Dispersion and decay of collective modes in neutron star cores

TL;DR

This paper calculates the frequencies of collective modes in neutron star cores, considering neutron, proton, and electron interactions, and assesses their impact on thermal conductivity, highlighting the effects of Landau damping and entrainment.

Contribution

It provides a detailed analysis of collective mode frequencies in neutron star cores, incorporating hydrodynamic and collisionless electron models, and evaluates their influence on thermal conductivity.

Findings

Landau damping significantly affects mode dispersion.

Mode frequencies are sensitive to neutron-proton entrainment.

Collective modes contribute notably to thermal conductivity.

Abstract

We calculate the frequencies of collective modes of neutrons, protons and electrons in the outer core of neutron stars. The neutrons and protons are treated in a hydrodynamic approximation and the electrons are regarded as collisionless. The coupling of the nucleons to the electrons leads to Landau damping of the collective modes and to significant dispersion of the low-lying modes. We investigate the sensitivity of the mode frequencies to the strength of entrainment between neutrons and protons, which is not well characterized. The contribution of collective modes to the thermal conductivity is evaluated.

Tables

Table 1. Table 1: Beta-equilibrium densities, entrainment parameter and thermodynamic derivatives. The derivatives E α ​ β subscript 𝐸 𝛼 𝛽 E_{\alpha\beta} are calculated from an equation of state based on chiral effective field theory HebelerEtal2013 . The entrainment parameter n n ​ p ( a ) superscript subscript 𝑛 𝑛 𝑝 𝑎 {n_{np}}^{(a)} is calculated from Fermi liquid theory and Skyrme energy functional SLy4, Eq. ( 49 ), while n n ​ p ( b ) superscript subscript 𝑛 𝑛 𝑝 𝑏 {n_{np}}^{(b)} is derived from considerations of nucleon effective masses, Eq. ( 56 ).

$n_B\left({\mathrm{fm}}^{-3}\right)$	$n_p\left({\mathrm{fm}}^{-3}\right)$	$n_n\left({\mathrm{fm}}^{-3}\right)$	$n_{np}^{}{}_{}{}^{\left(a\right)}\left({\mathrm{fm}}^{-3}\right)$	$n_{np}^{}{}_{}{}^{\left(b\right)}\left({\mathrm{fm}}^{-3}\right)$	$E_{pp}^{\mathrm{nuc}}$ (GeV fm3)	$E_{nn}$ (GeV fm3)	$E_{np}$ (GeV fm3)
0.08	$2.64\times {10}^{-2}$	$7.736\times {10}^{-2}$	$-3.205\times {10}^{-4}$	$-6.391\times {10}^{-4}$	$1.294$	0.1886	-0.5729
0.16	$7.746\times {10}^{-3}$	$1.523\times {10}^{-1}$	$-1.851\times {10}^{-3}$	$-3.692\times {10}^{-3}$	1.247	0.2684	-0.1891
0.24	$1.236\times {10}^{-2}$	$2.277\times {10}^{-1}$	$-4.418\times {10}^{-3}$	$-8.811\times {10}^{-3}$	1.325	0.3352	0.0811
0.32	$1.513\times {10}^{-2}$	$3.048\times {10}^{-1}$	$-7.239\times {10}^{-3}$	$-1.444\times {10}^{-2}$	1.433	0.3895	0.2972

Equations

$${\mathbf{p}}_{\alpha }=\hslash \nabla {\varphi }_{\alpha },$$

$$H$$

$$n_{\alpha \beta }=m\frac{{\partial }^{2}E}{\partial {\mathbf{p}}_{\alpha }\partial {\mathbf{p}}_{\beta }},$$

$$n_{pp}+n_{np}=n_p$$

$$n_{nn}+n_{np}=n_n,$$

$$\frac{n_{nn}}{2m}{p}_n^2+\frac{n_{pp}}{2m}{p}_p^2+\frac{n_{np}}{2m}{\mathbf{p}}_n\cdot {\mathbf{p}}_p$$

$$=\frac{n_n}{2m}{p}_n^2+\frac{n_p}{2m}{p}_p^2-\frac{n_{np}}{2m}({\mathbf{p}}_n-{\mathbf{p}}_p{)}^2.$$

$$n_{nn}=n_n\frac{m}{m_n^{\ast }}\left(1+N_n(0)\frac{f_1^{nn}}{3}\right),$$

$$n_{pp}=n_p\frac{m}{m_p^{\ast }}\left(1+N_p(0)\frac{f_1^{pp}}{3}\right),$$

$$n_{np}=\frac{m{k}_n^2{k}_p^2}{9{\pi }^4}{f}_1^{np}.$$

$${\partial }_t{n}_n+\nabla \cdot \left[\frac{n_n}{m}{\mathbf{p}}_n-\frac{n_{np}}{m}\left({\mathbf{p}}_n-{\mathbf{p}}_p\right)\right]=0$$

