Engineering topics

Szebehely, V.

Publications and source records attributed to Szebehely, V..

At least 19 records

NASA NTRS1993

Stability of outer planetary orbits around binary stars - A comparison of Hill's and Laplace's stability criteria

A comparison is made between the stability criteria of Hill and that of Laplace to determine the stability of outer planetary orbits encircling binary stars. The restricted, analytically determined results of Hill's method by Szebehely and coworkers and the general, numerically integrated results of Laplace's method by Graziani and Black (1981) are compared for varying values of the mass parameter mu. For mu = 0 to 0.15, the closest orbit (lower limit of radius) an outer planet in a binary system can have and still remain stable is determined by Hill's stability criterion. For mu greater than 0.15, the critical radius is determined by Laplace's stability criterion. It appears that the Graziani-Black stability criterion describes the critical orbit within a few percent for all values of mu.

Kubala, A.↗

NASA NTRS1983

Transformations of the perturbed two-body problem to unperturbed harmonic oscillators

Singular, nonlinear, and Liapunov unstable equations are made regular and linear through transformations that change the perturbed planar problem of two bodies into unperturbed and undamped harmonic oscillators with constant coefficients, so that the stable solution may be immediately written in terms of the new variables. The use of arbitrary and special functions for the transformations allows the systematic discussion of previously introduced and novel anomalies. For the case of the unperturbed two-body problem, it is proved that if transformations are power functions of the radial variable, only the eccentric and the true anomalies (with the corresponding transformations of the radial variable) will result in harmonic oscillators. The present method significantly reduces computation requirements in autonomous space operations.

Szebehely, V.↗

NASA NTRS1983

Stability and capture of asteroids

The problem of stability of asteroids is treated from the point of view of Hill's stability-concept and using Lyapunov's Characteristic Numbers. The quantitative measure of stability (S) introduced earlier is evaluated for over 300 asteroids and a surprisingly simple relation is established between the semi-major axes of some of the asteroids' orbits and S. A detailed analysis is presented of the Lyapunov Characteristic Numbers for two minor planets and the time-variation of these numbers is discussed. The technology of capture of asteroids is vitally dependent on their orbital stability, therefore, these two problems, i.e., capture and stability, are closely related. In fact, some predictable instabilities may be properly utilized to capture and/or change asteroidal orbits to accomplish collisions with the Earth.

Szebehely, V.↗

NASA NTRS1983

Regions of stability of asteroids

Using Hill's modified stability criterion, regions of orbital elements are established for conditions of stability. The model of the three-dimensional restricted problem of three bodies is used with the sun and Jupiter as the primaries. Four different cases are studied: direct and retrograde, outside and inside asteroidal orbits. The directions of the asteroidal orbits refer to the synodical reference frame and the positions refer to Jupiter's orbit. The orbital parameters of the asteroids are the semi-major axis (a), the eccentricity (e), and the inclination from Jupiter's orbital plane (i). The argument of the perihelion and the longitude of the ascending node are fixed at Omega = omega = 90 deg and the time of perihelion passage is T = 0 for all orbits.

Szebehely, V.↗

NASA NTRS1982

Global sensitivity to velocity errors at the libration points

The global effects on non-linear stability are investigated at and around the triangular libration points. The model of the circular restricted problem of three bodies is used, considering the earth and the moon as the primaries. Areas of the initial conditions in the configuration space leading to stability with zero initial velocity around the equilibrium points are compared to stable areas in the phase space with variable initial velocity and zero initial deviations at the equilibrium points. The behavior of the system concerning errors due to initial conditions in the configuration space is studied for 6 years and the effects of initial velocity errors are considered for 7.73 years.

Szebehely, V.↗

NASA NTRS1981

Approximations of satellite stability

Modifications and corrections are presented to relations obtained in an investigation conducted by Szebehely (1978), who has discussed the problem of Hill's (1878) stability of satellites in the restricted problem of three bodies. Attention is given to an approximation of the Jacobian constant for the satellite, the critical value of the Jacobian constant, and approximate solutions.

Markellos, V. V.↗

NASA NTRS1981

Determination of the potential in a synodic system

Determination of the potential field in a fixed (inertial) system may be accomplished by the solution of a homogeneous linear partial differential equation when a family of orbits of a body moving in the field is given. This partial differential equation was presented and thoroughly analyzed earlier. The present paper discusses the same problem in a rotating system where the centrifugal and Coriolis effects render the pertinent partial differential equation in general non-homogeneous and non-linear. A linear, though non-homogeneous, partial differential equation for the determination of the synodic potential is obtained only in the special case of iso-energetic families of orbits.

Szebehely, V.↗

NASA NTRS1981

Non-linear stability around the triangular libration points

The configuration space around the triangular libration points in the Earth-Moon system is partitioned according to the stability of the motion. The regions around L4 and L5 are established where particles placed with zero initial velocity will librate. The complexity of the partitioning is revealed.

