On the Distance Function Between Two Keplerian Elliptic Orbits

The problem of finding critical points of the distance function between two Keplerian elliptic orbits is reduced to the determination of all real roots of a trigonometric polynomial of degree 8. The coefficients of the polynomial are rational functions of orbital parameters. Using computer algebra methods we show that a polynomial of a smaller degree with such properties does not exist. This fact shows that our result cannot be improved and it allows us to construct an optimal algorithm to find the minimal distance between two Keplerian orbits. This revised version was published online in July 2006 with corrections to the Cover Date.     

On the collapse of a vertical packet of mixed patches. Laboratory and field studies

Visualization of a collapsing vertical packet of patches in a laboratory experiment and the results of acoustical sounding under field conditions allowed us to discriminate the stage of its evolution which was previously unknown. It is defined by the variations of the horizontal dimensions of patches and their effective scattering surface with a period ofT=150–250/N, whereN is the Brunt-Väisälä frequency. The horizontal dimensions of patches can reach 40% of their final dimensions at the evolution stage discriminated.Translated by Mikhail M. Trufanov.     

Relationship between the velocity ellipsoids of galactic-disk stars and their ages and metallicities

The dependences of the velocity ellipsoids of F-G stars of the thin disk of the Galaxy on their ages and metallicities are analyzed based on the new version of the Geneva-Copenhagen Catalog. The age dependences of the major, middle, and minor axes of the ellipsoids, and also of the dispersion of the total residual velocity, obey power laws with indices 0.25, 0.29, 0.32, and 0.27 (with uncertainties ±0.02). Due to the presence of thick-disk objects, the analogous indices for all nearby stars are about a factor of 1.5 larger. Attempts to explain such values are usually based on modeling relaxation processes in the Galactic disk. Elimination of stars in the most numerous moving groups from the sample slightly reduces the corresponding indices (0.22, 0.26, 0.27, and 0.24). Limiting the sample to stars within 60 pc of the Sun, so that the sample can be considered to be complete, leaves both the velocity ellipsoids and their age dependences virtually unchanged. With increasing age, the velocity ellipsoid increases in size and becomes appreciablymore spherical, turns toward the direction of the Galactic center, and loses angular momentum. The shape of the velocity ellipsoid remains far from equilibrium. With increasing metallicity, the velocity ellipsoid for stars of mixed age increases in size, displays a weak tendency to become more spherical, and turns toward the direction of the Galactic center (with these changes occurring substantially more rapidly in the transition through the metallicity [Fe/H]≈−0.25). Thus, the ellipsoid changes similarly to the way it does with age; however, with decreasing metallicity, the rotational velocity about the Galactic center monotonically increases, rather than decreases (!). Moreover, the power-law indices for the age dependences of the axes depend on the metallicity, and display a maximum near [Fe/H] ≈−0.1. The age dependences of all the velocity-ellipsoid parameters for stars with equal metallicity are roughly the same. It is proposed that the appearance of a metallicity dependence of the velocity ellipsoids for thin-disk stars, recorded from the close to the Sun, is most likely due to the radial migration of stars.     

Martian water vapor: Mars Express PFS/LW observations

We present the seasonal and geographical variations of the martian water vapor monitored from the Planetary Fourier Spectrometer Long Wavelength Channel aboard the Mars Express spacecraft. Our dataset covers one martian year (end of Mars Year 26, Mars Year 27), but the seasonal coverage is far from complete. The seasonal and latitudinal behavior of the water vapor is globally consistent with previous datasets, Viking Orbiter Mars Atmospheric Water Detectors (MAWD) and Mars Global Surveyor Thermal Emission Spectrometer (MGS/TES), and with simultaneous results obtained from other Mars Express instruments, OMEGA and SPICAM. However, our absolute water columns are lower and higher by a factor of 1.5 than the values obtained by TES and SPICAM, respectively. In particular, we retrieve a Northern midsummer maximum of 60 pr-μm, lower than the 100-pr-μm observed by TES. The geographical distribution of water exhibits two local maxima at low latitudes, located over Tharsis and Arabia. Global Climate Model (GCM) simulations suggest that these local enhancements are controlled by atmospheric dynamics. During Northern spring, we observe a bulge of water vapor over the seasonal polar cap edge, consistent with the northward transport of water from the retreating seasonal cap to the permanent polar cap. In terms of vertical distribution, we find that the water volume mixing ratio over the large volcanos remains constant with the surface altitude within a factor of two. However, on the whole dataset we find that the water column, normalized to a fixed pressure, is anti-correlated with the surface pressure, indicating a vertical distribution intermediate between control by atmospheric saturation and confinement to a surface layer. This anti-correlation is not reproduced by GCM simulations of the water cycle, which do not include exchange between atmospheric and subsurface water. This situation suggests a possible role for regolith-atmosphere exchange in the martian water cycle.     

