Dynamics of rotating triple systems

We investigate the dynamical evolution of 100 000 rotating triple systems with equal-mass components. The system rotation is specified by the parameter ω=?c²E, where c and E are the angular momentum and total energy of the triple system, respectively. We consider ω=0.1,1, 2, 4, 6 and study 20 000 triple systems with randomly specified coordinates and velocities of the bodies for each ω. We consider two methods for specifying initial conditions: with and without a hierarchical structure at the beginning of the evolution. The evolution of each system is traced until the escape of one of the bodies or until the critical time equal to 1000 mean system crossing times. For each set of initial conditions, we computed parameters of the final motions: orbital parameters for the final binary and the escaping body. We analyze variations in the statistical characteristics of the distributions of these parameters with ω. The mean disruption time of triple systems and the fraction of the systems that have not been disrupted in 1000 mean crossing times increase with ω. The final binaries become, on average, wider at larger angular momenta. The distribution of their eccentricities does not depend on ω and generally agrees with the theoretical law f(e)=2e. The velocities of the escaping bodies, on average, decrease with increasing angular momentum of the triple system. The fraction of the angles between the escaping-body velocity vector and the triple-system angular momentum close to 90° increases with ω. Escapes in the directions opposite to rotation and prograde motions dominate at small and large angular momenta, respectively. For slowly rotating systems, the angular momentum during their disruption is, on average, evenly divided between the escaping body and the final binary, whereas in rapidly rotating systems, about 80% of the angular momentum is carried away by the escaping component. We compare our numerical simulations with the statistical theory of triple-system disruption.     

Minimal energy configurations of gravitationally interacting rigid bodies

Consider a collection of n rigid, massive bodies interacting according to their mutual gravitational attraction. A relative equilibrium motion is one where the entire configuration rotates rigidly and uniformly about a fixed axis in \(\mathbb {R}^3\). Such a motion is possible only for special positions and orientations of the bodies. A minimal energy motion is one which has the minimum possible energy in its fixed angular momentum level. While every minimal energy motion is a relative equilibrium motion, the main result here is that a relative equilibrium motion of \(n\ge 3\) disjoint rigid bodies is never an energy minimizer. This generalizes a known result about point masses to the case of rigid bodies.     

Stability of equilibrium points in the generalized perturbed restricted three-body problem

This paper studies the existence and stability of equilibrium points under the influence of small perturbations in the Coriolis and the centrifugal forces, together with the non-sphericity of the primaries. The problem is generalized in the sense that the bigger and smaller primaries are respectively triaxial and oblate spheroidal bodies. It is found that the locations of equilibrium points are affected by the non-sphericity of the bodies and the change in the centrifugal force. It is also seen that the triangular points are stable for 0<μ<μ _c and unstable for \(\mu_{c}\le\mu <\frac{1}{2}\), where μ _c is the critical mass parameter depending on the above perturbations, triaxiality and oblateness. It is further observed that collinear points remain unstable.     

No hanging out in neighborhoods of infinity in the three-body problem

Consider the spatial Newtonian three-body problem at fixed negative energy and fixed angular momentum. The moment of inertia I provides a measure of the overall size of a three-body system. We will prove that there is a positive number \(I_0\) depending on the energy and angular momentum levels as well as the masses such that every solution at these levels passes through \(I\le I_0\) at some instant of time. Motivation for this result comes from trying to prove the impossibility of realizing a certain syzygy sequence in the zero angular momentum problem.     

Tidal synchronization of close-in satellites and exoplanets. A rheophysical approach

This paper presents a new theory of the dynamical tides of celestial bodies. It is founded on a Newtonian creep instead of the classical delaying approach of the standard viscoelastic theories and the results of the theory derive mainly from the solution of a non-homogeneous ordinary differential equation. Lags appear in the solution but as quantities determined from the solution of the equation and are not arbitrary external quantities plugged in an elastic model. The resulting lags of the tide components are increasing functions of their frequencies (as in Darwin’s theory), but not small quantities. The amplitudes of the tide components depend on the viscosity of the body and on their frequencies; they are not constants. The resulting stationary rotations (often called pseudo-synchronous) have an excess velocity roughly proportional to $6ne^2/(\chi ^2+\chi ^{-2})$ ( $\chi $ is the mean-motion in units of one critical frequency—the relaxation factor—inversely proportional to the viscosity) instead of the exact $6ne^2$ of standard theories. The dissipation in the pseudo-synchronous solution is inversely proportional to $(\chi +\chi ^{-1})$ ; thus, in the inviscid limit, it is roughly proportional to the frequency (as in standard theories), but that behavior is inverted when the viscosity is high and the tide frequency larger than the critical frequency. For free rotating bodies, the dissipation is given by the same law, but now $\chi $ is the frequency of the semi-diurnal tide in units of the critical frequency. This approach fails, however, to reproduce the actual tidal lags on Earth. In this case, to reconcile theory and observations, we need to assume the existence of an elastic tide superposed to the creeping tide. The theory is applied to several Solar System and extrasolar bodies and currently available data are used to estimate the relaxation factor $\gamma $ (i.e. the critical frequency) of these bodies.     

