Resonant Torques Research Articles

Analytical and Numerical Analysis of Circumbinary Disk Dynamics. I. Coplanar Systems

We present an analytical and numerical study of a system composed of a stellar binary pair and a massless, locally isothermal viscous accretion disk that is coplanar to the binary orbital plane. Analytically, we study the effect of the binary’s gravitational potential over short timescales through the stability of epicyclic orbits, and over long timescales by revisiting the concept of resonant torques. Numerically, we perform two-dimensional Newtonian simulations of the disk-binary system over a range of binary mass ratios. We find that the results of our simulations are consistent with those of previous numerical studies. We additionally show, by comparison of the analytical and numerical results, that the circumbinary gap is maintained on the orbital timescale through the driving of epicyclic instabilities, and does not depend on resonant torquing, contrary to the standard lore. While our results are applicable to any disk-binary system, we highlight the importance of this result in the search for electromagnetic and gravitational-wave signatures from supermassive black hole binaries.

Gravitational torque in circumbinary discs: global radial oscillations

ABSTRACT Circumbinary discs (CBDs) arise in many astrophysical settings, including young stellar binaries and supermassive black hole binaries. Their structure is mediated by gravitational torques exerted on the disc by the central binary. The spatial distribution of the binary torque density (so-called excitation torque density) in CBDs is known to feature global large-amplitude, quasi-periodic oscillations, which are often interpreted in terms of the local resonant Lindblad torques. Here, we investigate the nature of these torque oscillations using 2D, inviscid hydrodynamic simulations and theoretical calculations. We show that torque oscillations arise due to the gravitational coupling of the binary potential to the density waves launched near the inner cavity and freely propagating out in the disc. We provide analytical predictions for the radial periodicity of the torque density oscillations and verify them with simulations, showing that disc sound speed and the multiplicity of the density wave spiral arms are the key factors setting the radial structure of the oscillations. Resonant Lindblad torques play no direct role in determining the radial structure and periodicity of the torque oscillations and manifest themselves only by driving the density waves in the disc. We also observe the formation of vortices at the inner edge of the disc, which can provide a non-trivial contribution to the angular momentum transport in the CBD and may be involved in the development of a non-axisymmetric central cavity.

A survey of disc thickness and viscosity in circumbinary accretion: Binary evolution, variability, and disc morphology

ABSTRACTMuch of the parameter space relevant to the evolution of astrophysical circumbinary accretion discs remains unexplored. We have carried out a suite of circumbinary disc simulations surveying both disc thickness and kinematic viscosity, using both constant-ν and constant-α prescriptions. We focus primarily on disc aspect ratios between 0.1 and 0.033, and on viscosities between ν = 0.0005 and ν = 0.008 (in units of binary semimajor axis and orbital frequency), and specialize to circular equal-mass binaries. Both factors strongly influence the evolution of the binary semimajor axis: at ν = 0.0005, inspirals occur at aspect ratios ≲ 0.059, while at ν = 0.004 inspirals occur only at aspect ratios ≲ 0.04. Inspirals occur largely because of the increasingly strong negative torque on the binary by streams of material which lag the binary, with negligible contributions from resonant torques excited in the circumbinary disc. We find that reductions in accretion rate occur when simulations are initialized too far from the eventual quasi-steady state driven by interaction with the binary, rather than being intrinsically linked to the disc aspect ratio. We find not only that the cavity size increases as viscosity is decreased, but that thinner circumbinary discs become more eccentric. Our results suggest that supermassive black hole binaries should be driven, more rapidly than previous estimates, from ∼parsec separations to distances where gravitational waves drive their inspiral, potentially reducing the number of binaries observable by pulsar timing arrays.

Renewal of Transient Spiral Modes in Disk Galaxies

Abstract Spiral structure in disk galaxies could arise from transient modes that create conditions conducive for their regeneration; this is the proposal of Sellwood and Carlberg, based on simulations of stellar disks. The linear response of an axisymmetric stellar disk, to an adiabatic nonaxisymmetric transient mode, gives a final distribution function (DF) that is equal to the initial DF everywhere in phase space, except at the Lindblad and corotation resonances where the final DF is singular. We use the nonlinear theory of adiabatic capture into resonance to resolve the singularities and calculate the finite changes in the DF. These take the form of axisymmetric “scars” concentrated around resonances, whose DFs have simple general forms. Global changes in the physical properties are explored for a cool Mestel disk: we calculate the DFs of scars and estimate the changes in the disk angular momentum, surface density, and orbital frequencies leading to shifts in resonances. Resonant torques between disk stars and any new linear nonaxisymmetric mode are suppressed within a scar, as is epicyclic heating. Because all resonances of a linear mode with the same angular wavenumber and pattern speed as its precursor lie inside the scars of the precursor, it suffers less damping. Hence, scars filter the spectrum of noise-generated modes, promoting the renewal of a few select modes. Relic scars sustained by a galaxy disk, due to past tidal interaction with a passing companion, may still be active enablers of nonaxisymmetric modes, such as the two-armed “grand design” spiral patterns.

