The motion of a satellite in an axi-symmetric gravitational field

The equations for the variation of the osculating elements of a satellite moving in an axi-symmetric gravitational field are integrated to yield the complete first-order perturbations for the elements of the orbit. The expressions obtained include the effects produced by the second to eighth spherical harmonics. The orbital elements are presented in the most general form of summations by means of Hansen coefficients. Due to their general forms it is a simple matter to estimate the perturbations of any higher harmonic by simply increasing the index of summation. Finally, this paper gives the respective general expressions for the secular perturbations of the orbital elements. The formulae presented should be useful for the reductions of Earth-satellite observations and geopotential studies based on them.List of Symbols semi-major axis - C _jk ⁿ (, ) cosine functions of and - e eccentricity of the orbit - f acceleration vector of perturbing force - f sin²t - i inclination of the orbit - J _n coefficients in the potential expansion - M mean anomaly - n mean motion - p semi-latus rectum of the orbit - R, S, andW components of the perturbing acceleration - r radius-vector of satellite - r magnitude ofr - S _jk ⁿ (, ) sine functions of and - T time of perigee passage - u argument of latitude - U gravitational potential - true anomaly - V perturbing potential - G(M₊+m) (gravitational constant times the sum of the masses of Earth and satellite) - _n,k coefficients ofR component of disturbing acceleration (funtions off) - _n,k coefficients ofS andW components of disturbing acceleration (functions off) - mean anomaly at timet=0 - X ₀ ^n,m zero-order Hansen coefficients - argument of perigee - right ascension of the ascending node     

A method of computing the gravitational field of an axially symmetric flat galaxy

A method has been developed for computing the gravitational force field of an axially symmetric flat galaxy from its surface mass density. The method is simple to program, fast, and accurate. An inversion formula is derived that allows computation of surface density from rotation curves by use of any method that converts density to force. The method is compared with a method of Clutton-Brock that utilizes Hankel transforms of Laguerre functions.     

Dynamics of a satellite orbiting a planet with an inhomogeneous gravitational field

We study the dynamics of a satellite (artificial or natural) orbiting an Earth-like planet at low altitude from an analytical point of view. The perturbation considered takes into account the gravity attraction of the planet and in particular it is caused by its inhomogeneous potential. We begin by truncating the equations of motion at second order, that is, incorporating the zonal and the tesseral harmonics up to order two. The system is formulated as an autonomous Hamiltonian and has three degrees of freedom. After three successive Lie transformations, the system is normalised with respect to two angular co-ordinates up to order five in a suitable small parameter given by the quotient between the angular velocity of the planet and the mean motion of the satellite. Our treatment is free of power expansions of the eccentricity and of truncated Fourier series in the anomalies. Once these transformations are performed, the truncated Hamiltonian defines a system of one degree of freedom which is rewritten as a function of two variables which generate a phase space which takes into account all of the symmetries of the problem. Next an analysis of the system is achieved obtaining up to six relative equilibria and three types of bifurcations. The connection with the original system is established concluding the existence of various families of invariant 3-tori of it, as well as quasiperiodic and periodic trajectories. This is achieved by using KAM theory techniques.     

On static spherical symmetric solutions of the Bach-Einstein gravitational field equations

For field equations of 4th order, follwing from a Lagrangian “Ricci scalar plus Weyl scalar”, it is shown (using methods of non-standard analysis) that in a neighbourhood of Minkowski space there do not exist regular static spherically symmetric solutions. With that (besides the known local expansions about r = o nad r = ∞ resp.) for the first time a global statement on the existence of such solutions is given. Finally, this result will be discussed in connection with Einstein's particle programme.     

Restriction on the motion of a solid in a gravitational field

As is well known, the orbital and rotational motions of a solid are coupled, and the integrals of energy and angular momentum (in a gravitational field with spherical symmetry) impose restrictions on them. We study the regions allowed to the motion in configurational space. It turns out that even in the crudest model (planar motion of a triple rod) the restrictions on the libration angle and the orbital radius of the center of mass are coupled, so that excessive ellipticity of the orbit excludes stabilization in the neighbourhood of the spoke equilibrium position by gravitational forces only.Chargé de Cours.     

Elliptic-type motion in Fock's gravitational field

The elliptic-type motion in the gravitational field found by Fock as exact solution of Einstein's vacuum equations in the case of spherical symmetry (Solution called here Fock's gravitational field) is studied by means of a classic method based on the perturbation theory. Regarding the deviations of the orbit from a Kepleian orbit as perturbations, the first and second order variations of the Keplerian orbital elements over one nodal period as well as those of the nodal period itself are determined.     

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.     

Properties of motion in the gravitational field of a rotating bar

A qualitative study of the properties of motion (equilibrium points, regions of motion, periodic orbits) of a test particle in the gravitational field of a uniformly rotating solid bar is made. Two different models are used for the bar, a homogeneous ellipsoid and a homogeneous rectangular parallelepiped, and the dependence of the properties of motion on the specific choice of the model is investigated. It is found that stability properties, especially those of the equilibrium points on the long axis of the bar, are more pronounced in the case of the parallelepiped.     

