures 50 and 51). Therefore, the normal fluid does not carry any mass, i.e., its associated mass is equal to zero.
Entrainment effects, whereby momentum and velocity are not aligned, exist in any fluid mixtures owing to the microscopic interactions between the particles. But they are usually not observed in ordinary fluids due to the viscosity, which tends to equalize velocities. Even in superfluids like liquid Helium II, entrainment effects may be hindered at finite temperature^{11} by dissipative processes. For instance, when a superfluid is put into a rotating container, the presence of quantized vortices induces a mutual friction force between the normal and superfluid components (as discussed in Section 8.3.5). As a consequence, in the stationary limit the velocities of the two fluids become equal. Substituting υ = υ_{ N } in Equation (180) implies that p = mυ, as in the absence of entrainment.
8.3.7 Entrainment effects in neutron stars
A few years after the seminal work of Andreev & Bashkin [20] on superfluid ^{3}He −^{4}He mixtures, it was realized that entrainment effects could play an important role in the dynamic evolution of neutron stars (see, for instance, [363] and references therein). For instance, these effects are very important for studying the oscillations of neutron star cores, composed of superfluid neutrons and superconducting protons [14]. Mutual entrainment not only affects the frequencies of the modes but, more surprisingly, (remembering that entrainment is a nondissipative effect) also affects their damping. Indeed, entrainment effects induce a flow of protons around each neutron superfluid vortex line. The outcome is that each vortex line carries a huge magnetic field ∼ 10^{14} G [8]. The electron scattering off these magnetic fields leads to a mutual friction force between the neutron superfluid and the charged particles (see [18] and references therein). This mechanism, which is believed to be the main source of dissipation in the core of a neutron star, could also be at work in the bottom layers of the crust, where some protons might be unbound and superconducting (as discussed in Section 3.3).
9 Conductivity and Viscosity
9.1 Introduction
Except for the very outer envelope, the main carriers in the transport processes in the outer crust are electrons, and they scatter mainly off ions (exceptions will be mentioned at the end of the corresponding sections). Theoretical techniques for the calculation of the transport coefficients in neutron star crusts are to a large extent borrowed from solid state physics, the classical reference still remaining the book of Ziman [435]. However, one has to remember that the density/temperature conditions within neutron star crusts are tremendously different from those in terrestrial solids, so that special care concerning the approximations used should be taken.
9.2 Boltzmann equation for electrons and its solutions
The electron distribution function is f(p, r, t, s), where p and r are electron momentum and position vectors, respectively, t is time, and s is the electron spin projection on the spin quantization axis. The distribution function f(p, r, t, s) satisfies the Boltzmann equation (BE) for electrons. At first glance, the validity of the BE (originally derived for a gas of particles) for a super dense plasma of electrons may seem paradoxical. However, electrons are strongly degenerate, so that only electron states in a thin shell around the chemical potential μ_{ e } with energies |ϵ − μ_{ e }| ≲ k_{B}T are involved in the transport phenomena. In other words, the gas of “electron excitations” is dilute. Also, the infinite range of Coulomb interactions in vacuum is no longer a problem in dense electron-nuclear plasma because of screening. Moreover, as the kinetic Fermi energy of electrons is much larger than the Coulomb energy per electron, the Coulomb energy can be treated as a small perturbation and the electrons can be considered as a quasi-ideal Fermi gas.
The additivity of partial collision integrals is valid when the scatterers are uncorrelated. This assumption may seem surprising for a crystal. The electron scatters off a lattice by exciting it, i.e., transferring energy and momentum to the lattice. This process of electron-lattice interactions corresponds to the creation and absorption of phonons, which are the elementary excitations of the crystal lattice. In this way the electron-lattice interaction is equivalent to the scattering of electrons by phonons. At temperatures well below the Debye temperature, \(m_ \star ^{\rm{f}} \sim 10 - 15{m_n}\), the gas of phonons is dilute, and e-N scattering, represented by I_{eN}[f], is actually the electron scattering by single phonons. These phonons form a Bose gas, and their number density and mean energy depend on T.
In the absence of external forces, the solution of Equation (186) is the Fermi-Dirac distribution function, f^{(0)}, corresponding to full thermodynamic equilibrium. The collision integrals then vanish, I_{ ej }[f^{(0)}] = 0, with j = N, e, imp. We shall now show how to calculate the conductivities κ and σ. Let us consider small stationary perturbations characterized by gradients of temperature ∇T, of the electron chemical potential ∇μ_{ e }, and let us apply a weak constant electric field E. The plasma will become slightly nonuniform, with weak charge and heat currents flowing through it. We assume that the length scale of this nonuniformity is much larger than the electron mean free path. Therefore, any plasma element will be close to a local thermodynamic equilibrium. However, gradients of T and μ_{ e }, as well as E, will induce a deviation of f from f^{(0)} and will produce heat and charge currents.
This simple relaxation time approximation breaks down at T ≲ T_{pi}, when the quantum effects in the phonon gas become pronounced so that the typical energies transferred become ∼ k_{B}T (and the number of phonons becomes exponentially small). The dominance of electron-phonon scattering breaks down at very low T. Simultaneously, I_{ ee } has a characteristic low-T ≪ T_{Fe} behavior I_{ ee } ∝ (T/T_{Fe})^{2}. All this implies the dominance of the e-impurity scattering in the low-T limit, I_{imp} ≫I_{ ee },I_{eN}.
A second important approximation (after the relaxation time one) is expressed as the Matthiessen rule. In reality, the electrons scatter not only off nuclei (eN), but also by themselves (ee), and off randomly distributed impurities (imp), if there are any. The Matthiessen rule (valid under strong degeneracy of electrons) states that the total effective scattering frequency is the sum of frequencies on each of the scatterers.
Recently, the calculation of \(\nu _{e{\rm{N}}}^\sigma\) has been revised, taking into account the Landau damping of transverse plasmons [378]. This effect strongly reduces \(\nu _{ee}^\kappa\) for ultrarelativistic electrons at T < T_{pe}.
In the presence of a magnetic field B, transport properties become anisotropic, as briefly described in Section 9.5.
9.3 Thermal and electrical conductivities
It can be noted that for T < T_{m} “neutron excitations” scatter by the lattice phonons. Complete calculation of κ_{ n }, taking due account of the effect of the crystal lattice on neutron scattering and neutron superfluidity remains to be done.
Recent calculations of κ_{ ee }, taking into account the Landau damping of transverse plasmons, give a much larger contribution from ee scattering than the previous ones, using the static screening, on which Figures 54 and 55 are based. As shown by Shternin and Yakovlev [378], the Landau damping of transverse plasmons strongly reduces \(\kappa _n^{{\rm{conv}}}\) in the inner crust at T ≲ 10^{7} K.
The contribution of ions to κ was recently calculated by Chugunov and Haensel [100], who also quote older papers on this subject. As a rule, κ_{N} can be neglected compared to κ_{ e }. A notable exception, relevant for magnetized neutron stars, is discussed in Section 9.5.
9.4 Viscosity
In this section we consider the viscosity of the crust, which can be in a liquid or a solid phase. For strongly nonideal (Γ ≫ 1) and solid plasma the transport is mediated mainly by electrons. We will limit ourselves to this case only. For solid crust, we will assume a poly crystal structure, so that on a macroscopic scale the crust will behave as an isotropic medium.