$${\partial }_t{n}_p+\nabla \cdot \left[\frac{n_p}{m}{\mathbf{p}}_p-\frac{n_{np}}{m}\left({\mathbf{p}}_p-{\mathbf{p}}_n\right)\right]=0.$$

$$\hslash {\partial }_t{\varphi }_n=-{\mu }_n^{\mathrm{nuc}}$$

$$\hslash {\partial }_t{\varphi }_p=-{\mu }_p^{\mathrm{nuc}}-e\Phi ,$$

$${\mu }_{\alpha }^{\mathrm{nuc}}=\frac{\partial E^{\mathrm{nuc}}}{\partial n_{\alpha }}$$

$${\nabla }^2\Phi =-4\pi e(n_p-n_e),$$

$${\partial }_t{\mathbf{p}}_p=-\nabla \left({\mu }_p^{\mathrm{nuc}}+e\Phi \right)$$

$${\partial }_t{\mathbf{p}}_n=-\nabla {\mu }_n^{\mathrm{nuc}}.$$

$$\delta n_e={\chi }_{0}e\delta \Phi .$$

$${\chi }_0$$

$$\Phi =\frac{4\pi e}{k^2}(n_p-n_e),$$

$$\delta n_e=\frac{4\pi e^2}{k^2}{\chi }_0(\delta n_p-\delta n_e)$$

$$\delta n_e=\frac{(4\pi e^2/k^2){\chi }_0}{1+(4\pi e^2/k^2){\chi }_0}\delta n_p=\frac{(4\pi e^2/k^2){\chi }_0}{{ϵ}_e(\omega ,k)}\delta n_p.$$

$${ϵ}_e(\omega ,k)=1+(4\pi e^2/k^2){\chi }_0$$

$$\left(\begin{array}[]{cccc}-{m\omega}/{k}&0&n_{pp}&n_{np}\\ 0&-{m\omega}/{k}&n_{np}&n_{nn}\\ E_{pp}&E_{np}&-{\omega}/{k}&0\\ E_{np}&E_{nn}&0&-{\omega}/{k}\end{array}\right)\left(\begin{array}[]{c}\delta n_{p}\\ \delta n_{n}\\ \delta p_{p}\\ \delta p_{n}\end{array}\right)=0,$$

$$E_{pp}=E_{pp}^{\mathrm{nuc}}+\frac{4\pi e^2}{k^2{ϵ}_e(\omega ,k)}.$$

$${\omega }^2=\frac{{\Omega }_p^2}{{\epsilon }_e(\omega ,k)}+\frac{n_p{E}_{pp}^{\mathrm{nuc}}}{m}{k}^2\equiv v_{p0}^2{k}^2,$$

$${\chi }_0\simeq \frac{\partial n_e}{\partial {\mu }_e}\left(1+\mathrm{i}\frac{\pi }{2}s-s^2\right).$$

$$\left[m+\frac{{\mu }_e}{3c^2}\left(\frac{{\pi }^2}{4}-1\right)\right]{\omega }^{\prime 2}\simeq \frac{{\mu }_e}{3}{k}^2+n_p{E}_{pp}^{\mathrm{nucl}}{k}^2.$$

$${\omega }^{\prime \prime }\simeq -\frac{\pi }{12}\frac{{\mu }_e}{m{c}^2}ck.$$

$$\chi =\frac{\partial n_e}{\partial {\mu }_e}\frac{c_e^2{k}^2}{c_e^2{k}^2-{\omega }^2},$$

Full text

I Introduction

In the outer core of a neutron star, matter consists of neutrons, protons and electrons, with possibly other minority constituents. The protons are expected to be superconducting with pairs in the 1S0 state and gaps of order 1 MeV. The accuracy of present calculations does not permit an unambiguous answer to the question of whether or not neutrons are superfluid: calculations in which the internucleon interaction is taken to be the same as in free space predict pairing in the 3P2 state Hoffberg while estimates that take into account the modification of the interaction by the medium predict very small gaps SchwenkFriman and are consistent with there being no gap (for a review of pairing gaps in neutron star matter see Gezerlis ). Collective modes have previously been studied in Refs. Epstein ; BaldoDucoin2003 ; BedaqueReddy2014 ; KobyakovEtal2015, and the possible role of such modes in transporting heat in neutron stars has been discussed in Ref. Aguilera2009 . In this work, we shall assume that matter consists only of superfluid neutrons, superconducting protons and normal electrons. The superfluid critical temperatures for both neutrons and protons are estimated to be of the order of $1\,\mathrm{MeV}$, which is about two orders of magnitude larger than the typical temperature in the interior of neutron stars, and therefore it is a good first approximation to take the temperature to be zero.

In a recent work KobyakovCJP collective modes with wavelengths large compared with the Debye screening length for the electrons were studied under the assumption that the electrons could be treated in the hydrodynamic approximation. Here we extend that treatment to shorter wavelengths. When the wavelength is of order or shorter than the Debye screening length for electrons, the modes exhibit dispersion. In addition, modes are damped by collisions between electrons when the electrons are in the hydrodynamic regime and by Landau damping when the electrons are collisionless Landau1945 . On the basis of our results for the damping of collective modes, we calculate the contribution of collective modes to the thermal conductivity of the outer core.

