Resonance behaviour and partial averaging in a three-body system with gravitational radiation damping

In a previous investigation, a model of three-body motion was developed which included the effects of gravitational radiation reaction. The aim was to describe the motion of a relativistic binary pulsar that is perturbed by a third mass and look for resonances between the binary and third-mass orbits. Numerical integration of an equation of relative motion that approximates the binary gives evidence of such resonances. These ( m : n ) resonances are defined for the present purposes by the resonance condition,   m ω= 2 n Ω  , where m and n are relatively prime integers and ω and Ω are the angular frequencies of the binary orbit and third-mass orbit (around the centre of mass of the binary), respectively. The resonance condition consequently fixes a value for the semimajor axis a of the binary orbit for the duration of the resonance because of the Kepler relationship  ω= a ^−3/2  . This paper outlines a method of averaging developed by Chicone, Mashhoon and Retzloff, which renders a non-linear system that undergoes resonance capture into a mathematically amenable form. This method is applied to the present system and one arrives at an analytical solution that describes the average motion during resonance. Furthermore, prominent features of the full non-linear system, such as the frequency of oscillation and antidamping, accord with their analytically derived formulae.     

On the evolution of giant protoplanets forming in circumbinary discs

We present the results of hydrodynamic simulations of Jovian mass protoplanets that form in circumbinary discs. The simulations follow the orbital evolution of the binary plus protoplanet system acting under their mutual gravitational forces, and forces exerted by the viscous circumbinary disc. The evolution involves the clearing of the inner circumbinary disc initially, so that the binary plus protoplanet system orbits within a low density cavity. Continued interaction between disc and protoplanet causes inward migration of the planet towards the inner binary. Subsequent evolution can take three distinct paths: (i) the protoplanet enters the 4 : 1 mean motion resonance with the binary, but is gravitationally scattered through a close encounter with the secondary star; (ii) the protoplanet enters the 4 : 1 mean motion resonance, the resonance breaks, and the planet remains in a stable orbit just outside the resonance; (iii) when the binary has initial eccentricity   e _bin≥ 0.2  , the disc becomes eccentric, leading to a stalling of the planet migration, and the formation of a stable circumbinary planet.
These results have implications for a number of issues in the study of extrasolar planets. The ejection of protoplanets in close binary systems provides a source of 'free-floating planets', which have been discovered recently. The formation of a large, tidally truncated cavity may provide an observational signature of circumbinary planets during formation. The existence of protoplanets orbiting stably just outside a mean motion resonance (4 : 1) in the simulations indicate that such sites may harbour planets in binary star systems, and these could potentially be observed. Finally, the formation of stable circumbinary planets in eccentric binary systems indicates that circumbinary planets may not be uncommon.     

The dynamical stability of W Ursae Majoris-type systems

Theoretical study indicates that a contact binary system would merge into a rapidly rotating single star due to tidal instability when the spin angular momentum of the system is more than a third of its orbital angular momentum. Assuming that W Ursae Majoris (W UMa) contact binary systems rigorously comply with the Roche geometry and the dynamical stability limit is at a contact degree of about 70 per cent, we obtain that W UMa systems might suffer Darwin's instability when their mass ratios are in a region of about 0.076–0.078 and merge into the fast-rotating stars. This suggests that the W UMa systems with mass ratio   q ≤ 0.076  cannot be observed. Meanwhile, we find that the observed W UMa systems with a mass ratio of about 0.077, corresponding to a contact degree of about 86 per cent would suffer tidal instability and merge into the single fast-rotating stars. This suggests that the dynamical stability limit for the observed W UMa systems is higher than the theoretical value, implying that the observed systems have probably suffered the loss of angular momentum due to gravitational wave radiation (GR) or magnetic stellar wind (MSW).     

The magnetic SW Sextantis star RX J1643.7+3402

We present time-resolved spectroscopy and circular spectropolarimetry of the SW Sex star RX J1643.7+3402. We find significant polarization levels exhibiting a variability at a period of  19.38 ± 0.39  min. In addition, emission-line flaring is found predominantly at twice the polarimetric period. These two findings are strong evidences in favour of the presence of a magnetic white dwarf in the system. We interpret the measured periodicities in the context of our magnetic accretion model for SW Sex stars. In contrast with LS Pegasi – the first SW Sex star discovered to have modulated circular polarization – the polarization in RX J1643.7+3402 is suggested to vary at  2(ω−Ω)  , while the emission lines flare at  (ω−Ω)  . However, a  2ω/ω  interpretation cannot be ruled out. Together with LS Peg and V795 Her, RX J1643.7+3402 is the third SW Sex star known to exhibit modulated circular polarization.     

