Relative equilibria and the virial theorem

A solution of the Newtoniann-body problem for which the moment of inertia (with respect to the centre of mass) is constant is a solution of relative equilibrium.Research supported in part by NSF grant MCS-77-01716.     

A cosmic virial theorem

A new type of cosmic virial theorem is derived, representing a relation among the two-point and three-point galaxy correlation functions and the mean square relative peculiar velocity v ²(r) averaged over pairs of galaxies separated by distancer. The two correlation functions seem to be fairly well known, but v ²(r) at best is only very roughly estimated from available data. It might be possible to improve the estimate through a systematic program of redshift measurements in sample areas. If so, the cosmic virial theorem might yield an interesting measure of the mean mass density due to galaxies.Research supported in part by the National Science Foundation.     

Retrograde precession of stellar orbits and gravitational loss-cone instability in galactic centers

We consider disk and spherical subsystems of stars with nearly radial orbits under conditions when the well-known radial orbit instability is not possible. This requires that the precession of stellar orbits be retrograde, i.e., in the direction opposite to the orbital rotation of stars. We show that an instability that is an analogue of the loss-cone instability known in plasma physics can then develop in the presence of a “loss cone” in the angular momentum distribution of stars, which ensures a deficit or even absence of stars with low angular momenta. Examples of systems with a loss cone are the centers of galaxies or star clusters with massive black holes. The instability can produce a flux of stars onto the galactic center, i.e., it can serve as a mechanism of fueling the nuclear activity of galaxies. Mathematically, the problem is reduced to analyzing simple characteristic equations that describe small perturbations in a disk and a sphere of radially highly elongated stellar orbits. In turn, these characteristics equations are derived through a number of successive simplifications of the general linearized Vlasov equations (i.e., the system that includes the collisionless Boltzmann kinetic equation and the Poisson equation) in action—angle variables.     

Trojan-type orbits in the system of two gravitational centers with variable masses and separation

Trojan type orbits in the system of two gravitational centers with variable separation are studied within the framework of the restricted problem of three bodies. The backward numerical integration of the equations of motion of the bodies starting in the triangular libration pointsL ₄ andL ₅ (reverse problem) finds the breakdown of librations as the separation decreases because of the mass gain of the smaller component and an approach of the body of negligible, mass to the latter up to the distance below its sphere of action with a relative velocity approximately equal to the escape one on this sphere. The breakdown of librations aboutL ₅ occurs under the mass gain of the smaller component considerably larger than in the case ofL ₄ and implications are made for the asymmetry of the number of librators aboutL ₄ andL ₅ in the solar system (Greeks and Trojans). Other parameters of the libration motion near 1/1 commensurability are obtained, namely, the asymmetry of the libration amplitudes about the triangular points as well as the values of periods and amplitudes within the limits of those for real Trojan asteroids. Trojans could be supposedly, formed inside the Proto-jupiter and escape during its intensive mass loss.     

On meteorite orbits

The new method of calculation of meteorite orbits based on the analysis of meteorite showers scattering patterns is suggested. Elements of orbits of 14 meteorites are listed. Correlations between oribt type and petrological type of meteorite are absent.     

A new method for computing the spectrum of the gravitational perturbations on satellite orbits

Some aspects for efficient computation of the tidal perturbation due to the ellipticity effects of the Earth, the luni-solar potential on an Earth-orbiting satellite and the perturbations of the satellite's radial, transverse and normal position components due to the effects of the Earth's gravitational and ocean tide fields are presented. A straightforward method for computing the spectrum of the geopotential and the tidal-induced perturbations of the orbit elements and the radial, transverse and normal components is described.     

The virial theorem for subsystems: application to a globular cluster within a galaxy

A self‐similar evolution of a globular cluster within a galaxy, which implies a one‐component formulation of the virial theorem (Mouri & Taniguchi 2003), is extended to a two‐component formulation (Caimmi & Secco 2003). To this aim, the general case of an embedded sphere within an embedding sphere, both represented as truncated, singular isothermal spheres, is applied to the situation of interest. It is shown that, in the case under consideration, a two‐component formulation of the virial theorem reproduces the analytical results of a one‐component formulation. The process of energy change due to mass loss through the surface is analysed in detail, in connection with both a one‐component and a two‐component formulation of the virial theorem. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the possible resonant orbits of exoplanets

Within the framework of the restricted three-body problem, the possible orbits of a small-mass exoplanet in the system with a massive exoplanet on an elliptic orbit are investigated. Possible quasi-circular orbits are sought. The dependence of the Kozdai-Lidov effect (the Kozdai resonance) on the eccentricity of the orbit of a massive planet is discussed. The effect of the commensurabilities of the mean motions on the value of the eccentricity perturbations is considered.     

On the determination of double star orbits

It is proposed to improve the convergence of the determination of the elements of the orbits of visual binaries by using not only first, but second-order derivatives in the development of the appropriate equations of condition. Also, some improvements of the Kowalsky-Seeliger method are suggested which improve the accuracy of the orbit used as first approximation.     

