ALBERT

Notes: Abstract If a satellite orbit is described by means of osculating Jacobi α's and β's of a separable problem, the paper shows that a perturbing forceF makes them vary according to $$\dot \alpha _\kappa = {\text{F}} \cdot \partial {\text{r/}}\partial \beta _k {\text{ }}\dot \beta _k = {\text{ - F}} \cdot \partial {\text{r/}}\partial \alpha _k ,{\text{ (}}k = 1,2,3).{\text{ (A1)}}$$ Herer is the position vector of the satellite andF is any perturbing force, conservative or non-conservative. There are two special cases of (A1) that have been previously derived rigorously. If the reference orbit is Keplerian, equations equivalent to (A1), withF arbitrary, were derived by Brouwer and Clemence (1961), by Danby (1962), and by Battin (1964). IfF=−gradV 1(t), whereV 1 may or may not depend explicitly on the time, Equations (A1) reduce to the well known forms (e.g. Garfinkel, 1966) $$\dot \alpha _\kappa = {\text{ - }}\partial V_1 {\text{/}}\partial \beta _k {\text{ }}\dot \beta _k = \partial V_1 {\text{/}}\partial \alpha _k ,{\text{ (}}k = 1,2,3).{\text{ (A2)}}$$ holding for all separable reference orbits. Equations (A1) can of course be guessed from Equations (A2), if one assumes that $$\dot \alpha _k (t)$$ and $$\dot \beta _k (t)$$ depend only onF(t) and thatF(t) can always be modeled instantaneously as a potential gradient. The main point of the present paper is the rigorous derivation of (A1), without resort to any such modeling procedure. Applications to the Keplerian and spheroidal reference orbits are indicated.

Notes: Abstract The Schwarzschild field of a central massM is used to derive the general relativistic motion of a particle in a bounded orbit aroundM. A quadrature gives the central angle ϕ as a quasi-periodic function ϕ (f) of an effective true anomalyf. The linear term is an infinite series, whose second term yields the usual rate of advance of pericenter. For an artificial satellite this may be as large as 17″ of arc per year. The periodic part is a sine series, with coefficients containing the small parameter β≡2GM/c 2 p, wherep is closely approximated by the classical semi-latus rectum. The radius vectorr is a Kepler-like function off. The essentially new features of the calculation are the appropriate factoring of a certain cubic polynomialF(p/r), the use of the above effective true anomalyf, and the introduction of an effective eccentric anomalyE. The latter serves to reduce the differential equation forf as a function of the timet, obtained by combining the solution for ϕ(f) with the relativistic integrals of motion, to a Kepler equation forE. Knowing the constants of the motion, one can then solve successively forE(t), f(t), r(t), and ϕ(t). This is best done as a variational calculation, comparing the relativistic orbiter with a classical orbiter having the same initial conditions. The resulting variations agree with those of Lass and Solloway, but the present method is quite different from theirs and results in a simpler algorithm. The results show that the radial and transverse corrections, δr andr δϕ, arising from the Schwarzschild field, may be of the order of a kilometer for 1000 revolutions of an Earth satellite of orbital eccentricitye 0≈0.6. For bounded motion, the cubic polynomialF(p/r) has three positive real zeros, the two smaller ones corresponding to apocenter and pericenter. The third and apparently non-physical one occurs forr≈Schwarzschild radius. It may thus correspond to the incipient fall of the orbiter into the central body, if the latter is a black hole.

Notes: Abstract The paper derives the well known stabilities of free rotation of a rigid body about its principal axes of least and greatest moments of inertia directly from the constancy of the kinetic energy and of the square of the angular momentum. The resulting proof of Liapounov stability yields new quantitative measures of this stability. Involving only simple algebra, it depends on satisfying a weak sufficient condition that insures an unchanging sign of the main component of the angular velocity ω. The method cannot be used, however, to prove the well known instability of rotation about the intermediate axis. The quantitative results for the radii of the spheres in ω-space that occur in the Liapounov proof lead to a physical result that may be of interest. If the earth were truly a rigid body, rotating freely, the angular deviation of its instantaneous polar axis from the nearest principal axis could not increase from a given initial value by more than the factor ℚ2. These same quantitative results for the radii of the Liapounov spheres in ω-space also lead to suflicient conditions for the rotational stability of almost spherical bodies of various shapes, prolate or oblate. They may be pertinent in designing ‘spheres’ to be used in currently planned experiments to test general relativity by observing the rate of precession of a rotating sphere in orbit about the earth. The above results follow from restricted Liapounov stability alone. The last section contains the proof of general Liapounov stability.

