Chaos in Hill's generalized problem: from the solar system to black holes

We generalize the well‐known Hill's circular restricted three‐body problem by assuming that the primary generates a Schwarzschild‐type field of the form U = A/r + B/r³. The term in B influences the particle, but not the far secondary. Many concrete astronomical situations can be modelled via this problem. For the two‐body problem primary‐particle, a homoclinic orbit is proved to exist for a continuous range of parameters (the constants of energy and angular momentum, and the field parameter B > 0). Within the restricted three‐body system, we prove that, under sufficiently small perturbations from the secondary, the homoclinic orbit persists, but its stable and unstable manifolds intersect transversely. Using a result of symbolic dynamics, this means the existence of a Smale horseshoe, hence chaotic behaviour. Moreover, we find that Hill's generalized problem (in our sense) is nonintegrable. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

General relativity and satellite orbits: The motion of a test particle in the Schwarzschild metric

The motion of a satellite with negligible mass in the Schwarzschild metric is treated as a problem in Newtonian physics. The relativistic equations of motion are formally identical with those of the Newtonian case of a particle moving in the ordinary inverse-square law field acted upon by a disturbing function which varies asr ^?3. Accordingly, the relativistic motion is treated with the methods of celestial mechanics. The disturbing function is expressed in terms of the Keplerian elements of the orbit and substituted into Lagrange's planetary equations. Integration of the equations shows that a typical Earth satellite with small orbital eccentricity is displaced by about 17 cm from its unperturbed position after a single orbit, while the periodic displacement over the orbit reaches a maximum of about 3 cm. Application of the equations to the planet Mercury gives the advance of the perihelion and a total displacement of about 85 km after one orbit, with a maximum periodic displacement of about 13 km.     

Reconsidering Schwarzschild's original solution

We analyse the Schwarzschild solution in the context of the historical development of its present use, and explain the invariant definition of the Schwarzschild's radius as a singular surface, that can be applied to the Kerr‐Newman solution too.     

Problems connected with the tangential metric coefficient in the Schwarzschild metric: A possible solution

Four situations are shown where the Schwarzschild metric cannot be used or is subject to unsurmountable problems. The first is the question of a metric useful for PPN-formalism checking different gravitational theories. The second problem occurs in connection with Mach's principle, when the flatness of the spacetime inside a massive hollow sphere is a generally accepted solution. The metrical discontinuity on the same spherical shell is a third problem. The fourth one is the anisotropy of the mass-energy of a test particle in the gravitational field. Three principles for solution are proposed:

1. The space is not dilated, but rather contracted, in the gravitational field; then the measurement-rods are shorter and measured distances have greater magnitudes.
2. The potential energy is to be related to a potential level where a stationary observer is placed and the general relativistic potential must be used.
3. A new metric must be introduced which is distinct from the Schwarzschild metric, so that the space in the gravitational field is warped isotropically.

Then the problems stated are shown to be easily solvable.     

Discovery of Earth's quasi‐satellite

Abstract— The newly discovered asteroid 2003 YN₁₀₇ is currently a quasi‐satellite of the Earth, making a satellite‐like orbit of high inclination with apparent period of one year. The term quasi‐satellite is used since these large orbits are not completely closed, but rather perturbed portions of the asteroid's orbit around the Sun. Due to its extremely Earth‐like orbit, this asteroid is influenced by Earth's gravity to remain within 0.1 AU of the Earth for approximately 10 years (1997 to 2006). Prior to this, it had been on a horseshoe orbit closely following Earth's orbit for several hundred years. It will re‐enter such an orbit, and make one final libration of 123 years, after which it will have a close interaction with the Earth and transition to a circulating orbit. Chaotic effects limit our ability to determine the origin or fate of this object.     