On the third-body perturbations of high-altitude orbits

The long-term effects of a distant third-body on a massless satellite that is orbiting an oblate body are studied for a high order expansion of the third-body disturbing function. This high order may be required, for instance, for Earth artificial satellites in the so-called MEO region. After filtering analytically the short-period angles via averaging, the evolution of the orbital elements is efficiently integrated numerically with very long step-sizes. The necessity of retaining higher orders in the expansion of the third-body disturbing function becomes apparent when recovering the short-periodic effects required in the computation of reliable osculating elements.     

On the gravitational fields of Pandora and Prometheus

Utilizing topographic models of Saturn's F-ring shepherd satellites Prometheus (S16 1980S27) and Pandora (S15 1980S26), derived by Stooke (1994), and supposing that their mass density is constant, we derived basic geometrical and dynamical characteristics of the moons. They include the volume and mass, the mean radii, the tensor of inertia, and Stokes coefficients of the harmonic expansions of external gravitational potential. The best fitting ellipsoid approximations of the topography were calculated. A simple method of determining the gravitational potential on the surface of an irregular satellite is presented. Examples of equipotential surfaces of the satellites are shown     

Scope for a small circumsolar annular gravitational contribution to the Pioneer anomaly without affecting planetary orbits

All proposed gravitational explanations of the Pioneer anomaly must crucially face the Equivalence Principle. Thus, if Pioneers 10 and 11 were influenced by anomalous gravitational effects in regions containing other Solar System bodies, then those bodies should likewise be influenced, irrespective of their shape, composition or mass. Although the lack of any observed influence upon planetary orbits severely constrains such explanations, here we aim to construct by computer modeling, hypothetical gravitating annuli having no gravitational impact on planetary orbits from Mercury to Neptune. One model has a central zone, free of radial gravitation in the annular plane, and an ‘onset’ beyond Saturn’s orbit, where sunward annular gravitation increases to match the Pioneer anomaly data. Sharp nulls are included so that Uranus and Neptune escape this influence. Such models can be proportionately reduced in mass: a 1 % contribution to the anomaly requires an annulus of approximately 1 Earth mass. It is thus possible to comply with the JPL assessment of newly recovered data attributing 80 %, or more, of the anomaly to spacecraft heat, which appears to allow small contributions from other causes. Following the possibility of an increasing Kuiper belt density at great ranges, another model makes an outward small anomalous gravitation in the TNO region, tallying with an observed slight indication of such an effect, suggesting that New Horizons may slightly accelerate in this region.     

On the long-period evolution of the sun-synchronous orbits

The dynamic evolution of sun-synchronous orbits at a time interval of 20 years is considered. The numerical motion simulation has been carried out using the Celestial Mechanics software package developed at the Institute of Astronomy of the University of Bern. The dependence of the dynamic evolution on the initial value of the ascending node longitude is examined for two families of sun-synchronous orbits with altitudes of 751 and 1191 km. Variations of the semimajor axis and orbit inclination are obtained depending on the initial value of the ascending node longitude. Recommendations on the selection of orbits, in which spent sun-synchronous satellites can be moved, are formulated. Minimal changes of elements over a time interval of 20 years have been observed for orbits in which at the initial time the angle between the orbit ascending node and the direction of the Sun measured along the equator have been close to 90° or 270°. In this case, the semimajor axis of the orbit is not experiencing secular perturbations arising from the satellite’s passage through the Earth’s shadow.     

A physical interpretation of the Titius-Bode rule and its connection to the closed orbits of Bertrand's theorem

We consider the geometric Titius-Bode rule for the semimajor axes of planetary orbits. We derive an equivalent rule for the midpoints of the segments between consecutive orbits along the radial direction and we interpret it physically in terms of the work done in the gravitational field of the Sun by particles whose orbits are perturbed around each planetary orbit. On such energetic grounds, it is not surprising that some exoplanets in multiple-planet extrasolar systems obey the same relation. However,it is surprising that this simple interpretation of the Titius-Bode rule also reveals new properties of the bound closed orbits predicted by Bertrand's theorem, which has been known since 1873.     

Potential energy of continuous and discrete distributions of matter from the point of view of the application of the virial theorem

The potential energy of clusters of stars in which the distribution of matter is taken to be continuous is compared with that of static model clusters in which the distribution of matter is discrete, the comparison being made from the point of view of applying the virial theorem to estimate the masses of the clusters. There is good agreement on the average between the two cases as long as the stellar distribution is random. Systematic differences occur whenever there is any departure from randomness. However, reduction of the mass of a cluster as estimated by means of the virial theorem by even as much as a factor of 2 on the average would seem to require even greater departures from randomness in the stellar distribution than are considered here. As might be expected there are sometimes very large fluctuations in the potential energy from one cluster to the next in the discrete case.Contributions of the Louisiana State University Observatory, No. 30.     

On the gravitational radiation of gravitating objects

In the framework of unifying gravity and electromagnetism, we have shown that accelerating objects emit gravitational wave as those determined by Larmor formula for the accelerating charged particle. We have found new formulae for the power of Gravitational waves radiated by spinning and orbiting objects. The minimum wavelength of the gravitational wave emitted by an object of mass m and radius R is .