The plan of this paper is as follows. In Section II, we extend the formalism previously used to calculate the velocities of collective modes at long wavelengths KobyakovCJP to wavelengths shorter than the Debye screening length, when the plasma is no longer locally neutral electrically. Expressions for the real and imaginary parts of mode frequencies are given in Sec. III, and in Sec. IV we discuss the magnitude of the parameter determining the entrainment of neutrons by protons (and vice versa) and give numerical results for the mode frequencies derived using an equation of state based on chiral effective field theory HebelerEtal2013 . The thermal conductivity of matter is calculated in Sec. V, where we also comment on possible astrophysical effects. In the Appendix we demonstrate that the approach adopted in this paper, where we work in terms of the nucleon momenta per particle in the condensates, is equivalent to the traditional two-fluid formalism CJPChamelReddy ; ChamelPageReddy ; KobyakovPethick2013 , where one works in terms of the momentum per particle in the neutron condensate and the velocity of the charged particles.

II Basic formalism

We shall assume that the neutrons are superfluid, and that the protons are superconducting and we denote the phases of the pair wave functions by $2\phi_{n}$ and $2\phi_{p}$, respectively. Here we shall not take into account variations in the spatial orientation of possible anisotropic order parameters. We shall work in terms of the momenta per particle in the neutron and proton condensates, which are given by

[TABLE]

where the index $\alpha=n,p$ refers to neutrons and protons. In the conventional treatment of superfluid dynamics, one defines quantities ${\hbar}\bm{\nabla}\phi_{\alpha}/m$, where $m$ is the particle mass. These are referred to as superfluid velocities, even though they are covariant vectors, and therefore do not transform as velocities, which are contravariant vectors. To avoid confusion, we shall work almost exclusively in terms of the ${\bf p}_{\alpha}$. We shall neglect the difference between the neutron and proton rest masses since it is small compared with other effects, such as the contribution of interactions to the mass density, which we also neglect. Our approach is to start from the equations of motion for the nucleon densities and condensate phases, but we shall not impose the condition that the proton and electron densities are equal locally. Rather we shall include the Coulomb interaction between electrons and protons explicitly. For frequencies small compared with the pairing gaps and for wavelengths large compared with the coherence lengths, the neutrons and protons may be described by hydrodynamics (see KobyakovCJP and references therein), and the Hamiltonian for the system is thus

[TABLE]

Here the superfluid density tensor is given by

[TABLE]

where $E$ is the energy density. Here we include terms of second order in the velocities and neglect higher order contributions. With the help of the conditions (Mendell1991a, ; BorumandJoyntKluzniak1996, )

[TABLE]

and

[TABLE]

which follow from the Galilean invariance of the neutrons and protons when relativistic effects are neglected, the kinetic energy contributions to the Hamiltonian density take on the form

[TABLE]

When the condensation energy associated with pairing is small, the $n_{\alpha\beta}$ may be expressed in terms of Landau parameters BorumandJoyntKluzniak1996 :

[TABLE]

[TABLE]

and

[TABLE]

Here $k_{\alpha}$ is the Fermi wavenumber, $m_{\alpha}^{*}$ the effective mass and $N_{\alpha}(0)=m^{*}_{\alpha}k_{\alpha}/\pi^{2}\hbar^{2}$ the density of states at the Fermi surface of species $\alpha$, and $f_{1}^{\alpha\beta}$ is the $l=1$ component of the quasiparticle interaction. The equations of motion for the neutrons and protons may be obtained from Hamilton’s equations, since the particle density $n_{\alpha}$ and $\hbar\phi_{\alpha}$ are conjugate variables and therefore $\dot{n}_{\alpha}=\hbar^{-1}\delta H/\delta\phi_{\alpha}$ and $\dot{\phi}_{\alpha}=-\hbar^{-1}\delta H/\delta n_{\alpha}$ LL1980 . The continuity equations for the number densities of neutrons and protons have the same form as in Ref. KobyakovCJP :

[TABLE]

and

[TABLE]

In this paper we shall consider only linear modes. In this case, the time dependence of the phases is given by

[TABLE]

and

[TABLE]

where

[TABLE]

is the nucleon chemical potential in the absence of Coulomb contributions and in the absence of flows. The quantity $\Phi$ is the electric potential, which satisfies the Poisson equation,

[TABLE]

since the total charge density is $e(n_{p}-n_{e})$. Therefore the equations of motion for ${\mathbf{p}}_{\alpha}$ are

[TABLE]

and

[TABLE]