HydroSyMBA: A 1D Hydrocode Coupled with an N-body Symplectic Integrator

Abstract The numerical modeling of coexisting circumplanetary disks/rings and satellites is particularly challenging because each part of the system requires a very different approach. Disks are generally well represented by a fluid-like dense medium, whose evolution can be calculated by a hydrocode. On the other hand, the orbital evolution of satellites is generally performed using N-body integrators. We have developed a new numerical model that combines a one-dimensional hydrocode with the N-body integrator SyMBA. The disk evolves due to its viscosity, and resonant torques from satellites. The latter is applied to the satellites as an additional “kick” to their accelerations. The integrator also includes the ability to spawn new moonlets at the disk’s outer edge if the latter expands beyond a material-dependent Roche limit, as well as the effects of tidal dissipation in the planet and/or the satellite on the satellite orbits. The resulting integrator allows one to accurately model the evolution of an inner circumplanetary disk, and the formation of satellites by accumulation of disk material, all within a single self-consistent framework. Potential applications include the formation of Earth’s Moon, the evolution of the inner Saturn system, the Martian and uranian moons, and compact exoplanetary systems.

Induced resonant torques during the descent of a small asymmetric spacecraft in the atmosphere

This paper investigates an uncontrolled descent of an asymmetric spacecraft in the atmosphere. Suppose that the small spacecraft is a rigid body with small mass-inertial and small aerodynamic asymmetries. The aim of the research is to study small mechanical torques caused by the principal resonance at the non-resonant domains. The proposed study solves the following problems: revealing specific features of the mechanical torque, determining the magnitudes of non-resonant domains of activity of the torque and finding methods to measure the torque. One can conclude that the application of the averaging method makes it possible to research the small mechanical torques induced by the principal resonance. This torque leads to a significant angular velocity evolution of a small spacecraft in the non-resonant domains.

Linear analysis of the evolution of nearly polar low-mass circumbinary discs

In a recent paper Martin & Lubow showed through simulations that an initially tilted disc around an eccentric binary can evolve to polar alignment in which the disc lies perpendicular to the binary orbital plane. We apply linear theory to show both analytically and numerically that a nearly polar aligned low-mass circumbinary disc evolves to polar alignment and determine the alignment time-scale. Significant disc evolution towards the polar state around moderately eccentric binaries can occur for typical protostellar disc parameters in less than a typical disc lifetime for binaries with orbital periods of order 100 yr or less. Resonant torques are much less effective at truncating the inner parts of circumbinary polar discs than the inner parts of coplanar discs. For polar discs, they vanish for a binary eccentricity of unity. The results agree with the simulations in showing that discs can evolve to a polar state. Circumbinary planets may then form in such discs and reside on polar orbits.

EMRIs and the relativistic loss-cone: The curious case of the fortunate coincidence

Extreme mass ratio inspiral (EMRI) events are vulnerable to perturbations by the stellar background, which can abort them prematurely by deflecting EMRI orbits to plunging ones that fall directly into the massive black hole (MBH), or to less eccentric ones that no longer interact strongly with the MBH. A coincidental hierarchy between the collective resonant Newtonian torques due to the stellar background, and the relative magnitudes of the leading-order post-Newtonian precessional and radiative terms of the general relativistic 2-body problem, allows EMRIs to decouple from the background and produce semi-periodic gravitational wave signals. I review the recent theoretical developments [1] that confirm this conjectured fortunate coincidence [2], and briefly discuss the implications for EMRI rates, and show how these dynamical effects can be probed locally by stars near the Galactic MBH.

On the relationship between circumstellar disc size and X-ray outbursts in Be/X-ray binaries

We present long term H$\alpha$ monitoring results of five Be/X-ray binaries to study the Be disc size variations and their influence on Type II (giant) X-ray outbursts. The work is done in the context of the viscous decretion disc model which predicts that Be discs in binary systems are truncated by resonant torques induced by the neutron star in its orbit. Our observations show that type II outbursts are not correlated(nor anti-correlated) with the disc size, as they are seen to occur both at relatively small and large Be disc radii. We discuss these observations in context of alternate interpretation of Be disc behaviour, such as precession, elongation and density effects, and with cognisance of the limitations of our disc size estimates.

