Some remarks on the generalized many-body problem

This paper deals with the generalized problem of motion of a system of a finite number of bodies (material points).We suppose here that every point of the system acts on another one with a force (attractive or repulsive) directed along the straight line connecting these two points, and proportional to the product of their masses and a certain function of time, mutual distance and its derivatives of the first and second order (Duboshin, 1970).The laws of forces are different for different pairs of points and, generally speaking, the validity of the third axiom of dynamics (law of action and reaction) is not assumed in advance.With these general assumptions we find the conditions for the laws of the forces under which the problem admits the first integrals, analogous to the classic integrals of the many-body problem with the Newton's law of attraction.It is shown furthermore, that in this generalized problem it is possible to obtain an equation, analogous to the classic equation of Lagrange-Jacobi and deduce the conditions of stability or instability of the system in Lagrange's sense.The results obtained may be applied for the investigation of motion in some isolated stellar systems, where the laws of mechanics may be different from the laws in our solar system.     

The collapse of iron-oxygen stars: Physical and mathematical formulation of the problem and computational method

A statement of the problem of gravitational collapse and a computational method are described. The main feature of the collapse — its extremely high heterogeneity — is taken into account. The structure of a collapsing star is characterized by a dense and hot nucleon core which is opaque with respect to neutrino radiation and is embedded in to and extended envelope, almost transparent to neutrinos. The envelope is gradually being accreted onto the core. The enormous amount of energy, radiated in the form of neutrinos and antineutrinos, make us pay particular attention to relatively small absorption of neutrino radiation by extended envelope (so-called energy of deposition). The inclusion of the energy deposition in the calculations is of importance for the problem of transformation of an implosion into an explosion. The deposition is taken into consideration in the approximation of diluted neutrino radiation which escapes from neutrino photosphere and is partially absorbed in the envelope. Both the generation of energy due to deposition and the change of neutronto-proton ratio are taken into account. The increase of the mass of the core, which is opaque with respect to neutrino radiation, is fully taken into account in the calculations of the gravitational collapse.     

Note concerning gravitation and electromagnetism

The Maxwell equations for gravitational fields previously assumed by Sciama are derived from elementary considerations. The Lagrangian for a gravitating mass in a non-inertial coordinate system yields equations of motion leading to force definitions for a gravitational field intensity and a gravitational induction field. The non-inertial velocity of the coordinate system plays the role of a vector potential contributing to the generalized momenta of bodies moving in the system. A Lagrangian density constructed from the force-defined fields then lead to the source definitions of gravitational fields. It is found that positive field energy densities require repulsive gravitational forces, whereas attractive forces imply the violation of the conservation of energy. This paradox is resolved by representing gravitational quantities as pure-imaginary entities. Thus characterized, the equations which define gravitational fields become identical to Maxwell's equations but are pure-imaginary. This suggests a combined representation for gravitational and electromagnetic fields which, in covariant form, indicates both the well known equivalence of mass and energy and a possible equivalence of charge and energy. From orthogonality considerations, it is conjectured that this latter energy is gravitational, and that, whereas gravitational fields interact with electromagnetic energy, electromagnetic fields interact with gravitational energy. Parts of this work were completed at Air Force Cambridge Research Laboratories, Bedford, Mass., U.S.A.     

Expanding envelopes of binary stars

In this paper which is a continuation of Neutsch and Schmidt (1985a) (later on referred to as part I) we shall investigate the structure of binary envelopes under the hypothesis that at least one of the companions produces a very intense radiation field. The limiting case in which the forces due to the system's rotation (i.e., Coriolis and centrifugal forces) as well as gas pressure can be neglected is solved analytically using a classical result of Euler. Beyond this the velocity and density distributions in the envelope are determined.Under the assumptions mentioned above, the results are exact and are, therefore, of value as a reference model for later more advanced considerations (see, e.g., Neutsch and Schmidt, 1985b). Although the present approach is quite simple, it nevertheless shows some typical features of the general case. Furthermore, it serves as a helpful guide in guessing the system parameters to be initially adopted in physically sounder. calculations (for example: the Monte-Carlo approximation which will be described in Part III).     

