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It is noted that expressions for the apsidal precession rates to second order on J2 appear in the literature in at least three apparently mutually contradictory forms. The expressions are reconciled by accounting for subtle differences in the definitions of orbital elements.
Observations of facular regions on 35 days during 1982 obtained with the Extreme Limb Photometer are reported. The data were obtained at a wavelength of 0.53 microns with two apertures, No. 1 covering 36 arcsec and No. 2 covering 11 arcsec, inwards from the limb. The mean contrast of the faculae closer to the limb (aperture 2) is 1.51 + or - 0.23 times that from aperture No. 1. This contrast ratio can be fit to a 1/ms curve. These results are consistent with those from 1975 and 1979 observations and may be consistent with the facular limb-darkening function determined by Libbrecht and Kuhn (1984, 1985) if the data are normalized by the area of the solar surface. However, no calibrations or corrections are required to obtain the mean facular contrast presented here.
The orbital dynamics of micrometer-sized dust grains is explored numerically and analytically, treating the strongest perturbation forces acting on close circumplanetary dust grains: higher-order gravity, radiation pressure, and the electromagnetic force. The appropriate orbit-average equations are derived and applied to the E ring. Arguments are made for the existence of azimuthal and vertical asymmetries in the E ring. New understanding of the dynamics of E ring dust grains is applied to problems of the ring's breadth and height. The possibility for further ground-based and spacecraft observations is considered.
The most striking feature of Venus rotational state is its slow retrograde rotation which is apparently maintained by a balance between solid tidal friction and thermal tidal torques. Solid tides tend to drive the spin toward synchronous rotation while thermal tides drive it away. A balance is achieved at a specific rate because of the inverse frequency dependence of the thermal tide to the semi-diurnal heating. Atmospheric models have been constructed to estimate the thermal tidal torque based on ground heating. The solid friction dissipation factor Q approximately equal to 50 can be deduced assuming rotation has achieved steady state. The most perplexing feature of Venus orientation is its non-zero free obliquity epsilon approximately equal to 1.5 deg relative to its orbit. Although solid tides and perhaps atmospheric tides tend to increase the free obliquity on a time 1/K(sub t) approximately equal to 1 x 10(exp 8) yr, viscous friction (CMF) at a core-mantle boundary (CMB) resulting from the differential angular orientation Delta-epsilon of the core and mantle spin axes should have damped the free obliquity on a time scale as short as 10(exp 6) yr. One means of achieving a balance similar to that controlling rotation is to introduce a comparatively large CMB ellipticity e(sub c) to reduce Delta-epsilon such that there is a balance between solid-thermal tides and CMF. The balance depends not so much on the potential frequency dependence on the tides as on the quadratic dependence of CMF on Delta-epsilon if the layer is turbulent.
It is desired that the Nimbus meteorological satellite always cross the equator around local noon and, half-an-orbit later, cross the equator in the other direction around local midnight. The application of the phenomenon of nodal regression toward this end is discussed, and an analysis of the parameters angles of inclination, periods, and heights of such "ideal" circular orbits is presented. Also, the relative motion of the apparent versus the fictitious mean sun is briefly discussed.
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The bulge in the Earth at its equator has been shown to lead to a clustering of natural decays biased to occur towards the equator and away from the orbit's extreme latitudes. Such clustering must be considered when predicting the Expectation of Casualty (Ec) during a natural decay, because of the corresponding clustering of the human population in the lower latitudes. This study expands upon prior work, and formalizes in a single empirical equation the correction that must be made to the calculation of the average exposed population density as a result of this effect. The equation is represented as a function of ballistic number and inclination of the entering spacecraft over the credible range of ballistic numbers.
The bulge in the Earth at its equator has been shown to lead to a clustering of natural decays biased to occur towards the equator and away from the orbit's extreme latitudes. Such clustering must be considered when predicting the Expectation of Casualty (Ec) during a natural decay because of the clustering of the human population in the same lower latitudes. This study expands upon prior work, and formalizes the correction that must be made to the calculation of the average exposed population density as a result of this effect. Although a generic equation can be derived from this work to approximate the effects of gravitational and atmospheric perturbations on a final decay, such an equation averages certain important subtleties in achieving a best fit over all conditions. The authors recommend that direct simulation be used to calculate the true Ec for any specific entry as a more accurate method. A generic equation is provided, represented as a function of ballistic number and inclination of the entering spacecraft over the credible range of ballistic numbers.
