Relativistic positioning: four-dimensional numerical approach in Minkowski space-time

We simulate the satellite constellations of two Global Navigation Satellite Systems: Galileo (EU) and GPS (USA). Satellite motions are described in the Schwarzschild space-time produced by an idealized spherically symmetric non rotating Earth. The trajectories are then circumferences centered at the same point as Earth. Photon motions are described in Minkowski space-time, where there is a well known relation, (Coll et al. in Class. Quantum Gravit. 27:065013, 2010a), between the emission and inertial coordinates of any event. Here, this relation is implemented in a numerical code, which is tested and applied. The first application is a detailed numerical four-dimensional analysis of the so-called emission coordinate region and co-region. In a second application, a GPS (Galileo) satellite is considered as the receiver and its emission coordinates are given by four Galileo (GPS) satellites. The bifurcation problem (double localization) in the positioning of the receiver satellite is then pointed out and discussed in detail.     

Relativistic satellite orbits: central body with higher zonal harmonics

Satellite orbits around a central body with arbitrary zonal harmonics are considered in a relativistic framework. Our starting point is the relativistic Celestial Mechanics based upon the first post-Newtonian approximation to Einstein’s theory of gravity as it has been formulated by Damour et al. (Phys Rev D 43:3273–3307, 1991; 45:1017–1044, 1992; 47:3124–3135, 1993; 49:618–635, 1994). Since effects of order \((\mathrm{GM}/c^2R) \times J_k\) with \(k \ge 2\) for the Earth are very small (of order \( 7 \times 10^{-10}\,\times \,J_k\)) we consider an axially symmetric body with arbitrary zonal harmonics and a static external gravitational field. In such a field the explicit \(J_k/c^2\)-terms (direct terms) in the equations of motion for the coordinate acceleration of a satellite are treated first with first-order perturbation theory. The derived perturbation theoretical results of first order have been checked by purely numerical integrations of the equations of motion. Additional terms of the same order result from the interaction of the Newtonian \(J_k\)-terms with the post-Newtonian Schwarzschild terms (relativistic terms related to the mass of the central body). These ‘mixed terms’ are treated by means of second-order perturbation theory based on the Lie-series method (Hori–Deprit method). Here we concentrate on the secular drifts of the ascending node \(<\!{\dot{\Omega }}\!>\) and argument of the pericenter \(<\!{\dot{\omega }}\!>\). Finally orders of magnitude are given and discussed.     

Relativistic effects in the critical inclination problem in artificial satellite theory

It is well known that in artifical satellite theory special techniques must be employed to construct a formal solution whenever the orbital inclination is sufficiently close to the critical value cos^–1 (1/5). In this article the authors investigate the consequences of introducing certain relativistic effects into the motion of a satellite about an oblate primary. Particular attention is paid to the critical inclination(s), and for such critical motions an appropriate method of solution is formulated.     

Perturbations in satellite orbits due to variations in rotation of the central body

Variations in satellite orbital elements are derived due to perturbations in the external gravitational field of the central body caused by mass deformations of the body occurring from variations in its rotation; the central body is assumed to be perfectly elastic. General theory derived is applied to the actual Earth, as an example; possible resonances are discussed.     

On the perturbations of a close-Earth satellite due to lunar inequalities

The lunar disturbing function for a close-Earth satellite is expressed as a sum of products of harmonics of the satellite's position and harmonics of the Moon's position, and the latter are expanded about a rotating and precessing elliptic orbit inclined to the ecliptic. The deviations of the Moon from this approximate orbit are computed from Brown's lunar theory andthe perturbations in satellite orbital elements due to these inequalities are derived. Numerical calculations indicate that several perturbations in the position of the satellite's node and perigee have magnitudes on the order of one meter.The author is supported in part by a National Science Foundation Graduate Fellowship.     

Emission lines due to interstellar dust in the visible spectra of nebulae

It is suggested that certain sharp lines in the visible and near ultraviolet spectra of nebule may be due to fluorescence from grains. Evidence is presented that some of these lines could arise from MgO or CaO grains. Some sharp diffuse interstellar bands may also appear in the spectra of nebulae.     

