Secular Instability Research Articles

Gamma-ray burst afterglow plateaus and gravitational waves

The existence of a shallow decay phase in the early x-ray afterglows of gamma-ray bursts is a common feature. We discuss the possibility that such a feature is connected to the formation of a highly magnetized millisecond pulsar, pumping energy into the fireball via magnetic dipole emission, while undergoing a secular bar-mode instability. If this is the case, gravitational wave losses associated with the neutron star's ellipsoidal deformation, would affect the star's spin-down, possibly producing a gravitational wave signal detectable by the advanced LIGO and Virgo. Such a signal, being emitted in association with an observed x-ray light-curve plateau over relatively long timescales, could open a new interesting opportunity for multi-messenger studies to be carried out in coincidence with gamma-ray burst sources. We conclude that the hypothesis proposed here deserves further investigation.

Oscillations and instabilities of fast rotating neutron stars

We present our results of our study of the oscillations and instabilities of rapidly rotating relativistic stars in the Cowling approximation. This is the first study of its kind and provides information on the effect of fast rotation on the oscillation spectra while it offers the possibility in studying the complete problem by including spacetime perturbations. Our study includes both axisymmetric and non-axisymmetric perturbations and provides limits for the onset of the secular bar mode rotational instability. We also present approximate formulae for the dependence of the oscillation spectrum from rotation. The results suggest that it is possible to extract the relativistic star's parameters from the observed gravitational wave spectrum.

GAMMA-RAY BURST AFTERGLOW PLATEAUS AND GRAVITATIONAL WAVES: MULTI-MESSENGER SIGNATURE OF A MILLISECOND MAGNETAR?

The existence of a shallow decay phase in the early X-ray afterglows of gamma-ray bursts is a common feature. Here we investigate the possibility that this is connected to the formation of a highly magnetized millisecond pulsar, pumping energy into the fireball on timescales longer than the prompt emission. In this scenario, the nascent neutron star could undergo a secular bar-mode instability, leading to gravitational wave losses which would affect the neutron star spin-down. In this case, nearby gamma-ray bursts with isotropic energies of the order of 1050 ergs would produce a detectable gravitational wave signal emitted in association with an observed X-ray light-curve plateau, over relatively long timescales of minutes to about an hour. The peak amplitude of the gravitational wave signal would be delayed with respect to the gamma-ray burst trigger, offering gravitational wave interferometers such as the advanced LIGO and Virgo the challenging possibility of catching its signature on the fly.

Quasi-viscous accretion flow - I. Equilibrium conditions and asymptotic behaviour

In a novel approach to studying viscous accretion flows, viscosity has been introduced as a perturbative effect, involving a first-order correction in the $\alpha$-viscosity parameter. This method reduces the problem of solving a second-order nonlinear differential equation (Navier-Stokes equation) to that of an effective first-order equation. Viscosity breaks down the invariance of the equilibrium conditions for stationary inflow and outflow solutions, and distinguishes accretion from wind. Under a dynamical systems classification, the only feasible critical points of this "quasi-viscous" flow are saddle points and spirals. A linearised and radially propagating time-dependent perturbation gives rise to secular instability on large spatial scales of the disc. Further, on these same length scales, the velocity evolution equation of the quasi-viscous flow has been transformed to bear a formal closeness with Schr\"odinger's equation with a repulsive potential. Compatible with the transport of angular momentum to the outer regions of the disc, a viscosity-limited length scale has been defined for the full spatial extent over which the accretion process would be viable.

A NEW SECULAR INSTABILITY OF ECCENTRIC STELLAR DISKS AROUND SUPERMASSIVE BLACK HOLES, WITH APPLICATION TO THE GALACTIC CENTER

We identify a new secular instability of eccentric stellar disks around supermassive black holes. We show that retrograde precession of the stellar orbits, due to the presence of a stellar cusp, induces coherent torques that amplify deviations of individual orbital eccentricities from the average, and thus drive all eccentricities away from their initial value. We investigate the instability using N-body simulations, and show that it can drive individual orbital eccentricities to significantly higher or lower values on the order of a precession time-scale. This physics is relevant for the Galactic center, where massive stars are likely to form in eccentric disks around the SgrA* black hole. We show that the observed bimodal eccentricity distribution of disk stars in the Galactic center is in good agreement with the distribution resulting from the eccentricity instability and demonstrate how the dynamical evolution of such a disk results in several of its stars acquiring high (1-e < 0.1) orbital eccentricity. Binary stars on such highly eccentric orbits would get tidally disrupted by the SgrA* black hole, possibly producing both S-stars near the black hole and high-velocity stars in the Galactic halo.

