|
Zusammenfassung:
|
The problem of the minimal mutual distances for two confocal elliptical orbits (local
minima), in the literature known as the proximity calculation for minor planets and
recognised recently as Minimal Orbit Intersection Distance – MOID, occupies a very
important place in astronomical studies, not only because of the prediction of possible
collisions of asteroids and other celestial bodies, but also because of the fact that by
analysing the behaviour of asteroids during their encounters it is possible to determine
their masses, changes of orbital elements and other important characteristics. Dealing
with this problem in this thesis the author has analysed the distance function for two
elliptical confocal orbits of minor planets combining analytical and numerical methods
for proximity calculation.
A survey of all relevant results in this field from the middle of the XIX century till our
days indicates that the problem has been transformed from looking for a solution of two
transcendental equations by applying various methods and approximations of long
duration towards efficient and rapid solutions of vector equations of the system which
describes the problem. In the thesis a simple and efficient analytic-numerical method
has been developed, presented and applied. It finds out all the minima and maxima in
the distance function and, indirectly, makes it possible to determine also the inflection
points. The method is essentially based on Simovljevic’s (1974) graphical
interpretation and on transcendental equations developed by Lazovic (1993). The
present method has been examined on almost three million pairs of real elliptical
asteroid orbits and its possibilities and the computation results have been compared to
the algebraic solutions given by Gronchi (2005). The case of a pair of confocal orbits
with four proximities found by Gronchi (2002), who applied the method of random
samples and carried out numerous simulations with different values of orbital elements,
gave the motivation to try here to find out such a pair among the real pairs of asteroid
orbits. Thanks to the efficacy of the method developed in the thesis two such pairs have
been found and their parameters are presented.
In addition to the one meantioned above a further analysis of distance function through
simulations including more than 20 million different pairs of asteroid orbits has resulted
in several additional interesting solutions of the distance function. The results are given
in the form of tables and plots showing the diversity of solutions for the distance
function. |