To close this set of equations one needs to calculate the response of the electrons to an electric potential. We shall consider small disturbances from the uniform state and therefore one can use linear response theory. If the frequency, $\omega$, of the disturbance is large compared with the electron collision frequency and the wave number is large compared with the inverse mean free path, the electrons may be treated as collisionless. The mean free path of electrons is limited by scattering from electrons, and for calculating the properties of longitudinal modes, the relevant relaxation time is that for viscosity. This has been calculated in detail by Shternin and Yakovlev ShterninYakovlev2008 , who demonstrate that the dominant scattering process for electrons at temperatures less than $\sim~{}10^{9}$ K is via exchange of transverse photons, rather than the Coulomb interaction. For nuclear matter density and a temperature of $10^{8}$K, their calculations lead to a mean free path for viscosity of $\sim 10^{10}$ fm or $10^{-3}$ cm. This is intermediate between typical stellar length scales ($\sim$ km) and the wavelength scale of thermal excitations $\sim\hbar v/(k_{B}T)\approx 10^{4}(v/c)/T_{8}$ fm. Here $v$ is the velocity of the mode, $k_{B}$ is the Boltzmann constant, and $T_{8}$ is the temperature in units of $10^{8}$ K. In this paper, our primary interest is collective modes on a microscopic scale, so we shall assume that wavelengths are short compared with the electron mean free path and frequencies are large compared with the electron collision frequency and we shall henceforth neglect collisions.

In the absence of collisions, the response of electrons in dense matter is given to a very good approximation by the random phase approximation and for ultrarelativistic electrons the result is Jancovici1962

[TABLE]

Here

[TABLE]

with $s=\omega/(ck)$, $\partial n_{e}/\partial\mu_{e}=p_{e}^{2}/\pi^{2}\hbar^{3}c$ and $\Theta$ is the Heaviside step function. This form is valid for $\omega\ll\mu_{e}$ and $k\ll k_{e}$. In the absence of an external potential, the electric potential is given by

[TABLE]

and therefore to linear order

[TABLE]

or

[TABLE]

Here

[TABLE]

is the dielectric function of the electrons. Equation (22) enables us to eliminate $\delta n_{e}$ from the equations of motion.

III Collective modes

The properties of collective modes are obtained by solving Eqs. (10) – (13) together with Eqs. (20) and (22). Small longitudinal perturbations about a uniform state in which the nucleons and electrons are stationary satisfy the equation

[TABLE]

where

[TABLE]

Here, $E_{pp}^{\rm nuc}=\partial\mu_{p}^{\rm nuc}/\partial n_{p}$ and $E_{n\alpha}=\partial\mu_{n}^{\rm nuc}/\partial n_{\alpha}$. The first term on the right side of Eq. (25) represents the contribution from nuclear interactions and the second term is the Coulomb interaction between protons screened by electrons. We have dropped the superscript “nuc” on the other $E_{\alpha\beta}$ because there are no Coulomb contributions. Equation (24) reduces to the result of Ref.  KobyakovCJP if one neglects the frequency dependence of the electron response, and replaces $\epsilon_{e}$ by its asymptotic long-wavelength form $\simeq k_{TF}^{2}/k^{2}$, where $k_{TF}=\sqrt{4\pi e^{2}\partial n_{e}/\partial\mu_{e}}$ is the Thomas–Fermi screening wave number.

The dispersion relation for the collective modes is given by the condition $\det M=0$, where $M$ is the matrix in Eq. (24). In general, there are three collective modes, corresponding to the three components: two acoustic modes with in-phase and out-of-phase motions of the neutrons and the charged particles, and the plasma oscillation of the electrons, which has a nonzero frequency $\approx(4\pi n_{e}e^{2}c^{2}/\mu_{e})^{1/2}$ as $k\to 0$. The present formalism is inadequate to calculate the frequency shift of the electron plasma oscillation due to the motion of neutrons and protons because the frequency is high compared with the gap frequencies and consequently the hydrodynamic approximation for the neutrons and protons is not valid.

III.1 Uncoupled modes

It is instructive to investigate the case when there is no coupling between neutrons and protons ($E_{np}=0,n_{np}=0,n_{pp}=n_{p},$ and $n_{nn}=n_{n}$). The dispersion relation for the collective mode of electrons and protons is then

[TABLE]

where $\Omega_{p}=({4\pi e^{2}n_{p}}/{m})^{1/2}$ is the proton plasma frequency.

Because of the frequency- and wave number dependence of $\epsilon_{e}$, Eq. (26) has two solutions, the electron plasma oscillation and an acoustic mode, which has a velocity small compared with $c$. To calculate the frequency of the latter mode we may expand $\chi_{0}$ in powers of $s$,

[TABLE]

As we shall show, to take into account the nonzero effective mass of the electrons it is necessary to retain terms of order $s^{2}$. We write the frequency of the mode in terms of its real part $\omega^{\prime}$ and its imaginary part $\omega^{\prime\prime}$. In the long-wavelength limit, one finds

[TABLE]

The coefficient of $\omega^{2}$ in Eq. (28) shows that an electron contributes an amount $(\pi^{2}/4-1)\mu_{e}/3c^{2}\approx 0.49\mu_{e}/c^{2}$ to the effective mass of the charged particles.