A non-linear theory of vertical resonances in accretion discs

An important and widely neglected aspect of the interaction between an accretion disc and a massive companion with a coplanar orbit is the vertical component of the tidal force. As shown by Lubow, the response of the disc to vertical forcing is resonant at certain radii, at which a localized torque is exerted, and from which a compressive wave (p mode) may be emitted. Although these vertical resonances are weaker than the corresponding Lindblad resonances, the   m =2  inner vertical resonance in a binary star is typically located within the tidal truncation radius of a circumstellar disc.
In this paper I develop a general theory of vertical resonances, allowing for non-linearity of the response, and dissipation by radiative damping and turbulent viscosity. The problem is reduced to a universal, non-linear ordinary differential equation with two real parameters. Solutions of the complex non-linear Airy equation are presented to illustrate the non-linear saturation of the resonance and the effects of dissipation. It is argued that the   m =2  inner vertical resonance is unlikely to truncate the disc in cataclysmic variable stars, but contributes to angular momentum transport and produces a potentially observable non-axisymmetric structure.     

Magnetic field, dust and axisymmetrical mass loss on the asymptotic giant branch

I propose a mechanism for axisymmetrical mass loss on the asymptotic giant branch (AGB) that may account for the axially symmetric structure of elliptical planetary nebulae. The proposed model operates for slowly rotating AGB stars, having angular velocities in the range of 10⁻⁴ω _Kep  ω  10⁻² ω_Kep, where ω_Kep is the equatorial Keplerian angular velocity. Such angular velocities could be gained from a planet companion of mass  0.1  M _Jupiter, which deposits its orbital angular momentum to the envelope at late stages, or even from single stars that are fast rotators on the main sequence. The model assumes that dynamo magnetic activity results in the formation of cool spots, above which dust forms much more easily. The enhanced magnetic activity towards the equator results in a higher dust formation rate there, and hence higher mass-loss rate. As the star ascends the AGB, both the mass-loss rate and magnetic activity increase rapidly, and hence the mass loss becomes more asymmetrical, with higher mass-loss rate closer to the equatorial plane.     

Effect of solar pressure on the motion and stability of the system of two inter-connected satellites in an elliptical orbit

The effect of solar pressure on the two-dimensional motion of two cable-connected satellites in the Earth's central gravitational field of force for the elliptical orbit of the centre-of-mass of the system has been analysed. The equations of motion obtianed are nonlinear and non-autonomous.It is concluded with the aid of non-resonant solution that the system experiences resonance main as well as parametric. If the eccentricity is small, the system will always oscillate about the position of equilibrium with tight string like dumb-bell satellite with changing phase and constant amplitude.     

The 2/1 and 3/2 Resonant Asteroid Motion: A Symplectic Mapping Approach

A comparative study is made between the 2/1 and the 3/2 resonant asteroid motion, with the aim to understand their different behaviour (gap in the 2/1 resonance, group in the 3/2 resonance). A symplectic mapping model is used, for each of these two resonances, assuming the asteroid is moving in the three-dimensional space under the gravitational perturbation of Jupiter. It is found that these resonances differ in several points, and although there is, in general, more chaos in the phase space close to the 3/2 resonance, even in the model of circular orbit of Jupiter, there are regions, close to the secondary resonances, which are less chaotic in the 3/2 resonance compared to the 2/1 resonance, and consequently trapping can take place.     

Wave excitation in three-dimensional discs by external potential

We study the excitation of density and bending waves and the associated angular momentum transfer in gaseous discs with finite thickness by a rotating external potential. The disc is assumed to be isothermal in the vertical direction and has no self-gravity. The disc perturbations are decomposed into different modes, each characterized by the azimuthal index m and the vertical index n , which specifies the nodal number of the density perturbation along the disc normal direction. The   n = 0  modes correspond to the two-dimensional density waves previously studied by Goldreich & Tremaine and others. In a three-dimensional disc, waves can be excited at both Lindblad resonances (LRs; for modes with   n = 0, 1, 2, …  ) and vertical resonances (VRs; for the   n ≥ 1  modes only). The torque on the disc is positive for waves excited at outer Lindblad/vertical resonances and negative at inner Lindblad/vertical resonances. While the   n = 0  modes are evanescent around corotation, the   n ≥ 1  modes can propagate into the corotation region where they are damped and deposit their angular momenta. We have derived analytical expressions for the amplitudes of different wave modes excited at LRs and/or VRs and the resulting torques on the disc. It is found that for   n ≥ 1  , angular momentum transfer through VRs is much more efficient than LRs. This implies that in some situations (e.g. a circumstellar disc perturbed by a planet in an inclined orbit), VRs may be an important channel of angular momentum transfer between the disc and the external potential. We have also derived new formulae for the angular momentum deposition at corotation and studied wave excitations at disc boundaries.     