Radial,transverse and normal satellite position perturbations due to the geopotential

Perturbations in the position of a satellite due to the Earth's gravitational effects are presented. The perturbations are given in the radial, transverse (or alongtrack) and normal (or cross-track) components. The solution is obtained by projecting the Kepler element perturbations obtained by Kaula [Kaula, 1966] into each of the three components. The resulting perturbations are presented in a form analogous to the form of Kaula's solution which facilitates implementation and interpretation.     

The formation of commensurabilities in satellite systems due to the action of tidal forces

The five types of resonance possible between a pair of satellites at a 21 commensurability are described. By a modification of the method usually used in the restricted three-body problem, phase-plane diagrams are constructed for these resonances for the more general case where both satellite masses are non-zero. These phase-plane diagrams are used to discuss the different types of motion possible at the five resonances.It is shown that tidal forces can drive a pair of satellites towards a commensurability, and at the 21 commensurability it is possible for the satellites to be captured into a libration at any of the five resonances, the probability of capture depending on the eccentricities, inclinations, and masses of the satellites. The tidal hypothesis provides a reasonable explanation of the origin of the commensurabilities between Mimas and Tethys, and between Enceladus and Dione, in the satellite system of Saturn.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 27–September 2, 1972.     

Mode coupling due to twisting of magnetic field lines

A new approach to the theory of mode coupling in an inhomogeneous, birefringent medium is used to treat mode coupling in a magnetized plasma with a twisted magnetic field. The twist introduces a resonance, corresponding to the rate of twisting being equal to the rate of generalized Faraday rotation. When this resonance occurs it introduces a new regime of strong mode coupling. The implications of this regime are discussed in connection with the long-standing problem in solar radiophysics that mode coupling appears to be stronger than theory implies, but no obvious resolution of the problem is found.     

Perturbations of a close-earth satellite due to sunlight diffusely reflected from the Earth

A semianalytical method has been developed to calculate the radiation-pressure perturbations of a close-Earth satellite due to sunlight reflected from the Earth. It is assumed that the satellite is spherically symmetric and that the solar radiation is reflected from the Earth according to Lambert's Law. To account for the increasing reflectivity of the Earth toward the poles, its albedo is assumed to have a latitudinal dependence given bya=a ₀ +a ₂ sin². The effect of the terminator on the perturbations has been neglected. The perturbations within a particular revolution are given analytically, while the long-range perturbations are obtained by accumulation.     

Perturbations of a close-Earth satellite due to sunlight diffusely reflected from the Earth

A semianalytic method has been developed to calculate the radiation-pressure perturbations of a close-Earth satellite due to sunlight reflected from the Earth. The assumptions made are that the satellite is spherically symmetric and that the solar radiation is reflected from the Earth according to Lambert's Law with uniform albedo. By using expressions for the components of the radiation-pressure force due to Lochry, the expressions for the perturbations of the elements were developed into series in the true anomalyv. The perturbations within a given revolution can be obtained analytically by integrating with respect tov while holding all slowly varying quantities constant. The long-range perturbations are then obtained by accumulating the net perturbations at the end of each revolution.     

Perturbations of a spheroidal satellite due to direct solar radiation pressure

The orbital accelerations of certain balloon satellites exhibit marked oscillations caused by solar radiation impinging on the surface of the satellites, which, once spherical, have assumed a spheroidal shape producing a component of force at right-angles to the Sun-satellite direction. Given the characteristics and orientation of the satellite, the equations of force are determined by the formulae of Lucas. Otherwise the phase-angle and magnitude of the right-angle force are determined by trial and error, or best-fit techniques. Using a variation of the approach developed by Aksnes, a semi-analytical algorithm is presented for evaluating the perturbations of the Keplerian elements by direct solar radiation pressure on a spheroidal satellite. The perturbations are obtained by summing over the sunlit part of each orbit and allow for a linear variation in the phase-angle. The algorithm is used to determine the orbital accelerations of 1963-30D due to direct solar radiation pressure, and these results are compared to the observed values over two separate periods of the satellite's lifetime.     