Notes: Abstract Newtonian cosmology is developed with the assumption that the gravitational constantG diminishes with time. The functional form adopted forG(t), a modification of a suggestion of Dirac, isG=A(k+t) −1, wheret is the age of the Universe and a small constantk is inserted to avoid a singularity in the two-body problem. IfR is the scale factor, normalized to unity at an epoch time τ, the differential equation is then $$R^2 \ddot R = - (4\pi /3)G(t)\rho _0$$ . Here ρ0 is the mean density at the epoch time. With the above form forG(t), the solution is reducible to quadratures. The scale factorR either increases indefinitely or has one and only one maximum. LetH 0 be the present value of Hubble's ‘constant’ $$\dot R$$ /R and ρ0c the minimum density for a maximum ofR, i.e., for closure of the Universe. The conditions for a maximum lead to a boundary curve of ρ0c versusH 0 and the numbers indicate strongly that thisG-variable Newtonian model corresponds to an open universe. An upward estimate of the age of the Universe from 1010 yr to five times such a value would still lead to the same conclusion. The present Newtonian cosmology appears to refute the statement, sometimes made, that the Dirac model forG necessarily leads to the conclusion that the age of the Universe is one-third the Hubble time. Appendix B treats this point, explaining that this incorrect conclusion arises from using all the assumptions in Dirac (1938). The present paper uses only Dirac's final result, viz,G∼(k+t)−1, superposing it on the differential equation $$R^2 \ddot R = - (4\pi /3)G(t)\rho _0$$ .

Notes: Abstract 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.

Notes: Abstract Will (1971) has discussed a possible anisotropy in the gravitational constantG. Suppose that the attractive gravitational force between two particles of massesm 1 andm 2 is given by the usual expressionF=−Gm 1 m 2 r/r 3, wherer is the separation vector. Ifc is the velocity of light in vacuo and if 1 r ≡r/r, he expresses the anisotropy byG=G ∞[1+ε(v·1 r/c)2], whereG ∞ is a constant,v is identified practically as the velocity of the Sun around the galaxy, and ε≈1. Will's suggestion is to look for such an effect in the laboratory. The purpose of the present paper is to look for such an effect in the solar system, wherem 1 andm 2 become the masses of the Sun and a planet or of the Earth and the Moon. For simplicity I consider only those planets whose orbits are close to the ecliptic, so that the angle betweenv and the plane of the ecliptic is about 59°. With the above force, the resulting two-body problem is completely solvable. The results are these. If ε=1, there is an increase in mean motion of 7 parts in 108, a periodic fluctuation in true longitude with period half that of the orbit and amplitude ranging possibly from 0.01″ to 0.02″, and periodic fluctuations in the radius vector, with period also one half that for the orbit. The amplitudes are: 2.7 km for Mercury, 5.1 km for Venus, 7.0 km for Mars, 18 m for the Moon about the Earth, and 28 cm for a close artificial satellite with inclination 23°. The more conservative estimate ε〈0.0115 would reduce these values by the factor 70.

Notes: Abstract The paper represents the Earth's gravitational potentialV, outside a sphere bounding the Earth, by means of its difference δV from the author's spheroidal potential. The difference δV is in turn represented as arising from a surface density σ on the sphere bounding the Earth. Because of the slow decrease with ordern of the normalized coefficients in the spherical harmonic expansion ofV, the density anomalies from which the higher coefficients arise must occur in regions close to the Earth's surface. The surface density σ is thus an idealization of the product of the density anomaly Δϱ and the crustal thicknessb. Values of σ are computed from potential coefficients obtained from two sources, Rapp and the Smithsonian Astrophysical Observatory. The two sources give qualitative agreement for the values of σ and for its contour map. The numerical values obtained for σ are compatible with the idea that the responsible density anomalies are reasonably small, i.e., less than 0.05 g/cm3, and occur in the crust alone.

Notes: Summary The far zone field about an antenna of revolution is worked out according to the gap model. It is found to be the sum of three parts,.E 1,E 2 andE 3. The termE 1 depends on the electric field in the gap, but not explicitly on the antenna current. The termsE 2 andE 3 depend explicitly on the antenna current,E 2 being non-vanishing only if the antenna radius is variable, so that only the ends contribute to it in the case of a thick cylindrical antenna. The termE 3 reduces to the conventional expression for a line antenna when the antenna is sufficiently thin.