Incompatibility of QED/QCD and repulsive gravity,and implications for some recent approaches to dark energy

The measurement of the gravitational properties of antimatter is currently a hot research area in experimental physics. Using an outcome of QED calculations by Alves et al. (arXiv:0907.4110, 2009), this letter proves that QED and repulsive gravity are incompatible by showing that an extension of QED with the assumption of negative gravitational mass for antimatter yields a concrete prediction that is already falsified by the recent Eöt-Wash experiments: if repulsive gravity, and thus negative gravitational mass, would be observed by any of the upcoming experiments, then QED is thus experimentally falsified; the same goes for QCD. An immediate consequence is that virtual particle-antiparticle pairs from contemporary quantum theory cannot be a model for Hajdukovic’s virtual gravitational dipoles, nor for the dipolar medium of Blanchet and Le Tiec. There may be ways to reformulate quantum theory to restore consistency with experiment if repulsive gravity would be observed, but these involve a departure from the framework of four dimensions and four forces of nature: an observation of repulsive gravity would thus provide a reason to reject the quantum paradigm in its entirety and to search for new fundamental physics.     

Reduction the secular solution to periodic solution in the generalized restricted three-body problem

The aim of the present work is to find the secular solution around the triangular equilibrium points and reduce it to the periodic solution in the frame work of the generalized restricted thee-body problem. This model is generalized in sense that both the primaries are oblate and radiating as well as the gravitational potential from a belt. We show that the linearized equation of motion of the infinitesimal body around the triangular equilibrium points has a secular solution when the value of mass ratio equals the critical mass value. Moreover, we reduce this solution to periodic solution, as well as some numerical and graphical investigations for the effects of the perturbed forces are introduced. This model can be used to examine the existence of a dust particle near the triangular points of an oblate and radiating binary stars system surrounded by a belt.     

Disk wind in young binaries and the origin of the cyclic activity of young stars

We present the results of our numerical simulations of the cyclic brightness modulation in young binary systems with eccentric orbits and low-mass secondary components. We suggest that the binary components accrete matter from the remnants of the protostellar cloud, with the main accretor (according to current models) being the low-mass component. The brightness variations of the primary are attributable to the periodic extinction variations on the line of sight caused by the disk wind from the secondary and by the common envelope produced by this wind. The distribution of matter in the envelope was calculated in the ballistic approximation. When calculating the optical effects produced by the dust component of the disk wind, we adopted the dust-to-gas mass ratio of 1:100 characteristic of the interstellar medium and the optical parameters of the circumstellar dust typical of young stars. Our calculations show that the theoretical light curves for binaries with elliptical orbits exhibit a wider variety of shapes than those for binaries with circular orbits. In this case, the parameters of the photometric minima (their depth, duration, and shape of the light curve) depend not only on the disk-wind parameters and the orbital inclination of the binary to the line of sight, but also on the longitude of the periastron. We investigate the modulation of the scattered radiation from the common envelope with orbital phase in the single-scattering approximation. The modulation amplitude is shown to be at a maximum when the system is seen edge-on and to be also nonzero in binaries seen pole-on. We discuss possible applications of the theory to young stellar objects. In particular, several model light curves have been found to be similar to those of candidate FU Orionis stars (FUORs).     

Infrared photometry of the X-ray binary XTE J1118+480 in April 2000

We present the IR photometry of the X-ray binary XTE J1118+480 performed during seven nights in April and two nights in May–June 2000. A significant IR excess has been detected in the object, which may be due to the thermal radiation from a dust envelope/cloud. The observed energy distribution in the range 1.25–3.5 μm can be interpreted in terms of the sum of the fluxes from an accretion disk with a temperature of ～20 000 K and a dust envelope with grains heated to ～900 K. The distance to the X-ray binary estimated from the total flux from the dust envelope is no less than 0.6–3 kpc. The mean optical depth of the dust envelope for the accretion-disk radiation is about 0.06.     

The interaction between the secondary and the common convective envelope in a contact binary

It has been suggested that a contact system almost certainly cannot exist in static equilibrium undergoing periodic thermal relaxation oscillation. The energy transfer in a common convective envelope (CCE) makes the secondary have a complex structure, so the interaction between the secondary and CCE may play an important role in the structure and evolution of the contact system. The present paper tests the TRO theory and investigates this interaction with polytropic stellar model from the observational datum of 22 contact systems directly. It shows that the A-type systems are expanding with a velocity of 25.04 m yr^–1, and the W-type systems are contracting at velocity of 3.10 m yr^–1 by the calculations about these contact systems. Also, we calculate the ratio of energy transfer and the interaction coefficient for them. The HS (hot secondary) model is supported by our calculations. These results may help to understand the TRO theory and the W-phenomenon.     

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.     

