Central Gravitational Field Research Articles

Flame dynamics and consideration of deflagration-to-detonation transition in central gravitational field

Expanding reaction fronts are central to many terrestrial processes as well as such cosmic phenomena as the thermonuclear combustion in supernovae. While in terrestrial conditions the effect of intrinsic flamefront instabilities is generally believed to be supplementary to, say, external turbulence and chamber dynamics, at the astrophysical scale the role of flamefront instabilities in the flame acceleration is presumably dominant. Moreover, while in terrestrial systems we focus mainly on the hydrodynamic, Darrieus–Landau (DL) instability, the Rayleigh–Taylor (RT), body-force instability could be a key issue for astrophysical flames because of the enormous gravity and the associated large scales. Consequently, if RT effects dominate over DL effects, the globally-spherical flamefront can be replaced by an expanding bubble with a non-spherical shape, possibly that of digit “8”. In the present work we have developed a self-similar formulation describing a globally-spherical expanding flamefront corrugated due to the DL instability in a central gravitation field. The associated scenario of the flame acceleration, the evolution of the upstream flow, and the instant when a gas parcel ahead of the flamefront first explodes, hence initiating the deflagration-to-detonation transition (DDT), as well as the radial coordinate of this parcel, are determined. We have also compared the effects of DL and RT instabilities, estimating whether a globally-spherical, DL-corrugated flamefront is subsequently terminated by the formation of RT bubbles. It is shown how the instant of such a transition and the relevant global flame radius are coupled to various flame and flow parameters.

On Roto-Translatory Motion: Reductions and Radial Intermediaries

The roto-translational dynamics of an axial-symmetric rigid body is discussed in a central gravitational field. The six-degree-of-freedom Hamiltonian problem is formulated as a perturbation of the Kepler motion and torque-free rotation. A chain of canonical transformations is used to reduce the problem. First, the elimination of the nodes reduces the problem to a system of four degrees of freedom. Then, the elimination of the parallax simplifies the resulting Hamiltonian, which is shaped as a radial intermediary plus a remainder. Some features of this integrable intermediary are pointed out. The normalized first order system in closed form is also given, thus completing the solution. Finally the full reduction of the radial intermediary is constructed using the Hamilton-Jacobi equation.

Numerical investigation of some unsteady regimes of natural convection in a rotating spherical shell

The problem of natural convection in a rotating spherical shell in a central gravitational field is investigated. Three-dimensional governing equations are solved numerically using the control volume method. Numerical simulations of natural convection are performed for various angular velocities of the outer border of a spherical shell. Unsteady, periodicin time,non-axisymmetric solutions are obtained.

On stability of orbit of artificial moon moving around a small celestial body perturbed by an external body

We examine the stability of the orbit of an artificial moon of a small celestial body in the presence of an external massive perturbing body in terms of the restricted three-body problem. The orbit of this moon is shown to be dependent on the shape of the small body and central gravitational field of the external body. We study how these factors interact with each other and how they affect the stability of the orbit.

A new universal equation for the time of transfer between two points of the central gravitational field

The paper presents the derivation of two new equations for calculating the time of transfer between two points of the central gravitational field: for hyperbolic orbits and the universal equation for elliptical and hyperbolic orbits. In the paper we have used as an independent variable, instead of the linear elements (semimajor axis, focal parameter of orbit or a chord connecting the ends of boundary radii of transfer), the angular parameter—the angle between the radius vector of an initial point of transfer and the vector of initial velocity of transfer. Paper’s material is a continuation of that presented in the “Space research“ Journal, vol. 2, March–April, 2009.

