Extended analytical formulae for the perturbed Keplerian motion under low-thrust acceleration and orbital perturbations

This paper presents a collection of analytical formulae that can be used in the long-term propagation of the motion of a spacecraft subject to low-thrust acceleration and orbital perturbations. The paper considers accelerations due to: a low-thrust profile following an inverse square law, gravity perturbations due to the central body gravity field and the third-body gravitational perturbation. The analytical formulae are expressed in terms of non-singular equinoctial elements. The formulae for the third-body gravitational perturbation have been obtained starting from equations for the third-body potential already available in the literature. However, the final analytical formulae for the variation of the equinoctial orbital elements are a novel derivation. The results are validated, for different orbital regimes, using high-precision numerical orbit propagators.

A Hybrid-Precision Numerical Orbit Integration Technique for Next Generation Gravity Missions

<p>In Next Generation Gravity Missions (NGGM) the Laser Ranging Interferometer (LRI) is applied to measure inter-satellite range rate with nanometer-level precision. Thereby the precision of numerical orbit integration must be higher or at least same as that of LRI and the currently widely-used double-precision orbit integration technique cannot meet the numerical requirements of LRI measurements. Considering quadruple-precision orbit integration arithmetic is time consuming, we propose a hybrid-precision numerical orbit integration technique, in which the double- and quadruple-precision arithmetic is employed in the increment calculation part and orbit propagation part, respectively. Since the round-off errors are not sensitive to the time-demanding increment calculation but to the least time-consuming orbit propagation, the proposed hybrid-precision numerical orbit integration technique is as efficient as the double-precision orbit integration technique, and as precise as the quadruple-precision orbit integration. By using hybrid-precision orbit integration technique, the range rate precision is easily achieved at 10-12m/s in either nominal or Encke form, and furthermore the sub-nanometer-level range precision is obtainable in the Encke form with reference orbit selected as the best-fit one. Therefore, the hybrid-precision orbit integration technique is suggested to be used in the gravity field solutions for NGGM.</p>

Celestial Mechanics

Celestial mechanics abounds in interesting and counter-intuitive phenomena, such as descriptions of mass transfer between stars or optimal placements of satellites within the Solar System. Remarkably, many such features are already present in the restricted three-body problem, whose assumptions still allow for analytical understanding, and to which the second chapter is devoted. This ‘simplified’ system is discussed first in terms of forces (both gravitational and fictitious), and then using the Hamiltonian form. As well as traditional topics like stable and unstable Lagrange points and Roche lobes, a brief introduction to chaotic orbits is given. Additionally, readers are guided towards exploring on their own with numerical orbit integration.

SEARCH-BASED CHAOTIC PSEUDORANDOM BIT GENERATOR

This paper proposes the construction of a new chaotic pseudorandom bit generator, which forms the main building block of a chaotic stream cipher. The design of the algorithm is based on a single chaotic map whose numerical orbit indirectly contributes towards the generation of the keystream. The latter is produced from the numerical orbit by applying a technique that searches for iterates in specific intervals [a,b], for some real numbers a and b, and outputs 0 or 1 based on the iterate preceding the targeted iterate. The generator suggested here is built up from a quadratic map. We analyze the cycle length of the keystreams and investigate the resistance of the generator to well-known cryptanalytic attacks. Furthermore, the statistic characteristics of the keystreams are examined numerically using the NIST statistical test suite. The numerical and theoretical results demonstrate that the proposed technique results in generating keystreams possessing very good cryptographic properties and high level of security against existing cryptanalytic attacks. Empirical results show that the search technique leads to the generation of keystreams possessing good randomness properties when applied to any chaotic map whose orbits have good randomness properties such as the quadratic map, tent map and sawtooth map.

The Problem of Three Stars: Stability Limit

The problem of three stars arises in many connections in stellar dynamics: three-body scattering drives the evolution of star clusters, and bound triple systems form long-lasting intermediate structures in them. Here we address the question of stability of triple stars. For a given system the stability is easy to determine by numerical orbit calculation. However, we often have only statistical knowledge of some of the parameters of the system. Then one needs a more general analytical formula. Here we start with the analytical calculation of the single encounter between a binary and a single star by Heggie (1975). Using some of the later developments we get a useful expression for the energy change per encounter as a function of the pericenter distance, masses, and relative inclination of the orbit. Then we assume that the orbital energy evolves by random walk in energy space until the accumulated energy change leads to instability. In this way we arrive at a stability limit in pericenter distance of the outer orbit for different mass combinations, outer orbit eccentricities and inclinations. The result is compared with numerical orbit calculations.