On mechanisms separating stars into normal and chemically peculiar

The paper argues in favor of the assumption that magnetic and non-magnetic protostars, from which CP stars were formed, are the objects that had rotation velocities of the parent cloud V smaller than a critical value V _c . At V greater than the critical value, differential rotation emerges in the collapsing protostellar cloud, which twists magnetic lines of force into an’ invisible’ toroidal shape and disturbs the stability of the atmosphere. In magnetic protostars, the loss of angular momentum is due to magnetic braking, while in metallic protostars, the loss of rotation momentum occurs due to tidal interactions with a close component. HgMn stars are most likely not affected by some braking mechanism, but originated from the slowest protostellar rotators. The boundary of V _c where the differential rotation occurs is not sharp. The slower the protostar rotates, the greater the probability of suppressing the differential rotation and the more likely the possibility of CP star birth.     

Effect of perturbed potentials on the stability of libration points in the restricted problem

The location and the stability in the linear sense of the libration points in the restricted problem have been studied when there are perturbations in the potentials between the bodies. It is seen that if the perturbing functions satisfy certain conditions, there are five libration points, two triangular and three collinear. It is further observed that the collinear points are unstable and for the triangular points, the range of stability increases or decreases depending upon whetherP> or <0 wherep depends upon the perturbing functions. The theory is verified in the following four cases:

1. There are no perturbations in the potentials (classical problem).
2. Only the bigger primary is an oblate spheroid whose axis of symmetry is perpendicular to the plane of relative motion (circular) of the primaries.
3. Both the primaries are oblate spheroids whose axes of symmetry are perpendicular to the plane of relative motion (circular) of the primaries.
4. The primaries are spherical in shape and the bigger is a source of radiation.

     

The inclination changes in the problem of two triaxial rigid spheroids

The first-order perturbations of a system of two triaxial rigid spheroids under Hori-Lie transformation are investigated. The time dependence of the configuration of the three angular momentum vectors, two rotational and one orbital, is studied. The problem is simplified by the introduction of a new time parameter , such thatt is the hyperelliptic function of . The projectionsH ₁ andH ₂ of the rotational momentum vectors into the direction of the total angular momentum vector of the system are then harmonic or exponential functions of . The trajectory in theH ₁,H ₂ plane is a part of an ellipse or hyperbola respectively. If this conical section intersects a certain critical contourC, the system is bounced back along the original trajectory. The motion of the relative configuration of the angular momentum vectors is periodical except in a special aperiodic case. The expressions for the periods are given.     

A vectorial approach to determine frozen orbital conditions

Taking into consideration a probe moving in an elliptical orbit around a celestial body, the possibility of determining conditions which lead to constant values on average of all the orbit elements has been investigated here, considering the influence of the planetary oblateness and the long-term effects deriving from the attraction of several perturbing bodies. To this end, three equations describing the variation of orbit eccentricity, apsidal line and angular momentum unit vector have been first retrieved, starting from a vectorial expression of the Lagrange planetary equations and considering for the third-body perturbation the gravity-gradient approximation, and then exploited to demonstrate the feasibility of achieving the above-mentioned goal. The study has led to the determination of two families of solutions at constant mean orbit elements, both characterised by a co-planarity condition between the eccentricity vector, the angular momentum and a vector resulting from the combination of the orbital poles of the perturbing bodies. As a practical case, the problem of a probe orbiting the Moon has been faced, taking into account the temporal evolution of the perturbing poles of the Sun and Earth, and frozen solutions at argument of pericentre 0\(^{\circ }\) or 180\(^{\circ }\) have been found.     

Reduction,Brouwer's Hamiltonian,and the critical inclination

The reduction process is applied twice to the Brouwer Hamiltonian of artificial satellite theory to obain a reduced HamiltonianM _{H, L}on a reduced phase spaceP _{H, L}which is diffeomorphic to a twosphere. To first order in the oblateness ε, the reduced Hamiltonian has two nondegenerate critical points of index 2 and a nondegenerate critical circleC of index 0 at the critical inclinations ²-4/5. To second order in ε,M _H,Lis a Morse function onP _H,L,with three pairs of critical points of index 0.1, and 2, respectively.     