Mechanism of generation of the emission bands in the dynamic spectrum of the Crab pulsar

We show that the proportionately spaced emission bands in the dynamic spectrum of the Crab pulsar fit the oscillations of the square of a Bessel function whose argument exceeds its order. This function has already been encountered in the analysis of the emission from a polarization current with a superluminal distribution pattern: a current whose distribution pattern rotates (with an angular frequency ω) and oscillates (with a frequency  Ω > ω  differing from an integral multiple of ω) at the same time. Using the results of our earlier analysis, we find that the dependence on frequency of the spacing and width of the observed emission bands can be quantitatively accounted for by an appropriate choice of the value of the single free parameter  Ω/ω  . In addition, the value of this parameter, thus implied by Hankins & Eilek's data, places the last peak in the amplitude of the oscillating Bessel function in question at a frequency  (∼Ω³/ω²)  that agrees with the position of the observed ultraviolet peak in the spectrum of the Crab pulsar. We also show how the suppression of the emission bands by the interference of the contributions from differing polarizations can account for the differences in the time and frequency signatures of the interpulse and the main pulse in the Crab pulsar. Finally, we put the emission bands in the context of the observed continuum spectrum of the Crab pulsar by fitting this broad-band spectrum (over 16 orders of magnitude of frequency) with that generated by an electric current with a superluminally rotating distribution pattern.     

A Symplectic Mapping Model as a Tool to Understand the Dynamics of 2/1 Resonant Asteroid Motion

We present a 3-D symplectic mapping model that is valid at the 2:1 mean motion resonance in the asteroid motion, in the Sun-Jupiter-asteroid model. This model is used to study the dynamics inside this resonance and several features of the system have been made clear. The introduction of the third dimension, through the inclination of the asteroid orbit, plays an important role in the evolution of the asteroid and the appearance of chaotic motion. Also, the existence of the secondary resonances is clearly shown and their role in the appearance of chaotic motion and the slow diffusion of the elements of the orbit is demonstrated. This revised version was published online in July 2006 with corrections to the Cover Date.     

Multiple resonance in one problem of the stability of the motion of a satellite relative to the center of mass

We investigate the stability of the periodic motion of a satellite, a rigid body, relative to the center of mass in a central Newtonian gravitational field in an elliptical orbit. The orbital eccentricity is assumed to be low. In a circular orbit, this periodic motion transforms into the well-known motion called hyperboloidal precession (the symmetry axis of the satellite occupies a fixed position in the plane perpendicular to the radius vector of the center of mass relative to the attractive center and describes a hyperboloidal surface in absolute space, with the satellite rotating around the symmetry axis at a constant angular velocity). We consider the case where the parameters of the problem are close to their values at which a multiple parametric resonance takes place (the frequencies of the small oscillations of the satellite’s symmetry axis are related by several second-order resonance relations). We have found the instability and stability regions in the first (linear) approximation at low eccentricities.     

Hill stability of a triple system with an inner binary of large mass ratio

We determine the maximum dimensionless pericentre distance a third body can have to the barycentre of an extreme mass ratio binary, beyond which no exchange or ejection of any of the binary components can occur. We calculate this maximum distance, q '/ a , where q ' is the pericentre of the third mass to the binary barycentre and a is the semimajor axis of the binary, as a function of the critical value of   L ²  E   of the system, where L is the magnitude of the angular momentum vector and E is the total energy of the system. The critical value is obtained by calculating   L ²  E   for the central configuration of the system at the collinear Lagrangian points. In our case we can make approximations for the system when one of the masses is small. We compare the calculated values of the pericentre distance with numerical scattering experiments as a function of the eccentricity of the inner orbit, e , the mutual inclination i and the eccentricity of the outer orbit, e '. These show that the maximum observed value of   q '/ a   is indeed the critical q '/ a , as expected. However, when   e '→1  , the maximum observed value of q '/ a is equal to the critical value calculated when   e '=0  , which is contrary to the theory, which predicts exchange distances several orders of magnitude larger for nearly parabolic orbits. This does not occur because changes in the binding energy of the binary are exponentially small for distant, nearly parabolic encounters.     

On gravitational radiation from binary systems

We discuss gravitational radiation from a neutral mass particle within a bound orbit in the background Schwarzschild metric. We compare the power loss of gravitational radiation according to this formalism with the heuristic quadrupole radiation formula as applied to a binary system. There are evidence and compelling reasons to believe that the quadrupole formula is valid even in a fairly strong gravitational field, although its fully consistent analytical derivation is not yet known. In particular, we emphasize that the application of the quadrupole formula to the binary pulsar system PSR 1913+16 as well as other binary pulsars, which are weakly bound by gravity, is well justified.     

