Theory of an experiment in an orbiting space laboratory to determine the gravitational constant

The paper analyzes an experiment in an orbiting laboratory to determine the gravitational constantG. A massive sphere, according to a suggestion of L. S. Wilk, is to have three tunnels drilled through it along mutually perpendicular diameters. The sphere either floats in the orbiting laboratory, with its center held fixed by means of external jets issuing from the spacecraft, or is tethered to the spacecraft. In either case it is free to rotate; in the second case this freedom would be achieved by a system of gimbals.Each tunnel contains a small test object, which is held on the tunnel's axis by means of a suspension system, perhaps electrostatic, and held at rest relative to the sphere by slowly rotating the latter by means of inertia reaction wheels, governed by a servomechanism. Fundamentally, one balances the gravitational forces on the test objects by centrifugal force, determines the latter by measuring the components of angular velocity, and calculatesG from the resulting balance. It is better to use three tunnels than one because their use minimizes the effects of the Earth's gravity-gradient.Many other measurements and corrections are required. The latter arise from Earth gravity-gradient, aerodynamic drag (with the tethered sphere), gravitational forces produced by the spacecraft itself, and the force reductions produced by the empty space in all three tunnels. After the consideration of these effects there is a presentation and discussion of the equations required to reduce the observations to obtainG. There then follow the extra equations, not needed in the reduction, that are required for a computer simulation to investigate the possible extraction of a test object and to aid in designing the servomechanisms.In Appendix B, I have devised another version of the experiment, in which the sphere is kept intact, but has short thin hollow vestigial tunnels attached to the outside of the sphere, along perpendicular diameters. These external tunnels would contain the test objects and the suspension systems. The servomechanisms would then have to prevent collision of a test object with the sphere, as well as extraction. This second method could allow for some inhomogeneities in the sphere, would require no accurate drilling, and would make the suspension systems more accessible for construction and adjustment.This paper was prepared under the sponsorship of the National Aeronautics and Space Administration through NASA Contract NAS 9-8328.     

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.     

An orbiting clock experiment to determine the gravitational red shift

Plans are presented for an experiment to measure the gravitational red shift of the Earth by comparing a ground-based and satellite-borne hydrogen maser clock. The limiting accuracy is estimated to yield a determination of the red shift to 1 part in 10⁵, corresponding to a clock stability of 3 parts in 10¹⁵.This work was supported by the National Aeronautics and Space Administration.     

Astrophysical constraints on the gravitational constant

We study the astrophysical bounds on the change of the gravitational constant with time. We found that |/G|<10^–12yr^–1 is the condition that has to be satisfied in order not to cause a conflict with the observations. We find the condition to be in accord with the lower limits, the superstring theory predicts.     

Limits on the changes in the gravitational constant from nucleosynthesis

Due to the recent all-sky, high-precision measurement of microwave background anisotropies by WMAP, a value for baryon-to-photon ratio η was obtained. At the WMAP value for η, the ⁴HE abundance was predicted. In this article we use a simple semi-analytical method with ⁴He predicted and measured values to place a limit on the variation of the gravitational constant G. We find using a conservative range for the measured values for Y _p , that ΔG/G is constrained between −0.26 and 0.15. If we assume a monotonic power law time dependence G∼t ^−β then β values is constrained between −0.008 and 0.0038, which translate into . This compares well with results obtained by others using full numerical analysis.      

Constraints on the gravitational constant from the observations of white dwarfs

Recently some authors have questioned whether Newton's law of gravitation is actually true on scales less than 1 km. The available constraints on the gravitational constant show that is laboratory valueG ₀ may differ from the value at infinityG by 40%. Long (1976) reported experimental evidence for departures from Newton's law. In this note it is shown that the difference betweenG ₀ andG modifies the mass-radius relation of degenerate stars. The observations of white dwarfs are consistent with the theory of stellar evolution only ifG ₀ differs fromG by not more than 10%. This estimate may be improved by a higher accuracy of observations.     

A new approach to the theory with variable gravitational constant

A theory with variable gravitational constant, based on the bimetric formulation of general relativity, is suggested. It is in agreement with the Newtonian and post-Newtonian approximations of the theory of gravitation when the theory parameter ||4×10^–6. The statical spherical-symmetric distribution of gravitating masses is investigated. It is shown that the total action (including the own gravitational field of the system)S=–M, where is the time in the remote system of reference, relative to which the celestial body is in rest. In the Brans-Dicke theory ||1 andS–M. The Tolman formula for the massM in the Einstein theory is also valid in the suggestedGeneralized Bimetric Theory of Gravitation. External analytical and internal numerical solutions of the field equations are found for the case of incompressible matter. It is shown that static and supermassive configurations may exist if –0.13<0.     

Precise measurement of the solar gravitational red shift

We present the concept and the status of a multi-year project based on a new method to measure the Gravitational Red Shift of the Solar Spectrum with high precision. This project is aimed to conduct experimental verifications of the effect that the Einstein theory of General Relativity predicts for the frequencies of the Fraunhofer lines, that is, the light spectrum emitted by the Sun in its strong gravitational field. Previous determinations of such effect is limited to a precision of 2%. In order to discriminate between classical and relativistic explanations, we need to be sensitive to one part per million of the predicted effect. We have developed a new powerful technique, the Magneto-Optical Filter, that is able to provide far better precision and, for the future, possible space instrumentations able to extend our test to the second-order effect of the relativistic equivalence principle, never done before. The present paper is intended to describe the instrumentation, the procedure and the first encouraging results.     

