The I-Love-Q Relations for Superfluid Neutron Stars

The I-Love-Q relations are approximate equation-of-state independent relations that connect the moment of inertia, the spin-induced quadrupole moment, and the tidal deformability of neutron stars. In this paper, we study the I-Love-Q relations for superfluid neutron stars for a general relativistic two-fluid model: one fluid being the neutron superfluid and the other a conglomerate of all charged components. We study to what extent the two-fluid dynamics might affect the robustness of the I-Love-Q relations by using a simple two-component polytropic model and a relativistic mean field model with entrainment for the equation-of-state. Our results depend crucially on the spin ratio Ωn/Ωp between the angular velocities of the neutron superfluid and the normal component. We find that the I-Love-Q relations can still be satisfied to high accuracy for superfluid neutron stars as long as the two fluids are nearly co-rotating Ωn/Ωp≈1. However, the deviations from the I-Love-Q relations increase as the spin ratio deviates from unity. In particular, the deviation of the Q-Love relation can be as large as O(10%) if Ωn/Ωp differ from unity by a few tens of percent. As Ωn/Ωp≈1 is expected for realistic neutron stars, our results suggest that the two-fluid dynamics should not affect the accuracy of any gravitational waveform models for neutron star binaries that employ the relation to connect the spin-induced quadrupole moment and the tidal deformability.

Method of integral equations in the polytropic theory of stars with axial rotation. I. Polytropes n=0 and n=1

Calculations of characteristics of stars with axial rotation in the frame of polytropic model are based on the solution of mechanical equilibrium equation – differential equation of second order in partial derivatives. Different variants of approximate determinations of integration constants are based on traditional in the theory of stellar surface approximation, namely continuity of gravitational potential in the surface vicinity. We proposed a new approach, in which we used simultaneously differential and integral forms of equilibrium equations. This is a closed system and allows us to define in self-consistent way integration constants, the polytrope surface shape and distribution of matter over volume of a star. With the examples of polytropes n=0 and n=1, we established the existence of two rotation modes (with small and large eccentricities). It is proved that the polytrope surface is the surface of homogeneous rotational ellipsoid for the case n=0. The polytrope characteristics with n=1 in different approximations were calculated as the functions of angular velocity. For the first time it has been calculated the deviation of polytrope surface at fixed value of angular velocity from the surface of associated rotational ellipsoid.

Stirling-Engine in a Novel Alphagamma Configuration – a Key for Maintenance-free Operation

Since 2001, Frauscher Thermal Motors have been conducting research in the field of thermodynamic machines, in particular Stirling engines of various types. One important development step is the invention of a Stirling engine in an alphagamma® configuration. In this configuration, the expansion piston is designed as a differential piston with its ring surface connected to the cold volume. In this paper, the design advantages of the alphagamma® configuration in comparison with a traditional alpha configuration are shown analytically by using a polytropic model as a modification of the ideal adiabatic analysis. The findings were confirmed by also simulating the proposed alphagamma® configuration in a Sage model which was validated against experimental data with very good agreement. The results of both methods show that the counter-productive compression work can be reduced to almost zero – which makes the compression piston a displacer and explains the name alphagamma® – with the expansion work also reduced for the same net work output. As a consequence, the forces on the pistons, and thus, on the bearings can be significantly reduced, also leading to smaller piston side-loads, less friction and wear. The combination of all advantages allows the design of a mechanically sound and inexpensive machine.

A numerical method for computing optimum radii of host stars and orbits of planets, with application to Kepler-11, Kepler-90, Kepler-215, HD 10180, HD 34445 and TRAPPIST-1

In the so-called “global polytropic model”, we assume planetary systems in hydrostatic equilibrium and solve the Lane–Emden equation in the complex plane. We thus find polytropic spherical shells providing accommodation to planetary orbits. On the basis of this model, we develop a numerical method which can compute optimum values for the polytropic index of the global polytropic model that simulates the planetary system, for the orbits of the planets, and for the host star radius. We apply our method to the exoplanetary systems Kepler-11, Kepler-90, Kepler-215, HD 10180, HD 34445 and TRAPPIST-1.

What if the neutron star maximum mass is beyond ∼2.3 M⊙?

ABSTRACT By assuming the formation of a black hole soon after the merger event of GW170817, the maximum mass of non-rotating stable neutron star, MTOV ≃ 2.3 M⊙, is proposed by numerical relativity, but there is no solid evidence to rule out MTOV > 2.3 M⊙ from the point of both microphysical and astrophysical views. It is naturally expected that the equation of state (EOS) would become stiffer beyond a specific density to explain massive pulsars. We consider the possibility of EOSs with MTOV > 2.3 M⊙, investigating the stiffness and the transition density in a polytropic model, for two kinds of neutron stars (i.e. gravity-bound and strong-bound stars on surface). Only two parameters are input in both cases: (ρt, γ) for gravity-bound neutron stars, while (ρs, γ) for strong-bound strange stars, with ρt the transition density, ρs the surface density, and γ the polytropic exponent. In the matter of MTOV > 2.3 M⊙ for the maximum mass and 70 ≤ Λ1.4 ≤ 580 for the tidal deformability, it is found that the smallest ρt and γ should be ∼0.50 ρ0 and ∼2.65 for neutron stars, respectively, whereas for strange star, we have γ > 1.40 if ρs > 1.0 ρ0 (ρ0 is the nuclear saturation density). These parametric results could guide further research of the real EOS with any foundation of microphysics if a pulsar mass higher than 2.3 M⊙ is measured in the future, especially for an essential comparison of allowed parameter space between gravity-bound and strong-bound compact stars.