Radar observations of asteroid 1998 ML14

Abstract— Goldstone and Arecibo delay‐Doppler radar imaging of asteroid 1998 ML 14 shortly after its discovery reveals a 1 km diameter spheroid with prominent topography on one side and subdued topography on the other. The object's radar and optical properties are typical for S‐class near‐Earth asteroids. The gravitational slopes of a shape model derived from the images and assumed to have a uniform density are shallow, exceeding 30° over only 4% of the surface. If 1998 ML14's density distribution is uniform, then its orbital environment is similar to a planetary body with a spheroidal gravitational field and is relatively stable. Integration of a radar‐refined orbit reveals that the 1998 apparition was the asteroid's closest approach to Earth since at least 1100 and until 2283, when it approaches to within 2.4 lunar distances. Outside of that time interval, orbit uncertainties based on the present set of observations preclude reliable prediction.     

General Relativistic Machian Universe

The Machian Universe, is usually described with Newtonian Physics, We give an alternative General Relativistic picture for Mach’s Universe. As such, we show that, in the correct Machian limit, Schwarzschild’s metric is coherent with Robertson-Walker’s, on condition that there be a cosmological constant, or the Universe’s rotation—or both. It is now confirmed that the Universe is accelerating, so the former condition applies. The latter was also confirmed one more time with the recently discovered NASA space probes anomalies. From Kerr-Lense-Thirring solution, we find an inverse scale-factor dependent angular speed; we then, show that the cosmological “constant” may have Classically originated from a centrifugal acceleration field.     

Possible Alternatives to the Supermassive Black Hole at the Galactic Center

Now there are two basic observational techniques to investigate a gravitational potential at the Galactic Center, namely, (a) monitoring the orbits of bright stars near the Galactic Center to reconstruct a gravitational potential; (b) measuring the size and shape of shadows around black hole giving an alternative possibility to evaluate black hole parameters in mm-band with VLBI-technique. At the moment, one can use a small relativistic correction approach for stellar orbit analysis (however, in the future the approximation will not be precise enough due to enormous progress of observational facilities) while for smallest structure analysis in VLBI observations one really needs a strong gravitational field approximation. We discuss results of observations, their conventional interpretations, tensions between observations and models and possible hints for a new physics from the observational data and tensions between observations and interpretations. We discuss an opportunity to use a Schwarzschild metric for data interpretation or we have to use more exotic models such as Reissner–Nordstrom or Schwarzschild–de-Sitter metrics for better fits.     

Non-Existence Of <Emphasis Type="Italic">N</Emphasis>-Dimensional Static Plane Symmetric Solutions In Bimetric Relativity Theory

A problem of static plane symmetric metric in the perfect fluid, the mesonic massive scalar field and in their coupling is studied in Rosen’s (1973) bimetric theory of relativity. It was found that the matter field like either perfect fluid or mesonic massive scalar field or their coupling does not survive in bimetric theory of gravitation when the space–time is governed by n-dimensional static plane symmetric metric.     

Accretion disk spectra of super-luminal jet sources and ultra-luminous compact X-ray sources in nearby spiral galaxies

The Ultra-luminous Compact X-ray Sources (ULXs)in nearby spiral galaxies and the Galactic super-luminaljet sources sharethe common spectral characteristic that they haveextremely high disk temperatures which cannot be explainedin the framework of the standard accretion disk modelin the Schwarzschild metric. We have calculated an extreme Kerr disk model to examine if the Kerr disk model can instead explain the observed `too hot' accretion disk spectra.We found that the Kerr disk spectrum becomes significantly hardercompared to the Schwarzschild disk only when the disk is highlyinclined.For super-luminal jet sources, which are known to beinclined systems, the Kerr disk model may thuswork if we choose proper values for the black hole angular momentum. For the ULXs, however, the Kerr disk interpretation will be problematic,as is is highly unlikely that their accretion disks are preferentiallyinclined.     

