The fifth-order analytical solution of the equations of motion of a satellite in orbit around a non-spherical planet

A high-precise analytical theory of a satellite in orbit around a non-spherical planet has been developed. The Poisson's small parameter method has been used. All secular and short-periodic perturbations proportional up to and including a product of five arbitrary harmonic coefficients of the planetary potential expansion are calculated. Long-periodic perturbations are derived with the accuracy of up to the fourth-order, inclusive. The influence of the high-order perturbations on the motion of ETALON-1 satellite has been investigated. The results of comparison of the numerical and analytical integration of the equations of its motion over a five year interval are as follows:

Quadrature solution for the general relativistic motion of a satellite or a planet

The Schwarzschild field of a central massM is used to derive the general relativistic motion of a particle in a bounded orbit aroundM. A quadrature gives the central angle as a quasi-periodic function (f) of an effective true anomalyf. The linear term is an infinite series, whose second term yields the usual rate of advance of pericenter. For an artificial satellite this may be as large as 17 of arc per year. The periodic part is a sine series, with coefficients containing the small parameter 2GM/c ² p, wherep is closely approximated by the classical semi-latus rectum. The radius vectorr is a Kepler-like function off.The essentially new features of the calculation are the appropriate factoring of a certain cubic polynomialF(p/r), the use of the above effective true anomalyf, and the introduction of an effective eccentric anomalyE. The latter serves to reduce the differential equation forf as a function of the timet, obtained by combining the solution for (f) with the relativistic integrals of motion, to a Kepler equation forE.Knowing the constants of the motion, one can then solve successively forE(t), f(t), r(t), and (t). This is best done as a variational calculation, comparing the relativistic orbiter with a classical orbiter having the same initial conditions. The resulting variations agree with those of Lass and Solloway, but the present method is quite different from theirs and results in a simpler algorithm. The results show that the radial and transverse corrections, r andr , arising from the Schwarzschild field, may be of the order of a kilometer for 1000 revolutions of an Earth satellite of orbital eccentricitye ₀0.6.For bounded motion, the cubic polynomialF(p/r) has three positive real zeros, the two smaller ones corresponding to apocenter and pericenter. The third and apparently non-physical one occurs forrSchwarzschild radius. It may thus correspond to the incipient fall of the orbiter into the central body, if the latter is a black hole.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 27–September 2, 1972.Research sponsored by NASA Goddard Space Flight Center under Contract No. NAS 5-11909.     

Mass flow in a circumbinary disk with a gap around supermassive binary black holes

We study the interaction between supermassive binary black holes in an elliptical orbit and their surrounding disk with a gap. The gap in the disk is a low density region formed due to the tidal effects of the less massive black hole. The binary we have investigated has a sub-parsec separation and is coplanar with the disk. We find that the maximum variation of the surface density in the gap reaches 50% during an orbital period. However, in other regions of the disk, the density variation is much less than 1%. Furthermore, we calculate the corresponding variation of spectral energy distribution within a period, but little variation is found. The reason for these results is that the viscosity timescale of the disk at the binary radius is much longer than the orbital period of the binary.     

Computer simulation of planet formation in a binary star system: Terrestrial planets

A model of planetary formation in a binary system with a small relative mass of primary is computed on the assumption of a mass transfer from the less massive component to the more massive one with no mass and angular momentum carried away from the system under consideration. At the last stage of mass transfer the condensed Moon-like objects (planetoids) are ejected through the inner Lagrange point of the primary Roche lobe with the outflow of gaseous matter.The whole system is considered in the plane of binary star rotation. Newtonian equations of motion are integrated with the initial conditions for the planetoids referred to as the coordinates and velocity of the inner Lagrangian point at the moments of planetoid ejections, all the pairwise gravitational interactions being included in computations but without a gas-drag. The mass transfer ceases at the primary relative mass 10^–3 which corresponds to the present Sun-Jupiter system. The total mass of planetoids approximates that of the terrestrial planets. Those are formed through coagulation of the planetoids with the effective radius of capture cross-section as an input parameter in the computer simulation. When the minimum separation between the pair of bodies becomes less than this radius they coalesce into a single body with their masses and momenta summed. If the effective radius value is under a certain limit the computer simulation yields the planetary system like that of terrestrial planets of the present Sun system.Numerical computations reveal the division of the planetoids into 4 groups along their distances from the Sun. Further, each group forms a single planet or a planet and a less massive body at the nearest orbits. The parameters of simulated planet orbits are close to the present ones and the interplanetary spacings are in accord with the Titius-Bode law.     

Uncertainty maps for motion around binary asteroids

Celestial Mechanics and Dynamical Astronomy - Ballistic capture orbits offer safer Mars injection at longer transfer time. However, the search for such an extremely rare event is a computationally...     

Non-integrability of the problem of motion around an oblate planet

We provide a result of non-analytic integrability of the so-called J ₂-problem. Precisely by using the Lerman theorem we are able to prove the existence of a region of the phase space, where the dynamical system exhibits chaotic motions.     

