Degree Of Differential Rotation Research Articles

Equilibrium sequences of differentially rotating stars with post-merger-like rotational profiles

ABSTRACT We present equilibrium sequences of rotating relativistic stars, constructed with a new rotation law that was proposed by Uryū et al. (2017). We choose rotational parameters motivated by simulations of binary neutron star merger remnants, but otherwise adopt a cold, relativistic N = 1 polytropic EOS, in order to perform a detailed comparison to published equilibrium sequences that used the Komatsu, Eriguchi and Hachisu (1989) rotation law. We find a small influence of the choice of rotation law on the mass of the equilibrium models and a somewhat larger influence on their radius. The versatility of the new rotation law allows us to construct models that have a similar rotational profile and axial ratio as observed for merger remnants, while at the same time being quasi-spherical. More specifically, we construct equilibrium sequence variations with different degrees of differential rotation and identify type A and type C solutions, similar to the corresponding types in the classification of Ansorg, Gondek-Rosińska and Villain (2009). While our models are highly accurate solutions of the fully general relativistic structure equations, we demonstrate that for models relevant to merger remnants the IWM-CFC approximation still maintains an acceptable accuracy.

Erratum: Differential rotation in neutron stars with open and closed magnetic topologies

Analytic arguments have been advanced that the degree of differential rotation in a neutron star depends on whether the topology of the internal magnetic field is open or closed. To test this assertion, the ideal-magnetohydrodynamics solver PLUTO is employed to investigate numerically the flow of an incompressible, viscous fluid threaded by a magnetic field with open and closed topologies in a conducting, differentially rotating, spherical shell. Rigid body corotation with the outer sphere is enforced on the Alfven time-scale, along magnetic field lines that connect the northern and southern hemispheres of the outer sphere. Along other field lines, however, the behavior is more complicated. For example, an initial point dipole field evolves to produce an approximately closed equatorial flux tube containing at least one predominantly toroidal and approximately closed field line surrounded by a bundle of predominantly toroidal but open field lines. Inside the equatorial flux tube, the field-line-averaged magnetic tension approaches zero, and the fluid rotates differentially, adjusting its angular velocity on the viscous time-scale to match the boundary conditions on the flux tube's toroidal surface. Outside the equatorial flux tube the differential rotation increases, as the magnetic tension averaged along open field lines decreases.

Are hypermassive neutron stars stable against a prompt collapse?

AbstractDifferential rotation in neutron stars allows for significantly larger masses than rigid rotation. Some of those hypermassive objects are, however, unstable and collapse to a black hole immediately after formation. Yet, the exact threshold of dynamical stability is still unknown.In our work we explore the limits on masses of neutron stars with various degrees of differential rotation which could be stable against a prompt collapse to a black hole by using turning-point (j-constant) criterion. We considered both spheroidal and quasi-toroidal differentially rotating neutron stars described by the polytropic equation of state. We find that massive configurations could be temporarily stabilized by differential rotation. Such objects are important sources of gravitational waves. Our results are a starting point for more detailed studies of stability using hydrodynamical codes.

Differential rotation in neutron stars with open and closed magnetic topologies

ABSTRACT Analytic arguments have been advanced that the degree of differential rotation in a neutron star depends on whether the topology of the internal magnetic field is open or closed. To test this assertion, the ideal-magnetohydrodynamics solver pluto is employed to investigate numerically the flow of an incompressible, viscous fluid threaded by a magnetic field with open and closed topologies in a conducting, differentially rotating, spherical shell. Rigid body corotation with the outer sphere is enforced on the Alfvén time-scale, along magnetic field lines that connect the northern and southern hemispheres of the outer sphere. Along other field lines, however, the behaviour is more complicated. For example, an initial point dipole field evolves to produce an approximately closed equatorial flux tube containing at least one predominantly toroidal and approximately closed field line surrounded by a bundle of predominantly toroidal but open field lines. Inside the equatorial flux tube, the field-line-averaged magnetic tension approaches zero, and the fluid rotates differentially, adjusting its angular velocity on the viscous time-scale to match the boundary conditions on the flux tube’s toroidal surface. Outside the equatorial flux tube, the differential rotation increases, as the magnetic tension averaged along open field lines decreases.

