Libration Angle Research Articles

2H nuclear magnetic resonance study of supercooled toluene: Slow and fast processes above and below the glass transition

2H-NMR spin-lattice relaxation times T1 of phenyl ring deuterated toluene have been measured as a function of temperature (75–290 K) at Larmor frequencies between 13.8 and 55.8 MHz. The results are interpreted by assuming a Cole–Davidson rotational correlation time distribution for the α process dominating T1 at temperatures down to about 20 K above the glass transition (Tg=117 K) and one further process at lower temperatures. The latter is analyzed below Tg using a model with temperature dependent librational angles ΔΦ<13° and mean correlation times τ≳10−9 s. Self-diffusion coefficients are determined at temperatures down to 136 K (D=1.5×10−14 m2 s−1) using a static field gradient 1H-NMR method. The product of D and the mean rotational correlation time shows a tendency to increase at the lowest temperatures.

Tetrahedral rigid-body motion in silicates

The atomic displacement parameters (ADPs) obtained from published refinements of silicate crystals measured at or below room temperature are examined to determine ifT04 (T = AI,Si) tetrahedra display rigid-body thermal motion. In Inany cases, the correlation found among the ADPs is consistent with the TLS model of rigid-body motion. For these data, the translational motion is described by the ADPs of the central T atom, whereas both librational and translational motion are displayed in those of the surrounding 0 atoms. The libration angle for rigid tetrahedra is quadratically dependent on the difference between the isotropic equivalent displacement parameter of the T and 0 atoms, Beq(T) and Beq(O), respectively. The value of Beq(O) is on average twice that of Beq(T), with an observed maximum value of ,2.0 A2. Variation in the observed Si-O bond lengths of rigid tetrahedra in the silica polymorphs is related only to the fractional s-character of 0, h(O). ADPs that do not indicate rigid-body motion for a given tetrahedron may be used to identify crystals containing disorder or suggest problems with the refinement.

A Synthetic Theory of Motion for Titan-Hyperion

We present an iterative method allowing to synthetize a semi-numerical solution for the equations of motion of the resonant Saturn's satellites Titan-Hyperion (limited now to the planar problem). The current theory of Hyperion by Taylor, Sinclair & Message (1987) gives the greatest terms of the long-period part of the solution (depending on two angles: the libration angle τ, and the angular distance of the pericenters ζ). Using it as a first approximation, this solution is substituted numerically in the exact Lagrange equations of motion for Titan and Hyperion, computed for many values of the three angles: τ, ζ and ϕ (the mean synodic longitude). Then, a multivariable Fourier transform allows to reconstruct the equations in these three angles, that is in same form as the initial one with, in addition, the short-period terms. Then, a solution may be obtained and used as a better approximation in an iterative process. Besides a complete determination of the short-period perturbations of Hyperion obtained here completely for the first time, some long-period perturbations of Titan by Hyperion are also found which would be non negligible at the 10 km level.

Real time estimator for control of an orbiting single tether system

A real-time estimator is developed for the control of the Tether Dynamics Explorer (TDE) system. TDE is being used in a series of tethered satellite flight experiments whose purpose is to validate existing system models and test proposed control laws. Each experiment consists of an orbiting Delta II second stage which deploys toward the Earth a small box-shaped passive endbody at the end of a flexible tether. A discrete extended Kalman filter (DEKF) is presented which can operate in real time and in conjunction with control laws. This filter estimates the in-plane and out-of-plane tether libration angles and their rates from a proposed three-axis tether tension measurement device in the Delta second stage. The simulation results indicate that the DEKF can estimate the libration angle and their rates from a three-axis tension and length measurement with an acceptable error. The relatively simple computations required make this algorithm particularly well suited for real-time operation. >

Advantages of tether release of satellites from elliptic orbits

This paper examines the use of an elliptically orbiting tethered-dumbbell (Space Shuttle-satellite) system for satellite transfer to geosynchronou s altitude. The two-dimensional rigid-body equations of motion are derived using a Lagrangian method. Integration of these equations yields the system states, from tether deployment through payload release. The payload is given a swing zero release, i.e., the payload is released on a forward swing when the libration angle is instantaneously zero. By varying the predeployment true anomaly, the swing zero is caused to occur nearly simultaneously with periapse passage, yielding maximum savings in AK in comparison to Hohmann-type transfer from the same elliptic orbit to geosynchronous orbit. Deployment velocity and tether length were also varied so that their effect on A V savings could be determined. The maximum AF savings observed was 9%, corresponding to a payload mass gain of almost 23%.

