Studies in the application of recurrence relations to special perturbation methods

Using the rectangular equations of motion for the restricted three-body problem a comparison is made of the Encke and Cowell methods of integration. Each set of differential equations is integrated using Taylor series expansions where the coefficients of the powers of time are determined by recurrence relationships. It is shown that for fairly highly eccentric orbits in which the perturbing force is less than one thousandth of the two-body force the Encke method achieves a considerable saving in machine time. This is also true for almost circular orbits when low or moderate accuracy is required. When very high accuracy is required, however, the Cowell method is faster unless the perturbing force is less than 10^–6 of the two-body force. There is little difference in the accuracy of the two methods, the Cowell method being slightly more accurate when a low or moderate accuracy criterion is imposed.     

Implicit single-sequence methods for integrating orbits

The solutions of \(\ddot x = F(x,t)\) , and also \(\dot x = F(x,t)\) , are developed in truncated series in timet whose coefficients are found empirically. The series ending in thet ⁶ term yields a position at a final prechosen time that is accurate through 9th order in the sequence size. This is achieved by using Gauss-Radau and Gauss-Lobatto spacings for the several substeps within each sequence. This timeseries method is the same in principle as implicit Runge-Kutta forms, including some not described previously. In some orders these methods are unconditionally stable (A-stable). In the time-series formulation the implicit system converges rapidly. For integrating a test orbit the method is found to be about twice as fast as high-order explicit Runge-Kutta-Nyström-Fehlberg methods at the same accuracies. Both the Cowell and the Encke equations are solved for the test orbit, the latter being 35% faster. It is shown that the Encke equations are particularly well-adapted to treating close encounters when used with a single-sequence integrator (such as this one) provided that the reference orbit is re-initialized at the start of each sequence. This use of Encke equations is compared with the use of regularized Cowell equations.     

Numerical calculations in the orbital determination of an artificial satellite for a long arc

Numerical methods are usually used for the computation of ephemerides with perturbations for the precise orbital determination of an artificial satellite. But their numerical stability will be encountered in a long arc. In this case the use the improved Encke special perturbation methods has been suggested. The results of this paper show that Encke's method does indeed have a certain effectiveness, but cannot yet completely resolve the numerical stability, and the more efficient method is to use the energy integral or its variational relation to control the growth of the along-track error in general numerical calculations so that the aim of stabilization can be achieved.     

Error propagation in the numerical solutions of the differential equations of orbital mechanics

The relationship between the eigen values of the linearized differential equations of orbital mechanics and the stability characteristics of numerical methods is presented. It is shown that the Cowell, Encke, and Encke formulation with an independent variable related to the eccentric anomaly all have a real positive eigen value when linearized about the initial conditions. The real positive eigen value causes an amplification of the error of the solution when used in conjunction with a numerical integration method. In contrast an element formulation has zero eigen values and is numerically stable.     

Special perturbations employing osculating reference states

The concept of employing osculating reference position and velocity vectors in the numerical integration of the equations of motion of a satellite is examined. The choice of the reference point is shown to have a significant effect upon numerical efficiency and the class of trajectories described by the differential equations of motion. For example, when the position and velocity vectors on the osculating orbit at a fixed reference time are chosen, a universal formulation is yielded. For elliptical orbits, however, this formulation is unattractive for numerical integration purposes due to Poisson terms (mixed secular) appearing in the equations of motion. Other choices for the reference point eliminate this problem but usually at the expense of universality. A number of these formulations, including a universal one, are considered here. Comparisons of the numerical characteristics of these techniques with those of the Encke method are presented.     

