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1.
Giovanni Federico Gronchi 《Celestial Mechanics and Dynamical Astronomy》2009,103(4):301-326
Charlier’s theory (1910) provides a geometric interpretation of the occurrence of multiple solutions in Laplace’s method of
preliminary orbit determination, assuming geocentric observations. We introduce a generalization of this theory allowing to
take into account topocentric observations, that is observations made from the surface of the rotating Earth. The generalized
theory works for both Laplace’s and Gauss’ methods. We also provide a geometric definition of a curve that generalizes Charlier’s
limiting curve, separating regions with a different number of solutions. The results are generically different from Charlier’s:
they may change according to the value of a parameter that depends on the observations. 相似文献
2.
Contemporary surveys provide a huge number of detections of small solar system bodies, mostly asteroids. Typically, the reported astrometry is not enough to compute an orbit and/or perform an identification with an already discovered object. The classical methods for preliminary orbit determination fail in such cases: a new approach is necessary. When the observations are not enough to compute an orbit we represent the data with an attributable (two angles and their time derivatives). The undetermined variables range and range rate span an admissible region of solar system orbits, which can be sampled by a set of Virtual Asteroids (VAs) selected by an optimal triangulation. The attributable results from a fit and has an uncertainty represented by a covariance matrix, thus the predictions of future observations can be described by a quasi-product structure (admissible region times confidence ellipsoid), which can be approximated by a triangulation with each node surrounded by a confidence ellipsoid. The problem of identifying two independent short arcs of observations has been solved. For each VA in the admissible region of the first arc we consider prediction at the time of the second arc and the corresponding covariance matrix, and we compare them with the attributable of the second arc with its own covariance. By using the penalty (increase in the sum of squares, as in the algorithms for identification) we select the VAs which can fit together both arcs and compute a preliminary orbit. Even two attributables may not be enough to compute an orbit with a convergent differential corrections algorithm. The preliminary orbits are used as first guess for constrained differential corrections, providing solutions along the Line Of Variations (LOV) which can be used as second generation VAs to further predict the observations at the time of a third arc. In general the identification with a third arc will ensure a least squares orbit, with uncertainty described by the covariance matrix. 相似文献
3.
In addition to the detection of an asteroid moon or a binary asteroid, the knowledge of the satellite’s true orbit is of high importance to derive fundamental physical parameters of the binary system such as its mass and to shed light on its possible formation history and dynamical evolution (prograde/retrograde orbit, large/small eccentricity or inclination, etc.). A new methodology for preliminary orbit determination of binary asteroids – and visual binaries in general – is proposed. It is based on Thiele–Innes method combined with a ‘trial and error’ Monte-Carlo technique. This method provides the full set of solutions (bundle of orbits, with the 7 orbital elements) even for a reduced number of observations. The mass is a direct by-product of this orbit determination, from which one can next infer the bulk-density and porosity. In addition to the bundle of orbits, the method provides the marginal probability densities of the foreseen parameters. Such error analysis – since it avoids linear approximation – can be of importance for the prediction of the satellite’s position in the plane-of-sky during future stellar occultations or subsequent observations, but also for the analysis of the orbit’s secular evolution. After briefly describing the method, we present the algorithm and its application to some practical cases, with particular emphasis on asteroids binaries and applications on orbital evolution. 相似文献
4.
The process of calculating a good orbit from astrometric observations of the same object involves three main steps: preliminary orbit determination, least squares orbit fitting, and quality control assessing the orbit's uncertainty and reliability. For the next generation sky surveys, with much larger number density of observations, new algorithms, or at least substantial revisions of the classical ones, are needed. The classical theory of preliminary orbit algorithms was incomplete in that the consequences of the topocentric correction had not been fully studied. We show that it is possible to rigorously account for topocentric observations and that this correction may increase the number of alternate preliminary orbits without impairing the overall performance. We have developed modified least squares algorithms including the capability of fitting the orbit to a reduced number of parameters. The restricted fitting techniques can be used to improve the reliability of the orbit computing procedure when the observed arcs have small curvature. False identification (where observations of different objects are incorrectly linked together) can be discarded with a quality control on the residuals and a ‘normalization’ procedure removing duplications and contradictions. We have tested our algorithms on two simulations based on the expected performance of Pan-STARRS—one of the next generation all-sky surveys. The results confirm that large sets of discoveries can be handled very efficiently resulting in good quality orbits. In these tests we lost only 0.6 to 1.3% of the possible objects, with a false identification rate in the range 0.02 to 0.06%. 相似文献
5.