Resonances in retrograde circumbinary discs

We analyse the interaction of an eccentric binary with a circular coplanar circumbinary disc that rotates in a retrograde sense with respect to the binary. In the circular binary case, no Lindblad resonances lie within the disc and no Lindblad resonant torques are produced, as was previously known. By analytic means, we show that when the binary orbit is eccentric, there exist components of the gravitational potential of the binary which rotate in a retrograde sense to the binary orbit and so rotate progradely with respect to this disc, allowing a resonant interaction to occur between the binary and the disc. The resulting resonant torques distinctly alter the disc response from the circular binary case. We describe results of three-dimensional hydrodynamic simulations to explore this effect and categorize the response of the disc in terms of modes whose strengths vary as a function of binary mass ratio and eccentricity. These mode strengths are weak compared to the largest mode strengths expected in the prograde case where the binary and disc rotate in the same sense. However, for sufficiently high binary eccentricity, resonant torques open a gap in a retrograde circumbinary disc, while permitting gas inflow on to the binary via gas streams. The inflow results in a time varying accretion rate on to the binary that is modulated over the binary orbital period, as was previously found to occur in the prograde case.

Influence of externally applied magnetic perturbations on neoclassical tearing modes at ASDEX Upgrade

The influence of externally applied magnetic perturbations (MPs) on neoclassical tearing modes (NTM) and the plasma rotation in general is investigated at the ASDEX Upgrade tokamak (AUG). The low n resonant components of the applied field exert local torques and influence the stability of NTMs. The non-resonant components of the error field do not influence MHD modes directly but slow down the plasma rotation globally due to a neoclassical toroidal viscous torque (NTV). Both components slow down the plasma rotation, which in consequence increases the probability for the appearance of locked modes. To investigate the impact of externally applied MPs on already existing modes and the influence on the rotation profile, experimental observations are compared to modelling results. The model used here solves a coupled equation system that includes the Rutherford equation and the equation of motion, taking into account the resonant effects and the resistive wall. It is shown that the NTV torque can be neglected in this modelling. To match the experimental frequency evolution of the mode the MP field strength at the resonant surface has to be increased compared to the vacuum approximation. This leads to an overestimation of the stabilizing effect on the NTMs. The reconstruction of the entire rotation profile via the equation of motion including radial dependencies, confirms that the NTV is negligibly small and that small resonant torques at different resonant surfaces have the same effect as one large one. This modelling suggests that in the experiment resonant torques at different surfaces are acting and slowing down the plasma rotation requiring a smaller torque at the specific resonant surface of the NTM. This additionally removes the overestimated influence on the island stability, whereas the braking of the island's rotation is caused by the sum of all torques. Consequently, to describe the effect of MPs on the evolution of one island, all other islands and the corresponding torques must also be taken into account.

Error field detection in DIII-D by magnetic steering of locked modes

Optimal correction coil currents for the n = 1 intrinsic error field of the DIII-D tokamak are inferred by applying a rotating external magnetic perturbation to steer the phase of a saturated locked mode with poloidal/toroidal mode number m/n = 2/1. The error field is detected non-disruptively in a single discharge, based on the toroidal torque balance of the resonant surface, which is assumed to be dominated by the balance of resonant electromagnetic torques. This is equivalent to the island being locked at all times to the resonant 2/1 component of the total of the applied and intrinsic error fields, such that the deviation of the locked mode phase from the applied field phase depends on the existing error field. The optimal set of correction coil currents is determined to be those currents which best cancels the torque from the error field, based on fitting of the torque balance model. The toroidal electromagnetic torques are calculated from experimental data using a simplified approach incorporating realistic DIII-D geometry, and including the effect of the plasma response on island torque balance based on the ideal plasma response to external fields. This method of error field detection is demonstrated in DIII-D discharges, and the results are compared with those based on the onset of low-density locked modes in ohmic plasmas. This magnetic steering technique presents an efficient approach to error field detection and is a promising method for ITER, particularly during initial operation when the lack of auxiliary heating systems makes established techniques based on rotation or plasma amplification unsuitable.