Adiabatic invariants for the nonconservative Kepler's problem

This paper considers adiabatic invariants for the classical Kepler problem with resisting forces. The analysis is based on the theory of integrating factors and theory of adiabatic invariants in the Krylov-Bogoliubov-Mitropolski variables. The adiabatic invariants are series with respect to a small parameter. Also, for every particular case of nonconservative forces, it is shown that, with a complete set of adiabatic invariants, an approximate solution of the problem can be obtained. Four problems are analyzed in detail where approximate solutions are compared with numerical.     

Frequency analysis of the stability of asteroids in the framework of the restricted three-body problem

The stability of some asteroids, in the framework of the restricted three-body problem, has been recently proved in (Celletti and Chierchia, 2003), by developing an isoenergetic KAM theorem. More precisely, having fixed a level of energy related to the motion of the asteroid, the stability can be obtained by showing the existence of nearby trapping invariant tori existing at the same energy level. The analytical results are compatible with the astronomical observations, since the theorem is valid for the realistic mass-ratio of the primaries. The model adopted in (Celletti and Chierchia, 2003), is the planar, circular, restricted three-body model, in which only the most significant contributions of the Fourier development of the perturbation are retained. In this paper we investigate numerically the stability of the same asteroids considered in (Celletti and Chierchia, 2003), (namely, Iris, Victoria and Renzia). In particular, we implement the nowadays standard method of frequency-map analysis and we compare our investigation with the analytical results on the planar, circular model with the truncated perturbing function. By means of frequency analysis, we study the behaviour of the bounding tori and henceforth we infer stability properties on the dynamics of the asteroids. In order to test the validity of the truncated Hamiltonian, we consider also the complete expression of the perturbing function on which we perform again frequency analysis. We investigate also more realistic models, taking into account the eccentricity of the trajectory of Jupiter (planar-elliptic problem) or the relative inclination of the orbits (circular-inclined model). We did not find a relevant discrepancy among the different models.     

A binary merger model for the formation of the Supernova 1987A triple-ring nebula

We examine a binary merger model for the formation of the mysterious triple-ring nebula surrounding Supernova 1987A, which still has not been convincingly explained in detailed hydrodynamical calculations. During the merger of 15 and  5 M_⊙  binary systems, mass is ejected primarily at mid-latitudes for a sufficiently evolved primary, as demonstrated by Morris & Podsiadlowski. This material is swept up by the fast wind of the central star during its post-merger blue supergiant phase, leading to a density contrast of ∼150 in the outer rings at the time of the supernova. The equatorial ring probably formed later when the star contracted to become a blue supergiant. The asymmetry between the northern and southern outer rings can be explained by a 10 per cent asymmetry during the merger, perhaps due to a pulsational instability in the common envelope.
We present a parameter study from which we determine a mass-loss rate in the blue supergiant wind in the range  1.5–3 × 10⁻⁷ M_⊙ yr⁻¹  in agreement with previous estimates. The morphology of the best model is consistent with the well-known Hubble Space Telescope image at better than 5 per cent and is also in broad agreement with light-echo observations. The circumstellar environment on larger scales (up to 3 pc) is also investigated. We conclude with a brief discussion of the bipolar nebulae surrounding the Galactic stars, Sheridan 25, HD 168625 and η Carinae.     

Gravitational radiation damping and the three-body problem

A model of three-body motion is developed which includes the effects of gravitational radiation reaction. The radiation reaction due to the emission of gravitational waves is the only post-Newtonian effect that is included here. For simplicity, all of the motion is taken to be planar. Two of the masses are viewed as a binary system, and the third mass, whose motion will be a fixed orbit around the centre-of-mass of the binary system, is viewed as a perturbation. This model aims to describe the motion of a relativistic binary pulsar that is perturbed by a third mass. Numerical integration of this simplified model reveals that, given the right initial conditions and parameters, one can see resonances. These ( m , n ) resonances are defined by the resonance condition,   mω =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 of the binary orbit for the duration of the resonance; therefore the binary energy remains constant on average, while its angular momentum changes during the resonance.     