Robert M. L. Baker Jr Neil H. Jacoby Jr 《Celestial Mechanics and Dynamical Astronomy》1977,15(2):137-160
It has long been recognized and demonstrated in the astrodynamic literature that three observations of angular position are not always sufficient to determine a preliminary orbit. One reason for this is due to the fact that as the plane of the observer's motion approaches the plane of the orbit of the observed object, the determination of the orbit of the object becomes indeterminant. Merely changing the coordinate system will not eliminate the inherent indeterminacy or singularity. When the observed object is moving in the same plane as the observer, their relative motion is described in two dimensions rather than three. The problem reduces to defining two components of position and two of velocity given only three angular measures and no solution is possible. Although this singularity is a rather old, albeit infrequently arising problem in celestial mechanics, it has received renewed interest due to the advent of satellite observatories that observe other spacecraft. In this new circumstance the plane of the observer's motion is rather frequently near the plane of the object (12% to 35% of the time) and the co-planar singularity becomes a subject that deserves additional attention.It is the purpose of this paper to develop a practical and simple method of orbit determination using four observations. This method also allows one to avoid the problem of multiple orbit-determination solution roots, and provides numerical indices that are useful in assessing the degree of indeterminacy in any given observer/object geometry. This paper does not dwell at length on the theory of orbital singularities, since they have been already treated in celestial mechanics literature. Instead, the emphasis is on the details of a new computational technique, which has been found to be computationally more efficient than previous four-observation methods, and which is unique in being formulated in the geocentric system and involves only one scalar quantity in the correction process.The equations for the new method are developed and a numerical example is presented that demonstrates the efficiency of the method. 相似文献
6.
The dynamics of near-Earth asteroids near mean motion resonances with the Earth or other planets is considered. The probability domains of the motion of some near-Earth asteroids close to low-order resonances are presented. The investigations have been carried out by means of a numerical integration of differential equations, taking into account the perturbations from the major planets and the Moon. For each investigated object an ensemble of 100 test particles with orbital elements nearby those of the nominal orbit has been constructed and its evolution has been retraced over the time interval (–3000, +3000 years). The initial set of orbits has been generated on the basis of probable variations of the initial orbital elements obtained from the least square analysis of observations.This revised version was published online in October 2005 with corrections to the Cover Date. 相似文献
7.
We present a new method to solve the problem of initial orbit determination of any binary system. This method is mainly based
on the material available for an observer, for example relative positions at a given time of the couple in the “plane of sky”,
namely the tangent plane to the celestial sphere at the position of the primary component. The problem of orbit determination
is solved by splitting in successive stages in order to decorrelate the parameters of each other as much as possible. On one
hand, the geometric problem is solved using the first Kepler’s law from a single observing run and, on the other hand, dynamical
parameters are then inferred from the fit of the Kepler’s equation. At last, the final stage consists in determining the main
physical parameters involved in the secular evolution of the system, that is the spin axis and the J2 parameter of the primary if we assume that it is a quasi-spherical body. As a matter of fact there is no need to make too
restrictive initial assumptions (such as circular orbit or zero eccentricity) and initial guesses of parameters required by
a non-linear least-squares Levenberg–Marquardt algorithm are finally obtained after each stage. Such a protocol is very useful
to study systems like binary asteroids for which all of the parameters should be considered a priori as unknowns. As an example of application, we used our method to estimate the set of the Pluto–Charon system parameters from
observations collected in the literature since 1980. 相似文献
8.
Roberto Armellin Pierluigi Di Lizia Michele Lavagna 《Celestial Mechanics and Dynamical Astronomy》2012,112(3):331-352
A method for high-order treatment of uncertainties in preliminary orbit determination is presented. The observations consist
in three couples of topocentric right ascensions and declinations at three observation epochs. The goal of preliminary orbit
determination is to compute a trajectory that fits with the observations in two-body dynamics. The uncertainties of the observations
are usually mapped to the phase space only when additional observations are available and a least squares fitting problem
is set up. A method based on Taylor differential algebra for the analytical treatment of observation uncertainties is implemented.
Taylor differential algebra allows for the efficient computation of the arbitrary order Taylor expansion of a sufficiently
continuous multivariate function. This enables the mapping of the uncertainties from the observation space to the phase space
as high-order multivariate Taylor polynomials. These maps can then be propagated forward in time to predict the observable
set at successive epochs. This method can be suitably used to recover newly discovered objects when a scarce number of measurements
is available. Simulated topocentric observations of asteroids on realistic orbits are used to assess the performances of the
method. 相似文献
9.