Changes of the profiles of spectral lines due to mass transfer in close binaries

In this paper, the model of the ring envelope round the primary component and the stream of the gaseous mass flowing from the secondary component to the primary is constructed on the basis of theoretical computations concerning the exchange of the mass between the components of the binary. The paper studies the influence of the gaseous mass on the profiles of spectral lines before and after occultation; the influence of the stream on the profile in case the secondary is near elongation, is also investigated. The line profiles obtained by numerical computations show that their changes caused by outflowing mass should be well detectable from spectrograms taken at particular phase of the binary. Changes in the lines may influence the measurement of radial velocities. The method for distinguishing the influence of the stream from the influence of the ring is described.     

Relativistic corrections to the Kompaneets equation

We study the Sunyaev–Zeldovich effect for clusters of galaxies. We explore the relativistic corrections to the Kompaneets equation in terms of two different expansion approximation schemes, namely, the Fokker–Planck expansion approximation and delta function expansion approximation. We show that two expansion approximation formalisms are equivalent under the Thomson approximation, which is extremely good approximation for the CMB photon energies. This will clarify the situation for existing theoretical methods to analyse observation data.     

Relativistic blue‐ and red‐shifted absorption lines in AGNs

Current, accumulating evidence for (mildly) relativistic blue‐ and red‐shifted absorption lines in AGNs is reviewed. XMM‐Newton and Chandra sensitive X‐ray observations are starting to probe not only the kinematics (velocity) but also the dynamics (accelerations) of highly ionized gas flowing in‐and‐out from, likely, a few gravitational radii from the black hole. It is thus emphasized that X‐ray absorption‐line spectroscopy provides new potential to map the accretion flows near black holes, to probe the launching regions of relativistic jets/outflows, and to quantify the cosmological feedback of AGNs. Prospects to tackle these issues with future high energy missions are briefly addressed. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Assessment of the resonant perturbation errors in Galilean satellite ephemerides using precisely measured eclipse times

Astrometric satellite positions are derived from timings of their eclipses in the shadow of Jupiter. The 548 data points span 20 years and are accurate to about 0.006 arcsec for Io and Europa and about 0.015 arcsec or better for Ganymede and Callisto. The precision of the data set and its nearly continuous distribution in time allows measurement of regular oscillations with an accuracy of 0.001 arcsec. This level of sensitivity permits detailed evaluation of modern ephemerides and reveals anomalies at the 1.3 year period of the resonant perturbations between Io, Europa and Ganymede. The E5 ephemeris shows large errors at that period for all three satellites as well as other significant anomalies. The L1 ephemeris fits the observations much more closely than E5 but discrepancies for the resonant satellites are still apparent and the measured positions of Io are drifting away from the predictions. The JUP230 ephemeris fits the observations more accurately than L1 although there is still a measurable discordance between the predictions and observations for Europa at the resonance period.     

Long-periodic and secular perturbations to the orbit of a spherical satellite due to direct solar radiation pressure

A theory is developed for the perturbations to the orbit of a spherically symmetric satellite which accounts for the changes in the perigee and nodal positions and the variations of the Sun-Earth distance and direction over an orbital revolution. The theory is semi-analytical, the equations of motion being integrated with respect to time over the sunlit period of each orbital revolution. Long-periodic and short-periodic perturbations may be treated separately, and this is important for long-term analyses in terms of mean elements where short-period terms are averaged or omitted.     

Short-period and long-period perturbations of a spherical satellite due to direct solar radiation

On the basis of expressions derived by Kozai, and new ones developed here, a detailed, semianalytic algorithm is presented for calculating radiation-pressure perturbations in the Keplerian elements. Through some simple modifications, the algorithm is also made to hold whene=0 and/ori=0. The perturbations are obtained by summing over the sunlit segment of the satellite's orbit during each revolution or part thereof. The end points of this segment are evaluated numerically once per revolution. The effect of the inherent uncertainties in the boundaries of the Earth's shadow is discussed. The algorithm is tested by means of numerical integration of the equations of motion and through comparisons with observations of the balloon satellite 1963 30D during a 200-day interval.     

On the perturbation of the anomalistic period of a highly eccentric orbit satellite due to the zonal harmonics

In this note a simple formula is given for the perturbation of the anomalistic period of a highly eccentric orbit due to the zonal harmonics. This perturbation depends essentially only on the semi-major axisa, the eccentricitye (or pericentre radius r =a(1-e)) and the latitude of the pericentre.