Restriction on the motion of a solid in a gravitational field

As is well known, the orbital and rotational motions of a solid are coupled, and the integrals of energy and angular momentum (in a gravitational field with spherical symmetry) impose restrictions on them. We study the regions allowed to the motion in configurational space. It turns out that even in the crudest model (planar motion of a triple rod) the restrictions on the libration angle and the orbital radius of the center of mass are coupled, so that excessive ellipticity of the orbit excludes stabilization in the neighbourhood of the spoke equilibrium position by gravitational forces only.

Ar, N2, and CO2 in the structural cavities of cordierite, an optical and X-ray single-crystal study

Ar, N2 and CO2 were introduced into the structural cavities of channel-evacuated single-crystals of White Well cordierite with the composition: K0.01Na0.03(Mg1.91Fe0.09Mn0.01)Al3.98Si5.01O18. The gas refilling experiments were carried out in conventional hydrothermal bombs at 6–7 kbar and 600–700°C. The increase in the mean refractive indices for gas-treated crystals, as determined with a spindle-stage equipped microscope, was used along with point-dipole calculations to estimate the percentage of occupied structural cavities. The steep increase of the electronic polarizability parallel to the a-axis, which can be derived from the increase of the refractive index n γ (Z∥a) upon introduction of volatiles, indicates that N2 and CO2 are preferentially aligned parallel to the a-axis of cordierite. Single-crystal structure refinements at room temperature confirm these predictions. Additionally, decreased C–O and N–N bond lengths suggest a librational motion with a mean rotary oscillation angle of 35° (N2) and 25° (CO2) about a, where c is the rotation axis. Mean libration angles of 40° (N2) and 28° (CO2) were estimated from the electronic polarizability tensors of CO2 and N2. Site occupancy refinements of the channel position are in good agreement with the optically derived values for the volatile concentrations, both indicating about 70% and 60% filled cavities for Ar- and N2-cordierite, respectively. Chemical analyses and point-dipole calculations confirm that about 45% of the cavities are occupied in the CO2-treated crystal. The structural framework of cordierite is slightly but specifically altered by the various channel occupants.

Anharmonic effects in librational motion of N2-type crystals

The dynamics of librational motion in N2-type crystals (α-N2, α-CO, N2O, CO2) is treated by taking into account both anharmonic and correlation effects. The method used is similar to Tyablikov's method in the theory of magnetism. The main thermodynamic characteristics of the librational subsystem are calculated: the order parameter, rms librational angle, librational mode frequencies and corresponding Gruneisen parameters, librational heat capacity, and internal and free energies. The librational isotope effects for α-14N2 and α-15N2 are considered. An explanation of the anomalous isotope effects in the heat capacity is proposed. A theory of the phase transition into the orientationally disordered state is developed.

Satellite Librational Damping Using a Quasi-Feedback Solar Controller

The feasibility of developing a simple quasi-feedback solar attitude controller is explored. The case of arbitrarily-shaped satellites carrying two pairs of solar surfaces in circular orbits is considered. An approximate analytical approach is used to synthesize the suitable open-loop control relations enabling decay of the satellite librations. The quasi-feedback control scheme suggested here is found to be quite effective in damping the pitch motion. The proposed control approach appears to be particularly attractive as it requires knowledge of the librational angle and velocity only at a fixed satellite position in orbit.

Numerical model of the Moon's rotation

We have integrated numercially the differential equtions for the Moon's rotation with respect to an inertial coordinate system, and the variational equations for (i) the six initial conditions of the rotation; (ii) the moment-of-inertia ratios β and γ; and (iii) the coefficients of the third-degree gravitational harmonics. When these integrations are used in conjunction with our current lunar-orbit and Earth-rotation models, and all of the relevant initial conditions and parameters are adjusted to fit five years of McDonald Observatory lunar laser ranging observations, the root-mean-square (rms) of the postfit range residuals is 28 cm. When we adjust the lunar-rotation initial conditions separately to fit the physical libration angles given by the numerical model of Williams (1975), we find an rms orientation difference over a six-year interval of ∼0.03 arcsecond, after removal of a constant bias. A similar comparison of our model with the semi-analytical model of Eckhardt (1981) yields an rms orientation difference of ∼0.2 arcsecond.