Leif Kahl Kristensen 《Celestial Mechanics and Dynamical Astronomy》2009,105(4):275-287
Initial orbit determination by least squares of N observations is essentially a linear problem if the coordinates x
0 and x
1 at two standard epochs are used as elements. The orbit of a main belt object is approximated within the observational errors
by a third degree polynomial during a month. A 4-observation orbit is useful for the initial linking between two nights. Parallax
is treated rigorously and future simultaneous space and Earth based observations determine the critical distance directly.
The N-observation method is a great simplification of the classical 3-observation orbit followed by a differential correction by
N observations. 相似文献
10.
The behaviour of ‘resonances’ in the spin-orbit coupling in celestial mechanics is investigated in a conservative setting.
We consider a Hamiltonian nearly-integrable model describing an approximation of the spin-orbit interaction. The continuous
system is reduced to a mapping by integrating the equations of motion through a symplectic algorithm. We study numerically
the stability of periodic orbits associated to the above mapping by looking at the eigenvalues of the matrix of the linearized
map over the full cycle of the periodic orbit. In particular, the value of the trace of the matrix is related to the stability
character of the periodic orbit. We denote by ε*
(p/q) the value of the perturbing parameter at which a given elliptic periodic orbit with frequency p/q becomes unstable. A plot of the critical function ε*
(p/q) versus the frequency at different orbital eccentricities shows significant peaks at the synchronous resonance (for low eccentricities)
and at the synchronous and 3:2 resonances (at higher eccentricities) in good agreement with astronomical observations.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
11.
H. Ma G. Baù D. Bracali Cioci G. F. Gronchi 《Celestial Mechanics and Dynamical Astronomy》2018,130(10):70
We propose a method to account for the Earth oblateness effect in preliminary orbit determination of satellites in low orbits with radar observations. This method is an improvement of the one described in Gronchi et al. (Mon Not R Astron Soc 451(2):1883–1891, 2015b), which uses a pure Keplerian dynamical model. Since the effect of the Earth oblateness is strong at low altitudes, its inclusion in the model can sensibly improve the initial orbit, giving a better starting guess for differential corrections and increasing the chances to obtain their convergence. The input set consists of two tracks of radar observations, each one composed of at least four observations taken during the same pass of the satellite. A single observation gives the topocentric position of the satellite, where the range is very accurate, while the line-of-sight direction is poorly determined. From these data, we can compute by a polynomial fit the values of the range and range rate at the mean epochs of the two tracks. In order to obtain a preliminary orbit, we wish to compute the angular velocity, which is the rate of change of the line of sight. In the same spirit of Gronchi et al. (Mon Not R Astron Soc 451(2):1883–1891, 2015b), we also wish to correct the values of the angular measurements, so that they fit the selected dynamical model if the same holds for the radial distance and velocity. The selected model is a perturbed Keplerian dynamics, where the only perturbation included is the secular effect of the \(J_2\) term of the geopotential. 相似文献
12.
Preliminary data on the magnetic field structure of the unique magnetic star HD 45583 are obtained. The observational data
are well described by a configuration of two magnetic dipoles located on opposite sides relative to the star’s center, with
their axes directed roughly in a radial direction. The positive monopoles are closer to the surface and the negative, closer
to the star’s center. For this reason, there appear to be two positive magnetic poles on the star’s surface but no negative
poles. The need for further observations of this unique object is pointed out.
Translated from Astrofizika, Vol. 52, No. 1, pp. 127–133 (February 2009). 相似文献
13.