Maximum mass and universal relations of rotating relativistic hybrid hadron-quark stars

We construct equilibrium models of uniformly and differentially rotating hybrid hadron-quark stars using equations of state (EOSs) with a first-order phase transition that gives rise to a third family of compact objects. We find that the ratio of the maximum possible mass of uniformly rotating configurations - the supramassive limit - to the Tolman-Oppenheimer-Volkoff (TOV) limit mass is not EOS-independent, and is between 1.15 and 1.31,in contrast with the value of 1.20 previously found for hadronic EOSs. Therefore, some of the constraints placed on the EOS from the observation of the gravitational wave event GW170817 do not apply to hadron-quark EOSs. However, the supramassive limit mass for the family of EOSs we treat is consistent with limits set by GW170817, strengthening the possibility of interpreting GW170817 with a hybrid hadron-quark EOSs. We also find that along constant angular momentum sequences of uniformly rotating stars, the third family maximum and minimum mass models satisfy approximate EOS-independent relations, and the supramassive limit of the third family is approximately 16.5 % larger than the third family TOV limit. For differentially rotating spheroidal stars, we find that a lower-limit on the maximum supportable rest mass is 123 % more than the TOV limit rest mass. Finally, we verify that the recently discovered universal relations relating angular momentum, rest mass and gravitational mass for turning-point models hold for hybrid hadron-quark EOSs when uniform rotation is considered, but have a clear dependence on the degree of differential rotation.

The intimate relation between the low T/W instability and the corotation point

We study the low T/W instability associated with the f-mode of differentially rotating stars. Our stellar models are described by a polytropic equation of state and the rotation profile is given by the standard j-constant law. The properties of the relevant oscillation modes, including the instability growth time, are determined from time evolutions of the linearised dynamical equations in Newtonian gravity. In order to analyse the instability we monitor also the canonical energy and angular momentum. Our results demonstrate that the l=m=2 f-mode becomes unstable as soon as a co-rotation point develops inside the star (i.e. whenever there is a point where the mode's pattern speed matches the bulk angular velocity). Considering various degrees of differential rotation, we show that the instability grows faster deep inside the co-rotation region and deduce an empirical relation that correlates the mode frequency and the star's parameters, which captures the main features of the l=m=2 f-mode growth time. This function is proportional to the product of the kinetic to gravitational energy ratio and the gradient of the star's spin, strengthening further the relationship between the co-rotation point and the low T/W instability. We briefly consider also the l=m=2 r-mode and demonstrate that it never moves far inside the co-rotation region even for significant differential rotation.

Differentially-rotating neutron star models with a parametrized rotation profile

We analyze the impact of the choice rotation law on equilibrium sequences of relativistic differentially-rotating neutron stars in axisymmetry. The maximum allowed mass for each model is strongly affected by the distribution of angular velocity along the radial direction and by the consequent degree of differential rotation. In order to study the wide parameter space implied by the choice of rotation law, we introduce a functional form that generalizes the so called "j-const. law" adopted in all previous work. Using this new rotation law we reproduce the angular velocity profile of differentially-rotating remnants from the coalescence of binary neutron stars in various 3-dimensional dynamical simulations. We compute equilibrium sequences of differentially rotating stars with a polytropic equation of state starting from the spherically symmetric static case. By analyzing the sequences at constant ratio, T/|W|, of rotational kinetic energy to gravitational binding energy, we find that the parameters that best describe the binary neutron star remnants cannot produce equilibrium configurations with values of T/|W| that exceed 0.14, the criterion for the onset of the secular instability.

