The Stark problem in the Weierstrassian formalism

Abstract

We present a new general, complete closed-form solution of the three-dimensional Stark problem in terms of Weierstrass elliptic and related functions. With respect to previous treatments of the problem, our analysis is exact and valid for all values of the external force field, and it is expressed via unique formulae valid for all initial conditions and parameters of the system. The simple form of the solution allows us to perform a thorough investigation of the properties of the dynamical system, including the identification of quasi-periodic and periodic orbits, the formulation of a simple analytical criterion to determine the boundness of the trajectory, and the characterisation of the equilibrium points.

Theory for planetary exospheres: I. Radiation pressure effect on dynamical trajectories

“…However, they give the analytical solutions of all trajectories only in the 2D case (and for bounded trajectories in the 3D case), and their formulas have some issues as will be discussed later. Another analytical study proposed by Biscani and Izzo (2014) uses the Weierstrassian formulations to solve the motions for bounded and unbounded trajectories and to find periodic motions. Also, the motion can be approached numerically by developing the equations of motion in Taylor series but this leads to some issues for high eccentricities (Pellegrini et al, 2014).…”

“…In this first paper, we present the complete exact solutions of particles trajectories, which are not conics, under the influence of the solar radiation pressure. This problem was recently partly solved by Lantoine and Russell (2011) and completely by Biscani and Izzo (2014). We give here the full set of solutions, including solutions not previously derived, as well as simpler formulations for previously known cases and comparisons with recent works.…”

Non-conservative extension of Keplerian integrals and a new class of integrable system

“…This is the unified solution for all symmetric solutions, with z m = (θ − θ m )/k. Practical comments on the implementation of the Weierstrass elliptic functions can be found in Biscani and Izzo (2014), for example. Although ℘(z) = ℘(−z), the derivative ℘ ′ (z) is an odd function in z, ℘ ′ (−z) = −℘ ′ (z).…”

Comparison of two direct interplanetary trajectory optimization techniques

“…Эта задача называется задачей Штарка [13] и может быть решена как с помощью численного интегрирова-ния, так и аналитически. Основными современными и эффективными ана-литическими методами решения задачи Штарка являются методы, разрабо-танные Г. Лантуаном [14], Ф. Бискани [15] и Э. Пеллегрини [16]. Согласно исследованиям, проведенным Р. Расселом [17], использование аналитических методов помогает повысить быстродействие и точность вычислений.…”