Using the extended Hill's equations for elliptical orbits a method of path correction for satellites is presented. In part 1 the simplified differential equations for small eccentricities were used. In those governing equations the time was the independent variable. In this part the basic equations guilty for any elliptical orbit with the true anomaly as independent variable are used. Now, this time dependent orbital element may be used as variable instead of the transfer time. For such a chosen true anomaly at encounter, the two instantaneous velocity changes for a two-impulse transfer are determined. Additional the restrictions by singularities for this parameter are outlined by formulas and it is shown how the optimal parameter of minimal Delta nu, which is located between two singularities, will be computed. The objective function depends on one variable only, i.e. the true anomaly. Therefore, the minimum may be numerically determined straightforward by using an optimization program or more sophisticated by applying the Newton method. The method of path correction used here may be also applied to related problems of orbit mechanics.

Mit Hilfe der erweiterten Hillschen Gleichungen für Ellipsenbahnen mit kleiner Exzentrizität wird eine Methode zur Bahnkorrektur von Satelliten angegeben. Zu einem vorgegebenen Parameter, der Transferzeit, werden die erforderlichen beiden Geschwindigkeitskorrekturen für ein Zwei-Impuls-Manöver ermittelt. Außerdem werden die Einschränkungen für mögliche Transferzeiten angegeben. Es wird ferner gezeigt wie der optimale Parameter für einen minimalen Geschwindigkeitsbedarf zur Treibstoffoptimierung bestimmt werden kann. Die Zielfunktion dieser Optimierung hängt dabei nur von einer Variablen, nämlich der Transferzeit ab. Daher kann das Minimum mit einem Optimierungsverfahren oder noch einfacher mit dem Newton-Verfahren numerisch leicht bestimmt werden. Die angegebene Methode lässt sich auch auf verwandte Probleme der Bahnmechanik anwenden.

Die Relativbewegung benachbarter Satelliten lässt sich durch die Hillschen Gleichungen, auch Clohessy-Wiltshire-Gleichungen genannt, in einfacher und anschaulicher Weise beschreiben. Dabei wird angenommen, dass sich der erste Satellit auf einer Kreisbahn bewegt. Auf diese Einschränkung wird hier verzichtet und so werden für den Referenzsatelliten Ellipsenbahnen zugelasssen. Für die Relativbewegung des zweiten Satelliten werden ganz analog die linearisierten Bewegungsgleichungen aus den nichtlinearen Grundgleichungen für Keplerbahnen gewonnen und in Abhängigkeit der Anfangsbedingungen gelöst. Für Referenzbahnen mit kleiner Exzentrizität können auch hier explizite Funktionen der Zeit, wenn auch als Näherungslösungen, angegeben werden. In numerischen Beispielen wird das Bewegungsverhalten dargestellt und mit Lösungen der Hillschen Gleichungen quantitativ und qualitativ verglichen. Durch Vergleiche mit den exakten Lösungen für Keplerbahnen werden die gewonnenen Näherungen untermauert. Für allgemeine Ellipsenbahnen wird mit Hilfe der Lösungen von Lawden, die lediglich als Funktionen der wahren Anomalie gegeben sind, für die Driftbewegung des zweiten Satelliten eine lineare Zeitfunktion angegeben und durch numerische Beispiele bestätigt. Anwendungen finden die Gleichungen u. a. in der Mikrogravitation, bei Formationsflügen von Satelliten, bei Rendezvous-Problemen und beim Einsammeln von Weltraumschrott.

Es wird eine Methode vorgestellt um Datensätze mit Zusatzbedingungen in einem Intervall mittels Tschebyscheffpolynomen zu approximieren. Eine geeignete Segmentierung des Intervalls ermöglicht in einzelnen Teilintervallen eine Tschebyscheffapproximation von geringem Grad. Man kann dadurch insgesamt mit weniger Tschebyscheffkoeffizienten auskommen, als bei einer einzigen hochgradigen Approximation des Gesamtintervalls. An den inneren Segmentgrenzen fordern wir Stetigkeit sowie an den Intervallgrenzen der beiden äußeren Segmente die Annahme vorgegebener Werte bis zur zweiten Ableitung. Mit Hilfe von Lagrangefaktoren und der Methode der kleinsten Quadrate formulieren wir ein Minimumproblem mit Nebenbedingungen. Dies führt auf ein lineares Gleichungssystem, dessen Lösung die entsprechenden Tschebyscheffkoeffizienten liefert.