Commodities, Natural Resources and Growth: A Survey Through the History of Economics

The paper presents a survey of the complex relation between the production of primary commodities and economic growth throughout the history of economic thought. Some of the main topics investigated are: the apparent contrast between the theoretical advantages of natural wealth and the historical record; the impact of natural resource booms on the level of activity of other productive sectors of the economy; the different potential growth effects of the primary and secondary economic sectors; the secular trend and cyclical instability of terms of trade between primary and industrial goods; and the approaches to positive links between primary commodities exports and growth.

Quasiequilibrium models for triaxially deformed rotating compact stars

Quasiequilibrium models of rapidly rotating triaxially deformed stars are computed in general relativistic gravity, assuming a conformally flat spatial geometry (Isenberg-Wilson-Mathews formulation) and a polytropic equation of state. Highly deformed solutions are calculated on the initial slice covered by spherical coordinate grids, centered at the source, in all angular directions up to a large truncation radius. Constant rest mass sequences are calculated from nearly axisymmetric to maximally deformed triaxial configurations. Selected parameters are to model (proto-) neutron stars; the compactness is $M/R=0.001$, 0.1, 0.14, and 0.2 for polytropic index $n=0.3$ and $M/R=0.001$, 0.1, 0.12, and 0.14 for $n=0.5$, where $M/R$ refers to that of a nonrotating spherical star having the same rest mass. We confirmed that the triaxial solutions exist for these parameters as in the case of Newtonian polytropes. However, it is also found that the triaxial sequences become shorter for higher compactness, and those disappear at a certain large compactness for the $n=0.5$ case. In the scenario of the contraction of proto-neutron stars being subject to strong viscosity and rapid cooling, it is plausible that, once the viscosity driven secular instability sets in during the contraction, the proto-neutron stars are always maximally deformed triaxial configurations, as long as the compactness and the equation of state parameters allow such triaxial sequences. Detection of gravitational waves from such sources may be used as another probe for the nuclear equation of state.

Oscillations of rapidly rotating relativistic stars

Nonaxisymmetric oscillations of rapidly rotating relativistic stars are studied using the Cowling approximation. The oscillation spectra have been estimated by Fourier transforming the evolution equations describing the perturbations. This is the first study of its kind and provides information on the effect of fast rotation on the oscillation spectra while it offers the possibility of studying the complete problem by including space-time perturbations. Our study includes both axisymmetric and nonaxisymmetric perturbations and provides limits for the onset of the secular bar mode rotational instability. We also present approximate formulas for the dependence of the oscillation spectrum from rotation. The results suggest that it is possible to extract the relativistic star's parameters from the observed gravitational wave spectrum.

Resolving the Formation of Protogalaxies. II. Central Gravitational Collapse

Numerous cosmological hydrodynamic studies have addressed the formation of galaxies. Here we choose to study the first stages of galaxy formation, including nonequilibrium atomic primordial gas cooling, gravity, and hydrodynamics. Using initial conditions appropriate for the concordance cosmological model of structure formation, we perform two adaptive mesh refinement simulations of ~108 M☉ galaxies at high redshift. The calculations resolve the Jeans length at all times with more than 16 cells and capture over 14 orders of magnitude in length scales. In both cases, the dense, 105 solar mass, one parsec central regions are found to contract rapidly and have turbulent Mach numbers up to 4. Despite the ever decreasing Jeans length of the isothermal gas, we only find one site of fragmentation during the collapse. However, rotational secular bar instabilities transport angular momentum outward in the central parsec as the gas continues to collapse and lead to multiple nested unstable fragments with decreasing masses down to sub-Jupiter mass scales. Although these numerical experiments neglect star formation and feedback, they clearly highlight the physics of turbulence in gravitationally collapsing gas. The angular momentum segregation seen in our calculations plays an important role in theories that form supermassive black holes from gaseous collapse.