Analytical solution for the evolution of a binary with stable mass transfer from a giant

We derive a simple analytical solution for the evolution of a close binary with nuclear time-scale driven mass transfer from a giant. This solution is based on the well-known fact that the luminosity and the radius of a giant scale to a good approximation as simple power laws of the mass M _c of the degenerate helium core. Comparison with results of numerical calculations by Webbink, Rappaport & Savonije show the analytical solution and the power-law approximation to be quite accurate. The analytical solution presented does also allow (in parametrized form) for non-conservative mass transfer. Furthermore, it is shown that the near constancy of the mass-transfer rate over most of the mass-transfer phase seen in the results by Webbink, Rappaport & Savonije is not a generic feature of this type of evolution but rather a consequence of a particular choice of parameters. The analytical solution also demonstrates that the level of mass transfer is largely set by the core mass of the giant at the onset of mass transfer. Finally, we show that the model is self-consistent and discuss its applicability to low-mass X-ray binaries.     

Third-order secular Solution of the variational equations of motion of a satellite in orbit around a non-spherical planet

We constructed an analytical theory of satellite motion up to the third order relative to the oblateness parameter of the Earth (J ₂). Equations of secular variations was developed for the first three orbital elements (a, e, i) of an artificial satellite. The secular variations are solved in a closed form.     

Gaseous flows in the inner part of the circumbinary disk of the T Tauri star

We present results of 3D numerical simulations of the matter flow in the disk of a binary T Tauri star. It is shown that two bow-shocks caused by the supersonic motion of the binary components in the gas of the disk are formed in the system having parameters typical for T Tauri stars. These bow-shocks significantly change the flow pattern. In particular, for systems with circular orbits they determine the size and shape of the inner gap. We also show that the redistribution of the angular momentum due to the bow-shocks leads to occurrence of two matter flows propagating from the inner edge of the circumbinary disk to the components. Further redistribution of this matter between the components is considered.     

Stability of a planet in the HD 41004 binary system

The Hill stability criterion is applied to analyse the stability of a planet in the binary star system of HD 41004 AB, with the primary and secondary separated by 22 AU, and masses of 0.7 M_⊙ and 0.4 M_⊙, respectively. The primary hosts one planet in an S‐type orbit, and the secondary hosts a brown dwarf (18.64 M_J) on a relatively close orbit, 0.0177 AU, thereby forming another binary pair within this binary system. This star‐brown dwarf pair (HD 41004 B+Bb) is considered a single body during our numerical calculations, while the dynamics of the planet around the primary, HD 41004 Ab, is studied in different phase‐spaces. HD 41004 Ab is a 2.6 M_J planet orbiting at the distance of 1.7 AU with orbital eccentricity 0.39. For the purpose of this study, the system is reduced to a three‐body problem and is solved numerically as the elliptic restricted three‐body problem (ERTBP). The Hill stability function is used as a chaos indicator to configure and analyse the orbital stability of the planet, HD 41004 Ab. The indicator has been effective in measuring the planet's orbital perturbation due to the secondary star during its periastron passage. The calculated Hill stability time series of the planet for the coplanar case shows the stable and quasi‐periodic orbits for at least ten million years. For the reduced ERTBP the stability of the system is also studied for different values of planet's orbital inclination with the binary plane. Also, by recording the planet's ejection time from the system or collision time with a star during the integration period, stability of the system is analysed in a bigger phase‐space of the planet's orbital inclination, ≤ 90°, and its semimajor axis, 1.65–1.75 AU. Based on our analysis it is found that the system can maintain a stable configuration for the planet's orbital inclination as high as 65° relative to the binary plane. The results from the Hill stability criterion and the planet's dynamical lifetime map are found to be consistent with each other. (© 2016 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analytical model of compact star in low-mass X-ray binary with de Sitter spacetime

In this article, we suggest a Sitter model in favor of compact stars in low-mass X-ray binaries.Here, we have considered the presence of a cosmological constant(on a small scale) to investigate the stellar structure. We conclude that this approach is very suitable for the familiar physical model of compact stars in low-mass X-ray binaries. We calculate the probable radius, compactness(u) and surface redshift(Z_s) of six compact stars in low-mass X-ray binaries, namely Cyg X-2, V395 Carinae/2 S 0921–630, XTE J2123–058,X1822–371(V691 CrA), 4 U 1820–30 and GR Mus(XB 1254–690). We also offer a possible equation of state(EOS) for the stellar objects.     

Circular binary star and an interstellar interloper: The analytical solution

We consider an interstellar interloper moving at a relatively large distance from a circular binary star. We use the analytical method of separating rapid and slow subsystems, the rapid subsystem being the binary and the slow subsystem being the interstellar interloper. We show that due to the higher than geometrical symmetry of the problem, in addition to the conservation of the energy and the projection of the angular momentum on the axis of the rotation of the binary, the square of the angular momentum is also conserved. In the course of the time evolution, the vector of the angular momentum rotates about that axis at the constant angle to the axis. After obtaining this general counterintuitive result, we focus at the case where the interstellar interloper is coplanar with the binary. We provide an explicit equation of the motion of the interloper. Then we calculate analytically the angle of deflection of the interloper from the straight line. We analyze the difference in the angle of deflection between this three-body problem and the corresponding two-body problem: we show that this difference remains almost constant (a negative constant) at the range of the eccentricities of the interloper trajectory relatively close to unity and linearly increases (by the absolute value, remaining negative) with the eccentricity as the latter becomes much greater than unity.     