First, let us consider volume preserving flows, characterized by ∇ · U = 0. A schematic view of such a flow in the solid crust, characteristic of torsional oscillations of the crust, is shown in Figure 58. The dissipation resulting in the entropy production is determined by the shear viscosity η. In the outer crust, η is a sum of the electron and nuclei contributions, η = η_{ e } + η_{N}, but for ρ > 10^{5} g cm^{−3} η_{ e } ≫ η_{N} and η ≈ η_{ e }.^{13}. In the inner crust, an additional contribution from the normal component of the gas of dripped neutrons should be added.
The electrons are scattered on nuclei, on impurity nuclei, and on themselves, so that the effective frequency of their scattering is given by the sum \(\nu _e^\kappa\). However, as long as the temperature is not too low, the approximation ν_{ e } ≈ ν_{eN} can be used.
We are not aware of any calculations of the bulk viscosity of the crust, ζ. We just mention that it is generally assumed that the bulk viscosity of the crust is much smaller than the shear one, ζ ≪ η.
9.5 Transport in the presence of strong magnetic fields
We consider a surface magnetic field to be strong if B ≫ 10^{9} G. Such magnetic fields affect the accretion of plasma onto the neutron star and modify the properties of atoms in the atmosphere. On the contrary, a magnetic field B ≲ 10^{9} G, such as associated with millisecond pulsars or with most of the X-ray bursters, is considered to be weak. Typical pulsars are magnetized neutron stars, with the value of B near the magnetic pole ∼ 10^{12} G. Much stronger magnetic fields are associated with magnetars, ∼ 10^{14}⊥10^{15} G; such magnetic fields with B ≳ 10^{14} G are often called “super-strong”. These magnetic fields can strongly affect transport processes within neutron star envelopes. Electron transport processes in magnetized neutron star envelopes and crusts are reviewed in [338, 412]. In the present section we limit ourselves to a very brief overview.
9.5.1 Nonquantizing magnetic fields
9.5.2 Weakly-quantizing magnetic fields
9.5.3 Strongly-quantizing magnetic fields
Not only is k_{B}T < ħω_{ce}, but also most of the electrons are populating the ground Landau level. Both the values of σ_{ ij } and κ_{ ij } and their density dependence are dramatically different from those of the nonquantizing (classical) case. As shown by Potekhin [338], the formulae for σ_{ ij } and κ_{ ij } are still given by Equations (219). Analytical fitting formulae for \(\xi _{ij}^{\kappa, \sigma}\) and \(\tau _\parallel ^{\kappa, \sigma}\) are given in [338]. As seen in Figure 59, at T = 10^{6} K a field of 10^{12} G is strongly quantizing for ρ < 10^{4.2} g cm^{−3}
9.5.4 Possible dominance of ion conduction
Thermal conduction by ions is much smaller than that by electrons along B. However, the electron conduction across B is strongly suppressed. In outer neutron star crust, heat flow across B can be dominated by ion/phonon conduction [100]. This is important for the heat conduction across B in cooling magnetized neutron stars. Correct inclusion of the ion heat conductivity then leads to a significant reduction of the thermal anisotropy in the envelopes of magnetized neutron stars [100].
10 Macroscopic Model of Neutron Star Crusts
The understanding of many observed phenomena occurring in neutron stars (and briefly reviewed in Section 12, for instance, pulsar glitches or torsional oscillations in Soft Gamma Repeaters) requires modeling the dynamic evolution of the crust. So far theoretical efforts have been mainly devoted to modeling the dynamic evolution of the liquid core by considering a mixture of superfluid neutrons and superconducting protons (see, for instance, the recent review by Andersson & Comer [15]).
Macroscopic models of neutron star crusts, taking into account the presence of the neutron superfluid at ρ > ρ_{ND} (see Section 8), have been developed by Carter and collaborators. They have shown how to extend the two-fluid picture of neutron star cores [93] to the inner crust layers in the Newtonian framework [79, 94]. They have also discussed how to calculate the microscopic coefficients of this model [78, 77]. More elaborate models treating the crust as a neutron superfluid in an elastic medium and taking into account the effects of a frozen-in magnetic field have been very recently developed both in general relativity [73, 85] and in the Newtonian limit [73, 72]. All these models are based on an action principle that will be briefly reviewed in Section 10.1. We will consider a simple nonrelativistic two-fluid model of neutron star crusts in Section 10.2 using the fully-4D covariant formulation of Carter & Chamel [74, 75, 76]. Entrainment effects and superfluidity will be discussed in Sections 10.3 and 10.4, respectively.
10.1 Variational formulation of multi-fluid hydrodynamics
In the convective variational approach of hydrodynamics developed by Carter [70, 80], and recently reviewed by Gourgoulhon [174] and Andersson & Comer [15], the dynamic equations are obtained from an action principle by considering variations of the fluid particle trajectories. First developed in the context of general relativity, this formalism has been adapted to the Newtonian framework in the usual 3+1 spacetime decomposition by Prix [342, 343]. As shown by Carter & Chamel in a series of papers [74, 75, 76], the Newtonian hydrodynamic equations can be written in a very concise and elegant form in a fully-4D covariant framework. Apart from facilitating the comparison between relativistic and nonrelativistic fluids, this approach sheds a new light on Newtonian hydrodynamics following the steps of Elie Cartan, who demonstrated in the 1920’s that the effects of gravitation in Newtonian theory can be expressed in geometric terms as in general relativity.
The variational formalism of Carter provides a very general framework for deriving the dynamic equations of any fluid mixture and for obtaining conservation laws, using exterior calculus. In particular, this formalism is very well suited to describing superfluid systems, like laboratory superfluids or neutron star interiors, by making a clear distinction between particle velocities and the corresponding momenta (see the discussion in Section 8.3.6).
The dynamics of the system (either in relativity or in the Newtonian limit) is thus governed by a Lagrangian density Λ, which depends on the particle 4-currents \(\tau _ \bot ^{\kappa, \sigma}\), where n_{X} and \(n_{\rm{X}}^\mu = {n_{\rm{X}}}u_{\rm{X}}^\mu\) are the particle number density and the 4-velocity of the constituent X, respectively. We will use Greek letters for spacetime indices with the Einstein summation convention for repeated indices. The index X runs over the different constituents in the system. Note that repeated chemical indices X will not mean summation unless explicitly specified.
10.2 Two-fluid model of neutron star crust
In this section, we will review the simple model for neutron star crust developed by Carter, Chamel & Haensel [79] (see also Chamel & Carter [94]). The crust is treated as a two-fluid mixture containing a superfluid of free neutrons (index f) and a fluid of nucleons confined inside nuclear clusters (index c), in a uniform background of degenerate relativistic electrons. This model includes the effects of stratification (variation of the crust structure and composition with depth; see Section 3) and allows for entrainment effects (Section 8.3.7) that have been shown to be very strong [90]. However, this model does not take into account either the elasticity of the crust or magnetic fields. For simplicity, we will restrict ourselves to the case of zero temperature and we will not consider dissipative processes (see, for instance, Carter & Chamel [76], who have discussed this issue in detail and have proposed a three-fluid model for hot neutron star crust).