Maximum Entropy for Gravitational Wave Data Analysis: Inferring the Physical Parameters of Core‐Collapse Supernovae

The gravitational wave signal arising from the collapsing iron core of a Type II supernova progenitor star carries with it the imprint of the progenitor's mass, rotation rate, degree of differential rotation, and the bounce depth. Here, we show how to infer the gravitational radiation waveform of a core collapse event from noisy observations in a network of two or more LIGO-like gravitational wave detectors and, from the recovered signal, constrain these source properties. Using these techniques, predictions from recent core collapse modeling efforts, and the LIGO performance during its S4 science run, we also show that gravitational wave observations by LIGO might have been sufficient to provide reasonable estimates of the progenitor mass, angular momentum and differential angular momentum, and depth of the core at bounce, for a rotating core collapse event at a distance of a few kpc.

Nonaxisymmetric oscillations of differentially rotating relativistic stars

Nonaxisymmetric oscillations of differentially rotating stars are studied using both slow rotation and Cowling approximation. The equilibrium stellar models are relativistic polytropes where differential rotation is described by the relativistic j-constant rotation law. The oscillation spectrum is studied versus three main parameters: the stellar compactness $M/R$, the degree of differential rotation $A$, and the number of maximum couplings ${\ensuremath{\ell}}_{\mathrm{max}}$. It is shown that the rotational splitting of the nonaxisymmetric modes are strongly enhanced by increasing the compactness of the star and the degree of differential rotation. Finally, we investigate the relation between the fundamental quadrupole mode and the corotation band of differentially rotating stars.

The maximum mass of differentially rotating neutron stars

Binary neutron star merger lead to the formation of a massive, rapidly and differentially rotating neutron star (or a strange star) or to the prompt collapse to a black hole. The maximum mass of a differentially rotating remnant is crucial for distinguishing between these two final objects. We study the effect of differential rotation on the maximum mass of neutron stars (strange stars). We numerically construct stellar models using a highly accurate relativistic code based on a multi-domain spectral method. We find much larger mass increases for strange stars than for neutron stars for the same degree of differential rotation.

On the Structure and Properties of Differentially Rotating, Main‐Sequence Stars in the 1–2M⊙Range

We present models for chemically homogeneous, differentially rotating, main-sequence stars with masses in the range 1-2 M☉. The models were constructed using a code based on a reformulation of the self-consistent field method of computing the equilibrium stellar structure for a specified conservative internal rotation law. Relative to nonrotating stars of the same mass, these models all have reduced luminosities and effective temperatures, and flattened photospheric shapes (i.e., decreased polar radii) with equatorial radii that can be larger or smaller, depending on the degree of differential rotation. For a fixed ratio of the axial rotation rate to the surface equatorial rotation rate, increasingly rapid rotation generally deepens convective envelopes, shrinks convective cores, and can lead to the presence of a convective core (envelope) in a 1 (2) M☉ model, a feature that is absent in a nonrotating star of the same mass. The positions of differentially rotating models for a given mass in the H-R diagram can be shifted in such a way as to approximate the nonrotating ZAMS for lower mass stars. Implications of these results include (1) possible ambiguities arising from similarities between the properties of rotating and nonrotating models of different masses, (2) a reduced radiative luminosity for a young, rapidly rotating Sun, (3) modified rates of lithium destruction by nuclear processes in the layers beneath an outer convective envelope, and (4) the excitation of solar-like oscillations and the operation of a solar-like hydromagnetic dynamo in some 1.5-2 M☉ stars.

Effects of QCD phase transition on gravitational radiation from two-dimensional collapse and bounce of massive stars

We perform two-dimensional, magneto-hydrodynamical core-collapse simulations of massive stars accompanying the QCD phase transition. We study how the phase-transition affects the gravitational waveforms near the epoch of core-bounce. As for initial models, we change the strength of rotation and magnetic fields. Particularly, the degree of differential rotation in the iron core (Fe-core) is changed parametrically. As for the microphysics, we adopt a phenomenological equation of state above the saturation density, including two parameters to change the hardness before the transition. We assume the first order phase transition, where the conversion of bulk nuclear matter to a chirally symmetric quark-gluon phase is described by the MIT bag model. Based on these computations, we find that the phase transition can make the maximum amplitudes larger up to {approx}10 percents than the ones without the phase transition. On the other hand, when the degree of the differential rotation becomes larger, the maximum amplitudes become smaller up to {approx}10 percents owing to the phase transition. We find that even extremely strong magnetic fields {approx}10{sup 17} G in the protoneutron star do not affect these results.