The Mars Express Radio Science Experiment (MaRS) started regular operations in April 2004. The experiment employed radio occultation during two occultation seasons in April-August 2004 and December 2004 to sound the neutral martian atmosphere to derive vertical density, pressure and temperature profiles as functions of height, and to sound the ionosphere to derive vertical ionospheric electron density profiles. Both profile types were monitored as functions of time in order to determine diurnal variations and, in the case of the ionosphere, dependence on solar wind conditions. MaRS also determined the dielectric and scattering properties of the martian surface in specific target areas by using bistatic radar, determining gravity anomalies during pericentre passes at altitudes of 250 km for investigations of the structure and evolution of the crust and lithosphere, and sounding the solar corona during the superior conjunction of Mars with the Sun from mid-August to mid-October 2004. This chapter gives, as intended by the project, an overview of the observations from April 2004 to mid-2005, and presents examples and first results.

Still delivering ESA's Venus Express probe has been in orbit since April 2006. Eight research papers in this issue present new results from the mission, covering the atmosphere, polar features, interactions with the solar wind and the controversial matter of venusian lightning. Håkan Svedham et al . open the section with a review of the similarities and (mostly) differences between Venus and its 'twin', the Earth. Andrew Ingersoll considers the latest results, and also how the project teams plan to make the most of the probe's remaining six years of life.

The liquid–gas interface of a liquid in an annular container is subjected to a temperature gradient. Since shear stress on the free liquid surface depends on the temperature it shall create a variable shear stress on the surface which in turn yields by viscous traction a thermocapillary convection in the bulk of the liquid. For constant temperature T 2 > T 1 on the outer cylindrical wall and T 1 on the inner wall a steady convection shall emerge. The morphology of the ensuing flow is investigated as a function of the diameter ratio and the liquid height ratio and exhibits the growth of bottom rim vortex rings, which unite to single vortex rings of opposite flow direction to the original single strong vortex ring above it. The results also show how such a system depends on the magnitude of the diameter ratio and the liquid height ratio. The evaluation of the results exhibits the evolution of vortex rings.

The Venus Express Radio Science Experiment (VeRa) uses radio signals at wavelengths of 3.6 and 13 cm ("X"- and "S"-band, respectively) to investigate the Venus surface, neutral atmosphere, ionosphere, and gravity field, as well as the interplanetary medium. An ultrastable oscillator (USO) provides a high quality onboard reference frequency source; instrumentation on Earth is used to record amplitude, phase, propagation time, and polarization of the received signals. Simultaneous, coherent measurements at the two wavelengths allow separation of dispersive media effects from classical Doppler shift.VeRa science objectives include the following:(1) Determination of neutral atmospheric structure from the cloud deck (approximately 40 km altitude) to 100 km altitude from vertical profiles of neutral mass density, temperature, and pressure as a function of local time and season. Within the atmospheric structure, search for, and if detected, study of the vertical structure of localized buoyancy waves, and the presence and properties of planetary waves.(2) Study of the H2SO4 vapor absorbing layer in the atmosphere by variations in signal intensity and application of this information to tracing atmospheric motions. Scintillation effects caused by radio wave diffraction within the atmosphere can also provide information on small-scale atmospheric turbulence.(3) Investigation of ionospheric structure from approximately 80km to the ionopause (< 600km), allowing study of the interaction between solar wind plasma and the Venus atmosphere.(4) Observation of forward-scattered surface echoes obliquely reflected from selected high-elevation targets with anomalous radar properties (such as Maxwell Montes). More generally, such bistatic radar measurements provide information on the roughness and density of the surface material on scales of centimeters to meters.(5) Detection of gravity anomalies, thereby providing insight into the properties of the Venus crust and lithosphere.(6) Measurement of. the Doppler shift, propagation time, and frequency fluctuations along the interplanetary ray path, especially during periods of superior conjunction, thus enabling investigation of dynamical processes in the solar corona.(c) 2006 Elsevier Ltd. All rights reserved.