Hydrogeochemistry of the Koyna River basin,India

Hydrogeochemistry of the Koyna River basin, famous for the Koyna earthquake (magnitude 7) of 1967, has been studied. Basalt is the primary aquifer; laterites, alluvium, and talus deposits form aquifers of secondary importance. Groundwater generally occurs under water table conditions in shallow aquifers. Deeper aquifers are associated only with basalts. One hundred and 87 water samples were collected from various sources, such as dugwells, borewells, springs, and surface water, including 40 samples for analysis of iron. Only major constituents were analyzed. Analyses show that the concentrations of Ca²⁺ exceed that of Mg²⁺ in almost all water samples; the concentrations of Na⁺ are generally next to Ca²⁺ and are always higher than that of K⁺; and CO₃ ^2– and SO₄ ^2– are very low and are often negligible. Groundwater in borewells tapping deeper aquifers has higher mineralization compared to that in dugwells representing shallow aquifers. Majority of the water samples are dominated by alkaline earths (Ca²⁺, Mg²⁺) and weak acids (HCO₃ ⁻, CO₃ ^2–). Groundwater from shallow aquifers is generally calcium-bicarbonate type (53%) and calcium-magnesium-bicarbonate type (27%). In case of deeper aquifer, it is mostly calcium-magnesium-bicarbonate type (29%), sodium-bicarbonate type (24%), calcium-bicarbonate type (19%), calcium-magnesium-sodium-bicarbonate type (19%) and sodium-calcium-bicarbonate type (9%). Groundwater water is generally fit for drinking and irrigation purposes, except in the lower reaches of the Koyna River basin, which is affected by near water logging conditions.     

Particulate dispersion from sources within a forest

Particulate dispersion from sources within a 10- to 13-m tall pine forest was studied experimentally at Brookhaven National Laboratory using stained ragweed pollen and other tracers ranging from 14 to 58 m in size. Forty-seven continuous point source releases lasting from 22 to 55 min were made at heights from 1.75 to 14.0 m from locations having a long fetch through the forest. In most experiments, differently colored ragweed pollen were emitted simultaneously from three locations. In other tests, several particle types were released from a single point. The sampling network consisted of 119 rotoslide samplers at heights from 0.5 to 21.0 m at 57 positions within and at the edge of the forest. Deposition to the ground was sampled by greased microscope slides at each position. Meteorological measurements were taken in and near the forest.Data were classified by particle characteristics, source height and meteorological parameters. Concentration patterns were illustrated on scale diagrams of the sampling grid. Changes in centerline and crosswind integrated concentrations, plume width and height, mass flux, deposition and deposition velocity were studied as a function of distance, particle size and wind speed. Results were compared to those obtained from similar releases over open terrain.In the forest, vertical predominates over lateral dispersion and considerable interchange occurs through the canopy. Flow is channelled somewhat by vegetation density differences but is generally in the direction of the mean wind above the forest. No systematic turning of the wind with height was observed. Most particles are lost to the foliage rather than to the ground and large particles are lost more rapidly than smaller ones. Rate of change in mass flux is similar to that over open terrain and is greater with light than with stronger wind speeds.This research was carried out under the auspices of the New York State Museum and Science Service and the U.S. Atomic Energy Commission (now Energy Research and Development Administration) and was partially supported by Research Grant No. R-800677 from the Division of Meteorology, U.S. Environmental Protection Agency.     

New Models for Long Range Forecasts of Summer Monsoon Rainfall over North West and Peninsular India

Summary New models based on (a) Multivariate Principal Component Regression (PCR) (b) Neural Network (NN) and (c) Linear Discriminant Analysis (LDA) techniques were developed for long-range forecasts of summer monsoon (June–September) rainfall over two homogeneous regions of India, viz., North West India and Peninsular India. The PCR and NN models were developed with two different data sets. One set consisted 42 years (1958–1999) of data with 8 predictors and the other, 49 years (1951–1999) of data with 6 predictors. The predictors were subjected to the Principal Component Analysis (PCA) before model development. Two different neural networks were designed with 2 and 3 hidden neurons. To avoid the nonlinear instability, 20 ensemble runs were made while training the network and the ensemble mean results are discussed. The LDA model was developed with 42 years of data (1958–1999) for classifying three rainfall intervals with equal prior probability of 0.33. Both the PCR and NN models showed useful forecast skill for NW India and Peninsular India. Models with 8 predictors performed better than the models with only 6 predictors. The NN model with 3 hidden neurons performed better than model with 2 hidden neurons. For NW India, the NN model performed better than the PCR model. The RMSE of the NN model and PCR model with 8 predictors for NW India (Peninsular India) during the independent period 1984–99 was 12.5% (12.2%) and 12.6% (11.5%), respectively. Corresponding figures for the models with 6 predictors are 15.0% (13.0%) and 13.9% (11.4%) respectively. During the independent period, model errors were large in 1991, 1994, 1997 and 1999. However all the models showed deteriorating predictive skill after 1988, both for NW India and Peninsular India. The LDA model correctly classified 62% of grouped cases for NW India and Peninsular India. The LDA model showed better skill in classifying deficient rainfall (< − 8%) over NW India and excess rainfall (> 3%) over Peninsular India. Received October 2, 1999 Revised December 28, 1999