A Cosmological Framework for the Co‐evolution of Quasars, Supermassive Black Holes, and Elliptical Galaxies. II. Formation of Red Ellipticals

We develop and test a model for the cosmological role of mergers in the formation and quenching of red, early-type galaxies. By combining theoretically well-constrained halo and subhalo mass functions as a function of redshift and environment with empirical halo occupation models, we predict the distribution of mergers as a function of redshift, environment, and physical galaxy properties. Making the simple Ansatz that star formation is quenched after a gas-rich, spheroid-forming major merger, we demonstrate that this naturally predicts the turnover in the efficiency of star formation and baryon fractions in galaxies at ~ -->L* (without any parameters tuned to this value), as well as the observed mass functions and mass density of red galaxies as a function of redshift, the formation times of early-type galaxies as a function of mass, and the fraction of quenched galaxies as a function of galaxy and halo mass, environment, and redshift. Comparing our model to a variety of semianalytic models in which quenching is primarily driven by halo mass considerations or secular/disk instabilities, we demonstrate that our model makes unique and robust qualitative predictions for a number of observables, including the bivariate red fraction as a function of galaxy and halo mass, the density of passive galaxies at high redshifts, the emergence/evolution of the color-morphology-density relations at high redshift, and the fraction of disky/boxy (or cusp/core) spheroids as a function of mass. In each case, the observations favor a model in which some mechanism quenches future star formation after a major merger builds a massive spheroid. Models where quenching is dominated by a halo mass threshold fail to match the behavior of the bivariate red fractions, predict too low a density of passive galaxies at high redshift, and overpredict by an order of magnitude the mass of the transition from disky to boxy ellipticals. Models driven by secular disk instabilities also qualitatively disagree with the bivariate red fractions, fail to predict the observed evolution in the color-density relations, and predict order-of-magnitude incorrect distributions of kinematic types in early-type galaxies. We make specific predictions for how future observations, for example, quantifying the red fraction as a function of galaxy mass, halo mass, environment, or redshift, can break the degeneracies between a number of different assumptions adopted in present galaxy formation models. We discuss a variety of physical possibilities for this quenching and propose a mixed scenario in which traditional quenching in hot, quasi-static massive halos is supplemented by the strong shocks and feedback energy input associated with a major merger (e.g., tidal shocks, starburst-driven winds, and quasar feedback), which temporarily suppress cooling and establish the conditions of a dynamically hot halo in the central regions of the host, even in low-mass halos (below the traditional threshold for accretion shocks).

Secular Instability in Quasi‐viscous Disk Accretion

A first-order correction in the α-viscosity parameter of Shakura & Sunyaev has been introduced in the standard inviscid and thin accretion disk. A linearized time-dependent perturbative study of the stationary solutions of this quasi-viscous disk reveals the development of a secular instability on large spatial scales. This qualitative feature is equally manifest for two different types of perturbative treatment: a standing wave on subsonic scales, as well as a radially propagating wave. The stability of the flow is restored when the viscosity disappears.

Further Evidence for an Elliptical Instability in Rotating Fluid Bars and Ellipsoidal Stars

Using a three-dimensional nonlinear hydrodynamic code, we examine the dynamical stability of more than twenty self-gravitating, compressible, ellipsoidal fluid configurations that initially have the same velocity structure as Riemann S-type ellipsoids. Our focus is on ``adjoint'' configurations, in which internal fluid motions dominate over the collective spin of the ellipsoidal figure; Dedekind-like configurations are among this group. We find that, although some models are stable and some are moderately unstable, the majority are violently unstable toward the development of $m=1$, $m=3$, and higher-order azimuthal distortions that destroy the coherent, $m=2$ bar-like structure of the initial ellipsoidal configuration on a dynamical time scale. The parameter regime over which our models are found to be unstable generally corresponds with the regime over which incompressible Riemann S-type ellipsoids have been found to be susceptible to an elliptical strain instability \citep{LL96}. We therefore suspect that an elliptical instability is responsible for the destruction of our compressible analogs of Riemann ellipsoids. The existence of the elliptical instability raises concerns regarding the final fate of neutron stars that encounter the secular bar-mode instability and regarding the spectrum of gravitational waves that will be radiated from such systems.