A Moon orbiting observatory for the measurement of the elemental and isotopic composition in the primary cosmic radiation

We describe a project for the measurement of elemental composition of the primary cosmic radiation to be performed by a space observatory orbiting around the Moon. The absence of atmosphere and the low intrinsic magnetic field of the Moon give access to the very low energy component of the cosmic radiation, allowing the search for rare events.The quest for antinuclei, the determination of the lunar lepton albedo and the abundance measurement of galactic radioactive clocks (Be¹⁰, C¹⁴, Al²⁶) are the major tasks of the ANTARES apparatus (ANTimatter Assessment RESearch).We report details of the instrument design, the expected performance for single detectors, their capability to accomplish the proposed measurements and the characteristics of the space mission.     

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     

The rotational motion of an earth orbiting gyroscope according to the Einstein theory of general relativity

The classical Poisson equations of rotational motion are used to study the attitude motions of an Earth orbiting, rapidly spinning gyroscope perturbed by the effects of general relativity (Einstein theory). The center of mass of the gyroscope is assumed to move about a rotating oblate Earth in an evolving elliptic orbit which includes all first-order oblateness effects produced by the Earth.A method of averaging is used to obtain a transformation of variables, for the nonresonance case, which significantly simplifies the Poisson differential equations of motion of the gyroscope. Longterm solutions are obtained by an exact analytical integration of the simplified transformed equations. These solutions may be used to predict both the orientation of the gyroscope and the motion of its rotational angular momentum vector as viewed from its center of mass. The results are valid for all eccentricities and all inclinations not near the critical inclination.This paper represents a part of the author's Ph. D. dissertation for the Mathematics Department, Auburn University.     

The future evolution of white dwarf stars through baryon decay and time varying gravitational constant

Motivated by the possibility that the fundamental “constants” of nature could vary with time, this paper considers the long term evolution of white dwarf stars under the combined action of proton decay and variations in the gravitational constant. White dwarfs are thus used as a theoretical laboratory to study the effects of possible time variations, especially their implications for the future history of the universe. More specifically, we consider the gravitational constant G to vary according to the parametric relation G=G ₀(1+t/t _? )^?p , where the time scale t _? is the same order as the proton lifetime t _P . We then study the long term fate and evolution of white dwarf stars. This treatment begins when proton decay dominates the stellar luminosity, and ends when the star becomes optically thin to its internal radiation.     

Microwave remote sensing of Jupiter's atmosphere from an orbiting spacecraft

Microwave remote sounding from a spacecraft flying by or in orbit around Jupiter offers new possibilities for retrieving important and presently poorly understood properties of its atmosphere. In particular, we show that precise measurements of relative brightness temperature as a function of off-nadir emission angles, combined with absolute brightness temperature measurements, can allow us to determine the global abundances of water and ammonia and study the dynamics and deep circulations of the atmosphere in the altitude range from the ammonia cloud region to depths greater than 30 bars in a manner which would not be achievable with ground-based telescopes.     

A model-independent test of the spatial variations of the Newtonian gravitational constant in some extrasolar planetary systems

In this paper, we directly constrain possible spatial variations of the Newtonian gravitational constant G over the range  ≈ 0.01–5 au  in various extrasolar multiplanet systems. Using the third Kepler law, we determine the quantity  Γ_XY= G _X/ G _Y  for each couple of planets X and Y located at different distances from their parent star; deviations of the measured values of Γ from unity would signal variations of G . The obtained results for  η= 1 −Γ  are found to be very compatible with zero within the experimental errors  (η/δη≈ 0.2–0.3)  . We make a comparison with an analogous test previously performed in our Solar system.     

Radial mixing in the outer Milky Way disc caused by an orbiting satellite

The turbulent diffusion tensor describing the evolution of the mean concentration of a passive scalar is investigated for non-helically forced turbulence in the presence of rotation or a magnetic field. With rotation, the Coriolis force causes a sideways deflection of the flux of mean concentration. Within the magnetohydrodynamics approximation there is no analogous effect from the magnetic field because the effects on the flow do not depend on the sign of the field. Both rotation and magnetic fields tend to suppress turbulent transport, but this suppression is weaker in the direction along the magnetic field. Turbulent transport along the rotation axis is not strongly affected by rotation, except on shorter length-scales, i.e. when the scale of the variation of the mean field becomes comparable with the scale of the energy-carrying eddies. These results are discussed in the context of anisotropic convective energy transport in the Sun.     

A new limit on time dependence of the gravitational constant from gravity modified quantum electrodynamics

The time variation of the gravitational constantG is discussed in the light of the gravity modified form of quantum electrodynamics. From the experimental upper limit |a/| < 5 × 10^–15 yr^–1 on the time variation of the electromagnetic fine structure constant one finds |/G| < 5 × 10^–13 yr^–1.     

The need for an energy-dependent torsion-coupling constant in the early Universe

We present arguments to show that torsion-coupling constant (which depends on energy asE ^–2) can pass through the values of the coupling constants of the other interactions during the evolution of the Universe. An energy-dependent torsion-coupling constant helps in a natural way to understand the ratios of the coupling strengths of the different fundamental interactions.