Toroidal modeling of plasma response and resonant magnetic perturbation field penetration

The penetration dynamics of the resonant magnetic perturbation (RMP) field is simulated in the full toroidal geometry, under realistic plasma conditions in MAST experiments. The physics associated with several aspects of the RMP penetration—the plasma response and rotational screening, the resonant and non-resonant torques and the toroidal momentum balance—are highlighted. In particular, the plasma response is found to significantly amplify the non-resonant component of the RMP field for some of the MAST plasmas. A fast rotating plasma, in response to static external magnetic fields, experiences a more distributed electromagnetic torque due to the resonance with continuum waves in the plasma. At fast plasma flow (such as for the MAST plasma), the electromagnetic torque is normally dominant over the neoclassical toroidal viscous (NTV) torque. However, at sufficiently slow plasma flow, the NTV torque can play a significant role in the toroidal momentum balance, thanks to the precession drift resonance enhanced, so-called superbanana plateau regime.

LUNAR ACCRETION FROM A ROCHE-INTERIOR FLUID DISK

We use a hybrid numerical approach to simulate the formation of the Moon from an impact-generated disk, consisting of a fluid model for the disk inside the Roche limit and an N-body code to describe accretion outside the Roche limit. As the inner disk spreads due to a thermally regulated viscosity, material is delivered across the Roche limit and accretes into moonlets that are added to the N-body simulation. Contrary to an accretion timescale of a few months obtained with prior pure N-body codes, here the final stage of the Moon's growth is controlled by the slow spreading of the inner disk, resulting in a total lunar accretion timescale of ~10^2 years. It has been proposed that the inner disk may compositionally equilibrate with the Earth through diffusive mixing, which offers a potential explanation for the identical oxygen isotope compositions of the Earth and Moon. However, the mass fraction of the final Moon that is derived from the inner disk is limited by resonant torques between the disk and exterior growing moons. For initial disks containing < 2.5 lunar masses (ML), we find that a final Moon with mass > 0.8ML contains < 60% material derived from the inner disk, with this material preferentially delivered to the Moon at the end of its accretion.

The study about torsional vibration charateristics and its optimization of vehicle transmission system

The torsional vibration of vehicle transmission system is heavily concerned with the increase of vehicle speed. The whole powertrain system has to be matched according to the torsional vibration characteristics, especially in developing a new vehicle. The selection of proper elastic coupling has to be made for the torsional vibration match and some frequencies have to be moved out of engine's range . Thus the torsional vibration model of powertrain needs to be built. In the paper a new torsional vibration model is built, which is programmed in the form of a platform. The whole powertrain system torsional vibration model of a vehicle is built firstly with consideration of gear mesh stiffness and engine's excitation in it. The free torsional vibration mode analysis is made and the resonant torques of each lumped inertia in the transmission system are obtained. Secondly the forced vibration of transmission system with the engine's excitation is made and the dynamic torques of each lumped inertias are obtained. Thirdly the process for the torsional vibration analysis is integrated into the optimization process and the selection of elastic coupling for the transmission system is made according the optimization and match results. Fourthly in order to modify the design parameters in the structural design, the sensitivities of inertia and torsional stiffness with reference to eigenvalues are obtained. At last the evaluations of analysis results are made and some suggestions for structural modification for engineers are presented. According to the above study, the conclusion can be made that the new torsional modelling method, the elastic coupling selection method and integration optimization method in the paper are practical and reliabl and these methods play very important roles in torsional vibration analyzing, match and optimization of vehicle transmission system.

The dynamics of satellite disruption in cold dark matter haloes

We investigate the physical mechanisms of tidal heating and satellite disruption in cold dark matter host haloes using N-body simulations based on cosmological initial conditions. We show the importance of resonant shocks and resonant torques with the host halo to satellite heating. A resonant shock (torque) couples the radial (tangential) motion of a satellite in its orbit to its phase space. For a satellite on a circular orbit, an ILR-like resonance dominates the heating and this heating results in continuous satellite mass loss. We estimate the requirements for simulations to achieve these dynamics using perturbation theory. Both resonant shocks and resonant torques affect satellites on eccentric orbits. We demonstrate that satellite mass loss is an outside-in process in energy space; a satellite's stars and gas are thus protected by their own halo against tidal stripping. We simulate the evolution of a halo similar to the Large Magellanic Cloud (LMC) in our Galactic dark matter halo and conclude that the LMC stars have not yet been stripped. Finally, we present a simple algorithm for estimating the evolution of satellite mass that includes both shock heating and resonant torques.