The basic variables of the two-fluid model considered here are the particle 4-current vectors \(n_{\rm{X}}^i = {n_{\rm{X}}}\upsilon _{\rm{X}}^i\), \(n_{\rm{c}}^\mu\) and the number density n_{N} of nuclear clusters, which accounts for stratification effects. For clusters with mass number A, we have the relation n_{c} = An_{N}. In the following we will neglect the small neutron-proton mass difference and we will write simply m for the nucleon mass (which can be taken as the atomic mass unit, for example). The total mass density is thus given by ρ = m(n_{c} + n_{f}). The Lagrangian density Λ, which contains the microphysics of the system, has been derived by Carter, Chamel & Haensel [79, 78].
10.3 Entrainment and effective masses
10.4 Neutron superfluidity
Let us remark that the definition of neutrons that have to be counted as “free” is not unique and there is some arbitrariness in the above model. Nonetheless, it can be shown that the 4-momentum co-vector \(u_\upsilon ^\mu\) of the neutron superfluid is invariant under such “chemical” readjustments and the above superfluidity conditions are well defined [79]. Note also that these conditions are valid for both relativistic and nonrelativistic superfluids.
As discussed by Chamel & Carter [94], there are two cases, which are consistent with the nondissipative models considered here. The first possibility is that the neutron vortices are free and are co-moving with the superfluid, i.e. \(\pi _\mu ^{\rm{f}}\). On a sufficiently short time scale, it may be further assumed that the free neutron current is conserved, which implies from Equations (221) and (253) that the force \(u_\upsilon ^\mu = u_{\rm{f}}^\mu\) vanishes. Since no external force is supposed to be exerted on the system, the force acting on the confined nucleons also vanishes. However, on longer time scales, as discussed in Section 10.2, it would be more appropriate to replace Equation (240) by the equilibrium condition Equation (241). In this case, there will still be a force acting on the superfluid (hence, also a force, acting on the confined nucleons) owing to the conversion of free neutrons into confined protons and vice versa. The other possibility is that the vortices are pinned to the crust, so that \(f_\mu ^{\rm{f}}\). As shown by Chamel & Carter [94], the pinning condition \(u_\upsilon ^\mu = u_{\rm{c}}^\mu\) is equivalent to imposing that the individual vortices be subject to the corresponding Magnus force.
11 Neutrino Emission
11.1 Neutrino emission processes — an overview
There is a great wealth of neutrino emission processes in hot neutron star crusts. These processes are associated with weak interaction involving electrons, positrons, nuclei, and free nucleons. As soon as the crust becomes neutrino-transparent, which occurs in about a minute after neutron star birth, these neutrinos freely leave the crust, taking away thermal energy, and contributing in this way to crust cooling. Let us notice that, because of a finite thermal equilibration timescale, during the first few decades of a neutron star’s life the thermal evolution of the crust is decoupled from that of the liquid core.
Main neutrino emission processes in neutron star crusts. Symbols: γ stands for a photon or a plasmon; (A, Z) stands for a nucleus with charge number Z and mass number A; lepton symbol x = e, μ, τ; neutron quasiparticle (neutron-like elementary excitation) in superfluid neutron gas is denoted by ñ.
Process name | Reaction |
---|---|
e^{−}e^{+} pair annihilation | \({e^ -} + {e^ +} \rightarrow {v_x} + {{\bar v}_x}\) |
Plasmon decay | \(\gamma \rightarrow {v_x} + {{\bar v}_x}\) |
Photoneutrino emission | \(\gamma + {e^ -} \rightarrow {e^ -} + {v_x} + {{\bar v}_x}\) |
Electron synchrotron radiation | \({e^ -} \rightarrow {e^ -} + {v_x} + {{\bar v}_x}\) |
Electron-nucleus Bremsstrahlung | \({e^ -} + (A,Z) \rightarrow {e^ -} + (A,Z) + {v_x} + {{\bar v}_x}\) |
Cooper pair formation | \(\tilde n + \tilde n \rightarrow {v_x} + {{\bar v}_x}\) |
Processes reviewed in Sections 11.2–11.6 have been studied mainly at B = 0. The effects of B on Q_{ ν } were calculated only for some processes, and will be briefly mentioned at the end of the corresponding sections. The presence of B makes possible the synchrotron radiation of electrons, considered in Section 11.7.
A summary of the ρ − T dependence of different contributions to Q_{ ν }, and a discussion of their relative importance in different regions of the ρ − T plane, are presented in Section 11.9.
11.2 Electron-positron pair annihilation
The pair annihilation process can be affected by a strong magnetic field B. General expressions for \(Q_\nu ^{{\rm{pair}}}\) for arbitrary B were derived by Kaminker et al. [230, 228]. In these papers one can also find practical expressions for a hot, nondegenerate plasma in arbitrary B, as well as interpolating expressions for \(Q_\nu ^{{\rm{pair}}}\) in a plasma of any degeneracy and in any B. In a hot, nondegenerate plasma with T ≳ 10^{10} K, B ≫ 10^{15} G must be huge to affect \(Q_\nu ^{{\rm{pair}}}\). However, at T ≲ 10^{9} K, even B ∼ 10^{14} G may quantize the motion of positrons and increase substantially their number density. Consequently, B ∼ 10^{14} G strongly increases \(Q_\nu ^{{\rm{pair}}}\) al low densities. This is visualized in Figure 62.
11.3 Plasmon decay
The plasmon decay is influenced by a strong magnetic field, because B modifies the plasma dispersion relation (relation between plasmon frequency ω and its wavenumber k). In particular, new plasma modes may appear. The effects of magnetic fields are important if \(Q_\nu ^{{\rm{plas}}}\). At ρ ∼ 10^{11} g cm^{−3} this requires B ≳ 3 × 10^{15} G. The magnitude of B required to modify the plasmon dispersion relation grows as ρ^{2}/^{3}.
11.4 Photoneutrino emission
11.5 Neutrino Bremsstrahlung from electron-nucleus collisions
Things become more complicated at low temperatures. Then the electron states are no longer described by plane waves. Instead, the Bloch functions consistent with crystal symmetry should be used (see the review of band theory in Section 3.2.4). The electron energy spectrum is no longer continuous, but is formed of bands. At high T, the thermal motion of electrons “smears out” this band structure. However, the gaps between energy bands strongly suppress \(Q_\nu ^{{\rm{Brem}}}\) [231]. Detailed formulae valid for different domains of the density-temperature plane can be found in [428].
Strong magnetic fields affect the motion of electrons scattered off nuclei. However, the effect of B on \(Q_\nu ^{{\rm{Brem}}}\) has not been calculated.
11.6 Cooper pairing of neutrons
The actual value of the numerical prefactor in Equation (272) has recently become a topic of lively discussion. Recent calculations have shown that previous results may severely overestimate \(Q_\nu ^{{\rm{CP}}}\) (see, e.g., [368, 245], and references therein). However, the actual reduction of \(Q_\nu ^{{\rm{CP}}}\) is still a matter of debate.
