We consider sound waves in superfluid nucleon-hyperon matter of massive neutron-star cores. We calculate and analyze the speeds of sound modes and their damping times due to the shear viscosity and non-equilibrium weak processes of particle transformations. For that, we employ the dissipative relativistic hydrodynamics of a superfluid nucleon-hyperon mixture, formulated recently [M.E. Gusakov and E.M. Kantor, Phys. Rev. D78, 083006 (2008)]. We demonstrate that the damping times of sound modes calculated using this hydrodynamics and the ordinary (nonsuperfluid) one, can differ from each other by several orders of magnitude.
In recent years there is a growing interest in studies of neutron-star pulsations. This is related to a number of reasons. First of all, the recently discovered high frequency oscillations of electromagnetic radiation during giant flares may be associated with the pulsations of neutron stars [1,2]. Second, the gravitational-wave detectors, which will be able to detect gravitational radiation from isolated pulsating neutron stars, are under construction [3,4,5,6].
For interpretation of the observations, it is important to have a well-developed theory of neutron-star pulsations. The construction of such a theory is complicated by the fact that the baryons in neutron-star cores can be in superfluid state [7,8,9,10,11,12]. Thus, to study the pulsations, one has to use a hydrodynamics describing mixtures of superfluid liquids. There is a substantial body of literature, devoted to pulsations of superfluid neutron stars (see, e.g., Refs. [13,14,15,16,17,18,19,20,21,22,23,24,25,26]). All these papers deal with the nucleon npe(µ) matter composed of neutrons (n), protons (p), and electrons (e) with possible admixture of muons (µ). Most of them study pulsations at zero temperature (see, however, Refs. [15,16]).
In this paper we for the first time investigate dynamic properties of superfluid nucleon-hyperon matter in the cores of massive neutron stars, composed, in addition to neutrons, protons, electrons, and muons, of Λ and Σ -hyperons (Λ and Σ, respectively). For that, we employ the relativistic hydrodynamics [27], describing a superfluid nucleon-hyperon mixture at arbitrary temperature. We study the simplest pulsations in such matter -sound modes, how they travel and how they damp. Within this simple example we demonstrate, in particular, that the characteristic damping times of pulsations, calculated self-consistently in the frame of superfluid hydrodynamics, can differ by several orders of magnitude from those calculated using the nonsuperfluid hydrodynamics (in the latter case the effects of superfluidity are taken into account only at calculating kinetic coefficients).
The paper is organized as follows. In Sec. II we give an overview of the main reactions of particle mutual transformations in the nucleon-hyperon matter. In Sec. III we briefly discuss the relativistic dissipative hydrodynamics describing superfluid nucleon-hyperon mixture. In Sec. IV we analyze the sound modes in such matter neglecting dissipation. In Sec. V we calculate the damping times of sound modes due to the nonequilibrium reactions (1)-(4) (see Sec. II) and shear viscosity. Section VI presents summary.
Throughout the paper, unless otherwise stated, we use the system of units in which the Plank constant , the speed of light c and the Boltzmann constant k B equal unity, = c = k B = 1.
The most effective weak processes in nucleon-hyperon matter are the following nonleptonic reactions [28,29,30,31] n
The full thermodynamic equilibrium implies, in particular, the equilibrium with respect to these reactions, δµ 1 ≡ 2µ n0 -µ p0 -µ Σ0 = 0, (5) δµ 2 = δµ 3 = δµ 4 ≡ µ n0 -µ Λ0 = 0. (6) In this case the average number of direct reactions in unit volume per unit time is equal to the number of inverse reactions. In Eqs. ( 5) and ( 6) µ i0 are the chemical potentials of particle species i = n, p, Λ, and Σ, taken at equilibrium; δµ m (m = 1,. . ., 4) are the disbalances of the chemical potentials for the reactions (1)- (4). In what follows, we mark the equilibrium values of thermodynamic quantities with the subscript 0. Accordingly, the thermodynamic quantities without the subscript 0 (e.g., µ i ) refer to perturbed matter. Notice that, the equilibrium conditions for the reactions (2), (3), and (4) coincide.
Eqs. ( 5) and ( 6) do not hold in the perturbed matter (δµ 1 = 0, δµ 2 = δµ 3 = δµ 4 = 0), so that the numbers of direct and inverse reactions are not equal. The nonequilibrium reactions (1)-( 4) act to return the system to the equilibrium state. This leads to dissipation of mechanical energy, accumulated in the matter.
Along with the reactions (1)-( 4) there is a number of weak reactions with leptons. The leptonic reactions (e.g, the direct and modified Urca processes with electrons or muons) are much slower in comparison to the reactions (1)-( 4). In the interesting range of parameters (temperatures and pulsation frequencies), they cannot influence substantially chemical composition of the perturbed matter and will be neglected in what follows. However, we assume that the unperturbed matter satisfies the equilibrium conditions with respect to these reactions,
where µ e0 and µ µ0 are the equilibrium chemical potentials for electrons and muons, respectively. In addition to the processes described above, there is a fast nonleptonic reaction due to the strong interaction of baryons,
We assume that the perturbed matter is always in equilibrium with respect to this reaction [27,30],
It follows from the condition (9), that the chemical potenti