Planetary microlensing signals from the orbital motion of the source star around the common barycentre

With several detections, the technique of gravitational microlensing has proven useful for studying planets that orbit stars at Galactic distances, and it can even be applied to detect planets in neighbouring galaxies. So far, planet detections by microlensing have been considered to result from a change in the bending of light and the resulting magnification caused by a planet around the foreground lens star. However, in complete analogy to the annual parallax effect caused by the revolution of the Earth around the Sun, the motion of the source star around the common barycentre with an orbiting planet can also lead to observable deviations in microlensing light curves that can provide evidence for the unseen companion. We discuss this effect in some detail and study the prospects of microlensing observations for revealing planets through this alternative detection channel. Given that small distances between lens and source star are favoured, and that the effect becomes nearly independent of the source distance, planets would remain detectable even if their host star is located outside the Milky Way with a sufficiently good photometry (exceeding present-day technology) being possible. From synthetic light curves arising from a Monte Carlo simulation, we find that the chances for such detections are not overwhelming and appear practically limited to the most massive planets (at least with current observational set-ups), but they are large enough for leaving the possibility that one or the other signal has already been observed. However, it may remain undetermined whether the planet actually orbits the source star or rather the lens star, which leaves us with an ambiguity not only with respect to its location, but also to its properties.     

Perturbation theory for asteroid motion in the gravitational field of a migrating planet

A new perturbation method for the determination of proper elements of an asteroid in the gravitational field of a migrating planet is developed. The article is published in the original.     

Model for the explosion of a critical-mass neutron star in a binary system

A series of numerical models has been constructed for the three-dimensional explosion dynamics of a low-mass neutron star in a binary system that results from the collapse of the rotating iron core of a massive supernova. The numerical solution has been obtained by the particle method with an adaptive time step that allows the computational accuracy to be controlled automatically. The constructed numerical models include the proper motion of the massive component in the binary system of neutron stars, their finite sizes, the graduality of energy release during the explosive disruption of a critical-mass neutron star, and the nonuniform expansion velocity distribution of iron ejecta. The extent to which each of the listed parameters affects the explosion characteristics has been determined. The total explosion energy and the pulsar escape velocity have been estimated. A sizable fraction of the material of the exploded neutron star has been found to remain gravitationally bound to the massive component of the binary system. A further study of its dynamics is of interest in its own right, because the captured material can be considered as an additional source of muon neutrinos.     

Outline of a theory of planet formation in binary systems

A theory is presented for determining regions where planets may form in binary star systems. Planet formation by accretion is assumed possible if mean planetesimal collision velocities do not exceed a critical value. Collision velocities are increased by perturbations due to the companion star, treated by secular perturbation theory. Collision velocities are damped by aerodynamic drag within the solar nebula, taken as the linear case of Cameron and Pine.A general feature of planetary systems in binary stars is the existence of two zones. The inner zone has enough damping to permit unimpeded growth by accretion; in the outer zone, growth proceeds to a limited diameter, beyond which damping is insufficient. It is shown that the asteroids could not have failed to coalesce due to Jupiter perturbations in the primitive solar nebula. Binary star systems with semimajor axis < 30AU are not likely to have planets; these include Alpha Centauri and 70 Ophiuchi. Systems possibly possessing planets include Eta Cassiopeiae, 40 Eridani, and Σ 2398. Epsilon Eridani is a marginal case.     

The potential for Earth-mass planet formation around brown dwarfs

Recent observations point to the presence of structured dust grains in the discs surrounding young brown dwarfs, thus implying that the first stages of planet formation take place also in the substellar regime. Here, we investigate the potential for planet formation around brown dwarfs and very low-mass stars according to the sequential core accretion model of planet formation. We find that, for a brown dwarf mass 0.05 M_⊙, our models predict a maximum planetary mass of  ∼5   M_⊕  , orbiting with semimajor axis ∼ 1 au. However, we note that the predictions for the mass–semimajor axis distribution are strongly dependent upon the models chosen for the disc surface density profiles and the assumed distribution of disc masses. In particular, if brown dwarf disc masses are of the order of a few Jupiter masses, Earth-mass planets might be relatively frequent, while if typical disc masses are only a fraction of Jupiter mass, we predict that planet formation would be extremely rare in the substellar regime. As the observational constraints on disc profiles, mass dependencies and their distributions are poor in the brown dwarf regime, we advise caution in validating theoretical models only on stars similar to the Sun and emphasize the need for observational data on planetary systems around a wide range of stellar masses. We also find that, unlike the situation around solar-like stars, Type II migration is totally absent from the planet formation process around brown dwarfs, suggesting that any future observations of planets around brown dwarfs would provide a direct measure of the role of other types of migration.