Nongravitational forces and meteoroid streams

The action of the solar electromagnetic radiation on the moving interplanetary dust particles in its more complete form than the special case known as the Poynting-Robertson effect is theoretically discussed in application to meteoroid stream of comet Encke.Normal and transversal components of the perturbing nongravitational force are used due to the action of the solar electromagnetic radiation. It is shown that the normal component of the force is negligible. However, transversal component is very important: it can probably completely explain all the observed meteoroid streams situated along the orbit of comet Encke (and, possibly, some asteroids) as the product of the comet Encke alone. Much shorter time is required for producing such a meteoroid stream than is a general conception.If the idea about the significance of the transversal component of the nongravitational force (may be, not produced by electromagnetic radiation) is correct, it may have important consequences for our understanding of ageing of comets, global evolution of the cometary (and, partially, asteroidal) system, and, of course, for a long-term evolution of small interplanetary particles.     

GOCE: The Earth Gravity Field by Space Gradiometry

An artificial satellite, flying in a purely gravitational field is a natural probe, such that, by a very accurate orbit determination, would allow a perfect estimation of the field. A true satellite experiences a number of perturbational, non-gravitational forces acting on the shell of the spacecraft; these can be revealed and accurately measured by a spaceborne accelerometer. If more accelerometers are flown in the same satellite, they naturally eliminate (to some extent) the common perturbational accelerations and their differences are affected by the second derivatives of the gravity fields only (gradiometry). The mission GOCE is based on this principle. Its peculiar dynamical observation equations are reviewed. The possibility of estimating the gravity field up to some harmonic degree (200) is illustrated.     

Superosculating intermediate orbits: Fourth-and fifth-order tangencies to trajectories of perturbed motion

The theory of superosculating intermediate orbits previously suggested by the author is developed. A new class of orbits with a fourth-order tangency to the actual trajectory of a celestial body at the initial time is constructed. Orbits with a fifth-order tangency have been constructed for the first time. The motion in the constructed orbits is represented as a combination of two motions: the motion of a fictitious attracting center with a variable mass and the motion relative to this center. The first motion is generally parabolic, while the second motion is described by the equations of the Gylden—Mestschersky problem. The variation in the mass of the fictitious center obeys Mestschersky’s first and combined laws. The new orbits represent more accurately the actual motion in the initial segment of the trajectory than an osculating Keplerian orbit and other existing analogues. Encke’s generalized methods of special perturbations in which the constructed intermediate orbits are used as reference orbits are presented. Numerical simulations using the approximations of the motions of Asteroid Toutatis and Comet P/Honda—Mrkos—Pajdu?áková as examples confirm that the constructed orbits are highly efficient. Their application is particularly beneficial in investigating strongly perturbed motion.     

Steady state populations of Apollo-Amor objects

The steady-state distribution of orbits of Apollo-Amor objects is calculated for a variety of possible sources. These include asteroids near the inner edge of the belt, cometary orbits similar to Encke, and hypothetical extinct cometary orbits with perihelia larger than that of Encke. In all but one case, the steady-state distributions are similar for all these sources, and predict Amor/Apollo ratios of 1.5 to 3. These ratios are lower than those predicted by work in which the effects of the ν₆ secular resonance were not considered. These results are in general agreement with observation, although the higher (～3) Amor/Apollo ratios found for many of the sources may turn out to be unacceptably high. The absolute number of Apollo-Amors observed is found to require an injection rate of ～15 objects/(10⁶ years). This rate is easily achieved if the present existence of Encke is assumed to be a reasonably probable event, and if Encke becomes a ～1-km-diameter Apollo object following exhaustion of its volatile material; best estimates of the injection rate from the asteroid belt [～1.5/(10⁶ years)] are too low. Hence a dominant cometary component is suggested. The predicted number of Apollo objects in small (q < 1.0 AU, a < 1.4 AU orbits is in agreement with observation. Predicted lunar and terrestrial cratering rates agree approximately with observation. An unexplained difference between the lunar and terrestrial results is probably caused by uncertainties in the scaling laws or crater counts used. This discrepancy precludes an exact test of these calculations using cratering data.     