If one further neglects $E^{\rm nuc}_{pp}$ and considers the long-wavelength limit $k\to 0$. Eq. (28) is the analog for relativistic electrons of the expression for the sound speed in metals derived by Bohm and Staver BohmStaver . The leading contribution to $\omega^{\prime\prime}$ is

[TABLE]

This is due to Landau damping, the decay of the mode into electron–hole pairs Landau1945 .

If collisions between electrons are so frequent that conditions are hydrodynamic, the response function for the electrons has the form

[TABLE]

where $c_{e}=c/\sqrt{3}$ is the speed of hydrodynamic (first) sound in an ultrarelativistic Fermi gas. With this expression for $\chi$, one sees that the effective mass of an electron is $\mu_{e}/c^{2}$, or roughly twice the value for the collisionless limit. For $E_{pp}^{\rm nuc}\neq 0$, $\mu_{e}$ in Eqs. (28)-(29) is replaced by $\mu_{e}+3E_{pp}^{\rm nuc}/n_{p}$. Since $\mu_{e}\sim$ 100 MeV, the effects of the electron inertia in neutron star cores are at the 10% level and are therefore much more important than in terrestrial matter. However, the effects are comparable to the deviations of the energy per particle from the rest mass, which we have neglected in this article, where we have treated the nucleons nonrelativistically.

Due to incomplete screening at higher wave numbers, the velocity of the proton–electron mode depends on $k$. On solving the eigenvalue problem to first order in $\epsilon_{e}^{\prime\prime}=\Im\epsilon_{e}$ one finds

[TABLE]

and

[TABLE]

which shows that the velocity and damping of modes is markedly reduced for $k\gtrsim k_{TF}$. Equation (31) shows that both nucleons and electrons contribute to the bulk modulus of the medium.

The frequency of the neutron mode when coupling between neutrons and protons is neglected is given by

[TABLE]

III.2 The general case

The solution of the eigenvalue problem for Eq. (24) written in terms of the phase velocities $v=\omega/k$ of the modes has the form

[TABLE]

where

[TABLE]

where $v_{p0}$ and $v_{n0}$ are the velocities of the modes in the absence of coupling between neutrons and protons, given by Eqs. (26) and (33). To bring out the effects of the entrainment, it is useful to calculate mode frequencies in the absence of entrainment,

[TABLE]

The supercript $sc$ indicates that only the scalar coupling between components is present, and the vector coupling that gives rise to entrainment is absent. Equation (34) is algebraically equivalent to the results derived for collective mode velocities in the inner crust of a neutron star in Ref. (KobyakovPethick2013, , Sec. III) from the standard two-fluid model. The fact that, in the neutron star crust, the charged particles are not superfluid is irrelevant for the mode frquencies, because in that work the ions and electrons were treated using a hydrodynamic approtach. From Eq. (34) one sees that there is a zero frequency mode if $\det[E_{\alpha\beta}]$ or $\det[n_{\alpha\beta}]$ vanish. This reflects the fact that the uniform system with no flow of protons or neutrons is unstable to formation of a density wave if $\det[E_{\alpha\beta}]=0$ and to relative flow of the neutrons and protons if $\det[n_{\alpha\beta}]=0$.

In the long-wavelength limit the imaginary part of $E_{pp}$ can be obtained by combining Eqs. (23), (25) and (27) and Taylor-expanding $\chi_{0}^{-1}$: $E_{pp}^{\prime\prime}=E_{ee}^{\prime\prime}\approx-(\pi/2)E_{ee}^{\prime}\omega/ck$, where the real part of $E_{ee}$ is $E_{ee}^{\prime}=\partial\mu_{e}/\partial n_{e}$. Thus, the dispersion relation for the modes, the condition that the determinant of the matrix in Eq. (24) vanish, may be written

[TABLE]

where ${\tilde{v}}_{\pm}$ are the mode velocities when $E_{ee}^{\prime\prime}=0$. To first order in $E_{ee}^{\prime\prime}$ the real parts of the velocities are equal to $v_{\pm}^{\prime}$ and the imaginary parts of the velocities are thus given by

[TABLE]

and

[TABLE]

where the $v_{\pm}$ on the right sides are to be evaluated in the absence of damping.

III.3 Physical character of the modes

The coupled modes generally involve oscillations of both the neutron and proton densities. On eliminating the momentum variables $p_{p}$ and $p_{n}$ from the linear system given in Eq. (24), we find

[TABLE]

Therefore the ratio of the neutron and proton density variations in the eigenmode with velocity $v_{\pm}$ is given by

[TABLE]

It is convenient to define mixing angles by the equations

[TABLE]

and

[TABLE]

If $\gamma_{n+}=0$, the mode $v_{+}$ involves motion only of the charged particles. When $0<\gamma_{n+}<\pi/2$, the mode $v_{+}$ corresponds to an in-phase oscillation of the proton and neutron densities, and when $-\pi/2<\gamma_{n+}<0$, to an out-of-phase oscillation. Analogously, the mode with velocity $v_{-}$ is a pure neutron oscillation when $\gamma_{p-}=0$, an in-phase oscillation of the neutron and proton densities when $0<\gamma_{p-}<\pi/2$, and an out-of-phase oscillation when $-\pi/2<\gamma_{p-}<0$.