Turbulence in cosmology

The influence of short-wave turbulence on the expansion of a homogeneous and, on average, isotropic Universe was studied in Papers I–III. In the present paper we study the influence on the manner of expansion, for a complete spectrum of wavelengths, of scalar, tensor and vector perturbations. Ast»0, all waves become long (greater than the horizon); therefore, a knowledge of their influence on the averaged metric is required. It is shown that the long-wave modes of scalar and tensor perturbations which remain finite ast»0 deflect the metric for a homogeneous and, on average, isotropic Universe from the Friedmannian, giving it a form coinciding with the average quasi-isotropic solution of Lifshitz and Khalatnikov (1963). Ast»0 their contribution to the solution tends to zero. What remains to be determined is the contribution of those modes of scalar, tensor and vector perturbations which diverge ast»0. Att=0 the proposed solution for such modes becomes inapplicable. The behaviour of the metric of a homogeneous and, on average, isotropic Universe under the influence of all waves and all modes of perturbation is shown in Figure 1–3.     

Discovery of an asteroid and quasi‐satellite in an Earth‐like horseshoe orbit

Abstract— The newly discovered asteroid 2002 AA₂₉ moves in a very Earth‐like orbit that relative to Earth has a unique horseshoe shape and allows transitions to a quasi‐satellite state. This is the first body known to be in a simple heliocentric horseshoe orbit, moving along its parent planet's orbit. It is similarly also the first true co‐orbital object of Earth, since other asteroids in 1:1 resonance with Earth have orbits very dissimilar from that of our planet. When a quasi‐satellite, it remains within 0.2 AU of the Earth for several decades. 2002 AA₂₉ is the first asteroid known to exhibit this behavior. 2002 AA₂₉ introduces an important new class of objects offering potential targets for space missions and clues to asteroid orbit transfer evolution.     

The evolution of co‐orbiting material in the orbit of 2201 Oljato from 1980 to 2012 as deduced from Pioneer Venus Orbiter and Venus Express magnetic records

Asteroid 2201 Oljato passed through perihelion inside the orbit of Venus near the time of its conjunction with Venus in 1980, 1983, and 1986. During those three years, many interplanetary field enhancements (IFEs) were observed by the Pioneer Venus Orbiter (PVO) in the longitude sector where the orbit of Oljato lies inside Venus' orbit. We attribute IFEs to clouds of fine‐scale, possibly highly charged dust picked up by the solar wind after an interplanetary collision between objects in the diameter range of 10–1000 m. We interpret the increase rate in IFEs at PVO in these years as due to material in Oljato's orbit colliding with material in, or near to, Venus' orbital plane and producing a dust‐anchored structure in the interplanetary magnetic field. In March 2012, almost 30 yr later, with Venus Express (VEX) now in orbit, the Oljato‐Venus geometry is similar to the one in 1980. Here, we compare IFEs detected by VEX and PVO using the same IFE identification criteria. We find an evolution with time of the IFE rate. In contrast to the results in the 1980s, the recent VEX observations reveal that at solar longitudes in which the Oljato orbit is inside that of Venus, the IFE rate is reduced to the level even below the rate seen at solar longitudes where Oljato's orbit is outside that of Venus. This observation implies that Oljato not only lost its co‐orbiting material but also disrupted the “target material,” with which the co‐orbiting material was colliding, near Venus.     

Orbit of Comet C/1857 D1 (d'Arrest)

Comet C/ 1857 D1 (d'Arrest) is one of a large number of comets with parabolic orbits. Given that there are sufficient observations of the comet, 299 in right ascension and 279 in declination, it proves possible to calculate a better orbit. The calculations are based on a 12th order predictor‐corrector method. The comet's orbit is highly elliptical, e = 0.99982 and, from calculated mean errors, statistically different from a parabola. The comet will not return for at least 44000 years and thus represents no immediate NEO threat (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Particle motion in the Schwarzschild-Quintessence space-time

The influence of free static spherically symmetric quintessence on particle motion in the Schwarzschild-quintessence space-time has been studied by numerical calculation. In the Schwarzschild space-time, the particle motion can be determined by an effective potential. However, this potential is dependent on the quintessence’s state parameter w _q . We find that when the quintessence’s state parameter w _q is in the range of $[-\frac{1}{3},0]$ , the massive particle’s motion is just like that in the Schwarzschild space-time. And when $-1\leqslant w_{q}<-\frac{1}{3}$ , a maximum unstable circular orbit exists for every L, and no matter how small L is, the scattering state exists, which leads to the accelerating expansion of our universe. The exists of the maximum orbit can even explain why galaxies is in a ball.     