Global chaoticity in the Pythagorean three-body problem

The effects of small changes in the initial conditions of the Pythagorean three-body problem are investigated by computer simulations. This problem consists of three interacting bodies with masses 3, 4 and 5 placed with zero velocities at the apices of a triangle with sides 3, 4 and 5. The final outcome of this motion is that two bodies form a binary and the third body escapes. We attempt to establish regions of the initial positions which give regular and chaotic motions. The vicinity of a small neighbourhood around the standard initial position of each body defines a regular region. Other regular regions also exist. Inside these regions the parameters of the triple systems describing the final outcome change continuously with the initial positions. Outside the regular regions the variations of the parameters are abrupt when the initial conditions change smoothly. Escape takes place after a close triple approach which is very sensitive to the initial conditions. Time-reversed solutions are employed to ensure reliable numerical results and distinguish between predictable and non-predictable motions. Close triple approaches often result in non-predictability, even when using regularization; this introduces fundamental difficulties in establishing chaotic regions.     

On the formation and evolution of black hole binaries

We present the results of a systematic study of the formation and evolution of binaries containing black holes and normal-star companions with a wide range of masses. We first reexamine the standard formation scenario for close black hole binaries, where the progenitor system, a binary with at least one massive component, experienced a common-envelope phase and where the spiral-in of the companion in the envelope of the massive star caused the ejection of the envelope. We estimate the formation rates for different companion masses and different assumptions about the common-envelope structure and other model parameters. We find that black hole binaries with intermediate- and high-mass secondaries can form for a wide range of assumptions, while black hole binaries with low-mass secondaries can only form with apparently unrealistic assumptions (in agreement with previous studies).
We then present detailed binary evolution sequences for black hole binaries with secondaries of 2 to 17 M_⊙ and demonstrate that in these systems the black hole can accrete appreciably even if accretion is Eddington-limited (up to 7 M_⊙ for an initial black hole mass of 10 M_⊙) and that the black holes can be spun up significantly in the process. We discuss the implications of these calculations for well-studied black hole binaries (in particular GRS 1915+105) and ultraluminous X-ray sources of which GRS 1915+105 appears to represent a typical Galactic counterpart. We also present a detailed evolutionary model for Cygnus X-1, a massive black hole binary, which suggests that at present the system is most likely in a wind mass-transfer phase following an earlier Roche-lobe overflow phase. Finally, we discuss how some of the assumptions in the standard model could be relaxed to allow the formation of low-mass, short-period black hole binaries, which appear to be very abundant in nature.     

The accelerated Kepler problem

The accelerated Kepler problem (AKP) is obtained by adding a constant acceleration to the classical two-body Kepler problem. This setting models the dynamics of a jet-sustaining accretion disk and its content of forming planets as the disk loses linear momentum through the asymmetric jet-counterjet system it powers. The dynamics of the accelerated Kepler problem is analyzed using physical as well as parabolic coordinates. The latter naturally separate the problem’s Hamiltonian into two unidimensional Hamiltonians. In particular, we identify the origin of the secular resonance in the AKP and determine analytically the radius of stability boundary of initially circular orbits that are of particular interest to the problem of radial migration in binary systems as well as to the truncation of accretion disks through stellar jet acceleration.     

A search for collision orbits in the free-fall three-body problem I. Numerical procedure

A numerical procedure is devised to find binary collision orbits in the free-fall three-body problem. Applying this procedure, families of binary collision orbits are found and a sequence of triple collision orbits are positioned. A property of sets of binary collision orbits which is convenient to search triple collision orbits is found. Important numerical results are formulated and summarized in the final section.     

The Caledonian Symmetrical Double Binary Four-Body Problem I: Surfaces of Zero-Velocity Using the Energy Integral

The Caledonian four-body problem introduced in a recent paper by the authors is reduced to its simplest form, namely the symmetrical, four body double binary problem, by employing all possible symmetries. The problem is three-dimensional and involves initially two binaries, each binary having unequal masses but the same two masses as the other binary. It is shown that the simplicity of the model enables zero-velocity surfaces to be found from the energy integral and expressed in a three dimensional space in terms of three distances r ₁, r ₂, and r ₁₂, where r ₁ and r ₂ are the distances of two bodies which form an initial binary from the four body systems centre of mass andr ₁₂ is the separation between the two bodies.     

On the double averaged three-body problem

The Hamiltonian of three point masses is averaged over fast variablel and ll (mean anomalies) The problem is non-planar and it is assumed that two of the bodies form a close pair (stellar three-body problem). Only terms up to the order of (a/á)⁴ are taken into account in the Hamiltonian, wherea andá are the corresponding semi-major axes. Employing the method of elimination of the nodes, the problem may be reduced to one degree of freedom. Assuming in addition that the angular momentum of the close binary is much smaller than the angular momentum of the motion of the binary around a third body, we were able to solve the equation for the eccentricity changes in terms of the Jacobian elliptic functions.