The bulge in the Earth at its equator has been shown to lead to a clustering of natural decays biased to occur towards the equator and away from the orbit's extreme latitudes. Such clustering must be considered when predicting the Expectation of Casualty (Ec) during a natural decay, because of the corresponding clustering of the human population in the lower latitudes. This study expands upon prior work, and formalizes in a single empirical equation the correction that must be made to the calculation of the average exposed population density as a result of this effect. The equation is represented as a function of ballistic number and inclination of the entering spacecraft over the credible range of ballistic numbers.
Data from Earth sensor assemblies (ESA's) often are used in the attitude determination (AD) for both spinning and Earth-pointing spacecraft. The ESA's on previous such spacecraft for which the ground-based AD operation was performed by the Flight Dynamics Division (FDD) used the Earth scanning method. AD on such spacecraft requires a model of the shape of the Earth disk as seen from the spacecraft. AD accuracy requirements often are too severe to permit Earth oblateness to be ignored when modeling disk shape. Section 2 of this paper reexamines and extends the methods for Earth disk shape modeling employed in AD work at FDD for the past decade. A new formulation, based on a more convenient Earth flatness parameter, is introduced, and the geometric concepts are examined in detail. It is shown that the Earth disk can be approximated as an ellipse in AD computations. Algorithms for introducing Earth oblateness into the AD process for spacecraft carrying scanning ESA's have been developed at FDD and implemented into the support systems. The Tropical Rainfall Measurement Mission (TRMM) will be the first spacecraft with AD operation performed at FDD that uses a different type of ESA - namely, a static one - containing four fixed detectors D(sub i) (i = 1 to 4). Section 3 of this paper considers the effect of Earth oblateness on AD accuracy for TRMM. This effect ideally will not induce AD errors on TRMM when data from all four D(sub i) are present. When data from only two or three D(sub i) are available, however, a spherical Earth approximation can introduce errors of 0.05 to 0.30 deg on TRMM. These oblateness-induced errors are eliminated by a new algorithm that uses the results of Section 2 to model the Earth disk as an ellipse.
The problem of the influence of faculae on apparent solar shape is examined in light of general facular contrast models. The diagonal components of the oblateness signal from the faculae observed are calculated, and it is found that the faculae can contribute a signal whose time dependence is similar to that of the Dicke and Goldenberg (1967) oblateness signal, allowing an acceptable fit to the oblateness measurements for a facular contrast within the range of acceptable values.
Wind tunnel tests were made for spheres of various sizes over a range of Mach numbers and Reynolds numbers. The results indicated some conditions where the drag was affected by changes in the afterbody pressure due to a shock reflection from the tunnel wall. This effect disappeared when the Mach number was increased for a given sphere size or when the sphere size was decreased for a given Mach number. Drag measurements and Schlieren photographs show the possibility of obtaining inaccurate data when tests are made with a sphere too large for the test section size and Mach number. Tests were also made of an oblate spheroid. The results indicated a region at high Mach numbers where inherent positive static stability might occur with the oblate-face forward. The drag results are compared with those for a sphere and those for various other shapes. The drag results for the oblate spheriod and the sphere are also compared with some calculated results.
The shape of Mercury, particularly when combined with its geoid, provides clues to the planet’s internal structure, thermal evolution, and rotational history. Elevation measurements of the northern hemisphere acquired by the Mercury Laser Altimeter on the MESSENGER spacecraft, combined with 378 occultations of radio signals from the spacecraft in the planet’s southern hemisphere, reveal the low-degree shape of Mercury. Mercury’s mean radius is 2439.36±0.02 km, and there is a 0.14-km offset between the planet’s centers of mass and figure. Mercury is oblate, with a polar radius 1.65 km less than the mean equatorial radius. The difference between the semimajor and semiminor equatorial axes is 1.25 km with the long axis oriented 15° west of Mercury’s dynamically defined principal axis. Mercury’s geoid is also oblate and elongated, but it deviates from a sphere by a factor of 10 less than Mercury’s shape, implying compensation of elevation variations on a global scale.