Two-craft tether formation relative equilibria about circular orbits and libration points

The relative equilibria of a two spacecraft tether formation connected by line-of-sight elastic forces moving in the context of a restricted two-body system and a circularly restricted three-body system are investigated. For a two spacecraft formation moving in a central gravitational field, a common assumption is that the center of the circular orbit is located at the primary mass and the center of mass of the formation orbits around the primary in a great-circle orbit. The relative equilibrium is called great-circle if the center of mass of the formation moves on the plane with the center of the gravitational field residing on it; otherwise, it is called a nongreat-circle orbit. Previous research shows that nongreat-circle equilibria in low Earth orbits exhibit a deflection of about a degree from the great-circle equilibria when spacecraft with unequal masses are separated by 350 km. This paper studies these equilibria (radial, along-track and orbit-normal in circular Earth orbit and Earth–Moon Libration points) for a range of inter-craft distances and semi-major axes of the formation center of mass. In the context of a two-spacecraft Coulomb formation with separation distances on the order of dozens of meters, this paper shows that the equilibria deflections are negligible (less than 10 −6°) even for very heterogeneous mass distributions. Furthermore, the nongreat-circle equilibria conditions for a two spacecraft tether structure at the Lagrangian libration points are developed.

The asymptotic properties of the motions of satellites in a central field due to internal dissipation

A satellite in the form of a system of bodies that does not have the property of a gyrostat in the general case is considered. An algorithm for determining all the equilibrium configurations of the system that correspond to steady motions in a central gravitational field and an algorithm for analysing their stability are given. A method based on Routh's first theorem is used to investigate the asymptotic stability of the steady motions in the unconstrained problem. Three effects caused by internal dissipation are established in a model example: stabilization of the satellites in a neighbourhood of rotations about a normal to the orbital plane, which is codirectional with the axis of the largest moment of inertia, evolution of elliptic orbits into circular orbits, and capture of the satellites in resonant oscillatory modes of motion.

Free boundary effects on stability of two phase planar fluid motion in annulus. Part I: Migration of the stable mode

We study the linearized stability of a planar dynamical model describing two-phase perfect fluid circulating around a circle with a sufficiently large radius within a central gravitational field. The model is associated with the spatial and temporal structure of the zonally averaged global-scale atmospheric longitudinal circulation around the Earth. Two cases are studied separately; in the first one, the simulations were carried out using the rigid lid approximation at the upper boundary of the outer atmospheric layer. In the second one, the free boundary nonlinear conditions (kinematic and dynamic) were assumed on the outer atmospheric layer. For the both cases, a certain family of steady, explicit solutions which have circular streamlines was considered. The governing equations were linearized at these solutions to find the typical wave numbers of the interfacial wave perturbation to the basic state at which the destabilizing effect of shear, which overcomes the stabilizing effect of stratification, occurs. It is shown that for the both cases, the model always have the same two potentially unstable wave modes while there always exist two wave modes which are stable for any wavelengths. The behavior of the stable and unstable modes were compared for the both cases to investigate the effects of the free boundary on the mixing process at the interface.

DYNAMICS OF IONIZED GAS AT THE GALACTIC CENTER: VERY LARGE ARRAY OBSERVATIONS OF THE THREE-DIMENSIONAL VELOCITY FIELD AND LOCATION OF THE IONIZED STREAMS IN SAGITTARIUS A WEST