The Arecibo UHF radar is able to detect the head-echos of micron-sized meteoroids up to velocities of 75 km/s over a height
range of 80–140 km. Because of their small size there are many uncertainties involved in calculating their above atmosphere
properties as needed for orbit determination. An ab initio model of meteor ablation has been devised that should work over
the mass range 10−16 kg to 10−7 kg, but the faint end of this range cannot be observed by any other method and so direct verification is not possible. On
the other hand, the EISCAT UHF radar system detects micrometeors in the high mass part of this range and its observations
can be fit to a “standard” ablation model and calibrated to optical observations (Szasz et al. 2007). In this paper, we present a preliminary comparison of the two models, one observationally confirmable. Among the features
of the ab initio model that are different from the “standard” model are: (1) uses the experimentally based low pressure vaporization
theory of O’Hanlon (A users’s guide to vacuum technology, 2003) for ablation, (2) uses velocity dependent functions fit from
experimental data on heat transfer, luminosity and ionization efficiencies measured by Friichtenicht and Becker (NASA Special
Publication 319: 53, 1973) for micron sized particles, (3) assumes a density and temperature dependence of the micrometeoroids
and ablation product specific heats, (4) assumes a density and size dependent value for the thermal emissivity and (5) uses
a unified synthesis of experimental data for the most important meteoroid elements and their oxides through least square fits
(as functions of temperature, density, and/or melting point) of the tables of thermodynamic parameters given in Weast (CRC
Handbook of Physics and Chemistry, 1984), Gray (American Institute of Physics Handbook, 1972), and Cox (Allen’s Astrophysical
Quantities 2000). This utilization of mostly experimentally determined data is the main reason for calling this an ab initio
model and is made necessary by the fact that individual average meteoroid mass densities are now derivable from Arecibo observations. 相似文献
14.
The main problem in the orbit determination of the space debris population orbiting our planet is identifying which separate
sets of data belong to the same physical object. The observations of a given object during a passage above an observing station
are collectively called a Too Short Arc (TSA): data from a TSA cannot allow for a complete determination of an orbit. Therefore, we have to solve first the identification
problem, finding two or more TSAs belonging to the same physical object and an orbit fitting all the observations. This problem
is well known for the determination of orbits of asteroids: we shall show how to apply the methods developed for preliminary
orbit determination of heliocentric objects to geocentric objects. We shall focus on the definition of an admissible region for space debris, both in the case of optical observations and radar observations; then we shall outline a strategy to perform
a full orbit determination. 相似文献
15.
An explicit symplectic integrator is constructed for the problem of a rotating planetary satellite on a Keplerian orbit. The
spin vector is fixed perpendicularly to the orbital plane. The integrator is constructed according to the Wisdom-Holman approach:
the Hamiltonian is separated in two parts so that one of them is multiplied by a small parameter. The parameter depends on
the satellite’s shape or the eccentricity of its orbit. The leading part of the Hamiltonian for small eccentricity orbits
is similar to the simple pendulum and hence integrable; the perturbation does not depend on angular momentum which implies
a trivial ‘kick’ solution. In spite of the necessity to evaluate elliptic function at each step, the explicit symplectic integrator
proves to be quite efficient.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
16.
A star will become brighter and brighter with stellar evolution, and the distance of its habitable zone will become larger
and larger. Some planets outside the habitable zone of a host star during the main sequence phase may enter the habitable
zone of the host star during other evolutionary phases. A terrestrial planet within the habitable zone of its host star is
generally thought to be suitable for the existence of life. Furthermore, a rocky moon around a giant planet may be also suitable
for life to survive, provided that the planet–moon system is within the habitable zone of its host star. Using Eggleton’s
code and the boundary flux of the habitable zone, we calculate the habitable zone of our Solar system after the main sequence
phase. It is found that Mars’ orbit and Jupiter’s orbit will enter the habitable zone of the Solar system during the subgiant
branch phase and the red giant branch phase, respectively. And the orbit of Saturn will enter the habitable zone of Solar
during the He-burning phase for about 137 million years. Life is unlikely at any time on Saturn, as it is a giant gaseous
planet. However, Titan, the rocky moon of Saturn, may be suitable for biological evolution and become another Earth during
that time. For low-mass stars, there are similar habitable zones during the He-burning phase as our Solar, because there are
similar core masses and luminosities for these stars during that phase. 相似文献
17.
Toshio Fukushima 《Celestial Mechanics and Dynamical Astronomy》1999,75(3):201-226
We developed a procedure to solve a modification of the standard form of the universal Kepler’s equation, which is expressed
as a nondimensional equation with respect to a nondimensional variable. After reducing the domain of the variable and the
argument by using the symmetry and the periodicity of the equation, the method first separates the case where the solution
is so small that it is given an inverted series. Second, it separates the cases where the elliptic, parabolic, or hyperbolic
standard forms of Kepler’s equation are suitable. Here the separation is done by judging whether detouring these nonuniversal
equations will cause a 1-bit loss of information to their nonuniversal solutions or not. Then the nonuniversal equations are
solved by the author’s procedures to solve the elliptic Kepler’s equation (Fukushima, 1997a), Barker’s equation (Fukushima,
1998), and the hyperbolic Kepler’s equation (Fukushima, 1997b), respectively. And their nonuniversal solutions are transformed
back to the solution of the universal equation. For the rest of the case, we obtain an approximate solution by solving roughly
the approximated cubic equation as we did in solving Barker’s equation. Then the correction to the approximate solution is
obtained by Halley’s method precisely. There the special function appeared in the universal equation is rewritten into a combination
of similar special functions of small arguments, so that they are efficiently evaluated by their Taylor series. Numerical
measurements showed that, in the case of Intel Pentium II processor, the new method is 10–25 times as fast as Shepperd’s method
(Shepperd, 1985) and 7–13 times as fast as the standard Newton method.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
18.