Oscillatory secular modes: the thermal micropulses

Stars in the narrow mass range of about 2.5 and 3.5 solar masses can develop a thermally unstable He-burning shell during its ignition phase. We study, from the point of view secular stability theory, these so called thermal micropulses and we investigate their properties; the thermal pulses constitute a convenient conceptual laboratory to look thoroughly into the physical properties of a helium-burning shell during the whole thermally pulsing episode. Linear stability analyses were performed on a large number of 3 solar-mass star models at around the end of their core helium-burning and the beginning of the double-shell burning phase. The stellar models were not assumed to be in thermal equilibrium. The thermal mircopulses, and we conjecture all other thermal pulse episodes encountered by shell-burning stars, can be understood as the nonlinear finite-amplitude realization of an oscillatory secular instability that prevails during the whole thermal pulsing episode. Hence, the cyclic nature of the thermal pulses can be traced back to a linear instability concept.

Models of particle layers in the midplane of the solar nebula

In the absence of global turbulence, solid particles in the solar nebula tend to settle into a thin layer in the central plane. Shear between this layer and pressure-supported gas produces localized turbulence in the midplane; the thickness of the particle layer is determined by balance between settling and turbulent diffusion. A numerical model is described, which allows computation of the vertical structure of a layer of particles of arbitrary size, with self-consistent distributions of particle density, turbulent velocity, and radial fluxes of particles and gas. Effects of varying particle size and the abundances of solids and gas are evaluated. If the surface density of solids is increased by an order of magnitude over nominal solar abundance, the peak density within a layer of small particles can approach the critical value needed for gravitational instability. However, depletion of the nebular gas is much less effective for raising the density of such a layer to the critical value, due to decreased coupling of particles to the gas as the density of the gas decreases. The variation of radial particle flux with surface density of the particle layer is not consistent with secular instability of the layer driven by gas drag.

Mixing zones in magnetized differentially rotating stars

We study the secular instability of magnetized differentially rotating radiative zones taking account of viscosity and magnetic and thermal diffusivities. This considered instability generalizes the well-known Goldreich-Schubert-Fricke instability for the case of a sufficiently strong magnetic field. In magnetized stars, instability can lead to formation of non-spherical, unstable zones where weak turbulence mixes the material between the surface and interiors. Such unstable zones can manifest themselves by a non-spherical distribution of elements on the stellar surface.

Approximate dynamics of dark matter ellipsoids

The collapse of non-collisional dark matter and the formation of pancake structures in the Universe are investigated approximately. Collapse is described by a system of ordinary differential equations, in the model of a uniformly rotating, three-axis, uniform density ellipsoid. Violent relaxation, mass and angular momentum losses are taken into account phenomenologically. The formation of the equilibrium configuration, secular instability and the transition from a spheroid to a three-axis ellipsoid are investigated numerically and analytically in this dynamical model.

Global perturbation configurations in a composite disc system with an isopedic magnetic field