We present new results based on high-resolution observations of Sgr A West at the Galactic center with the Very Large Array (VLA) at 1.3 cm. By combining recent observations with those made at earlier epochs with the VLA at wavelengths of 1.3 and 3.6 cm, we measured proper motions for 71 compact H II components in the central 80 '' (3 pc, assuming D = 8 pc). Using VLA archival data for the H92 alpha radio recombination line, we also investigated radial velocities in the LSR velocity range from +200 to -415 km s(-1). Combining proper motion and radial velocity measurements, we have determined the three-dimensional velocity distribution in Sgr A West. We find that the three ionized streams (Northern Arm, Eastern Arm, and Western Arc) in the central 3 pc can be modeled with three bundles of Keplerian orbits around Sgr A*. Assuming that each of the observed streams of ionized gas follows a single orbit, we determined the five orbital parameters (a, e, Omega, omega, i) for each of them using least-square fitting to the loci of the streams. The degeneracy in the orbital solutions for both the direction of flow and the two mirror images can be further resolved using the information obtained from the velocity measurements. Our results confirm earlier results on the streams in the Western Arc and the Northern Arm to be in Keplerian orbits, suggesting that the stream in the Eastern Arm is also consistent with an elliptical orbit. All three are confined within the central 3 pc. Both the Northern and Eastern Arm streams have high eccentricities (e = 0.83 +/- 0.10 and 0.82 +/- 0.05, respectively), while the Western Arc stream is nearly circular, with e = 0.2 +/- 0.15. All three streams orbit around Sgr A* in a counterclockwise sense (viewed from the Earth) and have orbital periods in the range 4-8x10(4) yr. To verify the fit, the distributions of radial and transverse velocity vectors in Sgr A West were also computed using the Keplerian model and they show good agreement with both the proper motion and radial velocity data. In addition, the computed orbits suggest that the Northern and Eastern Arm streams may collide in the Bar region (a few arcsec south of Sgr A*) and that most of the orbiting ionized gas in the Bar region is located behind Sgr A*. We also report an ionized nebula associated with IRS 8, including a bow shock in radio continuum emission which shows excellent agreement with near infrared observations. From the H92a line data, we find evidence of substantial interaction between the IRS 8 nebula and the Northern Arm stream occurring in the bow-shock region. Other new morphological features revealed in our high-resolution image include (1) a helical structure in the Northern Arm, suggesting that MHD plays an important role in the motion of the ionized gas, in addition to the dynamics determined by the central gravitational field, and (2) a linear feature in the IRS 16 region, suggesting that the compressed edge of the Northern Arm may result from the collective winds and radiation pressure from the high mass stars in the IRS16 cluster.

Application of hodograph method to calculation of time of transfer in a central gravitational field

Application of hodograph method to calculation of time of transfer in a central gravitational field

The steady motions of a satellite with a two-degree-of-freedom powered gyroscope in a central gravitational field and their stability

The positions of relative equilibrium of a satellite carrying a two-degree-of-freedom powered gyroscope, in the axes of the framework of which only dissipative forces can act, are investigated within the limits of a restricted circular problem. For the case when the “satellite - gyroscope” system possesses the property of a gyrostat and the axis of the gyroscope frame is directed parallel to one of the principal central axes of inertia of the satellite, all the equilibrium positions are found as a function of the magnitude of the angular momentum of the rotor. It is established that the minimum number of equilibrium positions is equal to 32 and, in certain ranges of values of the system parameters, it can reach 80. All the positions satisfying the sufficient conditions for stability are also determined. The number of them is either equal to 4 or 8 depending on the values of the system parameters.

Nonsymmetric forms of envelopes of relativistic magnetic stars

Within the framework of a previously proposed relativistic model of a magnetized perfectly conducting material surface without internal stresses, an axisymmetrical equilibriumproblem is considered of the rotation of a stellar envelope, supported by a radial hypersonic stellar wind in a given central gravitational field and dipole magnetic field. In the absence of either the magnetic field or the stellar wind, the equations have an explicit solution; in the general case the solution is obtained numerically. Of special interest are the envelope shapes asymmetric with respect to the equator (relative to the direction of the dipole), which have the greatest stability, in particular, bottle-shaped structures.

The features of the motion of variable mass bodies in central gravitational field

The features of the motion of variable mass bodies in central gravitational field

Radially accelerated optimal feedback orbits in central gravity field with linear drag

The solution of a feedback optimal control problem arising in orbital mechanics is addressed in this paper. The dynamics is that of a massless body moving in a central gravitational force field subject also to a drag and a radial modulated force. The drag is linearly proportional to the velocity and inversely proportional to the square of the distance from the center of attraction. The problem is tackled by exploiting the properties of a suitably devised linearizing map that transforms the nonlinear dynamics into an inhomogeneous linear system of differential equations supplemented by a quadratic objective function. The generating function method is then applied to this new system, and the solution is back transformed in the old variables. The proposed technique, in contrast to the classical optimal control problem, allows us to derive analytic closed-loop solutions without solving any two-point boundary value problem. Applications are discussed.