When the observational data are not enough to compute a meaningful orbit for an asteroid/comet we can represent the data with an attributable, i.e., two angles and their time derivatives. The undetermined variables range and range rate span an admissible region of Solar System orbits, which can be sampled by a set of Virtual Asteroids (VAs) selected by means of an optimal triangulation [Milani, A., Gronchi, G.F., de' Michieli Vitturi, M., Kne?evi?, Z., 2004. Celest. Mech. Dyn. Astron. 90, 59-87]. The attributable 4 coordinates are the result of a fit and they have an uncertainty, represented by a covariance matrix. Two short arcs of observations, represented by two attributables, can be linked by considering for each VA (in the admissible region of the first arc) the covariance matrix for the prediction at the time of the second arc, and by comparing it with the attributable of the second arc with its own covariance. By defining an identification penalty we can select the VAs allowing to fit together both arcs and compute a preliminary orbit. Two attributables may not be enough to compute an orbit with convergent differential corrections. Thus the preliminary orbit is used in a constrained differential correction, providing solutions along the Line Of Variation which can be used as second generation VAs to further predict the observations at the time of a third arc. In general the identification with a third arc will ensure a well determined orbit, to which additional sets of observations can be attributed. To test these algorithms we use a large scale simulation and measure the completeness, the reliability and the efficiency of the overall procedure to build up orbits by accumulating identifications. Under the conditions expected for the next generation asteroid surveys, the methods developed in this and in the preceding papers are efficient enough to be used as primary identification methods, with very good results. One important property is that the completeness in finding the possible identifications is as good for comparatively rare orbits, such as the ones of Near-Earth Objects, as for main belt orbits. 相似文献
19.
The multicollinearity among regression variables is a common phenomenon in the reduction of astronomical data. The phenomenon
of multicollinearity and the diagnostic factors are introduced first. As a remedy, a new method, called adaptive ridge regression
(ARR), which is an improved method of choosing the departure constant θ in ridge regression, is suggested and applied in a
case that the Earth orientation parameters (EOP) are determined by lunar laser ranging (LLR). It is pointed out, via a diagnosis,
the variance inflation factors (VIFs), that there exists serious multicollinearity among the regression variables. It is shown
that the ARR method is effective in reducing the multicollinearity and makes the regression coefficients more stable than
that of using ordinary least squares estimation (LS), especially when there is serious multicollinearity.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
20.
We analyze the out-of-plane librations of a tethered satellite system that is nominally rotating in the orbit plane. To isolate
the librational dynamics, the system is modeled as two point masses connected by a rigid rod with the system mass center constrained
to an unperturbed circular orbit. For small out-of-plane librations, the in-plane motion is unaffected by the out-of-plane
librations and a solution for the in-plane motion is determined in terms of Jacobi elliptic functions. This solution is used
in the linearized equation for the out-of-plane librations, resulting in a Hill’s equation. Floquet theory is used to analyze
the Hill’s equation, and we show that the out-of-plane librations are unstable for certain ranges of in-plane spin rate. For
relatively high in-plane spin rates, the out-of-plane librations are stable, and the Hill’s equation can be approximated by
a Mathieu’s equation. Approximate solutions to the Mathieu’s equation are determined, and we analyze the dominant characteristics
of the out-of-plane librations for high in-plane spin rates. The results obtained from the analysis of the linearized equations
of motion are compared to numerical simulations of the nonlinear equations of motion, as well as numerical simulations of
a more realistic system model that accounts for tether flexibility. The instabilities discovered from the linear analysis
are present in both the nonlinear system and the more realistic system model. The approximate solutions for the out-of-plane
librations compare well to the nonlinear system for relatively high in-plane rotation rates, and also capture the significant
qualitative behavior of the flexible system. 相似文献