On the roto-translatory motion of a satellite of an oblate primary

We deal with the problem of the motion of a triaxial satellite of an oblate primary of larger mass. We show that the treatment is simplified by using a canonical set of variables previously introduced by the authors, that allows a drastic reduction in the expansions of the potential. A general method to avoid the appearance of virtual singularities when the angles between certain planes are small is designed. Our approach is applicable either to natural or artificial satellites.     

The regular motion of an axisymmetrical satellite in the field of a triaxial body

This paper investigates the regular motions of an axisymmetrical satellite in the field of Newton's attraction of a triaxial body. Both the orbital and the self rotational motions of the two bodies are taken into consideration. The exact solutions are discussed using Poincaré's method of small parameter. In the decomposition of the force function all the harmonic terms up to the third order are taken into account.The results show the existence of eight solutions. The stability of the new group of solutions is discussed using two methods to get the necessary and sufficient conditions required for the stability of these motions.     

Plane symmetric gravitational collapse and linear equation of state

This paper examines the gravitational collapse in plane symmetry with a perfect fluid using a linear equation of state p=kρ. We find a class of collapse models satisfying the Einstein field equations and also the regularity as well as energy conditions. For a given initial data, the outcome of the collapse turns out to be a black membrane or a naked singularity depending upon the equation of state parameter. We conclude that this parameter plays a crucial role in determining the final fate of the collapse.     

A static spherically symmetric gravitational field in the kaluza-klein theory

In the context of the Kaluza-Klein theory we obtain the external solution for the gravitational field generated by a static spherically symmetric distribution of neutral and charged matter. We obtain a formula that connects the mass of the central body with the integral of the pressure distribution inside it.Translated fromAstrofizika, Vol. 38, No. 3, 1995.     

On an analytical solution of the optimum trajectory problem in a gravitational field

Mayer's variational problem for a point with a limited mass flow rate is described by differential equations of the fourteenth order, allowing for a few first integrals. By reducing the equations to closed canonical form, these integrals are analyzed from the viewpoint of finding a possible solution to the problem via quadratures on zero, intermediate, and maximum thrust sections. In addition to confirming well-known cases of total integrability, this approach enabled us to establish that the essential difficulty of the solution of the space problem with intermediate thrust is reduced to finding one integral, and the solution of the problem with maximum thrust requires two integrals in involution. It is shown that these integrals can be applied to find particular solutions.     

A quasi-self-consistent axially symmetric model for the growth of a ring current through earthward motion from a pre-storm configuration

A quasi-self-consistent axially symmetric model of a storm-time ring current that has evolved quasi-adiabatically through inward motion from a prestorm state is presented in which the disturbance field as a function of geocentric distance, r, in the equatorial plane (including the value, H, at r = 0 from which Dst can be found), the beta of the plasma as a function of r, and the ring current magnetic moment are all given in terms of analytic expressions, having only two independent ring current parameters: the geocentric distance to the inner edge of the ring current, R, and the distance at which the beta of the plasma is unity, k—a constant of the model. The model is used to find H as a function of R at constant k, which corresponds to the growth of Dst as the ring current moves earthward, and to illustrate how the radial profile of the disturbance field varies also as the ring current moves earthward. Comparison with published S³-A storm-time, particle and field data shows reasonable agreement. The model is also able to fit the empirically determined relationship between Dst and the equatorward extension of quiet auroral arcs. This result applied to the great magnetic storm of September 1859 predicts a Dst of the order of 2000 nT, a value for which the sparse magnetometer data for that storm offer some support. Because the model (quasi) self-consistently determines the particle distribution function by allowing it to (quasi) adiabatically evolve in a field of its own making, the model avoids difficulties encountered by previous ring current models in which the particle distribution function had to be assumed. For example, the ring current magnetic moment is not limited to values less than the geomagnetic dipole moment, and null points in the field do not develop even in very high beta situations.     

On stationary axially symmetric solutions of a scalar-tensor theory of gravitation

In this paper we obtain the general axially symmetric space-times in scalar-tensor theory of gravitation proposed by Sáez and Ballester. The only possible solutions of these space-times in this theory are dust solutions. One of the obtained solutions here has a similar considerations to Van Stockum dust solution. A detailed study of geometric and kinematical properties of the obtained solutions are carried out. Furthermore, closed timelike curves are present in these solutions.     

Cases of a star's planar motion integrated in the gravitational field of a rotating system

The paper deals with some special cases of the existence of a local integral of motion in a two-dimensional potential field rotating with constant angular velocity. In this case the trajectories may be completely determined, which is not always possible in other cases with a local integral, in contrast to the cases with a true integral. Some cases where the trajectories can be determined analytically are trivial but there are also some new nontrivial cases.     

Similarity solutions for an axially symmetric explosion model in magnetogasdynamics. I

Similarity solutions, for a point explosion in a spheroid with axially symmetric density distribution obeying power laws in the presence of magnetic field, are obtained. A new technique suggested by Bhowmick (1978) has been utilised to study the character of flow variables behind the shock front in an axisymmetric model. The total energy of the wave is constant.