Gravitationless black holes

A gravitationless black hole model is proposed in accord with a five-dimensional fully covariant Kaluza-Klein (K-K) theory with a scalar field, which unifies the four-dimensional Einsteinian general theory of relativity and Maxwellian electromagnetic theory. It is shown that a dense compact core of a star, when it collapses to a critical density, suddenly turns off or shields its gravitational field. The core, if its mass exceeds an upper limit, directly collapses into a black hole. Otherwise, the extremely large pressure, as the gravity is turned off, immediately stops the collapse and drives the mantle material of supernova moving outward, which leads to an impulsive explosion and forms a neutron star as a remnant. A neutron star can further evolve into a black hole when it accretes enough matter from a companion star such that the total mass exceeds a lower limit. The black hole in the K-K theory is gravitationless at the surface because the scalar field is infinitely strong, which varies the equivalent gravitational constant to zero. In general, a star, at the end of its evolution, is relatively harder to collapse into a gravitationless K-K black hole than a strong gravitational Schwarzschild black hole. This is consistent with the observation of some very massive stars to form neutron stars rather than expected black holes. In addition, the gravitationless K-K black hole should be easier to generate jets than a Schwarzschild black hole.     

Orbit of comet C/1860 M1 (Great Comet of 1860)

Comet C/1860 M1 (Great Comet of 1860) is one of a large number of comets with parabolic orbits. Given that there are sufficient observations of the comet, 261 in right ascension and 251 in declination, it proves possible to calculate a better orbit. The comet's orbit is hyperbolic, and statistically different from a parabola. The comet, therefore, cannot be considered to be a Near Earth Oject. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Concerning the instantaneous mass and the extent of an expanding universe

In this article we want to answer the cosmologically relevant question what, with some good semantic and physical reason, could be called the massM _u of an infinitely extended, homogeneously matter‐filled and expanding universe. To answer this question we produce a space‐like sum of instantaneous cosmic energy depositions surrounding equally each spacepoint in the homogeneous universe. We calculate the added‐up instantaneous cosmic energy per volume around an arbitrary space point in the expanding universe. To carry out this sum we use as basic metrics an analogy to the inner Schwarzschild metric applied to stars, but this time applied to the spacepoint‐related universe. It is then shown that this leads to the added‐up proper energy within a sphere of a finite outer critical radius defining the point‐related infinity. As a surprise this radius turns out to be reciprocal to the square root of the prevailing average cosmic energy density. The equivalent mass of the universe can then also be calculated and, by the expression which is obtained here, shows a scaling with this critical radius of this universe, a virtue of the universe which was already often called for in earlier works by E. Mach, H. Thirring and F. Hoyle and others. This radius on the other hand can be shown to be nearly equal to the Schwarzschild radius of the so‐defined mass M _u of the universe. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On gravitational radiation from binary systems

We discuss gravitational radiation from a neutral mass particle within a bound orbit in the background Schwarzschild metric. We compare the power loss of gravitational radiation according to this formalism with the heuristic quadrupole radiation formula as applied to a binary system. There are evidence and compelling reasons to believe that the quadrupole formula is valid even in a fairly strong gravitational field, although its fully consistent analytical derivation is not yet known. In particular, we emphasize that the application of the quadrupole formula to the binary pulsar system PSR 1913+16 as well as other binary pulsars, which are weakly bound by gravity, is well justified.