11.7 Synchrotron radiation from electrons
11.8 Other neutrino emission mechanisms
There are many other mechanisms of neutrino emission. For example, there is the possibility of \(\nu \bar \nu\) pair Bremsstrahlung emission accompanying \(nn \to nn\nu \bar \nu\) scattering of dripped neutrons, and scattering of neutrons on nuclear clusters, \(n(A,Z) \to n(A,Z)\nu \bar \nu\). Moreover, in a newly-born neutron star beta processes involving electrons, positrons and nuclei, e.g., \({e^ -}(A,Z) \to (A,Z - 1){\nu _e},(A,Z - 1) \to (A,Z){e^ -}{\bar \nu _e}\), etc., are a source of neutrino emission. These are the famous Urca processes, proposed in the early 1940s; their intriguing history is described, e.g., in Section 3.3.5 of Yakovlev et al. [428]. One can also contemplate a photo-emission from nuclei, \(\gamma (A,Z) \to (A,Z)\nu \bar \nu\). Finally, we should also mention the interesting possibility of a very efficient neutrino emission by the direct Urca process in some “pasta layers” (see Section 3.3) near the bottom of the crust [274, 259, 260, 179]. As this mechanism, restricted to a bottom layer of the neutron star crust, could be a very efficient neutrino emission channel, we will describe it in more detail.
11.8.1 Direct Urca process in the pasta phase of the crust
11.9 Neutrinos from the crust — summary in the T − ρ plane
In this section we will summarize results for neutrino emission from a neutron star crust. We will limit ourselves to densities ρ ≳ 10^{7} g cm^{−3}, so that electrons will always be ultra-relativistic (see Section 3).
We start with the crust of a very young neutron star, with a temperature T = 3 × 10^{9} K (age ≲ 1 year), Figure 61. For density ρ ≲ 10^{8} g cm^{−3}, the contribution \(Q_\nu ^{{\rm{pair}}}\) is dominant. However, with increasing density, electrons become degenerate, and positrons disappear in the matter, so that \(Q_\nu ^{{\rm{pair}}}\) is strongly suppressed at ρ > 10^{9} g cm^{−3}. We also notice that \(Q_\nu ^{{\rm{phot}}}\) is never important in the inner crust, because of the strong electron degeneracy. \(Q_\nu ^{{\rm{plas}}}\) from the plasmon decay gives the dominant contribution to Q_{ ν } from ρ ≈ 10^{9} g cm^{−3} down to the bottom of the crust. We notice also that \(Q_\nu ^{{\rm{syn}}}\) behaves differently than the other contributions. Namely, at ρ ≳ 10^{9} g cm^{−3} its density dependence is very weak, and \(Q_\nu ^{{\rm{syn}}}\) scales approximately with the magnetic field B as ∝ B^{2}. Finally, one notices jumps of \(Q_\nu ^{{\rm{Brem}}}\) and \(Q_\nu ^{{\rm{plas}}}\), which result from jumps in Z and A in the ground-state matter. As we will see, this feature is even more pronounced at lower temperatures T.
Let us now consider the case of a colder crust at T = 10^{9} K, Figure 62. Except for \(Q_\nu ^{{\rm{syn}}}\), which is just scaled down due to the decrease of temperature, there is a dramatic change in the overall landscape. For a magnetic field B = 10^{14} G, \(Q_\nu ^{{\rm{syn}}}\) dominates in the lowest-density region. On the contrary, \(Q_\nu ^{{\rm{pair}}}\) is of marginal importance, and is influenced by B (increases with B). Moreover, contribution of \(Q_\nu ^{{\rm{phot}}}\) is negligible. Neglecting the effect of magnetic fields, one concludes that \(Q_\nu ^{{\rm{plas}}}\) dominates in the outer crust, while \(Q_\nu ^{{\rm{Brem}}}\) dominates in the inner crust. Let us notice that \(Q_\nu ^{{\rm{plas}}}\) reaches its maximum near 10^{10.5} g cm^{−3} and then decreases by four orders of magnitude when the density falls below ρ ∼ 10^{13} g cm^{−3}; this characteristic behavior is due to the exp(−T_{pe}/T) factor, Equation (265). On the contrary, \(Q_\nu ^{{\rm{Brem}}}\) rises steadily with increasing density.
Finally, in Figure 63 we consider an even colder crust at T = 3 × 10^{8} K. Pair and photoneutrino constributions have disappeared completely, while \(Q_\nu ^{{\rm{plas}}}\) dominates for ρ < 10^{9} g cm^{−3}, whereas at higher densities, \(Q_\nu ^{{\rm{Brem}}}\) is the main source of neutrino emission. At the magnetic field B = 10^{14} G, characteristic of magnetars, synchrotron radiation dominates in the density interval near ∼ 10^{10} g cm^{−3}, but then at ρ ≳ 10^{11} g cm^{−3}, \(Q_\nu ^{{\rm{Brem}}}\) becomes the strongest neutrino radiation mechanism.
Two general remarks are in order. First, as we have already mentioned, jumps in \(Q_\nu ^{{\rm{Brem}}}\) and \(Q_\nu ^{{\rm{plas}}}\) are due to specific factors involving Z^{2} and A and reflect the jumps in Z and A in the layered crust. For the other mechanisms, the electron chemical potential μ_{ e } with its smooth dependence on ρ plays the role of the crucial plasma parameter, and therefore no jumps are seen. Secondly, were the magnetic field B ≥ 10^{15} G, \(Q_\nu ^{{\rm{syn}}}\) would be overall dominant for T < 10^{9} K and ρ > 10^{9} g cm^{−3}.
The Cooper-pair mechanism of neutrino radiation differs fundamentally from the other mechanisms of neutrino cooling, discussed above, and therefore we consider it separately. \(Q_\nu ^{{\rm{CP}}}\) depends sensitively on the interplay between temperature T and the ^{1}S_{0} pairing gap Δ_{F} of the dripped neutrons. The gap itself depends on T, rising from zero at T = T_{ cn } to the asymptotic value Δ_{0} ≡ Δ_{F}(T = 0) for T ≪ T_{cn} (see Section 8.2.2). As we already discussed in Section 8.2.1, the dependence of Δ_{0} on the free neutron density, ρ_{ n }, is very poorly understood, and this introduces a large uncertainty in the calculated values of \(Q_\nu ^{{\rm{CP}}}\). Notice that the relation ρ_{ n }(ρ), needed to get \(Q_\nu ^{{\rm{CP}}}(\rho)\), depends on the model of the inner crust.
12 Observational Constraints on Neutron Star Crusts
12.1 Supernovae and the physics of hot dense inhomogeneous matter
The stellar evolution of massive stars with a mass M ∼ 10 − 20 M_{⊙} ends with the catastrophic gravitational collapse of the degenerate iron core (for a recent review, see, for instance, [219] and references therein). Photodissociation of iron nuclei and electron captures lead to the neutronization of matter. As a result, the internal pressure resisting the gravitational pull drops, thus accelerating the collapse, which proceeds on a time scale of ∼ 0.1 s. When the matter density inside the core reaches ∼ 10^{12} g cm^{−3}, neutrinos become temporarily trapped, thus hindering electron captures and providing additional pressure to resist gravity. However, this is not sufficient to halt the collapse and the core contraction proceeds until the central density reaches about twice the saturation density ρ ≃ 2.8 × 10^{14} g cm^{−3} inside atomic nuclei. After that, due to the stiffness (incompressibility) of nuclear matter, the collapse halts and the core bounces, generating a shock wave. The shock wave propagates outwards against the infalling material and eventually ejects the outer layers of the star, thus spreading heavy elements into the interstellar medium. A huge amount of energy, ∼ 10^{53} erg, is released, almost entirely (99%) in the form of neutrinos and antineutrinos of all flavors. The remaining energy is lost into electromagnetic and gravitational radiation. This scenario of core-collapse supernova explosion proved to be consistent with dense matter theory and various observations of the supernova 1987A in the Large Magellanic Cloud (discovered on February 23, 1987). In particular, the observation of the neutrino outburst provided the first direct estimate of the binding energy of the newly-born neutron star. With the considerable improvement of neutrino detectors and the development of gravitational wave interferometers, future observations of galactic supernova explosions would bring much more restrictive constraints onto theoretical models of dense matter. Supernova observations would indirectly improve our knowledge of neutron star crusts despite very different conditions, since in collapsing stellar cores and neutron star crusts the constituents are the same and are therefore described by the same microscopic Hamiltonian.