Restriction on the motion of a solid in a gravitational field

As is well known, the orbital and rotational motions of a solid are coupled, and the integrals of energy and angular momentum (in a gravitational field with spherical symmetry) impose restrictions on them. We study the regions allowed to the motion in configurational space. It turns out that even in the crudest model (planar motion of a triple rod) the restrictions on the libration angle and the orbital radius of the center of mass are coupled, so that excessive ellipticity of the orbit excludes stabilization in the neighbourhood of the spoke equilibrium position by gravitational forces only.Chargé de Cours.     

Effect of radiation on the stability of a retrograde particle orbit in different stellar systems

We have numerically investigated the stability of retrograde orbits/trajectories around Jupiter and the smaller of the primaries in binary systems RW-Monocerotis (RW-Mon) and Krüger-60 in the presence of radiation. A trajectory is considered as stable if it remains around the smaller mass for at least few hundred binary periods. In case of circular binary orbit, we find that the third order resonance provides the basis for reduction of stability region of retrograde motion of particle in RW-Mon and Sun-Jupiter system both in the presence and absence of radiation. Considering finite ellipticity in Sun-Jupiter system we find that for distant retrograde orbits, radiation from the Sun increases the width of the stable region and covers a significant portion of the region obtained in the absence of solar radiation. Further, due to solar radiation pressure, the stable region in the neighborhood of Jupiter has been found to shift much below the characteristic asymptotic line for the periodic retrograde orbits. In case of Krüger-60 we observe the distant retrograde orbits around the smaller of the primaries get affected considerably with increase in radiation parameter β₁. Further the range of velocities for which stable motion may persist narrows down for distant retrograde orbits in this system.     

Investigating cosmological weak lensing with the ray-bundle method

Using the ray-bundle method for calculating gravitational lens magnifications, we outline a method by which the magnification probability may be determined specifically in the weak lensing limit for cosmological models obtained from N -body simulations.
16 different models are investigated, which are variations on three broad classes of cold dark matter model: the standard model with  (Ω₀, λ ₀)=(1.0,0.0)  , the open model with  (Ω₀, λ ₀)=(0.3,0.0)  and the lambda model, which is a flat model with a cosmological constant  (Ω₀, λ ₀)=(0.3,0.7)  .
The effects of varying the Hubble parameter, H ₀, the power spectrum shape parameter, Γ, and the cluster mass normalization, σ ₈, are studied. It is shown that there is no signature of these parameters in the weak lensing magnification distributions. The magnification probability distributions are also shown to be independent of the numerical parameters such as the lens mass and simulation box size in the N -body simulations.     

Simulations of superhumps and superoutbursts

We numerically study the tidal instability of accretion discs in close binary systems using a two-dimensional SPH code. We find that the precession rate of tidally unstable, eccentric discs does not only depend upon the binary mass ratio q . Although the (prograde) disc precession rate increases with the strength of the tidal potential, we find that increasing the shear viscosity ν also has a significant prograde effect. Increasing the disc temperature has a retrograde impact upon the precession rate.   We find that motion relative to the binary potential results in superhump-like, periodic luminosity variations in the outer reaches of an eccentric disc. The nature and location of the luminosity modulation are functions of ν. Light curves most similar to observations are obtained for ν values appropriate for a dwarf nova in outburst.   We investigate the thermal–tidal instability model for superoutburst. A dwarf nova outburst is simulated by instantaneously increasing ν, which causes a rapid radial expansion of the disc. Should the disc encounter the 3: 1 eccentric inner Lindblad resonance and become tidally unstable, then tidal torques become much more efficient at removing angular momentum from the disc. The disc then shrinks and M _d increases. The resulting increase in disc luminosity is found to be consistent with the excess luminosity of a superoutburst.     

Influence of the r-mode instability on hypercritically accreting neutron stars

We have investigated the influence of the r-mode instability on hypercritically accreting neutron stars in close binary systems during their common envelope phases, based on the scenario proposed by Brown et al. On the one hand, neutron stars are heated by the accreted matter at the stellar surface, but on the other hand they are also cooled down by the neutrino radiation. At the same time, the accreted matter transports its angular momentum and mass to the star. We have studied the evolution of the stellar mass, temperature and rotational frequency.
The gravitational-wave-driven instability of the r-mode oscillation strongly suppresses spinning up of the star, the final rotational frequency of which is well below the mass-shedding limit, in fact typically as low as 10 per cent of that of the mass-shedding state. On a very short time-scale the rotational frequency tends to approach a certain constant value and saturates there, as long as the amount of accreted mass does not exceed a certain limit to collapse to a black hole. This implies that a similar mechanism of gravitational radiation to that in the so-called 'Wagoner star' may work in this process. The star is spun up by accretion until the angular momentum loss by gravitational radiation balances the accretion torque. The time-integrated dimensionless strain of the radiated gravitational wave may be large enough to be detectable by gravitational wave detectors such as LIGO II.