ABSTRACT
The main contribution to the effective shear modulus of neutron star crust can be calculated within Coulomb solid model and can be approximated by simple analytical expression for arbitrary ...(even multicomponent) composition. Here I consider correction associated with electron screening within Thomas–Fermi approximation. In particular, I demonstrate that for relativistic electrons (density ρ > 106 g cm−3) this correction can be estimated as $\delta \mu _\mathrm{eff}^\mathrm{V}= -9.4\times 10^{-4}\sum _Z n_Z Z^{7/3} e^2/a_\mathrm{e},$ where summation is taken over ion species, nZ is number density of ions with charge Ze, kTF is Thomas–Fermi screening wavenumber. Finally, ae = (4πne/3)−1/3 is electron sphere radius. Quasi-neutrality condition ne = ∑ZZnZ is assumed. This result holds true for arbitrary (even multicomponent and amorphous) matter and can be applied for neutron star crust and (dense) cores of white dwarfs. For example, the screening correction reduces shear modulus by ∼9 per cent for Z ∼ 40, which is typical for inner layers of neutron star crust.
ABSTRACT
I discuss elastic properties of neutron star crust in the framework of static Coulomb solid model when atomic nuclei are treated as non-vibrating point charges; electron screening is ...neglected. The results are also applicable for solidified white dwarf cores and other materials, which can be modelled as Coulomb solids (dusty plasma, trapped ions, etc.). I demonstrate that the Coulomb part of the stress–strain tensor has additional symmetry: contraction Bijil = 0. It does not depend on the structure (crystalline or amorphous) and composition. I show as a result of this symmetry the effective (Voigt averaged) shear modulus of the polycrystalline or amorphous matter to be equal to −2/15 of the Coulomb (Madelung) energy density at undeformed state. This result is general and exact within the model applied. Since the linear mixing rule and the ion sphere model are used, I can suggest a simple universal estimate for the effective shear modulus: $\sum _Z 0.12\, n_Z Z^{5/3}e^2 /a_\mathrm{e}$. Here summation is taken over ion species, nZ is number density of ions with charge Ze. Finally, ae = (4πne/3)−1/3 is electron sphere radius. Quasi-neutrality condition ne = ∑ZZnZ is assumed.
ABSTRACT
The elasticity of neutron star crust is important for adequate interpretation of observations. To describe elastic properties one should rely on theoretical models. The most widely used is ...Coulomb crystal model (system of point-like charges on neutralizing uniform background), in some works it is corrected for electron screening. These models neglect finite size of nuclei. This approximation is well justified except for the innermost crustal layers, where nuclei size becomes comparable with the inter-nuclear spacing. Still, even in those dense layers it seems reasonable to apply the Coulomb crystal result, if one assumes that nuclei are spherically symmetric: Coulomb interaction between them should be the same as interaction between point-like charges. This argument is indeed correct; however, as we point here, shear of crustal lattice generates (microscopic) quadrupole electrostatic potential in a vicinity of lattice cites, which induces deformation on the nuclei. We analyse this problem analytically within compressible liquid drop model. In particular, for ground state crust composition the effective shear modulus is reduced for a factor of $1-u^{5/3}/(2+3\, u-4\, u^{1/3})$, where u is the ratio of the nuclei volume to the volume of the cell. This result is universal, i.e. it does not depend on the applied nucleon interaction model within applied approach. For the innermost layers of inner crust u ∼ 0.2 leading to reduction of the shear modulus by $\sim 25{{\ \rm per\ cent}}$, which can be important for correct interpretation of quasi-periodic oscillations in the tails of magnetar flares.
Abstract
Dewitt et al. derived a useful relation between the plasma screening factor for a reaction of two fusing ions and their chemical potentials, based on the plasma pair distribution functions. ...We show that their result can be derived in a simpler, more straightforward way, by applying the principle of detailed balance, which also enables us to generalize the relation to reactions involving N fusing ions. In order to demonstrate the usefulness of applying the principle of detailed balance, we calculate the screening factor for the pep reaction, p + e + p → 2D +νe. For the plasma conditions near the centre of the Sun, the reaction is suppressed by roughly the same amount by which the reaction p + p → 2D + e+ + νe is enhanced. This effect may be measured in the near future.
With a spin frequency of 707 Hz, PSR J0952−0607 is the second fastest spinning pulsar known. It was discovered in radio by LOFAR in 2017 at an estimated distance of either 0.97 or 1.74 kpc and has a ...low-mass companion with a 6.42 hr orbital period. We report the discovery of the X-ray counterpart of PSR J0952−0607 using XMM-Newton. The X-ray spectra can be well-fit by a single power law (PL) model (Γ 2.5) or by a thermal plus PL model ( and Γ 1.4). We do not detect evidence of variability, such as that due to orbital modulation from pulsar wind and companion star interaction. Because of its fast spin rate, PSR J0952−0607 is a crucial source for understanding the r-mode instability, which can be an effective mechanism for producing gravitational waves. Using the high end of our measured surface temperature, we infer a neutron star core temperature of ∼107 K, which places PSR J0952−0607 within the window for the r-mode to be unstable unless an effect such as superfluid mutual friction damps the fluid oscillation. The measured luminosity limits the dimensionless r-mode amplitude to be less than ∼1 × 10−9.