The collapse of the supernova core and the formation of the proto-neutron star are governed by weak interaction processes and neutrino transport [250]. Numerical simulations generally show that as the shock wave propagates outwards, it loses energy due to the dissociation of heavy elements and due to the pressure of the infalling material so that it finally stalls around ∼ 10^{2} km, as can be seen, for instance, in Figure 66. According to the delayed neutrino-heating mechanism, it is believed that the stalled shock is revived after ∼ 100 ms by neutrinos, which deposit energy in the layers behind the shock front. The interaction of neutrinos with matter is therefore crucial for modeling supernova explosions. The microscopic structure of the supernova core has a strong influence on the neutrino opacity and, therefore, on the neutrino diffusion timescale. In the relevant core layers, neutrinos form a nondegenerate gas, with a de Broglie wavelength λ_{ ν } = 2πħc/E_{ ν }, where E_{ ν } ∼ 3k_{B}T ∼ 5 − 10 MeV. If λ_{ ν } > 2R_{ A }, where R_{ A } is the radius of a spherical cluster, then thermal neutrinos “do not see” the individual nucleons inside the cluster and scatter coherently on the A nucleons. Putting it differently, a neutrino couples to a single weak current of the cluster of A nucleons. If the neutrino scattering amplitude on a single nucleon is f, then the scattering amplitude on a cluster is Af, and the scattering cross section is \(\sigma _A^{{\rm{coh}}} = {A^2}|f{|^2}\) ([150], for a review, see [373]). Consider now the opposite case of λ_{ ν } ≪ 2R_{ A }. Neutrinos scatter on every nucleon inside the cluster. As a result, the scattering amplitudes add incoherently, and the neutrino-nucleus scattering cross section \(\sigma _A^{{\rm{incoh}}} = A|f{|^2}\), similar to that for a gas of A nucleons. In this way, \(\sigma _A^{{\rm{coh}}}/\sigma _A^{{\rm{incoh}}} \approx A \sim 100\). One therefore concludes, that the presence of clusters in hot matter can dramatically increase the neutrino opacity. The neutrino transport in supernova cores depends not only on the characteristic size of the clusters, but also on their geometrical shape and topology. In particular, the presence of an heterogeneous plasma (due to thermal statistical distribution of A and Z) in the supernova core [65] or the existence of nuclear pastas instead of spherical clusters [201, 384] have a sizeable effect on the neutrino propagation. The outcome is that the neutrino opacity of inhomogeneous matter is considerably increased compared to that of uniform matter.
12.2 Cooling of isolated neutron stars
Neutron stars are born in the core collapse supernova explosions of massive stars, as briefly reviewed in Section 12.1. During the first tens of seconds, the newly formed proto-neutron star with a radius of ∼ 50 km stays very hot with temperatures on the order of 10^{11} − 10^{12} K. In the following stage, the star becomes transparent to neutrinos generated in its interior via various processes (see Section 11). Within ∼ 10 − 20 s the proto-neutron star thus rapidly cools down by powerful neutrino emission and shrinks into an ordinary neutron star. The last cooling stage, after about 10^{4} −10^{5} years, is governed by the emission of thermal photons due to the diffusion of heat from the interior to the surface (for a recent review of neutron star cooling, see, for instance, [431, 318] and references therein). Neutron stars in X-ray binaries may be heated as a result of the accretion of matter from the companion star. Observational data and references have been collected on the UNAM webpage [212].
12.2.1 Thermal relaxation of the crust
12.2.2 Observational constraints from thermal X-ray emission
12.3 r-process in the decompression of cold neutron star crusts
12.4 Pulsar glitches
Since the discovery by Jocelyn Bell and Anthony Hewish in 1967 of highly-periodic radio sources soon identified with rotating neutron stars (Hewish was awarded the Nobel Prize in Physics in 1974 [305]), more than 1700 pulsars have been found at the time of writing (pulsar timing data are available online at [26]). Pulsars are the most precise clocks with rotation periods ranging from about 1.396 milliseconds for the recently discovered pulsar J1748−2446ad [195] up to several seconds. The periodicity of arrival time of pulses is extremely stable. The slight delays associated with the spin-down of the star are at most of a few tens of microseconds per year. Nevertheless, longterm monitoring of pulsars has revealed irregularities in their rotational frequencies.
Very soon after the observations of the first glitches in the Crab and Vela pulsars, superfluidity in the interior of neutron stars was invoked to explain the long relaxation times [41]. The possibility that dense nuclear matter becomes superfluid at low temperatures was suggested theoretically much earlier, even before the discovery of the first pulsars (see Section 8.2). Following the first observations, several scenarios were proposed to explain the origin of these glitches, such as magnetospheric instabilities, pulsar disturbance by a planet, hydrodynamic instabilities or collisions of infalling massive objects (for a review of these early models, see Ruderman [350]). Most of these models had serious problems. The most convincing interpretation was that of starquakes, as briefly reviewed in Section 12.4.1. However, large amplitude glitches remained difficult to explain. The possible role of superfluidity in pulsar glitches was first envisioned by Packard in 1972 [317]. Soon after, Anderson and Itoh proposed a model of glitches based on the motion of neutron superfluid vortices in the crust [10]. Laboratory experiments were carried out to study similar phenomena in superfluid helium [68, 408, 409]. It is now widely accepted that neutron superfluidity plays a major role in pulsar glitches. As discussed in Sections 12.4.2 and 12.4.3, the glitch phenomenon seems to involve at least two components inside neutron stars: the crust and the neutron superfluid. Section 12.4.4 shows how the observations of pulsar glitches can put constraints on the structure of neutron stars.
12.4.1 Starquake model
12.4.2 Two-component models
Due to the interior magnetic field, the plasmas of electrically charged particles inside neutron stars are strongly coupled and co-rotate with the crust on very long timescales on the order of the pulsar age [132], thus following the long-term spin-down of the star caused by the electromagnetic radiation. Besides, the crust and charged particles are rotating at the observed angular velocity of the pulsar due to coupling with the magnetosphere. In contrast, neutrons are electrically neutral and superfluid. As a consequence, they can rotate at a different rate by forming quantized vortex lines (Section 8.3.2). This naturally leads to the consideration of the stellar interior as a two-fluid mixture. A model of this kind was first suggested by Baym et al. [40] for interpreting pulsar glitches as a transfer of angular momentum between the two components. Following a sudden spin-up of the star after a glitch event, the plasma of charged particles readjusts to a new rotational frequency within a few seconds [133]. Moreover, as discussed in Section 8.3.7, neutron superfluid vortices carry magnetic flux giving rise to an effective mutual friction force acting on the superfluid. As a result, the neutron superfluid in the core is dynamically coupled to the crust and to the charged particles, on a time scale much shorter than the post-glitch relaxation time of months to years observed in pulsars like Vela, suggesting that glitches are associated with the neutron superfluid in the crust. This conclusion assumes that the distribution of proton flux tubes in the liquid core is uniform. Nevertheless, one model predicts that every neutron vortex line is surrounded by a cluster of proton flux tubes [369, 370]. In this vortex-cluster model, the coupling time between the core superfluid and the crust could be much longer than the previous estimates and could be comparable to the post glitch relaxation times.
