The space of Keplerian orbits and a family of its quotient spaces

Distance functions on the set of Keplerian orbits play an important role in solving problems of searching for parent bodies of meteoroid streams. A special kind of such functions are distances in the quotient spaces of orbits. Three metrics of this type were developed earlier. These metrics allow to disregard the longitude of ascending node or the argument of pericenter or both. Here we introduce one more quotient space, where two orbits are considered identical if they differ only in their longitudes of nodes and arguments of pericenters, but have the same sum of these elements (the longitude of pericenter). The function q is defined to calculate distance between two equivalence classes of orbits. The algorithm for calculation of ̺6 value is provided along with a reference to the corresponding program, written in C++ language. Unfortunately, ̺6 is not a full-fledged metric. We proved that it satisfies first two axioms of metric space, but not the third one: the triangle inequality does not hold, at least in the case of large eccentricities. However there are two important particular cases when the triangle axiom is satisfied: one of three orbits is circular, longitudes of pericenters of all three orbits coincide. Perhaps the inequality holds for all elliptic orbits, but this is a matter of future research.

GEANT 4 Simulation of the Helios E6 - Proton Contamination of Relativistic Electron Measurements

<p>The Helios mission consisted of two almost identical spacecraft in highly elliptic orbits launched in 1974 (Helios A) and 1976 (Helios B). Until Parker Solar Probes first perihelion, Helios B was the first spacecraft to reach a distance of 0.29 AU to the Sun. One of its instruments is the Experiment 6 (E6) which was designed and built at the Christian-Albrechts-University Kiel in order to measure ions (protons up to iron) in the energy range of 1.3 MeV/nucleon up to several GeV/nucleon and electrons in the energy range from 0.3 to about 8 MeV. The instrument relies on the dE/dx-E and on the dE/dx-Cherenkov method for stopping and penetrating particles, respectively. Electrons are separated from ions by the signal in the first 100 µm thick solid state detector. Any particle that does not trigger this detector is identified as an electron. Since the solid state detectors are not working perfectly, a significant part of protons is identified as electrons. Here, we present a new method to correct the electron measurements for the cross talk based on detailed instrument simulations.</p>

Nonsingular recursion formulas for third-body perturbations in mean vectorial elements

The description of the long-term dynamics of highly elliptic orbits under third-body perturbations may require an expansion of the disturbing function in series of the semi-major axes ratio up to higher orders. To avoid dealing with long series in trigonometric functions, we refer the motion to the apsidal frame and efficiently remove the short-period effects of this expansion in vectorial form up to an arbitrary order. We then provide the variation equations of the two fundamental vectors of the Keplerian motion by analogous vectorial recurrences, which are free from singularities and take a compact form useful for the numerical propagation of the flow in mean elements.