IV Calculation of mode frequencies

In this section we apply the formalism to calculate mode frequencies in the outer core. For the nucleon contributions to the quantities $E_{\alpha\beta}$ we use the results of Hebeler et al. HebelerEtal2013 , which were earlier employed in calculations of mode frequencies at long wavelengths KobyakovCJP . In our numerical calculations we use values of $E_{\alpha\beta}^{nuc}$ based on chiral effective field theory, because we regard these as the best available. However, since $f_{1}^{np}$ has not been calculated in detail for these interactions, we used for that the SLy4 value, Eq. (49), and estimates (56) and (59) based on arguments from nucleon effective masses obtained in chiral EFT models. The other important piece of input is the quantity $n_{np}$ that determines the strength of entrainment. Chamel and Haensel showed that values of $n_{np}$ obtained from Skyrme interactions had a large spread ChamelHaensel , and in Ref. KobyakovCJP the result for the SLy4 Skyrme interaction,

[TABLE]

which is representative of results found for many other Skyrme interactions, was adopted.

An alternative approach to estimating $n_{np}$ is to use information on nucleon effective masses, which are linked to the Landau parameters by the relations Sjoeberg

[TABLE]

and

[TABLE]

Thus, for symmetric nuclear matter at the saturation density, it follows from Eq. (50) that

[TABLE]

where $k_{0}=(3\pi^{2}n_{0}/2)^{1/3}$ is the Fermi wave number and $m^{*}_{0}$ the nucleon effective mass for matter under those conditions. To obtain information about $f_{1}^{nn}$, we make use of results for the limiting cases of low neutron densities and pure neutron matter. For low neutron densities, $f_{1}^{nn}$ tends to zero. Calculations for pure neutron matter indicate that $m^{*}_{n}/m\approx 1.0-1.2,~{}0.9-1.1$, and $0.8-1.0$ for densities $n=0.5n_{0},n_{0}$, and $1.5n_{0}$, respectively RGnmatt ; Kainmatt . These results are all consistent with an effective mass equal to the bare mass, which would imply that $f_{1}^{nn}$ for pure neutron matter is zero. Therefore the simplest assumption for the dependence of $f_{1}^{nn}$ on proton fraction is that it vanishes for all neutron densities, in which case Eq. (52) becomes

[TABLE]

and, from Eq. (9),

[TABLE]

If one assumes that $f_{1}^{np}$ is proportional to the product of the neutron and proton Fermi wavenumbers, as it is for the Skyrme interaction, one thus finds

[TABLE]

For symmetric nuclear matter, the nucleon effective mass is typically taken to be 0.7$m$ RGnmatt ; Kainmatt , but it could be larger. If one is conservative and takes the value to be $0.8m$, one finds

[TABLE]

roughly twice the value (49) obtained from the SLy4 Skyrme interaction.

We now estimate the changes in this result due to a nonzero value of $f_{1}^{nn}$. Let us denote the effective mass of neutrons in pure neutron matter at density $n_{0}$ by $m^{*}_{\rm NM}$. It then follows from Eq. (50) that

[TABLE]

where $k_{\rm NM}=(3\pi^{2}n_{0})^{1/3}$ is the Fermi wave number of neutron matter. If $f_{1}^{nn}$ scales with neutron density in the same way as it does for a Skyrme interaction ($\propto k_{n}^{2}$), this gives for this quantity for symmetric nuclear matter at density $n_{0}$,

[TABLE]

Therefore, from Eq. (52), one finds

[TABLE]

This shows that for $m_{0}=0.8\,m$, a neutron effective mass of $1.1m$ rather than $m$ for pure neutron matter would increase the magnitude of $f_{1}^{np}$, and hence also that of $n_{np}$, by $\sim 20\%$.

Throughout these arguments, we have assumed that the Landau parameters scale with Fermi momentum as they would for a Skyrme interaction. How good this assumption should be investigated using microscopic calculations. Results for low-density Fermi gases exhibit other scalings with density, e.g., for a spin-1/2 one component Fermi gas, $f_{1}$ scales as the Fermi wave number, not its square. This is due to the fact that this term comes from the induced interaction. However, because scattering lengths for nucleon–nucleon scattering are so large, we do not expect the scaling behavior predicted by the low-density gas to be relevant at most densities of interest in neutron stars.