The origin of pulsar glitches relies on a sudden release of stresses accumulated in the crust, similar to the starquake model. However, the transfer of angular momentum from the rapidly-rotating neutron superfluid to the magnetically-braked solid crust and charged constituents during a glitch allows much larger spin-up than that due solely to the readjustment of the stellar shape. Neutron superfluid is weakly coupled to a normal charged component by mutual friction forces and thus follows the spin-down of the crust via a radial motion of the vortices away from the rotation axis unless the vortices are pinned to the crust. In the latter case, the lag between the superfluid and the crust induces a Magnus force, acting on the vortices producing a crustal stress. When the lag exceeds a critical threshold, the vortices are suddenly unpinned. Vortex motion could also be initiated by a temperature perturbation, for instance the heat released after a starquake [267]. As a result, the superfluid spins down and, by the conservation of the total angular momentum, the crust spins up leading to a glitch [10]. If the pinning is strong enough, the crust could crack before the vortices become unpinned, as suggested by Ruderman [351, 352, 356, 353]. These two scenarios lead to different predictions for the internal heat released after a glitch event. It has been argued that observations of the thermal X-ray emission of glitching pulsars could thus put constraints on the glitch mechanism [251].
In the vortex creep model [9] a postglitch relaxation is interpreted as a motion of vortices due to thermal fluctuations. Even at zero temperature, vortices can become unpinned by quantum tunneling [268]. The vortex current increases with the growth of temperature and can prevent the accumulation of large crustal stress in young pulsars, thus explaining the low glitch activity of these pulsars. In the model of Alpar et al. [6, 7] a neutron star is analogous to an electric circuit with a capacitor and a resistor, the vortices playing the role of the electric charge carriers. The star is, thus, assumed to be formed of resistive regions, containing a continuous vortex current, and capacitive regions devoid of vortices. A glitch can then be viewed as a vortex “discharge” between resistive regions through capacitive regions. The permanent change in the spin-down rate observed in some pulsars is interpreted as a reduction of the moment of inertia due to the formation of new capacitive regions. A major difficulty of this model is to describe the unpinning and repinning of vortices.
Ruderman developed an alternative view based on the interactions between neutron vortices and proton flux tubes in the core, assuming that the protons form a type II superconductor [354, 355]. Unlike the vortex lines, which are essentially parallel to the rotation axis, the configuration of the flux tubes depends on the magnetic field and may be quite complicated. Recalling that the number of flux tubes per vortex is about 10^{13} (see Sections 8.3.3 and 8.3.4), it is therefore likely that neutron vortices and flux tubes are strongly entangled. As superfluid spins down, the vortices move radially outward dragging along the flux tubes. The motion of the flux tubes results in the build up of stress in the crust. If vortices are strongly pinned to the crust, the stress is released by starquakes fracturing the crust into plates like the breaking of a concrete slab reinforced by steel rods when pulling on the rods. These plates and the pinned vortices will move toward the equator thus spinning down superfluid and causing a glitch. Since the magnetic flux is frozen into the crust due to very high electrical conductivity, the motion of the plates will affect the configuration of the magnetic field. This mechanism naturally explains the increase of the spin-down rate after a glitch observed in some pulsars like the Crab, by an increase of the electromagnetic torque acting on the pulsar due to the increase of the angle between the magnetic axis and the rotation axis.
12.4.3 Recent theoretical developments
Other scenarios have recently been proposed for explaining pulsar glitches, such as transitions from a configuration of straight neutron vortices to a vortex tangle [324], and more exotic mechanisms invoking the possibility of crystalline color superconductivity of quark matter in a neutron star core [4]. These models, and those briefly reviewed in Section 12.4.2, rely on rather poorly known physics. The strength of the vortex pinning forces and the type of superconductivity in the core are controversial issues (for a recent review, see, for instance, [367] and references therein). Besides, it is usually implicitly assumed that superfluid vortices extend throughout the star (or at least throughout the inner crust). However, microscopic calculations show that the superfluidity of nuclear matter strongly depends on density (see Section 8.2). It should be remarked that even in the inner crust, the outermost and innermost layers may be nonsuperfluid, as discussed in Section 8.2.2. It is not clear how superfluid vortices arrange themselves if some regions of the star are nonsuperfluid. The same question also arises for magnetic flux tubes if protons form a type II superconductor.
Andersson and collaborators [16] have suggested that pulsar glitches might be explained by a Kelvin-Helmholtz instability between neutron superfluid and the conglomeration of charged particles, provided the coupling through entrainment (see Section 8.3.7) is sufficiently strong. It remains to be confirmed whether such large entrainment effects can occur. Carter and collaborators [81] pointed out a few years ago that a mere deviation from the mechanical and chemical equilibrium induced by the lack of centrifugal buoyancy is a source of crustal stress. This mechanism is always effective, independently of the vortex motion and proton superconductivity. In particular, even if the neutron vortices are not pinned to the crust, this model leads to crustal stress of similar magnitudes than those obtained in the pinned case. Chamel & Carter [94] have recently demonstrated that the magnitude of the stress is independent of the interactions between neutron superfluid and normal crust giving rise to entrainment effects. But they have shown that stratification induces additional crustal stress. In this picture, the stress builds up until the lag between neutron superfluid and the crust reaches a critical value, at which point the crust cracks, triggering a glitch. The increase of the spin-down rate observed in some pulsars like the Crab can be explained by the crustal plate tectonics of Ruderman [351, 352, 356, 353], assuming that neutron superfluid vortices remain pinned to the crust. Even in the absence of vortex pinning, Franco et al. [149] have shown that, as a result of starquakes, the star will oscillate and precess before relaxing to a new equilibrium state, followed by an increase of the angle between the magnetic and rotation axis (thus increasing the spin-down rate).
12.4.4 Pulsar glitch constraints on neutron star structure
Basing their work on the two-component model of pulsar glitches, Link et al. [269] derived a constraint on the ratio I^{f}/I of the moment of inertia I^{f} of the free superfluid neutrons in the crust to the total moment of inertia I of the Vela pulsar, from which they inferred an inequality involving the mass and radius of the pulsar. However, they neglected entrainment effects (see Sections 8.3.6 and 8.3.7), which can be very strong in the crust, as shown by Chamel [90, 91]. We will demonstrate here how the constraint is changed by including these effects, following the analysis of Chamel & Carter [94].
12.5 Gravitational wave asteroseismology
The development of gravitational wave detectors like LIGO [67], VIRGO [208], TAMA300 [301] and GEO600 [299] is opening up a new window of astronomical observations. With a central density on the order of ∼ 10^{15} g cm^{−3}, neutron stars are among the most compact objects in the universe, and could be efficient sources of gravitational waves. The existence of such waves predicted by Einstein’s theory of general relativity was beautifully confirmed by the observations of the binary pulsar PSR 1913+16 by Russel Hulse and Joseph Taylor (who were awarded the Nobel Prize in 1993 [306]). To compare the importance of general relativistic effects in binary pulsars with those around ordinary stars like the Sun, let us remember that the pulsar’s periastron in PSR 1913+16 advances every day by the same amount as Mercury’s perihelion advances in a century!