The disappearance of OH from comet P/Encke

Photoelectric observations of Comet P/Encke during its 1980 apparition are combined with other published data to relate molecular production rates to the visual lightcurve. In addition to a substantial asymmetry about perihelion which is already well known, there are shorter-term variations in specific molecules which have not been duplicated by models. The most dramatic of these fluctuations is a rapid decrease by more than a factor of 3 in the production of OH at 0.75 AU preperihelion.     

Elliptic Anomaly in Constructing Long-Term and Short-Term Dynamical Theories

The techniques of Brumberg and Brumberg (1999) based on the use of elliptic anomaly are specified in this paper in two aspects. The iteration technique (Broucke, 1969) to construct short-term semi-analytical theories of motion in rectangular coordinates in lines of Encke and Hill is reelaborated in terms of elliptic anomaly resulting in extending this technique for high-eccentricity orbits. In constructing long-term semi-analytical theories the key point is to integrate trigonometric functions of several angular arguments related to time by different differential expressions. This problem is reduced in the paper to linear algebraic recurrence relations admitting efficient solution by iterations.     

The isolated non-circular ringlets of Saturn

We show that the combined effect of electrodynamic and gravitational forces can account for a number of features observed by Voyagers 1 and 2 in the isolated fine dust rings of Saturn. These include (a) the appearance and disappearnce of the braids in the F-ring, (b) the eccentricities of the F-ring and the ringlets within the Encke and Cassine divisions and a gap in the C-ring, and (c) the kinks in the eccentric Encke ring. They may also account for the very existence of these rings.     

An efficient computational method for the approximate solution of nonlinear Lane-Emden type equations arising in astrophysics

The key purpose of this article is to introduce an efficient computational method for the approximate solution of the homogeneous as well as non-homogeneous nonlinear Lane-Emden type equations. Using proposed computational method given nonlinear equation is converted into a set of nonlinear algebraic equations whose solution gives the approximate solution to the Lane-Emden type equation. Various nonlinear cases of Lane-Emden type equations like standard Lane-Emden equation, the isothermal gas spheres equation and white-dwarf equation are discussed. Results are compared with some well-known numerical methods and it is observed that our results are more accurate.     

Infrared photometry of periodic comets Encke,Chernykh, Kearns-Kwee,Stephan-Oterma,and Tuttle

Nearly simultaneous photometry of the reflected and thermal infrared spectra of periodic comets Encke, Chernykh, Kearns-Kwee, Stephan-Oterma, and Tuttle are presented. The 10-μm, silicate emission feature has been detected for the first time in periodic comets and was observed in three of these objects. The albedo of the dust particles in the comae of these comets is calculted and compared to that of Comet Kohoutek. The peculiar behavior of the dust in Comet Encke is discussed.     

Computation of Partial Derivatives of Perturbed Motion Using the Arcs of Osculating and Superosculating Orbits

The methods for analytical determination of partial derivatives of the current parameters of motion with respect to their initial values are described. The methods take into account principal perturbations and are based on the use of the osculating and superosculating intermediate orbits constructed earlier by the author. These orbits ensure the first-, second-, and third-order contact to the real trajectory at the initial time. The solution for parameters of the intermediate motion and partial derivatives of these parameters is given in a universal closed form. The partial derivatives on long time intervals are computed using a step-by-step procedure combined with the Encke method of special perturbations, in which the intermediate orbits are used as the reference. The numerical results show that the new approach can be efficiently used for solving the problem of differential correction of orbits of asteroids and comets on the basis of observational data.     

The ultraviolet spectrum of periodic comet encke (1980 XI)

A comparison of the ultraviolet spectrum of periodic Comet Encke (1980 XI), recorded by the IUE between 24 October and 5 November 1980 with similar spectra of short- and long-period comets shows the gaseous composition of P/Encke to be nearly identical to that of the other comets observed by the IUE. If P/Encke is indeed the remains of a once giant comet, this similarity implies a homogeneous radial structure for the cometary ice nucleus. The OH brightness distribution shows a spatial variation similar to the visible fan-shaped image of the comet, suggestive of a nonuniform distribution of volatile ices on the surface of the nucleus. The total derived water production rate appears to be a factor of 5 higher than that derived from HI Lyman-α observations made during the 1970 apparition and shows a variation with heliocentric distance (r) as r^?3.3 over the range 0.81 to 1.02 AU.     