Twisted‐Torus Equilibrium Structures of Magnetic Fields in Magnetized Stars

We have investigated the of the twisted-torus equilibrium structures of the magnetic fields in deformed stars. The Tomimura-Eriguchi scheme for equilibrium structures of uniformly rotating magnetized stars is extended to treat equilibrium configurations of differentially rotating and meridional circulation-free magnetized polytropes with infinite conductivity. Characteristics of the magnetic fields obtained are summarized as follows: The exterior magnetic fields of the stars behave like dipolar poloidal fields, which decrease as r-3 when r → ∞. The interior magnetic fields are a mixture of poloidal and toroidal fields. Their geometric structures are tori of twisted field lines around the symmetry axis and of untwisted poloidal fields that penetrate the surface of the stars to continue to the exterior fields. It has been found that these structures of magnetic fields in stationary magnetic stars are robust and universal in the sense that they are nearly independent of the compressibility of the matter, rotation laws, and degrees of differential rotation. This universality is a consequence of the natural assumption that the electric current should be confined inside the star and of the integrability condition of the basic equation, i.e., that the electric current in the meridional plane is a function of the flux function, whose behavior is governed by a partial differential equation of the elliptic type.

The Supernova–Gamma-Ray Burst Connection

Observations show that at least some gamma-ray bursts (GRBs) happen simultaneously with core-collapse supernovae (SNe), thus linking by a common thread nature's two grandest explosions. We review here the growing evidence for and theoretical implications of this association, and conclude that most long-duration soft-spectrum GRBs are accompanied by massive stellar explosions (GRB-SNe). The kinetic energy and luminosity of well-studied GRB-SNe appear to be greater than those of ordinary SNe, but evidence exists, even in a limited sample, for considerable diversity. The existing sample also suggests that most of the energy in the explosion is contained in nonrelativistic ejecta (producing the supernova) rather than in the relativistic jets responsible for making the burst and its afterglow. Neither all SNe, nor even all SNe of Type Ibc produce GRBs. The degree of differential rotation in the collapsing iron core of massive stars when they die may be what makes the difference.

North‐South Neutrino Heating Asymmetry in Strongly Magnetized and Rotating Stellar Cores

We perform a series of two-dimensional magnetohydrodynamic simulations of supernova cores. Since the distributions of the angular momentum and the magnetic fields of strongly magnetized stars are quite uncertain, we systematically change the combinations of the strength of the angular momentum, the rotations law, the degree of differential rotation, and the profiles of the magnetic fields to construct the initial conditions. By so doing, we estimate how the rotation-induced anisotropic neutrino heating are affected by the strong magnetic fields through parity-violating effects and first investigate how the north-south asymmetry of the neutrino heating in a strongly magnetized supernova core could be. As for the microphysics, we employ a realistic equation of state based on the relativistic mean field theory and take into account electron captures and the neutrino transport via the neutrino leakage scheme. With these computations, we find that the parity-violating corrections reduce $ \lesssim 0.5 %$ of the neutrino heating rate than that without the magnetic fields in the vicinity of the north pole of a star, on the other hand, enhance about $ \lesssim 0.5 %$ in the vicinity of the south pole. If the global asymmetry of the neutrino heating in the both of the poles develops in the later phases, the newly born neutron star might be kicked toward the north pole in the subsequent time.

Differential rotation and star-spot evolution on HK Aqr in 2001 and 2002

We present images of the rapidly rotating (P rot = 0.431 d) M1.5Ve dwarf HK Aqr, from data obtained during 2001 and 2002. Star-spots are found distributed at a range of latitudes during both observing seasons, but unlike other solar-type rapid rotators of higher mass, HK Aqr does not show a strong polar cap at either epoch. In addition to the surface brightness images, we make use of the four-night time-base over which the 2002 observations were made, in order to derive an estimate of the latitudinal dependent rotation on HK Aqr. We find that the equator-lap-pole time lies in the range -1449 to +448 d. Although very small, we are therefore unable to determine whether the degree of differential rotation is in the solar or anti-solar sense.

