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Due to considerable difficulties of calculating L, B-coordinates, several simplified methods are currently used. The relative errors in L, B-coordinates calculated by approximate methods are given. It is suggested that different methods be used to calculate L, B-coordinates in the solution of various problems.
Simultaneous and individual camera observations of GEOS 1, GEOS 2, Pageos, and Midas 4 obtained during the International Satellite Geodesy Experiment are utilized to determine station coordinates. The Smithsonian Astrophysical Observatory Standard Earth III system of coordinates is used to tie the geometrical network to a geocentric system and as a reference for calculating satellite orbits. A solution for coordinates combining geometrical and dynamical methods is obtained, and a comparison between the solutions and terrestrial data is made. The radiation-pressure and earth-albedo perturbations for Pageos are very large, and Pageos' orbits are used to evaluate the analytical treatment of these perturbations. Residual effects, which are probably of interest to aeronomists, remain in the Pageos orbits.
The relativistic conversion between coordinate time and atomic time is reformulated to allow simpler time calculations relating analysis in solar system barycentric coordinates (using coordinate time) with earth-fixed observations (measuring 'earth-bound' proper time or atomic time). After an interpretation in terms of relatively well-known concepts, this simplified formulation, which has a rate accuracy of about 10 to the minus 15th, is used to explain the conventions required in the synchronization of a worldwide clock network and to analyze two synchronization techniques - portable clocks and radio interferometry. Finally, pertinent experimental tests of relativity are briefly discussed in terms of the reformulated time conversion.
A method of automatic body-fitted curvilinear coordinate generation is described and used to construct a finite-difference solution of the full incompressible time-dependent Navier-Stokes equations for the unsteady laminar viscous flow arbitrary two-dimensional airfoils or any other two-dimensional body. A method of controlling the spacing of the coordinate lines encircling the body is developed in order to treat high Reynolds number flows, since the coordinate lines must concentrate near the surface to a greater degree as the Reynolds number increases. Multiple airfoils and submerged hydrofoils are treated as illustrative examples. The solution shows good agreement with the Blasius boundary layer solution for the flow past a semi-infinite flat plate.
The current need for more precisely defined reference coordinate systems arises for geodynamics because the earth can certainly not be treated as a rigid body when measurement uncertainties reach the few-centimeter scale or its angular equivalent. At least two coordinate systems seem to be required. The first is a system defined in space relative to appropriate astronomical objects, suitable for ultimately expressing the dynamical equations of motion for the earth. The second coordinate system must be associated with the nonrigid earth in some well defined way so that the rotational motions of the whole earth are represented by the transformation parameters relating the earth system to the space-inertial system. The earth system should be defined so that the dynamical equations for relative motions of the various internal mechanical components of the earth and accurate measurements of these motions are conveniently expressed in this system.
Three distinct coordinate systems are required in the interpretation of the lunar range observations: a celestial frame and two-body-fixed frames. However, there is no coordinate system that is uniquely, or even preferentially, related to the observations themselves. Effectively, one specifies the coordinate systems by the procedures used in data reduction and parameter improvement. Each of the three systems affects the others in some way, and internal inconsistencies are quite possible. The discussion examines some of the more important aspects of this problem.
The decentralized stochastic control of a linear dynamic system consisting of several subsystems is considered. A two-level approach is used by the introduction of a coordinator who collects measurements from the local controllers periodically and in return transmits coordinating parameters. Two types of coordination are considered: open-loop feedback and closed loop. The resulting control laws are found to be intuitively attractive.
Knowledge of the long-wavelength features of the geopotential and the geocentric coordinates of satellite-tracking stations is significantly improved by the use of precision satellite tracking with lasers. Tracking data on nine satellites are combined with terrestrial gravimetry to obtain a spherical-harmonics representation of the geopotential complete through degree and order 24. Laser tracking data are used to determine the coordinates of tracking stations. This coordinate system is referred to an inertial reference frame by use of camera observations and observations of deep-space probes. Resulting geodetic parameters provide better satellite ephemerides and a reference for analyzing satellite-to-sea-surface altimetry.
A generalized curvilinear orthogonal coordinate system is presented which can be used for approximating various axisymmetric and two-dimensional body shapes of interest to aerodynamicists. Such body shapes include spheres, ellipses, spherically capped cones, flat-faced cylinders with rounded corners, circular disks, and planetary probe vehicles. A set of transformation equations is also developed whereby a uniform velocity field approaching a body at any angle of attack can be resolved in the transformed coordinate system. The Navier-Stokes equations are written in terms of a generalized orthogonal coordinate system to show the resultant complexity of the governing equations.
An application of a simple numerical technique which allows for the rapid construction of orthogonal coordinate systems about axisymmetric blunt bodies is presented. This technique can generate orthogonal meshes which have unequally spaced points in two directions. Relations are given for the numerical generation of the metric coefficients. Body shapes ranging from simple analytical bodies to complex reverse curvature bodies are presented together with their orthogonal coordinate systems. The relatively good accuracy of the technique is shown in tabular data describing coordinate line slopes and metric coefficients. The predictor-corrector numerical method used to generate these results is both simple in concept and easy to program, so that the application of the technique should be broader than the results presented.