Motion of particles on a z=2 Lifshitz black hole background in 3+1 dimensions

The object of study is the geodesic structure of a \(z=2\) Lifshitz black hole in 3+1 space–time dimensions, which is an exact solution to the Einstein-scalar-Maxwell theory. The motion of massless and massive particles in this background is researched using the standard Lagrangian procedure. Analytical expressions are obtained for radial and angular motions of the test particles, where the polar trajectories are given in terms of the \(\wp \) -Weierstraß elliptic function. It will be demonstrated that an external observer can see that photons with radial motion arrive at spatial infinity in a finite coordinate time. For particles with non-vanished angular momentum, the motion is studied on the invariant plane \(\phi = \pi /2\) and, it is shown that bounded orbits are not allowed for this space–time on this plane. These results are consistent with other recent studies on Lifshitz black holes.     

Supermassive black hole in an elliptical galaxy: Accretion of a hot gas with a low but finite angular momentum

The accretion of hot slowly rotating gas onto a supermassive black hole is considered. The important case where the velocities of turbulent pulsations at the Bondi radius r _B are low, compared to the speed of sound c _s, is studied. Turbulence is probably responsible for the appearance of random average rotation. Although the angular momentum at r _B is low, it gives rise to the centrifugal barrier at a depth r _c = l ² /GM _BH ? r _B, that hinders supersonic accretion. The numerical solution of the problem of hot gas accretion with finite angular momentum is found taking into account electron thermal conductivity and bremsstrahlung energy losses of two temperature plasma for density and temperature near Bondi radius similar to observed in M87 galaxy. The saturation of the Spitzer thermal conductivity was also taken into account. The parameters of the saturated electron thermal conductivity were chosen similar to the parameters used in the numerical simulations of interaction of the strong laser beam radiation with plasma targets. These parameters are confirmed in the experiments. It is shown that joint action of electron thermal conductivity and free-free radiation leads to the effective cooling of accreting plasma and formation of the subsonic settling of accreting gas above the zone of a centrifugal barrier. A toroidal condensation and a hollow funnel that separates the torus from the black hole emerge near the barrier. The barrier divides the flow into two regions: (1) the settling zone with slow subKeplerian rotation and (2) the zone with rapid supersonic nearly Keplerian rotation. Existence of the centrifugal barrier leads to significant decrease of the accretion rate ? in comparison with the critical Bondi solution for γ = 5/3 for the same values of density and temperature of the hot gas near Bondi radius. Shear instabilities in the torus and related friction cause the gas to spread slowly along spirals in the equatorial plane in two directions.As a result, outer (r > r _c) and inner (r < r _c) disks are formed. The gas enters the immediate neighborhood of the black hole or the zone of the internal ADAF flow along the accretion disk (r < r _c). Since the angular momentum is conserved, the outer disk removes outward an excess of angular momentum along with part of the matter falling into the torus. It is possible, that such outer Keplerian disk was observed by Hubble Space Telescope around the nucleus of the M87 galaxy in the optical emission lines. We discuss shortly the characteristic times during which the accretion of the gas with developed turbulence should lead to the changes in the orientation of the torus, accretion disk and, possibly, of the jet.     

About the problem of motion of n gyrostats: I. The first integrals

In this paper, the problem of the motion of n gyrostats in a central field, with relative momenta _k ^r (k=1, 2, ..., n) functions of the time, is considered.In these conditions, the existence of the linear and angular momentum integrals is established.Moreover, a sufficient condition for the existence of the Jacobi's Integral is given. This study generalizes the results given by Duboshin for n rigid bodies.     

Hydrodynamic processes of angular momentum transport in the interior of a rotating massive hydrogen-burning star

The evolution of a rotating star with a mass of 16M _⊙ at the hydrogen burning phase is considered together with the hydrodynamic processes of angular momentum transport in its interior. Shear turbulence is shown to limit the amplitude of the latitudinal variations in mean molecular weight on a surface of constant pressure in a layer with variable chemical composition. The resulting nonuniformity in the mean molecular weight distribution and the turbulent energy transport along the surface of constant pressure reduce the absolute value of the meridional circulation velocity. Nevertheless, meridional circulation remains the main mechanism of angular momentum transport in the radial direction in a layer with variable chemical composition. The intensity of the processes of angular momentum transport by meridional circulation and shear turbulence is determined by the angular momentum of the star. At a fairly high angular momentum, more specifically, at J = 3.69 × 10⁵² g cm² s^?1, the star during the second half of the hydrogen-burning phase in its convective core has characteristics typical of classical early Be stars.     