We construct stationary global configurations of both aligned and unaligned logarithmic spiral perturbations in a composite disc system of stellar and isopedically magnetized gaseous singular isothermal discs (SIDs) coupled by gravity. Earlier models are generalized to a more general theoretical framework. The thin gaseous SID is threaded across by a vertical magnetic field B z with a constant ratio of the surface gas mass density to B z . In reference to SID models of Shu & Li, Shu et al., Lou & Shen, Lou & Zou, Shen and Liu & Lou, there exist two classes of stationary magnetohydrodynamic (MHD) solutions with in-phase and out-of-phase density perturbations here. Relevant parameter regimes are explored numerically. For both aligned and unaligned cases with azimuthal periodicities |m| ≥ 2 (m is an integer), there may be two, one and no solution situations, depending on the chosen parameters. For the transition criteria from an axisymmetric equilibrium to aligned secular bar-like instabilities, the corresponding T/|W| ratio can be much lower than the oft-quoted value of T/|W| ∼ 0.14, where T is the total rotational kinetic energy and W is the total gravitational potential energy plus the magnetic energy. The T/|W| ratios for the two sets of solutions in different ranges are separated by m/(4m + 4). For the unaligned cases, we study marginal stabilities for axisymmetric (m = 0) and non-axisymmetric (m ¬= 0) disturbances. By including the gravitational influence of an axisymmetric dark matter halo on the background, the case of a composite partial magnetized singular isothermal discs (MSID) system is also examined. The global analytical solutions and their properties are valuable for testing and benchmarking numerical MHD codes. For astrophysical applications to large-scale galactic dynamics, our model analysis contains more realistic elements and offers useful insights into the structures and dynamics of disc galaxies consisting of stars and magnetized interstellar medium (ISM). In particular, in the presence of star burst activities, supernovae, hypernovae, superbubbles etc., our open magnetic field geometry in disc galaxies bears strong implications on circumnuclear and spiral galactic winds.

Secular Stability and Instability in Stellar Systems Surrounding Massive Objects

We examine the stability of a low-mass stellar system surrounding a massive central object. Examples of such systems include the centers of galaxies or star clusters containing a massive black hole, and the Oort comet cloud. If the self-gravity of the stellar system is the dominant non-Keplerian force, such systems may be subject to slowly growing (secular) lopsided instabilities. Stability to secular modes is largely determined by the dependence of the distribution function F on angular momentum J. If dF/dJ 0, as is expected if there is a loss cone at low angular momentum, all spherical systems in which F=0 at J=0 (an empty loss cone) are only neutrally stable, and flattened, non-rotating systems are generally unstable. These results suggest that secular instabilities may dominate the structure and evolution of the stellar systems in the centers of galaxies.

Streaming Instabilities in Protoplanetary Disks

Interpenetrating streams of solids and gas in a Keplerian disk produce a local, linear instability. The two components mutually interact via aerodynamic drag, which generates radial drift and triggers unstable modes. The secular instability does not require self-gravity, yet it generates growing particle density perturbations that could seed planetesimal formation. Growth rates are slower than dynamical, but faster than radial drift, timescales. Growth rates, like streaming velocities, are maximized for marginal coupling (stopping times comparable dynamical times). Fastest growth occurs when the solid to gas density ratio is order unity and feedback is strongest. Curiously, growth is strongly suppressed when the densities are too nearly equal. The relation between background drift and wave properties is explained by analogy with Howard's semicircle theorem. The three-dimensional, two-fluid equations describe a sixth order (in the complex frequency) dispersion relation. A terminal velocity approximation allows simplification to an approximate cubic dispersion relation. To describe the simplest manifestation of this instability, we ignore complicating (but possibly relevant) factors like vertical stratification, dispersion of particle sizes, turbulence, and self-gravity. We consider applications to planetesimal formation and compare our work to other studies of particle-gas dynamics.

Nonlinear Development of the Secular Bar‐Mode Instability in Rotating Neutron Stars

We have modeled the nonlinear development of the secular bar-mode instability that is driven by gravitational radiation reaction (GRR) forces in rotating neutron stars. In the absence of any competing viscous effects, an initially uniformly rotating axisymmetricn ¼ 1=2polytropic star with a ratio of rotational to gravitational potential energy T = W jj ¼0:181 is driven by GRR forces to a barlike structure, as predicted by linear theory. The pattern frequencyofthebarslowstonearlyzero,thatis,thebarbecomesalmoststationaryasviewedfromaninertialframe ofreferenceasGRRremovesenergyandangularmomentumfromthestar.Inthis‘‘Dedekind-like’’state,rotational energyisstoredasmotionofthefluidinhighlynoncircularorbitsinsidethebar.However,inlessthan10dynamical timesafteritsformationthe bar losesitsinitially coherent structure astheorderedflow insidethe baris disruptedby what appears to be a purely hydrodynamic short-wavelength ‘‘shearing’’-type instability. The gravitational waveforms generated by such an event are determined, and an estimate of the detectability of these waves is presented. Subject headingg gravitational waves — hydrodynamics — instabilities — stars: neutron