Entropy stable numerical approximations for the isothermal and polytropic Euler equations

In this work we analyze the entropic properties of the Euler equations when the system is closed with the assumption of a polytropic gas. In this case, the pressure solely depends upon the density of the fluid and the energy equation is not necessary anymore as the mass conservation and momentum conservation then form a closed system. Further, the total energy acts as a convex mathematical entropy function for the polytropic Euler equations. The polytropic equation of state gives the pressure as a scaled power law of the density in terms of the adiabatic index $$\gamma $$ γ . As such, there are important limiting cases contained within the polytropic model like the isothermal Euler equations ($$\gamma {=}1$$ γ = 1 ) and the shallow water equations ($$\gamma {=}2$$ γ = 2 ). We first mimic the continuous entropy analysis on the discrete level in a finite volume context to get special numerical flux functions. Next, these numerical fluxes are incorporated into a particular discontinuous Galerkin (DG) spectral element framework where derivatives are approximated with summation-by-parts operators. This guarantees a high-order accurate DG numerical approximation to the polytropic Euler equations that is also consistent to its auxiliary total energy behavior. Numerical examples are provided to verify the theoretical derivations, i.e., the entropic properties of the high order DG scheme.

Regularities in the spectrum of chaotic p-modes in rapidly rotating stars

Context. Interpreting the oscillations of massive and intermediate mass stars remains a challenging task. In fast rotators, the oscillation spectrum of p-modes is a superposition of sub-spectra which corresponds to different types of modes, among which island modes and chaotic modes are expected to be the most visible. This paper is focused on chaotic modes, which have not been thoroughly studied before. Aims. We study the properties of high frequency chaotic p-modes in a polytropic model. Unexpected peaks appear in the frequency autocorrelations of the spectra. Our goal is to find a physical interpretation for these peaks and also to provide an overview of the mode properties. Methods. We used the 2D oscillation code “TOP” to produce the modes and acoustic ray simulations to explore the wave properties in the asymptotic regime. Using the tools developed in the field of quantum chaos (or wave chaos), we derived an expression for the frequency autocorrelation involving the travel time of acoustic rays. Results. Chaotic mode spectra were previously thought to be irregular, that is, described only through their statistical properties. Our analysis shows the existence, in chaotic mode spectra, of a pseudo large separation. This means that chaotic modes are organized in series, such that the modes in each series follow a nearly regular frequency spacing. The pseudo large separation of chaotic modes is very close to the large separation of island modes. Its value is related to the sound speed averaged over the meridional plane of the star. In addition to the pseudo large separation, other correlations appear in the numerically calculated spectra. We explain their origin by the trapping of acoustic rays near the stable islands.

A Polytropic Approximation of Compressible Flow in Pipes With Friction

This paper demonstrates the usefulness of treating subsonic Fanno flow (adiabatic flow, with friction, of a perfect gas in a constant-area pipe) as a polytropic process. It is shown that the polytropic model allows an explicit equation for mass flow rate to be developed. The concept of the energy transfer ratio is used to develop a close approximation to the polytropic index. Explicit equations for mass flow rate and net expansion factor in terms of upstream properties and pressure ratio are developed for Fanno and isothermal flows. An approximation for choked flow is also presented. The deviation of the results of this polytropic approximation from the values obtained from a traditional gas dynamics analysis of subsonic Fanno flow is quantified and discussed, and a typical design engineering problem is analyzed using the new method.

Shock location and CME 3D reconstruction of a solar type II radio burst with LOFAR

Context. Type II radio bursts are evidence of shocks in the solar atmosphere and inner heliosphere that emit radio waves ranging from sub-meter to kilometer lengths. These shocks may be associated with coronal mass ejections (CMEs) and reach speeds higher than the local magnetosonic speed. Radio imaging of decameter wavelengths (20–90 MHz) is now possible with the Low Frequency Array (LOFAR), opening a new radio window in which to study coronal shocks that leave the inner solar corona and enter the interplanetary medium and to understand their association with CMEs. Aims. To this end, we study a coronal shock associated with a CME and type II radio burst to determine the locations at which the radio emission is generated, and we investigate the origin of the band-splitting phenomenon. Methods. Thetype II shock source-positions and spectra were obtained using 91 simultaneous tied-array beams of LOFAR, and the CME was observed by the Large Angle and Spectrometric Coronagraph (LASCO) on board the Solar and Heliospheric Observatory (SOHO) and by the COR2A coronagraph of the SECCHI instruments on board the Solar Terrestrial Relation Observatory(STEREO). The 3D structure was inferred using triangulation of the coronographic observations. Coronal magnetic fields were obtained from a 3D magnetohydrodynamics (MHD) polytropic model using the photospheric fields measured by the Heliospheric Imager (HMI) on board the Solar Dynamic Observatory (SDO) as lower boundary. Results. The type II radio source of the coronal shock observed between 50 and 70 MHz was found to be located at the expanding flank of the CME, where the shock geometry is quasi-perpendicular with θBn ~ 70°. The type II radio burst showed first and second harmonic emission; the second harmonic source was cospatial with the first harmonic source to within the observational uncertainty. This suggests that radio wave propagation does not alter the apparent location of the harmonic source. The sources of the two split bands were also found to be cospatial within the observational uncertainty, in agreement with the interpretation that split bands are simultaneous radio emission from upstream and downstream of the shock front. The fast magnetosonic Mach number derived from this interpretation was found to lie in the range 1.3–1.5. The fast magnetosonic Mach numbers derived from modelling the CME and the coronal magnetic field around the type II source were found to lie in the range 1.4–1.6.