The lower approximate natural frequencies of a spinning circular plate are determined for various boundary conditions. The plate is rotating with constant speed of spin about the axis of symmetry perpendicular to its plane. The boundary conditions treated here are: clamped, simply supported, free and guided. It is assumed that the normal displacement of the plate is small and that it does not affect during the oscillation the angular and radial variable stress distribution due to the rotating of the plate. This divides the problem at hand into two parts, of which the first part determines the radial and angular variable planar stresses in the plate, while the second part requires the solution of the partial differential equation with variable coefficients. The approximate lower natural frequencies are determined as functions of the speed of spin and exhibit in the plates with clamped, guided and simply supported boundary conditions instabilities.

The natural frequencies are obtained for elastic circular plates with mixed boundary conditions. The boundary conditions of the plate were combinations of clamped, simply supported, free, and guided. The natural frequencies are presented as a function of the angle over which the circumference of the plate is treated as one boundary and the remainder as another boundary condition.It was found that the axisymmetric modes exhibit continuous variation of the natural frequency from one pure condition to the other pure boundary condition, while the asymmetric modes show two branches of varying curvature and magnitude fluctuation, depending on the magnitude of the angles of mixed boundaries. (c) 2005 Elsevier Ltd. All rights reserved.

For a structure partially covering the free liquid surface of a two-dimensional rectangular container of infinite width, length l and a filling grade of height h hydroelastic vibrations have been treated for a frictionless liquid. It was found that with a partial covering of the free surface the lower coupled frequencies increase with structural area, yielding in addition a calmed down motion of the liquid. It also was found that the coupled frequencies stay always below those of the case of the rigid surface covering.

The axisymmetric fundamental natural damped frequency of the free surface in a slowly rotating cylindrical container is determined for incompressible viscous liquid. It was found that with increasing liquid height ratio h / a oscillation frequencies and decay magnitude both exhibit increased values. The influence of the surface tension parameter σ*=σ a /ϱν 2 shows for increasing magnitude increasing oscillation frequencies and decreasing decay magnitudes. For increasing Bond number Bo the oscillation frequency increases while the decay magnitude decreases. The effect of increasing the spin parameter ω 0 =Ω 0 a 2 /ν is manifested by an increase of the oscillation frequency and decay magnitude, i.e. for faster spin a liquid disturbance decays faster with larger oscillation frequency.

Für den EQUATOR-S-Satelliten wurde ein Nutationsdämpfersystem mit zwei gleichen symmetrisch angeordneten Flüssigkeitsdämpfern theoretisch untersucht und gebaut. Jeder dieser beiden Dämpfer besteht jeweils aus zwei Rohren mit kreisförmigem Querschnitt die durch zwei kurze Rohre mit wesentlich größerem kreisförmigem Querschnitt verbunden sind (kommunizierende Gefäße). Zu den vorgegebenen Trägheitsmomenten des Satelliten sowie dessen Spinfrequenz wurden dazu, unter Berücksichtigung gewisser Einschränkungen bezüglich der Geometrie der Dämpfer als auch der Wahl der Flüssigkeit, die Dämpferparameter analytisch so berechnet, daß eine gute Nutationsdämpfung mit sowohl eingeklappten als auch mit ausgeklappten Booms gewährleistet war. Die Dämpfer wurden nach den so berechneten Parametern gebaut und an einem luftgelagerten Pendel getestet. Hierzu wurde auch die Dämpfung am Pendel analytisch gerechnet und mit dem Experiment verglichen. Die Ergebnisse von Theorie und Experiment am Pendel stimmten dabei gut überein. Eine Aufzeichnung des abklingenden Nutationswinkels an Bord von EQUATOR-S zeigte ebenfalls gute Übereinstimmung mit der Theorie. Um diese gute Übereinstimmung zu erzielen, wurde jedoch, im Gegensatz zu früheren theoretischen Untersuchungen, eine Korrektur der bewegten Flüssigkeitsmasse im Dämpfer vorgenommen.