The effect of shape on the dynamics of suspensions of non-spherical heavy particles is examined by fully resolved numerical simulations of oblate and prolate spheroids, as well as spheres, for a density ratio of ten, volume fractions ranging from 0.5% to 5%, and Reynolds numbers between 20 and 30. The dynamics is determined both by the interactions of the particles with the fluid as well as by collisions, with the number and importance of collisions increasing with volume fractions. A single isolated oblate or prolate spheroid falling under gravity generally falls broadside on, for the governing parameters examined here, and at low-volume fractions, the majority of particles in a suspension fall that way. At higher-volume fractions, the orientation is more random. The slip velocity decreases as the volume fraction increases for all shapes, as expected, but the effect of the shape is much less than seen for a single particle. Furthermore, this seems to be due to two effects. For all volume fractions, the most deformed particles cluster more than spheres and less deformed particles, which increases their slip velocity. As the concentration increases, the increased particle interactions also causes more particles to fall short side-on, which reduces the frontal area and the resulting drag, increasing the slip velocity. This second effect is, of course, absent for spherical particles.
Here, the identification of the first multiphonon 𝛾-vibrational bands in an odd-odd neutron-rich nucleus of the nuclear chart is presented. These high-spin structures of hard to access $^{104}_{41}\textrm{Nb}_{63}$, produced in fission, were studied by combining a spectrometer with isotopic resolution coupled to a 𝛾-ray tracking array and independently from high-fold 𝛾 coincidence measurements. Triaxial projected shell model calculations for the high-spin states are in good agreement with the measured observables for the yrast, one-phonon, and two-phonon 𝛾 bands. The possibility of an oblate shape of an isomeric state and coexistence of triaxial and oblate configurations is investigated from the decay of the 141 keV isomer. The present Letter illustrates the robustness of vibration excitations in the presence of an odd valence proton and neutron as well as the possibly coexisting shapes beyond the 𝑁 = 60 transitional region.
In the special perturbation method, Cowell and variation-of-parameters formulations of the motion equations are implemented and numerically integrated. Variations in the orbital elements due to drag are computed using the 1970 Jacchia atmospheric density model, which includes the effects of semiannual variations, diurnal bulge, solar activity, and geomagnetic activity. In the general perturbation method, two-variable asymptotic series and automated manipulation capabilities are used to obtain analytical solutions to the variation-of-parameters equations. Solutions are obtained considering the effect of oblateness only and the combined effects of oblateness and drag. These solutions are then numerically evaluated by means of a FORTRAN program in which an updating scheme is used to maintain accurate epoch values of the elements. The atmospheric density function is approximated by a Fourier series in true anomaly, and the 1970 Jacchia model is used to periodically update the Fourier coefficients. The accuracy of both methods is demonstrated by comparing computed orbital elements to actual elements over time spans of up to 8 days for the special perturbation method and up to 356 days for the general perturbation method.
The matrizant of the two-body problem is developed in terms of elements that are free from singularities for zero eccentricities and zero- and ninety-degree inclinations. Retrograde equinoctial elements eliminate the singularity for inclinations near 180 degrees, with only minor changes in the expressions for the matrizant. The 'single-averaged' variation-of-parameters equations for these elements are developed for third-body, oblateness, and drag effects. Higher order terms are included in the expansions for the third-body and oblateness potential. A computer program that uses these equations to predict orbital evolution is described. Numerical results are given for a near-circular orbit.
Closed-form analytic expressions for the time variations of instantaneous orbital parameters and of the topocentric range and range rate of a spacecraft moving in the gravitational field of an oblate large body are derived using a first-order variation of parameters technique. In addition, the closed-form analytic expressions for the partial derivatives of the topocentric range and range rate are obtained, with respect to the coefficient of the second harmonic of the potential of the central body (J sub 2). The results are applied to the motion of a point-mass spacecraft moving in the orbit around the equatorially elliptic, oblate sun, with J sub 2 approximately equal to .000027.