Using the effective masses discussed above, we have evaluated the entrainment parameters defined in Eqs. (7), (8), and (9) and these are shown in Table 1 for characteristic densities in the neutron star outer core. The table also shows the derivatives of the energy density $E_{\alpha\beta}$ obtained using the expression for the energy density described in Ref. HebelerEtal2013 .

We write the mode velocity in terms of its real part, $v^{\prime}$, and its imaginary part $v^{\prime\prime}$ and results for these quantities are shown in Figs. 1 and 2, respectively. Only Landau damping is included in calculating the imaginary parts. This is valid when $k<2\Delta/v^{\prime}$, since for larger wave numbers pair-breaking is possible.

Equation (34) determines the real and imaginary parts of the eigenmode velocities and from them one may obtain the mean free paths,

[TABLE]

where $\tau_{k}=1/{2kv^{\prime\prime}}$ is the relaxation time of the hydrodynamic motion. The factor of $1/2$ comes from the fact that the relaxation of the hydrodynamic motion is linked to damping of the intensity of the mode, which is the amplitude squared, while the amplitude is exponentially damped on a time scale $1/{kv^{\prime\prime}}$. At nuclear matter density and at long wavelengths for a typical thermal phonon frequency $\omega=3k_{B}T/\hbar$ with $T=10^{8}$ K, the mean free path of the predominantly neutron mode is $3.022\times 10^{-9}$ cm for the case of no entrainment, $3.248\times 10^{-9}$ cm for $n^{(a)}_{np}$ and $3.321\times 10^{-9}$ cm for $n^{(b)}_{np}$, while for the predominantly charged particle mode the corresponding results are $7.003\times 10^{-10}$ cm, $6.454\times 10^{-10}$ cm, and $6.492\times 10^{-10}$ cm. The mean free paths for a thermal excitation vary as $1/T$ and numerically they are much shorter than those due to phonon–phonon scattering, which diverge as a higher inverse power of $T$ at low temperatures ManuelTolos2011 .

For weak mixing, $E_{np}^{2}\ll{E_{nn}E_{pp}}$ and $n_{np}^{2}\ll{n_{n}n_{p}}$, we can obtain from Eqs. (38) and (39) simple expressions for the damping of the modes. Here and in the remainder of this section $E_{pp}$ and $E_{ee}$ refer to the real parts of these quantities. For the predominantly charged-particle mode one finds

[TABLE]

and the mean free path is given by

[TABLE]

For the predominantly neutron mode the corresponding results are

[TABLE]

where

[TABLE]

and

[TABLE]

The results have limited utility for the situation considered in this paper because there is an avoided crossing of the two sound speeds, as one can see from Fig. 3. In the vicinity of the avoided crossing, the mixing is strong. Another complicating feature is that the quantity $n_{p}E_{pn}+n_{np}E_{nn}$ that enters in the expression for $\zeta$ passes through zero at a density close to that of the avoided crossing. This results in vanishing of the damping of the lower frequency mode (and a divergence of the mean free path) at a particular density, as may be seen in Fig. 4.

In earlier work, Bedaque and Reddy BedaqueReddy2014 , obtained a rough estimate of these mean free paths by neglecting effects due to entrainment. The range of values they predicted is compatible with the detailed estimates presented here. It is also worth noting that in the neutron star inner crust, where phonons of the neutron superfluid mix with the longitudinal lattice phonons, the predominantly neutron mode with thermal energy was found to have a mean free path in the range $10^{-6}$–$10^{-3}$ cm, much larger than that associated with lattice phonon modes, suggesting that heat transport due to phonons could play a role Aguilera2009 . However, revised estimates which included the effects due to entrainment lead to larger mixing and stronger damping of the neutron mode ChamelPageReddy .

V Astrophysical considerations and concluding remarks

The dispersion and decay of collective modes in superfluid and superconducting dense matter can influence observable aspects of neutron star evolution. At long wavelengths, these modes are relevant to understanding neutron star seismology. Short wavelength modes can contribute to the heat capacity and thermal conductivity, both of which play a role in the thermal evolution. In the following we shall explore some of the implications of the results obtained in the preceding section for neutron star phenomenology.

The temperatures of interest in isolated neutron stars (older than a few hundred years) are expected to be in the range $10^{7}$-$10^{9}$ K. In the core where nucleons condense to form Cooper pairs, the relevant contributions to the heat capacity are due to electrons and collective modes. The collective mode contribution to the heat capacity can be obtained from the dispersion relations we calculated in the preceding section. At low temperature when the typical momentum $k\ll k_{TF}$, the dispersion relations are approximately linear with $\omega\simeq v^{\prime}_{\pm}k$, and the associated heat capacity for each of the collective modes is given by

[TABLE]

In contrast the electron contribution in the core

[TABLE]

is significantly larger. The contribution from the collective modes is small because of the higher power of the temperature and the relatively high velocities of the modes, $\gtrsim 0.1c$.