Rezzolla et al. drew attention to the second-order secular drift associated with r-modes and claimed that it should lead to enhancement of the magnetic field and suppression of r-mode instability in ...magnetized neutron stars. We critically revise these results. We present a particular second-order r-mode solution with vanishing secular drift, thus refuting a widely believed statement that secular drift is an unavoidable feature of r-modes. This non-drifting solution is not affected by a magnetic field B, if B ≪ B
crit ≈ 1017 (ν/600 Hz) G (ν is the spin frequency) and does not lead to secular evolution of the magnetic field. For a general second-order r-mode solution, the drift does not necessarily vanish. The solution can be presented as a superposition of two solutions: one describes the evolution of differential rotation of a non-oscillating star (secular drift; for an unmagnetized star it is an arbitrary stationary rotation stratified on cylinders; for a magnetized star differential rotation evolves on the Alfvén timescale and may lead to enhancement of the magnetic energy), and the other is the non-drifting r-mode solution mentioned above. This representation allows us to conclude that enhancement of the magnetic field energy is limited by the initial energy of differential rotation, which is much less than the total energy of the r-mode (by a factor ∝ α2, where α is the mode amplitude). Hence, enhancement of the magnetic field by drift cannot suppress the r-mode instability. The results can be generalized for any oscillation mode in any medium, if this mode has a non-drifting solution for B = 0.
We consider an instability of rapidly rotating neutron stars in low-mass x-ray binaries (LMXBs) with respect to excitation of r modes (which are analogous to Earth's Rossby waves controlled by the ...Coriolis force). We argue that finite temperature effects in the superfluid core of a neutron star lead to a resonance coupling and enhanced damping (and hence stability) of oscillation modes at certain stellar temperatures. Using a simple phenomenological model we demonstrate that neutron stars with high spin frequency may spend a substantial amount of time at these "resonance" temperatures. This finding allows us to explain puzzling observations of hot rapidly rotating neutron stars in LMXBs and to predict a new class of hot, nonaccreting, rapidly rotating neutron stars, some of which may have already been observed and tentatively identified as quiescent LMXB candidates. We also impose a new theoretical limit on the neutron star spin frequency, which can explain the cutoff spin frequency ∼730 Hz, following from the statistical analysis of accreting millisecond x-ray pulsars. In addition to explaining the observations, our model provides a new tool to constrain superdense matter properties by comparing measured and theoretically predicted resonance temperatures.
Neutron stars are the densest objects in the Universe. They have a microscopically homogeneous core and heterogeneous crust. In particular, there may be a specific layer inside neutron stars, the ...mantle, which consists of substantially non-spherical nuclei immersed in a background of relativistic degenerate electrons and quasi-free neutrons. In this paper, we reconsider the transverse shear modulus for cylindrical phases of the mantle within the framework of the compressible liquid drop model. We demonstrate that transverse shearing affects the shape of nuclear clusters: their cross-section becomes elliptical. This effect reduces the respective elastic constant. Using a simple model, we perform all derivations analytically and obtain the expression for the transverse shear modulus, which can be useful for astrophysical applications.
At second order in perturbation theory, the unstable r-mode of a rotating star includes growing differential rotation whose form and growth rate are determined by gravitational-radiation reaction. ...With no magnetic field, the angular velocity of a fluid element grows exponentially until the mode reaches its nonlinear saturation amplitude and remains nonzero after saturation. With a background magnetic field, the differential rotation winds up and amplifies the field, and previous work where large mode amplitudes were considered L. Rezzolla, F. K. Lamb, and S. L. Shapiro, Astrophys. J. 531, L139 (2000)., suggests that the amplification may damp out the instability. A background magnetic field, however, turns the saturated time-independent perturbations corresponding to adding differential rotation into perturbations whose characteristic frequencies are of order the Alfvén frequency. As found in previous studies, we argue that magnetic-field growth is sharply limited by the saturation amplitude of an unstable mode. In contrast to previous work, however, we show that if the amplitude is small, i.e., ≲10−4, then the limit on the magnetic-field growth is stringent enough to prevent the loss of energy to the magnetic field from damping or significantly altering an unstable r-mode in nascent neutron stars with normal interiors and in cold stars whose interiors are type II superconductors. We show this result first for a toy model, and we then obtain an analogous upper limit on magnetic-field growth using a more realistic model of a rotating neutron star. Our analysis depends on the assumption that there are no marginally unstable perturbations, and this may not hold when differential rotation leads to a magnetorotational instability.
Nuclear pasta phases in the neutron stars mantle can affect the mechanical and transport properties of superdense matter, thus playing an important role in the dynamics and evolution of neutron ...stars. In this paper, we compare results obtained by the Extended Thomas–Fermi (ETF) method with the compressible liquid drop model (CLDM), based on the thermodynamically consistent description of the surface properties calculated for the two-phase plane interface and the same energy-density functional (for numerical illustration, we applied the Skyrme-type functional SLy4). Our ETF calculations found that pasta phases in cylindrical form cover a significant crustal region (both normal and inverse phases, aka spaghetti and bucatini are presented). Meanwhile, within the applied CLDM framework, which includes the thermodynamically required effect of neutron adsorption on the cluster’s surface but neglects curvature corrections, only the spaghetti phase was found to be energetically favorable in the small density range prior to crust–core transition. On the other hand, the recent CLDM of Dinh Thi et al., 2021, which, on the contrary, accounts for curvature term but neglects neutron adsorption, predicts pasta phase onset in better agreement with the ETF. This fact highlights the importance of the curvature effects and allows counting on the potential validity of the CLDMs as a convenient, transparent and accurate tool for investigation of the pasta-phase properties.
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