Analysis of solar radiation pressure induced coupled liberations of a gravity stabilized axisymmetric satellite

This paper presents an analysis of solar radiation pressure induced coupled librations of gravity stabilized cylindrical spacecraft with a special reference to geostationary communication satellites. The Lagrangian approach is used to obtain the corresponding equations of motion. The solar induced torques are assumed to be free of librational angles and are represented by their Fourier expansion. The response and periodic solutions are obtained through linear and nonlinear analyses, using the method of harmonic balance in the latter case. The stability conditions are obtained using Routh-Hurwitz criteria. To establish the ranges of validity the analytic response is compared with the numerical solution. Finally, values of the system parameters are suggested to make the satellite behave as desired. Among these is a possible approach to subdue the solar induced roll resonance. It is felt that the approximate analysis presented here should significantly reduce the computational efforts involved in the design and stability analysis of the systems.

The influence of satellite flexibility on orbital motion

The orbital perturbations induced by the librational motion and flexural oscillations are studied for satellites having large flexible appendages. Using a Lagrangian procedure, the equations for coupled motion are derived for a satellite having an arbitrary number of appendages in the nominal orbital plane and two flexible members normal to it. The formulation enables one to study the influence of flexibility on both the orbital and attitude motions. The orbital coordinates are expanded as perturbation series in e=(l/a 0)2,l anda 0 being a characteristic length of the satellite and unperturbed semi-major axis of the orbit, respectively. The first order perturbation equations are solved in terms of elastic deformations and librational angles using the WKBJ method in conjunction with the variation of parameter technique. Existence of secular perturbations is noted for certain librational flexural motions. Three specific examples, Alouette II, Radio Astronomy Explorer and Tethered Orbiting Interferometer, are considered subsequently and their possible secular drifts estimated.

Orientation of the moon by numerical integration

The differential equations of rotational motion of the Moon are solved by numerical integration methods. Euler's dynamical equations transformed to a convenient form are treated by techniques analogous to ordinary orbit determination procedures. The proposed method is fully consistent with the ephemeris of the Moon and can utilize a variety of observational material for the solution of the selected parameters. The parameters are grouped into three distinct groups, namely: --The physical libration angles of the Moon and their time rates at an arbitrary initial epoch. --Physical constants featuring the principal moments of ineria of the Moon. --Parameters associated with the particular observational material being used.

Dynamic Mechanical Properties of a Number of Elastomers and Related Polymers from 4 to 250 °K

Young's modulus and mechanical damping have been measured down to 4 °K for cis- and trans-polybutadienes, cis- and trans-polyisoprenes, butyl rubber, and random SBR copolymers. Trans-polybutadiene has two large damping peaks below Tg, trans-polyisoprene one peak, while the cis-polymers and butyl rubber show either no damping peaks or only very small ones below Tg. The low-temperature damping peaks of the trans-polymers might be explained by a type of crankshaft motion and by a special motion of the trans-monomeric unit which involves a simultaneous libration and bond angle deformation. The high Young's moduli of the polybutadienes compared to the polyisoprenes and the SBR copolymers are explained by the good molecular packing of the polybutadienes.

Gravity-Gradient Stabilization of Earth Satellites

The possibility of using the earth’s gravity field for vertical stabilization of near-earth satellites has intrigued theoreticians for many years. The very small stabilizing torque available, and the lack of a natural damping mechanism have been recognized as the major problems. The satellite 1963-22A launched in June 1963 was the first orbiting vehicle to successfully achieve passive gravity-gradient attitude stabilization. The initial spin of the satellite was removed by mechanical and magnetic despin devices. An internal electromagnet was then energized to align the satellite along the local magnetic field direction. Twelve hours later when the satellite passed over the north magnetic pole it was vertical with the correct side facing earthward. A 100 foot boom was then extended from the satellite and the magnetic dipole was turned off. Libration damping was provided by the combination of a lossy spring and magnetic hysteresis rods. Within a week after boom deployment the peak angle of oscillation of the satellite was damped below 15 degrees. By June 1964 four other satellites achieved gravity-gradient stabilization using passive damping techniques. The satellites 1963-38B and 1963-49B used the combination of a lossy spring and magnetic rods in a configuration that was virtually identical to the satellite 1963-22A. The satellite 1964-26A achieved gravity stabilization using only 0.6 pound of magnetic hysteresis rods to damp the peak libration angle from 40 degrees to 10 degrees in a period of approximately 4 days. This lightweight, simple and completely passive magnetic damping technique shows great promise for future applications for near-earth orbiting satellites. At satellite altitudes above 5000 miles, vector magnetometers with current amplifiers can be used to enhance the damping capability of the magnetic rods. This semi-passive technique with no moving parts should be suitable for satellites even at synchronous altitude. Thermal bending of the extendible booms has been observed on several satellites. A high frequency, dynamic boom bending has been observed for unplated booms with an amplitude of ±12 degrees. With silver plated booms, the dynamic thermal bending was reduced to less than ±1/4 degree.