Mckenzie, R.↗

NASA NTRS1981

Deformation of a line-element in the phase space at the triangular libration point

The flow in the projection of the phase space into the configuration space is presented in the neighborhood of a neutrally (or critically) stable equilibrium point in the restricted problem of three bodies. The projection is a line-element every point of which has zero initial velocity. After the elapse of various times the mapping (the rotations and elongations) of the line-element is described showing chaotic behavior.

Szebehely, V.↗

NASA NTRS1981

Stability of outer planetary systems

The conditions for stability in the Liapunov-Hill sense of outer planetary systems are given in terms of radii of planetary orbits. The outer planets of the solar system are found stable and the possible existence of other than the presently known planets between Jupiter and Pluto are indicated. The existence of other planetary systems with arbitrary mass ratios of the primaries is suggested, and the stability conditions for such systems are derived.

Szebehely, V.↗

NASA NTRS1981

Analysis of Lageos' altitude decrease

The paper treats the inverse problem of celestial mechanics which consists of determining the force field or potential from given or observed orbit(s). From the observational information, according to which Lageos loses approximately 1 mm altitude per day, a linear partial differential equation is formulated. The solution of this equation gives the field responsible for the above-mentioned, as yet unexplained, small but well established secular decrease in the semi-major axis. Note that the altitude-loss is not due to air-drag because of the very high altitude of this satellite.

Szebehely, V.↗

NASA NTRS1980

Potential in the central bar structure

The figure-eight orbits obtained by Miller and Smith (1979) inside the central bar structure of galaxies are used to establish possible potential functions which result in such orbits. It is shown that r to the -6th power type potentials are special cases of distance and angle-dependent potential functions.

Szebehely, V.↗

NASA NTRS1980

Stability of planetary orbits in binary systems

The possible existence of stable orbits is investigated in binary systems using Hill's method. Analytical stability conditions are established for satellites, for inner planets and for outer planets, allowing arbitrary values for the mass-ratio of the binary.

Szebehely, V.↗

NASA NTRS1980

On the capture of the Moon

This paper studies the possibility of lunar capture depending on variations of the solar mass under certain well specified conditions and assumptions regarding the behaviour of the three-body dynamical system formed by the Sun, Earth and Moon. It is found that a large amount of decrease in the solar mass (approximately 37%) would be required to allow capture if the model of the planar restricted problem of three bodies is assumed, if the masses of the Earth and Moon did not change and if the angular momentum of the Sun-Earth system did not change. Such large mass-changes of the Sun can not be associated with radiation mass losses only with catastrophic events, such as stellar close approaches.

Szebehely, V.↗

NASA NTRS1980

Long-time predictions in nonlinear dynamics

It is known that nonintegrable dynamical systems do not allow precise predictions concerning their behavior for arbitrary long times. The available series solutions are not uniformly convergent according to Poincare's theorem and numerical integrations lose their meaningfulness after the elapse of arbitrary long times. Two approaches are the use of existing global integrals and statistical methods. This paper presents a generalized method along the first approach. As examples long-time predictions in the classical gravitational satellite and planetary problems are treated.

Szebehely, V.↗

NASA NTRS1979

Stability of inner planetary systems

The stability of inner planetary systems with arbitrary mass ratios is studied on the basis of the model of the plane restricted three-body problem. A quantitative stability criterion is obtained in terms of the difference between the critical value of the Jacobi constants (at which bifurcation can occur) and the critical value corresponding to a planetary orbit. An orbit is stable if it cannot leave a region that contains only the larger central body (Hill). For small values of the mass parameter, the maximum dimensionless radius of a Hill-stable orbit is 1 minus 2.4 times the cube root of the mass parameter.

Szebehely, V.↗

NASA NTRS1978

Stability of artificial and natural satellites

A quantitative measure of stability based on Hill's definition is evaluated for direct and retrograde satellite orbits. These orbits are known as Poincare's first kind in the restricted problem of three bodies. Onsets of possible instabilities and captures are established. A critical (maximum) value of the satellites orbital radius is found for stability as a remarkably simple function of the mass-parameter. The results are applied to the natural satellites of the solar system.

Szebehely, V.↗

NASA NTRS1978

Comparison between stability limits for satellite motion

Three methods of obtaining stability information on satellite motion are compared by means of numerical and analytical computations. The model of the restricted problem of three bodies is used to describe the motion. Kuiper's (1961) approximate results, Szebehely's (1978) approximate results, and computer solutions obtained by successive iterations show close agreement regarding the maximum values of the orbital radii for stability. The lowest value, i.e., the most conservative estimate, is provided by the simplified form of Szebehely's formula.

Szebehely, V.↗