12.5.1 Mountains on neutron stars
12.5.2 Oscillations and precession
A large number of different nonaxisymmetric neutron-star-oscillation modes exists, for instance, in the liquid surface layers (“ocean”), in the solid crust, in the liquid core and at the interfaces between the different regions of the star. These oscillations can be excited by thermonuclear explosions induced by the accretion of matter from a companion star, by starquakes, by dynamic instabilities growing on a timescale on the order of the oscillation period or by secular instabilities driven by dissipative processes and growing on a much longer timescale. Oscillation modes are also expected to be excited during the formation of the neutron star in a supernova explosion. The nature of these modes, their frequency, their growing and damping timescales depend on the structure and composition of the star (for a review, see, for instance, [285, 244, 12]).
12.5.3 Crust-core boundary and r-mode instability
Of particular astrophysical interest are the inertial modes or Rossby waves (simply referred to as r-modes) in neutron star cores. They can be made unstable by the radiation of gravitational waves on short timescales of a few seconds in the most rapidly-rotating neutron stars [17]. However, the growth of these modes can be damped. One of the main damping mechanisms is the formation of a viscous boundary Ekman layer at the crust-core interface [49] (see also [164] and references therein). It has been argued that the heat dissipated in this way could even melt the crust [265]. The damping rate depends crucially on the structure of bottom layers of the crust and scales like (δυ/υ)^{2}, where δυ is the slippage velocity at the crust-core interface [263]. Let us suppose that the liquid in the core does not penetrate inside the crust, like a liquid inside a bucket. The slippage velocity in this case is very large δυ ∼ υ and as a consequence the r-modes are strongly damped. However, these assumptions are not realistic. First of all, the crust is not perfectly rigid, as discussed in Section 7. On the contrary, the crust is quite “soft” to shear deformations because μ/P ∼ 10^{−2}, where μ is the shear modulus and P the pressure. The oscillation modes of the liquid core are coupled to the elastic modes in the crust, which results in much smaller damping rates [263, 163]. Besides, the transition between the crust and the core might be quite smooth. Indeed, neutron superfluid in the core permeates the inner crust and the denser layers of the crust could be formed of nuclear “pastas” with elastic properties similar to those of liquid crystals (see Section 7.2). The slippage velocity at the bottom of the crust could, therefore, be very small δυ ≪ υ. Consequently, the Ekman damping rate of the r-modes could be much weaker than the available estimates. If the crust were purely fluid, the damping rate would be vanishingly small. However the presence of the magnetic field would also affect the damping time scale and should be taken into account [289, 238]. Besides the character of the core oscillation modes is likely to be affected by coupling with the crust. The role of the crust in the dissipation of the r-mode instability is, thus, far from being fully understood. Finally, let us mention that by far the strongest damping mechanism of r-modes, due to a huge bulk viscosity, may be located in the inner neutron star core, provided it contains hyperons (see, e.g., [182, 264] and references therein).
12.6 Giant flares from Soft Gamma Repeaters
The discovery of quasi-periodic oscillations (QPO) in X-ray flux following giant flares from Soft Gamma Repeaters (SGR) has recently triggered a burst of intense theoretical research. Oscillations were detected at frequencies 18, 26, 29, 92.5, 150, 626.5 and 1837 Hz during the spectacular December 27, 2004 giant flare (the most intense ever observed in our Galaxy) from SGR 1806−20 [214, 421, 397] and at 28, 54, 84 and 155 Hz during the August 27, 1998 giant flare from SGR 1900+14 [396]. Evidence has also been reported for oscillations at 43.5 Hz during the March 5, 1979 event in SGR 0526−66 [34]. Soft Gamma Repeaters (SGR) are believed to be strongly-magnetized neutron stars or magnetars endowed with magnetic fields as high as 10^{14} − 10^{15} G (for a recent review, see, for instance, [426] and references therein; also see the home page of Robert C. Duncan [128]). Giant flares are interpreted as crustquakes induced by magnetic stresses. Such catastrophic events are likely to be accompanied by global seismic vibrations, as observed by terrestrial seismologists after large earthquakes. Among the large variety of possible oscillation modes, torsional shear modes in the crust are the most likely [129]. Shear flow in the crust is illustrated in Figure 58. If confirmed, this would be the first direct detection of oscillations in a neutron star crust.
The effect of rotation on oscillation modes is to split the frequency of each mode with a given ℓ into 2ℓ + 1 frequencies. It has recently been pointed out that some of the resulting modes might, thus, become secularly unstable, according to the Chandrasekhar-Friedman-Schutz (CFS) criterion [411]. The study of oscillation modes becomes even more difficult in the presence of a magnetic field. Roughly speaking, the effects of the magnetic field increase the mode frequencies [129, 290, 328, 257, 385]. Simple Newtonian estimates lead to an increase of the frequencies by a factor \(\sqrt {1 + {{(B/{B_\mu})}^2}}\), where \({B_\mu} \equiv \sqrt {4\pi \mu}\) is expressed in terms of the shear modulus μ [129]. Sotani and et al. [385] recently carried out calculations in general relativity with a dipole magnetic field and found numerically that the frequencies are increased by a factor \(\sqrt {1 + {\alpha _{n,\ell}}{{(B/{B_\mu})}^2}}\), where α_{n, ℓ} is a numerical coefficient. However, it has been emphasized by Messios et al. [290] that the effects of the magnetic field strongly depend on its configuration. The most important effect is to couple the crust to the core so that the whole stellar interior vibrates during a giant flare [165]. Low frequency QPOs could, thus, be associated with magnetohydrodynamic (MHD) modes in the core [262].
Another important aspect to be addressed is the presence of neutron superfluid, which permeates the inner crust. The formalism for treating a superfluid in a magneto-elastic medium has been recently developed both in general relativity [85] and in the Newtonian limit [73, 72], based on a variational principle. This formalism has not yet been applied to study oscillation modes in magnetars. However, we can anticipate the effects of the neutron superfluid using the two-fluid description of the crust reviewed in Section 10.2. Following the same arguments as for two-fluid models of neutron star cores [13], two classes of oscillations can be expected to exist in the inner crust, depending on whether neutron superfluid is co-moving or countermoving with the crust. The countermoving modes are predicted to be very sensitive to entrainment effects, which are very strong in the crust [90, 91].
The neutron-star-oscillation problem deserves further theoretical study. The prospect of probing neutron star crusts by analyzing the X-ray emission of giant magnetar flares is very promising.
12.7 Low mass X-ray binaries
As reviewed in Section 4, the accretion of matter (mainly hydrogen and helium) onto the surface of neutron stars triggers thermonuclear fusion reactions. Under certain circumstances, these reactions can become explosive, giving rise to X-ray bursts. The unstable burning of helium ash produced by the fusion of accreted hydrogen, is thought to be at the origin of type I X-ray bursts. A new type of X-ray burst has been recently discovered. These superbursts are a thousand times more energetic than normal bursts and last several hours compared to a few tens of seconds, but occur much more rarely. These superbursts could be due to the unstable burning of ^{12}C accumulated from He burning. The mass of ^{12}C fuel has to be as high as 10^{−9} M_{⊙} to get E_{burst} ∼ 10^{42} erg. It can be seen from Figure 39 that ^{12}C ignition has to occur at ρ ∼ 10^{9} g cm^{−3} at a depth of ∼ 30 m. At the accretion rates characteristic of typical superbursters (Ṁ ∼ (1–3) × 10^{−9} M_{Ṁ}/y), this would correspond to recurrence times of a few years. It also seems that crustal heating might be quite important to getting such a relatively low ignition density of ^{12}C [178]. The ignition conditions are very sensitive to the thermal properties of the crust and core [104]. X-ray observations of low-mass X-ray binaries thus provide another way of probing the interior of neutron stars, both during thermonuclear bursting episodes and during periods of quiescence as discussed below.