Non-linear pulsations in differentially rotating neutron stars: mass-shedding-induced damping and splitting of the fundamental mode

We study small-amplitude, non-linear pulsations of uniformly and differentially rotating neutron stars employing a two-dimensional evolution code for general-relativistic hydrodynamics. Using Fourier transforms at several points inside the star, both the eigenfrequencies and two-dimensional eigenfunctions of pulsations are extracted. The centrifugal forces and the degree of differential rotation have significant effects on the mode eigenfunction. We find that near the mass-shedding limit, the pulsations are damped due to shocks forming at the surface of the star. This new damping mechanism may set a small saturation amplitude for modes that are unstable to the emission of gravitational waves. After correcting for the assumption of the Cowling approximation (used in our numerical code), we construct empirical relations that predict the range of gravitational-wave frequencies from quasi-periodic post-bounce oscillations in the core collapse of massive stars. We also find that the fundamental quasi-radial mode is split, at least in the Cowling approximation and mainly in differentially rotating stars, into two different sequences.

Linear Stability Analysis of Differentially Rotating Polytropes: New Results for them = 2f‐Mode Dynamical Instability

We have studied the f-mode oscillations of differentially rotating polytropes by making use of the linear stability analysis. We found that the critical values of T/|W| where the dynamical instability against the m = 2 f-mode oscillations sets in decrease down to T/|W| ~ 0.20 as the degree of differential rotation becomes higher. Here m is an azimuthal mode number and T and W are the rotational energy and gravitational potential energy, respectively. This tendency is almost independent of the compressibility of the polytropes. These are the first exact results of the linear stability analysis for the occurrence of the dynamical instability against the m = 2 f-modes.

Minimum velocity dispersion in stable stellar disks. Numerical simulations

N-body dynamical simulations are used to analyze the conditions for the gravitational stability of a three-dimensional stellar disk in the gravitational field of two rigid spherical components—a bulge and halo whose central concentrations and relative masses vary over wide ranges. The number of point masses N in the simulations varies from 40 to 500 000 and the evolution of the simulated systems is followed over 10–20 rotation periods of the outer edge of the disk. The initially unstable disks are heated and, as a rule, reach a quasi-stationary equilibrium with a steady-state radial-velocity dispersion c r over five to eight turns. The radial behavior of the Toomre stability parameter Q T(r) for the final state of the disk is estimated. Simple models are used to analyze the dependence of the gravitational stability of the disk on the relative masses of the spherical components, disk thickness, degree of differential rotation, and initial state of the disk. Formal application of existing, analytical, local criteria for marginal stability of the disk can lead to errors in cr of more than a factor of 1.5. It is suggested that the approximate constancy of Q T⋍1.2–1.5 for r⋍(1–2)×L (where L is the radial scale of disk surface density), valid for a wide range of models, can be used to estimate upper limits for the mass and density of a disk based on the observed distributions of the rotational velocity of the gaseous component and of the stellar velocity dispersion.

Gravitational Radiation Reaction Driven Secular Instability off‐Mode Oscillations in Differentially Rotating Newtonian Stars

We have carried out the linear stability analysis of equilibrium sequences of differentially rotating Newtonian polytropic stars. In particular, we have found critical points along the equilibrium sequences for the gravitational radiation reaction driven secular instability of f-mode oscillations with azimuthal wavenumbers m = 2, 3, and 4. We show that the critical values of T/|W| where the instability sets in strongly depend on the degree of differential rotation of the models. Here, T and W are the rotational kinetic energy and the gravitational energy of the star, respectively. We also find that as the degree of differential rotation is increased, the dependence of the critical value of T/|W| on the equations of state becomes weaker.