Statistical approach to triple systems in three-dimensional motion

This paper considers disruption of triple close approaches with low initial velocities and equal masses in the framework of statistical escape theory in a three-dimensional space. The statistical escape theory is based on the assumption that the phase trajectory of a triple system is quasi-ergodic. This system is described by allowing for both energy and angular momentum conservation in the phase space. In this paper, “possibility of escape” is derived with the formation of a binary on the basis of relative distances of the participating bodies. The complete statistical solutions (i.e. the semi-major axis \(a\), the distributions of eccentricity \(e\) of the binary, binary energy \({E}_{{b}}\), escape energy \({E}_{{s}}\) of escaper, and its escape velocity \({v}_{{s}}\)) of the system are derived from the allowable phase space volumes and are in good agreement with the numerical results in the range of perturbing velocities \({v}_{{i}}\)(\(10^{ - 1} \le {v}_{{i}} \le 10^{ - 10}\)) and directions of \({v}_{{i}}(0 \le \alpha _{{i}},\beta _{{i}},\gamma _{{i}} \le \pi )\), \({i} = 1,2,3\). In this paper, the double limit process has been applied to approximate the escape probability. Through this process, it is observed that the perturbing velocity \({v}_{{i}} \to 0^{ +} \), as the product of the semi-major axis \(a\) of the final binary and the square of the escape velocity \({v}_{{s}}\) approach 2/3, i.e. \({a} {v}_{{s}}^{2} \to 2 / 3\), whatever direction of \(\mathbf{v}_{{i}}\) may be.     

Periodic orbits in the general three-body problem and the relationship between them

We study the regions of finite motions in the vicinity of three simple stable periodic orbits in the general problem of three equal-mass bodies with a zero angular momentum. Their distinctive feature is that one of the moving bodies periodically passes through the center of mass of the triple system. We consider the dynamical evolution of plane nonrotating triple systems for which the initial conditions are specified in such a way that one of the bodies is located at the center of mass of the triple system. The initial conditions can then be specified by three parameters: the virial coefficient k and the two angles, φ₁ and φ₂, that characterize the orientation of the velocity vectors for the bodies. We scanned the region of variation in these parameters k∈(0, 1); φ₁, φ₂∈(0, π) at steps of δk=0.01; δφ₁=δφ₂=1° and identified the regions of finite motions surrounding the periodic orbits. These regions are isolated from one another in the space of parameters (k, φ₁, φ₂). There are bridges that correspond to unstable orbits with long lifetimes between the regions. During the evolution of these metastable systems, the phase trajectory can “stick” to the vicinity of one of the periodic orbits or move from one vicinity to another. The evolution of metastable systems ends with their breakup.     

The gas content in galactic disks: Correlation with kinematics

We consider the relationship between the total HI mass in late-type galaxies and the kinematic properties of their disks. The mass M_HI for galaxies with a wide variety of properties, from dwarf dIrr galaxies with active star formation to giant low-brightness galaxies, is shown to correlate with the product V_cR₀ (V_c is the rotational velocity, and R₀ is the radial photometric disks cale length), which characterizes the specific angular momentum of the disk. This correlation, along with the decrease in the relative mass of the gas in a galaxy with increasing V_c, can be explained in terms of the previous assumption that the gas density in the disks of most galaxies is maintained at a level close to the threshold (marginal) stability of a gaseous layer to local gravitational perturbations. In this case, the regulation mechanism of the star formation rate associated with the growth of local gravitational instability in the gaseous layer must play a crucial role in the evolution of the gas content in the galactic disk.     

Gravitational loss-cone instability: Monotonic and nonmonotonic angular momentum distributions of stars

Small perturbations of spherical star clusters around massive black holes are studied. The presence of a black hole gives rise to peculiar distributions that have no stars with low angular momenta (falling into the so-called “loss cone”). The stability of such a distribution has been found to depend significantly on whether it monotonically increases with angular momentum L (from the loss cone up to L = L _circ in circular orbits) or has a maximum at some intermediate L = L _*. In the case of spherical systems under consideration, the loss-cone instability is shown to be possible only for nonmonotonic distributions.     

On the spin distributions of ΛCDM haloes

We used merger trees realizations, predicted by the extended Press-Schechter theory, in order to study the growth of angular momentum of dark matter haloes. Our results showed that:

1. The spin parameter λ′ resulting from the above method, is an increasing function of the present day mass of the halo. The mean value of λ′ varies from 0.0343 to 0.0484 for haloes with present day masses in the range of 10⁹h^?1 M _⊙ to 10¹⁴h^?1 M _⊙.
2. The distribution of λ′ is close to a log-normal, but, as it is already found in the results of N-body simulations, the match is not satisfactory at the tails of the distribution. A new analytical formula that approximates the results much more satisfactorily is presented.
3. The distribution of the values of λ′ depends only weakly on the redshift.
4. The spin parameter of an halo depends on the number of recent major mergers. Specifically the spin parameter is an increasing function of this number.