Laser tracking of Lageos by the NASA and SAO laser tracking systems from its launch in May 1976 until December 1976 has been used to derive the coordinates of the tracking stations. The NASA tracking data from four systems in the United States had a precision of 10 to 15 cm and the SAO stations in North America, South America and Australia had precisions between 0.8 meters and 1.3 meters. Nearly 90,000 observations of Lageos were used in this analysis. Thirty-one orbital arcs, each five days in length, were derived which had orbital fits of 25 cm for the NASA data and at about 1 meter level for the SAO data. The coordinates of all eight stations were derived from this data set and the preliminary estimate of the overall accuracy of 50 cm in each coordinate. These results are in general agreement at about the 30 cm level with other results obtained from laser tracking of Beacon Explorer C.
The transformation permits a uniform mesh to be used in the computational coordinate which extends across the layer. This coordinate transformation uses the local value of the skin friction coefficient to scale the thickness of the wall layer region, and the local maximum value of turbulent viscosity to scale the boundary-layer thickness. Results are presented for two dimensional boundary layers in both positive and negative pressure gradients and comparisons are made with experimental data and conventional variable-grid results for low speed turbulent boundary-layers. The cases chosen illustrate the capability of this new transformation to capture the boundary layer growth over the full extent of laminar, transitional, and turbulent flow with no grid adjustment as well as its ability to consistently enlarge the wall layer region for accurate shear stress representation. Results of mesh refinement studies using the new coordinate transformation are presented.
A method is presented for computing orthogonal boundary fitted coordinate systems for geometries with coordinate distributions specified on all boundaries. The system which has found most extensive use in generating boundary fitted grids is made up of Poisson equations, of which the functions P and Q provide a means for controlling the spacing and density of grid lines in the coordinate system. While questions remain concerning the existence and uniqueness of orthogonal systems, the generating method presented adds to the available, useful techniques for constructing these systems.
Data from the OGO 6 satellite have been analyzed in magnetic latitude and magnetic local time coordinates for various seasons and magnetic activity levels. These measurements show considerable detail, particularly in the auroral regions where the energy inputs are well organized in this coordinate system. This detail is not readily observed in models based on geographic coordinates. Atomic oxygen and helium densities and a parameter related to the molecular nitrogen density were analyzed. The long-term averages of these quantities are presented in graphical form and as analytic functions to provide models of these thermospheric parameters. The atomic oxygen and helium densities show minima at high latitudes in the postmidnight sector for nearly all seasons and magnetic activity levels. The exospheric temperature inferred from the N2 density increases toward high latitudes for all seasons and all magnetic activity levels. This inferred temperature is about 300 deg K higher at the summer magnetic pole than at the winter magnetic pole.
During the planning stage for the International Magnetospheric Study (IMS), it was stressed that coordinated observations among various satellites and among satellite, ground-based, balloon, and rocket (GBR) experiments were essential in obtaining the required observational data base. In the course of operating the Satellite Situation Center (SSC), it was found to be desirable to assemble a problem-oriented digital data base, consisting of a large number of physical parameters obtained from satellite and GBR sensors, in a computer system which would permit a large number of scientists to manipulate, display, discuss, study and analyze the data together in a coordinated manner. It was felt that such a process might shorten the time required to gain full scientific understanding of the observations. This approach was called the Coordinated Data Analysis Workshop (CDAW) process. Attention is given to the preliminary concept, the the initial implementation of the CDAW process, and aspects of subsequent evolution.
Jovian coordinate systems are different from those employed in the case of the earth. Latitude and longitude coordinates are usually established relative to some solid surface. Because Jupiter does not have a solid surface (at least none which is visible through the clouds), arbitrary, but convenient, coordinate grids have been prescribed. A spin equator is made out from observations of cloud motion, and the direction of the planetary spin axis is, therefore, determined with relatively good accuracy. The problem in establishing a Jupiter longitude system is that the mean rotation period of the clouds is a function of latitude. The solution selected was to define two separate longitude grids. A third longitude system became necessary with the detection of radio signals which gave evidence for a rotating planetary magnetic field. Attention is also given to orbital phase angle and longitude conventions for satellites, and two latitude systems for Jupiter
Powerful local controls have been developed for mesh generation with the class of multisurface coordinate transformations. The controls come from local piecewise linear interpolants, and the resulting coordinates have continuous first derivatives. At the boundaries, the mesh controls provide the capability of joining distinct coordinate systems together with continuous derivatives. Away from boundaries, a local region can be given a particular mesh form to model internal objects or to simplify a problem. Applications of the controls are illustrated with a sequence of two-dimensional examples. For simplicity, each is generated about a NACA0012 airfoil which is of unit length and with leading edge pointed in a negative x-direction.
Coordinate generation techniques with precise local controls have been derived and analyzed for continuity requirements up to both the first and second derivatives, and have been projected to higher level continuity requirements from the established pattern. The desired local control precision was obtained when a family of coordinate surfaces could be uniformly distributed without a consequent creation of flat spots on the coordinate curves transverse to the family. Relative to the uniform distribution, the family could be redistributed from an a priori distribution function or from a solution adaptive approach, both without distortion from the underlying transformation which may be independently chosen to fit a nontrivial geometry and topology.