12.7.1 Burst oscillations
12.7.2 Soft X-ray transients in quiescence
The phenomenon of deep crustal heating appears to be relevant for the understanding of the thermal radiation observed in soft X-ray transients (SXTs) in quiescence, when the accretion from a disk is switched off or strongly suppressed. Typically, the quiescent emission is much higher than it would be in an old cooling neutron star. It has been suggested that this is because the interiors of neutron stars in SXTs are heated up during relatively short periods of accretion and bursting by the nonequilibrium processes associated with nuclear reactions taking place in the deep layers of the crust ([60], see also Section 4.3). The deep crustal heating model, combined with appropriate models of the neutron star atmosphere and interior, is used to explain measured luminosities of SXTs in quiescence. The luminosity in quiescence depends on the structure of neutron star cores, and particularly on the rate of neutrino cooling. This opens up the new possibility of exploring the internal structure and equation of state of neutron stars (see [102, 358, 429, 430] and references therein).
12.7.3 Initial cooling in quasi-persistent SXTs
13 Conclusion
The conditions prevailing inside the crusts of neutron stars are not so extreme as those encountered in the dense core. Nonetheless, they are still far beyond those accessible in terrestrial laboratories. The matter in neutron star crust is subject to very high pressures, as well as huge magnetic fields, which can attain up to 10^{14} − 10^{15} G in magnetars. For comparison, it is worth reminding ourselves that the strongest (explosive) magnetic fields ever produced on the Earth reach “only” 3 × 10^{7} G [298]. The description of such environments requires the interplay of many different branches of physics, from nuclear physics to condensed matter and plasma physics.
Considerable progress in the microscopic modeling of neutron star crusts has been achieved during the last few years. Yet the structure and properties of the crust remain difficult to predict, depending on the formation and subsequent cooling of the star. Even in the ground state approximation, the structure of the neutron star crust is only well determined at ρ ≲ 10^{11} g cm^{−3}, for which experimental data are available. Although all theoretical calculations predict the same large-scale picture of the denser layers of the crust, they do not quantitatively agree, reflecting the uncertainties in the properties of very exotic nuclei and uniform highly-asymmetric nuclear matter. The inner crust is expected to be formed of a lattice of neutron-rich nuclear clusters coexisting with a degenerate relativistic electron gas and a neutron liquid. The structure of the inner crust, its composition and the shape of the clusters are model dependent, especially in the bottom layers at densities ∼ 10^{14} g cm^{−3}. However, the structure of the inner crust is crucial for calculating many properties, such as neutrino emissivities, as well as transport properties like electric and thermal conductivities. Superfluidity of unbound neutrons in the crust seems to be well established, both observationally and theoretically. However, much remains to be done to understand its properties in detail and, in particular, the effects of the nuclear lattice.
The interpretation of many observed neutron star phenomena, like pulsar glitches, X-ray bursts in low-mass X-ray binaries, initial cooling in soft X-ray transients, or quasi-periodic oscillations in soft gamma repeaters, can potentially shed light on the microscopic properties of the crust. However, their description requires consistent models of the crust from the nuclear scale up to the macroscopic scale. In particular, understanding the evolution of the magnetic field, the thermal relaxation of the star, the formation of mountains, the occurrence of starquakes and the propagation of seismic waves, requires the development of global models combining general relativity, elasticity, magnetohydrodynamics and superfluid hydrodynamics. Theoretical modeling of neutron star crusts is very challenging, but not out of reach. A confrontation of these models with observations could hopefully help us to unveil the intimate nature of dense matter at subnuclear densities. The improvement of observational techniques, as well as the development of gravitational wave astronomy in the near future, open very exciting perspectives.
We are aware that some aspects of the physics of neutron star crusts have not been dealt with in this review. Our intention was not to write an exhaustive monograph, but to give a glimpse of the great variety of topics that are necessarily addressed by different communities of scientists. We hope that this review will be useful to the reader for his/her own research.
Footnotes
- 1.
Lattice energy including finite size effects is given by Equation (41). Even at the bottom of the outer crust finite size effects represent a small correction to the lattice energy, less than 1%.
- 2.
“free” means here that the electrons are not bound. However, they are interacting with other electrons and with the atomic lattice.
- 3.
We restrict ourselves to type I X-ray bursts. There are two X-ray bursters that are of type II, with bursts driven not by thermonuclear flashes on the neutron star surface, but originating in the accretion disk itself.
- 4.
In this section, by “energy” we will usually mean energy of a unit volume (i.e. energy density).
- 5.
Such pressures are very small in the context of neutron stars. For iron at room temperature they correspond to a density of about 8.2 g cm^{−3}, in comparison to 7.86 g cm^{−3} under normal atmospheric pressure [42].
- 6.
For a body-centered-cubic lattice, the lattice spacing a is related to the Wigner-Seitz radius R_{cell} by a = 2(π/3)^{1/3}R_{cell}.
- 7.
The Fermi surface is the surface in k-space defined by ϵ(k) = μ. Note that, in general, it is not spherical.
- 8.
In the case of fermionic superfluids, the superfluid particles are fermion pairs.
- 9.
For a proton superconductor, m = 2m_{ p } and q = 2e, where m_{ p } and e are the proton mass and proton electric charge, respectively.
- 10.
For instance at T = 0, n_{ S } = n while n_{Ψ} ≃ 0.1n.
- 11.
Entrainment effects disappear as T goes to zero since m_{★}(T = 0) = m so that p = mυ according to Equation (180).
- 12.
This assumption may not remain valid in the “nuclear pasta” layers at the bottom of the crust discussed in Section 3.3.
- 13.
Ion contribution to η can be important in the very outer layers with ρ < 10^{4}g cm^{−3}, where η ≈ η_{N}
- 14.
The gyromagnetic frequency of electrons should not be confused with the electron cyclotron frequency ω_{ce} = eB/m_{ e }c entering the formula for the energies of the Landau levels; see Section 2.
Notes
Acknowledgments
N.C. gratefully acknowledges financial support from a Marie Curie Intra-European grant (contract number MEIF-CT-2005-024660) and from FNRS (Belgium). P.H. was partially supported by the Polish MNiSW grant no. N20300632/0450.
We express our deep gratitude to D.G. Yakovlev for his help in the preparation of the present review. He read the whole manuscript and, during many hours of Skype sessions, he went over with one of us (P.H.) all the sections that required corrections, clarifications and improvement (they were legion). We thank him for his patience and expertise shared with us.
We are grateful to A.Y. Potekhin, who suggested several improvements in the sections referring to the effects of magnetic fields. He also suggested appropriate figures and made some new ones, which enriched the content of this review. Sometimes a good figure is better than hundreds of words.
We also thank Lars Samuelsson for his comments on elasticity and on axial modes in neutron stars.
We would like to address our gratitude to our colleagues, who contributed to this review through discussions or collaborations.
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