For the EQUATOR-S satellite a system of nutation dampers, consisting of two equal fluid nutation dampers was studied theoretically and finally constructed. Both dampers are tube-with-end pots dampers, composed caps. For given moments of inertia and the spin frequency of the satellite the parameters of the dampers were computed in such a way that, with some constraints as the geometry of the dampers and the choice of the liquid, an efficient nutation damping was ensured with booms in as well deployed and as in stored configuration. Based on these computed parameters, the dampers were built and tested on an air bearing pendulum. The theoretical damping of the system damper-pendulum was also computed and compared with the experiment. The results of theory and experiment agreed very well. An on-board recording of the nutation angle attenuation of EQUATOR-S also agreed very well with the theory. In contrast to former theoretical investigations, it was, however, necessary to introduce a correction factor for the moving liquid mass in the dampers.

For a liquid globule, partly captured in spherical caps of different meridian angles the thermo-capillary flow has been determined due to the change of temperature on the liquid surface. The variation of the applied temperature on the liquid surface causes a variation of the surface tension and therefore a tangential force moving from warm to cold. The streamlines are presented for linear and quadratic temperature distribution on the liquid surface. As the cap angle α increases the center of the ring vortex moves down and towards the free liquid surface. In addition the flow exhibits with increasing magnitude of cap coverage a reduced velocity distribution. For a quadratic temperature distribution the existence of two opposite vortex rings are shown for α<π/2. The upper vortex ring disappears as α becomes larger than π/2. In addition we notice the shift of the vortex core towards the free surface and bottom of the drop.

The thermo-capillary behavior or a viscous liquid in rectangular container configuration has been investigated for various liquid height ratios h/l. Streamlines and velocity distributions have been presented. The effect of a partial rigid surface cover has also been determined. With the increase of the liquid height ratio more vortices develop and exhibit decreasing strength towards the bottom of the container. An obstruction of the free liquid surface moves the vortices in the direction of the free liquid surface, exhibiting decreased strengths.

Forced oscillations of a viscous liquid with a free liquid surface in a circular cylindrical container have been treated by observing various boundary conditions. The response of the velocity distribution, the free surface elevation and response force of the liquid due to translational excitation of the tank have been determined. Three cases have been treated and compared with each other. One is the case of slipping liquid at the side wall, where the contact line is free to slip, while the other case considers an anchored contact line. Finally all adhesion conditions at the side wall of the container have been satisfied, while at the tank bottom only the normal velocity condition is satisfied. The first two cases are valid for small liquid filling rates, while the last treatment renders good results for large aspect ratio containers. It was found that a new phenomenon appears for low filling rates, where the liquid is only capable to perform an aperiodic motion.

During its interplanetary cruise to comet P/Wirtanen, the Rosetta spacecraft will encounter the asteroids 4979 Otawara and 140 Siwa on 11 July 2006 and 24 July 2008, respectively. The objective of the Rosetta Radio Science investigations (RSI) experiment at these flybys is a determination of the asteroid's mass and bulk density by analyzing the radio tracking data (Doppler and range) received from Rosetta before, during and after closet approach. The spacecraft's flyby trajectory will be gravitationally deflected by an amount proportional to the mass of the asteroid for a given flyby distance and velocity. An analysis of the Doppler noise sources indicates that the mass can be determined to an accuracy of 1% for 140 Siwa. The corresponding bulk density show be accurate to 20%. Unfortunately, a detectable trajectory perturbation seems to be hopeless for Otawara because of its small size and the large nominal flyby distance.