The coefficient of thermal conduction for a single species of bosonic mode with a dispersion relation $\omega_{k}$ is given by standard kinetic theory

[TABLE]

where $n_{k}=1/({\rm e}^{\hbar\omega_{k}/k_{B}T}-1)$ is the Bose distribution function, $\tau_{k}=1/(2\omega_{k}^{\prime\prime})$ is the relaxation time of the mode and $k_{B}$ is the Boltzmann constant. For wave numbers small compared with the electron screening wave number, the velocity of the modes may be taken to be constant and, on performing the sum in Eq. (68), one finds the simple result

[TABLE]

The thermal conductivity does not depend on the real part of the velocity of the mode, and consequently the modes with the smallest $v^{\prime\prime}$ contribute most to heat transport.

It is interesting to compare thermal conductivity of bosonic excitations in superfluid neutrons given in Eq. (69) with other contributions to the thermal conductivity. Thermal conductivity of fermionic excitations in superfluid neutrons and the electron contributions were calculated in ShterninYakovlev2007 , and found to be of the order of $10^{24}$ erg/(cm sec K), which shows that the bosonic excitations provide a negligible contribution to the thermal conductivity.

The frequency spectrum of the longitudinal modes that can be excited in the core is sensitive to the real part of the velocity in the long-wavelength limit and will in general differ from the sound velocity defined by $\sqrt{\partial P/\partial\rho}$, where $P$ is total pressure and $\rho c^{2}$ is the total energy density. In the parlance of stellar oscillations these modes are referred to as pressure modes, or simply p-modes. The existence of two acoustic modes in the long-wavelength limit is a unique feature of the superfluid-superconducting multi-component system that we have studied here, and it differs qualitatively from what is expected in a normal fluid where only one mode exists, with velocity $c_{s}$. Further, the difference in mode velocities is small and this may lead to unique neutron star seismology.

Perturbations either during accretion, thermonuclear x-ray bursts and superbursts, and neutron stars mergers may excite modes in the core but it is unclear if neutron star seismology can be probed observationally. One model of quasi-periodic oscillations observed in the tails of magnetar giant flares invokes a coupling between internal modes of the neutron star crust and the global magnetic field to provide a mechanism by which surface emission can be modulated at characteristic frequencies QPO . It would be interesting to explore if a similar coupling between the magnetic field and core p-modes can lead to observable effects.

Acknowledgments

D.N.K. gratefully acknowledges the hospitality of the Niels Bohr Institute, Copenhagen, and of the Ioffe Physical-Technical Institute, Saint Petersburg. This work was supported in part by the Russian Fund for Basic Research Grant 31 16-32-60023/15, the European Research Council Grant No. 307986 STRONGINT, the US Department of Energy Grant No. DE-FG02-00ER41132, and by NewCompStar, COST Action MP1304.

Appendix A Equivalence of two approaches to superfluid dynamics

In the traditional approach to describing superfluidity LLFluidMech one works in terms of the so-called superfluid velocity, ${\bf v}_{s}={\bm{\nabla}}\phi/m$, which is really a momentum, and the velocity of the normal component. This has previously been applied to the superfluid neutrons in the inner crust of neutron stars CJPChamelReddy ; KobyakovPethick2013 ; ChamelPageReddy . In the present context, when both neutrons and protons are superfluid, it is natural to treat them symmetrically, as we have done in the text. However, as we shall demonstrate, the equations of motion are equivalent to those in the traditional formalism. For simplicity we consider the case of small perturbations about a uniform state with both components at rest. We express the equations of motion (16) and (17) in terms of the average velocity of the protons, which is the proton current density divided by the proton number density:

[TABLE]

The continuity equations (11) and (10) for nucleons therefore become

[TABLE]

and

[TABLE]

Equations (71) and (72) have the form to be expected for a two-fluid model, even though in the present case the two components are both superfluid. The neutron current density is given by

[TABLE]

and therefore the superfluid neutron density in the notation of Refs. CJPChamelReddy ; KobyakovPethick2013 is given by

[TABLE]

while the density of neutrons entrained with the protons is given by

[TABLE]

In the traditional approach, the superscript $s$ refers to the superfluid component and $n$ to the normal component. In the present case, both components are superfluid, and the superscript $n$ refers to the component whose dynamics is described in terms of the average velocity of the component, rather than the momentum per particle in the condensate. The equation of motion for ${\mathbf{p}}_{n}$, Eq. (17) has the usual form as in the two-fluid model. We turn now to the equation of motion for $\bar{\mathbf{v}}_{p}$. From Eqs. (16) and (17) it follows that

[TABLE]

The total normal mass density is given by

[TABLE]

From Eq. (75) one thus sees that

[TABLE]

where the latter result is a consequence of the condition (4) for Galilean invariance. Thus Eq. (77) may be written in the form

[TABLE]

This result is consistent with KobyakovPethick2013 , Eq. 25, for the inner crust if one there neglects the shear rigidity of the solid. Equation (80) represents a generalization of earlier results CJPChamelReddy ; ChamelPageReddy ; KobyakovPethick2013 in that charge neutrality is not enforced locally.