A note on Le Verrier's expansion of the disturbing function

An examination of Le Verrier's seventh-order expansion of the disturbing function has revealed only one non-trivial error in Le Verrier's work. The error occurs in a sixth order term in the eccentricities.     

Average velocity components in a rotating infinitely flattened self-gravitating system near an equilibrium state

The average radial and angular velocity components are obtained for a rotating two-dimensional self-gravitating system near an equilibrium state. First-order perturbation configurations of flaring straight bars emanating from the center provide examples of such systems. In these systems the average velocity field to first order is incompressible and irrotational. The second-order effects on the product of the average velocity components with the spatial density are essentially independent of the angular coordinate.     

Profiles of spectral lines of an accretion ring around a Kerr black hole

Profiles of spectral lines emitted from an accretion ring around an object with strong gravitational field should be affected by Doppler shift, gravitational redshift, and deflection of light. Taking these effects into account, precise line profile of a Keplerian ring around a Kerr black hole for a distant observer is obtained by solving the kinetic equation of photons.     

The formal integrals of the restricted planar problem of the three bodies

The first integrals of motion of the restricted planar circular problem of three bodies are constructed as the formal power series in r^1/2, r being the distance of a moving particle from the primary. It is shown that the coefficients of these series are trigonometric polynomials of an angular variable. Some particular solutions have been found in a closed form. The proposed method for constructing the formal integrals can be generalized to a spatial problem of three bodies.     

Roughness of the lunar soil

In situ measurements at the lunar surface at millimeter resolution by the Apollo astronauts have been analyzed. Several statistical parameters have been determined for the landing site. The surface roughness has been found to be very nearly gaussian. The root-mean-square slopes have been obtained over scales between 0.5 mm and 5 cm. They steadily decrease with increasing scale length from 58° to 2° and are in reasonable agreement with radar-measured values. The autocorrelation coefficient of the height distribution has also been obtained. It has a scale-length of 0.7 mm.Adjunct Professor at the University of Massachusetts.Visiting Scholar at the University of Massachusetts.     

A new class of periodic solutions in the Kovalevskaya case of a rigid body in rotation about a fixed point

The equation of motion of a rigid body in Kovaleveskaya case is reduced to a plane motion. By using the method of small parameters introduced by Poincaré, the existence of a periodic solution is established.     

Observations on the ecological importance of salt marshes in the Cumberland Basin,a macrotidal estuary in the Bay of Fundy

The Cumberland Basin, a 118 km² estuary at the head of the Bay of Fundy which has an average tidal range of about 11m, contains large tracts of salt marsh (15% of the area below highest high water). Low marsh (below about 0·9 m above mean high water) is composed almost exclusively of Spartina alterniflora while the vegetation on high marsh is more diverse but dominated by Spartina patens. Because of its higher elevation, high marsh is flooded infrequently for short periods by only extreme high tides. Low marsh is inundated much more frequently by water as much as 4m deep for periods as long as 4 h per tide. Temporal variability in the occurrence of extreme tides influences the flooding frequency of high marsh for any given month and year. Using a modification of Smalley's method, the mean annual net aerial primary production (NAPP) of low and high marsh is estimated to be 272 and 172 g C m^?2, respectively. Vegetation turnover times average 1·0 and 2·0 y for low and high marsh, respectively. Because of abundant tidal energy, much of the low marsh production appears to be exported and distributed widely about the estuary. Since high levels of turbidity suppress phytoplankton production, salt marshes produce approximately half of the carbon fixed photosynthetically in the Cumberland Basin. It is concluded that salt marshes play a major ecological role in the Cumberland Basin.     

The effects of gravitational radiation on circular binary systems with spins

In any theory of gravitation that predicts the existence of gravitational waves, it is shown that, in the absence of mass exchange and mass loss and subjected to a condition, a circular binary system with spherically-symmetrical components cannot evolve in such a way that the spin angular velocity is always a linear function of the orbital velocity. Probably this relation between the angular velocities cannot be realized in any time interval. In particular, the system cannot remain in a synchronized state. Of eight special cases of evolution of this binary system, three cases, where the separation increases, are shown to be kinematically impossible while a restriction on the variation of a parameter governing the evaluation of the system can be made in another. In one of the kinematically allowed special cases the separation increases.     

Contact Binaries: A Study of the Proximity Effects and Gravity Darkening Based on Kopal’s Fourier Method

A new method for the determination of the proximity effects and gravity darkening exponents in contact binaries of W UMa type is presented. The method is based on Kopal’s method of Fourier analysis of the light changes of eclipsing variables in the Frequency Domain. The method was applied to 36 W UMa systems for which geometric and photometric elements have been derived by the most powerful techniques. The derived values are very close to those predicted by the existing theory of radiative transfer or convective equilibrium.