R- and J-band photometry of Comets 2P/Encke and 9P/Tempel 1

Near-simultaneous R- and J-band photometric measurements of the short-period Comets 2P/Encke and the Deep Impact mission target 9P/Tempel 1 were obtained. The resulting R-J colors are +0.82±0.08 mag and +1.46±0.13 mag for Encke and Tempel 1, respectively. Tempel 1's color is redder than the solar R-J color index of +0.76. The Tempel 1 observations directly detected the nucleus while the Encke observations likely suffered from coma contamination.     

Rapid changes of brightness in P/Halley

Scanner observations of the coma of periodic comet Encke (P/Encke) are presented for four nights in March 1984 during its post-perihelion period. The strong emission features of CN and C₂ molecules have been identified and the abundances of CN and C₂ are estimated. The production rates of these molecules have also been derived from their band luminosities. No trace of sodium emission has been found in this comet.     

The Christiansen Effect in Saturn’s narrow dusty rings and the spectral identification of clumps in the F ring

Stellar occultations by Saturn’s rings observed with the Visual and Infrared Mapping Spectrometer (VIMS) onboard the Cassini spacecraft reveal that dusty features such as the F ring and the ringlets in the Encke and the Laplace Gaps have distinctive infrared transmission spectra. These spectra show a narrow optical depth minimum at wavelengths around 2.87 μm. This minimum is likely due to the Christiansen Effect, a reduction in the extinction of small particles when their (complex) refractive index is close to that of the surrounding medium. Simple Mie-scattering models demonstrate that the strength of this opacity dip is sensitive to the size distribution of particles between 1 and 100 μm across. Furthermore, the spatial resolution of the occultation data is sufficient to reveal variations in the transmission spectra within and among these rings. In both the Encke Gap ringlets and F ring, the opacity dip weakens with increasing local optical depth, which is consistent with the larger particles being concentrated near the cores of these rings. The Encke Gap ringlets also show systematically weaker opacity dips than the F ring and Laplace Gap ringlet, implying that the former has a smaller fraction of grains less than ∼30 μm across. However, the strength of the opacity dip varies most dramatically within the F ring; certain compact regions of enhanced optical depth lack an opacity dip and therefore appear to have a greatly reduced fraction of grains in the few-micron size range. Such spectrally-identifiable structures probably represent a subset of the compact optically-thick clumps observed by other Cassini instruments. These variations in the ring’s particle size distribution can provide new insights into the processes of grain aggregation, disruption and transport within dusty rings. For example, the unusual spectral properties of the F-ring clumps could perhaps be ascribed to small grains adhering onto the surface of larger particles in regions of anomalously low velocity dispersion.     

Multiband photometry of Comets Kohoutek,Bennett, Bradfield,and Encke

Observations of Comets Kohoutek (1973f), Bradfield (1974b), and P/Encke have been made at a number of wavelengths between 0.55 and 18 μm. The silicate feature first observed in Comet Bennett (1969i) seems to be a common characteristic of cometary material. The comas of these comets radiate infrared with an effective temperature higher than the black-body temperature at the given distance from the Sun. The albedo of the dust particles is between 0.10 and 0.20. The particles in the coma and tail are small (diameter less than 2 μm), but the particles in the anti-tail of Comet Kohoutek must be larger than about 10 μm diameter. The observations give an absolute upper limit to the diameter of Comet Kohoutek of 30 km. A consistent interpretation would indicate that Comets Kohoutek and Bradfield have nuclear diameters of 5 to 10km, that Bennett was several times larger, and that P/Encke is 10 times smaller. The peculiar behavior of Bradfield showed that the coma of a single comet can abruptly change its dust composition.