Toroidal flow and radial particle flux in tokamak plasmas

Many effects influence toroidal flow evolution in tokamak plasmas. Momentum sources and radial plasma transport due to collisional processes and microturbulence-induced anomalous transport are usually considered. In addition, toroidal flow can be affected by nonaxisymmetric magnetic fields; resonant components cause localized electromagnetic toroidal torques near rational surfaces in flowing plasmas and nonresonant components induce “global” toroidal flow damping torque throughout the plasma. Also, poloidal magnetic field transients on the magnetic field diffusion time scale can influence plasma transport. Many of these processes can also produce momentum pinch and intrinsic flow effects. This paper presents a comprehensive and self-consistent description of all these effects within a fluid moment context. Plasma processes on successive time scales (and constraints they impose) are considered sequentially: compressional Alfvén waves (Grad–Shafranov equilibrium and ion radial force balance), sound waves (pressure constant along a field line and incompressible flows within a flux surface), and ion collisions (damping of poloidal flow). Finally, plasma transport across magnetic flux surfaces is induced by the many second order (in the small gyroradius expansion) toroidal torque effects indicated above. Nonambipolar components of the induced particle transport fluxes produce radial plasma currents. Setting the flux surface average of the net radial current induced by all these effects to zero yields the transport-time-scale equation for evolution of the plasma toroidal flow. It includes a combination of global toroidal flow damping and resonant torques induced by nonaxisymmetric magnetic field components, poloidal magnetic field transients, and momentum source effects, as well as the usual collision- and microturbulence-induced transport. On the transport time scale, the plasma toroidal rotation determines the radial electric field for net ambipolar particle transport. The ultimate radial particle transport is composed of intrinsically ambipolar fluxes plus nonambipolar fluxes evaluated at this toroidal-rotation-determined radial electric field.

Resonant effect of a background turbulence on the rotating magnetic island

Resonant interaction between wave packets of the background, short wavelength ion-temperature-gradient mode turbulence and macroscopic, rotating magnetic island is considered. Corresponding contribution to the toroidal torque balance is calculated. This resonant torque is comparable with nonresonant viscous torque produced by turbulent transport of toroidal momentum. Balance of resonant and viscous torques results in the equilibrium island rotation frequency residing in the gap where contribution of the polarization current to the modified Rutherford equation is stabilizing.

Accretion onto Pre-Main Sequence Binary Stars: The Circumstellar Environment

Observations show that mass accretion can occur at the surfaces of stars in young binaries of all separations. Given that resonant torques have been expected to clear gaps and dam the inward flow of material from circumbinary disks, this result begs the question of how mass accretion is supplied in binary systems. In this paper we address this question in the context of close binaries. Spectral energy distributions indicate that dynamical clearing is not complete for close binaries having massive circumbinary disks. High-resolution infrared spectra of CO rotational and ro-vibrational lines have also detected small amounts of gas which can be located in the region of dynamical clearing. Thus we have signatures of matter from the circumbinary environment through the predicted gap region to the stellar surfaces. We suggest that accretion streams may supply both dynamically cleared regions and circumstellar disks.

Turbine, generator, system modeling and impact of variable-frequency ripple currents on torsional stressing of generators in Poland and Sweden: Lithuania/Poland and Sweden/Poland HVDC links

This paper analyzes rectifier ripple current superimposed on the direct current of a 600 MW asynchronous bipolar link between Lithuania and Poland and a 600 MW asynchronous monopolar link between Poland and Sweden in evaluating possible torsional vibrations in steam turbine-generator-exciter shafts. The studies are performed for machines in Northeastern Poland and in Southern Sweden close to the inverter station which are most at risk. Analyses are performed for the generators in Poland for the Lithuania/Poland link and for the generators in Sweden for the Poland/Sweden link. The paper examines system scaling factors for the Lithuania/Poland Link. It then examines AC system scaling factors and generator scaling factors for modulation product harmonic currents impressed on generators connected to the Swedish Grid Network by the inverter. Amplitude of shaft torsional torque due to resonant excitation imposed by the link on the generators in Sweden is evaluated for different scenarios of system and generator load. First, theory which relates magnitude of harmonic AC current injected into the AC system by the inverter is summarized. Dispersion of the injected harmonic AC current to proportion the current to individual generators is then reviewed, considering impedance of the generators, loads, and other system elements. Significance of simple and detailed shaft train models of machine shafts as well as approximate (Id, Iq) and detailed (2d, 3q) generator damper models on resonant torques is reviewed. Then, effects of both power factor and generator load on generator scaling factors is illustrated.