About one problem of trajectory optimization

The optimization problem for trajectories of spacecraft flight from the Earth to an asteroid is considered in this paper. The flight is realized in the central Newtonian gravitational field of the Sun with a possibility of gravitational maneuvers near planets. Perturbation maneuvers are taken into account using the method of point area of action with a limitation on the flyby altitude. The spacecraft is controlled by changing the value and direction of the engine thrust. The problem is solved taking into account constraints on the launch time, flight duration, and minimum distance to the Sun.

Stability of plane oscillations of a satellite in a circular orbit

We study motions of a rigid body (a satellite) about the center of mass in a central Newtonian gravitational field in a circular orbit. There is a known particular motion of the satellite in which one of its principal central axes of inertia is perpendicular to the orbit plane and the satellite itself exhibits plane pendulum-like oscillations about this axis. Under the assumption that the satellite principal central moments of inertia A, B, and C satisfy the relation B = A + C corresponding to the case of a thin plate, we perform rigorous nonlinear analysis of the orbital stability of this motion.

Rotations of a near-symmetrical satellite in an elliptical orbit with Mercury-type resonance

The motion of a satellite, i.e., a rigid body, about to the centre of mass under the action of the gravitational moments of a central Newtonian gravitational field in an elliptical orbit of arbitrary eccentricity is investigated. It is assumed that the satellite is almost dynamically symmetrical. Plane periodic motions for which the ratio of the average value of the absolute angular velocity of the satellite to the average motion of its centre of mass is equal to 3/2 (Mercury-type resonance) are examined. An analytic solution of the non-linear problem of the existence of such motions and their stability to plane perturbations is given. In the special case in which the central ellipsoid of inertia of the satellite is almost spherical, the stability to spatial perturbations is also examined, but only in a linear approximation. ©2008.

Stability of a planar resonance satellite motion under spatial perturbations

The satellite motion relative to the center of mass in a central Newtonian gravitational field on an elliptic orbit is considered. The satellite is a rigid body whose linear dimensions are small compared with the orbit dimensions. We study a special case of planar motion in which the satellite rotates in the orbit plane and performs three revolutions in absolute space per two revolutions of the center of mass in the orbit. Perturbations are assumed to be arbitrary (they can be planar as well as spatial). In the parameter space of the problem, we obtain Lyapunov instability domains and domains of stability in the first approximation. In the latter, we construct third- and fourth-order resonance curves and perform nonlinear stability analysis of the motion on these curves. Stability was studied analytically for small eccentricity values and numerically for arbitrary eccentricity values.

Hamiltonian mechanics and relative equilibria of orbiting gyrostats

We use a noncanonical Hamiltonian approach to study the equilibrium attitudes of a rigid satellite with N rotors in a central gravitational field. The relative equilibria of this system of equations represent steady motions of the body as seen in the body frame, and correspond to stationary points of the Hamiltonian constrained by Casimir functions. This approach leads to an algorithm for computing the equilibria, and simultaneously providing direct stability information, since the calculations required to solve the constrained minimization problem are also involved in computing the positive definiteness of the constrained Hamiltonian for use as a Lyapunov function.

Nonlinear feedback control of low-thrust orbital transfer in a central gravitational field

This paper investigates the problem of continuous-thrust orbital transfer using orbital elements feedback from a nonlinear control standpoint, utilizing concepts of controllability, feedback stabilizability and their interaction. Gauss's variational equations (GVEs) are used to model the state-space dynamics of motion under a central gravitational field. First, the notion of accessibility is reviewed. It is then shown that the GVEs are globally accessible. Based on the accessibility result, a nonlinear feedback controller is derived which asymptotically steers a spacecraft form an initial elliptic orbit to any given elliptic orbit. The performance of the new controller is illustrated by simulating an orbital transfer between two geosynchronous Earth orbits. It is shown that the low-thrust controller requires less fuel than an impulsive maneuver for the same transfer time. Closed-form, analytic expressions for the new orbital transfer controller are given. Finally, it is proven, based on a topological nonlinear stabilizability test, that there does not exist a continuous closed-loop controller that can transfer a spacecraft onto a parabolic escape trajectory.